02 Automatic Control System Linearization

# 02 Automatic Control System Linearization -...

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System Linearization 11 Oct 16 Automatic Control

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Aims for this chapter For classic control theory we are going to discuss is based on the linear system. Therefore, it is not suitable for nonlinear systems. In control engineering, a normal operation of the system may be around an equilibrium point, and the signals may be considered small signals around the equilibrium. If the system operates around an equilibrium point and if the signals involved are small signals, then it is possible to approximate the nonlinear system by a linear system. Such a linear system can be approximated to the nonlinear system around the operation point.
System Linearization Why linearization process is needed? The advantages of using linear system model are 1). The principle of superposition can be applied. 2). Operational methods of analysis can be used. 3). The magnitude scale factor is preserved in a linear system. (The property of homogeneity) Linearization 1). Small signal linearization (in this lecture) 2). Linearization by feedback (in nonlinear control theory) 3). Transformation method (in nonlinear control theory) Linearization can help designers to use linear system theory to analyze the nonlinear system at certain interesting operation points.

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Function Linearization To obtain a linear mathematical model for a nonlinear system, we assume that the variables deviate only slightly from some operating condition. Note that the linearized models only describe local behavior. Consider a single input single output (SISO) nonlinear function (or a nonlinear system) whose input is x(t) and output is y(t). The relationship between y(t) and x(t) is given by If the normal operating point corresponds to The equation can be expanded into a Taylor series about this point. H.O.T
Equation Linearization Based on the Taylor series (on the point ) If the variation is small, the higher order terms can be neglected. where and Finally we got the linearized function H.O.T variation x K y output deviation input deviation

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