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��Chapter 13�Observers

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Course Overview

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

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Observers

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

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In Chapters 11 & 12, we assumed that the state was known.

Not realistic!

In this chapter, we assume only the measurement, and build an observer to estimate the state

State estimate

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Output model

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

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This is what we actually measure.

�We use y to estimate the state.

This is what the controller is driving to the reference.

It may be a non-measured quantity.

Sensor noise: assume zero mean Gaussian

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Luenberger Observer

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

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Luenberger Observer

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

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Observer Canonic Form (example)

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

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Observer Canonic Form (example)

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

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Observer Canonic Form (example)

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

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Observer Canonic Form

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Observer Canonic Form (example)

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

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Observer Canonic Form

Control Canonic Form

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Observer Canonic Form: General Form

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

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\[

Y(s) = \frac{b_ms^m + b_{m-1}s^{m-1} + \dots + b_1 s + b_0}{s^n + a_{n-1}s^{n-1} + \dots + a_1 s + a_0} U(s)

\]

�

\begin{align*}

\dot{x}_o

&= \underbrace{\begin{pmatrix} -a_{n-1} & 1 & \dots & 0 & 0 \\

-a_{n-2} & 0 & \dots & 0 & 0 \\

\vdots & & \ddots & & \vdots \\

-a_1 & 0 & \dots & 0 & 1 \\

-a_0 & 0 & \dots & 0 & 0 \end{pmatrix}}_{A_o} x_o

+ \underbrace{\begin{pmatrix} 0 \\ \vdots \\ 0 \\ b_m \\ \vdots \\ b_1 \\ b_0 \end{pmatrix}}_{B_o} u

\\

y &= \underbrace{\begin{pmatrix} 1 & 0 & \dots & 0 & 0\end{pmatrix}}_{C_o} x_o

\end{align*}

--

{\color{blue} In general:

\begin{align*}

A_o &= A_c^\top \\

B_o &= C_c^\top \\

C_o &= B_c^\top \\

D_o &= D_c

\end{align*}

}

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Observer Canonic Realization

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

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Observer Design Using Observer Canonic Form

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

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Closed loop system is

\begin{align*}

\dot{e}_o &= \left[\begin{pmatrix}

-a_{n-1} & 1 & \dots & 0 & 0 \\

-a_{n-2} & 0 & \dots & 0 & 0 \\

\vdots & & \ddots & & \vdots \\

-a_{1} & 0 & \dots & 0 & 1 \\

-a_{0} & 0 & \dots & 0 & 0

\end{pmatrix}

- \begin{pmatrix} \ell_1 \\ \ell_2 \\ \vdots \\ \ell_n \end{pmatrix}

\begin{pmatrix} 1 & 0 & \dots & 0 & 0\end{pmatrix} \right] \begin{pmatrix} e_1 \\ e_2 \\ \vdots \\ e_{n-1} \\ e_n \end{pmatrix} \\

&= \begin{pmatrix} -(a_{n-1}+\ell_1) & 1 & \dots & 0 & 0 \\

-(a_{n-2}+\ell_2) & 0 & \dots & 0 & 0 \\

\vdots & & \ddots & & \vdots \\

-(a_1+\ell_{n-1}) & 0 & \dots & 0 & 1 \\

-(a_0+\ell_n) & 0 & \dots & 0 & 0 \end{pmatrix} \begin{pmatrix} e_1 \\ e_2 \\ \vdots \\ e_{n-1} \\ e_n \end{pmatrix} \\

& = (A_o - L_o C_o) x_o

\end{align*}

�

With closed-loop characteristic equation

\[

\Delta_{cl}(s) = s^n + (a_{n-1}+\ell_1)s^{n-1} + \dots + (a_1+\ell_{n-1}) s + (a_0+\ell_n)

\]

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Observer Design Using Observer Canonic Form

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

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\begin{align*}

\text{Open-loop char eq:~~} \Delta_{ol}(s) &= s^n + a_{n-1}s^{n-1} + \dots + a_1 s + a_0 \\

\text{Observer char eq:~~} \Delta_{cl, ob}(s) &= s^n + (a_{n-1}+\ell_1)s^{n-1} + \dots + (a_1+\ell_{n-1}) s + (a_0+\ell_n) \\

\text{Desired observer char eq:~~} \Delta_{cl, ob}^d(s) &= s^n + \beta_{n-1}s^{n-1} + \dots + \beta_1 s + \beta_0

\end{align*}

Equating gives

\begin{align*}

a_{n-1} + \ell_1 = \beta_{n-1}, &\quad

a_{n-2} + \ell_2 = \beta_{n-2}, &\quad \dots &\quad

a_{1} + \ell_{n-1} = \beta_{1}, &\quad

a_0 + \ell_n = \beta_0

\end{align*}

Define

\begin{align*}

\mathbf{a}_A &= (a_{n-1}, a_{n-2}, \dots, a_1, a_0) \\

\boldsymbol{\beta} &= (\beta_{n-1}, \beta_{n-2}, \dots, \beta_1, \beta_0) \\

L_o &= (\ell_1, \ell_2, \dots, \ell_{n-1}, \ell_n)^\top

\end{align*}

Then

\[

L_o = (\boldsymbol{\beta}-\mathbf{a}_A)^\top

\]

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Observer Design Using Observer Canonic Form

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

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\par\noindent{\color{blue} Example:}

\[

\dot{x}_o = \begin{pmatrix}

-6 & 1 & 0 \\

11 & 0 & 1 \\

-8 & 0 & 0

\end{pmatrix} x_o + \begin{pmatrix} 0 \\ 2 \\ 3 \end{pmatrix} u

\]

Open-loop characteristic polynomial is given by

\[

\Delta_{ol}(s) = s^3 + 6s^2 - 11 s + 8, \qquad \text{Roots at $-7.5885$, $0.7942\pm j0.6507$ (unstable)}

\]

Suppose that the desired closed-loop characteristic polynomial is

\[

\Delta_{cl}^d(s) = s^3 + 50s^2 + 1200 s + 8000, \qquad \text{Roots at $-10$ and $-20\pm j20$ (stable)}

\]

Then

\begin{align*}

\mathbf{a}_A &= (6, -11, 8) \\

\boldsymbol{\beta} &= (50, 1200, 8000)

\end{align*}

The required feedback gains are

\[

L_o = (\boldsymbol{\beta}-\mathbf{a}_{A_o})^\top = \begin{pmatrix} 50 \\ 1200 \\ 8000 \end{pmatrix} - \begin{pmatrix} 6 \\ -11 \\ 8 \end{pmatrix} = \begin{pmatrix} 44 \\ 1211 \\ 7992 \end{pmatrix}

\]

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General Observer Design

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

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\par\noindent General State Feedback:

�

\vspace{1cm}

�

\par\noindent Find $K\in\mathbb{R}^{1\times n}$ to place eigenvalues of

\[

\underbrace{A}_{\text{known}} - \underbrace{B}_{\text{known}}\underbrace{K}_{\text{unknown}}

\]

Solution is

\[

K = (\boldsymbol{\alpha} - \mathbf{a}_A) \mathcal{A}_A^{-1}\mathcal{C}_{A,B}^{-1}

\]

---

\par\noindent General Observer:

�

\vspace{1cm}

�

\par\noindent Find $L\in\mathbb{R}^{n\times 1}$ to place eigenvalues of

\[

\underbrace{A}_{\text{known}} - \underbrace{L}_{\text{unknown}}\underbrace{C}_{\text{known}}

\]

Since $\text{eig}(M)=\text{eig}(M^\top)$, this is equivalent to place eigenvalues of

\[

\underbrace{A^\top}_{\text{known}} - \underbrace{C^\top}_{\text{known}}\underbrace{L^\top}_{\text{unknown}}

\]

Solution is

\begin{align*}

& L^\top = (\boldsymbol{\beta} - \mathbf{a}_A) \mathcal{A}_A^{-1}\mathcal{C}_{A^\top,C^\top}^{-1} \\

\implies & L = \mathcal{C}_{A^\top,C^\top}^{-\top} \mathcal{A}_A^{-\top} (\boldsymbol{\beta} - \mathbf{a}_A)^\top

\end{align*}

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General Observer Design

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

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General Observer Design

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

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Use place instead of acker because we will need multi-output, and place is more general

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Integral Control with Observer

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

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Separation Principle

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

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Separation Principle

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

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\par\noindent Change of variables:

\[

\begin{pmatrix} x \\ e \end{pmatrix} = \begin{pmatrix} I & \mathbf{0} \\ I & -I \end{pmatrix}\begin{pmatrix} x \\ \hat{x} \end{pmatrix}

\]

gives

\begin{align*}

\begin{pmatrix} \dot{x} \\ \dot{e} \end{pmatrix} &= \begin{pmatrix} I & \mathbf{0} \\ I & -I \end{pmatrix}\begin{pmatrix} \dot{x} \\ \dot{\hat{x}} \end{pmatrix}

= \begin{pmatrix} I & \mathbf{0} \\ I & -I \end{pmatrix}\begin{pmatrix} A & -BK \\ LC & A-BK-LC \end{pmatrix} \begin{pmatrix} I & \mathbf{0} \\ I & -I \end{pmatrix}^{-1} \begin{pmatrix} x \\ e \end{pmatrix}

\end{align*}

Noting that

\[

\begin{pmatrix} I & \mathbf{0} \\ I & -I \end{pmatrix}^{-1} = \begin{pmatrix} I & \mathbf{0} \\ I & -I \end{pmatrix}

\]

gives

\[

\begin{pmatrix} \dot{x} \\ \dot{e} \end{pmatrix} = \begin{pmatrix} A-BK & BK \\ \mathbf{0} & A-LC \end{pmatrix} \begin{pmatrix} x \\ e \end{pmatrix}.

\]

Since $\text{det}\begin{pmatrix}M & Q \\ 0 & N\end{pmatrix} = \text{det}(M)\text{det}(N)$:

\[

\text{eig}\begin{pmatrix} A-BK & BK \\ \mathbf{0} & A-LC \end{pmatrix} = \text{eig}(A-BK)\cup \text{eig}(A-LC)

\]

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Separation Principle

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

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Observer Design for Linearized Systems

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

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Design Study A

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

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Python Implementation:

https://github.com/randybeard/controlbook_public/tree/master/_A_arm/python

ctrlObserver.py

hw13_armSim.py

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Design Study B

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

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Python Implementation:

https://github.com/randybeard/controlbook_public/tree/master/_B_pendulum/python

ctrlObserver.py

hw13_pendulumSim.py

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Design Study C

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

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Python Implementation:

https://github.com/randybeard/controlbook_public/tree/master/_C_satellite/python

ctrlObserver.py

hw13_satelliteSim.py