TUTORIAL on

1
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.
2
Contents

• 1. Introduction General Features
• 2. NCS Modeling - the issue of network induced
  delays
• 3. Discretization of NCS dynamics
• 4. Decomposing the Uncertain Delay (nominal and
  uncertain parts) and the NCS dynamics
• 5. Robust Stability Analysis based on the
  augmented closed-loop vector (?)
• 6. Design of a Simple Output Tracking
  Controller
• 7. Investigation of Robust Tracking
  Performance via Simulation - Numerical
  Examples for Networked Stable and Unstable
  systems
• 8. Conclusions Topics for further study

3
Schematics of Networked Control Systems
Networked control systems (NCSs) are spatially
distributed systems for which the communication
between sensors, actuators, and controllers is
supported by a shared communication network.
Hespanha et al. Survey of Recent Results in
Networked Control Systems (Proceedings of the
IEEE, Vol. 95, No. 1, January 2007)
4
Motivation Some Benefits

• Easy and low cost installation, wiring,
  maintenance, configuration
• Distributed Controllers and Plant with low cost
  distributed sensors and actuators are all coupled
  over the same Real Time communications network
• The distributed nature of elements offers great
  flexibility of architectures.
• Applicable in a wide variety of fields such as
  Remote surgery, mobile sensor networks, UAVs,
  Space tele-operations and Robotics.

5
Distributed Networked System
6
Control networks are indicated by solid lines,
and diagnostics networks are indicated by dashed
lines.
7
1. 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.).

7
8
1. Introduction ?etwork Induced Delays

• Information flow in the control loop is delayed
  due to
• buffering,
• access contention (the time a node waits until it
  gets access to the network),
• computation delay (assume absorbed into tca
  (k) )
• propagation (transmission) delays.
• Network-induced delays in NCS appear in the
  information flow between (k denotes the
  dependence on the kth sampling period).
• A). The sensor and the controller tsc (k),
  (controller receives outdated information about
  process behavior)
• B). The controller and the actuator tca (k),
  (control action cannot be applied on time and
  the controller does not know the exact instance
  the calculated control signal will be received by
  the actuator)

9
1. Introduction ?etwork Induced Delays
When a static linear time invariant controller is
employed, can lump the delays tsc (k), tca (k),
into tk tsc (k)tca (k).
Network-induced delays in NCS between the sensor
and the controller tsc (k), and between the
controller and the actuator tca (k), (k
denotes the dependence on the kth sampling
period).
10
1.Introduction Tracking Control Design for NCS

• The Usual Approach for NCS Analysis Design
• design a controller ignoring the network, then
• analyze stability, performance and robustness
  with respect to the effects of
  network-delays and scheduling policy(usually via
  the selection of an appropriate scheduling
  protocol).
• The issue of Tracking Control over Networks
  has not been adequately met
• very limited published work on NCS Tracking
  !!!
• the majority of NCS publications concerns
  regulation , (design a controller which
  brings the output/state to 0 )
• many results on tracking for Time Delayed
  Systems (TDS) but cannot be applied as is
  to NCS due to the Network-centric
  nature of NCS e.g.
• special nature of delays in NCS
• the fundamental issue of Packet
  Loss/Drops
• Scheduling, Quality of Service, Middleware

11
1.Introduction Tracking Control Design for NCS

• Concerning NCS Robust Tracking Performance
• only preliminary results - no strict
  mathematical proofs
• yetuseful lessons learned through
  extensive simulations on S.I.S.O systems
• we investigate both constant unknown or
  time-varying uncertain delays with known bounds
• we do not take into account the network delays
  in the tracking controller design process
• a posteriori analysis of stability,
  performance and conservatism of results
• we do not take into account packet drops
• Analysis Synthesis in the continuous time
  domain
• No need to assume knowledge of the P.D.Fs (not a
  stochastic approach)

12
2. 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 continuoustime LTI plant with
  a discretetime controller.
• Time Invariant controller ? can lump tsc (k),
  tca (k), into tk tsc (k)tca (k).
• Single source of uncertainty and performance
  degradation ? the lumped transmission delay tk.
• No plant uncertainties or nonlinearities -
  No packet drops

13
2. NCS Modeling - Assumptions

• In Practice
• the dynamics of the NCS under investigation is a
  combination of a continuoustime
  uncertain/nonlinear plant with a discretetime
  (sampled-data) controller.
• The sampler is time-driven, whereas both
  controller and actuator are event-driven, (they
  update their outputs as soon as they receive a
  new sample).
• Some packets are lost or intentionally dropped
  (contain obsolete/useless info)

14
The delays tksc , tkca , tk lt h

• tksc tsc (k) is the delay experienced by a
  state or output sample x(kh), y(kh), sampled at
  time instance kh and presented after a delay
  tksc to the eventdriven remote controller for
  control computation purposes.
• tkca t ca (k) is the delay experienced by the
  controlaction, computed immediately after its
  reception at time instance kh tksc until it is
  transmitted via the network to the Z.O.H (and
  finally presented to the eventdriven actuator).
• The computation delay is absorbed into t kca

15
The delays tksc , tkca , tk lt h

• t k Total delay within the kth sampling period,
  
• i.e. the time from the instant when the sampling
  node samples sensor data from the plant to the
  instant when actuators exert a control action
  whose computation was based on this sample to
  the plant.
• tk tksc tkca
• (since a static time invariant control law is
  employed)
• Known Bounds
• 0 t min lt tk lt t max h

16
NCS Timing Diagram (tk lt h) for short (tk lt h)
bounded delay 0 t min lt tk lt t max h
17
2. NCS Modeling Difficulties in case of Discrete
Sampled Data Controller

• û(t) is the most recent control action
  presented to the eventdriven 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 t 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
  (r0)

18
2. NCS Modeling - the issue of network-induced
delays
NCS Timing Diagram form Zhang Branicky paper
(IEEE Control Systems Magazine, Febr.2001).
Possible misconceptions if symbols are not
adequately clarified Authors clarify that the
confusing symbol u(kh) denotes the actuation
that takes place at kh tk and its value is
u(kh) -Kx(kh)
Hence (unless tk is constant) it is not possible
to treat the ensuing NCS in a standard sampled
data or time-delayed setting. Instead a
hybrid setup should rather be used, as for
example the one presented in 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
Communication, Control, and Computing, 2006.
19
CONTENTS

• 1. Introduction General Features
• 2. NCS Modeling - the issue of network-induced
  delays
• 3. Discretization of NCS dynamics
• 4. Decomposing the Uncertain Delay (nominal and
  uncertain parts)
• 5. Robust Stability Analysis based on the
  closed-loop augmented vector (?)
• 6. Design of A Simple Output Tracking
  Controller
• 7. Investigation of Robust Tracking
  Performance via Simulation - Numerical
  Examples for Networked Stable and Unstable
  systems
• 8. Conclusions Topics for further study

20
3. Descretization of NCS state equation with
small delay tk lt h ? xk x(kh)
xk1 F xk G0(tk) ûk G1 (tk)
ûk-1 (S1)

• notation xk, xk-1, denotes the values x(kh),
  x(kh-h), of the periodically sampled
  discretetime signal coming out of the sampler.
  The same notation for yk, yk-1,
• We keep the hat notation for ûk , ûk-1 as a
  reminder of the asynchronous, (jump) nature
  of these signals.
• Õn is an n-column zero vector, In is the n x n
  identity matrix, 0n is the n x n zero matrix.
• MT is the transpose of a matrix. M gt 0 (lt 0)
  means that M is positive (negative) definite.

21
3. Discretization of state equation dynamics of
NCS (Comments)

• xk1 F xk G0(tk) ûk G1 (tk) ûk-1
  (S1)
• Exact Discretization between equidistant
  sampling instances ? finite dimensional
  dynamics
• The uncertain time varying delay tk can still
  take any (out of infinite) values within the
  allowable interval
• the uncertainty of tk ? generates an
  uncertainty in the actuation instance ?
• System matrices (G0(tk), G1(tk)) are uncertain
• Presence of a delayed input term ûk-1

22
Exact Discretization despite the jump nature
of û(t)xk x(kh), F exp(Ach)
?
?
23
Exact Discretization despite the jump nature
of û(t)xk x(kh), F exp(Ach)

• Similarly from the definition of G1, using the
  same change of variables as previously

24
Exact Discretization despite the jump nature
of û(t)xk x(kh), F exp(Ach)

• Notice that

(3.A).
(3.B).
25
Contents

• 1. Introduction General Features
• 2. NCS Modeling - the issue of network-induced
  delays
• 3. Descretization of NCS dynamics equation
• 4. Decomposing the Uncertain Delay (nominal and
  uncertain parts) and the NCS dynamics
• 5. Robust Stability Analysis based on the
  closed-loop augmented vector (?)
• 6. Design of A Simple Output Tracking
  Controller
• 7. Investigation of Robust Tracking
  Performance via Simulation - Numerical
  Examples for Networked Stable and Unstable
  systems
• 8. Conclusions Topics for further study

26
4. Decomposing the uncertain delay of the system
(into nominal uncertain part)

• Examples
• to t min
• to t max
• to t avg

• to is chosen as constant and known
  (semi-arbitrary)
• Use of Min Max techniques for selection of to
• The nominally delayed system, Stability
  Analysis and Controller Synthesis depend on the
  (users) choice of to

27
4. Decomposing the uncertain delay of the system
(into nominal uncertain part)-
(4.C).
28
4. Decomposing the uncertain delay of the system
(into nominal uncertain part) -
4.D
29
4. Decomposing the uncertain delay of the
system (into nominal uncertain part)
30
Contents
1. Introduction 2. NCS Modeling - the issue of
network-induced delays 3. Descretization of NCS
dynamics equation 4. Decomposing the Uncertain
Delay (nominal and uncertain parts) 5. Robust
Stability Analysis based on the augmented
closed-loop vector (?) 6. Design of Simple
Output Tracking Controller 7. Investigate Robust
Tracking Performance via Simulation
8. Conclusions Topics for further study
31
5. Robust Stability Analysis based on the
closed-loop vector augmented (?)- Closing the
loop

• THE AUGMENTED CLOSEDLOOP STATE VECTOR (ACLSV)
  ?

• xk1 F xk G0(tk) ûk G1 (tk) ûk-1
• Static State Feedback (SSF)
• ûk -Ksf xk ûk-1 -Ksfxk-1
• Closed Loop Dynamics
• xk1 F- G0(tk) Ksf xk - G1 (tk) Ksf xk-1
• only periodically sampled state vector values
  xk1, xk, xk-1 are present

32
5. Robust Stability Analysis based on the
closed-loop vector (?)

• Define the augmented
• sampled data
• closed-loop state vector

33
5. Robust Stability Analysis based on the
closed-loop vector (?)
The above matrix relation is manageable and
Robust Control Methods now can be used.
34
Contents

• 1. Introduction
• 2. NCS Modeling - the issue of network-induced
  delays
• 3. Discretization of NCS dynamics
• 4. Decomposing the Uncertain Delay (nominal and
  uncertain parts) and NCS dynamics
• 5. Robust Stability Analysis based on the
  augmented closed-loop vector (?)
• 6. Design of Simple Output Tracking Controller
• 7. Investigate Robust Tracking Performance
  via Simulation
• 8. Conclusions Topics for further study

35
6. Design of Simple (Set Point) Tracking
Controllers

• SPCT Set Point Tracking Controller(s)
• The reference signal to be tracked by the
  output is (piecewise) constant (a set
  point)
• Assumptionboth the plant and the
  controller under investigation are
  continuoustime LTI systems
• Since the controller is time invariant, can lump
  the delays tsc (k), tca (k), into tk tsc (k)tca
  (k).
• A naïve tracking controller consists of
  two parts Feedback Feedforward u(t)
  -Kx(t)Fr
• The feedback part (-Kx(t)) assures
  closed-loop stability
• The feedforward part (Fr) assures that the
  static gain is 1 (Stable Transfer Function
  from r to y)

36
6. 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)

37
Contents
1. Introduction 2. NCS Modeling - the issue of
network-induced delays 3. Descretization of NCS
dynamics equation 4. Decomposing the Uncertain
Delay (nominal and uncertain parts) 5. Robust
Stability Analysis based on the closed-loop
vector (?) 6. Design of Simple Output
Tracking Controller 7. Investigate Robust
Tracking Performance via Simulation
8. Conclusions Topics for further study
38
7. Robustness of Tracking PerformanceNumerical
Example 1 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 R1, Q1000I2
  
• u(t) -30.63 x1(t) - 30.63 x2(t)
  31.63r
• gives perfect tracking in the absence of
  delays

39
7. Robustness of Tracking PerformanceNmerical
Example 1 a networked stable minimum phase
system with constant delay

• The Networked Version with constant delay tk
• tsc tca 0.0131 s ? tk tsc tca0.0262s
• Assuming that tk 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)

40
7. Robustness of Tracking PerformanceNumerical
Example 1 a networked stable minimum phase
system with constant delay

• The Networked Version with constant delay tk
• tsc tca 0.0131 s ? tk tsc tca0.0262s
• 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
  systems time constants
• Simulation needs time for Instability to
  occur (see next Figs)

41
7. Robustness of Tracking PerformanceNumerical
Example 1 a networked stable minimum phase
system with constant delay
42
7. Robustness of Tracking PerformanceNumerical
Example 1 a networked stable minimum phase
system with constant delay
43
7. Robustness of Tracking PerformanceNumerical
Example 1 a networked stable minimum phase
system with uncertain (time-varying) delay

• The Networked Version with uncertain
  time-varying delay tk varying between
• tmin 0 and tmax
  0.0312s lt h
• corresponding to a sampling frequency of
  32 Hz
• Implementation used in simulations
• tk to d tunc , d lt 1
• with to tavg (tmax t min )/2 0.0156
  s being the mean value (a constant
  nominal delay) and dlt1 being a random
  variable of uniform distribution.

44
7. Robustness of Tracking PerformanceNumerical
Example 1 a networked stable minimum phase
system with uncertain (time-varying) delay
0tmin tk tmax 0.0312s

• An instance of the actual uncertainly varying
  delay used in simulations
• tk 0.0156 0.0156 d d lt 1

tk to d tunc , d lt 1 to tavg
(tmax t min )/2
45
7. Robustness of Tracking PerformanceNumerical
Example 1 a networked stable minimum phase
system with uncertain (time-varying) delay
tk 0.0156 0.0156 d d lt 1
46
7. Robustness of Tracking PerformanceNumerical
Example 1 a networked stable minimum phase
system with uncertain delay
47
7. Robustness of Tracking PerformanceNumerical
Example 2 a networked unstable system

• SPTC was designed via LQR with R1, Q100I2
  
• u(t) -9.05 x1(t) -10.78 x2(t) 10.05
  r
• gives perfect tracking in the absence of
  delays
• The Q matrix was selected small in
  order to avoid high feedback gainsand yet

48
7. Robustness of Tracking PerformanceNumerical
Example 2 a networked unstable system with
constant delay
tk tsc tca0.0155s
49
7. Robustness of tracking performance.some
comments

• Many more simulation results with different
  2nd order benchmark S.I.S.O systems from
  the literature (not shown)
• But.we can deduce useful conclusions (despite
  the lack of a mathematically rigorous
  approach)
• Clearly a more sophisticated approach is
  needed for the design of tracking
  controllers for NCS
• We cannot pretend that the delays are
  not there - must take them into account in
  the design phase.
• We can not compromise stability (avoid
  large gains) - rule of thump for
  Time-Delayed -Systems (mid 50s result !!!)

50
CONCLUSIONS AND FUTURE WORK

• 1. The constant delay case (contrary to
  intuition) is as detrimental to tracking
  performance as the varying delay case.
• 2. The feedback gain must be kept small.
• If an LQR design is employed extensive
  trial-and-error simulations with various Q
  matrices must be carried out for the entire delay
  range to ensure (at least) stability.
• Tracking for the case of unstable plants and/or
  lightly damped plants is not trivial.
• 3. For Unstable plants it is always difficult
  to enforce tracking (with or without delays).
• 4. When implementing the tracking
  controllers in discrete-time special attention
  is needed due to (1) the interplay between
  sampling period and delay and (2) the
  asynchronous / jump nature of the control
  signal

51
Last Minute Thoughts Dynamical Systems with
Time Delays

• Consider the time delay systems

52
(No Transcript)
53
CONCLUSIONS AND FUTURE WORK

• Generalize achieved results for
• MIMO NCS plants with multiple delays, Parametric
  Uncertainties Actuator constraints
• The use of Robust Control Methodologies (H8 or
  Guaranteed Cost) for the design of Feedback
  Gain
• The employment of Integral Action (apart from
  feedback and feedforward terms) in the tracking
  control Algorithm(s).
• Investigate Specific Applications Aerospace
  Robotics (Teleoperation)
• NCSs indeed constitute a very interesting and
  rich field of control systems both in theoretical
  results as well as in future applications.

54

• THANK YOU