1. Input-to-State Stability for Switched Systems
with Unstable Subsystems:
A Hybrid Lyapunov Construction
Guosong Yang and Daniel Liberzon
Coordinated Science Laboratory
University of Illinois at Urbana-Champaign
Urbana, IL 61801
11/8/2018
I am Guosong Yang from University of Illinois. Today I am going to talk about our work on input-to-state stability for switched systems. This is a joint work with my advisor, Prof. Daniel Liberzon. In this paper, we investigated a very general class of switched systems, namely, switched systems with non-ISS subsystems. While the sufficient condition established here has been proof in an earlier work via trajectory analysis, a new proof is given in this paper with the construction of a novel timer and a hybrid ISS Lyapunov function.

2. Outline Problem formulation and preliminaries
Assumptions and sufficient condition
Proof: auxiliary timer and hybrid ISS Lyapunov function
Simulation
Conclusion
I will start my presentation by formulating the problem we try to solve and introduce several definitions and preliminaries.

3. Problem Formulation Switched system is a family of dynamical systems
is the index set (arbitrary)
is the switching signal (piecewise constant, right- continuous)
doesn’t change at switches
Stability property of switched systems
Mathematical model of a switched system
For each p, the differential equation defines a dynamical system, called a subsystem (modes)
The set of all p is called the index set, calligraphic P, usually denoted by integers
Switching signal, which, as the definition suggests, specifies which subsystem is active
Piecewise continuous and conventionally right-continuous
In this work, the value of the state remains unchanged when a switch occurs.

4. Input-to-State Stability
A dynamical system is input-to-state stable (ISS) if
for all initial condition and all input .
: increasing in , , decreasing in and
: increasing, unbounded and
denotes the essential supremum norm on
For an autonomous system, ISS is equivalent to global asymptotic stability (GAS)
Input-to-state stability of a switched system
ISS if satisfies this formula, where
Increases w.r.t. the first argument, while decreases to zero w.r.t. the second argument
The norm used here is the ess.sup norm, an upper-bound of u on this interval except for a set of measure zero
Noteworthy that …, which is usually abbreviated as GAS

5. Problem Formulation Question: When is the switched system ISS?
Switched system with non-ISS subsystems
The index set
contains the indices of ISS subsystems
contains the indices of non-ISS subsystems
Question: When is the switched system ISS?
Obstacles:
Non-ISS subsystems
Switching
We study the input-to-state property of a switched system with non-ISS subsystems,
We have a partition of the index set P
Ps is a subset contains …
Pu …
Goal: sufficient condition which guarantees that the switched system is input-to-state stable
There are two type of undesired behaviors that prevent the system from being ISS
First non-ISS, which is clearly undesirable
Second switching, now explain

6. Stability Issue of Switching
Example:
This is an example which demonstrates the stability issue cause by switching.
There are two subsystems, both are stable as shown in the first two trajectory plots.
The third trajectory plot describes a switching signal under which the switched system is unstable
This example demonstrates that …
Asymptotic stability of subsystems are not sufficient
for stability of the switched system

7. Slow Switching Average dwell-time condition [Hespanha-Morse-99]
is the number of switches on
is called the average dwell-time
is the chatter bound
- no switch: cannot switch for any
- dwell-time: can only switch once for any
To handle this problem, a standard method is to constrain the switching to be slow enough
In this work, we adopted the ADT condition, which is introduced by Hespanha and Morse in 1999
This is the formula of an ADT condition …
Tau_a is called the average dwell-time, which, as the name suggests, describes how often the system may switch in average.
The constant N0 is the chatted bound.
We should notice that if N0 is zero, then there is no switch at all
If N0 is one, the ADT condition is equivalent to the dwell-time condition, which mean the time between any two switches should be at leat tau_a.

8. Slow Switching Theorem
Average dwell-time condition [Hespanha-Morse-99]
Theorem
is GAS if
The result in literature shows that, if all subsystem are GAS, which is specified by Lyapunov functions; and the Lyapunov function are compatible, then the switched system is GAS giving the ADT is large enough

9. Outline Problem formulation and preliminaries
Assumptions and sufficient condition
Proof: auxiliary timer and hybrid ISS Lyapunov function
Simulation
Conclusion
After talking about the problem formulation and preliminaries, in the next section I am going to talk about the assumptions we impose on the switched system and explain the sufficient condition we have for the ISS of the switched system.

10. Assumptions on dynamics
Switched system with non-ISS subsystem
Uniform ISS Lyapunov-type Constraint
s.t.
due to ISS
[Praly-Wang-96]
For each subsystem p, there exist a Vp, called ISS Lyapunov-type function.
The ISS Lyapunov-type functions Vp are supposed to satisfy the following … for all values of x.
The equations in this constraint may look complex, but the idea and the motivation behind are quite natural.
The first condition is the standard condition on the existence of comparison functions for ISS Lyapunov functions.
The second condition specifies the behavior of these ISS Lyapunov-type functions along solutions.
It means that, if the input u is not too large, which is described by this inequality with a class k infinity gain function.
For an ISS subsystems p_{s}, the Vp_{s} is decreasing along the solution. The idea of this assumption is that ISS is equivalent to the existence of an ISS Lyapunov functions, which is prove by Praly and Wong in 96.
For a non-ISS subsystems p_{u}, this means the Vp_{u} may increase along the solution, but the exponential growth rate is bounded. The idea of this assumption is that non-ISS subsystems are supposed to be forward complete, which makes sense as …. The existence of a function Vp_{u} for a forward complete system is proved by Angeli and Sontag in 99.
To be precise, ISS Lyapunov-type
While the value of the state x doesn’t change when the active subsystem switches from q to p, the values of the functions Vp, Vq may be different. The third condition means that the ratio of these values are bound for all x. It is somewhat restrictive. For example, it doesn’t hold is Vp is quadratic and Vq is square. But it is a considerable relaxation to a more restrictive condition, namely, the existence of common ISS Lyapunov function. If your interested in this condition, a more detailed discussion can be found in a paper …
All these three conditions hold for all subsystems in the corresponding class, therefore UNIFORM
forward complete
[Angeli-Sontag-99]
discussion in
[Vu-Chatterjee-L-07]

11. Assumptions on switching
Switched system with non-ISS subsystem
Average dwell-time constraint [Hespanha-Morse-99]
s.t.
Time-ratio constraint [Zhai et al.-00]
s.t.
is the total activation time of non-ISS subsystems on , i.e., when
First, we assume the switching is slow enough in the form of this ADT condition.
Second, we assume the switching signal satisfies the time-ratio constraint introduced by Zhai and coauthors in On an interval, the term on the left hand side denotes how long the active subsystem in non-ISS. Therefore, this constraint means that the active subsystem is not non-ISS for too long.

12. Sufficient condition for ISS
Switched system with non-ISS subsystem
Theorem [Müller-L-2012, Theorem 2]
Suppose all previous assumptions hold. Then the switched system is ISS if
Interpretation: average rate of exponential decay
due to ISS
due to non-ISS
due to switching

13. Outline Problem formulation and preliminaries
Assumptions and sufficient condition
Proof: auxiliary timer and hybrid ISS Lyapunov function
Simulation
Conclusion
Stability property of switched systems
Mathematical model of a switched system
For each p, the differential equation defines a dynamical system, called a subsystem
The set of all p is called the index set, calligraphic capital letter P, usually denoted by integers
Switching signal, which, as the definition suggests, specifies which subsystem is active
Piecewise continuous and conventationally right-continuous

14. Sketch of Proof ISS Lyapunov-type functions of subsystems
Concatenation of ISS Lyapunov-type functions

15. Sketch of Proof ISS Lyapunov-type functions of subsystems
Concatenation of ISS Lyapunov-type functions
should be
decrease when non-ISS
decrease at switches
Increase when ISS

16. Sketch of Proof ISS Lyapunov-type functions of subsystems
Concatenation of ISS Lyapunov-type functions
should be
decrease when non-ISS
decrease at switches
Increase when ISS

17. Sketch of Proof How? Auxiliary timer and hybrid system
ISS Lyapunov-type functions of subsystems
Concatenation of ISS Lyapunov-type functions
should be
decrease when non-ISS
decrease at switches
Increase when ISS
How?
Auxiliary timer and hybrid system

18. Auxiliary Timer
An auxiliary timer that models the constraints on the switching
increases when an ISS subsystem is active
decreases when a non-ISS subsystem is active
decreases when a switch occurs

19. Hybrid System Proposition 1
An auxiliary timer that models the constraints on the switching
A hybrid system with the state
Proposition 1
Suppose all assumptions hold. For any solution to the switched system, there is a complete solution to the hybrid system s.t.
Proof:
Construct a hybrid solution based on the dynamics of the timer
Assumptions guarantee that the solution is complete

20. Hybrid ISS Lyapunov Function
An auxiliary timer that models the constraints on the switching
A hybrid system with the state
A hybrid ISS Lyapunov function
Proposition 2
decreases along all hybrid solutions to the hybrid system
Proposition 3
Every complete solution to the hybrid system satisfies

21. Outline Problem formulation and preliminaries
Assumptions and sufficient condition
Proof: auxiliary timer and hybrid ISS Lyapunov function
Simulation
Conclusion
Stability property of switched systems
Mathematical model of a switched system
For each p, the differential equation defines a dynamical system, called a subsystem
The set of all p is called the index set, calligraphic capital letter P, usually denoted by integers
Switching signal, which, as the definition suggests, specifies which subsystem is active
Piecewise continuous and conventationally right-continuous

22. Simulation Subsystem 1 Subsystem 2: ISS Lyapunov functions
Switching signal and timer

23. Simulation Subsystem 1 Subsystem 2: ISS Lyapunov functions
Switching signal and timer

24. Outline Problem formulation and preliminaries
Assumptions and sufficient condition
Proof: auxiliary timer and hybrid ISS Lyapunov function
Simulation
Conclusion
Stability property of switched systems
Mathematical model of a switched system
For each p, the differential equation defines a dynamical system, called a subsystem
The set of all p is called the index set, calligraphic capital letter P, usually denoted by integers
Switching signal, which, as the definition suggests, specifies which subsystem is active
Piecewise continuous and conventationally right-continuous

25. Conclusion Contribution Timer Similar techniques in previous results:
[L-03] for switched system
[Hespanha-L-Teel-08] for impulsive system
[L-Nešić-Teel-14] for hybrid system
Advantage: our construction is able to handle both switches and non-ISS subsystems

26. Conclusion Contribution Timer Hybrid ISS Lyapunov function
Application: interconnected switched systems [Y-L-SCL]
For each switched system, construct a hybrid ISS Lyapunov function
Construction a Lyapunov function of the interconnected system based on
Global asymptotic stability of the interconnected system

27. Conclusion Summary ISS of a switched system with non-ISS subsystems
Constraints on dynamics
Constraints on switching
Sufficient condition for ISS
Auxiliary timer
Hybrid ISS Lyapunov function
In this paper, we investigated the input-to-state stability of a switched system with non-ISS subsystems.
We showed that, if the undesirable behaviors are not too severe, that is, the dynamics of the subsystems satisfy the uniform ISS Lyapunov-type constraint, and the switching is slow enough and the active subsystem is ISS for a enough fraction of time, then a sufficient condition can be formulated to guarantee the ISS of the switched. While this sufficient condition have be discover previously, a new proof is provided by a hybrid Lyapunov approach. Our major contributions include the construction of a timer which can handle multiple undesired behaviors, and the construction of a hybrid ISS Lyapunov function which is used later in the analysis of more complex systems.

28. Thank you for your attention!