12 Networked Control Systems: a Discrete-time Approach Nathan van - - PowerPoint PPT Presentation

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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

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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.)

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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

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Introduction

Cooperativerobotics Cooperative Adaptive CruiseControl NCS:Controlsystemsinwhichcontrollers,sensorsand actuatorsarecommunicatingoveranetwork Wireless/distributedcontrol

• fwaterdistributionnetworks

(EU-projectWIDE) WirelessMotionControl Etc.,etc. Wireless SensorNetworks

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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

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Introduction

...or not to network:

◮ Varying sampling/transmission interval ◮ Varying communication delays ◮ Packet loss ◮ Communication constraints through shared network ◮ Quantization

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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

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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

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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

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Network Control Systems: Modelling

NetworkedControlSystem

Sensor Controller Plant ZOH τ sc

k

τ ca

k

uk u∗(t) yk sk mk

Assumptions:

◮ Time-driven sensor

(sampling times: sk)

◮ Event-driven controller ◮ Event-driven actuator

Time-delays:

◮ Sensor-to-controller τsc,k ◮ Controller-to-actuator τca,k ◮ Computational delay τc,k ◮ τk = τsc,k + τca,k + τc,k

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Network Control Systems: Modelling

NetworkedControlSystem

Sensor Controller Plant ZOH τ sc

k

τ ca

k

uk u∗(t) yk sk mk

Network-induced uncertainties:

◮ Time-varying delays: τk ∈ [τmin, τmax] ◮ Time-varying sampling intervals:

hk = sk+1 − sk ∈ [hmin, hmax]

◮ Packet dropouts:

mk = 1, uk is dropped 0, uk is not dropped

◮ Maximum of ¯

δ subsequent dropouts:

k

• v=k−δ

mv ≤ δ

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Network Control Systems: Modelling

NetworkedControlSystem

Sensor Controller Plant ZOH τ sc

k

τ ca

k

uk u∗(t) yk sk mk

Assumptions:

◮ Time-varying delays: τk ∈ [τmin, τmax] ◮ Time-varying sampling intervals:

hk = sk+1 − sk ∈ [hmin, hmax]

◮ Packet dropouts:

mk = 1, uk is dropped 0, uk is not dropped

Problem: Howtoguaranteestability inthefaceofthese network-induceduncertainties

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Network Control Systems: Modelling

◮ Continuous-time (sampled-data) dynamics of the linear plant:

˙ x(t) = Ax(t) + Bu∗(t) u∗(t) = uk+ j−d−δ for t ∈ [sk + tk

j , sk + tk j+1),

where d := ⌊ τmin

hmax ⌋, d := ⌈ τmax hmin ⌉ and tk j ∈ [0, hk] the actuation update

instants

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Discrete-time Model

◮ Use an extended state vector: ξk =

• xT

k

uT

k−1

. . . uT

k−d−δ

T , xk := x(sk)

◮ Uncertain parameters: θk := (hk, tk

1, . . . , tk d+δ−d)

◮ Discrete-time uncertain NCS model:

ξk+1 = ˜ A(θk)ξk + ˜ B(θk)uk, where

˜ A(θk) =              (θk) Md+δ−1(θk) Md+δ−2(θk) . . . M1(θk) M0(θk) . . . I . . . . . . ... ... ... . . . . . . . . . ... ... . . . . . . I             

and

(θk) = eAhk , ˜ B(θk) =         Md+δ(θk) I . . .         , M j (θk) =        hk−tk j hk−tk j+1 eAs dsB if 0 ≤ j ≤ d + δ − d, if d + δ − d < j ≤ d + δ

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Network Control Systems: Modelling

BUT.....LET’SKEEP ITSIMPLE... Considerthesmall-delaycase,withaconstant samplingintervalandnopacketdropouts

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Network Control Systems: Modelling

◮ Continuous-time (sampled-data) dynamics of the linear plant:

˙ x(t) = Ax(t) + Bu∗(t) u∗(t) = uk for t ∈ [sk + τk, sk+1 + τk+1)

U Uk

k-1

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Discrete-time Model

◮ Use an extended state vector: ξk =

xT

k

uT

k−1

T , xk := x(sk)

◮ Uncertain parameters: θk := τk ◮ Discrete-time uncertain NCS model:

ξk+1 = ˜ A(τk)ξk + ˜ B(τk)uk where ˜ A(τk) =

• eAh

h

h−τk eAsdsB

• ,

˜ B(τk) = h−τk eAsdsB I

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Discrete-time Closed-loop Model

◮ Discrete-time uncertain NCS model:

ξk+1 = ˜ A(τk)ξk + ˜ B(τk)uk

◮ In closed-loop with the static discrete-time extended-state feedback

controller uk = −Kξk = − ¯ K xk − Kuuk−1: ξk+1 =

• ˜

A(τk) − ˜ B(τk)K

• ξk

=

• eAh −

h−τk eAsdsB ¯ K h

h−τk eAsdsB −

h−τk eAsdsBKu − ¯ K −Ku

• ξk

◮ Discrete-time linear system with exponential uncertainty : varying delay τk

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Stability Analysis

◮ Discrete-time uncertain closed-loop NCS model:

ξk+1 =

• ˜

A(τk) − ˜ B(τk)K

• ξk =: H(τk)ξk

◮ A first approach using a common quadratic Lyapunov function:

V (ξ) = ξ T Pξ, P = PT > 0

◮ Closed-loop NCS is globally asymptotically stable if there exists

P = PT > 0, 0 < γ < 1 such that H T(τ)PH(τ) − P < −γ P, ∀τ ∈ [τmin, τmax]

◮ Infinite set of Linear Matrix Inequalities (LMIs) ◮ How to arrive at a finite number of LMIs?

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Stability Analysis

◮ Basic idea: embed the uncertainty matrix set H(τ), τ ∈ [τmin, τmax]

in a polytopic set with generators (vertices) Hi, i = 1, . . . , N: {H(τ) | τ ∈ [τmin, τmax]} ⊆ convex hull(H1, . . . , HN)

H1 H2 H3 H4 H5 H(τ) convex hull(H1, . . . , HN)

◮ Discrete-time uncertain closed-loop NCS model

ξk+1 =H(τk)ξk is globally asymptotically stable if there exist P = PT > 0, 0 < γ < 1 such that the following finite set of LMIs are satisfied: H T

i PHi − P < −γ P, ∀i = 1, . . . , N

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Stability Analysis

◮ Basic idea: embed the uncertainty matrix set H(τ), τ ∈ [τmin, τmax]

in a polytopic set with generators (vertices) Hi, i = 1, . . . , N: {H(τ) | τ ∈ [τmin, τmax]} ⊆ convex hull(H1, . . . , HN)

H1 H2 H3 H4 H5 H(τ) convex hull(H1, . . . , HN)

◮ How to get such a polytopic overapproximation?

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Stability Analysis

◮ Methods for obtaining polytopic overapproximations based on

• Interval matrices

(Cloosterman, van de Wouw, Heemels, Nijmeijer, CDC 2006)

• Taylor series

(Hetel, Daafouz, Iung, TAC 2007)

• Real Jordan form

(Cloosterman, van de Wouw, Heemels, Nijmeijer, CDC 2007, ACC 2008, TAC 2009)

• Gridding (and norm bounding of approximation error)

(Fujioka, ACC 2008), (Suh, Automatica 2008), (Donkers, Hetel, Heemels, van de Wouw, HSCC 2009, TAC 2010), (Skaf, Boyd, TAC 2009)

• Cayley-Hamilton theorem

(Gielen et. al. Automatica 2010)

• Comparison of methods for polytopic overapproximation for NCS

(Heemels et. al., HSCC2010)

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Extensions & Remarks

Extensions & Remarks (Cloosterman et al, CDC 2007, ACC 2008, Automatica 2010), (Hetel et al, CDC2009)

◮ Stability of the intersample behavior (and therefore the sampled-data

NCS) is also guaranteed

◮ Variations in the sampling interval hk ∈ [hmin, hmax] and the large delay

case τk > hk

◮ Packet Dropouts: modeled as prolongation of maximal sampling interval,

maximal delay or separate (hybrid) model

◮ Usage of parameter-dependent Lyapunov functions, which can be proven

to be more general than discrete-time Lyapunov-Krasovskii methods

◮ LMI-based controller synthesis LMIs for (augmented) state feedbacks:

uk = −Kξk and uk = − ¯ K xk

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Motion Control Example

◮ Example from the document printing

domain with continuous-time dynamics: ˙ x = Ax + Bu∗(t), A = 1

• , B =

b

• ,

with b := nrR JM + n2JR , x = xs(t) ˙ xs(t)T

Lower roller Upper roller JR, rR xs n JM Motor u∗ ◮ State-feedback uk = − ¯

K xk

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Example with periodic delays

◮ Constant sampling interval: h = 1 ms ◮ Periodically varying delays: τa, τb, τa, τb, .... (τa = 0.2h, τb = 0.6h) ◮

¯ K = 50 1.18

◮ Also possible for varying sampling intervals hk showing similar effects

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Example with large delays

◮ Constant sampling interval: h = 1 ms ◮ τmin = 0, ¯

K = 50 K2

• ◮ Two methods for polytopic overapproximation:
• Method 1: based on interval matrices
• Method 2: based on the Jordan form approach

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Example with delays and packet dropouts

◮ Constant sampling interval: h = 1 ms ◮ τmin = 0, ¯

K = 50 Kb

• ◮ Packet dropouts, with the maximum number of subsequent dropouts

¯ δ = 0, 1, 2

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Approximate Tracking Control Problem

◮ Continuous-time (sampled-data) dynamics of the plant:

˙ x(t) = Ax(t) + Bu∗(t), x(0) = x0 u∗(t) = uk, for t ∈

• sk + τk, sk+1 + τk+1
• ◮ Time-varying sampling intervals, small delays, no packet dropouts

◮ Desired trajectory: xd(t) ◮ Approximate tracking control problem:

Design a discrete-time tracking controller for uk such that limt→∞ |x(t) − xd(t)| ≤ ε for some small ε > 0 Performance is about making ε small

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Controller Design for Tracking

◮ Controller Design (Feedforward + Feedback):

uk := u f f

e (sk)− ¯

K

• x(sk) − xd(sk)
• Exact feedforward u f f

e (t) induces the desired solution:

˙ xd(t) = Axd(t) + Bu f f

e (t)

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Controller Design for Tracking

◮ Controller Design (Feedforward + Feedback):

uk := u f f

e (sk)− ¯

K

• x(sk) − xd(sk)
• Exact feedforward u f f

e (t) induces the desired solution:

˙ xd(t) = Axd(t) + Bu f f

e (t)

◮ Sampled feedforward in face of ZOH, time-varying sampling times and delays in

u f f (t) = u f f

e (sk) − u f f e (t) for t ∈

• sk + τk, sk+1 + τk+1
• Time

Errordueto:

• Zero-orderhold
• Unknowndelays

s0 s1 s2 s0 + τ0 s1 + τ1 s2 + τ2

τ0 τ1 τ2

Sampled Feedforward +ZOH Exact feedforward

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Tracking Performance

◮ Sampled-data tracking error dynamics (e = x − xd):

˙ e(t) = Ae(t) − B ¯ Ke(sk) + Bu f f (t) (u f f (t) = feedforward error) for t ∈ sk + τk, sk+1 + τk+1

• ◮ We aim at input-to-state stability (ISS), i.e.

|e(t)| ≤ β(|¯ e(0)|, t) + γ ( sup

0≤s≤t

|u f f (s)|)

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Tracking Performance

◮ Sampled-data tracking error dynamics (e = x − xd):

˙ e(t) = Ae(t) − B ¯ Ke(sk) + Bu f f (t) (u f f (t) = feedforward error) for t ∈ sk + τk, sk+1 + τk+1

• ◮ We aim at input-to-state stability (ISS), i.e.

|e(t)| ≤ β(|¯ e(0)|, t) + γ ( sup

0≤s≤t

|u f f (s)|)

e(t) t γ |u f f (t)|

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Tracking Performance

◮ Sampled-data tracking error dynamics (e = x − xd):

˙ e(t) = Ae(t) − B ¯ Ke(sk) + Bu f f (t) (u f f (t) = feedforward error) for t ∈ sk + τk, sk+1 + τk+1

• ◮ We aim at input-to-state stability (ISS), i.e.

|e(t)| ≤ β(|¯ e(0)|, t) + γ ( sup

0≤s≤t

|u f f (s)|)

e(t) t β(|¯ e(0)|, t) γ |u f f (t)|

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Tracking Performance

◮ Sampled-data tracking error dynamics (e = x − xd):

˙ e(t) = Ae(t) − B ¯ Ke(sk) + Bu f f (t) (u f f (t) = feedforward error) for t ∈ sk + τk, sk+1 + τk+1

• ◮ We aim at input-to-state stability (ISS), i.e.

|e(t)| ≤ β(|¯ e(0)|, t) + γ ( sup

0≤s≤t

|u f f (s)|)

e(t) t β(|¯ e(0)|, t) γ |u f f (t)| β(|¯ e(0)|, t) + γ |u f f (t)|

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Tracking Performance

◮ Sampled-data tracking error dynamics (e = x − xd):

˙ e(t) = Ae(t) − B ¯ Ke(sk) + Bu f f (t) (u f f (t) = feedforward error) for t ∈ sk + τk, sk+1 + τk+1

• ◮ We aim at input-to-state stability (ISS), i.e.

|e(t)| ≤ β(|¯ e(0)|, t) + γ ( sup

0≤s≤t

|u f f (s)|)

◮ ISS gain function γ (·) determines ultimate bound on the tracking error

depending on 1) plant properties, 2) network properties, 3) controller

◮ Stability LMIs presented before also guarantee ISS!

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Steady-state Tracking Control Performance

Ultimate bound for the steady-state tracking error:

◮

|e(t)| ≤ γ sup |u f f (t)| based on:

• 1. The gain γ amplifying feedforward errors to tracking errors

γ depends on 1. plant, 2. network, 3. controller

• 2. Bound on the feedforward error:

sup |u f f (t)| ≤ (τmax + hmax) max

t∈R |∂u f f e (t)

∂t | depends on 1. network, 2. plant+desired trajectory

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Steady-state Tracking Control Performance

Ultimate bound for the steady-state tracking error:

◮

|e(t)| ≤ γ sup |u f f (t)| based on:

• 1. The gain γ amplifying feedforward errors to tracking errors

γ depends on 1. plant, 2. network, 3. controller

• 2. Bound on the feedforward error:

sup |u f f (t)| ≤ (τmax + hmax) max

t∈R |∂u f f e (t)

∂t | depends on 1. network, 2. plant+desired trajectory

◮ Steady-state performance depends on:

• 1. plant properties
• 2. network properties
• 3. controller
• 4. desired trajectory

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Motion Control Example

◮ Example from the document printing

domain with continuous-time dynamics: ˙ x = Ax + Bu∗(t), A = 1

• , B =

b

• ,

with b := nrR JM + n2JR , x = xs(t) ˙ xs(t)T

Lower roller Upper roller JR, rR xs n JM Motor u∗ ◮ Desired trajectory: xd(t) =

Ad sin(ωt) Adω cos(ωt)T , with Ad = 0.01 and ω = 2π

◮ Exact feedforward is given by u f f

e (t) = − Adω2 b

sin(ωt)

◮ Feedback gain matrix: ¯

K = 50 1.18

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Motion Control Example

◮ Constant sampling interval:

h = 5 × 10−3 s

◮ Network delays: τ ∈ [0, τmax]

Discrete-time approach Discrete-time approach (on sampling instants) Delay-impulsive approach

Tracking error bounds

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Motion Control Example

◮ Constant sampling interval:

h = 5 × 10−3 s

◮ Network delays: τ ∈ [0, τmax]

Discrete-time approach Discrete-time approach (on sampling instants) Delay-impulsive approach

Tracking error bounds Such plot can help to obtain insight in the tradeoff between

◮ Network Specs ◮ Control performance specs

For details on the delay-impulsive ap- proach, see van de Wouw et. al. IJRNC 2010

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NCS with communication constraints

◮ Communication constraints:

• Network is divided into sensor and actuator nodes
• Only one node can access the network simultaneously
• This gives rise to the problem of scheduling (communication protocols)

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NCS with communication constraints

Two approaches exist for NCS with communication constraints:

◮ Continuous-time approach:

• Work of Walsh, Neši´

c, Teel, Carnevale, Tabarra, Heemels, van de Wouw

• General nonlinear plants and (UGES) protocols
• Continuous-time controllers
• hmin = 0 = τmin = 0

◮ Discrete-time approach:

• Work of Donkers, Heemels, van de Wouw (to appear in TAC, 2010)
• Exploits linearity of plants and controllers
• Both continuous-time and discrete-time controllers
• hk ∈ [hmin, hmax], τk ∈ [τmin, τmax] with hmin = 0, τmin = 0
• Specific protocols

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Discrete-time NCS model with communication constraints

◮ Results in a discrete-time switched linear uncertain system

¯ xk+1 = ˜ Aσk,hk,τk ¯ xk with state ¯ x = (x p, xc, ey, eu) consisting of the plant state, controller state, and the network-induced errors on the sensor readings and actuator commands

• Uncertainty: unknown time-varying transmission intervals and

delays leads to exponential uncertainty terms

• Switching due to scheduling:

e.g. Round Robin or Try-Once-Discard protocols

◮ Polytopic overapproximations based on gridding method

to embed exponential uncertainties in a polytopic model

◮ LMI-based stability conditions based on parameter-dependent Lyapunov

functions

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Illustrative Example

Illustrative Example (batch reactor without delays) Linear plant, 2 sensor nodes

◮ Results on bounds on the transmission interval (given for TOD protocol

• nly)

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Illustrative Example

Illustrative Example (batch reactor with delays)

◮ Now: varying delays, varying transmission intervals & comm. constraints [17]: Heemels, W.P.M.H., Teel, A.R., van de Wouw, N., Neši´ c, D., "Networked Control Systems with Communication Constraints: Tradeoffs between Sampling Intervals, Delays and Performance", IEEE Transactionson Automatic Control, 55(8), p. 1781-1796, 2010

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Conclusions

◮ Discrete-time modelling framework for linear networked control systems

with

• time-varying sampling intervals
• time-varying delays
• packet dropouts
• communication constraints

◮ LMI-based conditions for stability ◮ LMI-based conditions for controller synthesis ◮ Results less generic than emulation framework by Neši´

c, Teel (nonlinear systems, generic classes of protocols)

◮ Approach is tailored to linear systems, discrete-time controller design,

particular protocols and can provide less conservative results

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Future Work

◮ Discrete-time approach for nonlinear NCS (CDC, 2010, joint work with

Dragan Nesic)

◮ Including quantisation in the discrete-time framework ◮ Decentralised/distributed controllers in a networked setting (EU-Project

WIDE)

◮ Application: Project ‘Connect & Drive’ on the Cooperative Adaptive Cruise

Control

◮ Work on a Matlab Toolbox for Networked Control Systems