Control-Lyapunov function

In control theory, a control-Lyapunov function (cLf)^[1][2][3][4] is an extension of the idea of Lyapunov function ${\displaystyle V(x)}$ to systems 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 ${\displaystyle x\neq 0}$ 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 asymptotically 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 asymptotically by applying the control u.

More formally, suppose we are given an autonomous dynamical system with inputs

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

where ${\displaystyle x\in \mathbb {R} ^{n}}$ is the state vector and ${\displaystyle u\in \mathbb {R} ^{m}}$ is the control vector, and we want to drive states to an equilibrium, let us ${\displaystyle x=0}$, from every initial state in some domain ${\displaystyle D\subset \mathbb {R} ^{n}}$.

This notion was introduced by E. D. Sontag in ^[5] who showed that the existence of a continuous cLf is equivalent to asymptotic stabilizability. It was later shown that every asymptotically controllable system can be stabilized by a (generally discontinuous) feedback.^[6] One may also ask when there is a continuous feedback stabilizer. For systems affine on controls, and differentiable cLf's, the definition translates as follows:

Definition. A control-Lyapunov function is a function ${\displaystyle V:D\rightarrow \mathbb {R} }$ 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 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 asymptotically to zero, that is to bring the system to a stop. This is made rigorous by Artstein's theorem, repeated here:

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. The Sontag's universal formula writes the feedback law directly in terms of the derivatives of the cLf.^[7][8] An alternative is to solve a static non-linear programming problem

${\displaystyle u^{*}(x)={\underset {u}{\operatorname {arg\,min} }}\nabla V(x)\cdot f(x,u)}$

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

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

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

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

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, ${\displaystyle {\ddot {q}}}$, gives

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

Solving for ${\displaystyle u}$ yields the control law

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

with ${\displaystyle \kappa }$ and ${\displaystyle \alpha }$, 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

${\displaystyle V=V(0)\exp(-\kappa t)}$

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:

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

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

${\displaystyle r=r(0)\exp \left(-{\frac {\kappa }{2}}t\right)}$

${\displaystyle {\dot {e}}+\alpha e=({\dot {e}}(0)+\alpha e(0))\exp \left(-{\frac {\kappa }{2}}t\right)}$

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

Notes[]

References[]

• Isidori, A. (1995). Nonlinear Control Systems. Springer. ISBN 978-3-540-19916-8.

• 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.
• Khalil, Hassan (2015). Nonlinear Control. Pearson. ISBN 9780133499261.
• Sontag, Eduardo (1998). Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition (PDF). Springer. ISBN 978-0-387-98489-6.

See also[]

• Artstein's theorem
• Lyapunov optimization
• Drift plus penalty