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

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.

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: 𝑥̇₁(𝑡)=tan⁻¹(𝑥₂)+𝜃(𝑡), 𝑥̇₂(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 into 𝑁 subsystems interconnected by their output terminals. The 𝑖th subsystem 𝑆_𝑖, 1≤𝑖≤𝑁, is shown as𝑥̇_𝑖1=𝑓_𝑖1𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑖𝑛𝑖+𝑑_𝑖1𝑦_𝑖𝑡−𝜏_𝑖1(𝑡)+Δ𝑓_𝑖1𝑦₁,𝑦₂,…,𝑦_𝑁+𝜃_𝑖1,𝑥̇_𝑖2=𝑓_𝑖2𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑖𝑛𝑖+𝑑_𝑖2𝑦_𝑖𝑡−𝜏_𝑖2(𝑡)+Δ𝑓_𝑖2𝑦₁,𝑦₂,…,𝑦_𝑁+𝜃_𝑖2⋮⋮𝑥̇_𝑖𝑛𝑖=𝑓_𝑖𝑛𝑖𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑖𝑛𝑖+𝑑_𝑖𝑛𝑖𝑦_𝑖𝑡−𝜏_𝑖𝑛𝑖(𝑡)+Δ𝑓_𝑖𝑛𝑖𝑦₁,𝑦₂,…,𝑦_𝑁+𝜃_𝑖𝑛𝑖+𝑔_𝑖𝑢_𝑖,𝑦_𝑖=𝑥_𝑖1, (1) where 𝑥_𝑖1(𝑡)𝑥_𝑖2(𝑡)⋯𝑥_𝑖𝑛𝑖(𝑡)^𝑇∈ℜ^𝑛𝑖, 𝑢_𝑖𝑛𝑖∈ℜ, and 𝑦_𝑖∈ℜ are the state vector, the input vector, and the output vector of subsystem 𝑆_𝑖, respectively. 𝑓_𝑖𝑛𝑖 and 𝑔_𝑖𝑛𝑖 are smooth nonlinear system functions. Δ𝑓_𝑖𝑛𝑖  represents the nonlinearity in the 𝑖th subsystem and the interconnection function between the 𝑖th subsystem and other subsystems. 𝜃_𝑖𝑛𝑖  is a bounded time-varying disturbances vector, and 𝜏_𝑖𝑛𝑖(𝑡) is a time-delay term.

Define 𝑁_𝑡≡^𝑁_𝑖=1𝑛_𝑖,𝑥₁≡𝑥₁₁,𝑥₂≡𝑥₁₂,…,𝑥_𝑛1≡𝑥_1𝑛1,𝑥_𝑛1+1≡𝑥₂₁,𝑥_𝑛1+2≡𝑥₂₂,…,𝑥_𝑛1+𝑛2≡𝑥_2𝑛2,⋮𝑥_{𝑛1+𝑛2+⋯+𝑛𝑁−1+1}≡𝑥_𝑁1,𝑥_{𝑛1+𝑛2+⋯+𝑛𝑁−1+2}≡𝑥_𝑁2,…,𝑥_{𝑛1+𝑛2+⋯+𝑛𝑁}≡𝑥_𝑁𝑛𝑁,𝑛₀𝑥≡0,_{𝑛0+𝑛1+⋯+𝑛𝑖+𝑗}≡𝑥_{(𝑖+1)(𝑗)},𝜏_{𝑛0+𝑛1+⋯+𝑛𝑖+𝑗}≡𝜏_{(𝑖+1)(𝑗)},Δ𝑓_{𝑛0+𝑛1+⋯+𝑛𝑖+𝑗}≡Δ𝑓_{(𝑖+1)(𝑗)},𝑑_{𝑛0+𝑛1+⋯+𝑛𝑖+𝑗}≡𝑑_{(𝑖+1)(𝑗)},𝐷_{𝑒(𝑛0+𝑛1+⋯+𝑛𝑖+𝑗)}≡𝑑_{(𝑖+1)(𝑗)}. (2)

The nonlinear time-delay large-scale system will be rewritten as𝑥̇₁𝑥̇₂⋮𝑥̇_𝑁𝑡=𝑓₁𝑥₁,𝑥₂,…,𝑥_𝑁𝑡𝑓₂𝑥₁,𝑥₂,…,𝑥_𝑁𝑡⋮𝑓_𝑁𝑡𝑥₁,𝑥₂,…,𝑥_𝑁𝑡+0_{(𝑛1−1)×1}𝑔₁𝑢₁0_{(𝑛2−1)×1}𝑔₂𝑢₂⋮0_{(𝑛𝑁−1)×1}𝑔_𝑁𝑢_𝑁+𝐷_𝑒1𝜏₁𝐷_𝑒2𝜏₂⋮𝐷_𝑒𝑁𝑡𝜏_𝑁𝑡+Δ𝑓₁𝑥₁,𝑥₂,…,𝑥_𝑁𝑡Δ𝑓₂𝑥₁,𝑥₂,…,𝑥_𝑁𝑡⋮Δ𝑓_𝑁𝑡𝑥₁,𝑥₂,…,𝑥_𝑁𝑡+^𝑁𝑡_𝑗=1𝑞^∗_𝑗𝜃_𝑗𝑑,𝑦₁𝑦₂⋮𝑦_𝑁≡ℎ₁ℎ₂⋮ℎ_𝑁=𝑥₁𝑥_𝑛1+1⋮𝑥_{𝑛1+𝑛2+⋯+𝑛𝑁+1}. (3a)(3b)That is, 𝑋̇(𝑡)=𝑓(𝑋(𝑡))+𝑔𝑢+𝐷_𝑒+Δ𝑓+^𝑁𝑡_𝑗=1𝑞^∗_𝑗𝜃_𝑗𝑑,𝑦(𝑡)=ℎ(𝑋(𝑡)), (4) where 𝑋(𝑡)≡𝑥₁(𝑡)𝑥₂(𝑡)⋯𝑥_𝑁𝑡(𝑡)^𝑇∈ℜ^𝑁𝑡 is the state vector, 𝑔𝑢≡0_{(𝑛1−1)×1}𝑔₁𝑢₁0_{(𝑛2−1)×1}𝑔₂𝑢₂⋯0_{(𝑛𝑁−1)×1}𝑔_𝑁𝑢_𝑁^𝑇∈ℜ^𝑁𝑡 is the input vector, 𝑦≡𝑦₁𝑦₂⋯𝑦_𝑁^𝑇∈ℜ^𝑁  is the output vector,  𝜃_𝑑≡𝜃_1𝑑(𝑡)𝜃_2𝑑(𝑡)⋯𝜃_𝑁𝑡𝑑(𝑡)^𝑇∈ℜ^𝑁𝑡  is a bounded time-varying disturbances vector, and Δ𝑓≡Δ𝑓₁Δ𝑓₂⋯Δ𝑓_𝑁𝑡∈ℜ^𝑁𝑡. 𝑓≡𝑓₁𝑓₂⋯𝑓_𝑁𝑡^𝑇∈ℜ^𝑁𝑡, 𝐷≡𝐷_𝑒1𝐷_𝑒2⋯𝐷_𝑒𝑁𝑡^𝑇∈ℜ^𝑁𝑡,  𝐷_𝑒≡∑^𝑁𝑡_𝑖=1𝑞^∗_𝑖𝐷_𝑒𝑖∈ℜ^𝑁𝑡,  and ℎ≡ℎ₁ℎ₂⋯ℎ_𝑁^𝑇∈ℜ^𝑁 are smooth vector fields. The nominal system is then defined as follows:𝑋̇(𝑡)(𝑋(𝑡𝑦(𝑡)(𝑋(𝑡.=𝑓))+𝑔𝑢,=ℎ)) (5a)(5b)

The nominal system of the form (5a) and (5b) is said to have the vector relative degree {𝑟₁,𝑟₂,…,𝑟_𝑁} [29] with the following properties for all 𝑋∈ℜ^𝑁𝑡:

(i) 𝐿_𝑔𝑗𝐿^𝑘_𝑓ℎ_𝑖(𝑋)=0,1≤𝑖≤𝑁,1≤𝑗≤𝑁,𝑘<𝑟_𝑖−1, (6) where the operator 𝐿 is the Lie derivative [30] and 𝑟₁+𝑟₂+⋯+𝑟_𝑁=𝑟;

(ii) the 𝑁×𝑁 matrix 𝐴_non≡𝐿_𝑔1𝐿^𝑟1_𝑓⁻¹ℎ₁(𝑋)⋯𝐿_𝑔𝑁𝐿^𝑟1_𝑓⁻¹ℎ₁(𝑋)𝐿_𝑔1𝐿^𝑟2_𝑓⁻¹ℎ₂(𝑋)⋯𝐿_𝑔𝑁𝐿^𝑟2_𝑓⁻¹ℎ₂(𝑋)𝐿⋮⋮_𝑔1𝐿^𝑟𝑁_𝑓⁻¹ℎ_𝑁(𝑋)𝐿_𝑔𝑁𝐿^𝑟𝑁_𝑓⁻¹ℎ_𝑁(𝑋) (7) is nonsingular.

The desired output trajectory 𝑦^𝑖_𝑑,1≤𝑖≤𝑁 and its first 𝑟_𝑖 derivatives are all uniformly bounded, and 𝑦^𝑖_𝑑,𝑦^𝑖_𝑑⁽¹⁾,…,𝑦^𝑖_𝑑^(𝑟𝑖)≤𝐵^𝑖_𝑑,1≤𝑖≤𝑁, (8) where 𝐵^𝑖_𝑑 is some positive real constant. Based on the existence of vector relative degree, it has been shown [30] that the function Ω∶ℜ^𝑁𝑡⟶ℜ^𝑁𝑡 (9) defined as 𝜉_𝑖≡𝜉^𝑖₁𝜉^𝑖₂⋮𝜉^𝑖_𝑟𝑖≡Ω^𝑖₁Ω^𝑖₂⋮Ω^𝑖_𝑟𝑖≡𝐿⁰_𝑓ℎ_𝑖(𝑋)𝐿¹_𝑓ℎ_𝑖(𝑋)⋮𝐿^𝑟𝑖_𝑓⁻¹ℎ_𝑖(𝑋)Ω,𝑖=1,2,…,𝑁,_𝑘(𝑋(𝑡))≡𝜂_𝑘(𝑡),𝑘=𝑟+1,𝑟+2,…,𝑁_𝑡,𝐿_𝑔𝑗Ω_𝑘(𝑋(𝑡))=0,𝑘=𝑟+1,𝑟+2,…,𝑁_𝑡,1≤𝑗≤𝑁 (10) is a diffeomorphism, if (i) the distribution 𝑔_span≡span𝑔₁,𝑔₂,…,𝑔_𝑁 (11) has involutive property; (ii) the vector fields 𝑍^𝑘_𝑗,1≤𝑗≤𝑁,1≤𝑘≤𝑟_𝑗 (12) are complete, where 𝑍^𝑘_𝑗≡(−1)^𝑘−1𝑎𝑑^𝑘−1𝑓𝑔_𝑗,1≤𝑗≤𝑁,1≤𝑘≤𝑟_𝑗,𝑓(𝑋)≡𝑓(𝑋)−𝑔(𝑋)𝐴⁻¹_non(𝑋)𝑠(𝑋),𝑠(𝑋)≡𝐿^𝑟1_𝑓ℎ₁(𝑋)𝐿^𝑟2_𝑓ℎ₂(𝑋)⋯𝐿^𝑟𝑁_𝑓ℎ_𝑁(𝑋)^𝑇,𝑔≡𝑔₁𝑔₂⋯𝑔_𝑁≡𝑔(𝑋)𝐴⁻¹_non(𝑋),𝑎𝑑^𝑘_𝑝𝑞≡𝑝𝑎𝑑_𝑝^𝑘−1𝑞,𝑝𝑞≡𝜕𝑞𝑝(𝑋)−𝜕𝑋𝜕𝑝𝑞(𝑋).𝜕𝑋 (13)

Denote the trajectory error to be 𝑒^𝑖_𝑗≡𝜉^𝑖_𝑗−𝑦_𝑑^{𝑖(𝑗−1)},𝑖=1,2,…,𝑁,𝑗=1,2,…,𝑟_𝑖,𝑒^𝑖≡𝑒^𝑖₁𝑒^𝑖₂⋯𝑒^𝑖_𝑟𝑖^𝑇∈ℜ^𝑟𝑖,𝑒^𝑖_𝑗≡𝜅^𝑗−1𝑒^𝑖_𝑗,𝑖=1,2,…,𝑁,𝑗=1,2,…,𝑟_𝑖,𝑒^𝑖≡𝑒^𝑖₁𝑒^𝑖₂⋯𝑒^𝑖_𝑟𝑖(𝑡)^𝑇∈ℜ^𝑟𝑖,𝑒≡𝑒¹𝑒²⋯𝑒^𝑁𝑇∈ℜ^𝑟,𝜉≡𝜉₁𝜉₂⋯𝜉_𝑟^𝑇∈ℜ^𝑟,𝜂(𝑡)≡𝜂_𝑟+1(𝑡)𝜂_𝑟+2(𝑡)⋯𝜂_𝑁𝑡(𝑡)^𝑇∈ℜ^{𝑁𝑡−𝑟},𝑞𝜉(𝑡),𝜂(𝑡)≡𝐿_𝑓𝜙_𝑟+1(𝑡)𝐿_𝑓𝜙_𝑟+2(𝑡)⋯𝐿_𝑓𝜙_𝑁𝑡(𝑡)^𝑇≡𝑞_𝑟+1𝑞_𝑟+2⋯𝑞_𝑁𝑡^𝑇. (14)(15)(16)(17)(18)(19)(20)(21)

Define a phase-variable canonical matrix 𝐴^𝑖_phase to be 𝐴^𝑖_phase≡010⋯0001⋯0⋮⋮000⋯1−𝑎^𝑖₁−𝑎^𝑖₂−𝑎^𝑖₃⋯−𝑎^𝑖_{𝑟𝑖𝑟𝑖×𝑟𝑖},1≤𝑖≤𝑁, (22) where 𝑎^𝑖₁,𝑎^𝑖₂,…,𝑎^𝑖_𝑟𝑖 are any chosen variable such that 𝐴^𝑖_phase is Hurwitz, and define the vector 𝐵^𝑖 to be 𝐵^𝑖≡00⋯01^𝑇_𝑟𝑖×1,1≤𝑖≤𝑁. (23)

Define the positive definite matrix 𝑄^𝑖 to be solution of the following Lyapunov equation: 𝐴^𝑖_phase^𝑇𝑄^𝑖+𝑄^𝑖𝐴^𝑖_phase𝜆=−𝐼,1≤𝑖≤𝑁,_max𝑄^𝑖≡themaximumeigenvalueof𝑄^𝑖𝜆,1≤𝑖≤𝑁,_min𝑄^𝑖≡theminimumeigenvalueof𝑄^𝑖𝜆,1≤𝑖≤𝑁,_∗max≡min𝜆_max𝑄¹,𝜆_max𝑄²,…,𝜆_max𝑄^𝑁,𝜆^∗_min≡min𝜆_min𝑄¹,𝜆_min𝑄²,…,𝜆_min𝑄^𝑁. (24)

Assumption 1. For all 𝑡≥0, 𝜂∈ℜ^{𝑁𝑡−𝑟}, and 𝜉∈ℜ^𝑟, there exists a positive constant 𝐿_𝑚 such that the following inequality holds: 𝑞₂₂𝑡,𝜂,𝑒−𝑞₂₂𝑡,𝜂,0≤𝐿_𝑚(‖𝑒‖), (25) where 𝑞₂₂(𝑡,𝜂,𝑒)≡𝑞(𝜉,𝜂).

For the sake of describing precisely the considered problem, denote 𝑑_𝑖𝑗≡𝐿_𝑔𝑗𝐿^𝑟𝑖_𝑓⁻¹ℎ_𝑖(𝑋)𝑐,1≤𝑖≤𝑁,1≤𝑗≤𝑁,_𝑖≡𝐿^𝑟𝑖_𝑓ℎ_𝑖(𝑋),1≤𝑖≤𝑁,𝑒^𝑖≡𝑎^𝑖₁𝑒^𝑖₁+𝑎^𝑖₂𝑒^𝑖₂+⋯+𝑎^𝑖_𝑟𝑖𝑒^𝑖_𝑟𝑖,1≤𝑖≤𝑁. (26)

Definition 2 (see [31]). Consider the nonlinear system 𝑧̇=𝑓(𝑡,𝑧,𝑛), where 𝑓∶[0,∞)×ℜ^𝑛×ℜ^𝑛→ℜ^𝑛 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 sup_𝑡≥𝑡0‖𝑛(𝑡)‖<𝜆₂, the system state has the following property: ‖𝑧(𝑡)‖≤𝛽𝑧𝑡₀,𝑡−𝑡₀+𝛾sup_{𝑡0≤𝜏≤𝑡}‖𝑛(𝜏)‖ (27) 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 law 𝑢 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 value 𝑥_𝑒0≡[𝑒(𝑡₀)𝜂(𝑡₀)]^𝑇, for any 𝑡≥𝑡₀, and for any 𝑡₀≥0, 𝑦(𝑡)−𝑦_𝑑(𝑡)≤𝛽₁₁𝑥𝑡₀,𝑡−𝑡₀+1𝛽₂₂𝛽₃₃sup_{𝑡0≤𝜏≤𝑡}‖𝜃(𝜏)‖_𝑡𝑡0,𝑦(𝜏)−𝑦_𝑑(𝜏)²≤1𝑑𝜏𝛽₄₄𝛽₅₅𝑥_𝑒0+^𝑡_𝑡0𝛽₃₃‖𝜃(𝜏)‖²𝑑𝜏, (28)(29)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 function 𝑇∶ℜ^{𝑁𝑡−𝑟}→ℜ⁺ such that the following properties hold for all  𝜂∈ℜ^{𝑁𝑡−𝑟}:Δ₁𝜂²≤𝑇𝜂≤Δ₂𝜂²,Δ₁,Δ₂∇>0,_𝑡𝑇+∇_𝜂𝑇^𝑇𝑞₂₂𝑡,𝜂,0≤−2𝛼_𝑥𝜂²,𝛼_𝑥∇>0,_𝜂𝑇≤Δ₃𝜂,Δ₃>0. (30a)(30b)(30c)Then the tracking problem with almost disturbance decoupling performance for time-delay large-scale system is globally solved by the control law: 𝑢=𝐴⁻¹_𝑛𝑜𝑛−𝑢_𝑏+𝑢_𝑣,𝑢_𝑏≡𝐿^𝑟1_𝑓ℎ₁𝐿^𝑟2_𝑓ℎ₂⋯𝐿^𝑟𝑁_𝑓ℎ_𝑁^𝑇,𝑢_𝑣≡𝑣₁𝑣₂⋯𝑣_𝑁^𝑇,𝑣_𝑖≡𝑦^𝑖_𝑑^(𝑟𝑖)−𝜅^−𝑟𝑖𝑎^𝑖₁𝐿⁰_𝑓ℎ_𝑖(𝑋)−𝑦^𝑖_𝑑−𝜅^1−𝑟𝑖𝑎^𝑖₂𝐿¹_𝑓ℎ_𝑖(𝑋)−𝑦^𝑖_𝑑⁽¹⁾−⋯−𝜅⁻¹𝑎^𝑖_𝑟𝑖𝐿^𝑟𝑖_𝑓⁻¹ℎ_𝑖(𝑋)−𝑦^𝑖_𝑑^{(𝑟𝑖−1)},1≤𝑖≤𝑁. (31)(32)(33)(34)
Moreover, the influence of disturbances on the 𝐿₂ norm of the tracking error can be arbitrarily reduced by adjusting parameter  𝑁₂>1:𝑘₁₁≡𝑘−2𝜅25𝑘²𝜅²𝜙¹_𝜉²𝑄¹²−⋯−25𝑘²𝜅²𝜙^𝑁_𝜉²𝑄^𝑁2𝑘−25,₂₂≡22𝛼_𝑥−1Δ100²³𝐿²_𝑚−25Δ²₃𝜙_𝜂²,𝑁₂≡min𝑘₁₁,𝑘₂₂,Ω^𝑖_𝜉(𝜅)≡𝜅𝜕ℎ𝜕𝑋^𝑖𝑞^∗₁𝜕⋯𝜅ℎ𝜕𝑋^𝑖𝑞^∗_𝑝𝜅⋮⋮_𝑟𝑖𝜕𝐿𝜕𝑋^{𝑟𝑖−1𝑓}ℎ_𝑖𝑞^∗₁⋯𝜅^𝑟𝑖𝜕𝐿𝜕𝑋^{𝑟𝑖−1𝑓}ℎ_𝑖𝑞^∗_𝑞,Ω1≤𝑖≤𝑁,_𝜂(𝜅)≡𝜕𝜙𝜕𝑋^𝑟+1𝑞^∗₁⋯𝜕𝜙𝜕𝑋^𝑟+1𝑞^∗_𝑝𝜕⋮⋮𝜙𝜕𝑋^𝑁𝑡𝑞^∗₁⋯𝜕𝜙𝜕𝑋^𝑁𝑡𝑞^∗_𝑞, (35a)(35b)(35c)(35d)(35e) where 𝑘(𝜅)∶ℜ⁺→ℜ⁺ is a continuous function satisfying lim_𝜀→0𝑘(𝜅)=0,lim_𝜅→0𝜅𝑘(𝜅)=0. (35f)

Proof. Applying the coordinate transformation (9) yields 𝜉̇¹₁=𝜕ℎ₁𝜕𝑋𝑓+^𝑁𝑡_𝑗=1𝜕ℎ₁𝑞𝜕𝑋^∗𝑗𝜃_𝑗𝑑+𝐷_𝑒𝑗=𝜉¹₂+^𝑁𝑡_𝑗=1𝜕ℎ₁𝑞𝜕𝑋^∗𝑗𝜃_𝑗𝑑+𝐷_𝑒𝑗,⋮𝜉̇¹_𝑟1−1=𝜕𝐿^𝑟1_𝑓⁻²ℎ₁𝜕𝑋𝑓+^𝑁𝑡_𝑗=1𝜕𝐿^𝑟1_𝑓⁻²ℎ₁𝑞𝜕𝑋^∗𝑗𝜃_𝑗𝑑+𝐷_𝑒𝑗=𝐿^𝑟1_𝑓⁻¹ℎ₁+^𝑁𝑡_𝑗=1𝜕𝐿^𝑟1_𝑓⁻²ℎ₁𝑞𝜕𝑋^∗𝑗𝜃_𝑗𝑑+𝐷_𝑒𝑗,𝜉̇¹_𝑟1=𝑐₁+𝑑₁₁𝑢₁+⋯+𝑑_1𝑁𝑢_𝑁+^𝑁𝑡_𝑗=1𝜕𝐿^𝑟1_𝑓⁻¹ℎ₁𝑞𝜕𝑋^∗𝑗𝜃_𝑗𝑑+𝐷_𝑒𝑗,⋮𝜉̇^𝑁₁=𝐿¹_𝑓ℎ_𝑁+^𝑁𝑡_𝑗=1𝜕ℎ₁𝑞𝜕𝑋^∗𝑗𝜃_𝑗𝑑+𝐷_𝑒𝑗=𝜉^𝑁₂+^𝑁𝑡_𝑗=1𝜕ℎ_𝑁𝑞𝜕𝑋^∗𝑗𝜃_𝑗𝑑+𝐷_𝑒𝑗,⋮𝜉̇^𝑁_𝑟𝑁−1=𝐿^𝑟𝑁_𝑓⁻¹ℎ_𝑁+^𝑁𝑡_𝑗=1𝜕𝐿^𝑟𝑁_𝑓⁻²ℎ_𝑁𝑞𝜕𝑋^∗𝑗𝜃_𝑗𝑑+𝐷_𝑒𝑗=𝜉^𝑁_𝑟𝑁+^𝑁𝑡_𝑗=1𝜕𝐿^𝑟𝑁_𝑓⁻²ℎ_𝑁𝑞𝜕𝑋^∗𝑗𝜃_𝑗𝑑+𝐷_𝑒𝑗,𝜉̇^𝑁_𝑟𝑁=𝑐_𝑁+𝑑_𝑁1𝑢₁+⋯+𝑑_𝑁𝑁𝑢_𝑁+^𝑁𝑡_𝑗=1𝜕𝐿^𝑟𝑁_𝑓⁻¹ℎ_𝑁𝑞𝜕𝑋^∗𝑗𝜃_𝑗𝑑+𝐷_𝑒𝑗,𝜂̇_𝑘(𝑡)=𝐿_𝑓Ω_𝑘+^𝑁𝑡_𝑗=1𝜕Ω_𝑘𝑞𝜕𝑋^∗𝑗𝜃_𝑗𝑑+𝐷_𝑒𝑗=𝑞_𝑘+^𝑁𝑡_𝑗=1𝜕Ω_𝑘𝑞𝜕𝑋^∗𝑗𝜃_𝑗𝑑+𝐷_𝑒𝑗. (36)
Since 𝑐_𝑖𝜉(𝑡),𝜂(𝑡)≡𝐿^𝑟𝑖_𝑓ℎ_𝑖(𝑋(𝑡𝑑)),1≤𝑖≤𝑁,_𝑖𝑗≡𝐿_𝑔𝑗𝐿^𝑟𝑖_𝑓⁻¹ℎ_𝑖(𝑋)𝑞,1≤𝑖≤𝑁,1≤𝑗≤𝑁,_𝑘𝜉(𝑡),𝜂(𝑡)=𝐿_𝑓Ω_𝑘(𝑋),𝑘=𝑟+1,𝑟+2,…,𝑁_𝑡. (37)(38)(39)The dynamic equations of systems (3a) and (3b) can be rewritten as follows: 𝜉̇¹_𝑖(𝑡)=𝜉¹_𝑖+1(𝑡)+^𝑁𝑡_𝑗=1𝜕𝐿𝜕𝑋^𝑓𝑖−1ℎ₁𝑞^∗_𝑗𝜃_𝑗𝑑+𝐷_𝑒𝑗,𝑖=1,2,…,𝑟₁𝜉̇−1,_1𝑟1(𝑡)=𝑐₁𝜉(𝑡),𝜂(𝑡)+𝑑₁₁𝜉(𝑡),𝜂(𝑡)𝑢₁+⋯+𝑑_1𝑁𝜉(𝑡),𝜂(𝑡)𝑢_𝑁+^𝑁𝑡_𝑗=1𝜕𝐿𝜕𝑋^{𝑟1𝑓−1}ℎ₁𝑞^∗_𝑗𝜃_𝑗𝑑+𝐷_𝑒𝑗,⋮𝜉̇^𝑁_𝑖(𝑡)=𝜉^𝑁_𝑖+1(𝑡)+^𝑁𝑡_𝑗=1𝜕𝐿𝜕𝑋^𝑓𝑖−1ℎ_𝑁𝑞^∗_𝑗𝜃_𝑗𝑑+𝐷_𝑒𝑗,𝑖=1,2,…,𝑟_𝑁𝜉̇−1,_𝑁𝑟𝑁(𝑡)=𝑐_𝑁𝜉(𝑡),𝜂(𝑡)+𝑑_𝑁1𝜉(𝑡),𝜂(𝑡)𝑢₁+⋯+𝑑_𝑁𝑁𝜉(𝑡),𝜂(𝑡)𝑢_𝑁+^𝑁𝑡_𝑗=1𝜕𝐿𝜕𝑋^{𝑟𝑁𝑓−1}ℎ_𝑁𝑞^∗_𝑗𝜃_𝑗𝑑+𝐷_𝑒𝑗,𝜂̇_𝑘(𝑡)=𝑞_𝑘𝜉(𝑡),𝜂(𝑡)+^𝑁𝑡_𝑗=1𝜕Ω𝜕𝑋^𝑘(𝑋)𝑞^∗_𝑗𝜃_𝑗𝑑+𝐷_𝑒𝑗,𝑘=𝑟+1,…,𝑁_𝑡,𝑦_𝑖(𝑡)=𝜉^𝑖₁(𝑡),1≤𝑖≤𝑁. (40)(41)(42)(43)(44)(45)
Applying (14), (34), (37), and (38) yields the controller as 𝑢=𝐴⁻¹_non−𝑢_𝑏+𝑢_𝑣. (46)
The dynamic equations of systems (3a) and (3b) can be rewritten as follows by substituting (46) into (41) and (43): 𝜉̇^𝑖₁(𝑡)𝜉̇^𝑖₂(𝑡)⋮𝜉̇^𝑖_𝑟𝑖−1(𝑡)𝜉̇^𝑖_𝑟𝑖(𝑡)=010⋯0𝜉0010⋯0⋮⋮000⋯1000⋯0^𝑖1(𝑡)𝜉^𝑖₂(𝑡)⋮𝜉^𝑖_𝑟𝑖−1(𝑡)𝜉^𝑖_𝑟𝑖(𝑡)+00⋮01𝑣_𝑖+^𝑁𝑡_𝑗∑₌₁𝜕ℎ𝜕𝑋^𝑖𝑞^∗_𝑗𝜃_𝑗𝑑+𝐷_{𝑒𝑗𝑁𝑡𝑗}∑₌₁𝜕𝐿𝜕𝑋^1𝑓ℎ_𝑖𝑞^∗_𝑗𝜃_𝑗𝑑+𝐷_{𝑒𝑗𝑁𝑡}⋮_𝑗∑₌₁𝜕𝐿𝜕𝑋^{𝑟𝑖𝑓−1}ℎ_𝑖𝑞^∗_𝑗𝜃_𝑗𝑑+𝐷_𝑒𝑗,𝜂̇_𝑟+1(𝑡)𝜂̇_𝑟+2(𝑡)⋮𝜂̇_𝑁𝑡−1(𝑡)𝜂̇_𝑁𝑡(𝑡)=𝑞_𝑟+1(𝑡)𝑞_𝑟+2(𝑡)⋮𝑞_𝑁𝑡−1(𝑡)𝑞_𝑁𝑡(𝑡)+^𝑁𝑡_𝑗∑₌₁𝜕Ω𝜕𝑋^𝑟+1𝑞^∗_𝑗𝜃_𝑗𝑑+𝐷_{𝑒𝑗𝑁𝑡𝑗}∑₌₁𝜕Ω𝜕𝑋^𝑟+2𝑞^∗_𝑗𝜃_𝑗𝑑+𝐷_{𝑒𝑗𝑁𝑡}⋮_𝑗∑₌₁𝜕Ω𝜕𝑋^𝑁𝑡−1𝑞^∗_𝑗𝜃_𝑗𝑑+𝐷_{𝑒𝑗𝑁𝑡𝑗}∑₌₁𝜕Ω𝜕𝑋^𝑁𝑡𝑞^∗_𝑗𝜃_𝑗𝑑+𝐷_𝑒𝑗,𝑦_𝑖=10⋯00_𝑟×1𝜉^𝑖₁(𝑡)𝜉^𝑖₂(𝑡)⋮𝜉^𝑖_𝑟𝑖−1(𝑡)𝜉^𝑖_𝑟𝑖(𝑡)_𝑟×1=𝜉^𝑖₁(𝑡),1≤𝑖≤𝑁. (47)
Combining (14), (16), (17), (22), and (34) yields the transformation of (47):𝜂̇(𝑡)=𝑞𝜉(𝑡),𝜂(𝑡)+Ω_𝜂𝜃_𝑑+𝐷≡𝑞₂₂𝑡,𝜂(𝑡),𝑒+Ω_𝜂𝜃_𝑑,𝜅̇+𝐷𝑒^𝑖(𝑡)=𝐴^𝑖_phase𝑒^𝑖+Ω^𝑖_𝜉𝜃_𝑑+𝐷,1≤𝑖≤𝑁, (48a)(48b)𝑦_𝑖(𝑡)=𝜉^𝑖₁(𝑡),1≤𝑖≤𝑁. (49)
We consider 𝐿(𝑒,𝜂) denoted by a weighted sum of 𝑇(𝜂) and 𝑊(𝑒),   and 𝐿𝑒,𝜂≡𝑇𝜂+𝑘(𝜅)𝑊(𝑒)≡𝑇𝜂+𝑘(𝜅)𝑊¹𝑒¹+⋯+𝑊^𝑁𝑒^𝑁, (50) where 𝑊(𝑒)≡𝑊¹𝑒¹+⋯+𝑊^𝑁𝑒^𝑁 (51) as a composite Lyapunov function of the subsystems (48a) and (48b) [33, 34], where 𝑊(𝑒^𝑖) satisfies 𝑊^𝑖𝑒^𝑖≡12𝑒^𝑖𝑇𝑄^𝑖𝑒^𝑖. (52)
Utilizing (14), (25) and (30a), (30b), and (30c) result in the derivative of 𝐿 along the trajectories of (48a) and (48b) as𝐿̇=∇_𝑡𝑇+∇_𝜂𝑇^𝑇𝜂̇+𝑘2^•𝑒^1𝑇𝑄¹𝑒¹+𝑒^1𝑇𝑄𝑒^1•1+⋯+𝑒^•𝑁𝑇𝑄^𝑁𝑒^𝑁+𝑒^𝑁𝑇𝑄𝑒^𝑁•𝑁≤∇_𝑡𝑇+∇_𝜂𝑇^𝑇𝑞₂₂𝑡,𝜂(𝑡),𝑒+∇_𝜂𝑉^𝑇Ω_𝜂𝜃_𝑑−𝑘+𝐷2𝜅𝑒^1𝑇𝑒¹+⋯+𝑒^𝑁𝑇𝑒^𝑁+𝑘𝜅𝜃_𝑑+𝐷Ω¹_𝜉𝑄¹𝑒¹+⋯+𝜃_𝑑+𝐷Ω^𝑁_𝜉𝑄^𝑁𝑒^𝑁≤−𝜂²2𝛼_𝑥−1Δ100²³𝐿²_𝑚−25Δ²₃Ω_𝜂²−‖𝑒‖²𝑘−2𝜅25𝑘²𝜅²Ω¹_𝜉²𝑄¹²−⋯−25𝑘²𝜅²Ω^𝑁_𝜉²𝑄^𝑁2+−25𝑁+1𝜃100^𝑑+𝐷². (53) That is, 𝐿̇≤−𝑘₁₁‖𝑒‖²−𝑘₂₂𝜂²+𝑁+1𝜃100^𝑑+𝐷². (54)
From (35c), we obtain 𝐿̇≤−𝑁₂‖𝑒‖²+𝜂²+𝑁+1𝜃100^𝑑+𝐷². (55)
Define 𝑒≡𝑒¹𝑒²⋮𝑒^𝑁≡𝑒¹₁𝑒¹_rem,𝑒¹_rem∈ℜ^𝑟−1. (56)
Hence 𝐿̇≤−𝑁₂𝜂²+𝑒¹₁²+𝑒¹_rem²+𝑁+1𝜃100^𝑑+𝐷². (57)
Utilizing (57) easily yields ^𝑡_𝑡0𝑦₁(𝜏)−𝑦¹_𝑑(𝜏)²≤𝐿𝑡𝑑𝜏₀𝑁₂+𝑁+1100𝑁₂^𝑡_𝑡0𝜃_𝑑(𝜏)+𝐷²𝑑𝜏. (58)
Similarly, it is easy to prove that ^𝑡_𝑡0𝑦_𝑖(𝜏)−𝑦^𝑖_𝑑(𝜏)²≤𝐿𝑡𝑑𝜏₀𝑁₂+𝑁+1100𝑁₂^𝑡_𝑡0𝜃_𝑑(𝜏)+𝐷²𝑑𝜏,2≤𝑖≤𝑁 (59) so that statement (29) is satisfied. From (55), we get𝐿̇≤−𝑁₂𝑦_total²+𝑁+1𝜃100^𝑑+𝐷², (60a) where 𝑦_total²≡‖𝑒‖²+𝜂². (60b)
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: Δ_min‖𝑒‖²+𝜂²≤𝐿≤Δ_max‖𝑒‖²+𝜂². (61) That is, Δ_min𝑦_total²≤𝐿≤Δ_max𝑦_total², (62) where Δ_min≡min{Δ₁,(𝑘/2)𝜆^∗_min} and Δ_max≡max{Δ₂,(𝑘/2)𝜆^∗_max}. From (55) and (62), we get 𝐿̇≤−𝑁₂Δ_max𝐿+𝑁+1100sup_{𝑡0≤𝜏≤𝑡}𝜃_𝑑(𝜏)+𝐷². (63)
Hence, 𝐿(𝑡)≤𝐿𝑡₀𝑒^{(−𝑁2/Δmax)(𝑡−𝑡0)}+Δ_max(𝑁+1)100𝑁₂sup_{𝑡0≤𝜏≤𝑡}𝜃_𝑑(𝜏)+𝐷²,𝑡≥𝑡₀ (64) which implies 𝑦₁(𝑡)−𝑦¹_𝑑(𝑡)≤2𝐿𝑡₀𝑘𝜆^∗_min𝑒^{(−𝑁2/2Δmax)(𝑡−𝑡0)}+Δ_max(𝑁+1)50𝑘𝜆^∗_min𝑁₂sup_{𝑡0≤𝜏≤𝑡}𝜃_𝑑(𝜏)+𝐷. (65)
Similarly, it is easy to prove that 𝑦_𝑖(𝑡)−𝑦¹_𝑑(𝑡≤)2𝐿𝑡₀𝑘𝜆^∗_min𝑒^{(−𝑁2/2Δmax)(𝑡−𝑡0)}+Δ_max(𝑁+1)50𝑘𝜆^∗_min𝑁₂×sup_{𝑡0≤𝜏≤𝑡}𝜃_𝑑(𝜏)+𝐷,2≤𝑖≤𝑁 (66)
so that statement (28) is proved, and then the tracking problem with almost disturbance decoupling is globally solved.

If the sum 𝑟₁+𝑟₂+⋯+𝑟_𝑚 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: 𝑢=𝐴⁻¹_𝑛𝑜𝑛−𝑢_𝑏+𝑢_𝑣,𝑢_𝑏≡𝐿^𝑟1_𝑓ℎ₁𝐿^𝑟2_𝑓ℎ₂⋯𝐿^𝑟𝑁_𝑓ℎ_𝑁^𝑇,𝑢_𝑣≡𝑣₁𝑣₂⋯𝑣_𝑁^𝑇,𝑣_𝑖≡𝑦^𝑖_𝑑^(𝑟𝑖)−𝜅^−𝑟𝑖𝑎^𝑖₁𝐿⁰_𝑓ℎ_𝑖(𝑋)−𝑦^𝑖_𝑑−𝜅^1−𝑟𝑖𝑎^𝑖₂𝐿¹_𝑓ℎ_𝑖(𝑋)−𝑦^𝑖_𝑑⁽¹⁾−⋯−𝜅⁻¹𝑎^𝑖_𝑟𝑖𝐿^𝑟𝑖_𝑓⁻¹ℎ_𝑖(𝑋)−𝑦^𝑖_𝑑^{(𝑟𝑖−1)},1≤𝑖≤𝑁. (67)
Moreover, the influence of disturbances on the 𝐿₂ norm of the tracking error can be arbitrarily reduced by adjusting parameter 𝑘₁₁>1: 𝑘₁₁≡𝑘−2𝜅25𝑘²𝜅²𝜙¹_𝜉²𝑄¹²−⋯−25𝑘²𝜅²𝜙^𝑁_𝜉²𝑄^𝑁2. (68)

3. Illustrative Example

Consider the two inverted pendulums coupled by a spring with disturbances as shown in Figure 1. The dynamic equations are given as follows:

Figure 1 

The two-inverted-pendulum system coupled by a spring.

𝑆₁ (inverted pendulum 1): 𝜃̈₁(𝑡)=−𝑘𝑟²4𝐽₁+𝑚₁𝑔𝑟𝐽₁sin𝜃₁(𝑡)+𝑘𝑟(𝑙−𝑏)2𝐽₁+𝑘𝑟²4𝐽₁sin𝜃₂(𝑡)+𝑢₁𝐽₁+𝜃₁𝑡−𝜏_1,2(𝑡)1+𝜃²₁𝑡−𝜏_1,2(𝑡),𝜏_1,2(𝑡)≡0.41+sin²(𝑡),𝑦₁=𝜃₁(𝑡), (69)

 𝑆₂ (inverted pendulum 2): 𝜃̈₂(𝑡)=−𝑘𝑟²4𝐽₂+𝑚₂𝑔𝑟𝐽₂sin𝜃₂(𝑡)+𝑘𝑟(𝑙−𝑏)2𝐽₂+𝑘𝑟²4𝐽₂sin𝜃̇₁(𝑡)+𝑢₂𝐽₂+𝜃₂𝑡−𝜏_2,2(𝑡)1+𝜃²₂𝑡−𝜏_2,2(𝑡),𝜏_2,2(𝑡)≡0.41+sin²(𝑡)𝑦₂=𝜃₂(𝑡), (70) where 𝑢_𝑖 is the torque input generated by the actuator for pendulum 𝑖 (𝑖=1,2), 𝜃_𝑖 is the angular displacement of pendulum 𝑖 (𝑖=1,2), 𝑚_𝑖 is the mass of pendulum 𝑖 (𝑖=1,2), 𝐽_𝑖 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≡𝜃̇₁(𝑡), 𝑥₃≡𝜃₂(𝑡), and 𝑥₄≡𝜃̇₂(𝑡) yields the following state space models:𝑥̇₁(𝑡)=𝑥₂(𝑡),𝑥̇₂(𝑡)=−𝑘𝑟²4𝐽₁+𝑚₁𝑔𝑟𝐽₁sin𝑥₁(𝑡)+𝑘𝑟(𝑙−𝑏)2𝐽₁+𝑘𝑟²4𝐽₁sin𝑥₃(𝑡)+𝑢₁𝐽₁+𝑥₁𝑡−𝜏_1,2(𝑡)1+𝑥²₁𝑡−𝜏_1,2(𝑡),𝑥̇₃(𝑡)=𝑥₄(𝑡),𝑥̇₄(𝑡)=−𝑘𝑟²4𝐽₁+𝑚₁𝑔𝑟𝐽₁sin𝑥₃(𝑡)+𝑘𝑟(𝑙−𝑏)2𝐽₁+𝑘𝑟²4𝐽₁sin𝑥₂(𝑡+𝑢₂)𝐽₂+𝑥₃𝑡−𝜏_1,2(𝑡)1+𝑥²₃𝑡−𝜏_1,2(𝑡),𝜏_1,2(𝑡)≡𝜏_2,2(𝑡)≡0.41+sin²(𝑡),𝑦₁=𝑥₁(𝑡),𝑦₂=𝑥₃(𝑡). (71)

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 𝑥̇₁(𝑡)=𝑥₂(𝑡),𝑥̇₂(𝑡)=1.935sin𝑥₁(𝑡)+0.025sin𝑥₃(𝑡)+𝑢₁+𝑥₁𝑡−0.41+sin²(𝑡)1+𝑥²₁𝑡−0.41+sin²(𝑡),𝑥̇₃(𝑡)=𝑥₄(𝑡),𝑥̇₄(𝑡)=2.425sin𝑥₃(𝑡)+𝑘𝑟(𝑙−𝑏)2𝐽₁+0.025sin𝑥₂(𝑡)+𝑢₂+𝑥₃𝑡−0.41+sin²(𝑡)1+𝑥²₃𝑡−0.41+sin²(𝑡),𝑦₁=𝑥₁(𝑡),𝑦₂=𝑥₃(𝑡). (72)

Now we will show how to explicitly construct a controller that tracks the desired signals 𝑦¹_𝑑=𝑦²_𝑑=0 and attenuates the disturbance’s effect on the output terminal to an arbitrary degree of accuracy. Let us arbitrarily choose 𝑎¹₁=𝑎¹₂=𝑎²₁=𝑎²₂=10, and the solution of Lyapunov equation is given as 𝐴¹_phase=𝐴²_phase=⁰¹₋₁₀₋₁₀,𝑄¹=𝑄²=^1.050.05_0.050.055, 𝜆_max(𝑄¹)=𝜆_max(𝑄²)=1.0525, 𝜆_min(𝑄¹)=𝜆_min(𝑄²)=0.0525, 𝜆^∗_max=1.0525,  and 𝜆^∗_min=0.0525. From (31), we obtain the desired tracking controllers 𝑢₁=−1000𝑥₁−100𝑥₂−1.935sin𝑥₁−0.025sin𝑥₃,𝑢₂=−1000𝑥₃−100𝑥₄−2.425sin𝑥₃−0.025sin𝑥₂. (73)

It can be verified that the relative conditions of Theorem 4 are satisfied with 𝜅=0.1,  𝐵¹_𝑑=𝐵²_𝑑=0,  𝑁₂=2.82,  𝑘=20√𝜅, 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 position 𝑥1(𝜃1) of the two inverted pendulums.

Figure 3 

The output angular position 𝑥3(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.