Solving state function of continuous-time LTI system

Solving state function of continuous-time LTI system

Continuous-time LTI system

Consider an LTI system with state equation, output equation and initial state

⎧⎩⎨⎪⎪λ˙(t)=Aλ(t)+Bx(t)y(t)=Cλ(t)+Dx(t)λ(0−)

Laplace transform of state function

Apply Laplace transform method to solve the state equation. Firstly, apply Laplace transform to both sides of the state equation,

sΛ(s)−λ(0−)=AΛ(s)+BX(s)

where Λ(s),X(s) are the Laplace transform of state vector λ(t) and signal vector x(t), respectively. In above equation, we aim to solve Λ(s), thus we have

(sI−A)Λ(s)=λ(0−)+BX(s)

if sI−A is invertiable, we have

Λ(s)=(sI−A)−1λ(0−)zero input (zi)+(sI−A)−1BX(s)zero state (zs)

where the first part of the right side is the Laplace transform of zero input response (ZIR) of the state vector and the second part is the Laplace transform of zero state response (ZSR) of the state vector. Then apply inverse Laplace transform to them, we have

{λzi(t)=λzs(t)=L−1{(sI−A)−1λ(0−)}=L−1{(sI−A)−1}λ(0−)=L−1{Φ(s)}λ(0−)L−1{(sI−A)−1BX(s)}=L−1{Φ(s)BX(s)}

where Φ(s)=(sI−A)−1 is the Laplace transform of the state transition matrix ϕ(t); λzi(t),λzs(t) are the ZIR and ZSR of the state vector, respectively.

Laplace transform of output function

Apply Laplace transform method to solve the ouput equation. Secondly, apply Laplace transform to both sides of the output equation,

Y(s)=CΛ(s)+DX(s)

The result of Φ(s) into above, we have

Y(s)==C(sI−A)−1λ(0−)+C(sI−A)−1BX(s)+DX(s)C(sI−A)−1λ(0−)zero input (zi)+[C(sI−A)−1B+D]X(s)zero state (zs)

where the first part of the right side is the Laplace transform of zero input response (ZIR) of the system and the second part is the Laplace transform of zero state response (ZSR) of the system. Then apply inverse Laplace transform to them, we have

{yzi(t)=yzs(t)=L−1{C(sI−A)−1λ(0−)}=L−1{CΦ(s)}λ(0−)L−1{[C(sI−A)−1B+D]X(s)}=L−1{[CΦ(s)B+D]X(s)}

where yzi(t),yzs(t) are the ZIR and ZSR of the system, respectively.

full response of the system

Finally we add up the ZIR and ZSR of the system and get the full response (FR) of the system

yf(t)=yzi(t)+yzs(t)

where yf(t) is the FR of the system.