Difference between revisions of "System Modeling"

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{{chaptertable left}}
 
{{chaptertable left}}
 
== Textbook Contents ==
 
== Textbook Contents ==
{{am05pdf|am07-modeling|20Jul07|System Modeling|}}
+
{{am05pdf|am08-modeling|28Sep12|System Modeling|}}
 
* 1. Modeling Concepts
 
* 1. Modeling Concepts
 
* 2. State Space Models
 
* 2. State Space Models
 +
** [[Balance systems]]
 +
** [[Predator prey]] (discrete time)
 
* 3. Modeling Methodology
 
* 3. Modeling Methodology
 
* 4. Modeling Examples
 
* 4. Modeling Examples
 +
** [[Vehicle steering]]
 +
** [[Queuing systems]]
 +
** [[Vectored thrust aircraft]]
 +
** [[Biological circuits]]
 
* 5. Further Reading
 
* 5. Further Reading
 
* Exercises
 
* Exercises
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== Supplemental Information ==
 
== Supplemental Information ==
 
* [[#Frequently Asked Questions|Frequently Asked Questions]]
 
* [[#Frequently Asked Questions|Frequently Asked Questions]]
 +
* [[#Exercises|Exercises]]
 +
* [[#MATLAB code|MATLAB code]]
 +
* [[#Errata|Errata]]
 
* Wikipedia entries: [[wikipedia:Mathematical_model|model]], [[wikipedia:Ordinary_differential_equation|ODEs]], [[wikipedia:State space (controls)|state space]]
 
* Wikipedia entries: [[wikipedia:Mathematical_model|model]], [[wikipedia:Ordinary_differential_equation|ODEs]], [[wikipedia:State space (controls)|state space]]
 
* [[#Additional Information|Additional Information]]
 
* [[#Additional Information|Additional Information]]
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This chapter provides an overview of the process and tools for modeling dynamical systems.
 
This chapter provides an overview of the process and tools for modeling dynamical systems.
 
<ol>
 
<ol>
<li> <p>A ''model'' is a mathematical representation of a system that can be used to answer question about that system.  The choice of the model depends on the questions one wants to ask.  Models for control systems are typically input/output models and combine techniques from mechanics and electrical engineering.  </p>
+
<li> <p>A ''model'' is a mathematical representation of a system that can be used to answer question about that system.  The choice of the model depends on the questions one wants to ask.  Models for control systems are typically input/output models and combine techniques from mechanics and electrical engineering.  </p></li>
  
<li><p> The ''state'' of a system is a collection of variables that summarize the past history of the system for the purpose of predicting the future.  A ''state space model'' is one that describe how the state of a system evolves over time.</p>
+
<li><p> The ''state'' of a system is a collection of variables that summarize the past history of the system for the purpose of predicting the future.  A ''state space model'' is one that describes how the state of a system evolves over time.</p></li>
  
<li><p>We can model the evolution of the state using a ''ordinary differential equations'' of the form
+
<li><p>We can model the evolution of the state using ''ordinary differential equations'' of the form
 
<center>
 
<center>
 
{|
 
{|
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|}
 
|}
 
</center>
 
</center>
where <amsmath>x</amsmath> represents the state of the system, <amsmath>\dot x</amsmath> is the time derivative of the state, <math>u</math> are the external inputs and <math>y</math> are the measured outputs.  For the linear form, <math>A</math>, <math>B</math>, <math>C</math> and <math>D</math> are matrices of the appropriate dimension and the model is ''linear time invariant'' (LTI).</p>
+
where <amsmath>x</amsmath> represents the state of the system, <amsmath>\dot x</amsmath> is the time derivative of the state, ''u'' are the external inputs and ''y'' are the measured outputs.  For the linear form, ''A'', ''B'', ''C'' and ''D'' are matrices of the appropriate dimension and the model is ''linear time invariant'' (LTI).</p></li>
  
 
<li><p>Another class of models for feedback and control systems is a ''difference equation'' of the form
 
<li><p>Another class of models for feedback and control systems is a ''difference equation'' of the form
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|}
 
|}
 
</center>
 
</center>
where <math>x_k</math> represents the state of the system at the <math>k</math>th time instant. </p>
+
where {{#tag:math|x_k}} represents the state of the system at the ''k''th time instant. </p></li>
  
<li><p>Three common questions that can be answered using state space models are (1) how the system state evolves from a given initial condition, (2) the stability of an equilibrium point from nearby initial conditions and (3) the steady state response of the system to sinusoidal forcing at different frequencies.  </p>
+
<li><p>Three common questions that can be answered using state space models are (1) how the system state evolves from a given initial condition, (2) the stability of an equilibrium point from nearby initial conditions and (3) the steady state response of the system to sinusoidal forcing at different frequencies.  </p></li>
  
<li><p>Models can be constructed from experiments by measuring the response of a system and determining the parameters in the model that correspond to features in the response.  Examples include measuring the period of oscillation, the rate of damping and the steady state amplitude of the response of a system to a step input.</p>
+
<li><p>Models can be constructed from experiments by measuring the response of a system and determining the parameters in the model that correspond to features in the response.  Examples include measuring the period of oscillation, the rate of damping and the steady state amplitude of the response of a system to a step input.</p></li>
  
<li><p>Schematic and block diagrams are common tools for modeling large, complex systems.  The following symbols are commonly used for modeling (linear) control systems:
+
<li><p>Schematic and block diagrams are common tools for modeling large, complex systems.  The following symbols are some of the ones commonly used for modeling control systems:
 
<center>[[Image:Modeling bdsym.png]]</center>
 
<center>[[Image:Modeling bdsym.png]]</center>
Computer packages such as [http://www.ni.com/labview LabView], [http://www.mathworks.com MATLAB/SIMULINK] and [http://www.modelica.org Modelica] can be used to construct models for complex, multi-component systems.</p>
+
Computer packages such as [http://www.ni.com/labview LabView], [http://www.mathworks.com MATLAB/SIMULINK] and [http://www.modelica.org Modelica] can be used to construct models for complex, multi-component systems.</p></li>
  
 
<li><p>Modeling examples (wiki-based):
 
<li><p>Modeling examples (wiki-based):
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| width=40% | <ncl>Modeling examples</ncl>
 
| width=40% | <ncl>Modeling examples</ncl>
 
|}
 
|}
</p>
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</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>System Modeling Exercises</ncl>
 
<ncl>System Modeling Exercises</ncl>
 
* Exercise: Consider the vehicle steering model in Section {{chnum|System Modeling}}.4. Derive the model for a vehicle with rear-wheel steering.
 
* Exercise: Consider the vehicle steering model in Section {{chnum|System Modeling}}.4. Derive the model for a vehicle with rear-wheel steering.
 
 
== Frequently Asked Questions ==
 
== Frequently Asked Questions ==
 
<ncl>System Modeling FAQ</ncl>
 
<ncl>System Modeling FAQ</ncl>
 +
{{chaptertable right}}
 +
== Errata ==
 +
<ncl>System Modeling errata v2.11b</ncl>
 +
* [[:Category:System Modeling errata|Full list of errata starting from first printing]]
 +
* {{submitbug}}
 +
<!-- Additional small typos: -->
 +
 +
== MATLAB code ==
 +
The following MATLAB scripts are available for producing figures that appear in this chapter.
 +
* Figure 2.2: {{matlabfile|modeling|springmass_nlplots.m}}, {{matlabfile|modeling|nlspringmass.m}}
 +
* Figure 2.4: {{matlabfile|modeling|ioresp.m}}
 +
* Figure 2.7: {{matlabfile|modeling|predprey_discrete.m}}
 +
* Figure 2.9: {{matlabfile|modeling|springmass_modeling.m}}
 +
* Figure 2.10: {{matlabfile|modeling|fresp.m}}
 +
* Figure 2.14: {{matlabfile|modeling|springmass_modeling.m}}
 +
* Figure 2.19: {{matlabfile|modeling|queue_sslength.m}}, {{matlabfile|modeling|queue_overload.m}}, {{matlabfile|modeling|queue_model.m}}
 +
* Figure 2.21: {{matlabfile|modeling|consensus.m}}
 +
* Figure 2.23: {{matlabfile|modeling|repressilator_plot.m}}, {{matlabfile|modeling|repressilator.m}}
 +
See the [[software|software page]] for more information on how to run these scripts.
 +
 
== Additional Information ==
 
== Additional Information ==
 
* [http://www.engin.umich.edu/group/ctm Control tutorials for MATLAB] (U. Michigan)
 
* [http://www.engin.umich.edu/group/ctm Control tutorials for MATLAB] (U. Michigan)
 
* [http://math.rice.edu/~dfield/index.html ODE tools for MATLAB] (Rice) - includes software for generating phase plots.
 
* [http://math.rice.edu/~dfield/index.html ODE tools for MATLAB] (Rice) - includes software for generating phase plots.
 +
 +
{{chaptertable end}}

Latest revision as of 21:40, 23 November 2012

Prev: Introduction Chapter 2 - System Modeling Next: Examples

A model is a precise representation of a system's dynamics used to answer questions via analysis and simulation. The model we choose depends on the questions we wish to answer, and so there may be multiple models for a single physical system, with different levels of fidelity depending on the phenomena of interest. In this chapter we provide an introduction to the concept of modeling, and provide some basic material on two specific methods that are commonly used in feedback and control systems: differential equations and difference equations.

Textbook Contents

System Modeling (pdf, 28Sep12)

Teaching Materials

Supplemental Information

Chapter Summary

This chapter provides an overview of the process and tools for modeling dynamical systems.

  1. A model is a mathematical representation of a system that can be used to answer question about that system. The choice of the model depends on the questions one wants to ask. Models for control systems are typically input/output models and combine techniques from mechanics and electrical engineering.

  2. The state of a system is a collection of variables that summarize the past history of the system for the purpose of predicting the future. A state space model is one that describes how the state of a system evolves over time.

  3. We can model the evolution of the state using ordinary differential equations of the form

    Nonlinear      Linear
    math math
    where math represents the state of the system, math is the time derivative of the state, u are the external inputs and y are the measured outputs. For the linear form, A, B, C and D are matrices of the appropriate dimension and the model is linear time invariant (LTI).

  4. Another class of models for feedback and control systems is a difference equation of the form

    Nonlinear      Linear
    math math
    where represents the state of the system at the kth time instant.

  5. Three common questions that can be answered using state space models are (1) how the system state evolves from a given initial condition, (2) the stability of an equilibrium point from nearby initial conditions and (3) the steady state response of the system to sinusoidal forcing at different frequencies.

  6. Models can be constructed from experiments by measuring the response of a system and determining the parameters in the model that correspond to features in the response. Examples include measuring the period of oscillation, the rate of damping and the steady state amplitude of the response of a system to a step input.

  7. Schematic and block diagrams are common tools for modeling large, complex systems. The following symbols are some of the ones commonly used for modeling control systems:

    Modeling bdsym.png
    Computer packages such as LabView, MATLAB/SIMULINK and Modelica can be used to construct models for complex, multi-component systems.

  8. Modeling examples (wiki-based):

The following exercises cover some of the topics introduced in this chapter. Exercises marked with a * appear in the printed text.

Additional Exercises

  • Exercise: Consider the vehicle steering model in Section 2.4. Derive the model for a vehicle with rear-wheel steering.

Frequently Asked Questions

Errata

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