Solution of Linear Statespace Systems


 Merryl Webb
 2 years ago
 Views:
Transcription
1 Solution of Linear Statespace Systems Homogeneous (u=0) LTV systems first Theorem (PeanoBaker series) The unique solution to x(t) = (t, )x 0 where The matrix function is given by is called the state transition function It satisfies the following properties 1. For every t and every t0, (t, ) is the unique solution of (t, )=I + d dt ẋ(t) =A(t)x(t), x( )=x 0 2 R n + A(s 1 ) ds 1 + (t, ) applez s1 A(s 1 ) applez s1 applez s2 (t, )=A(t) (t, ), (, )=I A(s 2 ) ds 2 ds 1 A(s 1 ) A(s 2 ) A(s 3 ) ds 3 ds 2 ds proof by substitution 2. For any s, (t, )= (t, s) (s, ) [the semigroup property] 3. For every t and t0, (t, ) is nonsingular and since I = (,t) (t, )= (t, ) (,t) 1
2 Solution to Nonhomogeneous LTV Systems Theorem (Variation of constants) The unique solution to is ẋ(t) =A(t)x(t)+B(t)u(t), y(t) =C(t)x(t)+D(t)u(t), x( )=x 0 x(t) = (t, )x 0 + Proof by substitution again Uniqueness follows by the global Lipschitz property of the mapping which is due to linearity The zeroinput response is y(t) =C(t) (t, )x 0 + (t, )B( ) d C(t) (t, )B( ) d + D(t)u(t) y zeroinput (t) =C(t) (t, )x 0 The zerostate response is y zerostate (t) = C(t) (t, )B( ) d + D(t)u(t) 2
3 The unique solution of is given by Discretetime LTV Systems where the discrete transition function, Properties of 1. (t, ) is the unique solution of 2. For every t s, (t, )= (t, s) (s, ) There is no requirement for (t, ) to be nonsingular will be singular if any A(s),, is singular The unique solution of is (t, ) x(t + 1) = A(t)x(t), x( )=x 0, t x(t) = (t, )x 0, t (t, ), is given by ( I, t = (t, )= A(t 1)A(t 2)...A( + 1)A( ), t > x(t) = (t, )x 0 + (t +1, )=A(t) (t, ), (, )=I (t, ) t>s Xt 1 y(t) =C(t) (t, )x 0 + x(t + 1) = A(t)x(t)+B(t)u(t), x( )=x 0 y(t) =C(t)x(t)+D(t)u(t) = (t, + 1)B( )u( ), t Xt 1 = C(t) (t, + 1)B( )u( )+D(t)u(t), t, t, 2 Z 3
4 The Matrix Exponential Solution of LTI Statespace Systems Define for M 2 R n n its exponential e M = also in From the PeanoBaker series for the ODE we see that the transition function for the LTI system (t, ) = I + A ds 1 + A 2 (t, t 0 )=I + A(s 1 ) ds 1 + A(s 1 ) applez s1 + A(s 1 ) A(s 2 ) applez s1 applez s1 applez s2 ds 2 ds 1 + A 3 x( )=x 0 2 R n A(s 2 ) ds 2 ds 1 applez s1 applez s2 (I M) 1 = I + M + M 2 + M = I + A(t )+ 1 2 A2 (t ) A3 (t ) 3 + = e A(t ) 1X k=0 1 k M k R n n A(s 3 ) ds 3 ds 2 ds is ds 3 ds 2 ds The solution to is x(t) =e A(t ) x 0 + y(t) =Ce A(t ) x 0 + e A(t Ce A(t ) Bu( ) d ) Bu( ) d 4
5 Properties of the Matrix Exponential 1. The function is the unique solution of 2. For every t, 2 R we have but in general 3. For every, e At is nonsingular and (unless A and B commute) 4. Using the CayleyHamilton theorem, for every A 2 R n n, there exist n scalar functions { i (t) :i =0,...,n 1} such that 5. For every, 6. so e At if e At e A = e A(t+ ) e At e Bt 6= e (A+B)t t 2 R e At 1 = e At e At = nx 1 i (t)a i, 8t 2 R i=0 A 2 R n n Ae At = e At A e At = L 1 (si A) 1 = L 1 1 [adj(si A)]T det(si A) = L 1 1 (s 1) m 1 (s 2) m 2...(s k) m [matrix of polynomials in s] k = [R 11 ]e 1t +[R 12 ]te 1t + +[R 1m1 ]t m 1 1 e 1 t +[R 21 ]e 2t +[R 22 ]te 2t + +[R 2m2 ]t m 2 1 e 2 t + +[R k1 ]e kt +[R k2 ]te kt + +[R kmk ]t m k 1 e k t e At 0 as t 1 Re [ i(a)] < 0, i=1,...,n d dt eat = Ae At, e A0 = I 5
6 7. Since for any invertible T More on Matrix Exponentials 8. Using a generalized eigenvector matrix X, X 1 AX = J (Jordan form) (T AT 1 ) k = T AT 1 T AT 1 T AT 1 = TA k T 1 1X e TAT 1t 1 1X = k (T AT 1 ) k t k 1 = T k Ak t k T 1 = Te At T 1 since J is block diagonal and upper triangular, so is This yields the same eigenvalue relationship as earlier 9. Discretetime equivalent Using the formal power series Z 1 (zi A) 1 = Z 1 z 1 (I z 1 A) 1 = Z 1 z 1 I + z 2 A + z 3 A = Z 1 1 [adj(zi A)]T det(zi A) we have k=0 e At = Xe Jt X 1 k=0 e Jt (I M) 1 = I + M + M 2 + M A k 0 as t 1if i(a) < 1, i=1,...n 6
7 Lyapunov Stability Definition An equilibrium, x*, of a system is stable in the sense of Lyapunov if given any > 0 there exists a ( ) > 0 such that for all x0 satisfying x x 0 < we have x x(t) < for all t>0 This is a definition applicable to all systems For linear systems it simplifies greatly x*=0 is the candidate equilibrium Lyapunov stability is a property of the state equation only Definition (Lyapunov stability for linear systems) ẋ(t) =A(t)x(t)+B(t)u(t), y(t) =C(t)x(t)+D(t)u(t) (i) is (marginally) stable in the sense of Lyapunov if for every the homogeneous state response x(t) = (t, )x 0, t is uniformly bounded (ii) is asymptotically stable in the sense of Lyapunov if, in addition, for every initial condition x( )=x 0 2 R n we have x(t) 0 as t 1 (iii) is exponentially stable if, in addition, there exist constant c, λ>0 such that for every initial condition x( )=x 0 2 R n we have kx(t)k applece (t ) kx 0 k for all t (iv) is unstable if it is not marginally stable in the sense of Lyapunov Lyapunov stability only treats the homogeneous system x( )=x 0 2 R n ẋ(t) =A(t)x(t) 7
8 Theorem (8.2) Lyapunov (internal) stability The HCLTI (homogeneous continuoustime linear timeinvariant system) Proof (i) is marginally stable if and only if all the eigenvalues of A have negative or zero real parts and al Jordan blocks corresponding to eignevalues with zero real parts are 1x1 (ii) is asymptotically stable if and only if all eigenvalues of A have strictly negative real parts (iii) is exponentially stable if and only if all eigenvalues of A have strictly negative real pats (iv) is unstable if and only if at least one eigenvalue of A has positive real part or has zero real part but the corresponding Jordan block is larger than 1x1 Follows from the properties of ẋ(t) =Ax(t), x 2 R n For LTI systems, asymptotic and exponential stability coincide e At This is NOT so for LTV systems  see any discussion of the Mathieu Equation ÿ(t)+[ + cos t]y(t) = 0 8
9 Stability of the Mathieu equation The Mathieu equation  an LTV system ÿ(t)+[ + cos t]y(t) = 0 9
10 Lyapunov Stability in General (Khalil) Nonlinear, timevarying, homogeneous, autonomous, continuoustime system where is locally Lipschitz from domain into The equilibrium x 2 D, where f(x )=0, is (i) stable if for every ε>0 there is a δ(ε)>0 such that (ii) unstable if it is not stable ẋ(t) =f(x) f : D R n D R n R n kx(0) x k < =) kx(t) x k <, 8 (iii) asymptotically stable if it is stable and δ can be chosen so that kx(0) x k < =) lim kx(t) x k =0 t1 (iv) globally asymptotically stable if it is asymptotically stable and δ may be chosen arbitrarily For nonautonomous systems introduce the ideas of uniform stability ẋ(t) =f(x, t) we have similar ideas, but we must Note that for linear systems any kind of stability is necessarily a global property 10
11 Lyapunov s theorem Let x=0 be an equilibrium of Let then x=0 is stable Moreover, if Lyapunov continued in the domain D be a continuously differentiable function such that then x=0 is asymptotically stable If D = R n and kxk 1 =) V (x) 1(radially unbounded) then x=0 is globally asymptotically stable V(x) is a scalar function of the nvector x V : D R ẋ(t) =f(x) V (0) = 0 and V (x) > 0 in D {0} (positive definite) V (x) < 0 in D {0} V (x) apple 0 in D (decrescent) It represents something akin to energy for a dissipative system Sometimes it is easier to find V than it is to prove stability directly There are many other uses for Lyapunov functions e.g. domain of attraction estimation 11
12 Lyapunov Stability in Linear Systems Consider the continuoustime LTI system The following statements are equivalent (i) The system is asymptotically stable (ii) The system is globally asymptotically stable (iii) The system is globally exponentially stable (iv) All the eigenvalues of A have negative real parts (v) For every symmetric positive definite matrix Q there exists a unique solution to the (continuoustime) Lyapunov Equation A T P + PA = Q Moreover, P is symmetric and positive definite (vi) There exists a symmetric positive definite matrix P for which the Lyapunov matrix inequality holds A T P + PA < 0 If A is asymptotically stable the unique solution of the Lyapunov Equation is P = Z 1 ẋ = Ax V(x)=x T Px is a Lyapunov function for this system 0 e AT t Qe At dt 12
13 Discretetime Linear Lyapunov Stability The discretetime LTV system (i) marginally stable in the sense of Lyapunov if for every initial condition x( )=x 0 the homogeneous state response x(t) = (t, )x 0 is uniformly (in t and t0) bounded (ii) asymptotically stable in the sense of Lyapunov if, in addition, we have for every initial condition x(t) 0 as t 1 (iii) exponentially stable if there exist constants c, λ such that for every initial condition kx(t)k applec (t t0) kx 0 k, 8t (iv) unstable if it is not marginally stable in the sense of Lyapunov x(t + 1) = A(t)x(t) The homogeneous discretetime LTI system x(t + 1) = Ax(t) is (i) marginally stable if and only if all the eigenvalues of A have magnitude less than or equal to one and all the Jordan blocks corresponding to eigenvalues equal to one have size 1x1 (ii) asymptotically and exponentially stable if and only if all of the eigenvalues of A have magnitude less than one (iii) unstable if and only if at least one eigenvalue of A has magnitude larger than one or magnitude equal to one and Jordan block larger than 1x1 is 13
14 Discretetime Linear Stability For the discretetime homogeneous LTI system following statements are equivalent (i) The system is asymptotically stable (ii) The system is exponentially stable (iii) All eigenvalues of A have magnitude less than one (iv) For every symmetric positive definite matrix Q there exists a unique solution P to the Stein Equation (Discretetime Lyapunov Equation) Moreover, P is symmetric and positive definite (v) There exists a symmetric positive definite matrix P for which the following Lyapunov matrix inequality holds A T PA P<0 For stable A, use the solution A T PA P = Q P = 1X A jt QA j V(x)=x T Px is a discretetime Lyapunov function for this system j=0 x(t + 1) = Ax(t) the 14
15 Discretetime Lyapunov Stability Consider the discretetime nonlinear homogeneous autonomous system Let x=0 be an equilibrium in D, as before Let then x=0 is stable Moreover, if asymptotically stable If V : D R x(t + 1) = f(x(t)) be a function such that along all solutions of the system V (0) = 0 and V (x) > 0 in D {0} (positive definite) V (x(t + 1)) = V (f(x(t))) apple V (x(t)) in D (decrescent) D = R n V (x(t)) = V (f(x(t))) V (x(t)) < 0 in D {0} and V(x) is radially unbounded then x=0 is g.a.s. Very similar ideas to continuous time Uses the Monotone Convergence Theorem then x=0 is 15
16 Discretetime Lyapunov (Stein) Equation  saying more Consider the discretetime LTI system Take as our candidate Lyapunov function V(x)=x T Px with P>0 satisfies positive definite and radially unbounded properties Suppose A T PA P = C T C where C is mxn so that the righthand side of the discrete Lyapunov equation is only nonpositive definite Then But x(t + 1) = Ax(t) V [x(t + 1)] V [x(t)] = x(t + 1) T Px(t + 1) x(t) T Px(t) = x(t) T A T PA P x(t) = x(t) T C T Cx(t) =[Cx(t)] T [Cx(t)] apple 0 V [x(t + n)] V [x(t)] = V [x(t + n)] V [x(t + n 1)] + V [x(t + n 1)] V [(x(t + n 2)] + + V [x(t + 1)] V [x(t)] = x(t + n 1) T C T Cx(t + n 1) x(t + n 2) T C T Cx(t + n 2) x(t) T C T h Cx(t) i = x(t) T A (n 1)T C T CA (n 1) + A (n 2)T C T CA (n 2) + + C T C x(t) = x(t) T [O T O]x(t)] < 0, if [C, A] is observable 16
17 A Grownups Version of the Lyapunov Equation The continuoustime LTI system ẋ = Ax is g.a.s. if there exists a positive definite solution P to the Lyapunov matrix equation with the matrix pair [A,C] observable The discretetime LTI system x(t + 1) = Ax(t) is g.a.s. if there exists a positive definite solution to the discretetime Lyapunov matrix equation with the matrix pair [A,C] observable We see observability arise again A T P + PA = C T C A T PA P = C T C It has to do with the only solutions to the equation which yield zero derivative (difference) over a time interval are those which are zero V is positive definite and decreasing, so it converges So ΔV converges to zero and this can only happen if x 0 17
18 The response of this system is InputOutput Stability where Lyapunov deals with the first term provided C(t) is uniformly bounded BoundedInput BoundedOutput (BIBO) stability deals with the second term Definition: The continuoustime LTV system is uniformly BIBO stable if there exists a constant g such that for every input u(t) its zerostate response satisfies sup ky zerostate (t)k appleg ku(t)k ẋ(t) =A(t)x(t)+B(t)u(t), y(t) =C(t)x(t)+D(t)u(t), x( )=x 0 y zeroinput (t) =C(t) (t, )x 0 y zerostate (t) = t2[,1) y(t) =y zeroinput (t)+y zerostate (t) C(t) (t, )B( ) d + D(t)u(t) sup t2[,1) Note that these are just (euclidean) vector norms not function norms The LTV system is BIBO stable if and only if every element of D(t) is uniformly bounded and sup t2[,1) [C(t) (t, )B( )] ij d < 1 18
19 The system Continuoustime LTI BIBO Stability ẋ = Ax + Bu, y = Cx + Du Z 1 0 is BIBO stable if and only if Ce A B ij d < 1 This condition is met if and only if every element of the transfer function matrix G(s) =D + C(sI A) 1 B only has poles with negative real part When the system is exponentially stable then it must also be BIBO stable The reason for the rather peculiar condition on the transfer function is that we need to be concerned with unstable polezero cancellations We saw some examples of systems with A matrices which were unstable while the transfer function was stable G(s) = A = , B = , C = i(a) ={ 3, 2, 1}, Lyapunov unstable s 2 1 s 3 +4s 2 + s 6 = (s + 1)(s 1) (s + 2)(s + 3)(s 1) = s +1 (s + 2)(s + 3) BIBO stable 19
20 Zerostate response Discretetime Inputoutput Stability The discretetime LTV system is uniformly BIBO stable if there exists a constant g such that sup ky zs (t)k appleg sup ku(t)k The system is uniformly BIBO stable if and only if For the discretetime LTI system the following statements are equivalent (i) the system is uniformly BIBO stable (ii) sup t2n 1X =0 x(t + 1) = A(t)x(t) + B(t)u(t), y zs (t) = Xt 1 =0 =0 y(t) = C(t)x(t) + D(t)u(t) Xt 1 C(t) (t, + 1)B( )u( )+ D(t)u(t), 8 t2n (iii) all the poles of all the elements of the transfer function matrix lie strictly inside the unit circle. t2n [C(t) (t, )B( )] ij < 1, for i =1,..., m, j =1,...,k x + = Ax + Bu, y = Cx + Du [CA B] ij < 1, for i =1,..., m, j =1,...,k 20
21 Summary The ideas of LTV systems specialize in the case of LTI systems Likewise, many ideas from linear systems give guidance how to proceed with nonlinear systems Lyapunov stability is one example The concepts of an energylike function and of dissipation are entirely generalizable Finding a Lyapunov function for a nonlinear system is often much simpler than proving stability by any other means The Lyapunov function can help find the region of attraction of an equilibrium Inputoutput stability differs dramatically from Lyapunov stability Ideas from normed function spaces come to the fore MAE281A, 281B begin to get a handle on all of this That is why we spent more time on these ideas 21
ME Fall 2001, Fall 2002, Spring I/O Stability. Preliminaries: Vector and function norms
I/O Stability Preliminaries: Vector and function norms 1. Sup norms are used for vectors for simplicity: x = max i x i. Other norms are also okay 2. Induced matrix norms: let A R n n, (i stands for induced)
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 informationChapter III. Stability of Linear Systems
1 Chapter III Stability of Linear Systems 1. Stability and state transition matrix 2. Timevarying (nonautonomous) systems 3. Timeinvariant systems 1 STABILITY AND STATE TRANSITION MATRIX 2 In this chapter,
More informationMath Ordinary Differential Equations
Math 411  Ordinary Differential Equations Review Notes  1 1  Basic Theory A first order ordinary differential equation has the form x = f(t, x) (11) Here x = dx/dt Given an initial data x(t 0 ) = x
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 informationLinear Systems. Linear systems?!? (Roughly) Systems which obey properties of superposition Input u(t) output
Linear Systems Linear systems?!? (Roughly) Systems which obey properties of superposition Input u(t) output Our interest is in dynamic systems Dynamic system means a system with memory of course including
More informationGeorgia Institute of Technology Nonlinear Controls Theory Primer ME 6402
Georgia Institute of Technology Nonlinear Controls Theory Primer ME 640 Ajeya Karajgikar April 6, 011 Definition Stability (Lyapunov): The equilibrium state x = 0 is said to be stable if, for any R > 0,
More informationCDS Solutions to the Midterm Exam
CDS 22  Solutions to the Midterm Exam Instructor: Danielle C. Tarraf November 6, 27 Problem (a) Recall that the H norm of a transfer function is timedelay invariant. Hence: ( ) Ĝ(s) = s + a = sup /2
More informationControl Systems. Laplace domain analysis
Control Systems Laplace domain analysis L. Lanari outline introduce the Laplace unilateral transform define its properties show its advantages in turning ODEs to algebraic equations define an Input/Output
More informationLinear ODEs. Existence of solutions to linear IVPs. Resolvent matrix. Autonomous linear systems
Linear ODEs p. 1 Linear ODEs Existence of solutions to linear IVPs Resolvent matrix Autonomous linear systems Linear ODEs Definition (Linear ODE) A linear ODE is a differential equation taking the form
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 StateSpace Internal Stability Solving the Equations Find the output given the input StateSpace:
More informationTopic # /31 Feedback Control Systems. Analysis of Nonlinear Systems Lyapunov Stability Analysis
Topic # 16.30/31 Feedback Control Systems Analysis of Nonlinear Systems Lyapunov Stability Analysis Fall 010 16.30/31 Lyapunov Stability Analysis Very general method to prove (or disprove) stability of
More informationThe goal of this chapter is to study linear systems of ordinary differential equations: dt,..., dx ) T
1 1 Linear Systems The goal of this chapter is to study linear systems of ordinary differential equations: ẋ = Ax, x(0) = x 0, (1) where x R n, A is an n n matrix and ẋ = dx ( dt = dx1 dt,..., dx ) T n.
More informationControl Systems Design
ELEC4410 Control Systems Design Lecture 13: Stability Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science Lecture 13: Stability p.1/20 Outline InputOutput
More informationLyapunov Stability Theory
Lyapunov Stability Theory Peter Al Hokayem and Eduardo Gallestey March 16, 2015 1 Introduction In this lecture we consider the stability of equilibrium points of autonomous nonlinear systems, both in continuous
More informationEEE582 Homework Problems
EEE582 Homework Problems HW. Write a statespace 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 informationModule 03 Linear Systems Theory: Necessary Background
Module 03 Linear Systems Theory: Necessary Background Ahmad F. Taha EE 5243: Introduction to CyberPhysical Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ taha/index.html September
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 informationNonlinear Observers. Jaime A. Moreno. Eléctrica y Computación Instituto de Ingeniería Universidad Nacional Autónoma de México
Nonlinear Observers Jaime A. Moreno JMorenoP@ii.unam.mx Eléctrica y Computación Instituto de Ingeniería Universidad Nacional Autónoma de México XVI Congreso Latinoamericano de Control Automático October
More information1. Find the solution of the following uncontrolled linear system. 2 α 1 1
Appendix B Revision Problems 1. Find the solution of the following uncontrolled linear system 0 1 1 ẋ = x, x(0) =. 2 3 1 Class test, August 1998 2. Given the linear system described by 2 α 1 1 ẋ = x +
More informationStability in the sense of Lyapunov
CHAPTER 5 Stability in the sense of Lyapunov Stability is one of the most important properties characterizing a system s qualitative behavior. There are a number of stability concepts used in the study
More informationNonlinear Systems and Control Lecture # 12 Converse Lyapunov Functions & Time Varying Systems. p. 1/1
Nonlinear Systems and Control Lecture # 12 Converse Lyapunov Functions & Time Varying Systems p. 1/1 p. 2/1 Converse Lyapunov Theorem Exponential Stability Let x = 0 be an exponentially stable equilibrium
More informationLecture 4. Chapter 4: Lyapunov Stability. Eugenio Schuster. Mechanical Engineering and Mechanics Lehigh University.
Lecture 4 Chapter 4: Lyapunov Stability Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture 4 p. 1/86 Autonomous Systems Consider the autonomous system ẋ
More informationIntro. Computer Control Systems: F8
Intro. Computer Control Systems: F8 Properties of statespace descriptions and feedback Dave Zachariah Dept. Information Technology, Div. Systems and Control 1 / 22 dave.zachariah@it.uu.se F7: Quiz! 2
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 informationNonlinear Control. Nonlinear Control Lecture # 8 Time Varying and Perturbed Systems
Nonlinear Control Lecture # 8 Time Varying and Perturbed Systems Timevarying Systems ẋ = f(t,x) f(t,x) is piecewise continuous in t and locally Lipschitz in x for all t 0 and all x D, (0 D). The origin
More informationIntroduction to Nonlinear Control Lecture # 3 TimeVarying and Perturbed Systems
p. 1/5 Introduction to Nonlinear Control Lecture # 3 TimeVarying and Perturbed Systems p. 2/5 Timevarying Systems ẋ = f(t, x) f(t, x) is piecewise continuous in t and locally Lipschitz in x for all t
More informationLinear System Theory
Linear System Theory Wonhee Kim Chapter 6: Controllability & Observability Chapter 7: Minimal Realizations May 2, 217 1 / 31 Recap State space equation Linear Algebra Solutions of LTI and LTV system Stability
More information1. The Transition Matrix (Hint: Recall that the solution to the linear equation ẋ = Ax + Bu is
ECE 55, Fall 2007 Problem Set #4 Solution The Transition Matrix (Hint: Recall that the solution to the linear equation ẋ Ax + Bu is x(t) e A(t ) x( ) + e A(t τ) Bu(τ)dτ () This formula is extremely important
More information1 Lyapunov theory of stability
M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 1 1 Lyapunov theory of stability Introduction. Lyapunov s second (or direct) method provides tools for studying (asymptotic) stability
More informationModule 09 From sdomain to timedomain From ODEs, TFs to StateSpace Modern Control
Module 09 From sdomain to timedomain From ODEs, TFs to StateSpace Modern Control Ahmad F. Taha EE 3413: Analysis and Desgin of Control Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/
More informationExamples include: (a) the Lorenz system for climate and weather modeling (b) the HodgkinHuxley system for neuron modeling
1 Introduction Many natural processes can be viewed as dynamical systems, where the system is represented by a set of state variables and its evolution governed by a set of differential equations. Examples
More informationControl Systems. Internal Stability  LTI systems. L. Lanari
Control Systems Internal Stability  LTI systems L. Lanari outline LTI systems: definitions conditions South stability criterion equilibrium points Nonlinear systems: equilibrium points examples stable
More informationIntroduction to Nonlinear Control Lecture # 4 Passivity
p. 1/6 Introduction to Nonlinear Control Lecture # 4 Passivity È p. 2/6 Memoryless Functions ¹ y È Ý Ù È È È È u (b) µ power inflow = uy Resistor is passive if uy 0 p. 3/6 y y y u u u (a) (b) (c) Passive
More informationLinear System Theory. Wonhee Kim Lecture 1. March 7, 2018
Linear System Theory Wonhee Kim Lecture 1 March 7, 2018 1 / 22 Overview Course Information Prerequisites Course Outline What is Control Engineering? Examples of Control Systems Structure of Control Systems
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 informationNonlinear Control. Nonlinear Control Lecture # 8 Time Varying and Perturbed Systems
Nonlinear Control Lecture # 8 Time Varying and Perturbed Systems Timevarying Systems ẋ = f(t,x) f(t,x) is piecewise continuous in t and locally Lipschitz in x for all t 0 and all x D, (0 D). The origin
More informationDiscrete and continuous dynamic systems
Discrete and continuous dynamic systems Bounded input bounded output (BIBO) and asymptotic stability Continuous and discrete time linear timeinvariant systems Katalin Hangos University of Pannonia Faculty
More informationLyapunov Stability Analysis: Open Loop
Copyright F.L. Lewis 008 All rights reserved Updated: hursday, August 8, 008 Lyapunov Stability Analysis: Open Loop We know that the stability of linear timeinvariant (LI) dynamical systems can be determined
More informationECE504: Lecture 8. D. Richard Brown III. Worcester Polytechnic Institute. 28Oct2008
ECE504: Lecture 8 D. Richard Brown III Worcester Polytechnic Institute 28Oct2008 Worcester Polytechnic Institute D. Richard Brown III 28Oct2008 1 / 30 Lecture 8 Major Topics ECE504: Lecture 8 We are
More informationState will have dimension 5. One possible choice is given by y and its derivatives up to y (4)
A Exercise State will have dimension 5. One possible choice is given by y and its derivatives up to y (4 x T (t [ y(t y ( (t y (2 (t y (3 (t y (4 (t ] T With this choice we obtain A B C [ ] D 2 3 4 To
More informationControl Systems I. Lecture 4: Diagonalization, Modal Analysis, Intro to Feedback. Readings: Emilio Frazzoli
Control Systems I Lecture 4: Diagonalization, Modal Analysis, Intro to Feedback Readings: Emilio Frazzoli Institute for Dynamic Systems and Control DMAVT ETH Zürich October 13, 2017 E. Frazzoli (ETH)
More informationStability lectures. Stability of Linear Systems. Stability of Linear Systems. Stability of Continuous Systems. EECE 571M/491M, Spring 2008 Lecture 5
EECE 571M/491M, Spring 2008 Lecture 5 Stability of Continuous Systems http://courses.ece.ubc.ca/491m moishi@ece.ubc.ca Dr. Meeko Oishi Electrical and Computer Engineering University of British Columbia,
More informationModule 06 Stability of Dynamical Systems
Module 06 Stability of Dynamical Systems Ahmad F. Taha EE 5143: Linear Systems and Control Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ataha October 10, 2017 Ahmad F. Taha Module 06
More informationModeling & Control of Hybrid Systems Chapter 4 Stability
Modeling & Control of Hybrid Systems Chapter 4 Stability Overview 1. Switched systems 2. Lyapunov theory for smooth and linear systems 3. Stability for any switching signal 4. Stability for given switching
More information7.1 Linear Systems Stability Consider the ContinuousTime (CT) Linear TimeInvariant (LTI) system
7 Stability 7.1 Linear Systems Stability Consider the ContinuousTime (CT) Linear TimeInvariant (LTI) system ẋ(t) = A x(t), x(0) = x 0, A R n n, x 0 R n. (14) The origin x = 0 is a globally asymptotically
More informationCONTROL DESIGN FOR SET POINT TRACKING
Chapter 5 CONTROL DESIGN FOR SET POINT TRACKING In this chapter, we extend the pole placement, observerbased output feedback design to solve tracking problems. By tracking we mean that the output is commanded
More informationTheorem 1. ẋ = Ax is globally exponentially stable (GES) iff A is Hurwitz (i.e., max(re(σ(a))) < 0).
Linear Systems Notes Lecture Proposition. A M n (R) is positive definite iff all nested minors are greater than or equal to zero. n Proof. ( ): Positive definite iff λ i >. Let det(a) = λj and H = {x D
More informationLinear Algebra, part 2 Eigenvalues, eigenvectors and least squares solutions
Linear Algebra, part 2 Eigenvalues, eigenvectors and least squares solutions AnnaKarin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2013 Main problem of linear algebra 2: Given
More informationModule 07 Controllability and Controller Design of Dynamical LTI Systems
Module 07 Controllability and Controller Design of Dynamical LTI Systems Ahmad F. Taha EE 5143: Linear Systems and Control Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ataha October
More informationChap 4. StateSpace Solutions and
Chap 4. StateSpace Solutions and Realizations Outlines 1. Introduction 2. Solution of LTI State Equation 3. Equivalent State Equations 4. Realizations 5. Solution of Linear TimeVarying (LTV) Equations
More informationInputtostate stability and interconnected Systems
10th Elgersburg School Day 1 Inputtostate stability and interconnected Systems Sergey Dashkovskiy Universität Würzburg Elgersburg, March 5, 2018 1/20 Introduction Consider Solution: ẋ := dx dt = ax,
More informationApplied Math Qualifying Exam 11 October Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems.
Printed Name: Signature: Applied Math Qualifying Exam 11 October 2014 Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems. 2 Part 1 (1) Let Ω be an open subset of R
More informationModeling and Analysis of Dynamic Systems
Modeling and Analysis of Dynamic Systems Dr. Guillaume Ducard Fall 2017 Institute for Dynamic Systems and Control ETH Zurich, Switzerland G. Ducard c 1 / 57 Outline 1 Lecture 13: Linear System  Stability
More informationAutonomous system = system without inputs
Autonomous system = system without inputs State space representation B(A,C) = {y there is x, such that σx = Ax, y = Cx } x is the state, n := dim(x) is the state dimension, y is the output Polynomial representation
More informationACM/CMS 107 Linear Analysis & Applications Fall 2016 Assignment 4: Linear ODEs and Control Theory Due: 5th December 2016
ACM/CMS 17 Linear Analysis & Applications Fall 216 Assignment 4: Linear ODEs and Control Theory Due: 5th December 216 Introduction Systems of ordinary differential equations (ODEs) can be used to describe
More informationConverse Lyapunov theorem and InputtoState Stability
Converse Lyapunov theorem and InputtoState Stability April 6, 2014 1 Converse Lyapunov theorem In the previous lecture, we have discussed few examples of nonlinear control systems and stability concepts
More informationHybrid Control and Switched Systems. Lecture #11 Stability of switched system: Arbitrary switching
Hybrid Control and Switched Systems Lecture #11 Stability of switched system: Arbitrary switching João P. Hespanha University of California at Santa Barbara Stability under arbitrary switching Instability
More informationNonlinear Control. Nonlinear Control Lecture # 6 Passivity and InputOutput Stability
Nonlinear Control Lecture # 6 Passivity and InputOutput Stability Passivity: Memoryless Functions y y y u u u (a) (b) (c) Passive Passive Not passive y = h(t,u), h [0, ] Vector case: y = h(t,u), h T =
More informationME 234, Lyapunov and Riccati Problems. 1. This problem is to recall some facts and formulae you already know. e Aτ BB e A τ dτ
ME 234, Lyapunov and Riccati Problems. This problem is to recall some facts and formulae you already know. (a) Let A and B be matrices of appropriate dimension. Show that (A, B) is controllable if and
More informationLinear System Theory
Linear System Theory Wonhee Kim Lecture 3 Mar. 21, 2017 1 / 38 Overview Recap Nonlinear systems: existence and uniqueness of a solution of differential equations Preliminaries Fields and Vector Spaces
More informationMa 227 Review for Systems of DEs
Ma 7 Review for Systems of DEs Matrices Basic Properties Addition and subtraction: Let A a ij mn and B b ij mn.then A B a ij b ij mn 3 A 6 B 6 4 7 6 A B 6 4 3 7 6 6 7 3 Scaler Multiplication: Let k be
More informationHybrid Systems Course Lyapunov stability
Hybrid Systems Course Lyapunov stability OUTLINE Focus: stability of an equilibrium point continuous systems decribed by ordinary differential equations (brief review) hybrid automata OUTLINE Focus: stability
More informationLecture Notes for Math 524
Lecture Notes for Math 524 Dr Michael Y Li October 19, 2009 These notes are based on the lecture notes of Professor James S Muldowney, the books of Hale, Copple, Coddington and Levinson, and Perko They
More informationELEC 3035, Lecture 3: Autonomous systems Ivan Markovsky
ELEC 3035, Lecture 3: Autonomous systems Ivan Markovsky Equilibrium points and linearization Eigenvalue decomposition and modal form State transition matrix and matrix exponential Stability ELEC 3035 (Part
More informationEE 380. Linear Control Systems. Lecture 10
EE 380 Linear Control Systems Lecture 10 Professor Jeffrey Schiano Department of Electrical Engineering Lecture 10. 1 Lecture 10 Topics Stability Definitions Methods for Determining Stability Lecture 10.
More information2 Lyapunov Stability. x(0) x 0 < δ x(t) x 0 < ɛ
1 2 Lyapunov Stability Whereas I/O stability is concerned with the effect of inputs on outputs, Lyapunov stability deals with unforced systems: ẋ = f(x, t) (1) where x R n, t R +, and f : R n R + R n.
More informationEECE Adaptive Control
EECE 574  Adaptive Control ModelReference Adaptive Control  Part I Guy Dumont Department of Electrical and Computer Engineering University of British Columbia January 2010 Guy Dumont (UBC EECE) EECE
More informationSYSTEMTEORI  ÖVNING 1. In this exercise, we will learn how to solve the following linear differential equation:
SYSTEMTEORI  ÖVNING 1 GIANANTONIO BORTOLIN AND RYOZO NAGAMUNE In this exercise, we will learn how to solve the following linear differential equation: 01 ẋt Atxt, xt 0 x 0, xt R n, At R n n The equation
More informationIntroduction to Modern Control MT 2016
CDT Autonomous and Intelligent Machines & Systems Introduction to Modern Control MT 2016 Alessandro Abate Lecture 2 Firstorder ordinary differential equations (ODE) Solution of a linear ODE Hints to nonlinear
More informationLMI Methods in Optimal and Robust Control
LMI Methods in Optimal and Robust Control Matthew M. Peet Arizona State University Lecture 15: Nonlinear Systems and Lyapunov Functions Overview Our next goal is to extend LMI s and optimization to nonlinear
More informationLyapunov stability ORDINARY DIFFERENTIAL EQUATIONS
Lyapunov stability ORDINARY DIFFERENTIAL EQUATIONS An ordinary differential equation is a mathematical model of a continuous state continuous time system: X = < n state space f: < n! < n vector field (assigns
More information21 Linear StateSpace Representations
ME 132, Spring 25, UC Berkeley, A Packard 187 21 Linear StateSpace Representations First, let s describe the most general type of dynamic system that we will consider/encounter in this class Systems may
More informationLinear Algebra Review (Course Notes for Math 308H  Spring 2016)
Linear Algebra Review (Course Notes for Math 308H  Spring 2016) Dr. Michael S. Pilant February 12, 2016 1 Background: We begin with one of the most fundamental notions in R 2, distance. Letting (x 1,
More informationControl Systems. Frequency domain analysis. L. Lanari
Control Systems m i l e r p r a in r e v y n is o Frequency domain analysis L. Lanari outline introduce the Laplace unilateral transform define its properties show its advantages in turning ODEs to algebraic
More informationControl Systems. Time response
Control Systems Time response L. Lanari outline zerostate solution matrix exponential total response (sum of zerostate and zeroinput responses) Dirac impulse impulse response change of coordinates (state)
More informationControl Systems. Dynamic response in the time domain. L. Lanari
Control Systems Dynamic response in the time domain L. Lanari outline A diagonalizable  real eigenvalues (aperiodic natural modes)  complex conjugate eigenvalues (pseudoperiodic natural modes)  phase
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 informationNumerical Methods for Differential Equations Mathematical and Computational Tools
Numerical Methods for Differential Equations Mathematical and Computational Tools Gustaf Söderlind Numerical Analysis, Lund University Contents V4.16 Part 1. Vector norms, matrix norms and logarithmic
More informationNonlinear Control Lecture 5: Stability Analysis II
Nonlinear Control Lecture 5: Stability Analysis II Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2010 Farzaneh Abdollahi Nonlinear Control Lecture 5 1/41
More informationMain topics and some repetition exercises for the course MMG511/MVE161 ODE and mathematical modeling in year Main topics in the course:
Main topics and some repetition exercises for the course MMG5/MVE6 ODE and mathematical modeling in year 04. Main topics in the course:. Banach fixed point principle. Picard Lindelöf theorem. Lipschitz
More informationControl Systems Design, SC4026. SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft
Control Systems Design, SC4026 SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft Lecture 4 Controllability (a.k.a. Reachability) vs Observability Algebraic Tests (Kalman rank condition & Hautus test) A few
More informationChapter 13 Internal (Lyapunov) Stability 13.1 Introduction We have already seen some examples of both stable and unstable systems. The objective of th
Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology 1 1 c Chapter
More informationECEEN 5448 Fall 2011 Homework #5 Solutions
ECEEN 5448 Fall 211 Homework #5 Solutions Professor David G. Meyer December 8, 211 1. Consider the 1dimensional timevarying linear system ẋ t (u x) (a) Find the statetransition matrix, Φ(t, τ). Here
More informationChapter 6 Nonlinear Systems and Phenomena. Friday, November 2, 12
Chapter 6 Nonlinear Systems and Phenomena 6.1 Stability and the Phase Plane We now move to nonlinear systems Begin with the firstorder system for x(t) d dt x = f(x,t), x(0) = x 0 In particular, consider
More informationNonlinear Systems and Elements of Control
Nonlinear Systems and Elements of Control Rafal Goebel April 3, 3 Brief review of basic differential equations A differential equation is an equation involving a function and one or more of the function
More information7 Planar systems of linear ODE
7 Planar systems of linear ODE Here I restrict my attention to a very special class of autonomous ODE: linear ODE with constant coefficients This is arguably the only class of ODE for which explicit solution
More informationNonlinear Control. Nonlinear Control Lecture # 3 Stability of Equilibrium Points
Nonlinear Control Lecture # 3 Stability of Equilibrium Points The Invariance Principle Definitions Let x(t) be a solution of ẋ = f(x) A point p is a positive limit point of x(t) if there is a sequence
More informationObservability. Dynamic Systems. Lecture 2 Observability. Observability, continuous time: Observability, discrete time: = h (2) (x, u, u)
Observability Dynamic Systems Lecture 2 Observability Continuous time model: Discrete time model: ẋ(t) = f (x(t), u(t)), y(t) = h(x(t), u(t)) x(t + 1) = f (x(t), u(t)), y(t) = h(x(t)) Reglerteknik, ISY,
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 information8 Periodic Linear Di erential Equations  Floquet Theory
8 Periodic Linear Di erential Equations  Floquet Theory The general theory of time varying linear di erential equations _x(t) = A(t)x(t) is still amazingly incomplete. Only for certain classes of functions
More informationFEL3210 Multivariable Feedback Control
FEL3210 Multivariable Feedback Control Lecture 8: Youla parametrization, LMIs, Model Reduction and Summary [Ch. 1112] Elling W. Jacobsen, Automatic Control Lab, KTH Lecture 8: Youla, LMIs, Model Reduction
More informationEE Control Systems LECTURE 9
Updated: Sunday, February, 999 EE  Control Systems LECTURE 9 Copyright FL Lewis 998 All rights reserved STABILITY OF LINEAR SYSTEMS We discuss the stability of input/output systems and of statespace
More informationNOTES ON LINEAR ODES
NOTES ON LINEAR ODES JONATHAN LUK We can now use all the discussions we had on linear algebra to study linear ODEs Most of this material appears in the textbook in 21, 22, 23, 26 As always, this is a preliminary
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 informationStability of Nonlinear Systems An Introduction
Stability of Nonlinear Systems An Introduction Michael Baldea Department of Chemical Engineering The University of Texas at Austin April 3, 2012 The Concept of Stability Consider the generic nonlinear
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 informationSolution of Additional Exercises for Chapter 4
1 1. (1) Try V (x) = 1 (x 1 + x ). Solution of Additional Exercises for Chapter 4 V (x) = x 1 ( x 1 + x ) x = x 1 x + x 1 x In the neighborhood of the origin, the term (x 1 + x ) dominates. Hence, the
More informationLecture 9 Nonlinear Control Design
Lecture 9 Nonlinear Control Design Exactlinearization Lyapunovbased design Lab 2 Adaptive control Sliding modes control Literature: [Khalil, ch.s 13, 14.1,14.2] and [GladLjung,ch.17] Course Outline
More informationMathematics 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