A Reduced-Order Estimator Design for Pendulum
Design a reduced-order estimator for the pendulum that has the error pole at $-10ω_0.$

Question

A Reduced-Order Estimator Design for Pendulum
Design a reduced-order estimator for the pendulum that has the error pole at [latex] −10ω_0.[/latex]

Accepted Answer

We are given the system equations

[latex] \begin{bmatrix} \dot x_1 \ \dot x_2 \end{bmatrix} = \begin{bmatrix} 0 & 1 \ -w^2_0 & 0 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix} + \begin{bmatrix} 0 \ 1 \end{bmatrix} u , [/latex]

[latex] y = \begin{bmatrix} 1 & 0 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix} [/latex]

The partitioned matrices are

[latex] \begin{bmatrix} F_{aa} & \pmb F_{ab} \ \pmb F_{ba} & \pmb F_{bb} \end{bmatrix} = \begin{bmatrix} 0 & 1 \−ω^2_0 & 0 \end{bmatrix}, [/latex]

[latex] \begin{bmatrix} G_a \ \pmb G_b \end{bmatrix} = \begin{bmatrix} 0 \ 1 \end{bmatrix}. [/latex]

From Eq. (7.158), we find the characteristic equation in terms of [latex] L:[/latex]

[latex] s − (0 − L) = 0. [/latex]

[latex] det[s \pmb I − ( \pmb F_{bb} − \pmb {LF}_{ab})] = 0. [/latex] (7.158)

We compare it with the desired equation,

[latex] α_e(s) = s + 10ω_0 = 0, [/latex]

which yields

[latex] L = 10ω_0. [/latex]

The estimator equation, from Eq. (7.161), is

[latex] \dot x_c = −10ω_0 \hat{x}_2 − ω^2_0y + u, [/latex]

[latex] \dot {\pmb x}_c = (\pmb F_{bb} − \pmb{LF}_{ab}) \hat{ \pmb x}_b + ( \pmb F_{ba} − \pmb LF_{aa})y + (\pmb G_b − \pmb LG_a)u, [/latex] (7.161)

and the state estimate, from Eq. (7.160), is

[latex] \hat{x}_2 = x_c + 10ω_0y. [/latex]

[latex] \pmb x_c riangleq \hat{ \pmb x}_b - \pmb L y . [/latex] (7.160)

We use the control law given in the earlier examples. The response of the estimator to a plant initial condition [latex] \pmb x_0 = [1.0, 0.0]^T [/latex] and an estimator initial condition [latex] x_{c0} = 0 [/latex] is shown in Fig. 7.33 for [latex]ω_0 = 1. [/latex]The response may be obtained using impulse or initial in MATLAB. Note the similarity of the initial-condition response to that of the full-order estimator plotted in Fig. 7.30.

Question 7.EX.26: A Reduced-Order Estimator Design for Pendulum Design a reduc...

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