Table of contents |

2 Analysis and control of Non-linear Systems 3 The Lur'e Problem |

- They do not follow the principle of superposition (linearity and homogenity).
- They may have multiple 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).

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

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**
**

- (A,B) is controllable and (C,A) is observable
- two real numbers a,b with a

We will discuss two main theorems concerning Lure's problem.

- The Circle Criterion
- The Popov Criterion.

where x ∈ R^{n}, &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

- 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.

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

inf

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.