Copyright © 2002 IFAC 15th Triennial World Congress, Barcelona, Spain
STABILIZING HYBRID CONTROL LAWS Patrick De Leenheer Henk Nijmeijer
Bram De Jager
Arizona State University Department of Mathematics Tempe, AZ 85287, USA email:
[email protected] Eindhoven University of Technology Department of Mechanical Engineering P.O. Box 513, 5600 MB Eindhoven, The Netherlands email:
[email protected],
[email protected]
Abstract: Hybrid control laws often naturally arise in stabilization problems with constrained control. These control laws are sometimes not smooth which can lead to difficulties in stability proofs. In this paper we show that for a certain class of hybrid systems the stability properties follow from the stability properties of an appropriate linearization. As an application we show that surge in a simple compressor model can be eliminated by a positive, hybrid feedback law. Keywords: Hybrid modes, Stabilizing controllers, Constraints, Linearization, Compressors.
1. INTRODUCTION In some control problems one is faced with finding stabilizing control laws that have to satisfy constraints. One type of constraint is boundedness of the control law, see e.g. (Teel, 1992). Another type is positive (or negative) control where the control law should only take positive (negative) values, see e.g. (Saperstone, 1973) for a related controllability result. In this paper we will briefly discuss an example of positive control, found in the study of surge control for compressors, see (Willems et al., 1999). In stabilization problems subject to constrained control it is natural to first disregard the constraint and find an unconstrained stabilizing control law. Then a constrained control law is easily found by using the stabilizing control law of the first step as long as the constraint is satisfied and putting the control equal to one of its extreme values (for example equal to zero in the case of positive control) when one of the con-
straints is violated. Obviously it remains to be shown that this new control law is stabilizing which might be hard or even impossible. This approach to stabilization with constrained control often leads to hybrid control laws because it is possible to distinguish different control modes in the resulting controller. For more on hybrid systems we refer to (Johansson and A. Rantzer, 1998) or (Branicky, 1998), while (Agrachev and Liberzon, 1999) considers the closely related switched systems. One of the problems in hybrid control is that the control laws result in nonsmooth controlled systems for which it may be very difficult to prove stability properties. As one of the simplest possible illustrations, consider a piecewise linear system with only two modes:
for for
(1)
and are closed where and
half spaces such that and for
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Section B for control of surge in compressors. Finally some conclusions are drawn in Section C .
2. PRELIMINARIES In this section we review homogeneous systems (and in particular a Converse Lyapunov Theorem for homogeneous systems proved in (Rosier, 1992)) and provide a simple technical Lemma to be used later. These two results are the only tools we need for the proof of our main result in the next section.
$ &+' ! % %* "!
Fig. 1. A piecewise linear system with two modes some plane containing the origin (see Figure , ). We assume throughout the rest of the paper that the / is continuous on . This vector field of system -.,0 1 implies in particular that for (of
all course this does not imply that , a situation which would make our results trivial). If this continuity assumption would not be satisfied, then system -2,3/ would not be modeled by an ordinary differential equation. Instead, a differential inclusion would be an appropriate model since to each point of would
and . correspond two different vectors The problem of establishing conditions (necessary and/or sufficient) for stability of system -.,0/ has received a lot of attention, yet it may not come as a surprise that it is very difficult. Until now it remains unsolved. Instead of adding to the body of literature on stability of the piecewise linear system -2,0/ we examine the relationship between the stability properties of system -.,0/ and those of the following system: 4
5
5 -
-
/
/
66
for for 66
(2)
where and are the same closed half spaces
featuring 57
58 in the system equations of system 57-. ,0/ and are 9 fields on with -;:::AF/ is called homogeneous if the following equality is satisfied:
5 H 6 ML
:
-
I
/
Examples Q
Q
An ordinary linear system R homogeneous of order zero.
Q
ION0P
5
-
with
/
S TUV
The scalar system WXZY7 [\\ neous of order two.
of (4)
is
is homoge-
Since by assumption the vector field of system -2,0/ is continuous on , it is homogeneous of order zero. L V^ Next we recall that a continuous function ] _ is called homogeneous of order G if the following equality is satisfied: HJIK
I I N ] ` / ]a- / (5) L 6^b If a continuous function ] is homogeneous of order G and of class 9 on , then its partial derivatives c=%0d e , f , hgigjgj .k are also homogeneous. c :
H 6
L
The order of homogeneity of the partial derivatives equals G*la, . This can be established by differentiating both sides of equation -;C:
