Hybrid Systems: Modeling and Analysis

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Hybrid Systems Modeling and Analysis

• David C. Conner
• dcconner_at_cmu.edu
• Robotics Institute
• Carnegie Mellon University

Center for the Foundations of Robotics
Microdynamic Systems Laboratory
Sensor Based Planning Lab
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Hybrid Systems

• Hybrid Systems combine both discrete and
  continuous dynamics.
• The dynamical evolution is given by equations of
  motion that depend on both continuous and
  discrete variables.
• Branicky et al., 1998

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Overview

• Hybrid Systems
• Examples
• Development History
• Modeling Formalisms
• Issues and Examples
• Hybrid Stability Analysis

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Hybrid Example

• Room with Thermostat and Heater

T Room temperature
T
T gt Tsp ?Tdeadband
Qheater
Heater Off
T
Heater On
Qloss
T lt Tsp
Qloss
Dynamics depend on both continuous and discrete
variables!
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Non-Hybrid Example

• Room with Lights

T
Switch Off
Lights Off
T
Lights On
Qloss
Switch On
Qloss
Dynamics depend only on continuous variables!
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Distinct Pedigrees

• Control Theory
• Stability and performance specifications
• Lyapunov theory for stability analysis Sastry,
  1999
• Linear and non-linear tools
• Variable structure with sliding mode control
  DeCarlo et al., 1988
• Computer Science
• Reachability and decidability
• Finite (discrete) Automata
• Predicate Logic
• Computational Tree Logic

Sliding Surface (? ) with Vector Fields
grad ?
f
f 0 ? f 0 (1-?) f 0
f 1
? 0
from DeCarlo et al., 1988
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Paradigm Convergence
Branicky, 1995
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Hybrid Phenomena
Continuous dynamics
Discrete phenomena arising in hybrid systems
Branicky, 1995
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Modeling Formalism
(Controlled) Dynamical System
or
X is an arbitrary topological space (state
space) ? is a transition semi-group
(time-like) ? is the extended transition map U
is the optional set of control inputs f is the
generator of ? ex. ??
Branicky, 1995
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Example Dynamical Systems
ODEs Finite Automata
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Modeling Formalism
Controlled General Hybrid Dynamical System
(CGHDS)
Q is the set of index states
Fj
Gi
G1
Hybrid State Space
Switching control parameters
From Branicky, 1995
Destination Set
Branicky, 1995
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Classification of Hybrid Systems

• Dynamical System (C)GHDS with Q 1 and A?
• Hybrid System (C)GHDS with Q countable,
  (or ?), Z
• Switched System Hybrid System with
• Continuous Switched System Switched System with

Z
i ?Q 1,,N,
each fi globally Lipschitz continuous,
finite switches in finite time (non-zeno).
constraint that the switched vector fields agree
at the switching time.
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Model Comparison
Multigraph
from Branicky, 1995
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Example Hybrid System Model

• Bouncing Ball

Q 1
Hybrid Automata
Branickys Hybrid System
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Issues for Hybrid Systems

• Reachability
• Can I get to a given state (vertex)?
• Decidability
• Given a specification,
• is the set of vertices satisfying the
  specification an empty set?
• (halting problem)
• Liveliness
• Deadlock or Zeno? (ex., bouncing ball is Zeno)
• Stability
• Is the resulting system stable?

Henzinger, 1996
Henzinger, 1996
Henzinger, 1996
Branicky, 1995
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Overview

• Hybrid Systems
• Hybrid Stability Analysis
• Motivating Examples
• Lyapunov Methods
• Lyapunov Analysis Review
• Single Lyapunov Functions
• Multiple Lyapunov Functions
• Partial Order Methods

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Hybrid Stability Motivating Example
Branicky, 1998
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Hybrid Stability Motivating Example
Branicky, 1998
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Hybrid Stability Motivating Example
Switched Linear System
1
2
1
2
Branicky, 1998
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Hybrid Stability Motivating Example
2
1
2
1
Branicky, 1998
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Hybrid Stability Motivating Example

• Take Home Message
• Stability of the underlying systems does not
  imply stability of the switched system

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Hybrid Stability Motivating Example
1
2
2
1
Branicky, 1998
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Hybrid Stability Motivating Example

• Take Home Message 2
• Instability of the underlying systems does not
  imply instability of the switched system

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Lyapunov Refresher

• Stability

(in the sense of Lyapunov)
x 0 is a stable equilibrium point of
if
such that
Uniform Stability
? independent of t0
Asymptotic Stability
x is attractive
lim
Sastry, 1999
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Lyapunov Refresher
Locally Positive Definite (lpdf)
?K
Positive Definite (pdf)
Decresent
NOTE Negative
Sastry, 1999
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Lyapunov Analysis of Hybrid Systems
Common Lyapunov Function

• Given
• If there exists a common Lyapunov function V(x)
  such that and , then the switched
  system is asymptotically
  stable for all switching strategies.

DeCarlo et al., 1998
Liberzon and Morse, 1999
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Lyapunov Analysis of Hybrid Systems
Lyapunov Contours
Linear Switched Systems
Common Lyapunov Function
Liberzon and Morse, 1999
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Lyapunov Analysis of Hybrid Systems

• Multiple Lyapunov Functions
• Given a switched system
• If there exist Lyapunov functions for each
  switching region, and if the Lyapunov value at

the time the system switches into a specific
region is less than the value at the time the
system last switched into the same region, then
the system is stable in the sense of
Lyapunov. Other theorems strengthen this to
asymptotic stability.
DeCarlo et al., 2000, Branicky, 1995
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Multiple Lyapunov Functions
1
2
2
1
1
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Partial Ordering

• If the switching strategy induces a partial
  ordering on the sequence of visited indices, and
  is guaranteed to switch to a stable controller
  last, the stability of the resulting composition
  is guaranteed.

Burridge, Rizzi, Koditschek used composition of
asymptotically stable controlled systems to
guarantee such a monotonic switching policy.
Burridge et al., 1999
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Conclusions

• Hybrid systems are here to stay
• Industrial processes are hybrid due to interlocks
  and safety systems
• Complex flight controllers are generally
  switched/hybrid systems
• Increasing prevalence of embedded controllers
• Conceptual framework allows us to begin analysis
• Choice of hybrid system or hybrid automata
  formalism depends mainly on user preference and
  what we want to prove
• Formalisms are interchangeable for most models
• Stability analysis is non-trivial
• Some results for stable switched systems
• More work needed for general systems
• Design tools are incomplete
• More work needed to enable design of
  switching/hybrid controllers
• Limited results for switched linear systems
  Liberzon and Morse, 1999

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References
Michael S. Branicky, Studies in Hybrid Systems
Modeling, Analysis, and Control, Ph.D.
Dissertation, Dept. of Elec. Eng. And Computer
Sci., MIT, June 1995.
Branicky et al., A unified framework for hybrid
control Model and optimal control theory, IEEE
Transactions on Automatic Control, Vol. 43, No.
1, January 1998.
Michael S. Branicky, Multiple Lyapunov functions
and other analysis tools for switched and hybrid
systems, IEEE Transactions on Automatic Control,
Vol. 43, No. 4, April 1998.
DeCarlo et al., Perspectives and results on the
stability and stabilizability of hybrid systems,
Proceedings of the IEEE, Vol. 88, No. 7, July
2000.
DeCarlo et al., Variable structure control of
nonlinear multivariable systems A tutorial,
Proceedings of the IEEE, Vol. 76, No. 3, March,
1988.
Thomas A. Henzinger, The theory of hybrid
automata, Proceedings of the 11th Annual
Symposium on Logic in Computer Science (LICS 96),
IEEE Computer Society Press, 1996, pp. 278-292.
An extended version appeared in Verification of
Digital and Hybrid Systems (M.K. Inan, R.P.
Kurshan, eds.), NATO ASI Series F Computer and
Systems Sciences, Vol. 170, Springer-Verlag,
2000, pp. 265-292.
Liberzon and Morse, Basic problems in stability
and design of switched systems, IEEE Control
Systems, October, 1999.
Shankar Sastry, Nonlinear Systems Analysis,
Stability, and Control, Springer-Verlag, New
York, NY, 1999.
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Groups,Semi-groups, and Monoids

• Group
• A group G is a finite or infinite set of
  elements together with a binary operation which
  together satisfy the four fundamental properties
  of closure, associativity, the identity property,
  and the inverse property.
• Elements A, B, C, ... with binary operation
  between A and B denoted AB form a group if
• 1. Closure If A and B are two elements in G,
  then the product AB is also in G.
• 2. Associativity The defined operation is
  associative. (AB)C A(BC)
• 3. Identity There is an identity element I. IA
  AI A
• 4. Inverse There must be an inverse or
  reciprocal of each element. A-1A AA-1 I

Semi-group A semi-group G is a finite or
infinite set of elements together with a binary
operation that is associative. (no other
restrictions)
Monoid A semi-group with an identity element.
www.mathworld.com
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Multiple Lyapunov Functions

• Theorem for Linear Switched System
• Given a system with equilibrium ,
  and a collection of Lyapunov functions

Suppose that . For i lt j,
let ti lt tj be switching times for which
, and suppose there exists such
that then the system with switching
p(t) is globally asymptotically stable
Peleties and DeCarlo, 1991
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Multiple Lyapunov Functions

• Theorem for General Switched Systems
• Given a system with equilibrium ,
  and a collection of Lyapunov functions as
  before.

Suppose that . Let p(t) be
switching sequence such that p(t) can take on
the value of i only if , and in
addition where denotes the kth time that
vector field fi is switched in, then
is stable in the sense of Lyapunov.
DeCarlo et al., 2000, Branicky, 1995