Controllability and Observability Consider a dynamic system described by G(s) = 2/(s + 4), which can be converted to state-space form as x1 = -4x1+2u, y = x1. a. A new state is added and the resulting state-space equation is x1 = -4x1 + 2u, x2 = -x2, y = x1 +3x2. Determine the transfer function

Question

Controllability and Observability

Consider a dynamic system described by [latex]G(s)=2 /(s+4)[/latex], which can be converted to state-space form as

[latex] \begin{aligned} & \dot{x}_{1}=-4 x_{1}+2 u, \ & y=x_{1} . \end{aligned} [/latex]

a. A new state is added and the resulting state-space equation is

[latex] \begin{aligned} & \dot{x}_{1}=-4 x_{1}+2 u, \ & \dot{x}_{2}=-x_{2}, \ & y=x_{1}+3 x_{2} . \end{aligned} [/latex]

Determine the transfer function for this new model.

b. Determine the transfer function for another model with state-space form

[latex] \begin{aligned} & \dot{x}_{1}=-4 x_{1}+2 u, \ & \dot{x}_{2}=-x_{2}+u, \ & y=x_{1} . \end{aligned} [/latex]

Accepted Answer

a. The system matrices [latex]\mathbf{A}, \mathbf{B}, \mathbf{C}[/latex], and [latex]D[/latex] are

[latex] \mathbf{A}=\left[\begin{array}{cc} -4 & 0 \ 0 & -1 \end{array} ight], \quad \mathbf{B}=\left[\begin{array}{l} 2 \ 0 \end{array} ight], \quad \mathbf{C}=\left[\begin{array}{ll} 1 & 3 \end{array} ight], \quad D=0 [/latex]

The transfer function is

[latex] G(s)=\left[\begin{array}{ll} 1 & 3 \end{array} ight]\left[\begin{array}{cc} s+4 & 0 \ 0 & s+1 \end{array} ight]^{-1}\left[\begin{array}{l} 2 \ 0 \end{array} ight]+0=\frac{\left[\begin{array}{ll} 1 & 3 \end{array} ight]\left[\begin{array}{cc} s+1 & 0 \ 0 & s+4 \end{array} ight]\left[\begin{array}{l} 2 \ 0 \end{array} ight]}{(s+1)(s+4)}=\frac{2}{s+4} [/latex]

b. Similarly, we have

[latex] \mathbf{A}=\left[\begin{array}{cc} -4 & 0 \ 0 & -1 \end{array} ight], \quad \mathbf{B}=\left[\begin{array}{l} 2 \ 1 \end{array} ight], \quad \mathbf{C}=\left[\begin{array}{ll} 1 & 0 \end{array} ight], \quad D=0 [/latex]

and

[latex] G(s)=\left[\begin{array}{ll} 1 & 0 \end{array} ight]\left[\begin{array}{cc} s+4 & 0 \ 0 & s+1 \end{array} ight]^{-1}\left[\begin{array}{l} 2 \ 1 \end{array} ight]+0=\frac{\left[\begin{array}{ll} 1 & 0 \end{array} ight]\left[\begin{array}{cc} s+1 & 0 \ 0 & s+4 \end{array} ight]\left[\begin{array}{l} 2 \ 1 \end{array} ight]}{(s+1)(s+4)}=\frac{2}{s+4} [/latex]

Note that the dynamic models in Parts (a) and (b) are second-order systems, with statespace forms that differ from the original first-order system. However, they both end up with the same transfer function as the given first-order system due to pole-zero cancellation. As seen in Part (a), the second state cannot be affected by the input matrix [latex]\mathbf{B}[/latex],

[latex] G(s)=\frac{[s+1 \quad 3(s+4)]}{(s+1)(s+4)}\left[\begin{array}{l} 2 \ 0 \end{array} ight] [/latex]

This implies that the second state is uncontrollable by the actuator defined by matrix [latex]\mathbf{B}[/latex]. Similarly, in Part (b), the second state cannot be observed by the output matrix [latex]\mathbf{C}[/latex],

[latex] G(s)=\left[\begin{array}{ll} 1 & 0 \end{array} ight] \frac{\left[\begin{array}{c} 2(s+1) \ s+4 \end{array} ight]}{(s+1)(s+4)} ext {. } [/latex]

This implies that the second state is unobservable by the sensor defined by matrix [latex]\mathbf{C}[/latex].

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