Digital Control Systems [MEE 4003] - Kckong.info

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2.2. LAPLACE TRANSFORMS AND TRANSFER FUNCTIONS 33 (a) Actual system under an arbitrary input u(t) t 0 ) 0 M & y + ky = u y( t) = y ( t) + ∫ g( t −τ ) u( τ dτ (b) Time domain analysis: g(t) is an impulse response of the system u(t) x & = Fx + Gu y = Hx x = e Ft x t F ( t−τ ) ( 0) + ∫ e Gu( τ ) dτ 0 (c) State space analysis U (s) 1 2 Ms + k (d) Laplace domain analysis 1 Y ( s) = U ( s) 2 Ms + k ω ω ω (e) Frequency domain analysis Figure 2.9: Various methods for analysis of an input-output relationship Digital Control Systems, Sogang University Kyoungchul Kong
2.2. LAPLACE TRANSFORMS AND TRANSFER FUNCTIONS 34 where x ∈ R n , y ∈ R, u ∈ R, F ∈ R n×n , G ∈ R n×1 , H ∈ R 1×n , and J ∈ R. Note that the feedforward matrix,J, is included for the sake of generality. Since the state space model above consists only of linear functions 2 , the Laplace transform can be applied 3 . Taking the Laplace transform, we obtain Assuming thatx(0) = 0, L{ẋ} = L{Fx+Gu} L{y} = L{Hx+Ju} sX(s) = FX(s)+GU(s) Y(s) = HX(s)+JU(s) Rearranging the equation above, the transfer function fromU(s) toY(s) is obtained: Y(s) U(s) = H(sI −F)−1 G+J (2.10) [Example 2-12] A state space model with state matrices, F = [ 0 1 −1 −2], G = [ 0 1 ], H = [ 1 0], J = 0, is converted into a transfer function as: G(s) = [ 1 0 ]( [ sI − 0 1 −1 −2 ]) −1 [ 0 1 ] +0 = 1 s 2 +2s+1 Recall that the dimensions of the state matrices are F ∈ R n×n , G ∈ R n×1 , H ∈ R 1×n , and J ∈ R, and a transfer function,G(s) is obtained uniquely via G(s) = H }{{} 1×n (sI −F) −1 } {{ } n×n }{{} G n×1 + }{{} J = b(s) a(s) 1×1 where b(s) is the numerator polynomial and a(s) is the denominator polynomial. Since (sI −F) −1 is reduced to (sI −F) −1 = 1 Adj(sI −F) det(sI −F) 2 In other words, the state space equation is a linear system. 3 This is not always true, if the state matrix includes nonlinear functions. For example, if ẋ = [ 0 1 −sin(t) −cos(t)] x+[ 0 1 ]u, the state space model cannot be transformed by the Laplace transform. Namely, the state space is capable of dealing with nonlinear or time-varying systems, while the Laplace transform can only be applied to linear systems. Digital Control Systems, Sogang University Kyoungchul Kong
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