1
Minimum-gain Pole Placement with Sparse Static Feedback
Vaibhav Katewa and Fabio Pasqualetti
Abstract—The minimum-gain eigenvalue assignment/pole system to perturbations in the system parameters, thereby placement problem (MGEAP) is a classical problem in LTI making the system robust. This is known as Robust Eigenvalue systems with static state feedback. In this paper, we study Assignment Problem (REAP). Alternatively, one can choose the MGEAP when the state feedback has arbitrary sparsity constraints. We formulate the sparse MGEAP problem as an the feedback matrix with minimum gain, thereby reducing equality-constrained optimization problem and present an an- the overall control effort. This is known as Minimum Gain alytical characterization of its solution in terms of eigenvector Eigenvalue Assignment Problem (MGEAP). Both of these matrices of the closed loop system. This result is used to problems have been studied extensively [2]- [3]. provide a geometric interpretation of the solution of the non- Recently, considerable attention has been given to the study sparse MGEAP, thereby providing additional insights for this classical problem. Further, we develop an iterative projected and design of sparse feedback control systems, where certain gradient descent algorithm to solve the sparse MGEAP using entries of the matrix F are required to be zero. Feedback a parametrization based on the Sylvester equation. We present sparsity typically arises in large scale systems in which the a heuristic algorithm to compute the projections, which also controllers do not have access to all the states of the system. provides a novel method to solve the sparse EAP. Also, a relaxed Sparsity may also be a result of the special structure of the version of the sparse MGEAP is presented and an algorithm is developed to obtain approximately sparse solutions to the control system which prohibits feedback from some states to MGEAP. Finally, numerical studies are presented to compare the controllers. the properties of the algorithms, which suggest that the proposed The feedback design problem with sparsity constraints is projection algorithm converges in almost all instances. considerably more difficult than the unconstrained case. There have been numerous studies to determine the optimal feed- back control law for H2/LQR/LQG problems with sparsity, I.INTRODUCTION particularly when the controllers have access to only local The Eigenvalue/Pole Assignment Problem (EAP) using information (see [4]–[7] and the references therein). While the static state feedback is one of the central problems in the optimal H2/LQR/LQG design problems with sparsity have a design of Linear Time Invariant (LTI) control systems. It plays rich history, studies on the REAP/MGEAP in the presence of a key role in system stabilization and shaping its transient arbitrary sparsity constrains are lacking. Even the problem of behavior. Given the following LTI system finding a particular (not necesary optimal) sparse feedback matrix that solves the EAP is not well studied. In this x(k) = Ax(k) + Bu(k), (1a) D paper, we study the EAP and MGEAP with arbitrary sparsity u(k) = F x(k), (1b) constraints on the feedback matrix F . We provide analytical n m characterization for the solution of sparse MGEAP and provide where x R is the state of the LTI system, u R is ∈ n×n n×m ∈ iterative algorithms to solve the sparse EAP and MGEAP. the control input, A R , B R , and denotes either the continuous time∈ differential∈ operator or theD discrete- Related work There have been numerous studies on the time shift operator, the EAP involves finding a real feedback optimal pole placement problem without sparsity constraints. m×n For the REAP, authors have considered optimizing different matrix F R such that the eigenvalues of the closed ∈ metrics which capture the sensitivity of the eigenvalues, such loop matrix Ac(F ) , A + BF coincide with a given set = S as the condition number of the eigenvector matrix [2], [8]– λ1, λ2, , λn that is closed under complex conjugation. { ··· } [11], departure from normality [12] and others [13], [14]. Most arXiv:1805.08762v1 [math.OC] 22 May 2018 It is well known that the existence of F depends on the controllability properties of the pair (A, B). Further, for single of these methods use gradient-based iterative procedures to input systems (m = 1), the feedback vector that assigns obtain the solutions. For surveys and performance comparisons the eigenvalues is unique and can be obtained using the of these REAP methods, see [8], [15], [16] and the references Ackermann’s formula [1]. On the other hand, for multi-input therein. systems (m > 1), the feedback matrix is not unique and there Early works for MGEAP, including [17], [18], presented exists a flexibility to choose the eigenvectors of the closed loop approximate solutions using low rank feedback and successive system. This flexibility can be utilized to choose a feedback pole placement techniques. Simultaneous robust and minimum matrix that satisfies some auxiliary control criteria in addition gain pole placement were studied in [11], [19]–[21]. For a to assigning the eigenvalues. For instance, the feedback matrix survey and performance comparison of these MGEAP studies, can be chosen to minimize the sensitivity of the closed loop see [3] and the references therein. The regional pole placement problem was studied in [22], [23], where the eigenvalues were This material is based upon work supported in part by ARO award assigned inside a specified region. While all these studies 71603NSYIP, and in part by NSF awards ECCS1405330 and BCS1631112. have provided useful insights on REAP/MGEAP, they do not Vaibhav Katewa and Fabio Pasqualetti are with the Department of Me- chanical Engineering, University of California at Riverside, {vkatewa, consider sparsity constraints on the feedback matrix. Further, fabiopas}@engr.ucr.edu. the techniques in these studies cannot be used or simply 2 extended to obtain a sparse solution. In contrast, we study Paper organization The remainder of the paper is organized the sparse EAP/MGEAP by explicitly including the sparsity as follows. In Section II we formulate the sparse MGEAP constraints into the problem. optimization problem. In Section III, we obtain the solution There have also been numerous studies on EAP with sparse of the optimization problem using the Lagrangian theory of dynamic LTI feedback. The concept of decentralized fixed optimization. We also provide a geometric interpretation for modes (DFMs) was introduced in [24] and later refined in the optimal solutions of the non-sparse MGEAP. In Section IV, [25]. Decentralized fixed modes are those eigenvalues of we present two heuristic algorithms for solving the sparse EAP. the system which cannot be shifted using a static/dynamic Further, we present a projected gradient descent algorithm to feedback with fully decentralized sparsity pattern (i.e. the case solve the sparse MGEAP and also an approximately-sparse where controllers have access only to local states). The re- solution algorithm for a relaxed version of the sparse MGEAP. maining eigenvalues of the system can be arbitrarily assigned. Section V contains numerical studies and comparisons of the However, this cannot be achieved in general using a static proposed algorithms. Finally, we conclude the paper in Section decentralized controller and requires the use of dynamic decen- VI. tralized controller [24]. Other algebraic characterizations of the DFMs were presented in [26], [27]. The notion of DFMs was II.SPARSE MGEAPFORMULATION generalized for an arbitrary sparsity pattern and the concept of A. Mathematical Notation and Properties structurally fixed modes (SFMs) was introduced in [28]. Graph theoretical characterizations of structurally fixed modes were We use the following notation throughout the paper: Spectral norm provided in [29], [30]. As in the case of DFMs, assigning the k · k2 non-SFMs also requires dynamic controllers. These studies on F Frobenius norm
d T d T d P.9 dX tr(AX)=A , dX tr(X X)=2X, dx (Ax)=A, eigenvalues of A which cannot be changed using LTI static P.10 d(X−1) = X−1dXX−1, (and also dynamic) state feedback, and are denoted by − P.11 d(A B) = dA B + A dB, ¯ ⊗ 2⊗ ⊗ Γf (A, B, F) , Γ(A + BF ). P.12 Let Dxf and Dxf be the gradient and Hessian of n T F : F F¯c = 0 f(x): R R. Then, df = (Dxf) dx and \ 2 →T 2 ◦ d f = (dx) (Dxf)dx, −1 T P.13 Tm,n = Tm,n = Tn,m. n We make the following assumptions regarding the fixed P.14 Projection of a vector y R on the null space of m×n ∈ + modes and feasibility of the optimization problem (2). A R is given by yp = [In A A]y. ∈ − A1: The fixed modes of the triplet (A, B, F¯) are included in the desired eigenvalue set , i.e., Γ (A, B, F¯) . S f ⊆ S B. Sparse MGEAP A2: There exists at least one feedback matrix F that satisfies The sparse MGEAP involves finding a real feedback matrix constraints (2a)-(2c) for the given . S F Rm×n with minimum norm that assigns the closed Assumption A1 is clearly necessary for the feasibility of loop∈ eigenvalues of (1a)-(1b) at some desired locations given the optimization problem (2). Assumption A2 is restrictive by set = λ1, λ2, , λn , and satisfies a given sparsity because, in general, it is possible that a static feedback matrix constraints.S Let{ F¯ ···0, 1 m}×n denote a binary matrix that with a given sparsity pattern cannot assign the closed loop specifies the sparsity∈ { structure} of the feedback matrix F . If eigenvalues to arbitrary locations (i.e. for an arbitrary set F¯ = 0 (respectively F¯ = 1), then the jth state is unavailable satisfying Assumption A1).1 In such cases, only a few ij ij S (respectively available) for calculating the ith input. Thus, (< n) eigenvalues can be assigned independently and other remaining eigenvalues are a function of them. To the best ¯ 0 if Fij = 0, and of our knowledge, there are no studies on characterizing Fij = ¯ (? if Fij = 1, conditions for the existence of a static feedback matrix for an arbitrary sparsity pattern F¯ and eigenvalue set [38] (although c where ? denotes a real number. Let F¯ 1 × F¯ denote S , m n − such characterization is available for dynamic feedback laws the complementary sparsity structure matrix. Further, (with a with arbitrary sparsity pattern [28], [31], [32], and static output slight abuse of notation, c.f. (1a)) let X , [x1, x2, , xn] feedback for decentralized sparsity pattern [39]). Thus, for n×n ··· ∈ C , xi = 0n denote the non-singular eigenvector matrix of 6 the purpose of this paper, we focus on finding the optimal the closed loop matrix Ac(F ) = A + BF . feedback matrix assuming that at least one such feedback The MGEAP can be mathematically stated as the following matrix exists. optimization problem
1 2 III.SOLUTION TO THE SPARSE MGEAP min F F (2) F,X 2 || || In this section we present the solution to the optimization s.t. (A + BF )X = XΛ, (2a) problem (2). We use the theory of Lagrangian multipliers for H equality constrained minimization problems to derive the op- xi xi = 1 i = 1, 2, , n and (2b) c ∀ ··· timality conditions. We begin with a remark on the formalism F¯ F = 0 × , (2c) ◦ m n involving complex variables. where Λ = diag([λ , λ , , λ ]T) is the diagonal matrix 1 2 n Remark 2. (Complex variables) If the desired eigenvalue set constructed from the entries··· of the desired eigenvalue set . contains complex entries, the corresponding eigenvectors Equations (2a) and (2c) represent the eigenvalue assignmentS inS X will also be complex. Thus, the eigenvalue-eigenvector and sparsity constraints, respectively. Further, the normaliza- constraint in (2a) is a complex-valued constraint, and it tion constraint in (2b) is included to uniquely determine the induces the following conjugate constraint: eigenvector matrix X. The constraint (2a) is not convex in (F,X) and, therefore, (A + BF )X∗ = X∗Λ∗. (3) the optimization problem (2) is non-convex. Consequently, multiple local minima may exist. This is a common feature In the optimization problem, we use the formalism wherein a in various minimum distance and eigenvalue assignment prob- complex number and its conjugate are treated as independent lems [37], including the non-sparse MGEAP. variables [40], [41] and, thus, we treat X and X∗ as independent variables. Remark 1. (Choice of norm) The Frobenius norm explicitly measures the element-wise gains of a matrix, which is infor- Remark 3. (Conjugate eigenvectors) We use the convention mative in sparsity constrained problems arising, for instance, that the right eigenvectors (xi, xj) corresponding to two in network control problems. Further, it is also convenient conjugate eigenvalues (λi, λj) are also conjugate. Thus, if ∗ ∗ for the analysis, as it allows us to analytically compute the λi = λj , then xi = xj . Similar convention is also used for derivatives of the cost function. the left eigenvectors.
Definition 1. (Fixed modes [24], [31]) The fixed modes of 1Although a sparse dynamic feedback law can assign the eigenvalues to (A, B) with respect to the sparsity constraints F¯ are those arbitrary locations under Assumption A1 [32]. 4
In the theory of equality constrained optimization, the first- Theorem III.2. (Optimality conditions) Let (X,ˆ Fˆ) (equiv- order optimality conditions are meaningful only when the alently zˆ = [ˆxT, xˆH, fˆT]T) satisfy the constraints (2a)-(2c). optimal point satisfies the following regularity condition: the Then, (X,ˆ Fˆ) is a local minimum of the optimization problem Jacobian of the constraints, defined by Jb, is full rank. This (2) if and only if regularity condition is mild and usually satisfied for most Fˆ = F¯ (BTLˆXˆ T), (7a) classes of problems [42]. Before presenting the main result, − ◦ we derive the Jacobian and state the regularity condition for ˆ ˆ the optimization problem (2). where X and L are the right and left eigenvector matrices of Ac(Fˆ), respectively, and satisfy Computation of Jb requires vectorization of the matrix constraints (2a)-(2c). For this purpose, let x , vec(X) n2 mn ∈ (A + BFˆ)Xˆ = XˆΛ (7b) C , f , vec(F ) R . Recalling Remark 2, let z , [xT, xH, f T]T be the∈ vector containing all the independent (A + BFˆ)TLˆ = LˆΛT and, (7c) variables of the optimization problem. Further, let ns denote Jb(ˆz) is full rank, (7d) the total number of feedback sparsity constraints (i.e. number P (ˆz)DPˆ (ˆz) > 0, (7e) of 1’s in F¯c): c c c where Dˆ is the Hessian defined as ns = (i, j): F¯ = [¯f ], ¯f = 1 . |{ ij ij }| ¯H Finally, note that the constraint (2c) can be equivalently written 0n2×n2 0n2×n2 L ˆ ¯T D , 0n2×n2 0n2×n2 L , (8) as (see the proof of Lemma III.1 for a formal definition of Q) ∗ L¯ L¯ 2Imn
Qf = 0ns , (4) ¯ T ˆ where L , Tn,m(B L In), and P (z) is the projection matrix for some binary matrix Q 0, 1 ns×mn. ⊗ ∈ { } of Jb(z) defined as Lemma III.1. (Jacobian of the constraints) The Jacobian of + P (z) = I 2 J (z)J (z). (9) the equality constraints (2a)-(2c) is given by 2n +mn − b b T T Proof. We prove the result using the Lagrange theorem for 2 2 Ac(F ) ( Λ ) 0n ×n X B n×n ∗ ⊕ − H H⊗ equality constrained minimization. Let L ,L ,M 2 2 C 0n ×n Ac(F ) ( Λ ) X B m×n n ∈ ∈ Jb(z)= ⊕∗ − ⊗ , (5) and h be the Lagrange multipliers associated x¯ x¯ 0 × R R n mn with constraints∈ (2a),(3),(2b) and (2c), respectively. Further, 0n ×n2 0n ×n2 Q s s let H = diag(h). The Lagrange function for the optimization where x¯ (1 xH) (I 1T). problem (2) is given by , n ⊗ ◦ n ⊗ n Proof. We construct the Jacobian J by rewriting the con- P.2 1 T 1 T b = tr(F F ) + 1 [L (A (F )X XΛ)]1 straints (2a)-(2c), (3) in vectorized form and taking their L 2 2 n ◦ c − n derivatives with respect to z. Constraint (2a) can be vectorized 1 T ∗ ∗ ∗ ∗ H + 1 [L (Ac(F )X X Λ )]1n + tr((X X In)H) in the following two different ways (using P.3 and P.4): 2 n ◦ − − T ¯c T +1m[M (F F )]1n [(A + BF ) ( Λ )]x = 0n2 , (6a) ◦ ◦ ⊕ − P.6,P.7 1 T 1 T T T = tr(F F ) + tr[L (A (F )X XΛ)] [A (Λ )]x + (X B)f = 0 2 . (6b) c ⊕ − ⊗ n 2 2 − 1 H ∗ ∗ ∗ H Differentiating (6a) w.r.t. x and (6b) w.r.t f yields the first + tr[L (Ac(F )X X Λ )] + tr((X X In)H) (block) row of J . Similar vectorizations of the conjugate 2 − − b +tr[(M F¯c)TF ]. eigenvalue constraint (3) and differentiation w.r.t. z yields the ◦ second (block) row of Jb. Differentiating constraint (2b) for We next derive the first-order necessary condition for a sta- i = 1, 2, , n w.r.t. x and x∗yields ··· tionary point. Differentiating l w.r.t. X and setting to 0, we H H T T get x1 x1 1 x1 0n 0n H − T H ··· T x x2 1 0 x 0 d 2 − n 2 ··· n d P.9 1 T T ∗ . = . . . . =x, ¯ = [Ac (F )L LΛ ] + X H = 0n×n. (10) dx ...... dX L 2 − H T T H T xnxn 1 0n 0n xn Pre-multiplying (10) by X , we get − ··· ∗ and x¯ , respectively. Finally, (2c) consists of ns non-trivial 1 T T T T T ∗ [X A (F )L X LΛ ] + X X H = 0 × , sparsity constraints, which can be written as (4) where Q = 2 c − n n T and supp vec F¯c is the [eq1 eq2 . . . eqns ] q1, . . . , qns = ( ( )) (2a) 1 T T T T T ∗ { } c [Λ X L X LΛ ] + X X H = 0 × . (11) set of indices indicating the ones in vec(F¯ ). Differentiating ⇒ 2 − n n (4) w.r.t. z yields the fourth (block) row of J , thus completing b Since Λ is diagonal and xHx = 1, the vector containing the the proof. i i diagonal elements of the left side of (11) is h. Therefore, h = Next, we provide the optimality conditions for the optimiza- 0n (and H = 0n×n) and from (10), we get (7c). Equation (7b) tion problem (2). is a restatement of (2a) for the optimal (F,ˆ Xˆ). 5
T T T Differentiating w.r.t. F , we get Corollary III.3. (Stationary point equation) Zˆ , [Xˆ , Lˆ ] L is a stationary point of the optimization problem (2) if and d P.9 T T c = F + Re(B LX ) + M F¯ = 0 × . (12) only if dF L ◦ m n Taking the Hadamard product of (12) with F¯c and using (2c), ¯ ˆ ˆ ¯ ¯T ¯ˆ ˆT ¯ ¯T ˆ we get (since F¯c F¯c = F¯c) AZ ZΛ = B1[F (B1 IZZ B2)]B2 Z, (14) ◦ − ◦ c T T c where, F¯ Re(B LX ) + M F¯ = 0 × (13) ◦ ◦ m n A 0 B 0 c ¯ n×n ¯ n×n Replacing M F¯ from (13) into (12), we get A , T , B1 , , 0 × A 0 × I ◦ n n n m n ¯ T T ¯ F = F Re(B LX ). ¯ In 0n×m F 0m×m − ◦ B2 , , F , T , and 0 × B 0 × F¯ n n n n Since the eigenvectors in X (and L , [l1, l2, .ln]) are ··· 0n×n In closed under complex conjugation (c.f. Remark 3), the product I¯ , . T n T In 0n×n LX = i=1 lixi is real and (7a) follows. Equation (7d) is the necessary regularity condition and follows from Lemma Proof. Combining (7b) and (7c) and using ΛT = Λ, we get III.1 P ˆ ˆ ˆ ˆ Next, we derive the second-order sufficient condition for a AX XΛ BF X T ˆ− ˆ = ˆT T ˆ local minimum by calculating the Hessian of . Taking the A L LΛ − F B L L − differential of twice, we get Fˆ 0 L A¯Zˆ ZˆΛ = B¯ B¯TZˆ ⇒ − − 1 0 FˆT 2 d2 = tr((dF )TdF ) + tr(LTBdF dX) + tr(LHBdF dX∗) ¯ T ˆ ˆ T L H F (B LX ) 0 T + 2tr((dX) dXH) = B¯1 ◦ B¯ Zˆ 0 F¯T (XˆLˆTB) 2 (P.2,H=0) T T T T T T T ∗ ∗ ◦ = df df + vec (dF B L)dx + vec (dF B L )dx T T LˆXˆ 0 T = B¯1 F B¯ B¯2 B¯ Zˆ (P.3,P.5) T T ¯ T ¯∗ ∗ ◦ 1 0 XˆLˆT 2 = df df + df Ldx + df L dx dx = B¯ F B¯TI¯(ZˆZˆT I¯)B¯ B¯TZˆ 1 1 1 2 2 = [dxH, dxT, df T]D dx∗ , ◦ ◦ 2 = B¯ F (nB¯TI¯ZˆZˆTB¯ ) B¯TZ,ˆo df 1{ ◦ 1 2 } 2 where D is the Hessian (c.f. P.12) defined in (8). The suf- where the above equations follow from direct manipulations ficient second-order optimality condition for the optimization using definition of Hadamard product. problem requires the Hessian to be positive definite in the Remark 5. (Partial spectrum assignment) The results of kernel of the Jacobian at the optimal point [43]. That is, Theorem III.2 and Corollary III.3 are also valid when only yTDy > 0, y : J (z)y = 0. This condition is equivalent to ∀ b p (p < n) eigenvalues need to be assigned and the remain- P (z)DP (z) > 0, since Jb(z)y = 0 if and only if y = P (z)s p×p 2 ing eigenvalues are unspecified. In this case, Λ C , for a s R2n +mn [42]. Since the projection matrix P (z) is ˆ n×p and ˆ n×p. Partial assignment∈ is useful ∈ X C L C symmetric, (7e) follows, and this concludes the proof. when∈ the sparsity constraints∈ allow only a limited number of eigenvalues to be assigned independently and the remaining Remark 4. (Eigenvector normalization constraints) From the eigenvalues are functionally related to them (c.f. discussion proof of Theorem III.2, we observe that the Lagrange multipli- below Assumption A2). Nevertheless, in this paper we focus ers associated with the eigenvector normalization constraints on assigning all the closed loop eigenvalues. in (2b) are zero (h = 0n). This implies that these constraints are redundant. Further, observe from (7a) that any pair of Remark 6. (General eigenstructure assignment) Although left and right eigenvector matrices LDˆ −1, XDˆ where D the optimization problem (2) is formulated by considering Λ is any diagonal matrix with non-zero{ (and possibly} complex) to be diagonal, the result in Theorem III.2 is valid for any diagonal entries, will result in the same feedback matrix Fˆ. general Λ satisfying Γ(Λ) = . For instance, we can choose Thus, we can always choose a suitable D to normalize Xˆ. Λ in a Jordan canonical form.S However, note that for a general Therefore, we ignore the eigenvector normalization constraints Λ, X will cease to be an eigenvector matrix. Thus, in such in the remainder of the paper. cases, we need to omit the normalization constraints (2b) in the optimization problem. Observe that the Hadamard product in (7a) guarantees that the feedback matrix satisfies the sparsity constraints given A solution of the optimization problem (2) can be obtained in (2c). However, note that the optimal sparse feedback Fˆ by numerically/iteratively solving the matrix equation (14), cannot be obtained by sparsification of the optimal non-sparse which resembles a Sylvester type equation with a non-linear feedback. The feedback optimality condition (7a) is an implicit right side, and using (7a) to compute the feedback matrix. condition in terms of the closed loop right and left eigenvector The regularity and local minimum of the solution can be ver- matrices. Next, we provide an explicit optimality condition in ified using (7d) and (7e), respectively. Since the optimization terms of L,ˆ Xˆ . problem is not convex, only local minima can be obtained { } 6 via this procedure. To improve upon the local solutions, the where (a) follows from the fact that Λ and LˆTXˆ commute. above procedure can be repeated for several different initial Sufficiency: From (16), we get conditions (for solving (14)). However, convergence to the global minimum is not guaranteed. tr(FˆT(Y ΛXˆ −1 XˆΛXˆ −1Y Xˆ −1)) = 0 − In this paper, instead of solving (14) directly, we use a (P.1) tr[(ΛXˆ −1FˆT Xˆ −1FˆTXˆΛXˆ −1)Y ] = 0. different approach based on the gradient descent procedure ⇒ − to obtain a local solution. Details of this approach and corre- Since the above equation is true for all Y Cn×n, we get sponding algorithms are presented in Section IV. ∈ −1 T −1 T −1 ΛXˆ Fˆ Xˆ Fˆ XˆΛXˆ = 0 × A. Results for non-sparse MGEAP − n n XˆΛXˆ −1FˆT = FˆTXˆΛXˆ −1 In this subsection, we present some results specific to the ⇒ case when the optimization problem (2) does not have any A (Fˆ)FˆT = FˆTA (Fˆ). ⇒ c c sparsity constraints (i.e. F¯ = 1m×n). Although the non-sparse T MGEAP has been studied previously, these results are novel Thus, Ac(Fˆ) and Fˆ commute and therefore, have common and further illustrate the properties of an optimal solution. left and right eigenspaces [44], i.e. FˆT = XGˆ Xˆ −1 = We begin by presenting a geometric interpretation of the XˆLˆT, where G is a diagonal matrix. This− completes the optimality conditions provided in Theorem III.2 under the as- proof.− sumption that B = In, i.e. all the entries of A can be perturbed independently. In this case, the optimization problem (2) can Next, we show the equivalence of the non-sparse MGEAP be written as (while ignoring the eigenvector normalization for two orthogonally similar systems. constraints; c.f. Remark 4): Lemma III.5. (Invariance under orthogonal transformation) 1 −1 2 ¯ min A XΛX F . (15) Let F = 1m×n and (A1,B1), (A2,B2) be two orthogonally X 2 || − || −1 similar systems such that A2 = PA1P and B2 = PB1, −1 Since A and R(X) , XΛX are elements (or vectors) of with P being an orthogonal matrix. Let an optimal solutions the matrix inner product space with Frobenius norm, a solution of (2) for the two systems be denoted by (Xˆ1, Lˆ1, Fˆ1) and of the optimization problem (15) is given by the projection of (Xˆ2, Lˆ2, Fˆ2), respectively. Then A on the manifold , R(X): X is non-singular . This M { } T projection can be obtained by solving the normal equation, Xˆ2 = P Xˆ1, Lˆ2 = P Lˆ1, Fˆ2 = Fˆ1P , and which states that the optimal error vector Fˆ = A XˆΛXˆ −1 (17) − Fˆ1 F = Fˆ2 F . should be orthogonal to the tangent plane of the manifold || || || || ˆ at the optimal point X. The next result shows that the Proof. From (7b), we have optimalityM conditions derived in Theorem III.2 are in fact the normal equations for the optimization problem (15). (A2 + B2Fˆ2)Xˆ2 = Xˆ2Λ ¯ −1 T Lemma III.4. (Geometric interpretation) Let F = 1m×n (PA1P + PB1Fˆ1P )P Xˆ1 = P Xˆ1Λ and B = I . Then, Equations (7a)-(7c) are equivalent to the ⇒ n (A + B Fˆ )Xˆ = Xˆ Λ. following normal equation: ⇒ 1 1 1 1 1
< F,ˆ M(Xˆ) >F = 0, (16) Similar relation can be shown between Lˆ1 and Lˆ2 using (7c). T Next, from (7a), we have where M(X) denotes the tangent space of at X. T M ˆ T ˆ ˆ T T ˆ ˆ T T ˆ T Proof. We begin by characterizing the tangent space M(X), F2 = B2 L2X2 = B1 L1X1 P = F1P . which is given by the first order approximation of theT matrix − − (P.1) function R(X): Finally, Fˆ 2 = tr(FˆTFˆ ) = tr(FˆTFˆ ) = Fˆ 2 , thereby || 1||F 1 1 2 2 || 2||F R(X + dX) = (X + dX)Λ(X + dX)−1 completing the proof. (P.10) = R(X) + dXΛX−1 XΛX−1dXX−1 − Recall from Remark 5 that Theorem III.2 is also valid for + higher order terms. MGEAP with partial spectrum assignment. Next, we consider Thus, the tangent space is given by the case when only one real eigenvalue needs to assigned for the MGEAP while the remaining eigenvalues are unspecified. −1 −1 −1 n×n M(X) = Y ΛX XΛX YX : Y C T { − ∈ } In this special case, we can explicitly characterize the global Necessity: Using Fˆ given by (7a), we get minimum of (2) as shown in the next result. T −1 −1 −1 ¯ < F,ˆ M(Xˆ) >= tr(Fˆ (Y ΛXˆ XˆΛXˆ Y Xˆ )) Corollary III.6. (One real eigenvalue assignment) Let F = T − T −1 T −1 −1 1m×n, Λ R, and B = In. Then, the global minima of = tr(XˆLˆ Y ΛXˆ ) + tr(XˆLˆ XˆΛXˆ Y Xˆ ) ∈ − the optimization problem (2) is given by Fˆgl = σmin(A (P.1) T T −1 T − − = tr(Lˆ Y Λ) + tr(Lˆ XˆΛXˆ Y ) ΛIn)uv , where u and v are unit norm left and right singular − (a) (P.1) vectors, respectively, corresponding to σmin(A ΛIn). Further, = tr(LˆTY Λ) + tr(ΛLˆTXˆXˆ −1Y ) = 0, − Fˆ = σmin(A ΛI ). − k glkF − n 7
ˆ n ˆ n ˆ 2 Proof. Since Λ R, X R , xˆ and L R , l. Let Fp always exists. Then, we can solve the modified MGEAP ˆ ˜ˆ ∈ ˆ ∈ ˆ∈ ˆ T with parameters F¯ . If is the sparse solution l = βl where β , l 2 > 0. Substituting F = lxˆ from (A+BFp,B, Λ, ) F (7a) into (7b)-(7c), wek k get − of this modified problem, then the solution of the original problem is Fp + F . ˆ T (2b) ˜ˆ (A lxˆ )ˆx =x ˆΛ (A ΛIn)ˆx = βl and, To avoid complex domain calculations in the algorithms, we − ⇒ − T ˆT ˆ ˆ T˜ˆ use the real counterpart of (2a). For two complex conjugate (A xˆl )l = lΛ (A ΛIn) l = βx.ˆ ∗ ∗ − ⇒ − eigenvalues (λi, λi ) and corresponding eigenvectors (xi, xi ), The above two equations imply that the unit norm vectors xˆ the following complex equation ˆ and ˜l are left and right singular vectors of A ΛI associated n ∗ ∗ λi 0 ˆ 2 − ˆT ˆ (A + BF ) xi xi = xi xi 0 λ∗ with the singular value β. Since F F = tr(F F ) = i tr ˆTˆ T 2, we pick as the minimumk k singular value h i (ˆxl lxˆ ) = β β is equivalent to the following real equation of A ΛIn, and the proof is complete. − Re(λi) Im(λi) (A+BF ) Re(xi) Im(xi) = Re(xi) Im(xi) We conclude this subsection by presenting a brief com- −Im(λi) Re(λi) parison of the non-sparse MGEAP solution with deflation h ∗ i Thus, for each complex eigenvalue, the columns xi x of techniques for eigenvalue assignment. For B = In, an i X are replaced by Re(x ) Im(x ) to obtain a real X0 and alternative method to solve the non-sparse EAP is via the i i λi 0 Re(λi) Im(λi) Wielandt deflation technique [45]. Wielandt deflation achieves the sub-matrix ∗ of Λ is replaced by to 0 λi −Im(λi) Re(λi) pole assignment by modifying the matrix A in n steps A obtain a real Λ0. Clearly, X0 is not the eigenvector matrix of → h i h i A1 A2 An. Step i shifts one eigenvalue of A + BF (c.f. Remark 6), and the latter can be easily obtained → → · · · → 0 Ai−1 to a desired location λi, while keeping the remaining by combining the columns of X . Also, the invertibility of X 0 eigenvalues of Ai−1 fixed. This is achieved by using the is equivalent to the invertibility of X . In the remainder of this feedback F i = (µ λ )v zT, where µ and v are any section, we assume (2a) to be in the real form. df − i − i i i i i eigenvalue and right eigenvector pair of Ai−1, and z is any T vector such that zi vi = 1. Thus, the overall feedback that n i A. Algorithms for non-sparse MGEAP solves the EAP is given as Fdf = i=1 Fdf . It is interesting to compare the optimal feedback expression In this subsection, we present two iterative gradient based ˆ n ˆ T P in (7a), F = i=1 lixˆi , with the deflation feedback. Both descent algorithms to obtain non-sparse and approximately- feedbacks are− sum of n rank 1 perturbations. However, the sparse solutions to the MGEAP, respectively. To develop the Wielandt deflationP has an inherent− special structure and a algorithms, we use the Sylvester equation based parametriza- restrictive property that, in each step, all except one eigenvalue tion [11], [46]. In this parametrization, instead of defining remain unchanged. Furthermore, each rank 1 term in Fˆ and (F,X) as free variables, we define a parameter G , FX − m×n ∈ Fdf involves the right/left eigenvectors of the closed and open R as the free variable. With this parametrization, the non- loop matrix, respectively. Due to these reasons, it clearly sparse MGEAP is stated as (while ignoring the eigenvector follows that Fˆ F . normalization constraints; c.f. Remark 4): || ||F ≤ || df ||F
OLUTION LGORITHMS IV. S A 1 2 min J = F F (18) In this section, we present an iterative algorithm to obtain G 2 || || a solution to the sparse MGEAP in (2). To develop the s.t. AX XΛ + BG = 0, (18a) algorithm, we first present two algorithms for computing non- − −1 sparse and approximately-sparse solutions to the MGEAP, F = GX . (18b) respectively. Next, we present two heuristic algorithms to Note that, for any given G, we can solve the Sylvester equation obtain a sparse solution of the EAP (i.e. any sparse solution, (18a) to obtain X. Assumption A4 guarantees that (18a) has which is not necessarily minimum-gain). Finally, we use these a unique solution [47]. Further, we can use (18b) to obtain a algorithms to develop the algorithm for sparse MGEAP. Note non-sparse feedback matrix F . Thus, (18) is an unconstrained that although our focus is to develop the sparse MGEAP optimization problem in the free parameter G. algorithm, the other algorithms presented in this section are The Sylvester equation based parametrization requires the novel in themselves to the best of our knowledge. unique solution X of (18a) to be non-singular, which holds We make the following assumptions: generically if (i) is controllable and (ii) ¯ (A, BG) (Λ, BG) A3: The triplet (A, B, F) has no fixed modes, i.e., is observable [48].− Since the system has no fixed− modes Γ (A, B, F¯) = . f ∅ (c.f. Assumption A3), (A, B) is controllable. Therefore, the A4: The open and closed loop eigenvalue sets are disjoint, i.e., above two conditions are mild and are satisfied for almost all Γ(A) Γ(Λ) = . instances as confirmed in our simulations (see Section V). ∩ A4∅ Assumption is not restrictive since if there are any The next result provides the gradient and Hessian of the common eigenvalues in A and Λ, we can use a preliminary cost J w.r.t. to the parameter g , vec(G). sparse feedback Fp to shift the eigenvalues of A to some other locations such that . Since 2 Γ(A + BFp) Γ(Λ) = Although the minimization cost of the modified MGEAP is 0.5||Fp + ∩ ∅ 2 the system has no fixed modes (c.f. Assumption A3), such a F ||F , it can be solved using techniques similar to solving MGEAP in (2). 8
Lemma IV.1. (Gradient and Hessian of J) The gradient and Clearly, the first-order optimality condition of the un- dJ Hessian of the cost J in (18) with respect to g is given by constrained optimization problem (18) is given by dg = Z(F,X)f = 0. The next result shows that this condition is dJ = (X−1 I )+(I BT)A˜−T(X−1 F T) f, (19) equivalent to the first-order optimality conditions of Theorem dg ⊗ m n ⊗ ⊗ h i III.2 without sparsity constraints. Z(F,X) , Corollary IV.2. (Equivalence of first-order optimality condi- 2 | {z } d J T tions) Let F¯ = 1 . Then, the first-order optimality condition , H(F,X) = Z(F,X)Z (F,X) m×n d2g Z(F,ˆ Xˆ)fˆ = 0 of (18) is equivalent to (7a)-(7c), where T T + Z1(F,X)Z (F,X) + Z(F,X)Z1 (F,X), (20) −T −1 T ˆl , vec(Lˆ) = A˜ (Xˆ Fˆ )f.ˆ (26) where Z (F,X) (I BT)A˜−T(X−1F T I )T , ⊗ 1 , n ⊗ ⊗ n m,n T Proof. The optimality condition (7b) follows from (18a)- and A˜ , A ( Λ ). ⊕ − (18b). Equation (7a) can be rewritten as FˆXˆ −T + BTLˆ = 0 Proof. Vectorizing (18a) using P.3 and taking the differential, and its vectorization using P.3 yields Z(F,ˆ Xˆ)fˆ = 0. Finally, we get vectorization of the left side of (7c) yields T T T T T Ax˜ + (I B)g = 0 vec[(A + BFˆ) Lˆ LˆΛ ] = vec[A Lˆ LˆΛ + (BFˆ) L] n ⊗ − − −1 P.3,P.5 T T dx = A˜ (In B)dg. (21) = A˜ ˆl + (I (BFˆ) )ˆl ⇒ − ⊗ n ⊗ (26) ˆ −1 ˆT ˆ ˆ T ˜−T ˆ −1 ˆT ˆ Note that due to Assumption A4, A˜ is invertible. Taking the = (X F )f + (In (BF ) )A (X F )f ⊗ ⊗ ⊗ differential of (18b) and vectorizing, we get P.=5 (I FˆT)Z(F,ˆ Xˆ)fˆ = 0. n ⊗ P.10 dF = dGX−1 GX−1 dXX−1 (22) ˆ − Thus, L is the right eigenvector matrix and the proof is F complete. P.3,P.4 −T −T df = (X Im|)dg{z } (X F )dx ⇒ ⊗ − ⊗ Using Lemma IV.1, we next present a steepest/Newton (21) −T −T −1 = [(X Im) + (X F )A˜ (In B)] dg. (23) descent algorithm to solve the non-sparse MGEAP (18) [42]. ⊗ ⊗ ⊗ In the algorithms presented in this section, we interchangeably P.=5 ZT(F,X) use the the matrices (G, F, X) and their respective vector- | {z } izations (g, f, x). The conversion of a matrix to the vector P.2 1 T T The differential of cost J = 2 f f is given by dJ=f df. (and vice-versa) is not specifically stated in the steps of the Using (23) and P.12, we get (19). To derive the Hessian, algorithms and is assumed wherever necessary. we compute the second-order differentials of the involved variables. Note that since g(G) is an independent variable, Algorithm 1: Non-sparse solution to the MGEAP 2 2 d g = 0(d G = 0) [36]. Further, the second-order differential Input: A, B, Λ,G . 2 0 of (18a) yields d X = 0. Rewriting (22) as dF X = dG Output: Local minimum (F,ˆ Xˆ) of (18). F dX, and taking its differential and vectorization, we get − Initialize: G ,X Solution of (18a), F G X−1 0 0 ← 0 ← 0 0 (d2F )X + (dF )(dX) = (dF )(dX) repeat − 2 −1 1 α Compute step size (see below); d F = 2(dF )(dX)X ← ⇒ − 2 P.4 g g αZ(F,X)f or; 2 −T ← − − d f = 2(X dF )dx. (24) 3 g g α[H(F,X) + V (F,X)] 1Z(F,X)f; ⇒ − ⊗ ← − Taking the second-order differential of J (and omitting the X Solution of Sylvester equation (18a); ← −1 parameter dependence notation), we get F GX ; until convergence;← d2J = (df)Tdf + f T(d2f) = (df)Tdf + (d2f)Tf return (F,X) (23),(24),P.5 = dgTZZTdg 2dxT (X−1 (dF )T)f − ⊗ Steps 2 and 3 of Algorithm 1 represent the steepest P.5 = vec((dF )TFX−T) and (damped) Newton descent steps, respectively. Since, in P.=5dgTZZTdg 2dxT(|X−1F T{z I )T} df general, the Hessian H(F,X) is not positive-definite, the − ⊗ n m,n (21),(23) T T Newton descent step may not result in a decrease of the cost. = dg ZZ dg Therefore, we add a Hermitian matrix V (F,X) to the Hessian + 2dgT(I BT)A˜−T(X−1F T I )T ZTdg . to make it positive definite [42]. We will comment on the n ⊗ ⊗ n m,n T T T choice of V (F,X) in Section V. In step 1, the step size α dg (Z1Z +ZZ1 )dg (25) can be determined by backtracking line search or Armijo’s | {z } rule [42]. For a detailed discussion of the steepest/Newton The Hessian in (20) follows from (25) and P.12 and the proof descent methods, the interested reader is referred to [42]. The is complete. computationally intensive steps in Algorithm 1 are solving the 9
Sylvester equation (18a) and evaluating the inverses of X and Algorithm 2: Approximately-sparse solution to the H + V . MGEAP Next, we present a relaxation of the optimization prob- Input: A, B, Λ, W, G0 lem (2) and a corresponding algorithm that provides Output: Local minimum (F,ˆ Xˆ) of (27). approximately-sparse solutions to the MGEAP. We remove Initialize: G ,X Solution of (18a), F G X−1 the explicit feedback sparsity constraints (2c) and modify the 0 0 0 0 0 repeat ← ← cost function to penalize it when these sparsity constraints are 1 α Update step size; violated. Using the Sylvester equation based parametrization, ← 2 g g αZ(F,X)W¯ f or; the relaxed optimization problem is stated as (while ignoring ← − − 3 g g α[H (F,X) + V (F,X)] 1Z(F,X)W¯ f; the eigenvector normalization constraints; c.f. Remark 4): ← − W W X Solution of Sylvester equation (18a); 1 2 ← −1 min JW = W F F (27) F GX ; G 2 || ◦ || until convergence;← s.t. (18a) and (18b) hold true, return (F,X) where W Rm×n is a weighing matrix that penalizes the cost for violation∈ of sparsity constraints, and is given by Lemma IV.4. (Feasibility of (F, X)) An invertible matrix X ¯ n×n ∈ 1 if Fij = 1, and R satisfies (2a) and (2c) if and only if Wij = ¯ ( 1 if Fij = 0. a˜(x) (B˜(X)) where, (30) ∈ R ˜ ˜ T ¯ As the penalty weights of W corresponding to the sparse a˜(x) , Ax, B(X) , (X B)PF¯,PF¯ , diag(vec(F)). entries of F increase, an optimal solution of (27) becomes − ⊗ Further, if holds true, then the set of sparse feedback more sparse and approaches towards the optimal solution of (30) matrices that satisfy and is given by (2). Note that the relaxed problem (27) corresponds closely to (2a) (2c) ˜ mn the non-sparse MGEAP (18). Thus, we use a similar gradient X = PF¯fns : B(X)fns =a ˜(x), fns R . (31) based approach to obtain its solution. F { ∈ } Proof. Any feedback f which satisfies the sparsity constraint mn Lemma IV.3. (Gradient and Hessian of JW) The gradient (2c) can be written as f = PF¯fns where fns R is a 3 ∈ and Hessian of the cost JW in (27) with respect to g is given non-sparse vector . Vectorizing (2a) using P.3 and P.4, and by substituting f = PF¯fns, we get dJW ˜ T = Z(F,X)W¯ f, (28) Ax = (X B)PF¯fns, (32) dg − ⊗ from which (30) and (31) follow. d2J W H (F,X) = Z(F,X)WZ¯ T(F,X) d2g , W Based on Lemma IV.4, we develop a heuristic iterative T T algorithm to obtain a sparse solution to the EAP. The algorithm +Z1,W (F,X)Z (F,X) + Z(F,X)Z1,W (F,X), (29) starts with a non-sparse EAP solution (F ,X ) that does not where W¯ diag(vec(W W )) and, 0 0 , ◦ satisfy (2c) and (30). Then, it takes repeated projections of T ˜−T −1 T Z1,W (F,X) ,(In B )A (X (W W F ) In)Tm,n a˜(x) on (B˜(X)) to update X and F , until a sparse solution ⊗ ◦ ◦ ⊗ R Proof. Since the constraints of optimization problems (18) and is obtained. (27) are same, Equations (21)-(24) from Lemma IV.1 also P.2,P.8 Algorithm 3: Sparse solution to EAP hold true for problem (27). Now, 1 vec JW = 2 ( (W ) ¯ T 1 T ¯ T ¯ ◦ Input: A, B, Λ, F,G0, itermax. f) (vec(W ) f) = 2 f W f. Thus, dJW = f W df and 2 ◦T T 2 Output: (F,X) satisfying (2a) and (2c). d JW = (df) W¯ df + f W¯ d f. Using the relation vec(W W F ) = W¯ f, the remainder of the proof is similar to proof◦ Initialize: G , X Solution of (18a), F G X−1, 0 0← ns,0 ← 0 0 of Lemma◦ IV.1. i 0 repeat← Using Lemma IV.3, we next present an algorithm to obtain 1 a˜(x) B˜(X)[B˜(X)]+a˜(x); an approximately-sparse solution to the MGEAP. ←− 2 x A˜ 1a˜(x); The step size rule and modification of the Hessian in ← Algorithm 2 is similar to Algorithm 1. 3 X Normalize X; i ←i + 1 until ←convergence or i > iter ; B. Algorithms for Sparse EAP max return (f X in (31),X) In this subsection, we present two heuristic algorithms to ∈ F obtain a sparse solution to the EAP (i.e. any sparse solution, which is not necessarily minimum-gain). This involves finding In step 1 of Algorithm 3, we update a˜(x) by projecting it on (B˜(X)). Step 2 computes x from a˜(x) using the fact that A˜ a pair (F,X) which satisfies the eigenvalue assignment and R sparsity constraints (2a), (2c). We begin with a result that 3 Since f satisfies (4), it can also be characterized as f = (Imn − + + combines these two constraints. Q Q)fns, and thus P¯F = Imn − Q Q. 10 is invertible (c.f. Assumption A4). Finally, the normalization a comprehensive simulation study in Subsection V-B suggests in step 3 is performed to ensure invertibility of X4. that Algorithm 4 converges in almost all instances. In contrast, Next, we develop a second heuristic algorithm for solv- Algorithm 3 converges in much fewer instances and its con- ing the sparse EAP problem using the non-sparse MGEAP vergence deteriorates considerably as the number of sparsity solution in Algorithm 1. The algorithm starts with a non- constraints increase (see Subsection V-B). sparse EAP solution (F0,X0). In each iteration, it sparsifies ¯ the feedback to obtain f = PF¯fns (or F = F Fns), and then C. Algorithm for sparse MGEAP solves the following non-sparse MGEAP ◦ In this subsection, we present an iterative projected gradient 1 2 algorithm to compute the sparse solutions of the MGEAP in min Fns F F (33) Fns,X 2 || − || (2). The algorithm consists of two loops. The outer loop is same as the non-sparse MGEAP Algorithm 1 (using steepest s.t. (A + BFns)X = XΛ, (33a) descent) with an additional projection step, which constitutes to update Fns that is close to the sparse F . Thus, using the the inner loop. Figure 1 represents one iteration of the algo- heuristics of repeated sparsification of the solution of non- rithm. First, the gradient Z(Fk,Xk)fk is computed at a current sparse MGEAP in (33), the algorithm obtains a sparse solution. point Gk (equivalently (Fk,Xk), where Fk is sparse). Next, ˆ Note that a solution Fns of the optimization problem (33) with the gradient is projected on the tangent plane of the sparsity ˆ ˆ ˆ parameters (A, B, Λ,F ) satisfies Fns = F + Kns, where Kns constraints (4), which is given by is a solution of the optimization problem (18) with parameters T (A + BF,B, Λ). Thus, we can use Algorithm 1 to solve (33). mn d(Qf) F = y R : y = 0 T ∈ dg ( ) Algorithm 4: Projection-based sparse solution to EAP mn T = y R : QZ (F,X)y = 0 . (34) ∈ Input: A, B, Λ, F¯,G0, itermax. From P.14, the projection of the gradient on is Output: (F,X) satisfying (2a) and (2c). TF given by PFk Z(Fk,Xk)fk, where PFk = Imn T + T − Initialize: G0, X0 Solution of (18a), [QZ (Fk,Xk)] [QZ (Fk,Xk)]. Next, a move is made −1 ← Fns,0 G0X0 , i 0 in the direction of the projected gradient to obtain ← ← repeat G (F ,X ). Finally, the orthogonal projection of ¯ ns,k ns,k ns,k F F Fns; is taken on the space of sparsity constraints to obtain ← ◦ Gns,k 1 (Kns,X) Algorithm 1(A + BF,B, Λ); . This orthogonal projection is equivalent ← Gk+1(Fk+1,Xk+1) Fns F + Kns; to solving (33) with sparsity constraints (2c), which in turn is ← i i + 1; equivalent to the original sparse MGEAP (2). Thus, the orthog- ← until convergence or i > itermax; onal projection step is as difficult as the original optimization return (F,X) problem. To address this issue, we use the heuristic Algo- rithm 4 to compute the projections. Although the projections obtained using Algorithm 4 are not necessarily orthogonal, Remark 7. (Comparison of EAP Algorithms 3 and 4) they are typically good (c.f. Remark 7). 1. Projection property: In general, Algorithm 3 results in a sparse EAP solution F that is considerably different from the initial non-sparse Fns,0. In contrast, Algorithm 4 provides a Zkfk sparse solution F that is close to Fns,0. This is due to the fact that Algorithm 4 updates the feedback by solving the optimiza- tion problem (33), which minimizes the deviations between Gns,k successive feedback matrices. Thus, Algorithm 4 provides a PFk Zkfk good (although not necessarily orthogonal) projection of a F T G given non-sparse EAP solution F on the space of sparse k ns,0 Algorithm 4 EAP solutions. Qf =0 Gk+1 2. Complexity: The computational complexity of Algorithm 4 is considerably larger than that of Algorithm 3. This is Fig. 1. A single iteration of Algorithm 5. because Algorithm 4 requires a solution of a non-sparse |{z} MGEAP problem in each iteration. In contrast, Algorithm 3 Algorithm 5 is computationally intensive due to the use only requires projections on the range space of a matrix in of Algorithm 4 in step 1 to compute the projection on the each iteration. Thus, Algorithm 3 is considerably faster as space of sparse matrices. In fact, the computational complexity compared to Algorithm 4. of Algorithm 5 is one order higher than that of non-sparse 3. Convergence: Although we do not formally prove the MGEAP Algorithm 1. However, a way to considerably reduce convergence of heuristic Algorithms 3 and 4) in this paper, the number of iterations of Algorithm 5 is to initialize it using the approximately-sparse solution obtained by Algorithm 2. In 4Since X is not an eigenvector matrix, we compute the eigenvectors from X, normalize them, and then recompute real X from the normalized this case, Algorithm 5 starts near the the local minimum and, eigenvectors. thus, its convergence time reduces considerably. 11
Algorithm 5: Sparse solution to the MGEAP As the iterations of Algorithm 4 progress, the sparse matrix F achieves eigenvalue assignment with increasing accuracy. Input: A, B, Λ, F¯,G0, itermax. Output: Local minimum (F,ˆ Xˆ) of (2). As a result, near the convergence of Algorithm 4, the eigen- values of A + BF and Λ are very close to each other. This Initialize: creates numerical difficulties when Algorithm 1 is used with (F ,X ) Algorithm 4(A, B, Λ, F¯,G , iter ), 0 0 ← 0 max parameters (A + BF,B, Λ) in step 1 (see Assumption A4). G0 F0X0, i 0 To avoid this issue, we run Algorithm 1 using a preliminary repeat← ← feedback Fp, as explained below Assumption A4. α Update step size; ← Finally, for Algorithm 5, we use the following convergence gns g αPF Z(F,X)f; dJ ← − criteria: PF dg < , 0 < 1. We choose stopping X Solution of Sylvester equation (18a); 2 ns −6 −5 ← −1 tolerance typically between 10 and 10 for all the Fns GnsXns ; ← algorithms. 1 (F,X) Algorithm 4(A, B, Λ, F¯,Gns, itermax); G FX←; ← i i + 1; B. Numerical study until ←convergence or ; i > itermax We begin this subsection with the following example: return (F,X) 3.7653 2.1501 0.3120 0.2484 −1.6789− 1.0374 0.5306− 1.3987 A = , 2.1829 2.5142 −1.2275 0.2833 V. SIMULATION STUDIES − − − 13.6811 9.6804 0.5242 2.9554 − − − In this section, we present the implementation details of T 1 1 2 5 1 1 0 0 the algorithms developed in Section IV and provide numerical B = , F¯ = , 1 3 4 2 1 0 1 1 simulations to illustrate their properties. = 2, 1, 0.5 j . S {− − − ± } A. Implementation aspects of the algorithms The eigenvalues of A are Γ(A) = 2, 1, 1 2j . Thus, the feedback F is required to move two{− unstable− ± eigenvalues} In the Newton descent step (step 3) of Algorithm 1, we into the stable region while keeping the other two stable eigen- need to choose (omitting the parameter dependence nota- values fixed. Table I shows the non-sparse, approximately- tion) V such that H + V is positive-definite. We choose sparse and sparse solutions obtained by Algorithms 1, 2 and T T V = δImn Z1Z ZZ1 where, 0 < δ 1. Thus, from 5, respectively, and Figure 2 shows a sample iteration run − − T T (20), we have: H + V = ZZ + Imn. Clearly, ZZ is of these algorithms. Since the number of iterations taken by positive-semidefinite and the small additive term Imn ensures the algorithms to converge depends on their starting points, that H + V is positive-definite. Note that other possible we report the average number of iterations taken over 1000 choices of V also exist. In step 1 of Algorithm 1, we use random starting points. Further, to obtain approximately-sparse the Armijo rule to compute the step size α. Finally, we solutions, we use the weighing matrix with Wij = w if dJ use dg < , 0 < 1 as the convergence criteria F¯ = 1. All the algorithms obtain three local minima, among 2 ij of Algorithm 1. For Algorithm 2, we analogously choose which the first is the global minimum. The second column in T T ˆ ˆ VW = δI mn Z1,W Z ZZ1,W and the same convergence X and L is conjugate of the first column (c.f. Remark 3). It criteria and step− size rule− as Algorithm 1. In both algorithms, can be verified that the non-sparse and sparse solutions satisfy if we encounter a scenario in which the solution X of (18a) the optimality conditions of Theorem III.2. is singular (c.f. paragraph below (18b)), we perturb G slightly For the non-sparse solution, the number of iterations taken such that the new solution is non-singular, and then continue by Algorithm 1 with steepest descent are considerably larger the iterations. We remark that such instances occur extremely than the Newton descent. This is because the steepest descent rarely in our simulations. converges very slowly near a local minimum. Therefore, we For the sparse EAP Algorithm 3, we use the convergence use Newton descent steps in Algorithms 1 and 2. Next, observe ˜ ˜ + criteria eX = [In2 B(X)(B(X)) ]˜a(x) 2 < , 0 < that the entries at the sparsity locations of the local minimum 1. For Algorithmk − 4, we use the convergencek criteria feedbacks obtained by Algorithm 2 have small magnitude. eF = F F¯ F F < , 0 < 1. Thus, the iterations Further, the average number of Newton descent iterations of thesek algorithms− ◦ k stop when x lies in a certain subspace and for convergence and the norm of the feedback obtained of when the sparsity error becomes sufficiently small (within the Algorithm 2 is larger as compared to Algorithm 1. This is specified tolerance), respectively. Further, note that Algorithm because the approximately-sparse optimization problem (27) 4 uses Algorithm 1 in step 1 without specifying an initial is effectively more restricted than its non-sparse counterpart condition G0 for the latter. This is because in step 1, we (18). effectively run Algorithm 1 for multiple initial conditions in Finally, observe that the solutions of Algorithm 5 are sparse. order to capture its global minima. We remark that the capture Note that Algorithm 5 involves the use of projection Algo- of global minima by Algorithm 1 is crucial for convergence rithm 4, which in turn involves running Algorithm 1 multiple of Algorithm 4. times. Thus, for a balanced comparison, we present the total 12
TABLE I 7 COMPARISON OF MGEAP SOLUTIONSBY ALGORITHMS 1, 2 AND 5 6 Non-sparse solutions by Algorithm 1 5 −0.0831 + 0.3288j −0.5053 0.2031 0.1919 − 0.4635j 0.5612 −0.2505 4 Xˆ = xˆ∗ 1 0.1603 + 0.5697j 1 0.5617 0.8441 0.3546 + 0.3965j −0.3379 0.4283 3 −0.5569 + 1.4099j −0.9154 2.5568 2 −0.2967 + 1.0244j ∗ −0.5643 1.7468 Lˆ1 = ˆl −0.0687 + 0.0928j 1 0.0087 0.2722 1 0.2259 − 0.2523j 0.0869 −0.4692 0 −0.1111 −0.1089 −0.0312 −0.4399 Fˆ = , kFˆ k = 0.5580 5 10 15 20 25 1 −0.1774 −0.2072 0.0029 0.1348 1 F 0.3817 −0.3349 0.7280 −0.2109 Fˆ = , kFˆ k = 1.1286 2 −0.0873 −0.4488 −0.4798 −0.0476 2 F Fig. 2. Optimization costs for a sample run of Algorithms 1, 2 and 5 (the algorithms converge to their global minima). 0.3130 2.0160 1.2547 −0.6608 Fˆ = , kFˆ k = 2.7972 3 0.0683 −0.7352 −0.0748 −1.0491 3 F Average # of Steepest/Newton descent iterations = 5402.1/15.5 0 0 Approximately-sparse solutions by Algorithm 2 with w = 30 -2 -2 0.9652 −1.3681 0.0014 −0.0021 -4 -4 Fˆ1 = , kFˆ1kF = 1.8636 0.4350 −0.0023 −0.6746 −0.1594 -6 -6 −1.0599 −1.7036 −0.0013 −0.0071 0 50 100 150 0 50 100 Fˆ = , kFˆ k = 2.0146 2 −0.1702 −0.0057 0.0263 −0.0582 2 F (a) (b) 3.3768 2.2570 0.0410 −0.0098 Fˆ = , kFˆ k = 5.7795 3 −1.4694 −0.0061 0.1242 −3.8379 3 F Fig. 3. Projection and sparsity errors for a sample run of (a) Algorithm 3 Average # of Newton descent iterations = 19.1 and (b) Algorithm 4, respectively. Sparse solutions by Algorithm 5 −0.3370 + 0.3296i −0.4525 0.5168 0.2884 − 0.1698i 0.1764 −0.3007 projection error e and the sparsity error e decrease as the Xˆ = xˆ∗ X F 1 −0.4705 + 0.5066i 1 −0.7478 0.7831 respective algorithms progress. 0.2754 + 0.3347i −0.4527 0.1711 Next, we provide an empirical verification of the conver- 25.2072 + 16.5227i 44.6547 82.3616 11.0347 + 11.8982i 12.5245 35.8266 gence of heuristic Algorithms 3 and 4. Let the sparsity ratio Lˆ = ˆl∗ 1 −12.2600 − 5.8549i 1 −24.5680 −39.1736 (SR) be defined as the ratio of the number of sparse entries to −0.6417 − 2.2060i −0.4383 −3.6464 Number of 00s in F¯ the total number of entries in F (i.e. SR = mn ). 0.9627 −1.3744 0.0000 0.0000 Fˆ = , kFˆ k = 1.8694 We perform 1000 random executions of both the algorithms. 1 0.4409 0.0000 −0.6774 −0.1599 1 F In each execution, n is randomly selected between 4 and 20 −1.0797 −1.7362 0.0000 0.0000 Fˆ = , kFˆ k = 2.0525 2 −0.1677 0.0000 0.0264 −0.0610 2 F and m is randomly selected between 2 and n. Then, matrices (A, B) are randomly generated with appropriate dimensions. 3.4465 2.2568 0.0000 0.0000 Fˆ = , kFˆ k = 6.0866 ¯ 3 −1.8506 −0.0000 0.3207 −4.0679 3 F Next, a binary sparsity pattern matrix F is randomly generated with the number of sparsity entries given by SR mn , Average # of Newton descent iterations: b × c 1. Using random initialization = 8231 where denotes rounding to the next lowest integer. To 2. Using initialization by Algorithm 2 = 715 ensure feasibilityb·c (c.f. Assumption A2 and discussion below), we pick the desired eigenvalue set randomly as follows: we S select a random Fr which satisfies the selected sparsity pattern ¯ number of Newton descent iterations of Algorithm 1 involved F, select = Γ(A + BFr). Finally, we set itermax = 1000 5 S in the execution of Algorithm 5. From Table I, we can and select a random starting point G0(F0,X0), and run both observe that Algorithm 5 involves considerably more Newton algorithms from the same starting point. Let Fsol denote the descent iterations compared to Algorithms 1 and 2, since it feedback solution obtained by Algorithms 3 and 4, respec- involves computationally intensive projection calculations by tively, and let dFsol,F0 Fsol F0 F denote the distance , k − k Algorithm 4. One way to reduce its computation time is to between the starting point F0 and the final solution. initialize is near the local minimum using the approximately sparse solution of Algorithm 2. TABLE II Figure 3 shows a sample run of EAP Algorithms CONVERGENCEPROPERTIESOF EAPALGORITHMS 3 AND 4 −1.0138 0.6851 −0.1163 0.8929 3 and 4 for G0 = −1.8230 −2.2041 −0.1600 0.7293 . SR Algorithm 3 Algorithm 4 Convergence instances = 427 Convergence instances = 997 The sparse feedback obtained by Algorithms 3 and 4 1/4 0.1528 −2.6710 0.0000 0.0000 Average d = 8.47 Average d = 1.28 are F = and F = Fsol,F0 Fsol,F0 −0.8382 0.0000 0.1775 −0.1768 Convergence instances = 220 Convergence instances = 988 4.2595 4.2938 0.0000 0.0000 1/2 −0.2519 0.0000 −2.1258 −1.3991 , respectively. Observe that the Average d = 4.91 Average d = 1.61 Fsol,F0 Fsol,F0 Convergence instances = 53 Convergence instances = 983 5 2/3 Although the number of outer iteration of Algorithm 5 are considerably Average dFsol,F0 = 6.12 Average dFsol,F0 = 1.92 less, for instance, 20 in Figure 2. 13
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