In this paper, control of linear differential-algebraic-equation systems, subject to general quadratic constraints, is considered. This setup, especially, includes the H∞ control problem and the design for strict passivity. Based on linear matrix inequality (LMI) analysis conditions, LMI synthesis conditions for the existence of linear output feedback controllers are derived by means of a linearizing change of variables. This approach is constructive: a procedure for the determination of controller parameterizations is given on the basis of the solution of the LMI synthesis conditions. A discussion of the possible applications of the presented results concludes the paper.
Differential-algebraic equation (DAE) systems (sometimes also referred to as singular, semistate or descriptor systems) describe a broad class of systems which are not only of theoretical interest but also have great practical significance. Models of chemical processes for example typically consist of differential equations describing the dynamic balances of mass and energy while additional algebraic equations account for thermodynamic equilibrium relations, steady-state assumptions, empirical correlations, etc. [7]. In mechanical engineering DAE system descriptions result from holonomic and non-holonomic constraints [16]. Also in electronics and even in economics DAE descriptionsare encountered [9].
DAE systems are able to describe system behaviors that cannot be captured by “non-DAE” systems (i.e. systems governed only by differential equations) [1]. Therefore, index reduction techniques (i.e. reduction of a DAE system to an ODE system) necessarily are connected to a loss of information for high index systems. Due to this fact much work has been focused on analysis and design techniques for linear DAE systems in recent years(see [8] for an overview). Even for index one DAE systems (i.e. DAE systems, which are equivalentto an ODE system) it is sometimes tedious or numerically not reliable to use the inversion of the algebraic equations in order to incorporate ODE based controller computation methods. This is especially the case for problem descriptions in chemical process control, where it is not uncommon to encounter much more algebraic equations than differential equations (e.g. in distillation control). Also for ODE process models an ODE based controller computationmay not be the natural method of choice: the actual control problem, typically, is given by the process model plus some weighting systems or filters plus the algebraic couplings between these systems, i.e. as a DAE system.
Quite recently LMI based analysis and synthesis methods have been introduced to DAE control problems, (but so far restricted to H2- and H∞-problems) [10], [19] and [13]. In this paper we consider the LMI approach to the generalized quadratic performance (GQP) control problem. In [15] this problem is solved for non-DAE systems. The idea is to control a generalized linear plant such that the closed loop transfer function is internally stable and such that the general quadratic constraint
equation(1)
is imposed on the external input and output functions w(·) and z(·) respectively. Here the notation “≪0” means: for a given quadratic scalar function Q(w,z), Q(w, z)≪0 is defined as ∃ϵ>0:Q(w, z)⩽−ϵwTw for all w. Analogously ∫0TQ(w(t)), z(t))dt≪0 means that ∫0TQ(w(t)), z(t))dt⩽−ϵ∫0Tw(t)w(t))dt holds for all w(·)∈L2 and some fixed ϵ>0.
The rather general GQP problem contains some important control problems as a special case if the objective parameters UP⩾0, VP=VPT, and WP are chosen accordingly. For example
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the H∞ constraint ||Gcl||∞<γ, if UP, VP, and WP are specified as UP=, VP=−γI, WP=0;
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the strict passivity constraint Gcl(jω)+Gcl(jω)*>0 for all ω∈ ∪{∞}, when UP, VP, WP are chosen as UP=0, VP=0, WP=−I;
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sector constraints of the form
equation(2)
for UP=I, VP=−αβI, WP=−(α+β)I.
The key to the synthesis problem is a modified version of the linearizing change of variables approach used in the corresponding case for ODE systems [15]. In contrast to a previous paper [11] we do not assume the DAE description to be in semi-explicit form. Especially, it is possible to include the standard ODE result without any additional fall differentiation. The paper is structured as follows: in the next section the necessary background on linear DAE systems is given and we discuss the generalization of the “internal stability” concept to DAE systems. Subsequently a LMI analysis result for general quadratic performance is given. By means of this result it is possible, for a given controller, to efficiently decide, whether or not a closed loop system in DAE form meets the performance requirements. Also the structure of the LMI solution connected to this problem will be examined. Based on these results a direct treatment of the synthesis problem in the next section is possible: with the controller being unknown, the analysis result formally becomes a nonlinear matrix inequality. However, the presented linearizing change of variables approach reveals, that the problem can be reduced to a strict LMI problem. We consider the computational implications of the derived synthesis result and finally discuss the possible range of applications.
We considered the GQP control problem for linear DAE systems. This comprises as special cases the H∞ control problem and the design for strict passivity. Based on an LMI analysis condition, sufficient and constructive conditions for the existence of a controller solvingthe GQP control problem in DAE form are given. These conditions essentially require the solution of LMIs, i.e. are numerically attractive.The controller renders the closed loop admissible (i.e. impulse-free and stable) and imposes a quadratic constraint on the input/output behavior of the closed loop system. In case of the H∞ control problem for DAE systems the derived LMI conditions are shown to be also necessary. In contrast to a previous solution [ 11] to the GQP control problem for DAE systems we do not require an explicit decomposition of the problem equations into an algebraic and a differential part. Therefore, it is possible to solve thecontroller computation problem directly on the basis of the DAE description, which, as we point out, is for almost all practical applications the appropriate formulation of the GQP control problem.
