Adaptive Delta Modulation in Networked Controlled Systems With bounded Disturbances

HAL Id: hal-00582444 https://hal.archives-ouvertes.fr/hal-00582444 Submitted on 1 Apr 2011 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Adaptive Delta Modulation in Networked Controlled Systems With bounded Disturbances Fabio Gomez-Estern, Carlos Canudas de Wit, Francisco Rubio To cite this version: Fabio Gomez-Estern, Carlos Canudas de Wit, Francisco Rubio. Adaptive Delta Modulation in Networked Controlled Systems With bounded Disturbances. IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 2011, 56 (1), pp.129-134. �10.1109/TAC.2010.2083370�. �hal-00582444� 1 Adaptive Delta Modulation in Networked Controlled Systems with bounded disturbances Fabio G´ omez-Estern, Carlos Canudas-de-Wit, and Francisco R. Rubio Abstract This paper investigates the closed-loop properties of the differential coding scheme known as Delta Modulation (∆-M) when used in feedback loops within the context of linear systems controlled through a communication network. We propose a new adaptive scheme with variable quantization step ∆, by defining an adaptation law exclusively in terms of information available at both the transmitter and receiver. With this approach, global asymptotic stability of the networked control system is achieved for a class of controllable (possible unstable) linear plants. Moreover, thanks to the globally defined switching policy, this architecture enjoys a disturbance rejection property that allows the system to recover from any finite–time unbounded disturbance or communication loss. Index Terms Differential Coding, Delta Modulation, stabilization of Networked Control Systems. I. INTRODUCTION DELTA Modulation (∆-M) is a well-known differential coding technique used for reducing the data rate required for voice communication, see [1]. The standard technique is based on synchronizing a state predictor on emitter and receiver and just sending a one–bit error signal corresponding to the innovation of the sampled data with respect to the predictor. The prediction is then updated by adding a positive or negative quantity (determined by the bit that has been transmitted) of absolute value ∆, a known parameter shared between emitter and receiver. Hence, C. Canudas-de-Wit is with The Laboratoire d’Automatique de Grenoble, UMR CNRS 5528, Grenoble, FRANCE. [email protected]. Fabio G´omez-Estern, Francisco R. Rubio are with the Department of Automatic Control and Systems Engineering at the University of Sevilla, Spain. Email: [email protected]s,[email protected], April 1, 2011 DRAFT 2 ∆can be regarded as the quantization step. This paper proposes an adaptive extension a of fixedgain differential coding scheme previously introduced by the same authors (see [2]) in the context of linear systems interconnected through some transmission network. The selection of ∆is a crucial issue on the quality of the decoded signals. It is well known in digital communications framework that large values of ∆will result in a high granular noise, while too small values of ∆will result in slope–overload distortion. In closed-loop configurations, as in the scenarios considered here, the choice of ∆is even more important because it may cause instability. The closed–loop stability properties of Delta Modulation coding with fixed or scheduled gains, has been studied in [2], [3], [4]. Delta modulation (∆-M) algorithm can also be understood as the coarsest two-level (1- bit) quantizer. Thus, this technique is a simple alternative to approaches concerning the use of quantizers in the context of NCS, i.e. [5], [6], [7], [8], [9], [10], [11] among others. For a Delta Modulation scheme with constant quantization step ∆it was shown in [2] that only a limited domain of attraction was obtained. In addition, the state was only guaranteed to converge asymptotically to a finite ball, being its size related to the parameters of the open-loop plant, and to the user-defined parameter ∆. By making ∆an adaptive quantity, more effective schemes of ∆-modulation have been already proposed in the communications community [1]. The idea is to design an update law for ∆, defined exclusively in terms of the information available both at the receiver and transmitter, aiming at improving the resolution of the differential coding by reducing the gain ∆for slowly varying signals, while enlarging ∆in case of rapid change of the input, and hence allowing for faster signal tracking and higher bandwidth of the transmitted signals. So far in the communications field, adaptation laws for ∆have been proposed under somewhat heuristic criteria, as little information is supposed to be available on the dynamics of the signal source. However, when dealing with feedback systems, the dynamical properties of the plant become very useful in designing the adaptive law. This problem, to which this paper is devoted. is framed as shown in Figure 1. The main paper contribution is the introduction of an adaptation mechanism consisting of varying the quantization interval ∆in terms of a minimal amount of information available at the transmitter and the receiver. This type of adaptation law, although well known in the communications field, is used and analyzed for the first time a feedback configuration shown April 1, 2011 DRAFT 3 Adaptive Delta Modulator Communication Channel Demodulator Adaptive xk+1 =Axk+Buk uk Kˆxk δk=±1 ˆxk xk Fig. 1. Block diagram of the problem set up studied in this paper. in Figure 1. It is also shown that this adaptive coding structure, modified as proposed in [2], is proven to yield closed-loop global asymptotic stability for a class of open-loop unstable linear systems. We provide also a comparison between our approach and existing ones, in order highlight the main advantage: a disturbance recovery property that guarantees that if the system state is driven temporarily away from the origin, due to some unattended unbounded finite–time disturbance, or possibly loss of communication, the system will ultimately stabilize the estimation error, and drive the state back to the origin. This is due to the globally defined switching policy (and not only in one time sense as happens in most of the referenced papers), and also allows a free initialization of the encoder without any previous information of the state. The results are presented in scalar form. The extensions to higher dimension are easy to develop in the case of diagonalizable system matrices, and will be spared for the sake of space. However, for non–diagonalizable systems, the analysis is involved, and the technical details have been presented in [12] for the fixed Delta Modulation algorithm, while the adaptive version is part of an ongoing research work. II. ADAPTIVE ∆-MCODING SCHEME The problem setup will be initially presented, and the fixed–step Delta Modulation networked controller [2] will be briefly recalled for the sake of completeness. Then we propose a ∆– adaptation law resulting in a global asymptotic convergence of the state estimation error and system states to zero. This is a significant achievement with respect to the fixed-gain scheme presented in [2]. April 1, 2011 DRAFT 4 A. Assumptions In the following, we will assume that the transmitted information is binary δk∈ {−1,1}, that only sensor–to–controller transmission is allowed, that a 100% reliable noiseless channel without transmission delays is used, and that the data is sent at a fixed rate (we select the sampling frequency in order to transmit only one δkat a time). B. Problem Setup Consider first the following one-dimensional discrete time-system, together with the control law, and the fixed–step differential coding law: Open-loop system, and encoder: Control law and decoder: xk+1 =axk+buk(1) ˆxk+1 = [a−bK]ˆxk+ ∆ ·δk(2) ˆx0= 0, δk △ =sgn(xk−ˆxk) ˆxk+1 = [a−bK]ˆxk+ ∆ ·δk(3) uk=−Kˆxk(4) ˆx0= 0 In that structure, x∈IR is the state of the plant, and a standard Delta Modulator is used to encode it. This structure uses a prediction of the state ˆxkgenerated by a model of the closed– loop dynamics (2) synchronized at both communication ends. When a new sample of the state xkarrives at the encoder, it is compared to the prediction ˆxk, and its difference is coded on a binary basis, i.e. the value of sgn(xk−ˆxk)is coded and transmitted as a binary digit. Stability of this system has been analyzed in [2]. Although it has interesting disturbance rejection properties, the main limitation of this approach is the fact that stability is semiglobal, in the sense that the quantization step must be chosen as a function of an upper bound of the initial state. This is a quite frequent fact in the literature (see [7]), and has been tackled (in that paper) by devising open–loop initialization mechanisms for estimating that bound. Another issue (also shown in [2]) is that using fixed–step Delta Modulation, the state does not converge strictly to zero, but to a finite ball around the origin, whose radius is proportional to ∆(there is chattering in steady state). As a consequence, when the gain ∆is fixed, there is an inherent trade–off between stability and precision. This motivates our search for other coding strategies with variant gains, as shown next. April 1, 2011 DRAFT 5 C. Adaptation law design Adaptive ∆–modulation in Digital Communications aims at improving the resolution of the differential coding scheme according to the size of the signals to be transmitted. In our case this signal is the system state, hence a reasonable approach is to enlarge ∆for large values of the estimated states, and decrease it for smaller values. Another requisite is the Equi–Memory property described in [13], which suggests that the quantization step must be the same at the transmitter and the receiver at any time. Hence, the adaptation law must be defined exclusively in terms of the shared information, i.e. of {δ0, δ1,...,δk}. Another condition is to keep the adaptation law as simple as possible, and also minimize its memory usage. With that aim we will propose a very heuristic and simple approach, and will further provide a detailed analysis to prove that it guarantees global asymptotic stability. In order to design an ∆–adaptive mechanism to achieve global asymptotic stability two opposite behaviors must be observed in the analysis of [2]. For large values of the estimation error ˜x, there is probably slope overload, so ∆kshould grow at a higher rate than the plant escape velocity. When the state is trapped into a domain of attraction for the present ∆k, the step size must decrease (for improving resolution) at a slower rate than the state convergence in order to prevent it from getting too small relative to the state. From this intuition, an adaptive scheme with minimal storage and computation requirements is proposed as follows: 1) If δk=δk−1then the state is assumed to be escaping, thus ∆kmust be increased. 2) If δk6=δk−1then the state is assumed to converge (oscillations close to zero) and ∆kmust be decreased. The following update law is proposed: ∆k+1 =φk+1∆k(5) φk+1 =     λ+if δk=δk−1 λ−if δk6=δk−1 (6) where 0< λ−<1is the exponential decay rate of ∆k, and λ+>1is the exponential growth rate. The proposed algorithm is shown in Fig.2. This adaptation law can be seen as a generalization of Jayant’s adaptation rule (1970) (see [1]). April 1, 2011 DRAFT 6 +- 1 -1 1 z−ac 1 z−ac ∆k ∆k ∆0 ∆0∆k+1 =φk∆k ∆k+1 =φk∆k ˆxk ˜xk xk × × φk φk gain selection gain selection channelxk+1 =Axk+Buk −Kˆxk δk=±1 δk=±1 ˆxk Fig. 2. Adaptive coding scheme. The figure shows the case of one–dimensional systems. The selection gain block toggles the value of φkaccording to Equation (6). D. Error equations The complete feedback system with the adaptive delta Delta Modulation coding scheme is then: Open-loop system (7), and encoder (8)-(10): Decoder (11)-(13) and control law (14): xk+1 =axk+buk(7) ˆxk+1 = [a−bK]ˆxk+ ∆k·δk(8) ∆k+1 =φk+1∆k(9) φk+1 =     λ+if δk=δk−1 λ−if δk6=δk−1 (10) ˆx0= 0,∆0>0freely assigned ˆxk+1 = [a−bK]ˆxk+ ∆k·δk(11) ∆k+1 =φk+1∆k(12) φk+1 =     λ+if δk=δk−1 λ−if δk6=δk−1 (13) uk=−Kˆxk(14) ˆx0= 0,∆0>0same as encoder With the above definitions, the closed–loop error dynamics become ˜xk+1 =a˜xk−∆k·δk ∆k+1 =φk+1∆k,∆0>0(15) φk+1 =λ−+1 2(λ+−λ−)|δk+1 +δk| The causality of the system is guaranteed because the computation of ˜xk+1 is only based on ˜xk April 1, 2011 DRAFT 7 and older values. The following Theorem states the stability of the closed–loop system. E. Main result (stability analysis) Theorem 1: The error trajectories ˜xkof system (15), resulting from the adaptive ∆-modulation coding scheme (7)-(14), globally asymptotically converge to zero as k→ ∞ if there exist parameters λ+>1,λ−∈(0,1) satisfying the following inequalities: λ+> a (16) λ−<(λ+)−β 2,(17) where β(a, λ−, λ+)△ = 1 + logρ 1 + a(a−λ−) (ρ−1) (λ−)2!and ρ△ =λ+ a Moreover, ∆kand hence xkalso converge to zero regardless the initial conditions (x0,∆0). Proof: The claim will be proved in two steps. First, a new variable will be defined in order to capture the ratio between ˜xkand ∆k, namely yk △ = ˜xk/∆k, and boundedness of that variable will be proved. Secondly, it will be shown that ∆kasymptotically converges to zero. Consequently, the convergence of ˜xktowards the origin is directly implied. Along the trajectories of (15), the variable ykevolves along the dynamics yk+1 =1 φk (ayk−sgn(yk)) .(18) Fact 1. Trajectories of ykcross the zero axis in finite time. This means that starting from any initial condition, ˜x0and ∆0, and thus y0, there must be a future time k0<∞such that yk0−1·yk0<0. This is easily shown by imagining a trajectory with no zero crossings on yk(hence on ˜xkas ∆kis always positive). Assuming initially positive ˜x, i.e. starting from y0>0, we have yk+1 = 1/λ+(ayk−1) and hence yk+1 −yk=1 λ+(ayk−1) −yk=a λ+−1yk−1 λ+<−1 λ+ for the given choice of λ. Then, starting from y0>0, we have yk< y0−k/λ+and hence there is some constant k0≤y0λ+such that yk0−1>0and yk0<0. Due to the symmetry of the system equations, a similar argument applies if the trajectory starts from y0<0. April 1, 2011 DRAFT 8 Fact 2. Trajectories of ykare bounded after finite time. Immediately after a zero crossing of the system (assuming + to - without loss of generality), yk0is bounded as follows yk0=1 λ+(ayk0−1−1) >−1 λ+(19) hence |yk0|<1 λ+, as yk0<0. Now if we search for bounds on subsequent samples we must update the growth factor of ∆to λ−and compute yk0+1 = 1/λ−(ayk0+ 1), which is positive, as from the bound 0> yk0>−1/λ+we have yk0+1 >1 λ−−a λ++ 1>0. Moreover, |yk0+1|<1/λ−. These observations are summarized in Fig. 3, where the necessity of two zero crossings after a set of positive values is illustrated, as well as the upper bounds inferred by the switching dynamics. Now as the switching policy is second order, the analysis is concluded by taking one further step. Using 0< yk0+1 <1/λ−, we have yk0+2 = 1/λ−(ayk0+1 −1), and hence −1 λ−< yk0+2 <1 λ−a λ−−1, As illustrated in Fig. 3 (left), two situations (a) and (b) are then identified, (a) −1 λ−< yk0+2 <0. This situation (including the norm bound) is exactly the one found at instant k0+ 1, then it has been already considered (b) yk0+2 >0. Then, we have two subsequent positive samples, i.e. the situation of yk0−1is recovered. In that case, the dynamic equation turns into (19) and along it the map yk→yk+1 is contracting, hence the norm will decrease until a future sign change. The previous analysis yields that after the first sign change, the state ykremains bounded as |yk|<1 λ−a λ−−1(20) With the above facts, the proof of the proposition reduces to show asymptotic convergence to zero of ∆k, and hence concluding convergence of the state ˜x. With this objective we will use the following definition, Definition 2.1: Given a sequence of positive (negative) samples of yk, the fly–time is the number of sampling instants elapsed between the two zero–crossings that enclose the signal. For its computation, the first and the last positive (negative) samples are considered. April 1, 2011 DRAFT 15 system and the controller. These results were extended to the case of adaptive ∆k. An explicit adaptation rule was proposed and the range of parameters were derived to ensure asymptotic stability. 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