# Module 03 Linear Systems Theory: Necessary Background

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1 Module 03 Linear Systems Theory: Necessary Background Ahmad F. Taha EE 5243: Introduction to Cyber-Physical Systems Webpage: taha/index.html September 2, 2015 Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 1 / 31

2 Module 03 Outline We will review in couple lectures the necessary background needed in linear systems theory and design. Outline of this module is as follows: 1 Computation of solution for an ODE, LTI systems 2 Stability of linear systems and Jordan blocks 3 Discrete dynamical systems 4 Controllability, observability, stabilizability, detectability 5 Design of controllers and observers 6 Linearization of nonlinear systems Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 2 / 31

3 LTI Systems LTI dynamical system: ẋ(t) = Ax(t) + Bu(t), x initial = x t0, (1) y(t) = Cx(t) + Du(t), (2) We now know that the solution is given by: x(t) = e A(t t 0) x t0 + Clearly the output solution is: t t t 0 e A(t τ) Bu(τ) dτ y(t) = C ( ) e A(t t0) x t0 + C e A(t τ) Bu(τ) dτ + Du(t) }{{} t 0 zero input response }{{} zero state response Question: how do I analytically compute the solution to (1)? Answer: you need to (a) integrate and (b) compute matrix exponentials (given A, B, C, D, x t0, u(t)) Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 3 / 31

4 Matrix Exponential 1 Exponential of scalar variable: e a = i=0 Power series converges a R a i i! = 1 + a + a2 2! + a3 3! + a4 4! + How about matrices? For A R n n, matrix exponential: e A = i=0 A i i! = I n + A + A2 2! What if we have a time-variable? e ta (ta) i = = I n + ta + (ta)2 i! 2! i=0 + A3 3! + (ta)3 3! + A4 4! + + (ta)4 4! + Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 4 / 31

5 Matrix Exponential Properties For a matrix A R n n and a constant t R: 1 Av = λv e At v = e λt v 2 1 det(e At ) = e (trace(a))t 3 (e At ) 1 = e At 4 e A t = (e At ) 5 If A, B commute, then: e (A+B)t = e At e Bt = e Bt e At 6 e A(t 1+t 2 ) = e At 1 e At 2 = e At 2 e At 1 1 Trace of a matrix is the sum of its diagonal entries. Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 5 / 31

6 When Is It Easy to Find e A? Method 1 Well...Obviously if we can directly use e A = I n + A + A2 2! + Three cases: A is nilpotent 2, i.e., A k = 0 for some k. Example: A = [ 2 2 ] 4 A is idempotent, i.e., A 2 = A. Example: A = A is of rank one: A = uv T for u, v R n A k = (v T u) k 1 A, k = 1, 2,... 2 Any triangular matrix with 0s along the main diagonal is nilpotent Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 6 / 31

7 Method 2 Jordan Canonical Form All matrices, whether diagonalizable or not, have a Jordan canonical form: A = T JT 1, then e At = T e Jt T 1 λ i 1 1. Generally, J =. J.. λ.. i, J i = Rn i n i, J p... 1 λi e Jit = e λ it te λ it... t n i 1 e λ i t (n i 1)!. 0 e λ it.. t n i 2 e λ i t (n i 2)! e λ it e At = T e J 1t... e Jot T 1 Jordan blocks and marginal stability Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 7 / 31

8 Example 1 Find e A(t t0) for matrix A given by: A = T JT 1 = [ ] v 1 v 2 v 3 v v 1 v 2 v 3 v Solution: = v 1 v 2 v 3 v 4 e A(t t0) = T e J(t t0) T t t v 1 v 2 v 3 v e (t t 0) e (t t 0) Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 8 / 31

9 Example 2 Quiz Time Consider a dynamical system defined by: [ 1 1 ] 0 [ 1 ] 0 A = 1 1 0, B = 0 0, C = D = 0, x(t 0) = x(1) = [0 1 1] 1 Determine y(t) if u(t) = 1 + (t) (t 1) 2 Solution: y(t) = t 1 MATLAB demo Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 9 / 31

10 From CTLTI to DTLTI Systems Given A, B, C, D for a continuous time, LTI system, what are the the equivalent matrices for the discrete dynamics? For a sampling period T, the equivalent representation is: ] Ã = e AT, B = [ (k+1)t kt e A((k+1)T τ) B dτ, C = C, D = D Dynamics: x(k + 1) = Ãx(k) + Bu(k), y(k) = Cx(k) + Du(k) MATLAB command: [Ad,Bd]=c2d(A,B,T) Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 10 / 31

11 LTI Discrete Systems LTI dynamical system: x(k + 1) = Ax(k) + Bu(k), x initial = x k0, (3) y(k) = Cx(k) + Du(k), (4) We now know that the solution is given by: x(k) = A k x k0 + k Clearly the output solution is: i=k 0 A k 1 i Bu(i) = A k x k0 + y(k) = C ( A k x k0 ) }{{} zero input response + C k k i=k 0 A i Bu(k 1 i) i=k 0 A k 1 i Bu(i) + Du(k) }{{} zero state response Question: how do I analytically compute the solution to (3)? Answer: you need to (a) evaluate summations and (b) compute matrix powers Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 11 / 31

12 Discrete LTI System Example Consider the following time-invariant discrete dynamics: 0 0 x(k + 1) = T T 1 1 x(k) + T u(k) Determine A k. Solution: A k k 0 = T k T 1 Find the zero-state state-response and x(9) given that u(k) = 0.5 k 1 + (k) Solution? Work it out and show it to me next time... Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 12 / 31

13 Controllability 1 A CTLTI system is defined as follows: ẋ = Ax + Bu, x(0) = x 0 Over the time interval [0, t f ], control input u(t) t [0, t f ] steers the state from x 0 to x tf : x(t f ) = e At f x 0 + tf 0 e A(t τ) Bu(τ) dτ Controllability Definition CTLTI system is controllable at time t f > 0 if for any initial state and for any target state (x tf ), a control input u(t) exists that can steer the system states from x(0) to x(t f ) over the defined interval. Reachable subspace: space of all reachable states DTLTI Controllability Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 13 / 31

14 Controllability 2 Controllability Test For a system with n states and m control inputs, the test for controllability is that matrix C = [ B AB A 2 B A n 1 B ] R n nm has full row rank (i.e., rank(c) = n). Theorem The test is equivalent for DTLTI and CTLTI systems The following statements are equivalent: 1 C is full rank 2 PBH Test: for any λ C, rank [λi A B] = n 3 Eigenvector Test: for any evector v C of A, v T B 0 4 For any t f > 0, the so-called Gramian matrix is nonsingular: W (t f ) = tf 0 e Aτ BB e A τ dτ Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 14 / 31

15 Observability 1 DTLTI system (n states, m inputs, p outputs): x(k + 1) = Ax(k) + Bu(k), x(0) = x 0, (5) y(k) = Cx(k) + Du(k), (6) Application: given that A, B, C, D, and u(k), y(k) are known k = 0 : 1 : k 1, can we determine x(0)? Solution: y(0) y(1). y(k 1) = C CA. CA k 1 x(0) + D CB D CA k 2 B... CB 0 u(0) u(1). u(k 1) Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 15 / 31

16 Observability 2 y(0) y(1). y(k 1) Or: = C CA. CA k 1 x(0) + D CB D CA k 2 B... CB 0 Y (k 1) = O k x(0) + T k U(k 1) O k x(0) = Y (k 1) T k U(k 1) u(0) u(1). u(k 1) Since O k, T k, Y (k 1), U(k 1) are all known quantities, then we can find a unique x(0) iff O k is full rank Observability Definition DTLTI system is observable at time k if the initial state x(0) can be uniquely determined from any given u(0),..., u(k 1), y(0),..., y(k 1). Unobservable subspace: null-space of O k Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 16 / 31

17 Observability 3 Observability Test For a system with n states and p outputs, the test for observability is that C CA matrix O = Theorem. CA n 1 Rnp n has full column rank (i.e., rank(c) = n). The test is equivalent for DTLTI and CTLTI systems The following statements are equivalent: 1 O is full rank, system is observable 2 λi A PBH Test: for any λ C, rank = n C 3 Eigenvector Test: for any evector v C of A, Cv 0 4 The matrices n 1 i=0 (A ) i C CA i for the DTLIT and t 0 ea τ C Ce Aτ dτ for the CTLTI are nonsingular Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 17 / 31

18 Example Consider a dynamical system defined by: [ 1 1 ] 0 [ 1 ] 0 A = 1 1 0, B = 0 0, C = Is this system controllable? Is this system observable? Answers: Yes, Yes! MATLAB commands: ctrb, obsv Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 18 / 31

19 Controller Design Open-loop control: design u(t) directly, i.e., through optimization or learning Closed-loop control: design u(t) as a function of state, i.e., u(t) = g(x, t) Linear state-feedback (LSF) control: design matrix K such that the control u(t) = Kx(t) + v(t) yields a desirable state-response Dynamics under LSF: ẋ(t) = (A BK)x(t) + Bv(t), v(t) is a reference signal Objective: design K such that eigenvalues of A BK are stable or at a certain location Fact: if the system is controllable, eig(a BK) can be arbitrarily reassigned Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 19 / 31

20 Stabilizability Stabilizability Definition DTLTI or CTLIT system, defined by (A, B), is stabilizable if there exists a matrix K such that A BK is stable. Stabilizability Theorem DTLTI or CTLIT system, defined by (A, B) is stabilizable if all its uncontrollable modes correspond to stable eigenvalues of A. Facts: A is stable (A, B) is stabilizable (A, B) is controllable (A, B) is stabilizable as well (A, B) is not controllable it could still be stabilizable Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 20 / 31

21 Observer Design Define dynamic estimation error: e(t) = x(t) ˆx(t) Error dynamics: ė(t) = ẋ(t) ˆx(t) = (A LC)(x(t) ˆx(t)) = (A LC)e(t) Hence, e(t) 0, as t if eig(a LC) < 0 Objective: design observer/estimator gain L such that eig(a LC) < 0 or at a certain location Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 21 / 31

22 Detectability Detectability Definition DTLTI or CTLIT system, defined by (A, C), is detectable if there exists a matrix L such that A LC is stable. Detectability Theorem DTLTI or CTLIT system, defined by (A, C) is detectable if all its unobservable modes correspond to stable eigenvalues of A. Facts: A is stable (A, C) is detectable (A, C) is observable (A, C) is detectable as well (A, B) is not observable it could still be detectable If system has some unobservable modes that are unstable, then no gain L can make A LC stable Observer will fail to track system state Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 22 / 31

23 Example Controller Design Given a system characterized by A = 1 3, B = 3 1 Is the system stable? What are the eigenvalues? Solution: unstable, eig(a) = 4, Find linear state-feedback gain K (i.e., u = Kx), such that the poles of the closed-loop controlled system are 3 and 5 Characteristic polynomial: λ 2 + (k 1 2)λ + (3k 2 k 1 8) = 0 x1 Solution: u = Kx = [10 11] = 10x x 1 11x 2 2 MATLAB command: K = place(a,b,eig desired) What if A =, B =, can we stabilize the system? Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 23 / 31

24 Example Observer Design Given a system characterized by A = 1 3, C = [ ] 3 1 Find linear state-observer gain L = [l 1 l 2] such that the poles of the estimation error are 5 and 3 Characteristic polynomial: λ 2 + ( 2 + l l 1)λ + ( l l 1) = 0 8 Solution: L = 6 MATLAB command: L = place(a,c,eig desired) Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 24 / 31

25 MATLAB Example x1, ˆx1 x2, ˆx x 1 ˆx Time (seeconds) x 2 ˆx Time (seeconds) A=[1-0.8; 1 0]; B=[0.5; 0]; C=[1-1]; % Selecting desired poles eig_desired=[.5.7]; L=place(A,C,eig_desired) ; % Initial state x=[-10;10]; % Initial estimate xhat=[0;0]; % Dynamic Simulation XX=x; XXhat=xhat; T=10; % Constant Input Signal UU=.1*ones(1,T); for k=0:t-1, u=uu(k+1); y=c*x; yhat=c*xhat; x=a*x+b*u; xhat=a*xhat+b*u+l*(y-yhat); XX=[XX,x]; XXhat=[XXhat,xhat]; end % Plotting Results subplot(2,1,1) plot(0:t,[xx(1,:);xxhat(1,:)]); subplot(2,1,2) plot(0:t,[xx(2,:);xxhat(2,:)]); Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 25 / 31

26 Observer-Based Control 1 Recall that for LSF control: u(t) = Kx(t) What if x(t) is not available, i.e., it can only be estimated? Solution: get ˆx by designing L Apply LSF control using ˆx with a LSF matrix K to both the original system and estimator Question: how to design K and L simultaneously? Poles of the closed-loop system? Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 26 / 31

27 Observer-Based Control 2 Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 27 / 31

28 Observer-Based Control 3 Closed-loop dynamics: ẋ(t) = Ax(t) BK ˆx(t) (7) ˆx(t) = Aˆx(t) + L(y(t) ŷ(t)) BK ˆx(t) (8) Or [ẋ(t) ] A BK x(t) = ˆx(t) LC A LC BK ˆx(t) x(t) x(t) I 0 x(t) Transformation: = = e(t) x(t) ˆx(t) I I ˆx(t) Hence, we can write: [ẋ(t) ] ė(t) A BK BK = 0 A LC }{{} A cl x(t) e(t) If the system is controllable & observable eig(a textcl ) can be arbitrarily assigned by proper K and L What if the system is stabilizable and detectable? Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 28 / 31

29 Nonlinear Systems Many dynamical systems are not originally linear To analyze the system (stability, observability, controllability), we need a linearized representation of the system For a nonlinear dynamical system ẋ(t) = f(x), follow this procedure to linearize: 1 Put the ODE in state-space vector form 2 Find all equilibrium points by solving f(x) = 0 3 List all the possible solutions: x 1 e, x2 e, x3 e,... 4 Find the Jacobian matrix of the nonlinear dynamics, Df(x) 5 Linearize using Taylor series expansion: ẋ(t) = f(x i e) + Df(x) x=x i e (x x i e) 6 Determine which equilibrium points are stable. If Df(x) equilibrium point is locally stable. x=x i e 0, the Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 29 / 31

30 Linearization Example ẋ 1 = x x 2 ẋ 2 = 3 x 2 x 3 ẋ 1 = 2 x 3 Find all the equilibrium points for the given nonlinear system and determine their corresponding local stability Solution: ] [ 1 x 1 e = 1, asymptotically stable 2 1 x 2 e = 1, unstable 2 Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 30 / 31

31 Questions And Suggestions? Thank You! Please visit engineering.utsa.edu/ taha IFF you want to know more Ahmad F. Taha Module 03 Linear Systems Theory: Necessary Background 31 / 31

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