An efficient algorithm to compute the real perturbation values of a matrix
|
|
- Doreen Cameron
- 6 years ago
- Views:
Transcription
1 An efficient algorithm to compute the real perturbation values of a matrix Simon Lam and Edward J. Davison 1 Abstract In this paper, an efficient algorithm is presented for solving the nonlinear 1-D optimization problem associated with computing the real perturbation values of an arbitrary complex matrix M C q l. The real perturbation values of a matrix is important in computing the real stability radius, the real controllability radius, the real decentralized fixed-mode radius, etc. of the control literature. This is because obtaining these radii requires solving a one or even two dimensional optimization problem involving real perturbation values. Hence, being able to quickly compute the real perturbation values of a matrix is crucial in such calculations. A numerical example is included to demonstrate the effectiveness of the proposed algorithm. 2 Introduction The real perturbation values of a general complex matrix were first introduced in [1]. From a perturbation theory point of view, real perturbation values are analogous to singular values except it focuses only on real perturbations, which arises in problems such as parametric robustness analysis in control theory, and the computation of real pseudospectra in numerical analysis. As examples of where such perturbation problems can arise, one is often This work has been supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada under grant No. A4396. S. Lam and E. J. Davison is with the Systems Control Group, Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON, Canada M5S 3G4 {simon,ted}@control.utoronto.ca
2 interested in the control literature in studying the robustness of various system properties with respect to parametric perturbations. Consider the following linear time-invariant (LTI) multivariable system ẋ = Ax + Bu y = Cx + Du (1) where x R n, u R m, and y R r are respectively the state, input, and output vectors, and A, B, C, and D are constant matrices with the appropriate dimensions for n 1, m 1, r 1, and max(r, m) n. It is well known that there exists a LTI controller that can assign the eigenvalues of the closed-loop system to any arbitrary spectrum if and only if the system is controllable and observable. However, when a controllable and observable system is subject to parametric perturbations (i.e. A A + A, B B + B, C C + C, and D D + D ), the system may be very close to becoming uncontrollable and/or unobservable. Hence, a continuous controllability (observability) measure, called the controllability (observability) radius [3, 4], is more informative than the traditional yes/no controllability (observability) metric, which simply determines whether a system is controllable (observable) or not. The same can be said about other system properties such as having decentralized fixed modes (DFM) [6], stability [8], minimum-phase [7], etc. Computing the real controllability radius (i.e. when the perturbations are real, A R n n, B R n m, C R r n, and D R r m ) requires optimizing in the complex plane the real perturbation values of a complex matrix. Hence, being able to compute the real perturbation values, which itself is a 1-D optimization problem, will improve the computation of the real controllability radius. The same is also true in computing the real DFM radius [6], the real stability radius [8], the real minimum-phase radius [7], etc. An algorithm is proposed in this paper to efficiently compute the real perturbation values of a matrix. The paper is organized as follows. In Section 3, the definition and formula for computing the real perturbation values of a matrix are reviewed. Then in Section 4, a few useful tools are developed, followed by a thorough description of the proposed algorithm in Section 5. Finally, Section 6 gives a numerical example to demonstrate the effectiveness of the algorithm. 3 Notation and Background In this paper, the field of real and complex numbers are denoted by R and C respectively. The i-th singular value of a matrix M C p m is denoted by σ i (M), where σ 1 (M) σ 2 (M). M denotes the spectral norm of a matrix M and is equal to σ 1 (M). Also, M, M T, and M H denote respectively the complex conjugate, transpose, and complex conjugate transpose of M. Furthermore, M H denotes ( M H) 1. The real and imaginary components of the matrix M are given by Re M and ImM respectively. Finally, the set of generalized eigenvalues for the matrix pair (A, B), where A C n n and B C n n, is denoted by sp(a, B); i.e. Ax = λbx for λ sp(a, B) and for some non-zero x C n. x is a generalized eigenvector of (A, B). The set of real generalized eigenvalues of (A, B) is denoted
3 by sp r (A, B), where sp r (A, B) sp(a, B). 3.1 Real Perturbation Values The following definition is made in [1]. Definition 3.1. Given M C q l, the i-th real perturbation values of the first kind of M are defined as: τ i (M) := 1 inf{ R l q, dimker(i l M) i} (2) and the real perturbation values of the second kind are defined as: where i N. τ i (M) := inf { R q l, rank(m ) < i } (3) The following result from [1] provides two formulas for computing both kinds of real perturbation values. Theorem 3.1 ([1]). Given M C q l and i N, τ i (M) = inf γ (0,1] σ 2i(P(γ, M)) (4) τ i (M) = sup σ 2i 1 (P(γ, M)) (5) γ (0,1] where [ P(γ, M) := Re M γ 1 Im M γ Im M Re M ] (6) Remark 3.1. The function to be minimized in (4) is quasiconvex for i = 1 (see [8]), while the function to be maximized in (5) is quasiconcave for i = min(q, l) (see [4]). Hence, for these two particular cases, the corresponding real perturbation values can be computed efficiently using, for example, golden-section search methods. For the general case, however, the functions to be minimized in (4), or maximized in (5), may have multiple local extrema. Hence, we cannot use such unimodal techniques in general. The algorithm proposed in this paper efficiently computes the real perturbation values of a matrix for the general case. Remark 3.2. Note that if M is a real matrix, then the real perturbation values of M are the same as the singular values of M. To avoid talking about this trivial case, we will assume in the rest of the paper that M is not a purely real matrix; i.e. M has at least one complex element.
4 4 Preliminary Results Before presenting the proposed algorithm for evaluating (4) and (5), the following theorem is first presented, which is used in the development of the algorithm. Theorem 1. Given M C q l and real x > 0, then, ([ ]) Re M γ Im M x σ γ 1 γ2 1 Im M Re M γ 2 sp(a(x, M),B(x, M)) (7) + 1 where σ( ) denotes the set of singular values of ( ) and [ ] 0 M A(x, M) := H M x 2 I M T M x 2 I 0 [ ] M B(x, M) := H M x 2 I 0 0 M T M x 2 I Proof. Let α = γ2 1 γ 2 +1 where for n N,. It can easily be shown (e.g. see [5]) that [ M 0 P(γ, M) = Tα,q 1 0 M [ T α,n := α I n ] T α,l (8) i 1 α I n 1 1+α I n i 1 α I n ] (9) ([ I αi and satisfies T α,n Tα,n H = αi I the following: ]) 1. The rest of the proof then follows from ([ ]) 0 P(γ, M) x σ(p(γ, M)) x sp P(γ, M) H 0 ([ ]) xi 2q P(γ, M) det P(γ, M) H = 0 xi 2l [ ] M 0 xt α,q Tα,q H det [ ] 0 M M H 0 0 M T xt H α,l T 1 α,l det ( xt α,q Tα,q H ) det ( xt H α,l T 1 α,l [ M H 0 ( det x = 0 ] [ ]) ( xtα,q 0 M T Tα,q H ) 1 M 0 = 0 0 M [ ] I αi + 1 [ ]) M H M αm H M αi I x αm T M M T = 0 M det(a(x, M) αb(x, M)) = 0
5 Lemma 2. The generalized eigenvalues of (A(x, M), B(x, M)) in (7) are symmetrical both about the imaginary axis and about the real axis. Proof. [ Let α ] C be a generalized eigenvalue of (A(x, M),B(x, M)), and define 0 I J 1 :=. Then, I 0 det(a(x, M) αb(x, M)) = 0 det ( J 1 (A(x, M) αb(x, M))J1 1 ) = 0 ([ ] [ 0 M det T M x 2 I M M H M x 2 α T M x 2 I 0 I 0 0 M H M x 2 I det(a(x, M) + αb(x, M)) = 0 ]) = 0 Hence, [ α is also ] a generalized eigenvalue of (A(x, M), B(x, M)). Similarly, define 0 I J 2 := and note that I 0 det(a(x, M) αb(x, M)) = 0 det(j 2 (A(x, M) αb(x, M)) J 2 ) = 0 Hence, α is also a generalized eigenvalue of (A(x, M),B(x, M)). Lemma 3. ([2]) Let A(γ) C n n and B(γ) C n n both be analytic symmetric matrix functions on an open set Γ R. Also, let (λ(γ),y(γ)) be a generalized eigenvalue and eigenvector pair such that A(γ) y(γ) = λ(γ) B(γ) y(γ) y(γ) T B(γ)y(γ) = 1 (10a) (10b) Then, dλ(γ) = y(γ) T ( A(γ) ) B(γ) λ(γ) y(γ) (11) Lemma 4. dw(γ) Given M C q l and w(γ) σ(p(γ, M)), then, ( ) γ 2γ 2 1 = w(γ)(γ 2 + 1) 2 y(γ) T γ 2 +1 I I ) y(γ) I I ( γ 2 1 γ 2 +1 ( ) where γ 2 1 γ 2 +1, y(γ) is a generalized eigenvalue and eigenvector pair of (A(w(γ),M), B(w(γ),M)), as defined in (7), such that A(w(γ),M)y(γ) = γ2 1 γ B(w(γ),M)y(γ) y(γ) T B(x, M)y(γ) = 1 1 (12)
6 Proof. Let α(γ) = γ2 1 γ Since w(γ) is a singular value of P(γ, M), then by Theorem 7, α(γ) is a generalized eigenvalue of (A(w(γ),M),B(w(γ),M)). Let y(γ) be a corresponding generalized eigenvector such that y(γ) T B(w(γ),M)y(γ) = 1 (such a y(γ) can always be obtained by the appropriate scaling of any eigenvector corresponding to α(γ)). Then by Lemma 3, ( dα(γ) da(w(γ),m) dw(γ) = y(γ)t α(γ) db(w(γ),m) ) y(γ) dw(γ) dw(γ) [ Noting that da(w(γ),m) 0 I dw(γ) = 2w(γ) I 0 then the proof follows from dw(γ) = dα(γ) ] dw(γ) dα(γ). [ and db(w(γ),m) I 0 dw(γ) = 2w(γ) 0 I ] 5 Algorithm An algorithm for computing the i-th real perturbation value (i.e. solving (4) and (5)) of a given M C q l, for i N, is now presented in this section. The following development focuses mainly on computing the real perturbation values of the second kind (i.e solving (5)). The algorithm to compute the real perturbation values of the first kind (i.e. to solve (4)) is similar. Firstly, let the global maximum of (5) be denoted by: ([ ]) r Re M γ Im M = sup σ 2i 1 γ (0,1] γ 1 Im M Re M with the global maximizer being γ (0, 1]; i.e. r = σ 2i 1 (P(γ, M)). Now suppose that at the k-th iteration, for k = 1, 2,..., we are given r k 1 and γ k 1 (i.e. obtained from the previous iteration), which are approximations of r and γ respectively, where r k 1 = σ 2i 1 (P(γ k 1, M)). The basic concept of the algorithm is then to use the information of r k 1 and γ k 1, along with Theorem 1 and Lemma 4, to obtain the so-called maximizing set (see [9]), S k, defined as: S k = {γ (0, 1] σ 2i 1 (P(γ, M)) > r k 1 } The global maximizer γ is therefore contained in S k. We then select a new point γ k S k, which achieves r k > r k 1, where r k = σ 2i 1 (P(γ k, M)). r k is then a new (and better) approximation of the global maximum r. The procedure is then repeated until the size of S k is smaller than an user-specified tolerance. 5.1 Obtaining the maximizing set, S k Let the current approximation of r and γ be r k 1 and γ k 1 respectively. Due to continuity, S k is a set of intervals in (0, 1], where the endpoints of these intervals are
7 all γ (0, 1] such that σ 2i 1 (P(γ, M)) = r k 1, and also possibly 0 and 1. Hence, S k is obtained by finding these endpoints. By Theorem 1, all γ (0, 1] that achieve a singular value of P(γ, M) equal to r k 1 (i.e. σ t (P(γ, M)) = r k 1 for some t N) can be obtained from the set of real generalized eigenvalues of (A(r k 1, M),B(r k 1, M)) as defined in (7). In particular, let Γ(r k 1 ) := {γ (0, 1] t N, σ t (P(γ, M)) = r k 1 } (13) { = γ (0, 1] γ 2 } 1 γ Λ r (by Theorem 1 and fact that γ is real) where we denote Λ r := sp r (A(r k 1, M),B(r k 1, M)). To determine which singular value of P(γ, M) equals r k 1 (i.e. to determine t such that σ t (P(γ, M)) = r k 1 ) for γ Γ(r k 1 ), one can compute the singular values of P(γ, M) and compare them with r k 1 for all γ Γ(r k 1 ). However, by Lemma 2, there can be at most min(q, l) elements in Γ(r k 1 ) for a given M C q l. Hence, computing the singular values of P(γ, M) for all γ Γ(r k 1 ) can be relatively costly. Instead, we use the derivative information at each γ Γ(r k 1 ) to determine which singular value of P(γ, M) equals r k 1. The procedure is described as follows, which is similar to a technique found in [9]. Let Γ(r k 1 ) be sorted in ascending order; i.e. Γ(r k 1 ) = {g 1, g 2,...} such that g 1 g 2. Also, let g w = γ k 1 for some w N (note that w always exists since r k 1 = σ 2i 1 (P(γ k 1, M)), which implies γ k 1 Γ(r k 1 )). Suppose that the derivative (as computed by (12) of Lemma 4) of the singular value of P(g w, M), denoted by dσ(p(gw,m)), is positive. Then, this implies that the plot of the (2i 1)-th singular value of P(γ, M) as a function of γ (0, 1], denoted y 2i 1 (γ) = σ 2i 1 (P(γ, M)), crosses upward at g w. Next, consider the derivative at g w+1 ; i.e. dσ(p(gw+1,m)). If the derivative is again positive, then this means that there is another upward crossing at g w+1. Such an upward crossing can only occur from the plot y 2i 2 (γ) = σ 2i 2 (P(γ, M)). This is because if one considers all the singular value plots y t (γ) = σ t (P(γ, M)) for t = 1,...,2 min(q, l), then by the ordering of the singular values, we have y t1 (γ) y t2 (γ) for all γ (0, 1] and for all t 1 t 2. In other words, the plot y t1 (γ) is always above (or equal to) the plot y t2 (γ) for all t 1 t 2. Hence, an upward crossing at g w+1 can only be by the next plot below y 2i 1 (γ); i.e. y 2i 2 (γ). On the other hand, if the derivative at g w+1 is negative, then there is a downward crossing at g w+1, which can only occur from y 2i 1 (γ) for a similar reason. Hence, starting from g w and working in the manner as described above, the upward and downward crossings are used to determine which singular value plot crosses at each g Γ(r k 1 ). Figure 1 illustrates a possible example scenario, where the + and signs depict an upward and downward crossing respectively. To obtain the maximizing set, S k, we are mainly interested in determining all γ Γ(r k 1 ) such that σ 2i 1 (P(γ, M)) = r k 1 ; i.e. all g Γ(r k 1 ) where there is a crossing by the y 2i 1 (γ) plot. Define such a set as Γ(r k 1 ) := {γ Γ(r k 1 ) σ 2i 1 (P(γ, M)) = r k 1 } (14)
8 2i 2i 1 2i 1 2i 2 2i 2 2i g w 2 g w 1 g w+1 g w+2 g w+3 g w Figure 1. Example of using the derivative information of g Γ(r k 1 ) to determine the maximizing set, S k 1. The maximizing set, S k, is hence bounded by the elements of Γ(r k 1 ) (and also possibly by 0 and 1); i.e. Γ(r k 1 ) contains the endpoints of the interval(s) in (0, 1] that achieves σ 2i 1 (P(γ, M)) greater than r k 1. The derivative information obtained previously can again be used to determine which elements in Γ(r k 1 ) are at the beginning of an interval and which are at the end. In particular, since we are searching for a maximum, all beginning endpoints will thus have an upward crossing and all ending endpoints will have a downward crossing. Referring back to the example scenario shown in Figure 1, the maximizing set is hence S k = (0, g w 1 ) (g w, g w+3 ) (assuming there are no more upward σ 2i 1 crossings in (0, g w 2 ) and (g w+3, 1]). 5.2 Updating the current maximum, r k By the construction of S k, any γ S k achieves σ 2i 1 (P(γ, M)) greater than r k 1. Hence, the next approximation of the global maximizer, γ k, can arbitrarily be chosen in S k, and then set r k = σ 2i 1 (P(γ k, M)) > r k 1. From experience, however, the performance of the algorithm seems to improve by choosing γ k as the maximizer of a cubic fit through the endpoints (and their derivatives) of the intervals of S k (see [9] and pseudo-code below). 5.3 Algorithm outline The algorithm can be summarized as follows: Algorithm 5.1 Computing the real perturbation value (of the second kind) of M Input: M C q l and i N Input tolerance: TOL Sk Output: r and γ, where r = σ 2i 1 (P(γ, M)) 1. Initialization: Choose arbitrary γ 0 (0, 1] and compute r 0 = σ 2i 1 (P(γ 0, M)). 2. Iteration k(= 1, 2,...) (a) Given r k 1 and γ k 1.
9 (b) Compute Λ r = sp r (A(r k 1, M),B(r k 1, M)). (c) Obtain the set (sorted in ascending order) { Γ(r k 1 ) = γ (0, 1] γ 2 } 1 γ Λ r (d) For each g Γ(r k 1 ), compute the derivative dσ(p(g,m)) via (12). (e) Using dσ(p(g,m)) for each g Γ(r k 1 ), determine Γ(r k 1 ) := {γ Γ(r k 1 ) σ 2i 1 (P(γ, M)) = r k 1 } (f) Find the maximizing set S k = {γ (0, 1] σ 2i 1 (P(γ, M)) > r k 1 } via Γ(r k 1 ) and dσ(p(g,m)) for each g Γ(r k 1 ). (g) Quit if the largest interval of S k is less than TOL Sk. (h) For each interval s j S k, Find cubic fit through the endpoints (and their derivatives) of s j. Define m j := maximizer of cubic fit within s j. Evaluate σ 2i 1 (P(m j, M)). Set γ k to whichever m j achieves a larger σ 2i 1 (P(m j, M)). 3. Update k k + 1 and repeat step Computational requirements To provide an idea of the computational requirements of using Algorithm 5.1 for computing the real perturbation values, the number of operations required in terms of the number of singular value (σ) and eigenvalue (λ) problems are noted in Table 1. As it can be seen from Table 1, only a small number of eigenvalue and singular value problems are solved at each iteration. In particular, only one generalized eigenvalue problem of size 2 min(q, l) is solved at step (2b). Then in step (2h), at least one singular value problem of size 2q 2l is solved in order to obtain the next approximation of the real perturbation value. However, since Γ(r k 1 ) in (2c) contains at most min(q, l) elements (by Lemma 2), then the maximizing set contains at most min(q,l) 2 min(q,l) intervals. Therefore, step (2h) requires at most + 1 singular value problems to be solved. Furthermore, it should be noted that the derivative information at step (2d) is computed at relatively low cost. As shown in (12), the derivatives are functions of the generalized eigenvectors of (A(r k 1, M),B(r k 1, M)) as defined in (7). Since these eigenvectors are already known from the eigendecomposition of (A(r k 1, M),B(r k 1, M)) in step (2b), the derivatives can be obtained very easily.
10 Table 1. Summary of operation count of Algorithm 5.1 Step # of (σ,λ) Size of matrix 1 1 σ 2q 2l 2(b) 1 λ 2(min(q, l)) 2(min(q, l)) 2(h) at least 1 σ 2q 2l at most ( min(q,l) ) σ Here, σ and λ denote the number of singular value and eigenvalue problem(s). 6 Numerical Example Consider the following matrix M = i i i where we are interested in computing τ 3 (M). Using a fine grid search, the third real perturbation value of M is found to be τ 3 (M) = , achieved at γ = Using the proposed algorithm with the (arbitrary) starting point γ 0 = 0.5, τ 3 (M) is found to within 3 significant figures in 1 iteration and to 9 significant figures in 2 iterations. Table 2 lists r k, γ k, and the maximizing set, S k, for each iteration k = 0,...,4. Figure 2 plots the maximizing set, S k, obtained at Iterations 1 and 2. It can be seen that in this example, the maximizing set is significantly reduced by the second iteration. Furthermore, the proposed algorithm only required solving a total of 4 generalized eigenvalue and 10 singular value problems for this example, which is much less demanding than performing a fine grid search. Table 2. Estimates of the global maximum (r k ), the maximizer (γ k ), and the maximizing set (S k ) at each iteration k Iteration (k) γ k r k S k ( , ) ( , ) ( , ) ( , ) ( , ) ( , )
11 Iteration σ 2i 1 (P(γ,M)) γ Iteration 2 σ 2i 1 (P(γ,M)) cubic fit maximizing set, S k σ 2i 1 (P(γ,M)) cubic fit maximizing set, S k 1.1 σ 2i 1 (P(γ,M)) Figure 2. Intermediate results at Iteration 1 (top) and Iteration 2 (bottom). 7 Conclusions In this paper, an efficient algorithm is presented for computing the real perturbation values of a given complex matrix. The algorithm discussed here focuses mainly on solving the real perturbation values of the second kind (i.e. solving the optimization in (5)). Modifying the algorithm to solve the real perturbation values of the first kind (4) is straightforward and is left for the reader. γ
12 Bibliography [1] B. Bernhardsson, A. Rantzer, and L. Qiu. Real perturbation values and real quadratic forms in a complex vector space. Linear Algebra and its Applications, 270: , [2] J. de Leeuw. Derivatives of Generalized Eigen Systems with Applications. Department of Statistics, UCLA, January Paper [3] R. Eising. Between controllable and uncontrollable. Systems and Control Letters, 4: , [4] G. Hu and E. J. Davison. Real controllability/stabilizability radius of LTI systems. IEEE Trans. Automat. Contr., 49(2): , [5] M. Karow. Geometry of Spectral Value Sets. PhD thesis, University of Bremen, Germany, [6] S. Lam and E. J. Davison. The real decentralized fixed mode radius of LTI systems. In 46 th IEEE Conference on Decision and Control, pages , New Orleans, LA, USA, December [7] S. Lam and E. J. Davison. The transmission zero at s radius and the minimum phase radius of LTI systems. In 17 th IFAC world congress, Seoul, Korea, July To appear. [8] L. Qiu, B. Bernhardsson, A. Rantzer, E. J. Davison, P. M. Young, and J. C. Doyle. A formula for computation of the real stability radius. Automatica, 31(6): , [9] J. Sreedhar, P. Van Dooren, and A. L. Tits. A fast algorithm to compute the real structured stability radius. In International Series of Numerical Mathematics, volume 121, pages Birkhäuser Verlag Basel, 1996.
A fast algorithm to compute the controllability, decentralized fixed-mode, and minimum-phase radius of LTI systems
Proceedings of the 7th IEEE Conference on Decision Control Cancun, Mexico, Dec. 9-, 8 TuA6. A fast algorithm to compute the controllability, decentralized fixed-mode, minimum-phase radius of LTI systems
More informationThe Transmission Zero at s Radius and the Minimum Phase Radius of LTI Systems
Proceedings of the 17th World ongress The International ederation of Automatic ontrol The Transmission Zero at s adius and the Minimum Phase adius of LTI Systems Simon Lam and Edward J. Davison Systems
More informationReal Robustness Radii and Performance Limitations of LTI Control Systems. Simon Sai-Ming Lam
Real Robustness Radii and Performance Limitations of LTI Control Systems by Simon Sai-Ming Lam A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department
More informationStability radii and spectral value sets for real matrix perturbations Introduction D. Hinrichsen and B. Kelb Institut fur Dynamische Systeme Universitat Bremen D-859 Bremen, F. R. G. dh@mathematik.uni-bremen.de
More information22.3. Repeated Eigenvalues and Symmetric Matrices. Introduction. Prerequisites. Learning Outcomes
Repeated Eigenvalues and Symmetric Matrices. Introduction In this Section we further develop the theory of eigenvalues and eigenvectors in two distinct directions. Firstly we look at matrices where one
More informationRecent Advances in Positive Systems: The Servomechanism Problem
Recent Advances in Positive Systems: The Servomechanism Problem 47 th IEEE Conference on Decision and Control December 28. Bartek Roszak and Edward J. Davison Systems Control Group, University of Toronto
More informationStructured singular values for Bernoulli matrices
Int. J. Adv. Appl. Math. and Mech. 44 2017 41 49 ISSN: 2347-2529 IJAAMM Journal homepage: www.ijaamm.com International Journal of Advances in Applied Mathematics and Mechanics Structured singular values
More informationGraph and Controller Design for Disturbance Attenuation in Consensus Networks
203 3th International Conference on Control, Automation and Systems (ICCAS 203) Oct. 20-23, 203 in Kimdaejung Convention Center, Gwangju, Korea Graph and Controller Design for Disturbance Attenuation in
More informationModel reduction via tangential interpolation
Model reduction via tangential interpolation K. Gallivan, A. Vandendorpe and P. Van Dooren May 14, 2002 1 Introduction Although most of the theory presented in this paper holds for both continuous-time
More informationSUCCESSIVE POLE SHIFTING USING SAMPLED-DATA LQ REGULATORS. Sigeru Omatu
SUCCESSIVE POLE SHIFING USING SAMPLED-DAA LQ REGULAORS oru Fujinaka Sigeru Omatu Graduate School of Engineering, Osaka Prefecture University, 1-1 Gakuen-cho, Sakai, 599-8531 Japan Abstract: Design of sampled-data
More informationPARAMETERIZATION OF STATE FEEDBACK GAINS FOR POLE PLACEMENT
PARAMETERIZATION OF STATE FEEDBACK GAINS FOR POLE PLACEMENT Hans Norlander Systems and Control, Department of Information Technology Uppsala University P O Box 337 SE 75105 UPPSALA, Sweden HansNorlander@ituuse
More informationA Novel Integral-Based Event Triggering Control for Linear Time-Invariant Systems
53rd IEEE Conference on Decision and Control December 15-17, 2014. Los Angeles, California, USA A Novel Integral-Based Event Triggering Control for Linear Time-Invariant Systems Seyed Hossein Mousavi 1,
More informationIMPULSIVE CONTROL OF DISCRETE-TIME NETWORKED SYSTEMS WITH COMMUNICATION DELAYS. Shumei Mu, Tianguang Chu, and Long Wang
IMPULSIVE CONTROL OF DISCRETE-TIME NETWORKED SYSTEMS WITH COMMUNICATION DELAYS Shumei Mu Tianguang Chu and Long Wang Intelligent Control Laboratory Center for Systems and Control Department of Mechanics
More informationPerturbation theory for eigenvalues of Hermitian pencils. Christian Mehl Institut für Mathematik TU Berlin, Germany. 9th Elgersburg Workshop
Perturbation theory for eigenvalues of Hermitian pencils Christian Mehl Institut für Mathematik TU Berlin, Germany 9th Elgersburg Workshop Elgersburg, March 3, 2014 joint work with Shreemayee Bora, Michael
More informationDecentralized control with input saturation
Decentralized control with input saturation Ciprian Deliu Faculty of Mathematics and Computer Science Technical University Eindhoven Eindhoven, The Netherlands November 2006 Decentralized control with
More informationTWELVE LIMIT CYCLES IN A CUBIC ORDER PLANAR SYSTEM WITH Z 2 -SYMMETRY. P. Yu 1,2 and M. Han 1
COMMUNICATIONS ON Website: http://aimsciences.org PURE AND APPLIED ANALYSIS Volume 3, Number 3, September 2004 pp. 515 526 TWELVE LIMIT CYCLES IN A CUBIC ORDER PLANAR SYSTEM WITH Z 2 -SYMMETRY P. Yu 1,2
More informationA small-gain type stability criterion for large scale networks of ISS systems
A small-gain type stability criterion for large scale networks of ISS systems Sergey Dashkovskiy Björn Sebastian Rüffer Fabian R. Wirth Abstract We provide a generalized version of the nonlinear small-gain
More informationState estimation of uncertain multiple model with unknown inputs
State estimation of uncertain multiple model with unknown inputs Abdelkader Akhenak, Mohammed Chadli, Didier Maquin and José Ragot Centre de Recherche en Automatique de Nancy, CNRS UMR 79 Institut National
More informationRepeated Eigenvalues and Symmetric Matrices
Repeated Eigenvalues and Symmetric Matrices. Introduction In this Section we further develop the theory of eigenvalues and eigenvectors in two distinct directions. Firstly we look at matrices where one
More informationThe Generalized Nyquist Criterion and Robustness Margins with Applications
51st IEEE Conference on Decision and Control December 10-13, 2012. Maui, Hawaii, USA The Generalized Nyquist Criterion and Robustness Margins with Applications Abbas Emami-Naeini and Robert L. Kosut Abstract
More informationStructural Properties of LTI Singular Systems under Output Feedback
Structural Properties of LTI Singular Systems under Output Feedback Runyi Yu Dept. of Electrical and Electronic Eng. Eastern Mediterranean University Gazimagusa, Mersin 10, Turkey runyi.yu@emu.edu.tr Dianhui
More informationSTABILITY OF 2D SWITCHED LINEAR SYSTEMS WITH CONIC SWITCHING. 1. Introduction
STABILITY OF D SWITCHED LINEAR SYSTEMS WITH CONIC SWITCHING H LENS, M E BROUCKE, AND A ARAPOSTATHIS 1 Introduction The obective of this paper is to explore necessary and sufficient conditions for stability
More informationThe Solvability Conditions for the Inverse Eigenvalue Problem of Hermitian and Generalized Skew-Hamiltonian Matrices and Its Approximation
The Solvability Conditions for the Inverse Eigenvalue Problem of Hermitian and Generalized Skew-Hamiltonian Matrices and Its Approximation Zheng-jian Bai Abstract In this paper, we first consider the inverse
More information1. Find the solution of the following uncontrolled linear system. 2 α 1 1
Appendix B Revision Problems 1. Find the solution of the following uncontrolled linear system 0 1 1 ẋ = x, x(0) =. 2 3 1 Class test, August 1998 2. Given the linear system described by 2 α 1 1 ẋ = x +
More informationarxiv: v1 [cs.sy] 29 Dec 2018
ON CHECKING NULL RANK CONDITIONS OF RATIONAL MATRICES ANDREAS VARGA Abstract. In this paper we discuss possible numerical approaches to reliably check the rank condition rankg(λ) = 0 for a given rational
More informationẋ n = f n (x 1,...,x n,u 1,...,u m ) (5) y 1 = g 1 (x 1,...,x n,u 1,...,u m ) (6) y p = g p (x 1,...,x n,u 1,...,u m ) (7)
EEE582 Topical Outline A.A. Rodriguez Fall 2007 GWC 352, 965-3712 The following represents a detailed topical outline of the course. It attempts to highlight most of the key concepts to be covered and
More informationPOLE PLACEMENT. Sadegh Bolouki. Lecture slides for ECE 515. University of Illinois, Urbana-Champaign. Fall S. Bolouki (UIUC) 1 / 19
POLE PLACEMENT Sadegh Bolouki Lecture slides for ECE 515 University of Illinois, Urbana-Champaign Fall 2016 S. Bolouki (UIUC) 1 / 19 Outline 1 State Feedback 2 Observer 3 Observer Feedback 4 Reduced Order
More informationDerivation of Robust Stability Ranges for Disconnected Region with Multiple Parameters
SICE Journal of Control, Measurement, and System Integration, Vol. 10, No. 1, pp. 03 038, January 017 Derivation of Robust Stability Ranges for Disconnected Region with Multiple Parameters Tadasuke MATSUDA
More informationStatic Output Feedback Stabilisation with H Performance for a Class of Plants
Static Output Feedback Stabilisation with H Performance for a Class of Plants E. Prempain and I. Postlethwaite Control and Instrumentation Research, Department of Engineering, University of Leicester,
More informationEigenvalues and Eigenvectors
Contents Eigenvalues and Eigenvectors. Basic Concepts. Applications of Eigenvalues and Eigenvectors 8.3 Repeated Eigenvalues and Symmetric Matrices 3.4 Numerical Determination of Eigenvalues and Eigenvectors
More informationOn the simultaneous diagonal stability of a pair of positive linear systems
On the simultaneous diagonal stability of a pair of positive linear systems Oliver Mason Hamilton Institute NUI Maynooth Ireland Robert Shorten Hamilton Institute NUI Maynooth Ireland Abstract In this
More informationRobust Stabilization of the Uncertain Linear Systems. Based on Descriptor Form Representation t. Toru ASAI* and Shinji HARA**
Robust Stabilization of the Uncertain Linear Systems Based on Descriptor Form Representation t Toru ASAI* and Shinji HARA** This paper proposes a necessary and sufficient condition for the quadratic stabilization
More informationMulti-Model Adaptive Regulation for a Family of Systems Containing Different Zero Structures
Preprints of the 19th World Congress The International Federation of Automatic Control Multi-Model Adaptive Regulation for a Family of Systems Containing Different Zero Structures Eric Peterson Harry G.
More informationLinear Quadratic Zero-Sum Two-Person Differential Games Pierre Bernhard June 15, 2013
Linear Quadratic Zero-Sum Two-Person Differential Games Pierre Bernhard June 15, 2013 Abstract As in optimal control theory, linear quadratic (LQ) differential games (DG) can be solved, even in high dimension,
More informationReal Eigenvalue Extraction and the Distance to Uncontrollability
Real Eigenvalue Extraction and the Distance to Uncontrollability Emre Mengi Computer Science Department Courant Institute of Mathematical Sciences New York University mengi@cs.nyu.edu May 22nd, 2006 Emre
More informationThe Kalman-Yakubovich-Popov Lemma for Differential-Algebraic Equations with Applications
MAX PLANCK INSTITUTE Elgersburg Workshop Elgersburg February 11-14, 2013 The Kalman-Yakubovich-Popov Lemma for Differential-Algebraic Equations with Applications Timo Reis 1 Matthias Voigt 2 1 Department
More informationThe ϵ-capacity of a gain matrix and tolerable disturbances: Discrete-time perturbed linear systems
IOSR Journal of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 11, Issue 3 Ver. IV (May - Jun. 2015), PP 52-62 www.iosrjournals.org The ϵ-capacity of a gain matrix and tolerable disturbances:
More informationMin-Max Output Integral Sliding Mode Control for Multiplant Linear Uncertain Systems
Proceedings of the 27 American Control Conference Marriott Marquis Hotel at Times Square New York City, USA, July -3, 27 FrC.4 Min-Max Output Integral Sliding Mode Control for Multiplant Linear Uncertain
More informationA Globally Stabilizing Receding Horizon Controller for Neutrally Stable Linear Systems with Input Constraints 1
A Globally Stabilizing Receding Horizon Controller for Neutrally Stable Linear Systems with Input Constraints 1 Ali Jadbabaie, Claudio De Persis, and Tae-Woong Yoon 2 Department of Electrical Engineering
More informationOn Positive Real Lemma for Non-minimal Realization Systems
Proceedings of the 17th World Congress The International Federation of Automatic Control On Positive Real Lemma for Non-minimal Realization Systems Sadaaki Kunimatsu Kim Sang-Hoon Takao Fujii Mitsuaki
More informationEnforcing Passivity for Admittance Matrices Approximated by Rational Functions
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 1, FEBRUARY 2001 97 Enforcing Passivity for Admittance Matrices Approximated by Rational Functions Bjørn Gustavsen, Member, IEEE and Adam Semlyen, Life
More informationBUMPLESS SWITCHING CONTROLLERS. William A. Wolovich and Alan B. Arehart 1. December 27, Abstract
BUMPLESS SWITCHING CONTROLLERS William A. Wolovich and Alan B. Arehart 1 December 7, 1995 Abstract This paper outlines the design of bumpless switching controllers that can be used to stabilize MIMO plants
More informationME 234, Lyapunov and Riccati Problems. 1. This problem is to recall some facts and formulae you already know. e Aτ BB e A τ dτ
ME 234, Lyapunov and Riccati Problems. This problem is to recall some facts and formulae you already know. (a) Let A and B be matrices of appropriate dimension. Show that (A, B) is controllable if and
More informationConvergence Rate of Nonlinear Switched Systems
Convergence Rate of Nonlinear Switched Systems Philippe JOUAN and Saïd NACIRI arxiv:1511.01737v1 [math.oc] 5 Nov 2015 January 23, 2018 Abstract This paper is concerned with the convergence rate of the
More informationModel reduction for linear systems by balancing
Model reduction for linear systems by balancing Bart Besselink Jan C. Willems Center for Systems and Control Johann Bernoulli Institute for Mathematics and Computer Science University of Groningen, Groningen,
More informationA note on a nearest polynomial with a given root
ACM SIGSAM Bulletin, Vol 39, No. 2, June 2005 A note on a nearest polynomial with a given root Stef Graillat Laboratoire LP2A, Université de Perpignan Via Domitia 52, avenue Paul Alduy F-66860 Perpignan
More informationRank Reduction for Matrix Pair and Its Application in Singular Systems
Rank Reduction for Matrix Pair Its Application in Singular Systems Jing Wang Wanquan Liu, Qingling Zhang Xiaodong Liu 1 Institute of System Sciences Northeasten University, PRChina e-mail: wj--zhang@163com
More informationand the nite horizon cost index with the nite terminal weighting matrix F > : N?1 X J(z r ; u; w) = [z(n)? z r (N)] T F [z(n)? z r (N)] + t= [kz? z r
Intervalwise Receding Horizon H 1 -Tracking Control for Discrete Linear Periodic Systems Ki Baek Kim, Jae-Won Lee, Young Il. Lee, and Wook Hyun Kwon School of Electrical Engineering Seoul National University,
More informationIterative Solution of a Matrix Riccati Equation Arising in Stochastic Control
Iterative Solution of a Matrix Riccati Equation Arising in Stochastic Control Chun-Hua Guo Dedicated to Peter Lancaster on the occasion of his 70th birthday We consider iterative methods for finding the
More informationNecessary and Sufficient Stability Condition of Discrete State Delay Systems
nternational Journal ecessary of Control, and Automation, Sufficient and Stability Systems, Condition vol, of no Discrete 4, pp State 50-508, Delay December Systems 004 50 ecessary and Sufficient Stability
More information2nd Symposium on System, Structure and Control, Oaxaca, 2004
263 2nd Symposium on System, Structure and Control, Oaxaca, 2004 A PROJECTIVE ALGORITHM FOR STATIC OUTPUT FEEDBACK STABILIZATION Kaiyang Yang, Robert Orsi and John B. Moore Department of Systems Engineering,
More informationPartial Eigenvalue Assignment in Linear Systems: Existence, Uniqueness and Numerical Solution
Partial Eigenvalue Assignment in Linear Systems: Existence, Uniqueness and Numerical Solution Biswa N. Datta, IEEE Fellow Department of Mathematics Northern Illinois University DeKalb, IL, 60115 USA e-mail:
More informationCDS Solutions to the Midterm Exam
CDS 22 - Solutions to the Midterm Exam Instructor: Danielle C. Tarraf November 6, 27 Problem (a) Recall that the H norm of a transfer function is time-delay invariant. Hence: ( ) Ĝ(s) = s + a = sup /2
More informationNetwork Reconstruction from Intrinsic Noise: Non-Minimum-Phase Systems
Preprints of the 19th World Congress he International Federation of Automatic Control Network Reconstruction from Intrinsic Noise: Non-Minimum-Phase Systems David Hayden, Ye Yuan Jorge Goncalves Department
More informationLinear Algebra: Matrix Eigenvalue Problems
CHAPTER8 Linear Algebra: Matrix Eigenvalue Problems Chapter 8 p1 A matrix eigenvalue problem considers the vector equation (1) Ax = λx. 8.0 Linear Algebra: Matrix Eigenvalue Problems Here A is a given
More informationOptimization Based Output Feedback Control Design in Descriptor Systems
Trabalho apresentado no XXXVII CNMAC, S.J. dos Campos - SP, 017. Proceeding Series of the Brazilian Society of Computational and Applied Mathematics Optimization Based Output Feedback Control Design in
More informationProblem 2 (Gaussian Elimination, Fundamental Spaces, Least Squares, Minimum Norm) Consider the following linear algebraic system of equations:
EEE58 Exam, Fall 6 AA Rodriguez Rules: Closed notes/books, No calculators permitted, open minds GWC 35, 965-37 Problem (Dynamic Augmentation: State Space Representation) Consider a dynamical system consisting
More informationHybrid Systems Techniques for Convergence of Solutions to Switching Systems
Hybrid Systems Techniques for Convergence of Solutions to Switching Systems Rafal Goebel, Ricardo G. Sanfelice, and Andrew R. Teel Abstract Invariance principles for hybrid systems are used to derive invariance
More informationLMI based output-feedback controllers: γ-optimal versus linear quadratic.
Proceedings of the 17th World Congress he International Federation of Automatic Control Seoul Korea July 6-11 28 LMI based output-feedback controllers: γ-optimal versus linear quadratic. Dmitry V. Balandin
More informationComputational Methods CMSC/AMSC/MAPL 460. Eigenvalues and Eigenvectors. Ramani Duraiswami, Dept. of Computer Science
Computational Methods CMSC/AMSC/MAPL 460 Eigenvalues and Eigenvectors Ramani Duraiswami, Dept. of Computer Science Eigen Values of a Matrix Recap: A N N matrix A has an eigenvector x (non-zero) with corresponding
More informationMultivariate Statistical Analysis
Multivariate Statistical Analysis Fall 2011 C. L. Williams, Ph.D. Lecture 4 for Applied Multivariate Analysis Outline 1 Eigen values and eigen vectors Characteristic equation Some properties of eigendecompositions
More informationDECENTRALIZED CONTROL DESIGN USING LMI MODEL REDUCTION
Journal of ELECTRICAL ENGINEERING, VOL. 58, NO. 6, 2007, 307 312 DECENTRALIZED CONTROL DESIGN USING LMI MODEL REDUCTION Szabolcs Dorák Danica Rosinová Decentralized control design approach based on partial
More informationA Multi-Step Hybrid Method for Multi-Input Partial Quadratic Eigenvalue Assignment with Time Delay
A Multi-Step Hybrid Method for Multi-Input Partial Quadratic Eigenvalue Assignment with Time Delay Zheng-Jian Bai Mei-Xiang Chen Jin-Ku Yang April 14, 2012 Abstract A hybrid method was given by Ram, Mottershead,
More informationIntro. Computer Control Systems: F8
Intro. Computer Control Systems: F8 Properties of state-space descriptions and feedback Dave Zachariah Dept. Information Technology, Div. Systems and Control 1 / 22 dave.zachariah@it.uu.se F7: Quiz! 2
More informationLinear Algebra Review
January 29, 2013 Table of contents Metrics Metric Given a space X, then d : X X R + 0 and z in X if: d(x, y) = 0 is equivalent to x = y d(x, y) = d(y, x) d(x, y) d(x, z) + d(z, y) is a metric is for all
More informationChapter 2 Formulas and Definitions:
Chapter 2 Formulas and Definitions: (from 2.1) Definition of Polynomial Function: Let n be a nonnegative integer and let a n,a n 1,...,a 2,a 1,a 0 be real numbers with a n 0. The function given by f (x)
More informationA NONSMOOTH, NONCONVEX OPTIMIZATION APPROACH TO ROBUST STABILIZATION BY STATIC OUTPUT FEEDBACK AND LOW-ORDER CONTROLLERS
A NONSMOOTH, NONCONVEX OPTIMIZATION APPROACH TO ROBUST STABILIZATION BY STATIC OUTPUT FEEDBACK AND LOW-ORDER CONTROLLERS James V Burke,1 Adrian S Lewis,2 Michael L Overton,3 University of Washington, Seattle,
More informationLecture 4 and 5 Controllability and Observability: Kalman decompositions
1 Lecture 4 and 5 Controllability and Observability: Kalman decompositions Spring 2013 - EE 194, Advanced Control (Prof. Khan) January 30 (Wed.) and Feb. 04 (Mon.), 2013 I. OBSERVABILITY OF DT LTI SYSTEMS
More informationTHE LARGE number of components in a power system
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 19, NO. 2, MAY 2004 857 Order Reduction of the Dynamic Model of a Linear Weakly Periodic System Part I: General Methodology Abner Ramirez, Member, IEEE, Adam Semlyen,
More informationOn linear quadratic optimal control of linear time-varying singular systems
On linear quadratic optimal control of linear time-varying singular systems Chi-Jo Wang Department of Electrical Engineering Southern Taiwan University of Technology 1 Nan-Tai Street, Yungkung, Tainan
More informationStrong stability of neutral equations with dependent delays
Strong stability of neutral equations with dependent delays W Michiels, T Vyhlidal, P Zitek, H Nijmeijer D Henrion 1 Introduction We discuss stability properties of the linear neutral equation p 2 ẋt +
More informationLinear Quadratic Zero-Sum Two-Person Differential Games
Linear Quadratic Zero-Sum Two-Person Differential Games Pierre Bernhard To cite this version: Pierre Bernhard. Linear Quadratic Zero-Sum Two-Person Differential Games. Encyclopaedia of Systems and Control,
More informationStatic Output Feedback Controller for Nonlinear Interconnected Systems: Fuzzy Logic Approach
International Conference on Control, Automation and Systems 7 Oct. 7-,7 in COEX, Seoul, Korea Static Output Feedback Controller for Nonlinear Interconnected Systems: Fuzzy Logic Approach Geun Bum Koo l,
More informationAPPENDIX A. Background Mathematics. A.1 Linear Algebra. Vector algebra. Let x denote the n-dimensional column vector with components x 1 x 2.
APPENDIX A Background Mathematics A. Linear Algebra A.. Vector algebra Let x denote the n-dimensional column vector with components 0 x x 2 B C @. A x n Definition 6 (scalar product). The scalar product
More informationEE363 homework 2 solutions
EE363 Prof. S. Boyd EE363 homework 2 solutions. Derivative of matrix inverse. Suppose that X : R R n n, and that X(t is invertible. Show that ( d d dt X(t = X(t dt X(t X(t. Hint: differentiate X(tX(t =
More informationDelay-Dependent Exponential Stability of Linear Systems with Fast Time-Varying Delay
International Mathematical Forum, 4, 2009, no. 39, 1939-1947 Delay-Dependent Exponential Stability of Linear Systems with Fast Time-Varying Delay Le Van Hien Department of Mathematics Hanoi National University
More informationRobust control of resistive wall modes using pseudospectra
Robust control of resistive wall modes using pseudospectra M. Sempf, P. Merkel, E. Strumberger, C. Tichmann, and S. Günter Max-Planck-Institut für Plasmaphysik, EURATOM Association, Garching, Germany GOTiT
More informationDenis ARZELIER arzelier
COURSE ON LMI OPTIMIZATION WITH APPLICATIONS IN CONTROL PART II.2 LMIs IN SYSTEMS CONTROL STATE-SPACE METHODS PERFORMANCE ANALYSIS and SYNTHESIS Denis ARZELIER www.laas.fr/ arzelier arzelier@laas.fr 15
More informationSimultaneous global external and internal stabilization of linear time-invariant discrete-time systems subject to actuator saturation
011 American Control Conference on O'Farrell Street, San Francisco, CA, USA June 9 - July 01, 011 Simultaneous global external and internal stabilization of linear time-invariant discrete-time systems
More informationSimultaneous State and Fault Estimation for Descriptor Systems using an Augmented PD Observer
Preprints of the 19th World Congress The International Federation of Automatic Control Simultaneous State and Fault Estimation for Descriptor Systems using an Augmented PD Observer Fengming Shi*, Ron J.
More informationControl Systems. Laplace domain analysis
Control Systems Laplace domain analysis L. Lanari outline introduce the Laplace unilateral transform define its properties show its advantages in turning ODEs to algebraic equations define an Input/Output
More informationc 2017 Society for Industrial and Applied Mathematics
SIAM J. MATRIX ANAL. APPL. Vol. 38, No. 3, pp. 89 923 c 207 Society for Industrial and Applied Mathematics CRISS-CROSS TYPE ALGORITHMS FOR COMPUTING THE REAL PSEUDOSPECTRAL ABSCISSA DING LU AND BART VANDEREYCKEN
More informationMASLOV INDEX. Contents. Drawing from [1]. 1. Outline
MASLOV INDEX J-HOLOMORPHIC CURVE SEMINAR MARCH 2, 2015 MORGAN WEILER 1. Outline 1 Motivation 1 Definition of the Maslov Index 2 Questions Demanding Answers 2 2. The Path of an Orbit 3 Obtaining A 3 Independence
More informationOn Linear Copositive Lyapunov Functions and the Stability of Switched Positive Linear Systems
1 On Linear Copositive Lyapunov Functions and the Stability of Switched Positive Linear Systems O. Mason and R. Shorten Abstract We consider the problem of common linear copositive function existence for
More informationRobust Control 2 Controllability, Observability & Transfer Functions
Robust Control 2 Controllability, Observability & Transfer Functions Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University /26/24 Outline Reachable Controllability Distinguishable
More informationminimize x x2 2 x 1x 2 x 1 subject to x 1 +2x 2 u 1 x 1 4x 2 u 2, 5x 1 +76x 2 1,
4 Duality 4.1 Numerical perturbation analysis example. Consider the quadratic program with variables x 1, x 2, and parameters u 1, u 2. minimize x 2 1 +2x2 2 x 1x 2 x 1 subject to x 1 +2x 2 u 1 x 1 4x
More information[k,g,gfin] = hinfsyn(p,nmeas,ncon,gmin,gmax,tol)
8 H Controller Synthesis: hinfsyn The command syntax is k,g,gfin = hinfsyn(p,nmeas,ncon,gmin,gmax,tol) associated with the general feedback diagram below. e y P d u e G = F L (P, K) d K hinfsyn calculates
More informationSystems of Algebraic Equations and Systems of Differential Equations
Systems of Algebraic Equations and Systems of Differential Equations Topics: 2 by 2 systems of linear equations Matrix expression; Ax = b Solving 2 by 2 homogeneous systems Functions defined on matrices
More informationChapter 1. Matrix Algebra
ST4233, Linear Models, Semester 1 2008-2009 Chapter 1. Matrix Algebra 1 Matrix and vector notation Definition 1.1 A matrix is a rectangular or square array of numbers of variables. We use uppercase boldface
More informationGramians based model reduction for hybrid switched systems
Gramians based model reduction for hybrid switched systems Y. Chahlaoui Younes.Chahlaoui@manchester.ac.uk Centre for Interdisciplinary Computational and Dynamical Analysis (CICADA) School of Mathematics
More informationADAPTIVE OUTPUT FEEDBACK CONTROL OF NONLINEAR SYSTEMS YONGLIANG ZHU. Bachelor of Science Zhejiang University Hanzhou, Zhejiang, P.R.
ADAPTIVE OUTPUT FEEDBACK CONTROL OF NONLINEAR SYSTEMS By YONGLIANG ZHU Bachelor of Science Zhejiang University Hanzhou, Zhejiang, P.R. China 1988 Master of Science Oklahoma State University Stillwater,
More informationOptimal Decentralized Control of Coupled Subsystems With Control Sharing
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 9, SEPTEMBER 2013 2377 Optimal Decentralized Control of Coupled Subsystems With Control Sharing Aditya Mahajan, Member, IEEE Abstract Subsystems that
More informationAn asymptotic ratio characterization of input-to-state stability
1 An asymptotic ratio characterization of input-to-state stability Daniel Liberzon and Hyungbo Shim Abstract For continuous-time nonlinear systems with inputs, we introduce the notion of an asymptotic
More informationResearch Article State-PID Feedback for Pole Placement of LTI Systems
Mathematical Problems in Engineering Volume 211, Article ID 92943, 2 pages doi:1.1155/211/92943 Research Article State-PID Feedback for Pole Placement of LTI Systems Sarawut Sujitjorn and Witchupong Wiboonjaroen
More informationBlock companion matrices, discrete-time block diagonal stability and polynomial matrices
Block companion matrices, discrete-time block diagonal stability and polynomial matrices Harald K. Wimmer Mathematisches Institut Universität Würzburg D-97074 Würzburg Germany October 25, 2008 Abstract
More information1 Matrix notation and preliminaries from spectral graph theory
Graph clustering (or community detection or graph partitioning) is one of the most studied problems in network analysis. One reason for this is that there are a variety of ways to define a cluster or community.
More information, some directions, namely, the directions of the 1. and
11. Eigenvalues and eigenvectors We have seen in the last chapter: for the centroaffine mapping, some directions, namely, the directions of the 1 coordinate axes: and 0 0, are distinguished 1 among all
More informationOn the Stabilization of Neutrally Stable Linear Discrete Time Systems
TWCCC Texas Wisconsin California Control Consortium Technical report number 2017 01 On the Stabilization of Neutrally Stable Linear Discrete Time Systems Travis J. Arnold and James B. Rawlings Department
More informationOn Dwell Time Minimization for Switched Delay Systems: Free-Weighting Matrices Method
On Dwell Time Minimization for Switched Delay Systems: Free-Weighting Matrices Method Ahmet Taha Koru Akın Delibaşı and Hitay Özbay Abstract In this paper we present a quasi-convex minimization method
More informationLecture 02 Linear Algebra Basics
Introduction to Computational Data Analysis CX4240, 2019 Spring Lecture 02 Linear Algebra Basics Chao Zhang College of Computing Georgia Tech These slides are based on slides from Le Song and Andres Mendez-Vazquez.
More information