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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 cannot be described as a linear function of the state of that system or the input variables to that system. For linear systems, there are many well-established control techniques, for example root-locus, Bode plot, Nyquist criterion, state-feedback, pole-placement etc.

Contents 1 Properties of non-linear systems 2 Analysis and control of non-linear systems 3 Nonlinear Feedback Analysis - The Lur'e problem 3.1 Absolute stability problem 3.1.1 Popov criterion 4 References 5 See also

Some properties of non-linear dynamic systems are

• They do not follow the principle of superposition (linearity and homogeneity).
• They may have multiple isolated equilibrium points (linear systems can have only one).
• They may exhibit properties such as limit-cycle, bifurcation, chaos.
• Finite escape time: The state of an unstable nonlinear system can go to infinity in finite time.
• For a sinusoidal input, the output signal may contain many harmonics and sub-harmonics with various amplitudes and phase differences (a linear system's output will only contain the sinusoid at the output).

There are several well-developed techniques for analyzing nonlinear feedback systems:

• Describing function method
• Phase plane method
• Lyapunov stability analysis
• Singular perturbation method
• Popov criterion (described in The Lur'e Problem below)
• Center manifold theorem
• Small-gain theorem
• Passivity analysis

Control design techniques for non-linear systems also exist. These can be subdivided into techniques which attempt to treat the system as a linear system in a limited range of operation and use (well-known) linear design techniques for each region:

• Gain scheduling
• Adaptive control

Those that attempt to introduce auxiliary nonlinear feedback in such a way that the system can be treated as linear for purposes of control design:

• Feedback linearization

And Lyapunov based methods:

• Lyapunov Redesign
• Back-stepping
• Sliding mode control

An early non-linear feedback system analysis problem was formulated by A.I. Lur'e. Control systems described by the Lur'e problem have a forward path that is linear and time-invariant, and a feedback path that contains a memory-less, possibly time-varying, static non-linearity.

The linear part can be characterized by four matrices (A,B,C,D), while the non-linear part is Φ(y) with (a sector non-linearity).

Consider:

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(s) and {a,b} such that x=0 is a globally uniformly asymptotically stable equilibrium of the system. This is known as the Lur'e problem.

There are two main theorems concerning the problem:

• The Circle criterion
• The Popov criterion.

The sub-class of Lur'e systems studied by Popov is described by:

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;

The transfer function from u to y is given by

Theorem: Consider the system (1)-(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 system is globally asymptotically stable if there exists a number r>0 such that
inf_{ω ∈ R} Re[(1+jωr)h(jω)] > 0 .

Things to be noted:

• The Popov criterion is applicable only to autonomous systems
• The system studied by Popov has a pole at the origin and there is no direct pass-through from input to output
• Non-linearity Φ belongs to an open sector

• 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.
• A. Isidori, Nonlinear Control Systems, third edition, Springer Verlag, London, 1995.
• H. K. Khalil, Nonlinear Systems, third edition, Prentice Hall, Upper Saddle River, New Jersey, 2002.

• Phase-locked loop