System Modeling

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

   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