State-space feedback linearization

The limitation of transfer function representation becomes plain 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 modem control, the method of choice is state space or state variables in 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 modem 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, we will put the explanation in examples as much as possible. The actual state space control has to be delayed until after tackling classical transfer function feedback systems. 

We now return to the use of state space representation that was introduced in Chapter 4. As you may have guessed, we want to design control systems based on state space analysis. State feedback controller is very different from the classical PID controller. Our treatment remains introductory, and we will stay with linear or linearized SISO systems. Nevertheless, the topics here should enlighten( ) us as to what modem control is all about. 

Since, in process control, input-output linearization techniques are usually preferred to state-space approaches, mostly due to the higher complexity of the latter, in the following, only input-output feedback linearization basic concepts are briefly reviewed. 

Most of the current MFC research is based on state-space models, because they provide an important theoretical advantage, namely, a unified framework for both linear and nonlinear control problems. State-space models are also more convenient for theoretical analysis and facilitate a wider range of output feedback strategies (Rawlings, 2000, Maciejowski, 2002 Qin and Badgwell, 2003). 

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