The limitation of transfer function representation becomes obvious as we tackle more complex problems. For complex systems with multiple inputs and outputs, transfer function matrices can become very clumsy. In the so-called modern control, the method of choice is state-space or state variables in the time domain – essentially a matrix representation of the model equations. The formulation allows us to make use of theories in linear algebra and differential equations. It is always a mistake to tackle modern control without a firm background in these mathematical topics. For this reason, we will not overreach by doing both the mathematical background and the control together. Without a formal mathematical framework, the explanation is made by means of examples as much as possible. The actual state-space control has to be delayed until after we tackle classical transfer function feedback systems.

What Are We Up to?

• Learning how to write the state-space representation of a model.

• Understanding the how a state-space representation is related to the transfer function representation.

State-Space Models

Just as we are feeling comfortable with transfer functions, we now switch gears totally. Nevertheless, we are still working with linearized differential equation models in this chapter. Whether we have a high-order differential equation or multiple equations, we can always rearrange them into a set of first-order differential equations. Bold statements indeed! We will see that when we go over the examples.