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Unformatted text preview: STABILIZATION THROUGH HYBRID CONTROL Jo˜ ao P. Hespanha , Department of Electrical and Computer Engineering, University of Cali- fornia, Santa Barbara, CA 93106-9560, USA. Keywords: Hybrid Systems; Switched Systems; Supervisory Control; Stability Contents 1. Introduction 2. Switched Systems 2.1. Stability under Arbitrary Switching 2.2. Stability under Slow Switching 2.3. Stability under State-dependent Switching 3. Supervisors 3.1. Dwell-time Supervisors 3.2. Hysteresis-based Supervisors 4. Case Studies 4.1. Vision-based Control of a Flexible Manipulator 4.2. Hybrid Adaptive Set-point Control Summary This chapter addresses the problem of controlling a dynamical process using a hybrid controller , i.e., a controller that combines continuous dynamics with discrete logic. Typically, the discrete logic is used to effectively switch between several continuous controls laws and is called a supervisor . We review several tools that can be found in the literature to design this type of hybrid controllers and to analyze the resulting closed-loop system. We illustrate how these tools can be utilized through two case studies. 1. Introduction The basic problem considered here is the control of complex systems for which traditional control methodologies based on a single continuous controller do not provide satisfactory per- formance. In hybrid control , one builds a bank of alternative candidate controllers and switches among them based on measurements collected online. The switching is orchestrated by a spe- cially designed logic that uses the measurements to decide which controller should be placed in the feedback loop at each instant of time. Figure 1 shows the basic architecture employed by hybrid control. In this figure u represents the control input, d an exogenous disturbance and/or measurement noise, and y the measured output. The dashed box is a conceptual representation of a switching controller. In practice, switching controllers are implemented differently. Suppose that we desire to switch among a family C of controllers parameterized by some variable q ∈ Q . For example, we could have C := ˙ z q = F q ( z q , y ) , u = G q ( z q , y ) : q ∈ Q , To appear in the UNESCO Encyclopedia of Life Support Systems. Version date: May 31, 2004. This material is based upon work supported by the National Science Foundation under Grant No. ECS-0093762. process supervisor controller 1 controller 2 controller m u y d σ σ Figure 1: Hybrid control where the set Q that parameterizes the functions F q ( · ), G q ( · ), q ∈ Q can be finite, infinite but countable, or not even countable (e.g., a ball in R k ). Switching among the controllers in C can then be accomplished using the following multi-controller : ˙ x C = F σ ( x C , y ) , u = G σ ( x C , y ) , (1) where σ : [0 , ∞ ) → Q is a piecewise constant signal—called the switching signal —that effec- tively determines which controller is in the loop at each instant of time. The points of discon-tively determines which controller is in the loop at each instant of time....
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