A nonlinear strategy to control unstable underactuated mechanical systems with underactuation > 1. Applications to control augmentations

Depósito de investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an article published IET Control Theory & Applications available at: 10.1049/iet-cta.2008.0583 © 2010. Personal use of this material is permitted. Permission from 10.1049/ietcta.2008.0583 must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other Works” Nonlinear Sliding Mode Surfaces for a Class of Underactuated Mechanical Systems M. L´opez-Mart´ınez, J. ´ A. Acosta and J.M. Cano ∗ Abstract This paper presents a new nonlinear sliding surface for a class of underactuated mechanical systems with uncertainty in their physical parameters. The nonlinear surface is designed through a fictitious output, which provides a minimum-phasesystem, and allows to prove stability using Lyapunovtheory. The attained nonlinear surface is a function of the fictitious output and a nonlinear external controller designed using the Lyapunovtheory. The present approach assures exponential stability of the equilibrium point and robust stability to parametric uncertainties, avoiding the appearance of some typical phenomena, like limit cycles due to uncertainties. Two pendulum-like examples inside the class are thoroughly analyzed, i.e. the pendulum on a cart and the inertia wheel pendulum. Performance, time response and parametric robustness are shown through simulations. Keywords: Sliding Mode Control, Feedback linearisation, Underactuated systems, 1 INTRODUCTION This paper presents a control scheme where three classical methods, feedback linearisation, Lyapunov design and sliding mode control, are combined to achieve the robust stabilization of the equilibrium point of a class of underactuated mechanical systems. On the one hand sliding mode control (SMC) is a powerful and robust control methodology that has been widely studied in the last thirty years [1, 2, 3], and still is an area under research as the work of [4] illustrates, where new construction methods are designed. The sliding mode control approach is an efficient tool to design robust controllers for systems operating under uncertainty conditions. The main advantage respect to other ∗The authors are with Dept. de Ingenier´ıa de Sistemas y Autom´atica, Escuela T´ecnica Superior de Ingenieros, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092, Sevilla, Spain. {mlm,jaar}@esi.us.es 1 control techniques is the low sensitivity to error estimation in the parameters of the plant and to disturbances, which removes the necessity of exact modelling. When plants include sufficient information about the uncertainty, such as an upper bound, a robust control is normally designed. In this research line, the stabilisation problem has been studied, in recent years, for different classes of systems with uncertainties, see [5] and [6]. Other additional advantage of SMC, is that control actions are discontinuous functions, which can be used to stabilize some classes of nonlinear systems, which are not stabilisable by continuous state feedback laws, as stated in [7]. On the other hand, most control design approaches using sliding surfaces are based upon Lyapunov and Jacobian linearisation methods. The complexity of the Lyapunov approach, together with the ease of Jacobian linearisation, make useful to combine techniques to take the best of each approach. For example, in this paper we use an approach next to the combination of sliding mode control and flatness, in the sense that we use a fictitious output to stabilize the system, as done in [8, 9, 10, 11]. The main difference in the present approach, is the combination of Lyapunov design and feedback linearisation. The advantage of this approach with respect to the method based on flatness, is that the latter is only valid if the relative degree of the system with respect to the output, is the same than the order of the system. That is, we cannot use flatness if there exist zero dynamics associated to the output, but we can use feedback linearisation instead. In fact, the control scheme proposed in this paper is the following: firstly a nonlinear fictitious output, which guarantees a minimum phase system, is designed; secondly the external controller through a Lyapunov function is proposed; and finally a redesign using a nonlinear sliding surface is made to cope with parametric uncertainties. Respect to the stability issues, the present approach assures, on the one hand exponential stability of the equilibrium point, which implies rejection to external disturbances, and on the other hand robust stability to parametric uncertainties because of the SMC scheme. The paper is organized as follows: next section is devoted to review some background concepts concerning SMC. Section 3 is devoted to the output and the external controller proposed. In section 4 the main contribution of the paper is presented, i.e. the design of a new nonlinear sliding mode surfaces. Finally, section 5 shows two applications and a last section where the major conclusions are summarized. 2 2 BACKGROUND: Sliding Mode Control Consider the SISO system1 x(n)=f(x) + g(x)u(1) where u∈Ris the control input, x∈Ris the output, and x= [x, ˙x,...,x(n−1)]⊤∈Rnis the state vector. We assume that the function f(x)is not completely known, but its uncertainty is upper bounded by a continuous function of x, that is known; in its turn, the control gain g(x)is of known sign and is bounded by a continuous function of x, that is also known. The general control problem is to make the state vector xtrack a specific time varying state xd= [xd,˙xd,...,x(n−1) d]⊤, in the presence of model uncertainty on f(x)and g(x). For example, usually, the inertia of a mechanical system is only known with limited accuracy. Let ˜x=x−xdbe the error of the output x, and let ˜x=x−xd= [˜x, ˙ ˜x, . . . , ˜x(n−1)]⊤be the error vector. Besides, let us define a surface in the state-space Rnby s(x, t) = d dt +λn−1 ˜x= 0 (2) where, if n= 2 ,s(x, t) = ˙ ˜x+λ˜x= 0 . The simplified problem of keeping the scalar sat zero can be achieved by choosing the control law uof the system (1) such that outside of the surface the expression 1 2 d dts2≤ −κ|s|(3) is verified, where κ > 0. Equation (3) constrains trajectories to evolve towards the surface s(x, t) = 0. Once on the surface, the system trajectories remain on it. That is to say, if the sliding condition (3) is satisfied, the surface becomes invariant2with respect to the system (1). The surface verifying (3) is called sliding surface, and the behavior of the system once on it is known as the sliding mode. The behavior of the system when the sliding condition (3) is satisfied is shown in Fig. 1 for n = 2. In this case, the sliding surface is a line in the phase portrait, which contains the point xd= [xd,˙xd]⊤. Starting from any initial condition, the state trajectory reaches the surface in a finite time and then moves through the surface towards xdexponentially. 1see e.g. [12] for more details on sliding mode control. 2A set Sis called positively invariant if for the system ˙x=f(x), x(0) ∈Simplies that x(t)∈Sfor all t≥0. We will call it just invariant (see [13]). 3 x ɺ 0s= x ( ) dtx · Finite-time reaching phase Sliding mode exponential convergence Slope - λ Figure 1: Graphical interpretation for n=2. To sum up, the idea behind equations (2) and (3) is to choose a function, s(x, t), whose behavior is known, and then design the feedback control law ufor (1), such that V=1 2s2shapes as a Lyapunov function of the closed-loop system, even in the presence of disturbances and parametric uncertainties. 3 PRELIMINARIES: Feedback linearisation In this paper, we will address the control problem of a class of underactuated mechanical systems, with two degrees of freedom (n= 2) and only one control input (m= 1). The Lagrange’s equations read M(q)¨q+C(q, ˙q) ˙q+∇U(q) = [0 1]⊤τ(4) where M∈R2×2, is the symmetric and positive definite inertia matrix, U∈Ris the potential function, the matrix C∈R2×2contains the Coriolis and centrifugal forces, and and τ∈R, the number of independent control inputs. In the following, we divide the set of generalized coordinates into q= (z, x)⊤∈R×R, where the z–coordinate represents the unactuated degree of freedom and the x–coordinate the actuated one. After this partition the Lagrange’s equations of motion can be written as   m11 m12 m12 m22  ¨q+  f1(q, ˙q) f2(q, ˙q) =  0 1 τ, (5) 4 where now m11 ∈R,m12 ∈R,m22 ∈Rand we have introduced the scalar functions f1(q, ˙q)∈Rand f2(q, ˙q)∈R, respectively. The first step of the approach presented here, is to linearize the actuated part of the equations of motion (4), as done in [14] where it was called collocated partial feedback. Indeed, following some calculations given in [14], it is easy to see that the partially feedback–linearized system takes the affine in control form: ¨q=  −1 m11(q)f1(q, ˙q) O +  −m12(q) m11(q) 1 u, (6) where the scalar function f1(q, ˙q)was defined as f1(q, ˙q),e⊤ 1[C(q, ˙q) ˙q+∇U(q)],(7) and, as usual, eidenotes the i–th vector of the n–dimensional Euclidean basis. Besides, the inner loop reads τ=−m12 m11 1  f1(q, ˙q) f2(q, ˙q) +m22 −m2 12 m11 u. (8) Now, we proceed to define the class of mechanical systems, for which we can explicitly solve the control problem. The class considered refers to systems with two degrees of freedom and only one control input (underactuation degree one), with the following additional properties. Assumption 1 It will be assumed that: •The elements of the inertia matrix m11 and m12 depends only on the unactuated coordinates z, and also verifies that m11(z)>0and m12(z)6= 0. •m22 satisfies that either it is a constant scalar or a function of the actuated coordinates, i.e. ∂m22 ∂z = 0. •The potential function is of the form U(q),V(z) + Φ(x), where V(z)≥0,Φ(x)≥0, and such that d2V dz2(0) <0and dV dz 6= 0,∀z6= 0,which means that there exists an isolated open-loop unstable equilibrium. For this class of systems3, it is possible to design a fictitious output and a preliminar control law4, based on feedback input-output linearisation. The result is summarized in the following proposition, where for the sake of clarity, it is stated in terms of q= (z, x)⊤and ˙q= ( ˙z, ˙x)⊤, being zthe unactuated coordinate. 3For a wider class of systems, it is necessary to relax some of the assumptions imposed here, as done in [16]. 4This preliminar design does not take into account parametric uncertainties yet. 5 Proposition 1 The system (6) is input-output linearisable through the output ˙η(z, ˙q) = [k1m12(z) 1] ˙q+k2Zz 0 m12(µ)dµ, ˙η(·)∈R,(9) together with the inner and external controller given respectively by u= ∆(z)−1k1dm12(z) dz ˙z2−m12(z) m11(z)f1+k2m12(z) ˙z−ν(10) ∆(z) = k1 m2 12(z) m11(z)−1∈R+,with k1>m11(z) m2 12(z)∈R+, k2∈R+, ν=−k3(m12(z) ˙z+p˙η), k3∈R+, p ∈R+.(11) Moreover, the origin (z, ˙z, ˙η) = (0,0,0) of the closed-loop system is Lyapunov asymptotically and locally exponentially stable (LES). Proof. The first claim of the proposition can be proved by applying to the output the method of feedback input-output linearisation. Thus, from (9) and computing the derivative with respect to time along the system’s trajectories yields ¨η(z, ˙q) = k1 dm12(z) dz ˙z0˙q+ [k1m12(z) 1] ¨q+k2m12(z) ˙z, (12) Substituting ¨qfrom eq. (6) the last expression reads ¨η(z, ˙q) = k1 dm12(z) ˙z dz 0˙q+ [k1m12(z) 1]    −1 m11(z)f1(q, ˙q) O +  −m12(z) m11(z) 1 u +k2m12(z) ˙z, =k1 dm12(z) dz ˙z2−k1 m12(z) m11(z)f1(q, ˙q)−k1 m2 12(z) m11(z)−1u+k2m12(z) ˙z. (13) Thus, the linearising control law is obtained by isolating uand taking ¨η=ν, giving rise to u=k1 m2 12(z) m11(z)−1−1k1 dm12(z) dz ˙z2−k1 m12(z) m11(z)f1(q, ˙q) + k2m12(z) ˙z−ν.(14) The second claim of the proposition can be proved choosing as a Lyapunov function candidate W(z, ˙z, ˙η) = 1 2m11(z) ∆(z) ˙z2+V0−V(z) + 1 2p˙η2,(15) for some positive constant V0and psuch that W≥0. Thus, computing the derivative respect to time along the system’s trajectories, and after some algebraical manipulations, yields ˙ W=−k2m2 12 ˙z2+ (m12 ˙z+p˙η)ν, (16) 6 and hence taking (11) the derivative gives rise to ˙ W=−k2m2 12 ˙z2−k3(m12 ˙z+p˙η)2≤0.(17) Since, the derivative is semidefinite negative we prove that the largest invariant set is the equilibrium point, i.e. ˙ W= 0 if and only if (z, ˙z, ˙η) = (0,0,0). For, note that if ˙ W= 0, then ˙z= 0 and ˙η= 0, and by definition of ˙ηthis implies that ˙x= 0, and hence ˙q=O. Therefore, studying the residual dynamics with ˙q=O yields ¨z= 0 = 1 m11∆f1(q, ˙q)−k1m⊤ 12 dm12 dz ˙z2˙q=O =1 m11∆ dV dz (18) and equality holds only for z= 0 by assumption 1, and then this part of the proof is concluded. The fact that the equilibrium point is also locally exponentially stable is straightforward from the linearisation of the closed-loop dynamics. This leads on one hand, that at the equilibrium point d2V dz2kz=0 <0and on the other hand that ∆(0) >0, which are both satisfied by assumption 1, hence the equilibrium point is LES. ⊳ Remark 1 In the case that there exists a value of z=zfsuch that m12(zf) = 0, assumption 1 is not verified. Then zfrepresents the limit of the feasibility region5where the proposed solution is not valid, because ∆(zf)< 0for any finite k1. In that case, there exits a set Ω,{(z, ˙z)∈R×R|∆(z)>0}, that can be tuned by k1, where the control law is well-defined. Thus, the set Ωc={(z, ˙z, ˙η)∈Ω×R|W(·)≤c}is an estimation of the attraction basin of the equilibrium point, based on the Lyapunov function defined in eq. (15). 4 MAIN RESULT The main contribution of this paper, is the design of a new nonlinear sliding surface, for the class of underactuated mechanical systems to which the constructive feedback linearisation has been previously applied. The sliding surface will allow to cope with the uncertainty in the cancelation of terms, done in the feedback linearisation. The design method can be summarized in the following proposition. Proposition 2 Given the output dynamics (13) expressed as ¨η= ¯α(z, ˙q)−∆(z)u, with ¨η∈R(19) 5see [15] for more details on this issue. 7 it will be assumed that the function ¯α(z, ˙q)is not exactly known, being estimated by the function α(z, ˙q), where the error of estimation is bounded by a positive definite function A(z, ˙q)as A(z, ˙q)≥ |¯α(z, ˙q)−α(z, ˙q)|>0. Then, the sliding surface s= ˙η+k3pη +k3Zz 0 m12(µ)dµ, with s∈R,(20) where ˙ηis given by (9) defines a modified linearising law given by u=uL+ ∆−1k4(z, ˙q)sign(s),where k4(z, ˙q)> A(z, ˙q)(21) and where uLcorresponds to the preliminar linearising law given by (10), stabilises asymptotically the origin of the system (6) in the presence of unmodelled parametric uncertainties. Proof. We prove that the state stends asymptotically to zero. For that, we choose as a Lyapunov function candidate V=1 2s2, with sgiven by (20). Computing the derivative of Vrespect to time and taking into account the definition of νin (11), just for compactness, yields ˙ V=s˙s=s(¨η−ν).(22) From (19) and considering a parametric uncertainty in α(z, ˙q)bounded by |¯α−α| ≤ A, we can rewrite (22) as ˙ V=s(¯α(z, ˙q)−∆(z)u−ν).(23) At this point, notice from (21) that the linearising law has been redefined using an extra term (sign(·)) in order to dominate the effect of the parametric uncertainty. Thus, the derivative of Vreads ˙ V=s(¯α−α−k4sign(s)) =s(¯α−α)−k4|s| ≤(A−k4)|s|,(24) and then taking k4> A ≥ |¯α−α|we ensure that ˙ V < 0and hence stends to zero asymptotically. The proof ends noting that, since by proposition 1 the origin is LES and then boundedness of trajectories are guaranteed in a ball around it, and therefore the sliding surface (20) is well defined. ⊳ 8 5.2 Inertia Wheel Pendulum (IWP) This system is thoroughly described in [20], where the friction is not taken into account. In this section, the friction model is in fact included, yielding ¨z=asin z−kf˙z−bu ¨x=u. (40) with a=mgl J1+J2,b=J2 J1+J2, where Jiare inertia moments. Following the same procedure as in the preceding example, we can attain the linearising law uL= ∆−1k1 J2 J1+J2 (mgl sin z−kf˙z) + k2J2˙z−ν ∆ = k1 J2 2 J1+J2 −1 ν=−k3(Pη +J2˙z),(41) which corresponds to the particular solution of eq. (10)-(11). Notice that, in this example, the friction term kf can be dominated by k2, which could be used to extend the class of systems in proposition 1 (see [16] for more details). The final law to be applied, using the sliding surface, is given by (26), yielding u=uL+ ∆−1k4(z, ˙q)sign(s),where k4(z, ˙q)≥B˜α+˜ ∆uLand B > ∆ ¯ ∆(42) where k4(z, ˙q)has to be estimated. For the sake of simplicity, we will assume that only the parameter J1is not well estimated. In that case, we can get to the following parameters expressions, B > ¯ J1+J2 J1+J2 ˜ ∆ = k1J2 21 ¯ J1+J2 −1 J1+J2 ˜α=k1J2mgl sin z1 ¯ J1+J2 −1 J1+J2(43) 15 Simulations In this section, we have followed the same procedure that in the previous example. For the simulations, the following values of the parameters are considered: ¯ J1= 4,J1= 2,¯ J2=J2= 1,mgl = 1. Notice that the relative error of ¯ J1is (¯ J1−J1)/¯ J1= 50%. Figure 5 illustrates how the sliding mode control is able to cope with bounded parametric uncertainties when they affect to both terms αand ∆, even when the uncertainty is big. 0 50 100 150 −10 0 10 20 0 50 100 150 −10 −5 0 5 10 0 50 100 150 −100 −50 0 50 100 0 50 100 150 −1 −0.5 0 0.5 −2 −1 0 1 2 −1 −0.5 0 0.5 0 50 100 150 −10 0 10 20 s u(N.m) Time (sec) Time (sec)Time (sec) Time (sec)Time (sec) θ θ(degrees) ˙ θ ˙ θ˙x Figure 5: IWP: Simulations with parametric uncertainties, using the control law (41) (equivalent to (42) with k4= 0), and the augmented controller (42) with k4≥B > 0. 6 CONCLUSIONS In this paper, we present a new nonlinear sliding mode surface for a class of underactuated mechanical systems. The surface has been designed by means of a fictitious output, its derivative and an extra nonlinear term from a Lyapunov redesign. This surface ensures, not only robust performance in the sense of parametric uncertainties, but also in perturbation rejection, because the origin is (locally) exponentially stable. Finally, simulations have illustrated the good performance of the controller, applied to two well-known examples, such as the pendulum on a cart and the inertia wheel pendulum, with uncertainty in the inertial terms. 16 ACKNOWLEDGMENTS This work was supported by the MICINN (Ministerio de ciencia e innovaci´on espa˜nol) under grant DPI2007- 64697 and by the Consejer´ıa de Innovaci´on Ciencia y Empresa of The Junta de Andaluc´ıa under IAC programme. References [1] Utkin, V.I.:’Sliding Modes and Their Application to Variable Structure Systems’, MIR Publishers, 1978. [2] Zinober, A.: ’Variable Structure and Lyapunov Control’, Springer, 1994. [3] Agrachev, A.A., Morse, A.S., Sontag, E.D., Sussmann, H.J., and Utkin, V.I.:’Nonlinear and Optimal Control Theory’, LNM 1932, Springer, 2004. 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