Nonlinear control systems part 2

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Nonlinear control systems part 2
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Linearity and superposition
Definition If all initial conditions in the
system are zero, that is, if the system is
completely at rest, then the system is a linear
system if it has the following property a. If
an input u1(t) produces an output y1 (t), and b.
an input u2(t) produces an output y2 (t), c.
then input c1u1 (t) c2u2 (t) produces an
output c1y1 (t) c2y2 (t) for all pairs of
inputs u1 (t) and u2 (t) and all pairs of
constants c1 and c2.
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Nonlinear systems

• Any system that does not satisfy this definition
  is nonlinear.

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Nonlinear systems

• The major difficulty with nonlinear systems,
  especially those described by nonlinear ordinary
  differential or difference equations, is that
  analytical or closed-form solutions are available
  only for very few special cases, and these are
  typically not of practical interest in control
  system analysis or design.

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• Unlike linear systems, for which free and forced
  responses can be determined separately and the
  results superimposed to obtain the total
  response, free and forced responses of nonlinear
  systems normally interact and cannot be studied
  separately, and superposition does not generally
  hold for inputs or initial conditions.

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A block diagram of a nonlinear feedback
The nonlinear block N has the transfer
characteristic f(e) defined in Figure (b). Such
nonlinearities are called (piecewise-linear)
saturation functions.
The linear block is represented by the transfer
function G2 1/D( D 1), where D d/dt is the
differential operator.
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LINEARIZED AND PIECEWISE-LINEARIZED
The system is approximated by a piecewise-linear
system that is, the system is described by the
linear equation M(d2x/dt2) kx 0 when x lt x1,
and by the equations M(d2x/dt2) - F1 0 when
x gt x1. The sign is used if xgt x1 and the -
sign if xlt -x1.
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The motion of a pendulum
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Taylor series
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Example

• Suppose y( t) f u( t) represents a nonlinear
  system with input u(t) and output y(t), where
  tgtt0 for some t0, and df/du exists for all u. If
  the normal operating conditions for this system
  are defined by the input u and output y, then
  small changes ?y(t) in output operation in
  response to small changes in the input ?u( t) can
  be expressed by the approximate linear relation

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System equations

• A property common to all basic laws of physics is
  that certain fundamental quantities can be
  defined by numerical values.
• The physical laws define relationships between
  these fundamental quantities and are usually
  represented by equations.

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Taylor Series for Vector Processes

• are readily generalized for nonlinear m-vector
  functions of n-vector arguments, f(x), where

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• For m1 and n2 equation reduces to

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Linearization of Nonlinear Differential Equations
Nonlinear Differential Equations
Nonlinear output equations
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The linearized versions
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PHASE PLANE METHODS

• Phase plane methods are developed for analyzing
  nonlinear differential equations in state
  variable form, without the need for
  linearization.

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A second-order differential equation
can be rewritten as a pair of first-order
differential equations,
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Solution - a trajectory in the phase plane

• If we eliminate time as the independent variable
  we obtain the first-order differential equation

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• By solving this equation for various initial
  conditions on x1 and x2 and examining the
  resulting phase plane trajectories, we can
  determine the behavior of the second-order
  system.

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Example
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The phase plane trajectory
Its direction in the phase plane is determined by
noting that dx2 /dt -x22lt 0 for all x2 ? 0.
Therefore x2 always decreases and we obtain the
trajectory shown.