Wenqiang Ji and Jianbin Qiu

(Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin 150001, China)

1 Introduction

Last two decades have witnessed that the networked control systems (NCSs) have drawn a great deal of attention from industrial engineering to control community because of their considerably productive applications in broad fields, for instance, power grids, water distribution networks, transportation networks, mobile sensor networks, unmanned aerial vehicles[1-3]. Nevertheless, new challenges appear when the networks are exploited in feedback loops as a communication medium continuously because of their limited transmission capacity. Specifically, in synthesis of networked control systems, quantization issue is of great importance to be addressed because signals are always needed to be quantized before communicated in a network environment[4-5].

On another fruitful frontier, the T-S fuzzy-model-based method is characterized to be a considerably efficient strategy to control many nonlinear systems[6-8]. The underlying fuzzy model is composed by a set of local plants, that are blended via fuzzy rules smoothly[9-13]. Thus it possesses competence to approximate a complicated nonlinear system via choosing appropriate fuzzy rules to arbitrary degrees of accuracy. See the references[14-16]for the latest developments on this topic. Recently, some results on nonlinear NCSs control were reported in the open literature via fuzzy-model-based approach[17-22]. For instance, in Ref.[20], both fault detection filter and state-feedback controller were designed for time-delay fuzzy NCSs with considering of quantized output and packet dropouts phenomena.

Nevertheless, the aforementioned results for NCSs were mainly derived in the framework of CQLF. Aiming to reduce the conservatism, there have also been some results on control of nonlinear NCSs on the basis of PQLFs[15,23-25]. Via a stochastic approach, the data-loss phenomenon of the network system existed in both uplink and downlink channels can be represented accurately. The authors in Ref.[23] proposed an asynchronous SOF controller for fuzzy discrete-time NCSs. Utilizing quantized measurements, Ref.[24] was concerned with the observer-based output-feedback control problem for fuzzy NCSs. Nevertheless, few results have been reported for fixed-order piecewise-affine (PWA) DOF controller design for fuzzy networked systems in framework of PQLFs.

Motivated by the above-mentioned statements, we propose a robustHfixed-order piecewise-affine (PWA) DOF controller design approach for discrete-time fuzzy networked control systems with quantized measurements. Via a state-input augmentation method, one attains the closed-loop system to be a descriptor form, such that couplings between the piecewise affine controller gains and system matrices are eliminated. Through using S-procedure and some convexification techniques, sufficient conditions can be presented in terms of LMIs. Finally, two simulation studies will be conducted to verify the feasibility of the aforementioned control scheme.

2 Preliminaries

2.1 T-S Fuzzy Affine Models

Consider the discrete-time fuzzy-affine model which is characterized by IF-THEN rules as follows:

(1)

Remark2.1It can be seen that the systems in Eq.(1) are in affine models. These models are with more powerful competence to approximate nonlinear systems.

Then denoteμl[ζ(t)] the normalized membership function (MF) of the fuzzy set

and

Based upon Eq.(2), then one has,

(2)

where

Following a similar process to Ref.[22], one can decompose the premise-variable space into crisp regions and fuzzy regions with

Consequently, one can reformulate the system(2) as follows:

(3)

where

For each subspaceSi,i∈Ithe indices utilized in the interpolation within that region are included in the setn(i). It is noted that for a crisp region, only one element is contained inn(i). Let the decomposed regions be divided into two classesI=I0∪I1whereI1stands for the index set of regions that do not contain the origin, whileI0stands for the corresponding ones which contain the origin. Thus, for all,i∈I0,ai=0.

The boundaries partitionS?Rgas:

0≤μl[ζ(t)+δ]<1,?0<|δ|?1,

l∈{1,2,…,r}}

We introduce a setΓto describe region transitions with all possibilities

For (i,j)∈Γand in the context ofi=j, the system states evolve in the same subspaceSiat the timet. Otherwise, the states are to jump from the subspaceSitoSjat that time.

Specifically, we assume that there exist matricesFiandfisuch that

This outer approximation is considerably useful especially whenSiare slab regions.

Particularly, onceSiare slabs with

whereαi,βi∈R,θi∈Rnx, and then one can precisely describe each region via a degenerate ellipsoid with

Then one has the following relationship,

(4)

wherei∈I.

2.2 Measurements Quantization

Here, the quantized measurements can be considered as:

yq(t)=?[y(t)]=[?[y1(t)],…,?[yny(t)]]T

(5)

where ?[.] stands for a logarithmic time-invariant and symmetric quantizer satisfying ?[-y(t)]=-?[y(t)]. In addition, one has the following relationships for each output channel,

(6)

whereρiis the quantization density.vijstands for the quantization level. Withσi=(1+ρi)/(1-ρi) the associated quantizer ?[.] for the logarithmic quantizer can be denoted as follows:

(7)

Noticing Eqs.(5)-(7) and utilizing a sector bound method shown in Ref.[24], one has:

-σjyj(t)≤?[yj(t)]-yj(t)≤σjyj(t)

(8)

where 1≤j≤ny.

On the basis of Eq.(8), it can be obtained that:

(9)

2.3 Fixed-Order DOF Controller

For system (3) with measurements quantization, we have the DOF controller as,

(10)

wherexc(t)∈Rnc,0≤nc≤nxis the controller states,Aci∈Rnc×nc,Bci∈Rnc×ny,Kci∈Rnu×nc,Dci∈Rnc×ny,aci∈Rnc,kci∈Rnuare controller gains to be determined. One can easily conclude thataci=0 andkci=0 fori∈I0. It is also worth mentioning that whennc=nx, Eq.(10) is characterized as a full-order DOF controller. Whilenc

With consideration of quantized measurements Eq.(9), combine the controller Eq.(5) with Eq.(3), and then one has that,

(11)

(12)

where

Via the state-input augmentation scheme, one can decouple the controller gains from the system matrices. This feature makes it easier to develop the proposed DOF controller (10) according to a convex optimization framework on the basis of LMI techniques.

Thus, we aim to synthesize a DOF controller such that the closed-loop system is asymptotically stable with robust performanceγas

‖z‖2<γ‖w‖2

where

3 Main Results

We will present several novel results on the robustHfixed-order DOF controller synthesis for system (1) in this section.

3.1 Analysis and Synthesis of Fixed-Order DOF Controller

wherem∈M(i),i∈I0,(i,j)∈Γ,

(13)

wherem∈M(i),i∈I1,(i,j)∈Γ, and

Moreover, we can attain the controller gains as:

(14)

ProofWithout loss of generality, we only prove the more complicated situation wheni∈I1in the following. Take the PQLF as follows:

(15)

On the basis of Eq.(15), if inequality Eq.(15) holds for (i,j)∈Γ,i∈I1,

zT(t)z(t)-γ2ωT(t)ω(t)<0

(16)

closed-loop system in Eq.(12) is asymptotically stable with robustHperformanceγ.

Define an augmented vector as,

(17)

and then with (i,j)∈Γ, Eq.(18) implies (17),

(18)

Noticing the space partition (4) and using S-procedure, then it yields:

and then one has,

(19)

where

(20)

where

Combining Eq.(20) and Eq.(19), then one has,

where

Extracting the fuzzy membership functions, the following inequality implies ,

where

Based on Lemma 1, forεi>0, one has:

(21)

Applying Schur complement, and then Eq.(21) can be reformulated as,

(22)

For the numerical tractability, the slack variable matricesGi1,Gi2,Gi3,Gi4, are with the following structure,

(23)

whereδ1,δ2,δ3,ρ1,ρ2,ρ3are scalar parameters.

Define

(24)

With consideration of Eq.(24), substituting the matrices defined in Eq.(23) into Eq.(22), it leads to Eq.(13).

Furthermore, the conditions in Eq.(13) indicates thatGi1(4)andGi1(5)are invertible. Consequently, one can attain the controller gains via Eq.(14).

Thus the proof is completed.

Remark3.1It is worth pointing out that the results shown in Theorem 3.1 are deduced based upon a PWA DOF controller as in Eq.(10). However the proposed control method can be further synthesized as a fuzzy PWA controller with quantization measurements as follows:

3.2 A Special Case for Static Output Feedback (SOF) Control Method Synthesis with Measurements Quantization

Ifnc=0, the proposed DOF controller (10) declines to a SOF one as,

u(t)=Dciyq(t)+kci,i∈I

(25)

whereDci∈Rnu×nyandkci∈Rnuare controller gains. It is easy to see thatkci=0 fori∈I0.

where

Based upon Theorem 3.1, one has the synthesis conditions of the aforementioned SOF controller (25).

wherem∈M(i),i∈I0,(i,j)∈Γ.

(26)

wherem∈M(i),i∈I1,(i,j)∈Γ, and

Moreover, one can attain the controller gains via

The rest derivation procedures are omitted for it is similar to Theorem 3.1.

4 Simulations

Example4.1Considering a discrete-time fuzzy-affine system as follows,

Plant RuleRl: IFx2(t) isFll, THEN

x(t+1)=Alx(t)+al+Blu(t)+Dl1w(t)

y(t)=Clx(t)+Dl2(t)w(t)

z(t)=Llx(t)+Nlu(t),l={1,2,3}

and the system matrices are given as,

[D12|D22|D32]=[0.02|0.02|0.03]

Nl=0.5,l={1,2,3}

whered1=50,d2=300. The normalized MFs and regionsSi,i=1,2,3 are presented in Fig.1.

Fig.1 MFs in Example 4.1

Quantization denseρ=0.8 is selected for the quantizer, and utilizing Theorem 3.1 withρ1=ρ3=1,ρ2=3,δ1=10，δ2=1, andδ3=-1, for the full-order (2-order) case, one can attain the feasible solutions withγmin=0.970 9. Consequently, the DOF controller gains are:

[Ac1|ac1|Bc1]=

kc1=0.743 9,Dc1=-1.250 9

[Ac2|ac2|Bc2]=

kc1=0,Dc2=-1.425 4

[Ac3|ac3|Bc3]=

kc3=-60.221 7,Dc3=-0.900 4

For the reduced-order (1-order) case, theHperformance isγmin=0.975 3 with controller gains,

[Ac1|ac1|Bc1]=[0.131 5|7.452 8|-0.162 5]

[Kc1|kc1|Dc1]=[3.047 2|1.582 8|-1.243 2]

[Ac2|ac2|Bc2]=[0.285 6|0.000 0|-0.173 8]

[Kc2|kc2|Dc2]=[2.426 7|0.000 0|-1.422 6]

[Ac3|ac3|Bc3]=[0.347 0|-0.393 0|-0.099 1]

[Kc1|kc1|Dc1]=[2.838 4|-65.640 8|-0.882 6]

In addition, by using Corollary 3.1, for the SOF controller case, we can attain feasible solution withγmin=1.185 9, and the controller gains are:

It is worth mentioning that whenki≡0, the piecewise-affine controller (10) reduces to a piecewise-linear one. Table 1 compares the robust performance between piecewise-affine (PWA) controllers with piecewise-linear (PWL) controllers.

Table1ComparisonofrobustHperformanceforpiecewise-affine/piecewise-linearcontrollers

ControllersFull-orderReduced-order0-orderPWA controllers0.970 90.975 31.185 9PWL controllers1.113 81.167 82.352 2

One can attain a better robustHperformance with a higher order controller in piecewise-affine form.

With the initial conditionsx0=[1.5 -1]T, and the external disturbancesw=270e-2t·sin(2t), the state trajectories are presented in Fig.2.

TheHperformance can be verified in Fig.3 with zero initial conditions. The ratio

is about 0.168, that is lower thanγmin=0.970 9.

Fig.2 States of the closed-loop system in Example 4.1

Fig.3 Response of the ratio in Example 4.1

In view of the practical industrial application, another simulation study is to be considered.

Example4.2Considering a nonlinear networked continuous stirred tank reactor (CSTR) system as follows,

where the system statesx1andx2stand for the reactor temperature and the concentration, respectively.urefers to the coolant stream’s temperature, which is also the system control input andwstands for the external disturbance. In this simulation example, Table 2 presents the nominal plant parameters, and

b0=F/V,b1=-ΔH/(ρCp),b2=E/R,b3=(UA)/(VρCp)

Table2NominalparametersofCSTRsysteminExample4.2

F(L/min)Cao(mol/L)Tfs(K)V(L)ρ(g/L)10013501001 000Cp(J/gk)-ΔH(J/mol)E/R(K)k0(l/min)UA(J/min·k)0.2395×1048 7507.2×10105×104

Plant RuleRl: IFx1(t) isF1l, THEN

where

Dl2=0.02

Nl=0.5

l∈{1,2,3}

Fig.4 shows the normalized MFs withd1=-50,d2=-25.6,d3=-20.65, andd4=50.

Fig.4 MFs in Example 4.2

Based on Fig.4, one can divide the premise-variable space into three subspaces,

One has that regionsS1andS3are crisp while regionS2is fuzzy.

For brevity, based on Theorem 3.1, only the full-order DOF controller will be calculated in this example.

With the quantization denseρ=0.64 for the quantizer, consequently, one can attain the controller gains:

[Ac1|ac1|Bc1]=

kc1=174.613 0,Dc1=-0.978 3

kc1=0,Dc2=-4.816 4

kc3=-189.862 2,Dc3=-0.070 6

with performance indexγmin=0.684 8.

Fig.5 States of the closed-loop system in Example 4.2

Fig.6 Response of the ratio in Example 4.2

5 Conclusions

This paper develops a new robustHfixed-order DOF controller for discrete-time fuzzy NCSs utilizing quantized measurements. Via a descriptor system transformation method and several convexification procedures, it is easy to see that one can attain the PWA fixed-order DOF controller gains via solving a set of LMIs. Two simulation studies have been conducted to illustrate the feasibility of the aforementioned strategy.

Appendix

Lemma1[15]For two real matricesMandNwith appropriate dimensions, the inequality

MTΔ(t)N+NTΔ(t)M≤εMTM+ε-1NTN

holds for a scalarε>0.

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