In this part of the book we will explore the use of MATLAB for dynamical systems and control system design in a number of technologies. In each area we will derive the equations of motion for the system. A system is defined by its state equations, states and parameters. The equations of motion are the equations of the states of a system. The state variables are the set of variables that evolve with time that completely define the current state of the system and allow for future prediction of the state without any knowledge of the past. We also need parameters that are independent of the states to fully define the system along with the inputs to the system. The state vector will always be represented by an n-by-1 MATLAB array.

In the equations that we present, we will use the dot notation for derivatives, i.e.    State equations are of the form    x is the state and is an n × 1 vector represented by an n row by 1 column MATLAB array. u is the input matrix and is m × 1. y is the measurement. a relates the state to the state derivative and is an n × n array. b is the input array and is n × m where the number of inputs, u, is m. c relates the state the to measurement and d relates the input to the measurement.

We are not going to delve into control theory in detail. That would require a complete textbook by itself, or many textbooks if you wanted to explore control system design in depth. We will provide an intuitive approach to allow you to get control systems up and running quickly without too much code!