Control-Lyapunov function explained

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 inputswhere

x\in R ^{n} is the state vector and u\in R ^{m} is the control vector. Suppose our goal is to drive the system to an equilibrium x _{*} \in R ^{n} from every initial state in some domain D\subset R ^{n} . Without loss of generality, suppose the equilibrium is at x _{*=0} (for an equilibrium x _{* ≠} 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 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 R ^{n} (x ≠ 0 ), there exists u\in R ^{m} such that • 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 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:where

f : R ^{n} \to R ^{n} and g _{i} : R ^{n} \to R ^{n} for i = 1 , ... , m .

Theorems

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 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 : R ^{n} \to R ^{m} directly in terms of the derivatives of the CLF.^[4] In the special case of a single input system (m =1 ) , Sontag's formula is written ask(x) = \begin{cases} \displaystyle - (L _{f} V(x) + \sqrt{\left[L _{f} V(x)\right] ^{2} + \left[L _{g} V(x)\right] ^{4})/() } & \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 problemu ^{*(x)} = \underset{u}{\operatorname{argmin}} \nabla V(x) ⋅ 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 q} + K _{0q+K} ^{3= u} ^{1 q} Now given the desired state, q _{d} , and actual state, q , with error, e = q _{d} - q , define a function r as• r = e + \alpha e A Control-Lyapunov candidate is thenr \mapsto V(r) : = (1)/(2) r ^{2} which is positive for all r \ne 0 .

Now taking the time derivative of

V • V = r • r • V = ( • e + \alpha e)(\ddot{e} + \alpha • e )

The goal is to get the time derivative to be

• V =- \kappa V which is globally exponentially stable if V is globally positive definite (which it is).

Hence we want the rightmost bracket of

• V , (\ddot{e} + \alpha • e ) = (\ddot{q} _{d-\ddot{q}+\alpha} • e ) to fulfill the requirement(\ddot{q} _{d-\ddot{q}+\alpha} • e ) =- (\kappa)/(( 2) • e + \alpha e) which upon substitution of the dynamics, \ddot{q} , gives\left(\ddot{q} _{d - (u - K _{0 q - K} ^{• 3- b q} ^{1 q})/(m( 1+ q ^{2)})} + \alpha • e \right) =- (\kappa)/(( 2) • e + \alpha e) Solving for u yields the control lawu = m( 1+ q ^{2 )\left(\ddot{q}} _{d} + \alpha • e + (\kappa)/(2) r\right) + K _{0q+K} ^{• 3+ b q} ^{1 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

• V =- \kappa V which is a linear first order differential equation which has solutionV = V( 0 )\exp( - \kappa t)

And hence the error and error rate, remembering that

V = (1)/(( 2) • 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 r =- (\kappa)/(2) r ^{2} • r =- (\kappa)/(2) r r = r( 0 )\exp\left( - (\kappa)/(2) t\right) • e + \alpha e = ( • e ( 0 ) + \alpha e( 0 ))\exp\left( - (\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

Notes and References

1. Book: Isidori, A. . 1995 . Nonlinear Control Systems . Springer . 978-3-540-19916-8.
2. Book: Freeman , Randy A. . Petar V. Kokotović. Robust Nonlinear Control Design. Robust Control Lyapunov Functions. https://link.springer.com/chapter/10.1007/978-0-8176-4759-9_3. Birkhäuser. 2008. 33–63. 10.1007/978-0-8176-4759-9_3. illustrated, reprint. 978-0-8176-4758-2. 2009-03-04.
3. Book: Khalil, Hassan . 2015 . Nonlinear Control. Pearson . 9780133499261.
4. Book: Sontag, Eduardo . Eduardo D. Sontag . 1998 . Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition . Springer . 978-0-387-98489-6 .
5. E.D. . Sontag . A Lyapunov-like characterization of asymptotic controllability. SIAM J. Control Optim.. 21 . 3 . 1983 . 462–471. 10.1137/0321028 . 450209 .
6. F.H.. Clarke . Y.S.. Ledyaev . E.D.. Sontag . A.I.. Subbotin . Asymptotic controllability implies feedback stabilization . IEEE Trans. Autom. Control. 42 . 10 . 1997 . 1394–1407. 10.1109/9.633828 .

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