A fast algorithm to compute the controllability, decentralized fixed-mode, and minimum-phase radius of LTI systems

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1 Proceedings of the 7th IEEE Conference on Decision Control Cancun, Mexico, Dec. 9-, 8 TuA6. A fast algorithm to compute the controllability, decentralized fixed-mode, minimum-phase radius of LTI systems Simon Lam Edward J. Davison Abstract In this paper, an efficient algorithm is presented for solving the -D optimization problem which is associated with computing the robustness measures of a number of linear time-invariant LTI) system properties such as the controllability radius, the decentralized fixed-mode radius, the minimum-phase radius. Unlike methods such as gradient search methods, the proposed algorithm works well independent of the initial trial point, obtains the minimum with a high level of confidence that it is indeed global. A numerical example is included. I. INTRODUCTION Consider the following LTI multivariable system ẋ = Ax + Bu y = Cx + Du ) where x R n, u R m, y R r are respectively the state, input, output vectors, A, B, C, D are constant matrices with the appropriate dimensions for n, m, r, maxr,m) n. It is well known that there exists a LTI controller that can assign the eigenvalues of the closed-loop system to any arbitrary spectrum if only if the system is controllable observable. However, when a controllable system is subject to parametric perturbations i.e. A A + A B B + B ), the system may be very close to becoming uncontrollable. Hence, a continuous controllability measure is more informative than the traditional yes/no controllability metric, which simply determines whether a system is controllable or not. The same can be said about other system properties such as observability, stability, minimum-phase, etc. In the current literature, various continuous measures have been proposed to measure the robustness of various system properties with respect to parametric perturbations. In particular, the controllability radius introduced in [, [ measures how close a controllable system is to being uncontrollable, the stability radius in [ measures how close a stable system is to an unstable one, the decentralized fixed-mode DFM) radius, [) measures how close a system with no DFMs is to having one. More recently, such robustness measures have been extended to characterize the robustness of a system s transmission zero properties. For instance, the minimum-phase radius in [6 measures how close a minimum-phase system is to becoming non-minimum phase. This work has been supported by NSERC under grant No. A96. S. Lam E. J. Davison is with the Systems Control Group, Department of Electrical Computer Engineering, University of Toronto, Toronto, ON, Canada MS G {simon,ted}@control.utoronto.ca Computing the stability radius mentioned above ) requires solving only a -D optimization problem, a fast algorithm to solve this is proposed in [7. On the other h, computing the other radii mentioned above i.e. except the stability radius) requires solving a -D optimization problem. As of now, the -D problem is solved using nonlinear search methods such as gradient search methods. A disadvantage with using such nonlinear techniques is that the minimum obtained may be a local minimum. Hence selecting the initial trial points are critical, a large number of initial points are generally tested in order to achieve a certain level of confidence that the obtained minimum is global. In this paper, we propose a more efficient algorithm for solving the -D optimization problem. The proposed algorithm has some similar features as the one in [7 in particular, the use of a so-called minimizing set ), has the advantage that it is not dependent on the initial test values, obtains the global minimum with a high degree of confidence. The paper is organized as follows. In Section II, the controllability radius is reviewed the -D optimization problem to be solved is described. Then in Section III, the few useful tools are developed, followed by a thorough description of the algorithm in Section IV. Finally, Section V gives a numerical example to demonstrate the effectiveness of the algorithm. II. NOTATION AND BACKGROUND In this paper, the field of real complex numbers are denoted by R C respectively. C + C U denotes the closed right half the closed upper half of the complex plane respectively, R + denotes the nonnegative real line [,+ ). The i-th singular value of a matrix M C p m is denoted by σ i M), where σ M) σ M). M denotes the spectral norm of a matrix M is equal to σ M). Also, M, M T, M H, M + denote respectively the complex conjugate, transpose, complex conjugate transpose, Moore-Penrose pseudoinverse of M. The real imaginary components of the matrix M are given by RM IM respectively. Finally, the set of eigenvalues of a square matrix A C n n is denoted by λa). A. Controllability radius The following definition is made. Definition.: Given a LTI system ), the controllability radius, rf c, is defined to be: r c FA,B) = inf{ [ A, B A F n n, B F n m, A + A,B + B ) is uncontrollable} ) /8/$. 8 IEEE 8

2 7th IEEE CDC, Cancun, Mexico, Dec. 9-, 8 TuA6. where F {C, R}. Similarly, the stabilizability radius, r c+ F, is defined as: r c+ F A,B) = inf{ [ A, B A F n n, B F n m, A + A,B + B ) is unstabilizable } ) In [ [, the complex real controllability radii are respectively shown to be given by: r c CA,B) = min s C σ n A sin B ) ) rra,b) c = min τ n A sin B ) ) s C where for W C p q, ) Re W γ Im W τ n W) := sup σ n γ, γ Im W Re W 6) For notational convenience in the paper, denote [ Re W γ ImW Pγ,W) := γ 7) ImW Re W Similarly, the complex real stabilizability radii are respectively shown to be: r c+ C A,B) = min σ n A sin B ) 8) s C + r c+ R A,B) = min τ n A sin B ) 9) s C + B. -D optimization problem It can be seen from ) 9) that a -D optimization problem in the complex plane is required to be solved when computing the complex real controllability/stabilizability radii. A similar -D problem is to be solved when computing the DFM radius, [), the minimum-phase radius 6). In this paper, a fast algorithm is proposed to solve the two stard general problems associated with such -D optimization problems; i.e. for given constant real matrices A,B,C,D) i {,...,n + minr,m)}: Problem : Find rcc,a,b,d) i A si B = min σ i ) s C Problem : Find rrc,a,b,d) i A si B = min τ i ) s C [ A si B For the sake of simplicity, denote Gs) :=. To avoid the trivial case where the radius is zero for all s C, we will assume that the two problems ) ) satisfy the following assumption: A B rank i ) To distinguish which field is consider, rc c i.e. F = C) is called the complex controllability radius rr c is called the real controllability radius. τ nw) is referred to as the n-th real perturbation value of W see [8). III. PRELIMINARY RESULTS Before presenting an efficient algorithm for solving ) ), some preliminary tools are needed; in particular Theorem. &. given below. Definition. The H matrix): Given real x, A R n n, B R n m, C R r n, D R r m, nonsingular matrices C n n, E C n+r) n+r), F C n+m) n+m), define: Hx,A,B,C,D,,E,F) := Ã B D ) C ) [ [ A B where Ã = A T + T, B = C T + T, [ [ C D C = B T + T, D = D T + T, [ [ T T = H, where T := satisfies T T [ [ T T x E = P H E ) T T l x FF H) P r I n for the permutation matrices P l = I n I r I m P r = I n I r I n I m. The following preliminary result [ is obtained. A s B Theorem.: Let M := E F, where s R, E C n+r) n+r), F C n+m) n+m), C n n are all nonsingular. Then, for given x R +, x σm) s λhx,a,b,c,d,,e,f)) ) Proof: M x σm) x λ M H xin+r M det M H = xi n+m [ ) xin+r M det P l P M H xi r = n+m A s B det A T s H C T C D B T D T [ ) T T + = T T Ã s B det = ) C D ) ) det D detã s B D C = s λ Ã B D )) C 9

3 7th IEEE CDC, Cancun, Mexico, Dec. 9-, 8 TuA6. Remark.: If D in ) is singular, then by ), ) can be modified such that s in ) is[ a generalized ) eigenvalue of the matrix pair Ã B C D, instead; [ Ã B i.e. s is in the set of λ C such that C D x = [ λ x for some non-zero eigenvector x C n+m+r. The following result is also obtained. Theorem.: Let γ,, θ [,π real x be given. Then, for all s R +, x σ P γ,g se iθ))) )) s λ H x,â, ˆB,Ĉ, ˆD, ˆ,T γ,n+rp nr,p nm T γ,n+m [ [ [ A B C where Â =, ˆB =, Ĉ =, [ A [ B C D e ˆD =, ˆ iθ I = n D e iθ, P I nr = n I n I r I n I r T γ,n := γ, P mr = I n I n I m I m [ + γ I n iγ + γ I n + γ I n iγ + γ I n., Proof: It can easily be verified e.g. see [8) that T γ,n+r P γ,g se iθ)) [ ) G se Tγ,n+m iθ = Gse iθ ) A se iθ I n B = P nr A se iθ I n B C D P nm C D Therefore, P γ,g se iθ)) [ = Tγ,n+rP Â sˆ nr Ĉ ˆB ˆD P nm T γ,n+m the proof follows immediately from Theorem.. IV. ALGORITHM An efficient iterative algorithm for solving ) ) for given constant real matrices A,B,C,D) i {,...,n + minm,r)} is now presented in this section. The following development focuses mainly on solving ), but it will be shown later that with a slight modification, the algorithm also applies to solving ). Firstly, let the global minimum of ) be denoted by: r = min τ igs)) = min s C s C sup γ, σ i Pγ,Gs))) is achieved at s C γ,, i.e. r = σ i Pγ,Gs ))). Now suppose that at the k-th iteration, for k =,,..., we are given r k, γ k, s k i.e. obtained from the previous iteration), which are approximations of r, γ, s respectively, where r k = σ i Pγ k,gs k ))). Furthermore, suppose we are also given the so-called minimizing set, S k, which is a closed) set in the complex plane that contains the global minimizer s. The basic idea of the algorithm is then to use the information of r k, γ k, s k, together with Theorem., to reduce the size of S k ; i.e. to reduce the region containing the global minimizer s. Denoting the reduced set by S k S k ), we then search within S k for a new point s k S k such that τ i Gs k )) < r k assign r k = τ i Gs k )). Also, assign γ k to be the value of γ that achieves r k ; i.e. r k = σ i Pγ k,gs k ))). Note that since r k < r k, then r k is a new better approximation of the global minimum r. The procedure is then repeated until either the size of S k is smaller than an user-specified tolerance, or until r k becomes very close to zero more on this later). A. The minimizing set, S k In the algorithm, the minimizing set is defined to be the union of particular sectors of interest in the closed upper half of the complex plane, can be described by S k [,π. The reason why the minimizing set is constrained to the closed upper half of the complex plane is due to the following e.g. see [6): A si σ i C A si τ i C ) B D ) B D A si B = σ i A si B = τ i ) ) In other words, the radii in ) ) are both symmetrical with respect to the ; hence one only needs to search within either the closed upper or lower half of the complex plane for the global minimum. It should be pointed out that S k is a subset of [,π, that the underlying minimizing set actually consists of the sectors in the complex plane described by S k ; i.e. the minimizing set is {s C s S k }, where s denotes the angle of s C. For convenience, however, we will sometimes refer to the minimizing set as S k. To reduce the size of S k based on given r k, γ k, s k, where r k = σ i Pγ k,gs k ))), we first construct the set R k : R k = {s C U σ i Pγ k,gs))) < r k } The significance of R k is that for all points not in R k, the radius can never be smaller than r k. This is because for all s / R k, τ i Gs)) = sup γ, σ i Pγ,Gs))) σ i Pγ k,gs))) r k Hence, one only needs to search within R k to find points in the complex plane that achieve a radius smaller than r k. Note that R k can also be a tighter) minimizing set. For ease of implementation, however, we chose the minimizing set to be the sectors containing R k i.e. as described by S k ) instead.

4 7th IEEE CDC, Cancun, Mexico, Dec. 9-, 8 TuA6. To obtain R k, we first fix a particular θ [,π, then find the set R θ k = { w R + σ i P γk,g we iθ))) < r k } 6) Therefore, R k = θ [,π R θ k 7) Note that Rk θ for a particular θ [,π can be obtained by applying Theorem.. In particular, by { Theorem., } the values along the ray, R θ := se iθ C s R +, that achieve a radius equal to r k are among the real nonnegative eigenvalues) of H r k,â, ˆB,Ĉ, ˆD, ˆ,T γ k,n+rp nr,p nm T γk,n+m. Hence, by solving for Λ )) = λ H r k,â, ˆB,Ĉ, ˆD, ˆ,T γ k,n+rp nr,p nm T γk,n+m then finding all real nonnegative s Λ such that σ i P γk,g se iθ))) = r k 8) we obtain the endpoints of the intervals along the ray R θ where σ i P γk,g se iθ))) < r k where σ i P γk,g se iθ))) > r k. To determine which interval is the former i.e. the one of interest), one can simply pick a trial point, w, within a particular interval evaluate σ i P γk,g we iθ))). If there are no real nonnegative eigenvalues, s Λ, such that σ i P γk,g se iθ))) = r k, then either Rk θ = R +, or Rk θ =. Again, a straightforward test using a trial point can confirm which is true. If Rk θ = is true for a particular θ [,π, then this implies that all the values along the ray R θ achieve a radius larger than r k, hence θ can be eliminated from the minimizing set, S k. Therefore the minimizing set for the next iteration is updated as follows. Let Θ k [,π be the set of intervals that describes the smallest union of sectors that contains R k ; i.e. Then update S k = S k Θ k. Θ k = { θ [,π R θ k } 9) B. Updating the current minimum, r k To obtain a radius smaller than r k i.e. a better approximation of the global minimum r ), one can perform a search e.g. a grid or rom search) within the set { R θ k θ S k }. From experience though, it is found that one can often obtain r k by letting s k = w m e iθm i.e. r k = τ i Gs k ))), where θ m is the midpoint of the largest interval of S k, w m is the midpoint of the largest interval of R θm k. C. Stopping criteria The algorithm has two main stopping criteria. Firstly, the algorithm stops when the size of the minimizing set, S k, the size of R k in 7) are both respectively smaller than user-specified tolerances, TOL Sk TOL Rk, where the size of S k is chosen to be the length of the largest interval in S k, the size of R k is chosen to be: sizer k ) = max θ { length of largest interval in R θ k } The second stopping criteria hles the special case when the radius is zero in spite of the fact that assumption ) is satisfied. For example, in [6, it is shown that the complex real minimum-phase radius are both when D = ; i.e. r n+minr,m) F A,B,C,) = where F {C, R}, this is achieved as real) s. Therefore, to prevent the algorithm from searching off into infinity, the second stopping criteria is added to quit when r k is smaller than an user-specified tolerance, TOL rk. D. Algorithm outline The algorithm can be summarized as follows: Algorithm.: Input: A,B,C,D) i, where ) is satisfied Input tolerances: TOL Sk, TOL Rk TOL rk Output: r, s, γ, where r = σ i Pγ,Gs ))) ) Initialization: Set S = [,π. Choose an arbitrary s C U, compute r γ, where r = σ i Pγ,Gs ))). ) Iteration k=,,...) a) Given r k, γ k, s k, S k. b) Reset Θ k =. c) For θ S k θ can be discretized steps of S k ) Compute Rk θ by finding all real nonnegative s λ H r k,â, ˆB,Ĉ, ˆD, ˆ,T γ k,n+rp nr, )) P nm T γk,n+m such that σ i P γk,g se iθ))) = r k. If Rk θ, then update Θ k Θ k θ. d) Update S k = S k Θ k. e) Quit if the length of the largest interval of S k < TOL Sk sizer k ) < TOL Rk. f) Find one point s k Rk θ such that r k < r k, where θ S k, r k = τ i Gs k )). ) Quit if r k < TOL rk. ) Update k k + goto step. E. Algorithm for solving problem ) To solve problem ), only two slight modifications of Algorithm. are needed, which actually results in a simpler algorithm. Firstly, instead of computing the eigenvalues ) of H r k,â, ˆB,Ĉ, ˆD, ˆ,T γ k,n+rp nr,p nm T γk,n+m in step -c) to obtain Rk θ, one is only required to compute the eigenvalues of H ) r k,a,b,c,d,e iθ I n,i n+r,i n+m the proof is trivial follows directly from Theorem.). Furthermore, when solving problem ), all the values within R k actually achieve a smaller radius than r k all the values outside R k do not). This is not true when solving problem ), where R k only achieves a bound. Hence, step -f) for solving problem ) can very easily be accomplished by selecting any point in R k.

5 7th IEEE CDC, Cancun, Mexico, Dec. 9-, 8 F. Algorithm for computing the real stabilizability radius To compute the real stabilizability radius, the real unstable DFM radius, the real minimum-phase radius, where the search is for a minimum in the closed right half of the complex plane i.e. by solving the following problem r RA,B,C,D) i A si B = min τ i ) s C + for given A,B,C,D) i {,...,n + minr,m)} only one very simply modification of Algorithm. is needed. In particular, one only needs to change the initialization step ) of Algorithm. from S = [,π to S = [, π the proof is trivial). G. Operations count To provide an idea of the computational requirements of using Algorithm., the number of operations required in terms of the number of singular value σ), real perturbation value τ), eigenvalue λ) problems are noted in Table I. It is to be noted that every time the radius is evaluated at a particular s C, a real perturbation value problem is solved. This occurs in step ) -f). In step ), the real perturbation value problem is evaluated only once, but step -f) may require multiple evaluations, depending on the number of trials needed before r k < r k is obtained. However, it is found from experiment that the number of trials is typically small i.e. close to when using the method outlined in Section IV-B). It is also to be noted that the main portion of the total operations count is from computing Rk θ for all θ S k in step -c). In terms of implementation, S k is discretized into a number of points. From experiment, it is found that it is generally sufficient to discretize S k into to points, or a minimum of. i.e. a maximum of 6 = π. ) points), whichever results in a smaller angular division. For a given θ S k, computing Rk θ requires: i) solving for the eigenvalues, Λ, of the n n matrix ) H r k,â, ˆB,Ĉ, ˆD, ˆ,T γ k,n+rp nr,p nm T γk,n+m ; ii) evaluating a number of n + r) n + m)-sized singular value problems to verify which real nonnegative eigenvalues, s Λ, there are a maximum of n such values) have the property that they satisfy 8). Furthermore, since there are at most n real nonnegative eigenvalues s Λ that may satisfy 8), Rk θ contains at most n + intervals. Hence, to determine which intervals in Rk θ achieve a radius less than r k, at most n + points are tested, resulting in n + additional n + r) n + m)-sized singular value problems. Remark.: It should be noted that Algorithm. solves only a limited number of real perturbation value problems, which by itself is a -D optimization problem involving multiple n+r) n+m)-sized singular value problems. Hence, this proves to be an advantage of Algorithm., as compared to say, a gradient search method, which requires solving a real perturbation value problem at each trial point, at each step involved with computing the steepest gradient. TABLE I SUMMARY OF OPERATIONS COUNT OF ALGORITHM. Step # of σ,τ,λ) Size of matrix τ n + r) n + m) -c for each λ n n θ S k ) at most n + ) σ n + r) n + m) -f at least τ n + r) n + m) Here, we denote σ, τ, λ as singular value, real perturbation value, eigenvalue problems. V. NUMERICAL EXAMPLE The following example can be found in [, where the real controllability radius of A =. B =. is found to be.9, is achieved at s =.97+j.98. Using Algorithm. with the arbitrary) starting point s = j, the same radius global minimizer are found to within significant figures in iterations to 6 significant figures in 7 iterations. Table II lists r k, s k, the minimizing set, S k for each iteration k =,...,9. Figure provides a grid plot of the real controllability radius with respect to a given s C. Figures to plot R k outlined by the dots) obtained at Iterations,,, 8 respectively, superimposed on a contour plot of Figure. The straight lines originating from the origin depict S k. It is interesting to note that on a computer with a Pentium IV.GHz processor, MB of RAM, MATLAB 7., Algorithm. took a total of about 6 sec. to complete, which is approximately the same amount of time a gradient search method requires to obtain a possibly local) minimum from a single starting point. Hence, if the gradient search method has to test initial points in order to achieve a certain level of confidence that the obtained minimum is global, then the gradient search method will take approximately times longer to run than Algorithm.. VI. CONCLUSIONS In this paper, an efficient algorithm is presented for solving the general -D optimization problems ) ), which are essential for computing the complex TABLE II ESTIMATES OF THE GLOBAL MINIMUM RADIUS r k ), THE MINIMIZER s k ), AND THE MINIMIZING SET S k ) AT EACH ITERATION k TuA6. Iter. k) r k s k S k.767 j [., j. [., [., j.86 [., j.97 [., j.98 [.7, j.9879 [., j.98 [.7, j.989 [.9, j.9897 [.,.

6 7th IEEE CDC, Cancun, Mexico, Dec. 9-, 8 TuA6. Grid plot of real controllability radius with respect to s Iteration radius Fig.. Grid plot of the controllability radius with respect to a given s C Iteration..... Fig.. Iteration Iteration Fig.. Iteration real controllability radius, the minimum-phase radius, the DFM radius, etc. The algorithm works by iteratively reducing the size of the so-called minimizing set which contains the global minimizer. Unlike methods such as the gradient search method, the choice of the initial point of the proposed Fig.. Iteration Iteration 8 Fig.. Iteration 8 algorithm is not crucial, the minimum is obtained with a high degree of confidence that it is indeed global. REFERENCES [ R. Eising, Between controllable uncontrollable, Systems Control Letters, vol., pp. 6 6, 98. [ G. Hu E. J. Davison, Real controllability/stabilizability radius of LTI systems, IEEE Trans. Automat. Contr., vol. 9, no., pp. 7,. [ L. Qiu, B. Bernhardsson, A. Rantzer, E. J. Davison, P. M. Young, J. C. Doyle, A formula for computation of the real stability radius, Automatica, vol., no. 6, pp , 99. [ A. F. Vaz E. J. Davison, A measure for the decentralized assignability of eigenvalues, Systems & Control Letters, vol., pp. 9 99, 988. [ S. Lam E. J. Davison, The real decentralized fixed mode radius of LTI systems, in 6 th IEEE Conference on Decision Control, New Orleans, LA, USA, December - 7, pp. 6. [6, The transmission zero at s radius the minimum phase radius of LTI systems, in 7 th International Federation of Automatic Control IFAC) world congress, Seoul, Korea, July 6-8, pp [7 J. Sreedhar, P. V. Dooren, A. L. Tits, A fast algorithm to compute the real structured stability radius, in International Series of Numerical Mathematics. Birkhäuser Verlag Basel, 996, vol., pp. 9. [8 B. Bernhardsson, A. Rantzer, L. Qiu, Real perturbation values real quadratic forms in a complex vector space, Linear Algebra its Applications, vol. 7, pp., 998.

An efficient algorithm to compute the real perturbation values of a matrix

An efficient algorithm to compute the real perturbation values of a matrix Simon Lam and Edward J. Davison 1 Abstract In this paper, an efficient algorithm is presented for solving the nonlinear 1-D optimization