symbolic control design of cyber-physical systems · paulo tabuada université joseph fourier,...
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Symbolic control design of cyber-physical systems
Giordano Pola Center of Excellence for Research DEWS
University of L’Aquila
Basilica di Santa Maria di Collemaggio, 1287, L’Aquila
Group at DEWS:
Maria Domenica Di Benedetto
Pierdomenico Pepe
Giordano Pola
Alessandro Borri (now at CNR-IASI, Italy)
External Collaborations:
University of California at Los Angeles, USA
Paulo Tabuada
Université Joseph Fourier, France
Antoine Girard
Delft University of Technology, The Netherlands
Manuel Mazo Jr. and Majid Zamani
Projects:
Hycon 2: Highly-complex and networked control systems
EU FP7 NoE, 2010-2014
Group, Collaborations & Projects
Cyber-Physical Systems
London CPS Workshop, 21st October 2012
Cyber-physical systems (CPS) are physical, biological and engineered systems whose operations are monitored, coordinated, controlled and integrated by a computing and communication core
Key features of CPS are:
tight integration of the cyber and the physical parts
cyber capability (i.e. networking and computational capability) in every physical component
networked at multiple scales
complex at multiple spatial scales
dynamically reorganizing and reconfiguring
control loops are closed at each spatial scale. Maybe human in the loop.
…
Cyber-Physical Systems
Correct-by-design embedded control software: 1. Construct a finite model T*(P) of the plant system P 2. Design a finite controller C that solves the specification S for T*(P) 3. Design a controller C’ for P on the basis of C
Advantages:
Integration of software and hardware constraints in the control design of purely continuous or hybrid processes
Use of computer science techniques to address complex specifications
Homogenizing heterogeneities …
Symbolic domain
Continuous or hybrid domain
Plant: Continuous or Hybrid system
Symbolic model Finite controller Software & hardware
Hybrid controller
approximate bisimulation
[Girard & Pappas,IEEE-TAC-2007]
incremental stability
[Angeli,IEEE-TAC-2002]
Research at DEWS
stable control systems
[Automatica-2008]
stable switched systems
[IEEE-TAC-2010]
stable time-delay systems
[SCL-2010]
stable control systems
with disturbances
[SICON-2009]
[IJC-2012] efficient control
algorithms
[IEEE-TAC-2012]
stable time-varying
delay systems
[IEEE-CDC-2010]
unstable control
systems
[IEEE-TAC-2012]
networked
control systems
[HSCC-2012]
[IEEE-CDC-2012]
A Labelled Transition System (LTS) is a tuple
T = (Q,L, ,O,H)
where:
Q is the set of states
L is the set of labels
Q × L × Q is the transition relation
O is the set of outputs
H: Q O is the output function We denote (q,l,p) by q p T is said to be: Countable, if Q and L are countable Symbolic, if Q and L are finite Metric, if O is equipped with a metric
l
Dealing with heterogeneity
q3 q2 q1
o1 o2
l2
o1
l2
l1 l2
Bisimulation equivalence
R.J. van Glabbeek, The linear time- branching time spectrum, CONCUR’90
Quantifying accuracy
Bisimulation equivalence
R.J. van Glabbeek, The linear time- branching time spectrum, CONCUR’90
Quantifying accuracy
Bisimulation Equivalence [Milner; Prentice-Hall, 89] [Park; Proc. 5th GI Conf. on TCS, 81] It preserves important properties on LTSs as for example Linear Temporal Logic (LTL) properties [Clarke, Grumberg, Peled; Model Checking, 99]
2
4 1
3
6 7 5
Bisimulation Equivalence [Milner; Prentice-Hall, 89] [Park; Proc. 5th GI Conf. on TCS, 81] Approximate Bisimulation Equivalence [Girard, Pappas; IEEE-TAC-07]
Bisimulation equivalence
R.J. van Glabbeek, The linear time- branching time spectrum, CONCUR’90
Quantifying accuracy
2
4 1
3
6 7 5
Bisimulation Equivalence [Milner; Prentice-Hall, 89] [Park; Proc. 5th GI Conf. on TCS, 81] Approximate Bisimulation Equivalence [Girard, Pappas; IEEE-TAC-07]
Bisimulation equivalence Quantifying accuracy
2
4 1
3
6 7 5
[Angeli; IEEE-TAC-02]
A nonlinear control system dx/dt= f(x,u) is
incrementally input-to-state stable
(-ISS) if there exist a KL function and a K
function such that
||x(t,y,u) - x(t,y’,u’)|| ≤ (||y–y’||,t) + (||u–u’||)
y’
y x(.,y’,u’)
x(.,y,u)
Incremental stability
≤ (||u–u’||)
Symbolic models for control systems
Theorem [Pola, Girard, Tabuada; Automatica-08] For any incrementally stable nonlinear control system with compact state and input spaces it is possible to construct a symbolic system that approximates the control system in the sense of approximate bisimulation with any desired accuracy
Author's personal copy
Given a plant P, a specification Q expressed as a non-deterministic automaton, and a desired precision > 0, find a symbolic controller C that implements Q up to the precision and that is non-blocking when interacting with P
Plant system P
A/D
D/A
Symbolic Controller C
xp
u ≼
Symbolic control design
q1 q3
Specification Q
q2
… Approximate similarity game!
Given a plant P, a specification Q expressed as a non-deterministic automaton, and a desired precision > 0, find a symbolic controller C that implements Q up to the precision and that is non-blocking when interacting with P
Plant system P
A/D
D/A
Symbolic Controller C
xp
u ≼
Symbolic control design
q1 q3
Specification Q
q2
… Approximate similarity game!
Generalizations
Dealing with unstable dynamics
In [Zamani et al.; IEEE-TAC-12], symbolic models for -FC control systems
Dealing with disturbances
In [Pola, Tabuada; SICON-09], symbolic models for -GAS control systems with disturbances
In [Borri et al.; IJC-12], symbolic models for -ISS control systems with disturbances
Dealing with delays In [Pola et al.; SCL-10], symbolic models for -ISS time-delay systems with
constant delays In [Pola et al.; IEEE-CDC-10], symbolic models for -IDSS time-delay systems with
time-varying delays Dealing with heterogeneous dynamics In [Girard et al.; IEEE-TAC-10], symbolic models for -GAS switched systems Dealing with imperfect communication channels: Networked control systems In [Borri et al.; HSCC-12] and [Borri et al.; IEEE-CDC-12], symbolic models and
symbolic control for -GAS and -FC networked control systems
Symbolic control design of cyber-physical systems
Giordano Pola Center of Excellence for Research DEWS
University of L’Aquila
Basilica di Santa Maria di Collemaggio, 1287, L’Aquila