**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.

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.