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Chapter 8 Nonlinear Systems Theory
8.1 Overview of Nonlinear Systems
8.2 Typical Nonlinear Characteristics and Mathematical Description
8.3 Describing Function Approach
8.4 Phase Plane Analysis
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Interesting Examples of Nonlinearity
Lorenz Chaotic Attractor
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Interesting Examples of Nonlinearity
Acrobot Robots
Biomimetic robots
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§8.1 Overview of Nonlinear Systems
Such as：
Thinking: How to distinguish nonlinearity？
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1 Significance of Studying Non-linear Systems
1）There are no systems without nonlinearity.
(Dead-zone characteristics of measurement element、 saturation characteristics of amplification element、 Dead-zone and saturation characteristics of actuator、 Gap Characteristics of actuating unit and so on）。
2）The intrinsic nonlinearities make the linear system theory cannot be applied in analyzing the actual systems. The influences of nonlinear factors can not be explained by linear system theory.
3）The nonlinear characteristics do not always have negative impacts on systems. Optimal control laws are often nonlinear laws, such as the relay and waveform generator.
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2. Features of Nonlinear Systems
Comparing with linear control systems，non-linear systems have many new features：
1. A linear systems satisfies the principle of superposition, while a non-linear system does not.
（1）additivity：
additivity
A linear function satisfies the principle of
superposition obviously
Thinking：
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Superposition Theorem
（2） multiplicativity ：
Nonlinear systems may be additive, but it is not multiplicative.
filter I
filter II
non-linear equipment I
non-linear equipment II
X1+X2
Y1＋Y2
Fig. 8－1 Nonlinear systems with filters
additivity：
multiplicativity ：
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2.The stabilities of non-linear systems depend on not only the inherent structure and parameters of control systems, but also the initial conditions and the inputs.
have two equilibrium points obviously,x1=0 and x2=1.The equation equals to
Example：An nonlinear systems described by the
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