1 Formal Models for Stability Analysis : Verifying Average Dwell Time* Sayan Mitra MIT,CSAIL [email protected]Research Qualifying Exam 20 th December 2004 Joint work with Daniel Liberzon (UIUC) and Nancy Lynch (MIT) * Full version of the paper has been sent for
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1 Formal Models for Stability Analysis : Verifying Average Dwell Time * Sayan Mitra MIT,CSAIL [email protected] Research Qualifying Exam 20 th December.
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1
Formal Models for Stability Analysis : Verifying Average Dwell Time*
Joint work with Daniel Liberzon (UIUC) and Nancy Lynch (MIT)
* Full version of the paper has been sent for journal review.
2Verifying Average Dwell Time
A common math model (HIOA) Expressive: few constraints on continuous and discrete behavior
Compositional: analyze complex systems by looking at parts
Structured: inductive verification
Compatible: application of CT results e.g. stability, synthesis
Motivation: Macro
Control Theory: Dynamical system with boolean variables
Stability
Controllability
Controller design
Computer Science: State transition systems with continuous dynamics
Safety verification model checking theorem proving
Hybrid Systems
3Verifying Average Dwell Time
Motivation: Micro
Analysis of mobile algorithms (CT view) nodes: plant with continuous motion, disturbance
algorithm: controller maintaining some structure
Complexity
Stability and Robustness
4Verifying Average Dwell Time
Outline
1. Background
2. Stability under slow switching
3. Formal Model
4. Invariant Approach
5. MILP Approach
6. Conclusions
5Verifying Average Dwell Time
Switching and Stability
M1
M2
M1M2
M2 M1
M3
6Verifying Average Dwell Time
Stability Under Slow Switchings
Theorem [Hespanha]: Assuming Lyapunov functions for the individual modes exist, global asymptotic stability is guaranteed if τa is large enough.
),( Tt# of switches on average dwell time (ADT)
t1 12 2
)()( tV t decreasing sequence
--- (1)
7Verifying Average Dwell Time
Problem Statement
If all the executions of the hybrid system satisfy Equation (1), then the
system is said to have ADT τa .
Q: Given hybrid system A, does it have ADT τa ? or, what is the largest τa that is ADT for A ?
8Verifying Average Dwell Time
V: set of variables, types, valuations val(V), dtypes Q: set of states, Q val(V) : start states A: set of actions D Q A Q: discrete transitions. (v,a,v) є D is written in
short as
T: set of trajectories for V, functions describing continuous
evolution
A trajectory : J val(V)
T is closed under prefix, suffix, and concatenation
Formal Definitions: Hybrid Automata
[Lynch,Segala,Vaandrager]
9Verifying Average Dwell Time
Every variable is either discrete or continuous V = Vc U Vc
A set F of state models for the continuous variables Vc
A state model is a locally Lipschitz function f such that the solution to the system of differential equation d(v) = f(v) are in the dtypes of the corresp. continuous variables
A mode switching function
So, we have only continuous variables changing over trajectories: