Elsevier

Automatica

Volume 42, Issue 12, December 2006, Pages 2143-2150
Automatica

Brief paper
Generalized pole placement via static output feedback: A methodology based on projections

https://doi.org/10.1016/j.automatica.2006.06.021Get rights and content

Abstract

This paper presents an algorithm for solving static output feedback pole placement problems of the following rather general form: given n subsets of the complex plane, find a static output feedback that places in each of these subsets a pole of the closed-loop system. The algorithm presented is iterative in nature and is based on alternating projection ideas. Each iteration of the algorithm involves a Schur matrix decomposition, a standard least-squares problem and a combinatorial least-squares problem. While the algorithm is not guaranteed to always find a solution, computational results are presented demonstrating the effectiveness of the algorithm.

Introduction

There has been a great deal of research done on the problems of pole placement and stabilization via static output feedback. An overview of theoretical results, existing algorithms and historical developments can be found in Byrnes (1989), Rosenthal and Willems (1998), Syrmos, Abdallah, Dorato, and Grigoriadis (1997), de Oliveira and Geromel (1997), Rosenthal and Sottile (1998) and Eremenko and Gabrielov (2002). Our main interest here is in algorithms, and in this regard, for pole placement, the survey paper Rosenthal and Willems (1998) states that existing sufficiency conditions are mainly theoretical in nature and that there are no good numerical algorithms available in many cases when a problem is known to be solvable. Despite the great deal of work that has been done in this area, new algorithms for these important problems are still of interest.

In this paper we will actually consider the following generalized static output feedback pole placement problem.

Problem 1

Given ARn×n, BRn×m, CRp×n and closed subsets C1,,CnC, find KRm×p such that λi(A+BKC)Cifori=1,,n.

Here λi(A+BKC) denotes the ith eigenvalue of A+BKC.

Problem 1 encompasses many types of pole placement problems. Here are some examples:

  • (1)

    Classical pole placement: Ci={ci},ciC.

  • (2)

    Stabilization type problems for continuous time and discrete time systems:C1==Cn={zC|Re(z)-α},α>0and C1==Cn={zC||z|α},0<α<1,respectively.

  • (3)

    Relaxed classical pole placement: Ci={zC||z-ci|ri}.Here each region Ci is a disk centered at ciC with radius ri0.

  • (4)

    Hybrid problems: For example, problems of the type shown in Fig. 1. Here cC and the aim is to place a pair of poles at c and c¯, and to place the remaining poles in the truncated cone C: C1={c},C2={c¯},C3==Cn=C.

As far as the authors are aware, pole placement in the generality presented in Problem 1 has not previously been considered. The most closely related results from the existing literature are as follows. Early work in Gutman and Jury (1981) considers pole placement in a single region specified by polynomials. While a Lyapunov-type necessary and sufficient condition is given for a matrix to have its eigenvalues in such a region, this condition is polynomial in the matrix in question and hence not readily amenable to controller design. In Chilali and Gahinet (1996), LMI conditions are presented that are sufficient though not necessary for pole placement in various convex regions. These results cover state feedback and full-order dynamic output feedback but not static output feedback. Another LMI-based approach, again for a single, though this time, possibly disconnected region, is considered in Bosche, Bachelier, and Mehdi (2004). A method for placing poles in distinct convex regions (each region is specified using linear programming constraints or second-order cone constraints) is given in Hassibi, How, and Boyd (1999); however, the method is based on eigenvalue perturbation results and hence appears largely limited to cases where the open-loop poles are already quite close to the desired poles. In Satoh, Okubo, and Sugimoto (2003), pole placement in distinct convex regions (each region is a disk or a half plane) is achieved via a rank constrained LMI approach though the results are only for state feedback.

This paper presents an algorithm for Problem 1. The approach employed here is quite different to each of the approaches mentioned above. Problem 1 is shown to be equivalent to finding a point in the intersection of two particular sets, one of which is a simple convex set and the other a rather complicated nonconvex set. The algorithm is iterative in nature and is based on an alternating projection like scheme between these two sets. Each iteration of the algorithm involves a Schur matrix decomposition and a standard least-squares problem. If the Ci's are not all equal, each iteration also requires a combinatorial least-squares matching step.

Alternating projection type ideas have been employed previously for output feedback stabilization, see in particular Grigoriadis and Skelton (1996), Grigoriadis and Beran (1999) and Orsi, Helmke, and Moore (2006).1 A distinguishing feature of our algorithm is that, unlike these methods, our algorithm does not involve LMIs. (A further technical difference is that rather than solving feasibility problems that involve symmetric matrices, the problem solved by the algorithm is a feasibility problem involving nonsymmetric matrices.)

The algorithm can be applied to problems with rather general choices for the Ci regions. In fact the only formal requirement is the following, which we state in the form of an assumption.

Assumption 1

It is possible to calculate projections onto each of the Ci's: given zC it is possible to find ziCi such that |z-zi||z-c| for all cCi.

In particular, the Ci's must be closed sets though they need not be convex or even connected.

For given A, B, C and Ci's, Problem 1 may or may not have a solution. Indeed, one would expect that determining whether a particular instance of Problem 1 is solvable is in general difficult. For example, the problem of determining whether the classical pole placement problem is solvable for particular A, B, C and desired poles has recently been shown to be NP-hard (Fu, 2004). Given the difficulty of Problem 1, an efficient (i.e., polynomial time) algorithm that is able to correctly solve all instances of the problem cannot be expected, and while the algorithm presented here is often quite effective in practice, it is not guaranteed to find a solution even if a solution exists.

The structure of the paper is as follows. The remainder of this section lists some notation which is used in the rest of the paper. Projections play a key part in the algorithm and Section 2 contains general properties of projections and recalls how alternating projections can be used to find a point in the intersection of a finite number of closed (convex) sets. Section 3 presents the solution methodology. To motivate the solution methodology we first restrict our attention to systems that have a symmetric state space representation (A=AT, C=BT) and present an algorithm for this easier class of problems. The general problem is then considered. Section 4 contains computational results of applying the algorithm to various instances of Problem 1. The paper ends with some concluding remarks.

Notation: Sr is the set of real symmetric r×r matrices. diag(v) for vCr denotes the r×r diagonal matrix whose ith diagonal term is vi. For ZCr×s, vec(Z)Crs consists of the columns of Z stacked below each other. YZ denotes the Kronecker product of Y and Z. For ZCr×s,Re(Z)Rr×s and Im(Z)Rr×s denote, respectively, the real and imaginary parts of Z.

Section snippets

Projections

Projections play a key part in the algorithm. This section contains some general properties of projections that are used throughout the paper.

Let x be an element in a Hilbert space H and let D be a closed (possibly nonconvex) subset of H. Any d0D such that x-d0x-d for all dD will be called a projection of x onto D. In the cases of interest here, namely that H is a finite dimensional Hilbert space, there is always at least one such point for each x. If D is convex as well as closed then

The symmetric problem

Systems with a symmetric state space realization, that is, systems with state space matrices satisfying A=AT, C=BT, occur in various contexts, for example RC-networks. In order to motivate our solution methodology, we first consider the following special case of Problem 1 for symmetric systems.2

Problem 5

Given ASn, B

Computational results

This section contains computational results of applying the algorithm to various instances of Problem 1. We include results for classical pole placement, continuous time stabilization, and a hybrid problem. (Additional results can be found in Yang & Orsi (2005) and Yang, Orsi, & Moore (2004).)

The algorithms for each problem were implemented in Matlab 6.5 and all results were obtained using a 3.06 GHz Pentium 4 machine.

Throughout this section a randomly generated matrix will be a matrix whose

Conclusion

In this paper a new methodology for solving a broad class of output feedback pole placement problems has been presented. While the methodology is not guaranteed to find a solution, numerical experiments presented in the paper demonstrate it can be quite effective in practice. A particular strength of the algorithm is that, in addition to being able to solve classical pole placement and stabilization problems, it can also be used to solve less standard pole placement problems.

Acknowledgments

The authors thank Iven Mareels and Robert Mahony for their helpful guidance to various parts of the pole placement literature.

The second author acknowledges the support of the Australian Research Council through Grant DP0450539. National ICT Australia is funded through the Australian Government's Backing Australias Ability initiative, in part through the Australian Research Council.

Kaiyang Yang was born in Shenyang, PR China, in 1979. She received her B.E. degree in Control Science and Engineering from Zhejiang University, PR China, in 2002. From July 2002 to June 2003, she worked for Jinboyuan System Integration as a system engineer. Ms. Yang started her Ph.D. studies at the Australian National University in June 2003. In early 2005 she spent three months as a visiting student at the IBM Tokyo Research Laboratory. Her current research interests are in optimization

References (27)

  • A. Eremenko et al.

    Pole placement by static output feedback for generic linear systems

    SIAM Journal of Control Optimization

    (2002)
  • M. Fu

    Pole placement via static output feedback is NP-hard

    IEEE Transactions on Automatic Control

    (2004)
  • Grigoriadis, K. M., & Beran, E. B. (1999). Alternating projection algorithms for linear matrix inequalities problems...
  • Cited by (0)

    Kaiyang Yang was born in Shenyang, PR China, in 1979. She received her B.E. degree in Control Science and Engineering from Zhejiang University, PR China, in 2002. From July 2002 to June 2003, she worked for Jinboyuan System Integration as a system engineer. Ms. Yang started her Ph.D. studies at the Australian National University in June 2003. In early 2005 she spent three months as a visiting student at the IBM Tokyo Research Laboratory. Her current research interests are in optimization algorithm development with applications to control design and related problems.

    Robert Orsi received his Ph.D. in 2000 from the University of Melbourne, Australia. Since then he has held positions with Voyan Technology, the Australian National University, and National ICT Australia. Currently, since July 2004, he is the recipient of an Australian Research Council Australian Postdoctoral Fellowship, which he has taken up at the Australian National University. His current research interests include optimization methods for engineering problems, convex optimization, rank constrained LMIs, and inverse eigenvalue problems.

    A preliminary version of this paper was presented at the 16th IFAC World Congress. This paper was recommended for publication in revised form by Associate Editor Yasumasa Fujisaki under the direction of Editor Roberto Tempo.

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