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