System Modeling

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Chapter 3 - System Modeling

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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 dynamical 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 present some basic material on two specific methods commonly used in feedback and control systems: differential equations and difference equations.

Chapter Summary

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
   ${\begin{aligned}{\dot {x}}&=f(x,u)\\y&=h(x,u)\end{aligned}}$		${\begin{aligned}{\dot {x}}&=Ax+Bu\\y&=Cx+Du\end{aligned}}$

   where $x$ represents the state of the system, ${\dot {x}}$ 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
   ${\begin{aligned}x_{k+1}&=f(x+k,u_{k})\\y_{k}&=h(x_{k},u_{k})\end{aligned}}$		${\begin{aligned}x_{k+1}&=Ax_{k}+Bu_{k}\\y_{k}&=Cx_{k}+Du_{k}\end{aligned}}$

   where $x_{k}$ 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:

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

8. Modeling examples (wiki-based):

   Cruise control

Teaching Materials None available Additional Exercises None available Frequently Asked Questions Question: How can I go from a continuous time linear ODE to a discrete time representation? Question: How can we tell from the phase plots if the system is oscillating? Question: How do you know when your model is sufficiently complex? Question: In the predator prey example, where is the fox birth rate term? Question: What is "closed form"? Question: What is a state? How does one determine what is a state and what is not? Question: What is a stochastic system? Question: What is the advantage of having a model? Question: Why does the effective service rate f(x) go to zero when x = 0 in the example on queuing systems? Question: Why isn't there a term for the rabbit death rate besides being killed by the foxes? Errata None reported

Python Code The following Python scripts are available for producing figures that appear in this chapter. Figure 3.2: Illustration of a state model Figure 3.4: Input/output response of a linear system Figure 3.8: Discrete-time simulation of the predator–prey model Figure 3.11: Simulation of the forced spring–mass system with different simulation time constants Figure 3.12: Frequency response computed by measuring the response of individual sinusoids Figure 3.22: Queuing dynamics Figure 3.24: Consensus protocols for sensor networks Figure 3.26: The repressilator genetic regulatory network Figure 3.28: Response of a neuron to a current input See the software page for more information on how to run these scripts. Additional Information First Edition Supplemental Information