The norms can also be characterized in terms of Riccati inequalities.
|
|
- Kerry Boyd
- 6 years ago
- Views:
Transcription
1 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 are equivalent: 1. A is stable, and G < 1 2. A is stable, and the matrix A BB T C T C A T has no imaginary axis eigenvalues. 3. A is stable and there exists a matrix X R n n, such that X = X T, A + BB T X is stable, and A T X + XA + XBB T X + C T C = 0 Proof: 1 2 follows directly from the homework. Using Theorem 19, it is clear that 2 3. The norms can also be characterized in terms of Riccati inequalities. Theorem 86 The following statements are equivalent: 1. A is stable, and G < 1 2. A is stable, and for some ǫ > 0, C (si A 1 B < 1 ǫi 166
2 3. A is stable and there exists a matrix X R n n, such that X = X T, A + BB T X is stable, and A T X + XA + XBB T X + C T C < 0 4. A is stable and there exists a matrix X R n n, such that X = X T, and A T X + XA + XBB T X + C T C < 0 5. There exists a matrix X R n n such that X = X T > 0 and A T X + XA + XBB T X + C T C < 0 6. There exists a matrix X R n n such that X = X T > 0 and A T X + XA + C T C XB B T X I < 0 7. There exists a matrix X R n n such that X = X T > 0 and Proof: A T X + XA XB C T B T X I 0 C 0 I 1 2 Since A is stable (si A 1 B is finite. Hence, for some ǫ > 0, the condition holds. 2 3 Use Theorem 85, that 1 3, to conclude that there exists an X = X T such that 3 4 Obvious. 4 5 Let W > 0 such that < 0 A T X + XA + XBB T X + C T C = ǫ 2 I < 0. A T X + XA = XBB T X C T C W 167
3 Since A is stable, and is observable, it follows from Theorem 3 that X > 0. A, B T X C W There are two easy ways to do this, each involve a completion of the square in either time-domain or frequency-domain. This is in a later homework Schur complements... For systems with a D term, the Hamiltonian matrix from the homework needs to be modified, and you should take some time to do this. In this case, the correct version of the inequality theorem is: Theorem 87 The following statements are equivalent: 1. The continuous-time system described by ẋ = Ax + Bu y = Cx + Du is internally stable, and has T yu < σ (D < 1, A is stable, and there is a matrix X = X T such that A + BB T X + BD T ( I DD T 1 ( C + DB T X is stable and (A + BD T ( I DD T 1 C T X + X (A + BD T ( I DD T 1 C +XB [I + D T ( I DD T ] 1 D B T X + C T ( I DD T 1 C = 0 3. σ (D < 1 and there exists a matrix X = X T > 0 such that (A + BD T ( I DD T 1 C T X + X (A + BD T ( I DD T 1 C +XB [I + D T ( I DD T ] 1 D B T X + C T ( I DD T 1 C <
4 4. There exists a matrix X R n n such that X = X T > 0 and A T X + XA XB C T B T X I D T C D I < 0 169
5 State Feedback/Full Information H Riccati Inequality Formulation The material is motivated from earlier work, namely Petersen (IEEE Transactions on Automatic Control, 1987, vol. 32, pp , Zhou and Khargonekar (Systems and Control Letters, 1988, vol. 11, pp , Becker (ACC 93, PhD and SCL 1994, vol. 23, pp , and Gahinet (CDC 92, ACC 93. There are two problems considered in this section: State-Feedback and Full-Information. In this section, we also use some very special orthogonality assumptions which make the formulae much cleaner. Later, we will show how to relax the orthogonality assumptions. The generalized plant for the State-Feedback problem is ẋ e y = A B 1 B 2 C 1 0 D 12 I 0 0 x d u (9.37 The orthogonality assumptions are that D T 12 D 12 = I nu, D T 12 C 1 = 0 The generalized plant for the Full-Information problem is ẋ e y 1 y 2 = A B 1 B 2 C 1 0 D 12 I I 0 Again, we impose the restrictions that D12D T 12 = I nu, D12C T 1 = 0 x d (9.38 u 170
6 Synthesis Result Theorem 88 Consider the generalized plant P SF (state-feedback case in equation (9.37 and P FI (full information in equation (9.38. The following statements are equivalent: 1. For the plant P SF, there exists a linear, constant-gain feedback law u(t = F 1 y(t = F 1 x(t that achieves closed-loop internal stability and T ed < 1 2. For the plant P FI, there exists a linear, dynamic feedback law u = K 1 y 1 + K 2 y 2 = K 1 x + K 2 d that achieves closed-loop internal stability and T ed < 1 3. There exists a matrix X R n n, X = X T > 0 such that A T X + XA + X ( B 1 B T 1 B 2 B T 2 X + C T 1 C 1 < 0 4. There exists a matrix Y R n n, Y = Y T > 0 such that Y A T + AY + ( B 1 B T 1 B 2 B T 2 C 1 Y Y C T 1 I < 0 5. There exists a matrix Y R n n, Y = Y T > 0 such that Y A T + AY + ( B 1 B T 1 B 2 B T 2 + Y C T 1 C 1 Y < 0 Remark: Hence, for this special H control design problem, it is of no advantage to use dynamic controllers, or even to use the information in the disturbance measurement. This is essentially due to the fact that the disturbance affects the error only through the state (D 11 = 0, and hence the state contains all of the useful information about the disturbance. 171
7 Proof 2 3 Let the controller state-space matrix be ẋ c u = A c B 1c B 2c C c D 1c D 2c x c y 1 y 2 where the state dimension of the controller is n c 0. The closed-loop system matrix is simply ẋ e ẋ c = A 0 B nc 0 C B I nc D 12 0 D 1c C 1c D 2c B 1c A c B 2c x x c By assumption, the closed-loop system is stable, and has < 1. Hence, by the main analysis lemma, there is a matrix X R (n+n c (n+n c, X = X T > 0 such that for matrices N, L, R and K ss defined as we have N := L := X X A nc [ [ [ + B T 1 0 [ B I nc 0 0 D 12 0 ] C 1 0 ] K ss := A T 0 ] 0 0 nc X X B 1 0 C T 1 0 X I nd 0 ], R := D 2c C 1c D 1c B 2c A c B 1c N + LK ss R + R T K T ssl T < 0 0 I ne I n+nc I nd 0 Note that L is not full column rank. Hence, for this matrix to be negative definite, by proper choice of X and K ss, it follows that some portion 172 d
8 of N must be negative definite already (since L can t affect it in all directions. What is orthogonal to L? Let Y := X 1 and partition X and Y as X 11 X 12 Y 11 Y 12 X = X T 12 X 22 It is easy to show (do it that L :=, Y = 0 0 Y Y12 T I nd D 12 D 12 B T 2 Y T 12 Y 22 is full column rank, and spans the orthogonal complement of the span of L. Hence, we must have L T NL = L T ( N + LKss R + R T K T ssl T L < 0 Multiply this out (remember that D T 12C 1 = 0, which also implies that C 1 is in the span of D 12, so that C T 1 D 12 D T 12 C 1 = C T 1 C 1. Then, do an obvious Schur complement to get that Y 11 A T + AY 11 + Y 11 C T 1 C 1Y 11 + B 1 B T 1 B 2B T 2 < 0 In doing this, you need to use the fact that X 11 Y 11 +X 12 Y12 T = I n, which follows from XY = I. Then, multiply both sides by Y11 1 to get the desired Riccati inequality (ie., the X R n n in the desired Riccati inequality is just Y Verify that u(t = B2 T Xx(t works. 1 2 Use the constant-gain state-feedback, K 1 := F 1, K 2 := Y := X 1 and Schur complement. 4 5 Schur complement. 173
9 Maximal solutions Riccati equations For reference, the results from Gohberg, Lancaster and Rodman are restated, for real (and hence symmetric hermitian solutions. Suppose that A R n n, C R n n, D R n n, with D = D T 0, C = C T and (A, D stabilizable. Consider the matrix equation XDX A T X XA C = 0 (9.39 Definition 89 A symmetric solution X + = X T + of (9.39 is maximal if X + X for any other symmetric solution X of (9.39. Theorem 90 If there is a symmetric solution to (9.39, then there is a maximal solution X +. The maximal solution satisfies max i Reλ i (A DX + 0 Theorem 91 If there is a symmetric solution to (9.39, then there is a sequence {X j } j=1 such that for each j, X j = X T j and 0 X j DX J A T X j X j A C X j X for any X = X T solving 9.39 A DX j is stable lim j X j = X + Theorem 92 If there are symmetric solutions to (9.39, then for every symmetric C C, there exists symmetric solutions (and hence a maximal solution X + to XDX A T X XA C = 0 Moreover, the maximal solutions are related by X + X
10 Inequalities and Equalities Theorem 93 Suppose that A R n n, B R n m and Q = Q T R n n, with (A, B stabilizable. Let a matrix function R : R n n R n n be defined as R(X := A T X + XA XBB T X + Q Then the following statements are equivalent: 1. exists an X R n n such that X = X T and R(X There exists a unique X + R n n such that X + = X T +, R(X + = 0, and A BB T X + is stable 3. The matrix A Q has no imaginary-axis eigenvalues. BB T A T Moreover, if any of these are satisfied, then for any Y = Y T with R(Y 0, it must be that Y X +. Also, if any of the three conditions are satisfied, then there exists a sequence {X j } j=1 such that R (X j 0, lim j X j = X + 175
11 Proof 1 2 By hypothesis, there is an X = X T R n n such that A T X + XA XBB T X + Q =: W 0 Hence, X is a symmetric solution to XBB T X A T X XA (Q W = 0 (9.40 By [GohLR], there is a maximal solution X + = X T + to the equation X + BB T X + A T X + X + A Q = 0 (9.41 (since C := Q Q W =: C, and X + X for any symmetric solution X to equation (9.40. Now, note that A T X + + X + A X + BB T X + + Q = 0 A T X + XA XBB T X + Q = W Subtracting these equations gives A T (X + X + (X + X A X + BB T X + + XBB T X = W (9.42 Define := X + X. Equation (9.42 can be rearranged into ( A BB T X + T + ( A BB T X + = BB T W The right-hand side is clearly negative definite, and since = X + X 0, it follows by standard Lyapunov theory that A BB T X + is stable. In fact, it then follows that 0, so that X + X. 2 3 Obvious, by early results on Riccati equations 3 1 By continuity, there is an ǫ > 0 such that A BB T (Q ǫi A T has no imaginary-axis eigenvalues. Since (A, B is stabilizable, Theorem 17 applies, and there exists a matrix X = X T R n n such that A T X + XA XBB T X + (Q ǫi = 0 (
12 with A BB T X stable. Note that (9.43 is rewritten as as desired. R(X = ǫi 0 177
13 Additional Results Finally, if Y = Y T has R(Y 0, then following the same argument gives ( A BB T X + T Y + Y ( A BB T X + = Y BB T Y R(Y where Y := X + Y. Since ( A BB T X + is stable, and the right-hand side is negative semi-definite, it follows that Y 0. For the last statement, choose γ > 0 such that for all γ (0, γ], the matrix A (Q γi BB T A T has no imaginary axis eigenvalues. This is possible by continuity of eigenvalues. Hence, for each such γ, the 3rd condition of the theorem is true, and all conditions are true. Let X γ + be the unique symmetric matrix satisfying A T X γ + X γ +A + X γ +BB T X γ + (Q γi = 0 Re [ λ i ( A BB T X γ +] < 0 Note that for each γ, R (X+ γ = γi 0, hence X+ γ X +. But also, if 0 < γ 2 γ 1 γ, then X γ 2 + X γ 1 + Hence, the function X γ + (on (0, γ] is bounded above by X +, and is increasing as γ decreases. Hence, there is a limit, X satisfying lim γ 0 Xγ + = X X = X T R ( X = 0 X X + Re ( A BB T X 0 But the eigenvalue distribution of A Q BB T A T 178
14 implies that there are no eigenvalues on the imaginary axis, hence it must in fact be that Re ( A BB T X < 0 so that X = X
15 H Control Riccati Equalities The generalized plant for the State-Feedback problem is ẋ e y = A B 1 B 2 C 1 0 D 12 I 0 0 x d u (9.44 The generalized plant for the Full-Information problem is ẋ e y 1 y 2 = A B 1 B 2 C 1 0 D 12 I I 0 x d u (9.45 In both problems, we impose orthogonality assumptions D T 12D 12 = I nu, D T 12C 1 = 0. The theorem about the solvability of the H synthesis problem for these generalized plants stated that the control synthesis problem is solvable if and only if there exists a matrix X R n n, X = X T 0 such that A T X + XA + X ( B 1 B T 1 B 2B T 2 X + C T 1 C 1 < 0 Using Theorem 93, we can connect these theorems to Riccati equalities, rather than inequalities. 180
16 Connect Connect Theorem 94 Consider the state-feedback control problem described earlier. Suppose that (A, C 1 is observable. The following statements are equivalent: 1. There exists a matrix X R n n with X = X T 0 and A T X + XA + X ( B 1 B T 1 B 2 B T 2 X + C T 1 C 1 < 0 2. There exists a matrix X R n n with X = X T 0 such that A T ( X + X A + X B1 B1 T B 2 B2 T [ ( Reλ i A + B1 B1 T B 2B2 T ] X < 0 X + C T 1 C 1 = 0 3. The state-feedback or full-information synthesis problem (as stated in Theorems are solvable with constant gain, or dynamic gain linear feedback laws. Moreover, if either of these conditions are true, then for any ǫ > 0, there exists a matrix X ǫ > 0 such that A T X ǫ + X ǫ A + X ǫ ( B1 B T 1 B 2 B T 2 and X ǫ X, and σ (X ǫ X < ǫ. Xǫ + C T 1 C 1 < 0 Proof: 1 2 By hypothesis, there is a matrix X = X T 0 such that A T X + XA + X ( B 1 B1 T B 2B2 T X + C T 1 C 1 < 0 Multiply on the left and right by X 1 =: Y to give Y A T + AY + B 1 B1 T B 2 B2 T + Y C1 T C 1 Y < 0 Multiply by 1, Y ( A T + ( AY Y C1 T C 1Y + ( B 2 B2 T B 1B1 T 0 181
17 In terms of Theorem 93, let Theorem 93 X := Y A := A T B := C 1 Current Data Q := ( B 2 B T 2 B 1B T 1 Since (A, C 1 is observable, it follows that ( A T, C T 1 C 1 is detectable. Apply Theorem 93, since Y = Y T. Hence there is a matrix Y + = Y T + solving Y + ( A T + ( AY + Y + C T 1 C 1 Y + + ( B 2 B T 2 B 1 B T 1 = 0 (9.46 with ( A T C T 1 C 1 Y + stable, and Y + Y = X 1 0. Manipulating the Riccati equation (9.46 gives A + ( B 1 B T 1 B 2 B T 2 which shows that A+ ( B 1 B1 T B 2B2 T X := Y+ 1 completes the proof. ( Y 1 + = Y + A T C1 T C 1 Y + Y 1 + Y Exercise: see Lemma 6, page 836 of [DoyGKF]. 3 1 Follows easily from Theorems is indeed stable. Defining The final statement follows straightforwardly from the convergent sequence described (and constructed at the end of Theorem
ME 234, Lyapunov and Riccati Problems. 1. This problem is to recall some facts and formulae you already know. e Aτ BB e A τ dτ
ME 234, Lyapunov and Riccati Problems. This problem is to recall some facts and formulae you already know. (a) Let A and B be matrices of appropriate dimension. Show that (A, B) is controllable if and
More information6.241 Dynamic Systems and Control
6.241 Dynamic Systems and Control Lecture 24: H2 Synthesis Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology May 4, 2011 E. Frazzoli (MIT) Lecture 24: H 2 Synthesis May
More informationModern Optimal Control
Modern Optimal Control Matthew M. Peet Arizona State University Lecture 19: Stabilization via LMIs Optimization Optimization can be posed in functional form: min x F objective function : inequality constraints
More informationw T 1 w T 2. w T n 0 if i j 1 if i = j
Lyapunov Operator Let A F n n be given, and define a linear operator L A : C n n C n n as L A (X) := A X + XA Suppose A is diagonalizable (what follows can be generalized even if this is not possible -
More informationThe Kalman-Yakubovich-Popov Lemma for Differential-Algebraic Equations with Applications
MAX PLANCK INSTITUTE Elgersburg Workshop Elgersburg February 11-14, 2013 The Kalman-Yakubovich-Popov Lemma for Differential-Algebraic Equations with Applications Timo Reis 1 Matthias Voigt 2 1 Department
More informationISOLATED SEMIDEFINITE SOLUTIONS OF THE CONTINUOUS-TIME ALGEBRAIC RICCATI EQUATION
ISOLATED SEMIDEFINITE SOLUTIONS OF THE CONTINUOUS-TIME ALGEBRAIC RICCATI EQUATION Harald K. Wimmer 1 The set of all negative-semidefinite solutions of the CARE A X + XA + XBB X C C = 0 is homeomorphic
More informationStatic Output Feedback Stabilisation with H Performance for a Class of Plants
Static Output Feedback Stabilisation with H Performance for a Class of Plants E. Prempain and I. Postlethwaite Control and Instrumentation Research, Department of Engineering, University of Leicester,
More informationLMI based output-feedback controllers: γ-optimal versus linear quadratic.
Proceedings of the 17th World Congress he International Federation of Automatic Control Seoul Korea July 6-11 28 LMI based output-feedback controllers: γ-optimal versus linear quadratic. Dmitry V. Balandin
More information[k,g,gfin] = hinfsyn(p,nmeas,ncon,gmin,gmax,tol)
8 H Controller Synthesis: hinfsyn The command syntax is k,g,gfin = hinfsyn(p,nmeas,ncon,gmin,gmax,tol) associated with the general feedback diagram below. e y P d u e G = F L (P, K) d K hinfsyn calculates
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science : Dynamic Systems Spring 2011
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.4: Dynamic Systems Spring Homework Solutions Exercise 3. a) We are given the single input LTI system: [
More informationIterative Solution of a Matrix Riccati Equation Arising in Stochastic Control
Iterative Solution of a Matrix Riccati Equation Arising in Stochastic Control Chun-Hua Guo Dedicated to Peter Lancaster on the occasion of his 70th birthday We consider iterative methods for finding the
More informationFEL3210 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 informationOutline. Linear Matrix Inequalities in Control. Outline. System Interconnection. j _jst. ]Bt Bjj. Generalized plant framework
Outline Linear Matrix Inequalities in Control Carsten Scherer and Siep Weiland 7th Elgersburg School on Mathematical Systems heory Class 3 1 Single-Objective Synthesis Setup State-Feedback Output-Feedback
More informationA Simple Derivation of Right Interactor for Tall Transfer Function Matrices and its Application to Inner-Outer Factorization Continuous-Time Case
A Simple Derivation of Right Interactor for Tall Transfer Function Matrices and its Application to Inner-Outer Factorization Continuous-Time Case ATARU KASE Osaka Institute of Technology Department of
More informationonly 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 informationDenis ARZELIER arzelier
COURSE ON LMI OPTIMIZATION WITH APPLICATIONS IN CONTROL PART II.2 LMIs IN SYSTEMS CONTROL STATE-SPACE METHODS PERFORMANCE ANALYSIS and SYNTHESIS Denis ARZELIER www.laas.fr/ arzelier arzelier@laas.fr 15
More informationNetwork Reconstruction from Intrinsic Noise: Non-Minimum-Phase Systems
Preprints of the 19th World Congress he International Federation of Automatic Control Network Reconstruction from Intrinsic Noise: Non-Minimum-Phase Systems David Hayden, Ye Yuan Jorge Goncalves Department
More informationLecture 10: Linear Matrix Inequalities Dr.-Ing. Sudchai Boonto
Dr-Ing Sudchai Boonto Department of Control System and Instrumentation Engineering King Mongkuts Unniversity of Technology Thonburi Thailand Linear Matrix Inequalities A linear matrix inequality (LMI)
More informationRiccati Equations and Inequalities in Robust Control
Riccati Equations and Inequalities in Robust Control Lianhao Yin Gabriel Ingesson Martin Karlsson Optimal Control LP4 2014 June 10, 2014 Lianhao Yin Gabriel Ingesson Martin Karlsson (LTH) H control problem
More informationSemidefinite Programming Duality and Linear Time-invariant Systems
Semidefinite Programming Duality and Linear Time-invariant Systems Venkataramanan (Ragu) Balakrishnan School of ECE, Purdue University 2 July 2004 Workshop on Linear Matrix Inequalities in Control LAAS-CNRS,
More informationRobust Multivariable Control
Lecture 2 Anders Helmersson anders.helmersson@liu.se ISY/Reglerteknik Linköpings universitet Today s topics Today s topics Norms Today s topics Norms Representation of dynamic systems Today s topics Norms
More informationChapter 3. LQ, LQG and Control System Design. Dutch Institute of Systems and Control
Chapter 3 LQ, LQG and Control System H 2 Design Overview LQ optimization state feedback LQG optimization output feedback H 2 optimization non-stochastic version of LQG Application to feedback system design
More informationHomework 1 Elena Davidson (B) (C) (D) (E) (F) (G) (H) (I)
CS 106 Spring 2004 Homework 1 Elena Davidson 8 April 2004 Problem 1.1 Let B be a 4 4 matrix to which we apply the following operations: 1. double column 1, 2. halve row 3, 3. add row 3 to row 1, 4. interchange
More informationGrammians. Matthew M. Peet. Lecture 20: Grammians. Illinois Institute of Technology
Grammians Matthew M. Peet Illinois Institute of Technology Lecture 2: Grammians Lyapunov Equations Proposition 1. Suppose A is Hurwitz and Q is a square matrix. Then X = e AT s Qe As ds is the unique solution
More informationLecture 15: H Control Synthesis
c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 1/14 Lecture 15: H Control Synthesis Example c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control
More informationModern Optimal Control
Modern Optimal Control Matthew M. Peet Arizona State University Lecture 22: H 2, LQG and LGR Conclusion To solve the H -optimal state-feedback problem, we solve min γ such that γ,x 1,Y 1,A n,b n,c n,d
More informationDecentralized control with input saturation
Decentralized control with input saturation Ciprian Deliu Faculty of Mathematics and Computer Science Technical University Eindhoven Eindhoven, The Netherlands November 2006 Decentralized control with
More informationZeros and zero dynamics
CHAPTER 4 Zeros and zero dynamics 41 Zero dynamics for SISO systems Consider a linear system defined by a strictly proper scalar transfer function that does not have any common zero and pole: g(s) =α p(s)
More informationModern Optimal Control
Modern Optimal Control Matthew M. Peet Arizona State University Lecture 21: Optimal Output Feedback Control connection is called the (lower) star-product of P and Optimal Output Feedback ansformation (LFT).
More informationCONTROL DESIGN FOR SET POINT TRACKING
Chapter 5 CONTROL DESIGN FOR SET POINT TRACKING In this chapter, we extend the pole placement, observer-based output feedback design to solve tracking problems. By tracking we mean that the output is commanded
More informationTopic # Feedback Control Systems
Topic #20 16.31 Feedback Control Systems Closed-loop system analysis Bounded Gain Theorem Robust Stability Fall 2007 16.31 20 1 SISO Performance Objectives Basic setup: d i d o r u y G c (s) G(s) n control
More informationRobust 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 informationStationary trajectories, singular Hamiltonian systems and ill-posed Interconnection
Stationary trajectories, singular Hamiltonian systems and ill-posed Interconnection S.C. Jugade, Debasattam Pal, Rachel K. Kalaimani and Madhu N. Belur Department of Electrical Engineering Indian Institute
More informationControl, Stabilization and Numerics for Partial Differential Equations
Paris-Sud, Orsay, December 06 Control, Stabilization and Numerics for Partial Differential Equations Enrique Zuazua Universidad Autónoma 28049 Madrid, Spain enrique.zuazua@uam.es http://www.uam.es/enrique.zuazua
More informationObservability and state estimation
EE263 Autumn 2015 S Boyd and S Lall Observability and state estimation state estimation discrete-time observability observability controllability duality observers for noiseless case continuous-time observability
More informationEE363 homework 7 solutions
EE363 Prof. S. Boyd EE363 homework 7 solutions 1. Gain margin for a linear quadratic regulator. Let K be the optimal state feedback gain for the LQR problem with system ẋ = Ax + Bu, state cost matrix Q,
More informationMapping MIMO control system specifications into parameter space
Mapping MIMO control system specifications into parameter space Michael Muhler 1 Abstract This paper considers the mapping of design objectives for parametric multi-input multi-output systems into parameter
More informationLinear-quadratic control problem with a linear term on semiinfinite interval: theory and applications
Linear-quadratic control problem with a linear term on semiinfinite interval: theory and applications L. Faybusovich T. Mouktonglang Department of Mathematics, University of Notre Dame, Notre Dame, IN
More informationLecture 19 Observability and state estimation
EE263 Autumn 2007-08 Stephen Boyd Lecture 19 Observability and state estimation state estimation discrete-time observability observability controllability duality observers for noiseless case continuous-time
More informationThe 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 informationA Globally Stabilizing Receding Horizon Controller for Neutrally Stable Linear Systems with Input Constraints 1
A Globally Stabilizing Receding Horizon Controller for Neutrally Stable Linear Systems with Input Constraints 1 Ali Jadbabaie, Claudio De Persis, and Tae-Woong Yoon 2 Department of Electrical Engineering
More informationTRACKING AND DISTURBANCE REJECTION
TRACKING AND DISTURBANCE REJECTION Sadegh Bolouki Lecture slides for ECE 515 University of Illinois, Urbana-Champaign Fall 2016 S. Bolouki (UIUC) 1 / 13 General objective: The output to track a reference
More informationECEN 420 LINEAR CONTROL SYSTEMS. Lecture 6 Mathematical Representation of Physical Systems II 1/67
1/67 ECEN 420 LINEAR CONTROL SYSTEMS Lecture 6 Mathematical Representation of Physical Systems II State Variable Models for Dynamic Systems u 1 u 2 u ṙ. Internal Variables x 1, x 2 x n y 1 y 2. y m Figure
More informationMarcus 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 informationTo 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 informationLinear 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 informationFINITE 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 informationPrashant Mhaskar, Nael H. El-Farra & Panagiotis D. Christofides. Department of Chemical Engineering University of California, Los Angeles
HYBRID PREDICTIVE OUTPUT FEEDBACK STABILIZATION OF CONSTRAINED LINEAR SYSTEMS Prashant Mhaskar, Nael H. El-Farra & Panagiotis D. Christofides Department of Chemical Engineering University of California,
More information16.31 Fall 2005 Lecture Presentation Mon 31-Oct-05 ver 1.1
16.31 Fall 2005 Lecture Presentation Mon 31-Oct-05 ver 1.1 Charles P. Coleman October 31, 2005 1 / 40 : Controllability Tests Observability Tests LEARNING OUTCOMES: Perform controllability tests Perform
More informationLinear Algebra. P R E R E Q U I S I T E S A S S E S S M E N T Ahmad F. Taha August 24, 2015
THE UNIVERSITY OF TEXAS AT SAN ANTONIO EE 5243 INTRODUCTION TO CYBER-PHYSICAL SYSTEMS P R E R E Q U I S I T E S A S S E S S M E N T Ahmad F. Taha August 24, 2015 The objective of this exercise is to assess
More informationReduced-order Model Based on H -Balancing for Infinite-Dimensional Systems
Applied Mathematical Sciences, Vol. 7, 2013, no. 9, 405-418 Reduced-order Model Based on H -Balancing for Infinite-Dimensional Systems Fatmawati Department of Mathematics Faculty of Science and Technology,
More informationRobust Anti-Windup Controller Synthesis: A Mixed H 2 /H Setting
Robust Anti-Windup Controller Synthesis: A Mixed H /H Setting ADDISON RIOS-BOLIVAR Departamento de Sistemas de Control Universidad de Los Andes Av. ulio Febres, Mérida 511 VENEZUELA SOLBEN GODOY Postgrado
More informationLMI Methods in Optimal and Robust Control
LMI Methods in Optimal and Robust Control Matthew M. Peet Arizona State University Lecture 4: LMIs for State-Space Internal Stability Solving the Equations Find the output given the input State-Space:
More informationLinear Quadratic Gausssian Control Design with Loop Transfer Recovery
Linear Quadratic Gausssian Control Design with Loop Transfer Recovery Leonid Freidovich Department of Mathematics Michigan State University MI 48824, USA e-mail:leonid@math.msu.edu http://www.math.msu.edu/
More informationDiscussion on: Measurable signal decoupling with dynamic feedforward compensation and unknown-input observation for systems with direct feedthrough
Discussion on: Measurable signal decoupling with dynamic feedforward compensation and unknown-input observation for systems with direct feedthrough H.L. Trentelman 1 The geometric approach In the last
More informationChapter 5. Standard LTI Feedback Optimization Setup. 5.1 The Canonical Setup
Chapter 5 Standard LTI Feedback Optimization Setup Efficient LTI feedback optimization algorithms comprise a major component of modern feedback design approach: application problems involving complex models
More informationMA 527 first midterm review problems Hopefully final version as of October 2nd
MA 57 first midterm review problems Hopefully final version as of October nd The first midterm will be on Wednesday, October 4th, from 8 to 9 pm, in MTHW 0. It will cover all the material from the classes
More informationOPTIMAL CONTROL SYSTEMS
SYSTEMS MIN-MAX Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University OUTLINE MIN-MAX CONTROL Problem Definition HJB Equation Example GAME THEORY Differential Games Isaacs
More informationIntro. Computer Control Systems: F8
Intro. Computer Control Systems: F8 Properties of state-space descriptions and feedback Dave Zachariah Dept. Information Technology, Div. Systems and Control 1 / 22 dave.zachariah@it.uu.se F7: Quiz! 2
More information1 Similarity transform 2. 2 Controllability The PBH test for controllability Observability The PBH test for observability...
Contents 1 Similarity transform 2 2 Controllability 3 21 The PBH test for controllability 5 3 Observability 6 31 The PBH test for observability 7 4 Example ([1, pp121) 9 5 Subspace decomposition 11 51
More informationOutput Stabilization of Time-Varying Input Delay System using Interval Observer Technique
Output Stabilization of Time-Varying Input Delay System using Interval Observer Technique Andrey Polyakov a, Denis Efimov a, Wilfrid Perruquetti a,b and Jean-Pierre Richard a,b a - NON-A, INRIA Lille Nord-Europe
More informationECEN 605 LINEAR SYSTEMS. Lecture 7 Solution of State Equations 1/77
1/77 ECEN 605 LINEAR SYSTEMS Lecture 7 Solution of State Equations Solution of State Space Equations Recall from the previous Lecture note, for a system: ẋ(t) = A x(t) + B u(t) y(t) = C x(t) + D u(t),
More informationOn Positive Real Lemma for Non-minimal Realization Systems
Proceedings of the 17th World Congress The International Federation of Automatic Control On Positive Real Lemma for Non-minimal Realization Systems Sadaaki Kunimatsu Kim Sang-Hoon Takao Fujii Mitsuaki
More informationAn 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 informationOptimization problems on the rank and inertia of the Hermitian matrix expression A BX (BX) with applications
Optimization problems on the rank and inertia of the Hermitian matrix expression A BX (BX) with applications Yongge Tian China Economics and Management Academy, Central University of Finance and Economics,
More informationBalanced Truncation 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.242, Fall 2004: MODEL REDUCTION Balanced Truncation This lecture introduces balanced truncation for LTI
More informationFull State Feedback for State Space Approach
Full State Feedback for State Space Approach State Space Equations Using Cramer s rule it can be shown that the characteristic equation of the system is : det[ si A] 0 Roots (for s) of the resulting polynomial
More informationControl Systems Design
ELEC4410 Control Systems Design Lecture 18: State Feedback Tracking and State Estimation Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science Lecture 18:
More informationClosed-Loop Structure of Discrete Time H Controller
Closed-Loop Structure of Discrete Time H Controller Waree Kongprawechnon 1,Shun Ushida 2, Hidenori Kimura 2 Abstract This paper is concerned with the investigation of the closed-loop structure of a discrete
More informationDampening Controllers via a Riccati Equation. Approach. Pod vodarenskou vez Praha 8. Czech Republic. Fakultat fur Mathematik
Dampening Controllers via a Riccati Equation Approach J.J. Hench y C. He z V. Kucera y V. Mehrmann z y Institute of Information Theory and Automation Academy of Sciences of the Czech Republic Pod vodarenskou
More informationMultivariable MRAC with State Feedback for Output Tracking
29 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 1-12, 29 WeA18.5 Multivariable MRAC with State Feedback for Output Tracking Jiaxing Guo, Yu Liu and Gang Tao Department
More informationIdentification Methods for Structural Systems
Prof. Dr. Eleni Chatzi System Stability Fundamentals Overview System Stability Assume given a dynamic system with input u(t) and output x(t). The stability property of a dynamic system can be defined from
More informationState Feedback and State Estimators Linear System Theory and Design, Chapter 8.
1 Linear System Theory and Design, http://zitompul.wordpress.com 2 0 1 4 State Estimator In previous section, we have discussed the state feedback, based on the assumption that all state variables are
More informationH 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 informationOn Solving Large Algebraic. Riccati Matrix Equations
International Mathematical Forum, 5, 2010, no. 33, 1637-1644 On Solving Large Algebraic Riccati Matrix Equations Amer Kaabi Department of Basic Science Khoramshahr Marine Science and Technology University
More informationAlgebraic Properties of Solutions of Linear Systems
Algebraic Properties of Solutions of Linear Systems In this chapter we will consider simultaneous first-order differential equations in several variables, that is, equations of the form f 1t,,,x n d f
More information10 Transfer Matrix Models
MIT EECS 6.241 (FALL 26) LECTURE NOTES BY A. MEGRETSKI 1 Transfer Matrix Models So far, transfer matrices were introduced for finite order state space LTI models, in which case they serve as an important
More informationNonlinear Control Systems
Nonlinear Control Systems António Pedro Aguiar pedro@isr.ist.utl.pt 5. Input-Output Stability DEEC PhD Course http://users.isr.ist.utl.pt/%7epedro/ncs2012/ 2012 1 Input-Output Stability y = Hu H denotes
More informationTopics in control Tracking and regulation A. Astolfi
Topics in control Tracking and regulation A. Astolfi Contents 1 Introduction 1 2 The full information regulator problem 3 3 The FBI equations 5 4 The error feedback regulator problem 5 5 The internal model
More informationDiscrete Riccati equations and block Toeplitz matrices
Discrete Riccati equations and block Toeplitz matrices André Ran Vrije Universiteit Amsterdam Leonid Lerer Technion-Israel Institute of Technology Haifa André Ran and Leonid Lerer 1 Discrete algebraic
More informationProblem Set 5 Solutions 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Problem Set 5 Solutions The problem set deals with Hankel
More informationLow Gain Feedback. Properties, Design Methods and Applications. Zongli Lin. July 28, The 32nd Chinese Control Conference
Low Gain Feedback Properties, Design Methods and Applications Zongli Lin University of Virginia Shanghai Jiao Tong University The 32nd Chinese Control Conference July 28, 213 Outline A review of high gain
More informationLMIs for Observability and Observer Design
LMIs for Observability and Observer Design Matthew M. Peet Arizona State University Lecture 06: LMIs for Observability and Observer Design Observability Consider a system with no input: ẋ(t) = Ax(t), x(0)
More informationMATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators.
MATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators. Adjoint operator and adjoint matrix Given a linear operator L on an inner product space V, the adjoint of L is a transformation
More informationAN EVENT-TRIGGERED TRANSMISSION POLICY FOR NETWORKED L 2 -GAIN CONTROL
4 Journal of Marine Science and echnology, Vol. 3, No., pp. 4-9 () DOI:.69/JMS-3-3-3 AN EVEN-RIGGERED RANSMISSION POLICY FOR NEWORKED L -GAIN CONROL Jenq-Lang Wu, Yuan-Chang Chang, Xin-Hong Chen, and su-ian
More informationAppendix A Solving Linear Matrix Inequality (LMI) Problems
Appendix A Solving Linear Matrix Inequality (LMI) Problems In this section, we present a brief introduction about linear matrix inequalities which have been used extensively to solve the FDI problems described
More informationH 2 Optimal State Feedback Control Synthesis. Raktim Bhattacharya Aerospace Engineering, Texas A&M University
H 2 Optimal State Feedback Control Synthesis Raktim Bhattacharya Aerospace Engineering, Texas A&M University Motivation Motivation w(t) u(t) G K y(t) z(t) w(t) are exogenous signals reference, process
More informationHankel Optimal Model Reduction 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.242, Fall 2004: MODEL REDUCTION Hankel Optimal Model Reduction 1 This lecture covers both the theory and
More informationEEE582 Homework Problems
EEE582 Homework Problems HW. Write a state-space realization of the linearized model for the cruise control system around speeds v = 4 (Section.3, http://tsakalis.faculty.asu.edu/notes/models.pdf). Use
More informationObservability. It was the property in Lyapunov stability which allowed us to resolve that
Observability We have seen observability twice already It was the property which permitted us to retrieve the initial state from the initial data {u(0),y(0),u(1),y(1),...,u(n 1),y(n 1)} It was the property
More informationAn Introduction to Linear Matrix Inequalities. Raktim Bhattacharya Aerospace Engineering, Texas A&M University
An Introduction to Linear Matrix Inequalities Raktim Bhattacharya Aerospace Engineering, Texas A&M University Linear Matrix Inequalities What are they? Inequalities involving matrix variables Matrix variables
More informationAnalysis of undamped second order systems with dynamic feedback
Control and Cybernetics vol. 33 (24) No. 4 Analysis of undamped second order systems with dynamic feedback by Wojciech Mitkowski Chair of Automatics AGH University of Science and Technology Al. Mickiewicza
More informationand the nite horizon cost index with the nite terminal weighting matrix F > : N?1 X J(z r ; u; w) = [z(n)? z r (N)] T F [z(n)? z r (N)] + t= [kz? z r
Intervalwise Receding Horizon H 1 -Tracking Control for Discrete Linear Periodic Systems Ki Baek Kim, Jae-Won Lee, Young Il. Lee, and Wook Hyun Kwon School of Electrical Engineering Seoul National University,
More informationPrinciples of Optimal Control Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 16.323 Principles of Optimal Control Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 16.323 Lecture
More informationMath 489AB Exercises for Chapter 2 Fall Section 2.3
Math 489AB Exercises for Chapter 2 Fall 2008 Section 2.3 2.3.3. Let A M n (R). Then the eigenvalues of A are the roots of the characteristic polynomial p A (t). Since A is real, p A (t) is a polynomial
More information2 The Linear Quadratic Regulator (LQR)
2 The Linear Quadratic Regulator (LQR) Problem: Compute a state feedback controller u(t) = Kx(t) that stabilizes the closed loop system and minimizes J := 0 x(t) T Qx(t)+u(t) T Ru(t)dt where x and u are
More informationStability and Inertia Theorems for Generalized Lyapunov Equations
Published in Linear Algebra and its Applications, 355(1-3, 2002, pp. 297-314. Stability and Inertia Theorems for Generalized Lyapunov Equations Tatjana Stykel Abstract We study generalized Lyapunov equations
More informationLecture 4 and 5 Controllability and Observability: Kalman decompositions
1 Lecture 4 and 5 Controllability and Observability: Kalman decompositions Spring 2013 - EE 194, Advanced Control (Prof. Khan) January 30 (Wed.) and Feb. 04 (Mon.), 2013 I. OBSERVABILITY OF DT LTI SYSTEMS
More informationResearch 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 informationOptimal discrete-time H /γ 0 filtering and control under unknown covariances
International Journal of Control ISSN: 0020-7179 (Print) 1366-5820 (Online) Journal homepage: http://www.tandfonline.com/loi/tcon20 Optimal discrete-time H filtering and control under unknown covariances
More information