Parametrization of All Strictly Causal Stabilizing Controllers of Multidimensional Systems single-input single-output case

Size: px
Start display at page:

Download "Parametrization of All Strictly Causal Stabilizing Controllers of Multidimensional Systems single-input single-output case"

Transcription

1 Parametrization of All Strictly Causal Stabilizing Controllers of Multidimensional Systems single-input single-output case K. Mori Abstract We give a parametrization of all strictly causal stabilizing controllers of single-input singleoutput multidimensional systems with the structural stability. The approach used is the factorization approach. 1 Introduction The objective of this paper is to present a parametrization method of all strictly causal stabilizing controllers for single-input and single-output multidimensional systems in the framework of the factorization approach[1, 2, 3. A transfer function of the factorization approach is considered as the ratio of two stable causal transfer functions. Further the set of the stable causal transfer functions is considered as a commutative ring. Since the stabilizing controllers are not unique in general, the choice of the stabilizing controllers is important for the resulting closed loop. In the single-input and single-output case, there exist the coprime factorizations of the given stabilizable plant (Corollary of [4). Thus the stabilizing controllers can be parameterized by the method called Youla- School of Computer Science and Engineering, The University of Aizu, Aizu-Wakamatsu , JAPAN Kazuyoshi.MORI@IEEE.ORG

2 Kučera-parametrization[1, 5, 6, 3, 2, 7. However, it is also known that such parametrization may include a stabilizing controllers which is not causal and may result a direct loop (see Section 2). Thus we present an alternative parametrization of stabilizing controllers, that is the parametrization of all strictly causal stabilizing controllers. This parametrization will include neither any non-causal stabilizing controller nor any direct loop. Fact 4.9 of [8 gives the existence of a strictly causal stabilizing controller of a causal stabilizable plant. On the other hand, this paper gives all strictly causal stabilizing controllers with n-parameters (for n-dimensional systems). 2 Motivation Let us consider the feedback system shown in Figure 1 of the classical discrete-time system. Although this is a 1-dimensional system, the observation here can be applied to multidimensional systems analogously. Let c p Figure 1. Feedback system. A = {f(z) R(z) f(z) has no poles on or inside the unit circle in the complex plane} (1) (z denotes the delay operator), which is the set of stable causal transfer functions. Now we let p = z + 1. Then we have the Bézout identity (z + 1)y + x = 1 over A. Using the Youla-Kučera-parametrization, these y and x can be parametrized as y = r, x = 1 (z + 1)r with a parameter r of A. By letting r = 1, we have y = 1 and x = z. In this case, the stabilizing controller obtained from y and x is 1/z. Unfortunately this is not causal. Otherwise, by letting r = 1, we have y = 1 and x = z + 2. In this case, the stabilizing controller obtained from y and x is 1/(z + 2). This results a direct loop, that is, the current input affects the whole closed feedback system immediately as shown in Figure 2. This loop is normally unsafe even if the total feedback system is stable. To avoid these situation, we will present the parametrization of all strictly causal stabilizing controllers, (i) which does not include any non-causal stabilizing controller and (ii) which does not result any direct loop.

3 3 Preliminaries We employ the factorization approach [1, 6, 3, 2 and the symbols used in [9 and [10. The reader is referred to Appendix A of [3 for algebraic preliminaries if necessary. For a moment let us consider a general linear systems. Denote by A a commutative ring that is the set of stable causal transfer functions. The total ring of fractions of A is denoted by F; that is, F = {n/d n, d A, d is a nonzero divisor}. This F is considered to be the set of all possible transfer functions. Let Z be a prime ideal of A with Z A. Further, let P = {a/b F a A, b A Z} and P s = {a/b F a Z, b A Z}. A transfer function f is said to be causal (strictly causal) if and only if f is in P (P s ). We here apply the factorization approach to the multidimensional systems with structural stability. To do so, we need to give (i) the set A of stable causal transfer functions and (ii) the ideal Z for the causality; They are given as follows: A = {a/b a, b R[z 1,..., z n, b 0 in U n }, (2) n Z = z i A = {a/b A a, b R[z 1,..., z n, i=1 the constant term of a is zero. }, (3) where U n denotes the closed unit polydisc. Throughout the paper, the plant we consider is causal, single-input and single-output, and its transfer function, which is also called a plant itself, is denoted by p and belongs to P. We consider the feedback system Σ [3, Ch.5, Figure 5.1 shown in Figure 3. In the figure, c denotes a controller and is a transfer function of F. The stabilization problem, considered in this paper, follows the one developed in [1, 3, 2. For details, the reader is referred to [3, 11, 10, 4. Let H(p, c) denote the transfer matrix from [ u 1 u 2 t to [ e 1 e 2 t of the feedback system Σ, that is, [ (1 + pc) 1 p(1 + pc) H(p, c) := 1 c(1 + pc) 1 (1 + pc) 1 F 2 2 (4) provided that 1 + pc is nonzero. We also use H(P, C) and W (P, C) for transfer matrices P and C. For these notations, the reader is referred to [3. We say that the plant p is stabilizable, p is stabilized by c, and c is a stabilizing controller of p if and only if 1 + pc is nonzero and H(p, c) A 2 2. In the definition above, we do not mention the causality of the stabilizing controller. However, it is known that if a causal plant is stabilizable, there always exists a causal stabilizing controller of the plant [11. Because we investigate the set of some kind of stabilizing controllers, we introduce some notations as follows (P P n m ) : S(P ) := {C F m n H(P, C) A (m+n) (m+n) } (5) = the set of all stabilizing controllers of P, SP(P ) := S(P ) P m n (6) = the set of all causal stabilizing controllers of P,

4 SP s (P ) := S(P ) P m n s (7) = the set of all strictly causal stabilizing controllers of P. We further introduce the set of all causal stabilizing controllers of P including a postcompensator as follows: SP s,post (P ; Ψ) := {ΨC 0 SP s (P ) C 0 P m n }, (8) where Ψ is a diagonal matrix and every entry of Ψ is an element of a minimal number of generators of Z (such as z 1 and z 1 6z 2 z 3 ). For strictly causal plants, we know the following proposition. PROPOSITION 1. (cf. Proposition 1 of [12) Let P be a strictly causal plant. Then any stabilizing controller of P is causal, that is, SP(P ) = S(P ). 4 Main Result The following is the main result of this paper, which leads the parametrization of all strictly causal stabilizing controllers of a multidimensional system. THEOREM 2. Let p 0 be a stabilizable plant of P. Further let P = [ n {}}{ p 0 p 0 p 0 t, (9) and Then, where c i P. Ψ = Diag(z 1, z 2,, z n ). n SP s (p 0 ) = { c i z i [ c 1 c 2 c n SP(ΨP )}, (10) i=1 Once we have Theorem 2, we can obtain the parametrization of all strictly causal stabilizing controllers of a multidimensional system as follows. Because p 0 is stabilizable, SP s (p 0 ) is not empty. Thus SP(ΨP ) is also not empty. Let (N, D) and ( D, Ñ) be right- and left-coprime factorization of ΨP with ΨP = ND 1 = D 1 Ñ, Ỹ N + XD = I 1, ÑY + DX = I n, where Ỹ, X, Y, and X are matrices over A with appropriate sizes (N, Ñ An 1, D, X A 1 1, D, X A n n, Ỹ, Y A1 n ). Then by the Youla-Kučera-parametrization, we have SP(ΨP ) = {( X RÑ) 1 (Ỹ + R D) R A 1 n }. (11)

5 By combining (10) and (11), we obtain the parametrization of all strictly causal stabilizing controller of p 0 as follows: { n SP s (p 0 ) = c i z i i=1 [ c 1 c 2 c n := {( X RÑ) 1 (Ỹ + R D), R A 1 n}. 5 Stabilizing Controllers with Post-compensators In this section, we investigate the set of stabilizing controllers with a postcompensator. We denote by ζ 1, ζ 2,, ζ n the minimal number of generators of the ideal Z (typical example is (ζ 1, ζ 2,, ζ n ) = (z 1, z 2,, z n )). THEOREM 3. Let P F n m, C F m n, and Ψ = diag(ζ 1,, ζ n ). Then P is causal and H(ΨP, C) is over A if and only if C is causal and H(P, CΨ) is over A. Before proving Theorem 3, we give a lemma. LEMMA 4. Let A be a matrix over P with size n m. Let Ψ be as in Theorem 3. Then A is over A if and only if ΨA is over A. Proof. This is obvious so the proof is omitted. Proof of Theorem 3: Let [ Ha11 H a12 := H(ΨP, C) H a21 H a22 [ (I + ΨP C) 1 ΨP (I + CΨP ) = 1 C(I + ΨP C) 1 (I + CΨP ) 1 and [ Hb11 H b12 := H(P, CΨ) H b21 H b22 [ (I + P CΨ) 1 P (I + CΨP ) = 1 CΨ(I + P CΨ) 1 (I + CΨP ) 1. Only If part. Suppose that P is causal and H(ΨP, C) is over A, that is H aij (1 i, j 2) is over A. Because P is causal, ΨP is strictly causal. By Proposition 1, we see that C is causal. We now show that H(P, CΨ) is over A, that is H bij (1 i, j 2) is over A. H b11. Recall first that H a11 = I ΨP (I + CΨP ) 1 C, and H b11 = I P (I + CΨP ) 1 CΨ. Because H a11 is over A, the matrix ΨP (I + CΨP ) 1 C is also over A.

6 By applying Lemma 4 twice, we see that the matrix P (I + CΨP ) 1 CΨ is over A. Hence H b11 is over A. H b12. Because H a12 is over A, by Lemma 4, the matrix P (I + CΨP ) 1 is also over A, which is equal to H b12. H b21. H b21 is equal to H a21 Ψ. Because H a21 is over A, by Lemma 4 again, the matrix H b21 is also over A. H b22. Because of H b22 = H a22, H b22 is over A. If part. Suppose that C is causal and H(P, CΨ) is over A. Recall first that [ [ Im O H(A, B) = H(B t, A t ) t Im O O I n O I n. By using this equation, we have [ [ H(ΨC t, P t Im O ) = H(P, CΨ) t Im O O I n O I n. (12) Applying the result of Only if part to H(ΨC t, P t ) in (12), we have that P t is causal and H(C t, P t Ψ) is over A. Hence P is causal and H(ΨP, C) is over A. THEOREM 5. Let P be a plant of P n m. Let Ψ = diag(ζ 1,, ζ n ). Then the following (i) and (ii) hold. (i) SP s,post (P ; Ψ) = {C 0 Ψ C 0 SP(ΨP )}. (13) (ii) Assume that P is stabilizable. Let N, D, Ñ, D, Y, X, Ỹ, X be matrices over A such that ΨP = ND 1 = D 1 N, Ỹ N + XD = I m, ÑY + DX = I n. (14) Then SP s,post (P ; Ψ) ={( X RÑ) 1 (Ỹ + R D)Ψ R A m n } (15) ={(Y + DS)(X NS) 1 Ψ S A m n }. (16) Proof. (i) We prove (13) by showing and. : Let C 0 SP(ΨP ). Because P is causal, by Theorem 3, C 0 Ψ SP(P ). Because every entry of Ψ is in Z, C 0 Ψ is strictly causal, that is, C 0 Ψ SP s (P ). Hence C 0 Ψ SP s,post (P ; Ψ). : Let C SP s,post (P ; Ψ) and C 0 P m n with C = C 0 Ψ. Then H(P, C 0 Ψ) is over A. Because C 0 is causal, by Theorem 3 again, H(ΨP, C 0 ) is over A. Hence C 0 is an element of SP(ΨP ).

7 (ii) We have N, D, Ñ, D, Y, X, Ỹ, X such that the equations in (14) hold. Hence the Youla-Kučera-parametrization can be applied. By noting that ΨP is strictly causal, we have SP(ΨP ) = {( X RÑ) 1 (Ỹ + R D) R A m n }. (17) By (17) and Theorem 3, we have (15). Analogously (16) can be obtained. 6 Proof of Theorem 2 Before proving Theorem 2, we present a proposition below. PROPOSITION 6. Let p 0 be a plant of P. Further let P = [ n {}}{ p 0 p 0 p 0 t. Suppose that C is a stabilizing controller of P. Then [ n {}}{ C is a stabilizing controller of p 0. That is, if we decompose C as [ c 1 c 2 c n := C, (18) where c i F, then n i=1 c i is a stabilizing controller of p 0. Proof. Let C be a stabilizing controller of P. Decompose it as in (18). Let us consider W (P, C). This W (P, C) is over A, because C is a stabilizing controller of P. Let Let n V = [ I 2 O 2 (n 1) W (P, C). W 11 = C(I n + P C) 1, W 12 = CP (I 1 + CP ) 1, W 21 = P C(I n + P C) 1, W 22 = P (I 1 + CP ) 1,

8 that is Let [ W11 W W (P, C) = 12 W 21 W 22 I 1,n = [ n {}}{ t. and V 11 = W 11 I 1,n, V 12 = W 12, V 21 = [ 1 O 1 (n 1) W 21 I 1,n, V 22 = [ 1 O 1 (u 1) W 22. Now we have and and let Then Further let P = I 1,n P 0 [ p 0 = [ 1 O 1 (n 1) P [ c 0 = CI 1,n. I 1 + CP = [ 1 + c 0 p 0. Then we have V 11 = (1 + c 0 p 0 ) 1 c 0, V 12 = c 0 p 0 (1 + c 0 p 0 ) 1, V 21 = p 0 (1 + c 0 p 0 ) 1 c 0, V 22 = p 0 (1 + c 0 p 0 ) 1. [ V11 V W (p 0, c 0 ) = 12 V 21 V 22 which is over A. That is, C 0 is a stabilizing controller of P 0. Now that we have obtained Proposition 6, we can prove Theorem 2. Proof of Theorem 2: We prove (10) by showing and. : Let C := [ c 1 c 2 c n SP(ΨP ). Then by Theorem 3, CΨ is in SP(P ). Applying Proposition 6, n i=1 c iz i is a stabilizing controller of p 0. The strict causality of n i=1 c iz i is obvious. : Let p 0 be in P. Let c 0 be a strictly causal stabilizing controller of P. Let (n 0, d 0 ) be a coprime factorization of p 0 with y 0 n 0 + x 0 d 0 = 1,,

9 where y 0 and x 0 are of A with c 0 = y 0 /x 0. Then a right-coprime factorization (N, D) of P can be given as Ỹ N + XD = I 1, where N = [ n 0 n 0 n 0 t A n 1, D = [ d 0 A 1 1, Ỹ = [ y 01 z 1 y 02 z 2 y 0n z n A 1 n, X = [ x 0 A 1 1 such that n y 0 = y 0i z i (y 0i A m n ) i=1 (Note that because c 0 is strictly causal, y 0 can be expressed as above). Let c i be y i /x i and C := [ c 1 c 2 c n. Hence CΨ is in SP(P ). By Theorem 3, C is causal and C is in SP(ΨP ). 7 Conclusion This paper has been given a method to parametrize all strictly causal stabilizing controllers. In this paper, we have focused on the single-input single-output case. The multi-input multi-output case should also be investigated.

10 z+2 1 z+1 Figure 2. Direct loop. c u 2 p u 1 e 1 y 1 e 2 y 2 Figure 3. Feedback system Σ.

11 Bibliography [1 C. A. Desoer, R. W. Liu, J. Murray, and R. Saeks, Feedback system design: The fractional representation approach to analysis and synthesis, IEEE Trans. Automat. Contr., vol. AC-25, pp , [2 M. Vidyasagar, H. Schneider, and B. A. Francis, Algebraic and topological aspects of feedback stabilization, IEEE Trans. Automat. Contr., vol. AC-27, pp , [3 M. Vidyasagar, Control System Synthesis: A Factorization Approach, Cambridge, MA: MIT Press, [4 S. Shankar and V. R. Sule, Algebraic geometric aspects of feedback stabilization, SIAM J. Control and Optim., vol. 30, no. 1, pp , [5 V. Kučera, Stability of discrete linear feedback systems, in Proc. of the IFAC World Congress, 1975, Paper No [6 V.R. Raman and R. Liu, A necessary and sufficient condition for feedback stabilization in a factor ring, IEEE Trans. Automat. Contr., vol. AC-29, pp , [7 D.C. Youla, H.A. Jabr, and J.J. Bongiorno, Jr., Modern Wiener-Hopf design of optimal controllers, Part II: The multivariable case, IEEE Trans. Automat. Contr., vol. AC-21, pp , [8 Z. Lin, Output feedback stabilizability and stabilization of linear n-d systems, in Multidimensional Signals, Circuits and Systems, K. Galkowski and J. Wood, Eds. 2001, pp , New York, NY: Taylor & Francis. [9 K. Mori, Parameterization of stabilizing controllers over commutative rings with application to multidimensional systems, IEEE Trans. Circuits and Syst. I, vol. 49, pp , [10 V. R. Sule, Feedback stabilization over commutative rings: The matrix case, SIAM J. Control and Optim., vol. 32, no. 6, pp , [11 K. Mori and K. Abe, Feedback stabilization over commutative rings: Further study of coordinate-free approach, SIAM J. Control and Optim., vol. 39, no. 6, pp , 2001.

12 [12 K. Mori, Elementary proof of controller parametrization without coprime factorizability, IEEE Trans. Automat. Contr., vol. AC-49, pp , [13 J. Cadzow, Discrete-Time Systems: An Introduction with Interdisciplinary Applications, NY, Prentice Hall, 1973.

arxiv:math.oc/ v2 14 Dec 1999

arxiv:math.oc/ v2 14 Dec 1999 FEEDBACK STABILIZATION OVER COMMUTATIVE RINGS: arxiv:math.oc/9902124 v2 14 Dec 1999 FURTHER STUDY OF THE COORDINATE-FREE APPROACH KAZUYOSHI MORI AND KENICHI ABE June 16, 2006 Abstract. This paper is concerned

More information

The parameterization of all. of all two-degree-of-freedom strongly stabilizing controllers

The parameterization of all. of all two-degree-of-freedom strongly stabilizing controllers The parameterization stabilizing controllers 89 The parameterization of all two-degree-of-freedom strongly stabilizing controllers Tatsuya Hoshikawa, Kou Yamada 2, Yuko Tatsumi 3, Non-members ABSTRACT

More information

On a generalization of the Youla Kučera parametrization. Part II: the lattice approach to MIMO systems

On a generalization of the Youla Kučera parametrization. Part II: the lattice approach to MIMO systems Math. Control Signals Systems (2006) 18: 199 235 DOI 10.1007/s00498-005-0160-9 ORIGINAL ARTICLE A. Quadrat On a generalization of the Youla Kučera parametrization. Part II: the lattice approach to MIMO

More information

A GENERALIZATION OF THE YOULA-KUČERA PARAMETRIZATION FOR MIMO STABILIZABLE SYSTEMS. Alban Quadrat

A GENERALIZATION OF THE YOULA-KUČERA PARAMETRIZATION FOR MIMO STABILIZABLE SYSTEMS. Alban Quadrat A GENERALIZATION OF THE YOULA-KUČERA ARAMETRIZATION FOR MIMO STABILIZABLE SYSTEMS Alban Quadrat INRIA Sophia Antipolis, CAFE project, 2004 Route des Lucioles, B 93, 06902 Sophia Antipolis cedex, France.

More information

Notes on n-d Polynomial Matrix Factorizations

Notes on n-d Polynomial Matrix Factorizations Multidimensional Systems and Signal Processing, 10, 379 393 (1999) c 1999 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. Notes on n-d Polynomial Matrix Factorizations ZHIPING LIN

More information

A Design Method for Smith Predictors for Minimum-Phase Time-Delay Plants

A Design Method for Smith Predictors for Minimum-Phase Time-Delay Plants 00 ECTI TRANSACTIONS ON COMPUTER AND INFORMATION TECHNOLOGY VOL., NO.2 NOVEMBER 2005 A Design Method for Smith Predictors for Minimum-Phase Time-Delay Plants Kou Yamada Nobuaki Matsushima, Non-members

More information

Information Structures Preserved Under Nonlinear Time-Varying Feedback

Information Structures Preserved Under Nonlinear Time-Varying Feedback Information Structures Preserved Under Nonlinear Time-Varying Feedback Michael Rotkowitz Electrical Engineering Royal Institute of Technology (KTH) SE-100 44 Stockholm, Sweden Email: michael.rotkowitz@ee.kth.se

More information

A Convex Characterization of Multidimensional Linear Systems Subject to SQI Constraints

A Convex Characterization of Multidimensional Linear Systems Subject to SQI Constraints IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 62, NO 6, JUNE 2017 2981 A Convex Characterization of Multidimensional Linear Systems Subject to SQI Constraints Şerban Sabău, Nuno C Martins, and Michael C

More information

A DESIGN METHOD FOR SIMPLE REPETITIVE CONTROLLERS WITH SPECIFIED INPUT-OUTPUT CHARACTERISTIC

A DESIGN METHOD FOR SIMPLE REPETITIVE CONTROLLERS WITH SPECIFIED INPUT-OUTPUT CHARACTERISTIC International Journal of Innovative Computing, Information Control ICIC International c 202 ISSN 349-498 Volume 8, Number 7(A), July 202 pp. 4883 4899 A DESIGN METHOD FOR SIMPLE REPETITIVE CONTROLLERS

More information

A Method to Teach the Parameterization of All Stabilizing Controllers

A Method to Teach the Parameterization of All Stabilizing Controllers Preprints of the 8th FAC World Congress Milano (taly) August 8 - September, A Method to Teach the Parameterization of All Stabilizing Controllers Vladimír Kučera* *Czech Technical University in Prague,

More information

On the simultaneous stabilization of three or more plants

On the simultaneous stabilization of three or more plants On the simultaneous stabilization of three or more plants Christophe Fonte, Michel Zasadzinski, Christine Bernier-Kazantsev, Mohamed Darouach To cite this version: Christophe Fonte, Michel Zasadzinski,

More information

Lecture 4 Stabilization

Lecture 4 Stabilization Lecture 4 Stabilization This lecture follows Chapter 5 of Doyle-Francis-Tannenbaum, with proofs and Section 5.3 omitted 17013 IOC-UPC, Lecture 4, November 2nd 2005 p. 1/23 Stable plants (I) We assume that

More information

A Convex Characterization of Distributed Control Problems in Spatially Invariant Systems with Communication Constraints

A Convex Characterization of Distributed Control Problems in Spatially Invariant Systems with Communication Constraints A Convex Characterization of Distributed Control Problems in Spatially Invariant Systems with Communication Constraints Bassam Bamieh Petros G. Voulgaris Revised Dec, 23 Abstract In this paper we consider

More information

Research Article Minor Prime Factorization for n-d Polynomial Matrices over Arbitrary Coefficient Field

Research Article Minor Prime Factorization for n-d Polynomial Matrices over Arbitrary Coefficient Field Complexity, Article ID 6235649, 9 pages https://doi.org/10.1155/2018/6235649 Research Article Minor Prime Factorization for n-d Polynomial Matrices over Arbitrary Coefficient Field Jinwang Liu, Dongmei

More information

3 Stabilization of MIMO Feedback Systems

3 Stabilization of MIMO Feedback Systems 3 Stabilization of MIMO Feedback Systems 3.1 Notation The sets R and S are as before. We will use the notation M (R) to denote the set of matrices with elements in R. The dimensions are not explicitly

More information

On syzygy modules for polynomial matrices

On syzygy modules for polynomial matrices Linear Algebra and its Applications 298 (1999) 73 86 www.elsevier.com/locate/laa On syzygy modules for polynomial matrices Zhiping Lin School of Electrical and Electronic Engineering, Nanyang Technological

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

Optimizing simultaneously over the numerator and denominator polynomials in the Youla-Kučera parametrization

Optimizing simultaneously over the numerator and denominator polynomials in the Youla-Kučera parametrization Optimizing simultaneously over the numerator and denominator polynomials in the Youla-Kučera parametrization Didier Henrion Vladimír Kučera Arturo Molina-Cristóbal Abstract Traditionally when approaching

More information

THE PARAMETERIZATION OF ALL ROBUST STABILIZING MULTI-PERIOD REPETITIVE CONTROLLERS FOR MIMO TD PLANTS WITH THE SPECIFIED INPUT-OUTPUT CHARACTERISTIC

THE PARAMETERIZATION OF ALL ROBUST STABILIZING MULTI-PERIOD REPETITIVE CONTROLLERS FOR MIMO TD PLANTS WITH THE SPECIFIED INPUT-OUTPUT CHARACTERISTIC International Journal of Innovative Computing, Information Control ICIC International c 218 ISSN 1349-4198 Volume 14, Number 2, April 218 pp. 387 43 THE PARAMETERIZATION OF ALL ROBUST STABILIZING MULTI-PERIOD

More information

Control for stability and Positivity of 2-D linear discrete-time systems

Control for stability and Positivity of 2-D linear discrete-time systems Manuscript received Nov. 2, 27; revised Dec. 2, 27 Control for stability and Positivity of 2-D linear discrete-time systems MOHAMMED ALFIDI and ABDELAZIZ HMAMED LESSI, Département de Physique Faculté des

More information

MRAGPC Control of MIMO Processes with Input Constraints and Disturbance

MRAGPC Control of MIMO Processes with Input Constraints and Disturbance Proceedings of the World Congress on Engineering and Computer Science 9 Vol II WCECS 9, October -, 9, San Francisco, USA MRAGPC Control of MIMO Processes with Input Constraints and Disturbance A. S. Osunleke,

More information

We are IntechOpen, the world s leading publisher of Open Access books Built by scientists, for scientists. International authors and editors

We are IntechOpen, the world s leading publisher of Open Access books Built by scientists, for scientists. International authors and editors We are IntechOpen, the world s leading publisher of Open Access books Built by scientists, for scientists 3,500 08,000.7 M Open access books available International authors and editors Downloads Our authors

More information

Infinite elementary divisor structure-preserving transformations for polynomial matrices

Infinite elementary divisor structure-preserving transformations for polynomial matrices Infinite elementary divisor structure-preserving transformations for polynomial matrices N P Karampetakis and S Vologiannidis Aristotle University of Thessaloniki, Department of Mathematics, Thessaloniki

More information

A Notion of Zero Dynamics for Linear, Time-delay System

A Notion of Zero Dynamics for Linear, Time-delay System Proceedings of the 17th World Congress The International Federation of Automatic Control A Notion of Zero Dynamics for Linear, Time-delay System G. Conte A. M. Perdon DIIGA, Università Politecnica delle

More information

Spectral factorization and H 2 -model following

Spectral factorization and H 2 -model following Spectral factorization and H 2 -model following Fabio Morbidi Department of Mechanical Engineering, Northwestern University, Evanston, IL, USA MTNS - July 5-9, 2010 F. Morbidi (Northwestern Univ.) MTNS

More information

2 Problem formulation. Fig. 1 Unity-feedback system. where A(s) and B(s) are coprime polynomials. The reference input is.

2 Problem formulation. Fig. 1 Unity-feedback system. where A(s) and B(s) are coprime polynomials. The reference input is. Synthesis of pole-zero assignment control law with minimum control input M.-H. TU C.-M. Lin Indexiny ferms: Control systems, Pules and zeros. Internal stability Abstract: A new method of control system

More information

Decentralized Control Subject to Communication and Propagation Delays

Decentralized Control Subject to Communication and Propagation Delays Decentralized Control Subject to Communication and Propagation Delays Michael Rotkowitz 1,3 Sanjay Lall 2,3 IEEE Conference on Decision and Control, 2004 Abstract In this paper, we prove that a wide class

More information

MIT Algebraic techniques and semidefinite optimization February 16, Lecture 4

MIT Algebraic techniques and semidefinite optimization February 16, Lecture 4 MIT 6.972 Algebraic techniques and semidefinite optimization February 16, 2006 Lecture 4 Lecturer: Pablo A. Parrilo Scribe: Pablo A. Parrilo In this lecture we will review some basic elements of abstract

More information

Prime and irreducible elements of the ring of integers modulo n

Prime and irreducible elements of the ring of integers modulo n Prime and irreducible elements of the ring of integers modulo n M. H. Jafari and A. R. Madadi Department of Pure Mathematics, Faculty of Mathematical Sciences University of Tabriz, Tabriz, Iran Abstract

More information

Robust Control. 1st class. Spring, 2017 Instructor: Prof. Masayuki Fujita (S5-303B) Tue., 11th April, 2017, 10:45~12:15, S423 Lecture Room

Robust Control. 1st class. Spring, 2017 Instructor: Prof. Masayuki Fujita (S5-303B) Tue., 11th April, 2017, 10:45~12:15, S423 Lecture Room Robust Control Spring, 2017 Instructor: Prof. Masayuki Fujita (S5-303B) 1st class Tue., 11th April, 2017, 10:45~12:15, S423 Lecture Room Reference: [H95] R.A. Hyde, Aerospace Control Design: A VSTOL Flight

More information

8. Prime Factorization and Primary Decompositions

8. Prime Factorization and Primary Decompositions 70 Andreas Gathmann 8. Prime Factorization and Primary Decompositions 13 When it comes to actual computations, Euclidean domains (or more generally principal ideal domains) are probably the nicest rings

More information

A MODEL-THEORETIC PROOF OF HILBERT S NULLSTELLENSATZ

A MODEL-THEORETIC PROOF OF HILBERT S NULLSTELLENSATZ A MODEL-THEORETIC PROOF OF HILBERT S NULLSTELLENSATZ NICOLAS FORD Abstract. The goal of this paper is to present a proof of the Nullstellensatz using tools from a branch of logic called model theory. In

More information

274 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 2, FEBRUARY A Characterization of Convex Problems in Decentralized Control

274 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 2, FEBRUARY A Characterization of Convex Problems in Decentralized Control 274 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 51, NO 2, FEBRUARY 2006 A Characterization of Convex Problems in Decentralized Control Michael Rotkowitz Sanjay Lall Abstract We consider the problem of

More information

QUANTITATIVE L P STABILITY ANALYSIS OF A CLASS OF LINEAR TIME-VARYING FEEDBACK SYSTEMS

QUANTITATIVE L P STABILITY ANALYSIS OF A CLASS OF LINEAR TIME-VARYING FEEDBACK SYSTEMS Int. J. Appl. Math. Comput. Sci., 2003, Vol. 13, No. 2, 179 184 QUANTITATIVE L P STABILITY ANALYSIS OF A CLASS OF LINEAR TIME-VARYING FEEDBACK SYSTEMS PINI GURFIL Department of Mechanical and Aerospace

More information

Copyrighted Material. Type A Weyl Group Multiple Dirichlet Series

Copyrighted Material. Type A Weyl Group Multiple Dirichlet Series Chapter One Type A Weyl Group Multiple Dirichlet Series We begin by defining the basic shape of the class of Weyl group multiple Dirichlet series. To do so, we choose the following parameters. Φ, a reduced

More information

Dynamic Model Predictive Control

Dynamic Model Predictive Control Dynamic Model Predictive Control Karl Mårtensson, Andreas Wernrud, Department of Automatic Control, Faculty of Engineering, Lund University, Box 118, SE 221 Lund, Sweden. E-mail: {karl, andreas}@control.lth.se

More information

A connection between number theory and linear algebra

A connection between number theory and linear algebra A connection between number theory and linear algebra Mark Steinberger Contents 1. Some basics 1 2. Rational canonical form 2 3. Prime factorization in F[x] 4 4. Units and order 5 5. Finite fields 7 6.

More information

Splitting sets and weakly Matlis domains

Splitting sets and weakly Matlis domains Commutative Algebra and Applications, 1 8 de Gruyter 2009 Splitting sets and weakly Matlis domains D. D. Anderson and Muhammad Zafrullah Abstract. An integral domain D is weakly Matlis if the intersection

More information

1030 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 5, MAY 2011

1030 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 5, MAY 2011 1030 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 56, NO 5, MAY 2011 L L 2 Low-Gain Feedback: Their Properties, Characterizations Applications in Constrained Control Bin Zhou, Member, IEEE, Zongli Lin,

More information

Control of Vibrations

Control of Vibrations Control of Vibrations M. Vidyasagar Executive Vice President (Advanced Technology) Tata Consultancy Services Hyderabad, India sagar@atc.tcs.com www.atc.tcs.com/ sagar Outline Modelling Analysis Control

More information

Structural Stability and Equivalence of Linear 2D Discrete Systems

Structural Stability and Equivalence of Linear 2D Discrete Systems Proceedings of the 6th IFAC Symposium on System Structure and Control, Istanbul, Turkey, June -4, 016 FrM11. Structural Stability and Equivalence of Linear D Discrete Systems Olivier Bachelier, Ronan David,

More information

Part IX. Factorization

Part IX. Factorization IX.45. Unique Factorization Domains 1 Part IX. Factorization Section IX.45. Unique Factorization Domains Note. In this section we return to integral domains and concern ourselves with factoring (with respect

More information

Zero controllability in discrete-time structured systems

Zero controllability in discrete-time structured systems 1 Zero controllability in discrete-time structured systems Jacob van der Woude arxiv:173.8394v1 [math.oc] 24 Mar 217 Abstract In this paper we consider complex dynamical networks modeled by means of state

More information

THIS paper deals with robust control in the setup associated

THIS paper deals with robust control in the setup associated IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 10, OCTOBER 2005 1501 Control-Oriented Model Validation and Errors Quantification in the `1 Setup V F Sokolov Abstract A priori information required for

More information

Complex Laplacians and Applications in Multi-Agent Systems

Complex Laplacians and Applications in Multi-Agent Systems 1 Complex Laplacians and Applications in Multi-Agent Systems Jiu-Gang Dong, and Li Qiu, Fellow, IEEE arxiv:1406.186v [math.oc] 14 Apr 015 Abstract Complex-valued Laplacians have been shown to be powerful

More information

Robust and Optimal Control, Spring A: SISO Feedback Control A.1 Internal Stability and Youla Parameterization

Robust and Optimal Control, Spring A: SISO Feedback Control A.1 Internal Stability and Youla Parameterization Robust and Optimal Control, Spring 2015 Instructor: Prof. Masayuki Fujita (S5-303B) A: SISO Feedback Control A.1 Internal Stability and Youla Parameterization A.2 Sensitivity and Feedback Performance A.3

More information

POLYNOMIAL EMBEDDING ALGORITHMS FOR CONTROLLERS IN A BEHAVIORAL FRAMEWORK

POLYNOMIAL EMBEDDING ALGORITHMS FOR CONTROLLERS IN A BEHAVIORAL FRAMEWORK 1 POLYNOMIAL EMBEDDING ALGORITHMS FOR CONTROLLERS IN A BEHAVIORAL FRAMEWORK H.L. Trentelman, Member, IEEE, R. Zavala Yoe, and C. Praagman Abstract In this paper we will establish polynomial algorithms

More information

The 4-periodic spiral determinant

The 4-periodic spiral determinant The 4-periodic spiral determinant Darij Grinberg rough draft, October 3, 2018 Contents 001 Acknowledgments 1 1 The determinant 1 2 The proof 4 *** The purpose of this note is to generalize the determinant

More information

Michael Rotkowitz 1,2

Michael Rotkowitz 1,2 November 23, 2006 edited Parameterization of Causal Stabilizing Controllers over Networks with Delays Michael Rotkowitz 1,2 2006 IEEE Global Telecommunications Conference Abstract We consider the problem

More information

1. Partial Fraction Expansion All the polynomials in this note are assumed to be complex polynomials.

1. Partial Fraction Expansion All the polynomials in this note are assumed to be complex polynomials. Partial Fraction Expansion All the polynomials in this note are assumed to be complex polynomials A rational function / is a quotient of two polynomials P, Q with 0 By Fundamental Theorem of algebra, =

More information

A design method for two-degree-of-freedom multi-period repetitive controllers for multiple-input/multiple-output systems

A design method for two-degree-of-freedom multi-period repetitive controllers for multiple-input/multiple-output systems A design method for two-degree-of-freedom multi-period repetitive controllers for multiple-input/multiple-output systems Zhongxiang Chen Kou Yamada Tatsuya Sakanushi Iwanori Murakami Yoshinori Ando Nhan

More information

ROBUST CONSTRAINED REGULATORS FOR UNCERTAIN LINEAR SYSTEMS

ROBUST CONSTRAINED REGULATORS FOR UNCERTAIN LINEAR SYSTEMS ROBUST CONSTRAINED REGULATORS FOR UNCERTAIN LINEAR SYSTEMS Jean-Claude HENNET Eugênio B. CASTELAN Abstract The purpose of this paper is to combine several control requirements in the same regulator design

More information

1. Factorization Divisibility in Z.

1. Factorization Divisibility in Z. 8 J. E. CREMONA 1.1. Divisibility in Z. 1. Factorization Definition 1.1.1. Let a, b Z. Then we say that a divides b and write a b if b = ac for some c Z: a b c Z : b = ac. Alternatively, we may say that

More information

The Tuning of Robust Controllers for Stable Systems in the CD-Algebra: The Case of Sinusoidal and Polynomial Signals

The Tuning of Robust Controllers for Stable Systems in the CD-Algebra: The Case of Sinusoidal and Polynomial Signals The Tuning of Robust Controllers for Stable Systems in the CD-Algebra: The Case of Sinusoidal and Polynomial Signals Timo Hämäläinen Seppo Pohjolainen Tampere University of Technology Department of Mathematics

More information

Chapter 5. Modular arithmetic. 5.1 The modular ring

Chapter 5. Modular arithmetic. 5.1 The modular ring Chapter 5 Modular arithmetic 5.1 The modular ring Definition 5.1. Suppose n N and x, y Z. Then we say that x, y are equivalent modulo n, and we write x y mod n if n x y. It is evident that equivalence

More information

REDUCTION OF NOISE IN ON-DEMAND TYPE FEEDBACK CONTROL SYSTEMS

REDUCTION OF NOISE IN ON-DEMAND TYPE FEEDBACK CONTROL SYSTEMS International Journal of Innovative Computing, Information and Control ICIC International c 218 ISSN 1349-4198 Volume 14, Number 3, June 218 pp. 1123 1131 REDUCTION OF NOISE IN ON-DEMAND TYPE FEEDBACK

More information

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution

More information

Decompositions Of Modules And Matrices

Decompositions Of Modules And Matrices University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Faculty Publications, Department of Mathematics Mathematics, Department of 1973 Decompositions Of Modules And Matrices Thomas

More information

Convex Functions and Optimization

Convex Functions and Optimization Chapter 5 Convex Functions and Optimization 5.1 Convex Functions Our next topic is that of convex functions. Again, we will concentrate on the context of a map f : R n R although the situation can be generalized

More information

NONBLOCKING CONTROL OF PETRI NETS USING UNFOLDING. Alessandro Giua Xiaolan Xie

NONBLOCKING CONTROL OF PETRI NETS USING UNFOLDING. Alessandro Giua Xiaolan Xie NONBLOCKING CONTROL OF PETRI NETS USING UNFOLDING Alessandro Giua Xiaolan Xie Dip. Ing. Elettrica ed Elettronica, U. di Cagliari, Italy. Email: giua@diee.unica.it INRIA/MACSI Team, ISGMP, U. de Metz, France.

More information

HOMOLOGY AND COHOMOLOGY. 1. Introduction

HOMOLOGY AND COHOMOLOGY. 1. Introduction HOMOLOGY AND COHOMOLOGY ELLEARD FELIX WEBSTER HEFFERN 1. Introduction We have been introduced to the idea of homology, which derives from a chain complex of singular or simplicial chain groups together

More information

Nonlinear Q-Design for Convex Stochastic Control

Nonlinear Q-Design for Convex Stochastic Control 2426 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 54, NO 10, OCTOBER 2009 Nonlinear Q-Design for Convex Stochastic Control Joëlle Skaf and Stephen Boyd, Fellow, IEEE Abstract In this note we describe a

More information

Scott Taylor 1. EQUIVALENCE RELATIONS. Definition 1.1. Let A be a set. An equivalence relation on A is a relation such that:

Scott Taylor 1. EQUIVALENCE RELATIONS. Definition 1.1. Let A be a set. An equivalence relation on A is a relation such that: Equivalence MA Relations 274 and Partitions Scott Taylor 1. EQUIVALENCE RELATIONS Definition 1.1. Let A be a set. An equivalence relation on A is a relation such that: (1) is reflexive. That is, (2) is

More information

Block companion matrices, discrete-time block diagonal stability and polynomial matrices

Block 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 information

However another possibility is

However another possibility is 19. Special Domains Let R be an integral domain. Recall that an element a 0, of R is said to be prime, if the corresponding principal ideal p is prime and a is not a unit. Definition 19.1. Let a and b

More information

Rings With Topologies Induced by Spaces of Functions

Rings With Topologies Induced by Spaces of Functions Rings With Topologies Induced by Spaces of Functions Răzvan Gelca April 7, 2006 Abstract: By considering topologies on Noetherian rings that carry the properties of those induced by spaces of functions,

More information

arxiv: v1 [math.ac] 15 Jul 2011

arxiv: v1 [math.ac] 15 Jul 2011 THE KAPLANSKY CONDITION AND RINGS OF ALMOST STABLE RANGE 1 arxiv:1107.3000v1 [math.ac] 15 Jul 2011 MOSHE ROITMAN DEPARTMENT OF MATHEMATICS UNIVERSITY OF HAIFA MOUNT CARMEL, HAIFA 31905, ISRAEL Abstract.

More information

Rings, Integral Domains, and Fields

Rings, Integral Domains, and Fields Rings, Integral Domains, and Fields S. F. Ellermeyer September 26, 2006 Suppose that A is a set of objects endowed with two binary operations called addition (and denoted by + ) and multiplication (denoted

More information

MINIMAL POLYNOMIALS AND CHARACTERISTIC POLYNOMIALS OVER RINGS

MINIMAL POLYNOMIALS AND CHARACTERISTIC POLYNOMIALS OVER RINGS JP Journal of Algebra, Number Theory and Applications Volume 0, Number 1, 011, Pages 49-60 Published Online: March, 011 This paper is available online at http://pphmj.com/journals/jpanta.htm 011 Pushpa

More information

ThM06-2. Coprime Factor Based Closed-Loop Model Validation Applied to a Flexible Structure

ThM06-2. Coprime Factor Based Closed-Loop Model Validation Applied to a Flexible Structure Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii USA, December 2003 ThM06-2 Coprime Factor Based Closed-Loop Model Validation Applied to a Flexible Structure Marianne Crowder

More information

ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ.

ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ. ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ. ANDREW SALCH 1. Hilbert s Nullstellensatz. The last lecture left off with the claim that, if J k[x 1,..., x n ] is an ideal, then

More information

Parameterization of Stabilizing Controllers for Interconnected Systems

Parameterization of Stabilizing Controllers for Interconnected Systems Parameterization of Stabilizing Controllers for Interconnected Systems Michael Rotkowitz G D p D p D p D 1 G 2 G 3 G 4 p G 5 K 1 K 2 K 3 K 4 Contributors: Sanjay Lall, Randy Cogill K 5 Department of Signals,

More information

PARAMETERIZATION OF STATE FEEDBACK GAINS FOR POLE PLACEMENT

PARAMETERIZATION 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 information

Counterexamples to pole placement by static output feedback

Counterexamples to pole placement by static output feedback Counterexamples to pole placement by static output feedback A. Eremenko and A. Gabrielov 6.11.2001 Abstract We consider linear systems with inner state of dimension n, with m inputs and p outputs, such

More information

Acyclic Digraphs arising from Complete Intersections

Acyclic Digraphs arising from Complete Intersections Acyclic Digraphs arising from Complete Intersections Walter D. Morris, Jr. George Mason University wmorris@gmu.edu July 8, 2016 Abstract We call a directed acyclic graph a CI-digraph if a certain affine

More information

On common quadratic Lyapunov functions for stable discrete-time LTI systems

On common quadratic Lyapunov functions for stable discrete-time LTI systems On common quadratic Lyapunov functions for stable discrete-time LTI systems Oliver Mason, Hamilton Institute, NUI Maynooth, Maynooth, Co. Kildare, Ireland. (oliver.mason@may.ie) Robert Shorten 1, Hamilton

More information

A generalization of Serre s conjecture and some related issues

A generalization of Serre s conjecture and some related issues Linear Algebra and its Applications 338 (2001) 125 138 www.elsevier.com/locate/laa A generalization of Serre s conjecture and some related issues Zhiping Lin a,,n.k.bose b,1 a School of Electrical and

More information

Chapter 2 Application to DC Circuits

Chapter 2 Application to DC Circuits Chapter 2 Application to DC Circuits In this chapter we use the results obtained in Chap. 1 to develop a new measurement based approach to solve synthesis problems in unknown linear direct current (DC)

More information

The upper triangular algebra loop of degree 4

The upper triangular algebra loop of degree 4 The upper triangular algebra loop of degree 4 Michael Munywoki Joint With K.W. Johnson and J.D.H. Smith Iowa State University Department of Mathematics Third Mile High Conference on Nonassociative Mathematics

More information

Minimal-span bases, linear system theory, and the invariant factor theorem

Minimal-span bases, linear system theory, and the invariant factor theorem Minimal-span bases, linear system theory, and the invariant factor theorem G. David Forney, Jr. MIT Cambridge MA 02139 USA DIMACS Workshop on Algebraic Coding Theory and Information Theory DIMACS Center,

More information

Binary Convolutional Codes of High Rate Øyvind Ytrehus

Binary Convolutional Codes of High Rate Øyvind Ytrehus Binary Convolutional Codes of High Rate Øyvind Ytrehus Abstract The function N(r; ; d free ), defined as the maximum n such that there exists a binary convolutional code of block length n, dimension n

More information

Section III.6. Factorization in Polynomial Rings

Section III.6. Factorization in Polynomial Rings III.6. Factorization in Polynomial Rings 1 Section III.6. Factorization in Polynomial Rings Note. We push several of the results in Section III.3 (such as divisibility, irreducibility, and unique factorization)

More information

arxiv: v3 [math.oc] 1 Sep 2018

arxiv: v3 [math.oc] 1 Sep 2018 arxiv:177.148v3 [math.oc] 1 Sep 218 The converse of the passivity and small-gain theorems for input-output maps Sei Zhen Khong, Arjan van der Schaft Version: June 25, 218; accepted for publication in Automatica

More information

CONSIDER the linear discrete-time system

CONSIDER the linear discrete-time system 786 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 42, NO. 6, JUNE 1997 Optimal Damping of Forced Oscillations in Discrete-Time Systems Anders Lindquist, Fellow, IEEE, Vladimir A. Yakubovich Abstract In

More information

Available online at ScienceDirect. Procedia Engineering 100 (2015 )

Available online at   ScienceDirect. Procedia Engineering 100 (2015 ) Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 100 (015 ) 345 349 5th DAAAM International Symposium on Intelligent Manufacturing and Automation, DAAAM 014 Control of Airflow

More information

1 Differentiable manifolds and smooth maps

1 Differentiable manifolds and smooth maps 1 Differentiable manifolds and smooth maps Last updated: April 14, 2011. 1.1 Examples and definitions Roughly, manifolds are sets where one can introduce coordinates. An n-dimensional manifold is a set

More information

ON THE BOUNDEDNESS BEHAVIOR OF THE SPECTRAL FACTORIZATION IN THE WIENER ALGEBRA FOR FIR DATA

ON THE BOUNDEDNESS BEHAVIOR OF THE SPECTRAL FACTORIZATION IN THE WIENER ALGEBRA FOR FIR DATA ON THE BOUNDEDNESS BEHAVIOR OF THE SPECTRAL FACTORIZATION IN THE WIENER ALGEBRA FOR FIR DATA Holger Boche and Volker Pohl Technische Universität Berlin, Heinrich Hertz Chair for Mobile Communications Werner-von-Siemens

More information

An asymptotic ratio characterization of input-to-state stability

An 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 information

ADAPTIVE FEEDBACK LINEARIZING CONTROL OF CHUA S CIRCUIT

ADAPTIVE FEEDBACK LINEARIZING CONTROL OF CHUA S CIRCUIT International Journal of Bifurcation and Chaos, Vol. 12, No. 7 (2002) 1599 1604 c World Scientific Publishing Company ADAPTIVE FEEDBACK LINEARIZING CONTROL OF CHUA S CIRCUIT KEVIN BARONE and SAHJENDRA

More information

August 23, 2017 Let us measure everything that is measurable, and make measurable everything that is not yet so. Galileo Galilei. 1.

August 23, 2017 Let us measure everything that is measurable, and make measurable everything that is not yet so. Galileo Galilei. 1. August 23, 2017 Let us measure everything that is measurable, and make measurable everything that is not yet so. Galileo Galilei 1. Vector spaces 1.1. Notations. x S denotes the fact that the element x

More information

Mathematics for Control Theory

Mathematics for Control Theory Mathematics for Control Theory Outline of Dissipativity and Passivity Hanz Richter Mechanical Engineering Department Cleveland State University Reading materials Only as a reference: Charles A. Desoer

More information

OUTER MEASURES ON A COMMUTATIVE RING INDUCED BY MEASURES ON ITS SPECTRUM. Dariusz Dudzik, Marcin Skrzyński. 1. Preliminaries and introduction

OUTER MEASURES ON A COMMUTATIVE RING INDUCED BY MEASURES ON ITS SPECTRUM. Dariusz Dudzik, Marcin Skrzyński. 1. Preliminaries and introduction Annales Mathematicae Silesianae 31 (2017), 63 70 DOI: 10.1515/amsil-2016-0020 OUTER MEASURES ON A COMMUTATIVE RING INDUCED BY MEASURES ON ITS SPECTRUM Dariusz Dudzik, Marcin Skrzyński Abstract. On a commutative

More information

Divisibility. Chapter Divisors and Residues

Divisibility. Chapter Divisors and Residues Chapter 1 Divisibility Number theory is concerned with the properties of the integers. By the word integers we mean the counting numbers 1, 2, 3,..., together with their negatives and zero. Accordingly

More information

1 Overview and revision

1 Overview and revision MTH6128 Number Theory Notes 1 Spring 2018 1 Overview and revision In this section we will meet some of the concerns of Number Theory, and have a brief revision of some of the relevant material from Introduction

More information

Controller Parameterization for Disturbance Response Decoupling: Application to Vehicle Active Suspension Control

Controller Parameterization for Disturbance Response Decoupling: Application to Vehicle Active Suspension Control IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 3, MAY 2002 393 Controller Parameterization for Disturbance Response Decoupling: Application to Vehicle Active Suspension Control Malcolm C.

More information

Lyapunov Stability of Linear Predictor Feedback for Distributed Input Delays

Lyapunov Stability of Linear Predictor Feedback for Distributed Input Delays IEEE TRANSACTIONS ON AUTOMATIC CONTROL VOL. 56 NO. 3 MARCH 2011 655 Lyapunov Stability of Linear Predictor Feedback for Distributed Input Delays Nikolaos Bekiaris-Liberis Miroslav Krstic In this case system

More information

DESCENT OF THE CANONICAL MODULE IN RINGS WITH THE APPROXIMATION PROPERTY

DESCENT OF THE CANONICAL MODULE IN RINGS WITH THE APPROXIMATION PROPERTY PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 6, June 1996 DESCENT OF THE CANONICAL MODULE IN RINGS WITH THE APPROXIMATION PROPERTY CHRISTEL ROTTHAUS (Communicated by Wolmer V. Vasconcelos)

More information

Boolean Algebra CHAPTER 15

Boolean Algebra CHAPTER 15 CHAPTER 15 Boolean Algebra 15.1 INTRODUCTION Both sets and propositions satisfy similar laws, which are listed in Tables 1-1 and 4-1 (in Chapters 1 and 4, respectively). These laws are used to define an

More information

Stabilization of Networked Control Systems with Clock Offset

Stabilization of Networked Control Systems with Clock Offset Stabilization of Networked Control Systems with Clock Offset Masashi Wakaiki, Kunihisa Okano, and João P Hespanha Abstract We consider the stabilization of networked control systems with time-invariant

More information

A Partial Order Approach to Decentralized Control

A Partial Order Approach to Decentralized Control Proceedings of the 47th IEEE Conference on Decision and Control Cancun, Mexico, Dec. 9-11, 2008 A Partial Order Approach to Decentralized Control Parikshit Shah and Pablo A. Parrilo Department of Electrical

More information