Non-linear control

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

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

Absolute Stability Problem

1. (A,B) is controllable and (C,A) is observable
2. 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.

   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.

   Referenced By

   Control theory | Kalman filtering | List of dynamical system and differential equation topics | List of dynamical system topics | List of mathematical topics (M-O) | Open loop control