Brief PaperA fast algorithm for the computation of an upper bound on the μ-norm☆
Introduction
In the context of robust control analysis and synthesis a quantity of great interest is the structured singular-value norm, or μ-norm, of the system. Consider a feedback connection of a continuous-time system with real coefficients as in Fig. 1. Let P(s)=C(sI−A)−1B be an m×m stable transferfunction matrix and let Δ(s) be a structured perturbation constrained to lie in the setwhere is the set of all real-rational, proper, stable, m×m transfer matrices and the uncertainty setis defined for integers m1,…,mF. Let σ1(·) denote the maximum singular value of its matrix argument and define the complex structured singular value for a constant matrix as (see Zhou, Doyle & Glover, 1995 for a complete discussion of the structured singular value)unless det(I−MΔ)≠0 for all in which case . Finally, define the “μ-norm”3 asIt has been shown that the computation of is NP-hard (see Toker & Özbay, 1995), in consequence no efficient algorithms are likely to exist for its computation. In practice, a standard upper bound is used in its place. This upper bound is computed as follows. DefineThen, for a constant complex matrix M, an upper bound for isSubstituting this into the expression for , an upper bound on the μ-norm of the system P(s) is obtained as the optimally frequency-dependent scaled -norm4It is well-known that if P(s) is stable, then is necessary and sufficient (thus is sufficient) for uniform robust stability of the P−Δ loop (Fig. 1) for any linear time-invariant structured Δ(s) of L2-gain no greater than one (see, e.g., Corollary 3 in Tits & Balakrishnan, 1998). Further, it has recently been shown that is necessary and sufficient for uniform robust stability of the P−Δ loop for any linear, arbitrarily slowly time-varying structured Δ of L2-gain no greater than one (Poolla & Tikku, 1995).
Algorithms for the efficient computation of , for given ω, have long been available. In fact, given ω,which is a linear matrix inequality (LMI) problem. Efficient algorithms exist for obtaining global solutions to such problems, e.g., Boyd, El Ghaoui, Feron and Balakrishnan (1994). Note that minimizers (or approximate minimizers) D for σ1(DMD−1) are related to the minimizers (or approximate minimizers) D̃ for (1) by . is usually computed via a “frequency sweep”, i.e. choose a set of frequencies and use the approximationThe drawbacks of this approach are obvious. First, a large number of computations are required. Second, an upper bound on is not necessarily obtained. Finally, the result can be arbitrarily bad, i.e. it is difficult to bound the error.
Another approach that has been used to compute an upper bound to the μ-norm is based on the Main Loop Theorem (see, e.g., Packard & Doyle, 1993) and the extension to μ of the Maximum Modulus Theorem (see, e.g., top of p. 1201 in Tits & Fan, 1995). In the discrete-time case, given P(z)=C(zI−A)−1B+D, with A stable, and a positive scalar γ, it can be seen that the μ-norm of P is less than γ if and only if , whereand is an “augmented” uncertainty structure given byThe original idea is due to Doyle and Packard (1987). The continuous-time case can be reduced to the discrete-time case by means of a bilinear transformation (see Section 10.2 in Zhou et al., 1995 for details). Repeated evaluation of the upper bound according to a bisection search over γ yields an upper bound on the μ-norm of P. On the down side, note that this upper bound is generally less tight than that obtained by gridding, because the augmented uncertainty structure involves an additional block which, to make things worse, is of the “repeated” type. The bisection search can be done away with, as shown by Ferreres and Fromion (1997), by invoking the “skewed μ” proposed and studied by Fan and Tits (1992).5 Computation then entails the solution of a single LMI constrained quasi-convex generalized eigenvalue minimization problem (GEVP; see Boyd et al., 1994).6 The approach proposed in this paper improves on the scheme of Ferreres and Fromion (1997) in that it computes the same (tighter) upper bound as the gridding approach. As far as computational cost is concerned, the tradeoff is that of a sequence of LMIs (μ upper bound computations with respect to the original, non-augmented uncertainty structure) versus a single GEVP with a larger number of variables and constraints. Due to the “repeated” block, the complexity of the latter is strongly affected by the dimension of the state space.
Recently, an efficient algorithm has been proposed (Boyd & Balakrishnan, 1990; Bruinsma & Steinbuch, 1990) for the computation of the -normThis algorithm makes use of the well-known fact that a given scalar ξ>0 is a singular value of if and only if jω is an eigenvalue of the related Hamiltonian matrixGiven any , a set of frequency intervals may thus be computed where maximizers for (2) are known to lie. At step k, ξ is selected as and ωk is chosen as the mid-point of the largest among the intervals thus determined. A quadratic rate of convergence ensues.
In this paper, using the idea just outlined as a stepping stone, an algorithm is constructed for the fast computation of . (A similar algorithm can be used for the computation of the real stability radius; see Sreedhar, Van Dooren & Tits, 1996.)
Section snippets
Key ideas
LetThe goal of the algorithm is to maximize over . Defineandassuming for now that such a unique maximizer exists. Finally, for a fixed , define the curveWhile reading the following, it may be helpful to refer to Fig. 2. The main idea of the algorithm is as follows (more details are given in Section 3). At iteration k, suppose ξk is the best-known lower bound to thus far, and let ωk be the current trial
Details of the algorithm
The computation of is now considered in greater detail. The pure imaginary eigenvalues of the Hamiltonian matrixtell us the frequencies at which one of the singular values of takes the value ξk+1. That is, if is an imaginary eigenvalue of Hk, thenfor some r∈{1,…,m} (see Fig. 3). Of course, the frequencies of interest are those for which r=1. Suppose the Schur decomposition of the Hamiltonian matrix Hk has been
Convergence
Theorem 1, Theorem 2 below are proved for the case when the mid-point rule is used throughout to select the next trial point ωk+1. Assumption 1 The choice of ω0 is such that
Assumption 1 is necessary in order for the algorithm to be well-defined. Typically, ω0=0 will satisfy this assumption; if not, a random search will provide a suitable ω0 as long as P(s) is not identically 0. Under this assumption, is bounded and, in view of the fact that , a simple induction argument shows that the
Numerical experiments
The algorithm was implemented10 in MATLABTM. In the implementation, a mid-point rule is used until is reduced to just one interval and enough information has been accumulated to compute an interpolating function. All numerical experiments were run on a Sun UltraSparc 10, machine with of RAM, running Solaris 2.5 operating system.
The performance of the new algorithm was first compared to that of the “skewed μ” approach mentioned in
Extension to mixed-μ
The algorithm is readily extended to compute an upper bound on the mixed-μ norm, that is, the μ-norm for systems with mixed dynamic and real parametric uncertainty. (See Feron (1997) for a sophisticated grid-based approach to this computation, accounting for possible discontinuities.) Two expressions for the widely used “D-G” upper bound are as follows (see, e.g., Chapter 18 in Zhou & Doyle, 1998)
Concluding remarks
The algorithm just outlined in this note can be refined in various ways. For instance, in the process of minimizing νD(ωk) to evaluate , if it is found that the minimum value is less than ξk, then it is unnecessary to compute it (or Dk) with great accuracy.
The new algorithm can be extended to discrete-time systems. The Hamiltonian eigenvalue problem H(ξ,A,B,C) is replaced by a simplectic eigenvalue problem S(ξ,A,B,C) which will have an eigenvalue ejω on the unit circle if and only if ξ is
Craig T. Lawrence received the B.S. degree in Electrical Engineering and the B.S. degree in Mathematics in 1993, the M.S. degree in Electrical Engineering in 1996, and the Ph.D. in 1998, all at the University of Maryland, College Park. Since graduating, he has been with ALPHATECH, Inc., in Arlington, Virginia working on algorithms for intelligence, surveillance, and reconnaissance. His research interests are in the areas of optimization and robust control theory.
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Craig T. Lawrence received the B.S. degree in Electrical Engineering and the B.S. degree in Mathematics in 1993, the M.S. degree in Electrical Engineering in 1996, and the Ph.D. in 1998, all at the University of Maryland, College Park. Since graduating, he has been with ALPHATECH, Inc., in Arlington, Virginia working on algorithms for intelligence, surveillance, and reconnaissance. His research interests are in the areas of optimization and robust control theory.
André L. Tits was born in Verviers, Belgium on April 13, 1951. He received the ‘Ingenieur Civil’ degree from the University of Liege, Belgium and the M.S. and Ph.D. degrees from the University of California, Berkeley, all in Electrical Engineering, in 1974, 1979, and 1980, respectively. Since 1981, Dr. Tits has been with the University of Maryland, College Park. Currently, he is a Professor of Electrical Engineering and he holds a permanent joint appointment with the Institute for Systems Research. He has held visiting positions at the University of California, Berkeley, at the Lund Institute of Technology, at INRIA, at the Catholic University of Louvain at Louvain-la-Neuve, Belgium and at the Australian National University.
Dr. Tits received a 1985 NSF Presidential Young Investigator Award. He is a Fellow of the Institute of Electrical and Electronics Engineers. He is an Associate Editor of Automatica and the Editor for Technical Notes and Correspondence of the IEEE Transactions on Automatic Control. His main research interests lie in various aspects of numerical optimization, optimization-based system design and robust control with emphasis on numerical methods.
Paul M. Van Dooren received the engineering degree in computer science and the doctoral degree in applied sciences, both from the Katholieke Universiteit te Leuven, Belgium, in 1974 and 1979, respectively. He held research and teaching positions at the Katholieke Universiteit te Leuven (1974–1979), the University of Southern California (1978–1979), Stanford University (1979–1980), the Australian National University (1984), Philips Research Laboratory Belgium (1980–1991), the University of Illinois at Urbana-Champaign (1991–1994), and the Université Catholique de Louvain (1980–1991, 1994–now) where he is currently a professor of Mathematical Engineering.
Dr. Van Dooren received the Householder Award in 1981 and the Wilkinson Prize of Numerical Analysis and Scientific Computing in 1989. He is an Associate Editor of Journal of Computational and Applied Mathematics, Numerische Mathematik, SIAM Journal on Control and Optimization, Applied Mathematics Letters, Linear Algebra and its Applications, Journal of Numerical Algorithms, the Electronic Transactions of Numerical Analysis, Applied and Computational Control, Signals and Systems, Mathematics of Control, Signals and editor in chief of SIAM Journal on Matrix Analysis and Applications. His main interests lie in the areas of numerical linear algebra, systems and control theory, digital signal processing, and parallel algorithms. He is currently the Chairman of the SIAM Special Interest Group on Linear Algebra. He has published over 60 journal papers and 80 refereed conference papers.
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This paper has appeared, in slightly different form, in the Proceedings of the 1996 IFAC World Congress (Lawrence, Tits & Van Dorren, 1996). This paper was recommended for publication in revised form by Associate Editor P-O. Gutman under the direction of Editor T. Basar.
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Supported in part by NSF's Engineering Research Center Program, under grant NSFD-CDR-88-03012.
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Supported by the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister's Office for Science, Technology and Culture. The scientific responsibility rests with the authors.