Elsevier

Automatica

Volume 42, Issue 7, July 2006, Pages 1217-1222
Automatica

Brief paper
Robust output feedback model predictive control of constrained linear systems

https://doi.org/10.1016/j.automatica.2006.03.005Get rights and content

Abstract

This paper provides a solution to the problem of robust output feedback model predictive control of constrained, linear, discrete-time systems in the presence of bounded state and output disturbances. The proposed output feedback controller consists of a simple, stable Luenberger state estimator and a recently developed, robustly stabilizing, tube-based, model predictive controller. The state estimation error is bounded by an invariant set. The tube-based controller ensures that all possible realizations of the state trajectory lie in a simple uncertainty tube the ‘center’ of which is the solution of a nominal (disturbance-free) system and the ‘cross-section’ of which is also invariant. Satisfaction of the state and input constraints for the original system is guaranteed by employing tighter constraint sets for the nominal system. The complexity of the resultant controller is similar to that required for nominal model predictive control.

Introduction

Model predictive control has received considerable attention driven largely by its ability to handle hard constraints as well as nonlinearity. An inherent problem is that model predictive control normally requires full knowledge of the state (Findeisen et al., 2003, Mayne et al., 2000). Whereas, in many control problems, not all states can be measured exactly. In practice this problem is often overcome by employing ‘certainty equivalence’. For linear systems not subject to disturbances that employ an observer and linear control, stability of the closed-loop can be guaranteed by the separation principle. However, when state and output disturbances are present and the system or its controller is nonlinear (as is the case with model predictive control of constrained systems), stability of the closed-loop cannot, in general, be ensured by simply combining a stable estimator with a stable state feedback controller (Atassi and Khalil, 1999, Teel and Praly, 1995). Robust stability is obtained if the nominal system is inherently robust and the estimation errors are sufficiently small; however, predictive controllers are not always inherently robust (Grimm et al., 2004, Scokaert et al., 1997).

An appealing approach for overcoming this drawback is to use robust controller design methods that take the state estimation error directly into account. In the late sixties, Witsenhausen, 1968a, Witsenhausen, 1968b studied robust control synthesis and set-membership estimation of linear dynamic systems subject to bounded uncertainties; this work was followed by Schweppe (1968), Bertsekas and Rhodes, 1971a, Bertsekas and Rhodes, 1971b, Glover and Schweppe (1971) and by related work on viability (Aubin, 1991, Kurzhanski and Vályi, 1997, Kurzhanski and Filippova, 1993). Further results include: a paper by Blanchini (1990) that deals with linear, constrained, uncertain, discrete-time systems, an anti-windup scheme that employs invariant sets in a similar fashion to their use in this paper (Shamma, 2000), output feedback control for uncertain dynamic systems with scalar controls (Shamma & Tu, 1998), recursive dynamic programming equations for optimal control of dynamic systems subject to uncertainty and imperfect measurements (Moitié, Quincampoix, & Veliov, 2002), and optimal control of systems subject to both observable and unknown disturbances (Quincampoix & Veliov, 2005). Several authors, e.g. Bemporad and Garulli (2000) advocate set-membership estimation for model predictive control; to reduce complexity in such controllers, Chisci and Zappa (2002) employs efficient computational methods. For a similar system class, Löfberg (2002) proposes joint state estimation and feedback using min-max optimization. For linear systems with input constraints, the method in Lee and Kouvaritakis (2001) achieves stability by using invariant sets for an augmented system. By constructing an invariant set for the observer error, Kouvaritakis, Wang, and Lee (2000) adapts the model predictive controller in Cannon, Kouvaritakis, Lee, and Brooms (2001) to ensure closed-loop asymptotic stability. In Fukushima and Bitmead (2005) the authors use a comparison system to establish ultimate boundedness of the trajectories of a closed-loop system consisting of an observer and a model predictive controller.

In this paper we consider the output feedback problem for constrained linear discrete-time systems subject to state and measurement disturbances. The basic idea is to consider the state estimation error as an additional, unknown but bounded uncertainty that can be accounted for in a suitably modified robust model predictive controller that controls the nominal observer state rather than the unknown but bounded system state. The state estimation error is bounded by a simple, pre-computed, invariant set. The controller uses a tube, the center of which is obtained by solving a conventional (disturbance-free) model predictive control problem that yields the center of the tube; the tube is obtained by ‘adding’ to its center another simple, pre-computed, invariant set. Tighter constraints in the optimal control problem solved online ensure that all realizations of the state trajectory satisfy the state and control constraints.

In comparison to previous work on output model predictive control, simplicity is achieved by using a Luenberger observer (that is less complex than a set-membership state estimator), an invariant set that bounds the estimation error, as in Shamma (2000), and a simple tube-based robust model predictive controller that is nearly as simple to implement as a nominal linear model predictive controller. Robust exponential stability of a “minimal” robust invariant set is established.

Nomenclature: In the following N{0,1,2,}, N+{1,2,} and Nq{0,1,,q}. A polyhedron is the (convex) intersection of a finite number of open and/or closed half-spaces and a polytope is the closed and bounded polyhedron. Given two sets U and V, such that URn and VRn, Minkowski set addition is defined by UV{u+v|uU,vV} and Minkowski (Pontryagin) set difference by UV{x|xVU}. A set URn is a C set if it is a compact, convex set that contains the origin in its (non-empty) interior. Let d(z,X)inf{|z-x|p|xX} where |·|p denotes the p vector norm. Let ρ(A) denote spectral radius of a given matrix ARn×n.

Section snippets

Proposed control methodology

We consider the following uncertain discrete-time linear time-invariant system:x+=Ax+Bu+w,y=Cx+v,where xRn is the current state, uRm is the current control action, x+ is the successor state, wRn is an unknown state disturbance, yRp is the current measured output, vRp is an unknown output disturbance, (A,B,C)Rn×n×Rn×m×Rp×n, the couple (A,B) is assumed to be controllable and the couple (A,C) observable. The state and additive disturbances w and v are known only to the extent that they lie,

Bounding the estimation and control errors

We recall the following standard definitions (Blanchini, 1999, Kolmanovsky and Gilbert, 1998):

Definition 1

A set ΩRn is positively invariant for the system x+=f(x) and the constraint set X if ΩX and f(x)Ω,xΩ. A set ΩRn is robust positively invariant for the system x+=f(x,w) and the constraint set (X,W) if ΩX and f(x,w)Ω,wW, xΩ.

Our next step is to establish that the estimation and control errors (respectively, x˜ and e) can be bounded by robust positively (control) invariant sets. The difference

Robust output feedback MPC

Theorem 1 provides the ingredients we require to obtain output model predictive control of the original system by employing the model predictive controller in Mayne, Seron, and Raković (2005) to control robustly the observer system (4):x^+=Ax^+Bu+δ¯,δ¯LCx˜+Lv,where δ¯Δ¯=LCS˜LV. Let the cost VN(x¯,u¯) be defined byVN(x¯,u¯)i=0N-1(x¯(i),u¯(i))+Vf(x¯(N)),where N is the horizon and Vf(·) is the terminal cost function and (·) is the stage cost defined by(x,u)(1/2)[xQx+uRu],Vf(x)=(1/2)xPx,

Brief example

Our illustrative example is a double integrator: x+=1101x+11u+w,y=[11]x+vwith additive disturbances (w,v)W×V where W{wR2||w|0.1} and V{vR||v|0.05}. The state and control constraints are (x,u)X×U where X{xR2|x1[-50,3],x2[-50,3]} and U{uR||u|3} (xi is the ith coordinate of a vector x). The control matrix K and the output injection matrix L are K=[11], L=[11]. The cost function is defined by (16)–(17) with Q=I, R=0.01; the terminal cost Vf(x) is the value function (1/2)xPfx for

Conclusions

This paper presents a simple output feedback model predictive controller for constrained linear systems with input and output disturbances. Simplicity is achieved by using a Luenberger observer (that is less complex than a set-membership state estimator), an invariant set that bounds the estimation error and a tube-based robust model predictive controller that is almost as simple to implement as a nominal linear model predictive controller. Robust exponential stability is established.

David Mayne received the Ph.D. and D.Sc degrees from the University of London, and the degree of Doctor of Technology, honoris causa, from the University of Lund, Sweden. He has held posts at the University of the Witwatersrand, the British Thomson Houston Company, University of California, Davis and Imperial College London where he is now Senior Research Fellow. His research interests include optimization, optimization based design, nonlinear control and model predictive control.

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    David Mayne received the Ph.D. and D.Sc degrees from the University of London, and the degree of Doctor of Technology, honoris causa, from the University of Lund, Sweden. He has held posts at the University of the Witwatersrand, the British Thomson Houston Company, University of California, Davis and Imperial College London where he is now Senior Research Fellow. His research interests include optimization, optimization based design, nonlinear control and model predictive control.

    Sasa V. Raković received the B.Sc. degree in Electrical Engineering from the Technical Faculty Cačak, University of Kragujevac (Serbia and Montenegro), the M.Sc degree in Control Engineering and the Ph.D. degree in Control Theory from Imperial College London. He is currently employed as a Research Associate in the Control and Power Research Group at Imperial College London. His research interests include set invariance, robust model predictive control and optimization based design.

    Rolf Findeisen received the M.Sc. degree in Chemical Engineering from the University of Wisconsin, Madison and the Diploma degree in Engineering Cybernetics and the Dr.-Ing. from the University of Stuttgart. He is currently Habilitand (assistant professor) at the University of Stuttgart. His research interests include nonlinear system theory and control, particularly predictive and optimization based control, with applications in mechatronics, process control, and biology.

    Frank Allgöwer is the director of the Institute for Systems Theory and Automatic Control at the University of Stuttgart. He studied Engineering Cybernetics and Applied Mathematics in Stuttgart and at UCLA, respectively, and received his Ph.D. degree in chemical engineering from the University of Stuttgart. He is an editor of Automatica, associated editor of a number of journals and organizer and co-organizer of several international conferences. His main areas of interest are in nonlinear, robust and predictive control, with application to a wide range of fields.

    This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Franco Blanchini under the direction of Editor Roberto Tempo. Research supported by the Engineering and Physical Sciences Research Council, UK.

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