Non-linear control

From Free net encyclopedia

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.

Properties of non-linear systems

Some properties of non-linear systems are

• They do not follow the principle of superposition (linearity and homogeneity)
• They may have multiple isolated equilibrium points
• They may exhibit properties such as limit-cycle, bifurcation, chaos
• 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 input)

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
• Stability

The Lur'e problem

Control systems exhibiting the Lur'e problem (after A.I.Lur'e) have a forward path that is linear and time-invariant, and a feedback path that contains a memory-less, and possibly time-varying, non-linearity.

Image:Lure.png

The linear part can be characterized by four matrices (A,B,C,D), while the non-linear part is Φ ∈ [a,b], a<b (a sector non-linearity).

Absolute stability problem

Consider:

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 Φ ∈ [a,b]. This is known the Lur'e problem.

There are two main theorems concerning the problem

• The Circle criterion
• The Popov criterion.

Popov criterion

The class of systems studied by Popov is described by

<math>

\begin{matrix} \dot{x}&=&Ax+bu \\ \dot{\xi}&=&u \\ y&=&cx+d\xi \quad (1) \end{matrix} </math>

<math> u = -\phi (y) \quad (2) </math>

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; (3)

The transfer function from u to y is given by

<math> h(s) = \frac{d}{s} + c(sI-A)^{-1}b \quad \quad (4)</math>

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 throughput from input to output
• Non-linearity Φ belongs to an 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,∞)

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

• 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 loopeo:Vikipedio:Projekto matematiko/Ne-lineara rego