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Non-linear control is a sub-division of control engineering which deals with the control of non-linear systems. The behaviour of a non-linear system is not expressible as a linear function of its state or input variables. For linear systems, there are many well-established control techniques like root-locus, Bode plot, Nyquist criterion, state-feedback, pole-placement etc.

## Properties of non-linear systems Edit

• They do not follow the principle of superposition (linearity and homogeneity).
• They may have multiple isolated equilibrium points.
• They may 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).

## The Lur'e problem Edit

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.

Lure Problem Block Diagram

The linear part is characterized by four matrices (A,B,C,D). The non-linear part is Φ ∈ [a,b], a<b, is a sector non-linearity.

### Absolute stability problem Edit

Consider:

1. (A,B) is controllable and (C,A) is observable
2. two real numbers a, b with a<b.

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 Φ ∈ [a,b]. This is also known as Lure's problem.

There are two main theorems concerning Lure's problem.

• The Circle criterion
• The Popov criterion.

### Popov criterion Edit

The class of systems studied by Popov is described by

$\begin{matrix} \dot{x}&=&Ax+bu \\ \dot{\xi}&=&u \\ y&=&cx+d\xi \quad (1) \end{matrix}$

$u = -\phi (y) \quad (2)$

where x ∈ Rn, ξ,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, ∞). This means that

Φ(0) = 0, y Φ(y) > 0, ∀ y ≠ 0; (3)

The transfer function from u to y is given by

$h(s) = \frac{d}{s} + c(sI-A)^{-1}b \quad \quad (4)$

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 Φ belongs to a open sector.

Theorem: Consider the system (1) and (2) and suppose

1. A is Hurwitz
2. (A,b) is controllable
3. (A,c) is observable
4. d>0 and
5. Φ ∈ (0,∞)

then the above system is globally asymptotically stable if there exists a number r>0 such that
infω ∈ R Re[(1+jωr)h(jω)] > 0

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