# Controller Design of Multiinput Multioutput Time-Delay Large-Scale System

Authors: Chia-Wei Lin; Tzuu-Hseng S. Li; Chung-Cheng Chen
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The paper presents a novel feedback linearization controller of nonlinear multiinput multioutput time-delay large-scale systems to obtain both the tracking and almost disturbance decoupling (ADD) performances. The significant contribution of this paper is to build up a control law such that the overall closed-loop system is stable for given initial condition and bounded tracking trajectory with the input-to-state-stability characteristic and almost disturbance decoupling performance. We have simulated the two-inverted-pendulum system coupled by a spring for networked control systems which has been used as a test bed for the study of decentralized control of large-scale systems.


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## Body

## 1. Introduction

Recently, robust stabilization of system with time delay has been of a challenging and interesting problem [1].  As we know, in general, the existence of time-delay degrades the control performance and sometimes makes the closed-loop stabilization difficult, especially for nonlinear systems.  Appropriate mathematical descriptions incorporating the time delay are the differential-difference equations, that is, differential equations with deviating arguments.  Several related reports have shown that differential-difference equations have been widely applied in theory of automatic control, the theory of self-oscillating systems, the study of problems connected with combustion in rocket motion, the problem of long-range planning in economics, a series of biological problems, and in many other areas of science and technology [2, 3].  In the past, there have been a number of interesting developments in stability criteria and controller designs for time-delay control systems but mostly were restricted to linear cases; see, for example, [1, 4–6]. In general, the global stability test of control systems with time delays is not as easy, even in the linear case, as without time delays.  It requires some disgusting tasks as solving nonlinear matrix equations [7].  It is clear that the investigation of nonlinear time-delay systems is worthwhile.  In this paper, the globally tracking and almost disturbance decoupling problem of a general class of nonlinear time-delay control systems is investigated.A large-scale system is organized as some interconnected subsystems, such as industrial control system, power systems, computer, biomedical networks, district heating systems [8], and wind farm power control [9].  Due to the inherited complexity of control approach and physical constraints on information communication between subsystems, it is necessary to design for each subsystem a decentralized controller depending on local data, even if to construct an object for the whole large-scale system.  Recently networked control systems have attracted many research interests.  The significant characteristics of networked control systems are controller decentralization, interconnected diagnostics, convenient maintenance, and low cost [10].  The application of networked control architecture will increase the performances of interconnected applications via reduced maintenance costs [11].  The major limitations in the existing networked control systems research are that (a) most researches are limited to linear cases and (b) the significant applications of networked control systems are for distributed control systems.  However, some theoretical results are not valid for distributed control systems and then have little application value [12, 13].  We have proposed a study of two-pendulum system coupled by a spring for networked control systems which has been used as a test bed for the investigation of decentralized control of large-scale systems.Recently, variable control method has been utilized to investigate nonlinear industrial system.  However, nasty chattering characteristics may excite unmodeled high frequency and even force system into instable state for variable control method [14].  The backstepping approach has been an important method for designing controller for nonlinear industrial systems.  However, a disadvantage with the backstepping approach is the intricacy which is generated by the complex duplicated differentiations of some nonlinear functions [15].  The output regulation control method [16] is utilized to the industrial system in which the output terminals are derived with an exosystem.  However, the output regulation problem should solve the complicated solution of partial-differential equation and the problem of creating the transient tracking errors.  The nonlinear  H∞ control method generally has to find out the solution for the complex Hamilton-Jacobi equation [17].  However, it is unfortunate that we can solve a closed-form solution only for some special industrial control systems [18].  The internal-model-principle method, solving a partial-differential center manifold equation [19], transforms the original tracking problem to output regulation structure.  Only for some particular industrial systems and desired trajectories, the asymptotic solutions can be found out [20].  The H∞ adaptive fuzzy control approach has been utilized to systematically address some industrial control systems [1].  Its shortcoming is that the troublesome parameter update law makes the method impractical.  During the past decade, the feedback linearization control has been the significant research direction for nonlinear systems [21] and has been addressed successfully to investigate many industrial control systems including the shunt hybrid power filter [22], high-power self-commutated voltage-source converter and current-source converter [23], and differential-drive wheeled mobile robots [24].The almost disturbance decoupling control problem, originally developed for linear and nonlinear control systems by [25, 26], respectively, exploits the research that a controller reduces the effect of the disturbance on the outputs to an arbitrary degree of accuracy, and then many important researches have been proposed for nonlinear industrial control systems [27, 28].  The authors of [26] exploit the fact that the almost disturbance decoupling performance cannot be achieved for the following control system: x˙1(t)=tan-1(x2)+θ(t), x˙2(t)=u, y=x1, where u, y are the input and output, respectively, and θ is the disturbance term.  It is fortunate that this example can be easily solved via the proposed approach in this study.  In order to exploit the significant industrial applicability, this study has favorably designed the almost disturbance decoupling controller for a two-inverted-pendulum system.


## 2. Tracking and Almost Disturbance Decoupling Controller Design

Consider the following nonlinear time-delay large-scale system with disturbance that is organized intoN subsystems interconnected by their output terminals.  The ith subsystem Si, 1≤i≤N, is shown as(1)x˙i1=fi1(xi1,xi2,…,xini)+di1(yi(t-τi1(t)))dddd+Δfi1(y1,y2,…,yN)+θi1,x˙i2=fi2(xi1,xi2,…,xini)+di2(yi(t-τi2(t)))dddd+Δfi2(y1,y2,…,yN)+θi2ix˙i2⋮ix˙i2⋮x˙ini=fini(xi1,xi2,…,xini)+dini(yi(t-τini(t)))dddd+Δfini(y1,y2,…,yN)+θini+giui,syi=xi1,
where [xi1(t)xi2(t)⋯xini(t)]T∈ℜni, uini∈ℜ, and yi∈ℜ are the state vector, the input vector, and the output vector of subsystem Si, respectively.  fini and gini are smooth nonlinear system functions. Δfini  represents the nonlinearity in the ith subsystem and the interconnection function between the ith subsystem and other subsystems.  θini  is a bounded time-varying disturbances vector, and τini(t) is a time-delay term.Define(2)Nt≡∑i=1Nni,x1≡x11,x2≡x12,…,xn1≡x1n1,xn1+1≡x21,xn1+2≡x22,…,xn1+n2≡x2n2,⋮xn1+n2+⋯+nN-1+1≡xN1,xn1+n2+⋯+nN-1+2≡xN2,…,xn1+n2+⋯+nN≡xNnN,n0≡0,xn0+n1+⋯+ni+j≡x(i+1)(j),τn0+n1+⋯+ni+j≡τ(i+1)(j),Δfn0+n1+⋯+ni+j≡Δf(i+1)(j),dn0+n1+⋯+ni+j≡d(i+1)(j),De(n0+n1+⋯+ni+j)≡d(i+1)(j).The nonlinear time-delay large-scale system will be rewritten as(3a)[x˙1x˙2⋮x˙Nt]=[f1(x1,x2,…,xNt)f2(x1,x2,…,xNt)⋮fNt(x1,x2,…,xNt)]+[0(n1-1)×1g1u10(n2-1)×1g2u2⋮0(nN-1)×1gNuN]+[De1(τ1)De2(τ2)⋮DeNt(τNt)]+[Δf1(x1,x2,…,xNt)Δf2(x1,x2,…,xNt)⋮ΔfNt(x1,x2,…,xNt)]+∑j=1Ntqj*θjd,(3b)[y1y2⋮yN]≡[h1h2⋮hN]=[x1xn1+1⋮xn1+n2+⋯+nN+1].That is,
(4)X˙(t)=f(X(t))+gu+De+Δf+∑j=1Ntqj*θjd,y(t)=h(X(t)),
where X(t)≡[x1(t)x2(t)⋯xNt(t)]T∈ℜNt is the state vector, gu≡[0(n1-1)×1g1u10(n2-1)×1g2u2⋯0(nN-1)×1gNuN]T∈ℜNt is the input vector, y≡[y1y2⋯yN]T∈ℜN  is the output vector,  θd≡[θ1d(t)θ2d(t)⋯θNtd(t)]T∈ℜNt  is a bounded time-varying disturbances vector, and Δf≡[Δf1Δf2⋯ΔfNt]∈ℜNt. f≡[f1f2⋯fNt]T∈ℜNt,  D≡[De1De2⋯DeNt]T∈ℜNt,  De≡∑i=1Ntqi*Dei∈ℜNt,  and h≡[h1h2⋯hN]T∈ℜN are smooth vector fields.  The nominal system is then defined as follows:(5a)X˙(t)=f(X(t))+gu,(5b)y(t)=h(X(t)).The nominal system of the form (5a) and (5b) is said to have the vector relative degree {r1,r2,…,rN} [29] with the following properties for all X∈ℜNt:(i)(6)LgjLfkhi(X)=0,1≤i≤N,hhhhhh1≤j≤N,k<ri-1,
where the operator L is the Lie derivative [30] and r1+r2+⋯+rN=r;(ii)  theN×N matrix
(7)Anon≡[Lg1Lfr1-1h1(X)⋯LgNLfr1-1h1(X)Lg1Lfr2-1h2(X)⋯LgNLfr2-1h2(X)⋮⋮Lg1LfrN-1hN(X)LgNLfrN-1hN(X)]
is nonsingular.The desired output trajectoryydi,1≤i≤N and its first ri derivatives are all uniformly bounded, and
(8)∥[ydi,ydi(1),…,ydi(ri)]∥≤Bdi,1≤i≤N,
where Bdi is some positive real constant.  Based on the existence of vector relative degree, it has been shown [30] that the function
(9)Ω:ℜNt⟶ℜNt
defined as
(10)ξi≡[ξ1iξ2i⋮ξrii]≡[Ω1iΩ2i⋮Ωrii]≡[Lf0hi(X)Lf1hi(X)⋮Lfri-1hi(X)],i=1,2,…,N,Ωk(X(t))≡ηk(t),k=r+1,r+2,…,Nt,LgjΩk(X(t))=0,k=r+1,r+2,…,Nt,1≤j≤N
is a diffeomorphism, if (i) the distribution
(11)gspan≡span{g1,g2,…,gN}
has involutive property; (ii) the vector fields
(12)Zjk,1≤j≤N,1≤k≤rj
are complete, where
(13)Zjk≡(-1)k-1adf~k-1g~j,1≤j≤N,1≤k≤rj,f~(X)≡f(X)-g(X)Anon-1(X)s(X),s(X)≡[Lfr1h1(X)Lfr2h2(X)⋯LfrNhN(X)]T,g~≡[g~1g~2⋯g~N]≡g(X)Anon-1(X),adpkq≡[padpk-1q],[pq]≡∂q∂Xp(X)-∂p∂Xq(X).Denote the trajectory error to be(14)eji≡ξji-ydi(j-1),i=1,2,…,N,j=1,2,…,ri,(15)ei≡[e1ie2i⋯erii]T∈ℜri,(16)eji¯≡κj-1eji,i=1,2,…,N,j=1,2,…,ri,(17)ei¯≡[e1i¯e2i¯⋯erii¯(t)]T∈ℜri,(18)e¯≡[e1¯e2¯⋯eN¯]T∈ℜr,(19)ξ≡[ξ1ξ2⋯ξr]T∈ℜr,(20)η(t)≡[ηr+1(t)ηr+2(t)⋯ηNt(t)]T∈ℜNt-r,(21)q(ξ(t),η(t))≡[Lfϕr+1(t)Lfϕr+2(t)⋯LfϕNt(t)]T≡[qr+1qr+2⋯qNt]T.Define a phase-variable canonical matrixAphasei to be
(22)Aphasei≡[010⋯0001⋯0⋮⋮000⋯1-a1i-a2i-a3i⋯-arii]ri×ri,1≤i≤N,
where a1i,a2i,…,arii are any chosen variable such that Aphasei is Hurwitz, and define the vector Bi to be
(23)Bi≡[00⋯01]Tri×1,1≤i≤N.Define the positive definite matrixQi to be solution of the following Lyapunov equation:
(24)(Aphasei)TQi+QiAphasei=-I,1≤i≤N,λmax(Qi)≡themaximumeigenvalueofQi,1≤i≤N,λmin(Qi)≡theminimumeigenvalueofQi,1≤i≤N,λmax*≡min{λmax(Q1),λmax(Q2),…,λmax(QN)},λmin*≡min{λmin(Q1),λmin(Q2),…,λmin(QN)}.Assumption 1.
For allt≥0, η∈ℜNt-r, and ξ∈ℜr, there exists a positive constant Lm such that the following inequality holds:
(25)∥q22(t,η,e-)-q22(t,η,0)∥≤Lm(∥e-∥),
where q22(t,η,e-)≡q(ξ,η).For the sake of describing precisely the considered problem, denote(26)dij≡LgjLfri-1hi(X),1≤i≤N,1≤j≤N,ci≡Lfrihi(X),1≤i≤N,ei¯¯≡a1ie1i¯+a2ie2i¯+⋯+ariierii¯,1≤i≤N.Definition 2 (see [31]).
Consider the nonlinear systemz˙=f(t,z,n), where f:[0,∞)×ℜn×ℜn→ℜn is piecewise continuous in t and has the locally Lipschitz property in z and n.  This nonlinear system is said to has the input-to-state stable property if there exist a class KL function β, a class K function γ, and positive real constants λ1 and λ2 such that, for given initial state z(t0) with ∥z(t0)∥<λ1 and any bounded noise n(t) with supt≥t0∥n(t)∥<λ2, the system state has the following property:
(27)∥z(t)∥≤β(∥z(t0)∥,t-t0)+γ(supt0≤τ≤t∥n(τ)∥)
for all t≥t0≥0.Definition 3 (see [32]).
The tracking problem with almost disturbance decoupling performance is denoted to be globally solved by the control lawu for the overall system, if the control law u has the following characteristics.(i)
The control system is input-to-state stable for disturbance inputs.(ii)
For any given initial valuex-e0≡[e-(t0)η(t0)]T, for any t≥t0, and for any t0≥0,
(28)|y(t)-yd(t)|≤β11(∥x(t0)∥,t-t0)+1β22β33(supt0≤τ≤t∥θ(τ)∥),(29)∫t0t[y(τ)-yd(τ)]2dτ≤1β44[β55(∥x-e0∥)+∫t0tβ33(∥θ(τ)∥2)dτ],where β22,  β44 are positive real constants,  β33, β55 are class K functions, and β11 is a class KL function.Theorem 4.
Assume that there exists a continuously differentiable functionT:ℜNt-r→ℜ+ such that the following properties hold for all  η∈ℜNt-r:(30a)Δ1∥η∥2≤T(η)≤Δ2∥η∥2,Δ1,Δ2>0,(30b)∇tT+(∇ηT)Tq22(t,η,0)≤-2αx∥η∥2,αx>0,(30c)∥∇ηT∥≤Δ3∥η∥,Δ3>0.Then the tracking problem with almost disturbance decoupling performance for time-delay large-scale system is globally solved by the control law:
(31)u=Anon-1{-ub+uv},(32)ub≡[Lfr1h1Lfr2h2⋯LfrNhN]T,(33)uv≡[v1v2⋯vN]T,(34)vi≡ydi(ri)-κ-ria1i[Lf0hi(X)-ydi]-κ1-ria2i[Lf1hi(X)-ydi(1)]-⋯-κ-1arii[Lfri-1hi(X)-ydi(ri-1)],1≤i≤N.
Moreover, the influence of disturbances on theL2 norm of the tracking error can be arbitrarily reduced by adjusting parameter  N2>1:(35a)k11≡k2κ-25k2κ2∥ϕξ1∥2∥Q1∥2-⋯-25k2κ2∥ϕξN∥2∥QN∥2-25,(35b)k22≡22αx-1100Δ32Lm2-25Δ32∥ϕη∥2,(35c)N2≡min{k11,k22},(35d)Ωξi(κ)≡[κ∂∂Xhiq1*⋯κ∂∂Xhiqp*⋮⋮κri∂∂XLfri-1hiq1*⋯κri∂∂XLfri-1hiqq*],ddddddddddddddddddd20031≤i≤N,(35e)Ωη(κ)≡[∂∂Xϕr+1q1*⋯∂∂Xϕr+1qp*⋮⋮∂∂XϕNtq1*⋯∂∂XϕNtqq*],
where k(κ):ℜ+→ℜ+ is a continuous function satisfying
(35f)limε→0k(κ)=0,limκ→0κk(κ)=0.Proof.
Applying the coordinate transformation (9) yields
(36)ξ˙11=∂h1∂Xf+∑j=1Nt∂h1∂Xqj*(θjd+Dej)=ξ21+∑j=1Nt∂h1∂Xqj*(θjd+Dej),⋮ξ˙r1-11=∂Lfr1-2h1∂Xf+∑j=1Nt∂Lfr1-2h1∂Xqj*(θjd+Dej)=Lfr1-1h1+∑j=1Nt∂Lfr1-2h1∂Xqj*(θjd+Dej),ξ˙r11=c1+d11u1+⋯+d1NuN+∑j=1Nt∂Lfr1-1h1∂Xqj*(θjd+Dej),⋮ξ˙1N=Lf1hN+∑j=1Nt∂h1∂Xqj*(θjd+Dej)=ξ2N+∑j=1Nt∂hN∂Xqj*(θjd+Dej),⋮ξ˙rN-1N=LfrN-1hN+∑j=1Nt∂LfrN-2hN∂Xqj*(θjd+Dej)=ξrNN+∑j=1Nt∂LfrN-2hN∂Xqj*(θjd+Dej),ξ˙rNN=cN+dN1u1+⋯+dNNuN+∑j=1Nt∂LfrN-1hN∂Xqj*(θjd+Dej),η˙k(t)=LfΩk+∑j=1Nt∂Ωk∂Xqj*(θjd+Dej)=qk+∑j=1Nt∂Ωk∂Xqj*(θjd+Dej).
Since(37)ci(ξ(t),η(t))≡Lfrihi(X(t)),1≤i≤N,(38)dij≡LgjLfri-1hi(X),1≤i≤N,1≤j≤N,(39)qk(ξ(t),η(t))=LfΩk(X),k=r+1,r+2,…,Nt.The dynamic equations of systems (3a) and (3b) can be rewritten as follows:
(40)ξ˙i1(t)=ξi+11(t)+∑j=1Nt∂∂XLfi-1h1qj*(θjd+Dej),ggggggggggggi=1,2,…,r1-1,(41)ξ˙r11(t)=c1(ξ(t),η(t))+d11(ξ(t),η(t))u1+⋯+d1N(ξ(t),η(t))uN+∑j=1Nt∂∂XLfr1-1h1qj*(θjd+Dej),⋮(42)ξ˙iN(t)=ξi+1N(t)+∑j=1Nt∂∂XLfi-1hNqj*(θjd+Dej),ggggggggggggggggi=1,2,…,rN-1,(43)ξ˙rNN(t)=cN(ξ(t),η(t))+dN1(ξ(t),η(t))u1+⋯+dNN(ξ(t),η(t))uN+∑j=1Nt∂∂XLfrN-1hNqj*(θjd+Dej),(44)η˙k(t)=qk(ξ(t),η(t))+∑j=1Nt∂∂XΩk(X)qj*(θjd+Dej),dddddddddddddddddddddk=r+1,…,Nt,(45)yi(t)=ξ1i(t),1≤i≤N.
Applying (14), (34), (37), and (38) yields the controller as
(46)u=Anon-1[-ub+uv].
The dynamic equations of systems (3a) and (3b) can be rewritten as follows by substituting (46) into (41) and (43):
(47)[ξ˙1i(t)ξ˙2i(t)⋮ξ˙ri-1i(t)ξ˙rii(t)]=[010⋯00010⋯0⋮⋮000⋯1000⋯0][ξ1i(t)ξ2i(t)⋮ξri-1i(t)ξrii(t)]+[00⋮01]vi+[∑j=1Nt∂∂Xhiqj*(θjd+Dej)∑j=1Nt∂∂XLf1hiqj*(θjd+Dej)⋮∑j=1Nt∂∂XLfri-1hiqj*(θjd+Dej)],[η˙r+1(t)η˙r+2(t)⋮η˙Nt-1(t)η˙Nt(t)]=[qr+1(t)qr+2(t)⋮qNt-1(t)qNt(t)]+[∑j=1Nt∂∂XΩr+1qj*(θjd+Dej)∑j=1Nt∂∂XΩr+2qj*(θjd+Dej)⋮∑j=1Nt∂∂XΩNt-1qj*(θjd+Dej)∑j=1Nt∂∂XΩNtqj*(θjd+Dej)],yi=[10⋯00]r×1[ξ1i(t)ξ2i(t)⋮ξri-1i(t)ξrii(t)]r×1=ξ1i(t),1≤i≤N.
Combining (14), (16), (17), (22), and (34) yields the transformation of (47):(48a)η˙(t)=q(ξ(t),η(t))+Ωη(θd+D)≡q22(t,η(t),e-)+Ωη(θd+D),(48b)κei¯.(t)=Aphaseiei¯+Ωξi(θd+D),1≤i≤N,(49)yi(t)=ξ1i(t),1≤i≤N.
We considerL(e-,η) denoted by a weighted sum of T(η) and W(e-),   and
(50)L(e-,η)≡T(η)+k(κ)W(e-)≡T(η)+k(κ)(W1(e1¯)+⋯+WN(eN¯)),
where
(51)W(e-)≡W1(e1¯)+⋯+WN(eN¯)
as a composite Lyapunov function of the subsystems (48a) and (48b) [33, 34], where W(ei¯) satisfies
(52)Wi(ei¯)≡12ei¯TQiei¯.
Utilizing (14), (25) and (30a), (30b), and (30c) result in the derivative of L along the trajectories of (48a) and (48b) as(53)L˙=[∇tT+(∇ηT)Tη˙]+k2[(e1¯•)TQ1e1¯+(e1¯)TQ1(e1¯•)ffffgff+⋯+(eN¯•)TQNeN¯+(eN¯)TQN(eN¯•)]≤[∇tT+(∇ηT)Tq22(t,η(t),e¯)+(∇ηV)TΩη(θd+D)]-k2κ[(e1¯)Te1¯+⋯+(eN¯)TeN¯]+kκ[∥eN¯∥∥(θd+D)∥∥Ωξ1∥∥Q1∥∥e1¯∥ddjjjjjdd+⋯+∥(θd+D)∥∥ΩξN∥∥QN∥∥eN¯∥]≤-∥η∥2[2αx-1100Δ32Lm2-25Δ32∥Ωη∥2]-∥e-∥2[k2κ-25k2κ2∥Ωξ1∥2∥Q1∥2ddddddddd-⋯-25k2κ2∥ΩξN∥2∥QN∥2-25]+N+1100∥(θd+D)∥2.
That is,
(54)L˙≤-k11∥e-∥2-k22∥η∥2+N+1100∥(θd+D)∥2.
From (35c), we obtain
(55)L˙≤-N2(∥e-∥2+∥η∥2)+N+1100∥(θd+D)∥2.
Define(56)e¯≡[e1¯e2¯⋮eN¯]≡[e11¯erem1¯],erem1¯∈ℜr-1.
Hence(57)L˙≤-N2(∥η∥2+∥e11¯∥2+∥erem1¯∥2)+N+1100∥(θd+D)∥2.
Utilizing (57) easily yields
(58)∫t0t(y1(τ)-yd1(τ))2dτ≤L(t0)N2+N+1100N2∫t0t∥(θd(τ)+D)∥2dτ.
Similarly, it is easy to prove that(59)∫t0t(yi(τ)-ydi(τ))2dτ≤L(t0)N2+N+1100N2∫t0t∥(θd(τ)+D)∥2dτ,ddddddddddddddddhdddddd2≤i≤N
so that statement (29) is satisfied.  From (55), we get(60a)L˙≤-N2(∥ytotal∥2)+N+1100∥(θd+D)∥2,
where
(60b)∥ytotal∥2≡∥e-∥2+∥η∥2.
Utilizing [31, Theorem 5.2] and (60a) implies the input-to-state stable property for the overall system.  Furthermore, it is easy to obtain the following inequality:
(61)Δmin(∥e-∥2+∥η∥2)≤L≤Δmax(∥e-∥2+∥η∥2).
That is,
(62)Δmin(∥ytotal∥2)≤L≤Δmax(∥ytotal∥2),
where Δmin≡min{Δ1,(k/2)λmin*} and Δmax≡max{Δ2,(k/2)λmax*}.  From (55) and (62), we get
(63)L˙≤-N2ΔmaxL+N+1100(supt0≤τ≤t∥(θd(τ)+D)∥)2.
Hence,(64)L(t)≤L(t0)e(-N2/Δmax)(t-t0)+Δmax(N+1)100N2(supt0≤τ≤t∥(θd(τ)+D)∥)2,sssssssssssssssssffssssssssssgsssssssst≥t0
which implies
(65)|y1(t)-yd1(t)|≤2L(t0)kλmin*e(-N2/2Δmax)(t-t0)+Δmax(N+1)50kλmin*N2(supt0≤τ≤t∥(θd(τ)+D)∥).
Similarly, it is easy to prove that(66)|yi(t)-yd1(t)|≤2L(t0)kλmin*e(-N2/2Δmax)(t-t0)+Δmax(N+1)50kλmin*N2×(supt0≤τ≤t∥(θd(τ)+D)∥),2≤i≤N
so that statement (28) is proved, and then the tracking problem with almost disturbance decoupling is globally solved.If the sumr1+r2+⋯+rm is equal to the system dimension n, then Theorem 4 will be reduced to the following simplified version with cancelling Assumption 1 and (30a), (30b), and (30c).Theorem 5.
The tracking problem with almost disturbance decoupling performance for time-delay large-scale system is globally solved by the control law:(67)u=Anon-1{-ub+uv},ub≡[Lfr1h1Lfr2h2⋯LfrNhN]T,uv≡[v1v2⋯vN]T,vi≡ydi(ri)-κ-ria1i[Lf0hi(X)-ydi]-κ1-ria2i[Lf1hi(X)-ydi(1)]-⋯-κ-1arii[Lfri-1hi(X)-ydi(ri-1)],1≤i≤N.
Moreover, the influence of disturbances on theL2 norm of the tracking error can be arbitrarily reduced by adjusting parameter k11>1:
(68)k11≡k2κ-25k2κ2∥ϕξ1∥2∥Q1∥2-⋯-25k2κ2∥ϕξN∥2∥QN∥2.


## 3. Illustrative Example

Consider the two inverted pendulums coupled by a spring with disturbances as shown in Figure1.  The dynamic equations are given as follows:Figure 1
The two-inverted-pendulum system coupled by a spring.S
1 (inverted pendulum 1):
(69)θ¨1(t)=(-kr24J1+m1grJ1)sin(θ1(t))+kr(l-b)2J1+kr24J1sin(θ2(t))+u1J1+θ1(t-τ1,2(t))1+θ12(t-τ1,2(t)),τ1,2(t)≡0.4(1+sin2(t)),y1=θ1(t),S2 (inverted pendulum 2):
(70)θ¨2(t)=(-kr24J2+m2grJ2)sin(θ2(t))+kr(l-b)2J2+kr24J2sin(θ˙1(t))+u2J2+θ2(t-τ2,2(t))1+θ22(t-τ2,2(t)),τ2,2(t)≡0.4(1+sin2(t))y2=θ2(t),
where ui is the torque input generated by the actuator for pendulum i (i=1,2), θi is the angular displacement of pendulum i (i=1,2), mi is the mass of pendulum i (i=1,2), Ji is the moment of inertia (i=1,2), k is the spring constant, r is the pendulum height, l is the length of pendulum, g is the gravitational acceleration constant, and b is the distance between the pendulums. Defining the state variables x1≡θ1(t), x2≡θ˙1(t), x3≡θ2(t), and x4≡θ˙2(t) yields the following state space models:(71)x˙1(t)=x2(t),x˙2(t)=(-kr24J1+m1grJ1)sin(x1(t))+kr(l-b)2J1+kr24J1sin(x3(t))+u1J1+x1(t-τ1,2(t))1+x12(t-τ1,2(t)),x˙3(t)=x4(t),x˙4(t)=(-kr24J1+m1grJ1)sin(x3(t))+kr(l-b)2J1+kr24J1sin(x2(t))+u2J2+x3(t-τ1,2(t))1+x32(t-τ1,2(t)),τ1,2(t)≡τ2,2(t)≡0.4(1+sin2(t)),y1=x1(t),y2=x3(t).The following physical parameters are chosen in our simulation:m1=2kg,  m2=2.5kg,  J1=1kg/m2,  J2=1kg/m2,  k=10N/m,  r=0.1m,  l=0.5m,  g=9.8m/s2, and b=0.5m. Hence the mathematical model can be rewritten as
(72)x˙1(t)=x2(t),x˙2(t)=1.935sin(x1(t))+0.025sin(x3(t))+u1+x1(t-0.4(1+sin2(t)))1+x12(t-0.4(1+sin2(t))),x˙3(t)=x4(t),x˙4(t)=2.425sin(x3(t))+kr(l-b)2J1+0.025sin(x2(t))+u2+x3(t-0.4(1+sin2(t)))1+x32(t-0.4(1+sin2(t))),y1=x1(t),y2=x3(t).Now we will show how to explicitly construct a controller that tracks the desired signalsyd1=yd2=0 and attenuates the disturbance’s effect on the output terminal to an arbitrary degree of accuracy.  Let us arbitrarily choose a11=a21=a12=a22=10, and the solution of Lyapunov equation is given as Aphase1=Aphase2=[01-10-10],Q1=Q2=[1.050.050.050.055],  λmax(Q1)=λmax(Q2)=1.0525, λmin(Q1)=λmin(Q2)=0.0525, λmax*=1.0525,  and λmin*=0.0525.  From (31), we obtain the desired tracking controllers
(73)u1=-1000x1-100x2-1.935sinx1-0.025sinx3,u2=-1000x3-100x4-2.425sinx3-0.025sinx2.It can be verified that the relative conditions of Theorem4 are satisfied with κ=0.1,  Bd1=Bd2=0,  N2=2.82,  k=20κ, and k11=2.82. Hence the tracking controllers will steer the output tracking errors of the closed-loop system, starting from any initial value to be asymptotically attenuated to zero by virtue of  Theorem 4.  The complete trajectories of the outputs are depicted in Figures 2 and 3.Figure 2
The output angular positionx1(θ1) of the two inverted pendulums.Figure 3
The output angular positionx3(θ2) of the two inverted pendulums.


## 4. Conclusion

In this paper, we propose a novel control design which globally solves the almost disturbance decoupling problem for multiinput multioutput nonlinear time delay large-scale system via the fuzzy feedback linearization approach.  The investigation of feedback linearization of nonlinear time-delay large-scale control systems by diffeomorphism has been proposed.  Moreover, a practical industrial system of two inverted pendulums coupled by a spring demonstrates the applicability of the proposed feedback linearization method.  Simulation results exploit the fact that the proposed methodology is successfully utilized to solve the feedback linearization problem and achieve the desired almost disturbance decoupling performance of the overall system.



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*Source: 286043-2013-09-15.xml*