This paper considers the problem of finding a perturbation matrix with the least spectral norm such that a matrix-valued function becomes singular, where the dependence of the function on the perturbation is allowed to be nonlinear. It is proved that such a problem can be approximated by a smooth unconstrained minimization problem with compact sublevel sets. A computational procedure proposed based on this result is demonstrated to be effective in both linear and nonlinear cases.published_or_final_versio

Lam, J

Yan, WY

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Title On the nonlinearly structured stability radius problemAuthor(s) Yan, WY; Lam, JCitation American Control Conference, Philadelphia, USA, 24-26 June1998, v. 6, p. 3622-3626Issued Date 1998URL http://hdl.handle.net/10722/46640Rights©1998 IEEE. Personal use of this material is permitted. However,permission to reprint/republish this material for advertising orpromotional purposes or for creating new collective works forresale or redistribution to servers or lists, or to reuse anycopyrighted component of this work in other works must beobtained from the IEEE.Proceedings of the American Control Conference Philadelphia, Pennsylvania June 1998 On the Nonlinearly Structured Stability Radius Problem Wei-Yong Yan* James Lam+§ * School of Electrical and Electronic Engineering Nanyang Technological University Nanyang Avenue, Singapore 639798 t Department of Mechanical Engineering University of Hong Kong Pokfulam Road, Hong Kong Abstract This paper considers the problem of finding a pertur- bation matrix with the least spectral norm such that a matrix-valued function becomes singular, where the dependence of the function on the perturbation is al- lowed to be nonlinear. It is proved that such a prob- lem can be approximated by a smooth unconstrained minimization problem with compact sublevel sets. A computational procedure proposed based on this result is demonstrated to be effective in both linear and non- linear cases. 1 Introduction Many control systems are subject to perturbations in terms of uncertain parameters. An important quantita- tive measure of stability robustness of a system against such perturbations is what is called the real or complex stability radius, depending on the nature of perturba- tions in concern. The computation of a stability radius is a subject which has attracted a lot of interest over recent decades, see e.g. [l, 2, 3, 4, 5 ,  61 and references therein. Given a stable matrix A ,  the general stability radius problem is to determine a real or complex perturbation matrix A with a minimum spectral norm such that the perturbed matrix A + S(A)  becomes unstable, where S(A) is a mapping describing the structure of the per- turbation. It is well known that the basic subproblem of this problem is to determine a perturbation matrix A with a minimum spectral norm such that A - j w + S ( A )  becomes singular at each given frequency w. This leads to the general linear algebra problem of determining a A with a minimum spectral norm such that R(A) becomes singular for a given mapping R(A). Although the special case where R(A) is of the form I - A M  has been successfully and completely treated recently by Qiu et al. in [5 ] ,  the problem has yet to be §Supported by RGC HKU 5441963. solved in the general case, including such a simple case where R(A) is of the form I + GlAH1 + G2AH2. It should also be mentioned that a nonlinear R(A) natu- rally arises when the dependence of a system matrix on uncertain parameters is nonlinear. 2 Problem Formulation Throughout this section, it is assumed that R(A) : RmxP H Pxn is a continuous mapping with h(A) 4 det(R(A)R*(A)) (2.1) The problem to be considered in this paper can be pre- cisely stated as follows: Problem P: minimize J (A)  4 IlAll over C 4 {A E Rmxp I h(A) = 0) This problem is obviously very general as the map- ping R(A) is allowed to take various forms. A typical linear form of R(A) could be the following N R(A) = I + C G i A H i  i=l In the special case where R(A) = I - AM, the problem has been tackled in [5] by converting it into the problem of minimizing the second largest singular value of a cer- tain matrix defined on the interval (0, 11. Specifically, it has been shown there that min{llAll I det(I  - AM) = 0} 1 where a 2 ( . )  denotes the second largest singular value. A procedure has also been proposed for computing a worst real perturbation A after a minimum is found. 0-7803-4530-4198 $10.00 0 1998 AACC 3622 One nonlinear example of R(A) is given by which implies that A; E ck. Thus, one has N R(A) = I + ( G ~ A H ~ ) '  i = I  although more complicated forms can easily be envis- aged. It should be emphasized that there is no loss of gen- erality in restricting A to be real in the above problem formulation. This is because a complex A can be rep- resented by a real matrix of larger size formed by the real and imaginary parts of A. As such, Problem P is closely related to the general complex stability radius problem as well. Quite obviously, Problem P is a nonsmooth and con- strained minimization problem, which is generally dif- ficult to solve. On the other hand, a global minimum for the problem does exist unless the set C is empty. Our main result to be presented below shows that Prob- lem P can be arbitrarily approximated by a smooth and unconstrained minimization problem. ' Theorem 2.1 Suppose that R(A) is continuous in PmxP with E nonempty. Let a function Jk(A) : P m x P  C )  P be defined by Jk(A) fk(A) + kh(A) (2.2) with fk(A) A {trace [(A'A)k]};" (2.3) If J* is the global minimum for Problem P, then there holds J* = lim min J k  (A) k-tm Proof. It is known that for all k 2 1 and A E PmxP, Ji 2 fk(A;) 2 J(A;) 2 J(Ak) This together with (2.5) establishes that J(Ak) 2; Jl 5 fk(A*) (2.7) Quite obviously, { J(&)} is a nondecreasing sequence bounded by J(A*)  from above as ckl 2 C k Z  for kl < kz. As a consequence, there exists a convergent subse- quence {Ank}, whose lirnit is denoted by A,. From h(A,,k) I a/nk and the continuity of h(A), it follows that Am E C, leading to lim J(&) = lim J(Ank)  = J(Am) 2 J(A*) k-tm k-tm Hence, there holds The theorem is immediately concluded from (2.4), From the above proof', we immediately conclude the following result, which says that an optimal solution to Problem P can be arbitrarily approximated by a global minimum point of Jk (A)i provided k is sufficiently large. (2.7), and (2.8). 0 Corollary 2.1 Adopt the same assumption and nota- tion as an Theorem 2.1. If A; is a global minimum point of Jk(A), then there hold lim J(A;) = J' (2.9) lim h(A;) = 0 (2.10) k - t m  k + c c  J(A;) 5 minJk(A), V k  2 1 (2.11) NOW with A* denoting an optimal solution to Prob- lem P and J i  = min Jk (A) ,  it is apparent that In this section, R(A) vdl be assumed to be smooth, which is usually the case in practical applications. This smoothness will be exploited to suggest an efficient pro- cedure for solving Problem P, whose optimal solution Ji 5 fk(A*) (2.5) ..a can be approached by minimizing the cost function Jk(A) with sufficiently ]large k by Theorem 2.1. In two Next, let A; E I w m X p  be such that Ji = Jk(Ai) and let Ak be a global minimum Point Of J(A) Over the set specific cases, we will giYYe a closed-form formula for the gradient of Jk (A).  Jk(A) can be found to Ibe Ck {A E Rmxp 1 h(A) 5 Cr/k} By straightforward calculations, the gradient of where a is such a constant that fk(A*) 5 0, v k  2 1 Combining (2.5) with (2.6) yields h(A;) 5 Ji/k 5 alk Of course, the formula for the gradient Vh(A) cannot be obtained unless R(A) is known. In the following two cases which each cover the case treated in [5]: 3623 Case 1: R(A) = F + xEl G ~ A H ~  Case 2: R(A) = F + AM1 + AM2AT it can be established that Vh(A) = @(A) + G(A) where N @(A) = xGfadj(R(A)R*(A))R(A)H; (3.1) i=l for Case 1 and Q(A) = adj(R(A)R*(A))R(A)(M; + AM;) + R*(A)adj (R(A)R* (A)) AM, (3.2) The gradient vJk(A) can be used to form the gradi- for Case 2. ent flow 1 A(t) = - [fk(A)]1-2k A (ATA)k-l - kVh(A) I (3.3) which enjoys several nice properties listed below. 0 The ODE (3.3) has a unique solution A( t )  on the 0 Any solution A(t) to (3.3) converges to a connected interval [0, CO) for any initial condition. set of critical points as t goes to infinity. 0 The cost function Jk(A) is nonincreasing along any solution A(t) to (3.3) and is strictly decreasing when A(0) is not a critical point of Jk(A). These properties, which follow from the compactness of all the sublevel sets of Jk(A), suggest that a mini- mum of Jk(A) can be found by solving the ODE (3.3). Whether the obtain minimum is local or global may depend on the choice of an initial condition for (3.3). However, the global minimum must result regardless of the choice of a starting point if there is no other mini- mum. Certainly, the ODE (3.3) can be easily integrated us- ing an appropriate numerical routine, e.g., in Matlab on a digital computer as only standard matrix operations are involved. In the meantime, analog computing has recently gained renewed interest in view of advances in neural networks which allow massively parallel process- ing. As a result, it has recently become increasingly ac- ceptable to make use of ODES for solving various prob- lems such as optimization and linear algebra problems, see e.g. [7, 81 and references therein. We end this section by proposing the following gen- eral procedure for solving Problem P. The convergence of the algorithm is essentially guaranteed by Theo- rem 2.1. Step 1 Choose an initial integer k and a point A0 E E%mxp. Step 2 Seek a minimum point A of the cost function Jk(A) by finding a limiting solution to the ODE (3.3) with the initial condition A(0) = Ao. Step 3 If both I det(R(A))l and fk(d) - J(&) are less than a preset value, stop; otherwise, go back to step 2 with a larger k and A0 = A. Remark 3.1 In the second step, a general-purpose globally convergent search algorithm given in [9] could alternatively be employed to  f ind a minimum point A of the cost function Jk (A). Remark 3.2 In order t o  achieve computational e f i -  ciency as well 0s t o  avoid numerical problems associated with large I C ,  the gradient of fk(A) should be computed as follows where A is assumed to  have the singular value decompo- sition A = USVT and s k  results f r o m  replacing every diagonal element si of S by (&) 2k-1.  4 Examples Example 1: Assume that R(A) = I - A M  where M = [  2 + j  1 2:j1 As was indicated in [5], this case leads to the nonsmooth behavior of the cost function introduced there, which is defined to be the second largest singular vaule of an augmented matrix associated with M .  Let us implement the algorithm proposed in the last section to find a worst real perturbation which makes R(A) singular. The details of the implementation are described as follows. We use a randomly generated ma- trix A0 as an initial condition for solving the ODE with IC = 50. Figure 1 depicts the norm of the solution A1 ( t )  to the ODE as t goes from 0 to 0.14. By using the fi- nal value A1(0.14) as an initial condition, we continue to solve the ODE with k = 100 to obtain the solution A,(t) on [0, 0.51. The norm IlA,(t)ll as a function o f t  is depicted in Figure 2. With A2(0.5) as a new starting point, the two solutions &(t)  defined on [0, 0.0071 for k = 1000 and A4(t) on [0, 0.0031 for k = 3000 are ob- tained by repeating the same process. Figure 3 shows a concatenation of the two functions IlA,(t)ll and IlA4(t)ll on the combined interval [0, 0.011. The initial value A0 and the values of the four solutions at the end point of the respective intervals are given in Table 1 along with their norms as well as their corresponding determinants of R(A). As is seen, the last matrix in Table 1 is iden- tical with the worst perturbation obtained in [ 5 ] .  The 3624 total time taken to complete the whole simulation on an HP workstation is 12 seconds. Table 1. Properties of initial, intermediate, and final perturbation matrices. R I IlAll I Idet(R(A))l 1 A r.3332 0.2357 -0.23551 3 32 0.2357 0.3333 1 0.3333 -0.2357 1 (“‘“ ~ : ~ ~ ~ ~ l ,  1 0.7650 1 1.6394 1 0.0330 0.2822 -0.1649 o,4538 0.3385 0.2471 o.4082 0.0002 0.0001 o.4082 0.0023 Example 2: Consider R(A) = I - A M  where taken from [5]. There, an optimal solution to Problem P was found to be 1 0.3333 -0.2981 0.2981 0.3333 A * =  [ and its norm i.e. the optimum is 0.4472. To test our algorithm, we use the randomly generated matrix A, = p.4337 0.1160] 0.7092 0.0781 as a starting point and numerically solve the ODE (3.3) successively for IC = 50,280,560,840 to obtain a se- quence of 148 perturbation matrices. Figure 4 displays three curves associated with I det(R(A))l, llA - A*ll, and IlAll - IlA*ll, respectively. It is clear that the se- quence converges to the optimal solution while the se- quence of corresponding norms converges to the mini- mum. The actual time taken to complete the simulation is about 10 seconds. Example 3: Consider a nonlinear R(A) given by l + j  2 R ( A ) = I - A  Note that without the quadratic term the minimum for Problem P was found to be 0.4472 in [5]. By invok- ing our algorithm with the randomly generated starting point 0.07‘745 0.10237 0.76290 0.57027 1 a worst real perturbation is found to be A* = p.02751 -0.337751 0.37673 0.03563 with IlA*li = 0.37854 and 1 det(R(A*))l = 5 x low6 5 Conclusions This paper has proposed. a unified approach to solv- ing the real and complex: stability radius problems in the face of nonlinearly structured perturbations. It has been proved and verified with numerical examples that an optimal solution can be approached by solving a smooth and unconstraine’d minimization problem. References R. V. Patel and M. ’lroda, “Quantitative measures of robustness for multivariable systems,” in Proc. Automat. Contr. Conf., (San Francisco), 1980. R. K. Yedavalli, “Improved measures of stability robustness for linear state space models,” IEEE Trans. Autom. Control, vol. 30, pp. 557-579, 1985. K. Zhou and P. P. Khargonekar, “Stability ro- bustness bounds for linear state-space models with structured uncertainty,” IEEE Trans. Autom. Control, vol. 32, pp. 621-623, 1987. R. Genesio and A. ‘Tesi, “Results on the stabil- ity robustness of systems with state space pertur- bations,” Systems and Control Letters., vol. 11, L. Qiu, B. Bernhardsson, A. Rantzer, E. Davison, P. Young, and J. Doyle, “A foumula for compu- tation of the real stability radius,” Automatica, J. Sreedhar, P. V. IDooren, and A. L. Tits, “A fast algorithm to compute the real structured sta- bility radius,” in Stability Theory, pp. 219-230, Birkhauser Verlag Bissel, 1996. R. W. Brockett, “Dynamical systems that sort lists, diagonalise matrices and solve linear pro- gramming problems,” Linear Algebra and its Ap- plications, vol. 146, pp. 79-91, 1991. U. Helmke and J. B. Moore, Optimization and Dy- namical Systems. London: Spring-Verlag, 1994. J. E. Dennis Jr. and R. B. Schnabel, Numerical methods for unconstrained optimization and non- linear equations. Englewood Cliffs: Prentice-Hall, 1983. pp. 39-46, 1988. vol. 31, pp. 879-890, 1995. 3625 0.35' I Time Figure 1: The norm of the ODE solution versus t E [0, 0.141 for ,k = 50. 0.4L ' 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Time Figure 2: The norm of the ODE solution versus t E [0, 0.51 for IC = 100. I 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 Time Figure 3: The norm of the combined ODE solution ver- sus t E [0, 0.011 for IC = 1000, 3000. 10' 1 Io" t i .* ... ..*.... I... ....*.....* ....,. -....*.* *...* I 50 100 150 10% Iterations Figure 4: Convergence of a sequence of ODE solutions to  a minimum point. 3626 

On the nonlinearly structured stability radius problem

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