# Module 09 From s-domain to time-domain From ODEs, TFs to State-Space Modern Control

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1 Module 09 From s-domain to time-domain From ODEs, TFs to State-Space Modern Control Ahmad F. Taha EE 3413: Analysis and Desgin of Control Systems Webpage: taha April 14, 2016 Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 1 / 38

2 Modern Control Readings: Ogata; Dorf & Bishop In the previous modules, we discussed the analysis and design of control systems via frequency-domain techniques Root locus, PID controllers, compensators, state-feedback control, etc... These studies are considered as the classical control theory based on the s-domain This module: we ll introduce time-domain techniques Theory is based on State-Space Representations modern control Why do we need that? Many reasons Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 2 / 38

3 ODEs & Transfer Functions For linear systems, we can often represent the system dynamics through an nth order ordinary differential equation (ODE): y (n) (t) + a 1 y (n 1) (t) + a 2 y (n 2) (t) + + a n 1 ẏ(t) + a n y(t) = b 0 u (n) (t) + b 1 u (n 1) (t) + b 2 u (b 2) (t) + + b n 1 u(t) + b n u(t) The y (k) notation means we re taking the kth derivative of y(t) Input: u(t); Output: y(t) What if we have MIMO system? Given that ODE description, we can take the LT (assuming zero initial conditions for all signals): H(s) = Y (s) U(s) = b 0s n + b 1 s n b n 1 s + b n s n + a 1 s n a n 1 s + a n Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 3 / 38

4 ODEs & TFs H(s) = Y (s) U(s) = b 0s n + b 1 s n b n 1 s + b n s n + a 1 s n a n 1 s + a n This equation represents relationship between one system input and one system output This relationship, however, does not show me the internal states of the system, nor does it explain the case with multi-input system For that (and other reasons), we discuss the notion of system state Definition: x(t) is a state-vector that belongs to R n : x(t) R n x(t) is an internal state of a system Examples: voltages and currents of circuit components Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 4 / 38

5 ODEs, TFs to State-Space Representations H(s) = Y (s) U(s) = b 0s n + b 1 s n b n 1 s + b n s n + a 1 s n a n 1 s + a n State-space (SS) theory: representing the above TF of a system by a vector-form first order ODE: ẋ(t) = Ax(t) + Bu(t), x initial = x t0, (1) y(t) = Cx(t) + Du(t), (2) x(t) R n : dynamic state-vector of the LTI system, u(t): control input-vector, n = order of the TF/ODE y(t): output-vector and A, B, C, D are constant matrices For the above transfer function, we have one input U(s) and one output Y (s), hence the size of y(t) and u(t) is only one (scalars) Module Objectives: learn how to construct matrices A, B, C, D given a transfer function Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 5 / 38

6 State-Space Representation 1 H(s) = Y (s) U(s) = b 0s n + b 1 s n b n 1 s + b n s n + a 1 s n a n 1 s + a n Given the above TF/ODE, we want to find ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) The above two equations represent a relationship between the input and output of the system via the internal system states The above 2 equations are nothing but a first order differential equation Wait, WHAT? But the TF/ODE was an nth order ODE. How do we have a first order ODE now? Well, because this equation is vector-matrix equation, whereas the ODE/TF was a scalar equation Next, we ll learn how to get to these 2 equations from any TF Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 6 / 38

7 State-Space Representation 2 [Ogata, P. 689] Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 7 / 38

8 State-Space Representation 3 [Ogata, P. 689] Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 8 / 38

9 State-Space Representation 4 [Ogata, P. 689] Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 9 / 38

10 Final Solution Combining equations (9-74,75,76), we can obtain the following vector-matrix first order differential equation: ẋ 1 (t) x 1 (t) 0 ẋ 2 (t) x 2 (t) 0 ẋ(t) =. = u(t) ẋ n 1 (t) x n 1 (t) 0 ẋ n (t) a n } a n 1 a n 2 {{ a 1 x n (t) } 1 }{{ } Ax(t) Bu(t) x 1 (t) x 2 (t) y(t) = [ ] b n a n b 0 b n 1 a n 1 b 0 b 1 a 1 b 0. x n 1 (t) } {{ x n (t) } Cx(t) + b 0 u(t) }{{} Du(t) Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 10 / 38

11 Remarks For any TF with order n (order of the denominator), with one input and one output: A R n n, B R n 1, C R 1 n, D R Above matrices are constant system is linear time-invariant (LTI) If one term of the TF/ODE (i.e., the a s and b s) change as a function of time, the matrices derived above will also change in time system is linear time-varying (LTV) The above state-space form is called the controllable canonical form You can come up with different forms of A, B, C, D matrices given a different transformation Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 11 / 38

12 State-Space and Block Diagrams From the derived eqs. before, you can construct the block diagram An integrator block is equivalent to a 1 s, the inputs and outputs of each integrator are the derivative of the state ẋ i (t) and x i (t) A system (TF/ODE) of order n can be constructed with n integrators (you can construct the system with more integrators) Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 12 / 38

13 Example 1 Find a state-space representation (i.e., the state-space matrices) for the system represented by this second order transfer function: Y (s) U(s) = s + 3 s 2 + 3s + 2 Solution: look at the previous slides with the matrices: H(s) = Y (s) U(s) = b 0s n + b 1 s n b n 1 s + b n s n + a 1 s n a n 1 s + a n = b 0 {}}{ First, n = 2 A R 2 2, B R 2 1, C R 1 2, D R [ ] [ ẋ(t) = x(t) + u(t) 2 3 1] }{{}}{{} A B y(t) = [ 3 1 ] x(t) + 0 u(t) }{{} C }{{} D b 1 {}}{ 0 s s + s 2 + }{{} 3 b 2 {}}{ 3 s + }{{} 2 a 1 Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 13 / 38 a 2

14 Other State-Space Forms Given a TF/ODE 1 Observable Canonical Form: 1 Derivation from Ogata, but similar to the controllable canonical form. Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 14 / 38

15 Block Diagram of Observable Canonical Form Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 15 / 38

16 2 This factorization assumes that the TF has only distinct real poles. Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 16 / 38 Other State-Space Forms Given a TF/ODE Diagonal Canonical Form 2 :

17 Block Diagram of Diagonal Canonical Form Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 17 / 38

18 Example 1 Solution for other Canonical Forms Find the observable and diagonal forms for Y (s) U(s) = b 0 {}}{ b 1 {}}{ 0 s s + s 2 + }{{} 3 b 2 {}}{ 3 s + }{{} 2 a 1 Solution: look at the previous slides with the constructed state-space matrices: Observable Canonical Form: [ ] [ ] ẋ(t) = x(t) + u(t), y(t) = [ 0 1 ] x(t) }{{}}{{} 0 u(t) }{{}}{{} D C A B Diagonal Canonical Form: [ ] [ ] ẋ(t) = x(t)+ u(t), y(t) = [ 2 1 ] x(t) }{{}}{{} 0 u(t) }{{}}{{} D C A B Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 18 / 38 a 2

19 State-Space to Transfer Functions Given a state-space representation: ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) can we obtain the transfer function back? Yes: Y (s) U(s) = C(sI A) 1 B + D Example: find the TF corresponding for this SISO system: [ ] [ ẋ(t) = x(t)+ u(t), y(t) = 0 2 1] [ 2 1 ] x(t)+ }{{}}{{} 0 u(t) }{{}}{{} D C A B Solution: Y (s) U(s) = C(sI n A) 1 B+D = [ 2 1 ] ( [ ] [ ]) 1 [ ] s s + 3 = s 2, that s the TF from the previous example! + 3s + 2 Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 19 / 38

20 MATLAB Commands ss2tf(a,b,c,d,iu) tf2ss(num,den) Demo Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 20 / 38

21 Important Remarks So why do we want to go from a transfer function to a time-representation, ODE form of the system? There are many benefits for doing so, such as: 1 Stability analysis for MIMO systems becomes way easier 2 We have powerful mathematical tools that helps us design controllers 3 RL and compensator designs were relatively tedious design problems 4 With state-space representations, we can easily design controllers 5 Nonlinear systems: cannot use TFs for nonlinear systems 6 State-space is all about time-domain analysis, which is far more intuitive than frequency-domain analysis 7 With Laplace transforms and TFs, we had to take inverse Laplace transforms. In many cases, the Laplace transform does not exist, which means time-domain analysis is the only way to go We will learn how to get a solution for y(t) for any given u(t) from the state-space representation of the system without Laplace transform via ODE solutions for matrix-vector equations Before that, we need to introduce some linear algebra preliminaries Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 21 / 38

22 Linear Algebra Revision Eigenvalues/Eigenvectors of a matrix Evalues/vectors are only defined for square 3 matrices For a matrix A R n n, we always have n evalues/evectors Some of these evalues might be distinct, real, repeated, imaginary To find evalues(a), solve this equation (I n : identity matrix of size n) det(λi n A) = 0 or det(a λi n ) = 0 a 0 λ n + a 1 λ n a n = 0 [ ] a b Example: det = ad bc. c d Eigenvectors: A number λ and a non-zero vector v satisfying Av = λv (A λi n )v = 0 are called an eigenvalue and an eigenvector of A λ is an eigenvalue of an n n-matrix A if and only if λi n A is not invertible, which is equivalent to det(a λi n ) = 0. 3 A square matrix has equal number of rows and columns. Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 22 / 38

23 Matrix Inverse Inverse of a generic 2by2 matrix: [ ] 1 [ ] [ ] A 1 a b 1 d b 1 d b = = = c d det(a) c a ad bc c a Notice that A 1 A = AA 1 = I 2 Inverse of a generic 3by3 matrix: a b c A 1 = d e f g h i 1 = 1 det(a) A B C D E F G H I T = 1 det(a) A D G B E H C F I A = (ei fh) D = (bi ch) G = (bf ce) B = (di fg) E = (ai cg) H = (af cd) C = (dh eg) F = (ah bg) I = (ae bd) Notice that A 1 A = AA 1 = I 3 det(a) = aa + bb + cc. Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 23 / 38

24 Linear Algebra Example 1 Find the eigenvalues, eigenvectors, and inverse of matrix [ ] 1 4 A = 3 2 Eigenvalues: λ 1,2 = 5, 2 Eigenvectors: v 1 = [ 1 1 ], v 2 = [ ] [ ] Inverse: A = Write A in the matrix diagonal transformation, i.e., A = TDT 1 where D is the diagonal matrix containing the eigenvalues of A: λ 1 A = [ ] λ 2 [ ] 1 v 1 v 2 v n v 1 v 2 v n... λn Only valid for matrices with distinct, real eigenvalues Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 24 / 38

25 Rank of a Matrix Rank of a matrix: rank(a) is equal to the number of linearly independent rows ([ or columns ]) Example 1: rank =? Example 2: rank =? Rank computation: reduce the matrix to a simpler form, generally row echelon form, by elementary row operations Example 2 Solution: r 1 + r r 1 + r r 2 + r r 2 + r rank(a) = For a matrix A R m n : rank(a) min(m, n) Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 25 / 38

26 Null Space of a Matrix A = The Null Space of any matrix A is the subspace K defined as follows: N(A) = Null(A) = ker(a) = {x K Ax = 0} Null(A) has the following three properties: Null(A) always contains the zero vector, since A0 = 0 If x Null(A) and y Null(A), then x + y Null(A) If x Null(A) and c is a scalar, then cx Null(A) [ 1 0 1/ /8 0 Example: Find N(A) [ ] ] [ ] a b = c a = 1 16 c, b = 13 8 c [ ] [ ] a 1/16 1 b = α 13/8 = α 26 c 1 16 Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 26 / 38

27 Linear Algebra Example 2 Find the determinant, rank, and null-space set of this matrix: B = det(b) = 0 rank(b) = 2 3 null(b) = α 2, α R 1 Is there a relationship between the determinant and the rank of a matrix? Yes! Matrix drops rank if determinant = zero 1 zero evalue True or False? AB = BA for all A and B FALSE! A and B are invertible (A + B) is invertible FALSE! Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 27 / 38

28 Matrix Exponential 1 Exponential of scalar variable: e a = i=0 a i i! = 1 + a + a2 2! + a3 3! + a4 4! + Power series converges a R How about matrices? For A R n n, matrix exponential: e A = i=0 A i i! = I n + A + A2 2! + A3 3! + A4 4! + What if we have a time-variable? e ta = (ta) i i=0 i! = I n + ta + (ta)2 2! + (ta)3 3! + (ta)4 4! + Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 28 / 38

29 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 4 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(t1+t2) = e At1 e At2 = e At2 e At1 4 Trace of a matrix is the sum of its diagonal entries. Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 29 / 38

30 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 for Method 1 Case 1 A is nilpotent 5, i.e., A k = 0 for some k. Example: A = Case 2 A is idempotent, i.e., A 2 = A. Example: A = Case 3 A is of rank one: A = uv T for u, v R n A k = (v T u) k 1 A, k = 1, 2,... 5 Any triangular matrix with 0s along the main diagonal is nilpotent Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 30 / 38

31 Method 2 Jordan Canonical Form All matrices, whether diagonalizable or not, have a Jordan canonical form: A = TJT 1, then e At = T e Jt T 1 Generally, J = J 1... J p, λ i 1. J i = λ.. i... 1 Rni ni, λi e λi t te λi t t... n i 1 e λ i t (n i 1)! e Ji t 0 e = λi t... t n i 2 e λ i t (n i 2)! e At = T e J1t e λi t e Jot T 1 Jordan blocks and marginal stability Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 31 / 38

32 Examples 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: e A(t t0) = T e J(t t0) T 1 e (t t0) = [ ] v 1 v 2 v 3 v t t 0 0 [ ] v 1 v 2 v 3 v e (t t0) Find e A(t t0) for matrix A given by: [ ] 1 0 A 1 = and A = [ ] Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 32 / 38

33 Solution to the State-Space Equation In the next few slides, we ll answer this question: what is a solution to this vector-matrix first order ODE: ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) By solution, we mean a closed-form solution for x(t) and y(t) given: An initial condition for the system, i.e., x(t initial ) = x(0) A given control input signal, u(t), such as a step-input (u(t) = 1), ramp (u(t) = t), or anything else Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 33 / 38

34 The Curious Case of Autonomous Systems Case 1 Let s assume that we seek solution to this system first: ẋ(t) = Ax(t), x(0) = x 0 = given y(t) = Cx(t) This means that the system operates without any control input autonomous system (e.g., autonomous vehicles) First, let s look at ẋ(t) = Ax(t) what s the solution to this first order ODE? First case: A = a is a scalar x(t) = e at x 0 Second case: A is a matrix x(t) = e At x 0 y(t) = Cx(t) = Ce At x 0 Exponential of scalars is very easy, but exponentials of matrices can be very challenging Hence, for an nth order system, where n 2, we need to compute the matrix exponential in order to get a solution for the above system we learned that in the linear algebra revision section Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 34 / 38

35 Example (Case 1) x(t) = e At x 0, y(t) = Cx(t) = Ce At x 0 Find the solution for these two autonomous systems separately: [ ] 1 0 A 1 =, C = [ 1 2 ] [ ], x (1) 1 0 = 2 Solution: A 2 = [ ] 0 1, C = [ 2 0 ], x (2) 0 = [ ] 1 1 Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 35 / 38

36 Case 2 Systems with Inputs MIMO (or SISO) LTI dynamical system: ẋ(t) = Ax(t) + Bu(t), x(t 0 ) = x t0 = given y(t) = Cx(t) + Du(t) The to the above ODE is given by: x(t) = e A(t t0) x t0 + t Clearly the output solution is: ) ( t y(t) = C (e A(t t0) x t0 + C }{{} t 0 zero input response t 0 e A(t τ) Bu(τ) dτ ) e A(t τ) Bu(τ) dτ + Du(t) } {{ } zero state response Question: how do I analytically compute y(t) and x(t)? Answer: you need to (a) integrate and (b) compute matrix exponentials (given A, B, C, D, x t0, u(t)) Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 36 / 38

37 Example (Case 2) t x(t) = e A(t t0) x t0 + e A(t τ) Bu(τ) dτ t 0 ) ( t y(t) = C (e A(t t0) x t0 + C }{{} t 0 zero input response ) e A(t τ) Bu(τ) dτ + Du(t) } {{ } zero state response Find the solution for these two LTI systems with inputs: [ ] [ ] A 1 =, B =, C 1 1 = [ 1 2 ] [ ], x (1) 1 0 =, D 2 1 = 0, u 1 (t) = 1 A 2 = [ ] [ 0 1 1, B =, C 1] 2 = [ 2 0 ], x (2) 0 = Solution: [ ] 1, D 1 2 = 1, u 2 (t) = 2e 2t Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 37 / 38

38 Questions And Suggestions? Thank You! Please visit engineering.utsa.edu/ taha IFF you want to know more Ahmad F. Taha Module 09 From s-domain to Time-Domain; From ODEs, TFs to State-Space 38 / 38

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