PROBLEM: Design an observer for the plant G(s) = 1/(s+7)(s+8)(s+9) whose estimated plant is represented in state space in cascade form as z = Az +Bu =[-7 1 0 0 -8 1 0 0 -9]z +[0 0 1]u y = Cx =[1 0 0]z The closed-loop step response of the observer is to have 10% overshoot with a 0.1 second settling

Question

PROBLEM: Design an observer for the plant

[latex]G(s)=\frac{1}{(s+7)(s+8)(s+9)} [/latex]

whose estimated plant is represented in state space in cascade form as

[latex]\dot{\hat{z}}=A\hat{z}+Bu=\left[\begin{matrix} -7 & 1 & 0 \ 0 & -8 & 1 \ 0 & 0 & -9 \end{matrix} ight] \hat{z}+\left[\begin{matrix} 0 \ 0 \ 1 \end{matrix} ight] u [/latex]

[latex]\hat{y}=C\hat{x}=\left[\begin{matrix} 1 & 0 & 0\end{matrix} ight] \hat{z} [/latex]

The closed-loop step response of the observer is to have 10% overshoot with a 0.1 second settling time.

Accepted Answer

The system is represented in cascade form by the following state and output equations:

[latex]\dot{z}=\left[\begin{matrix} -7 & 1 & 0 \ 0 & -8 & 1 \ 0 & 0 & -9 \end{matrix} ight] z+\left[\begin{matrix} 0 \ 0 \ 1 \end{matrix} ight] u[/latex]

[latex]y =\left[\begin{matrix} 1 & 0 & 0 \end{matrix} ight] z[/latex]

The observability matrix is [latex]O_{Mz} =\left[\begin{matrix} C_{z} \ C_{z}A_{z} \ C_{z}A^{2}_{z} \end{matrix} ight] =\left[\begin {matrix} 1 & 0 & 0 \ -7 & 1 & 0 \ 49 & -15 & 1 \end{matrix} ight][/latex], where [latex]A^{2}_{z}=\left[\begin{matrix} 49 & -15 & 1 \ 0 & 64 & -17 \ 0 & 0 & 81 \end{matrix} ight] [/latex]. Since [latex]G(s) =\frac{1}{(s+7)(s+8)(s+9)}=\frac{1}{s^{3}+24s^{2}+191s+504 }[/latex], we can write the observable canonical form as

[latex]\dot{x}=\left[\begin{matrix} -24 & 1 & 0 \ -191 & 0 & 1 \ -504 & 0 & 0 \end{matrix} ight] x+\left[\begin{matrix} 0 \ 0 \ 1 \end{matrix} ight] u[/latex]

[latex]y=\left[\begin{matrix} 1 & 0 & 0 \end{matrix} ight] x[/latex]

The observability matrix for this form is [latex]O_{Mx} =\left[\begin {matrix} C_{x} \ C_{x}A_{x} \ C_{x}A^{2}_{x} \end{matrix} ight] =\left[\begin {matrix} 1 & 0 & 0 \ -24 & 1 & 0 \ 385 & -24 & 1 \end{matrix} ight][/latex],

where

[latex]A^{2}_{x}=\left[\begin{matrix} 385 & -24 & 1 \ 4080 & -191 & 0 \ 12096 & 504 & 0 \end{matrix} ight] [/latex].

We next find the desired characteristic equation. A 10% overshoot requires [latex]\zeta =\frac{-\log \left(\frac{\%}{100} ight) }{\sqrt{\pi ^{2}+\log ^{2}\left(\frac{\%}{100} ight) } } =0.591 [/latex]. Also, [latex]\omega _{n} =\frac{4}{\zeta T_{s} } =67.66 [/latex] rad/s. Thus, the characteristic equation is [latex]s^{2}+2 \zeta\omega _{n}s+\omega ^{2}_{n} =s^{2}+80s+4578.42[/latex]. Adding a pole at -400, or 10 times the real part of the dominant second-order poles, yields the resulting desired characteristic equation, [latex](s^{2}+80s+4578.42)(s+400)=s^{3}+480s^{2}+36580s +1.831x 10^{6}[/latex]. For the system represented in observable canonical form [latex]e_{\dot{x}} =(A_{x}-L_{x}C_{x} )e_{x}=\left[\begin {matrix} -(24+l_{1}) & 1 & 0 \ -(191+l_{2}) & 0 & 1 \ -(504+l_{3}) & 0 & 0 \end{matrix} ight] e_{x}[/latex]. The characteristic polynomial is given by [latex]\left|\left[sI-(A_{x}-L_{x}C_{x}) ight] ight| = s^{3}+(24+l_{1})s^{2}+(191+l_{2})s+(504+l_{3})[/latex]. Equating coefficients of the desired characteristic equation to the system’s characteristic equation yields [latex]L_{x}=\left[\begin{matrix} 456 \ 36,389 \ 1,830,496 \end{matrix} ight][/latex].

Now, develop the transformation matrix between the observer canonical and cascade forms.

[latex]P=x^{-1}_{Mz}O_{Mz} =\left[\begin{matrix} 1 & 0 & 0 \ -7 & 1 & 0 \ 49 & -15 & 1 \end{matrix} ight] ^{-1} \left[\begin{matrix} 1 & 0 & 0 \ -24 & 1 & 0 \ 385 & -24 & 1 \end{matrix} ight][/latex]

[latex]=\left[\begin{matrix} 1 & 0 & 0 \ 7 & 1 & 0 \ 56 & 15 & 1 \end{matrix} ight]\left[\begin{matrix} 1 & 0 & 0 \ -24 & 1 &0 \ 385 & -24 & 1 \end{matrix} ight] [/latex]

[latex]=\left[\begin{matrix} 1 & 0 & 0 \ -17 & 1 & 0 \ 81 & -9 & 1 \end{matrix} ight] [/latex]

Finally,

[latex]L_{z}=PL_{z}=\left[\begin{matrix} 1 & 0 & 0 \ -17 & 1 & 0 \ 81 & -9 & 1 \end{matrix} ight] \left[\begin{matrix} 456 \ 36,389 \ 1,830,496 \end{matrix} ight] =\left[\begin{matrix} 456 \ 28,637 \ 1,539,931 \end{matrix} ight]\approx \left[\begin{matrix} 456 \ 28,640 \ 1,540,000 \end{matrix} ight][/latex].

Question 12.SE.6: PROBLEM: Design an observer for the plant G(s) = 1/(s+7)(s+8...

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