STATE FEEDBACK CONTROL OF UNCERTAIN NETWORKED CON- TROL SYSTEMS

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CHAPTER 2 Modelling of Networked Control Systems

 Formulation of Networked Control Systems

In this chapter, the modelling procedure of NCSs will be presented. Before pro-ceeding to the modelling procedure, the following assumptions will be used throughout this thesis
† The sensor is time-driven: the states of the plant are sampled periodically.
† The controller is event-driven: the control signal is calculated as soon as a new sensor data arrives at the controller.
† The actuator is event-driven: the control signal is applied to the plant as soon as a new controller data arrives at the actuator.
In this thesis, we will study the networked control system of which the generalized plant setup is depicted in Figure 2.1. Plant’s measurement signals are denoted as y(t) while input signals are denoted as u(t). Samplers S1 and S2 are time-driven while two zero-order-holds are event-driven. Furthermore, samplers S1 and S2 are not assumed to have the same sampling time. The plant outputs are sampled with a sampling interval hs 22 and sent through the network at times khs; k 2 N while the control signals are sampled with a sampling interval ha.
As shown in Figure 1.5, when two data messages sent by the sensor side arrive at the controller side during the same sampling time period, only the most recent data packet is used and the previous one is then discarded. This situation is referred to as data packet dropout. Therefore, it is not hard to see that the data packet dropouts can be viewed as prolonged network-induced delays that are at least longer than one sampling period.
In Figure 2.1, it can be noted that the measurement signals fy(khs); k 2 Ng are received by the controller side at times khs + ¿ks where ¿ks is the delay that the measurement sent at khs experiences. When there are data packet dropouts in the communication channel, the signals that the controller receives can be described as follows:
y^(t) = y(khs); 8t 2 [khs + ¿ks; (k + 1 + ns)hs + ¿ks+ns+1); (2.1)
where y(khs) is equal to the last successfully received measurement signal and ns is the number of consecutive dropouts.
The controller generates control signals using the information of y^(t). The control signals are then sampled with a sampling interval ha and sent through the network at times, equal to fu^(lha); l 2 Ng. These signals, arrive at the plant side at time lha + ¿la where ¿la is the delay that the control signal sent at lha experiences. Along with time- delay, we also include the data packet dropouts into consideration. This leads to the following digital control law:
u(t) = u^(lha); 8t 2 [lha + ¿la; (l + 1 + na)ha + ¿la+na+1): (2.2)
where na is the number of consecutive dropouts.
Now we apply a technique of modelling of continuous-time systems with digital (sampled-data) control in the form of continuous-time systems with delayed control input that was introduced by Mikheev et al. [101], Astrom and Wittenmark [43], and further developed by Fridman [102, 42]. After minor adaption of the original model.

Modelling of Random Network-induced Delays

A stochastic process has the Markov property if the conditional probability distri-bution of future states of the process, given the present state and all past states, depends only upon the present state and not on any past states, i.e. it is conditionally indepen-dent of the past states. In [53], a Markov chain is utilised to model these network delays. Modes of the Markov chain are deflned as difierent network load conditions. This def-inition using Markov property is realistic in industry network applications as network tra–c and network load conditions are rather of random nature, either in spatial or tem-poral sense. This can be re°ected in Figures 2.5 and 2.6, which respectively describes a monthly Internet tra–c and the NACA Internet tra–c grid.
In this thesis, for each mode in the Markov chain, a corresponding delay is assumed to be time-varying but upper bounded by a known constant.
Following the same line as [53], we use two Markov chains f•1(t)g and f•2(t)g to model ¿ks and ¿la, respectively. f•1(t)g is a continuous-time discrete-state Markov process taking values in a flnite set S = f1; 2; ¢ ¢ ¢ ; sg. In some small increment of time from t.

Problem Formulation and Preliminaries

A class of uncertain linear systems under consideration is described by the following equation:
x(t) = (A + ¢A)x(t) + (B + ¢B)u(t) (3.1)
where x(t) 2 Rn and u(t) 2 Rm are the plant states and control inputs, respectively. Matrices A and B are known matrices of appropriate dimensions. Matrices ¢A and ¢B characterise the uncertainties in the system and satisfy the following assumption:
ASSUMPTION 3.1 [¢A ¢B] = HF (t)[E1 E2]
where H; E1 and E2 are known real constant matrices of appropriate dimensions, and F (t) is an unknown matrix function with Lebesgue-measurable elements and satisfles F (t)T F (t) • I, in which I is the identity matrix of appropriate dimension.
Following the modelling procedure in Chapter 2, therefore, we have the model of a networked control system to be investigated, of which the setup is depicted in Figure 2.4, where ¿(•1(t); t) ‚ 0 is the sensor-to-controller delay and ‰(•2(t); t) ‚ 0 is the controller-to-actuator delay. For each mode in the Markov chain, a corresponding delay is assumed to be time-varying but upper bounded by a known constant, that is, ¿(•1(t); t) • ¿⁄(i) and ‰(•2(t); t) • ‰⁄(k).
Using the modelling procedure described in Chapter 2, the NCS can be expressed as follows:
Plant: x(t) = (A + ¢A)x(t) + (B + ¢B)u(t ¡ ‰(•2(t); t) (3.2)
Controller: u(t) = K(•1(t); •2(t))x(t ¡ ¿(•1(t); t)); (3.3)
where K(•1(t); •2(t)) is the mode-dependent controller gain and yet to be determined. Substituting (3.3) into (3.2) yields.

1 . INTRODUCTION
1.1 Introduction of Networked Control Systems
1.2 Fundamental Issues with NCSs
1.3 Recent Works on Networked Control Systems
1.3.1 Control techniques applied in networked control systems
1.4 Research Motivation
1.5 Contribution of the Thesis
1.6 Thesis Outline
2 . MODELLING OF NETWORKED CONTROL SYSTEMS
2.1 Formulation of Networked Control Systems
2.2 Modelling of Random Network-induced Delays
PART I : LINEAR UNCERTAIN NETWORKED CONTROL SYSTEMS 
3 . STATE FEEDBACK CONTROL OF UNCERTAIN NETWORKED CON- TROL SYSTEMS
3.1 Introduction
3.2 Problem Formulation and Preliminaries
3.3 Main Result
3.4 Numerical Example
3.5 Conclusion
4 . DYNAMIC OUTPUT FEEDBACK CONTROL FOR UNCERTAIN NET- WORKED CONTROL SYSTEMS
4.1 System Description and Problem Formulation
4.2 Main Result
4.3 Numerical Example
4.4 Conclusion
5 . ROBUST DISTURBANCE ATTENUATION FOR UNCERTAIN NETWORKED CONTROL SYSTEMS WITH RANDOM TIME-DELAYS
5.1 Introduction
5.2 System Description and Problem Formulation
5.3 Main Result
5.4 Numerical Example
5.5 Conclusion
6 . ROBUST FAULT ESTIMATOR DESIGN FOR UNCERTAIN NETWORKED CONTROL SYSTEMS
6.1 Problem Formulation and Preliminaries
6.2 Main Result
6.3 Numerical Example
6.4 Conclusion
PART II : NONLINEAR UNCERTAIN NETWORKED CONTROL SYSTEMS 96
7 . TAKAGI-SUGENO FUZZY CONTROL SYSTEM
7.1 Takagi-Sugeno Fuzzy Modelling
7.2 Takagi-Sugeno Fuzzy Controller
8 . STATE FEEDBACK CONTROLLER DESIGN FOR UNCERTAIN NON- LINEAR NETWORKED CONTROL SYSTEMS
8.1 Introduction
8.2 Problem Formulation and Preliminaries
8.3 Main Result
8.4 Numerical Example
8.5 Conclusion
9 . DYNAMIC OUTPUT FEEDBACK CONTROLLER DESIGN FOR UNCER- TAIN NONLINEAR NETWORKED CONTROL SYSTEMS
9.1 Problem Formulation and Preliminaries
9.2 Main Result
9.3 Numerical Example
9.4 Conclusion
10 . ROBUST DISTURBANCE ATTENUATION FOR UNCERTAIN NONLIN- EAR NETWORKED CONTROL SYSTEMS
10.1 Problem Formulation and Preliminaries
10.2 Main Result
10.3 Numerical Example
10.4 Conclusion
11 . FAULT ESTIMATION FOR UNCERTAIN NONLINEAR NETWORKED CONTROL SYSTEMS
11.1 Problem Formulation and Preliminaries
11.2 Main Result
11.3 Numerical Example
11.4 Conclusion
12 . CONCLUSIONS
12.1 Summary of Thesis
12.2 Future Research Work
APPENDIX
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