Symbolic Approximate Time-Optimal Control

SYMBOLIC APPROXIMATE TIME-OPTIMAL CONTROL MANUEL MAZO JR AND PAULO TABUADA Abstract. There is an increasing demand for controller design techniques ca- pable of addressing the complex requirements of todays embedded applications. This demand has sparked the interest in symbolic control where lower complex- ity models of control systems are used to cater for complex specifications given by temporal logics, regular languages, or automata. These specification mech- anisms can be regarded as qualitative since they divide the trajectories of the plant into bad trajectories (those that need to be avoided) and good trajecto- ries. However, many applications require also the optimization of quantitative measures of the trajectories retained by the controller, as specified by a cost or utility function. As a first step towards the synthesis of controllers reconciling both qualitative and quantitative specifications, we investigate in this paper the use of symbolic models for time-optimal controller synthesis. We con- sider systems related by approximate (alternating) simulation relations and show how such relations enable the transfer of time-optimality information between the systems. We then use this insight to synthesize approximately time-optimal controllers for a control system by working with a lower com- plexity symbolic model. The resulting approximately time-optimal controllers are equipped with upper and lower bounds for the time to reach a target, describing the quality of the controller. The results described in this paper were implemented in the Matlab Toolbox Pessoa [1] which we used to workout several illustrative examples reported in this paper. 1. Introduction Symbolic abstractions are simpler descriptions of control systems, typically with finitely many states, in which each symbolic state represents a collection or aggre- gate of states in the control system. The power of abstractions has been exploited in the computer science community over the years, and only recently started to gather the attention of the control systems community. In the present paper we analyze the suitability of symbolic abstractions of control systems to synthesize controllers enforcing both qualitative and quantitative specifications. Qualitative specifications require the controller to preclude certain undesired trajectories from the system to be controlled. The term qualitative refers to the fact that all the desired trajectories are treated as being equally good. Examples of qualitative specifications include requirements given by means of temporal-logics, ω-regular languages, or automata on infinite strings. These specifications are hard (if not impossible) to address with classical control design theories. In practice, This work has been partially supported by the National Science Foundation CAREER award 0717188. M. Mazo Jr is with INCAS3, Assen and the Department of Discrete Technology and Production Automation, University of Groningen, The Netherlands, m.mazo@rug.nl P. Tabuada is with the Department of Electrical Engineering, University of California, Los Angeles, CA 90095-1594,tabuada@ee.ucla.edu. 1 arXiv:1004.0763v2 [math.OC] 3 Feb 2011 2 MANUEL MAZO JR AND PAULO TABUADA most solutions to such problems are obtained through hierarchical designs with supervisory controllers on the top layers. Such designs are usually the result of an ad-hoc process for which correctness guarantees are hard to obtain. Moreover, these kinds of designs require a certain level of insight that just the most experienced system designers posses. Recent work in symbolic control [2, 3, 4] has emerged as an alternative to ad-hoc designs. In many practical applications, while there are plant trajectories that must be eliminated, there is also a need to select the best of the remaining trajectories. Typically, the best trajectory is specified by means of a cost or utility associated to each trajectory. The control design problem then requires the removal of the undesirable trajectories and the selection of the minimum cost or maximum utility trajectory. As a first step towards our objective of synthesizing controllers enforcing qualitative and quantitative objectives, we consider in the present paper the syn- thesis of time-optimal controllers for reachability specifications. A problem of this kind, widely studied in the robotics literature, is that of optimal kinodynamic mo- tion planning. Such problem is known to easily become computationally hard [5]. We discuss in Section 4.4 where the complexity of solving this kind of problems resides when following our methods. Since the illustrious seminal contributions in the 50’s by Pontryagin [6] and Bell- man [7], the design of optimal controllers has remained a standing quest of the controls community. Despite the several advances since then, solving optimal con- trol problems with complex geometries on the state space, constraints in the input space, and/or complex dynamics is still a daunting task. This has

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