Fixed-switching frequency sliding mode control applied to power converters

Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters Thesis submitted in partial fulfillment of the requirement for the PhD Degree issued by the Universitat Polit´ecnica de Catalunya, in its Electronic Engineering Program. V´ıctor Repecho Del Corral Advisor: Domingo Biel Sol´e Barcelona, December 2017 Aknowlegdements This thesis is the result of almost four years of intensive work, dedication and efforts. All the hours spent in the laboratory, the discussions around a blackboard or the realization of scientific articles have led to the present thesis, of which I am proud of. I would like to show my gratitude to all people that have helped and supported me during this exciting time, without whom the development of this thesis would not have been possible. First of all, i would like to express my grateful thanks to my advisor Dr. Domingo Biel, who not only gave me the opportunity to become a part of the electronics and control laboratory in the University, but also have managed this thesis with dedication, effort and patience. I feel truly grateful to Rafel Cardoner, for his valuable advices about power electronics implementations and micro controllers programming, which have been indispensable for the experimentation evaluations of this work. Many thanks also to Enric Mir´o for his help in the daily work at the laboratory. I would like to express my gratitude towards Rafael Ramos, who provided me his knowledge about FGPA programming required in one of the experimental tests. I am also grateful to the Advanced Control of Energy Systems (ACES) group, which have contributed to this work with the support of its research members. Specifically, i would like to show my gratitude to Josep Mar´ıa Olm, Antoni Arias, Enric Fossas, Robert Gri˜n´o and Arnau D`oria for their collaboration in several aspects of the thesis development. Finally, i want to thank my girl Pili for her understanding and encouragement and, most of all, i would like to thank my family and specially my parents, Isidro and Daniela, to whom this thesis is dedicated. V´ıctor Repecho del Corral, Barcelona, January 2018 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters iii Abstract The application of the sliding mode control in power converters has a well-known inconvenient from the practical point of view, which is to obtain fixed switching frequency implementations. This thesis deals with the development of a hysteresis band controller in charge of fixing the switching frequency of the sliding motions in power electronics applications. The proposed control measures the switching period of the control signal and modifies the hysteresis band of the comparator in order to regulate the switching frequency of the sliding motion. The proposed structure becomes in an additional control loop aside from the main control loop implementing the sliding mode controller. In the first part of the thesis, the switching frequency control system is modelled and a design criteria for the control parameters are derived for guaranteeing closed-loop stability, under different approaches and taking into account the most expectable working scenarios. In the second part of the thesis, the proposed strategies are validated in several power converters. Specifically, DC-to-DC and DC-to-AC power converters are assembled and the experimental results are shown. In this part, the procedures used for implementing the controllers are also deeply discussed. Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters v Contents Contents vii I Introduction and Problem Statement 1 1 Introduction 3 1.1 The switched power converters .......................... 3 1.2 Control techniques in Switched Power Converters ............... 5 1.3 Ideal Sliding Motion ............................... 7 1.4 Real Sliding Motion ................................ 9 1.5 Proposed Solutions to the Variable Switching Frequency Problem ...... 11 1.5.1 Variable Hysteresis Band .......................... 12 1.5.2 External Synchronization Signal ...................... 12 1.5.3 Zero Average Dynamics .......................... 13 1.5.4 PWM-Based SMC ............................. 15 1.5.5 Additional Control Loop .......................... 15 1.6 Thesis objectives ................................. 16 1.7 Thesis structure .................................. 18 II Theoretical Analysis of the Proposed Solution and Study of the resulting Sliding Dynamics 21 2 Modelling and stability analysis of the switching frequency control loop 23 2.1 Open loop case: The fixed hysteresis band comparator ............ 23 2.2 Closed-loop case: A discrete-time modelling. .................. 25 2.2.1 The regulation case ............................. 27 2.2.2 The tracking case .............................. 29 2.3 Closed-loop case: A continuous-time modelling. ................ 34 3 A case of study: SFC design and simulation results 39 3.1 Design of the sliding mode controller ...................... 39 3.2 The regulation case in the discrete-time approach ............... 41 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters vii CONTENTS 3.3 The regulation case in the continuous-time approach .............. 43 3.4 The tracking case ................................. 46 4 Real sliding dynamics in a switching frequency control loop 53 4.1 The regular form approach ............................ 54 4.2 Fixed hysteresis band ............................... 56 4.3 Case study: the Buck converter ......................... 59 4.3.1 Mathematical model ............................ 59 4.3.2 Simulation results .............................. 60 4.4 Time-varying hysteresis band .......................... 64 4.4.1 The regulation case ............................. 64 4.4.2 The tracking case .............................. 68 4.4.3 Simulation results .............................. 70 III Application of the Proposed Solution to Power Converters. 77 5 Voltage Regulation in a Buck Converter. 79 5.1 The Buck Converter ............................... 79 5.2 Sliding mode control of the Output voltage ................... 80 5.2.1 Switching surface design .......................... 80 5.2.2 Sliding dynamics .............................. 81 5.2.3 Control law ................................. 82 5.3 Switching frequency regulation .......................... 82 5.3.1 Evaluation of ρ± k.............................. 82 5.3.2 SFC design ................................. 83 5.4 Implementation Details .............................. 83 5.5 Experimental results ............................... 86 5.6 Conclusions .................................... 92 6 Voltage Regulation in a Multiphase Buck Converter. 93 6.1 The multiphase converter ............................. 93 6.2 Interleaved sliding mode control of the output voltage ............. 95 6.2.1 Master switching surface design ...................... 95 6.2.2 Sliding dynamics of the Master phase ................... 96 6.2.3 Master phase control law .......................... 97 6.2.4 Slaves switching surfaces design. Interleaved Sliding Mode ....... 97 6.3 Switching frequency regulation .......................... 98 6.3.1 Evaluation of ρ± k.............................. 98 6.3.2 SFC design ................................. 99 6.4 Implementation Details .............................. 100 6.5 Experimental results ............................... 102 6.6 Conclusions .................................... 108 viii Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CONTENTS 7 Voltage Regulation in a Boost Converter. 109 7.1 The Boost Converter ............................... 109 7.2 Sliding mode control of the output voltage ................... 110 7.2.1 Switching surface design .......................... 110 7.2.2 Sliding dynamics .............................. 111 7.2.3 Control law ................................. 113 7.3 Switching frequency regulation .......................... 113 7.3.1 Evaluation of ρ± k.............................. 113 7.3.2 SFC design ................................. 113 7.4 Implementation Details .............................. 114 7.5 Experimental results ............................... 116 7.6 Conclusions .................................... 120 8 Voltage Tracking in a Voltage Source Inverter. 121 8.1 The voltage source inverter ............................ 121 8.2 Sliding mode tracking of the output voltage .................. 123 8.2.1 Switching surface design .......................... 123 8.2.2 Ideal sliding dynamics ........................... 126 8.2.3 Control law ................................. 127 8.2.4 Sliding dynamics for pure resistive load .................. 127 8.2.5 Sliding dynamics for reactive linear load ................. 131 8.2.6 Sliding dynamics for nonlinear load .................... 134 8.3 Switching frequency regulation .......................... 139 8.4 Implementation details .............................. 141 8.4.1 Effects of the hysteresis comparator discretization ............ 141 8.4.2 Digital emulation of the ideal hysteresis comparator ........... 143 8.4.3 Switching function digitalization issues .................. 145 8.4.4 Digital emulation of the SFC for a tracking control task ......... 146 8.4.5 Estimation of switching function slopes .................. 147 8.4.6 Controller implementation ......................... 149 8.4.7 Assembled converter and devices employed ................ 150 8.5 Experimentation results ............................. 152 8.5.1 SFC performance .............................. 152 8.5.2 SMC performance .............................. 154 8.5.3 Nonlinear load test ............................. 159 8.6 Conclusions .................................... 161 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters ix CHAPTER 1. INTRODUCTION Pin Switched Control Input Pout u d Power Converter fc PWM Figure 1.2: Switched power converter scheme, where the real control input, u, and the averaged control input, d, are depicted. For most of the applications, the PWM-based linear control theory is enough to provide a desirable performance in the systems, but, for specific applications, a different control technique could be necessary. Mainly, the drawbacks of the PWM-based linear control techniques come from the used linearized model. Since the considered models for the controllers design are local, when the power converter moves away from the nominal conditions the performance can be compromised. Additionally, the averaged models assume known the elements affecting the systems dynamics as inductors and capacitors. Whatever drift in these values from the nominal ones can also negatively affect the control performance. The Sliding Mode Control (SMC) [7] constitutes an alternative to the classical PWM-based linear controllers. This methodology, which can be classified as a nonlinear control technique [8,9], provides benefits to the power converters control. Besides the wellknown robustness, the order reduction and a fast transient response, the first order SMC uses as control outputs non-continuous signals with two allowable states, which perfectly matches with the physical structure of the power converters, transistors acting as switches. Moreover, the methodology works with the system state space equations without linearization process, and as a consequence, the provided stability conditions are, in general, more global. Of course, it exists an important inconvenient that historically has limited its practical application: how to achieve a bounded switching frequency of operation. This chapter briefly introduces the main idea of the sliding mode control theory. This analysis not only help us to demonstrate that in a real application the switching frequency under sliding motion becomes variable but also to justify the basis of the new proposal in order to get a fixed switching frequency. Finally, a short list of the proposed approaches for solving this inconvenient found in the literature until now is reviewed. 6 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 1. INTRODUCTION 1.3 Ideal Sliding Motion First of all, an affine nonlinear system is considered ˙x=f(x) + g(x)u(1.1) where the control signal utakes values from a discrete set with two values {u−, u+}. The state space equation matches with whatever switched power converter with a single control input. Indeed, such equation could represent arbitrary systems, not only power converters. The basis of the SMC is to design a switching function, σ(x), which depends on the state vector, and enforces this function to be in a state space region where the desired system dynamics is achieved, in general σ(x) = 0. The most simple switching function definition is σ(x) = ex, where ex=x∗−xis the tracking error, being x∗the desired value of the state space vector. It is immediate to demonstrate that if the control guarantees that σ(x) = 0, the desired and real state space vector are equal. Therefore, the control objective is to ensure that the function σ(x) = x∗−xalways converges to the space region defined by σ(x) = 0. This condition is satisfied when ˙σ(x)σ(x)<0 (1.2) is fulfilled. The previous consideration can be proved just taking the Lyapunov function candidate V(x)=0.5σ(x)2, where the first time derivative yields ˙ V(x) = ˙σ(x)σ(x). As the function V(x)=0.5σ(x)2fulfils all the conditions to be a Lyapunov function [8], the condition defined in (1.2) is enough to ensure that σ(x) = 0 is an attractive region for σ(x). When the switching function σ(x) is on the desired switching surface σ(x) = 0, it is called that the system is under sliding motion. Nevertheless, according to the system relative degree [7], the construction of the surface may include higher orders elements of the tracking error. It is simple to figure out that the way to impose the condition ˙σ(x)σ(x)<0 will be through ˙σ(x), since σ(x) does not depend on control. In other words, in order to enforce sliding motion on σ(x) = 0, the control action, u, has to appear in ˙σ(x). The equation (1.3) shows the case where the switching function includes the tracking error and its first time derivative σ(x) = φ1ex+φ2˙ex(1.3) where φ1, φ2are strictly positive constants. In the initial case, where σ(x) = ex, when the system is under sliding motion automatically implies that ex= 0 and xtracks x∗without dynamics. This fact does not happen for the switching function defined in (1.3), since, when the switching function is on the surface σ(x) = 0, the following first order linear differential equation governs the error dynamics: φ1ex+φ2˙ex= 0 (1.4) Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 7 CHAPTER 1. INTRODUCTION which has ex= 0 as unique asymptotically stable equilibrium point. Therefore, it is clear that the error dynamics under sliding motion will be determined by the selected switching surface. Regardless of the switching function order, the control law of uhas to ensure that condition (1.2) always holds. The previous condition will be guaranteed by a discontinuous control law, taking the available discrete values, previously defined (u+and u−). It is very common to use a sign function as the control, since this function generates the values 1 or -1, which match with values for u+and u−for several converters. In general, the control law will be of the form u=sign(σ(x)) (1.5) where the function sign() provides 1 when σ≥0 and -1 when σ < 0. Nevertheless, some converters have control input states that do not correspond to 1 or -1, being for instance 1 or 0. In these cases, the control law will be modified accordingly. In any case, since uis discontinuous, from a theoretical point of view, this signal will switch at infinite switching frequency when σ(x) is zero. This assumption allows to analyse the ideal sliding dynamics, since at infinite switching frequency the switching function will be placed exactly on σ(x) = 0. Studying the ideal sliding dynamics is the process to derive the dynamics of the system defined in (1.1) with the dynamics imposed by σ(x) = 0. This task can be carried out using the equivalent control method proposed by Professor Vadim Utkin [7]. The equivalent control, ueq, is defined as the solution of the equations σ= 0,˙σ(x, ueq) = 0.(1.6) Taking as switching function an expression depending on the error, ex=x−x∗, as: σ(x) = µ(ex),(1.7) the switching function first time derivative is found as: ˙σ(x) = ∂σ(x) ∂x f(x) + ∂σ(x) ∂x g(x)u. (1.8) According to (1.6), the equivalent control can be obtained solving the equation 0 = ∂σ(x) ∂x f(x) + ∂σ(x) ∂x g(x)ueq,(1.9) which yields ueq =−∂σ(x) ∂x g(x)−1∂σ(x) ∂x f(x).(1.10) Notice that (1.10) implies a continuous-time solution, which ucannot attain. Finally, u is replaced by ueq in (1.1) in order to find the ideal sliding dynamics. Placing (1.10) into (1.1), one gets: ˙x=f(x) + g(x)ueq.(1.11) 8 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 1. INTRODUCTION Furthermore, according to [7], the space region where the sliding motion occurs, which is called sliding domain, is characterized by the inequality u−< ueq(x, t)< u+.(1.12) 1.4 Real Sliding Motion The real sliding motion can be understood as the system dynamics that arises when the control action, u, is forced to operate at finite switching frequency. The employment of a sign function in real systems leads to a finite switching frequency, due to unmodelled dynamics and delays. However, this switching frequency can be too high for some systems, like power converters. Substitute the sign function on (1.5) by a comparator with hysteresis, allows to bound the switching frequency. The control law, therefore, will be of the form: u=u+if σ < −∆, u−if σ > ∆,(1.13) being ∆ the hysteresis width. This approach can be found in sliding mode control literature in order to study the real sliding dynamics [10–12]. From a rigorous point of view, the previous control law is not defined within the hysteresis bands, so an alternative expression can be used instead: u=u+if σ < −∆ or (|σ|<∆ & ˙σ > 0) u−if σ > ∆ or (|σ|<∆ & ˙σ < 0) .(1.14) The different behaviour of the system trajectories on the phase plane can be observed in Figure 1.3. ex ˙ex ex ˙ex (a) (b) σ(x) = 0 σ(x) uu σ(x) = 0 σ(x) ∆ −∆ σ(x)σ(x) ∆ −∆ Figure 1.3: Phase plane system trajectories. (a) Ideal sliding motion. (b) Real sliding motion within a boundary layer ∆. Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 9 CHAPTER 1. INTRODUCTION As Figure 1.3 shows in the right side, the real switching function, σ(x), is not on the sliding surface, σ= 0, but chatters in its vicinity. In this scenario, the condition |σ(x)|<∆ holds. It is a designer task to keep the hysteresis band value small enough, in the way that this dynamics can be neglected with respect to the low frequency dynamics, determined by the equivalent control. Analysing the high frequency dynamics of σ(x) provides the expression that allows to study and control the switching frequency of the control action. A good approach consists in defining the control action as a continuous low frequency component, which is ueq, plus a high frequency component, uhf , which takes values in the set {u−, u+}. u=ueq +uhf (1.15) Let us just combine the expressions (1.15) and (1.8). ˙σ(x) = ∂σ(x) ∂x f(x) + ∂σ(x) ∂x g(x)ueq +∂σ(x) ∂x g(x)uhf (1.16) By definition, ˙σ(x, ueq) = 0, therefore ˙σ(x) = ∂σ(x) ∂x g(x)uhf (1.17) The motion of σ(x) in the vicinity of σ(x) = 0 is governed by (1.17). From this expression it is clear that this dynamics depends on the system. Of course, the dynamics of the state space vector can be also stated as: ˙x=f(x) + g(x)ueq +g(x)∂σ(x) ∂x g(x)−1 ˙σ(x).(1.18) The switching function trajectories within the hysteresis band are governed by the last term of the right hand side of equation (1.18). Figure 1.4 depicts the expected behaviour. σ(x)=0 ∆ −∆ t1t2 t T σ(x) ˙σu=u+ ˙σu=u− Figure 1.4: Switching function behaviour within the hysteresis band in the vicinity of σ(x) = 0. 10 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 1. INTRODUCTION From Figure 1.4 the switching period of the control action is derived: T=t1+t2=2 ∆ ˙σu=u+−2 ∆ ˙σu=u− (1.19) Expression (1.19) can be written in different ways. One interesting approach is to formulate it as a function of the equivalent control. Recalling (1.15) and (1.17), one gets: ˙σ(x) = ∂σ(x) ∂x g(x)(u−ueq) (1.20) and, therefore, assuming that u+= 1 and u−=−1, (1.19) boils down to T= 4 ∆ 1 (1 −u2 eq)∂σ(x) ∂x g(x)−1 .(1.21) Notice that from (1.21) the hysteresis band of the comparator can be calculated as a function of the equivalent control and the desired switching period, T∗, as: ∆ = ∂σ(x) ∂x g(x)T∗ 4(1 −u2 eq).(1.22) However, this approach has an important limitation, which will be discussed later. Moreover, expression (1.21) confirms the well-known problem of the SMC applied to switched power converters, the switching frequency is variable and system dependent, as (1.21) states. A very important aspect should be remarked at this point. In the previous methodology, it has been assumed that the slopes of σ(x) remain constant along the switching interval, that is ¨σ(x) = 0. This is a very reasonable assumption that is often taken in the sliding mode control literature [11,13,14]. The basis of this hypothesis relies in consider the switching period of the control action small enough with respect to the system time constants. If this consideration holds, it is reasonable to assume that the switching function slopes are locally constant during a switching interval. Indeed, the switching frequency in the power converters are designed as high as possible, due to the reason introduced in Section 1.1, which perfectly fits with this assumption. 1.5 Proposed Solutions to the Variable Switching Frequency Problem At this stage, some solutions proposed in the literature in order to set the switching frequency in the power converters under SMC are reviewed. Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 11 CHAPTER 1. INTRODUCTION 1.5.1 Variable Hysteresis Band An intuitive solution is to adjust the comparator hysteresis band using the expression (1.22), modifying its level in accordance with the system state. Such approach has been proposed by different authors [15–21]. An equivalent scheme is depicted in Figure 1.5. uEquation (1.1) x∗ x σ(x) x ∆ ∆ −∆ Equation (1.22)T∗ ueq Figure 1.5: System for operating at fixed switching period under sliding motion based on a variable hysteresis band. From Figure 1.5, and as equation (1.22) shows, the hysteresis computation requires a high level of the plant knowledge since the term ∂σ(x) ∂x g(x) should be known. Furthermore, the measurement of the equivalent control or, alternatively, its estimation based on expression (1.10) should be also performed, which entails an evident complexity. The procedure provides good results in general, although it is difficult to achieve the desired switching frequency accurately, due to the complexity for generating the hysteresis. This complexity is, in fact, its main drawback. Additionally, it is evident that if in the hysteresis calculation some system parameters are assumed known and constant, the system will not be robust in the face of parametric variations. In order to improve the robustness, additional sensors and/or observers have to be included to get a proper adaptation of the hysteresis band amplitude, leading to a further system complexity, decreasing the reliability and increasing the cost up to unmanageable levels. 1.5.2 External Synchronization Signal Fixed switching frequency can also be achieved by using an external signal to force the switching instants [22,23]. The idea is sketched at Figure 1.6, specifically in the left side. The synchronization signal, D(t), is added to the switching function in the way that such signal controls the switching events. As it is depicted in the Figure, the peak values of signal D(t) should be higher than the hysteresis values, ∆, as D(t) can control the 12 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 1. INTRODUCTION switching events. Through a complicated tuning process, it is possible to adjust the signal D(t) fixing the switching frequency of the control action. However this process is extremely sensitive, and it is usual that steady-state errors appear in the system when the working conditions change. The right part of Figure 1.6 tries to show this phenomenon. Notice how the switching function (grey lines) does not reach the upper bound of the hysteresis band, entailing an average value different from zero of σ(x), leading to the aforementioned steady-state error. ∆ σ(t) D(t) ut Tk 2∆ -∆ t Tk σ(t) D(t) Figure 1.6: External Signal Synchronization scheme for a fixed frequency sliding mode control. Moreover, this approach needs some additional hardware on the controller in order to generate the synchronization signal. As a consequence, the usage of this method is, in general, not recommended. 1.5.3 Zero Average Dynamics As it was introduced in the previous Section, and noted in different works available in the literature, a switching function describing piecewise linear behaviour within a symmetric hysteresis band comparator implies that its average value, along the switching interval, is zero. This is the concept exploited by the Zero Average Dynamics (ZAD), which was presented in [24]. The method computes a duty cycle that guarantees zero T-periodic mean value of the switching function, with Tdenoting the switching period. The Figure 1.7 shows this idea. From the Figure, it is possible to develop the formulation delivering the duty cycle to be used in the next switching interval that leads to a zero averaged value of σ(x). The results are shown in Table 1.1. Therefore, fixed switching frequency is reached in the steady-state, and the averaged behaviour is close to the ideal sliding mode one. The ZAD strategy has been successfully implemented in [25]. The results presented therein show a good performance of the ZAD, but also point out the requirement of a fast digital processor to solve the complex calculations involved in the duty cycle computation, as expressions in Table 1.1 corroborate. In fact, such computation complexity constitutes the main drawback of ZAD-based SMC fixed switching frequency implementations. Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 13 CHAPTER 1. INTRODUCTION T/2T/2 ˙σ(x, t)u=u+ ˙σ(x, t)u=u− dKT σ(x, t) Tk−1Tk Figure 1.7: Zero Average Dynamics trajectories detail. Table 1.1: Zero Average Dynamics control action formulas σ(x(Tk), Tk) and σ(x(Tk), Tk) + T 2˙σ|k,u+≥0u(Tk) = u+;dk= 1 σ(x(Tk), Ttk) and σ(x(Tk), Tk) + T 2˙σ|k,u+<0u(Tk0) = u+;dk= 1 −v u u u t˙σ|(K,u+)−2|σ[x(Tk, Tk)]| T ˙σ|(K,u+)+˙σ|(K,u−) σ(x(Tk), Tk) and σ(x(Tk), Tk) + T 2˙σ|k,u+≤0u(Tk) = u−;dk= 1 σ(x(Tk), Tk) and σ(x(Tk), Tk) + T 2˙σ|k,u+>0u(Tk0) = u−;dk= 1 −v u u u t˙σ|(K,u+)−2|σ[x(Tk, Tk)]| T ˙σ|(K,u+)+˙σ|(K,u−) 14 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 1. INTRODUCTION 1.5.4 PWM-Based SMC This proposal is based on the use of PWM at fixed switching frequency to implement the so-called PWM-SMC. Initially proposed in [26,27], the method directly implements the equivalent control and obtains the switching instants by comparing the equivalent control with the fixed frequency sawtooth waveform at the PWM. The method is equivalent to the traditional PWM implementation according to a linear controller design, but using the equivalent control ueq instead of the duty cycle provided by the linear controller. The corresponding scheme is depicted in Figure 1.8. + - ueq T T u Figure 1.8: PWM-based sliding mode controller. It should be noted that the expression for the equivalent control depends on the system state and on its parameters for a given working conditions, as (1.10) states. It is important to notice that this equation holds from the ideal point of view. The results presented in [28] show overall good performance, but it should be highlighted that the same solution can also be derived by calculating the required duty cycle to obtain the desired system dynamics. Indeed, from our point of view, the equivalent control is more a theoretical concept than a practical method. From its definition, the equivalent control is the continuous control action that places the system trajectories exactly on the sliding surface. In a practical implementation, this includes whatever unmodelled dynamics, delays and uncertainties in the power converter. Besides, power converters commonly suffer external disturbances, which generally can only be bounded. In other words, there is no way to determine the equivalent control a priori. The equivalent control could be measured low pass filtering the hysteretic control action enforcing sliding motion in a certain switching surface (|σ(x)|<∆), but not vice versa. Hence, equation (1.10) should be used only for theoretical issues. Moreover, in this method, some sliding mode properties, such as order reduction or robustness in the face of disturbances, could be lost. 1.5.5 Additional Control Loop Hysteretic controllers are often applied to power converters without using SMC theory. However, like occur in SMC approaches, the switching frequency becomes variable. Indeed, in most of the hysteretic controllers, sliding motion naturally occurs. Under this research field, some interesting solutions in order to fix the switching frequency were provided [29–32]. These proposals are based on adding an additional loop in order to properly adjust the hysteresis band, as Figure 1.9 shows. The system uses a period sensor or a Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 15 Chapter 2 Modelling and stability analysis of the switching frequency control loop This chapter deals with the modelling and the analysis of the structure for the SFC presented in the Chapter 1. As it was previously introduced, the control methodology is based on the adjustment of the hysteresis band value of the comparator through an additional control loop, as Figure 1.10 shows. Since the control loop will vary such hysteresis band, the analysis of the switching period presented in Section 1.4 needs to be revisited. Furthermore, this structure has an additional peculiarity: the SFC structure used to adjust the hysteresis band value will affect the relation between the applied hysteresis and the corresponding switching period. In this thesis two different approaches have been applied, namely: 1. A discrete-time approach, where the hysteresis band can be updated just once per switching cycle. 2. A continuous-time approach, assuming that the hysteresis band will be a time-varying signal. The chapter is structured as follows: In Section 2.1, the analysis of the time invariant hysteresis case done in Section 1.4 is revisited, being used to define some essential parameters for the analysis performed hereafter of the SFC structure. In Section 2.2 the discrete-time approach is presented and the two most expectable working scenarios in power converters, the regulation and the tracking case problems, are analysed. Finally, in Section 2.3 a continuous-time approach for the SFC, particularizing it only to the regulation control problem, is presented. 2.1 Open loop case: The fixed hysteresis band comparator The time invariant hysteresis band amplitude case was already analysed in the Section 1.4. However, the resulting expression for the switching period (see equations (1.19), (1.21)) is Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 23 CHAPTER 2. MODELLING AND STABILITY ANALYSIS OF THE SWITCHING FREQUENCY CONTROL LOOP redefined at this point, including some important definitions. Assume that a switching function, σ(x), has been defined in order to enforce sliding motion in the subspace σ(x) = 0, with the objective to control the dynamics of a power converter (voltage, current,..). Suppose also that the control law provided by the SMC theory, which guarantees convergence of σ(x) to σ(x) = 0, is implemented using a fixed band hysteresis comparator, as (1.13) states. Once the sliding motion is reached, the expected behaviour of σ(x) within the hysteresis band is shown in Figure 2.1. Notice that, as it was justified in Chapter 1, the switching function σ(x) has been represented by straight lines. Under these conditions, the k-th switching period, Tk, can be deduced from Figure 2.1. The subindex kholds for any switching interval that will occur. Tk=T+ k+T− k= 2∆ ρ+ k−ρ− k,(2.1) where ρ+ kand ρ− kare defined as the inverses of ˙σ(x) for each control input state: ρ+ k:= 1 ˙σ(x)+ k , ρ− k:= 1 ˙σ(x)− k .(2.2) σ(x)=0 ∆ −∆T+ kT− k t Tk σ(x) ˙σku=u+ ˙σku=u− Figure 2.1: Switching function behaviour within a time invariant hysteresis band. The time derivative of σ(x) was already shown in the Chapter 1, specifically by equation (1.20). From the definitions of ρ± k, it is required to evaluate ˙σ(x)±at any switching interval kfor the two possible control input states, u+and u−. Therefore the expressions for ˙σ(x)± k respond to ˙σ(x)+ k=∂σ(x) ∂x k g(x)ku+−ueq(x)k(2.3) ˙σ(x)− k=∂σ(x) ∂x k g(x)ku−−ueq(x)k(2.4) where the subindex kholds for the corresponding sampled counterparts of the original 24 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 2. MODELLING AND STABILITY ANALYSIS OF THE SWITCHING FREQUENCY CONTROL LOOP signals in the specific time instant. In Chapter 1, it was introduced that for the sliding motion to exist, the equivalent control, ueq, has to be within u−and u+(see inequality (1.12)). As a consequence, the right hand sides of expressions (2.3) and (2.4) are always different from zero. Furthermore, taking into account the fulfilment of the transversality condition (see for details [33]), which defines ∂σ(x) ∂x g(x)6= 0,(2.5) it is obvious that condition ˙σ(x)± k6= 0 ∀kholds under sliding motion. Therefore, from (2.3), (2.4) and (2.5) it results that the values of ρ+ kand ρ− kalways exist under sliding motion, according to their definitions in (2.2). Assuming that (2.5) is not only non null but positive, and u+>0 and u−<0, it follows immediately that ρ+ k>0, ρ− k<0,∀k≥0.(2.6) The analyses developed in the next subsections assume that ρ+ kis always positive definite, meanwhile ρ− kis negative definite, as the inequalities in (2.6) state. However, in some applications the term ∂σ(x) ∂x g(x) could result negative definite. In these cases, in order to keep the methodologies developed in this work, a slightly different definition for ˙σ(x)± kmust be used as: ˙σ(x)+ k=∂σ ∂xk g(x)ku−−ueq(x)k(2.7) ˙σ(x)− k=∂σ ∂xk g(x)ku+−ueq(x)k.(2.8) Which such alternative definitions, the inequalities defined in (2.6) always hold. 2.2 Closed-loop case: A discrete-time modelling. According to (2.1), the switching period becomes in a series of discrete-time measurements. Thus, it is reasonable to modify the hysteresis band amplitude only at the beginning of each switching period. Assuming that the hysteresis band amplitude is updated at the beginning of each switching interval and remains constant up to the next switching event, the expected switching function behaviour within the hysteresis band can be depicted as the Figure 2.2 shows. In order to stablish a standard methodology, the beginning of the switching interval will be considered when the slope of σ(x) changes from the negative value to the positive one, as Figures 2.1,2.2 display, regardless of the control action state. As it was commented in the previous Section, the values of ρ+ kwill be always strictly positive and ρ− kstrictly Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 25 CHAPTER 2. MODELLING AND STABILITY ANALYSIS OF THE SWITCHING FREQUENCY CONTROL LOOP σ(x)=0 ∆k −∆k T+ kT− k t Tk σ(x) ˙σku=u+ ˙σku=u− ∆k−1 −∆k−1 Figure 2.2: Switching function behaviour within a time-varying hysteresis band. negative. Therefore the beginning of the switching interval is fully determined. Since, in this approach, the hysteresis value will change between two consecutive switching intervals, the expression found in (2.1) should be revisited. From Figure 2.2 the new relation between ∆kand Tkcan be found Tk=T+ k+T− k=ρ+ k(∆k+ ∆k−1)−2ρ− k∆k= ˆρk∆k+ (˜ρk−ˆρk) ∆k−1,(2.9) with ˆρk:= ρ+ k−2ρ− k, ˜ρk:= 2 ρ+ k−ρ− k. Expression (2.9) constitutes the discrete-time model for the switching frequency control loop, and relates the k-th switching period and the hysteresis band value. Due to the used methodology to update the hysteresis band, such relation includes the hysteresis band values in the interval kand k−1. In Figure 2.3 the resulting model of the SFC, which includes the model found in (2.9), is shown. From Figures 2.2,2.1 an important aspect to be taken into account can be deduced. It is easy to figure out that the measured value of Tk will not be available until the k-th switching interval ends. This phenomenon is included to the system through an asynchronous discrete delay, z−1(see bottom part of Figure 2.3). Tk ∆k Tk−1 Equation (2.9) z−1 T∗ SFC Figure 2.3: Detail of the switching frequency control loop. Discrete-time model. At this point, using the loop shown in Figure 2.3, the closed-loop behaviour of the 26 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 2. MODELLING AND STABILITY ANALYSIS OF THE SWITCHING FREQUENCY CONTROL LOOP system can be studied. Let us define the switching period error as e:= T∗−T, where T∗ is the reference switching period. From (2.9), the error equation of the SFC can be easily found as ek−ek−1= ˆρk(∆k−1−∆k) + ρ+ k−1(∆k−2−∆k−1) + (˜ρk−1−˜ρk) ∆k−1.(2.10) Once the discrete-time model is achieved, the control objective will be the proper design of the SFC in such a way that ekconverges to zero, thus implying Tktends to T∗. The controller design procedure and the corresponding stability condition derivation will be carried out in the next subsections particularizing expression (2.10) in two different working conditions, namely: the regulation case and the tracking case. The analysis will be based on to substitute the expression provided by the SFC, which will be of the form ∆k=f(ek), in equation (2.10) and analyse the conditions for ekto converge to zero as k→ ∞. It is clear that a specific controller expression, ∆k=f(ek), could deliver arbitrary values in presence of power converter transients, compromising the SMC itself (∆kcould become too high or negative). Moreover, in Chapter 1it was explained that in order to approximate the real sliding motion to the ideal one, it is required that the ∆kvalue should be small enough (see Section 1.4). As a consequence, it is important to define an allowable range of the hysteresis band values used by the SFC in order to preserve the good performance of the SMC. This idea is expressed in the Remark 1. Remark 1. Arbitrarily hysteresis band values may take the system far away from the real sliding regime. Hence, a specific range I∆:= [∆min,∆max]such that ∆k∈ I∆,∀k≥0, has to be defined for preserving the existence of the sliding motion. In order to establish the suitable hysteresis range for a certain system, the following design criterion is proposed: ∆min may be obtained from the maximum allowable switching frequency, while the maximum acceptable ripple for the state variables would be used to set ∆max and, consequently, the minimum switching frequency. In turn, the switching frequency reference should be accordingly selected within these limiting values. 2.2.1 The regulation case In the power electronics field, it is very common to regulate the output voltage of a given converter to a DC value. In some applications the output current of a converter is also controlled to a fixed value, acting the converter as a constant current source. These two cases correspond to a regulation case. In the regulation case the state vector reference, x∗, is constant. Assume that a SMC has been designed to regulate such voltage or current, and that through a proper design of the I∆(see Remark 1), the oscillations of σ(x) in the vicinity of σ(x) = 0 are small, such that x≃x∗holds. Therefore, considering the state vector xconstant, the switching function derivatives and their inverses are also constant under steady-state sliding motion. Hence, from a certain discrete-time instant k0the following relations are fulfilled: ρ± k=ρx∗, u±:= ρ± ∗,ˆρk:= ˆρ∗,˜ρk:= ˜ρ∗,∀k≥k0,(2.11) Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 27 CHAPTER 2. MODELLING AND STABILITY ANALYSIS OF THE SWITCHING FREQUENCY CONTROL LOOP for a given steady-state working point. With these definitions, (2.10) can be simplified up to the following expression: ek−ek−1= ˆρ∗(∆k−1−∆k) + ρ+ ∗(∆k−2−∆k−1).(2.12) Now the goal is to properly design the SFC generating the value for ∆kas a function of the measured switching period error, ek, in order to get the convergence of the error to zero. Let us try as SFC controller a pure discrete-time integrator, which corresponds to: ∆k= ∆k−1+γek−1,(2.13) being γ > 0 the integral constant. Replacing the control action, (2.13), in the closed-loop error dynamics, (2.12), one gets ek= (1 −γˆρ∗)ek−1−γρ+ ∗ek−2.(2.14) The arisen equation in (2.14) is an homogeneous linear difference equation, which implies that the only solution of the equilibrium is ek= 0. The integral gain γ, is the design parameter which should be selected to get the equilibrium point of (2.14) stable. In order to rigorously define the conditions for γto achieve a stable behaviour of the equilibrium point ek= 0, the Assumption Ais defined. Assumption A. A real sliding motion has been enforced over a specific switching surface, σ(x) = 0, in such a way |σ(x)|<∆with a given control law defined as (1.5)depicts. Moreover, after the sliding mode transient, the system reaches the steady-state where x≃ x∗, and the relations shown in equations (2.11)are fulfilled. Once the Assumption Ais defined, the Theorem 1states the condition for γto achieve stability of the SFC. Theorem 1. Let Assumption A be fulfilled, and let the hysteresis band amplitude, ∆k, be updated according to (2.13). If γfulfils that 0< γ < min nρ+ ∗ −1,ρ− ∗ −1o,(2.15) with ρ± ∗defined in (2.11), then the switching period, Tk, converges asymptotically to the reference value, T∗, in the steady-state. Proof. It follows applying Jury stability criterion [34] to the characteristic polynomial associated to the difference equation (2.14), which results in p(z) = z2+z(γˆρ∗−1) + γρ+ ∗.(2.16) The conditions to be fulfilled, according to the criterion, are: 1. p(1) >0→γ > 0 28 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 2. MODELLING AND STABILITY ANALYSIS OF THE SWITCHING FREQUENCY CONTROL LOOP 2. p(−1) >0→γ < |ρ− ∗|−1 3. |γˆρ∗−1|<|γρ+ ∗| → γ < |ρ+ ∗|−1 In this case, since the values of ρ± ∗can be considered parameters, the ztransform has been used in order to find the stability conditions of (2.14). Finally, the equivalent model in the zdomain is shown in Figure 2.4. Notice how the inherent delay produced when the switching period is measured has been considered for the proper implementation of (2.13). Tk eq (2.9) T∗ γ∆k Tk−1z−1 z−1 Equivalent to (2.13) Figure 2.4: Equivalent model in the zdomain of the control loop for the discrete-time approach in the regulation case. 2.2.2 The tracking case The power converters usually have to work connected to the AC grid, sometimes injecting power from a renewable plant, sometimes consuming power from the grid. In these cases, the power converters are working with references (of voltage or currents) that vary with time. These applications are classified as tracking control problems. When the SMC is tracking a time-varying reference, x∗=f(t), it exists a variation of the switching function time derivatives with k, i.e. ρ+ k6=ρ+ k−1,ρ− k6=ρ− k−1. As a consequence, the simplification made in (2.11) does not apply. Before progressing with the tracking case stability analysis, an important consideration about the variations of ρ± kwith kis discussed in Remark 2. Remark 2. Although in the tracking case it is assumed that the values of ρ± kvary with k, such variations must be sufficiently slow so that these values can be considered locally constant during a switching interval. This fact will be achieved when the effective switching period of the control action is small enough with respect to the system time constants. This hypotheses is equivalent to assume that the behaviour of σ(x)in Figures 2.1,2.2 can be considered piecewise linear even when the system is tracking a time-varying reference x∗=f(t). Therefore, the switching period has to be small enough as the condition: df(t) dt ≈f(Tk)−f(Tk−1) Tk→0 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 29 CHAPTER 2. MODELLING AND STABILITY ANALYSIS OF THE SWITCHING FREQUENCY CONTROL LOOP is met. Notice how in this case the last term on the right side in equation (2.10)does not vanish. The chosen SFC structure for the regulation case was a pure discrete-time integrator (equation (2.13)), which was characterized by: ∆k= ∆k−1+γek−1. Keeping such structure as SFC, and without the aforementioned simplifications for ρ± k, the new expression for the error dynamics in the tracking case is found. The dynamics is governed by (2.17): ek= (1 −γˆρk)ek−1−γρ+ k−1ek−2+ ∆k−1(˜ρk−1−˜ρk).(2.17) The difference between expression (2.12) and (2.17) is the last term on the right side of expression (2.17). It should be highlighted that this term makes the expression (2.17) non-homogeneous and, as a consequence, it losses ek= 0 as equilibrium solution. Thus, in order to recover the homogeneous characteristic of expression (2.14), this work proposes the design of a feedforward action in order to cancel this term in the closed-loop dynamics. This idea is sketched in Figure 2.5. Tk eq (2.9) T∗ γ∆k Tk−1z−1 z−1 Equivalent to (2.19) Ψk Ωk Figure 2.5: Switching frequency regulation control loop with feedforward action. The inherent time delay due to the switching period measurement is represented by z−1. The analysis is simple, the new control action includes the feedforward term Ωk, as ∆k= Ψk+ Ωk,(2.18) where Ψkkeeps the structure of a discrete-time integrator Ψk= Ψk−1+γek−1.(2.19) The value for Ωkis derived placing (2.18) in (2.10) and equalling all the terms that do not depend on ekto zero. The result of the proposed procedure is given by equation (2.20): Ωk=ˆρk−1−ρ+ k ˆρk Ωk−1+ρ+ k−1 ˆρk Ωk−2+˜ρk−1−˜ρk ˆρk Ψk−1.(2.20) 30 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 2. MODELLING AND STABILITY ANALYSIS OF THE SWITCHING FREQUENCY CONTROL LOOP The proposed modified structure for the tracking control problem is depicted in Figure 2.6, according to (2.18), (2.19) and (2.20), where the detail of Ωkcomputation has been highlighted. Tk eq (2.9) T∗ γ∆k Tk−1z−1 z−1 Equivalent to (2.19) Ψk Ωk ρ± k eq (2.20) Figure 2.6: Switching frequency regulation control loop with feedforward action. Detail of new controller structure including the feedforward action. From the obtained expression for Ωk, some considerations should be taken into account, which are mainly noted in Remark 3. Remark 3. An important remark should be made at this point with regards to the expression found for Ωkin (2.20). Looking carefully the expression, it is simple to note how the k-th value for Ωkdepends on the k-th values of ˆρ,˜ρand ρ+. Such result is not realizable from a practical point of view since it depends on samples that are not available yet. Besides, the values ˆρ,˜ρand ρ+should be properly estimated. As a consequence, some approximations will be required when the proposed controller is implemented, both in the simulation and in the experimentation cases, which will be discussed later. Assuming that the expression for Ωkcan be properly obtained, the final error dynamics for the proposed controller with feedforward action can be evaluated. It follows replacing (2.18), (2.19) and (2.20) in (2.10), which provides the resulting closed-loop error dynamics in (2.21). ek= (1 −γˆρk)ek−1−γρ+ k−1ek−2.(2.21) Now the equation of the switching period error boils down to an homogeneous time-varying discrete-time linear system, recovering ek= 0 as the equilibrium solution. It should be noted that ˆρk,ρ+ k−1cannot be treated as constant parameters in this case. Although the ρ± kare not constant with k, it can be defined an expectable range of values related to a certain working conditions. Under steady-state sliding motion, the state vector profile x∗=f(t) will produce the following time-varying values: ρ+ k=ρk(x∗(t), u+) = ρ+ ∗k, ρ− k=ρk(x∗(t), u−) = ρ− ∗k, ˆρk= ˆρk(x∗(t)) := ˆρ∗ k, ˜ρk= ˜ρk(x∗(t)) := ˜ρ∗ k, (2.22) Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 31 Chapter 3 A case of study: SFC design and simulation results In this Section, a very simple system is introduced in order to show the complete process for the SFC loop design, including the aforementioned practical approaches. The design and simulation of the proposed strategies will be tested in a linear system. Firstly, an sliding mode controller will be designed using the classical approach, through the equivalent control method, as it was introduced in Sections 1.3 and 1.4. Such SMC controller will be applied for both regulation and tracking tasks. The next step shows the derivation of the expressions for the ρ±, allowing to properly design the SFC in the different cases, namely: the discrete-time approach for regulation and tracking cases, and the continuoustime approach in a regulation scenario. Finally, the simulations results of all the designed approaches are presented. 3.1 Design of the sliding mode controller Let us introduce the single-input single-output linear system ˙x1=−x1+x2,(3.1) ˙x2=−x1+Mu, (3.2) where Mis a system parameter, which could be interpreted as the system gain and uis the discontinuous input, taking values in the discrete set {−1,1}. The control objective is to control the dynamics of x2to a desired behaviour, x∗ 2(t). Taking into account the relative degree of the system (3.1), (3.2) (see Section 1.3), a switching surface based on the tracking error is enough. Therefore, the switching surface is chosen as (3.3) states. σ(x, t) := x2−x∗ 2(t) = 0.(3.3) Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 39 CHAPTER 3. A CASE OF STUDY: SFC DESIGN AND SIMULATION RESULTS The desired dynamics for x2is x∗ 2(t) := A+Bsin ωt, (3.4) with A, B ≥0. It should be noticed that the switching function can create a regulation control case, (B= 0), or a tracking control case, (B6= 0). The sliding motion will be enforced in |σ(x, t)|<∆ with a hysteretic control law of the type shown in equation (1.13), being u+= 1 and u−=−1. As it was presented in Section 1.3, the sliding mode control design is based on the first time derivative of the switching function. Once the first time derivative is found, the equivalent control can also be derived (see expressions (1.9) and (1.10)). These operations result in: ˙σ=Mu −(x1+ ˙x∗ 2),(3.5) and, hence, ueq =1 M(x1+ ˙x∗ 2).(3.6) From (3.5) the control law can be derived. The control law enforcing σ˙σ < 0 is: u=u+if σ < −∆, u−if σ > ∆,(3.7) Recalling the sliding mode domain condition defined in (1.12), the ideal sliding motion on σ= 0 (or the real sliding motion on |σ(x, t)|<∆) can be enforced if −1<1 M(x1+ ˙x∗ 2)<1 holds. The ideal sliding dynamics is determined placing the equivalent control in the state space equations. Therefore using (3.6) in (3.1), (3.2) one gets x2=x∗ 2(t),(3.8) ˙x1=−x1+x∗ 2(t).(3.9) The dynamic of x2is directly imposed by the sliding motion, which is equal to x∗ 2(t). The remaining dynamics characterizing x1on sliding motion, x∗ 1, is asymptotically stable. Hence, it is also possible to find the steady-state solution of the differential equation (3.9) in the time domain, x∗ 1(t), which is x∗ 1(t) = A+B 1 + ω2(sin ωt −ωcos ωt).(3.10) Once the ideal steady-state sliding regime x∗= (x∗ 1, x∗ 2)>has been reached, the switching function time derivatives allow to find the values for ρ± ∗, which can be derived using (3.4), 40 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 3. A CASE OF STUDY: SFC DESIGN AND SIMULATION RESULTS (3.5) and (3.10), resulting in ρ± ∗(t) = ±M−A−B 1 + ω2sin ωt +ω3cos ωt−1 (3.11) for the tracking case. The regulation case is determined by B= 0, and therefore ρ± ∗=1 ±M−A.(3.12) Notice that the previous values, ρ± ∗, are the key for the SFC design, as it was presented in Chapter 2. Thereupon, the simulation results for the system defined in (3.1),(3.2) with the different approaches presented in Chapter 2for the SFC will be shown. Specifically, the regulation control problem is simulated for the discrete-time approach (Section 2.2.1) and for the continuous-time one (Section 2.3). Analogously, the tracking case is simulated for the discrete-time approach using the feedforward action (Section 2.2.2). The simulations have been performed with Matlab-Simulink using the following parameters: M= 3, A= 1, and ω= 2π·0.02, while the desired switching period is T∗= 0.1 s. The parameter Bwill be selected depending on the used approach. 3.2 The regulation case in the discrete-time approach First of all, the design of the SFC for the discrete-time regulation control case is performed. In this case, the value of Bis set to 0. For the design of the γvalue of the SFC, it is required to use the values of ρ±at the corresponding steady-state sliding motion. As it was derived in (3.12), the values of ρ±at the specific steady-state sliding motion, ρ± ∗, can be evaluated with the given data for Mand A. Therefore, it stems from (3.12) that ρ+ ∗= 0.5 and ρ− ∗=−0.25. Recalling Theorem 1, the characteristic polynomial detailed in (2.16), for the case of study is found: p(z) = z2+z(γ−1) + 0.5γ. (3.13) Hence, following Theorem 1and equation (2.15), under the SFC with the structure given by (2.13), the closed-loop system of the SFC is stable for values in the range 0 < γ < 2. Furthermore, using the equation (3.13), the root locus for different values of γcan be studied. Figure 3.1 depicts the poles placement in the complex plane for different γvalues, together with the responses, in the time domain, of the system in front of a step reference (from T∗= 0.05 s to T∗= 0.1 s). From the Figure 3.1, it can be seen how values below γ= 0.3 deliver responses with the poles in the real axis, being such responses underdamped for values from 0.3 to 2. For γ > 2 the SFC loop reaches the unstable region. Notice that the response for γ= 2 is omitted due to the oscillating behaviour, showing the one for γ= 1.8 instead. It is obvious that, for this case, the system is close to the stability limit. In Figure 3.2 it can be seen the response of the SFC and the SMC for two values of γand T∗, specifically for γ= 0.1 and γ= 1 with a reference value that changes, again, Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 41 CHAPTER 3. A CASE OF STUDY: SFC DESIGN AND SIMULATION RESULTS -1 01 -1 1 Real Part Imaginary Part γ– + γ γ= 0.1 γ= 0.3 γ= 1 γ= 2 γ= 0.1 γ= 0.3 γ= 1 γ= 1.8 0.1 0.05 11.512 12.5 13 13.5 14 14.5 15 15.5 t(s) T∗, Tk 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -0.5 0.5 16 Figure 3.1: Left Plot: Root locus of p(z) for the simulation case of study. Right Plot: response in the time domain for different values of γ. 0.5 0 2 4 6 8 10 12 14 16 6810 12 14 16 0.995 1 Tk sx1sur σ(x), ∆k t(s) γ= 0.1 Tk Tm k x1 γ= 0.1γ= 1 T∗= 0.1sT∗= 0.05 sT∗= 0.1s 1.005 0.4 0.2 0 1 0.5 0 0.2 0 -0.2 Figure 3.2: Regulation: performance of the SFC, with the discrete-time approach, for different values of γand T∗. 42 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 3. A CASE OF STUDY: SFC DESIGN AND SIMULATION RESULTS between T∗= 0.1 s and T∗= 0.05 s. The values that take T∗and γduring the simulation are depicted in the top of Figure 3.2. The top plot of the Figure contains the switching period of the simulated system (3.1),(3.2), i.e. Tk, as well as the switching period arising from the model developed in Section 2.2.1 and detailed in Figure 2.4, which is labelled as Tm k. The mid plot shows the resulting dynamics of x1, while the x2one is directly derived from σ(see equation (3.3)), which is plotted at the bottom plot of the Figure. Notice in the zoomed view of this mid plot how the ripple of x1changes as the switching period does. This ripple is the consequence of the different value of ∆ generated by the SFC for different desired switching periods. However, in all the cases such ripples are small with respect to the DC value of x1, as expected. The third plot illustrates how ∆kis updated until Tkattains the reference value T∗, confirming that the desired switching period is achieved in steady-state. Moreover, the model response, Tm k, always fits with the real switching period, Tk, except for the start-up, where the steady-state sliding motion is still not reached and, as a consequence, ρ±6=ρ± ∗. Once the system reaches the steady-state sliding motion, i.e. ρ±=ρ± ∗, both responses perfectly match, even under switching period reference variations occurring at t=8s and t= 12 s; thus validating the mathematical model of the switching period behaviour presented in Section 2.2. Finally, it has to be noted that the hysteresis band amplitude achieves a constant value in the steady-state, as it can be inferred from (2.13). The asymptotic convergence of the state variable x1to the reference one x∗ 1is illustrated in the mid part of the Figure 3.2, thus corroborating an overall good performance of the full system (SMC+SFC). 3.3 The regulation case in the continuous-time approach In the continuous-time approach, the main difference with respect to the analysis performed in the previous section are focused in the SFC implementation. The analysis of the SMC is shared by the two approaches, since the control goal is the same for both, that is to regulate x2to x∗ 2. Even the evaluation of the parameters ρ± ∗are exactly the same, because such values are related to the same steady-state sliding motion. The analysis differences rely on the SFC approach. In this case, the stability conditions and the design guideline of the SFC were defined in Section 2.3. It should be remembered that in the continuous-time approach, an important assumption was taken (Assumption B) in order to develop the equivalent model of the SFC loop. Thus, the equivalent diagram in the sdomain shown in Figure 2.9 only can be used under certain conditions, when Assumption Bis met. At this stage, we assume that the error of the switching period in the simulation performed hereunder, will be limited by a known value, being this bound |ek|<0.05 s. Let us study the grade of fulfilment of the Assumption B, which is: γL|ek|<< min nρ+ ∗ −1,ρ− ∗ −1o,∀k≥k0. Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 43 CHAPTER 3. A CASE OF STUDY: SFC DESIGN AND SIMULATION RESULTS In this case, a factor of 20 is proposed. This implies the following relation 20 ·|ek|γL20 = min nρ+ ∗ −1,ρ− ∗ −1o. According to Section 3.2, the values for ρ± ∗are ρ+ ∗= 0.5 and ρ− ∗=-0.25, therefore: 20 ·|ek|γL20 = 2. Hence, γL20 =2 20 ·0.05 = 2. Thus, ensuring that the designed value for γLis within the range 0 < γL< γL20 , the model of Figure 2.9 holds. Additionally, Theorem 3allows to check if the equivalent model is stable. Since in the simulations presented hereafter the switching period sensor is ideal, the stability conditions must be determined by (2.43). This stability condition is stated as: γL<2 λT∗, being λ= 2 (ρ+ ∗−ρ− ∗), yielding γL<2 2 (0.5+0.25) ·0.1= 13.3. In the following simulation the switching period reference will be also varied to T∗= 0.05 s, thus the corresponding stability condition is also checked γL<2 2 (0.5+0.25) ·0.05 = 26.667. From this result, the stability for values of γL≤2 are guaranteed. Notice that in this case, it does not make sense to show the resulting root locus for different values of γLsince the model not always applies. The simulation shown in Figure 3.3 corresponds to the same simulation set-up of the previous Section, but in this case, the SFC is implemented by means of a continuous-time integrator, as it has already commented. In the test, two different values of γLare used, specifically γL= 1 and γL= 10, corresponding one to a value within the values where Assumption Bis fulfilled, and another outside this range. In the simulation, the ideal response of the switching period according to Figure 2.9, labelled as Tm k, is shown. During the simulations, two transients in the reference switching period are applied, specifically from T∗=0.1 s to T∗= 0.05 s and vice versa. In fact, from these transients arise the bound for the period error, previously used in the analysis performed over the γLvalues fulfilling Assumption B. During the start-up, the real period, Tk, and the ideal one, Tm k, do not match since the steady-state sliding motion has not been established, and ρ± k6=ρ± ∗. Once the system 44 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 3. A CASE OF STUDY: SFC DESIGN AND SIMULATION RESULTS 0 0 1 0 2 4 6 8 10 12 14 16 0 Tk σ(x) t(s) γL= 1 x1 T∗= 0.1sT∗= 0.1s T∗= 0.05 s Tk Tm k 6810 12 14 16 1 12 14 13 1.005 0.995 0.05 1.05 1 γL= 1 γL= 10 8910 0.1 0.06 -0.2 0.2 0.5 Figure 3.3: Regulation: performance of the SFC, with the continuous-time approach, for different values of γand T∗. reaches the steady-state, the responses Tk,Tm kconverge. Despite of the start-up, the behaviour during the transients at time instant t= 8 s and t= 12 s deserve a special attention. In the first transient (reference change at t= 8 s), the value for integral gain is γL= 1. During this transient, it is evident that Tkand Tm kmatch, thus validating the model and also confirming that, in this case, the Assumption Bis met. It should be noted that since the response is underdamped, the fulfilment of the hypothesis in the initial condition (|e0|<0.05 s) implies that the transients are exactly the same for Tkand Tm k. However, this does not happen in the second transient (reference change at t= 12 s), where a gain of γ= 10 is used, and, as expected, the responses of Tkand Tm kdo not overlap. Indeed, since the initial error does not meet the Assumption B, the responses Tk and Tm kwill not fit along the transient, as it could be observed from the zoomed view of the top plot. As a conclusion, with a reasonable known bound for |ek|∀k, it is possible to successfully design the SFC in the continuous-time approach, obtaining the expected dynamics for Tk. Notice that the no fulfilment of Assumption Bdoes not imply an unstable response, it just means that the developed model is not valid. Despite of the range of application of the model developed in Section 2.3, the switching period is properly regulated (even when γL=10) in the entire test, being negligible the impact over the SMC which regulates x2, and as a consequence, x1. In the zoomed area of the mid plot, it can be appreciated the different ripple levels in x1according to the variation in the switching period, being in both cases negligible when comparing with the DC value. Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 45 CHAPTER 3. A CASE OF STUDY: SFC DESIGN AND SIMULATION RESULTS 3.4 The tracking case A tracking performance is obtained when B6= 0. It should be noticed again that the SMC design is the same as the one presented in Section 3.1 so, it is omitted here for the sake of brevity. Since the reference for x∗ 2is a time-varying signal, the values for ρ∗ kwill also have dependency with k. It is simple to evaluate the expected values for ρ± ∗(t) just replacing the given values of M,A,Band ωin (3.11). The values for the ρ± ∗kwill be the corresponding ones of ρ± ∗(t) at the specific time instant k. As a result, the functions ˜ρ∗ kand ˆρ∗ k, can be derived, following their definitions in Section 2.2 (equation (2.9)). The expression for ρ± ∗(t) are: ρ± ∗(t) = ±M−A−B 1 + ω2sin ωt +ω3cos ωt−1 The values for M,Aand ωwere already defined, being the selected amplitude of the reference signal B= 0.5. Therefore: ρ± ∗(t) = ±3−1−0.5 1 + (2π·0.02)2sin (2π·0.02) t+ (2π·0.02)3cos (2π·0.02) t−1 . The previous expressions are the tools required for the SFC design in the tracking case. From these expressions ˜ρ∗ kand ˆρ∗ kare computed, and the stable range for γaccording to Theorem 2is found. With the goal of clarifying the result provided by Theorem 2, and how to apply it, a graphical approach is shown in Figure 3.4. In this plot can be seen, from top to bottom, the reference signal, x∗ 2(t), the resulting signals ρ+ ∗k,ρ− ∗kand the set of solutions obtained applying Theorem 2. The mid plot corroborates the expected variations of ρ± ∗kwith kand also shows that the time evolution of such values are slow with respect to the desired switching period. Once such signals are plotted, the solution according to Theorem 2can be graphically found, finding the minimum value and the maximum one of the corresponding solutions (see bottom plot of Figure 3.4). Notice how in the bottom plot the minimum value, γm, and the maximum one, γM, delimiting the stable range for γare graphically determined. These values result in 0.314 < γ < 1.0315. It is worth remarking that numerical simulations show that stability is indeed guaranteed for 0 < γ < 1.6. The reader has to keep in mind that the stability condition provided by Theorem 2is sufficient but not necessary. Once the stable range for γis found, the remaining task is to implement the feedforward action. Recalling the Subsection 2.2.2, the SFC for tracking cases includes a feedforward action Ω (see equation (2.18)). As it was commented in Remark 3, the implementation of such function implies an important limitation, since due to the inherent delay of the switching period measurement, it is not possible to exactly implement the theoretical value found for Ωk. Besides, the feedforward action needs to estimate the values of ρ± ∗k(see equation (2.20)) which also inherit the problem of the measuring delay of Tk. In order to 46 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 3. A CASE OF STUDY: SFC DESIGN AND SIMULATION RESULTS overcome these problems the following approximation is used: Ωk≃Ωk−1,(3.14) where the corresponding values for ρ± k−1are estimated using the expressions: ρ+ k−1=T+ k−1 ∆k−1+ ∆k−2 ;ρ− k−1=T+ k−1 2 ∆k−1 . The basis of this approximation is again the idea expressed in Remark 2, in the sense that the system dynamics is slow enough with respect to the switching period, and, hence, the variation between consecutive ksamples is small. Such phenomena will be highlighted lately in the simulation results. 100 110 120 130 140 150 160 170 180 190 200 0 1 2 0 1 0 1 -1 1.5 0.5 t(s) ρ+ k∗(t), ρ− k∗(t) Solutions of Theorem 2x∗ 2 ρ+ k∗(t) ρ− k∗(t) for γM for γm γMγm Figure 3.4: From top to bottom: Reference signal x∗ 2; temporal evolution of the signals ρ+ k, ρ− k, measured from the system; solutions according to Theorem 2. For the evaluation of the performance of the designed SFC for the tracking case, different tests are performed. The results are detailed in the simulations shown in Figures 3.5, 3.6 and 3.7. The selected value for the integral gain of the SFC is γ= 0.4. Figure 3.5 compares the performance of the system without and with SFC action, and highlights the effect of the SFC. The SMC operates with fixed hysteresis band for the first 150 s. The SFC with γ=0.4 is enabled at t=150 s. It should be noticed that it is not until the activation of the SFC that Tkis able to attain the reference value T∗= 0.1 s. Indeed, the switching period with a fixed band value is time-varying, as expected from (1.19), taking into account that the values of ˙σ± kvary with k. Notice how, in the same way, Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 47 CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP 1.4, equation (1.18)). Such description, based on the regular form, will be used in this Chapter in order to analyse the impact of a hysteresis band on the real sliding dynamics. The chapter is structured as follows. In Section 4.1, the approach based on the regular form is briefly introduced. In Section 4.2, the case ∆ 6→ 0 is considered, leading to an interesting result regarding the mean value of the switching function and its piecewise linear behaviour. Some simulation results are introduced in Section 4.3 with the aim to validate the result provided in Section 4.2. Finally, in Section 4.4 the case for time-varying ∆ is considered, leading to some conclusions about the real sliding mode in systems using SFC, which will be corroborated through numerical simulations as well. 4.1 The regular form approach Consider the following linear system shown in (4.1). ˙x=Ax +Bu +g(t),(4.1) where x∈Rn,A∈Mn(R), B∈Mn×1(R), g∈Rnis a vector function representing a smooth external disturbances, and u∈Ris the control action. An associated switching surface to this system would be: σ(x) = Cx = 0,(4.2) where C∈M1×n(R). Let the state space equation of the system given by (4.1) be expressed as it is shown in (4.3),(4.4) through a specific state transformation: ˙x1=A11x1+A12x2+g1(t),(4.3) ˙x2=A21x1+A22x2+B1u+g2(t).(4.4) This description is call the regular form ( [37]). Taking into account the performed change of variables, the switching function, σ(x), can also be written in the transformed state space as σ(x1, x2) = C1x1+C2x2,(4.5) where x1∈Rn−m,x2∈Rm, and matrices Aij, Ci, B1have appropriate dimensions. Then, the computation of ˙σ(x1, x2) (from now on ˙σ) yields: ˙σ=C1˙x1+C2˙x2=C1(A11x1+A12x2+g1(t)) + +C2(A21x1+A22x2+B1u+g2(t)) , 54 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP which allows uto be expressed as follows: u=−B−1 1C−1 2C1(A11x1+A12x2+g1(t)) + −B−1 1(A21x1+A22x2+g2(t)) + B−1 1C−1 2˙σ. (4.6) Notice that the previous expression for the control law includes both the low frequency component (related to the equivalent control) and the high frequency one (related to the real sliding motion). The equivalent control is easily found assuming that ˙σ= 0, according to its definition in Section 1.3, equation (1.6), as: ueq =−B−1 1C−1 2C1(A11x1+A12x2+g1(t)) + −B−1 1(A21x1+A22x2+g2(t)) .(4.7) Then, (4.6) can be updated up to: u=ueq +B−1 1C−1 2˙σ. (4.8) As a consequence, the real dynamics is governed by the reduced order system: ˙x1=A11x1+A12x2+g1(t), σ=C1x1+C2x2, It is obvious to see that if (4.6) is replaced in (4.3)-(4.4), the second state space equation becomes the desired dynamics imposed by (4.5), since it consists of a linear combination of x1and x2. Hence, the previous system can also be expressed as: ˙x1=A11 −A12C−1 2C1x1+g1(t) + A12C−1 2σ, (4.9) x2=−C−1 2C1x1+C−1 2σ. (4.10) The ideal dynamics can be obtained setting σ=0in(4.9),(4.10), which results in the ideal sliding dynamics for x1, x2, denoted as x? 1, x? 2: ˙x? 1=A11 −A12C−1 2C1x? 1+g1(t),(4.11) x? 2=−C−1 2C1x? 1.(4.12) Notice that now the input signal in the real dynamics is σ=σ(t), where σ(t) is the real evolution of the switching function chattering in the vicinity of σ= 0. Meanwhile, in the approach presented in Section 1.4 this role was played by ˙σ(see equation (1.18)). Remark 5. Notice that initially, in Section 1.4 a more general nonlinear system was considered in equation (1.1). In this chapter a linear system has been considered in the analysis. As a consequence, the results found hereafter only hold for linear systems. Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 55 CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP 4.2 Fixed hysteresis band Both on the regular approach and on the classical one, it has been proved that the real and the ideal trajectories of the sliding mode are related in such a way that kx(t)−x?(t)k ≤ N∆, being Na positive number (see [38], [37] for details). Moreover, as the switching function is enforced to chatter around the desired space region σ(x) = 0, it is reasonable to think that x(t) will chatter around the ideal response x?(t). Assuming that in the real sliding dynamics x(t) = x?(t) cannot be attained, let us see if at least, under certain conditions, the average values of both responses match. One could think that if the average value of the switching function, σ(x), is zero, the average value of x(t)−x?(t) should converge to zero as well. The analysis of such hypotheses is developed at this stage using the regular form approach shown in the previous section. First of all, the average value of the switching function in the case of a fixed hysteresis band is analysed through the averaging operator defined in (4.13). h(·)i=1 TZt t−T (·)dt. (4.13) Recalling Figure 2.1, and assuming a piecewise linear behaviour of σ, its time evolution can be modelled as: σ(t) = σ(0) + ˙σ(x)+ ktfor 0 ≤t < T+ k σ(t) = σ(T+ k) + ˙σ(x)− k(t−T+ k) for T+ k≤t < Tk(4.14) Applying the averaging operator, the mean value of σ(t) is found integrating for the two time intervals: 0 ≤t < T+ kand T+ k≤t<Tk: hσi=1 Tk" ZT+ k 0σ(0) + ˙σ(x)+ ktdt!+ ZTk T+ kσ(T+ k) + ˙σ(x)− kt−T+ kdt!#.(4.15) Solving the previous integrals one gets: hσi=1 Tk σ(0)t+˙σ(x)+ k 2t2T+ k 0 +σ(T+ k)t+˙σ(x)− k 2(t−T+ k)2Tk T+ k!.(4.16) According to Figure 2.1 the following relations hold: σ(0) = −∆, σ(T+ k) = ∆ and σ(Tk) = −∆. Moreover, it is also obvious that T+ k= 2∆/˙σ(x)+ k,T− k=−2∆/˙σ(x)− kand Tk= T+ k+T− k. Therefore, the resulting mean value is: hσi=1 Tk−2∆2 ˙σ(x)+ k +2∆2 ˙σ(x)+ k +2∆2 ˙σ(x)+ k−2∆2 ˙σ(x)− k +2∆2 ˙σ(x)− k−2∆2 ˙σ(x)+ k= 0.(4.17) Expression (4.17) confirms the suggested result with regard to the average value of σ 56 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP when it is confined within a fixed and symmetric hysteresis band. Notice that this result holds assuming a piecewise linear behaviour of σ. Once the null average value of the switching function is confirmed, the next step is to see if hσi= 0 entails that hxi=hx?iin steady-state. For such purpose, the following assumption is made: Assumption C. Once the sliding mode steady-state has been reached, the switching function, σ, becomes T-periodic, with T∈R+,from a certain time instant. Theorem 4. Assume that equations (4.9),(4.10)and (4.11),(4.12)characterize the real sliding dynamics and the ideal sliding dynamics, respectively. Consider also that Assumption Cis fulfilled and that Matrix A1, defined as A1=A11 −A12C−1 2C1(see equations (4.9),(4.11)), is a Hurwitz matrix. Then, both systems admit asymptotically stable, Tperiodic solutions ˜x,˜x?, respectively, such that h˜x?i= Γ?hg1(t)i,(4.18) h˜xi=h˜x?i+ Γ hσ(x)i,(4.19) where Γ?=−A−1 1 C−1 2C1A−1 1, Γ = −A−1 1A12C−1 2 C−1 2C1A−1 1A12C−1 2+I. then, since hσi= 0,h˜xi=h˜x?iholds, with h˜x?igiven by (4.18). The proof of Theorem 4is outlined hereunder [38]. Proof. As A1is Hurwitz and Assumption Cis fulfilled, the existence of T-periodic, asymptotically stable solutions ˜x>=˜x> 1,˜x> 2, ˜x?>=˜x?> 1,˜x?> 2for (4.9), (4.10) and (4.11), (4.12), respectively, is ensured by basic linear systems theory ( [39]). In turn, these solutions satisfy: ˙ ˜x1=A1˜x1+g1(t) + A12C−1 2σ, (4.20) ˜x2=−C−1 2C1˜x1+C−1 2σ, (4.21) and ˙ ˜x? 1=A1˜x? 1+g1(t),(4.22) ˜x? 2=−C−1 2C1˜x? 1.(4.23) Applying the averaging operator (4.13) to (4.20), (4.22) while taking into account its linearity and the fact that ˙ ˜x1= 0, ˙ ˜x? 1= 0, because neither ˙ ˜x1nor ˙ ˜x? 1have continuous Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 57 CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP component but just zero averaged terms, it results that 0 = A1h˜x1i+hg1(t)i+A12C−1 2hσi, 0 = A1h˜x? 1i+hg1(t)i. Then, it follows immediately from the Hurwitz character of A1that h˜x? 1i=−A−1 1hg1(t)i, h˜x1i=h˜x? 1i−A−1 1A12C−1 2hσi and, subsequently from (4.21),(4.23), h˜x? 2i=C−1 2C1A−1 1hg1(t)i, h˜x2i=h˜x? 2i+C−1 2C1A−1 1A12C−1 2+Ihσi. Now, gathering terms appropriately, (4.18), (4.19) follow immediately. The previous result confirms that if hσi= 0, and σis T-periodic, for a linear system the relation hxi=hx?iis fulfilled in steady-state. This result can be applied as long as the average value of σbecomes null in a T-periodic window. It is evident that for regulation cases, i.e. x∗=ct, once the steady-state sliding motion has been achieved and the switching period has been successfully regulated to the desired switching period T∗, appears a T∗-periodic behaviour of σ(x). Since under these conditions ∆ becomes constant, the mean value of σis null and the previous result applies. Equivalently, in the tracking case scenario, this result could also be applied, even when the switching function does not have a T∗-periodic behaviour it exists a T-periodic behaviour of σin a larger time window, Tw. In order to fit the conditions of Theorem 4, at steady-state sliding motion and with the SFC properly regulating the switching period, Twmust be a multiple of T∗and the mean value of σduring Twshould be also zero. However, notice that the previous result does not apply in transients, neither under regulation task nor in the tracking case. Section 4.4 deals with this case, in order to see its impact in the real sliding mode. Remark 6. The null mean value of σobtained in (4.17)and used in Theorem 4takes an important relevance in this thesis, due to the fact that such null mean value was achieved under the assumption of piecewise linear behaviour of the switching function within a fixed hysteresis band. In all the theoretical developments shown in Chapter 2, the piecewise linear characteristic of σwas the hypothesis supporting the validity of the developed models. Hence, with the last result, a switching function with constant slopes in steady-state is not only useful for the validity of the SFC developments but also to guarantee that x?−x, has a null mean value under certain conditions. 58 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP 4.3 Case study: the Buck converter In order to highlight the importance of the result provided by Theorem 4, some simulations are included below using a similar system than the proposed in (3.1), (3.2). 4.3.1 Mathematical model The proposed system is given by (4.24), (4.25), where the factor βhas been included and the control action, u, takes values in the se {0,1}, being u+= 1, u−= 0. ˙x1=−βx1+x2,(4.24) ˙x2=−x1+Mu, (4.25) where Mand βare real positive system parameters. Unlike the simulation shown in Chapter 3, in this case the regulated state variable is x1instead of x2. According to the relative degree, the switching function includes the regulation error and its first time derivative: σ=σ(x) = α(x∗ 1−x1) + d dt (x∗ 1−x1),(4.26) being α > 0 a surface parameter and x∗ 1the reference value. Assume at this point that, following the procedure discussed in Section 3.1, the control law for uensuring sliding motion on |σ|<∆ has been properly designed. Let us directly follow with the application of the regular form method introduced in Section 4.1. The new variables are based on the errors e= (e1, e2)>, defined as: e1=x∗ 1−x1, e2=βx∗ 1−x2.(4.27) The selection of the new variables arise from x1,x2related to the steady-state equilibrium point of (4.24), (4.25), which are x1=x∗ 1and x2=βx∗ 1. Therefore, the error dynamics can be found as: ˙e1=e2−βe1(4.28) ˙e2=−e1−M u +x∗ 1,(4.29) and the switching function becomes: σ(e1, e2)=(α−β)e1+e2.(4.30) System (4.28), (4.29) is already in regular form and matches (4.3), (4.4), and so does the switching function (4.30) with (4.5). Hence, following the procedure detailed in Section 4.1, the real dynamics (4.9), (4.10) reads in this case as: ˙e1=−αe1+σ, (4.31) e2= (β−α)e1+σ, (4.32) Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 59 CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP while the ideal dynamics (4.11), (4.12) boils down to ˙e? 1=−αe? 1,(4.33) e? 2= (β−α)e? 1.(4.34) Notice that, in this case, g(t) = 0 and α > 0; consequently, when σ=σ(t) is Tperiodic within the hysteresis bandwidth (|σ|<∆), the hypotheses of Theorem 4, including Assumption C, are fulfilled. Hence, the solutions of (4.31), (4.32) tend asymptotically to the periodic solution ˜e= (˜e1,˜e2)>, with ˜e1,˜e2related by (4.32). As for ˜e1, a T-periodic solution for (4.31) is given by (see, for example, [40]): ˜e1(t) = e−αt eαT −1ZT 0 eατ σ(τ)dτ +e−αt Zt 0 eατ σ(τ)dτ. (4.35) In turn, the solutions of (4.33), (4.34) tend asymptotically to the equilibrium point ˜e?= 0. Then, according to Theorem 4, since the disturbance vector g(t) is null in this case, (4.18) becomes h˜e?i= 0, while h˜eiis fulfilling (4.19), i.e. h˜e1i=1 αhσi,(4.36) h˜e2i=β αhσi.(4.37) Therefore, in case that hσi= 0, (4.36), (4.37) yield h˜ei= 0. This implies that hx1i=x∗ 1 and hx2i=βx∗ 1, meaning that the average value of the real signal is exactly equal to the reference one. 4.3.2 Simulation results A numerical analysis of system (4.24), (4.25) has been carried out with the parameter values M=β= 3. For the switching function in (4.26) we have chosen α= 1, while the reference value for x1has been set to x∗ 1= 1. In accordance with (4.27), this selection entails that the ideal steady-state value for x2is x∗ 2= 3. The simulation consists in implementing the control law of the form denoted in equation (1.13), with three different values for the hysteresis bandwidth, ∆ =    0.1 when 0 ≤t < 16s, 0.04 when 16s≤t < 30s, 0.01 when 30s≤t < 40s, and checking that (4.36), (4.37) are fulfilled, i.e. that the average steady-state errors he1i, he2iverify: he1i=hσi,he2i= 3 hσi. Let us assume that during the last instants of the time windows in which each of the three 60 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP values of ∆ is active the variables have achieved a steady-state. The tests have carried out with the software package MATLAB/Simulink (R2016b) using an ODE 5 solver with a fixed step of 10−5. As for the average values, they have been extracted using a low-pass filter with transfer function H(s) = 100 s2+ 18s+ 10010 . 0 5 10 15 20 25 30 35 40 -0.2 0 0.2 0 5 10 15 20 25 30 35 40 0 0.5 1 0 5 10 15 20 25 30 35 40 0.99 0.995 1 ∆ =0.1 ∆ =0.04 ∆ =0.01 time (s) x? 1 x1 x? 1 x1 σ Figure 4.1: System response with different hysteresis band values. Top plot: switching function. Mid plot: state variable x1and its reference, x? 1. Bottom plot: zoom of the mid plot. The top plot in Figure 4.1 depicts the switching function, σ. As expected, the chattering amplitude is higher when ∆ = 0.1, and decreases while ∆ does. In turn, the mid plot depicts the behaviour of the state variable x1with respect to its reference value and the ideal sliding dynamics one, x? 1. The zoom in the bottom plot reveals that x1stabilizes closer to the reference, i.e. with less average steady-state error, and also with decreasing chattering amplitude, for lower values of ∆. From here it is confirmed how x1→x? 1as ∆→0. According to Theorem 4, if this exists a regulation error in x1with respect to x∗ 1 (as Figure 4.1 depicts, mainly when ∆ = 0.1), the mean value of σshould not be zero. Let us see such result with detailed zoomed views of the three cases appeared in Figure 4.1. The top plots in Figures 4.2,4.3 and 4.4 show the switching function and its mean value for ∆ = 0.1, ∆ = 0.04 and ∆ = 0.01. Notice that the signal envelope allows an easy Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 61 CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP 15 15.5 16 -0.2 0 0.2 15 15.5 16 3 4 510-3 15 15.5 16 0 1 2 ∆ = 0.1 time(s) 10-2 hσi he1i 3hσi he2i σ Figure 4.2: Performance for ∆ = 0.1 in the steady-state. Top plot: switching function. Mid plot: he1imatching hσi. Bottom plot: he2imatching 3 hσi. identification of the current hysteresis value. It is also worth emphasizing that, as we are in a regulation control problem and the hysteresis band is symmetric with respect to zero, the switching period of the control action achieves a constant value, T, in the steady-state, and the switching function becomes T-periodic, thus meeting Assumption C. It is clear from the top plot in Figure 4.2 that, for the highest value of the hysteresis, namely ∆ = 0.1, the switching function does not show a piecewise linear behaviour. Consequently, its mean value is not zero, as confirmed by the mid plot, where it is shown to match that of e1, this resulting in the steady-state error for x1observed in the first part of the bottom plot in Figure 4.1. In turn, one can observe in Figures 4.3 and 4.4 that lower values of ∆ enforce the piecewise linear character of σwithin the hysteresis band, this yielding lower mean values for the respective switching functions and also for the steady-state errors of x1arising in Figure 4.1. In all these cases he1imatches hσi. In turn, he2ialways coincides with 3 hσi, as expected. Hence, this confirms the theoretical predictions of Theorem 4. When the piecewise linear assumption for σ(x) is closer to be fulfilled, the average values of σtend to zero, and so do the average state errors, he1i,he2i. Conversely, steady-state errors appear in the state variables when the piecewise linear assumption for σdoes not hold and hσi 6= 0. In any case he1i,he2i, match the expected values hσiand 3 hσi, respectively. 62 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP 29.5 29.75 30 -0.1 0 0.1 29.5 29.75 30 5 5.5 610-4 29.5 29.75 30 0 1 2 310-3 time (s) hσi he1i 3hσi he2i ∆ = 0.04 σ Figure 4.3: Performance for ∆ = 0.04 in the steady-state. Top plot: switching function. Mid plot: he1imatching hσi. Bottom plot: he2imatching 3 hσi. 39.9 39.95 40 -0.02 0 0.02 39.9 39.95 40 2.5 3 3.5 10 39.9 39.95 40 0 1 210 time (s) hσi he1i 3hσi he2i ∆ = 0.01 σ -4 -5 Figure 4.4: Performance for ∆ = 0.01 in the steady-state. Top plot: switching function. Mid plot: he1imatching hσi. Bottom plot: he2imatching 3 hσi. Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 63 CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP dynamics of x. As a consequence, whatever mean value appearing on xdue to the variation of the hysteresis band, as (4.49)states, will be negligible with respect to x∗. 4.4.3 Simulation results In order to show the effect of the SFC transients in the SMC performance, a simple simulation is done. Using the system introduced in Section 4.3.2, a sudden variation of the switching period reference is applied enforcing a SFC transient. The values of γ(γL) will be designed in order to show how higher values of γ(γL) will imply a higher impact in the state variables behaviours. The simulation depicts the transient detail of Tkfollowing the variation of T∗. This transient will be tested with two different values of γ(γL), designed so that one of them produces a soft transient in Tk, and the other an underdamped one. Let us calculate the values of ρ± ∗(related to the steady-state sliding motion) for the simulated system in Section 4.3.2. Using (4.26) the first time derivative of σfor the steady-state sliding motion can be found assuming that x1=x∗ 1and x2=βx∗ 1: ˙σ∗=x∗ 1−Mu. (4.51) Using the values provided in Section 4.3.2 for x∗ 1and M, and according to the ρ± ∗definition in (2.2), one gets: ρ+ ∗:= 1 ˙σ∗|u=0 , ρ− ∗:= 1 ˙σ∗|u=1 . Hence, ρ+ ∗= 1, ρ− ∗=−0.5. Regulation case: the discrete-time approach Firstly, the discrete-time approach is simulated (Section 2.2.1, equation (2.13)).The characteristic polynomial of the SFC for this case is (according to (2.16)): p(z) = z2+z(2γ−1) + γ, and the selected values for γfor the aforementioned purpose are: γ= 0.05 and γ= 0.5, which lead to the following closed-loop poles: γ= 0.05; pz1= 0.8405, pz2= 0.0595 γ= 0.5; pz1= 0.7071i, pz2=−0.7071i. Notice how the chosen values for γprovide poles in the real axis for a soft transient response for the case γ= 0.05, becoming complex for γ= 0.5. The resulting poles are within the unit circle in both cases, thus confirming stable SFC responses. The simulation for both cases are shown in Figures 4.7,4.8. The signals depicted in the Figures are structured as follows. In the top plot the switching function, σ, and the hysteresis bandwidth generated 70 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP by the SFC are depicted. In the mid plot the switching period reference, T∗, and real switching period, Tk, are shown. Finally, the bottom plot presents the behaviour of x1and the mean value of σ. In this case, the low pass filter detailed in Section 4.3.2 measuring the average value is substituted by the operator shown in (4.13) computed one time per switching period. As a consequence, such mean value is delayed one switching interval. 14 15 16 17 18 19 20 0.05 0.1 14 15 16 17 18 19 20 -0.05 0 0.05 14 15 16 17 18 19 20 -2 0 210-3 T∗ Tk x∗ 1−x1 hσi σ ∆ Time (s) Figure 4.7: Mean value impact on σin the face of a switching period transient with the discrete-time approach. γ= 0.05. First of all, it should be noted that sliding motion is preserved during the entire test, since σis perfectly confined within the hysteresis bandwidth during the transients (see top plots of Figures 4.7 and 4.8). Similarly, the proper regulation of Tkto T∗in the transients is also confirmed from the mid plots of Figures 4.7,4.8. From the results on the bottom plots, it is obvious that the signal x1is highly perturbed when the transient in the SFC is faster (Figure 4.8,γ=0.5), being such perturbation smaller when γ=0.05 is used (Figure 4.7). This result coincides with the result obtained in (4.39), as the mean value of σis proportional to the used γ. Furthermore, it is worth remarking that the simulations in Figures 4.7 and 4.8 corroborate the result of Theorem 4, since the mean values of σand x∗ 1−x1only match at steady-state, when ∆ is constant and σbecomes T∗-periodic. Regulation case: the continuous-time approach The same simulation has been performed using the SFC in the continuous-time approach (see Section 2.3, equation (2.35)). The previous calculus for ρ± ∗hold for this approach, which yields α= 2 ( ˙σ+ ∗= 1, ˙σ− ∗= -2). The initial condition of ek−1is 0.05 and the values Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 71 CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP Time (s) 14 15 16 17 18 19 20 0.05 0.1 14 15 16 17 18 19 20 -0.05 0 0.05 14 15 16 17 18 19 20 -2 0 210-3 T∗ Tk x∗ 1−x1 hσi σ ∆ Figure 4.8: Mean value impact on σin the face of a switching period transient with the discrete-time approach. γ= 0.5. for γLin order to produce a smooth and a fast transient in Tkare γL= 0.5 and γL= 2.5, respectively. From the direct comparison of the results in Figures 4.9,4.10, the analysis made in Section 4.4.1 can be validated, since the use of a higher value of γLincreases the mean value of σduring the transient. The tracking case The system simulated in Section 4.3.2 is now tested with a time-varying reference for x1, in order to check the mean value of σ. In this case, the simulation results are focused in the steady-state operation, as the transient effects are essentially the same than the ones shown in Figures 4.7,4.8, because the approach to adjust the hysteresis values is the same. The objective here is to check if the mean value of σduring a larger time interval, Tw, presents a periodic behaviour, and it could become zero when averaging it in Tw. The time-varying reference for this case is: x∗ 1= 2 sin (2 ·0.02 π t), keeping T∗= 0.1 s as the desired switching period. For the tracking case, the system is slightly modified, including a control action taking u+= 1 and u−=−1 (see (4.24), (4.25)). With this configuration, the system is able to track references without offset. The SFC structure is the corresponding to a tracking approach, (Section 2.2.2, Figure 2.6, Equations 72 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP 14 15 16 17 18 19 20 14 15 16 17 18 19 20 14 15 16 17 18 19 20 10 -0.5 1 0.05 -0.05 0 -3 2 -2 0 Time (s) T∗ Tk x∗ 1−x1 hσi Figure 4.9: Mean value impact on σin the face of a switching period transient with the continuous-time approach. γL= 0.5. 14 15 16 17 18 19 20 14 15 16 17 18 19 20 14 15 16 17 18 19 20 10 -0.5 1 0.05 -0.05 0 -3 2 -2 0 Time (s) T∗ Tk x∗ 1−x1 hσi Figure 4.10: Mean value impact on σin the face of a switching period transient with the continuous-time approach. γL= 2.5. Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 73 CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP (2.18), (2.19) and (2.20)). The control gain is set to γ= 0.5 according to Theorem 2. 100 120 140 160 180 200 0 100 120 140 160 180 200 0 100 120 140 160 180 200 -2 0 2 x10-3 x∗ 1−x1 hσi ∆ σ T∗ T Time (s) 0.1 0.2 0.1 -0.1 Figure 4.11: Steady-state mean value of σin the tracking case with SFC, γ= 0.5. Figure 4.11 shows: the switching function and the hysteresis band in the top plot, the desired and real switching periods in the mid plot and the tracking error together with the average value of σin the bottom plot. It should be noticed that the mean value of σ has been calculated using the operator shown in (4.13) at any switching period, T. From Figure 4.11, the proper SFC function is confirmed, since T∗and Tperfectly fit (mid plot). The interesting result is located in the bottom plot, where a periodic behaviour of hσiis observed (the showed time interval corresponds exactly to a two periods of Tw= 50 s). Due to the symmetry of the signal, it is not unreasonable to assume that the mean value of σalong a time interval of Tw= 50 s is null. Notice how hσiis close to the averaged tracking error defined as ev=x∗ 1−x1, since the expression enforced by the sliding motion (see (4.26)): σ=αev+ ˙ev, always holds. Performing the Laplace transform, the previous expression reveals that hσi is the result of low-pass filtering ev. Finally, it has been considered interesting to show the same result in Figure 4.11 but with a fixed hysteresis band. The results are shown in Figure 4.12. The difference, aside that a fixed hysteresis value produces time-varying switching periods, relies in the different mean values produced in σ. By direct comparison with Figure 4.11, it can be noted that the maximum value of the mean value is slightly higher when the hysteresis band is fixed. Again, the mean value of σmatches the averaged tracking error. 74 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 100 120 140 160 180 200 100 120 140 160 180 200 100 120 140 160 180 200 10-3 x∗ 1−x1 hσi 0.1 -0.1 0 0.1 0.2 0 x 2 -2 0 T∗ T ∆ σ Time (s) Figure 4.12: Steady-state mean value of σin the tracking case with fixed hysteresis band, ∆ = 0.05. Part III Application of the Proposed Solution to Power Converters. Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 77 Chapter 5 Voltage Regulation in a Buck Converter. At this stage, the discrete-time approach of the SFC is experimentally tested in a DC-DC Buck converter for evaluation purpose. The state space equations modelling the Buck converter behaviour correspond to a linear system. In this case, the SMC will regulate the converter output voltage, becoming a regulation task for a time invariant linear system. The SFC will be implemented by a mid-range micro-controller (µC), whereas the SMC will be assembled by means of analog circuitry. The Chapter is structured as follows: in the first Section the power converter is presented, where its state space equations and parametric values are introduced. Then, the SMC controller is designed for regulating the output voltage. The next step develops the studies required for properly tuning the SFC. Finally, the implementation issues and the experimental results are presented. 5.1 The Buck Converter The Buck converter circuit scheme is shown in Figure 5.1. Table 5.1 lists the parameter values of the experimental prototype. E M1 u= 1 u= 0 L CRvc il M2 Figure 5.1: Buck converter. Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 79 CHAPTER 5. VOLTAGE REGULATION IN A BUCK CONVERTER. In that case, the computing time results in tc= 1.4 µs (empirically measured using an oscilloscope), while the expected value of T+ kis 2.5 µs, thus confirming the usefulness of (2.1). Finally, and additional aspect about this implementation methodology is discussed. It exists a relation between the resolutions of the used TIM and DAC devices [29]. If the resolution of the DAC is higher than the resolution of the TIM, the hysteresis values could oscillate between two values in steady-state. For this reason, is advisable to use a TIM with a higher resolution than the DAC one in order to ensure that the hysteresis value becomes constant in steady-state. This effect is also related with the integral gain γ, since the grade of change of ∆ ( ∆k−∆k−1) is, from (2.13), ekγ, and ekinherits the timer resolution. Therefore, in order to avoid oscillations in the system due to the resolutions of the used peripheral, the inequation (5.14) has to be fulfilled: DACR> γ TIMR,(5.14) being DACRand TIMRthe corresponding peripheral resolutions. 5.5 Experimental results The experimental results are presented at this stage. In the following oscilloscope captures, the switching period, Tk, appears converted to voltage with a rate of 0.35 V/µs. Due to implementation aspects, σhas an offset of 2.5 V (as it was shown in Figure 5.2). Unless otherwise noted, the SFC gain is set to γ= 2 ·104. 1. SMC performance The star-up of the converter is illustrated for two different initial values of ∆ in Figures 5.5 and 5.6. In these cases the output voltage is regulated to 12 V. The initial values of ∆ have been selected as they become smaller and higher than the steady-state one. Notice that both vcand Tkattain their references with a good transient response, while the hysteresis band is adapting till Tkreaches T∗. From both responses, the expected overdamped characteristic can also be confirmed, according to the designs developed at Sections 5.2,5.3. The switching period, Tk, can be easily measured observing the switching function in the zoomed windows (green signals at the bottom parts of the Figures). As it is noted in (2.13), once Tkreaches the desired value, the hysteresis band ∆kbecomes constant. In Figure 5.7 a voltage regulation from 12 to 24 V and vice-versa, with R= 4 Ω, is tested. In this case v∗ cis step changed and, consequently, the switching function suddenly drops the hysteresis amplitude value and recovers it again in less than one switching period, with a brief and smooth transient of Tk. Again, the expected overdamped behaviour of vcis confirmed. 86 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 5. VOLTAGE REGULATION IN A BUCK CONVERTER. vcTk σ ∆ Figure 5.5: Start-up for v∗ c= 12 V with R= 2 Ω and ∆0lower than the steady-state value. vc: blue; σ: green; |∆k|: magenta; Tk: red. vcTk σ ∆ Figure 5.6: Start-up for v∗ c= 12 V with R= 2 Ω and ∆0higher than the steady-state value. vc: blue; σ: green; |∆k|: magenta; Tk: red. Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 87 CHAPTER 5. VOLTAGE REGULATION IN A BUCK CONVERTER. Tk vc v∗ c σ Figure 5.7: Voltage reference, v∗ c, variation from 12 V to 24 V and vice-versa, with R= 4 Ω. vc: blue; σ: green; v∗ c: magenta; Tk: red. 2. SFC performance In order to evaluate the proper performance of the SFC, several reference step variations are tested in the following. First of all, Figure 5.8 shows the result for a step change of T∗between 12.5 µs and 8.3 µs with a desired output voltage of 12 V and R=4 Ω. The Figure confirms a good performance of the SFC with an overdamped behaviour, as expected with the used value of γ(γ= 2 ·104). Tk vc ∆ σ Figure 5.8: Overdamped responses for γ= 2 ·104with R=4 Ω and a T∗variation from 12.5 µs to 8.3 µs. vc: blue; σ: green; |∆k|: magenta; Tk: red. Figure 5.9 shows the result for the same test (step change of T∗), when the integral gain is selected close to the unstable values (γ= 2 ·105, being the stability limit of 88 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 5. VOLTAGE REGULATION IN A BUCK CONVERTER. 2.07 ·105). In this case T∗is varied from 12.5 µs to 14 µs and Ris kept at 4 Ω. The modification of the step values is required to avoid the effects of ∆ saturation and the computing time influence that would modify the expected dynamics (see Section 5.4). As γis now closer to the upper stability limit, both Tkand ∆kexhibit underdamped transient responses with very low damping ratio, thus confirming the theoretical prediction. It is evident that such response fits with the expected one, as Figure 3.2 denoted in Section 3.1 Tk vc ∆ σ Figure 5.9: Underdamped responses for γ= 2 ·105with R=4 Ω and a T∗variation from 12.5 µs to 14 µs. vc: blue; σ: green; |∆k|: magenta; Tk: red. Finally, the previous test is repeated with different values of γ, in order to confirm the validity of the model developed in Chapter 2. In Figure 5.10 the root locus of the SFC in the case of 12 V is shown, where the poles placement for different values of γare depicted. Related with such poles, the corresponding responses in the time domain are also shown. All the test are made with a step of T∗from 12.5 µs and 8.3 µs, except for the γ= 2 ·105case, which is from 12.5 µs to 14 µs. The switching period (red signal in Figure 5.10) is measured with an analog sensor, which adds some dynamics to the measure. The performance has to be analysed from the ∆ response (magenta signal), which is generated through a DAC by the µC, without any delay. From the results shown in Figure 5.10, the SFC model developed at Chapter 2is fully corroborated. 3. System robustness Finally, a load transient test is performed at the converter output, in order to confirm that one of the main benefits of the SMC (its robustness) is preserved under the SFC operation. The test consists of suddenly variations of the linear load applied, from 0 to 6 A in the 12 V and 24 V cases. In the Figures 5.11,5.12, the variable ioresponds Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 89 CHAPTER 5. VOLTAGE REGULATION IN A BUCK CONVERTER. γ= 2 ·105 γ= 1 ·105 γ= 6 ·104 γ= 3 ·104 γ= 2 ·104 γ= 1 ·104 γ= 0 Imaginary Part Real Part 1 0.4 0.8 0.6 0.2 0 1 -0.4 -0.8 -0.6 -0.2 1 -0.6 0 -1 -0.2 0.2 0.6 Root locus zp1= 0.86, zp2= 0.05 zp1= 0.7, zp2= 0.14 zp1,2= 0.38 ±0.025i zp1,2= 0.26 ±0.47i zp1,2= 0.1±0.69i zp1,2= 0.3±0.93i Figure 5.10: Poles placement in the complex plane and the corresponding time domain step responses, with R=4 Ω. vc: blue; σ: green; |∆k|: magenta; Tk: red. 90 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 5. VOLTAGE REGULATION IN A BUCK CONVERTER. to: io=vc R. vc io Tk σ Figure 5.11: Load Transient: responses for γ= 2 ·104with Rfrom no load to 2 Ω, T∗=10 µs and v∗ c=12 V. vc: blue; σ: green; io: magenta; Tk: red. From Figures 5.11 and 5.12, the good transient response of the SMC is confirmed, since the regulated output voltage is hardly disturbed. In the same way, it is also confirmed that in this design the values of ρ± ∗do not depend on the output current, since before and after the transient, the steady-state hysteresis values are essentially the same. This fact confirms the theoretical values found for ρ± ∗in Section 5.3.1, as expression (5.12) does not depend on the output load R. Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 91 CHAPTER 5. VOLTAGE REGULATION IN A BUCK CONVERTER. vc io Tk σ Figure 5.12: Load Transient: responses for γ= 2·104with Rfrom no load to 2Ω, T∗= 10µs and v∗ c=24 V. vC: blue; σ: green; io: magenta; Tk: red. 5.6 Conclusions In this Chapter the design and implementation steps of the SMC and SFC have been described, and several experimental results have been presented corroborating the validity of the proposed procedures, and the models developed in Section 2.2.1. Additionally, a digital implementation of the SFC has been developed using a µC from ST Microelectronics (STM32F407) allowing to demonstrate the expected behaviour of the SFC in the discrete-time approach. The experimental results confirm the output voltage regulation, the operation at fixed switching period, and the system robustness with respect to load and output voltage variations. 92 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters Chapter 6 Voltage Regulation in a Multiphase Buck Converter. The continuous-time approach of the SFC, found in Section 2.3, is applied to a multiphase synchronous Buck converter in the following. The Chapter is structured as follows, in the first Section the power converter data and the sliding mode interleaving operation are briefly introduced. Then, the SMC controller is designed for regulating the output voltage with interleaving operation, followed by the design of the SFC in the continuous-time approach. Finally, the implementation details and the experimental results are presented. 6.1 The multiphase converter The multiphase synchronous converter is made up by the parallel connection of mBuck converters. Such topology is shown in Figure 6.1. Since this topology is based on the connection of several synchronous Buck converters with their outputs joined in parallel, the corresponding state space equations are equivalent to the ones presented in Chapter 5, extended to a multi-input case. E Mm M1 Mm M1 L1 Lm mC R il1 ilm io m m vc Figure 6.1: Circuit scheme of the m-phase synchronous Buck converter. Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 93 CHAPTER 6. VOLTAGE REGULATION IN A MULTIPHASE BUCK CONVERTER. Therefore, for the multiphase structure the equations result in: Ldik dt =−vc+E uk;k= 1, .., m (6.1) Cdvc dt = m X k=1 ik−vc R(6.2) where the control actions uktake values from the set {0,1},Eis the input voltage, vc the output voltage, ikis the current flowing through the k-th phase and L,C,Rare the inductance, the capacitance and the resistive load, respectively. An important benefit of the multiphase topology is the possibility to implement interleaving operation. This technique is based on phase shifting the control actions of each converter in such a way that the high frequency current ripple of each inductor are cancelled in the common output connection, thus generating an ideally free ripple current to the output voltage. This technique allows to reduce considerably the output capacitor value, since, in the ideal case, there is no high frequency current ripple to be filtered at the output. Moreover, the distribution of the power through different converters permits the reduction of the component features, as the allowable conduction current of the power switches, or the required heat sinks for losses dissipation, which in some cases can be directly removed from the system. The reduction of the current flowing by the switches also allows to increase the switching frequency, which in turn, would lead to an additional reduction of the value of the reactive components. As a consequence, the multiphase structures have gained interest within the industrial community for different applications due to their high efficiency, good power density, fast transient response, and ability of interleaving operation [47–51]. The control objective is again the output voltage regulation, with the additional task of guaranteeing the interleaving among the phases, which is also controlled by a sliding mode technique. The SMC and the SFC will be implemented by means of analog circuitry, meanwhile the interleaving control is implemented with an FPGA. The parametric data of the assembled multiphase converter are detailed in Table 6.1. 94 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 6. VOLTAGE REGULATION IN A MULTIPHASE BUCK CONVERTER. Table 6.1: Multiphase Buck Converter Parameters Parameter Symbol Value Input Voltage E48 V Desired Output Voltage Range v∗ c12-24 V Output Capacitor C100 µF Phase Inductance L22 µH±10% Number of phases m8 Load Range Io0-65 A Desired switching period T∗10 µs Current transformer parameters Lx,M800 µH, 6.4 µH Current transformer burden resistor Rb10 Ω 6.2 Interleaved sliding mode control of the output voltage The interleaved sliding mode control corresponds to a Master-Slave strategy. One of the Buck converters regulates the output voltage, the Master converter (or Master phase), while the rest of the phases track the control signal of the Master one with the proper phase shifting. 6.2.1 Master switching surface design The Master switching surface σMfor output voltage regulation is designed as: σM:= ψ1(vc−v∗ c) + ψ2xM= 0,(6.3) where ψ1,2are the switching surface constants and v∗ cis the desired output voltage. Comparing it with the surface designed for the single Buck converter (see Section 5.2), the first time derivative of ecis replaced by the signal xM. As it was discussed in the Section 5.4, it is usual to employ the measured current flowing by the output capacitor instead of time differentiate the output voltage in the implementation set-up. In this case, it is not possible to use the current ripple of the output capacitor since it is ideally cancelled by the interleaving operation. As a consequence, a current transformer is placed in series with the Master phase inductor. The signal xMis the output of this current transformer, which Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 95 CHAPTER 6. VOLTAGE REGULATION IN A MULTIPHASE BUCK CONVERTER. circuits, one of them is in charging of synchronizing the capture of the voltage value of the capacitor in a rising edge of the control signal u, while the second one controls the switch that resets this voltage just after the value has been acquired. This behaviour is depicted in the left block of Figure 6.5, called Period Sensor. The system works as a sample and hold circuit synchronized with uM, holding the last measurement until the circuit is triggered again. Although the system delivers discrete-time measurements, the time used to capture the voltage value (1 µs) together with the finite value of the sample and hold capacitor (CSH = 1 nF) add dynamics to the measurement, which can be modelled by a first order response. This model is characterized by τ, which was empirically found to 65 ·10−6and used in the previous stability analysis. Such designed circuit is intended to generate 5 V when the switching period is of 10 µs. The parts called Integrator and Period Error make up the SFC. Indeed, as it can be inferred form the Figure, they use standard configuration based on AO for computing the period error and the integral action. Notice how due to the electronics used are unipolar, all the circuits are polarized with a 5 V offset. Despite of the integral action itself, such circuit includes a hardware saturator for the hysteresis value, conformed by a Schottky diode in anti series with a zener diode. This structure implements the maximum increment of ∆, ∆max, being the block called Hysteresis generator who fixes the minimum value of ∆, ∆min. Again using AO the ∆ and −∆ are generated, ensuring their symmetrical characteristic. 6.5 Experimental results Finally, the experimental results obtained in the laboratory with the built prototype are shown at this Section. Firstly, the measured system features as efficiency, line regulation and load regulation are summarized in Table 6.2. Table 6.2: Experimental results of the Multiphase converter Efficiency (%) Load Regulation (%) Line Regulation (%) Line Regulation (%) @65A, vc=12/24. E=48 V, vc=12/24. @vc=24 V, Pout=1 kW. @vc=24 V, No load. E=48 V 0 - 65 A E=36 to 55 V E=36 to 55 V 94.8/97.1 ≤0.97/1.1 ≤1.7 ≤0.5 The data shown at Table 6.2 certify a good performance of the prototype, achieving good levels of efficiency. It should be remarked that such efficiency includes all the power consumption of the system (including, FPGA, drivers, etc). The system also presents a good robustness, which can be inferred from the line and load regulation results. In order to check the expected features provided by the designed controllers, several tests have been performed in the following. They are organized in four groups, namely: 102 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 6. VOLTAGE REGULATION IN A MULTIPHASE BUCK CONVERTER. SMC performance, interleaving, SFC performance and system robustness. 1. SMC performance The first test consists in step-change variations of the reference voltage from 12 V to 24 V and reversely, with a resistive load of 1 Ω connected at the output. Figure 6.6 portrays the responses of the output voltage, the load current, the switching function and the measured switching period (scaled by 0.5 V/µs). The bottom windows show a zoom view of the transient behaviour when the output voltage reference changes from 12 V to 24 V (left window) and when it decreases from 24 V to 12 V (right window). Notice how the output voltage behaves with a smooth transient response, which corresponds to the ideal sliding motion, and the hysteresis values are adapted such that the switching period reaches the desired value at steady-state. vc Tk σM io Figure 6.6: Reference change 12-24-12 V with a load of 1 Ω at the output. 2. Interleaving Figure 6.7 shows the behaviour of the current transformer signals of the 8 phases in the start-up, when the converter supplies a load of 21 A and the output voltage is regulated to 24 V. As it can be seen in the oscilloscope capture, the interleaving operation is started from the second switching period (see current waveforms on the left bottom window) and achieves interleaving at the desired switching frequency of 100 kHz in the steady-state (see right bottom window). Figure 6.8 shows the steadystate behaviours of the current transformer signals of the 8 phases for a load of 65 A with an output voltage of 24 V. From the previous Figures, the proper interleaving operation is fully corroborated. 3. SFC performance The two following tests are designed with the purpose to validate the SFC operation. In Figure 6.9 the start-up of the multiphase converter for an output voltage reference Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 103 CHAPTER 6. VOLTAGE REGULATION IN A MULTIPHASE BUCK CONVERTER. Figure 6.7: Start-up of the current transformer signals of the 8 phases with a load of 21 A for an output voltage of 24 V. Figure 6.8: Steady-state of the current transformer signals of the 8 phases with a load of 65 A for an output voltage of 24 V. 104 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 6. VOLTAGE REGULATION IN A MULTIPHASE BUCK CONVERTER. of 24 V delivering 21 A to the load (at steady-state) is presented. Notice that the initial value of ∆ is far away from the steady-state one. Again, the Figure shows the behaviours of the output voltage, the switching function, the Master control signal and the measured switching period. The bottom windows detail the waveforms in the transient state (left window) and in the steady-state (right window). As it can be seen in the Figure, the output voltage reaches the desired voltage with a smooth transient and with a small overshoot. Furthermore, the hysteresis bands are adapted such that the steady-state switching frequency achieves the desired value of 100 kHz with the theoretically predicted overdamped response in the switching period transient (see Section 6.3). vc Tk σM uM Figure 6.9: Start-up with a load of 21 A for a desired output voltage of 24 V. The second test is devoted to highlight the switching period tracking of a step-type reference. The switching period reference varies from 8 µs to 12 µs and vice-versa. The output voltage is regulated to 24 V and there is no load at the output. Figure 6.10 shows the behaviours of the output voltage ripple, ∆vc, the switching function, the switching period, and the switching period reference. The SFC adjusts the hysteresis band value in order to achieve the desired steady-state switching period with the expected motion according to the model derived in Section 6.3. From the Figure, it can be seen how the expected settling time of around 2.5 ms of Tkis qualitatively fulfilled in the real system, validating the developed models and assumptions taken. Besides, the output voltage is not affected by the switching period reference variation, implying that the real sliding mode is not being perturbed by the action of the SFC. This effect can be also inferred from the low output voltage ripple, ∆vc, observed during the entire test. The waveforms detailed in the bottom windows correspond to the steady-state dynamics at 8 µs (left window) and at 12 µs (right window). Such Figures also confirm the assumption of the piecewise linear behaviour of the switching function. Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 105 CHAPTER 6. VOLTAGE REGULATION IN A MULTIPHASE BUCK CONVERTER. ∆vc Tk σM T∗ Figure 6.10: Switching period variation from 8 µs to 12 µs with a desired output voltage of 24 V and no load. 4. System robustness Finally, the robustness of both controllers are evaluated through sudden variations of the load applied at the output. The following Figures depict the responses of the output voltage, the switching function and the switching period (scaled by 0.5 V/µs) when the load changes from 21 A to 65 A (Figure 6.11) and from 65 A to 21 A (Figure 6.12). In both cases, the output voltage reference is set to 24 V. From these Figures it can be inferred how the converter recovers the desired output voltage after a smooth transient and the switching period is not affected by the load changes. Moreover, it is confirmed that the values of ρ± ∗and λare almost insensitive to load changes, since the hysteresis value, ∆, is nearly the same in both cases. 106 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 6. VOLTAGE REGULATION IN A MULTIPHASE BUCK CONVERTER. vc Tk σM io Figure 6.11: Load change from 21 A to 65 A for a regulated output voltage of 24 V. vc Tk σM io Figure 6.12: Load change from 65 A to 21 A for a regulated output voltage of 24 V. Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 107 CHAPTER 6. VOLTAGE REGULATION IN A MULTIPHASE BUCK CONVERTER. 6.6 Conclusions Through the experimental results shown above, the proper performance of the SMC together with the SFC is confirmed in a multi input linear system. The expected properties of the full system as output voltage regulation, interleaving operation, steady-state fixed switching period and robustness with respect to load variations have been confirmed. Moreover, the continuous-time approach of the SFC has been successfully implemented by analog circuitry, thus confirming the expected dynamics according to the developed model in Section 2.3. 108 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters Chapter 7 Voltage Regulation in a Boost Converter. The converters implemented in the previous Chapters (Chapters 5and 6) respond to linear systems with respect to the control input. However, there are power converters that do not fit with this description, as the Boost converter. The equations describing the Boost converter respond to a nonlinear ones, and it is interesting to evaluate the performance of the SFC in this type of structure. Therefore, the SFC is implemented for regulating the switching period of a Boost converter. The Chapter is structured as follows: firstly the nonlinear equations of the converter and the parametric data of the Boost converter are presented. Then, the SMC will be designed to regulate its output voltage. Next, the design of the SFC in the continuoustime approach is detailed. Finally, the implementation details and the experimental results will be shown. 7.1 The Boost Converter The circuit scheme of a Boost converter is shown in Figure 7.1. The Boost converter can be understood like a Buck converter where the input is used as the output, and this one as the input. Nevertheless, the state space equations derived from this structure become nonlinear. The equations describing the dynamics of the Boost converter are shown in (7.1), (7.2). Ldil dt =E−vc(1 −u),(7.1) Cdvc dt =il(1 −u)−vc R,(7.2) where Eis the input voltage, vcthe output voltage and L,C,Rare the inductance, the capacitance and the resistive load, respectively. In the previous equations, the discontinuous control input, u, takes again the values {0,1}.M1 and M2 work in a complementary Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 109 CHAPTER 7. VOLTAGE REGULATION IN A BOOST CONVERTER. E M1 u= 0 u= 1 L CRvc + - il M2 io Figure 7.1: Boost converter. manner, remaining one closed while the other one is open and vice versa. The parametric data of the converter built in the laboratory are shown in Table 7.1. Table 7.1: Boost converter parameters Parameter Symbol Value Input voltage E12 V Output voltage reference v∗ c48 V Output capacitor C132 µF Inductance L20 µH Nominal resistive load R20 Ω Switching period reference T∗10 µs The Boost converter, due to its step-up voltage conversion property, is extensively employed in different industrial applications [54–57]. In general, the control of this topology (and other similar ones) involves a complex task due to its nonlinear characteristics, which becomes a challenge from the control point of view. 7.2 Sliding mode control of the output voltage 7.2.1 Switching surface design The sliding mode controller is designed for regulating the output voltage of the Boost converter. Due to the non minimum phase property of the Boost converter, the direct output voltage regulation using the natural switching function σ(vc) := v∗ c−vcis not possible, since it results in an unstable behaviour of the inductor current in sliding motion 110 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 7. VOLTAGE REGULATION IN A BOOST CONVERTER. [10]. Alternatively, an indirect output voltage regulation is proposed with the sliding surface: σ(vc, il) := κ1ev+κ2Zev(τ)dτ −κ3il= 0,(7.3) where the voltage error has been defined as ev=v∗ c−vcand the switching surface parameters κ1,2,3are assumed positive. Some readers could identify the previous surface as the typical control structure with two nested loops, the inner one regulating the inductor current, and the outer one generating the proper inductor current reference for the inner loop, in order to keep the output voltage at the desired level. In that sense, the terms in (7.3) depending on evconstitute the outer controller, delivering the aforementioned current reference. In these types of control structures, in order to study the system stability, the inner loop is assumed to be much more faster than the outer one [7]. The difference between that analysis and the one performed here resides in the fact that such hypothesis is not employed. 7.2.2 Sliding dynamics The time derivative of the switching function is ˙σ(vc, il) = −ψ1(vc, il) + (1 −u)ψ2(vc, il),(7.4) where ψ1(vc, il) = Eκ3 L−κ1 R C vc−κ2ev, ψ2(vc, il) = κ3 Lvc−κ1 Cil.(7.5) From (7.4) it is clear that the sliding mode exists when |ψ2(vc, il)|>|ψ1(vc, il)|. Using the equivalent control method [7], it is possible to find the ideal sliding dynamics. The ueq, which is found with ˙σ(vc, il) = 0 and σ(vc, il) = 0, is determined by: ueq =ψ2(vc, il)−ψ1(vc, il) ψ2(vc, il),(7.6) leading to an existence range of the sliding mode as: 0<ψ1(vc, il) ψ2(vc, il)<1.(7.7) The sliding mode dynamics are derived replacing the equivalent control in (7.1)-(7.2): Ldil dt =E−vc ψ1(vc, il) ψ2(vc, il),(7.8) Cdvc dt =il ψ1(vc, il) ψ2(vc, il)−vc R,(7.9) Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 111 CHAPTER 7. VOLTAGE REGULATION IN A BOOST CONVERTER. u σ Tk ∆vc Figure 7.5: Switching period regulation for a step change from T∗= 8 µs to T∗= 12 µs for v∗ c= 48 V and R = 20 Ω. ∆vc: red, T: blue, σ: magenta, u: green. 3. System robustness The two last results show the robustness of the controllers in front of load transients at the converter output. Again, the responses of the output voltage, vc, the load current, io=vc R, the switching function, σ(vc, il), and the switching period (scaled by 0.5 V/µs), Tk, are shown in Figures 7.6 and 7.7, when the resistive load changes from R= 20 Ω to R= 100 Ω, and from R= 100 Ω to R= 20 Ω, respectively. In these Figures, one can see how the converter recovers the desired output voltage after a very smooth transient (with a maximum deviation of 2 V around the desired value of 48 V), as expected from the design of Section 7.2, while the switching frequency is only slightly affected by the load change. 118 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 7. VOLTAGE REGULATION IN A BOOST CONVERTER. io σ Tk vc Figure 7.6: Load variation from R= 20 Ω to R= 100 Ω for v∗ c= 48 V. vc: red, io: blue, σ: magenta, T: green. vc io σ Tk Figure 7.7: Load variation from R= 100 Ω to R= 20 Ω for v∗ c= 48 V. vc: red, io: blue, σ: magenta, T: green. Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 119 CHAPTER 7. VOLTAGE REGULATION IN A BOOST CONVERTER. 7.6 Conclusions The experimental results confirm an overall good performance of the system under the control of the SMC and SFC, and most importantly, they corroborate that the SFC is able to regulate the switching frequency in steady-state without degrading the well-known features of the SMC, as the robustness or the high transient response. The dynamics observed in the laboratory results, highlight the usefulness of the theoretical developments described along the Chapter, for both SMC and SFC controllers. Moreover, the developed model in Section 2.3 is validated for a nonlinear system as the Boost converter is. 120 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters Chapter 8 Voltage Tracking in a Voltage Source Inverter. The applications of the SFC presented in the previous chapters belong to cases where the SMC is under a regulation control task. In this Chapter, the application of the switching frequency regulation strategy in a tracking control problem is set out. Specifically, a voltage source inverter (VSI) is assembled for experimental evaluation in the laboratory. A sliding mode control will be designed in order to generate a sinusoidal voltage at the converter output, thus leading to a tracking control problem. In this experimentation, both controllers, the SMC and the SFC, are digitally implemented by a micro-controller. Moreover, with the purpose of showing results as realistic as possible, the VSI developed in the laboratory has been designed with a medium power-handling capability (up to 2.2 kW). Another important characteristic of the work presented hereafter is that the SMC has been designed with the aim of operating with linear and nonlinear loads connected at the VSI output. Such property allows us to test further the performance of the SFC under a new working scenario. The Chapter is organized as follows: in the first Section the VSI structure, its parametric data and the corresponding state space equations are introduced. The second Section tackles the sliding mode generation of the AC signal (220 V RMS / 50 Hz) at the VSI output, supporting linear and nonlinear loads. Then, the SFC is designed according to the theory developed at Section 2.2.2. Subsequently, the implementation of both controllers in a digital platform is addressed and discussed. Lastly, the main experimental results are shown in the last Section. 8.1 The voltage source inverter The VSI circuit scheme is depicted in Figure 8.1. This circuit is commonly employed to generate a sinusoidal signal at its output and it is classified as DC/AC converter. With regard to its structure, the VSI can be understood as a traditional Buck structure with a full bridge of switches. As a consequence, the VSI is able to generate voltages at its output Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 121 CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. between Eto −E. The VSI dynamics are described by the following state space equations: Eu= 1 L CZ + - il M4 vc u=−1 M3 M2 M1 u=−1u= 1 io Figure 8.1: Voltage source inverter structure. Cdvc dt =il−io,(8.1) Ldil dt =−vc+E u, (8.2) where ilis the inductor current, vcis the output voltage, iois the load current, Lis the inductance, Cis the capacitor and Eis the input voltage. The discontinuous control input utakes values in the discrete set {−1,1}. This strategy corresponds to the well-known two level modulation in the pulse width modulation (PWM) techniques [60]. The power switches are represented by M1, M2, M3,and M4. As it is shown in Figure 8.1,M1and M4 are short circuited when u= 1, and remain open when u=−1, whereas M2and M3work in a complementary way. Table 8.1 presents the specific values of the converter parameters used in the experimental setup. The voltage source inverter is the most used DC/AC converter in the industry [61–65]. Because of its simple structure, the converter has the capability to work with high voltage and manage high powers. As an example of application, VSIs are used in the photovoltaic plants injecting the power generated by the solar cells to the AC power grid. Another application of VSIs can be found in the uninterruptible power supply (UPS) systems, where is common the employment of back-to-back structures [66], being the VSI one of its most important parts. Similarly, the VSI converter is the preferable option for AC machine drives. The AC motors are used, among other, in air conditioner’s compressors, refrigerators, water pumps, electric saw, conveyor belts, electric traction in trains and, increasingly, in the growing market of the electric vehicle. 122 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. Table 8.1: Voltage Source Inverter parameters. Parameter Symbol Value Input voltage E420 V Desired output voltage amplitude A220√2 V Output voltage frequency f50 Hz Inductor L440 µH Output capacitor C100 µF Nominal output power (Linear Load) P2.2 kW Peak output power (Linear Load) Pp3.3 kW Switching period reference T∗50 µs 8.2 Sliding mode tracking of the output voltage 8.2.1 Switching surface design In this case, the control objective is to track a time-varying reference voltage at the output. The signal to be tracked is: v∗ c=Asin ωt. (8.3) Once the functionality of the VSI has been defined, the following task is to design a switching function that fulfils the desired performance. In the traditional SMC schemes applied to this converter, since the relative degree of the output voltage, vc, with respect to the control, u, is two, the following first order linear switching surface is typically used [46]: σ(vc,˙vc) = ψ1ev+ψ2C˙ev= 0,(8.4) where ev=vc−v∗ c. Notice how (8.4) contains the first time derivative of the output voltage. As it was explained in Section 5.4, it is usual to take the current flowing by the output capacitor as the first time derivative of the output voltage, avoiding the direct differentiation of the measured voltage which always brings noise problems (even in digital differentiation). Such measured current is properly replaced in the switching function written in (8.4), leading to an equivalent expression. However, this technique usually provides good results only with linear loads, degrading its performance with other types of loads. Moreover, from the sliding mode control design, the substitution of the first time derivative by the output capacitor current can compromise the piecewise linear behaviour of the switching function when the load is not pure resistive. As a consequence, in order to design a switching function less sensitive to the type of load applied at the output, an alternative switching function is proposed. The new switching function uses the output signal generFixed-Switching Frequency Sliding Mode Control Applied To Power Converters 123 CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. ated by a Current Transformer (CT), measuring the inductor current, il. Specifically, the proposed switching function is: σ(vc, xM) := −ψ1ev+ψ2C˙v∗ c−ψ2 Lx MRb xM(8.5) where again v∗ cis the desired output voltage, xMthe signal coming from the current transformer output and ψ1, ψ2>0 are the switching function parameters. Lx,Mand Rb are CT parameters, and Cis the capacitance of the output VSI filter. As it was already introduced in Section 6.2, the state space equation generated by the CT inclusion is: Lx dxM dt =−RbxM+RbMdil dt .(8.6) In order to illustrate the lack of piecewise linear behaviour in the switching function if ic is used as C˙vc, a simulation of the system designed and experimented in this Chapter is early introduced at this step. Figure 8.2 depicts the simulation result for a nonlinear load, where the signal icand ˆxMare shown, being ˆxM: ˆxM=xM Lx MRb . Therefore, expression (8.5) can be rewritten as: σ(vc, xM) := −ψ1ev+ψ2C˙v∗ c−ψ2ˆxM. The behaviour of the CT can be understood as a high pass filter, since, from (8.6), it can be derived that: Il(s) = sLx+Rb sRbMXM(s). The gain of the previous filter in the band pass can be found solving the following limit: lim s→∞ sLx+Rb sRbM=Lx RbM, and hence ˆxMis proportional to the high frequency ripple of il. The nonlinear load used in the simulation corresponds to a diode rectifier with a filtering capacitor supplying a resistor (see Figure 8.10). When the diodes are closed (|vc| ≥ Vdc), the load connected at the VSI is a capacitor in parallel with a resistor (assuming ideal diodes). When the diodes stay open (|vc| ≤ Vdc), the VSI is in no load condition. From the result in Figure 8.2, it is clear that when the diodes are switched on and there is current flowing to the load, icloses the piecewise linear behaviour while ˆxMdoes not. Such results justify the selection of the switching function shown in (8.5). 124 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. 0.0658 0.066 -14 -10 -6 -2 0 4 8 0.0298 0.03 -30 -20 -10 0 10 20 -400 0 400 0 0.02 0.04 0.06 0.08 -40 0 40 vc 10 ·io t(s) ˆxM ic ˆxM ic ˆxM ic Figure 8.2: Details of the signals icand ˆxMwhen the inverter is loaded by a nonlinear load. Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 125 CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. 8.2.2 Ideal sliding dynamics The SMC design is derived from the first time derivative of the switching function: ˙σ(vc, xM) = −ψ1˙ev+ψ2C¨v∗ c+ψ2 MxM−ψ2 L[E u −vc].(8.7) Moreover, the dynamics enforced by the sliding motion on σ(vc, xM) = 0 can be found from (8.5) as: xM=MRb ψ2Lx [ψ1(v∗ c−vc) + ψ2C˙v∗ c],(8.8) and, therefore ˙σ(vc, xM) = −ψ1˙ev+ψ2C¨v∗ c−β(ψ1ev−ψ2C˙v∗ c)−ψ2 L[E u −vc] (8.9) where β=Rb Lx. Thanks to the inclusion of xMin (8.5), the relative degree between the switching function and the control input is one, as it corroborates the fact that uappears in ˙σ(vc, xM). Arranging terms in (8.9) one gets (notice that ev=vc−v∗ c): ˙σ(vc, xM) = f∗+fe−ψ2 LEu (8.10) where f∗=ψ2C¨v∗ c+βψ2C˙v∗ c+ψ2 Lv∗ c,fe=ψ2 L−βψ1ev−ψ1˙ev. From (8.10) it can be figured out that a sliding motion can be enforced in σ(vc, xM) = 0 when the term depending on control dominates the rest of terms, or ψ2 LE > |f∗+fe|. From (8.10) the equivalent control is easily found: ueq = (f∗+fe)L ψ2E.(8.11) Once the equivalent control and the sliding mode equation have been found, it is time to analyse the resulting sliding dynamics in order to check if the desired tracking of v∗ cby vc is achieved. Replacing the equivalent control, (8.11), in the original system (8.1), (8.2), the state space equations that arise are: ψ2 dil dt =f∗+fe−ψ2 Lvc,(8.12) Cdvc dt =il−io.(8.13) Let us analyse the sliding dynamics for the tracking error ev. Combining equations 126 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. (8.12) and (8.13), the resulting dynamics for the tracking error, ev=vc−v∗ c, can be found: C¨ev+ψ1 ψ2 ˙ev+ψ1Rb ψ2Lx ev=RbC Lx ˙v∗ c−˙ io.(8.14) As expected, the sliding mode dynamics of the output voltage includes the evolution of the load current. In this thesis, three types of loads are studied, namely: resistive loads, reactive loads and nonlinear loads. Each of those cases are analysed separately in the coming Sections 8.2.4,8.2.5 and 8.2.6 for further understanding. 8.2.3 Control law The control law that ensures sliding motion in σ(vc, xM) = 0 regardless of the type of load applied to the VSI, is obtained from (8.10) as: u=−1 if σ < −∆kor (|σ|<∆k& ˙σ > 0) 1 if σ > ∆kor (|σ|<∆k& ˙σ < 0) .(8.15) 8.2.4 Sliding dynamics for pure resistive load The specific case for a pure resistive load is characterized by io=vc R, which, using (8.14), boils down to the equation: C¨ev+ψ1 ψ2 ˙ev+ψ1Rb ψ2Lx ev=RbC Lx ˙v∗ c−˙vc R.(8.16) The dynamics in (8.16) depends on the VSI parameters and the controller gains. The selection procedure of Land Cvalues is omitted for the sake of brevity, but, summarizing, it follows setting the cut-off frequency of the output LC filter at least a decade below the switching frequency, thus reducing the output voltage ripple [2]. Therefore, the parameters to design are ψ1,ψ2,Lxand Rb. Notice that the CT parameters are treated as switching surfaces gains, including the sensor dynamics in the design. Let us define the following parameters in order to clarify the future developments: α=ψ1 ψ2 , β =Rb Lx , γ =1 RC , Replacing those definitions in the sliding mode dynamics, (8.16) results in: C¨ev+ [α+γ C] ˙ev+αβev=C˙v∗ c[β−γ].(8.17) From (8.17) it can be easily derived that if β=γa perfect tracking of the output voltage is achieved, since (8.17) has ev= 0 as an asymptotic stable equilibrium point. The problem is that the output load is not fixed and can vary under different situations. As a consequence, the strategy followed in this study is to adjust βfor the worst load case, which corresponds Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 127 CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. with positive coeficients, implying that G(s) is Hurwitz [69] and the conditions 1 and 2 of Definition 2are meet. Let us check the third condition in Definition 2. The evaluation of the real part of G(jω) is presented as follows: G(jω) = jω −ω2C+jωψ1 ψ2+ψ1Rb ψ2Lx = ω2ψ1 ψ2+jω ψ1Rb ψ2Lx−ω2C ψ1Rb ψ2Lx−ω2C2+ω2ψ2 1 ψ2 2 (8.27) being the real part of the complex impedance, Re G(jω) = ω2ψ1 ψ2 ψ1Rb ψ2Lx−ω2C2+ω2ψ2 1 ψ2 2 ,(8.28) always positive. Therefore, the condition 3 of the Definition 2is also met and G(s) is PR. As a consequence, the phase of G(s)H(s), fulfils |argG(jω)H(jω)| ≤ π. The stability of the resulting system can also be found through the Nyquist stability criterion. Since |argG(jω)H(jω)| ≤ π, the resulting path of G(jω)H(jω) in the complex plane describing the contour −jω to jω, with 0 ≤ω≤ ∞, cannot cross from the second to the third quadrant or vice versa, and it is therefore impossible that the point −1 + j0 be encircled [6]. In Figure 8.7 a table with different configurations of loads, Z(s), connected to the VSI is shown. The objective is to demonstrate that the Nyquist diagrams of G(s)H(s) with these loads do not encircle the point −1+j0. Such results are depicted in Figure 8.8. With the aim of assigning values to the different load configurations in Figure 8.7, the loads have been designed to provide an approximate apparent power between 0.5 and 1.5 kVA at 50 Hz in all the cases. Figure 8.8 confirms that the Nyquist curves in the complex plane do not encircle the point −1 + j0. For the derivation of the plots shown in Figure 8.8, the designed values for ψ1,ψ2and βobtained in Section 8.2.4 have been used. Notice, however, that such values could be redesigned in order to achieve a optimized tracking performance for a specific impedance connected at the VSI. Finally, the bode diagrams of the frequency responses of the transfer functions, T(s), defined in (8.25) are shown in Figure 8.9. From such frequency responses, it can be corroborated that in all the cases the tracking errors are small in magnitude, being around of 2.5 % in the worst case for the load Z4(s). 8.2.6 Sliding dynamics for nonlinear load There are different types of loads that exhibit a nonlinear consumption from the power source they are connected. In the power electronics field, some rectifiers (converters from AC to DC voltages) produce nonlinear currents, as the uncontrolled rectifier (diode rectifier), which is the nonlinear load studied in this Section. The diode rectifier, shown in Figure 8.10, is widely used in industry, appliances, etc. The rectifier provides a DC voltage 134 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. + - vcio rs RL + - vcio rs RL CL LL + - vcio RL CL + - vc ioLL RL Z1(s) = RL+rs+sCLRLrs 1 + sCLRL Z2(s) = rsRL+sLL(rs+RL) RL+sLL Z3(s) = 1 + sCLrs sCL Z4(s) = rs+sLL + - vc io LL RL Z5(s) = s2CLLL+sRLLL+ 1 sCL CL RL= 8 Ω rs= 30 Ω CL= 0.1mF RL= 10 Ω rs= 30 Ω LL= 2 mH RL= 60 Ω CL= 0.1mF RL= 50 Ω LL= 50 mH RL= 40 Ω LL= 2 mH CL= 6 mF Figure 8.7: Several configurations of output impedance, Z(s), connected at the VSI output with their parametric values. Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 135 CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. Nyquist Diagram Real Axis Imaginary Axis -0.01 0 0.01 0.02 0.03 0.04 -0.02 -0.01 0 0.01 0.02 Zo1 Zo2 Zo3 Zo4 Zo5 Zo6 Figure 8.8: Nyquist diagram for the different conditions for the output impedance shown in Figure 8.7. The case of Zo6is the pure resistive case with R= 25 Ω. 0 0.02 0.04 0.06 0.08 Magnitude (abs) 100101102103104105 -90 -45 0 45 90 Phase (deg) Bode Diagram Frequency (Hz) 50 T1(s) T2(s) T3(s) T4(s) T5(s) T6(s) Figure 8.9: Bode responses of the tracking errors of the SMC with the output impedances, Z(s), shown on the table of Figure 8.7. The case of T6(s) is the case for the pure resistive load of R= 25 Ω. 136 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. from an AC input voltage, which in this case will be the output voltage of the VSI designed in this Chapter. CLRL + - Vdc io + - vc D1D3 D2D4 rs Figure 8.10: Diode rectifier topology. The nonlinear characteristic of this structure comes from the behaviour of the diodes together with the output capacitor, CL. The diodes act as switches, taking two discrete states, closed or open. According to this behaviour, the rectifier dynamics can be divided in two alternating topologies: one when the load is connected to the output of the VSI through the diodes (diodes ON) and other when the diodes remain in open circuit and the VSI is in no load condition (diodes OFF). The resulting topologies of the VSI with the rectifier connected at its output for the two states of the diodes are shown in Figure 8.11. When the diodes are closed (ON), the resulting load is linear and reactive, and the analysis developed in Section 8.2.5 can be applied. When the diodes remain open (OFF), the equivalent output resistor is R=∞, which fits with the analysis of Section 8.2.4. Since the generated output voltage, vc, is a sinusoidal waveform, the aforementioned topologies occur two times per cycle. Specifically, there exist 4 different time intervals, two of them corresponding to a no load condition (io= 0 A) and two corresponding to a reactive load (see top plot of Figure 8.2). From the stability conditions stated in Sections 8.2.4 and 8.2.5, it is clear that the closedloop systems of the resulting topologies are stable. Therefore, the nonlinear tracking error in steady-state will be bound by known values if the settling times of each topology, with respect to the duration of the expected time interval, are fast enough. In the case that only one of the settling times is fast enough to achieve the steady-state error, such tracking error at least will reach a known value two times per cycle, corresponding to a tracking error which can be also bounded. In the case when both topologies present slow settling times with respect to their time intervals, the stability of the system is not guaranteed. Notice that the found stability conditions depend on several aspects as the expected topologies, their corresponding settling times and the time interval application of each Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 137 CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. CLRL + - Vdc + - vc rs Diodes ON Eu= 1 L C il M4 u=−1 M3 M2 M1 u=−1u= 1 io CLRL + - Vdc + - vc rs Diodes OFF Eu= 1 L C il M4 u=−1 M3 M2 M1 u=−1u= 1 io= 0 Figure 8.11: Resulting topologies with the VSI and a diode rectifier connected as output load. 138 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. topology. In that sense, the stability conditions should be checked for a certain system. Let us analyse the rectifier used in the laboratory. The values of the nonlinear load are (see Figure 8.10): RL= 132 Ω, rs= 1 Ω, CL= 6.6mF. For the calculations, the diodes are assumed ideal. According to the equivalent systems in sliding motion obtained in (8.18), (8.25) the expected tracking errors, et, and settling times at 95 % of the final value, ts95% , for each state are found (it should be noticed that the parameters αand βhave been already designed to α=1, β=680): Diode OFF →et= 2.7% ts95% = 4.1 ms, Diode ON →et= 30% ts95% = 30 ms. The case for the diodes connected produces a very slow transient and a considerable tracking error, due to the load capacitor, CL, is really big in value. As a consequence, taking into account that the period of the output voltage will be 20 ms, it is clear that this state will not reach the steady-state behaviour. The corresponding case to the diodes switched off provides a faster settling time, producing also a smaller amplitude error. This second settling time is of 4.1 ms, and assuming that the signal period is of 20 ms, it is reasonable to figure out that the resulting time interval for this topology would be larger than 4.1 ms, making possible to achieve the steady-state amplitude error of 2.7%. This result will be corroborated in the experimental part. 8.3 Switching frequency regulation The next step is to design the SFC controller in order to provide steady-state fixed switching frequency to the VSI. Again, the parameters ρ±should be derived for the steady-state sliding motion. Since in this case the control problem belongs to a tracking scheme, the SFC structure defined in Section 2.2.2 applies. Firstly, the expression yielding the evolution of ρ± kfor the steady-state sliding motion, ρ± k∗, are found from (8.10). In the previous Sections, it has been explained that the perfect tracking performance can be only achieved when a resistive load of a specific value, Rp, is connected at the output. Anyhow, it was also demonstrated that although the load varies, even including elements as capacitors and inductors at the output load, it is possible to adjust the control parameters such as the tracking error becomes negligible. As a consequence, in the following study, it is assumed that, under steady-state sliding motion, vc=v∗ cholds. Recalling (8.10), the first time derivative of the switching function at steady-state sliding motion becomes: ˙σ(v∗ c, xM) = ωβCA cos(ωt)−ω2ψ2CA sin(ωt) + ψ2 L(Asin(ωt)−Eu).(8.29) From (8.29), it is immediate to derive the expressions for ρ± ∗just replacing the two Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 139 CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. possible values of the control signal u, which are u+= 1 and u−=−1. Notice that once the discontinuous control input is replaced by one of its possible states, the expressions for ρ± ∗become continuous functions of time, ρ(t)± ∗. The sampling of these functions at any switching period interval, T∗, yields: ρ+ ∗k=˙σ(v∗ c, xM)u=−1−1=ωβCA cos(ωkT∗)−ω2ψ2CA sin(ωkT ∗) + ψ2 L(Asin(ωkT ∗) + E)−1 ρ− ∗k= [ ˙σ(v∗ c, xM)u=1]−1=ωβCA cos(ωkT∗)−ω2ψ2CA sin(ωkT∗) + ψ2 L(Asin(ωkT ∗)−E)−1 . Once the expressions for ρ± k∗are found, the stability values for γcan be found applying time (s) v∗ c(t) ρ(t)± ρ(t)+ ρ(t)− γm Roots of γM for γM for γm (2.24), (2.25) Figure 8.12: From top to bottom. 1Desired output voltage, v∗ c. 2Dynamic evolution of ρ+ ∗kand ρ− ∗k. 3Roots of the conditions set in Theorem 2. Theorem 2(Section 2.2.2), which is sketched in Figure 8.12. Specifically, in the top plot, Figure 8.12 shows the desired output voltage, v∗ c, and the dynamic evolution of ρ(t)+, ρ(t)− in the mid plot. Finally, the set of solutions of the condition stated at Theorem 2for the resulting values of ρ+ ∗k, ρ− ∗kare presented in the bottom plot. With such signals, it is straightforward to find the maximum and minimum values guaranteeing stability of the SFC. The exact values that define the stability margin are numerically found in γM= 1.76 ·107and γm= 9.98 ·106, i.e. 9.98 ·106< γ < 1.76 ·107. However, through simulations and experimental testing, the stability of the system with values below the minimum value, γm, has been confirmed. It is worth remarking here that 140 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. Theorem 2gives sufficient but not necessary stability conditions. Indeed, γvalues below the range provide a much more reliable performance in practise. As a consequence, in the experimental evaluation, some different values of γwill be tested, including values within and outside of the range. Remark 8. In the early developed analysis, it has been assumed that under steady-state sliding motion the output voltage, vc, perfectly tracks the reference signal v∗ c. As it was explained in the previous Section, just for a certain load condition such perfect tracking occurs. However, the switching surface parameters are designed to ensure that the output voltage tracking error can be neglected in all the load conditions, as Figures 8.5,8.9 depict. As a consequence, the applied load does not influence in the switching functions slopes, and the aforementioned stability condition applies for whatever load applied to the VSI. 8.4 Implementation details In this Section, the methodologies used to implement the designed controllers are explained. It is important to remark here that the implementation is based on a micro-controller (µC). The digitalization of the switching function and of the hysteresis comparator will deserve a special attention. Hence, this Section will be organized in several parts divided as follows: in the first Subsection the effects of sampling the hysteresis comparator are presented; in the second Subsection, a strategy able to emulate the operation of the ideal hysteresis comparator using digital devices is introduced, followed by the third Subsection, where the discretization of the switching function is detailed. The fourth Subsection deals with the digital implementation of the SFC in the tracking case. Fifthly, a Subsection describing the available procedures in order to estimate the switching functions slopes (and their inverse values, ρ± k) is presented. Finally, the details of the assembled system in the laboratory are given. 8.4.1 Effects of the hysteresis comparator discretization As it has been explained in Section 2.1, a sliding motion enforced in a fixed hysteresis band comparator provides a known switching frequency. From the implementation point of view, the hysteresis comparator can be accurately built using analog circuitry, as it was made in the experimental evaluations of Chapters 5,6and 7. However, the implementation in digital processors degrades its performance if the sampling effect is not taken into account. Figure 8.13 shows the behaviour of a switching function, σ, working with an ideal hysteresis comparator, being Tits related switching period. Analogously, the expected performance of the same switching function under the effect of discretization is sketched through σz, and its corresponding switching period, Tz. Notice that the sampling process includes a least a delay equal to the sampling time, ts, since the required actions to be performed cannot be executed up to the next sampling period. This contemplates that the computing time, tc, which is the time spent by the µC Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 141 CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. σ= 0 ∆ −∆ σ ˙σ+˙σ− t TTz σz T+ ˙σ+ z˙σ− z −∆−δ T− ∆ + δ ts ∆z Figure 8.13: Hysteresis comparator discretization effect on the switching surface and related switching period. to evaluate a certain algorithm according to the sampled signals, is lower than ts. If this tcis higher than ts, a higher delay should be considered. This effect produces that the switching function is no longer confined within the hysteresis band in sliding motion, inducing stationary errors in the state variables. In this scenario, as it is illustrated in Figure 8.13, a new and larger boundary layer appears, defined as |σz|<∆ + δ. According to Figure 8.13, in the best case the system will switch with a delay of one sampling period (just one sample within δ). In the worst scenario the delay will be of 2ts. Notice that the expected switching period will not be constant even at steady-state, due to such discretization effect. However, it is possible to qualitatively analyse these errors. The maximum and minimum values of δare defined in (8.30). δ+ min =ts˙σ+;δ− min =−ts˙σ− δ+ max = 2ts˙σ+;δ− max =−2ts˙σ−,(8.30) where ˙σ= ˙σzhas been assumed. Then, the maximum and the minimum ∆zvalues will be: ∆zmax = 2∆ + δ+ max −δ− max ∆zmin = 2∆ + δ+ min −δ− min.(8.31) Replacing (8.30) in (8.31) the maximum and minimum values are: ∆zmax = 2∆ + 2ts[ ˙σ++ ˙σ−] ∆zmin = 2∆ + ts[ ˙σ++ ˙σ−].(8.32) Using the previous expressions, the maximum and minimum switching periods in a fixed hysteresis band could be found applying equation (2.1), placing ∆zmax and ∆zmin instead of ∆, respectively. Even though it is possible to bound the switching periods, such bounds 142 Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. depend on ˙σ+,˙σ+, which are time-varying signals in a tracking control case, leading to a nondesirable situation. Remark 9. From now on, the subindex nwill identify the sampling related with the µC operation’s, ts, keeping kfor the ones related to the switching periods events, T∗. 8.4.2 Digital emulation of the ideal hysteresis comparator In order to recover the ideal switching period derived in Section 2(see expressions (2.1), (2.9)) for an ideal hysteresis comparator, an emulation of this ideal behaviour is developed using a discrete-time algorithm. The control law is properly modified such that the response shown in Figure 2.1 is recovered in a discretized system. The procedure is able to deliver the proper control action u(t), in such a way the switching function, σ, changes its slope sign just when it hits the hysteresis band ∆. The desired performance is illustrated in Figure 8.14. Since the µC is able to perform actions only at the sampling time instants, tn+1, tn+2, .., in order to make the switching action at time instant t=t1+tn+1, a pulse width modulated (PWM) control signal is applied. The definition of the desired values of u(t) in the time interval tnto tn+3 are (see Figure 8.14): u(t) =        −1 for tn< t < tn+1 −1 for tn+1 < t < t1+tn+1 1 for t1+tn+1 < t < tn+2 1 for tn+2 < t < tn+3 .(8.33) The following duty cycles, computed according to the values presented in ((8.33)), have to be updated in the pulse width modulator at time instants (tn, tn+1, tn+2), as: d=   dn= 0 at t=tn dn+1 =t1/tsat t=tn+1 dn+2 = 1 at t=tn+2 (8.34) The duty ratio dn+1 updated at t=tn+1 makes possible the commutation at the desired time instant t=tn+1 +t1. This value, according to Figure 8.14, is defined as: dn+1 =∆−σn+1 σn+2 −σn+1 .(8.35) It is worth remarking that dn+1 depends on the future sample σn+2. At this point, a prediction of the value of σn+2 is required. Furthermore, an additional delay should be taken into account, due to the computing time tc. This means that the duty cycle which can be applied at t=tn+1, has to be calculated with the available information at t=tn. Referred to the equation (8.35), this implies a prediction of σn+1 and σn+2 at time instant t=tn. Fixed-Switching Frequency Sliding Mode Control Applied To Power Converters 143