Finite-time resilient control for uncertain periodic piecewise polynomial time-varying systems

Abstract

This study explores the problem of finite-time resilient control for periodic piecewise polynomial time-varying systems in the face of parameter uncertainties, time-varying state delays and external disturbances. Particularly, the considered system is characterized by dividing the fundamental period of periodic systems into numerous subintervals, each of which can be expressed by using matrix polynomial functions. The foremost intention of this work is to lay out a resilient controller such that the resulting closed-loop system is finite-time bounded and satisfies a mixed $H_{\infty}$ and passivity performance index. Furthermore, by constructing a periodic piecewise time-varying Lyapunov–Krasovskii functional, a delay-dependent sufficient condition is established in line with Wiritinger’s inequality and matrix polynomial lemma to guarantee the needed outcomes of the system under study. Following this, the gain matrix of the devised controller can be calculated by solving the established constraints. As a final step, we conclude with a numerical example that validates the potential and importance of the theoretical discoveries and the developed control scheme.

Keywords

Periodic piecewise polynomial systems time-varying delays finite-time boundedness parameter uncertainties resilient control

Introduction

In recent times, a great deal of emphasis has been placed on periodic systems due to the periodic nature of many real-world problems such as hypersonic cruise vehicles, mechanical vibrations, electrical circuits and wind turbine systems (Zhou, 2020; Zhou and Huang, 2020; Zhu et al., 2021). Specifically, the majority of control problems in continuous-time periodic systems are difficult to solve because of their time-varying properties compared with their discrete-time counterparts. Since Floquet theory and lifting technique make it easier to analyse discrete-time periodic systems, they lack the ability to do the same for continuous-time periodic systems because their solutions are in closed form. In this context, a limited number of studies have focused on addressing the aforementioned problem by employing approximation methodologies. Consequently, it leads to the development of piecewise methods for reworking continuous-time periodic systems, and as a result, periodic piecewise systems (PPSs) are introduced (Fan et al., 2021; Xie et al., 2019; Zhu et al., 2023). Since then, PPSs have become one of the primary priorities in the scientific community over the past several years (Li et al., 2018; Xie et al., 2021a; Yang et al., 2021). It is important to note, however, that in reality, PPSs are not always linear, so it is more appropriate to describe them in polynomial form, which is intrinsically more precise. Thereupon, the continuous-time periodic systems have been approximated by a number of polynomial subsystems varying over time, yielding periodic piecewise polynomial time-varying systems (PPPTVSs) (Li et al., 2019b). Furthermore, the studies on PPPTVSs have advanced remarkably across scholar communities owing to their inherent potential and relevant literature may be found in the work by Li et al. (2019a) and Xie et al. (2020a, 2021b). Despite their useful features, PPPTVSs have received very little academic attention; this gap in research inspired this study.

Besides, parameter uncertainties exist in several real-world models as an outcome of external interference, modelling errors, communication problems and so on. Under those scenarios, the system may become unstable. Therefore, it is practically vital to take into account the parameter perturbations in the system model. As a result, various articles have been reported on the stabilization problem of diverse dynamical systems with parameter uncertainties (Chen et al., 2022; Feng and Hao, 2021; Liu et al., 2020; Luo and Zhao, 2015; Xie et al., 2018a); however, it should be noted that work on considering the parameter uncertainties in PPPTVSs is in its initial stages. On the contrary, the occurrence of time delays is ineluctable in almost all practical systems, and their presence typically results in adverse dynamic behaviour such as performance degradation, oscillations or perhaps even instability. As a consequence, during the past few decades, one of the most hotly debated concerns in control problems has been the stability analysis for time delay systems (Park et al., 2015; Tan et al., 2018; Xie et al., 2018b; Xie and Lam, 2018). On the contrary, in the realm of PPSs, only a small number of studies have investigated time delays within the system (Xie et al., 2018b; Xie and Lam, 2018). But when the time delay arises in PPPTVSs, the complexity of stability analysis is outdistanced. Therefore, PPPTVSs with time delays have to be addressed, which motivates this study.

Furthermore, it is worthwhile to observe that most controllers have been implemented on the assumption that controller gains can be accurately calculated. However, in many practical situations, a controller operation’s inaccuracy cannot be avoided due to round-off errors of numeric values, parameter drifts, accuracy problems and so on. These factors bring about an inaccurate realization of the implemented controller, and these inaccuracies can be classified as additive and multiplicative gain fluctuations. Due to this fact, it becomes necessary to design a controller that is insensitive to gain variations. As a result, it led to the study of resilient controller characterizations, which has garnered attention from the research community (Liu et al., 2019; Xie et al., 2020a,b; Xiong et al., 2020). For instance, Liu et al. (2019) studied the stabilization problems of delayed PPSs, wherein a resilient controller is developed to handle the sensitivity of gain variations. Nevertheless, the issue of additive and multiplicative gain fluctuations in PPPTVSs remains unreported, serving as the primary impetus for the present investigation. In practice, the plethora of external factors may cause disturbances and have a negative impact on the system’s performance. Furthermore, this kind of disturbance is one of the primary factors in regard to instabilities and performance degradation. To alleviate the effect of disturbances, a mixed $H_{\infty}$ and passivity disturbance attenuation control method was employed, which in turn maintains the performance of the dynamical system by suppressing the footprints of the disturbance to a predetermined level (Li et al., 2022; Shen et al., 2019).

Aside from the foregoing, it should be emphasized that for both the asymptotic and exponential stabilization techniques, to meet the required stability criteria, an infinite time period is needed. However, in real-world situations, the behaviour of the system has to always be analysed in terms of predefined or finite-time (FT) rather than the infinite time frame. At this instant, the FT notion has been deployed, which concentrates on the fast cognizance of the system’s state and also assures that states are in specific limitation over a FT interval. As a result of its practical merits, the exploration of the FT stability notion has piqued significant interest among the scholar community and substantial pieces of literature have been reported. For instance, the FT stability theory has been implemented in neutral systems (Sakthivel et al., 2018), periodic piecewise time-invariant systems (Xie et al., 2017) and nonlinear systems (Wang et al., 2018). Due to these factors, it is of paramount significance to gain insight into the FT stabilization issue for PPPTVSs, which is a topic that has not been done before in the realm of previously published studies.

Prompted by the aforesaid discussions, the FT boundedness problem of PPPTVSs with parameter uncertainties, time-varying delays and external disturbances is examined via resilient control in this article. Moreover, the significant aspects of this work are listed below:

A unifying framework integrating the FT scheme, time-varying delays, parameter uncertainties and external disturbances is postulated for the stabilization of PPPTVSs.

Predominantly, a periodic piecewise polynomial time-varying controller is conjured up for the intent of procuring FT boundedness. Furthermore, a mixed $H_{\infty}$ and passivity disturbance attenuation technique has been made use of to deal with the fallout of environmental disturbances.

Moreover, both additive and multiplicative gain perturbations are factored into the controller design to increase the resiliency of the controller.

Specifically, by blending periodic piecewise polynomial Lyapunov–Krasovskii functional (LKF) and FT stability theory, an adequate delay-dependent criterion is acquired under the frame of linear matrix inequalities (LMIs), which ascertains the system’s stability in a FT with an endorsed disturbance attenuation index.

Subsequently, the requisite resilient control gain matrix can be computed by solving the derived LMI based criteria. Ultimately, a simulation study is conducted to exemplify the efficacy and viability of the theoretical discoveries.

Problem formulation

In this portion, uncertain periodic systems are modelled with external disturbances and time-varying delays, which can be characterized by the following ordinary differential equations

{\begin{matrix} \overset{\cdot}{x} (t) = (A (t) + Δ A (t)) x (t) + (A_{h} (t) + Δ A_{h} (t)) x (t - h (t)) \\ + B (t) u (t) + C (t) w (t), \\ z (t) = D (t) x (t), \\ x (t_{0}) = ψ (t_{0}), \forall t_{0} \in [- h, 0] \end{matrix}

(1)

where $x (t) \in R^{m}$ is the state vector; $z (t) \in R^{n}$ denotes output vector; $u (t) \in R^{p}$ represents the control input; $w (t) \in L_{2} [0, \infty)$ is the external disturbance; $h (t)$ is the time-varying delay that satisfies the settings $0 < h (t) \leq h$ and $\overset{\cdot}{h} (t) \leq ν < 1$ with known positive scalars $h$ and $ν$ ; $ψ (t_{0})$ is the system’s initial condition; $A (t)$ , $A_{h} (t)$ , $B (t)$ , $C (t)$ and $D (t)$ are the periodic system matrices that take the form $A (t + T_{n})$ , $A_{h} (t + T_{n})$ , $B (t + T_{n})$ , $C (t + T_{n})$ and $D (t + T_{n})$ , respectively, $\forall t \geq 0$ with fundamental period $T_{n}$ .

Here, the continuous-time periodic system (equation (1)) can be reconstructed as PPPTVSs by using piecewise approximation strategy. To be specific, each time segment within the fundamental period $T_{n}$ is partitioned into $S$ subintervals, that is, $[ℓ T_{n} + t_{i - 1}, ℓ T_{n} + t_{i})$ , $i \in N ≜ {1, 2, . . . S}, ℓ = 0, 1, \dots$ , with $T_{i} = t_{i} - t_{i - 1}$ , $t_{0} = 0$ and $t_{S} = T_{n}$ . Based on the preceding criteria, the periodic system (equation (1)) can be rewritten as

{\begin{matrix} \overset{\cdot}{x} (t) = (A_{i} (t) + Δ A_{i} (t)) x (t) + (A_{hi} (t) + Δ A_{hi} (t)) x (t - h (t)) \\ + B_{i} (t) u (t) + C_{i} (t) w (t), \\ z (t) = D_{i} (t) x (t), \\ x (t_{0}) = ψ (t_{0}), \forall t_{0} \in [- h, 0] \end{matrix}

(2)

where $A_{i} (t), A_{hi} (t), B_{i} (t),$ $C_{i} (t)$ and $D_{i} (t)$ are time-varying matrices of the $i^{th}$ subsystem which can be portrayed in the polynomial format as follows

\begin{matrix} [\begin{matrix} A_{i} (t) & A_{hi} (t) & B_{i} (t) & C_{i} (t) & D_{i} (t) \end{matrix}] \\ = \sum_{j = 0}^{m} [φ_{i} (t)]^{j} [\begin{matrix} A_{i, j} & A_{hi, j} & B_{i, j} & C_{i, j} & D_{i, j} \end{matrix}] \end{matrix}

(3)

here $φ_{i} (t) = \frac{t - ℓ T_{n} - t_{i - 1}}{T_{i}}$ $\in [0, 1)$ ; $A_{i, j}$ , $A_{hi, j}$ , $B_{i, j}$ , $C_{i, j}$ and $D_{i, j}$ are the suitable dimensioned known constant matrices; $Δ A_{i} (t)$ and $Δ A_{hi} (t)$ signify the parameter uncertainty matrices having the form $[Δ A_{i} (t) Δ A_{hi} (t)] = H_{i} F_{i} (t) [E_{Ai} E_{Bi}]$ with appropriate dimensioned constant matrices $H_{i}, E_{Ai}$ and $E_{Bi}$ , and $F_{i} (t)$ is the time-varying function satisfying the condition $F_{i}^{T} (t) F_{i} (t) \leq I$ .

Moreover, in some cases, the gain matrices are extremely susceptible to their own inevitable variations or perturbations, resulting in undesirable behaviours in the system. So, it is important to consider the fragility of the gain matrix to achieve better performance. In light of this, the following resilient state feedback controller is devised

u (t) = (K_{i} (t) + Δ K_{i} (t)) x (t)

(4)

where $K_{i} (t)$ denotes the periodic piecewise gain matrix, which has to be reckoned later; $Δ K_{i} (t)$ stands for the time-varying gain perturbations having the following forms:

Norm-bounded fluctuations in the additive form

Δ K_{i} (t) = H_{ki} F_{i} (t) E_{ki}

(5)

Norm-bounded fluctuations in the multiplicative form

Δ K_{i} (t) = H_{ki} F_{i} (t) E_{ki} K_{i} (t)

(6)

Here, $H_{ki}$ and $E_{ki}$ are proper dimensioned constant matrices.

Then, we obtain the subsequent closed-loop system by fusing the devised controller (equation (4)) in the PPPTVSs (equation (2))

{\begin{matrix} \overset{\cdot}{x} (t) = (A_{i} (t) + Δ A_{i} (t)) x (t) + (A_{hi} (t) + Δ A_{hi} (t)) x (t - h (t)) \\ + B_{i} (t) (K_{i} (t) + Δ K_{i} (t)) x (t) + C_{i} (t) w (t), \\ z (t) = D_{i} (t) x (t), \\ x (t_{0}) = ψ (t_{0}), \forall t_{0} \in [- h, 0] \end{matrix}

(7)

Before proceeding to the next section, the following assumption, definitions and lemmas are presented, which are indispensable to procure the desired results.

Assumption 1. For any positive scalar $η$ , the external disturbance signal $w (t)$ is presumed to satisfy the requirement $\int_{0}^{T_{f}} w^{T} (s) w (s) ds \leq η$ .

Definition 1. For given constants $C_{1} > 0$ , $C_{2} > 0$ , $η > 0$ , $T_{f} > 0$ with $C_{1} < C_{2}$ and a positive definite matrix $G > 0$ , the PPPTVSs (equation (7)) are FT bounded (FTB) in regard to $(C_{1}, C_{2}, G, T_{f}, η)$ if (Xie et al., 2017)

sup_{- h \leq t_{0} \leq 0} {x^{T} (t_{0}) G x (t_{0})} \leq C_{1} = \Rightarrow x^{T} (t) G x (t) < C_{2}, \forall t \in T_{f} .

Definition 2. Under zero initial condition, the closed-loop PPPTVSs (equation (7)) satisfy mixed $H_{\infty}$ and passivity performance index $γ$ under Assumption 1 for given weighting parameter $θ \in [0, 1]$ , such that (Shen et al., 2019)

\int_{0}^{t} (- γ^{- 1} θ z^{T} (k) z (k) + 2 (1 - θ) z^{T} (k) w (k)) dk \geq \int_{0}^{t} - γ w^{T} (k) w (k) dk

for all $t > 0$ and any non-zero $w (t) \in L_{2} [0, \infty)$ .

Lemma 1: For a given matrix $J > 0$ , the inequality $\int_{a}^{b} φ^{T} (s) J φ (s) ds \geq \frac{1}{b - a} ς^{T} J ς$ holds for all continuously differentiable function $φ : [a, b] \to R^{m}$ , where $J = [\begin{matrix} 4 J - \frac{6 J}{b - a} \\ \frac{12 J}{{(b - a)}^{2}} \end{matrix}]$ and $ς^{T} = [\begin{matrix} \int_{a}^{b} φ^{T} (s) ds & \int_{a}^{b} \int_{s}^{b} φ^{T} (u) duds \end{matrix}]$ (Park et al., 2015).

Lemma 2: For a given positive definite matrix $G$ and continuously differentiable function $Ω : [a, b] \to R^{m}$ , the inequality $\int_{a}^{b} \int_{s}^{b} Ω^{T} (v) G Ω (v) dvds \geq Σ^{T} G Σ$ holds, where $G = [\begin{matrix} 3 G \frac{- 6 G}{(b - a)} \\ \frac{18 G}{{(b - a)}^{2}} \end{matrix}]$ and $Σ^{T} = [\begin{matrix} \int_{a}^{b} \int_{s}^{b} Ω^{T} (v) dvds & \int_{a}^{b} \int_{s}^{b} \int_{u}^{b} Ω^{T} (u) dudvds \end{matrix}]$ (Park et al., 2015).

Lemma 3: Let $f (ξ_{1}, ξ_{1}, \dots, ξ_{m})$ be a bounded matrix polynomial function defined as $f (ξ_{1}, ξ_{1}, \dots, ξ_{m}) = Ξ_{0} + ξ_{1} Ξ_{1} + ξ_{1} ξ_{2}$ $Ξ_{2} + \dots + (\prod_{k = 1}^{m} ξ_{k}) Ξ_{m}$ , where $m \in Z^{+}$ and $Ξ_{j}, (j = 0, 1, \dots m)$ are real symmetric matrices, $ξ_{k}, (k = 0, 1, 2, \dots m)$ are variables and $ξ_{k} \in [0, 1]$ . If $\sum_{k = 0}^{r} Ξ_{k} < 0, r = 0, 1, \dots m,$ then the matrix polynomial $f (ξ_{1}, ξ_{2}, \dots ξ_{m}) < 0 .$ (Li et al., 2019a).

Main results

The principal objective of this part is to draft a solution to the FT boundedness problem proffered in the former section. Precisely, the notions of FT stability theory and Lyapunov stability theory, along with Wirtinger’s inequality, are deployed to confirm the FT boundedness of the assayed system. First, the FT boundedness criteria are derived in Theorem 1 utilizing known controller gain matrix and additive gain perturbations. Next, based on the defined constraints, the design approach of the resilient control gain matrix is given in Theorem 2, where the controller gains are considered as unknown. Conclusively, by following the foregoing results, the pertinent stability criteria in the face of multiplicative gain perturbations are given in Theorem 3.

Before heading on to the theorem aspect, the periodic piecewise positive definite matrix $P_{i} (t)$ that will be employed in the LKF is constructed as follows

P_{i} (t) = P_{i} + φ_{i} (t) (P_{i + 1} - P_{i})

(8)

for $t \in [ℓ T_{n} + t_{i - 1}, ℓ T_{n} + t_{i}), i \in N$ . Moreover, the derivative of $P_{i} (t)$ is given by ${\overset{\cdot}{P}}_{i} (t) = \frac{P_{i + 1} - P_{i}}{T_{i}}$ .

FT boundedness analysis

In what follows, the befitting criteria in the frame of LMIs affirming the FT boundedness of PPPTVSs are formulated in the below theorem.

Theorem 1: Consider a symmetric matrix $G$ and positive scalars $α_{i}, h, μ, η, T_{f}$ . The closed-loop PPPTVS (equation (7)) is FTB with regard to $(C_{1}, C_{2}, G, T_{f}, η)$ satisfying Assumption 1, if there exist positive definite matrices $P_{i} (t), R_{r}, (r = 1, 2, 3, 4)$ and positive scalars $δ_{1 i}, δ_{2 i}, δ_{3 i}$ , $(i \in N)$ , such that the following relations hold

[Φ] < 0

(9)

(\tilde{ζ} C_{1} + η (1 - e^{α_{i} T_{f}})) \leq λ_{1} C_{2} e^{α_{i} T_{f}}

(10)

where $Φ_{1, 1} = sym {P_{i} (t) A_{i} (t) + P_{i} (t) B_{i} (t) (K_{i} (t))} + \frac{P_{i + 1} - P_{i}}{T_{i}} + α_{i}$ $P_{i} (t) + R_{1} + R_{2} + h R_{3} + \frac{h^{4}}{4} R_{4}$ , $Φ_{1, 2} = P_{i} (t) A_{hi} (t)$ , $Φ_{1, 7} = P_{i}$ $(t) C_{i} (t)$ , $Φ_{1, 8} = P_{i} (t) δ_{1 i} H_{i}$ , $Φ_{1, 9} = E_{Ai}^{T}$ , $Φ_{1, 10} = P_{i} (t) δ_{2 i} H_{i}$ , $Φ_{1, 12} = δ_{3 i} P_{i} (t) B_{i} (t) H_{ki}$ , $Φ_{1, 13} = E_{ki}^{T} (t)$ , $Φ_{2, 2} = - (1 - μ) R_{1}$ , $Φ_{2, 11} = E_{Bi}^{T}$ , $Φ_{3, 3} = - R_{2}$ , $Φ_{4, 4} = - \frac{4}{h} R_{3}$ , $Φ_{4, 5} = \frac{6}{h^{2}} R_{3}$ , $Φ_{5, 5} = - \frac{12}{h^{3}} R_{3} - 3 R_{4}$ , $Φ_{5, 6} = \frac{6}{h} R_{4}$ , $Φ_{6, 6} = - \frac{18}{h^{2}} R_{4}$ , $Φ_{7, 7} = - α_{i} I$ , $Φ_{8, 8} = Φ_{9, 9} = - δ_{1 i} I$ , $Φ_{10, 10} = Φ_{11, 11} = - δ_{2 i} I$ , $Φ_{12, 12} = Φ_{13, 13} = - δ_{3 i} I$ , $λ_{1} = λ_{\min} (P_{i}), λ_{2} = λ_{\max} (P_{i}), λ_{3} = λ_{\max} (R_{1}),$ $λ_{4} = λ_{\max} (R_{2}), λ_{5} = λ_{\max} (R_{3}), λ_{6} = λ_{\max} (R_{4}) and \tilde{ζ} = λ_{2} + h λ_{3} +$ $h λ_{4} + \frac{h^{2}}{2} λ_{5} + \frac{h^{5}}{12} λ_{6} .$

Proof: To demonstrate the FT boundedness criteria, the periodic piecewise LKF of the sequel form is conjured up

\begin{matrix} V (t) = x^{T} (t) P_{i} (t) x (t) + \int_{t - h (t)}^{t} x^{T} (v) R_{1} x (v) dv + \int_{t - h}^{t} x^{T} (v) R_{2} x (v) dv \\ + \int_{t - h}^{t} \int_{s}^{t} x^{T} (v) R_{3} x (v) dvds + \frac{h^{2}}{2} \int_{t - h}^{t} \int_{s}^{t} \int_{v}^{t} x^{T} (u) R_{4} x (u) dudvds \end{matrix}

(11)

Now, by reckoning the time-derivative of LKF, we obtain

\begin{matrix} \overset{\cdot}{V} (t) \leq sym {x^{T} (t) P_{i} (t) [(A_{i} (t) + Δ A_{i} (t)) + (A_{hi} (t) + Δ A_{hi} (t)) \\ x (t - h (t)) + B_{i} (t) (K_{i} (t) + Δ K_{i} (t)) x (t) + C_{i} (t) w (t)]} \\ + x^{T} (t) \frac{P_{i + 1} - P_{i}}{T_{i}} x (t) + x^{T} (t) R_{1} x (t) - (1 - μ) \\ x^{T} (t - h (t)) R_{1} x (t - h (t)) + x^{T} (t) R_{2} x (t) \\ - x^{T} (t - h) R_{2} x (t - h) + h x^{T} (t) R_{3} x (t) \\ - \int_{t - h}^{t} x^{T} (s) R_{3} x (s) ds + \frac{h^{4}}{4} x^{T} (t) R_{4} x (t) \\ - \frac{h^{2}}{2} \int_{t - h}^{t} \int_{s}^{t} x^{T} (v) R_{4} x (v) dvds \end{matrix}

(12)

Next, by employing Lemmas 1 and 2, the single and double integral portions in the preceding equation can be recast as

- \int_{t - h}^{t} x^{T} (s) R_{3} x (s) ds \leq {\hat{β}}^{T} (t) [\begin{matrix} \frac{- 4 R_{3}}{h} & \frac{6 R_{3}}{h^{2}} \\ * & - \frac{12 R_{3}}{h^{3}} \end{matrix}] \hat{β} (t)

(13)

- \frac{h^{2}}{2} \int_{t - h}^{t} \int_{s}^{t} x^{T} (v) R_{4} x (v) dvds \leq {\bar{β}}^{T} (t) [\begin{matrix} - 3 R_{4} & \frac{6 R_{4}}{h} \\ * & - \frac{18 R_{4}}{h^{2}} \end{matrix}] \bar{β} (t)

(14)

where ${\hat{β}}^{T} (t) = [β_{1} β_{2}]$ , ${\bar{β}}^{T} (t) = [β_{2} β_{3}]$ , $β_{1} = \int_{t - h}^{t} x^{T} (s) ds$ , $β_{2} = \int_{t - h}^{t} \int_{s}^{t} x^{T} (v) dvds$ and $β_{3} = \int_{t - h}^{t} \int_{s}^{t} \int_{u}^{t} x^{T} (u) dudvds$ .

Subsequently, by merging the relations (equations (12)–(14)), we acquire the below expression

\dot{V} (t) + α_{i} V (t) - w^{T} (t) α_{i} w (t) \leq {\overset{⌣}{φ}}^{T} (t) \bar{Φ} \overset{⌣}{φ} (t)

(15)

where ${\overset{⌣}{φ}}^{T} (t) = [\begin{matrix} x^{T} (t) & x^{T} (t - h (t)) & x^{T} (t - h) & β_{1} & β_{2} & β_{3} & w^{T} (t) \end{matrix}]$ , ${\bar{Φ}}_{1, 1} = sym {P_{i} (t) (A_{i} (t) + Δ A_{i} (t)) + P_{i} (t) B_{i} (t) (K_{i} (t) + Δ K_{i} (t))} +$ $\frac{P_{i + 1} - P_{i}}{T_{i}} + α_{i} P_{i} (t) + R_{1} + R_{2} + h R_{3} + \frac{h^{4}}{4} R_{4}$ , ${\bar{Φ}}_{1, 2} = P_{i} (t) (A_{hi} (t) + Δ A_{hi} (t))$ , ${\bar{Φ}}_{1, 7} = P_{i} (t) C_{i} (t)$ , ${\bar{Φ}}_{2, 2} = - (1 - μ) R_{1}$ , ${\bar{Φ}}_{3, 3} = - R_{2}$ , ${\bar{Φ}}_{4, 4} = - \frac{4}{h} R_{3}$ , ${\bar{Φ}}_{4, 5} = \frac{6}{h^{2}} R_{3}$ , ${\bar{Φ}}_{5, 5} = - \frac{12}{h^{3}} R_{3} - 3 R_{4}$ , ${\bar{Φ}}_{5, 6} = \frac{6}{h} R_{4}$ , ${\bar{Φ}}_{6, 6} = - \frac{18}{h^{2}} R_{4}$ and ${\bar{Φ}}_{7, 7} = - α_{i} I$ .

Furthermore, by applying the Schur complement lemma and S procedure to $\bar{Φ}$ in equation (15), we thus arrive at the matrix $[Φ]$ specified in the theorem’s statement. Moreover, if the inequality (equation (9)) holds, then it is easy to attain the following relation

e^{α_{i} T_{f}} \overset{\cdot}{V} (t) + e^{α_{i} T_{f}} α_{i} V (t) < w^{T} (t) α_{i} w (t)

(16)

Now, by integrating the former inequality (equation (16)) from $[0, T_{f}]$ , we get the following constraint

V (t) e^{α_{i} T_{f}} < V (0) + α_{i} \int_{0}^{T_{f}} e^{α_{i} s} w^{T} (s) w (s) ds,

V (t) < e^{- α_{i} T_{f}} [V (0) + η (1 - e^{α_{i} T_{f}})]

(17)

By assuming the criteria ${\hat{P}}_{i} = G^{1 / 2} P_{i} G^{1 / 2},$ ${\hat{R}}_{r} = G^{1 / 2} R_{r} G^{1 / 2}, (r = 1, 2, 3, 4),$ it is easy to generate the following expression

\begin{matrix} V (t) \geq x^{T} (t) P_{i} x (t) = x^{T} (t) G^{1 / 2} {\hat{P}}_{i} G^{1 / 2} x (t) \\ \geq λ_{\min} ({\hat{P}}_{i}) x^{T} (t) G x (t) = λ_{1} x^{T} (t) G x (t) \end{matrix}

(18)

On a related note, from equation (11), we have

\begin{matrix} V (0) = x^{T} (0) P_{i} (0) x (0) + \int_{- h (0)}^{0} x^{T} (v) R_{1} x (v) dv + \int_{- h}^{0} x^{T} (v) R_{2} x (v) dv \\ + \int_{- h}^{0} \int_{s}^{0} x^{T} (v) R_{3} x (v) dvds + \frac{h^{2}}{2} \int_{- h}^{0} \int_{s}^{0} \int_{v}^{0} x^{T} (u) R_{4} x (u) dudvds, \\ V (0) \leq (λ_{2} + h λ_{3} + h λ_{4} + \frac{h^{2}}{2} λ_{5} + \frac{h^{5}}{12} λ_{6}) sup_{- h \leq θ \leq 0} \\ {x^{T} (θ) G x (θ)} \leq \tilde{ζ} C_{1} \end{matrix}

(19)

Furthermore, from equations (17)–(19), we can easily get

x^{T} (t) G x (t) \leq \frac{V (t)}{λ_{1}} \leq e^{- α_{i} T_{f}} \frac{[\tilde{ζ} C_{1} + η (1 - e^{α_{i} T_{f}})]}{λ_{1}}

Thus, if the criterion (equation (10)) is met, it is easy to manifest that $x^{T} (t) G x (t) < C_{2},$ for all $t \in [0, T_{f}]$ . Then, in accordance with Definition 1, the closed-loop PPPTVSs are FTB with regard to $(C_{1}, C_{2}, T_{f}, G, η)$ , which brings an end to the theorem’s proof.

Mixed $H_{\infty}$ and passivity-based FT resilient controller design

Following the preceding subsection, we present a sequel theorem demonstrating the FT boundedness of the PPPTVS by taking mixed $H_{\infty}$ and passivity performance into account, wherein the gain matrix is taken as unknown.

Theorem 2. For given positive scalars $α_{i}, h, μ, η$ , $σ$ , $T_{f}$ , $γ$ , $θ$ and matrix $G = G^{T}$ , the closed-loop PPPTVSs (equation (7)) are FTB with regard to $(C_{1}, C_{2}, G, T_{f}, η)$ under Assumption 1, if the matrices $X_{i} > 0, Q_{r} > 0, (r = 1, 2, 3, 4)$ , $Y_{i}$ and positive scalars $δ_{1 i}, δ_{2 i}, δ_{3 i}$ , ( $i \in N$ ) exist, so that the criteria underneath met

\sum_{j = 0}^{k} [F_{i, j}]_{20 \times 20} < 0, k = 0, 1, \dots m + 1

(20)

X_{S + 1} = X_{1}, Y_{S + 1} = Y_{1}

(21)

{\frac{1}{σ} + h [\frac{4}{σ^{2}} + \frac{h}{σ^{2}} + \frac{h^{4}}{6 σ^{2}}]} C_{1} + η [1 - e^{α_{i} T_{f}}] < C_{2} e^{α_{i} T_{f}}

(22)

where the elements of $F_{i, 0}$ are listed hereunder:

\begin{matrix} F_{i, 0}^{1, 1} = sym {A_{i, 0} X_{i} + B_{i, 0} Y_{i}} - \frac{X_{i + 1} - X_{i}}{T_{i}} + α_{i} X_{i}, F_{i, 0}^{1, 2} \\ = A_{hi, 0} X_{i}, F_{i, 0}^{1, 7} = C_{i, 0} - (1 - θ) X_{i} D_{i, 0}^{T}, \\ F_{i, 0}^{1, 8} = \sqrt{θ} X_{i} D_{i, 0}^{T}, F_{i, 0}^{1, 9} = δ_{1 i} H_{i}, F_{i, 0}^{1, 10} = X_{i} E_{Ai}^{T}, \\ F_{i, 0}^{1, 11} = δ_{2 i} H_{i}, F_{i, 0}^{1, 13} = δ_{3 i} B_{i, 0} H_{ki}, F_{i, 0}^{1, 14} = X_{i} E_{ki}^{T}, \\ F_{i, 0}^{1, 15} = F_{i, 0}^{1, 16} = X_{i}^{T}, F_{i, 0}^{1, 17} = \sqrt{h} X_{i}, F_{i, 0}^{1, 18} = \frac{h^{2}}{2} X_{i}, F_{i, 0}^{2, 2} \\ = 2 X_{i} + \frac{Q_{1}}{1 - μ}, F_{i, 0}^{2, 12} = X_{i} E_{Bi}^{T}, \\ F_{i, 0}^{3, 3} = - 2 X_{i} + Q_{2}, F_{i, 0}^{4, 4} = - 2 X_{i} + \frac{h}{4} Q_{3}, F_{i, 0}^{4, 19} = \frac{\sqrt{6}}{h} X_{i}, F_{i, 0}^{5, 5} \\ = - 4 X_{i} + \frac{h^{3}}{12} Q_{3} + \frac{1}{3} Q_{4}, F_{i, 0}^{5, 20} = \sqrt{\frac{6}{h}} X_{i}, \\ F_{i, 0}^{6, 6} = - 2 X_{i} + \frac{h^{2}}{18} Q_{4}, F_{i, 0}^{7, 7} = F_{i, 0}^{8, 8} = - γ I, F_{i, 0}^{9, 9} = F_{i, 0}^{10, 10} \\ = - δ_{1 i} I, F_{i, 0}^{11, 11} = F_{i, 0}^{12, 12} = - δ_{2 i} I, \\ F_{i, 0}^{13, 13} = F_{i, 0}^{14, 14} = - δ_{3 i} I, F_{i, 0}^{15, 15} = - Q_{1}, F_{i, 0}^{16, 16} = - Q_{2}, F_{i, 0}^{17, 17} \\ = - Q_{3}, F_{i, 0}^{18, 18} = - Q_{4}, F_{i, 0}^{19, 19} = - Q_{3}, and F_{i, 0}^{20, 20} = - Q_{4} . \end{matrix}

The entries $F_{i, 1}$ are provided as follows:

\begin{matrix} F_{i, 1}^{1, 1} = sym {A_{i, 1} X_{i} + A_{i, 0} (X_{i + 1} - X_{i}) + B_{i, 1} Y_{i} + B_{i, 0} (Y_{i + 1} - Y_{i})} \\ + α_{i} (X_{i + 1} - X_{i}) \\ F_{i, 1}^{1, 2} = A_{hi, 1} X_{i} + A_{hi, 0} (X_{i + 1} - X_{i}), F_{i, 1}^{1, 7} = C_{i, 1} - (1 - θ) X_{i} D_{i, 1}^{T} \\ + (1 - θ) (X_{i + 1} - X_{i}) D_{i, 0}^{T}, \\ F_{i, 1}^{1, 8} = \sqrt{θ} X_{i} D_{i, 1}^{T} + \sqrt{θ} (X_{i + 1} - X_{i}) D_{i, 0}^{T}, F_{i, 1}^{1, 10} \\ = (X_{i + 1} - X_{i}) E_{Ai}^{T}, F_{i, 1}^{1, 13} = δ_{3 i} B_{i, 1} H_{ki}, \\ F_{i, 1}^{1, 14} = (X_{i + 1} - X_{i}) E_{ki}^{T}, F_{i, 1}^{1, 15} = F_{i, 1}^{1, 16} = (X_{i + 1} - X_{i}), F_{i, 1}^{1, 17} \\ = \sqrt{h} (X_{i + 1} - X_{i}), F_{i, 1}^{1, 18} = \frac{h^{2}}{2} (X_{i + 1} - X_{i}), \\ F_{i, 1}^{2, 2} = - 2 (X_{i + 1} - X_{i}), F_{i, 1}^{2, 12} = (X_{i + 1} - X_{i}) E_{Bi}^{T}, F_{i, 1}^{3, 3} \\ = - 2 (X_{i + 1} - X_{i}), F_{i, 1}^{4, 4} = - 2 (X_{i + 1} - X_{i}), \\ F_{i, 1}^{4, 19} = \frac{\sqrt{6}}{h} (X_{i + 1} - X_{i}), F_{i, 1}^{5, 5} = - 4 (X_{i + 1} - X_{i}), F_{i, 1}^{5, 20} \\ = \sqrt{\frac{6}{h}} (X_{i + 1} - X_{i}) and F_{i, 1}^{6, 6} = - 2 (X_{i + 1} - X_{i}) . \end{matrix}

The elements corresponding to $F_{i, m}$ are displayed as follows:

\begin{matrix} F_{i, m}^{1, 1} = sym {A_{i, m} X_{i} + A_{i, m - 1} (X_{i + 1} - X_{i}) + B_{i, m} Y_{i} \\ + B_{i, m - 1} (Y_{i + 1} - Y_{i})} \\ F_{i, m}^{1, 2} = A_{hi, m} X_{i} + A_{hi, m - 1} (X_{i + 1} - X_{i}), F_{i, m}^{1, 7} \\ = C_{i, m} - (1 - θ) X_{i} D_{i, m}^{T} - (1 - θ) (X_{i + 1} - X_{i}) D_{i, m - 1}^{T}, \\ F_{i, m}^{1, 8} = \sqrt{θ} X_{i} D_{i, m}^{T} + \sqrt{θ} (X_{i + 1} - X_{i}) D_{i, m - 1}^{T} and F_{i, m}^{1, 13} = B_{i, m} H_{ki} . \end{matrix}

The entries of $F_{i, m + 1}$ are specified as follows:

\begin{matrix} F_{i, m + 1}^{1, 1} sym {A_{i, m} (X_{i + 1} - X_{i}) + B_{i, m} (Y_{i + 1} - Y_{i})} = F_{i, m + 1}^{1, 2} \\ = A_{hi, m} (X_{i + 1} - X_{i}), \\ F_{i, m + 1}^{1, 7} = - (1 - θ) (X_{i + 1} - X_{i}) D_{i, m}^{T} and F_{i, m + 1}^{1, 8} \\ = \sqrt{θ} (X_{i + 1} - X_{i}) D_{i, m}^{T} . \end{matrix}

Furthermore, through the following connection, time-varying gain matrices can be calculated for a resilient state feedback controller

K_{i} (t) = Y_{i} (t) X_{i}^{- 1} (t)

(23)

here $X_{i} (t) = X_{i} + φ_{i} (t) (X_{i + 1} - X_{i}) and Y_{i} (t) = Y_{i} + φ_{i} (t) (Y_{i + 1} - Y_{i}) .$

Proof: Following up on the previous theorem, we will use the same LKF (equation (11)) in an effort to prove this claim. Then, by incorporating the mixed $H_{\infty}$ and passivity performance as in Definition 2 and following the same procedure as in Theorem 1, we obtain a matrix $[ℵ]_{14 \times 14}$ and its elements will be as follows:

\begin{matrix} ℵ_{1, 1} = sym {P_{i} (t) A_{i} (t) + P_{i} (t) B_{i} (t) K_{i} (t)} + \frac{P_{i + 1} - P_{i}}{T_{i}} \\ + α_{i} P_{i} (t) + R_{1} + R_{2} + h R_{3} + \frac{h^{4}}{4} R_{4}, \\ ℵ_{1, 2} = P_{i} (t) A_{hi} (t), ℵ_{1, 7} = P_{i} (t) C_{i} (t) - (1 - θ) D_{i}^{T} (t), \\ ℵ_{1, 8} = \sqrt{θ} D_{i}^{T} (t), ℵ_{1, 9} = P_{i} (t) δ_{1 i} H_{i}, \\ ℵ_{1, 10} = E_{Ai}^{T}, ℵ_{1, 11} = P_{i} (t) δ_{2 i} H_{i}, ℵ_{1, 13} = P_{i} (t) B_{i} (t) δ_{3 i} H_{ki}, \\ ℵ_{1, 14} = E_{ki}^{T}, ℵ_{2, 2} = - (1 - μ) R_{1}, \\ ℵ_{2, 12} = E_{Bi}^{T}, ℵ_{3, 3} = - R_{2}, ℵ_{4, 4} = - \frac{4}{h} R_{3}, ℵ_{4, 5} = \frac{6}{h^{2}} R_{3}, \\ ℵ_{5, 5} = - \frac{12}{h^{3}} R_{3} - 3 R_{4}, ℵ_{5, 6} = \frac{6}{h} R_{4}, \\ ℵ_{6, 6} = - \frac{18}{h^{2}} R_{4}, ℵ_{7, 7} = ℵ_{8, 8} = - γ I, ℵ_{9, 9} = ℵ_{10, 10} = - δ_{1 i} I, \\ ℵ_{11, 11} = ℵ_{12, 12} = - δ_{2 i} I and ℵ_{13, 13} = ℵ_{14, 14} = - δ_{3 i} I . \end{matrix}

Subsequently, pre- and post-multiply the resultant matrix by $diag {\underset{6}{\underset{︸}{X_{i} (t), \dots, X_{i} (t)}}, \underset{8}{\underset{︸}{I, \dots, I}}}$ under the connections $X_{i} (t) {\overset{\cdot}{P}}_{i} (t) X_{i} (t) = - {\overset{\cdot}{X}}_{i} (t)$ , $X_{i} (t) = P_{i}^{- 1} (t)$ , $R_{r} = Q_{r}^{- 1} (r = 1, 2, 3, 4),$ ${\hat{P}}_{i} (t) = X_{i} (t) P_{i} (t) X_{i} (t),$ ${\hat{R}}_{r} = X_{i} (t) R_{r} X_{i} (t)$ and $- X_{i} (t) Q_{r}^{- 1} X_{i} (t) \leq - 2 X_{i} (t) + Q_{r}$ and deploy the Schur complement lemma. As a result, we acquired the matrix $[Ξ]_{20 \times 20}$ , and their elements are presented below:

\begin{matrix} Ξ_{1, 1} = sym {A_{i} (t) X_{i} (t) + B_{i} (t) Y_{i} (t)} - {\overset{\cdot}{X}}_{i} (t) + α_{i} X_{i} (t), \\ Ξ_{1, 2} = A_{hi} (t) X_{i} (t) \\ Ξ_{1, 7} = C_{i} (t) - (1 - θ) X_{i} (t) D_{i}^{T} (t), Ξ_{1, 8} = \sqrt{θ} X_{i} (t) D_{i}^{T} (t), \\ Ξ_{1, 9} = δ_{1 i} H_{i}, Ξ_{1, 10} = X_{i} (t) E_{Ai}^{T}, \\ Ξ_{1, 11} = δ_{2 i} H_{i}, Ξ_{1, 13} = δ_{3 i} B_{i} (t) H_{ki}, Ξ_{1, 14} = X_{i} (t) E_{ki}^{T}, \\ Ξ_{1, 15} = X_{i} (t), Ξ_{1, 16} = X_{i} (t), \\ Ξ_{1, 17} = \sqrt{h} X_{i} (t), Ξ_{1, 18} = \frac{h^{2}}{2} X_{i} (t), Ξ_{2, 2} = - 2 X_{i} (t) + \frac{Q_{1}}{1 - μ}, \end{matrix}

\begin{matrix} Ξ_{2, 12} = X_{i} (t) E_{Bi}^{T}, Ξ_{3, 3} = - 2 X_{i} (t) + Q_{2}, \\ Ξ_{4, 4} = - 2 X_{i} (t) + \frac{h}{4} Q_{3}, Ξ_{4, 19} = \frac{\sqrt{6}}{h} X_{i} (t), \\ Ξ_{5, 5} = - 4 X_{i} (t) + \frac{h^{3}}{12} Q_{3} + \frac{1}{3} Q_{4}, Ξ_{5, 20} = \sqrt{\frac{6}{h}} X_{i} (t), \\ Ξ_{6, 6} = - 2 X_{i} (t) + \frac{h^{2}}{18} Q_{4}, Ξ_{7, 7} = Ξ_{8, 8} = - γ I, \\ Ξ_{9, 9} = Ξ_{10, 10} = - δ_{1 i} I, Ξ_{11, 11} = Ξ_{12, 12} = - δ_{2 i} I, \\ Ξ_{13, 13} = Ξ_{14, 14} = - δ_{3 i} I, Ξ_{15, 15} = - Q_{1}, \\ Ξ_{16, 16} = - Q_{2}, Ξ_{17, 17} = - Q_{3}, Ξ_{18, 18} = - Q_{4}, \\ Ξ_{19, 19} = - Q_{3} and Ξ_{20, 20} = - Q_{4} . \end{matrix}

Moreover, the aforementioned matrix $Ξ$ is expressed in the ensuing equivalent frame

\begin{matrix} Ξ = F_{i, 0} + φ_{i} (t) F_{i, 1} + [φ_{i} (t)]^{2} F_{i, 2} + \dots + \\ [φ_{i} (t)]^{m} F_{i, m} + [φ_{i} (t)]^{m + 1} F_{i, m + 1} \end{matrix}

(24)

where $F_{i, 0}$ , $F_{i, 1}$ , $F_{i, 2}$ , $F_{i, m}$ and $F_{i, m + 1}$ are defined in the theorem statement. Therefore, in light of Lemma 3, if the condition (equation (20)) holds, it is apparent that the condition $[Ξ] < 0$ is satisfied, which directly implies that $\overset{\cdot}{V} (t) < 0$ .

On the contrary, by notating ${\hat{X}}_{i} = G^{1 / 2} X_{i} G^{1 / 2}$ , ${\bar{Q}}_{r}^{- 1} = G^{1 / 2} Q_{r}^{- 1} G^{1 / 2}$ , $(r = 1, 2, 3, 4)$ and considering the relation $λ_{\max} ({\hat{X}}_{i}) = \frac{1}{λ_{\min} ({\hat{P}}_{i})},$ we get $1 < G^{- 1 / 2} P_{i} G^{- 1 / 2} < \frac{1}{σ} I,$ $λ_{1} = λ_{\min} ({\hat{P}}_{i}) > 1, λ_{2} = λ_{\max} ({\hat{P}}_{i}) < \frac{1}{σ}$ . Furthermore, it follows from equation (20) that $0 < G^{- 1 / 2} {\bar{Q}}_{r}^{- 1} G^{- 1 / 2} < \frac{2}{σ^{2}} I$ , which implies that $λ_{3} < \frac{2}{σ^{2}}, λ_{4} < \frac{2}{σ^{2}},$ $λ_{5} < \frac{2}{σ^{2}}, λ_{6} < \frac{2}{σ^{2}}$ . From the aforesaid relation, the condition (equation (22)) can be guaranteed. Thus, the PPPTVSs (equation (7)) are FTB, which ends the proof.

Following the foregoing results, in this theorem, adequate conditions proving the addressed system to be FTB in the face of multiplicative gain perturbations are offered.

Theorem 3: Let the symmetric matrix $G$ and positive scalars $α_{i}$ , $h$ , $μ$ , $η$ , $T_{f}$ , $σ$ and $θ$ be predefined. Under Assumption 1, the closed-loop PPPTVS (equation (7)) is FTB with regard to $(C_{1}, C_{2}, G, T_{f}, η)$ with gratified mixed $H_{\infty}$ and passivity index $γ > 0$ , if the positive definite matrices $X_{i} > 0, Q_{r}, (r = 1, 2, 3, 4)$ , $Y_{i}$ and positive scalars $δ_{1 i}, δ_{2 i}, δ_{3 i}$ , ( $i \in N$ ) exist such that the following criteria along with the relations (equations (21) and (22)) are satisfied

\sum_{j = 0}^{k} {[{\overset{⌣}{F}}_{i, j}]}_{20 \times 20} < 0, k = 0, 1, \dots m + 1

(25)

where the element of matrix ${[{\overset{⌣}{Ϝ}}^{i, j}]}_{20 \times 20} = [Ϝ^{i, j}]_{20 \times 20}$ except ${\overset{⌣}{Ϝ}}_{1, 14}^{i, j}$ and the elements of ${\overset{⌣}{Ϝ}}_{1, 14}^{i, j}$ are given as ${\overset{⌣}{Ϝ}}_{1, 14}^{i, 0} = Y_{i}^{T} E_{k i}^{T}$ and ${\overset{⌣}{Ϝ}}_{1, 14}^{i, 1} = (Y_{i + 1} - Y_{i})^{T} E_{k i}^{T}$ .

Proof: By adhering to the same procedure as the previous theorem and letting $Δ K_{i} (t) = H_{ki} F_{i} (t) E_{ki} K_{i} (t)$ , the argument of this theorem is generated.

Validation of methodology

In this segment, we present a numerical example and its simulation to validate and demonstrate the importance of the analytical conclusions in the preceding section. For the sake of conciseness, we consider PPPTVSs (equation (2)) with two degrees and three subsystems. Furthermore, for $ℓ = 0, 1, 2, \dots$ , the respective system parameters with $T_{1} = 2, T_{2} = 2,$ $T_{3} = 1$ and fundamental period $T_{n} = 5$ are opted as follows:

Subsystem 1:

\begin{matrix} A_{1, 0} = [\begin{matrix} - 0.1 & 2 \\ 0.1 & 0.5 \end{matrix}], A_{1, 1} = [\begin{matrix} - 1 & 2 \\ 2 & 0.2 \end{matrix}], \\ A_{1, 2} = [\begin{matrix} - 0.1 & 0.2 \\ 0.5 & - 5 \end{matrix}], \end{matrix}

\begin{matrix} A_{h 1, 0} = [\begin{matrix} 0.1 & - 0.2 \\ 1 & 0.5 \end{matrix}], A_{h 1, 1} = [\begin{matrix} - 0.1 & 0.3 \\ 2 & 0.2 \end{matrix}], \\ A_{h 1, 2} = [\begin{matrix} - 1 & 2 \\ 0.5 & - 0.5 \end{matrix}], \end{matrix}

B_{1, 0} = [\begin{matrix} - 1 & 0.2 \\ 0.1 & 1 \end{matrix}], B_{1, 1} = [\begin{matrix} - 0.5 & 0.2 \\ 0.1 & 2 \end{matrix}], B_{1, 2} = [\begin{matrix} - 1 & 0.2 \\ 0.1 & 1 \end{matrix}],

\begin{matrix} C_{1, 0} = {[\begin{matrix} 0.5 - 0.2 \end{matrix}]}^{T}, C_{1, 1} = {[\begin{matrix} - 0.25 - 0.1 \end{matrix}]}^{T}, \\ C_{1, 2} = {[\begin{matrix} - 0.150.21 \end{matrix}]}^{T}, \end{matrix}

\begin{matrix} D_{1, 0} = [\begin{matrix} 0.12 - 0.12 \end{matrix}], D_{1, 1} = [\begin{matrix} - 0.31 0.12 \end{matrix}], \\ D_{1, 2} = [\begin{matrix} - 0.21 - 0.23 \end{matrix}], \end{matrix}

\begin{matrix} H_{1} = [\begin{matrix} - 0.4 & 0.2 \\ 0.3 & 0.2 \end{matrix}], E_{A 1} = [\begin{matrix} 0.2 & - 0.2 \\ 0.3 & 0.7 \end{matrix}] and \\ E_{B 1} = [\begin{matrix} 0.2 & - 0.12 \\ 0.3 & 0.2 \end{matrix}] . \end{matrix}

Subsystem 2:

\begin{matrix} A_{2, 0} = [\begin{matrix} - 2.1 & 0.6 \\ 0.1 & 2.2 \end{matrix}], A_{2, 1} = [\begin{matrix} - 2.1 & 2.6 \\ 0 & 1.5 \end{matrix}], \\ A_{2, 2} = [\begin{matrix} - 1 & 0.6 \\ 1 & 1.5 \end{matrix}], \end{matrix}

\begin{matrix} A_{h 2, 0} = [\begin{matrix} - 0.3 & 0.3 \\ 1 & 0.2 \end{matrix}], A_{h 2, 1} = [\begin{matrix} - 0.3 & 0.3 \\ 1 & 0.2 \end{matrix}], \\ A_{h 2, 2} = [\begin{matrix} - 0.1 & 0.2 \\ 0.5 & - 1.2 \end{matrix}], \end{matrix}

\begin{matrix} B_{2, 0} = [\begin{matrix} 0.5 & 0.2 \\ 0.1 & 1 \end{matrix}], B_{2, 1} = [\begin{matrix} - 0.5 & 0.3 \\ 0.2 & 1 \end{matrix}], \\ B_{2, 2} = [\begin{matrix} - 0.5 & - 0.2 \\ 0.2 & - 1 \end{matrix}], \end{matrix}

C_{2, 0} = {[\begin{matrix} - 0.5 - 2 \end{matrix}]}^{T}, C_{2, 1} = {[\begin{matrix} - 0.51 \end{matrix}]}^{T}, C_{2, 2} = {[\begin{matrix} - 0.140.11 \end{matrix}]}^{T},

D_{2, 0} = [\begin{matrix} 0.12 - 0.12 \end{matrix}], D_{2, 1} = [\begin{matrix} - 0.110.12 \end{matrix}], D_{2, 2} = [\begin{matrix} - 0.26 - 0.25 \end{matrix}],

\begin{matrix} H_{2} = [\begin{matrix} 0.2 & 0.2 \\ 0.3 & 0.2 \end{matrix}], E_{A 2} = [\begin{matrix} 0.11 & 0.2 \\ 0.3 & 0.12 \end{matrix}] and \\ E_{B 2} = [\begin{matrix} - 0.11 & 0.12 \\ 0.3 & 0.4 \end{matrix}] . \end{matrix}

Subsystem 3:

\begin{matrix} A_{3, 0} = [\begin{matrix} - 2.1 & 0.6 \\ 0.2 & 1.5 \end{matrix}], A_{3, 1} = [\begin{matrix} - 2.1 & 0.6 \\ 0.1 & 1.5 \end{matrix}], \\ A_{3, 2} = [\begin{matrix} - 1 & 1 \\ 0.3 & 5 \end{matrix}], \end{matrix}

\begin{matrix} A_{h 3, 0} = [\begin{matrix} - 0.1 & - 0.2 \\ 1 & 0.5 \end{matrix}], A_{h 3, 1} = [\begin{matrix} - 0.1 & - 0.4 \\ - 0.9 & 0.5 \end{matrix}], \\ A_{h 3, 2} = [\begin{matrix} - 0.1 & 0 \\ - 0.1 & 0.1 \end{matrix}], \end{matrix}

\begin{matrix} B_{3, 0} = [\begin{matrix} - 0.7 & 0.3 \\ 0.4 & 1 \end{matrix}], B_{3, 1} = [\begin{matrix} - 0.05 & 0.1 \\ - 0.3 & 1 \end{matrix}], \\ B_{3, 2} = [\begin{matrix} - 0.5 & - 0.1 \\ 0.3 & - 1 \end{matrix}], \end{matrix}

C_{3, 0} = {[\begin{matrix} 0.5 - 0.1 \end{matrix}]}^{T}, C_{3, 1} = {[\begin{matrix} - 0.5 - 0.2 \end{matrix}]}^{T}, C_{3, 2} = {[\begin{matrix} - 0.5 1 \end{matrix}]}^{T},

D_{3, 0} = [\begin{matrix} 0.12 - 0.12 \end{matrix}], D_{3, 1} = [\begin{matrix} - 0.11 0.12 \end{matrix}], D_{3, 2} = [\begin{matrix} 0.26 - 0.25 \end{matrix}],

\begin{matrix} H_{3} = [\begin{matrix} - 0.2 & 0.02 \\ 0.3 & 0.2 \end{matrix}], E_{A 3} = [\begin{matrix} 0.8 & 0.25 \\ 0.3 & 0.32 \end{matrix}] and \\ E_{B 3} = [\begin{matrix} 0.6 & 0.5 \\ 0.5 & 0.6 \end{matrix}] . \end{matrix}

Moreover, along with the aforementioned parameters, to conduct a feasible test, the time-varying state delay is considered as $h (t) = 0.1 + 0.6 \sin (t)$ with $h = 0.7$ , $μ = 0.6$ . Furthermore, the values associated with the FT stability are chosen as $C_{1} = 0.1, C_{2} = 10, G = 1, T_{f} = 5$ , $η = 0.7$ , and the remaining parameters are opted as $α_{i} = 0.5$ , $α_{\max} = 0.5$ and $θ = 0.5$ . Apart from this, as we mentioned in ‘Problem formulation’ section, perturbations in the devised controller can be categorized as

Case 1: Additive fluctuations: $Δ K_{i} (t) = H_{ki} F_{i} (t) E_{ki} (t),$

Case 2: Multiplicative fluctuations: $Δ K_{i} (t) = H_{ki} F_{i} (t) E_{ki} (t) K_{i} (t) .$

Following this, by substituting the above-mentioned parameters in the conditions (equations (20)–(22)) and through the medium of the LMI toolbox in MATLAB, the feasible solutions are computed with the disturbance attenuation index $γ = 0.76$ . After the feasibility analysis is performed, the gain values for the devised controller mechanism are computed by dint of equation (23). Alongside, the pictorial delineation of the computed gain matrix and its relevant parameters are presented in Figure 1.

Figure 1.

Variation of norm for $K (t)$ , $X (t)$ and $Y (t)$ for Case 1 and Case 2.

For the simulation purposes, hereunder, we opted the external disturbance signal and initial condition, respectively, as

w (t) = {\begin{matrix} 0.2 + 0.3 \sin (0.2 t), & 0 \leq t < 2, \\ 0.01 \sin (0.02 π t), & otherwise, \end{matrix} and x (t_{0}) = [0.1 - 0.1]^{T} .

By the stand of prior settings, the simulation is carried out with the help of the MATLAB Simulink programme, and the graphs are provided in Figures 2 –8. Specifically, in Figure 2, the state trajectory under the developed controller is offered, in which the assayed system’s convergence rate is slower for Case 2 than for Case 1. Furthermore, Figure 3 illustrates the state responses of PPPTVS without designating a controller, and it can be clearly seen that state trajectories fail to converge. Hence, it is evident from Figures 2 and 3 that the developed resilient control protocol is effective in stabilizing the considered system, even though time delays, parameter uncertainties, external disturbances, additive and multiplicative gain fluctuations occur in the assayed system. Furthermore, the response of state trajectories under both cases of perturbations for various delay bounds is illustrated in Figure 4. It is apparent from this figure that the presence of a time delay has a substantial amount of influence on the stability of the addressed system; that is, when the scale of the time delay is increased, the convergence rate of the undertaken system is slower. From this figure, the impact of time delays on the system under discussion is abundantly evident. In addition, designed controller responses under additive and multiplicative perturbations are showcased in Figure 5.

Figure 2.

Evolution of closed-loop trajectories.

Figure 3.

Evolution of open-loop trajectories.

Figure 4.

State responses of PPPTVSs under distinct delay bounds.

Figure 5.

Control responses for two cases.

Figure 6.

Time history of $x^{T} (t) G x (t)$ in the presence and absence of proposed control (a) with control and (b) without control.

Figure 7.

Evolution of output trajectories: (a) Case 1 and (b) Case 2.

Figure 8.

Evolution of $x^{T} (t) G x (t)$ under conventional control.

Furthermore, in Figure 6, the time evolution of $x^{T} (t) G x (t)$ in the presence and absence of a controller has been provided. Therein, when $u (t) \neq 0$ , the states of PPPTVSs taking the initial values inside the optimal boundary of $C_{1}$ are not exceeding the optimal bound value $C_{2}$ in the simulation process. Subsequently, in the case when $u (t) = 0$ , the curve $x^{T} (t) G x (t)$ exceeds the specified range of $C_{2}$ , indicating that the open-loop PPPTVSs do not possess FT boundedness. The presented figure purportedly demonstrates the efficacy and impact of the outlined controller in the examined system.

Eventually, the response of system output has been supplied in Figure 7 to show off the potential benefits of the proposed resilient control in comparison with the traditional controller. The outcomes indicate that the designed controller delivers far better performance for the system than the conventional controller. Furthermore, the response of $x^{T} (t) G x (t)$ under a conventional controller is showcased in Figure 8. In this figure, we can see that the trajectory of $x^{T} (t) G x (t)$ exceeds the optimal bound value $C_{2}$ , and it can be concluded that the conventional controller cannot cope with the gain fluctuations.

As such, the simulation results presented above clearly show that the devised control design is resilient against gain fluctuations and ensures the FT boundedness of the investigated system despite the time-varying delays in the state, uncertainties and exogenous disruptions.

Conclusion

In this investigation, a FT stabilization problem has been addressed for a class of PPPTVSs in the existence of unpredictable factors such as time-varying state delay, parameter uncertainties and external disturbances by utilizing the resilient control scheme. More precisely, additive and multiplicative gain fluctuations have been scrutinized making the control more appropriate for practical scenarios and it is rendered by periodic piecewise characteristics. By exploiting LKF methodology along with FT stability notion and matrix polynomial lemma, a set of adequate constraints has been established to assure the FT boundedness of the considered system. Furthermore, relying on the established settings, a precise relation for reckoning the desired periodic piecewise time-varying resilient control gain matrix has been presented. Finally, a numerical analysis was carried out in which the intrinsic potential of the derived theoretical conclusions was confirmed using simulations.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship and/or publication of this article: This work of R Sakthivel was financially supported by Science and Engineering Research Board (SERB), Department of Science and Technology (DST) and Government of India under Core Research Grant (CRG), Sanction No. CRG/2020/002844.

Data availability statement

Data sharing is not applicable to this article as no data sets were generated or analysed during this study.

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Finite-time resilient control for uncertain periodic piecewise polynomial time-varying systems

Abstract

Keywords

Introduction

Problem formulation

Main results

FT boundedness analysis

Mixed H ∞ and passivity-based FT resilient controller design

Validation of methodology

Conclusion

Footnotes

Declaration of conflicting interests

Funding

Data availability statement

References

Mixed $H_{\infty}$ and passivity-based FT resilient controller design