Question 10.20: Controllability and Observability Consider a dynamic system ...
Controllability and Observability
Consider a dynamic system described by G(s)=2 /(s+4), which can be converted to state-space form as
\begin{aligned} & \dot{x}_{1}=-4 x_{1}+2 u, \\ & y=x_{1} . \end{aligned}
a. A new state is added and the resulting state-space equation is
\begin{aligned} & \dot{x}_{1}=-4 x_{1}+2 u, \\ & \dot{x}_{2}=-x_{2}, \\ & y=x_{1}+3 x_{2} . \end{aligned}
Determine the transfer function for this new model.
b. Determine the transfer function for another model with state-space form
\begin{aligned} & \dot{x}_{1}=-4 x_{1}+2 u, \\ & \dot{x}_{2}=-x_{2}+u, \\ & y=x_{1} . \end{aligned}
a. The system matrices \mathbf{A}, \mathbf{B}, \mathbf{C}, and D are
\mathbf{A}=\left[\begin{array}{cc} -4 & 0 \\ 0 & -1 \end{array}\right], \quad \mathbf{B}=\left[\begin{array}{l} 2 \\ 0 \end{array}\right], \quad \mathbf{C}=\left[\begin{array}{ll} 1 & 3 \end{array}\right], \quad D=0
The transfer function is
G(s)=\left[\begin{array}{ll} 1 & 3 \end{array}\right]\left[\begin{array}{cc} s+4 & 0 \\ 0 & s+1 \end{array}\right]^{-1}\left[\begin{array}{l} 2 \\ 0 \end{array}\right]+0=\frac{\left[\begin{array}{ll} 1 & 3 \end{array}\right]\left[\begin{array}{cc} s+1 & 0 \\ 0 & s+4 \end{array}\right]\left[\begin{array}{l} 2 \\ 0 \end{array}\right]}{(s+1)(s+4)}=\frac{2}{s+4}
b. Similarly, we have
\mathbf{A}=\left[\begin{array}{cc} -4 & 0 \\ 0 & -1 \end{array}\right], \quad \mathbf{B}=\left[\begin{array}{l} 2 \\ 1 \end{array}\right], \quad \mathbf{C}=\left[\begin{array}{ll} 1 & 0 \end{array}\right], \quad D=0
and
G(s)=\left[\begin{array}{ll} 1 & 0 \end{array}\right]\left[\begin{array}{cc} s+4 & 0 \\ 0 & s+1 \end{array}\right]^{-1}\left[\begin{array}{l} 2 \\ 1 \end{array}\right]+0=\frac{\left[\begin{array}{ll} 1 & 0 \end{array}\right]\left[\begin{array}{cc} s+1 & 0 \\ 0 & s+4 \end{array}\right]\left[\begin{array}{l} 2 \\ 1 \end{array}\right]}{(s+1)(s+4)}=\frac{2}{s+4}
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 \mathbf{B},
G(s)=\frac{[s+1 \quad 3(s+4)]}{(s+1)(s+4)}\left[\begin{array}{l} 2 \\ 0 \end{array}\right]
This implies that the second state is uncontrollable by the actuator defined by matrix \mathbf{B}. Similarly, in Part (b), the second state cannot be observed by the output matrix \mathbf{C},
G(s)=\left[\begin{array}{ll} 1 & 0 \end{array}\right] \frac{\left[\begin{array}{c} 2(s+1) \\ s+4 \end{array}\right]}{(s+1)(s+4)} \text {. }
This implies that the second state is unobservable by the sensor defined by matrix \mathbf{C}.