AUTOMATIC CONTROL THEORY II Slovak University of Technology Faculty of Material Science and Technology in Trnava

Nonlinear Systems in mathematics, a nonlinear system is a system which is not linear a system which does not satisfy the superposition principle any problem where the variable(s) to be solved for cannot be written as a linear sum of independent components

Nonlinear Systems a nonhomogenous system, which is linear apart from the presence of a function of the independent variables  is nonlinear according to a strict definition  but are usually studied alongside linear systems because they can be transformed to a linear system as long as a particular solution is known

Nonlinear Systems nonlinear equations are difficult to solve and give rise to interesting phenomena such as chaos in mathematics, a linear function (or map) f(x) is one which satisfies both of the following properties  additivity  homogeneity

Nonlinear Systems an equation written as is called linear if  f(x) is linear  and nonlinear otherwise x does not need to be a scalar (can be a vector, function, etc) C must not depend on x

Nonlinear Systems the equation is called homogeneous if C = 0 nonlinear algebraic problems are often exactly solvable the equation may be written as

Nonlinear Systems it is nonlinear because f(x) satisfies  neither additivity  nor homogeneity the nonlinearity is due to the x 2 this example may be solved exactly (via the quadratic formula) it is very well understood

Nonlinear Systems a nonlinear recurrence relation defines successive terms of a sequence as a nonlinear function of preceding terms examples of nonlinear recurrence relations are  the logistic map  the relations that define the various Hofstadter sequences

Nonlinear Systems problems involving nonlinear differential equations are extremely diverse methods of solution or analysis are very problem dependent the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions

Nonlinear Systems common methods for the qualitative analysis of nonlinear ordinary differential equations include  examination of any conserved quantities, especially in Hamiltonian systems  examination of dissipative quantities analogous to conserved quantities  linearization via Taylor expansion  change of variables into something easier to study  bifurcation theory  perturbation methods (can be applied to algebraic equations too)

Nonlinear Systems the most common basic approach to studying nonlinear partial differential equations is to change the variables the resulting problem is simpler (possibly even linear) the equation may be transformed into one or more ordinary differential equations

Nonlinear Systems another tactic is to use scale analysis to simplify a general, natural equation in a certain specific boundary value problem the nonlinear Navier-Stokes equations can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe

Nonlinear Systems the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation other methods include examining the characteristics and using the methods outlined above for ordinary differential equations

Nonlinear Systems types of nonlinear behaviors  indeterminism - the behavior of a system cannot be predicted  multistability - alternating between two or more exclusive states  aperiodic oscillations - functions that do not repeat values after some period (otherwise known as chaotic oscillations or chaos)

Nonlinear Systems examples of nonlinear equations  AC power flow model  Bellman equation for optimal policy  Boltzmann transport equation  General relativity  nonlinear optics  nonlinear Schrödinger equation  Richards equation for unsaturated water flow  Robot unicycle balancing