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 equations define an Input/Output representation of the system through the transfer function exploring the structure of the transfer function solve a realization problem Lanari: CS - Frequency domain analysis 2
Laplace transform f(t) L F (s) complex valued function in the complex variable t R s C F (s) = f(t)e st dt Laplace (unilateral) transform for f with no impulse at t = F (s) =L[f(t)] Region of convergence Lanari: CS - Frequency domain analysis 3
Laplace transform Laplace transform of the impulse F (s) = f(t)e st dt L[δ(t)] = δ(t)e st dt = δ(t)e st dt = e s =1 Linearity L[αf(t)+βg(t)] = αl[f(t)] + βl[g(t)] Lanari: CS - Frequency domain analysis 4
Inverse Laplace transform f(t) L F (s) L 1 f(t) =L 1 [F (s)] = 1 2πj α j α j F (s)e st ds unique for one-sided functions (one-to-one) or f(t) t defined for f(t) = for t< Lanari: CS - Frequency domain analysis 5
Laplace transform Derivative property L df (t) dt = sl[f(t)] f() can be applied iteratively L f(t) = s 2 L[f(t)] sf() f() very useful in model derivation Lanari: CS - Frequency domain analysis 6
LTI systems also useful here ẋ(t) =Ax(t)+Bu(t) (proof): linearity + derivative property algebraic solution X(s) =(si A) 1 x +(si A) 1 BU(s) and therefore Y (s) =C(sI A) 1 x + C(sI A) 1 B + D U(s) Lanari: CS - Frequency domain analysis 7
LTI systems state ZIR transform state ZSR transform X(s) =(si A) 1 x +(si A) 1 BU(s) x(t) =e At x + t e A(t τ) Bu(τ)dτ and therefore, comparing L e At =(si A) 1 Heaviside step function L e at = 1 s a L [δ 1 (t)] = 1 s 1 δ 1 (t) 1 for t for t< Lanari: CS - Frequency domain analysis 8 t
LTI systems y(t) =Ce At x + t w(t τ)u(τ)dτ with transform w(t) =Ce At B + D δ(t) W (s) =C(sI A) 1 B + D Y (s) = C(sI A) 1 x + C(sI A) 1 B + D U(s) = C(sI A) 1 x + W (s) U(s) Convolution theorem L t w(t τ)u(τ)dτ = W (s) U(s) being L[δ(t)] = 1 the output response transform corresponding to u(t) = ±(t) is indeed W(s) same for H(s) Lanari: CS - Frequency domain analysis 9
Transfer function Input/Output behavior x = (ZSR) y ZS (t) = t w(t τ)u(τ)dτ Y ZS (s) =W (s) U(s) Transfer function W (s) = C(sI A) 1 B + D = Y ZS(s) U(s) = L [w(t)] Input/Output behavior independent from state choice? independent from state space representation? Lanari: CS - Frequency domain analysis 1
Transfer function (A, B, C, D) ẋ = Ax + Bu y = Cx + Du z = Tx det(t ) = ( A, B, C, D) ż = Az + Bu y = Cz + Du A = TAT 1 B = TB C = CT 1 D = D W (s) =C(sI A) 1 B + D = C(sI A) 1 B + D For a given system, the transfer function is unique Lanari: CS - Frequency domain analysis 11
Shape of the transfer function W (s) =C(sI A) 1 B + D inverse through the adjoint (transpose of the cofactor matrix) 1 C (adjoint of (si A)) B + D det(si A) cofactor(i, j) =( 1) i+j minor(i, j) polynomial of order n - 1 polynomial of order n - 1 1 det(si A) C (cofactor of (si A))T B + D polynomial of order n rational function (strictly proper) strictly proper rational function: proper rational function: degree of numerator < degree of denominator degree of numerator = degree of denominator Lanari: CS - Frequency domain analysis 12
Shape of the transfer function W (s) = strictly proper rational function + D proper rational function W (s) = N(s) D(s) D = D = strictly proper rational function proper rational function W (s) = N(s) D(s) roots zeros poles (for coprime N(s) & D(s)) i.e. no common roots from previous analysis the poles are a subset of the eigenvalues of A {poles} {eigenvalues} more on this later Lanari: CS - Frequency domain analysis 13
Laplace transform (other properties) Integral property L t f(τ)dτ = 1 s L [f(t)] = 1 s F (s) Time shifting property L [f(t T )δ 1 (t T )] = e st L [f(t)δ 1 (t)] = e st F (s) N(s) and D(s) are said to be coprime if they have no common factor Lanari: CS - Frequency domain analysis 14
Laplace transform tables δ(t) 1 sin ωt ω s 2 + ω 2 = 1/2j s jω 1/2j s + jω δ 1 (t) 1 s cos ωt s s 2 + ω 2 = 1/2 s jω + 1/2 s + jω e at 1 s a sin(ωt + ϕ) s sin ϕ + ω cos ϕ s 2 + ω 2 t k k! 1 s k+1 e at sin ωt ω (s a) 2 + ω 2 t k 1 k! eat (s a) k+1 e at cos ωt (s a) (s a) 2 + ω 2 Lanari: CS - Frequency domain analysis 15
Realizations (A, B, C, D) ok W (s) =C(sI A) 1 B + D how? W (s) =C(sI A) 1 B + D realization infinite solutions (A, B, C, D) state dimension? how can we easily find one state space representation (A, B, C, D)? may be complicated for MIMO systems (here SISO) we see only one, obtainable directly from the coefficients of the transfer function (others are obtainable by simple similarity transformations) Lanari: CS - Frequency domain analysis 16
Realizations W (s) = N(s) D(s) coprime N(s) & D(s) - find D W (s) = b n 1 s n 1 + b n 2 s n 2 + + b 1 s + b s n + a n 1 s n 1 + a n 2 s n 2 + + a 1 s + a + D - state with dimension n - one possible realization for A, B, C 1 1 A c = 1 a a 1 a 2 a n 1 C c = b b 1 b 2 b n 1 B c =. 1 Controller canonical form (useful for eigenvalue assignment) Lanari: CS - Frequency domain analysis 17