Observer Design by Transformation

PROBLEM: Design an observer for the plant
$G(s)=\frac{1}{(s+1)(s+2)(s+5)}$
represented in cascade form. The closed-loop performance of the observer is governed by the characteristic polynomial used in Example 12.5: $s^{3}+120s^{2}+2500s+50,000$ .

Question

Observer Design by Transformation

PROBLEM: Design an observer for the plant

[latex]G(s)=\frac{1}{(s+1)(s+2)(s+5)}[/latex]

represented in cascade form. The closed-loop performance of the observer is governed by the characteristic polynomial used in Example 12.5: [latex]s^{3}+120s^{2}+2500s+50,000[/latex] .

Accepted Answer

First represent the plant in its original cascade form.

[latex]\dot{z} = Az+Bu=\left[\begin{matrix} -5 & 1 & 0 \ 0 & -2 & 1 \ 0 & 0 & -1 \end{matrix} ight] z+\left[\begin{matrix} 0 \ 0 \ 1 \end{matrix} ight] u[/latex]

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

The observability matrix,[latex]O_{Mz} [/latex],is

[latex]O_{Mz}=\left[\begin{matrix} C \ CA \ CA^{2} \end{matrix} ight] = \left[\begin{matrix} 1 & 0 & 0 \ -5 & 1 & 0 \ 25 & -7 & 1 \end{matrix} ight][/latex]

whose determinant equals 1. Hence, the plant is observable. The characteristic equation for the plant is

[latex]\det (sI-A)=s^{3}+ 8s^{2}+17s+10=0[/latex]

We can use the coefficients of this characteristic polynomial to form the observer canonical form

[latex]\dot{x}= A_{x} x+B_{x} x[/latex]

[latex]y = C_{x} x[/latex]

where

[latex]A_{x} =\left[\begin{matrix} -8 & 1 & 0 \ -17 & 0 & 1 \ -10 & 0 & 0 \end{matrix} ight] ;[/latex] [latex]C_{x} =\left[\begin{matrix} 1 & 0 & 0\end{matrix} ight][/latex]

The observability matrix for the observer canonical 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 \ -8 & 1 & 0 \ 47 & -8 & 1 \end{matrix} ight] [/latex]

We now design the observer for the observer canonical form. First form [latex](A_{x}-L_{X}C_{X})[/latex].

[latex]A_{x}-L_{x}C_{x}= \left[\begin{matrix} -8 & 1 & 0 \ -17 & 0 & 1 \ -10 & 0 & 0 \end{matrix} ight]-\left[\begin{matrix} l_{1} \ l_{2} \ l_{3} \end{matrix} ight] \left[\begin{matrix} 1 & 0 & 0\end{matrix} ight] = \left[\begin{matrix} -(8+l_{1} ) & 1 & 0 \ -(17+l_{2} ) & 0 & 1 \ -(10+l_{3} ) & 0 & 0 \end{matrix} ight][/latex]

whose characteristic polynomial is

[latex]\det \left[sI-(A_{x}-L_{x}C_{x}) ight]=s^{3}+(8+l_{1} )s^{2}+(17+l_{2})s+(10+l_{3})[/latex]

Equating this polynomial to the desired closed-loop observer characteristic equation, [latex]s^{3}+120s^{2}+2500s+50,000[/latex], we find

[latex]L_{x}=\left[\begin{matrix} 112 \ 2483 \ 49,990 \end{matrix} ight][/latex]

Now transform the design back to the original representation. Using Eq. (12.89),

[latex]P=O_{Mz} ^{-1} O_{Mz}[/latex]

the transformation matrix is

[latex]P=O_{Mz} ^{-1} O_{Mz}=\left[\begin{matrix} 1 & 0 & 0 \ -3 & 1 & 0 \ 1 & -1 & 1 \end{matrix} ight][/latex]

Transforming [latex]L_{x}[/latex] to the original representation, we obtain

[latex]L_{z}=PL_{x}=\left[\begin{matrix} 112 \ 2147 \ 47,619 \end{matrix} ight][/latex]

The final configuration is shown in Figure 12.18.

A simulation of the observer is shown in Figure 12.19(a). To demonstrate the effect of the observer design, Figure 12.19(b) shows the reduced speed if the observer is simply a copy of the plant and all observer feedback paths are disconnected.

Students who are using MATLAB should now run ch12p6 in Appendix B. You will learn how to use MATLAB to design an observer for a plant
not represented in observer canonical form. You will see that MATLAB does not require transformation to observer canonical form. This exercise solves Example 12.8 using MATLAB.

Question 12.8: Observer Design by Transformation PROBLEM: Design an observe...

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