A note on calculation of polytopic-invariant feasible sets for linear continuous-time systems

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Abstract

A common way of guaranteeing stability in model predictive control (MPC) is to add into the open-loop optimization problem a terminal constraint set which is invariant and feasible. This note presents an algorithm for calculation of low-complexity polytopic-invariant sets for linear continuous-time systems with complex conjugate closed-loop eigenvalues. Asymmetrical amplitude and rate control input constraints are considered.

Introduction

Any locally stable time-invariant dynamical system admits some domains in its state space from which any state-vector trajectory cannot escape. These domains are called positively invariant sets of the system and are presently widely used in MPC for designing of terminal constraint sets—target sets—as a one tool imperative for guarantee of the system closed-loop stability De Dona et al., 2002, Kouvaritakis et al., 2002, Marruedo et al., 2002, Mayne et al., 2000. The existence, characterization and practical calculation of positively invariant sets of dynamical systems is therefore a basic issue for many constrained MPC schemes utilizing dual mode prediction strategies. The main idea is to determine an invariant set in the state space having the property that no constraints violation occurs as long as the state remains in such a set. More recent contributions Blanchini, 1994, Blanchini, 1999 have proposed constructive methods to deal with the problem.

Among the candidate invariant sets in literature there are two kind of families which have been essentially considered, ellipsoidal and polyhedral sets. Ellipsoids are classical invariant sets in control theory closely related to quadratic Lyapunov functions. Polyhedral sets have been involved in the solution of control problem starting from the 1970s. Their importance is due to the fact that they are often natural expression of physical constraints on control, state or output variables. As invariant sets, they can have significantly larger volumes than the invariant ellipsoidal ones, also. Their shape is in some sense more flexible than that of the ellipsoids, this fact leads to their better ability in the approximation of reachability sets and domains of attraction of dynamical systems. The trade off of this flexibility is that they have in general a more complex representation.

This note deals with a problem of calculation of low-complexity polyhedral invariant sets for linear continuous-time systems with complex conjugate closed-loop eigenvalues. It is assumed that the system is subject to in general asymmetric amplitude and rate control input constraints, what is the most frequent case in practice. The extension to state or output constraints is straightforward and will not be considered here. Compare to maximal admissible sets (Kouvaritakis, Lee, Tortora, & Cannon, 2000), the low complexity invariant sets are more numerically tractable and their definition does not depend on higher powers of the system closed-loop matrix.

Section snippets

Definitions and notations

Generally a polyhedral set (polyhedron) can be introduced in the form of plane or vertex representation. In the plane representation the polyhedral set can be defined as an intersection of a finite number of halfspaces as follows: S(G,w1,w2)={xRn;−w1Gxw2}where the matrix GRg×n, and vectors w1,w2Rg. According to the selected triple [G,w1,w2], S(G,w1,w2) can be any type of the polyhedral set—bounded or unbounded, including or not including the origin point, x=0, in the state space. In

Invariant and feasible terminal sets

An invariant and feasible set of states is such a set that, given the state feedback control law u(t)=Fx(t) operating under the second mode of the dual-mode MPC scheme, satisfies control input constraints , and makes the state trajectory of the closed-loop system remain in the set continuously. The set can then be used as the terminal or target set for the first mode of the MPC scheme.

A polyhedral invariant set of a dynamical system can be described as an intersection of a number of invariant

An algorithm for calculation of invariant and feasible sets

The task now is to construct a possibly largest polytope P(W̃,w1,w2) satisfying positive invariance and feasibility conditions of Lemma 3.4 that means to design algorithms for solving the following problem:

Problem 4.1

Find for the given state feedback gain F optimal values ŵ1, ŵ2 of w1,w2, maximizing in a defined sense the polytope P(W̃,w1,w2) fulfilling simultaneously conditions of Lemma 3.4.

Usually the aim is to design a procedure intended to enlarge the polytope volume. Based on linear and convex

Conclusions

In this note the construction of low-complexity invariant and feasible polytopic sets as terminal constraint sets for stable predictive control of continuous-time linear systems with amplitude and rate control input constraints is discussed. An algorithm has been suggested trying to maximize volumes of these sets. The Theorem 3.1 defines a condition which is necessary for its solvability in the case of closed-loop complex conjugate eigenvalues. The theorem requires to select the state feedback

Acknowledgements

The investigation reported in the paper was supported by the Slovak Scientific Grant Agency VEGA—project No. 1/9278/02.

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Presented as an invited session paper at the 2nd IFAC Conference on Control Systems Design, Bratislava, Slovak Republic, September 2003.

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