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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 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.
This chapter provides an overview of the process and tools for modeling dynamical systems.
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.
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.
We can model the evolution of the state using a ordinary differential equations of the form
Another class of models for feedback and control systems is a difference equation of the form
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.
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.
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 and Modelica can be used to construct models for complex, multi-component systems.
Modeling examples (wiki-based):
- Exercise: Insect flight control modeling
- Exercise: Modeling and simulation of an exothermic reaction
- Exercise: Properties of linear discrete time systems
- Exercise: Traffic light simulation
- Exercise: Vehicle powertrain modeling and cruise control
- Exercise: Vehicle suspension system modeling and input response
- Exercise: Consider the vehicle steering model in Section 2.4. Derive the model for a vehicle with rear-wheel steering.
Frequently Asked Questions
- FAQ: Can we get more information about state space formulation?
- FAQ: How can I go from a continuous linear ODE to a discrete representation?
- FAQ: How can we tell from the phase plots if the system is oscillating?
- FAQ: How do we learn how to translate MATLAB equations into the Simulink diagrams?
- FAQ: How do you know when your model is sufficiently complex?
- FAQ: In Exercise 2.10, what do the variables phi represent?
- FAQ: In the predator prey example, where is the fox birth rate term?
- FAQ: What is a state? How does one determine what is a state and what is not?
- FAQ: What is a stochastic system?
- FAQ: What is closed form?
- FAQ: What is the advantage of having a model?
- FAQ: Why does the effective service rate f(x) go to zero when x = 0 in Example 2.10 on queuing systems?
- FAQ: Why is the parameter "a" in the predator-prey problem used as both death of rabbit and birth of foxes?
- FAQ: Why isn't there a term for the rabbit death rate besides being killed by the foxes?
- Errata: In discussion on disturbance signals, "can" should be "cannot"
- Errata: In Exercises 2.4 and 5.9, the (2,1) entry of the dynamics matrix should be ab-b
- Errata: In Exercise 2.7, reference to inverted pendulum may be confusing
- Errata: In Example 2.1, the state space equations for the balance system are missing a divisor of J t in one of the terms
- Errata: In Figure 2.7, the coefficients are for an update period of one day
- Errata: In Figure 2.9, "solid" and "dashed" should be swapped
- Errata: In Example 2.5, expression for the step response is missing some factors
- Errata: In Example 2.7, q should be replaced by p (three places) and theta (one place)
- Errata: In Figure 2.16, theta should be O
- Errata: In Exercise 2.4, a = 0.25 should be a = 0.75