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 (Lyapunov) stable or (more restrictively) asymptotically stable. Lyapunov stability means that if the system starts in a state ${\displaystyle x\neq 0}$ 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 ${\displaystyle x=0}$. 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 ${\displaystyle u(x,t)}$ 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.

Consider an autonomous dynamical system with inputs

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

(1)

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

Definition. A control-Lyapunov function (CLF) is a function ${\displaystyle V:D\to \mathbb {R} }$ that is continuously differentiable, positive-definite (that is, ${\displaystyle V(x)}$ is positive for all ${\displaystyle x\in D}$ except at ${\displaystyle x=0}$ where it is zero), and such that for all ${\displaystyle x\in \mathbb {R} ^{n}(x\neq 0),}$ there exists ${\displaystyle u\in \mathbb {R} ^{m}}$ such that

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

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

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:

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

(2)

where ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}$ and ${\displaystyle g_{i}:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}$ for ${\displaystyle i=1,\dots ,m}$.

Eduardo Sontag showed that for a given control system, there exists a continuous CLF if and only if the origin is asymptotic stabilizable.^[5] It was later shown by Francis H. Clarke, Yuri Ledyaev, Eduardo Sontag, and A.I. Subbotin that every asymptotically controllable system can be stabilized by a (generally discontinuous) feedback.^[6] 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).

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 ${\displaystyle k:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}$ directly in terms of the derivatives of the CLF.^{[4]: Eq. 5.56} In the special case of a single input system ${\displaystyle (m=1)}$, Sontag's formula is written as

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

where ${\displaystyle L_{f}V(x):=\langle \nabla V(x),f(x)\rangle }$ and ${\displaystyle L_{g}V(x):=\langle \nabla V(x),g(x)\rangle }$ are the Lie derivatives of ${\displaystyle V}$ along ${\displaystyle f}$ and ${\displaystyle g}$, respectively.

For the general nonlinear system (1), the input ${\displaystyle u}$ can be found by solving 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.

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

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

which is positive for all ${\displaystyle r\neq 0}$.

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.

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