Optimal regulation of homogeneous systems☆,☆☆
Introduction
Through the last two decades, investigations on asymptotic stabilization of continuous-time (CT) homogeneous control systems have yielded a wealth of results. See, for instance, Grüne (2000), Hermes (1995), Kawski (1988), and references therein. Although not as much, discrete-time (DT) homogeneous systems were also studied and results emulating their CT counterparts were generated, see Grüne and Wirth (1999) and Hammouri and Benamor (1999). The starting point for this work, however, has been Tuna and Teel (2004) where an offline numerical algorithm was presented to solve an infinite horizon optimization problem and compute a look-up table type feedback law that globally stabilizes the attractor of a DT homogeneous (with respect to a generalized dilation with degree zero) system that is asymptotically controllable. In this paper we combine the feedback generating algorithm of Tuna and Teel (2004) with the idea of sample and hold with state dependent hold interval policy and propose a method to robustly regulate the origin of any asymptotically controllable CT homogeneous system (with arbitrary degree of homogeneity) under generic assumptions.
The DT model obtained from a CT homogeneous system via sample and hold applied in a standard fashion, where the hold intervals are fixed, is not necessarily homogeneous. However, if one adopts a sample and hold policy where the hold duration is chosen proportional to the value of a positive definite homogeneous measure of the state (at the instant of sampling) raised to the power equal to the negative of the degree of homogeneity of the system being sampled, then the resulting DT model turns out to be homogeneous with degree zero. Moreover, if the proportionality constant is small enough then the DT model inherits the global asymptotic controllability of the CT system, provided, of course, that the CT system is in fact globally asymptotically controllable to the origin. Exploiting these facts, we propose the following procedure to regulate the origin of a given CT homogeneous system. First, obtain an asymptotically controllable DT homogeneous model using an appropriate sample and hold policy. Then, generate a globally stabilizing feedback law for the DT model by numerically solving an infinite horizon optimization problem (using the recursive algorithm presented in Tuna & Teel, 2004) on a bounded set of initial conditions. Finally, close the loop for the CT system applying the computed feedback law through the sample and hold policy used in obtaining the DT model.
In the sample and hold policy that we require in order to realize the method, the lengths of hold intervals need to get arbitrarily small as the state approaches the origin for systems with negative degree of homogeneity or as the state moves away from the origin for systems with positive degree of homogeneity. Since in practice we always have a positive lower bound on the length of the hold interval, the stability of the closed loop is semiglobal (in the minimum possible hold interval) for positive degree systems and practical (in the minimum possible hold interval) for negative degree systems. For degree zero homogeneous systems, on the other hand, a fixed hold interval works and results in global exponential convergence to the origin.
Another practical issue we would like to point out is that the proposed procedure depends on numerical calculations that are performed via gridding a bounded surface of dimension one less than the order of the system. The number of points on the grid increases exponentially in the order of the system which in turn implies that the amount of time and memory required for computations increases exponentially in the number of states system has. However, the computations are carried offline. That is, once they are completed (if they can ever be), we no longer have to worry about their being lengthy.
Part of our approach bears some similarities with (part of) the works Grüne (2000), Junge and Osinga (2004), and Kreisselmeier and Birkhölzer (1994) from certain aspects. In Grüne (2000, Section 5) it is shown that for a given CT homogeneous system, a control Lyapunov function and an associated feedback law can be numerically computed by solving an optimization problem. There, too, the computations are performed on a bounded set (the unit sphere) and the resulting feedback law when applied for a state dependent hold interval is shown to stabilize the origin of the system. In Junge and Osinga (2004), a numerical method to compute an optimization problem, the result of which are a control Lyapunov function and a local feedback law for a given DT system, is shown. There, the problem is cast as a shortest path computation for a finite directed graph with weight of the edges determined by the stage cost function used to construct the optimization problem. In Kreisselmeier and Birkhölzer (1994), a method, with numerical solvability, to generate a local feedback law to stabilize a given DT system is presented where the feedback law comes as a result of an optimization problem solved recursively.
The flow of the paper is as follows. In Section 2 we give the notation and basic definitions. We also describe the sample and hold policy with state dependent hold interval to obtain a degree zero homogeneous DT model out of a CT homogeneous system with arbitrary degree of homogeneity. In Section 3 we expound how to numerically generate an offline-calculated, globally stabilizing feedback law for a DT homogeneous system. In Section 4 we tie the knot by combining the results from Sections 2–3 and show that the feedback law computed for the DT model by solving an infinite horizon optimization problem can be used to robustly regulate the origin of the CT homogeneous system. In Section 5 we show that the proposed method of Section 4 can be used to generate feedback laws that robustly globally exponentially regulate the origin of chained systems and systems in power form. We also show that the method can easily be adapted to handle input saturation for these special classes of systems in order to preserve the robust global asymptotic behavior of stability of the closed loop. We demonstrate our results via simulations for the fourth-order chained system. Finally, we conclude.
Section snippets
Preliminaries
For the ease of analysis we benefit from the extended real line, defined as . The following two definitions can be found in Grüne (2000). Definition 1 A dilation is an operator satisfying, for all and , that with fixed . Definition 2 A vector field (or a transition map) is said to be homogeneous with respect to dilation pair if , where is fixed and called the degree of homogeneity.
We consider the systemwhere
Homogeneous optimization in discrete time
In this section we consider the right-hand side and the DT system which we suppose is homogeneous with respect to the dilation pair with degree zero and asymptotically controllable to the origin. Given , we define the value function, for , asNote that, from Bellman's principle of optimality, we can write, for all , We adopt the notation . Let be two points in
Regulation in continuous time
Definition 5 A function is proper if for at least one and for all . Definition 6 A function is lower semicontinuous at if It is lower semicontinuous on if this holds for every . Definition 7 A function is level-bounded in u locally uniformly in x if for each and there exists a neighborhood of x such that the set is bounded on .
The following result resides in Rockafellar and Wets (1998, Theorem 1.17). Lemma 5 For
Chained systems and systems in power form
We devote this section to two particular classes of systems, namely chained systems and systems in power form, that have received much attention from the literature and that happen to fit into the framework we have discussed. See, for instance, Valtolina and Astolfi (2003) and Luo and Tsiotras (1998) to get exposed to the subject. We briefly talk about how the method we described in the previous sections can be used in regulation of the origin of a chained system or a system in power form. We
Conclusion
We proposed a method to robustly regulate the origin of continuous-time homogeneous systems. The method can be summarized in three steps. First obtain a discrete-time degree zero homogeneous model out of the continuous-time system via an appropriate sample and hold policy with state dependent hold intervals. Second get an asymptotically stabilizing feedback law for the discrete-time model by numerically solving an optimization problem of infinite horizon. Third apply the calculated feedback law
S. Emre Tuna received a B.S. degree in electrical and electronics engineering from Orta Dogu Teknik Universitesi, Ankara, in 2000. He has since been a Ph.D. student in electrical and computer engineering at the University of California, Santa Barbara.
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S. Emre Tuna received a B.S. degree in electrical and electronics engineering from Orta Dogu Teknik Universitesi, Ankara, in 2000. He has since been a Ph.D. student in electrical and computer engineering at the University of California, Santa Barbara.
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This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Zhihua Qu under the direction of Editor Hassan Khalil.
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Research supported in part by AFOSR Grant number F49620-03-1-0203; NSF Grant number ECS 0324679; NIH Grant number R21 AI057071.