Stabilization of fixed modes in expansions of LTI systems

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Abstract

The objective of this note is to propose a method for stabilization of structurally fixed modes in expansions of LTI dynamic systems in the scope of overlapping decentralized control design based on the expansion/contraction framework, enabling successful design for a broader class of problems than considered so far. The method is based on a judicious choice of complementary matrices in the expanded space. Numerical examples are provided to illustrate simplicity and efficiency of the proposed approach.

Introduction

It has been often found advantageous, for either conceptual or computational reasons, to decompose large complex systems into overlapping subsystems sharing common parts, and to apply decentralized control strategies which offer satisfactory performance at minimal communication cost [14], [18], [9], [10], [13], [1], [7], [19]. To design the control law, the designer expands the system into a larger space where the subsystems appear as disjoint, designs decentralized controllers in the expanded space using standard methods, and then contracts the local controllers to the original space for implementation. The mathematical framework for expansion and contraction has become known as the inclusion principle [16], [18], and has been applied in fields as diverse as electric power systems [20], mechanical structures [1], applied mathematics [18], [2], automated highway systems [23] and formations of unmanned arial vehicles [24].

One of the obstacles in the design of decentralized overlapping control with information structure constraints is the presence of structurally fixed modes in the expanded models [17], [14], [18]. It has been found that instability of these modes can cause a conflict between contractibility and stability requirements. Either contractibility of the expanded control law is guaranteed but stabilizability is not due to unstable fixed modes in the expanded space or stabilizability is achieved but the control law is not contractible to the original space [1], [2], [3], [4], [9], [10], [12], [13], [14], [15], [16], [20], [21], [22], [23], [24]. One of the references that discusses the expansion/contraction paradigm in this context is the book [18], which contains a tutorial-like presentation of the structurally fixed mode problem in a two-area power system interconnected by a tie-line (Section 8.3; in particular, Examples 8.17 and 8.22). Recently, an LMI approach has been proposed [25], which addresses the joint contractibility–stability problem at the price of numerical difficulties in computing design parameters involving stabilizing gain matrices.

The objective of this note is to demonstrate that the contractibility–stability dilemma can be resolved by choosing appropriately the complementary matrices at the outset of the expansion. It will be shown that the proposed method represents a simple and efficient tool for overlapping decentralized control design using restriction (extension) [9], [10], [12], [13], [8] within the LMI framework. The proposed approach is also in the spirit of the work in [2], [3], [4], exploiting flexibility of complementary matrices to achieve a desired overlapping structure for control system design.

Section snippets

Overlapping decompositions: a generalization

Consider a linear time-invariant (LTI) dynamic system with the state modelS:x˙=Ax+Bu,x(t0)=x0,in which matrices A=(Aij) and B=(Bij) are decomposed into compatible block-matrices Aij and Bij with dimensions (ni×nj) and (ni×mj), respectively (i,j=1,2,,N); accordingly, the state and input vectors x and u can be represented as x=[x1Tx2TxNT]T and u=[u1Tu2TuNT]T, respectively, with dimxi=ni, dimui=mi, i=1Nni=n, i=1Nmi=m. The assumed structure of S can be induced either by specific structural

Stabilization of fixed modes

Our goal is to apply to S˜ standard control design procedures, such as the LMI-based methods, and by stabilizing S˜ to guarantee stabilization of S after contraction, having in mind that only then we have a transparent effect of the choice of the overlapping decentralized feedback on stability of the overall contracted system. Feasibility of this approach may be violated by unstable fixed modes in S˜, in spite of the fact that these modes are eliminated after contraction, and the contracted

Conclusion

In this note an efficient general method for stabilization of fixed modes in expansions of LTI systems is proposed. Starting from the expansion/contraction paradigm and the inclusion principle, it is shown that overlapping decentralized control design based on expansions satisfying restriction conditions can suffer from the problem of instability of fixed modes. A precise elaboration leads to the formulation of a general, simple and efficient method for overcoming this problem, enabling, in

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