The model reduction algorithm proposed is based on an iterative two-step LMI scheme. The convergence of the algorithm is not analyzed but examples sho

Size: px
Start display at page:

Download "The model reduction algorithm proposed is based on an iterative two-step LMI scheme. The convergence of the algorithm is not analyzed but examples sho"

Transcription

1 Model Reduction from an H 1 /LMI perspective A. Helmersson Department of Electrical Engineering Linkoping University S Linkoping, Sweden tel: fax: andersh@isy.liu.se September 6, ECC Abstract This paper treats the problem of approximating an nth order continuous system by a system of order k < n. Optimal solutions minimizing the H1 norm exist for k = 0 n; 1. This paper presents an iterative two-step LMI method for improving the H1 model error compared to Hankel norm reduction. As an intermediate step, the algorithm nds a generalized balanced realization such thatthen;k Hankel singular values coincide. Keywords: LMIs, model reduction, H-innity. 1 Introduction Truncated balanced realization and Hankel norm reduction [6] are well-behaved algorithms for nding reduced order models. In this paper the selection of model is performed using an unweighted H 1 norm as criterion. When reducing a model of order n to order k, the Hankel norm reduced model is optimal if k = n ; 1. In other cases the H 1 model error is bounded by thesum of the n ; k smallest Hankel singular values. If these values are close to each other, one can expect that the Hankel norm reduction can be improved in some cases. During recent years linear matrix inequalities (LMIs) have attracted a lot of interrest for solving H 1 problems. Since the model reduction problem can be considered as a special case of H 1 design with conned degree of the controller, it seems natural to adopt LMIs for model reduction. The LMI formulation does not yield a convex problem when the degree of the controller is conned, and it is not guaranteed that the solution will converge to a (global) minimum. However, we can expect improvement compared to the Hankel model reduction in most cases (if k<n; 1). 1

2 The model reduction algorithm proposed is based on an iterative two-step LMI scheme. The convergence of the algorithm is not analyzed but examples show that both linear (geometric) and sub-linear convergence can occur. The paper is outlined as follows. In section the results on the optimal Hankel norm reduction is summarized and in section the LMI formulation for H 1 problems is given. The algorithm is presented in section and it is applied to a number of examples in section Notations X T denotes the transpose of X X > () 0 a symmetric, positive denite (semidenite) matrix diag [X 1 X ] a block-diagonal matrix composed of X 1 and X rank X denotes the rank of the matrix X (X) the maximal singular value of X and k:k 1 the H 1 norm. Optimal Hankel Norm Reduction One well-behaved method for model reduction is the Optimal Hankel-norm approximation (see [6] for a comprehensive treatment). It is based on balanced realizations and Hankel singular values. The Hankel-norm model reduction starts by nding the balanced realization of the system..1 Balanced Realizations The Hankel norm reduction and the truncated balanced realization are both basedonthebalanced realization of a system G(s) =D + C(sI ; A) ;1 B. A balanced realization is characterized by having observability and controllability gramians equal and diagonal, that is A T +A + C T C =0 A T + A+BB T =0 with = diag [ 1 ::: k n I n;k ]. Hankel singular values.. Upper and lower bounds The diagonal elements in are called According to the theory for Hankel-norm model reduction [6], the H 1 model error, here denoted, is bounded by k+1 + nx i=k+r+1 i (1) where we have assumed that the Hankel singular values i are ordered 1 ::: k = ::: = k+r k+1 ::: n 0: () For any reduced model of order k we have that k+1 () Thus, the Hankel-norm model reduction is optimal in the H 1 sense if k = n;1, but not necessarily so in the general case.

3 . Optimal Hankel Norm Reduction The optimal Hankel norm reduction (see [6] for a complete treatment) is based on the balanced realization as is truncated model reduction. The H 1 model error is in this case improves to n when n ; k states are removed assuming these correspond to coinciding Hankel singular values. The Hankel model reduction can be performed by the following manipulations. First introduce 1 R kk dened by and a diagonal matrix ; 1 dened by diag [ 1 I n;k ]= () ; 1 = 1 ; I k : (5) We can now write the reduced model ^G(s) = ^D + ^C(sI ; ;1 ^A) ^B, where ^A ^B ; = ; A A T 11 1 B 1 ^C ^D 0 I C 1 1 D ; 1 A 1 + A T 1 C ;A + A T ;1 A1 1 + A T 1 B ; ; I (6) where A B C and D have been partitioned into a structure corresponding to k (A 11 R kk ) 6 A 11 A 1 B 1 A 1 A B C 1 C D " 5 = A C B D # : () It is now straight-forward to show that (9) is satised with = n with P = 1 0 ;; 1 0 I n;k 0 ;; > 0: (8) Note that it is not necessary to compute the balanced realization explicitly. A numerically better method is given in [10]. Theory And Algorithms.1 The Bounded Real Lemma Model reduction can be seen as a special case of a H 1 problem with conned degree. The original system G(s) = D + C(sI ; A) ;1 B of order n and is to be approximated by alower-order system ^G(s) = ^D + ^C(sI ; ;1 ^A) ^B of degree k<nsuch thatkg ; ^Gk1. Using the Bounded Real Lemma (see e.g. [11]) this is equivalent of the existence of a P > such that 6 ~A T P + P ~ A P ~ B ~ C T ~B T P ;I ~ D T ~C D ;I 5 0 (9)

4 with " ~A ~ B ~C ~ D # = 6 A 0 B 0 ^A ^B C ; ^C D ; ^D 5 : (10). A First Algorithm As can be seen this matrix inequality is bilinear in P and ( ^A ^B ^C ^D) and can not be solved directly using an LMI solver. The following iterative two-step algorithm can be used even if it does not guarantee (global) convergence. Algorithm.1 Assume that S P = N T N L (i) Start with a ^G obtained from, for example, Hankel model reduction or truncated balanced realization. (ii) Keeping ( ^A ^B) constant, minimize subject to (9) with respect to (P ^C ^D) (iii) Keeping (N L) constant, minimize subject to (9) with respect to (S ^A ^B ^C ^D) (iv) Repeat (ii) and (iii) until the solution converges given some criterion. This algorithm has been tried on a number of examples of low order (n 5), see [].. Solvability Conditions The following lemma plays an important role for both LMI-based H 1 synthesis and model reduction. Lemma.1 Given a symmetric matrix R mm and two matrices U and V of column dimension m and full column rank, consider the problem of nding some matrix of compatible dimensions and of full column rank, such that +UV T + V T U T < 0: (11) Denote by U? and V? any matrices such that U T? U =0 VT? V =0and [U U?] [V V? ] square and invertible. Then (11) is solvable for if and only if U T? U? < 0 V T? V? < 0: (1) Proof. Necessity of(1)isclear:for instance, U T? U = 0 implies U T? U? < 0 when pre- and post-multiplying (11) by U T? and U? respectively. For details on the suciency part, see [].

5 Using this lemma, step (ii) in Algorithm.1 can be simplied by removing ^C and ^D. To achieve thiswe start by rewriting (9) as 6 ~A T P + P A ~ P B ~ C T 0 ~B T P ;I 0 C 0 0 ;I 5 + U ^C ^D V T + V ^CT ^D T U T 0 (1) T 0 I 0 0 where U = ;I and V = I Using U? = 6 5 and V? = 6 5, I I I we obtain the following lemma I I T. Lemma. Let G ~ = G ; ^G with given ^A and ^B. Then there exists an approximation ^G such that k Gk1 ~ if and only if there existap > 0 satisfying " # ~A T P + P A ~ P B ~ ~B T 0 (1) P ;I. Model Reduction from an H 1 Control Perspective The problem of nding a reduced-order model minimizing the H 1 error can be considered as nding a controller of order k of the corresponding H 1 control problem. Using the results based on the LMI approach on the (augmented) plant dened by 6 A B 0 C D ;I 0 I 0 5 : (15) Using the LMI-based parametrization of all H 1 controllers given in [], we arrive at the following equivalence to the existence of a reduced model with an H 1 error less than or equal to : AR + RA T B 0 (16) ;I B T A T S + SA T C R I I S C T ;I 0 (1) 0 (18) and rank (I ; RS) k: (19) 5

6 The inequalities (16) and (1) are equivalent to the Lyapunov inequalities and AR + RA T + ;1 BB T 0 (0) A T S + SA + ;1 C T C 0 (1) respectively. We interpret R and S as generalized controllability and observability gramians. With the exception of the rank condition these are linear and, thus, the solution set is convex. If we can nd solutions R and S, we can also nd a generalized balanced realization using a similarity transformation such that R = S = ;1 =. The rank condition (19) now implies that (the diagonal) has n ; k elements equal to one. We cannow use (6) to nd the reduced order model of order k, see also [9, 8]. Thus, we have Theorem.1 Assume G is of order n. Then the existence ofa ^G of order k, such that kg ; ^Gk1 is equivalent to the existence of a generalized balanced realization of G A T +A + C T C 0 A+A T + BB T 0 with = diag [ 1 ::: k I n;k ] > 0. We can now make the following observations. Except for the trivial case with k = n ( ^G = G), we can discern two special cases of the model reduction problem: k =0andk = n ; 1. These two can be solved as non-iterative LMI problem since the explicit rank condition can be eliminated. The case k = n ; 1 has already been discussed. For the case k = 0 the rank condition (19) implies that R ;1 = S and since (16) can be rewritten as A T R ;1 + R ;1 A R ;1 B B T R ;1 0 () ;I which together with (1) is an LMI problem with respect to S. The general problem is harder to solve since no non-iterative method using LMIs has been found. In the algorithm proposed previously, steps (ii) and (iii) are modied. Algorithm. Assume that S P = N T N L (i) Start with a ^G obtained from, for instance, Hankel model reduction or truncated balanced realization. (ii) Let ~A = A 0 0 ^A () 6

7 and ~B = B^B : () With constant ( A ~ B) ~ minimize subject to (1) and ~A T P + P A ~ P B ~ ~B T 0 (5) P ;I with respect to S P = N T N L > 0: This step nds an appropriate N, which spans the subspace of the original state-space tobe kept in the reduced controller (iii) Keeping N constant, minimize subject to (1) and () with respect to R ;1 > 0 and L ;1 > 0 with S = R ;1 + NL ;1 N T. This step nds the best reduced approximation of the original system given the subspace to be kept (dened byn). The reduced order model ^G = ^D + ^C(sI ; ^A) ^B is found via the (generalized) balanced realization and the Hankel model reduction (6). (iv) Repeat (ii) and (iii) until the solution converges given some criterion. Note that the exact choice of N is immaterial as long as it spans the same subspace. The degree of freedom in the subspace dened by N is k(n ; k). This also explains why we can solve the case k = 0 (and the trivial k = n) more easily since the degree of freedom in N disappears. In the case k = n ; 1the subspace to be truncated is uniquely dened by the balanced realization..5 Computational Complexity The computational complexity per iteration is mainly determined by the number of variables involved in the two LMI problems. The rst LMI nds a P to the augmented system with n + k states and thus has N =(n + k)(n + k +1)= free variables to be solved for in addition to. According to [] the computational complexity for solving an LMI is typically O(N :1 ) for a xed number of LMI inequalities. The second LMI has R ;1 and L ;1 as free variables, that is n(n + 1)= +k(k +1)= variables. Both LMIs have roughly between n = and n variables the second LMI has always less variables than the rst one. Thus the complexity per iteration is O(n : ). This has to be compared with other typical matrix operations such as solving a Lyapunov equation, matrix multiplication and inversion, which all are in the order of O(n ). Examples We will illustrate the algorithm on a number of examples and compare it with the Hankel norm reduction. Four examples are given. The rst one shows how to approximate a system with a constant. This only requires one step and is

8 done without iterations. The second example reduces a third order system to rst order. For this problem the convergence is quite fast. The third example shows a case when the converge is slow, which indicates that the algorithm may fail to convergence in some applications. The fourth example shows a case when the improvement to the standard Hankel norm reduction is more signicant..1 Implementation In these examples a Matlab implementation employing the LMI-lab by Gahinet and Nemirovskii [5] has been made. This package has a user-friendly and exible interface for Matlab. The LMI-based algorithm has been compared to Hankel norm reduction. The following commands in the -Analysis and Synthesis Toolbox [1] has been used for obtaining the Hankel norm reduced model >> [bal, sv] = sysbal (sys) where bal is the balanced realization of the system sys and sv contains the Hankel singular values. Then the Hankel-norm reduced model is obtained using the command >> hank = hankmr (bal, sv, k, 'd') where k is the order of the reduced system hank.. Example 1: A Non-Iterative Solution We start by giving a very simple example of model reduction. We want to replace a dynamical system with a constant (D-matrix). This can be obtained without iterations since R ;1 = S. We consider the system G(s) = 1 (s + 1)(s +5) with Hankel singular values and Hankel-norm model reduction yields ^GH (s) = 0:1, while the LMI method gives ^G0 (s) = 0:0851. The model errors are 0.1 and respectively.. Example : A Third-order System We rst consider the third-order system G(s) = 1 s s + s + We will now try to reduce this to a rst order system. The Hankel singular values are 0.1, , Thus, by reducing the model order to one the model error can not be lower than =0:1911. The Hankel-norm model reduction guarantees a maximum H 1 error of P i, where the sum is taken over the removed states. In this case we get The Hankel-norm reduced model is ^G =1:51 1 ; 0:10651s 1+0:8516s 8

9 10 gamma convergence gamma gamma* iteration Figure 1: -convergence of Example. The LMI reduced model is ^G 1 (s) =1:1 1 ; 0:1109s 1+0:81918s : The model errors are and respectively. The LMI reduced model is only marginally better than the Hankel-norm reduced one (about.6%). The Hankel-norm model reduced model is in this case quite close to its theoretical bound and much improvements using other techniques cannot be obtained. In Figure 1 the error convergence versus iteration is shown. The plot gives ; where is the minimum value obtained during the iterative search. Usually it is also the value obtained in the last iteration but in this example we have run the algorithm to the extremes of the numerical precision of the computer. The convergence is linear (or geometric) in this case, that is the remaining error is scaled by an approximately constant factor less than one in each iteration. In this case the factor is about 0.5. The convergence rate depends on how well the LMIs are performed in each iterative step. If the LMIs, which are also iterative, are run into higher accuracy by modifying the stopping criterion, then the convergence rate of the model reduction iterations are faster. However, it is not known which trade-o oers the best overall computational performance. In Figure the model errors are shown for the Hankel-norm reduced (dashed line) and the LMI reduced (solid line) models.. Example : Slow converge The next example is somewhat more dicult. The system to be reduced is G(s) = 1 (s +1)(s + s +10) 9

10 0. Model error error magnitude frequency Figure : Model errors of Example. The Hankel model error is given as a dashed line and the LMI reduced model error as a solid line. The Hankel-norm singular values are , and The Hankel model reduction yields ^G H (s) =0:101 1 ; 0:686s 1+1:80116s with a model error of The LMI method reduces the error close to 0.05, which apparently is the optimal value. The LMI model converges to ^G 1 (s) = 1 1 ; s 0 1+s The convergence rate is in this case very slow, probably due to the fact that the original system and the optimally reduced system have a common pole. The convergence rate seems to be sublinear, at least during the rst 100 iterations (see Figure ). After about 50 iterations the error norm is not reduced any longer due to numerical errors. The accuracy in the LMI solver was set to 10 ;1. The reason for the diculty is not understood, but one reason could be that the model error converges towards an all-pass second order system G(s) ; ^G1 (s) = 1 0 s ; s +10 s + s +10 which is of order only compared to 5 of the augmented system ~ G..5 Example : A Higher-order Example Next we will look at a 11th order system: 10 1 ; 0:05s 1 G(s) = 1+0:05s 1+0:1s 10

11 10 gamma convergence gamma gamma* iteration Figure : Model error convergence versus iteration of Example The Hankel singular values are: 0.916, 0.896, 0.885, 0.6, , 0.50, 0.56, 0.68, 0.18, , and Since the singular values are of the same order, we can expect problems using the Hankel norm reduction due to the fact that the upper and lower bounds are far apart. Reducing the system to fth order with Hankel model reduction yields an H 1 error of LMI reduction yields after the rst iteration and after ve iterations..6 Computational Load The computational load was determined by running a number of LMI reductions of dierent orders. Each system was reduced to half that order, rst by Hankel reduction and then by one iteration of the LMI reduction. The CPU time for the two LMI steps were clocked. The computations were performed on a Sparcstation 5.. Summary The model errors obtained for the dierent examples are summarized in the table below. As can be seen the improvement in error norm by using LMI reduction is usually not signicant. However, the LMI model comes very close to the theoretical bound set by the Hankel singular values. Even if the the proposed model reduction algorithm is iterative, most of the improvement from Hankel norm reduction is achieved already after the rst iteration. 11

12 10 CPU time vs. model order 10 CPU time in seconds model order n Figure : CPU time per iteration versus model order n. The reduced model order k = n=. The test was performed on a Sparc-station 5 Example n! k Hankel LMI k+1 1! ! ! ! ! ! ! Conclusions A model reduction algorithm using linear matrix inequalities (LMIs) have been implemented and studied on a number of examples. The algorithm starts by using the Hankel-norm reduced model and performs improvements by applying an iterative two-step LMI scheme. The special case of reducing the model to a constant (matrix) with no states (k = 0) can be performed optimally in one step without iterations. The well-known Hankel norm reduction is noniterative and optimal for n ; 1. Generally, k [1 n; ], convergence to the global optimum can not be guaranteed, but higher and lower bounds are provided by the Hankel singular values. In all examples studied a better approximation in the H 1 sense have been found provided that the reduced model is of order n ; or less, where n is the order of the original model. Often, the improvement is not signicant (in the order of ten percents) but some cases with more improvement have been found. The computational load with the LMI algorithm is higher than Hankel norm reduction and truncated balanced realization: about O(n : ) (per iteration) 1

13 compared to O(n ). It is usually not worth-while to employ the LMI algorithm unless optimal performance is a necessity. In the examples studied the LMI reduced model has been close to the theoretical limit dened by the largest Hankel singular value that was removed ( k+1 ). The convergence of the iterative algorithm has not been analyzed. In the examples studied both linear (geometric) and sub-linear convergence occur. However, already in the rst step most of the improvement compared to the Hankel norm reduction is achieved. References [1] G. Balas, J. Doyle, K. Glover, A. Packard, and R. Smith. -Analysis and Synthesis Toolbox for Use with Matlab, User's Guide. The MathWorks, Inc., 199. [] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. SIAM Studies in Applied Mathematics. SIAM, 199. [] P. Gahinet and P. Apkarian. A linear matrix inequality approach toh 1 control. International Journal of Robust and Nonlinear Control, 1:656{661, October 199 (to appear in). [] P. Gahinet and P. Apkarian. An LMI-based parametrization of all H 1 controllers with applications. In IEEE Proceedings of the nd Conference on Decision and Control, volume 1, pages 656{661, San Antonio, Texas, December 199. [5] P. Gahinet and A. Nemirovskii. LMI-lab - A Package for Manipulating and Solving LMI's. Version [6] K. Glover. All optimal Hankel-norm approximations of linear multivariable systems and their L 1 -error bounds. International Journal of Control, 9(6):1115{119, 198. [] A. Helmersson. Model reduction using LMIs. In Proceedings of the Conference on Decision and Control, 199.accepted paper. [8] D. Kavranoglu. A computational scheme for H 1 model reduction. IEEE Transactions on Automatic Control, 9():1{151, July 199. [9] D. Kavranoglu and M. Bettayeb. Characterization of the solution to the optimal H 1 model reduction problem. Systems & Control Letters, 0():99{ 10, February 199. [10] M. Safonov, R. Chiang, and D Limebeer. Optimal Hankel model reduction for nonminimal systems. IEEE Transactions on Automatic Control, 5():96{50, April [11] C. Scherer. The Riccati Inequality and State-Space H 1 -Optimal Control. Ph. D. Dissertation, Universitat Wurtzburg, Germany,

1.1 Notations We dene X (s) =X T (;s), X T denotes the transpose of X X>()0 a symmetric, positive denite (semidenite) matrix diag [X 1 X ] a block-dia

1.1 Notations We dene X (s) =X T (;s), X T denotes the transpose of X X>()0 a symmetric, positive denite (semidenite) matrix diag [X 1 X ] a block-dia Applications of mixed -synthesis using the passivity approach A. Helmersson Department of Electrical Engineering Linkoping University S-581 83 Linkoping, Sweden tel: +46 13 816 fax: +46 13 86 email: andersh@isy.liu.se

More information

LMI Based Model Order Reduction Considering the Minimum Phase Characteristic of the System

LMI Based Model Order Reduction Considering the Minimum Phase Characteristic of the System LMI Based Model Order Reduction Considering the Minimum Phase Characteristic of the System Gholamreza Khademi, Haniyeh Mohammadi, and Maryam Dehghani School of Electrical and Computer Engineering Shiraz

More information

FEL3210 Multivariable Feedback Control

FEL3210 Multivariable Feedback Control FEL3210 Multivariable Feedback Control Lecture 8: Youla parametrization, LMIs, Model Reduction and Summary [Ch. 11-12] Elling W. Jacobsen, Automatic Control Lab, KTH Lecture 8: Youla, LMIs, Model Reduction

More information

Static Output Feedback Stabilisation with H Performance for a Class of Plants

Static 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 information

KTH. Access to the published version may require subscription.

KTH. Access to the published version may require subscription. KTH This is an accepted version of a paper published in IEEE Transactions on Automatic Control. This paper has been peer-reviewed but does not include the final publisher proof-corrections or journal pagination.

More information

Rank-one LMIs and Lyapunov's Inequality. Gjerrit Meinsma 4. Abstract. We describe a new proof of the well-known Lyapunov's matrix inequality about

Rank-one LMIs and Lyapunov's Inequality. Gjerrit Meinsma 4. Abstract. We describe a new proof of the well-known Lyapunov's matrix inequality about Rank-one LMIs and Lyapunov's Inequality Didier Henrion 1;; Gjerrit Meinsma Abstract We describe a new proof of the well-known Lyapunov's matrix inequality about the location of the eigenvalues of a matrix

More information

June Engineering Department, Stanford University. System Analysis and Synthesis. Linear Matrix Inequalities. Stephen Boyd (E.

June Engineering Department, Stanford University. System Analysis and Synthesis. Linear Matrix Inequalities. Stephen Boyd (E. Stephen Boyd (E. Feron :::) System Analysis and Synthesis Control Linear Matrix Inequalities via Engineering Department, Stanford University Electrical June 1993 ACC, 1 linear matrix inequalities (LMIs)

More information

To appear in IEEE Trans. on Automatic Control Revised 12/31/97. Output Feedback

To appear in IEEE Trans. on Automatic Control Revised 12/31/97. Output Feedback o appear in IEEE rans. on Automatic Control Revised 12/31/97 he Design of Strictly Positive Real Systems Using Constant Output Feedback C.-H. Huang P.A. Ioannou y J. Maroulas z M.G. Safonov x Abstract

More information

Parameterized Linear Matrix Inequality Techniques in Fuzzy Control System Design

Parameterized Linear Matrix Inequality Techniques in Fuzzy Control System Design 324 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 9, NO. 2, APRIL 2001 Parameterized Linear Matrix Inequality Techniques in Fuzzy Control System Design H. D. Tuan, P. Apkarian, T. Narikiyo, and Y. Yamamoto

More information

Simultaneous State and Fault Estimation for Descriptor Systems using an Augmented PD Observer

Simultaneous 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 information

Linear Systems with Saturating Controls: An LMI Approach. subject to control saturation. No assumption is made concerning open-loop stability and no

Linear Systems with Saturating Controls: An LMI Approach. subject to control saturation. No assumption is made concerning open-loop stability and no Output Feedback Robust Stabilization of Uncertain Linear Systems with Saturating Controls: An LMI Approach Didier Henrion 1 Sophie Tarbouriech 1; Germain Garcia 1; Abstract : The problem of robust controller

More information

only nite eigenvalues. This is an extension of earlier results from [2]. Then we concentrate on the Riccati equation appearing in H 2 and linear quadr

only nite eigenvalues. This is an extension of earlier results from [2]. Then we concentrate on the Riccati equation appearing in H 2 and linear quadr The discrete algebraic Riccati equation and linear matrix inequality nton. Stoorvogel y Department of Mathematics and Computing Science Eindhoven Univ. of Technology P.O. ox 53, 56 M Eindhoven The Netherlands

More information

Linear Matrix Inequality (LMI)

Linear Matrix Inequality (LMI) Linear Matrix Inequality (LMI) A linear matrix inequality is an expression of the form where F (x) F 0 + x 1 F 1 + + x m F m > 0 (1) x = (x 1,, x m ) R m, F 0,, F m are real symmetric matrices, and the

More information

LMI based output-feedback controllers: γ-optimal versus linear quadratic.

LMI 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 information

Convex Optimization Approach to Dynamic Output Feedback Control for Delay Differential Systems of Neutral Type 1,2

Convex Optimization Approach to Dynamic Output Feedback Control for Delay Differential Systems of Neutral Type 1,2 journal of optimization theory and applications: Vol. 127 No. 2 pp. 411 423 November 2005 ( 2005) DOI: 10.1007/s10957-005-6552-7 Convex Optimization Approach to Dynamic Output Feedback Control for Delay

More information

On the solving of matrix equation of Sylvester type

On the solving of matrix equation of Sylvester type Computational Methods for Differential Equations http://cmde.tabrizu.ac.ir Vol. 7, No. 1, 2019, pp. 96-104 On the solving of matrix equation of Sylvester type Fikret Ahmadali Aliev Institute of Applied

More information

Research Article An Equivalent LMI Representation of Bounded Real Lemma for Continuous-Time Systems

Research Article An Equivalent LMI Representation of Bounded Real Lemma for Continuous-Time Systems Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 28, Article ID 67295, 8 pages doi:1.1155/28/67295 Research Article An Equivalent LMI Representation of Bounded Real Lemma

More information

Linköping studies in science and technology. Dissertations No. 406

Linköping studies in science and technology. Dissertations No. 406 Linköping studies in science and technology. Dissertations No. 406 Methods for Robust Gain Scheduling Anders Helmersson Department of Electrical Engineering Linkoping University S-581 8 Linkoping, Sweden

More information

where m r, m c and m C are the number of repeated real scalar blocks, repeated complex scalar blocks and full complex blocks, respectively. A. (D; G)-

where m r, m c and m C are the number of repeated real scalar blocks, repeated complex scalar blocks and full complex blocks, respectively. A. (D; G)- 1 Some properties of an upper bound for Gjerrit Meinsma, Yash Shrivastava and Minyue Fu Abstract A convex upper bound of the mixed structured singular value is analyzed. The upper bound is based on a multiplier

More information

H State-Feedback Controller Design for Discrete-Time Fuzzy Systems Using Fuzzy Weighting-Dependent Lyapunov Functions

H State-Feedback Controller Design for Discrete-Time Fuzzy Systems Using Fuzzy Weighting-Dependent Lyapunov Functions IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL 11, NO 2, APRIL 2003 271 H State-Feedback Controller Design for Discrete-Time Fuzzy Systems Using Fuzzy Weighting-Dependent Lyapunov Functions Doo Jin Choi and PooGyeon

More information

An Exact Stability Analysis Test for Single-Parameter. Polynomially-Dependent Linear Systems

An Exact Stability Analysis Test for Single-Parameter. Polynomially-Dependent Linear Systems An Exact Stability Analysis Test for Single-Parameter Polynomially-Dependent Linear Systems P. Tsiotras and P.-A. Bliman Abstract We provide a new condition for testing the stability of a single-parameter,

More information

Linear Regression and Its Applications

Linear Regression and Its Applications Linear Regression and Its Applications Predrag Radivojac October 13, 2014 Given a data set D = {(x i, y i )} n the objective is to learn the relationship between features and the target. We usually start

More information

LOW ORDER H CONTROLLER DESIGN: AN LMI APPROACH

LOW ORDER H CONTROLLER DESIGN: AN LMI APPROACH LOW ORDER H CONROLLER DESIGN: AN LMI APPROACH Guisheng Zhai, Shinichi Murao, Naoki Koyama, Masaharu Yoshida Faculty of Systems Engineering, Wakayama University, Wakayama 640-8510, Japan Email: zhai@sys.wakayama-u.ac.jp

More information

w 1... w L z 1... w e z r

w 1... w L z 1... w e z r Multiobjective H H 1 -Optimal Control via Finite Dimensional Q-Parametrization and Linear Matrix Inequalities 1 Haitham A. Hindi Babak Hassibi Stephen P. Boyd Department of Electrical Engineering, Durand

More information

Stability of linear time-varying systems through quadratically parameter-dependent Lyapunov functions

Stability of linear time-varying systems through quadratically parameter-dependent Lyapunov functions Stability of linear time-varying systems through quadratically parameter-dependent Lyapunov functions Vinícius F. Montagner Department of Telematics Pedro L. D. Peres School of Electrical and Computer

More information

Vector Space Basics. 1 Abstract Vector Spaces. 1. (commutativity of vector addition) u + v = v + u. 2. (associativity of vector addition)

Vector Space Basics. 1 Abstract Vector Spaces. 1. (commutativity of vector addition) u + v = v + u. 2. (associativity of vector addition) Vector Space Basics (Remark: these notes are highly formal and may be a useful reference to some students however I am also posting Ray Heitmann's notes to Canvas for students interested in a direct computational

More information

Robust Anti-Windup Compensation for PID Controllers

Robust Anti-Windup Compensation for PID Controllers Robust Anti-Windup Compensation for PID Controllers ADDISON RIOS-BOLIVAR Universidad de Los Andes Av. Tulio Febres, Mérida 511 VENEZUELA FRANCKLIN RIVAS-ECHEVERRIA Universidad de Los Andes Av. Tulio Febres,

More information

Optimization based robust control

Optimization based robust control Optimization based robust control Didier Henrion 1,2 Draft of March 27, 2014 Prepared for possible inclusion into The Encyclopedia of Systems and Control edited by John Baillieul and Tariq Samad and published

More information

On Computing the Worst-case Performance of Lur'e Systems with Uncertain Time-invariant Delays

On Computing the Worst-case Performance of Lur'e Systems with Uncertain Time-invariant Delays Article On Computing the Worst-case Performance of Lur'e Systems with Uncertain Time-invariant Delays Thapana Nampradit and David Banjerdpongchai* Department of Electrical Engineering, Faculty of Engineering,

More information

Robust Multivariable Control

Robust Multivariable Control Lecture 1 Anders Helmersson anders.helmersson@liu.se ISY/Reglerteknik Linköpings universitet Addresses email: anders.helmerson@liu.se mobile: 0734278419 http://users.isy.liu.se/rt/andersh/teaching/robkurs.html

More information

FINITE HORIZON ROBUST MODEL PREDICTIVE CONTROL USING LINEAR MATRIX INEQUALITIES. Danlei Chu, Tongwen Chen, Horacio J. Marquez

FINITE HORIZON ROBUST MODEL PREDICTIVE CONTROL USING LINEAR MATRIX INEQUALITIES. Danlei Chu, Tongwen Chen, Horacio J. Marquez FINITE HORIZON ROBUST MODEL PREDICTIVE CONTROL USING LINEAR MATRIX INEQUALITIES Danlei Chu Tongwen Chen Horacio J Marquez Department of Electrical and Computer Engineering University of Alberta Edmonton

More information

A Singular Value Decomposition Based Closed Loop Stability Preserving Controller Reduction Method

A Singular Value Decomposition Based Closed Loop Stability Preserving Controller Reduction Method A Singular Value Decomposition Based Closed Loop Stability Preserving Controller Reduction Method Kin Cheong Sou and Anders Rantzer Abstract In this paper a controller reduction method which preserves

More information

Chapter 6 Balanced Realization 6. Introduction One popular approach for obtaining a minimal realization is known as Balanced Realization. In this appr

Chapter 6 Balanced Realization 6. Introduction One popular approach for obtaining a minimal realization is known as Balanced Realization. In this appr Lectures on ynamic Systems and Control Mohammed ahleh Munther A. ahleh George Verghese epartment of Electrical Engineering and Computer Science Massachuasetts Institute of Technology c Chapter 6 Balanced

More information

Outline Introduction: Problem Description Diculties Algebraic Structure: Algebraic Varieties Rank Decient Toeplitz Matrices Constructing Lower Rank St

Outline Introduction: Problem Description Diculties Algebraic Structure: Algebraic Varieties Rank Decient Toeplitz Matrices Constructing Lower Rank St Structured Lower Rank Approximation by Moody T. Chu (NCSU) joint with Robert E. Funderlic (NCSU) and Robert J. Plemmons (Wake Forest) March 5, 1998 Outline Introduction: Problem Description Diculties Algebraic

More information

Marcus Pantoja da Silva 1 and Celso Pascoli Bottura 2. Abstract: Nonlinear systems with time-varying uncertainties

Marcus Pantoja da Silva 1 and Celso Pascoli Bottura 2. Abstract: Nonlinear systems with time-varying uncertainties A NEW PROPOSAL FOR H NORM CHARACTERIZATION AND THE OPTIMAL H CONTROL OF NONLINEAR SSTEMS WITH TIME-VARING UNCERTAINTIES WITH KNOWN NORM BOUND AND EXOGENOUS DISTURBANCES Marcus Pantoja da Silva 1 and Celso

More information

Chapter 3 Least Squares Solution of y = A x 3.1 Introduction We turn to a problem that is dual to the overconstrained estimation problems considered s

Chapter 3 Least Squares Solution of y = A x 3.1 Introduction We turn to a problem that is dual to the overconstrained estimation problems considered s Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology 1 1 c Chapter

More information

Fast Algorithms for SDPs derived from the Kalman-Yakubovich-Popov Lemma

Fast Algorithms for SDPs derived from the Kalman-Yakubovich-Popov Lemma Fast Algorithms for SDPs derived from the Kalman-Yakubovich-Popov Lemma Venkataramanan (Ragu) Balakrishnan School of ECE, Purdue University 8 September 2003 European Union RTN Summer School on Multi-Agent

More information

The norms can also be characterized in terms of Riccati inequalities.

The norms can also be characterized in terms of Riccati inequalities. 9 Analysis of stability and H norms Consider the causal, linear, time-invariant system ẋ(t = Ax(t + Bu(t y(t = Cx(t Denote the transfer function G(s := C (si A 1 B. Theorem 85 The following statements

More information

H controller design on the COMPLIB problems with the Robust Control Toolbox for Matlab

H controller design on the COMPLIB problems with the Robust Control Toolbox for Matlab H controller design on the COMPLIB problems with the Robust Control Toolbox for Matlab Didier Henrion 1,2 Draft of August 30, 2005 1 Introduction The COMPLIB library is a freely available Matlab package

More information

Dept. of Aeronautics and Astronautics. because the structure of the closed-loop system has not been

Dept. of Aeronautics and Astronautics. because the structure of the closed-loop system has not been LMI Synthesis of Parametric Robust H 1 Controllers 1 David Banjerdpongchai Durand Bldg., Room 110 Dept. of Electrical Engineering Email: banjerd@isl.stanford.edu Jonathan P. How Durand Bldg., Room Dept.

More information

A new robust delay-dependent stability criterion for a class of uncertain systems with delay

A new robust delay-dependent stability criterion for a class of uncertain systems with delay A new robust delay-dependent stability criterion for a class of uncertain systems with delay Fei Hao Long Wang and Tianguang Chu Abstract A new robust delay-dependent stability criterion for a class of

More information

Mixed Parametric/Unstructured LFT Modelling for Robust Controller Design

Mixed Parametric/Unstructured LFT Modelling for Robust Controller Design 2 American Control Conference on O'Farrell Street, San Francisco, CA, USA June 29 - July, 2 Mixed Parametric/Unstructured LFT Modelling for Robust Controller Design Harald Pfifer and Simon Hecker Abstract

More information

An LQ R weight selection approach to the discrete generalized H 2 control problem

An LQ R weight selection approach to the discrete generalized H 2 control problem INT. J. CONTROL, 1998, VOL. 71, NO. 1, 93± 11 An LQ R weight selection approach to the discrete generalized H 2 control problem D. A. WILSON², M. A. NEKOUI² and G. D. HALIKIAS² It is known that a generalized

More information

REGLERTEKNIK AUTOMATIC CONTROL LINKÖPING

REGLERTEKNIK AUTOMATIC CONTROL LINKÖPING QC Synthesis based on nertia Constraints Anders Helmerssonn Department of Electrical Engineering Linkoping University, SE-581 83 Linkoping, Sweden www: http://www.control.isy.liu.se email: andersh@isy.liu.se

More information

Chapter 6 Balanced Realization 6. Introduction One popular approach for obtaining a minimal realization is known as Balanced Realization. In this appr

Chapter 6 Balanced Realization 6. Introduction One popular approach for obtaining a minimal realization is known as Balanced Realization. In this appr Lectures on ynamic Systems and Control Mohammed ahleh Munther A. ahleh George Verghese epartment of Electrical Engineering and Computer Science Massachuasetts Institute of Technology c Chapter 6 Balanced

More information

PARAMETERIZATION OF MODEL VALIDATING SETS FOR UNCERTAINTY BOUND OPTIMIZATIONS. K.B. Lim, D.P. Giesy y. NASA Langley Research Center.

PARAMETERIZATION OF MODEL VALIDATING SETS FOR UNCERTAINTY BOUND OPTIMIZATIONS. K.B. Lim, D.P. Giesy y. NASA Langley Research Center. PARAMETERIZATION OF MODEL VALIDATING SETS FOR UNCERTAINTY BOUND OPTIMIZATIONS KB Lim, DP Giesy y NASA Langley Research Center Hampton, Virginia Abstract Given experimental data and a priori assumptions

More information

UNCERTAINTY MODELING VIA FREQUENCY DOMAIN MODEL VALIDATION

UNCERTAINTY MODELING VIA FREQUENCY DOMAIN MODEL VALIDATION AIAA 99-3959 UNCERTAINTY MODELING VIA FREQUENCY DOMAIN MODEL VALIDATION Martin R. Waszak, * NASA Langley Research Center, Hampton, Virginia Dominick Andrisani II, Purdue University, West Lafayette, Indiana

More information

arxiv: v1 [math.oc] 17 Oct 2014

arxiv: v1 [math.oc] 17 Oct 2014 SiMpLIfy: A Toolbox for Structured Model Reduction Martin Biel, Farhad Farokhi, and Henrik Sandberg arxiv:1414613v1 [mathoc] 17 Oct 214 Abstract In this paper, we present a toolbox for structured model

More information

APPLICATION OF D-K ITERATION TECHNIQUE BASED ON H ROBUST CONTROL THEORY FOR POWER SYSTEM STABILIZER DESIGN

APPLICATION OF D-K ITERATION TECHNIQUE BASED ON H ROBUST CONTROL THEORY FOR POWER SYSTEM STABILIZER DESIGN APPLICATION OF D-K ITERATION TECHNIQUE BASED ON H ROBUST CONTROL THEORY FOR POWER SYSTEM STABILIZER DESIGN Amitava Sil 1 and S Paul 2 1 Department of Electrical & Electronics Engineering, Neotia Institute

More information

Robust Observer for Uncertain T S model of a Synchronous Machine

Robust Observer for Uncertain T S model of a Synchronous Machine Recent Advances in Circuits Communications Signal Processing Robust Observer for Uncertain T S model of a Synchronous Machine OUAALINE Najat ELALAMI Noureddine Laboratory of Automation Computer Engineering

More information

Model reduction for linear systems by balancing

Model 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 information

A MULTIGRID ALGORITHM FOR. Richard E. Ewing and Jian Shen. Institute for Scientic Computation. Texas A&M University. College Station, Texas SUMMARY

A MULTIGRID ALGORITHM FOR. Richard E. Ewing and Jian Shen. Institute for Scientic Computation. Texas A&M University. College Station, Texas SUMMARY A MULTIGRID ALGORITHM FOR THE CELL-CENTERED FINITE DIFFERENCE SCHEME Richard E. Ewing and Jian Shen Institute for Scientic Computation Texas A&M University College Station, Texas SUMMARY In this article,

More information

On the Application of Model-Order Reduction Algorithms

On the Application of Model-Order Reduction Algorithms Proceedings of the 6th International Conference on System Theory, Control and Computing Sinaia, Romania, October 2-4, 22 IEEE Catalog Number CFP236P-CDR, ISBN 978-66-834-848-3 On the Application of Model-Order

More information

On Bounded Real Matrix Inequality Dilation

On Bounded Real Matrix Inequality Dilation On Bounded Real Matrix Inequality Dilation Solmaz Sajjadi-Kia and Faryar Jabbari Abstract We discuss a variation of dilated matrix inequalities for the conventional Bounded Real matrix inequality, and

More information

A Comparative Study on Automatic Flight Control for small UAV

A Comparative Study on Automatic Flight Control for small UAV Proceedings of the 5 th International Conference of Control, Dynamic Systems, and Robotics (CDSR'18) Niagara Falls, Canada June 7 9, 18 Paper No. 13 DOI: 1.11159/cdsr18.13 A Comparative Study on Automatic

More information

While using the input and output data fu(t)g and fy(t)g, by the methods in system identification, we can get a black-box model like (In the case where

While using the input and output data fu(t)g and fy(t)g, by the methods in system identification, we can get a black-box model like (In the case where ESTIMATE PHYSICAL PARAMETERS BY BLACK-BOX MODELING Liang-Liang Xie Λ;1 and Lennart Ljung ΛΛ Λ Institute of Systems Science, Chinese Academy of Sciences, 100080, Beijing, China ΛΛ Department of Electrical

More information

Robust linear optimization under general norms

Robust linear optimization under general norms Operations Research Letters 3 (004) 50 56 Operations Research Letters www.elsevier.com/locate/dsw Robust linear optimization under general norms Dimitris Bertsimas a; ;, Dessislava Pachamanova b, Melvyn

More information

Math Camp Notes: Linear Algebra I

Math Camp Notes: Linear Algebra I Math Camp Notes: Linear Algebra I Basic Matrix Operations and Properties Consider two n m matrices: a a m A = a n a nm Then the basic matrix operations are as follows: a + b a m + b m A + B = a n + b n

More information

Robust Output Feedback Controller Design via Genetic Algorithms and LMIs: The Mixed H 2 /H Problem

Robust Output Feedback Controller Design via Genetic Algorithms and LMIs: The Mixed H 2 /H Problem Robust Output Feedback Controller Design via Genetic Algorithms and LMIs: The Mixed H 2 /H Problem Gustavo J. Pereira and Humberto X. de Araújo Abstract This paper deals with the mixed H 2/H control problem

More information

RECENTLY, there has been renewed research interest

RECENTLY, there has been renewed research interest IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 49, NO 12, DECEMBER 2004 2113 Distributed Control of Heterogeneous Systems Geir E Dullerud Raffaello D Andrea Abstract This paper considers control design for

More information

Model Reduction for Unstable Systems

Model Reduction for Unstable Systems Model Reduction for Unstable Systems Klajdi Sinani Virginia Tech klajdi@vt.edu Advisor: Serkan Gugercin October 22, 2015 (VT) SIAM October 22, 2015 1 / 26 Overview 1 Introduction 2 Interpolatory Model

More information

H 2 optimal model reduction - Wilson s conditions for the cross-gramian

H 2 optimal model reduction - Wilson s conditions for the cross-gramian H 2 optimal model reduction - Wilson s conditions for the cross-gramian Ha Binh Minh a, Carles Batlle b a School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Dai

More information

Multi-Model Adaptive Regulation for a Family of Systems Containing Different Zero Structures

Multi-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 information

Multiobjective Optimization Applied to Robust H 2 /H State-feedback Control Synthesis

Multiobjective Optimization Applied to Robust H 2 /H State-feedback Control Synthesis Multiobjective Optimization Applied to Robust H 2 /H State-feedback Control Synthesis Eduardo N. Gonçalves, Reinaldo M. Palhares, and Ricardo H. C. Takahashi Abstract This paper presents an algorithm for

More information

Model reduction of interconnected systems

Model reduction of interconnected systems Model reduction of interconnected systems A Vandendorpe and P Van Dooren 1 Introduction Large scale linear systems are often composed of subsystems that interconnect to each other Instead of reducing the

More information

Here, u is the control input with m components, y is the measured output with k componenets, and the channels w j z j from disturbance inputs to contr

Here, u is the control input with m components, y is the measured output with k componenets, and the channels w j z j from disturbance inputs to contr From Mixed to Multi-Objective ontrol arsten W. Scherer Mechanical Engineering Systems and ontrol Group Delft University of Technology Mekelweg, 8 D Delft, The Netherlands Paper ID: Reg Abstract. We revisit

More information

A Gain-Based Lower Bound Algorithm for Real and Mixed µ Problems

A Gain-Based Lower Bound Algorithm for Real and Mixed µ Problems A Gain-Based Lower Bound Algorithm for Real and Mixed µ Problems Pete Seiler, Gary Balas, and Andrew Packard Abstract In this paper we present a new lower bound algorithm for real and mixed µ problems.

More information

Filter Design for Linear Time Delay Systems

Filter Design for Linear Time Delay Systems IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 11, NOVEMBER 2001 2839 ANewH Filter Design for Linear Time Delay Systems E. Fridman Uri Shaked, Fellow, IEEE Abstract A new delay-dependent filtering

More information

Linear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) 1.1 The Formal Denition of a Vector Space

Linear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) 1.1 The Formal Denition of a Vector Space Linear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) Contents 1 Vector Spaces 1 1.1 The Formal Denition of a Vector Space.................................. 1 1.2 Subspaces...................................................

More information

Discrete time Generalized Cellular. Johan A.K. Suykens and Joos Vandewalle. Katholieke Universiteit Leuven

Discrete time Generalized Cellular. Johan A.K. Suykens and Joos Vandewalle. Katholieke Universiteit Leuven Discrete time Generalized Cellular Neural Networks within NL q Theory Johan A.K. Suykens and Joos Vandewalle Katholieke Universiteit Leuven Department of Electrical Engineering, ESAT-SISTA Kardinaal Mercierlaan

More information

A Linear Matrix Inequality Approach to Robust Filtering

A Linear Matrix Inequality Approach to Robust Filtering 2338 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 9, SEPTEMBER 1997 A Linear Matrix Inequality Approach to Robust Filtering Huaizhong Li Minyue Fu, Senior Member, IEEE Abstract In this paper, we

More information

Global Optimization of H problems: Application to robust control synthesis under structural constraints

Global Optimization of H problems: Application to robust control synthesis under structural constraints Global Optimization of H problems: Application to robust control synthesis under structural constraints Dominique Monnet 1, Jordan Ninin 1, and Benoit Clement 1 ENSTA-Bretagne, LabSTIC, IHSEV team, 2 rue

More information

u - P (s) y (s) Figure 1: Standard framework for robustness analysis function matrix and let (s) be a structured perturbation constrained to lie in th

u - P (s) y (s) Figure 1: Standard framework for robustness analysis function matrix and let (s) be a structured perturbation constrained to lie in th A fast algorithm for the computation of an upper bound on the -norm Craig T. Lawrence, y Andre L. Tits y Department of Electrical Engineering and Institute for Systems Research, University of Maryland,

More information

Some applications ofsmall gain theorem to interconnected systems

Some applications ofsmall gain theorem to interconnected systems Available online at www.sciencedirect.com Systems & Control Letters 5 004 63 73 www.elsevier.com/locate/sysconle Some applications ofsmall gain theorem to interconnected systems Zhisheng Duan, Lin Huang,

More information

A Survey Of State-Feedback Simultaneous Stabilization Techniques

A Survey Of State-Feedback Simultaneous Stabilization Techniques University of New Mexico UNM Digital Repository Electrical & Computer Engineering Faculty Publications Engineering Publications 4-13-2012 A Survey Of State-Feedback Simultaneous Stabilization Techniques

More information

In: Proc. BENELEARN-98, 8th Belgian-Dutch Conference on Machine Learning, pp 9-46, 998 Linear Quadratic Regulation using Reinforcement Learning Stephan ten Hagen? and Ben Krose Department of Mathematics,

More information

Gary J. Balas Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN USA

Gary J. Balas Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN USA μ-synthesis Gary J. Balas Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455 USA Keywords: Robust control, ultivariable control, linear fractional transforation (LFT),

More information

New concepts: Span of a vector set, matrix column space (range) Linearly dependent set of vectors Matrix null space

New concepts: Span of a vector set, matrix column space (range) Linearly dependent set of vectors Matrix null space Lesson 6: Linear independence, matrix column space and null space New concepts: Span of a vector set, matrix column space (range) Linearly dependent set of vectors Matrix null space Two linear systems:

More information

Linear Algebra. Preliminary Lecture Notes

Linear Algebra. Preliminary Lecture Notes Linear Algebra Preliminary Lecture Notes Adolfo J. Rumbos c Draft date April 29, 23 2 Contents Motivation for the course 5 2 Euclidean n dimensional Space 7 2. Definition of n Dimensional Euclidean Space...........

More information

State feedback gain scheduling for linear systems with time-varying parameters

State feedback gain scheduling for linear systems with time-varying parameters State feedback gain scheduling for linear systems with time-varying parameters Vinícius F. Montagner and Pedro L. D. Peres Abstract This paper addresses the problem of parameter dependent state feedback

More information

The servo problem for piecewise linear systems

The servo problem for piecewise linear systems The servo problem for piecewise linear systems Stefan Solyom and Anders Rantzer Department of Automatic Control Lund Institute of Technology Box 8, S-22 Lund Sweden {stefan rantzer}@control.lth.se Abstract

More information

A Bisection Algorithm for the Mixed µ Upper Bound and its Supremum

A Bisection Algorithm for the Mixed µ Upper Bound and its Supremum A Bisection Algorithm for the Mixed µ Upper Bound and its Supremum Carl-Magnus Fransson Control and Automation Laboratory, Department of Signals and Systems Chalmers University of Technology, SE-42 96

More information

Absolute Value Programming

Absolute Value Programming O. L. Mangasarian Absolute Value Programming Abstract. We investigate equations, inequalities and mathematical programs involving absolute values of variables such as the equation Ax + B x = b, where A

More information

New Lyapunov Krasovskii functionals for stability of linear retarded and neutral type systems

New Lyapunov Krasovskii functionals for stability of linear retarded and neutral type systems Systems & Control Letters 43 (21 39 319 www.elsevier.com/locate/sysconle New Lyapunov Krasovskii functionals for stability of linear retarded and neutral type systems E. Fridman Department of Electrical

More information

IN THIS paper, we will consider the analysis and synthesis

IN THIS paper, we will consider the analysis and synthesis 1654 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 42, NO. 12, DECEMBER 1997 Robustness Analysis and Synthesis for Nonlinear Uncertain Systems Wei-Min Lu, Member, IEEE, and John C. Doyle Abstract A state-space

More information

Complexity Reduction for Parameter-Dependent Linear Systems

Complexity Reduction for Parameter-Dependent Linear Systems Complexity Reduction for Parameter-Dependent Linear Systems Farhad Farokhi Henrik Sandberg and Karl H. Johansson Abstract We present a complexity reduction algorithm for a family of parameter-dependent

More information

Departement Elektrotechniek ESAT-SISTA/TR About the choice of State Space Basis in Combined. Deterministic-Stochastic Subspace Identication 1

Departement Elektrotechniek ESAT-SISTA/TR About the choice of State Space Basis in Combined. Deterministic-Stochastic Subspace Identication 1 Katholieke Universiteit Leuven Departement Elektrotechniek ESAT-SISTA/TR 994-24 About the choice of State Space asis in Combined Deterministic-Stochastic Subspace Identication Peter Van Overschee and art

More information

Intrinsic diculties in using the. control theory. 1. Abstract. We point out that the natural denitions of stability and

Intrinsic diculties in using the. control theory. 1. Abstract. We point out that the natural denitions of stability and Intrinsic diculties in using the doubly-innite time axis for input-output control theory. Tryphon T. Georgiou 2 and Malcolm C. Smith 3 Abstract. We point out that the natural denitions of stability and

More information

DECENTRALIZED CONTROL DESIGN USING LMI MODEL REDUCTION

DECENTRALIZED 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 information

Linear Algebra (part 1) : Matrices and Systems of Linear Equations (by Evan Dummit, 2016, v. 2.02)

Linear Algebra (part 1) : Matrices and Systems of Linear Equations (by Evan Dummit, 2016, v. 2.02) Linear Algebra (part ) : Matrices and Systems of Linear Equations (by Evan Dummit, 206, v 202) Contents 2 Matrices and Systems of Linear Equations 2 Systems of Linear Equations 2 Elimination, Matrix Formulation

More information

Hybrid Systems Course Lyapunov stability

Hybrid Systems Course Lyapunov stability Hybrid Systems Course Lyapunov stability OUTLINE Focus: stability of an equilibrium point continuous systems decribed by ordinary differential equations (brief review) hybrid automata OUTLINE Focus: stability

More information

Complexity Reduction for Parameter-Dependent Linear Systems

Complexity Reduction for Parameter-Dependent Linear Systems 213 American Control Conference (ACC) Washington DC USA June 17-19 213 Complexity Reduction for Parameter-Dependent Linear Systems Farhad Farokhi Henrik Sandberg and Karl H. Johansson Abstract We present

More information

Solution Set 3, Fall '12

Solution Set 3, Fall '12 Solution Set 3, 86 Fall '2 Do Problem 5 from 32 [ 3 5 Solution (a) A = Only one elimination step is needed to produce the 2 6 echelon form The pivot is the in row, column, and the entry to eliminate is

More information

M. Matrices and Linear Algebra

M. Matrices and Linear Algebra M. Matrices and Linear Algebra. Matrix algebra. In section D we calculated the determinants of square arrays of numbers. Such arrays are important in mathematics and its applications; they are called matrices.

More information

Linear Algebra, 4th day, Thursday 7/1/04 REU Info:

Linear Algebra, 4th day, Thursday 7/1/04 REU Info: Linear Algebra, 4th day, Thursday 7/1/04 REU 004. Info http//people.cs.uchicago.edu/laci/reu04. Instructor Laszlo Babai Scribe Nick Gurski 1 Linear maps We shall study the notion of maps between vector

More information

A New Strategy to the Multi-Objective Control of Linear Systems

A New Strategy to the Multi-Objective Control of Linear Systems Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 25 Seville, Spain, December 12-15, 25 TuC8.6 A New Strategy to the Multi-Objective Control of Linear

More information

Null controllable region of LTI discrete-time systems with input saturation

Null controllable region of LTI discrete-time systems with input saturation Automatica 38 (2002) 2009 2013 www.elsevier.com/locate/automatica Technical Communique Null controllable region of LTI discrete-time systems with input saturation Tingshu Hu a;, Daniel E. Miller b,liqiu

More information

Finding Succinct. Ordered Minimal Perfect. Hash Functions. Steven S. Seiden 3 Daniel S. Hirschberg 3. September 22, Abstract

Finding Succinct. Ordered Minimal Perfect. Hash Functions. Steven S. Seiden 3 Daniel S. Hirschberg 3. September 22, Abstract Finding Succinct Ordered Minimal Perfect Hash Functions Steven S. Seiden 3 Daniel S. Hirschberg 3 September 22, 1994 Abstract An ordered minimal perfect hash table is one in which no collisions occur among

More information

1. The Polar Decomposition

1. The Polar Decomposition A PERSONAL INTERVIEW WITH THE SINGULAR VALUE DECOMPOSITION MATAN GAVISH Part. Theory. The Polar Decomposition In what follows, F denotes either R or C. The vector space F n is an inner product space with

More information

Chapter 8 Gradient Methods

Chapter 8 Gradient Methods Chapter 8 Gradient Methods An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Introduction Recall that a level set of a function is the set of points satisfying for some constant. Thus, a point

More information