Non-linear control

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.

Properties of Non-linear systems

• 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).

Analysis and control of Non-linear Systems

• 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

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<b, is a sector non-linearity.

Absolute Stability Problem

Given the

• 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 &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.

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

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

then the above system is globally asymptotically stable if there exists a number r>0 such that
inf_{ω ∈ 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.

See also:

• Phase-locked loop