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[Advanced Control Systems Group] [Autonomous Mobile Robotics Group]





Hybrid systems

In their most general form hybrid systems are characterized by the interaction of continuous-valued components (governed by differential or difference equations) and logic rules. Hybrid systems are very common in engineering and many systems encountered in every day life can be effectively modeled as hybrid systems as well. A simple example of a hybrid system would be a car. The dynamics of the car switch when a gear shift occurs, either because the driver moves the stick shift (input event) or because the state variable ``speed'' exceeds a specified threshold (state event) in the case of an automatic transmission.

Another special example of a hybrid system would be a linear system under feedback control with actuator constraints. When the actuator hits a constraint the dynamics change.

We focus on the class of discrete-time piecewise affine (PWA) systems that are defined by the partition of the extended state+input space into polyhedral regions and a set of different affine state update equations associated with those regions.

Discrete-time PWA models can describe a large number of processes, such as discrete-time linear systems with static piecewise linearities, discrete-time linear systems with logic states and inputs or switching systems where the dynamic behavior is described by a finite number of discrete-time linear models, together with a set of logic rules for switching among these models. Moreover, PWA systems can approximate nonlinear discrete-time dynamics via multiple linearizations at different operating points.

Even though hybrid systems are a special class of nonlinear systems most of the nonlinear system and control theory does not apply because it requires certain smoothness assumptions. For the same reason we also cannot simply use linear control theory in some approximate manner to design controllers for PWA systems. Different methods for the analysis and design of controllers for hybrid systems have emerged over the last few years. Among them, the class of optimal controllers is one of the most studied. The approaches differ greatly in the hybrid models adopted, in the formulation of the optimal control problem and in the method used to solve it.

The optimal control problem for a PWA model of the system can, in general, be restated as a mixed-integer program - a non-convex mathematical problem that is too hard to solve in real-time for practical problems. However, by using multi-parametric mathematical programming, combined with polyhedra manipulation, the closed-form of the optimal control law can be computed off-line for the whole admissible range of the system states.

It turns out that the optimal control law has a piecewise affine form even for the PWA models. Therefore it can be rather easily implemented on-line, on a cheap hardware, even for the fast-sampling plants.

Optimal control law


Value function

Algorithms for computation of the control law for various optimal control problems for linear and PWA systems are implemented in the Multi-Parametric Toolbox (MPT). The MPT, toolbox in development of which we also participate, is available from http://control.ee.ethz.ch/~mpt.

Case studies and obtained experimental results

Our goal is to make the use of explicit MPC suitable for current industrial practice. To this aim, as a case study we implemented explicit optimal controller for the electronic throttle a fast sampling and extremely nonlinear plant from automotive industry.


Electronic throttle

Electronic throttle control system in cars


We approximated two major nonlinearities present (friction, limp-home) in a PWA form and computed the explicit optimal controller based on the PWA model and the manufacturer's as well as safety constraints on the throttle variables. The goal of the electronic throttle control system is to maintain the valve opening angle at the desired reference value. Using MPC, the angle transient was made more than two times faster than by a tuned PID control with feedforward nonlinearities compensation, still without overshoot and steady-state error.

Another experimental setup on which we test the optimal controllers is the magnetic levitation system.

Identification of systems in the PWA form

A systematic procedure to identify a plant in the PWA form is an additional challenge that arises when it comes to the industrial implementation of such a controller. Namely, for the fast control system commissioning a PWA model should be derived based on the input-output data collected directly from the plant. For the electronic throttle, we identified a PWA model based on the collected input-output data using the procedure that combines least-squares parameter estimation, data clustering and linear classification.

Robust optimal control of PWA systems

In practice there always exists certain mismatch between a PWA model and the real plant. This discrepancy is taken into account as an unknown bounded uncertainty that is added to the model. Outcome of the clustering based PWA model identification procedure are both the model and its uncertainty bounds, and both should be used in consequent off-line controller computations.

To guarantee constraints satisfaction and optimal system performance even in the case when the uncertainty is present in the model, the robust optimal control problem needs to be solved to find the explicit robust model predictive controller. Although the basic procedure how to do those computations is known, many computational geometry algorithms used thereby should be made more efficient to make the overall computations tractable in reasonable time for fairly complex systems. Our current research efforts are going in this direction.

 

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