Over the past few decades, the control problem of nonlinear systems has been extensively addressed [1]. In practical systems, there are often inherent nonlinearities that must be considered to more precisely describe the dynamics of control systems. Additionally, systems in descriptor form have been shown to provide a powerful tool for enhancing the completeness in describing practical control systems [2,3]. It is worth noting that not only is the implicit relation included to represent the interrelations in many systems, but there also exist purely static relations featuring identity correlations among different dynamic variables. Because of this reason, the description of descriptor systems includes both differential and algebraic equations, thus being referred to as differential-algebraic systems [4]. Descriptor systems can also be applied to situations involving singularities, singular perturbations, and noncausalities. Notably, singular systems have been widely discussed due to their use in various control scenarios, such as acceleration and its derivative, autonomous ships, as well as bio-economic systems [5–7]. However, the algebraic constraints in descriptor systems also lead to the issue of infinite poles and impulse behaviors [8,9], as the static relationships must be satisfied at a given instant. Taking advantage of large-scale systems, the control and analysis of nonlinear descriptor systems have attracted significant attention [10]. Nevertheless, the nonlinearities and algebraic constraints still pose formidable challenges to the development of controller design approaches.

Consequently, ensuring the absence of impulse modes has emerged as a critical consideration in the controller design of nonlinear descriptor systems. To address this issue, a proportional and derivative feedback concept has been introduced to transform the closed-loop system into a regular system [11,12]. That is, the feedback signal guarantees non-impulsiveness and causality for the system. For discrete-time systems, this concept is referred to as Proportional-Difference (P-D) feedback [13]. Due to the nonlinearities, the development of controller design approaches for nonlinear descriptor systems is confronted with significant challenges and complexities. Over the past decades, the Takagi-Sugeno Fuzzy Model (T-SFM) has proven to be a powerful tool for tackling the control problem of nonlinear systems [14]. By characterizing a nonlinear system as several linear subsystems, the controller design problem is efficiently reduced to a linear problem. In other words, the linear P-D feedback technique can be utilized in the control problem of nonlinear descriptor systems. Leveraging the T-SFM and Parallel Distributed Compensation (PDC) framework, some researchers have successfully developed fuzzy P-D controllers to stabilize the nonlinear descriptor systems [15,16]. Moreover, some simulations have been implemented to explore practical applications [17–18]. Nevertheless, disturbance and uncertainty issues must be resolved to enhance the reliability of the control approach. The trade-off between convergence rate and control cost should also be considered while meeting practical control requirements.

In most industrial systems, external disturbances, such as power quality disturbances for generators [19], road roughness for electric power steering systems [20], and winds and waves for ships [21], are inevitable and persistent factors. In view of disturbance problem, many researchers have contributed to the development of anti-disturbance control approaches for nonlinear descriptor systems [22,23]. From the energy perspective, passive performance has been introduced to achieve the dissipativity of external energy [24]. A system can be called a dissipative system because the energy dissipated inside is less than the energy supplied by the external source [24,25]. Therefore, several researchers have successfully extended the concept of dissipativity to the disturbance problem, where disturbance energy is regarded as external energy [25,26]. In [25,27], the researchers have also presented the energy supply function such that the system consumption energy is always greater than the external energy to achieve this disspativity. Passivity theory has long been recognized as an effective tool for the analysis of electrical networks and nonlinear systems. It is worth noting that the passive performance constraint provides a general framework for various considerations, including H_∞, strictly input passivity, and other related forms, depending on the setting of the energy supply function. Moreover, uncertainties pose a significant challenge in enhancing control performance for practical applications [28,29]. Fuzzy theory is widely recognized as an effective approach to addressing uncertainties in daily life and engineering systems [30]. Nevertheless, the T-SFM constructed for actual nonlinear systems may involve modeling errors [31]. Based on the T-SFM, some researchers have improved the identification and control issues using adaptive control under system uncertainties [32]. Extending the control issue, some researchers have also solved the controller design problem for uncertain nonlinear descriptor systems [33]. By employing proportional and derivative feedback techniques, fuzzy controllers have also been successfully developed to handle uncertainty and disturbance issues [34]. However, the development of tracking controllers using the P-D fuzzy control scheme for discrete-time descriptor systems remains an unresolved challenge.

More importantly, the control effort required to ensure both the passive constraint and robustness may increase significantly. For specific objectives such as trajectory tracking, the responses of the controlled variables could become excessively drastic, which is not suitable for practical control applications. Furthermore, the control cost could potentially exceed practical requirements, leading to inefficiency. The H₂ performance index has been introduced to evaluate the energy of state and input [35]. It has also been recognized as a powerful performance metric in the control of autonomous vehicles [36]. The application of the H₂ performance index is often accompanied by a minimization constraint, which aims to optimize the energy associated with the index. Nowadays, some researchers have successfully extended the H₂/H_∞ constraint to the fuzzy controller design for the tracking problem in nonlinear systems under the T-S fuzzy model framework [37]. Based on the descriptor form, the T-S Fuzzy Descriptor Model (T-SFDM) based fuzzy controller design have been investigated with disturbance and uncertainty problems in several papers [38]. Considering the control efficiency while affected by the external disturbances, an input shaping approach has been developed under H_∞ performance for the fuzzy tracking control problem [39]. Nonetheless, there is lack of existing papers discussing the multiple-performance-constrained problem for discrete-time nonlinear descriptor systems, even for their tracking issue. However, the difficulty of the tracking controller design problem subject to multiple constraints increases significantly due to the nonlinearities and descriptor form. By leveraging the linear representation of subsystems in T-SFDM, the complexity of the design process can be effectively reduced.

The reason outlined above motivated this research to propose a fuzzy tracking synthesis for discrete-time nonlinear descriptor systems that satisfies multiple performance criteria, including robustness, passivity, and optimal H₂. First, the T-SFDM is established for a class of nonlinear descriptor systems. Considering the existence of modeling errors, the T-SFDM is also constructed for the target trajectory with uncertain factors. Next, the T-S Fuzzy Descriptor Error Model (T-SFDEM) is derived by subtracting the target model from the T-SFDM. With the same premise as the T-SFDEM, a PDC-based fuzzy controller is developed using the P-D error feedback signals. For the closed-loop T-SFDEM, a definition is provided to examine the regularity and causality. Additionally, the constraints of robust control, passive performance, and H₂ performance are introduced separately. Based on the multiple performance constraints and Lyapunov theory, a stability criterion is proposed to ensure the stability of the error dynamics and achieve the trajectory tracking objective. It is worth noting that the free-weighting matrix technique is integrated into the analysis process to further reduce the conservativeness caused by the multiple constraints, particularly the minimization of H₂ performance. Consequently, the stability criteria are derived into the Linear Matrix Inequality (LMI) problem, enabling efficient solutions via computational tools. Finally, two simulation examples, including a nonlinear DC-motor system and a ship steering system, are provided to demonstrate the feasibility and effectiveness of the proposed P-D fuzzy tracking controller.

This paper is organized as follows. In Section 2, the T-SFDM and the corresponding T-SFDEM are obtained. In Section 3, a P-D feedback fuzzy tracking synthesis is proposed for the tracking purpose. In Section 4, two simulation examples, including a nonlinear DC motor system and a ship steering system, are provided. Based on the results, some conclusions are given in Section 5.

In this section, the T-SFDMs of nonlinear descriptor systems and target trajectory are respectively established. Then, the T-SFDEM is obtained from two T-SFDMs. According to the PDC design concept, the P-D fuzzy tracking controller is presented. Some definitions and lemmas related to multiple performance requirements are also introduced. First, the T-SFDM is presented as follows.

Model Rules α:

If ϑ1(k) is ϕα1 and ϑ2(k) is ϕα2 and…and ϑℓ(k) is ϕαℓ, then (1){T⌢α(k)x(k+1)=A⌢α(k)x(k)+B⌢α(k)u(k)+Wαw(k)y(k)=Cαx(k)+Vαw(k)where model matrices T⌢α(k)=Tα+ΔTα(k), A⌢α(k)=Aα+ΔAα(k), B⌢α(k)=Bα+ΔBα(k) are constructed with the time-varying uncertainties, x(k)∈Rs, u(k)∈Ri, y(k)∈Ro and w(k)∈Rd respectively are the system’s state, input, output and disturbance vectors, ϑ1(k) to ϑℓ(k) are the premise variables, ϕα1 to ϕαℓ are the fuzzy sets, α=1,2,…,τ and ς=1,2,…,ℓ are the numbers of premise variables and fuzzy rules. Note that the matrix Tα, with or without full rank, can be used to describe the singular and typical systems in descriptor form.

In the T-SFDM (1), ΔTα(k), ΔAα(k) and ΔBα(k) represent the time-varying uncertain factors that may arise from modeling parameter errors and other relevant factors. Then, these matrices are further decomposed into the following structure. (2)ΔTα(k)=HαΔα(k)RTα,ΔAα(k)=HαΔα(k)RAα,andΔBα(k)=HαΔα(k)RBαwhere Δα(k) denotes the time-varying uncertain item, Hα, RTα, RAα and RBα denote the structure of uncertainties, with appropriate dimensions.

Then, the following overall T-SFDM is obtained by blending with all fuzzy subsystems in (1) with membership functions. (3){∑α=1τΦα(ϑ(k)){T⌢α(k)x(k+1)}=∑α=1τΦα(ϑ(k)){A⌢α(k)x(k)+B⌢α(k)u(k)+Wαw(k)}y(k)=∑α=1τΦα(ϑ(k)){Cαx(k)+Vαw(k)}where Φα(ϑ(k))=∏ς=1ℓϕας(ϑς(k))/∑α=1τ∏ς=1ℓϕας(ϑς(k)), which satisfies the situation ∑α=1τΦα(ϑ(k))>0 and Φα(ϑ(k))≥0, ς=1,2,…,ℓ is the number of premise variables. To simplify the expressions in this research, the time-varying terms involving premise variables and uncertainties, denoted as T⌢α(k), A⌢α(k), B⌢α(k), Φα(ϑ(k)), are reduced to T⌢α, A⌢α, B⌢α and Φα throughout the remainder of this paper.

For the tracking purpose, the T-SFDM is also constructed as follows, with consideration of the parameter uncertainty caused by modeling errors, similar to the process described in Eqs. (1)–(3). (4){∑α=1τΦα{T⌢αxd(k+1)}=∑α=1τΦα{A⌢αxd(k)}yd(k)=∑α=1τΦα{Cαxd(k)}where xd(k) and yd(k) denote the desired state and output trajectories.

Then, the tracking T-SFDEM is obtained as follows from (3) and (4). (5){∑α=1τΦα{T⌢αex(k+1)}=∑α=1τΦα{A⌢αex(k)+B⌢αu(k)+Wαw(k)}ey(k)=∑α=1τΦα{Cαex(k)+Vαw(k)}where ex(k)=x(k)−xd(k) and ey(k)=y(k)−yd(k) are the tracking errors of the system’s states and outputs.

According to the T-SFDEM (5), the following P-D error feedback fuzzy controller is developed by the PDC framework to complete the tracking task. (6)u(k)=∑α=1τΦα{Tαpex(k)−Tαdex(k+1)}where Tαp and Tαd are the gains for the proportional and difference feedback terms. It is observed that Φα of the premise part is designed in the same way as the T-SFDEM (5) to comply with the PDC concept. To make the application of P-D feedback concept more meaningful, Remark 1 is presented to illustrate the use of feedback signals.

Remark 1. Building on the results of research on linear discrete-time descriptor systems, including but not limited to [40,41], k + 1 is considered the current time instant, which can be derived from the previous time instant k when there is a design of the state observer. However, observer design is not the primary focus of this research; the main discussion is centered on the design of the multi-performance P-D fuzzy tracking controller. For this reason, all states of the control systems are assumed to be available for the implementation of the P-D feedback technique.

Therefore, the following closed-loop T-SFDEM is obtained by substituting the P-D fuzzy tracking controller (6) into T-SFDEM (5). (7){∑α=1τ∑β=1τΦαΦβ{Q⌢αβdex(k+1)}=∑α=1τ∑β=1τΦαΦβ{Q⌢αβpex(k)+Wαw(k)}ey(k)=∑α=1τΦα{Cαex(k)+Vαw(k)}where Q⌢αβd=T⌢α+B⌢αTβd and Q⌢αβp=A⌢α+B⌢αTβp are the closed-loop matrices.

Conducting an examination of regularity and causality, Definition 1 is presented for the T-SFDEM.

Definition 1. If the following situations are satisfied, the regularity and causality of the closed-loop T-SFDEM (7) are guaranteed, and the system’s responses will be impulsive-free.

(a) The system is said to be regular when det(zT⌢α−A⌢α)≠0.

(b) The system is said to be causal when deg(det(zT⌢α−A⌢α))=rank(T⌢α).

where det(⋅) and deg(⋅) denote the determinant and its degree.

It is obvious that the two situations are certainly ensured for the typical system, where T⌢α is of full rank. With the P-D fuzzy tracking controller (6), the conditions in Definition 1 can also be guaranteed for the closed-loop T-SFDEM (7) even when T⌢α is a singular matrix. The P-D feedback technique can effectively transform the singular system into a regular form by ensuring that the matrix Q⌢αβd is of full rank. Aiming to address the uncertainty problems, an inequality for robust control is introduced in Lemma 1.

Lemma 1. [42]. For a scalar ω, the following result is satisfied by assigning the constant matrices H and R of appropriate dimension. (8)HΔ(k)R+RTΔT(k)HT≤ωHHT+ω−1RTRwhere time-varying uncertain term Δ(k) satisfies Δ(k)ΔT(k)≤I.

In the context of the energy concept, the following passive performance constraint in Lemma 2 is employed to achieve disturbance dissipativity by appropriately selecting the parameters for the general form in [25].

Lemma 2. The T-SFDEM (7) is called strictly input passive if the following relationship is satisfied with a positive scalar μ provided. (9)2∑k=0kTeyT(k)w(k)>μ∑k=0kTwT(k)w(k)where kT denotes the terminal time.

Therefore, the energy consumption of the T-SFDEM (7) is greater than the disturbance energy by ensuring the relationship (9). However, the control effort is expected to increase significantly when the inequality conditions in (8) and (9) must be satisfied simultaneously. Considering practical control scenarios, the trade-off between state and input responses should also be carefully evaluated. Therefore, the optimal H₂ performance index is stated in Definition 2.

Definition 2. If the following condition is satisfied for the energy of state errors and control inputs, the H₂ performance index is optimized. (10)mink{∑k=0kTexT(k)Ωeex(k)+uT(k)Ωuu(k)}where Ωe and Ωu are the index matrices whose dimensions correspond to the states and inputs.

Thus, the control effort for the tracking purpose of the T-SFDEM (7) can be ensured not to become excessively large when multiple performance requirements are met. Therefore, the main tracking problem of nonlinear descriptor systems is to ensure the asymptotic stability of the tracking error ex(k) by applying the PDC-based P-D fuzzy tracking controller (6). Based on the closed-loop T-SFDEM (7), stability criteria can be proposed for the purpose limk→∞ex(k)=0 so that the system state x(k) of nonlinear descriptor systems can approach the desired state xd(k). Moreover, the disturbance dissipativity is guaranteed while achieving a better balance between tracking performance and control effort by satisfying the passivity constraint (9) and minimizing the H₂ performance (10) in the criteria.

The following section presents the P-D fuzzy tracking synthesis and stability analysis process for nonlinear descriptor systems.

By incorporating the robust control stated in Lemma 1, the passive constraint stated in Lemma 2, and the minimized H₂ performance stated in Definition 2, the stability criteria are proposed based on Lyapunov theory and the closed-loop T-SFDEM (7). First, the sufficient conditions are derived in the following theorem to ensure the tracking objective and multiple performances.

Theorem 1. The T-SFDEM (7) achieves the asymptotical stability, passive constraint (9) and minimization of H₂ performance (10) if there exist the positive definite matrices P, Ωe, Ωu, the free-weighting matrices Ξp, Ξd and the minimized scalar ρ such that the following sufficient conditions are satisfied with the given positive scalar μ. (11)[Ψ11Ψ12ΞpWα−CαT∗Ψ22ΞdWα∗∗μ−Vα−VαT]<0for β=α=1,2,…,τ (12)[Ψ~11Ψ~12Ξp(Wα + Wβ2)−(Cα + Cβ2)T∗Ψ~22Ξd(Wα + Wβ2)∗∗μ−(Vα + Vβ2)−(Vα + Vβ2)T]<0for β>α (13)[−ρexT(0)∗−P−1]<0where * denotes the transpose item in the matrix inequalities, sym{Λ} denotes the sum of a matrix and its transpose Λ+ΛT, and the symbols in the conditions (11) and (12) are stated below.

Ψ11=−P+sym{ΞpQ⌢ααp}+Ωe+TαpTΩuTαp, Ψ12=−ΞpQ⌢ααd+Q⌢ααpTΞdT−TαpTΩuTαd,

Ψ22=P−sym{ΞdQ⌢ααd}+TαdTΩuTαd, Ψ~11=−P+sym{Ξp(Q⌢αβp + Q⌢βαp2)} + Ωe + TαpTΩuTαp + TβpTΩuTβp2,

Ψ~12=−Ξp(Q⌢αβd + Q⌢βαd2)+(Q⌢αβp + Q⌢βαp2)TΞdT−TαpTΩuTαd + TβpTΩuTβd2,

and Ψ~22=P−sym{Ξd(Q⌢αβd + Q⌢βαd2)}+TαdTΩuTαd + TβdTΩuTβd2.

Proof of Theorem 1. To implement the stability analysis, the Lyapunov function is selected as (14)V(k)=exT(k)Pex(k)

Thus, the first forward difference of the Lyapunov function (14) is derived with the following process. (15)ΔV(k)=exT(k+1)Pex(k+1)−exT(k)Pex(k)

It is well known that the Lyapunov function (14) is defined in terms of the energy of the tracking error. As a result, the energy change is expressed by the derivative in (15). The stability analysis aims to achieve the tracking purpose by ensuring a negative energy change for (15). In addition, the technique of free-weighting matrix is also considered in the analysis process of this research to reduce the conservativeness. The equation for the free-weighting matrix is defined according to closed-loop T-SFDEM (7) as follows: (16)0=2(exT(k)Ξp+exT(k+1)Ξd)(∑α=1τ∑β=1τΦαΦβ{−Q⌢αβdex(k+1)+Q⌢αβde(k)+Wαw(k)})

Then, the following results are obtained by integrating (15) with the free-weighting Eq. (16). (17)∑α=1τ∑β=1τΦαΦβ{exT(k)[−P+sym{ΞpQ⌢αβp}−ΞpQ⌢αβd+Q⌢αβpTΞdTΞpWα∗P−sym{ΞdQ⌢αβd}ΞdWα∗∗0]ex(k)}where ex(k)=[ex(k)ex(k+1)w(k)]T.

With respect to the strictly input passive constraint (9), the cost function for the passivity analysis is defined as follows: (18)C(k)=∑k=0kT(μwT(k)w(k)−2eyT(k)w(k))

Considering ΔV(k) derived in the form of (17), the following relationship is obtained from the cost function (18) and the H₂ performance constraint under the zero-initial condition. (19)C(k)=∑k=1kT(μwT(k)w(k)−2eyT(k)w(k)+ΔV(k))−V(kT)≤∑k=0kT(μwT(k)w(k)−2eyT(k)w(k)+ΔV(k))≤∑k=0kT(wT(k)w(k)−2eyT(k)w(k)+ΔV(k)+exT(k)Ωeex(k)+uT(k)Ωuu(k))

Based on the P-D fuzzy tracking controller (6), the following result is obtained from the right-hand side ofinequality (19). (20)∑α=1τ∑β=1τΦαΦβ{exT(k)[−P+sym{ΞpQ⌢αβp}+Ωe+TβpTΩuTβp−ΞpQ⌢αβd+Q⌢αβpTΞdT−TβpTΩuTβdΞpWα−CαT∗P−sym{ΞdQ⌢αβd}+TβdTΩuTβdΞdWα∗∗μ−Vα−VαT]ex(k)}

It is obvious that (20) becomes negative definite if the conditions (11) and (12) are satisfied by Theorem 1 according to the PDC concept. Then, the following results are also derived for the stability, passivity and H₂ performance.

From the relationship in (19), the cost function can be made negative (C(k)<0) when (20) is ensured to be negative definite. Compared with the strictly passive constraint in Lemma 2, the inequality (9) is satisfied accordingly. This also means that the T-SFDEM (7) has the ability to dissipate external disturbance energy. Then, the asymptotic stability can be proved by considering w(k)=0. Therefore, Eq. (20) is reduced to the following form based on the setting. (21)∑α=1τ∑β=1τΦαΦβ{[ex(k)ex(k+1)]T[−P+sym{ΞpQ⌢αβp}+ Ωe+TβpTΩuTβp−ΞpQ⌢αβd+Q⌢αβpTΞdT−TβpTΩuTβd∗P−sym{ΞdQ⌢αβd}+ TβdTΩuTβd][ex(k)ex(k+1)]}

It is observed that the negative definite of (21) also can be guaranteed by the conditions (11) and (12) in Theorem 1. Based on the properties of the free-weighting Eq. (16), w(k)=0, and the property exT(k)Ωeex(k)+uT(k)Ωuu(k)>0 of H₂ performance, the result of ΔV(k)<0 is thus achieved for (15). According to the Lyapunov theory, the asymptotic stability of the closed-loop T-SFDEM (7) is proved such that the convergence of tracking error energy can be achieved.

Consequently, the minimization of H₂ performance can be verified as follows. If the stability is ensured by Theorem 1, the following relationship can be derived from (15), (16) and (21). (22)ΔV(k)+exT(k)Ωeex(k)+uT(k)Ωuu(k)<0

The following result is undoubtedly achieved with (22). (23)exT(k)Ωeex(k)+uT(k)Ωuu(k)<−ΔV(k)

The following relationship is derived by employing the cumulative sum on both sides of inequality (23) due to the convergence of the state error dynamic. (24)∑k=0kT(exT(k)Ωeex(k)+uT(k)Ωuu(k))<exT(0)Pex(0)

Then, the following relationship can be obtained from inequality (24) if the condition (13) is satisfied by Theorem 1. (25)∑k=0kT(exT(k)Ωeex(k)+uT(k)Ωuu(k))<exT(0)Pex(0)<min(ρ)

From the relationship in (25), it can be observed that the H₂ performance index is minimized under condition (13) by Theorem 1. This indicates that the trade-off between state and input responses can be optimized while ensuring stability and disturbance dissipativity. □

However, the sufficient conditions in Theorem 1 remain in a non-LMI form and involve time-varying uncertainties. To transform Theorem 1 into an LMI problem solvable by computational programs, the Schur complement and Lemma 1 are applied in the following theorem.

Theorem 2. The T-SFDEM (7) achieves the asymptotical stability, passive constraint (9), robustness and minimization of H₂ performance (10) if there exist the positive definite matrices Z, ℧e, ℧u, the matrices σ1, σ2, Yβ, Gβ, the positive scalar ω and the minimized scalar ρ such that the following sufficient conditions are satisfied with the given positive scalar μ. (26)[−ZΘ12−ZCαTYαTσ1TZΘ17∗Θ22−Wα−GαTσ2T0Θ27∗∗μ−Vα−VαT0000∗∗∗−Ωu−1000∗∗∗∗−Z00∗∗∗∗∗−Ωe−10∗∗∗∗∗∗−ω]<0for β=α=1,2,…,τ (27)[−ZΘ~12−Z(Cα + Cβ2)T(Yα + Yβ2)Tσ1TZΘ~17∗Θ~22−(Wα + Wβ2)−(Gα + Gβ2)Tσ2T0Θ~27∗∗μ−(Vα + Vβ2)−(Vα + Vβ2)T0000∗∗∗−℧u000∗∗∗∗−Z00∗∗∗∗∗−℧e0∗∗∗∗∗∗−ω]<0for β>α (28)[−ρexT(0)∗−Z]<0where ℧e=Ωe−1, ℧u=Ωu−1, and the symbols in the conditions (26) and (27) are stated below.

Θ12=−ZAαT−YαTBαT+σ1TEαT, Θ22=sym{σ2TEαT+GαTBαT}+ωHαHαT,Θ17=−ZRAαT−YαTRBαT+σ1TRTαT, Θ27=σ2TRTαT+GαTRBαT, Θ~12=−Z(Aα + Aβ2)T−(YβTBαT + YαTBβT2)+σ1T(Eα + Eβ2)T,

Θ~22=sym{σ2T(Eα + Eβ2)T+(GβTBαT + GαTBβT2)} + ω(HαHαT + HβHβT2),

Θ~17=−Z(RAα + RAβ2)T−(YβTRBαT + YαTRBβT2)+σ1T(RTα + RTβ2)T,

Θ~27=σ2T(RTα + RTβ2)T+(GβTRBαT + GαTRBβT2),Yα=TαpZ−Tαdσ1, Yβ=TβpZ−Tβdσ1, Gα=Tαdσ2, Gβ=Tβdσ2, σ1=−Ξd−TΞpTZ, σ2=−Ξd−T and Z=P−1.

Proof of Theorem 2. In the proof of Theorem 2, the derivation process for the condition (26) is only presented as an example. In other words, the condition (27) can be derived through a similar process. According to the Schur complement lemma, the following inequality is obtained from (26) if the condition (26) is satisfied by the stability criteria in Theorem 2. (29)[−ZΘ12−ZCαTYαTσ1TZ∗sym{σ2TEαT+GαTBαT}−Wα−GαTσ2T0∗∗μ−Vα−VαT000∗∗∗−Ωu−100∗∗∗∗−Z0∗∗∗∗∗−Ωe−1]+ω[0Hα0000][0HαT0000]+ω−1[Θ17Θ270000][Θ17TΘ27T0000]<0

Then, the following condition is also satisfied by condition (26) due to (29) and inequality (8) in Lemma 1. (30)[−Z−ZQ⌢αβpT+σ1TQ⌢αβdT−ZCαTYαTσ1TZ∗sym{σ2TQ⌢αβdT}−Wα−GαTσ2T0∗∗μ−Vα−VαT000∗∗∗−Ωu−100∗∗∗∗−Z0∗∗∗∗∗−Ωe−1]<0

It is evident that the time-varying terms in the uncertainties of (30) can be successfully transformed into a constant form in (29) for LMI analysis through Lemma 1. With the application Schur complement, the condition (30) is further transferred into the following form. (31)[−Z+ZΩeZ+σ1TZ−1σ1+YαTΩuYα−ZQ⌢αβpT+σ1TQ⌢αβdT+σ1TZ−1σ2−YαTΩuGα−ZCαT∗sym{σ2TQ⌢αβdT}+σ2TZ−1σ2+GαTΩuGα−Wα∗∗μ−Vα−VαT]<0

According to the definitions of Yα, Gα, σ1, σ2, and Z, the left-hand side of inequality (31) is obtained from condition (11) by pre- and post-multiplying the following matrices: (32)[P−1−P−1ΞpΞd−100−Ξd−1000I]and[P−1−P−1ΞpΞd−10−Ξd−TΞpTP−1−Ξd−T000I]

As a result, it is derived that the condition (11) in Theorem 1 can be satisfied if the inequality (31) holds, which can be guaranteed by the condition (26) in Theorem 2 following the analysis process from (29) to (32). Analogously, the condition (12) in Theorem 1 can be ensured by the condition (27) in Theorem 2. In compliance with the analysis process of Theorem 1 and Lyapunov theory, the asymptotic stability of the tracking error subject to multiple constraints is achieved by satisfying conditions (11)–(13), which are ensured by the LMI conditions (26)–(28). □

Therefore, it is concluded that the closed-loop T-SFDEM (7) achieves the asymptotic stability, passivity, robustness and minimized H₂ performance simultaneously if the conditions (26)–(28) are all satisfied by the P-D fuzzy tracking synthesis of Theorem 2. Moreover, the conditions (26)–(28) presented in LMI form are solved and the feasible solutions are conveniently derived using the convex optimization algorithm in MATLAB or other software. As a final point, the computational complexity of the proposed design process is discussed in the following remark.

Remark 2. The computational complexity of the T-SFDM-based fuzzy control and P-D feedback technique in tracking synthesis for nonlinear descriptor systems can be summarized in the following points:

[I] The application of the T-SFDM (1), (2) and T-SFDEM (5) within the T-SFM framework can represent nonlinear descriptor systems as several linear subsystems. Therefore, the control problem of nonlinear descriptor systems can be tackled using linear techniques. This feature reduces the complexity and computational requirements in designing a tracking synthesis.

Although the tracking synthesis under the T-SFM and PDC framework can efficiently reduce the computational complexity, some questions still remain to be discussed. From [43], it is known that the computational complexity may gradually increase while ensuring the approximation accuracy. This is due to the increased number of variables in the antecedent part and fuzzy rules. However, the number of these factors is often not large for practical nonlinear systems, which is observed in the later section.

Before the simulation begins, the following algorithm is provided to make the application of the proposed fuzzy tracking synthesis clearer.

From Lemma 2, it can be known that a larger parameter μ leads to a more conservative analysis process, although better disturbance dissipativity can be achieved. Moreover, the preservation of control effort is somewhat contrary to the ability to handle disturbance effects. This characterization makes the control problem in Theorem 2 more difficult to solve. Therefore, the value of the parameter is required to be decreased if feasible solutions cannot be obtained by the solver in Step 3. Based on the design process introduced in Algorithm 1, two simulation examples are provided to demonstrate the feasibility and applicability of Theorem 2.

The tracking control problem for nonlinear descriptor systems is addressed by a fuzzy tracking synthesis proposed in Theorem 2, which satisfies multiple requirements, including asymptotic stability, passivity, robustness, and the optimization between state responses and control efforts. Based on the representation of the T-SFDEM (7), two simulation examples are provided. The first example involves a singular-type system based on the equivalent circuit model of a DC motor. Moreover, the comparison results with the existing research are also provided to demonstrate the effectiveness of combining multiple performance constraints. To verify the effectiveness of the controller design approach based on descriptor form, the second example examines a general-type system derived from the discretized ship steering model. First, the simulation results of the DC motor compared with [44] are provided as follows.

Example 1. Referring to [45,46], the singular nonlinear mathematical model is given as follows for the equivalent circuit system of the DC motor, with the consideration of uncertainties and disturbances. (33)(3+Δe1(k))x1(k+1)=2.85x1(k)+x3(k)−2u1(k)−0.8w(k) (34)(3+Δe2(k))x2(k+1)=0.0225x1(k)x2(k)+(1.875+Δa22(k))x2(k)+3u1(k)+0.5w(k) (35)x3(k)=(0.03+Δa32(k))x22(k)−0.8w(k)−u1(k)−(4+Δb32(k))u2(k)where x1(k) is the angular velocity, x2(k) is the current of equivalent circuit, and the expression x3(k)=(Km⋅Lf⋅tJ)x22(k)−u1(k)−4u2(k) represents the algebraic constraint associated with the current energy characteristics of the DC motor in [45], Km, Lf, t, J respectively denote the torque/back emf constant, the field winding inductance, the sampling time and the moment of inertia, u1(k) is the load torque and u2(k) is the input voltage.

Remark 3. For the control issue of the nonlinear DC motor (33)–(35), the uncertain factors Δa22(k)=0.01875sin(k), Δa32(k)=0.0003sin(k), Δb32(k)=0.04sin(k) and Δe1(k)=Δe2(k)=0.3sin(k) are considered to account for the effect of temperature on the circuit’s current. Additionally, the disturbance w(k) is introduced to represent external influences from the environment or human interaction.

Based on references [45,46] and related research [47], it is known that the mathematical model (33)–(35) has been established for a series DC motor. This type of motor has been efficiently applied in electric traction systems, such as electric buses and subways, which require large initial torque. However, these applications often face uncertainties such as wear and tear, as well as disturbances from rough roads. As a result, the control efficiency of series DC motors must be improved.

Considering the operating range x2(k)∈[−3.5,3.5] for the nonlinearities, the T-SFDM of the nonlinear descriptor system (33)–(35) is constructed as follows: (36){∑α=12Φα{T⌢αx(k+1)}=∑α=12Φα{A⌢αx(k)+B⌢αu(k)+Wαw(k)}y(k)=∑α=12Φα{Cαx(k)+Vαw(k)}where x(k)=[x1(k)x2(k)x3(k)]T, u(k)=[u1(k)u2(k)]T and the model matrices are presented as follows: (37)T1=T2=[300030000,] A1=[2.85030.07881.875000−0.1050−1], A2=[2.8503−0.07881.8750000.1050−1]B1=B2=[−2030−1−4], C1=C2=[001], W1=W2=[0.80.50.8] and V1=V2=0.