Control Synthesis in Hybrid Systems with Finsler Dynamics

This paper is concerned with a symbolic-based synthesis of feedback control policies for hybrid and continuous dynamic systems. A key step in our synthesis procedure is a new method to solve the following dynamic programming problem: ∂V ∂t (z, t) = min v ∂V ∂z (z, t)ρv(z, v)v ż = ρ(z, v), t ∈ [0, T ] (1) V (z, T ) = Ψ(z, T ) (2) Here V (z, t) is the cost-to-go function associated with a certain type of homogeneous calculus of variations problem on a Finsler manifold and (z, v) is a positive homogeneous function of degree one in v. This optimization problem is at the core of the control synthesis procedure for many hybrid control problems [1], [2]. 1. Dedication We dedicate this paper to Professor Chern with gratitude, and for good reason. Nerode learned his differential geometry from Chern’s classes at the University of Chicago in 1951. As a lifelong mathematical logician, Nerode never expected to use this knowledge, especially knowledge of Finsler Geometry, which he had later acquired from his friend and fellow student of Chern’s, Louis Auslander. When developing feedback policies for optimal control systems, we discovered that these can be modeled as Finsler geodesic fields and connections. One never knows in advance what mathematics one may need, and it is nice to be in a position to recognize what is needed. A great virtue of the University of Chicago program, of which Chern was one of its founders, was that general knowledge of broad fields was an explicit aim. Kohn learned differential geometry at MIT and found its tremendous potential in control applications. He has been an avid student 353 354 WOLF KOHN, VLADIMIR BRAYMAN, AND ANIL NERODE of Professor Chern’s work. Vladimir Brayman who is completing his PhD in Electrical Engineering is working on the application of the differential geometric methods to enterprise control problems. 2. Introduction The discipline of hybrid systems emerged in the decade 1990-2000 as an important science at the interface of control theory, electrical engineering, and computer science. Hybrid systems are systems that incorporate (discrete) logical control programs that interact with continuous physical plants in a changing environment. In this context, a plant is thought of as an evolving vector field of plant states. Before the Hybrid System approach, the standard models for such systems tended to ignore either the continuous or the discrete aspects of the system. There are many cases where non-hybrid approaches are not adequate to develop optimal control laws. An enterprise control system is an example. The discipline of hybrid systems attempts to build and analyze models in which both the continuous and discrete aspects of the control problem are taken into account in a combined continuous field encoding both the discrete (event driven) and continuous dynamics of the system, with a transformation to the discrete domain when needed. A good place to see examples of these systems, is in the four volumes of hybrid systems which have appeared in the Lecture Notes in Computer Science series by Springer-Verlag, see references [1], [2]. There have been many conferences on this subject worldwide since the publication of these seminal volumes. We design logical control laws for hybrid systems to force the evolution of the plant states to satisfy a performance specification, even when subjected to disturbances in the environment, and in the presence of unmodeled plant dynamics. A control law is implemented by a real time control architecture. The way a control architecture module operates is as follows: when the plant enters a certain prescribed region of the state manifold, the event is sensed by the control architecture, which triggers the control program to change the control parameters for the plant actuators, thereby changing the plant evolution vector field to a new vector field. The plant state trajectory is thus a piecewise differentiable path. Therefore, the discontinuities in the direction of the trajectory take place at the times of the control law intervention. In other words, the hybrid control programs are event-driven finite automata, switching the plant from one vector field evolution to another for the purpose of enforcing plant performance specifications. The use of hybrid control programs extends the range of applications of CONTROL SYNTHESIS IN HYBRID SYSTEMS WITH FINSLER DYNAMICS 355 conventional continuous control theory to complex non-linear non-homogeneous systems. The problem is, how does one develop such a program? The fundamental problem of hybrid systems is to produce control algorithms and control implementation architectures that enforce the performance specification for the system, given the plant state models. In our approach, we introduce a manifold on which evolution of plant state trajectory y(t) take place. The manifold is determined by the constraints. The control policies are formulated as functions γ(y(t), ẏ(t)) that determine the direction ẏ(t) = ρ(y(t), u(t)) at time t when the plant state is y(t). In practice, the control effort should depend on the current state y and rate ẏ, and not on the current time, because the unknown disturbances and inaccuracies in the modelling parameters may lead to timing and positional inaccuracies. We introduce a suitable non-negative Lagrangian L̃(y, u, t) and a goal set G on the manifold. We rephrase the performance specification so that the requirement on the control policy can be chosen such that the plant state trajectory y(t) remains on the manifold. The trajectory y(t) leads from current state to the goal set G, and minimizes ∫ L̃(y, γ(y, ẏ), t)dt among all trajectories y(t) arising from admissible control policies. The discussion is confined to the case when the goal is a single point. Therefore, a control policy indicates, given the current state, the optimal direction to go in order to end at the goal point. We allow the tangent field along the optimal state trajectory to be measure-valued. This is done to ensure that mathematically optimal trajectories exist. Furthermore, this implies that we allow control policies that are generalized curves u(t) in the sense of L.C. Young [7]. Measure-valued optimal control policies are generally not physically realizable, but there are close approximations that are realizable. Given a positive ̄, we can generate algorithms that allow one to compute a piecewise constant approximation to γ(y, ẏ) to an optimal control policy which brings ∫ L̃(y, γ(y, ẏ), t)dt within ̄ of its minimum among all admissible trajectories. If the states y and the directions ẏ are then discretized, the approximate control policy can be implemented as a logical control program that is a hybrid control automaton [4]. Unlike the true optimal control policy, this approximate control policy can be implemented in such a way as to guarantee the ̄-optimality. This automaton is easily realizable in a generic form with Horn clauses [6], [5]. The Pontryagin School of optimal control was based on necessary, not sufficient, conditions. In this approach, one solves the necessary conditions, and among the feasible solutions to the plant equations one finds candidates to an optimal 356 WOLF KOHN, VLADIMIR BRAYMAN, AND ANIL NERODE solution. In our Young-based sufficiency approach, we approximate to an optimal weak solution already known to exist by the convexity properties of Finsler spaces. Finsler manifolds enter through the Caratheodory-Cartan reformulation of a Hamiltonian variational problem as a feedback control extraction problem. Following the example of Weierstrass for ordinary differential equations and variational calculus, time is introduced as an additional explicit variable, replacing y by x = (y, t). With the transformed variables and Finsler Lagrangian, the positive homogeneity condition that L(x, λẋ) = λL(x, ẋ) for positive λ is produced. An optimal control policy yields a path from the present position x to the goal point minimizing the cost-to-go function ∫ L(x, ẋ)dt. When (gij(x, ẋ)) = ( ∂2(L2(x,ẋ) ∂ẋi∂ẋj ) is positive definite, the Finsler interpretation of the cost-to-go function ∫ L(x, ẋ)dx measures the ”Finsler length” of the curve x(t). The Finsler fundamental ground form Σijgij(x, ẋ)dxidxj represents infinitesimal length as a function of position x and velocity ẋ. Integrating this quantity along curves gives its Finsler length. Thus the optimal plant state trajectories are Finsler geodesics. One may take as admissible control policies functions u(t) = γ(x, ẋ), giving the velocity ẋ = ρ(x, γ(x, ẋ)) at each position x, with ρ a positive homogeneous function of degree one in the second argument. ρ is the Euler-Lagrange form of the Lagrangian L. The generalized control policies γ(x, ẋ) may be interpreted as having probability measures on sets of values of ẋ and x. These generalized control policies yield a Lebesgue measurable plant state trajectory on which the expected value of ẋ(t) is almost always the velocity of the actual plant state trajectory x(t). When the apparently artificial homogeneity in ẋ is introduced, where no homogeneity was originally present, the Finsler manifold structure introduced allows one to compute these control policies explicitly. But one only sees the control policies as geodesic fields or connections when this transformation is carried out. Reading the introduction to Cartan’s book and Finsler’s thesis, the transformation of variational problems to Finsler form seems to have been the inspiration for developing Finsler geometry in the first place. It is fitting that this source of Finsler spaces is now found to be very powerful for computing optimal control policies. Another inspiration for Finsler’s thesis was undoubtedly Caratheodory’s famous ”Golden Path” to the calculus of variations, which was an axiomatic treatment of the relation between a geodesic field and its family of wave front hypersurfaces. Finsler’s work made this Caratheodory relation arise automatically from Lagrangian problems recast in Finsler form. Now much of the structure of control policies can be seen clearly through the duality between the finite dimensional tangent space unit sphere and its cotangent space. CONTROL SYNTHESIS IN HYBRID SYSTEMS WITH FINSLER DYNAMICS 357 The process for extracting digital programs to control continuous physical systems breaks into several stages. First, the control problem is reinterpreted as an optimal control problem, by the inverse problem method of the calculus of variations, and then the optimal control problems are translated to the corresponding Finsler Manifold. On the Finsler manifold, the control problem becomes one of computing a geodesic field. This amounts to finding the connection matrix in the Cartan sense. Connections or geodesic fields are the required control policies. They often exist only as weak limits of sectionally smooth geodesic fields if one uses the sufficient conditions of the calculus of variations attributed to L. C. Young. Weak limits are usually not physically realizable, but they can be approximated by digital control programs that are real-time finite automata. Thus for any ̄, a sectionally smooth trajectory that will produce a result within ̄ of the minimum for the Lagrangian involved, and a discrete real time digital control program that issues control orders to achieve such a trajectory can be generated. The digital control program arises by decomposing the Finsler tangent manifold into a finite number of regions, and when a new region is entered, the system communicates this fact to the control automaton, which issues the appropriate control to the actuators, usually a chattering control. Digital control programs can achieve near optimal behavior when they arise from continuous models by discretization. In this paper, we do not delve into the discretization process. It is more appropriate for this volume to describe the classical differential geometry tools used to compute the needed control policies. These are in the tradition of Cartan, and we use his notation. At the Hynomics corporation, the needed algorithms have been implemented in symbolic software. The first commercialization of this software is for agent based supply chain programming. Included here are some of the tools used. 3. An Approach to Control Synthesis Our approach for designing adaptive feedback control laws is not the usual one. We begin by describing the desired behavior of the intended closed-loop system. This desired behavior is a trajectory on a suitable constructed manifold, called the carrier manifold, generated by a variational formulation