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Definition

Consider an autonomous dynamical system with inputs

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

1

where x ∈ R n {\displaystyle x\in \mathbb {R} ^{n}} is the state vector and u ∈ R m {\displaystyle u\in \mathbb {R} ^{m}} is the control vector. Suppose our goal is to drive the system to an equilibrium x ∗ ∈ R n {\displaystyle x_{*}\in \mathbb {R} ^{n}} from every initial state in some domain D ⊂ R n {\displaystyle D\subset \mathbb {R} ^{n}} . Without loss of generality, suppose the equilibrium is at x ∗ = 0 {\displaystyle x_{*}=0} (for an equilibrium x ∗ ≠ 0 {\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 V : D → R {\displaystyle V:D\to \mathbb {R} } that is continuously differentiable, positive-definite (that is, V ( x ) {\displaystyle V(x)} is positive for all x ∈ D {\displaystyle x\in D} except at x = 0 {\displaystyle x=0} where it is zero), and such that for all x ∈ R n ( x ≠ 0 ) , {\displaystyle x\in \mathbb {R} ^{n}(x\neq 0),} there exists u ∈ R m {\displaystyle u\in \mathbb {R} ^{m}} such that

V ˙ ( x , u ) := ⟨ ∇ V ( x ) , f ( x , u ) ⟩ < 0 , {\displaystyle {\dot {V}}(x,u):=\langle \nabla V(x),f(x,u)\rangle <0,}

where ⟨ u , v ⟩ {\displaystyle \langle u,v\rangle } denotes the inner product of u , v ∈ R n {\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:

x ˙ = f ( x ) + ∑ i = 1 m g i ( x ) u i {\displaystyle {\dot {x}}=f(x)+\sum _{i=1}^{m}g_{i}(x)u_{i}}

2

where f : R n → R n {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{n}} and g i : R n → R n {\displaystyle g_{i}:\mathbb {R} ^{n}\to \mathbb {R} ^{n}} for i = 1 , … , m {\displaystyle i=1,\dots ,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 (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 : R n → R m {\displaystyle k:\mathbb {R} ^{n}\to \mathbb {R} ^{m}} directly in terms of the derivatives of the CLF.⁷: Eq. 5.56  In the special case of a single input system ( m = 1 ) {\displaystyle (m=1)} , Sontag's formula is written as

k ( x ) = { − L f V ( x ) + [ L f V ( x ) ] 2 + [ L g V ( x ) ] 4 L g V ( x )  if  L g V ( x ) ≠ 0 0  if  L g V ( x ) = 0 {\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 L f V ( x ) := ⟨ ∇ V ( x ) , f ( x ) ⟩ {\displaystyle L_{f}V(x):=\langle \nabla V(x),f(x)\rangle } and L g V ( x ) := ⟨ ∇ V ( x ) , g ( x ) ⟩ {\displaystyle L_{g}V(x):=\langle \nabla V(x),g(x)\rangle } are the Lie derivatives of V {\displaystyle V} along f {\displaystyle f} and g {\displaystyle g} , respectively.

For the general nonlinear system (1), the input u {\displaystyle u} can be found by solving a static non-linear programming problem

u ∗ ( x ) = a r g m i n u ∇ V ( x ) ⋅ f ( x , u ) {\displaystyle 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 ) q ¨ + b q ˙ + K 0 q + K 1 q 3 = u {\displaystyle m(1+q^{2}){\ddot {q}}+b{\dot {q}}+K_{0}q+K_{1}q^{3}=u}

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

r = e ˙ + α e {\displaystyle r={\dot {e}}+\alpha e}

A Control-Lyapunov candidate is then

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

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

Now taking the time derivative of V {\displaystyle V}

V ˙ = r r ˙ {\displaystyle {\dot {V}}=r{\dot {r}}} V ˙ = ( e ˙ + α e ) ( e ¨ + α e ˙ ) {\displaystyle {\dot {V}}=({\dot {e}}+\alpha e)({\ddot {e}}+\alpha {\dot {e}})}

The goal is to get the time derivative to be

V ˙ = − κ V {\displaystyle {\dot {V}}=-\kappa V}

which is globally exponentially stable if V {\displaystyle V} is globally positive definite (which it is).

Hence we want the rightmost bracket of V ˙ {\displaystyle {\dot {V}}} ,

( e ¨ + α e ˙ ) = ( q ¨ d − q ¨ + α e ˙ ) {\displaystyle ({\ddot {e}}+\alpha {\dot {e}})=({\ddot {q}}_{d}-{\ddot {q}}+\alpha {\dot {e}})}

to fulfill the requirement

( q ¨ d − q ¨ + α e ˙ ) = − κ 2 ( e ˙ + α e ) {\displaystyle ({\ddot {q}}_{d}-{\ddot {q}}+\alpha {\dot {e}})=-{\frac {\kappa }{2}}({\dot {e}}+\alpha e)}

which upon substitution of the dynamics, q ¨ {\displaystyle {\ddot {q}}} , gives

( q ¨ d − u − K 0 q − K 1 q 3 − b q ˙ m ( 1 + q 2 ) + α e ˙ ) = − κ 2 ( e ˙ + α e ) {\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 u {\displaystyle u} yields the control law

u = m ( 1 + q 2 ) ( q ¨ d + α e ˙ + κ 2 r ) + K 0 q + K 1 q 3 + b q ˙ {\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 κ {\displaystyle \kappa } and α {\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

V ˙ = − κ V {\displaystyle {\dot {V}}=-\kappa V}

which is a linear first order differential equation which has solution

V = V ( 0 ) exp ⁡ ( − κ t ) {\displaystyle V=V(0)\exp(-\kappa t)}

And hence the error and error rate, remembering that V = 1 2 ( e ˙ + α e ) 2 {\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 V {\displaystyle V} and solve for e {\displaystyle e} . This is left as an exercise for the reader but the first few steps at the solution are:

r r ˙ = − κ 2 r 2 {\displaystyle r{\dot {r}}=-{\frac {\kappa }{2}}r^{2}} r ˙ = − κ 2 r {\displaystyle {\dot {r}}=-{\frac {\kappa }{2}}r} r = r ( 0 ) exp ⁡ ( − κ 2 t ) {\displaystyle r=r(0)\exp \left(-{\frac {\kappa }{2}}t\right)} e ˙ + α e = ( e ˙ ( 0 ) + α e ( 0 ) ) exp ⁡ ( − κ 2 t ) {\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.

See also

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

References

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

2. Freeman, Randy A.; Petar V. Kokotović (2008). "Robust Control Lyapunov Functions". Robust Nonlinear Control Design (illustrated, reprint ed.). Birkhäuser. pp. 33–63. doi:10.1007/978-0-8176-4759-9_3. ISBN 978-0-8176-4758-2. Retrieved 2009-03-04. 978-0-8176-4758-2 ↩

3. Khalil, Hassan (2015). Nonlinear Control. Pearson. ISBN 9780133499261. 9780133499261 ↩

4. Sontag, Eduardo (1998). Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition (PDF). Springer. ISBN 978-0-387-98489-6. 978-0-387-98489-6 ↩

5. Sontag, E.D. (1983). "A Lyapunov-like characterization of asymptotic controllability". SIAM J. Control Optim. 21 (3): 462–471. doi:10.1137/0321028. S2CID 450209. /wiki/Doi_(identifier) ↩

6. Clarke, F.H.; Ledyaev, Y.S.; Sontag, E.D.; Subbotin, A.I. (1997). "Asymptotic controllability implies feedback stabilization". IEEE Trans. Autom. Control. 42 (10): 1394–1407. doi:10.1109/9.633828. /wiki/Doi_(identifier) ↩

7. Sontag, Eduardo (1998). Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition (PDF). Springer. ISBN 978-0-387-98489-6. 978-0-387-98489-6 ↩