Nonlinear control theory is the area of
control theory
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
which deals with systems that are
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
,
time-variant, or both. Control theory is an interdisciplinary branch of engineering and
mathematics that is concerned with the behavior of
dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
s with inputs, and how to modify the output by changes in the input using
feedback,
feedforward, or
signal filtering. The system to be controlled is called the "
plant
Plants are predominantly photosynthetic eukaryotes of the kingdom Plantae. Historically, the plant kingdom encompassed all living things that were not animals, and included algae and fungi; however, all current definitions of Plantae exclu ...
". One way to make the output of a system follow a desired reference signal is to compare the output of the plant to the desired output, and provide
feedback to the plant to modify the output to bring it closer to the desired output.
Control theory is divided into two branches.
Linear control theory applies to systems made of devices which obey the
superposition principle. They are governed by
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s. A major subclass is systems which in addition have parameters which do not change with time, called ''
linear time invariant
In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly defin ...
'' (LTI) systems. These systems can be solved by powerful
frequency domain
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a s ...
mathematical techniques of great generality, such as the
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...
,
Fourier transform,
Z transform,
Bode plot,
root locus
In control theory and stability theory, root locus analysis is a graphical method for examining how the roots of a system change with variation of a certain system parameter, commonly a gain within a feedback system. This is a technique used as a ...
, and
Nyquist stability criterion
In control theory and stability theory, the Nyquist stability criterion or Strecker–Nyquist stability criterion, independently discovered by the German electrical engineer at Siemens in 1930 and the Swedish-American electrical engineer Harry ...
.
Nonlinear control theory covers a wider class of systems that do not obey the superposition principle. It applies to more real-world systems, because all real control systems are nonlinear. These systems are often governed by
nonlinear differential equations. The mathematical techniques which have been developed to handle them are more rigorous and much less general, often applying only to narrow categories of systems. These include
limit cycle
In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity o ...
theory,
Poincaré maps,
Lyapunov stability theory, and
describing functions. If only solutions near a stable point are of interest, nonlinear systems can often be
linearized by approximating them by a linear system obtained by expanding the nonlinear solution in a
series
Series may refer to:
People with the name
* Caroline Series (born 1951), English mathematician, daughter of George Series
* George Series (1920–1995), English physicist
Arts, entertainment, and media
Music
* Series, the ordered sets used in ...
, and then linear techniques can be used. Nonlinear systems are often analyzed using
numerical methods on
computers, for example by
simulating their operation using a
simulation language
A computer simulation language is used to describe the operation of a simulation on a computer.Fritzson, Peter, and Vadim Engelson.Modelica—A unified object-oriented language for system modeling and simulation" European Conference on Object-Orie ...
. Even if the plant is linear, a nonlinear controller can often have attractive features such as simpler implementation, faster speed, more accuracy, or reduced control energy, which justify the more difficult design procedure.
An example of a nonlinear control system is a
thermostat-controlled heating system. A building heating system such as a furnace has a nonlinear response to changes in temperature; it is either "on" or "off", it does not have the fine control in response to temperature differences that a proportional (linear) device would have. Therefore, the furnace is off until the temperature falls below the "turn on" setpoint of the thermostat, when it turns on. Due to the heat added by the furnace, the temperature increases until it reaches the "turn off" setpoint of the thermostat, which turns the furnace off, and the cycle repeats. This cycling of the temperature about the desired temperature is called a ''
limit cycle
In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity o ...
'', and is characteristic of nonlinear control systems.
Properties of nonlinear systems
Some properties of nonlinear dynamic 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
In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity o ...
,
bifurcation
Bifurcation or bifurcated may refer to:
Science and technology
* Bifurcation theory, the study of sudden changes in dynamical systems
** Bifurcation, of an incompressible flow, modeled by squeeze mapping the fluid flow
* River bifurcation, the ...
,
chaos
Chaos or CHAOS may refer to:
Arts, entertainment and media Fictional elements
* Chaos (''Kinnikuman'')
* Chaos (''Sailor Moon'')
* Chaos (''Sesame Park'')
* Chaos (''Warhammer'')
* Chaos, in ''Fabula Nova Crystallis Final Fantasy''
* Cha ...
.
* Finite escape time: Solutions of nonlinear systems may not exist for all times.
Analysis and control of nonlinear systems
There are several well-developed techniques for analyzing nonlinear feedback systems:
*
Describing function method
*
Phase plane method
*
Lyapunov stability
Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. ...
analysis
*
Singular perturbation method
* The
Popov criterion and the
circle criterion for absolute stability
*
Center manifold theorem In the mathematics of evolving systems, the concept of a center manifold was originally developed to determine stability of degenerate equilibria. Subsequently, the concept of center manifolds was realised to be fundamental to mathematical modelling ...
*
Small-gain theorem
*
Passivity analysis
Control design techniques for nonlinear 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
In control theory, gain scheduling is an approach to control of non-linear systems that uses a family of linear controllers, each of which provides satisfactory control for a different operating point of the system.
One or more observable variable ...
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
*
Control-Lyapunov function In control theory, a control-Lyapunov function (CLF) is an extension of the idea of Lyapunov function V(x) to systems with control inputs. The ordinary Lyapunov function is used to test whether a dynamical system is ''(Lyapunov) stable'' or (more ...
*
Nonlinear damping
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportionality (mathematics), proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, m ...
*
Backstepping
*
Sliding mode control
In control systems, sliding mode control (SMC) is a nonlinear control method that alters the dynamics of a nonlinear system by applying a discontinuous control signal (or more rigorously, a set-valued control signal) that forces the system to ...
Nonlinear feedback analysis – The Lur'e problem
An early nonlinear 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 nonlinearity.
The linear part can be characterized by four matrices (''A'',''B'',''C'',''D''), while the nonlinear part is Φ(''y'') with