 ControlLyapunov function

In control theory, a controlLyapunov function V(x,u) ^{[1]}is a generalization of the notion of Lyapunov function V(x) used in stability analysis. The ordinary Lyapunov function is used to test whether a dynamical system is stable (more restrictively, asymptotically stable). That is, whether the system starting in a state in some domain D will remain in D, or for asymptotic stability will eventually return to x = 0. The controlLyapunov function is used to test whether a system is feedback stabilizable, that is whether for any state x there exists a control u(x,t) such that the system can be brought to the zero state by applying the control u.
More formally, suppose we are given a dynamical system
where the state x(t) and the control u(t) are vectors.
Definition. A controlLyapunov function is a function V(x,u) that is continuous, positivedefinite (that is V(x,u) is positive except at x = 0 where it is zero), proper (that is as ), and such that
The last condition is the key condition; in words it says that for each state x we can find a control u that will reduce the "energy" V. Intuitively, if in each state we can always find a way to reduce the energy, we should eventually be able to bring the energy to zero, that is to bring the system to a stop. This is made rigorous by the following result:
Artstein's theorem. The dynamical system has a differentiable controlLyapunov function if and only if there exists a regular stabilizing feedback u(x).
It may not be easy to find a controlLyapunov function for a given system, but if we can find one thanks to some ingenuity and luck, then the feedback stabilization problem simplifies considerably, in fact it reduces to solving a static nonlinear programming problem
for each state x.
The theory and application of controlLyapunov functions were developed by Z. Artstein and E. D. Sontag in the 1980s and 1990s.
Contents
Example
Here is a characteristic example of applying a Lyapunov candidate function to a control problem.
Consider the nonlinear system, which is a massspringdamper system with spring hardening and position dependent mass described by
Now given the desired state, q_{d}, and actual state, q, with error, e = q_{d} − q, define a function r as
A ControlLyapunov candidate is then
which is positive definite for all , .
Now taking the time derivative of V
The goal is to get the time derivative to be
which is globally exponentially stable if V is globally positive definite (which it is).
Hence we want the rightmost bracket of ,
to fulfill the requirement
which upon substitution of the dynamics, , gives
Solving for u yields the control law
with κ and α, both greater than zero, as tunable parameters
This control law will guarantee global exponential stability since upon substitution into the time derivative yields, as expected
which is a linear first order differential equation which has solution
 V = V(0)e ^{− κt}
And hence the error and error rate, remembering that , exponentially decay to zero.
If you wish to tune a particular response from this, it is necessary to substitute back into the solution we derived for V and solve for e. This is left as an exercise for the reader but the first few steps at the solution are:
which can then be solved using any linear differential equation methods.
Notes
 ^ Freeman (46)
References
 Freeman, Randy A.; Petar V. Kokotović (2008) (in English). Robust Nonlinear Control Design (illustrated, reprint ed.). Birkhäuser. pp. 257. ISBN 0817647589. http://books.google.com/books?id=_eTb4Yl0SOEC. Retrieved 20090304.
See also
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