Difference between revisions of "Linear Systems"

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{{chaptertable left}}
 
{{chaptertable left}}
 
== Textbook Contents ==
 
== Textbook Contents ==
{{am05pdf|am07-linsys|20Jul07|Linear Systems|}}
+
{{am05pdf|am08-linsys|28Sep12|Linear Systems|}}
 
* 1. Basic Definitions
 
* 1. Basic Definitions
 
* 2. The Matrix Exponential
 
* 2. The Matrix Exponential
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== Supplemental Information ==
 
== Supplemental Information ==
 
* [[#Frequently Asked Questions|Frequently Asked Questions]]
 
* [[#Frequently Asked Questions|Frequently Asked Questions]]
 +
* [[#Errata|Errata]]
 
* Wikipedia entries: [http://en.wikipedia.org/wiki/Linear_system linear system], [http://en.wikipedia.org/wiki/LTI_system_theory LTI system theory]
 
* Wikipedia entries: [http://en.wikipedia.org/wiki/Linear_system linear system], [http://en.wikipedia.org/wiki/LTI_system_theory LTI system theory]
 
* [[#Additional Information|Additional Information]]
 
* [[#Additional Information|Additional Information]]
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This chapter introduces the analysis tools for linear input/output systems.
 
This chapter introduces the analysis tools for linear input/output systems.
 
<ol>
 
<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> <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 <amsmath>u(t) = \alpha u_1(t) + \beta u_2(t)</amsmath> with zero initial condition, the corresponding output will be <amsmath>y(t) = \alpha y_1(t) + \beta y_2(t)</amsmath>, where <amsmath>y_i</amsmath> is the output associated with the input <amsmath>u_i</amsmath>.  This propery is called linear ''superposition''.</p></li>
  
<li><p>A differential equation of the form
+
<li><p><span id=SISO>A differential equation of the form
 
<center><amsmath>
 
<center><amsmath>
 
\begin{aligned}
 
\begin{aligned}
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\end{aligned}
 
\end{aligned}
 
</amsmath></center>
 
</amsmath></center>
is a ''single-input, single-output'' (SISO) ''linear differential equation''.  Its solution can be written in terms of the ''[[matrix exponential]]''
+
is a ''single-input, single-output'' (SISO) ''linear differential equation''</span>.  Its solution can be written in terms of the ''[[matrix exponential]]''
 
<center>
 
<center>
 
<amsmath>
 
<amsmath>
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</p></li>
 
</p></li>
  
<li><p>A linear system <center><amsmath>\dot x = A x</amsmath></center> 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 strickly 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>A linear system <center><amsmath>\dot x = A x</amsmath></center> is [[Dynamic Behavior#stability|asymptotically stable]] if and only if all eigenvalues of <amsmath>A</amsmath> all have strictly negative real part and is {{unstable}} if any eigenvalue of <amsmath>A</amsmath> 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 <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>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>
 
<li><p>
 
The step response is characterized by the following parameters:
 
The step response is characterized by the following parameters:
* The ''steady state value'', <math>y_{ss}</math>, of a step response is the final level of the output, assuming it converges.
+
* The ''steady state value'', <amsmath>y_{ss}</amsmath>, of a step response is the final level of the output, assuming it converges.
* The ''rise time'', <math>T_r</math>, is the amount of time required for the signal to go from 10% of its final value to 90% of its final value.
+
* The ''rise time'', <amsmath>T_r</amsmath>, is the amount of time required for the signal to go from 10% of its final value to 90% of its final value.
* The ''overshoot'', <math>M_p</math>, is the percentage of the infal value by which the signal initially rises above the final value.
+
* The ''overshoot'', <amsmath>M_p</amsmath>, is the percentage of the initial value by which the signal initially rises above the final value.
* The ''settling time'', <math>T_s</math>, is  the amount of time required for the signal to stay within 5% of its final value for all future times.
+
* The ''settling time'', <amsmath>T_s</amsmath>, is  the amount of time required for the signal to stay within 5% of its final value for all future times.
 
</p></li>
 
</p></li>
  
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           t}}_{\text{steady state}},
 
           t}}_{\text{steady state}},
 
</amsmath></center>
 
</amsmath></center>
where <amsmath>\cos\omega t = \frac{1}{2} \left(e^{j \omega t} + e^{-j \omega t}\right)</amsmath> and <math>s = j \omega</math>.  The gain and phase of the frequency response are given by
+
where <amsmath>\cos\omega t = \frac{1}{2} \left(e^{j \omega t} + e^{-j \omega t}\right)</amsmath> and <amsmath>s = j \omega</amsmath>.  The gain and phase of the frequency response are given by
 
<center><amsmath>
 
<center><amsmath>
 
   \text{gain}(\omega) = \frac{A_y}{A_u} = M \qquad
 
   \text{gain}(\omega) = \frac{A_y}{A_u} = M \qquad
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\end{aligned}
 
\end{aligned}
 
</amsmath></center>
 
</amsmath></center>
is a single-input, single-output (SISO) nonlinear system.  It can be linearized about an equibrium point <math>x = x_e</math>, <math>u = u_e</math>, <math>y = y_e</math> by defining new variables
+
is a single-input, single-output (SISO) nonlinear system.  It can be linearized about an equibrium point <amsmath>x = x_e</amsmath>, <amsmath>u = u_e</amsmath>, <amsmath>y = y_e</amsmath> by defining new variables
 
<center>
 
<center>
 
<amsmath>z = x - x_e \qquad v = u - u_e \qquad w = y - h(x_e, u_e)</amsmath>.
 
<amsmath>z = x - x_e \qquad v = u - u_e \qquad w = y - h(x_e, u_e)</amsmath>.
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</amsmath></center>
 
</amsmath></center>
 
The equilibrium point for a nonlinear system is locally asymptotically stable if the real part of the eigenvalues of the linearization about that equilibrium point have strictly negative real part.
 
The equilibrium point for a nonlinear system is locally asymptotically stable if the real part of the eigenvalues of the linearization about that equilibrium point have strictly negative real part.
</p>
+
</p></li>
  
 
</ol>
 
</ol>
  
== Exercises ==
+
{{chaptertable begin}}
 +
{{chaptertable left}}
 +
 
 +
<!--
 +
The following exercises cover some of the topics introduced in this chapter.  Exercises marked with a * appear in the printed text.
 +
== Additional Exercises ==  
 
<ncl>Linear Systems Exercises</ncl>
 
<ncl>Linear Systems Exercises</ncl>
 +
-->
 
== Frequently Asked Questions ==
 
== Frequently Asked Questions ==
 
<ncl>Linear Systems FAQ</ncl>
 
<ncl>Linear Systems FAQ</ncl>
== Additional Information ==
+
== Errata ==
 +
<ncl>Linear Systems errata v2.11b</ncl>
 +
* [[:Category:Linear Systems errata|Full list of errata starting from first printing]]
 +
* {{submitbug}}
 +
Minor corrections:
 +
* After equation (5.31), the parameter <amsmath>c</amsmath> listed in the text should be <amsmath>b_g</amsmath>
 +
{{chaptertable right}}
  
 +
== MATLAB code ==
 +
The following MATLAB scripts are available for producing figures that appear in this chapter.
 +
* Figure 5.1: {{matlabfile|linsys|linearity.m}}, {{matlabfile|.|springmass.m}}
 +
* Figure 5.2: {{matlabfile|linsys|piecewise.m}}, {{matlabfile|.|springmass.m}}
 +
* Figure 5.5: {{matlabfile|linsys|modes.m}}
 +
* Figure 5.6: {{matlabfile|linsys|pulses.m}}
 +
* Figure 5.8: {{matlabfile|linsys|transient.m}}, {{matlabfile|.|springmass.m}}
 +
* Figure 5.9, 5.11: {{matlabfile|linsys|response.m}}
 +
* Figure 5.10: {{matlabfile|linsys|compartment_stepresp.m}}
 +
* Figure 5.12: {{matlabfile|linsys|opamp_bandbode.m}}
 +
* Figure 5.13: {{matlabfile|linsys|afm_freqresp.m}}, {{matlabfile|afm|afm_data.m}}
 +
* Figure 5.14: {{matlabfile|linsys|cruisepi_lin_nl_ch5.m}}, {{matlabfile|.|cruise_conpar.m}}, {{matlabfile|.|cruise_lin.m}}
 +
See the [[software|software page]] for more information on how to run these scripts.
 +
 +
== Additional Information ==
 
* [[Other Time Domain Specifications]]
 
* [[Other Time Domain Specifications]]
 +
{{chaptertable end}}

Latest revision as of 22:45, 4 November 2012

Prev: Dynamic Behavior Chapter 5 - Linear Systems Next: State Feedback

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.

Textbook Contents

Linear Systems (pdf, 28Sep12)

  • 1. Basic Definitions
  • 2. The Matrix Exponential
  • 3. Input/Output Response
  • 4. Linearization
  • 5. Further Reading
  • Exercises

Lecture Materials

Supplemental Information

Chapter Summary

This chapter introduces the analysis tools for linear input/output systems.

  1. 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 with zero initial condition, the corresponding output will be math, where math is the output associated with the input math. This propery is called linear superposition.

  2. A differential equation of the form

    math

    is a single-input, single-output (SISO) linear differential equation. Its solution can be written in terms of the matrix exponential

    math

    The solution to the differential equation is given by the convolution equation

    math

  3. A linear system

    math
    is asymptotically stable if and only if all eigenvalues of math all have strictly negative real part and is unstable if any eigenvalue of math has strictly positive real part. For systems with eigenvalues having zero real-part, stability is determined by using the 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.

  4. 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 and the frequency response, which is the response of the system to a sinusoidal input at a given frequency.

  5. The step response is characterized by the following parameters:

    • The steady state value, math, of a step response is the final level of the output, assuming it converges.
    • The rise time, math, is the amount of time required for the signal to go from 10% of its final value to 90% of its final value.
    • The overshoot, math, is the percentage of the initial value by which the signal initially rises above the final value.
    • The settling time, math, is the amount of time required for the signal to stay within 5% of its final value for all future times.

  6. The frequency response is given by

    math

    where math and math. The gain and phase of the frequency response are given by

    math

  7. A nonlinear system of the form

    math

    is a single-input, single-output (SISO) nonlinear system. It can be linearized about an equibrium point math, math, math by defining new variables

    math.

    The dynamics of the system near the equilibrium point can then be approximated by the linear system

    math

    where

    math

    The equilibrium point for a nonlinear system is locally asymptotically stable if the real part of the eigenvalues of the linearization about that equilibrium point have strictly negative real part.

Frequently Asked Questions

Errata

Minor corrections:

  • After equation (5.31), the parameter math listed in the text should be math

MATLAB code

The following MATLAB scripts are available for producing figures that appear in this chapter.

See the software page for more information on how to run these scripts.

Additional Information