Question 26.4: Consider the plant Go(s) = Gon(s)[GoD(s)]^-1 (26.4.2) where ...
Consider the plant
G_{o}(s)=G_{oN}(s)[G_{oD}(s)]^{-1} (26.4.2)
where
G_{oN}(s)=\begin{bmatrix} -5 & s^{2} \\ 1 & -0.0023 \end{bmatrix}; G_{oD}(s)=\begin{bmatrix} 25s+1 & 0 \\ 0 & s(s+1)^{2} \end{bmatrix} (26.4.3)
(i) Convert to state space form and evaluate the zeros.
(ii) Design a pre-stabilizing controller to give static decoupling for reference signals.
(iii) Design a prefilter to give full dynamic decoupling for reference signals.
(i)
If we compute det(G_{oN}(s)) we find that this is a non-minimum phase system with zeros at s = ±0.1072. We wish to design a controller which achieves dynamic decoupling. We proceed in a number of steps.
Step 1. State space model
The state space model for the system is
\dot{x} _{p}(t)=A_{o}x_{p}(t)+B_{o}u(t) (26.4.4)
y(t)=C_{o}x_{p}+D_{o}u(t) (26.4.5)
where
A_{o}=\left [ \begin{matrix} -0.04 & 0 & 0 &0 \\0 & -2 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 &0 \end{matrix} \right ] B_{o}=\left [ \begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \\0 & 0 \end{matrix} \right ] (26.4.6)
C_{o}=\left [ \begin{matrix} -0.2 & 1 & 0 & 0 \\ 0.04 & 0 & 0 & -0.0023 \end{matrix} \right ] D_{o}=0 (26.4.7)
We will design a stabilizing controller using the architecture shown in Figure 26.4 We design an observer for the state x_{p}(t) given the output y(t). This design uses Kalman filter theory with Q = B_{o}B_{o}^{T} and R = 0.05I_{2×2}
The optimal observer gains turn out to be
J=\left [ \begin{matrix} -3.9272 & 1.3644 \\ 2.6120 & 0.1221 \\ -0.6379 & 0.1368 \\ -2.7266 & -4.6461 \end{matrix} \right ] (26.4.8)
We wish to have zero steady state errors in the face of step input disturbances. We therefore follow the procedure of section §22.13 and introduce an integrator with transfer function I/s at the output of the system (after the comparator). That is we add²
\dot{\bar{z}}=-y(t)=-C_{o}x_{p}(t) (26.4.9)
We can now define a composite state vector \bar{x}(t)=[x^{T }_{p} z^{T}(t) ]^{T} leading to the composite model
\dot{\bar{x}}(t) =\bar{A}\bar{x}(t) +\bar{B}u(t) (26.4.10)
where
\bar{A}=\begin{bmatrix} A_{o} & 0 \\ C_{o} & 0 \end{bmatrix}; \bar{B}=\begin{bmatrix} B_{o} \\ 0 \end{bmatrix} (26.4.11)
We next consider the composite system and design a state controller via LQR theory.
We choose
\Psi =\left [ \begin{matrix} C_{o}^{T}C_{o} & 0 & 0 \\ 0 & 0.005 & 0 \\ 0 & 0 & 0.1 \end{matrix} \right ]; \Phi =2I_{2\times 2} (26.4.12)
leading to the feedback gain K = [K1 K2], where
K_{1}=\left [ \begin{matrix} 0.1807 & -0.0177 & 0.1011 & -0.0016 \\ -0.0177 & 0.1496 & 0.0877 & 0.0294 \end{matrix} \right ]; K_{2}=\begin{bmatrix} 0.0412 & -0.1264 \\ 0.0283 & 0.1844 \end{bmatrix} (26.4.13)
This leads to the equivalent closed loop shown in Figure 26.5 (where we have ignored the observer dynamics since these disappear in steady state).
The resulting closed loop responses, for unit step references, are shown in Figure 26.6, where r_{1}(t) = µ(t − 1) and r_{2}(t) = −µ(t − 501). Note that, as expected, the system is statically decoupled but significant coupling occurs during transients, especially following the step in the second reference.
(iii)
The closed loop has transfer function
T_{o}(s)=(I+\tilde{G}(s) )^{-1}\tilde{G} (s) (26.4.14)
\tilde{G}(s)= C_{o}(sI-A_{o}+B_{o}K_{1})^{-1}B_{o}K_{o}\frac{1}{s} (26.4.15)
This is a stable proper transfer function. Note, however, that this is nonminimum phase since the original plant was non-minimum phase.
We use the techniques of subsection §26.2.2 to design an inverse that retains dynamic decoupling in the presence of non-minimum phase zeros. To use those techniques the equivalent plant is the closed loop system with transfer function (26.4.14) and with state space model given by the 4-tuple (A_{e}, B_{e}, C_{e}, 0), where
A_{e}=\begin{bmatrix} A_{o}-B_{o}K_{1} & B_{o}K_{2} \\ -C_{o} & 0 \end{bmatrix}; B_{e}=\left [ \begin{matrix} 0 & I \end{matrix} \right ]^{T}; C_{e}=\left [ \begin{matrix} C_{o} & 0 \end{matrix} \right ] (26.4.16)
An interactor for this closed loop system is
\xi _{L}(s)=\begin{bmatrix} (s+\alpha )^{2} & 0 \\ 0 & (s+\alpha )^{2} \end{bmatrix}; α = 0.03
This leads to an augmented system having state space model (A^{\prime }_{e}, B^{\prime }_{e}, C^{\prime }_{e}, D^{\prime }_{e}) with
A^{\prime }_{e} = A_{e}B^{\prime }_{e} = B_{e}
C^{\prime }_{e} = \alpha ^{2}C_{e} +2\alpha C_{e}A_{e}+C_{e}A_{e}^{2}
D^{\prime }_{e} = C_{e}A_{e}B_{e}
The exact inverse then has state space model (A_{\lambda }, B_{\lambda }, C_{\lambda }, D_{\lambda }) where
A_{\lambda }=A^{\prime }_{e}-B^{\prime }_{e}[D^{\prime }_{e}]^{-1}C^{\prime}_{e}B_{\lambda }=B^{\prime }_{e}[D^{\prime }_{e}]^{-1}C^{\prime}_{e}
C_{\lambda }=-[D^{\prime }_{e}]^{-1}C^{\prime}_{e}
D_{\lambda }=[D^{\prime }_{e}]^{-1}
We now form the two subsystems as in subsection §26.2.2. We form a minimal realization of these two systems which we denote by (A_{1}, B_{1}, C_{1}, D_{1}) and (A_{2}, B_{2}, C_{2}, D_{2}). We determine stabilizing feedback for these two systems using LQR theory with
\Psi _{1}= C_{1}^{T}C_{1} \Phi _{1}=10^{6}
\Psi _{2}= C_{2}^{T}C_{2} \Phi _{2}=10^{7}
We then implement the precompensator as in Figure 26.2 where we choose
t_{1}(s)=\alpha ^{2}\frac{1-K_{1}[A_{1}]^{-1}B_{1}}{(s+\alpha )^{2}}t_{2}(s)=\alpha ^{2}\frac{1-K_{2}[A_{2}]^{-1}B_{2}}{(s+\alpha )^{2}}
where K_{1}, K_{2} now represent the stabilizing gains for the two subsystems as in subsection §26.2.2.
The resulting closed loop responses for step references are shown in Figure 26.7, where r_{1}(t) = µ(t − 1) and r_{2}(t) = −µ(t − 501). Note that, as expected, the system is now fully decoupled from the reference to the output response.
