Benchmarks for the Numerical Solution of Algebraic Riccati Equations

Transcription

1 Benchmarks for the Numerical Solution of Algebraic Riccati Equations Peter Benner Alan J. Laub Volker Mehrmann Two collections of benchmark examples are presented for the numerical solution of continuous-time and discrete-time algebraic Riccati equations. These collections may serve for testing purposes in the construction of new numerical methods, but may also be used as a reference set for the comparison of methods. Introduction The solution of algebraic Riccati equations (discrete-time or continuous-time) is an important task in many applications. Among those are the linear-quadratic regulator problem, Kalman filtering, H control, total least-squares problems, canonical factorization, solution of two-point boundary value problems, model reduction problems, and many others. See the monographs [3, 15, 36, 43] and the references therein. Due to its importance in practice, numerous numerical methods that differ with respect to accuracy, numerical stability, efficiency, implementability on parallel computers, etc., have been proposed for the solution of these problems. See [15, 16, 39, 43] and the references therein. Currently further methods are being developed in order to address the needs in large scale problems, the use of modern computer architectures, and also to reflect recent theoretical developments in this area. In such a situation it is extremely important to have a set of examples to test and compare different methods and their implementation. Such a set of benchmark examples should be well analyzed, standardized and freely available to serve the scientific community as well as industrial users. We have collected and implemented such a benchmark collection for the numerical solution of continuous-time algebraic Riccati equations (CARE) of the form 0 = Q + A T X + XA XGX (1) and discrete-time algebraic Riccati equations (DARE) of the form 0 = A T XA X (A T XB + S)(R + B T XB) 1 (B T XA + S T ) + Q. (2) Zentrum für Technomathematik, FB3-Mathematik und Informatik, Universität Bremen, D Bremen, FRG. benner@numerik.uni-bremen.de Supported by Deutsche Forschungsgemeinschaft, research grant Me 790/7-1 Singuläre Steuerungsprobleme. Dean, College of Engineering, 1050 Engineering II University of California, Davis, CA , USA. e- mail:laub@ucdavis.edu This author has been supported by National Science Foundation Grant ECS and Air Force Office of Scientific Research Grant F DEF. Fakultät für Mathematik, Technische Universität Chemnitz Zwickau, D Chemnitz, FRG. mehrmann@mathematik.tu-chemnitz.de Supported by Deutsche Forschungsgemeinschaft, research grant Me 790/7-1 Singuläre Steuerungsprobleme. 1

2 Here A, G, Q, X IR n n, B, S IR n m, and R IR m m. The matrices Q = Q T and G = G T may be given in factored form Q = C T QC, G = BR 1 B T with C IR p n, Q = QT IR p p, and B, R as before. The algebraic Riccati equations (ARE) (1) and (2) usually have many solutions but in applications often a particular solution is required. The properties of the algebraic Riccati equations arising in the several areas mentioned above and of the required solutions differ slightly, but the exisiting numerical methods can usually be used to solve the AREs in all these problem classes as they are mostly oriented towards the forms given in (1) and (2) rather than the actual physical background. One of the most important applications of (1) is the solution of linear-quadratic control problems. As the properties (both theoretical and numerical) of AREs and their solution methods can most appropriately be described using this mathematical model, we focus here on this problem class. In the continuous-time case, the autonomous linear-quadratic optimal control problem is given by: Minimize J(x 0, u) = 1 ( ) y(t) T Qy(t) + u(t) T Ru(t) dt (3) 2 subject to the dynamics 0 ẋ(t) = Ax(t) + Bu(t), x(0) = x 0, (4) y(t) = Cx(t). (5) Under some typical controllability or stabilizability assumptions, the solution of the optimal control problem (3) (5) is given by the feedback law u(t) = R 1 B T Xx(t), (6) where X is the unique stabilizing, positive semidefinite solution of (1) (see e.g. [43, 60]). Analogously the DARE (2) arises from a discrete-time linear-quadratic control problem: Minimize J(x 0, u) = 1 2 subject to the difference equation (yk T Qy ) k + 2x T k Su k + u T k Ru k k=0 x k+1 = Ax k + Bu k, k = 0, 1,..., x 0 = ξ, (8) y k = Cx k, k = 0, 1,.... (9) (7) Again, under suitable assumptions the solution of the optimal control problem (7) (9) is given by the feedback law u k = (R + B T XB) 1 (A T XB + S) T x k, k = 0, 1,..., where X is the unique stabilizing positive semidefinite solution of (2) (see, e.g., [43, 55]). 2

3 Most of the examples contained in the benchmark collection arise from such linear-quadratic optimal control problems. It will be pointed out whenever we give an example with a different physical background. One common approach to solve (1) or (2) is to compute the Lagrange invariant subspaces of the Hamiltonian matrix H, constructed from the coefficients [ ] [ ] A G A BR 1 B T H = Q A T = C T QC A T IR 2n 2n, in the continuous-time case or the symplectic pencil [ ] [  0 In G L λm = ˆQ λ I n 0  T ]. (10) in the discrete time case, respectively, where  := A B T R 1 S, ˆQ := Q S T R 1 S. In the discrete-time case, if  is invertible, the symplectic pencil is equivalent to the symplectic matrices [ ] Z = M 1  + G  L = T ˆQ G  T, (11) or Z = L 1 M = [  T ˆQ  1 ˆQ 1  T  1 G ˆQ 1 G + ÂT ]. (12) Actually, H and L λm, Z are the system matrices of the Euler-Lagrange equations of the optimal control problems, see [36, 43]. In the continuous-time case, the assumption that R is nonsingular is necessary as otherwise, the CARE (1) can not be formulated. (The optimal control problem (3) (5) may still have a solution, though.) On the other hand, in the discrete-time case, this assumption is not necessary and it is not uncommon for R being singular, e.g., in stochastic realization problems where R represents the measurement noise covariance matrix. In that case, the symplectic pencil (10) cannot be formed, but the DARE has a close relationship to the extended symplectic pencil (ESP) L λ M = A 0 B Q I n S S T 0 R λ I n A T 0 0 B T 0. (13) The relationship between the Riccati equations and the Lagrange invariant subspaces is as follows. Consider the invariant subspace corresponding to the eigenvalues of H in the open left half plane, or the deflating subspace of the symplectic pencil L λm (the ESP L λ M) corresponding to the eigenvalues inside the open unit disk, respectively. If this subspace is [ ] T [ ] T spanned by U1 T U2 T (or U1 T U2 T U3 T for the ESP (13)) and U1 is invertible, then X = U 2 U 1 1 (14) 3

4 is the stabilizing solution of (1) or (2), respectively, e.g. [16, 38, 39, 43, 60]. All other solutions are obtained via different Lagrange invariant (deflating) subspaces, too. At this point it should be noted that it is possible to transform a continuous-time algebraic Riccati equation (CARE) into a DARE (and vice versa) via a (generalized) Cayley transformation, i.e., the Hamiltonian matrix corresponding to the CARE is transformed into a symplectic matrix/pencil. From this symplectic pencil it is possible to derive the coefficient matrices of a corresponding DARE under certain regularity assumptions; see [44]. In this way, it is possible to create a DARE example from a CARE example and vice versa. We do not use this approach here, though, and restrict ourselves to data arising naturally in a discrete-time setting and/or highlighting some of the properties of discrete-time algebraic Riccati equations. In the sequel, we denote by σ(a) the set of eigenvalues or spectrum of a matrix A. The spectral norm of a matrix is given by A = max{ λ : λ σ(a T A)} and the given matrix condition numbers are based upon the spectral norm, κ(a) = A A 1, A IR l l. All numerical values of norms and condition numbers were computed by the MATLAB functions norm and cond. In the testing of numerical methods it is not only important to know the exact solution of the problem, but also the sensitivity of the problem under small perturbations (such as roundoff errors), measured by the condition number of the problem. The right condition number of algebraic Riccati equations is still an open problem. In the continuous-time case one usually refers to the condition number which was introduced by Arnold [4], Arnold and Laub [5], and Byers [17], and the estimates and bounds given by Kenney and Hewer [31]. Let Ă, Ğ, and Q be real n n matrices near A, G, and Q with respect to the 2-norm and in addition, assume G, Q to be symmetric positive semidefinite. (Ă, Ğ, Q may be considered as perturbed input data.) Define A = Ă A, G = Ğ G, and Q = Q Q. Then denoting the stabilizing solution of the CARE (1) by X, the CARE condition number presented in [17, 31] is defined by { X K CARE = lim sup δ 0 δ X } : A δ A, G δ G, Q δ Q. Let Z i, i = 0, 1, 2, be the solutions of the Lyapunov equations and define (A GX) T Z i + Z i (A GX) = X i, i = 0, 1, 2, (15) K U = Z 0 Q + 2 Z 0 Z 2 A + Z 2 G, X (16) K L = Z 0 Q + 2 Z 1 A + Z 2 G. X (17) 4

5 Then one can prove (see [31]) that 1 3 K L K CARE K U. In most cases K L is close to K U, i.e., this provides a very good approximation to K CARE. (Note that in [31] a more refined lower bound is given which in some cases is closer to K U.) Since for the analysis of the conditioning the upper bound is more important and since the bounds are almost always very close, we only give the upper bound K U as an approximation for K CARE. In the discrete-time case the situation is similar, see, e.g., [25, 34, 50, 57]. For simplicity, we use here the condition number proposed in [25]. This condition number measures the sensitivity of the stabilizing DARE solution with respect to first-order perturbations by means of the Fréchet derivative of the DARE at X. In [25] it is shown that, assuming Q 0, R > 0, such a condition number is given by K DARE = [ Z 1, Z 2, Z 3 ] X F, (18) where ( ) Z 1 = A F P 1 I n Fd T X + (Fd T X I n )T, (19) Z 2 = G F P 1 ( Â T X(I n + GX) 1 ÂT X(I n + GX) 1), (20) Z 3 = Q F P 1. (21) Here, denoting the jth unit vector by e j, the permutation matrix T is defined by T = n i,j=1 e i e T j e j e T i, where denotes the Kronecker product. Furthermore, P is the matrix representation of the Stein (discrete Lyapunov) operator and Ω(Z) = Z F T d ZF d, F d = Â (R + BT XB) 1 B T XÂ is the feedback matrix in the discrete-time case. We will give the condition numbers K CARE and K DARE for each of the benchmark examples together with some other relevant values that are important in order to check the behaviour of a Riccati solver under certain circumstances. Acquiring the Benchmark Collections All presented examples of the collections for the CARE (or DARE) may be generated by FORTRAN 77 subroutines CAREX.F (or DAREX.F) and the MATLAB 1 functions carex.m (or darex.m), respectively. 1 MATLAB is a trademark of The MathWorks, Inc. 5

6 A detailed description of all the examples and the data, prologs of the subroutines and further reference tables is given in the two technical reports [11, 12]. The FORTRAN 77 codes can be obtained via netlib (e.g., at the ftp sites netlib.att.com, elib.zib-berlin.de, or any netlib mirror site) from the directory netlib/control. The MATLAB codes are available at the The MathWorks, Inc. ftp site, ftp.mathworks.com and its mirror sites from the directory control/carex (CARE benchmark collection) and control/darex (DARE benchmark collection). In the future it is planned to extend the collections whenever new and interesting examples are available. The authors appreciate any comments regarding the collected examples or suggestions for new examples. Benchmark Collections The benchmark collection for both cases is split into four parts: parameter-free examples of fixed dimension, parameter-dependent problems of fixed size, parameter-free examples of scalable size, and scalable examples with parameters. For all parameters needed in the examples there exist default values that are also given in the tables. These default values are chosen in such a way that the collection of examples can be used as a testset for the comparison of methods. For each continuous-time example, we provide norms and condition numbers of the stabilizing solution X and the Hamiltonian matrix H. One possible source for loss of accuracy in the numerical solution of AREs is the condition number of U 1 in (14). If the columns of [U T 1, U T 2 ]T are orthonormal, then it can be shown (see [32, 25]) that κ(u 1 ) 1 + X. We do not show κ(u 1 ) in the tables as it is not given uniquely and may be different if U 1 is computed by different methods. But X may be used as an indicator for κ(u 1 ) due to the bound given above and moreover, if U 2 is invertible, then κ(u 1 ) κ(x)/κ(u 2 ) such that a large κ(x) may also signal a large κ(u 1 ). A large condition number of H may indicate that one can expect problems with methods like the sign function method [18, 23, 53], that depend on inversions of H. On the other hand, a large condition number may also be due to a large norm of H rather than to ill conditioning with respect to inversion. We also give the absolute value of the real part of the eigenvalue of H closest to the imaginary axis and denote it by where λ re min := min{ Re(λ) : λ σ(f c ) }, F c := A BR 1 B T X is the closed-loop matrix in the continuous-time case. Note that the eigenvalues of F c are the eigenvalues of H in the open left half plane. (In some applications one is also interested in Riccati solutions if H has eigenvalues on the imaginary axis; in that case, obviously λ re min = 0.) A small λre min indicates eigenvalues close to te imaginary axis. Most of the numerical methods for the solution of CAREs have problems handling such eigenvalues. In particular, methods that do not respect the special structure of the problem may fail as small perturbations can cause eigenvalues with small real part to cross the imaginary axis; see, e.g., [59]. For computing the system properties we use the exact stabilizing solution if an analytical solution is available. Otherwise, we computed approximations by the multishift algorithm [2] 6

7 and the Schur vector method [38]. If possible, these approximations were refined by Newton s method [33] possibly combined with an exact line search [9, 10] to achieve the highest possible accuracy. We then chose the approximate solution with smallest residual norm to compute the properties of the example. (Note that this is not necessarily the most accurate solution!) In one example we used the sign function method to compute a first approximation to the solution which was then refined by Newton s method combined with exact line search. For each discrete-time example, we provide the condition number κ(â) which shows if the symplectic matrix Z in (11) can be formed safely (though it is still preferable to use the pencil approach and thereby avoid a matrix inversion unless  1 is easy to form). The column λ C max indicates the absolute value of the closed-loop eigenvalue of largest modulus, i.e., where λ C max := max{ λ : λ σ(f d ) } F d := A B(R + B T XB) 1 (B T XA + S T ) is the feedback matrix in the discrete-time case. The eigenvalues of F d are the (generalized) eigenvalues of the symplectic matrix (pencils) in (10), (11) inside the unit circle. Further, we give norms and condition numbers of the stabilizing solution X. The remarks concerning the correspondence of κ(u 1 ) and X, κ(x) do also apply here as well as the discussion of λ re min if we replace the imaginary axis by the unit circle and λ re min by λc max. For examples without an analytical solution available, we computed approximations by the generalized Schur vector method [5, 46]. If possible, these approximations were refined by Newton s method [5, 27, 43] to achieve the highest possible accuracy. We then chose the approximate solution with smallest residual norm and recomputed the solution using the optimal scaling strategy proposed in [25]. This computed solution was then used to determine the properties of the example. We do not describe or investigate the properties of the given examples beyond the ARE setting as our focus is the numerical solution of AREs. For those examples that have an actual control-theoretic background we refer the reader to the original sources for properties, problems, and significance of the given data in control-theoretic terms. We also want to emphasize that part of the examples, are contrived to highlight some of the properties of AREs and the numerical methods to solve them. These may be pathological examples from the point of view of control engineering, but in developing numerical methods it is also of importance to study the behaviour of an algorithm in extreme situations in order to derive a robust solution method and to detect possible limitations of the applicability of the algorithm. Reference Tables and Special Features The description of each example in the collection contains a table with some of the system properties. We refrain here from describing in detail the whole collection but instead we will point out important features of the examples. For the actual data, please consult the technical reports [11] and [12]. The following values describing the examples are given in the tables: n the order of the Riccati equation, i.e., X IR n n ; 7

8 m interpreting the data as data from a linear-quadratic optimal control problem, m denotes the number of inputs of the system; p interpreting the data as data from a linear-quadratic optimal control problem, p denotes the number of outputs of the system; default default setting for the parameters of the examples; X availability of an analytical solution ( yes = +, no = ); X 2-norm of the stabilizing solution of the CARE/DARE; κ(x) 2-norm condition number of the stabilizing solution of the CARE/DARE; H 2-norm of the Hamiltonian matrix corresponding to the CARE; κ(h) 2-norm condition number of the Hamiltonian matrix corresponding to the stabilizing solution of the CARE; κ(a) 2-norm condition number of the state matrix A or Â, respectively; λ re min absolute value of the real part of eigenvalue(s) of H closest to the imaginary axis; λ C max absolute value of the eigenvalue of the (extended) symplectic matrix/pencil inside the unit disk closest to the unit circle; K U upper bound for the CARE condition number as given in (16); K DARE DARE condition number given in (18). Moreover, indicates that a value cannot be computed (e.g., if the data has properties such that the corresponding magnitude is not defined); the value for a matrix condition number indicates that the matrix is singular. Besides, we use the notation a (b) to denote the real number a 10 b in order to save some space in the tables. Parameter-free problems of fixed size CARE examples no. n m p X λ re min H κ(h) X κ(x) K U (3) (3) (3) (8) 1.5 (10) 3.6 (3) 3.7 (9) Table 1: CARE examples with fixed size without parameters. 1. Source: [38, Example 1]. This example can be used for a first verification of any solver for (1), since the solution may be computed by hand. There are no particular properties or difficulties. 2. Source: [38, Example 2]. The underlying system is stabilizable-detectable, but uncontrollable-unobservable. 8

9 3. Source: [8]. Here the system matrices describe a mathematical model of an L-1011 aircraft. The constant coefficient Q in (1) has one negative, real eigenvalue of order O(10 3 ) although from theory, Q is supposed to be positive semidefinite. This may reflect a small perturbation in the input data. The computed stabilizing solution is nevertheless positive definite with eigenvalues greater than one. 4. Source: [13]. The data come from a mathematical model of a binary distillation column with condenser, reboiler, and nine plates. The constant term Q of (1) is indefinite and the computed stabilizing solution is indefinite, too. 5. Source: [48]. This is the data for a 9th-order continuous state space model of a tubular ammonia reactor. It should be noted that the underlying model includes a disturbance term which is neglected in this context. 6. Source: [20]. This example is a well-studied problem in control theory. The data come from a control problem for a J 100 jet engine as a special case of a multivariable servomechanism problem. In the cost functional of the linear-quadratic optimal control problem as given in (3), R and Q are chosen as identities of appropriate size such that G = BB T, Q = C T C. The data of this example differ by 10 orders of magnitude and the norm and condition number of H are quite large. The matrix H can be considered as badly balanced with respect to balancing unsymmetric matrices as proposed in [47]. The eigenvalues of H vary in magnitude from O(10 1 ) to O(10 2 ). From the structure of the control problem it is clear that the Hamiltonian matrix corresponding to (1) has isolated eigenvalues. These are given by eigenvalues at ±20 with algebraic multiplicity three and at ±33.3 with algebraic multiplicity one. DARE examples 1. Source: [38, Example 2]. This is an example of stabilizable-detectable, but uncontrollable-unobservable data. 2. Source: [38, Example 3], [35, Example 6.15]. This example illustrates a simple discrete-time linear-quadratic optimal control problem as defined by (7) (9). 3. Source: [58, Example II]. In this example, the matrix R in (2) is singular and therefore, the symplectic pencil (10) cannot be formed. In order to solve the DARE by a deflating subspace method, we thus have to work with the extended symplectic pencil as given in (13). Due to the singularity of R, the condition number K DARE is not defined here (represented by in the table). This example can be used, e.g., as a first test of any solver to deal with a singular weighting/measurement noise covariance matrix. 9

10 no. n m p X κ(a) λ C max X κ(x) K DARE ( 2) (3) (12) 5.1 (4) (6) Table 2: DARE examples with fixed size without parameters. 4. Source: [29]. This is another example with a singular R matrix. Furthermore, we have a nonzero S matrix. Again, the DARE condition number K DARE can not be computed due to the singular matrix R. 5. Source: [30], [44, Example 2]. This example shows one of the major differences between the properties of continuoustime algebraic Riccati equations and their discrete counterparts. This DARE has only two solutions, one positive definite and one indefinite. In the case of a continuous-time system, controllability ensures the existence of a positive semidefinite and a negative semidefinite solution. But here, despite the fact that (A, B) is controllable, no negative semidefinite solution exists. See the analysis in [44] for details. The stabilizing solution in the control-theoretic sense is the positive definite solution. 6. Source: [1]. The data of this example represent a very simple control problem for a satellite. The system is given by equations describing the small angle attitude variations about the roll and yaw axes of a satellite in circular orbit. These equations originally form a second-order differential equation. A first-order realization of this model and sampling every 0.1 seconds yields a standard system as given in (8) (9). 7. Source: [41]. This is a simple example of a control system having slow and fast modes. One complex conjugate pair of the computed closed-loop eigenvalues is located on a circle with radius 0.99 around the origin, i.e., relatively close to the unit circle. Requiring that this distance should not cause problems for any DARE solver seems to be reasonable. 8. Source: [42, Example 4.3]. 10

11 All the generalized eigenvalues of the corresponding symplectic pencil L λm are real. The distance of the largest closed-loop eigenvalue from the unit circle is of order The problem is designed so that κ(l + M) = O(10 11 ). Due to this large condition number and the eigenvalues close to the unit circle, problems with the convergence of the iteration process are to be expected when the DARE is solved via a method based on the sign function iteration (e.g., the method presented in [23]). Note that K DARE signals only a very mild ill-conditioning of the DARE. 9. Source: [22, Section 2.7.4]. The fifth-order linearized state-space model of a chemical plant presented in [24, 56] is discretized by sampling every 0.5 seconds, yielding a discrete-time linear-quadratic control problem of the form (7) (9). 10. Source: [21]. The data for this example were obtained from a slightly more general dynamical system than the one given in (8) (9) as the output depends explicitly upon the control, i.e., y(t) = Cx(t) + Du(t), t 0. The corresponding optimal control problem can be solved using the DARE (2) by setting Q = C T QC, R = R + D T QD, and S = C T QD. The system properties are benign. The system can easily be transformed to a standard system as in (10). Therefore, this example is helpful for a first verification of any DARE solver based on the extended formulation given in (13), since the results can be compared to those obtained by any other solver based on the formulation by the symplectic pencil (10). 11. Source: [48]. This is the data for a 9th-order discrete state-space model of a tubular ammonia reactor. It should be noted that the underlying model includes a disturbance term which is neglected in this context. The continuous state-space model of this problem yields the data of CARE Example 5. The discrete-time model is obtained by sampling every 30 seconds. In the discrete model, only the first and fifth state variables are used as outputs. Besides, the weighting matrices in (7) differ from the ones chosen for the continuous-time problem. Parameter-dependent problems of fixed size CARE examples 7. Source: [5, Example 1]. Here, the matrix B in (4) and thus the matrix G in (1) depend upon a parameter ε. As ε 0, the matrix pair (A, B) becomes unstabilizable and the norm of the stabilizing solution X tends to. Condition esitmators in numerical methods for computing the stable invariant subspace of H yield estimates for U 1 of order 1/ε 2. This is in accordance with the theoretical bound κ(x) 8/ε 2. 11

12 no. n m p default X λ re min H κ(h) X κ(x) K U ε = (12) 8.0 (12) ε = (6) 1.5 (8) 9.3 (3) 9.4 (6) 6.7 (9) ε = (2) 1.0 (6) 1.0 (6) 1.4 (3) 2.0 (6) 8.7 (5) ε = ( 7) (14) (7) 3.8 (3) ε = ε = (6) 3.5 (6) (12) ε = (12) 5.7 (13) (8) 4.1 (13) ε = ( 13) (13) Table 3: CARE examples with fixed size and parameters. 8. Source: [5, Example 3]. This is an example with increasingly ill-conditioned control weighting matrix R. (R has a parameter ε.) If ε < 1, then κ(r) = O(1/ε) and as ε 0, the elements of G = BR 1 B T become increasingly large such that the CARE becomes increasingly ill balanced in the sense of [52]. 9. Source: [32, Example 2] In this example, the matrix A contains a parameter ε. As ε grows, X increases like ε and the Riccati equation becomes ill conditioned in terms of KCARE and K U. One of the closed-loop eigenvalues approaches zero as ε 0. In this case, κ(x) = O(1/ε) and κ(h) = O(1/ε 2 ). Hence, this example may be used to test the ability of a CARE solver to deal with bad scaling due to the A matrix and mild ill conditioning (ε + ), as well as with closed-loop eigenvalues close to the imaginary axis and very ill-conditioned Hamiltonian matrices (ε 0). 10. Source: [7]. The coefficient matrices A and Q in (1) depend upon a parameter ε. As ε 0, then H becomes increasingly ill conditioned, i.e., κ(h) = O(1/ε 2 ), whereas κ(x) behaves like 1/ε. Note that for ε = 10 7, then K L = 2.0 which is three orders of magnitude smaller than K U. This shows that K U, K L may sometimes be far apart and thus, K U may overestimate K CARE by orders of magnitude. 11. Source: [28]. This example represents a type of algebraic Riccati equation arising in H control problems as presented, e.g., in [40]. The stabilizing solution X of (1) for ε > 0 is constant and solves (1) for arbitrary ε. In particular, for ε = 0 it is still the solution obtained by an H invariant Lagrangian subspace, i.e., the required solution in the sense of H control. The spectrum of H is {±ε ± j}. Hence the closed-loop eigenvalues approach the imaginary axis as ε 0. Note that for ε = 0, the condition number K CARE as given in [31] is not defined. 12

13 12. Source: [52]. Here, all three coefficient matrices of the CARE (1) depend upon a parameter ε. For growing ε, the CARE becomes more and more badly scaled which leads to a significant loss of accuracy in all CARE solvers based on eigenvalue methods. This demonstrates the need to use an appropriate scaling as proposed in [32, 45]. 13. Source: [19]. The data of this example describe a magnetic tape control problem. The matrices A and B in the underlying system contain a parameter ε. As ε 0, the pair (A, B) becomes unstabilizable and the condition numbers κ(x), κ(h), and K CARE (as measured by K L and K U ) increase. The CARE (1) and the Hamiltonian matrix H become very badly scaled. 14. Source: [5, Example 2]. In this example, the data can be interpreted as a single input/single output linearquadratic optimal control problem of the form (3) (5). The matrix A depends upon a parameter ε. As ε 0, a pair of complex conjugate eigenvalues of the Hamiltonian matrix H approaches the imaginary axis (λ ± ε2 2 ±ı, where ı = 1), (A, B) gets close to an unstabilizable system, and the CARE becomes fairly ill conditioned as measured by K U. DARE Examples no. n m p default X κ(a) λ C max X κ(x) K DARE ε = (12) 1.0 (12) ε = (6) τ = 10 8, D = 1.0 K = 1.0, r = (7) 3.1 (7) 1.8 (8) Table 4: DARE examples with fixed size and parameters. 12. Source: modified from [25, Example 2]. Here, the matrix A has a parameter ε. As ε, this becomes an example of a DARE which is badly scaled in the sense of [51] due to the fact that A F G F, Q F. The norm (and condition) of the stabilizing solution grow like ε 2 whereas the DARE condition number K DARE remains constant. 13. Source: [51]. In this example, the blocks G and Q of the symplectic pencil in (10) contain a parameter ε. For growing ε, the symplectic pencil becomes more and more badly scaled which leads to a significant loss of accuracy in all DARE solvers based on eigenvalue methods. This demonstrates the need to use an appropriate scaling as proposed in [25]. 14. Source: [14, 46] The underlying dynamical system describes a very simple process control of a paper 13

14 machine. The continuous-time model with time delay is sampled at intervals of length D, yielding a singular transition matrix A. The time delay is equal to the length of three sampling intervals. The other parameters defining the system are a first-order time constant τ and the steady-state gain K. Due to the variety of parameters in this example, it is possible to investigate DAREs with critical properties in many aspects. Since these properties merely rely on τ and r, these effects can be produced by keeping K and D constant and varying τ (and r). For τ D, K the largest closed-loop eigenvalue approaches the unit circle. For τ D, K the norm and condition of the stabilizing solution X become large and the DARE becomes ill conditioned with respect to K DARE. Examples of scalable size without parameters CARE examples no. n m p default X λ re min H κ(h) X κ(x) K U 15 2N 1 N N 1 N = n n n n = Table 5: Scalable CARE examples without parameters. 15. Source: [38, Example 4], [6]. The matrices presented here describe a mathematical model of position and velocity control for a string of high-speed vehicles. (This problem is also known as smart highway or intelligent highway.) If N vehicles are to be controlled, the size of the system matrices will be n = 2N 1. The stabilizing solution is singular (rank(x) = n 1). The system does not have any particular bad properties for growing n. All condition numbers only grow very mildly. The closed-loop eigenvalues are all of magnitude O(1). Hence, this example is particularly well suited for testing how an algorithm behaves when the dimension of the problem increases. Moreover, the corresponding Hamiltonian matrix is very sparse and can therefore also be used to test numerical methods for large and sparse Hamiltonian matrices. 16. Source: [38, Example 5]. In this example, all system matrices and the solution of (1) are circulant. Most eigenvalues of the Hamiltonian matrix have algebraic multiplicity 2 (but all eigenvalues have geometric multiplicity 1). Growth of the problem size n does not influence norms and condition numbers. All the closed-loop modes λ are real and of magnitude O(1). Therefore, this example is perfectly suited to test the behavior of algorithms for growing problem size. The CARE can be solved using an inverse discrete Fourier transformation and the theory of circulant matrices such that an exact solution is available. 14

15 Parameter-dependent examples of scalable size CARE examples no. n m p default X λ re min H κ(h) X κ(x) K U 17 n 1 1 n = 21 q = r = n 1 1 n = 100, a = 0.01 b = c = 1.0 β 1 = γ 1 = 0.2 β 2 = γ 2 = l 2 2l l = 30, κ = 1.0 µ = δ = ( 2) (9) 1.3 (9) (3) 1.4 (5) 7.1 ( 4) 1.0 (4) 6.2 ( 3) (5) (3) 20 2l 1 l l l = ( 2) 4.1 (11) 5.5 (14) 7.2 (7) 1.4 (22) 3.0 (8) Table 6: Scalable CARE examples with parameters. 17. Source: [38, Example 6]. This example describes a system of n integrators connected in series and a feedback controller is supposed to be applied to the nth system. The resulting dynamics are given by a single input/single output system. (For more details about the physical background see [38].) The eigenvalues of the Hamiltonian matrix are the roots of λ 2n + ( 1) n qr = 0, where in the cost functional in (7), we have Q = q and R = r, q, r IR. It is known that x 1n = qr (note the correction from [38]). Therefore, the relative error in x 1n, i.e., x 1n qr, may be used as an indicator of the accuracy of the results. qr The difficulty in this example lies in the fact that U 1 becomes extremely ill-conditioned with respect to inversion as n increases and the elements of X become very large in magnitude. Observe that the condition number of U 1 increases for increasing values of q and r whereas K U remains constant. This reflects the fact that both values may (or may not) signal some kind of ill conditioning of the CARE. 18. Source: [54]. The data of this example come from a linear-quadratic control problem of one-dimensional heat flow. This problem is described in terms of infinite-dimensional operators on a Hilbert space and the optimal control can be obtained solving an operator Riccati equation. Using a standard finite element approach based on linear B splines, a finitedimensional approximation to the problem may be obtained by the solution of algebraic Riccati equations (1). The order of the CARE increases when reducing the meshsize in the grid for the underlying approximation scheme. The other parameters are used to define the operators in the underlying infinite-dimensional mathematical model. 15

16 19. Source: modified from [26, Example 3]. The data in this example come from a stabilization/optimal control problem for a second-order system which is given by a model of a string consisting of coupled springs, dashpots, and masses. The inputs are two forces, one of them at each side of the string. The parameters in this example are the number of springs and some constants defining the physical properties of the system. 20. Source: modified from [49]. As in Example 19, the underlying mechanical system is described by a second-order ordinary differential equation and an output equation which is transformed to a standard, first-order system as in (4) (5). The data come from a problem arising in power plants. The resulting system is neither observable nor detectable. We may overcome this problem by eliminating the unobservable state variable, see [11] for details. The weighting matrix Q in the cost functional (3) is chosen to normalize the rows of C, i.e., Q = W T C W C, where W C is a diagonal weighting matrix such that the rows of W C C have unit length. The control weighting matrix R is chosen as identity. As default values we use data provided by [37] corresponding to a generator axle in a hydroelectric power plant. For the default data, the Hamiltonian matrix has a very large norm and condition number despite the scaling of the output matrix. (Without the scaling corresponding to W C, these values are even larger by about 10 orders of magnitude.) This is due to the large entries in A. The reference solution was computed by the sign function method as proposed in [18] where the defect correction was performed using Newton s method combined with exact line search [10]. Due to the bad scaling of this example, it was necessary to scale the Lyapunov equation (15) by A when computing K U. DARE examples no. n m p default X κ(a) λ C max X κ(x) K DARE 15 n 1 n n = 100, r = Table 7: Scalable DARE examples with parameters. 15. Source: modified from [46, Example 3]. In this example, the closed-loop eigenvalues are all zero, that is, the spectrum of the symplectic pencil L λm in (10) is given by the generalized eigenvalues λ 1 =... = λ n = 0 and λ n+1 =... = λ 2n =. This example can be used to test any DARE solver for growing dimension of the problem. The DARE condition number K DARE increases only slowly. The stabilizing solution of (2) has the very simple form X = diag (1, 2,..., n) and for any order n of the DARE, X = κ(x) = n. Here, R = r IR. Note that the choice of r does not influence the stabilizing solution X but for r < 1, the condition number K DARE behaves like 1/r. 16

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