Chapter 6 Feedback Linearization

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Chapter 6Feedback Linearization

• Central idea
• To algebraically transform a nonlinear system
  dynamics into a (fully or partly) linear one
• linear control techniques can be applied.
• Feedback linearization techniques
• Ways of transforming original system models into
  equivalent models of a simpler form.

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6.1 Intuitive Concepts
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Feedback linearization amounts to canceling the
nonlinearities in a nonlinear system so that the
closed loop dynamics is a linear form
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6.1.1 Feedback Linearization and the Canonical
Form

• Example 6.1Controlling the fluid level in a
  tank
• Dynamic model of the tank is
• A(h) cross section of the tank
• a cross section of the outlet pipe.

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• initial level
• desired level
• The dynamics (6.1) can be rewritten as
• If
• with v being an "equivalent input" to be
• specified, the resulting dynamics is linear

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• Choosing
• with level error, and ? gt
  0
• Then
• as
• Thus, the nonlinear control law

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• If the desired level is a known time-varying
  function , set
• as

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Applying Feedback linearization to companion form
or controllable canonical form system
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Companion Form, or Controllable Canonical Form
System A companion form system u scalar
control input x scalar output of interest
state vector f(x) and b(x)
nonlinear function of states
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Using v
(6.7)Then, the control law with
has all its roots strictly in the left-half
complex plane exponentially stable
dynamics i.e. as
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Tracking Desired Output xd(t) the control law
(6.8)
(where e(t) x(t) -xd(t)) exponentially
convergent tracking.
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One interesting application of the above control
design idea
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Example 6.2Feedback linearization of a two-link
robot - Tracking control problems arise when a
robot hand is required to move along a specified
path, e. g., to draw circles. - Control
objective To make the joint positions q1 and q2
follow desired position histories qd1(t) and
qd2(t) specified by the motion planning system.
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• Using Lagrangian equations in classical dynamics,
  the dynamics of the robot

two joint angles
joint inputs
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• Compact expression
• Multiplying both sides by Companion form

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6.1.2 Input-State Linearization
(6.11a) (6.11b)

• Linear control design can stabilize the system in
  a small region around (0, 0)
• What controller can stabilize it in a larger
  region.
• - Note that the nonlinearity in the first
  equation cannot be canceled by the control input
  u.

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• Consider the new set of state variables
• (6.12a)
• (6.13a)
• 
  (6.13b)
• With equilibrium point at (0,0).

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• Using state feedback control law can place the
  poles
• anywhere with proper choices of feedback gains.

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• The inner loop achieves the linearization of
  the input-state relation.
• The outer loop achieves the stabilization of
  the closed-loop dynamics.

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• Remarks
• The result is not global.
• The control law is not well defined when x1
• ( ), k 1, 2,

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• Different from a Jacobian linearization for
• small range operation.
• The new state components (z1, z2) must be
• available. If they cannot be measured
  directly,
• the original state x must be measured and used
  
• to compute them from (6.12)
• Uncertainty in the model, e. g., uncertainty on
• the parameter a, will cause error in the
• computation of z and u.

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(6.23)
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(6.24)
with and being positive constants
(6.25)
If otherwise, e(t) converges to zero
exponentially.
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Internal Dynamics

• A part of the system dynamics (described by
• one state component) has been rendered
• "unobservable" in the input-output
• linearization, because it cannot be seen from
• the external input-output relationship (6.21).

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- For the above example, the internal state can
be chosen to be x3, and the internal dynamics is
represented by
(6.26)
- If this internal dynamics is stable (by which
we actually mean that the states remain bounded
during tracking, i. e., stability in the BIBO
sense), our tracking control design problem has
indeed been solved.
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• Otherwise, the above tracking controller is
  practically meaningless, because the instability
  of the internal dynamics would imply undesirable
  phenomena such as the burning-up of fuses or the
  violent vibration of mechanical members.
• The effectiveness of the above control design,
  based on the reduced-order model (6.21), hinges
  upon the stability of the internal dynamics.

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(6.28)
substitute into (6.27b)
(6.29)
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The dynamics of cannot be observed from
(6.27b). Applying u to the second dynamic
equation
(6.30)
If where D gt 0, then (perhaps after a
transient) in the internal dynamic is
bounded (6.28) is satisfactory
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• The Internal Dynamics of Linear Systems
• In general, it is very difficult to directly
  determine the stability of the internal dynamics
  because it is nonlinear, non-autonomous, and
  coupled to the "external" closed-loop dynamics.
• Q How to seek simpler ways of determining the
  stability of the internal dynamics.

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- For all linear system, the internal dynamic is
stable if the plant zeros are in the left-half
plane.
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• Input u appears in the second differentiation
• The required number of differentiations (the
  relative degree) the excess of poles over zeros
• Since the input-output relation of y to u is
  independent of the choice of state variables, it
  would also take two differentiations for u to
  appear if we used the original state-space
  equations (6.34)).

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-Since y is bounded (y e yd), we see that the
stability of the internal dynamics depends on the
location of the pole of the internal dynamics,
which is the zero -b0 / b1 of the original
transfer function in (6.35). -If the systems
zero is in the left-half plane, which implies
that the internal dynamics (6.39) is stable,
independently of the initial conditions and of
the magnitudes of the desired yd , , yd (r)
(where r is the relative degree).
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• Motivation for Zero Dynamics
• Previously discussed transfer functions, on
  which linear system zeros are based, cannot be
  defined for nonlinear systems.
• Zeros are intrinsic properties of a linear
  plant, while for nonlinear systems the stability
  of the internal dynamics may depend on the
  specific control input.

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(6.45)
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-For linear systems, the stability of the
zero-dynamics implies the global stability of the
internal dynamics The left-hand side of (6.39)
completely determines its stability
characteristics, given that the right-hand side
tends to zero or is bounded. -For nonlinear
systems, it can be shown for local asymptotic
stability of the zero-dynamics is enough to
guarantee the local asymptotical stability of the
internal dynamics. Though we will not prove it
here.
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To summarize, control design based on
input-output linearization can be made in three
steps A. differentiate the output y until the
input u appears B. choose u to cancel the
nonlinearities and guarantee tracking
convergence C. study the stability of the
internal dynamics