Chapter 6 Feedback 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:
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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
Example 6.1:Controlling 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.
: desired level
The dynamics (6.1) can be rewritten as
with v being an "equivalent input" to be
specified, the resulting dynamics is linear
with : level error, and > 0
Thus, the nonlinear control law
Applying Feedback linearization to 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
= v (6.7)Then, the control law
with has all its roots strictly in the left-half complex plane exponentially stable dynamics
the control law
(where e(t) = x(t) -xd(t))
exponentially convergent tracking.
- 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.
Figure 6.2: A two-link robot- Each joint equipped with a motor to provide input torque, an encoder to measure joint position, and a tachometer to measure joint velocity
: two joint angles : joint inputs
Using the control law (computed torque)
with v = [v1v2 ]T being the equivalent input, being the tracking error and a positive number. The tracking error then satisfies the equation
and converges to zero exponentially.
With equilibrium point at (0,0).
v : an equivalent input to be designed
- Linear and controllable
anywhere with proper choices of feedback gains.
in the stable closed-loop dynamics
The control law
Since both z1 and z2 converge to zero, the original state x converges to zero.
The inner loop: achieves the linearization of the input-state relation.
Consider the system
Objective : make y(t)track a desired trajectory
yd(t) while keeping the whole state bounded
yd(t) : and its nth order derivative are known
Q: how to design the input u ?
Consider the third-order system
v : a new input to be determined.
with and being positive constants
If otherwise, e(t) converges to zero exponentially.
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).
- For the above example, the internal state can be chosen to be x3, and the internal dynamics is represented by
- 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.
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.
Consider the nonlinear system
substitute into (6.27b)
The dynamics of cannot be observed from (6.27b). Applying u to the second dynamic equation :
If where D > 0, then
(perhaps after a transient)
in the internal dynamic is bounded (6.28)
Consider the simple controllable and observable linear system
(where e = y -yd) and the internal dynamics
- and is bounded and u are bounded (6.32) is a satisfactory tracking controller for system (6.31).
for system (6.33),
- For all linear system, the internal dynamic is stable if the plant zeros are in the left-half plane.
To keep notations simple, let us consider a third-order linear system in state space form (6.34)
and having one zero (i.e. two more poles than zeros).
- The system's input-output linearization can be facilitated if we first transform it into companion form.
- To do this, we note from linear control that the input/output behavior of this system can be expressed in the form
Thus, if we define
where e = y - yd , yields an exponentially stable tracking error with and :
which is a second order dynamics
-The internal dynamics can be described by only one state equation. Since we can show x1, y, are related to x1, x2, and x3 through a one-to-one transformation (and thus can serve as states for the system).
-The internal dynamics can be found from (6.36a) and (6.36b):
-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 system’s 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).
The zero-dynamics (6.45) is asymptotically stable (by using the Lyapunov function V = x22 ).
-Similarly, for the linear system (6.34), the zero-dynamics is (from (6.39))
-Thus, in this linear system, the poles of the zero-dynamics are exactly the zeros of the system.
-This result is general for linear systems, and therefore, in linear systems, having all zeros in the left-half complex plane guarantees the global asymptotic stability of the zero-dynamics.
-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.
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