Output Feedback - Revision history

Murray at 16:29, 24 November 2024

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Murray at 13:28, 7 April 2024

2024-04-07T13:28:01Z

Murray at 05:15, 7 October 2021

2021-10-07T05:15:00Z

Murray at 21:33, 28 August 2021

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Murray at 21:27, 28 August 2021

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Murray at 05:48, 28 December 2020

2020-12-28T05:48:48Z

Murray at 00:48, 28 December 2020

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Murray at 00:41, 28 December 2020

2020-12-28T00:41:29Z

Murray: Created page with "{{Chapter |Chapter number=8 |Previous chapter=State Feedback |Next chapter=Transfer Functions }}"

2020-12-28T00:09:59Z

Created page with "{{Chapter |Chapter number=8 |Previous chapter=State Feedback |Next chapter=Transfer Functions }}"

New page

{{Chapter
|Chapter number=8
|Previous chapter=State Feedback
|Next chapter=Transfer Functions
}}

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 |Chapter contents=# Frequency Domain Modeling
 # Determining the Transfer Function
 # Laplace Transforms
 # Block Diagrams and Transfer Functions
 # Zero Frequency Gain, Poles, and Zeros
 # The Bode Plot
 # Further Reading
 :: Exercises

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 :: Exercises
 }}
 == Chapter Summary ==
 <ol>
 <li> <p> A linear system with dynamics

← Older revision		Revision as of 05:15, 7 October 2021
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	:: Exercises		:: Exercises
	}}		}}
		+
		+	== Chapter Summary ==
		+
		+	This chapter describes how to estimate the state of a system through measurements of its inputs and outputs:
		+	<ol>
		+	<li> <p> A linear system with dynamics
		+	<center><math>
		+	\begin{aligned}
		+	\dot x &= A x + B u & \quad x &\in {R}^n, u \in {R} \\
		+	y &= C x + D u & y &\in {R}
		+	\end{aligned}
		+	</math></center>
		+	is said to be ''observable'' if we can determine the state of the system through measurements of the input <math>u(t)</math> and the output <math>y(t)</math> over a time interval <math>[0, T]</math>.
		+	</p></li>
		+
		+	<li><p>The ''observability matrix'' for a linear system is given by
		+	<center><math>
		+	W_o = \begin{bmatrix} C \\ CA \\ \vdots \\ C A^{n-1} \end{bmatrix}.
		+	</math></center>
		+	A linear system is observable if and only if the observability matrix <math>W_o</math> is full rank. Systems that are not reachable have "hidden" states that cannot be determined by looking at the inputs and outputs.
		+	</p></li>
		+
		+	<li><p>A linear system of the form
		+	<center><math>
		+	\begin{aligned}
		+	\frac{dz}{dt} &= \begin{bmatrix}
		+	-a_1 & 1 & 0 & \cdots & 0 \\
		+	-a_2 & 0 & 1 & & 0\\
		+	\vdots \\
		+	-a_{n-1} & 0 & 0 & & 1 \\
		+	-a_n & 0 & 0 & & 0\\
		+	\end{bmatrix} z+ \begin{bmatrix}
		+	b_1 \\ b_2 \\ \vdots \\ b_{n-1} \\ b_n
		+	\end{bmatrix} u \\
		+	y &= \begin{bmatrix}
		+	1 & 0 & 0 \cdots & 0
		+	\end{bmatrix} z+ Du.
		+	\end{aligned}
		+	</math></center>
		+	is said to be in ''observable canonical form''. A system in this form is always observable and has a characteristic polynomial given by
		+	<center><math>
		+	\det(sI-A) = s^n+a_1 s^{n-1} + \cdots + a_{n-1}s + a_n,
		+	</math></center>
		+	An observable linear system can be transformed into observable canonical form through the use of a coordinate transformation <math>z = T x</math>.
		+	</p></li>
		+
		+	<li><p>An ''observer'' is a dynamical system that estimates the state of another system through measurement of inputs and outputs. For a linear system, the observer given by
		+	<center><math>
		+	\frac{d\hat x}{dt} = A \hat x + B u + L(y - C \hat x)
		+	</math></center>
		+	generates an estimate of the state that converges to the actual state if <math>A - LC</math> is has eigenvalues with negative real part.
		+	If a system is observable, then there exists a an ''observer gain'' <math>L</math> such that the observer error is governed by a linear differential equation with an arbitrary characteristic polynomial. Hence the eigenvalues of the error dynamics for an observable linear system can be placed arbitrarily through the use of an appropriate observer gain.
		+	</p></li>
		+
		+	<li><p>A state feedback controller and linear observer can be combined to form a stabilizing controller for a reachable and observable linear system by using the estimate of the state in the feedback control law. The resulting controller is given by
		+	<center><math>
		+	\begin{aligned}
		+	\frac{d\hat x}{dt} &= A \hat x + B u + L(y - C \hat x) \\
		+	u &= -K \hat x + K_r r
		+	\end{aligned}
		+	</math></center>
		+	</p></li>
		+
		+	<li><p>A discrete time, linear process with noise is given by
		+	<center><math>
		+	\begin{aligned}
		+	x(k+1) &= A x(k) + B u(k) + v(k) &\quad x &\in R^n, u \in R \\
		+	y(k) &= C x(k) + D u(k) + w(k) & y &\in R
		+	\end{aligned}
		+	</math></center>
		+	where <math>v</math> is a vector, white, Gaussian random process with mean 0, autocovariance <math>R_w</math>, <math>w</math> is a white, Guassian random process with mean 0, variance <math>R_v</math>. We take the initial condition to be random with mean 0 and covariance <math>P_0</math>. The optimal estimator is given by
		+	<center><math>
		+	\hat x(k+1) = A \hat x(k) + B u(k) + L(y(k) - C \hat x(k))
		+	</math></center>
		+	where the observer gain satisfies
		+	<center><math>
		+	\begin{aligned}
		+	P(k+1) &= A^T P(k) A^T + R_v - A P(k)C^T(R_w + C P C^T)^{-1} C P^T(k) A^T \\
		+	P(0) &= P_0 \\
		+	L &= A^T P(k) C^T (R_w + C P C^T)^{-1}
		+	\end{aligned}
		+	</math></center>
		+	This estimator is an example of a ''Kalman filter''.
		+	</p></li>
		+	</ol>
		+
		+	{{Chapter footer}}

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 |Previous chapter=State Feedback
 |Next chapter=Transfer Functions
-|First edition URL=
+|First edition URL=https://www.cds.caltech.edu/~murray/amwiki/index.php?title=Output_Feedback#Frequently_Asked_Questions
 |Chapter summary=In this chapter we show how to use output feedback to modify the dynamics of the system through the use of observers. We introduce the concept of observability and show that if a system is observable, it is possible to recover the state from measurements of the inputs and outputs to the system. We then show how to design a controller with feedback from the observer state. A general controller with two degrees of freedom is obtained by adding feedforward. We illustrate by outlining a controller for a nonlinear system that also employs gain scheduling.
 |Chapter contents=# Frequency Domain Modeling

@@ Line 4: / Line 4: @@
 |Previous chapter=State Feedback
 |Next chapter=Transfer Functions
 |Chapter summary=In this chapter we show how to use output feedback to modify the dynamics of the system through the use of observers. We introduce the concept of observability and show that if a system is observable, it is possible to recover the state from measurements of the inputs and outputs to the system. We then show how to design a controller with feedback from the observer state. A general controller with two degrees of freedom is obtained by adding feedforward. We illustrate by outlining a controller for a nonlinear system that also employs gain scheduling.
 |Chapter contents=# Frequency Domain Modeling

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 {{Chapter
 |Chapter number=8
 |Previous chapter=State Feedback
 |Next chapter=Transfer Functions