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��Chapter 12�Integrator with �Full State Feedback

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Course Overview

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

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State feedback with and without Integrator

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

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State Feedback �without integrator

State Feedback �with integrator

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State feedback with Integrator

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

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State feedback with Integrator

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

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Controllability

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

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Pole Placement with Integrator

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

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DC-Gain

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

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\begin{align*}

\begin{pmatrix} \dot{x} \\ \dot{x}_I \end{pmatrix} &= \begin{pmatrix} A & \mathbf{0} \\ -C_r & 0\end{pmatrix} \begin{pmatrix} x \\ x_I \end{pmatrix} - \begin{pmatrix} B \\ 0 \end{pmatrix} \begin{pmatrix} K & k_I \end{pmatrix}\begin{pmatrix} x \\ x_I \end{pmatrix} + \begin{pmatrix} \mathbf{0} \\ 1 \end{pmatrix} r \\

&= \begin{pmatrix} A-BK & -Bk_I \\ -C_r & 0\end{pmatrix}\begin{pmatrix} x \\ x_I \end{pmatrix} +

\begin{pmatrix} \mathbf{0} \\ 1 \end{pmatrix} r

\end{align*}

where the reference output equation is

\[

y_r = \begin{pmatrix} C_r & 0 \end{pmatrix}

\begin{pmatrix} x \\ x_I \end{pmatrix}

\]

Therefore, the transfer function from $R(s)$ to $Y_r(s)$ is

\[

Y_r(s) = \begin{pmatrix} C_r & 0 \end{pmatrix} \left(sI - \begin{pmatrix} A-BK & -Bk_I \\ -C_r & 0\end{pmatrix}\right)^{-1} \begin{pmatrix} \mathbf{0} \\ 1 \end{pmatrix} R(s)

\]

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DC-Gain

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

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Summary: State Feedback with Integrator

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

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{\color{blue}

\par\noindent{\bf Problem Statement:}

\par\noindent For the open-loop system

\begin{align*}

\dot{x} &= Ax + Bu \\

y_r &= C_r x

\end{align*}

Use state feedback control with an integrator

\[

u(t) = -Kx(t) - k_I\int_{-\infty}^t (r(\tau)-y_r(\tau))d\tau

\]

Find $K_1=\begin{pmatrix} K & k_I \end{pmatrix}$ to place the closed-loop poles at

$p_1, p_2, \dots, p_n$, $p_I$.

}

---

\par\noindent{\bf Step 1.} Augment the state evolution equation as

\[

\begin{pmatrix}\dot{x} \\ \dot{x}_I \end{pmatrix}

= \underbrace{\begin{pmatrix} A & \mathbf{0} \\ -C_r & 0 \end{pmatrix}}_{A_1} \begin{pmatrix}x \\ x_I\end{pmatrix} + \underbrace{\begin{pmatrix}B \\ 0\end{pmatrix}}_{B_1}u + \begin{pmatrix}\mathbf{0} \\ 1\end{pmatrix}r

\]

�

\par\noindent{\bf Step 2.} Use pole placement:

\[

K_1 = \text{place}\left(A_1, B_1, [p_1, \dots, p_n, p_I]\right)

\]

�

\par\noindent{\bf Step 3.} Find the gains $K$ and $k_I$ as

\begin{align*}

K &= K_1(1:n) \\

k_I &= K_1(n+1).

\end{align*}

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Summary: State Feedback with Integrator

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

ctrlStateFeedbackIntegrator.py

hw12_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

ctrlStateFeedbackIntegrator.py

hw12_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

ctrlStateFeedbackIntegrator.py

hw12_satelliteSim.py