Elsevier

Automatica

Volume 106, August 2019, Pages 315-326
Automatica

On the well-posedness in the solution of the disturbance decoupling by dynamic output feedback with self bounded and self hidden subspaces

https://doi.org/10.1016/j.automatica.2019.04.038Get rights and content

Abstract

This paper studies the disturbance decoupling problem by dynamic output feedback with required closed-loop stability, in the general case of nonstrictly-proper systems. We will show that the extension of the geometric solution based on the ideas of self boundedness and self hiddenness presents structural differences with respect to the strictly proper case. The most crucial aspect that emerges in the general case is the issue of the well-posedness of the feedback interconnection, which obviously has no counterpart in the strictly proper case. A fundamental property of the feedback interconnection that has so far remained unnoticed in the literature is investigated in this paper: the well-posedness condition is decoupled from the remaining solvability conditions. An important consequence of this fact is that the well-posedness condition written with respect to the supremal output nulling and infimal input containing subspaces does not need to be modified when we consider the solvability conditions of the problem with internal stability (where one would expect the well-posedness condition to be expressed in terms of supremal stabilizability and infimal detectability subspaces), and also when we consider the solution which uses the dual lattice structures of Basile and Marro.

Introduction

The disturbance decoupling problem (DDP) played a central role in the development of the geometric approach in systems and control theory. Indeed, from the pioneering papers (Basile and Marro, 1969, Wonham and Morse, 1970), it was recognized that geometry is a natural language for this type of problems; consequently, the solvability conditions of the first disturbance decoupling problems considered in the literature were expressed by means of inclusions involving certain subspaces.

The basic decoupling problem, consisting of the rejection of a disturbance from the output of a system by means of a static state-feedback, was solved in Basile and Marro (1969) and, independently, in Wonham and Morse (1970), via the introduction of controlled invariant subspaces. These subspaces were then found to be powerful tools in the understanding of many system-theoretic properties of linear time-invariant (LTI) systems and in the solution of several control problems. The disturbance decoupling problem by static state feedback with the extra requirement of internal stability of the closed-loop was taken into account in Wonham and Morse (1970) with the introduction of stabilizability subspaces. An alternative solution to the same problem was suggested in Basile and Marro (1982), relying on the concept of self bounded controlled invariance, which, unlike the stabilizability subspaces of Wonham and Morse (1970), does not require eigenspace computation; in other words, the solution with self boundedness remains at the fundamental level of finite arithmetics.

A key contribution to the understanding of the advantages deriving from the adoption of self bounded controlled invariant subspaces in the solution of the disturbance decoupling problem by static state-feedback was given in Malabre, Martinez-Garcia, and Del-Muro-Cuéllar (1997), where it was shown that in the solution of this problem there is a number of closed-loop eigenvalues that are fixed for any feedback matrix which solves the decoupling problem; these unassignable eigenvalues are called the fixed poles of the decoupling problem. It is shown in Malabre et al. (1997) that choosing a particular self bounded subspace, denoted by Vm in Basile and Marro (1992), is the best choice in terms of pole assignment, because it ensures that the maximum number of eigenvalues of the closed-loop can be freely assigned. For systems whose state is not accessible, a state-feedback decoupling filter cannot be implemented. This led to the formulation of the disturbance decoupling problem by dynamic output feedback. The first paper which provided a solution to this problem is Schumacher (1980). Around the same time, the same problem with the additional requirement of internal stability was addressed in Imai and Akashi (1981) and Willems and Commault (1981). In Basile, Marro, and Piazzi (1987), an alternative geometric solution was proposed for this problem which uses self bounded subspaces, as well as their duals, the so-called self hidden subspaces. Again, the importance of this solution lies in the fact that it does not require eigenspace computation. Even more importantly, in Del-Muro-Cuéllar and Malabre (2001) it was proved that this solution based on the idea of self boundedness and self hiddenness, is still the best in terms of assignability of the closed-loop dynamics, see also Del-Muro-Cuéllar (1997) and Del-Muro-Cuéllar and Malabre (2003).

Most of the literature in geometric control has been developed for strictly proper systems, i.e., for those systems which have zero feedthrough between the input and the output. For a systematic and well-organized extension of the geometric approach for systems with a possibly non-zero direct feedthrough term we refer to the monograph (Trentelman, Stoorvogel, & Hautus, 2001). The disturbance decoupling problem with dynamic output feedback and nonzero feedthrough has been completely solved in terms of stabilizability and detectability subspaces in Stoorvogel and van der Woude (1991). More recently, the approach based on self boundedness and self hiddenness has been generalized in Ntogramatzidis (2008) for the disturbance decoupling problem with static state-feedback. In Ntogramatzidis (2008), the result of Malabre et al. (1997) on the fixed poles was also generalized to nonstrictly proper systems.

A significantly more challenging task is the solution of the disturbance decoupling problem by dynamic output feedback for nonstrictly proper systems using the concepts of self boundedness and self hiddenness. An issue of well-posedness arises in the case where the feedthrough between the control input and the measurement output is non-zero. It was observed in Stoorvogel and van der Woude (1991) that the solvability conditions, when dealing with the problem in its full generality, need to take into account the well-posedness: this results in a condition that cannot be expressed as the typical subspace inclusion of most control/estimation problems for which a geometric solution is available. In this paper, we study the role that the well-posedness condition plays in the disturbance decoupling problem by dynamic output feedback. We prove, in particular, that this condition is invariant with respect to the stabilizing pair of self bounded and self hidden subspaces involved in the solution of the disturbance decoupling problem. In other words, we show that the well-posedness condition is disjoint, and therefore independent, from the remaining solvability conditions of the decoupling problem. This new property is the key to a full generalization of the solution of the disturbance decoupling problem by dynamic output feedback, as it shows that the fundamental requirement of stability does not reduce the set of well-posed feedback interconnections; therefore, choosing self bounded and self hidden subspaces does not impact on the solvability of the disturbance decoupling problem by dynamic output feedback. Furthermore, it also implies that the solution of Stoorvogel and van der Woude (1991) can be conveniently re-written with a well-posedness condition for the supremal output nulling and infimal input containing subspaces instead of the corresponding stabilizability and detectability subspaces.

Notation. Given a vector space X, we denote by 0X the origin of X. The image and the kernel of matrix A are denoted by imA and kerA, respectively. When A is square, we denote by σ(A) the spectrum of A. If A:XY is a linear map and if JX, the restriction of the map A to J is denoted by A|J. If X=Y and J is A-invariant, the eigenstructure of A restricted to J is denoted by σ(A|J). If J1 and J2 are A-invariant subspaces and J1J2, the mapping induced by A on the quotient space J2J1 is denoted by A|J2J1, and its spectrum is denoted by σ(A|J2J1). Given a map A:XX and a subspace S of X, we denote by AS the smallest A-invariant subspace of X containing S and by SA the largest A-invariant subspace contained in S. Consider two vector spaces X and P. Let S be a subspace of XP. The linear operators p,i are defined as p(S)=defxX|pP:xpS and i(S)=defxX|x0S, where p(S) is referred to as the projection of S on X and i(S) is the intersection of S with X. It is easy to see that p(S) and i(S) are subspaces of X. Both operators preserve addition and intersection, and p(W)=(i(W)), see Basile and Marro (1992, Prop. 5.1.3).

Section snippets

Problem statements

In what follows, the time index set of any signal is denoted by T, which represents either R+ in the continuous time or N in the discrete time. The symbol g denotes either the open left-half complex plane in the continuous time or the open unit disc in the discrete time. A matrix MRn×n is said to be asymptotically stable if σ(M)g. Finally, we say that λ is stable if λg. The operator D denotes either the time derivative in the continuous time, i.e., Dx(t)=ẋ(t), or the unit time

Geometric background

Consider a quadruple (A,B,C,D) associated with the non-strictly proper LTI system Dx(t)=Ax(t)+Bu(t)y(t)=Cx(t)+Du(t). A subspace V is said to be an (A,B)-controlled invariant subspace if, for any initial state x0V, there exists a control function u such that the state trajectory generated by the system remains identically on V; equivalently, V is (A,B)-controlled invariant if the subspace inclusion AVV+imB holds. The control function that maintains the trajectory on V can always be expressed

Dual lattice structures

The following results extend the classic results that relate the concepts of output nullingness and input containingness, see Basile and Marro (1992, Chpt. 5).

Lemma 4.1

Let V be an (A,B,C,D) -output nulling subspace and let S be an (A,B,C,D) -input containing subspace. Then, SBkerD and VC1imD .

Theorem 4.1

Let V be an (A,B,C,D) -output nulling subspace and let S be an (A,B,C,D) -input containing subspace. Then:

  • VS is an (A,B,C,D) -output nulling subspace;

  • V+S is an (A,B,C,D) -input containing subspace.

We now

Solution of Problem 2.1

We begin by first presenting the following result, see Stoorvogel and van der Woude (1991, Lemma 3.2). It can be carried out along the same lines of the proof of Trentelman et al. (2001, Lemma 6.3). The next few preliminary results involve integers n1,n2,m,pN{0}, a field K, a subspace of Kn2 and a subspace N of Kn1. Let ÃKn1×n2, B̃Kn1×m and C̃Kp×n2.

Lemma 5.1

There holds ÃN+imB̃ and Ã(kerC̃)N if and only if there exists KRm×p such that (Ã+B̃KC̃)N .

Lemma 5.2

Let V be an (A,B,E,Dz) -output nulling

Solution of Problem 2.2

We now consider Problem 2.2. Two necessary solvability conditions are the asymptotic stabilizability of the pair (A,B) and the asymptotic detectability of the pair (C,A) (Trentelman et al., 2001 Thm. 3.40). These are, therefore, standing assumptions for this section. The following result provides a solution to Problem 2.2 in terms of the largest (A,B,E,Dz)-stabilizability subspace and of the smallest (A,H,C,Gy)-detectability subspace, see Stoorvogel and van der Woude (1991, Thm. 4.1).

Theorem 6.1

Problem 2.2

Concluding remarks

In this paper, we have developed a geometric solution to the disturbance decoupling by dynamic output feedback for systems which are not necessarily strictly proper. The building blocks of this solution do not require eigenspace computations that are at the basis of a solution involving stabilizability and detectability subspaces: the solution given here remains in the realm of finite arithmetics. The crucial issue in the extension of the classical theory to the nonstrictly proper case is the

Fabrizio Padula holds a M.Sc. degree in Industrial Automation Engineering (2009) and a Ph.D. degree in Computer Science and Automatic Control (2013), both from the University of Brescia in Italy. Currently, he is a Research Fellow in the School of Electrical Engineering, Computing and Mathematical Sciences at Curtin University, Perth, Australia. Fabrizio’s research focuses primarily on fractional control, tracking control, anesthesia control. He also has a keen interest in mechatronics and

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Fabrizio Padula holds a M.Sc. degree in Industrial Automation Engineering (2009) and a Ph.D. degree in Computer Science and Automatic Control (2013), both from the University of Brescia in Italy. Currently, he is a Research Fellow in the School of Electrical Engineering, Computing and Mathematical Sciences at Curtin University, Perth, Australia. Fabrizio’s research focuses primarily on fractional control, tracking control, anesthesia control. He also has a keen interest in mechatronics and industry related research, and he has worked with many companies.

Lorenzo Ntogramatzidis received the ”Laurea” degree, cum laude, in Computer Engineering in 2001 from the University of Bologna, Italy. He received the Ph.D. degree in Control and Operations Research in 2005. From 2005 to 2008, he was a post-doctoral Research Fellow at the Department of Electrical and Electronic Engineering, The University of Melbourne, Australia. Since 2009, he has been with the Department of Mathematics and Statistics at Curtin University, Perth, Australia, where he is currently Associate Professor. His research interests are in the area of systems and geometric control theory.

Partially supported by the Australian Research Council under the grant DP160104994. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Graziano Chesi under the direction of Editor Richard Middleton.

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