Control-Lyapunov Function

In control theory, a control-Lyapunov function (CLF) is an extension of the idea of Lyapunov function $V(x)$ to systems with control inputs. The ordinary Lyapunov function is used to test whether a dynamical system is ''(Lyapunov) stable'' or (more restrictively) ''asymptotically stable''. Lyapunov stability means that if the system starts in a state $x \ne 0$ in some domain ''D'', then the state will remain in ''D'' for all time. For ''asymptotic stability'', the state is also required to converge to $x = 0$. A control-Lyapunov function is used to test whether a system is ''asymptotically 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 asymptotically by applying the control ''u''.

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

Definition

Consider an autonomous dynamical system with inputs

where $x\in\mathbb^n$ is the state vector and $u\in\mathbb^m$ is the control vector. Suppose our goal is to drive the system to an equilibrium $x_* \in \mathbb^n$ from every initial state in some domain $D\subset\mathbb^n$. Without loss of generality, suppose the equilibrium is at $x_*=0$ (for an equilibrium $x_*\neq 0$, it can be translated to the origin by a change of variables).

Definition. A control-Lyapunov function (CLF) is a function $V : D \to \mathbb$ that is continuously differentiable, positive-definite (that is, $V(x)$ is positive for all $x\in D$ except at $x=0$ where it is zero), and such that for all $x \in \mathbb^n (x \neq 0),$ there exists $u\in \mathbb^m$ such that
:$\dot(x, u) := \langle \nabla V(x), f(x,u)\rangle < 0,$
where $\langle u, v\rangle$ denotes the inner product of $u, v \in\mathbb^n$.

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.

Some results apply only to control-affine systems—i.e., control systems in the following form:

where $f : \mathbb^n \to \mathbb^n$ and $g_i : \mathbb^n \to \mathbb^$ for $i = 1, \dots, m$.

Theorems

E. D. Sontag showed that for a given control system, there exists a continuous CLF if and only if the origin is asymptotic stabilizable. It was later shown by Francis H. Clarke that every asymptotically controllable system can be stabilized by a (generally discontinuous) feedback.
Artstein proved that the dynamical system () has a differentiable control-Lyapunov function if and only if there exists a regular stabilizing feedback ''u''(''x'').

Constructing the Stabilizing Input

It is often difficult to find a control-Lyapunov function for a given system, but if one is found, then the feedback stabilization problem simplifies considerably. For the control affine system (), ''Sontag's formula'' (or ''Sontag's universal formula'') gives the feedback law $k : \mathbb^n \to \mathbb^m$ directly in terms of the derivatives of the CLF.Sontag (1998). ''Mathematical Control Theory'', Equation 5.56 In the special case of a single input system $(m=1)$, Sontag's formula is written as
:$k(x) = \begin \displaystyle -\frac & \text L_ V(x) \neq 0 \\ 0 & \text L_ V(x)=0 \end$
where $L_f V(x) := \langle \nabla V(x), f(x)\rangle$ and $L_g V(x) := \langle \nabla V(x), g(x)\rangle$ are the Lie derivatives of $V$ along $f$ and $g$, respectively.

For the general nonlinear system (), the input $u$ can be found by solving a static non-linear programming problem
:$u^*(x) = \underset \nabla V(x) \cdot f(x,u)$
for each state ''x''.

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
:$m(1+q^2)\ddot+b\dot+K_0q+K_1q^3=u$
Now given the desired state, $q_d$, and actual state, $q$, with error, $e = q_d - q$, define a function $r$ as
:$r=\dot+\alpha e$
A Control-Lyapunov candidate is then
:$V=\fracr^2$
which is positive definite for all $q \ne 0$, $\dot \ne 0$.

Now taking the time derivative of $V$
:$\dot=r\dot$
:$\dot=(\dot+\alpha e)(\ddot+\alpha \dot)$

The goal is to get the time derivative to be
:$\dot=-\kappa V$
which is globally exponentially stable if $V$ is globally positive definite (which it is).

Hence we want the rightmost bracket of $\dot$,
:$(\ddot+\alpha \dot)=(\ddot_d-\ddot+\alpha \dot)$
to fulfill the requirement
:$(\ddot_d-\ddot+\alpha \dot) = -\frac(\dot+\alpha e)$
which upon substitution of the dynamics, $\ddot$, gives
:$\left(\ddot_d-\frac+\alpha \dot\right) = -\frac(\dot+\alpha e)$
Solving for $u$ yields the control law
:$u= m(1+q^2)\left(\ddot_d + \alpha \dot+\fracr\right)+K_0q+K_1q^3+b\dot$
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
:$\dot=-\kappa V$
which is a linear first order differential equation which has solution
:$V=V(0)\exp(-\kappa t)$

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

:$r\dot=-\fracr^2$
:$\dot=-\fracr$
:$r=r(0)\exp\left(-\frac t\right)$
:$\dot+\alpha e= (\dot(0)+\alpha e(0))\exp\left(-\frac t\right)$
which can then be solved using any linear differential equation methods.

Notes

References

*

*
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*{{cite book , last = Sontag , first = Eduardo , author-link = Eduardo D. Sontag , year = 1998 , title = Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition , publisher = Springer , url = http://www.sontaglab.org/FTPDIR/sontag_mathematical_control_theory_springer98.pdf , isbn = 978-0-387-98489-6

See also

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

Stability theory