![Page 1: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/1.jpg)
Provably efficient algorithms
for Hybrid Systems
Raphaël Jungers
(UCLouvain, Belgium)
ADHS’18
Oxford, July 2018
![Page 2: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/2.jpg)
Models of complex systems
Modelling power
Solvability/efficiency
![Page 3: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/3.jpg)
Models of complex systems
Modelling power
Solvability/efficiency
If I have seen further it is by standing on the shoulders of Giants.
Thoralf Skolem1887-1963
Stephen Smale
1930-
Fields Medal 1966
Wolf prize 2007
Emil Artin
1898-1962
Giancarlo Rota
1932-1999
![Page 4: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/4.jpg)
Outline
• Two motivations
– Consensus
– Wireless control networks
• Three techniques
– Observability/controllability of hybrid systems
– Guaranteed accuracy for stability analysis
– Data-driven/blackbox control
• Discussion
![Page 5: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/5.jpg)
Outline
• Two motivations
– Consensus
– Wireless control networks
• Three techniques
– Observability/controllability of hybrid systems
– Guaranteed accuracy for stability analysis
– Data-driven/blackbox control
• Discussion
![Page 6: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/6.jpg)
Smart cities
![Page 7: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/7.jpg)
Autonomous robots
![Page 8: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/8.jpg)
Gossip algorithms
0.20.5
0.1
0.2
S. Boyd, A. Ghosh, B. Prabhakar, D. Shah IEEE Transactions on Information Theory, Special issue of IEEE Transactions on Information Theory and IEEE ACM Transactions on Networking, June 2006, 52(6):2508-2530.
![Page 9: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/9.jpg)
Gossip algorithms
0.20.5
0.1
0.2
S. Boyd, A. Ghosh, B. Prabhakar, D. Shah IEEE Transactions on Information Theory, Special issue of IEEE Transactions on Information Theory and IEEE ACM Transactions on Networking, June 2006, 52(6):2508-2530.
![Page 10: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/10.jpg)
Gossip algorithms
S. Boyd, A. Ghosh, B. Prabhakar, D. Shah IEEE Transactions on Information Theory, Special issue of IEEE Transactions on Information Theory and IEEE ACM Transactions on Networking, June 2006, 52(6):2508-2530.
0.350.35
0.1
0.2
![Page 11: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/11.jpg)
Gossip algorithms
S. Boyd, A. Ghosh, B. Prabhakar, D. Shah IEEE Transactions on Information Theory, Special issue of IEEE Transactions on Information Theory and IEEE ACM Transactions on Networking, June 2006, 52(6):2508-2530.
0.350.35
0.1
0.2
(0.35+0.1)/2=0.225
![Page 12: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/12.jpg)
Gossip algorithms
S. Boyd, A. Ghosh, B. Prabhakar, D. Shah IEEE Transactions on Information Theory, Special issue of IEEE Transactions on Information Theory and IEEE ACM Transactions on Networking, June 2006, 52(6):2508-2530.
0.350.225
0.225
0.2
![Page 13: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/13.jpg)
Consensus of multi-agent systemsOur setting: We are given a set of stochastic matrices, representing different connectivity topologies
Problem: Do all products of these matrices converge to consensus?(that is, all rows are equal)
![Page 14: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/14.jpg)
This is actually a stability problem
• Property:
Any consensus state is an equilibrium
with
• Proposition [Jadbabaie 03] : After projection along the
(1,1,…,1) vector, convergence to consensus becomesconvergence to zero
Do all the products converge to zero?
Paz, Wolfovitz, Blondel-Olshevsky, Chevalier,..
P-Y Chevalier
![Page 15: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/15.jpg)
Switching systems
Point-to-point Given x0 and x*, is there a product (say, A0 A0 A1 A0 … A1) for which x*=A0 A0 A1 A0 … A1 x0?
Boundedness Is the set of all products {A0, A1, A0A0, A0A1,…} bounded?
Mortality Is there a product that gives the zero matrix?
![Page 16: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/16.jpg)
Switching systems
Global convergence to the origin Do all products of the type A0 A0 A1 A0 … A1 converge to zero? (GUAS)
The joint spectral radius of a set of matrices is given by
All products of matrices in converge to zero iff
The spectral radius of a matrix A controls the growth or decay of powers of A
The powers of A converge to zero iff
[Rota, Strang, 1960]
![Page 17: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/17.jpg)
Outline
• Two motivations
– Consensus
– Wireless control networks
• Three techniques
– Observability/controllability of hybrid systems
– Guaranteed accuracy for stability analysis
– Data-driven/blackbox control
• Discussion
![Page 18: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/18.jpg)
Wireless Control Networks
Industrial automation
Environmental Monitoring,
Disaster Recovery and
Preventive Conservation
Supply Chain and
Asset Management
Physical Security
and Control
![Page 19: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/19.jpg)
Wireless Control Networks
![Page 20: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/20.jpg)
[Jungers D’Innocenzo Di Benedetto, TAC 2015]
Wireless Control Networks
![Page 21: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/21.jpg)
[Jungers Kundu Heemels, 2016]
Wireless Control Networks
![Page 22: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/22.jpg)
V(t) u(t)
Controllability with packet dropouts
u(0) u(0)
1 or 0
Wireless Control Networks are subject to packet dropouts
A data loss signal determines the packet dropouts
…this is a switching system!
![Page 23: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/23.jpg)
V(t) u(t)
Controllability with packet dropouts
U(1)
1 or 0
Wireless Control Networks are subject to packet dropouts
A data loss signal determines the packet dropouts
…this is a switching system!
![Page 24: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/24.jpg)
V(t) u(t)
Controllability with packet dropouts
U(1)
1 or 0
Wireless Control Networks are subject to packet dropouts
A data loss signal determines the packet dropouts
…this is a switching system!
![Page 25: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/25.jpg)
V(t) u(t)
Controllability with packet dropouts
U(2)
1 or 0
Wireless Control Networks are subject to packet dropouts
A data loss signal determines the packet dropouts
…this is a switching system!
![Page 26: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/26.jpg)
V(t) u(t)
Controllability with packet dropouts
U(2)
1 or 0
Wireless Control Networks are subject to packet dropouts
A data loss signal determines the packet dropouts
…this is a switching system!
![Page 27: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/27.jpg)
V(t) u(t)
Controllability with packet dropouts
Wireless Control Networks are subject to packet dropouts
A data loss signal determines the packet dropouts
…this is a switching system!
u(4) u(4)
1 or 0
![Page 28: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/28.jpg)
The switching signal
We are interested in the controllability of such a system
Of course we need an assumption on the switching signal
The switching signal is constrained by an automatonExample:
Bounded number ofconsecutive dropouts (here, 3)
![Page 29: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/29.jpg)
Constrained switching systemsA more general model
The switching sequence is constrained by a graph (AKA an automaton)
Many natural applications-Communication networks-Markov Chains-Supervisory control
The dynamics is subject to switchings:
![Page 30: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/30.jpg)
Outline
• Two motivations
– Consensus
– Wireless control networks
• Three techniques
– Observability/controllability of hybrid systems
– Guaranteed accuracy for stability analysis
– Data-driven/blackbox control
• Discussion
Joint work withA. Kundu, M. Heemels
![Page 31: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/31.jpg)
Controllability with Packet Dropouts
We are given a pair (A,b) and an automaton
The controllability problem: for any starting point x(0), and any target x*, does there exist, for any admissible switching signal, a control signal u(.) and a time T such that x(T)=x* ?
Theorem: Deciding controllability of switching systems is undecidable in general (consequence of [Blondel Tsitsiklis, 97])
![Page 32: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/32.jpg)
We are given a pair (A,b) and an automaton
Controllability with Packet Dropouts
Proposition: The system is controllable iff the generalized controllability matrix
is bound to become full rank at some time t
The controllability problem: for any starting point x(0), and any target x*, does there exist, for any admissible switching signal, a control signal u(.) and a time T such that x(T)=x* ?
![Page 33: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/33.jpg)
Standing on Giants shoulders…
Theorem ([Skolem 34]): Given a matrix A and two vectors b,c, the set of values n such that
is eventually periodic.Example: 1 0 0 1 0 1 1 1 0 0 1 0 0 1 0 0 1 0 0 …
Algorithm for deciding controllability
Now, how to optimally chose the control signal, if one does not know the switching signal in advance?
Thoralf Skolem1887-1963Axioms, set theory, lattices, first order logic
Jungers, Kundu, Heemels, TAC 2017
![Page 34: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/34.jpg)
The dual observability problem
Observability under intermittent outputs is algebraically equivalent
V(t) u(t)
PY(k)Network
O
![Page 35: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/35.jpg)
• Safety critical
• Extremely costly
• A network of medical centers,
• with big data lying out there
Beamline
Line magnets
Scanning magnetsQA/DosiDetectors
Equipment settings Beam diagnosticsMachine learning
techniques
Convergence Algorithms
Protontherapy
![Page 36: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/36.jpg)
Outline
• Two motivations
– Consensus
– Wireless control networks
• Three techniques
– Observability/controllability of hybrid systems
– Guaranteed accuracy for stability analysis
– Data-driven/blackbox control
• Discussion
![Page 37: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/37.jpg)
Switching systems stability criteria
The joint spectral radius of a set of matrices is given by
All products of matrices in converge to zero iff
[Rota, Strang, 1960]
[Goebel, Hu, Teel 06]
[Daafouz Bernussou 01]
[Lee and Dullerud 06] …
[Bliman Ferrari-Trecate 03]
AKA computing theJoint Spectral Radius
![Page 38: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/38.jpg)
Path-complete stability criteria
Theorem: The LMIs are a sufficient condition for stability IFF
their representation G is path-complete .
Results valid beyond the LMI framework
Path complete (generates all the possible words)
Sufficient conditionfor stability
[J. Ahmadi Parrilo Roozbehani 17]
[Ahmadi J. Parrilo Roozbehani 14]
![Page 39: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/39.jpg)
Theorem: Build your own Lyapunov function.
Results valid beyond the LMI framework
Path complete (generates all the possible words)
Sufficient conditionfor stability
[J. Ahmadi Parrilo Roozbehani 17]
[Ahmadi J. Parrilo Roozbehani 14]
Theorem: We provide a hierarchy of criteria that reachesarbitrary accuracy
Path-complete stability criteria
![Page 40: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/40.jpg)
• Examples:
– CQLF
– Example 1
This type of graph gives a max-of-quadratics
Lyapunov function (i.e. intersection of ellipsoids)
– Example 2
Some examples
![Page 41: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/41.jpg)
Standing on Giants shouldersSymbolic dynamics
Stephen Smale
1930-
Fields Medal 1966
Wolf prize 2007
Gustav Hedlund
1904-1993
Marston Morse
1892-1977
Emil Artin
1898-1962
Symbolic dynamicsM Morse, GA Hedlund - American Journal of
Mathematics, 1938 - JSTOR
![Page 42: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/42.jpg)
A. Ahmadi (Princeton),P. Parrilo, M. Roozbehani (MIT)
David Angeli (Imperial) Matthew Philippe,
Nikos Athanasopoulos
Guillaume Berger
F. Forni and R.Sepulchre (Cambridge)
Paulo Tabuada(UCLA)
Benoit Legat
SICON’14, TAC’16Path-complete characterization
TAC’18Equivalent Common Lyapunov Function
ADHS’18Generalization to Invariant sets computation
IFAC WC’16Generalization to delta-ISS certificates
Geir Dullerud
And Ray Essick (UIUC) Matthew Philippe Automatica’16Generalization to Constrained switching systems
Path-complete techniques
![Page 43: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/43.jpg)
Outline
• Two motivations
– Consensus
– Wireless control networks
• Three techniques
– Observability/controllability of hybrid systems
– Guaranteed accuracy for stability analysis
– Data-driven/blackbox control
• Discussion
Joint work withJ. Kenanian, A. Balkan,P. Tabuada
![Page 44: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/44.jpg)
Control in the industry
Often, models can be nonlinear, complex, hybrid, heterogeneous, with look-up tables, pieces of code, proprietary softwares, old legacy components…
Termination of Computer programs
![Page 45: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/45.jpg)
Control in the industry
![Page 46: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/46.jpg)
Data-driven stability analysis of SS
xt+1= A0 xt
A1 xt
Global convergence to the origin Do all products of the type A0 A0 A1 A0 … A1 converge to zero? (GUAS)
The joint spectral radius of a set of matrices is given by
All products of matrices in converge to zero iff
[Rota, Strang, 1960]
![Page 47: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/47.jpg)
Data-driven stability analysis of SS
Setting: we sample N points at random in the state space, and observe their image by one (unknown) of the modes
Question: Is the system stable?
![Page 48: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/48.jpg)
Data-driven stability analysis of SS
Observation: One thing we can to is to check for the existence of a Common Lyapunov function for the N sampled points
![Page 49: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/49.jpg)
Standing on giants shoulders…
Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random homogeneous N-sampling of the infinite number of constraints Z. For any desired correctness level one can guarantee
Measure of ‘good’ N-Samplings
Measure of violated
constraints
Lower bound on the measure of good
samplings (Tends to 1 when N grows)
![Page 50: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/50.jpg)
From a partial guarantee toa formal upper bound
With some level of confidence, we have:
Not much!Already for a linear system, this does not imply much:
The challenge: translate this guarantee on a subset of the statespace into a global guarantee (on the whole statespace)
![Page 51: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/51.jpg)
From a partial guarantee toa formal upper bound
Theorem:
For sure
With probability beta,
![Page 52: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/52.jpg)
Outline
• Two motivations
– Consensus
– Wireless control networks
• Three techniques
– Observability/controllability of hybrid systems
– Guaranteed accuracy for stability analysis
– Data-driven/blackbox control
• Discussion
![Page 53: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/53.jpg)
Models of complex systems
Modelling power
Solvability/efficiency
![Page 54: Provably efficient algorithms for Hybrid Systems · Standing on giants shoulders… Theorem [adapted from Campi, Calafiore]: Consider the optimization problem below, where is a random](https://reader033.vdocuments.site/reader033/viewer/2022042322/5f0cf60a7e708231d437fcb5/html5/thumbnails/54.jpg)
Ads
References: http://perso.uclouvain.be/raphael.jungers/
Thanks!Questions?
Joint work with A.A. Ahmadi (Princeton), D. Angeli (Imperial), N. Athanasopoulos
(UCLouvain), V. Blondel (UCL), G. Dullerud (UIUC), F. Forni (Cambridge), M. Heemels (TU/e), B. Legat (UCLouvain), P. Parrilo (MIT), M. Philippe
(UCLouvain), V. Protasov (Moscow), M. Roozbehani (MIT), R. Sepulchre (Cambridge), P. Tabuada (UCLA)…
The JSR Toolbox: http://www.mathworks.com/matlabcentral/fileexchange/33202-the-jsr-toolbox[Van Keerberghen, Hendrickx, J. HSCC 2014]The CSS toolbox, 2015
Several open positions:[email protected]