Stability and stabilization of hybrid systemsMikael Johansson
School of Electrical Engineering
KTH
Stability and stabilization of hybrid systems
Mikael Johansson
School of Electrical Engineering
KTH
Goals and class structure
Three lectures:
Stability theory
Computational tools for piecewise linear systems
Applications
Goal: After these lectures, you should
Know some basic theory for stability and stabilization of hybrid systems
Be familiar with the computational methods for piecewise linear systems
Understand how the tools can be applied to (relatively) practical systems
Part I – Stability theory
Acknowledgements: M. Heemels, TU/e
Outline:
A hybrid systems model and stability concepts
Lyapunov theory for smooth systems
Lyapunov theory for stability and stabilization of hybrid systems
A hybrid systems model
Unless stated otherwise, we will assume that is piecewise continuous
(i.e., that there is only a finite number of mode changes per unit time)
The discrete state indexes vector fields while
is the transition function describing the evolution of the discrete state. For now, disregard issues with sliding modes, zeno, … (see refs for details) We consider hybrid systems on the form
where
Example: a switched linear system
(numerical values for matrices Ai are given in notes for Lecture 2)
Stability concepts
Focus: stability of equilibrium point (in continuous state-space)
Global asymptotic stability (GAS): ensure that
Global uniform asymptotic stability (GUAS): ensure that
(i.e., uniformly in )
Three fundamental problems
Problem P3: determine if a given switched system
is globally asymptotically stable. Problem P2: Given vector fields , design strategy :
is globally asymptotically stable. Problem P1: Under what conditions is
GAS for all (piecewise continuous) switching signals ?
Part I – Stability theory
Outline:
A hybrid systems model and stability concepts
Lyapunov theory for smooth systems
Lyapunov theory for stability and stabilization of hybrid systems
Aim: establishing common grounds by reviewing fundamentals.
Lyapunov theory for smooth systems
Interpretation: Lyapunov function is abstract measure of system energy, system energy should decrease along all trajectories.
Converse theorem
Under appropriate technical conditions (mainly smoothness of vector fields)
Consequence: worthwhile to search for Lyapunov functions
Remaining challenge: how to perform Lyapunov function search?
Stability of linear systems
Partial proof
Stability of discrete-time systems
Interpretation: energy should decrease at each sampling instant (event)
Performance analysis
Lyapunov techniques also useful for estimating system performance.
Part I – Stability theory
Outline:
A hybrid systems model and stability concepts
Lyapunov theory for smooth systems
Lyapunov theory for stability and stabilization of hybrid systems
Content:
Guaranteeing stability independent of switching strategy
Design a stabilizing switching strategy
Prove stability for a given switching strategy
Switching between stable systems
Q: does switching between stable dynamics always create stable motions?
A: no, not necessarily.
Subsystems are stable and share the same eigenvalues,
but stability depends on switching!
P1: Stability for arbitrary switching signals
Claim: only if each subsystem
admits a radially unbounded Lyapunov function.
(can you explain why?)
Problem: when is the switched system
GAS for all (piecewise continuous) switching signals ?
The common Lyapunov function approach
In fact, if the submodels are smooth, the following results hold.
Hence, common Lyapunov functions necessary and sufficient.
Switched linear systems
For switched linear systems
it is natural to look for a common quadratic Lyapunov function
is a common Lyapunov function if
Such a Lyapunov function can be found by solving linear matrix inequalities
(systems that admit quadratic V(x) are called quadratically stable)
Infeasibility test
It is also possible to prove that there is no common quadratic Lyapunov fcn:
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