DISSIPATIVE ANALYSIS FOR RANDOM TIME VARYING DELAY SAMPLED-DATA CONTROLLER DESIGN FOR SINGULAR NETWORKED CASCADE CONTROL SYSTEM

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DISSIPATIVE ANALYSIS FOR RANDOM TIME

VARYING DELAY SAMPLED-DATA CONTROLLER

DESIGN FOR SINGULAR NETWORKED CASCADE

CONTROL SYSTEM

1

M. Vijaya Kumar,

S.Saravanakumar

S.Nagarani

1.2.3.

Department of Mathematics, Sri Ramakrishna Institute of Technology – Coimbatore (India)

ABSTRACT

In this paper we investigated the problem on dissipative analysis using sample data controller for a class of singular networked cascade control systems (NCCS) with random time varying delays and external disturbances. A new stochastic variable with a Bernoulli distribution has been introduced and the information of probability distribution of the varying delay is measured and transformed into a new deterministic time-varying delay. A new set of Lyapunov-Krasovskii functions are constructed to verify that the NCCS obeyed regular, impulse-free, stable and strictly (Q, S, R) - dissipative. The derived LKF conditions are framed by linear matrix inequalities (LMIs) and can be verified by using MATLAB LMI Toolbox. At the end a Numerical simulations are given to express the efficiency of the derived results.

Index Terms: Singular Systems; Networked Cascade Control; Sampled-Data; Convex

Combination.

I. INTRODUCTION

Singular systems in other word descriptor systems have been studied by many researchers because of its

valuable applications in various fields like aerospace systems, electrical circuits, power systems and mechanical

systems and so on. In recent works number of standard state space are converted to singular systems. More

preciously, the singular systems are more complicated to compare with other standard type systems. On recent

days a large number of research article based on stability analysis are solved by many researchers.

In recent days number of improvements related to cascade control systems have been developed due its

importance in many fields. [1] Developed the concept of H-infinity state feedback control for a class of

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class of nonlinear systems and Performance assessment of cascade control systems. Cascade control of

superheated steam temperaturewith neuro-pid controller been developed by [3].

Robust dissipativity for delayed neural networks with random uncertainties has been modeled by [12]. Reliable

dissipative control of discrete-time singular systems with time delays studied by [13]. Dissipative analysis for

Markovian jump systems with time-varying delays, neural networks with time-varying delay and randomly

occurring uncertainties, sliding mode control of switched stochastic systems and discrete-Time T-S fuzzy

stochastic systems with time-varying delay been discussed by [14]-[16].

Motivated by the above consideration, this is the first attempt to consider the problem of dissipative analysis for

singular NCCSs with random time varying delay and external disturbances via sampled-data cascade control. By

implementing Wirtinger’s inequality, a new set of sufficient conditions are developed which ensure that the resulting closed-loop system is admissible.

Notations: Throughout this paper, Superscripts”T” and “(-1)” stand for matrix transposition and matrix inverserespectively. Rn denotes the n-dimensional Euclidean space. Z+ denotes the set of positive integers. Rn n

denotes the set of all n×n real matrices. P >0 (respectively P <0) means that P is positive definite (respectively, negative definite). I and 0 represent identity matrix and zero matrix with compatible dimension.

II. PROBLEM FORMULATION AND PRELIMINARIES

In this section, we first present the system model for the NCCSs with a singular plant. The considered system is

based on a class of singular plant with the data packets are transmitted via networks. The communication

network are considered with the network-induced delay and packet loss simultaneously and the controllers

depend on state vectors of the respective plants.

The cascade system under consideration have the primary plant given by the following equation:

wherex1(t) and y1(t) are the state vector and output vector of the primary plant. A1, B1, C1 and C3 are known real

constant matrices with appropriate dimensions.

Further, a continuous-time system secondary plant is formulated using a class of linear time invariant

singularsystem with time-varying state delay and external disturbances, given by

where is the state vector; is the control input vector; is the disturbance with

limited energy; is the output of the secondary plant; matrix E may be singular and it is assumed that rank

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and are the connection weights with appropriate dimensions. The time-varying delay in

(2) is assumed to satisfy 0≤h(t)≤h with h˙(t) =µ <1, h is a positive constant representing maximum time delay.

The state feedback controller in secondary plant is described as;

u

(t) = u

(t) + K

x

(t),

₍₃₎

whereu1(t) is the control input of the primary plant.

It is noted that, in networked cascade control system the data packets are transmitted via network, so there is a

possibility of the control signals depend on the sampling communication and performing control operations

according to the sampling results. Especially, the sampled data control problem will be formulated through an

input delay approach. By considering this point, we consider the control input of primary plant of the form

u

(t) = K

x

(t

), t

≤ t < t

, k = 0, 1, 2, . . . ,

whereK1 is the gain matrix to be determined. The state variables of system are measured at time instants

. . . , tk, tk+1, . . . , that is, only x1(tk)are available for intervaltk ≤ t < tk+1. The interval between any twosampling

instants is assumed to be bounded by τ for any k≥ 0, tk+1−tk = τk≤τ always hold, where τ is the maximum upper

bound of the sampling interval. Usually, in existing literature, the input delay due to sampling is assumed as τ(t) = t−tk then the sampling interval can be written as tk = t− (t−tk) = t−τ(t).

[A1:] Let the time delay τ(t) can be bounded by 0≤τ(t)≤τ2 and its probability distribution is assumed as follows;

Suppose τ(t) takes values in [0 :τ1] or (τ1, τ2] and prob{τ(t)∈[0 :τ1)}=δ0 or prob{τ(t)∈(τ1:τ2]}= 1−δ0 where τ1, τ2

are integers satisfying 0≤τ(t)≤τ2 and 0≤δ0≤1. Further, in order to describe the probability distribution of the time

delay, define two sets as follows;

D1= {t/τ(t)∈[0 : τ1]} and D2= {t/τ(t)∈(τ1: τ2]} . (4)

[A2:] Time varying delays τ1(t) and τ2(t) satisfying the condition

0 ≤τ

(t) ≤τ

, τ˙

(t) = 1 and τ

≤τ

(t) ≤τ

, τ˙

(t) = 1

₍₅₎

Further it follows from D1∪D2=z >0, D1∩D2=ϕ, where ϕ is the empty set. It is easy to check that

t∈ D1implies that event τ(t)∈[0 : τ1]occurs and t ∈ D2implies that event τ(t)∈(τ1: τ2]occurs.

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Now to introduce time varying delays τ1(t) and τ2(t) such that

Remark 2.1: Under the Assumption II and above equation, it can be seen thatδ(t)is a Bernoulli distributedwhite sequence with P rob{δ(t) = 1}=E[δ(t)] =δ0 and P rob{δ(t) = 0}=E[δ(t)] = 1−δ0. Furthermore, we can show that

E[δ(t)−δ0] = 0 and E[(δ(t)−δ0)2] =δ0(1−δ0).

Combining (1), (2), and (3), the closed-loop model of the singular networked cascade control system with time

Varying delay and external disturbance can be described as;

Definition 2.2: 1. The pair(E, A2)is said to be regular ifdet(sE−A2)is not identically zero.2. The pair (E, A2) is

said to be impulse free if deg(det(sE−A2)) =rank(E).

3. The unforced singular system is said to be regular and impulse free, if the pair (E, A2) is regular and impulse

free.

Definition 2.3: Given scalarθ >0, matricesQ,RandSwithQandRreal symmetric, system (8) is strictly

(Q,S,R) dissipative if fort> 0 under zero initial state, the following condition is satisfied:

−Q

Without loss of generality, we assume that the matrix

and

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Passive cases are shown in Figs 1, 7 and 4, respectively. Further, the corresponding controller performance

areshown in Figs 2, 5 and 8, respectively. From the simulation results, it is easy to see that the obtained

controllerdesigns are suitable to make sure the state trajectories are converging well. Also, it is noticed that the

optimized minimum gamma obtained through the results shows that the LMI based conditions in Theorems are

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V. CONCLUSION

The problems of stability and dissipative analysis of NCCS have been investigated. In this paper for a time

varying random delay technique, some novel LyapunovKrasovskii functional candidates were introduced for

admissibility and dissipative of NCCS. The derived results are tabulated. At the end, a numerical example were

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