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12 Networked Control Systems: a Discrete-time Approach Nathan van de Wouw Dynamics and Control Group, Department of Mechanical Engineering, Eindhoven University of Technology, the Netherlands Universit Catholique de Louvain, October 5th,

  1. 12 Networked Control Systems: a Discrete-time Approach Nathan van de Wouw Dynamics and Control Group, Department of Mechanical Engineering, Eindhoven University of Technology, the Netherlands Université Catholique de Louvain, October 5th, 2010 1/34

  2. Acknowledgement of Collaborators ◮ Maurice Heemels, Tijs Donkers, Henk Nijmeijer (Eindhoven University of Technology, the Netherlands) ◮ Marieke Cloosterman (ASML Research, the Netherlands) ◮ Laurentiu Hetel (University of Lille, France) ◮ Jamal Daafouz (University of Nancy, France) ◮ Payam Naghsthabrizi (Ford Research, U.S.A.) ◮ Joao Hespanha (University of California, Santa Barbara, U.S.A.) 2/34

  3. Contents ◮ Introduction on Networked Control Systems (NCS) ◮ Discrete-time Modelling of linear NCS: • Time-varying sampling intervals • Communication delays • Packet dropouts ◮ Stability analysis of linear NCS ◮ Tracking control of linear NCS ◮ NCS including communication protocols ◮ Conclusions & Outlook on Future Work 3/34

  4. Introduction Cooperative Adaptive Cooperative�robotics Cruise�Control Wireless Wireless/distributed�control Sensor�Networks of�water�distribution�networks (EU-project�WIDE) Wireless�Motion�Control Etc.,�etc. NCS:�Control�systems�in�which�controllers,�sensors�and actuators�are�communicating�over�a�network 4/34

  5. Introduction To network ... ◮ Ease of installation and maintenance ◮ Large flexibility (especially with WSN) ◮ Lower costs ◮ Less wires (less wear, less disturbances, less weight!) in case of WSN ◮ Control of physically distributed systems 5/34

  6. Introduction ...or not to network: ◮ Varying sampling/transmission interval ◮ Varying communication delays ◮ Packet loss ◮ Communication constraints through shared network ◮ Quantization 5/34

  7. Network Control Systems: Modelling ◮ Existing approaches towards modelling/stability analysis: 1. Emulation approach (Neši´ c, Teel, Carnevale, Tabarra, Heemels, van de Wouw) : • Time-varying sampling intervals, SMALL delays • Communication constraints, general classes of scheduling protocols • Nonlinear systems • Continuous-time control synthesis based on continuous-time model 6/34

  8. Network Control Systems: Modelling ◮ Existing approaches towards modelling/stability analysis: 1. Emulation approach (Neši´ c, Teel, Carnevale, Tabarra, Heemels, van de Wouw) : • Time-varying sampling intervals, SMALL delays • Communication constraints, general classes of scheduling protocols • Nonlinear systems • Continuous-time control synthesis based on continuous-time model 2. Modelling in terms of delay-impulsive differential equations (Naghsthabrizi, Hespanha, Teel, van de Wouw) : • Time-varying sampling intervals, LARGE delays • Linear systems • LMI-based stability analysis and controller synthesis 6/34

  9. Network Control Systems: Modelling ◮ Existing approaches towards modelling/stability analysis: 1. Emulation approach (Neši´ c, Teel, Carnevale, Tabarra, Heemels, van de Wouw) : • Time-varying sampling intervals, SMALL delays • Communication constraints, general classes of scheduling protocols • Nonlinear systems • Continuous-time control synthesis based on continuous-time model 2. Modelling in terms of delay-impulsive differential equations (Naghsthabrizi, Hespanha, Teel, van de Wouw) : • Time-varying sampling intervals, LARGE delays • Linear systems • LMI-based stability analysis and controller synthesis 3. Discrete-time modelling (Zhang, Hetel, Fujioka, Garcia, Cloosterman, van de Wouw, Heemels, Donkers) : • Time-varying sampling intervals, LARGE delays, packet dropouts • Communication constraints, particular classes of scheduling protocols • Linear systems • LMI-based stability analysis and controller synthesis 6/34

  10. Network Control Systems: Modelling s k u k u ∗ ( t ) y k ZOH Plant Sensor τ sc τ ca k k Controller m k Networked�Control�System Time-delays: Assumptions: ◮ Sensor-to-controller τ sc , k ◮ Time-driven sensor (sampling times: s k ) ◮ Controller-to-actuator τ ca , k ◮ Event-driven controller ◮ Computational delay τ c , k ◮ Event-driven actuator ◮ τ k = τ sc , k + τ ca , k + τ c , k 7/34

  11. Network Control Systems: Modelling s k u k u ∗ ( t ) y k ZOH Plant Sensor τ sc τ ca k k Controller m k Networked�Control�System Network-induced uncertainties: ◮ Time-varying delays: τ k ∈ [ τ min , τ max ] ◮ Maximum of ¯ δ subsequent ◮ Time-varying sampling intervals: dropouts: h k = s k + 1 − s k ∈ [ h min , h max ] k ◮ Packet dropouts: � m v ≤ δ � 1 , v = k − δ u k is dropped m k = 0 , u k is not dropped 7/34

  12. Network Control Systems: Modelling s k u k u ∗ ( t ) y k ZOH Plant Sensor τ sc τ ca k k Controller m k Networked�Control�System Assumptions: ◮ Time-varying delays: τ k ∈ [ τ min , τ max ] ◮ Time-varying sampling intervals: Problem : h k = s k + 1 − s k ∈ [ h min , h max ] How�to�guarantee�stability ◮ Packet dropouts: in�the�face�of�these � 1 , network-induced�uncertainties u k is dropped m k = 0 , u k is not dropped 7/34

  13. Network Control Systems: Modelling ◮ Continuous-time (sampled-data) dynamics of the linear plant: x ( t ) Ax ( t ) + Bu ∗ ( t ) ˙ = for t ∈ [ s k + t k j , s k + t k u ∗ ( t ) u k + j − d − δ = j + 1 ), h min ⌉ and t k j ∈ [0 , h k ] the actuation update where d := ⌊ τ min h max ⌋ , d := ⌈ τ max instants 8/34

  14. Discrete-time Model � T � x T u T u T ◮ Use an extended state vector: ξ k = . . . , k − 1 k k − d − δ x k := x ( s k ) ◮ Uncertain parameters: θ k := ( h k , t k 1 , . . . , t k d + δ − d ) ◮ Discrete-time uncertain NCS model: A (θ k )ξ k + ˜ B (θ k ) u k , ξ k + 1 = ˜ where Md + δ − 1 (θ k ) Md + δ − 2 (θ k ) M 1 (θ k ) M 0 (θ k )  �(θ k ) . . .  0 0 0 0 0 . . .   0 0 0 0  I  . . .     . . .  ... ... ...  A (θ k ) = ˜  . . .   . . .      . ... ...   .  0 0  .   0 0 0 I . . . . . . and Md + δ(θ k )   � hk − tk I  j   eAs dsB  0 if 0 ≤ j ≤ d + δ − d ,    �(θ k ) = eAhk ,  B (θ k ) = , M j (θ k ) = hk − tk ˜     j + 1 .    . 0    if d + δ − d < j ≤ d + δ   .  0 9/34

  15. Network Control Systems: Modelling BUT.....LET’S�KEEP IT�SIMPLE... Consider�the�small-delay�case,�with�a�constant sampling�interval�and�no�packet�dropouts 10/34

  16. Network Control Systems: Modelling ◮ Continuous-time (sampled-data) dynamics of the linear plant: x ( t ) Ax ( t ) + Bu ∗ ( t ) ˙ = u ∗ ( t ) u k for t ∈ [ s k + τ k , s k + 1 + τ k + 1 ) = U k U k-1 11/34

  17. Discrete-time Model � T , x k := x ( s k ) � x T u T ◮ Use an extended state vector: ξ k = k − 1 k ◮ Uncertain parameters: θ k := τ k ◮ Discrete-time uncertain NCS model: ξ k + 1 = ˜ A (τ k )ξ k + ˜ B (τ k ) u k where � h �� h − τ k � e Ah h − τ k e As dsB � e As dsB � 0 A (τ k ) = ˜ B (τ k ) = ˜ , 0 0 I 12/34

  18. Discrete-time Closed-loop Model ◮ Discrete-time uncertain NCS model: ξ k + 1 = ˜ A (τ k )ξ k + ˜ B (τ k ) u k ◮ In closed-loop with the static discrete-time extended-state feedback controller u k = − K ξ k = − ¯ K x k − K u u k − 1 : � � A (τ k ) − ˜ B (τ k ) K ˜ ξ k + 1 = ξ k � h − τ k � h � h − τ k � � e Ah − e As dsB ¯ h − τ k e As dsB − e As dsBK u K 0 0 = ξ k − ¯ K − K u ◮ Discrete-time linear system with exponential uncertainty : varying delay τ k 13/34

  19. Stability Analysis ◮ Discrete-time uncertain closed-loop NCS model: � � A (τ k ) − ˜ B (τ k ) K ξ k =: H (τ k )ξ k ˜ ξ k + 1 = ◮ A first approach using a common quadratic Lyapunov function: V (ξ) = ξ T P ξ, P = P T > 0 ◮ Closed-loop NCS is globally asymptotically stable if there exists P = P T > 0 , 0 < γ < 1 such that H T (τ) PH (τ) − P < − γ P , ∀ τ ∈ [ τ min , τ max ] ◮ Infinite set of Linear Matrix Inequalities (LMIs) ◮ How to arrive at a finite number of LMIs? 14/34

  20. Stability Analysis ◮ Basic idea: embed the uncertainty matrix set H (τ), τ ∈ [ τ min , τ max ] in a polytopic set with generators (vertices) H i , i = 1 , . . . , N : { H (τ) | τ ∈ [ τ min , τ max ] } ⊆ convex hull ( H 1 , . . . , H N ) convex hull ( H 1 , . . . , H N ) H 1 H 5 H 2 H (τ) H 4 H 3 ◮ Discrete-time uncertain closed-loop NCS model ξ k + 1 = H (τ k )ξ k is globally asymptotically stable if there exist P = P T > 0 , 0 < γ < 1 such that the following finite set of LMIs are satisfied: H T i PH i − P < − γ P , ∀ i = 1 , . . . , N 15/34

  21. Stability Analysis ◮ Basic idea: embed the uncertainty matrix set H (τ), τ ∈ [ τ min , τ max ] in a polytopic set with generators (vertices) H i , i = 1 , . . . , N : { H (τ) | τ ∈ [ τ min , τ max ] } ⊆ convex hull ( H 1 , . . . , H N ) convex hull ( H 1 , . . . , H N ) H 1 H 5 H 2 H (τ) H 4 H 3 ◮ How to get such a polytopic overapproximation? 15/34

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