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Non-linear control is a sub-division of control engineering which deals with the control of Non-linear Systems. Non-linear systems are those systems whose input-output behaviour are very much unpredictable. For linear systems, we have a lot of well-established control techniques like root-locus , Bode plot, Nyquist criterion , state-feedback , pole-placement etc. 1 Properties of Non-linear systems
- They do not follow the principle of superposition (linearity and homogenity).
- They may have multiple isolated equilibrium points.
- They exhibit properties like limit-cycle, bifurcation, chaos.
- For a sinusoidal input, the output signal may contain many harmonics and sub-harmonics with various amplitudes and phase differences. While for a linear system, we know that for u= A sin(ωt), output y = B sin(ωt+ φ).
2 Analysis and control of Non-linear Systems
3 The Lur'e Problem
In this section, we will study the stability of an important class of control systems namely feedback systems whose forward path contains a linear time-invariant subsystem and whose feedback path contains a memory-less and possibly time-varying non-linearity. This class of problem is named for A. I. Lur'e .
The linear part is characterized by four matrices (A,B,C,D). The non-linear part is &Phi &isin [a,b], a
3.1 Absolute Stability Problem
Given that
- (A,B) is controllable and (C,A) is observable
- two real numbers a,b with a
The problem is to derive conditions involving only the transfer matrix H(.) and the numbers a,b, such that x=0 is a globally uniformly asymptotically stable equilibrium of the system (1)-(3) for every function &Phi &isin [a,b]. This is also known as Lure's problem.
We will discuss two main theorems concerning Lure's problem.
- The Circle Criterion
- The Popov Criterion.
3.2 Popov Criterion
The class of systems studied by Popov is described by
-
where x ∈ Rn, &xi,u,y are scalars and A,b,c,d have commensurate dimensions. The non-linear element Φ: R → R is a time-invariant nonlinearity belonging to open sector (0, &infin). This means that
&Phi(0) = 0, y &Phi(y) > 0, &forall y &ne 0; (3)
The transfer function from u to y is given by
-
Things to be noted
- Popov criterion is applicable only to autonomous systems.
- The system studied by Popov has a pole at the origin and there is no throughput from input to output.
- Non-linearity &Phi belongs to a open sector.
Theorem:
Consider the system (1) and (2) and suppose
- A is Hurwitz
- (A,b) is controllable
- (A,c) is observable
- d>0 and
- Φ ∈ (0,&infin)
then the above system is globally asymptotically stable if there exists a number r>0 such that
infω ∈ R Re[(1+jωr)h(j&omega)] > 0
Further reading:
- A. I. Lur’e and V. N. Postnikov, "On the theory of stability of control systems," Applied mathematics and mechanics, 8(3), 1944, (in Russian).
- M. Vidyasagar, "Nonlinear Systems Analysis, second edition, Prentice Hall, Englewood Cliffs, New Jersey 07632.
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