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 $x\neq 0$ 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

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

(1)

where $x\in \mathbb {R} ^{n}$ is the state vector and $u\in \mathbb {R} ^{m}$ is the control vector. Suppose our goal is to drive the system to an equilibrium $x_{*}\in \mathbb {R} ^{n}$ from every initial state in some domain $D\subset \mathbb {R} ^{n}$. 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 $V:D\to \mathbb {R}$ 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 $x\in \mathbb {R} ^{n}(x\neq 0),$ there exists $u\in \mathbb {R} ^{m}$ such that

${\dot {V}}(x,u):=\langle \nabla V(x),f(x,u)\rangle <0,$

where $\langle u,v\rangle$ denotes the inner product of $u,v\in \mathbb {R} ^{n}$.

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:

${\dot {x}}=f(x)+\sum _{i=1}^{m}g_{i}(x)u_{i}$

(2)

where $f:\mathbb {R} ^{n}\to \mathbb {R} ^{n}$ and $g_{i}:\mathbb {R} ^{n}\to \mathbb {R} ^{n}$ 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 (2) 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 (2), Sontag's formula (or Sontag's universal formula) gives the feedback law $k:\mathbb {R} ^{n}\to \mathbb {R} ^{m}$ directly in terms of the derivatives of the CLF. In the special case of a single input system $(m=1)$, Sontag's formula is written as

$k(x)={\begin{cases}\displaystyle -{\frac {L_{f}V(x)+{\sqrt {\left[L_{f}V(x)\right]^{2}+\left[L_{g}V(x)\right]^{4}}}}{L_{g}V(x)}}&{\text{ if }}L_{g}V(x)\neq 0\\0&{\text{ if }}L_{g}V(x)=0\end{cases}}$

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 (1), the input $u$ can be found by solving a static non-linear programming problem

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

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

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

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

$r\mapsto V(r):={\frac {1}{2}}r^{2}$

which is positive for all $r\neq 0$.

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

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

Solving for $u$ yields the control law

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

with $\kappa$ 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

$V=V(0)\exp(-\kappa t)$

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:

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

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

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

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

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

See also

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