Generalized Output Regulation for a Class of Nonlinear Systems via the Robust Control Approach

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1 Generalized Output Regulation for a Class of Nonlinear Systems via the Robust Control Approach L. E. RAMOS-VELASCO, S. CELIKOVSKÝ, V. LÓPEZ-MORALES AND V. KU CERA Centro de Investigación en Tecnologías de Información y Sistemas Universidad Autónoma del Estado de Hidalgo Carr. Pachuca-Tulancingo, Km. 4.5 C.P , Pachuca, Hgo. MEXICO Institute of Information Theory and Automation Academy of Sciences of the Czech Republic. P.O. Box 18, Prague CZECH REPUBLIC Abstract: - We address the problem of generalized output regulation for nonlinear systems in the presence of unknown parameters in the full information case. We generalize the classical output regulation problem in order to expand the class of reference or disturbance signals. Under appropriate sufcient conditions, a state feedback regulator is built for a class of nonlinear systems where the term including the unknown parameter is assumed to satisfy a matching condition. Our study of the above problem, further referred to as the so-called generalized output regulation problem, combines the approach based on the well-known notion of the regulator equation with the classical concept of the invariant distribution and on the Lyapunov theory. Key-Words: - Nonlinear systems, output regulation, center manifold theory, Lyapunov function, disturbance rejection. 1 Introduction 1 A central problem in control theory and applications is to design a control law to achieve asymptotic tracking with disturbance rejection in a nonlinear system. When a class of reference inputs and disturbances are generated by an autonomous differential equation, this problem is called nonlinear output regulation problem, or alternatively, nonlinear servomechanism problem 10. The corresponding autonomous differential equation is usually called as the exogeneous system and is supposed to be neutraly stable. In the sequel, the above setting will be referred to as the classical output regulation problem. In other words, the classical output regulation problem treats a possible unknown reference signal and/or disturbances generated by the known neutrally stable autonomous exosystem with possible unknown initial states. For linear systems the classical output regulation was extensively studied in 2, 3, 4. For nonlinear systems, the problem was rst studied in 7, and solutions to the output regulation of nonlinear 1 Supported by PROMEP under research program 103.5/03/1130 systems have been presented in 8, 10 using full information" which includes the measurements of exogenous signals as well as of the system state. The necessary and sufcient conditions for the existence of a local full information solution of the classical output regulation problem are given in 10, 8; they basically mean that the linearized system is stabilizable and there exists a certain invariant manifold. The classical output regulation via error feedback has been solved in 1, 9 by application of system immersion technique. The plant uncertainty parametrized by unknown constant parameters is treated as a special case of exogenous signals and the solution, extended from the error feedback regulation, is referred to as the structurally stable regulation in 1. Some of the recent results in the robust control eld 13, 14, 18 are based on the center manifold theory and the related nonlinear regulator theory; in the adaptive control eld. However, the main limitation of the classical regulation scheme is that a precise model of the system that generates all exogenous inputs must be available, to be replicated in the control law. This limitation becomes immediately evident in the problem of rejecting a sinusoidal disturbances, not only of unknown am-

2 . 2 / plitude and phase, but also of unknown frequency. Therefore, an alternative formulation would be to require asymptotic tracking of known reference trajectories in spite of unmodelled disturbances acting on the exosystem. To unify this alternative formulation with the classical output regulation concept, the so-called generalized output regulation may be considered. The generalized output regulation was rst posed and solved in 17 for linear systems both continuous and discrete time in terms of necessary and suf- cient geometric conditions involving the classical notions of disturbance decoupling. The corresponding design procedure presented in 17 handles the unmodelled bounded disturbances generated by the known nonautonomous linear system driven by an unknown bounded reference signal. The question arises how the results established by 17 can be extended to the general case in which the plant is described by a nonlinear equation. The generalized output regulation problem resembles somehow the asymptotic model matching problem (AMM) considered in 19, nevertheless, the crucial difference is that the input of the reference model in AMM is supposed to be known and is then used in the corresponding feedback compensator. Therefore, AMM solution cannot be directly applied to the generalized output regulation problem, considering the exogeneous system driven by an unknown signal as the analogue of the reference model in AMM problem. In 15, we were able to characterize the solvability of the state feedback generalized output regulation problem for nonlinear systems in terms of the solvability of the regulator equation with the classical concept of the invariant distribution. The state feedback generalized almost output regulation problem for a class of nonlinear systems is solved in 16. The purpose of this paper is to point out that by combining the approach to classical output regulation presented in 9 with the design method for generalized output regulation via state feedback presented in 15, it is easy to address the problem of generalized output regulation for a signicant class of nonlinear systems, in the presence of unknown parameters. We establish a link between the two approaches; as a matter of fact, in this work, an adaptive controller, including a particular Lyapunov function is determined by exploiting concepts of the classical nonlinear regulation theory. The paper is organized as follows. In Section 2 we outline the generalized nonlinear output regulation problem which, as anticipated above, is based on the introduction of a driving signal to the exosystem. In Section 3, we state the assumptions necessary to the well-posedness of the problem. In Section 4 an adaptive controller solving the state feedback generalized output regulation problem for a specic class of nonlinear systems including an unknown vector parameter, is determined. We show how this specic problem is related to the standard problems of disturbance decoupling for nonlinear systems which have been recently studied in 9, 12. Finally, Section 5 draws conclusions and outlines some future research. 2 Problem formulation Following the linear concept of the generalized output regulation problem 17, the generalized output regulation problem for nonlinear systems can be introduced via the conguration provided by the master-slave block diagram of Figure 1. The task! " # ' y( ) *,+-. /10 3 $! 4 0 / Figure 1: Conguration of the output regulation schemes. (a) r 0, generalized output regulation. (b) r 0, classical output regulation. of the controller is to generate u so that the tracking error e is converging to zero for all initial conditions of both plant and exosystem and all external signals r(t) from a suitable functional class. The plants (slaves) we consider in this paper are afne multiinput multi-output (MIMO) nonlinear systems, described by the equations of the form ẋ = f(x) + % & m g i (x)u i + l p i (x)w i (1) := f(x) + g(x)u + p(x)w, y i = h i (x), i = 1,..., p (2) where (1) describes the plant with state x, dened on a neighborhood X of the origin of R n, input u R m and output y R p, subjected to the effect of a disturbance represented by the vector elds

3 p i (x), i = 1,..., l. It is assumed that the vector eld f( ), and the columns of g( ) and p( ) are smooth vector elds, while each component of h i ( ) is a smooth function, with f(0) = 0, g(0) = 0 and h(0) = 0. We only consider reference outputs to be tracked and perturbations to be rejected which both are generated by an unknown exosystem as follows. Assumption 1 (Exosystem): Exosystem is the following nonautonomous system with output ẇ = s(w) + d(w)r(t), w R l (3) y ref = q(w), r R ρ, d := d 1... d ρ. Here, s( ) and the columns of d( ) are smooth vector elds with s(0) = 0 and q(0) = 0, q, q(0) = 0, is smooth function, while r(t) is unknown external driving signal. Further, it is assumed that r(t) is limited to a functional subclass of L Rρ where it holds for all solutions of (3) and some class-k functions α, β that w(t) L R l α( w(0) R l) + β( r(t) L R ρ ). Roughly speaking, the Assumption 1 restricts the class of exogenous inputs to those signals which do not decay to zero and do not tend to innity as time goes to innity for any signal r. The controller is to be designed so that the slave obeys the master, namely the error signal e(t) = y(t) q(w(t)) (4) converges asymptotically to zero as t. In the following statement we give a precise formulation of the control problem under consideration. Denition 1 (State Feedback Generalized Output Regulation Problem (SFGORP)): Given the reference output y ref generated by an exosystem (3), the SFGORP consists in nding a state feedback controller u = γ(x, w) where γ(, ) is a C k (k 2) mapping, with γ(0, 0) = 0 such that: S F I The equilibrium x = 0 of ẋ = f(x) + g(x)γ(x, 0) is asymptotically stable in the rst approximation. R F I There exists a neighborhood U R n R l of (0, 0) such that, for each initial condition on U and for any signal r (piecewise continuous), the solution of the closed loop system ẋ = f(x) + g(x)γ(x, w) + p(x)w (5) ẇ = s(w) + d(w)r (6) and error (4) satises lim e(t) = 0. t In 15 the following result is proved: Theorem 1 Consider the system given in (1)-(2). Let Assumption 1 be satised. Then, the generalized output regulation problem via state feedback regulator is solvable if (a) there exist C k (k 2) mappings x = π(w), u = c(w), π : R l R n, c : R l R m, ρ : R l R ρ, dened locally in a neighborhood of the origin W 0 R l, with π(0) = 0, c(0) = 0, ρ(0) = 0 satisfying the so-called generalized regulation equation π(w) (s(w) + d(w)ρ(w)) w = f(π(w)) +g(π(w))c(w) + p(π(w))w (7) 0 = h(π(w)) q(w) (8) (b) there exist a C k (k 2) mapping ũ(x, w), ũ : R n R l R m, ũ(0, 0) = 0, dened locally in a neighborhood of the origin U 0 R n R l and a regular involutive distribution in T R n such that (b 1 ) the linear approximation of f + gũ(x, 0) is Hurwitz, (b 2 ũ(π(w), w) = c(w), (b 3 ) for all w, f(x) + g(x)ũ(x, w) + p(x)w, ker dh, (b 4 ) dπ( w ) w, w = span d 1 (w),..., d ρ (w) T R l. 3 Standing assumptions In this section we consider a plant modelled by equations of the form ẋ = f(x) + + m g i (x)u i + m v j (x)θ j j=1 l p i (x)w i = f(x) + g(x)u + p(x)w + v(x)θ, (9) y i = h i (x), i = 1,..., p (10)

4 where θ is a vector of unknown parameters and the same considerations presented in the previous section with regard to system (1)-(2), apply. Our goal is to solve the SFGORP for the system (9)-(10) without knowing the parameter θ. In the following, we set φ(t) = ˆθ(t) θ (11) where ˆθ(t) and θ represent the current estimate and the exact value of the unknown parameter, respectively. The main assumptions needed to solve the problem are, for convenience, listed in the following and then briey justied: Assumption 2 (Matching condition): There exists a vector of m functions k(x) such that v(x) = g(x)k(x) (12) Assumption 3 (Local decomposition): There exists a regular involutive distribution in T R n such that (b 3 ) and (b 4 ) hold. Assumption 4 (Nominal regulator): There exists a mapping ũ(x, w) with ũ(0, 0) = 0, satisfying (b 1 ) and (b 2 ). Assumption 5 (Solution of the regulator equation): There exist two smooth mappings x = π(w), u = c(w), ρ(w), with π(0) = 0, c(0) = 0, ρ(0) = 0 satisfying (7)-(8). As specied in Assumption 2 we consider unknown parameter whose values enter to the plant in a particular form. Assumption 3, namely the existence of a regular involutive distribution and, in turn, the existence of the internal triangular decomposition (16)- (18). The state feedback controller, which solves the regulation problem when the unknown parameter is set to zero, is established in Assumption 4. Finally Assumption 5 is standard in classical output regulation of nonlinear systems Regulator via state feedback Although the main goal of paper is the design of an state feedback regulator, this section is devoted to briey discuss the solution when the state (x, w) is available. This preliminary discussion will make the presentation of the general solution more meaningful. The rst step consists of choosing a control law to force, despite the presence of the unknown parameter θ, the state behavior. In view of this, we consider the static feedback regulator u = ũ(x, w) k(x)ˆθ (13) The closed loop system (9)-(10) with (13), taking into account the Assumptions 2, has the form ẋ = f(x) + g(x)ũ(x, w) g(x)k(x)ˆθ +p(x)w + v(x)θ (14) = f(x) + g(x)ũ(x, w) + p(x)w v(x)φ y i = h i (x), i = 1,..., p. (15) By the Assumption 3, there exists a regular involutive distribution with dim = n n 1 for some n 1 < n. Let ζ = Φ(x) be new coordinates, which exist by virtue of the Frobenius Theorem 10, such that = span{ ζ,, n1 +1 ζ n }. Using the compact notation ζ = (ζ 1, ζ 2 ) T with ζ 1 = (ζ 1,..., ζ n1 ) and ζ 2 = (ζ n1 +1,..., ζ n ), we have that in the ζ- coordinates the system (14)-(15) takes the following form ζ 1 = f 1 (ζ) + g 1 (ζ)ũ(φ 1 (ζ), w) + p 1 (ζ)w ṽ 1 (ζ)φ by (b3) := p(ζ 1, w) ṽ 1 (ζ 1 )φ (16) ζ 2 = f 2 (ζ) + g 2 (ζ)ũ(φ 1 (ζ), w) + p 2 (ζ)w ṽ 2 (ζ)φ (17) y i = hi (ζ 1 ) i = 1,..., p. (18) Further, let π(w), c(w) be a solution of the regulator equation (7)-(8). Straightforward computations show that π π(w) = 1 (w) π 2 = Φ(π(w)), (w) c(w) = ũ(φ 1 (π(w)), w), with dim π 1 = n 1, dim π 2 = n n 1 are solutions to the following equations π 1 (w) w s(w) = f 1 ( π(w)) + g 1 ( π(w)) c(w) + p 1 ( π(w))w ṽ 1 ( π(w))φ by (b3) := p( π 1 (w), w) ṽ 1 ( π 1 (w))φ (19) π 2 (w) w s(w) = f 2 ( π(w)) + g 2 ( π(w)) c(w) + p 2 ( π(w))w ṽ 2 ( π(w))φ (20) 0 = h( π 1 (w)) q(w). (21)

5 Moreover, by Assumption 3 Further, let π 1 (w) d(w) 0. (22) w ζ = ζ π(w) (23) where π(w) is a solution of the system (19)-(20) and denote ζ = ( ζ 1, ζ 2 ) T with ζ 1 = ( ζ 1,..., ζ n1 ) and ζ 2 = ( ζ n1 +1,..., ζ n ). Then the system (16)-(17) can be rewritten in the form where ζ = F 1 ( ζ 1, w) F 2 ( ζ, w) + G 1 ( ζ 1, w) G 2 ( ζ, w) φ = F ( ζ, w) + G( ζ, w)φ (24) ẇ = s(w) + d(w)r (25) F 1 ( ζ 1, w) = p( ζ 1 + π 1 (w), w) p( π 1 (w), w) (26) F 2 ( ζ, w) = f 2 ( ζ + π(w)) + g 2 ( ζ + π(w)) ũ(φ 1 ( ζ + π(w)), w) + + p 2 ( ζ + π(w))w f 2 ( π(w)) g 2 ( π(w)) c(w) p 2 ( π(w))w (27) G 1 ( ζ 1, w) = ṽ 1 ( ζ 1 + π 1 (w)) + ṽ 1 ( π 1 (w)) (28) G 2 ( ζ, w) = ṽ 2 ( ζ + π(w)) + ṽ 2 ( π(w)) (29) Since ũ(x, w) solves the State Feedback Generalized Output Regulator Problem for the system (9)- (10) considered with θ = 0, the equilibrium ζ = 0 of the system ζ = = ζ1 ζ 2 = F 1 ( ζ 1, w) F 2 ( ζ, w) p( ζ 1, w) f 2 ( ζ) + g 2 ( ζ)ũ(φ 1 ( ζ, 0) (30) is exponentially stable. Moreover, the equilibrium w = 0 of the exosystem (25) is stable by Assumption 1. The above properties imply 11 that the equilibrium ( ζ, w) = (0, 0) of the system ζ = F ( ζ, w) (31) ẇ = s(w) + d(w)r (32) is stable. Then, the theory of stability 6, 11 assures the existence of a Lyapunov function V ( ζ, w) of a neighborhood of the origin U such that V (0, 0) = 0 (33) V ( ζ, w) > 0 ( ζ, w) U (0, 0) (34) V ( ζ, w) = V ζ F ( ζ, w) + V w (s(w) +d(w)r) 0 ( ζ, w) U (35) Let R be the set of all points where V ( ζ, w) = 0, and M be the largest invariant set (with respect to the motion of the system (24)-(25)) in R. Assumption 6 (Largest invariant): Let P be the following set with P = {( ζ, w) : ζ = 0} (36) M P. Now, we present the following theorem on the generalized output regulation problem via state feedback regulator. Theorem 2 Consider the system given in (9)-(10). Assume Assumptions 1-6. Then, the controller together with the update law u = ũ(x, w) k(x)ˆθ (37) φ = V ζ G( ζ, w) (38) where G( ζ, w) and V ( ζ, w) are the previously de- ned functions, solves the State Feedback Generalized Output Regulation Problem. Proof: Assuming Assumptions 1-6 hold, we aim to show that Denition 1 is satised with γ(x, w) = ũ(x, w) k(x)ˆθ. Obviously, by Assumption 4 the condition S F I of Denition 1 holds. To prove the condition R F I consider the Lyapunov function W ( ζ, φ, w) = V ( ζ, w) φ2 (39) Taking into account (24)-(25)) and (38), the derivative of the function (39) is such that W (0, 0, 0) = 0 (40) W ( ζ, φ, w) > 0 ( ζ, w) U (0, 0), φ (41) Ẇ ( ζ, φ, w) = V ζ F ( ζ, w) + V w (s(w) +d(w)r) 0 ( ζ, w) U, φ (42)

6 From (40)-(42), it can be deduced that the origin of the closed loop system is stable. Moreover, by applying the invariance principle of LaSalle's Theorem 6, 11 it is possible to claim that the motion of the system (24)-(25) originated in a point U, asymptotically converge to the largest invariant subset characterized by Ẇ ( ζ, φ, w) = 0, that is, due to Assumption 6, by ζ = 0. From (23), this implies, that every motion ζ(t), originated in U, asymptotically converges to the center manifold π(w), i.e. we can see from (4) and (8) that error e(t) tends to zero as time tends to innity. In other words, the condition (R F I ) of Denition 1 holds. 5 Conclusions The solution of the problem of generalized regulation with nonautonomous exosystem for a large class of nonlinear systems characterized by the presence of an unknown parameter, has been presented. The output regulation is achieved by linking concepts of the robust control theory (center manifold), with concepts of the adaptive control theory (Lyapunov function). References: 1 C. I. Byrnes, F. Delli Priscoli, A. Isidori and W. Kang, Structurally stable output regulation of nonlinear systems, Automatica, Vol. 33, 1997, pp E.J. Davison, The output control of linear time-invariant multi-variable systems with unmeasured arbitrary disturbances, IEEE Transactions on Automatic Control, Vol. 17, 1972, pp B.A. Francis and W. Murray Wonham, The internal model principle of control theory, Automatica, Vol. 12, 1976, pp B.A. Francis, The linear multivariable regulator problem, SIAM Journal on Control and Optimization, Vol. 15, 1977, pp R.A. Freeman and P.V. Kokotovic, Tracking controllers for systems linear in unmeasured states, Automatica, Vol. 32, 1996, pp W. Hahn, Stability of motion, Springer-Verlag, J.S.A. Hepburn and W.M. Wonham, Error feedback and internal models on differentiable manifolds, IEEE Transactions on Automatic Control, Vol. 29, 1981, pp J. Huang and W.J. Rugh, Stabilization on zeroerror manifolds and the nonlinear servomechanism problem, IEEE Transactions on Automatic Control, Vol. 37, 1992, pp A. Isidori, Nonlinear Control Systems, 3rd ed., New York: Springer-Verlag, A. Isidori and C. I. Byrnes, Output regulation of nonlinear systems, IEEE Transactions on Automatic Control, Vol. 35, 1990, pp H.K. Khalil, Nonlinear Systems, 2nd ed. New York: MacMillan, R. Marino and P. Tomei, Nonlinear Control Design-Geometric, Adaptive and Robust. London, U.K.: Prentice-Hall, J.B. Pomet and L. Praly, Adaptive nonlinear regulation: Equation error from the Lyapunov function, 28th Conference on Decision and Control, Tampa, F.D. Priscoli, Adaptive control of a class of nonlinear systems via the robust control approach, IFAC NOLCOS92, Vol. 1, 1992, pp L.E. Ramos, S. Celikovský and V. Kucera, Generalized Output Regulation Problem for a Class of Nonlinear Systems with Nonautonomous Exosystem," IEEE Transactions on Automatic Control, accepted. 16 L.E. Ramos, S. Celikovský, V. Kucera and J. Ruíz, Almost Output Regulation of A Class of Nonlinear Systems with Nonautonomous Exosystem," Latin American Control Conference. Guadalajara, México, A. Saberi, A.A. Stoorvogel and P. Sannuti, On output regulation for linear systems, International Journal of Control, Vol. 74, 2001, pp A.R Tell, Robust and adaptive nonlinear output regulation, European Control Conference, Grenoble, M. Yokomichi and M. Shima, Another approach to asymptotic model matching problem for nonlinear systems, International Journal of Robust and Nonlinear Control, Vol. 8, 1998, pp