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In , a control-Lyapunov function is a
{\displaystyle V(x)}
for a system with control inputs. The ordinary Lyapunov function is used to test whether a
is stable (more restrictively, asymptotically stable). That is, whether the system starting in a state
{\displaystyle x\neq 0}
in some domain D will remain in D, or for asymptotic stability will eventually return to
{\displaystyle 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
{\displaystyle 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
{\displaystyle {\dot {x}}=f(x,u)}
{\displaystyle x\in \mathbf {R} ^{n}}
is the state vector and
{\displaystyle u\in \mathbf {R} ^{m}}
is the control vector, and we want to feedback stabilize it to
{\displaystyle x=0}
in some domain
{\displaystyle D\subset \mathbf {R} ^{n}}
Definition. A control-Lyapunov function is a function
{\displaystyle V:D\rightarrow \mathbf {R} }
that is continuously differentiable, positive-definite (that is
{\displaystyle V(x)}
is positive except at
{\displaystyle x=0}
where it is zero), and such that
{\displaystyle \forall x\neq 0,\exists u\qquad {\dot {V}}(x,u)=\nabla V(x)\cdot f(x,u)&0.}
The last 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
{\displaystyle 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
in the 1980s and 1990s.
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
{\displaystyle m(1+q^{2}){\ddot {q}}+b{\dot {q}}+K_{0}q+K_{1}q^{3}=u}
Now given the desired state,
{\displaystyle q_{d}}
, and actual state,
{\displaystyle q}
, with error,
{\displaystyle e=q_{d}-q}
, define a function
{\displaystyle r}
{\displaystyle r={\dot {e}}+\alpha e}
A Control-Lyapunov candidate is then
{\displaystyle V={\frac {1}{2}}r^{2}}
which is positive definite for all
{\displaystyle q\neq 0}
{\displaystyle {\dot {q}}\neq 0}
Now taking the time derivative of
{\displaystyle V}
{\displaystyle {\dot {V}}=r{\dot {r}}}
{\displaystyle {\dot {V}}=({\dot {e}}+\alpha e)({\ddot {e}}+\alpha {\dot {e}})}
The goal is to get the time derivative to be
{\displaystyle {\dot {V}}=-\kappa V}
which is globally exponentially stable if
{\displaystyle V}
is globally positive definite (which it is).
Hence we want the rightmost bracket of
{\displaystyle {\dot {V}}}
{\displaystyle ({\ddot {e}}+\alpha {\dot {e}})=({\ddot {q}}_{d}-{\ddot {q}}+\alpha {\dot {e}})}
to fulfill the requirement
{\displaystyle ({\ddot {q}}_{d}-{\ddot {q}}+\alpha {\dot {e}})=-{\frac {\kappa }{2}}({\dot {e}}+\alpha e)}
which upon substitution of the dynamics,
{\displaystyle {\ddot {q}}}
{\displaystyle ({\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
{\displaystyle u}
yields the control law
{\displaystyle 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}}}
{\displaystyle \kappa }
{\displaystyle \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
{\displaystyle {\dot {V}}=-\kappa V}
which is a linear first order differential equation which has solution
{\displaystyle V=V(0)e^{-\kappa t}}
And hence the error and error rate, remembering that
{\displaystyle 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
{\displaystyle V}
and solve for
{\displaystyle e}
. This is left as an exercise for the reader but the first few steps at the solution are:
{\displaystyle r{\dot {r}}=-{\frac {\kappa }{2}}r^{2}}
{\displaystyle {\dot {r}}=-{\frac {\kappa }{2}}r}
{\displaystyle r=r(0)e^{-{\frac {\kappa }{2}}t}}
{\displaystyle {\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.
Freeman (46)
Freeman, Randy A.; Petar V. Kokotovi? (2008).
(illustrated, reprint ed.). Birkh?user. p. 257.  .