Linear Systems - Revision history

Murray at 16:28, 24 November 2024

2024-11-24T16:28:21Z

Murray: /* Chapter Summary */

2024-04-07T13:27:25Z

Chapter Summary

Murray at 21:33, 28 August 2021

2021-08-28T21:33:02Z

Murray at 21:26, 28 August 2021

2021-08-28T21:26:50Z

Murray at 05:48, 28 December 2020

2020-12-28T05:48:20Z

Murray: /* Chapter Summary */

2020-12-28T04:21:33Z

Chapter Summary

Murray: /* Chapter Summary */

2020-12-28T04:20:18Z

Chapter Summary

Murray: /* Chapter Summary */

2020-12-28T04:15:47Z

Chapter Summary

Murray at 00:40, 28 December 2020

2020-12-28T00:40:30Z

Murray at 00:16, 28 December 2020

2020-12-28T00:16:07Z

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 |Chapter summary=Previous chapters have focused on the dynamics of a system with relatively little attention to the inputs and outputs.  This chapter gives an introduction to input/output behavior for linear systems and shows how a nonlinear system can be approximated near an equilibrium point by a linear model.
 |Chapter contents=# Reachability
 # Stabilization by State Feedback
 # Design Considerations
 # Integral Action
 # Linear Quadratic Regulators
 # Further Reading

@@ Line 25: / Line 25: @@
 }}
 == Chapter Summary ==
 <ol>
 <li> <p>A ''linear system'' is one in which the output is jointly linear in the intitial condition for the system and the input to the system.  In particular, a linear system has the property that if we apply an input <math>u(t) = \alpha u_1(t) + \beta u_2(t)</math> with zero initial condition, the corresponding output will be <math>y(t) = \alpha y_1(t) + \beta y_2(t)</math>, where <math>y_i</math> is the output associated with the input <math>u_i</math>.  This propery is called linear ''superposition''.</p></li>

@@ Line 4: / Line 4: @@
 |Previous chapter=Dynamic Behavior
 |Next chapter=State Feedback
-|First edition URL=
+|First edition URL=https://www.cds.caltech.edu/~murray/amwiki/index.php?title=Linear_Systems#Frequently_Asked_Questions
 |Chapter summary=Previous chapters have focused on the dynamics of a system with relatively little attention to the inputs and outputs.  This chapter gives an introduction to input/output behavior for linear systems and shows how a nonlinear system can be approximated near an equilibrium point by a linear model.
 |Chapter contents=# Reachability

@@ Line 4: / Line 4: @@
 |Previous chapter=Dynamic Behavior
 |Next chapter=State Feedback
 |Chapter summary=Previous chapters have focused on the dynamics of a system with relatively little attention to the inputs and outputs.  This chapter gives an introduction to input/output behavior for linear systems and shows how a nonlinear system can be approximated near an equilibrium point by a linear model.
 |Chapter contents=# Reachability

@@ Line 1: / Line 1: @@
 {{Chapter
 |Chapter number=6
 |Previous chapter=Dynamic Behavior
 |Next chapter=State Feedback

@@ Line 96: / Line 96: @@
    B &= \left.\frac{\partial f(x, u)}{\partial u}\right|_{x_\text{e}, u_\text{e}} \\
    C &= \left.\frac{\partial h(x, u)}{\partial x}\right|_{x_\text{e}, u_\text{e}} &\quad
-   D &= \left.\frac{\partial y(x, u)}{\partial u}\right|_{x_e, u_e}
+   D &= \left.\frac{\partial y(x, u)}{\partial u}\right|_{x_\text{e}, u_\text{e}}
 \end{aligned}
 </math></center>

@@ Line 75: / Line 75: @@
 <center><math>
 \begin{aligned}
-   \dot x &= f(x, u) &\quad x &\in R^n, u \in R \\
+   \dot x &= f(x, u) &\quad x &\in \mathbb{R}^n, u \in \mathbb{R} \\
-   y &= h(x, u) & y &\in R
+   y &= h(x, u) & y &\in \mathbb{R}
 \end{aligned}
 </math></center>
 is a single-input, single-output (SISO) nonlinear system.  It can be linearized about an equibrium point <math>x = x_\text{e}</math>, <math>u = u_\text{e}</math>, <math>y = y_\text{e}</math> by defining new variables
 <center><math>
-z = x - x_e \qquad v = u - u_e \qquad w = y - h(x_e, u_e).
+z = x - x_\text{e}e \qquad v = u - u_\text{e} \qquad w = y - h(x_\text{e}, u_\text{e}).
 </math></center>
 The dynamics of the system near the equilibrium point can then be approximated by the linear system

@@ Line 48: / Line 48: @@
 is [[Dynamic Behavior#stability|asymptotically stable]] if and only if all eigenvalues of <math>A</math> all have strictly negative real part and is [[Dynamic Behavior#stability|unstable]] if any eigenvalue of <math>A</math> has strictly positive real part.  For systems with eigenvalues having zero real-part, stability is determined by using the ''[http://en.wikipedia.org/wiki/Jordan_normal_form Jordan normal form]'' associated with the matrix.  A system with eigenvalues that have no strictly positive real part is stable if and only if the Jordan block corresponding to each eigenvalue with zero part is a scalar (1x1) block.</p></li>
-<li><p>The input/output response of a (stable) linear system contains a transient region portion, which eventually decays to zero, and a steady state portion, which persists over time.  Two special responses are the ''step response'', which is the output corresponding to an step input applied at <amsmath>t = 0</amsmath> and the ''frequency response'', which is the response of the system to a sinusoidal input at a given frequency.</p></li>
+<li><p>The input/output response of a (stable) linear system contains a transient region portion, which eventually decays to zero, and a steady state portion, which persists over time.  Two special responses are the ''step response'', which is the output corresponding to an step input applied at <math>t = 0</math> and the ''frequency response'', which is the response of the system to a sinusoidal input at a given frequency.</p></li>
 <li><p>

@@ Line 1: / Line 1: @@
 {{Chapter
 |Chapter number=6
 |Next chapter=State Feedback
 |Chapter summary=Previous chapters have focused on the dynamics of a system with relatively little attention to the inputs and outputs.  This chapter gives an introduction to input/output behavior for linear systems and shows how a nonlinear system can be approximated near an equilibrium point by a linear model.
-|Chapter contents=# Basic Definitions
+|Chapter contents=# Reachability
-# The Matrix Exponential
+#* Definition of Reachability
-# Input/Output Response
+#* Testing for Reachability
-# Linearization<br>
+#* Reachable Canonical Form
-# Further Reading</br>
+# Stabilization by State Feedback
 :: Exercises
 }}