Non-linear control

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 is not expressible as a linear function of its state or input variables. For linear systems, there are many 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 homogeneity).
They may have multiple isolated equilibrium points.
They may 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.

Lure Problem Block Diagram

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

Absolute stability problem[]

Consider:

(A,B) is controllable and (C,A) is observable
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 also known as Lure's problem.

There are 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

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

$u = -\phi (y) \quad (2)$

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

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

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 Φ belongs to a open sector.

Theorem: Consider the system (1) and (2) and suppose

A is Hurwitz
(A,b) is controllable
(A,c) is observable
d>0 and
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.