**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).

## Analysis and control of non-linear systems Edit

- Describing function method
- Phase plane method
- Lyapunov based methods
- Input-output stability
- Feedback linearization
- Back-stepping
- Sliding mode control
- Singular perturbation

## 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.

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:

- (A,B) is controllable and (C,A) is observable
- 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 ∈ R^{n}, 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

- A is Hurwitz
- (A,b) is controllable
- (A,c) is observable
- d>0 and
- Φ ∈ (0,∞)

then the above system is globally asymptotically stable if there exists a number r>0 such that

inf_{ω ∈ R} Re[(1+jωr)h(jω)] > 0

## References Edit

- 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.