A Generalization of Barbalat s Lemma with Applications to Robust Model Predictive Control Fernando A. C. C. Fontes 1 and Lalo Magni 2 1 Officina Mathematica, Departamento de Matemática para a Ciência e Tecnologia, Universidade do Minho, 4800-058 Guimarães, Portugal (e-mail:ffontes@mct.uminho.pt). 2 Dipartimento di Informatica e Sistimistica, Università degli Studi di Pavia, via Ferrata 1, 27100 Pavia, Italy. (e-mail: lalo.magni@unipv.it). Abstract Barbalat s lemma is a well-known and powerful tool to deduce asymptotic stability of nonlinear systems, specially time-varying systems, using Lyapunov-like approaches. Although simple variants of this lemma have already been used successfully to prove stability results for Model Predictive Control (MPC) of nonlinear and time-varying systems, further modifications are needed to address systems allowing uncertainty. The generalization proposed here can be used to guarantee that the state trajectory (resulting from the MPC algorithm) asymptotically approaches some set containing the origin, if there is a function that coincides with the trajectory at a sequence of instants of time and satisfies some boundedness and smoothness conditions. We discuss here the way the proposed lemma can help to establish robust stability results for MPC of nonlinear systems subject to bounded disturbances. keywords: Barbalat s lemma, robust stability, nonlinear systems, predictive control, receding horizon. 1 Introduction Barbalat s lemma is a well-known and powerful tool to deduce asymptotic stability of nonlinear systems, specially time-varying systems, using Lyapunov-like approaches (see e.g. [4] for a discussion and applications). Simple variants of this lemma have been used successfully to prove stability results for Model Predictive Control (MPC) of nonlinear and time-varying systems [3, 1]. A recent work on robust MPC of nonlinear systems [2] used a generalization of Barbalat s lemma as an important step to prove stability of the algorithm. Due to lack of space, the use of such lemma was not much explored there.
However, is our believe that the generalization of the lemma presented might provide a useful tool to analyse stability in other robust continuous-time MPC approaches. Barbalat s lemma can be used to guarantee that the state trajectory converges asymptotically to the origin, if some smoothness and boundedness conditions involving the trajectory are satisfied. The generalization proposed here can be used to guarantee that the state trajectory asymptotically approaches some set containing the origin, if there is a function that coincides with the trajectory at a sequence of instants of time and satisfies some boundedness and smoothness conditions. When the stability properties of a continuous-time MPC framework allowing uncertainty are analysed, a difficulty encountered is that the predicted trajectory only coincides with the resulting trajectory at specific sampling instants. Stability properties of the resulting trajectory can be obtained using information on the behavior of the predicted trajectory through the application of the lemma proposed here. 2 Barbalat s Lemma and Variants A standard resuln Calculus states thaf a function is lower bounded and decreasing, then it converges to a limit. However, we cannot conclude whether its derivative will decrease or not. For instance, the function f(t) = e t sin(e 2t ) converges to zero as t, but f (= e t sin(e 2 t )+2 e t cos(e 2 t )) is unbounded. If we want to guarantee that f(t) 0 as t, we musmpose some smoothness property on f(t). Thas, we must require that f is uniformly continuous. We have in this way a well-known form of the Barbalat s lemma (see e.g. [4]). Lemma 2.1 (Barbalat s lemma) Let t F (t) be a differentiable function with a finite limit as t. If F is uniformly continuous, then F (t) 0 as t. A simple modification that has been useful in some MPC (nominal) stability results [3, 1] is the following. Lemma 2.2 Let M be a continuous, positive definite function and x be an absolutely continuous function on IR. If x( ) L <, ẋ( ) L <, and T lim T M(x(t)) dt <, then x(t) 0 as t. 0 Now, suppose that due to disturbances we have no means of guaranteeing that all the hypothesis of the lemma are satisfied for the trajectory x we want to analyse. Instead some hypothesis are satisfied on a neighbouring trajectory ˆx that coincides with the former at a sequence of instants of time. These are the conditions the following lemma, which is the main result here. Lemma 2.3 (A generalization of Barbalat s lemma) Let A be subset of IR n containing the origin, and M : IR n IR be a continuous function such that M(x) > 0 for all x / A and M(x) = 0 for some x A. Let d A (x) be the distance function from a point x IR n to the set A. Consider also functions x and ˆx from IR to IR n coinciding at the points of a sequence π = { } i IN with +1 = + δ. 2
If x ( ) L (0, ) <, ẋ ( ) L (0, ) <, ˆx( ) L (0, ) <, ˆx( ) L ([,+δ)) <, and then for some δ > 0 T lim M(ˆx(t)) dt <, T 0 d A (x (t)) 0 as t. 3 Application to a Robust MPC framework The application of this lemma to prove stability for a Robust MPC framework is now discussed. Consider the following nonlinear system subject to disturbances ẋ(t) = f(x(t), u(t), d(t)) a.e. t 0, (1) x(0) = x 0 X 0, x(t) X for all t 0, u(t) U a.e. t 0, d(t) D a.e. t 0. Here X 0 IR n is the set of possible initial states, X IR n is the set of possible states modelling state constraints, U IR m is the set of possible control values modelling input constraints, and D IR p is the set of possible disturbance values. An MPC algorithm to control this system drive it to a target set A is based on repeatedly solving at a sequence of instants of time t 0, t 1,... a Min-max optimal control problem P(x tk, T ): Min u Max d k+t subject to: t k L(x(s), u(s))ds + W (x(t k + T )) x(t k ) = x tk ẋ(s) = f(x(s), u(s), d(s)) a.e. s [t k, t k + T ] u(s) U for all s [t k, t k + T ] x(s) X for all s [t k, t k + T ] x(t k + T ) S. Here, the functions L and W, the horizon T and the terminal set S are design parameters to be tuned according to some stability condition. The solution to each optimal control problem P(x tk, T ) is a pair trajectory/control defined on the interval [t k, t k + T ) and is denoted by ( x k, ū k ). The MPC algorithm performs according to the following receding horizon strategy: 3
1. Measure the current state of the plant x tk. 2. Solve problem P(x tk, T ), obtaining the optimal control ū k on the interval [t k, t k + T ). 3. Apply to the plant the control ū k on the interval [t k, t k + δ) (and discard all the remaining for t t k + δ). 4. Repeat the procedure from (1.) for the next sampling instant t k+1 = t k +δ. Let x represent the actual trajectory resulting from the MPC strategy. Let ˆx be the concatenation of predicted trajectories x k for each optimization problem. Thas for k 0 ˆx(t) = x k (t) for all t [t k, t k + δ). Note that ˆx coincides with x at all sampling instants. A stability analysis can be carried out (see [2]) to show thaf the design parameters are conveniently selected, then a certain value function V is decreasing. More precisely, for some δ > 0 small enough and for any t > t > 0 V (t, x (t )) V (t, x (t )) t M(ˆx(s))ds. where M is a continuous function satisfying M(x) > 0 for all x / A and M(x) = 0 for some x A. We can then write that for any t t 0 0 V (t, x (t)) V (t 0, x (t 0 )) t 0 M(ˆx(s))ds. Since V (t 0, x (t 0 )) is finite, we conclude that the function t V (t, x (t)) is bounded and then that t t 0 M(ˆx(s))ds is also bounded. Therefore t ˆx(t) is bounded and, since f is continuous and takes values on bounded sets of (x, u, d), t ˆx is also bounded. Using the fact that x is absolutely continuous and coincides with ˆx at all sampling instants, we may deduce that t ẋ (t) and t x (t) are also bounded. We are in the conditions to apply the modification of Barbalat s lemma, yielding that the trajectory asymptotically converges to the set A. 4 Proof of Lemma 2.3 Assume, contradicting the assertion of the lemma, that x (t) fails to converge to A as t. Then for some c > 0 there exists a sequence {s k } k IN tending to such that d A (x (s k )) 2c for all k IN. Since ẋ ( ) L is bounded, we can find δ > 0 such that for some subsequence of π, { } i I (with I some infinite countable subset of IN) such that d A (x ( )) c for all i I. Since A contains the origin, we also have x ( ) d A (x ( )) c. The same conclusions are valid for ˆx at the points I, i.e. ˆx( ) d A (ˆx( )) c. Let K be a positive number satisfying ˆx( ) L ([, +δ)) K and c/(2δ) < K. (The lasnequality guarantees that the intervals [, + c/(2k)] are nonoverlapping and contained in [, + δ).) 4
Let R be positive number such that ˆx( ) L R and the set B = {x IR n : d A (x) c/2, x R} is non-empty. Since M is continuous, M(x) > 0 for all x B, and B is compact, then there exists m > 0 such that m M(x) for all x B. Note that for all t [, + c/(2k)] we have ˆx(t) ˆx( ) ˆx(s) ds [c/(2k)]k c/2 Then, by the triangle inequality d A (ˆx(t)) d A (ˆx( )) ˆx(t) ˆx( ) c/2 Therefore M(ˆx(t)) m for all t [, + c/(2k)], and i+1 M(ˆx(s))ds i+c/(2k) M(ˆx(s))ds mc/(2k). This would imply that M(ˆx(s))ds as t, contradicting the hypothesis and thereby completing the 0 proof. References [1] F. A. C. C. Fontes. A general framework to design stabilizing nonlinear model predictive controllers. Systems & Control Letters, 42:127 143, 2001. [2] F.A.C.C. Fontes and L. Magni. Min-max model predictive control of nonlinear systems using discontinuous feedbacks. IEEE Transactions on Automatic Control, 48(10):1750 1755, 2003. [3] H. Michalska and R. B. Vinter. Nonlinear stabilization using discontinuous moving-horizon control. IMA Journal of Mathematical Control and Information, 11:321 340, 1994. [4] J. E. Slotine and W. Li. Applied Nonlinear Control. Prentice Hall, New Jersey, 1991. 5