Discretize Continuous State Space Model of LTI System

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Discretize_Continuous_State_Space_Model_of_LTI_System.pdf

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Continuous State Space Model of Linear Time invariant System

$$\begin{gathered} \dot{x}=Ax(t)+Bu(t) \\ y(t)=Cx(t)+Du(t),t\ge 0 \end{gathered}$$, (1)

 

The matrix exponential function as

$\frac{d}{dt}{e}^{At}=A{e}^{At}=e^{At}A$, (2)

 

Multiplying both sides of (1) by

$e^{-At}$, we obtain(3)

$e^{-At}\dot{x}(t)=e^{-At}Ax(t)+e^{-At}Bu(t)$, (3)

 

Substituting (2) into (3)

$e^{-At}\frac{d}{dt}x(t)=-\frac{d}{dt}{e}^{-At}x(t)+e^{-At}Bu(t)$

$\to \frac{d}{dt}[e^{-At}x(t)]=e^{-At}Bu(t)$, (4)

 

Integrating the result of multiplying

$dt$ both side of (4)

$e^{-At}x(t){|}_0^t=e^{-At}x(t)-x(0)={\int }_0^t{e}^{-A\tau }Bu(\tau )d\tau$, (5)

 

Rewriting the equation (5), we get the solution of the state-space eq of LTI system.

$x(t)=e^{At}x(0)+{\int }_0^t{e}^{A(t-\tau )}Bu(\tau )d\tau$, (6)

 

Discretize the continuous state space model

In an analog control system, the control input is updated continuously, while in a digital control system, it is held constant within each sampling period.

To accurately represent this behavior in a system model, discretization must be applied. we use the Zero-Order-Hold(ZOH) method in this contents.

$x(t)=e^{A(t-t_0)}x(t_0)+{\int }_{t_0}^t{e}^{A(t-\tau )}Bu(\tau )d\tau$, (1)

 

To transition to a discrete-time representation, we replace the continuous-time variable

$t,t_0$ with the discrete-time variable

$T(k+1),Tk$

$t=T(k+1),t_0=Tk,T:samplingtime$

 

Rewriting equation (1) under the assumption that the input

$u$ remained during sampling time. so input

$u$ is constant.

$x(T(k+1))=e^{AT}x(t_0)+{\int }_{Tk}^{T(k+1)}{e}^{A(T(k+1)-\tau )}Bu(k)d\tau$ (2)

 

For simplicity, introduce a change of variable

$\sigma =T(k+1)-\tau$

 

Thus, by substituting

$d\tau =-d\sigma$, equation (2) transforms into (3)

$x(T(k+1))=e^{AT}x(t_0)+{\int }_T^0{e}^{A\sigma }Bu(\tau )(-d\sigma )$ (3)

 

Rearranging (3), we get the final form in continuous-time

$x(T(k+1))=e^{AT}x(t_0)+{\int }_0^T{e}^{A\sigma }d\sigma Bu(k)$ (4)

 

$$\begin{gathered} A_d=e^{AT}=I+AT+\frac{(AT{)}^2}{2!}... \\ {B}_d={\int }_0^T{e}^{A\sigma }d\sigma B=A^{-}1[e^{AT}-I]B=T+\frac{A{T}^2}{2!}+\frac{A^2{T}^3}{3!}... \end{gathered}$$

 

We can represent the eq (4) in discrete-time.

→

$x(k+1)=A_{d}x(k)+B_{d}u(k)$