Linköping Studies in Science and Technology.
Dissertations

No. 527

**Constructive Methods for Inequality
Constraints in Control**

Mats Jirstrand

**Akademisk avhandling**

som för avläggande av teknisk doktorsexamen vid Linköpings tekniska högskola kommer att offentligt försvaras i sal C3, Linköpings universitet, fredagen den 15 maj 1998, kl. 10.15. Opponent: Dr Per-Olof Gutman, Technion - Israel Institute of Technology, Israel

**Abstract**

In practical control problems there are always constraints on inputs and system variables. The constraints are often described by inequalities and there is a need for methods that take them into account. This thesis consists of three parts. In the first two parts a computer algebra approach is taken and in the third part we use methods based on linear matrix inequalities and convex optimization.

A large class of systems can be described by a set of polynomial differential equations and there are different approaches to analyze systems of this kind. In the first part of the thesis we utilize mathematical tools from commutative and differential algebra such as Gröbner bases and characteristic sets to study input-output relations of systems given in state space form. The state space representation of a system can be described by a finitely generated differential ideal. We show that this is not always true for the corresponding differential ideal of input-output relations. However, the input-output relations up to a fixed order can be computed and represented using non-differential tools. Using characteristic sets we also show how the above problem of finite representation of the input-output relations can be resolved.

Many problems in control theory can be reduced to finding solutions of
a system of polynomial equations, inequations, and inequalities, a so
called *real polynomial system*. The *cylindrical algebraic
decomposition* method is an algorithm that can be utilized to find
solutions to such systems. The extension of real polynomial systems to
expressions involving Boolean operators (and, or, not, implies) and
quantifiers (exists, for all) is called the first-order theory of real
closed fields. There are algorithms to perform *quantifier elimination*
in such expressions, i.e. to derive equivalent expressions without any
quantified variables.

We show how these algorithms can be used to solve problems in control such as stabilization of a system with real parametric uncertainties; feedback design of linear systems; computation of bounds on static nonlinearities in feedback systems that ensure stability; computation of equilibrium points for nonlinear systems subject to constraints on the control and state variables; and curve following. We also consider stabilization of systems by switching among a set of state feedback controllers. Furthermore, for a nonlinear system which is not affine in the control, quantifier elimination can be used to decide if the zero dynamics can be stabilized by switching between different controllers.

Invariant sets of dynamic systems play an important role for verification of control systems, i.e. to check if a designed controller meets performance and safety constraints. In part three of the thesis we have compiled a number of results on representations and computations on polyhedral and quadratic sets, i.e. sets defined by affine and quadratic inequalities. We also derive criteria for deciding when an affine dynamic system has polyhedral or quadratic invariant sets. These results are utilized to propose a method for computing invariant sets of a class of hybrid systems, i.e. systems that exhibit both continuous and discrete behavior. The methods for invariant set computations and verification are based on convex optimization techniques and linear matrix inequalities. Since there are no well established design procedures for hybrid systems, verification of heuristically designed controllers is of outmost importance.

Department of Electrical Engineering

Linköping University, S-581 83 Linköping, Sweden

Linköping 1998

ISBN 91-7210-187-2 ISSN 0345-7524