11

TUTORIAL on Networked Control Systems with Delay Cicsyn2010

2nd International Conference on Computational Intelligence, Communication Systems and Networks

Liverpool, UK, July 29th, 2010

Vasilis Tsoulkas: Center for Security Studies, Athens, Greece

& Dept. of Mathematics, University of Athens

Research Group: - Pantelous Athanasios., University of Liverpool, - Dritsas Leonidas., Hellenic Airforce Academy - Halikias George, City University, London, UK.

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ContentsContents

1.1. Introduction – General FeaturesIntroduction – General Features2.2. NCS Modeling - the issue of networkNCS Modeling - the issue of network induced delays induced delays 3.3. Discretization of NCS dynamicsDiscretization of NCS dynamics4.4. Decomposing the Uncertain Delay (nominal and Decomposing the Uncertain Delay (nominal and

uncertain parts) and the NCS dynamicsuncertain parts) and the NCS dynamics5.5. Robust Stability Analysis based on the augmented Robust Stability Analysis based on the augmented

closed-loop vector (“closed-loop vector (“ξξ”)”)6.6. Design of a Simple Output Tracking ControllerDesign of a Simple Output Tracking Controller7.7. Investigation of Robust Tracking Performance via Investigation of Robust Tracking Performance via

Simulation - Numerical Examples for Networked Simulation - Numerical Examples for Networked Stable and Unstable systemsStable and Unstable systems

8.8. Conclusions & Topics for further studyConclusions & Topics for further study

33

Schematics of Networked Control SystemsSchematics of Networked Control Systems

Networked control systems (NCSs) are spatiallyNetworked control systems (NCSs) are spatially distributed distributed systems for which the communication betweensystems for which the communication between sensors, sensors, actuators, and controllers is supported by a sharedactuators, and controllers is supported by a shared communication network.communication network. Hespanha et al.: “Survey of Recent Results in Networked Control Systems” (Proceedings of the IEEE, Vol. 95, No. 1, January 2007)

Motivation & Some BenefitsMotivation & Some Benefits

Easy and low cost installation, wiring, maintenance, Easy and low cost installation, wiring, maintenance, configurationconfiguration

Distributed Controllers and Plant with low cost Distributed Controllers and Plant with low cost distributed sensors and actuators are all coupled over distributed sensors and actuators are all coupled over the same Real Time communications networkthe same Real Time communications network

The distributed nature of elements offers great The distributed nature of elements offers great flexibility of architectures.flexibility of architectures.

Applicable in a wide variety of fields such as: Applicable in a wide variety of fields such as: Remote surgery, mobile sensor networks, UAV’s, Remote surgery, mobile sensor networks, UAV’s, Space tele-operations and Robotics.Space tele-operations and Robotics.

44

55

Distributed Networked SystemDistributed Networked System

66

Control networks are indicated by solid lines, Control networks are indicated by solid lines, and diagnostics networks are indicated by dashed lines.and diagnostics networks are indicated by dashed lines.

777

1. Introduction1. Introduction

Feedback control systems wherein the control loops are closed through a real-time network are called Networked Control Systems (NCSs)

Defining feature of NCS: Information (reference input, plant output, control input, etc.) is exchanged using a network among control system components (sensors, controllers, actuators, etc.).

88

1. Introduction1. Introduction ΝΝetwork Induced Delaysetwork Induced Delays

Information flow in the control loop is delayed due to Information flow in the control loop is delayed due to – buffering, buffering, – access contentionaccess contention (the time a node waits until it gets access to the (the time a node waits until it gets access to the

network),network),– computation computation delay delay (assume absorbed into {(assume absorbed into {ττca ca (k)(k)} )} )– propagation (“transmission”) delays.propagation (“transmission”) delays.

– Network-induced delays in NCS appear in the information flow Network-induced delays in NCS appear in the information flow betweenbetween (“k”(“k” denotes denotes the dependence on the the dependence on the kkthth sampling periodsampling period))..

A).A). TThe he sensorsensor and the and the controller controller {{ττsc sc ((kk))}}, , ( (controller controller receives “outdated” information about process behaviorreceives “outdated” information about process behavior))

B).B). TThe he controllercontroller and the and the actuatoractuator {{ττca ca ((kk))}}, , ((control control action cannot be applied “on time” and the controller does not action cannot be applied “on time” and the controller does not know the exact instance the calculated control signal will be know the exact instance the calculated control signal will be received by the actuatorreceived by the actuator))

99

1.1. IntroductionIntroduction ΝΝetwork Induced Delaysetwork Induced Delays

When a When a static linear time static linear time invariant controller invariant controller is is employed, employed, can can lumplump the the delaysdelays ττsc sc ((kk)),, ττca ca ((kk)),,

into into

ττkk== ττsc sc ((kk))++ττca ca ((kk).).

Network-induced delays in NCS Network-induced delays in NCS between the sensor and thebetween the sensor and the controller controller {{ττsc sc ((kk))}}, a, andnd between between the controller and the actuator the controller and the actuator {{ττca ca ((kk))}}, , (“k”(“k” denotes denotes the dependence on the the dependence on the kkthth sampling periodsampling period)). .

1010

1.Introduction1.Introduction Tracking Control Design for NCS Tracking Control Design for NCS

The Usual Approach for NCS Analysis & Design:The Usual Approach for NCS Analysis & Design: – design a controller ignoring the network, then design a controller ignoring the network, then – analyze stability, performance and robustness with respect to the analyze stability, performance and robustness with respect to the

effects of network-delays and scheduling policy…(usually via effects of network-delays and scheduling policy…(usually via the selection of an appropriate the selection of an appropriate scheduling protocolscheduling protocol))..

The issue of Tracking Control over Networks has not been The issue of Tracking Control over Networks has not been adequately metadequately met

– very limited published work on NCS Tracking !!!very limited published work on NCS Tracking !!!– the majority of NCS publications concerns the majority of NCS publications concerns regulationregulation , ,

(“design a controller which brings the output/state to “0” )(“design a controller which brings the output/state to “0” )– many results on tracking for many results on tracking for TTime ime DDelayed elayed SSystems (TDS) but ystems (TDS) but

cannot be applied “as is” to NCS due to the “Network-cannot be applied “as is” to NCS due to the “Network-centric” nature of NCS e.g.centric” nature of NCS e.g. special nature of delays in NCSspecial nature of delays in NCS the fundamental issue of “Packet Loss/Drops” the fundamental issue of “Packet Loss/Drops” Scheduling, Quality of Service, MiddlewareScheduling, Quality of Service, Middleware

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1.Introduction1.Introduction Tracking Control Design for NCSTracking Control Design for NCS

Concerning NCS Robust Tracking PerformanceConcerning NCS Robust Tracking Performance……

– only preliminary results - no strict mathematical proofsonly preliminary results - no strict mathematical proofs– yet…useful “lessons learned” through extensive simulations on yet…useful “lessons learned” through extensive simulations on S.I.S.O systems S.I.S.O systems– we investigate both we investigate both constant unknown constant unknown or or time-varying uncertain time-varying uncertain delaysdelays with known bounds with known bounds– we do not take into account the network delays in the tracking we do not take into account the network delays in the tracking controller design process… controller design process… – ““a posteriori” analysis of stability, performance and a posteriori” analysis of stability, performance and conservatism conservatism of resultsof results

– we do not take into account “packet drops”we do not take into account “packet drops”– Analysis & Synthesis in the Analysis & Synthesis in the continuous time domaincontinuous time domain– No need to assume knowledge of the P.D.FsNo need to assume knowledge of the P.D.Fs (not a stochastic (not a stochastic approach)approach)

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2. NCS Modeling2. NCS Modeling NCS with network-induced delays in the actuation and sensor path

Assumptions made …• the dynamics of the NCS under investigation is a combination of a the dynamics of the NCS under investigation is a combination of a continuous–time LTIcontinuous–time LTI plant with a plant with a discrete–time controllerdiscrete–time controller..• Time Invariant controller can lump τsc (k), τca (k), into τk= τsc (k)+τca

(k).• Single source of uncertainty and performance degradation the lumped transmission delay τk. • No plant uncertainties or nonlinearities - No packet drops

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2. NCS Modeling - Assumptions 2. NCS Modeling - Assumptions

In Practice…In Practice… the dynamics of the NCS under investigation is a the dynamics of the NCS under investigation is a

combination of a combination of a continuous–time uncertain/nonlinearcontinuous–time uncertain/nonlinear plant plant with a with a discrete–time (“sampled-data”) controllerdiscrete–time (“sampled-data”) controller..

The The sampler is time-drivensampler is time-driven, whereas , whereas both both controller and controller and actuator areactuator are event-drivenevent-driven, , (=(=they update their outputs as they update their outputs as soon as they receive a new samplesoon as they receive a new sample))..

Some packets are lost or intentionally dropped (contain Some packets are lost or intentionally dropped (contain obsolete/useless info)obsolete/useless info)

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The delays The delays ττkksc , sc , ττkk

caca ,, ττkk < h < h

ττkksc sc = = ττsc sc (k)(k) is the delay experienced by a state or output is the delay experienced by a state or output

sample x(kh), y(kh), sampled at time instance “kh” and sample x(kh), y(kh), sampled at time instance “kh” and presented –after a delay presented –after a delay ττkk

sc sc to the event–driven remote to the event–driven remote

controller for control computation purposes.controller for control computation purposes.

ττkkca ca ==τ τ ca ca (k)(k) is the delay experienced by the control–action, is the delay experienced by the control–action,

computed immediately after its reception at time instance computed immediately after its reception at time instance kh+ kh+ ττkk

sc sc until it is transmitted via the network to the Z.O.H until it is transmitted via the network to the Z.O.H

(and finally presented to the event–driven actuator). (and finally presented to the event–driven actuator).

The computation delay is absorbed into The computation delay is absorbed into τ τ kkcaca

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ττ k k : : Total delay within the kTotal delay within the kthth sampling period sampling period, ,

i.e. the time i.e. the time fromfrom the instant when the sampling node the instant when the sampling node samples sensor data from the plant samples sensor data from the plant toto the instant when the instant when actuators exert a control action –actuators exert a control action –whose computation whose computation was based on this samplewas based on this sample– to the plant.– to the plant.

ττkk== ττkksc sc + + ττkk

ca ca

(since a static time invariant control law is employed)(since a static time invariant control law is employed)

Known BoundsKnown Bounds::

0 ≤ 0 ≤ ττ minmin < < ττk k < < ττ maxmax ≤ h ≤ h

The delays The delays ττkksc , sc , ττkk

caca ,, ττkk < h < h

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NCS Timing Diagram (NCS Timing Diagram (ττkk < h < h) for “short” () for “short” (ττkk < h < h) ) + bounded delay + bounded delay 00 ≤≤ ττ minmin < < ττk k < < ττ max max ≤h≤h

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2. NCS Modeling 2. NCS Modeling

DDifficulties in case of Discrete “Sampled Data” Controllerifficulties in case of Discrete “Sampled Data” Controller

û(t) is the “most recent” control action presented to the event–driven actuator at the time instance t within a sampling period [kh, kh + h) & can take two values ûk or ûk-1

û(t) experiences a “jump” at the uncertain or unknown time instance kh+ τ k , changing from ûk-1 into ûk (uncertain actuation instance) Very Complicated Dynamics Impulse Delayed Systems, Asynchronous Dynamical Systems, Hybrid Systems, etc even for the regulation case (r=0)

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2. NCS Modeling - the issue of network-induced delays2. NCS Modeling - the issue of network-induced delays

NCS Timing Diagram form Zhang & Branicky paper (IEEE Control Systems Magazine, Febr.2001).

Hence (unless Hence (unless ττkk is constant) is constant) it is not possible to treat the ensuing NCS it is not possible to treat the ensuing NCS in a standard “sampled data” or “time-delayed” settingin a standard “sampled data” or “time-delayed” setting. Instead . Instead a a “hybrid” setup should rather be“hybrid” setup should rather be used used, , as for example the one presented inas for example the one presented in P. Naghshtabrizi and J. P. Hespanha, “Stability of network control systems with P. Naghshtabrizi and J. P. Hespanha, “Stability of network control systems with variable sampling and delays” in Proc. of the 44th Annual Allerton Conf. on variable sampling and delays” in Proc. of the 44th Annual Allerton Conf. on Communication, Control, and Computing, 2006.Communication, Control, and Computing, 2006.

Possible misconceptions if symbols are not adequately clarified…Authors clarify that the confusing symbol “u(kh)” denotes the actuation that takes place at kh+ τk and its value is u(kh) = -Kx(kh)

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CONTENTSCONTENTS

1.1. Introduction – General FeaturesIntroduction – General Features2.2. NCS Modeling - the issue of network-induced delays NCS Modeling - the issue of network-induced delays 3.3. Discretization of NCS dynamicsDiscretization of NCS dynamics4.4. Decomposing the Uncertain Delay (nominal and Decomposing the Uncertain Delay (nominal and

uncertain parts)uncertain parts)5.5. Robust Stability Analysis based on the closed-loop Robust Stability Analysis based on the closed-loop

augmented vector (“augmented vector (“ξξ”)”)6.6. Design of A Simple Output Tracking ControllerDesign of A Simple Output Tracking Controller7.7. Investigation of Robust Tracking Performance via Investigation of Robust Tracking Performance via

Simulation - Numerical Examples for Networked Simulation - Numerical Examples for Networked Stable and Unstable systemsStable and Unstable systems

8.8. Conclusions & Topics for further studyConclusions & Topics for further study

2020

33. . Descretization of NCS state equation withDescretization of NCS state equation with “small delay”“small delay”

ττkk < h < h xxkk = = x(kh)x(kh)

xxk+1k+1 = = ΦΦ x xk k + + ΓΓ00((ττkk) û) ûkk + + ΓΓ1 1 ((ττkk) û) ûk-1k-1 (Σ1)(Σ1)

notation notation xxkk, x, xk-1k-1,…,… denotes the values x(kh), x(kh-h), …denotes the values x(kh), x(kh-h), … of of the the periodically sampledperiodically sampled discrete–time signaldiscrete–time signal coming out of the coming out of the sampler. The same notation for sampler. The same notation for yykk, y, yk-1k-1,…,…

We keep the “hat” notation for ûWe keep the “hat” notation for ûk k , û, ûk-1 k-1 as a reminder of the as a reminder of the asynchronous,asynchronous, (“jump”) nature of these signals (“jump”) nature of these signals..

ÕÕnn … … is an is an nn-column zero vector, -column zero vector, IInn is the is the n x n n x n identity matrix, identity matrix, 00nn is the is the n x n n x n zero matrix.zero matrix.

MMTT is the transpose of a matrix. M > 0 (< 0) means that M is the transpose of a matrix. M > 0 (< 0) means that M is is positive (negative) definite.positive (negative) definite.

ˆ( ) ( ) ( ) and ( ) ( )c c cx t A x t B u t y t C x t

2121

33. Discretization. Discretization of state equation dynamics of NCS of state equation dynamics of NCS (Comments)(Comments)

xxk+1k+1 = = ΦΦ x xk k + + ΓΓ00((ττkk) û) ûkk + + ΓΓ1 1 ((ττkk) û) ûk-1k-1 (Σ1)(Σ1)

Exact Discretization between equidistantExact Discretization between equidistant sampling sampling instances instances finite dimensional finite dimensional dynamics…dynamics…

The uncertain time varying delay The uncertain time varying delay ττkk can still take any can still take any (out of infinite) values within the allowable interval (out of infinite) values within the allowable interval the uncertainty of the uncertainty of ττkk generates an uncertainty in the generates an uncertainty in the actuation instance actuation instance System matrices System matrices (Γ (Γ00((ττkk)), Γ, Γ11((ττkk)))) are uncertain are uncertain

Presence of a delayed input termPresence of a delayed input term ûûk-1k-1

22

Exact Discretization despite the “jump’’ nature of Exact Discretization despite the “jump’’ nature of û(t)xk = x(kh), Φ = exp(Ach)

-1

State Equation: (1) leads to

ˆ ˆ( ) exp( ) ( ) exp( ( - )) exp( ( - ))

ˆ( ) ( ) ( ), ( ) ( )k

k

kh kh h

c c c k c c k

kh kh

c c c

x kh h A h x kh A kh h s B u A kh h s B u ds

x t A x t B u t y t C x t

0 1

1 0

Define the three matrices , ,

exp( ), ( ) exp( ( - )) , ( ) exp( ( - ))kh kh h

kc c c c c

kh kh

A h A kh h s B ds A kh h s B ds

0 1

0 -

0 1 -1, where ( ) exp( ) , ( ) exp( )

we have used the following: - ( - since .) and changing the variable of integrat

: ˆ ˆ( ) ( ) ( ) ( )k

k

h hk k

c c c c

h

k kk k A B d A d

kh h s so d ds h const

x kh h x kh t u t u

0

0 -

00-

ion into

- , the new limits of integration are ( - ) and 0 so we get the simplified expression for Γ :

( ) exp( ( - )) exp( ) (- ) exp( )k

k

k

kh h hk

c c c c c c

kh h

d ds h

A kh h s B ds A d A d

λ λ

23

Exact Discretization despite the “jump’’ nature of Exact Discretization despite the “jump’’ nature of û(t)xk = x(kh), Φ = exp(Ach)

• Similarly from the definition of Γ1, using the same change of variables as

previously

-

1

-

( - - ) the new limits of integration are and ( - ) so we get :

( ) exp( ( - )) exp( ) (- ) exp( )

Moreover assuming there is no uncertainty on out

k

k

k

kh h hk

c c c c c c

kh h h

kh h s d ds h h

A kh h s B ds A d A d

put matrix we have : ( ) ( ) c k c ky kh C x kh or y C x

( ) ( )

0 0

A usefull identity for calculating integrals of matrix exponential functions

( ( )). exp( )

0 00

tX t Xr

n x n n x nk

m x n m x m

m x n m

X Y e e Ydrsuch as t

I

-( )

0

0

( ( - ) ) 0 0

0

1

To Compute ( ) we can use above identity as:

0( ) [ 0 ] .

1

0 0 the zero row vector with n columns

Ident ity I

c

k

hAk

c

Ac Bc Thk T

n

x n

e d

I e

where

24

• Notice that :

Exact Discretization despite the “jump’’ nature of Exact Discretization despite the “jump’’ nature of û(t)xk = x(kh), Φ = exp(Ach)

1

0( ) ( ) ( )

0- -

( )0

0

From 2nd equation : ( )

=

- ( )

c c c

k k

c

kh h

A A Ac c c

h h

hAk

c

e d e d e d

e d

(3.A).

(3.B).

2525

ContentsContents

1.1. Introduction – General FeaturesIntroduction – General Features2.2. NCS Modeling - the issue of network-induced delays NCS Modeling - the issue of network-induced delays 3.3. Descretization of NCS dynamics equationDescretization of NCS dynamics equation4.4. Decomposing the Uncertain Delay (nominal and Decomposing the Uncertain Delay (nominal and

uncertain uncertain parts) and the NCS dynamicsparts) and the NCS dynamics5.5. Robust Stability Analysis based on the closed-loop Robust Stability Analysis based on the closed-loop

augmented vector (“augmented vector (“ξξ”)”)6.6. Design of A Simple Output Tracking ControllerDesign of A Simple Output Tracking Controller7.7. Investigation of Robust Tracking Performance via Investigation of Robust Tracking Performance via

Simulation - Numerical Examples for Networked Simulation - Numerical Examples for Networked Stable and Unstable systemsStable and Unstable systems

8.8. Conclusions & Topics for further studyConclusions & Topics for further study

2626

4.4. Decomposing the uncertain delay of the system Decomposing the uncertain delay of the system (into nominal & uncertain part)(into nominal & uncertain part)

Examples: τo = τ min τo = τ max τo = τ avg

ττoo is chosen as constant and known is chosen as constant and known («(«semi-arbitrarysemi-arbitrary») ») Use ofUse of “Min Max” “Min Max” techniques for selection of techniques for selection of ττoo The nominally delayed system, The nominally delayed system, Stability Analysis and Stability Analysis and

Controller Synthesis depend on the (user’s)Controller Synthesis depend on the (user’s) choice of choice of ττoo

2727

4.4. Decomposing the uncertain delay of the system Decomposing the uncertain delay of the system (into nominal & uncertain part)(into nominal & uncertain part)--

0

0

( )0

0

( )

so the nominal part of ( ) is and

and the uncertain part of is ΔΓ ( )

( )

k

c

c

hA

c

hA

c

h

e d

e d

(4.C).

0 ( )k

2828

4.4. Decomposing the uncertain delay of the system (into Decomposing the uncertain delay of the system (into nominal & uncertain part)nominal & uncertain part) - -

1( )k

0

0

( )1

( )01

The Uncertain part is: ( ) and

the Nominal Part is : Γ ( )

c

k

c

hAk

c

h

hA

c

h

e d

e d

4.D

2929

4. Decomposing the uncertain delay of the system 4. Decomposing the uncertain delay of the system (into nominal & uncertain part)(into nominal & uncertain part)

0 1

( )0 01 0

0

The nominal part can be calculated from:

Moreover from previous ΔΓ ( ) and ( ) we observe the following relation

between the two uncertain matrices:

( ) ( ) c

k k

hA

ce d

max1 0 min ( ) ( ), [ , ] 4.Ek k k

3030

ContentsContents

1.1. IntroductionIntroduction

2.2. NCS Modeling - the issue of network-induced delays NCS Modeling - the issue of network-induced delays

3.3. Descretization of NCS dynamics equationDescretization of NCS dynamics equation4.4. Decomposing the Uncertain Delay (nominal and Decomposing the Uncertain Delay (nominal and

uncertain parts)uncertain parts)5.5. Robust Stability Analysis based on the augmented Robust Stability Analysis based on the augmented

closed-loop vector (“closed-loop vector (“ξξ”)”)6.6. Design of Simple Output Tracking ControllerDesign of Simple Output Tracking Controller7.7. Investigate Robust Tracking Performance via Investigate Robust Tracking Performance via

Simulation Simulation 8.8. Conclusions & Topics for further studyConclusions & Topics for further study

3131

5. Robust Stability Analysis5. Robust Stability Analysis based on the closed-loop based on the closed-loop vector augmented (“vector augmented (“ξξ”)”)

- Closing the loop - Closing the loop

THE AUGMENTED CLOSED–LOOP STATE VECTOR THE AUGMENTED CLOSED–LOOP STATE VECTOR (“ACLSV”)(“ACLSV”) “ “ξξ””

xxkk+1+1 = = ΦΦ x xkk + + ΓΓ00((ττkk) û) ûkk + + ΓΓ1 1 ((ττkk) û) ûk-1k-1

Static State Feedback Static State Feedback ((SSFSSF)) ûûkk = -= -KKsf sf xxkk ûûk-1k-1 = = -K-Ksfsfxxk-1k-1

Closed Loop DynamicsClosed Loop Dynamics xxk+1k+1= [= [Φ-Φ- ΓΓ00((ττkk) K) Ksfsf] x] xkk + [- + [- ΓΓ1 1 ((ττkk) K) Ksfsf ] x ] xk-1k-1

only periodically sampled state vector values {xonly periodically sampled state vector values {xk+1k+1, , xxkk, , xxk-1 k-1 } } are presentare present

3232

55. Robust Stability Analysis . Robust Stability Analysis based on the closed-loop vector based on the closed-loop vector (“(“ξξ”)”)

Define the augmentedDefine the augmented

“ “sampled data”sampled data”

closed-loop state vectorclosed-loop state vector

k

Assuming there is no uncertainty on output matrix it holds:

[ 0 ] [ 0]

0 is the zero square matrix (n x n).

ok k k c n n k c k

n

y C x C I C C

3333

5. Robust Stability Analysis 5. Robust Stability Analysis based on the closed-loop vector (“based on the closed-loop vector (“ξξ”)”)

2 x 2

k

sf

ο

The Closed loop Matrix A ( , , ) is a function of :

the uncertain delay τ

the preselected gain K and

the predetrmined nominal delay τ

k n nsf sf sfA K R

The above matrix relation is manageable and Robust Control Methods now can be used.

sf

0 1 0sf

n n

Using 4C, 4D, 4E : A can brake down to a nominal time invariant part A

and an uncertain part ( ) as

( ) - ( ) -ΔΓ ( ) -A +

I

:

0

osf

ksf

sf sf sf

1

n n

ΔΓ ( )

0 0

sf osf sfA

3434

ContentsContents

1.1. IntroductionIntroduction2.2. NCS Modeling - the issue of network-induced NCS Modeling - the issue of network-induced

delaysdelays3.3. Discretization of NCS dynamicsDiscretization of NCS dynamics4.4. Decomposing the Uncertain Delay (nominal and Decomposing the Uncertain Delay (nominal and

uncertain parts) and NCS dynamicsuncertain parts) and NCS dynamics5.5. Robust Stability Analysis based on the augmented Robust Stability Analysis based on the augmented

closed-loop vector (“closed-loop vector (“ξξ”)”)6.6. Design of Simple Output Tracking ControllerDesign of Simple Output Tracking Controller7.7. Investigate Robust Tracking Performance via Investigate Robust Tracking Performance via

Simulation Simulation 8.8. Conclusions & Topics for further studyConclusions & Topics for further study

3535

66.. Design of Simple (Set Point) Tracking Controllers Design of Simple (Set Point) Tracking Controllers SPCT = Set Point Tracking Controller(s) SPCT = Set Point Tracking Controller(s) The reference signal to be tracked by the output is (piecewise) The reference signal to be tracked by the output is (piecewise) constant (a “set point”)constant (a “set point”)

–AssumptionAssumption:both the plant and the controller under :both the plant and the controller under investigation are investigation are continuous–time LTI systemscontinuous–time LTI systems

–Since theSince the controller is time invariantcontroller is time invariant, , can can lumplump the delaysthe delays ττsc sc ((kk)),, ττca ca

((kk)),, into into ττkk== ττsc sc ((kk))++ττca ca ((kk)). . – A “naïve” tracking controller consists of two parts: Feedback & A “naïve” tracking controller consists of two parts: Feedback & Feedforward Feedforward u(t) = -Kx(t)+Fru(t) = -Kx(t)+Fr

The feedback part (The feedback part (-Kx(t)-Kx(t)) assures closed-loop stability) assures closed-loop stability The feedforward part (The feedforward part (FrFr) assures that the static gain is “1” ) assures that the static gain is “1” (Stable Transfer Function from r to y) (Stable Transfer Function from r to y)

3636

6. Design of Simple (Set Point) Tracking Controller6. Design of Simple (Set Point) Tracking Controller

•Suffers from three drawbacks (“naïve”):• the plant must not contain integrators (system matrix A is nonsingular)• cannot handle disturbances and/or model uncertainties (it is NOT Robust)• Number of inputs ≥ Number of outputs (“overactuation”)

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ContentsContents

1.1. IntroductionIntroduction2.2. NCS Modeling - the issue of network-induced NCS Modeling - the issue of network-induced

delaysdelays3.3. Descretization of NCS dynamics equationDescretization of NCS dynamics equation4.4. Decomposing the Uncertain Delay (nominal and Decomposing the Uncertain Delay (nominal and

uncertain parts)uncertain parts)5.5. Robust Stability Analysis based on the closed-loop Robust Stability Analysis based on the closed-loop

vector (“vector (“ξξ”)”)6.6. Design of Simple Output Tracking ControllerDesign of Simple Output Tracking Controller7.7. Investigate Robust Tracking Performance via Investigate Robust Tracking Performance via

Simulation Simulation 8.8. Conclusions & Topics for further studyConclusions & Topics for further study

3838

7. Robustness of Tracking Performance7. Robustness of Tracking PerformanceNumerical Example 1Numerical Example 1: a networked stable & minimum phase system: a networked stable & minimum phase system

• A “benevolent” stable & minimum phase (=zeros in LHP) system with infinite Gain Margin and…• “Lightly Damped” = stable poles close to the Imaginary axis “damping ratio” is small damped oscillative open-loop behaviour (typical in aerospace and “flexible space structure” applications)

•SPTC was designed via LQR with R=1, Q=1000*I2

u(t) = -30.63 x1(t) - 30.63 x2(t) + 31.63*rgives “perfect tracking” in the absence of delays

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7. Robustness of Tracking Performance7. Robustness of Tracking PerformanceNmerical Example 1: a networked stable & minimum phase system Nmerical Example 1: a networked stable & minimum phase system

with constant delaywith constant delay

The Networked Version with constant delay constant delay ττk k

ττsc sc ==ττca ca = 0.0131 s = 0.0131 s ττkk== ττsc sc ++ττcaca=0.0262s=0.0262s

•Assuming that τk ≤ h this delay corresponds (for the discrete time control case) to a sampling frequency of 38Hz a relatively “slow sampling”…

• “slow sampling” is typical for NCS (fast sampling increases # of packets increases network traffic increases chances for collisions packet loss/drops)

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7. Robustness of Tracking Performance7. Robustness of Tracking PerformanceNumerical Example 1Numerical Example 1: : a networked stable & minimum phase system a networked stable & minimum phase system

with constant delay with constant delay

The Networked Version with constant delay ττkk

ττsc sc ==ττca ca = 0.0131 s = 0.0131 s ττkk== ττsc sc ++ττcaca=0.0262s=0.0262s

• 7th order Pade Approximation used in simulations for the constant time-delay

•Reference Signal(s) r are (piecewise) constant:• combination of step functions or • square pulse with period slower than the system’s time constants

• Simulation needs time for Instability to occur (see next Figs)

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77. Robustness of Tracking Performance. Robustness of Tracking PerformanceNumerical Example 1Numerical Example 1: a networked stable & minimum phase system : a networked stable & minimum phase system with with constant delayconstant delay

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77. Robustness of Tracking Performance. Robustness of Tracking PerformanceNumerical Example 1Numerical Example 1: a networked stable & minimum phase system : a networked stable & minimum phase system with with constant delay constant delay

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7. Robustness of Tracking Performance7. Robustness of Tracking PerformanceNumerical Example 1Numerical Example 1: a networked stable & minimum phase system : a networked stable & minimum phase system

with with uncertain (time-varying) delay uncertain (time-varying) delay

• The Networked Version with uncertain time-varying delay ττk k

varying between ττminmin = 0 = 0 and ττmaxmax = 0.0312s < h = 0.0312s < h corresponding to a sampling frequency of 32 Hz

• Implementation used in simulations:ττkk = = ττo o + + δ τδ τunc unc , , ||δ| < 1δ| < 1

with ττoo = = ττavg avg = (= (ττmax max ++ ττ min min )/2 = 0.0156 s )/2 = 0.0156 s being the “mean value” (a constant “nominal” delay) and |δ|<1 being a random variable of uniform distribution.

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77. Robustness of Tracking Performance. Robustness of Tracking PerformanceNumerical Example 1Numerical Example 1: a networked stable & minimum phase system : a networked stable & minimum phase system with with uncertain (time-varying) delay uncertain (time-varying) delay 0=0=ττminmin ≤ ≤ ττk k ≤ ≤ ττmaxmax = 0.0312s = 0.0312s

ττkk = = ττo o + + δ τδ τunc unc , , ||δ| < 1δ| < 1

ττoo = = ττavg avg = (= (ττmax max ++ ττ min min )/2 )/2

An instance of the actual An instance of the actual uncertainly varying delay uncertainly varying delay used in simulations used in simulations

ττkk = 0.0156 = 0.0156 + 0.0156*+ 0.0156* δδ

||δ| < 1δ| < 1

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7. Robustness of Tracking Performance7. Robustness of Tracking PerformanceNumerical Example 1Numerical Example 1: a networked stable & minimum phase system : a networked stable & minimum phase system with with uncertain (time-varying) delay uncertain (time-varying) delay

ττkk = 0.0156 = 0.0156 ++ 0.01560.0156 **δδ ||δ| < 1δ| < 1

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7. Robustness of Tracking Performance7. Robustness of Tracking PerformanceNumerical Example 1Numerical Example 1: a networked stable & minimum phase system : a networked stable & minimum phase system with with uncertain delayuncertain delay

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7. Robustness of Tracking Performance7. Robustness of Tracking PerformanceNumerical Example 2Numerical Example 2: a networked unstable system: a networked unstable system

•SPTC was designed via LQR with R=1, Q=100*I2

u(t) = -9.05 x1(t) -10.78 x2(t) + 10.05 *rgives “perfect tracking” in the absence of delays

•The “Q” matrix was selected “small” in order to avoid high feedback gains…and yet

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7. Robustness of Tracking Performance7. Robustness of Tracking PerformanceNumerical Example 2Numerical Example 2: a networked unstable system with : a networked unstable system with constant constant

delaydelay

ττkk== ττsc sc ++ττcaca=0.0155s=0.0155s

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7. Robustness of tracking performance…….some 7. Robustness of tracking performance…….some commentscomments

Many more simulation results with different 2Many more simulation results with different 2ndnd order order “benchmark” S.I.S.O systems from the literature (not “benchmark” S.I.S.O systems from the literature (not shown)shown)

But….we can deduce useful conclusions (despite the lack of But….we can deduce useful conclusions (despite the lack of a mathematically rigorous approach)a mathematically rigorous approach)

Clearly a more sophisticated approach is needed for the Clearly a more sophisticated approach is needed for the design of tracking controllers for NCSdesign of tracking controllers for NCS

We cannot “pretend” that the “delays are not there” - must We cannot “pretend” that the “delays are not there” - must take them into account in the design phase.take them into account in the design phase.

We can not compromise stability (avoid large gains) - rule We can not compromise stability (avoid large gains) - rule

of thump for Time-Delayed -Systems (mid ’50s result !!!)of thump for Time-Delayed -Systems (mid ’50s result !!!)

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CONCLUSIONS AND FUTURE WORKCONCLUSIONS AND FUTURE WORK

1.1. The constant delay case (contrary to intuition) is as The constant delay case (contrary to intuition) is as detrimental to detrimental to tracking performance as the varying delay tracking performance as the varying delay case.case.

2.2. The feedback gain must be kept “small”. The feedback gain must be kept “small”.

–If an LQR design is employed : extensive trial-and-error If an LQR design is employed : extensive trial-and-error simulations with various “Q” matrices must be carried out for the simulations with various “Q” matrices must be carried out for the entire delay range to ensure (at least) stability.entire delay range to ensure (at least) stability.

–Tracking for the case of unstable plants and/or lightly damped Tracking for the case of unstable plants and/or lightly damped plants is not trivial. plants is not trivial.

3.3. For Unstable plants it is always difficult to enforce tracking For Unstable plants it is always difficult to enforce tracking (with or without delays).(with or without delays).

4.4. When implementing the tracking controllers in discrete-When implementing the tracking controllers in discrete-time special attention is needed due to (1) the interplay between time special attention is needed due to (1) the interplay between sampling period and delay and (2) the “asynchronous / jump” sampling period and delay and (2) the “asynchronous / jump” nature of the control signalnature of the control signal

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Last Minute Thoughts – Dynamical Systems with Time DelaysLast Minute Thoughts – Dynamical Systems with Time Delays

Consider the time delay systems:Consider the time delay systems:

2

n

1) )

and

( ) ( ), t [- , 0] ( ) ( ), t [- ,0]

state x(t) R (t) is the continuous I.C.

( ) ( ) ( ) (S ( ) ( ) ( ( )) (S

M

d d

x t t d x t t d

x t Ax t A x t d x t Ax t A x t d t

M

M

M

M:1. ( ) is a Cont. function satisfying

with d a constant positive scalar. It is the upper bound of d(t)

2. d(t) is a differ. function satisfying 0 d(t) d and

d is given

d(t) 1.

0 d(t) dAsumptions d t

m

previously and is the u.b. of d(t).

3. d(t) is a differ. function satisfying and d(t) .

, are given pos. scalars representing l. u. bounds of interval time varying

delay d(t) and

0 d ( )

m M

M

d d

d t d

ρ as given previously.

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1

T

Theorem 1.( ) is asymptotically stable if

there exist matrices P > 0 and Q > 0

A

: 0

such that

where the Lyapunov

-

d

Td

S

P PA Q PA

A P Q

V(t

- K

) =

rasovskii functional was used:

( ) ( ) ( ) ( )t

T T

t d

x t Px t x s Qx s ds

2

T

Theorem 2. Under assumption 2, (S ) is As. Stable if there exist matrices P > 0

andA

: 0

Q > 0 such that

-

(1-

the

) d

Td

P PA Q PA

A P Q

( )

Lyapunov - Krasovskii functional of the form

V(t) = ( ) ( ) ( ) ( ) is used.t

T T

t d t

x t Px t x s Qx s ds

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CONCLUSIONS AND CONCLUSIONS AND FUTURE WORKFUTURE WORK GGeneralizeneralizee achieved results for achieved results for

– MIMO NCS plants with multiple delays, Parametric MIMO NCS plants with multiple delays, Parametric Uncertainties & Actuator constraints Uncertainties & Actuator constraints

The use of The use of Robust Control Methodologies Robust Control Methodologies (H(H∞ ∞ or or “Guaranteed Cost”) for the design of Feedback Gain“Guaranteed Cost”) for the design of Feedback Gain

The employment of The employment of Integral Action Integral Action (apart from feedback (apart from feedback and feedforward terms) in the tracking control and feedforward terms) in the tracking control Algorithm(s).Algorithm(s).

Investigate Specific Investigate Specific ApplicationsApplications: Aerospace & : Aerospace & Robotics (Teleoperation)Robotics (Teleoperation)

NCS’s indeed constitute a very interesting and rich field NCS’s indeed constitute a very interesting and rich field of control systems both in theoretical results as well as in of control systems both in theoretical results as well as in future applications.future applications.

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THANK YOUTHANK YOU