Wikipedia / Control-Lyapunov function
Wikipedia / Control-Lyapunov function
In control theory, a control-Lyapunov function (CLF) is an extension of the idea of Lyapunov function 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 in some domain D, then the state will remain in D for all time. For asymptotic stability. A control-Lyapunov function is used to test whether a system is asymptotically stabilizable, that is whether for any state x 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 . Without loss of generality, it can be translated to the origin by a change of variables).
Definition. that is continuously differentiable such that
denotes the inner product.
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: .
Theorems
Eduardo 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, Yuri Ledyaev, Eduardo Sontag, and A.I. Subbotin 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 directly in terms of the derivatives of the CLF., Sontag's formula is written as
are the Lie derivatives satisfies the small control property is continuous.
For the general nonlinear system ( can be found by solving a static non-linear programming problem
.
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
as
A Control-Lyapunov candidate is then
.
Now taking the time derivative
The goal is to get the time derivative to be
is globally positive definite (which it is).
,
to fulfill the requirement
, gives
yields the control law
, 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
which is a linear first order differential equation which has solution
, exponentially decay to zero.
. This is left as an exercise for the reader but the first few steps at the solution are:
which can then be solved using any linear differential equation methods.