Modelling, Analysis and Design of Wireless Networked Control Systems

Alessandro D’Innocenzo Department of Information Engineering, Computer Science and Mathematics

Center of Excellence for Research DEWS, University of L’Aquila

EU FP7 NoE Hycon2: Highly-complex and networked control systems, 2010-2014 Total cost: 4.9M€

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Challenge: close the loop around wireless multi-hop

control networks.

Wireless control systems

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Challenge: close the loop around wireless multi-hop

control networks.

Wireless control systems

Controller

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Challenge: close the loop around wireless multi-hop

control networks.

Wireless control systems

Controller

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Cycle

Slot 1 Slot 1 Slot 2 Slot 3

WirelessHART MAC layer (scheduling)

time is divided in periodic frames, each divided in Π time slots, each of duration Δ

to avoid interference, a periodic scheduling allows each node to transmit data only in a subset of time slots

model impact of scheduling on the closed-loop dynamics

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Cycle

Slot 1 Slot 2 Slot 3 Slot 2

time is divided in periodic frames, each divided in Π time slots, each of duration Δ

to avoid interference, a periodic scheduling allows each node to transmit data only in a subset of time slots

model impact of scheduling on the closed-loop dynamics

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WirelessHART MAC layer (scheduling)

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Cycle

Slot 1 Slot 2 Slot 3 Slot 3

time is divided in periodic frames, each divided in Π time slots, each of duration Δ

to avoid interference, a periodic scheduling allows each node to transmit data only in a subset of time slots

model impact of scheduling on the closed-loop dynamics

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WirelessHART MAC layer (scheduling)

• Single path vs multi path routing

• Static vs Dynamic routing

• Redundancy in the data routing (flooding) and network coding

WirelessHART network layer (routing)

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Wireless control networks as switching systems

Mathematical model: 𝑥 𝑡 + 1 = 𝐴 𝜎 𝑡 𝑥 𝑡 + 𝐵 𝜎 𝑡 𝑢 𝑡 , 𝑡 ∈ ℕ, where 𝑥 𝑡 is

the plant state, 𝜎 𝑡 ∈ Σ depends on routing/scheduling and 𝑢 𝑡 = 𝐾(𝑡)𝑥(𝑡) is the control signal. The communication parameters are considered as a disturbance.

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𝑡

Wireless control networks as switching systems

Mathematical model: 𝑥 𝑡 + 1 = 𝐴 𝜎 𝑡 𝑥 𝑡 + 𝐵 𝜎 𝑡 𝑢 𝑡 , 𝑡 ∈ ℕ, where 𝑥 𝑡 is

the plant state, 𝜎 𝑡 ∈ Σ depends on routing/scheduling and 𝑢 𝑡 = 𝐾(𝑡)𝑥(𝑡) is the control signal. The communication parameters are considered as a disturbance.

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𝑡+1

Wireless control networks as switching systems

Mathematical model: 𝑥 𝑡 + 1 = 𝐴 𝜎 𝑡 𝑥 𝑡 + 𝐵 𝜎 𝑡 𝑢 𝑡 , 𝑡 ∈ ℕ, where 𝑥 𝑡 is

the plant state, 𝜎 𝑡 ∈ Σ depends on routing/scheduling and 𝑢 𝑡 = 𝐾(𝑡)𝑥(𝑡) is the control signal. The communication parameters are considered as a disturbance.

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𝑡+2

Mathematical model: 𝑥 𝑡 + 1 = 𝐴 𝜎 𝑡 𝑥 𝑡 + 𝐵 𝜎 𝑡 𝑢 𝑡 , 𝑡 ∈ ℕ, where 𝑥 𝑡 is

the plant state, 𝜎 𝑡 ∈ Σ depends on routing/scheduling and 𝑢 𝑡 = 𝐾(𝑡)𝑥(𝑡) is the control signal. The communication parameters are considered as a disturbance.

Problem 1: given 𝐾(𝑡), verify whether the closed loop systems is asymptotically stable,

i.e. the Joint Spectral Radius of 𝐴 𝜎 𝑡 + 𝐵 𝜎 𝑡 𝐾 𝑡𝜎 𝑡 ∈Σ

is smaller than 1.

Problem 2: Design a controller 𝐾(𝑡) such that the closed loop system is asymptotically stable.

Insight: stabilizability depends on our knowledge of the switching signal 𝜎 𝑡 :

• We cannot measure 𝜎 𝑡 : then 𝐾 𝑡 = 𝐾, ∀𝑡 ∈ ℕ

• We can measure and keep memory of 𝜎 𝑡 , 𝐾 𝑡 = 𝐾 𝜎 𝑡 − 𝑑 ⋯ 𝜎 𝑡

• We also have a finite horizon knowledge of future 𝑁 switching singnals

𝜎 𝑡 : 𝐾 𝑡 = 𝐾 𝜎 𝑡 − 𝑑 ⋯ 𝜎 𝑡 + 𝑁

Wireless control networks as switching systems

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Collaborations: Raphael Jungers (Université Catholique de Louvain) Nicola Guglielmi (University of L’Aquila) George Pappas (University of Pennsylvania)

Selected Publications:

R.M. Jungers, A. D'Innocenzo, M.D. Di Benedetto. Feedback stabilization of dynamical systems with switched delays. 51 IEEE Conf. on Decision and Control, Maui, Hawaii, December 10-13 2012.

R. Alur, A. D'Innocenzo, K.H. Johansson, G.J. Pappas, G. Weiss. Compositional Modeling and Analysis of Multi-Hop Control Networks. IEEE Transactions on Automatic Control, 56(10):2345-2357, 2011.

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Wireless control networks as switching systems

Fault tolerant stabilizability of wireless control networks

Mathematical model: 𝑥 𝑡 + 1 = 𝐴𝑥 𝑡 + 𝐵𝑢 𝑡 𝑦 = 𝐶𝑥 𝑡 , 𝑡 ∈ ℕ

𝑡

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Fault tolerant stabilizability of wireless control networks

Mathematical model: 𝑥 𝑡 + 1 = 𝐴𝑥 𝑡 + 𝐵𝑢 𝑡 𝑦 = 𝐶𝑥 𝑡 , 𝑡 ∈ ℕ

𝑡+1

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Fault tolerant stabilizability of wireless control networks

Mathematical model: 𝑥 𝑡 + 1 = 𝐴𝑥 𝑡 + 𝐵𝑢 𝑡 𝑦 = 𝐶𝑥 𝑡 , 𝑡 ∈ ℕ

𝑡+2

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⋯

Cycle n+2

Fault tolerant stabilizability of wireless control networks

~ ~ ~ ~

Mathematical model: 𝑥 𝑡 + 1 = 𝐴𝑥 𝑡 + 𝐵𝑢 𝑡 𝑦 = 𝐶𝑥 𝑡 , 𝑡 ∈ ℕ

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𝐹 = 2𝐸ℛ∪𝐸𝒪 set of all configurations of links subject to a failure or a malicious intrusion

Assumption: Failures are slow with respect to plant time constants

M Mf

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Fault tolerant stabilizability of wireless control networks

Problem 1: Design the communication system parameters of 𝐺ℛ and 𝐺𝒪 (topology, scheduling, routing) to guarantee existence of a stabilizing controller for any failure

Problem 2: Design the communication system parameters to guarantee the existence of a Fault Detection and Isolation (FDI) systems

Insight: Translate classical stabilizability and FDI conditions on LTI systems to conditions on the communication system parameters

Mf

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Fault tolerant stabilizability of wireless control networks

𝐹 = 2𝐸ℛ∪𝐸𝒪 set of all configurations of links subject to a failure or a malicious intrusion

Assumption: Failures are slow with respect to plant time constants

Selected Publications:

A. D'Innocenzo, M.D. Di Benedetto, F. Smarra. Fault detection and isolation of malicious nodes in MIMO Multi-hop Control Networks. Submitted to the 52nd IEEE Conference on Decision and Control, 2013.

A. D'Innocenzo, M.D. Di Benedetto, E. Serra. Fault Tolerant Control of Multi-Hop Control Networks. IEEE Transactions on Automatic Control, 58(6):1377-1389, 2013.

F. Smarra, A. D'Innocenzo, M.D. Di Benedetto. Optimal co-design of control, scheduling and routing in multi-hop control networks. 51st IEEE Conference on Decision and Control, Maui, Hawaii, December 10-13 2012.

F. Smarra, A. D'Innocenzo, M.D. Di Benedetto. Fault Tolerant Stabilizability of MIMO Multi-Hop Control Networks. 3rd IFAC Workshop on Estimation and Control of Networked Systems (NecSys'12), Santa Barbara, CA, September 14-15, 2012.

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Fault tolerant stabilizability of wireless control networks

Co-analysis and co-design of wireless control systems using finite probabilistic abstractions

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Mathematical model: 𝒮: 𝑥 𝑡 + 1 = 𝑓 𝑥 𝑡 , 𝜎 𝑡 , 𝑢 𝑡 , 𝑝(𝑡) , 𝑡 ∈ ℕ:

𝑥 𝑡 is the plant state, 𝑓 ∙ is a non-linear function

𝜎 𝑡 ∈ Σ is a Markov Chain depending on routing, scheduling, and packet losses

𝑢 𝑡 is the plant control signal

𝑝 𝑡 ∈ 𝑃 is the transmission power control signal

Co-analysis and co-design of wireless control systems using finite probabilistic abstractions

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Mathematical model: 𝒮: 𝑥 𝑡 + 1 = 𝑓 𝑥 𝑡 , 𝜎 𝑡 , 𝑢 𝑡 , 𝑝(𝑡) , 𝑡 ∈ ℕ:

𝑥 𝑡 is the plant state, 𝑓 ∙ is a non-linear function

𝜎 𝑡 ∈ Σ is a Markov Chain depending on routing, scheduling, and packet losses

𝑢 𝑡 is the plant control signal

𝑝 𝑡 ∈ 𝑃 is the transmission power control signal

Problem 1: given the control signals 𝑢 𝑡 , 𝑝(𝑡), verify whether the closed loop system satisfies a probabilistic property (e.g. unsafe with probability < 10−9) Insight: Derive a Markov Chain abstraction of the controlled stochastic process 𝒮 with precision 𝜀 and use Model Checking techniques. Need models to handle both non-determinism and stochasticity: Markov-set chains, Interval Markov Chains. Problem 2: given 𝒮, design a control laws for 𝑢 𝑡 , 𝑝 𝑡 such that the closed loop system satisfies a probabilistic property.

Insight: Derive a Markov Decision Process (MDP) abstraction of the stochastic process 𝒮 with precision 𝜀 and use design techniques for MDPs.

Collaborations: Alessandro Abate (University of Oxford) Joost-Pieter Katoen (RWTH Aachen University)

Claudia Rinaldi and Fortunato Santucci (University of L’Aquila)

Selected Publications:

A. D'Innocenzo, C. Rinaldi, M.D. Di Benedetto and F. Santucci. Hybrid power control on a wireless networked control system. 4th IFAC Conference on Analysis and Design of Hybrid Systems, Eindhoven, The Netherlands. June 6-8, 2012.

A. D'Innocenzo, A. Abate and J.-P. Katoen. Robust PCTL Model Checking. Proceedings of the 15th ACM international conference on Hybrid Systems: Computation and Control, Pages 275-286, ACM New York, NY, USA ©2012.

A. Abate, A. D'Innocenzo, M.D. Di Benedetto. Approximate Abstractions of Stochastic Hybrid Systems. IEEE Transactions on Automatic Control, 56(11):2688-2694, 2011.

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Wireless control networks as switching systems