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-- Stability in Nonlinear systems

posted Apr 23, 2011, 4:13 AM by Javad Taghia   [ updated Apr 24, 2011, 12:35 AM ]
The aim of control is making system stable around one operational point. In the best design we make error zero by controller. 

Let's define stability for a linear and nonlinear system.
In general there are two types of systems. Linear and nonlinear systems. 
Linear systems have one position of rest or infinite position of rest or one position of rest. 
In contrast nonlinear systems can have one or finite number position of rest(s). What is position of rest? 
Definitions are from book Fuzzy Control (By: Kai MichelsRudolf KruseFrank Klawonn):
Def. 1: Position of rest:
For a given constant vector u0 as the input, a dynamic system is at a position of rest - characterized by the state vector xR - if and only if the state variables remain constant, i. e. if  0=X'= f (xR,u0) holds. For example for a linear system which is described by x'R=AxR+Bu0. If |A|=0 then there are infinite answer for making x'R=0. for our linear system; so there are infinite position of rest. 
For a nonlinear system as a pendulum, there are two position of rests. One when it is at lower potential energy and other when it's at the highest potential energy. 
Def. 2: Stability in sense of Lyapunov: 
A position of rest xR is said to be stable for a given constant input value u0, if and only if for  every E>0 region a Delta>0 region can be found such that |x(0)-xR|>Delta the condition |x(t)-xR| < E (t>+0) satisfied. 
It means if we start at a point in Delta surrounding we never cross E surrounding. So the response of our system is LIMITED. 
For our pendulum the position of rest at highest potential energy is not stable in sense of Lyapunov. Because we are not able to define this regions. For the point at lowest potential energy this surroundings are definable and we are able to know this point stable. 
Def. 3: Asymptotically stability:   
If we can define a Beta region instead of Delta region for our system that after constant input activation u0, system will converge to the position of rest xR. If the domain is limited by technical restrictions the asymptotically stability occurs  in large, . If the domain is the entire state space it is global asymptotically stable. It means if we start from one position of rest in finite time we finish at the same position of rest not a new one. So, for the example of pendulum, in general it is asymptotically stable in large. Because if we do not restrict the movement of pendulum for example by a kind of ceiling we may reach to the other position of rest at highest potential energy. 

We can extend this idea to trajectories. We can define two major trajectories. One is Steady Oscillation another is Limit Cycle. 
Steady Oscillation: It is a stable trajectory it means in sense of Lyapunov it is stable; it means if we consider one oscillation freq. after applying constant input we reach to new oscillation or the original one. If we reach the new one which is in the boundary of E then it is stable if we return to the original one we have asymptotically stable we call it LIMIT CYCLE. Limit cycles are a frequency which our trajectory after transient behavior converges to.