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