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The theory and application of control-Lyapunov functions were developed by Z. Artstein and [[Eduardo D. Sontag|E. D. Sontag]] in the 1980s and 1990s.
==Example==
Here is a characteristic example of applying a Lyaponov candidate function to a control problem.
Consider the non-linear system, which is a mass-spring-damper system with spring hardening and position dependant mass described by
:<math>
m(1+q^2)\ddot{q}+b\dot{q}+K_0q+K_1q^3=u
</math>
Now given the desired state, <math>q_d</math>, and actual state, <math>q</math>, with error, <math>e = q_d - q</math>, define a function <math>r</math> as
:<math>
r=\dot{e}+\alpha e
</math>
A Control-Lyaponov candidate is then
:<math>
V=\frac{1}{2}r^2
</math>
which is positive definite for all <math> q \ne 0</math>, <math>\dot{q} \ne 0</math>.
Now taking the time derivitive of <math>V</math>
:<math>
\dot{V}=(\dot{e}+\alpha e)(\ddot{e}+\alpha \dot{e})
</math>
The goal is to get the time derivitive to be
:<math>
\dot{V}=-\kappa V
</math>
which is globally exponentially stable if <math>V</math> is globally positive definite (which it is).
Hence we want the rightmost bracket of <math>\dot{V}</math>,
:<math>
(\ddot{e}+\alpha \dot{e})=(\ddot{q}_d-\ddot{q}+\alpha \dot{e})
</math>
to fulfill the requirement
:<math>
(\ddot{q}_d-\ddot{q}+\alpha \dot{e}) = -\frac{\kappa}{2}(\dot{e}+\alpha e)
</math>
which upon substitution of the dynamics, <math>\ddot{q}</math>, gives
:<math>
(\ddot{q}_d-\frac{u-K_0q-K_1q^3-b\dot{q}}{m(1+q^2)}+\alpha \dot{e}) = -\frac{\kappa}{2}(\dot{e}+\alpha e)
</math>
Solving for <math>u</math> yields the control law
:<math>
u= m(1+q^2)(\ddot{q}_d + \alpha \dot{e}+\frac{\kappa}{2}r )+K_0q+K_1\dot{q}+b\dot{q}
</math>
with <math>\kappa</math> and <math>\alpha</math> as tunable parameters
This control law will guarentee global exponential stability since upon substitution into the time derivitive yields, as expected
:<math>
\dot{V}=-\kappa V
</math>
which is a linear first order differential equation which has solution
:<math>
V=V(0)e^{-\kappa t}
</math>
And hence the error and error rate, remembering that <math>V=\frac{1}{2}(\dot{e}+\alpha e)^2</math>, exponentially decay to zero.
This example makes explicit use of [[feedback linearisation]] and contains a feedforward and feedback component
==Notes==
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