Control-Lyapunov function

In control theory, a control-Lyapunov function^[1] is a Lyapunov function $V(x)$ for a system with control inputs. 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 $x\neq 0$ in some domain D will remain in D, or for asymptotic stability will eventually return to $x=0$ . The control-Lyapunov 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 an autonomous dynamical system

{\dot {x}}=f(x,u)

where $x\in\mathbf{R}^n$ is the state vector and $u\in \mathbf {R} ^{m}$ is the control vector, and we want to feedback stabilize it to $x=0$ in some domain $D\subset \mathbf {R} ^{n}$ .

Definition. A control-Lyapunov function is a function $V:D\rightarrow \mathbf {R}$ that is continuously differentiable, positive-definite (that is $V(x)$ is positive except at $x=0$ where it is zero), and such that

\forall x\neq 0,\exists u\qquad {\dot {V}}(x,u)=\nabla V(x)\cdot f(x,u)<0.

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 control-Lyapunov function if and only if there exists a regular stabilizing feedback u(x).

It may not be easy to find a control-Lyapunov 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 non-linear programming problem

u^{*}(x)=\operatorname {*} {argmin}_{u}\nabla V(x)\cdot f(x,u)

for each state x.

The theory and application of control-Lyapunov functions were developed by Z. Artstein and E. D. Sontag in the 1980s and 1990s.

Example

Here is a characteristic example of applying a Lyapunov candidate function to a control problem.

Consider the non-linear system, which is a mass-spring-damper system with spring hardening and position dependent mass described by

m(1+q^{2}){\ddot {q}}+b{\dot {q}}+K_{0}q+K_{1}q^{3}=u

Now given the desired state, $q_d$ , and actual state, $q$ , with error, $e=q_{d}-q$ , define a function $r$ as

r={\dot {e}}+\alpha e

A Control-Lyapunov candidate is then

V={\frac {1}{2}}r^{2}

which is positive definite for all $q\neq 0$ , ${\dot {q}}\neq 0$ .

Now taking the time derivative of $V$

{\dot {V}}=r{\dot {r}}

{\dot {V}}=({\dot {e}}+\alpha e)({\ddot {e}}+\alpha {\dot {e}})

The goal is to get the time derivative to be

{\dot {V}}=-\kappa V

which is globally exponentially stable if $V$ is globally positive definite (which it is).

Hence we want the rightmost bracket of ${\dot {V}}$ ,

({\ddot {e}}+\alpha {\dot {e}})=({\ddot {q}}_{d}-{\ddot {q}}+\alpha {\dot {e}})

to fulfill the requirement

({\ddot {q}}_{d}-{\ddot {q}}+\alpha {\dot {e}})=-{\frac {\kappa }{2}}({\dot {e}}+\alpha e)

which upon substitution of the dynamics, ${\ddot {q}}$ , gives

({\ddot {q}}_{d}-{\frac {u-K_{0}q-K_{1}q^{3}-b{\dot {q}}}{m(1+q^{2})}}+\alpha {\dot {e}})=-{\frac {\kappa }{2}}({\dot {e}}+\alpha e)

Solving for $u$ yields the control law

u=m(1+q^{2})({\ddot {q}}_{d}+\alpha {\dot {e}}+{\frac {\kappa }{2}}r)+K_{0}q+K_{1}q^{3}+b{\dot {q}}

with $\kappa$ and $\alpha$ , 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

{\dot {V}}=-\kappa V

which is a linear first order differential equation which has solution

V=V(0)e^{-\kappa t}

And hence the error and error rate, remembering that $V={\frac {1}{2}}({\dot {e}}+\alpha e)^{2}$ , 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:

r{\dot {r}}=-{\frac {\kappa }{2}}r^{2}

{\dot {r}}=-{\frac {\kappa }{2}}r

r=r(0)e^{-{\frac {\kappa }{2}}t}

{\dot {e}}+\alpha e=({\dot {e}}(0)+\alpha e(0))e^{-{\frac {\kappa }{2}}t}

which can then be solved using any linear differential equation methods.

Notes

↑ Freeman (46)

References

Freeman, Randy A.; Petar V. Kokotović (2008). Robust Nonlinear Control Design (illustrated, reprint ed.). Birkhäuser. p. 257. ISBN 0-8176-4758-9. Retrieved 2009-03-04.

Control-Lyapunov function

Example

Notes

References

See also