Hybrid Control System Design Methodologies

The methodologies of [4,5,6] begin by modeling the systems dynamics at the continuous level. Two factors affect the system evolution at this level. The first is the control, and the second is the disturbances that enter the system, over which we assume no control. We distinguish three classes of disturbances: class 1: exogenous signals, such as unmodeled forces, sensor noise, etc.; class 2: unmodeled dynamics, and class 3: actions of other agents, in a multi-agent setting. Disturbances of classes 1 and 2 are standard in control theory, but class 3 will be the most interesting one from the point of view of hybrid control. Recall that at this stage we are merely modeling the plant, therefore we assume no cooperation between the agents. As a result, each agent views the actions of its neighbors as uncontrollable disturbances. Acceptable performance can be encoded by means of thresholds on certain cost functions. Our objective is to derive a continuous design for the control inputs that guarantees performance despite the disturbances. If it turns out that the disturbance is such that the specifications cannot be met for any controller, the design fails. The only way to salvage the situation is to somehow limit the disturbance, and for disturbances of class 3 this may be possible by means of communication and coordination between the agents. Thus, our approach to hybrid controller design consists of determining continuous control laws and conditions under which they satisfy the closed loop requirements. Then a discrete design is constructed to ensure that these conditions are satisfied.

Optimal Control Encoding of Multi-Agent Multi-Objective Designs: The MAHCA Framework
One of our team members (Nerode) has in collaboration with Kohn proposed the so-called Kohn-Nerode Multiple Agent Hybrid Control Architecture (MAHCA). MAHCA defines a plant, or system of plants called plant processes, where each plant process has an agent for its digital control program and plant process controller. That agent monitors its plant process and its plant process controller and its digital control program, passes and receives messages to and from other agents for other plant processes, and occasionally computes and installs a new finite automaton for its plant process controller whenever performance and stability criteria are violated. There is no central supervisor. The Kohn-Nerode approach models hybrid systems states as points on an appropriate Finsler manifold and extracts controls for hybrid systems as finite automata approximations to connections on the manifold. Methods are from relaxed variational calculus, automata theory, differential geometry, Lie Algebras, and linear hyperbolic PDEs. there is distributed message passing of information about Lagrangians from agent to agent. See Kohn-Nerode [7] [8] on modeling hybrid systems and Kohn-Nerode [9] [10] on distributed autonomous hybrid control. Examples of problems which may be addressed by this approach include:

1. Large rule based state estimation for hybrid systems. The problem here is to extract estimates of the state from sensor observations using the MAHCA variational apparatus and deductive inference. There are algorithms that build automata (real time processors) that reconstruct the state from sensor history, given that constraints on the state are included in the rule base [10]).
2. Rule based detection theory for hybrid systems. The problem addressed here is to express queries as goals and use the MAHCA variational framework to instantiate parameters of the goal.
3. Reduction of model uncertainty in distributed hybrid systems. When a hybrid system is observed to fail to obey its performance specification, its model may need more radical revisions than traditional parameter adjustment. The MAHCA framework may be used to encode this form of online learning.

Game-Theoretic Approach to Hybrid Control Design
We will conduct the design of continuous control laws and interfaces between these laws and the discrete world, using game theory ([11]) : in this framework, the control and the disturbances are viewed as adversaries in a game. Roughly speaking, the designer has to find the best possible control and the worst possible disturbance. If the requirements are met for this pair, it is possible to obtain a satisfactory design (one such design is the ``best possible'' control). If the requirements are not satisfied, the problem can not be solved as is, since there exists a choice of disturbance for which, no matter what the controller does, the closed loop system will fail to satisfy the requirements. In case of disturbances of class 3, coordination will have to be used to limit the disturbance.

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