Introduction

In reality, one of the most significant tasks in the analysis and synthesis of control systems is the development of an accurate mathematical model, which can properly represent the dynamic behavior of systems. In this context, reaction–diffusion systems (RDSs), as an effective model for characterizing the performance of space-time related dynamic systems, appear in recent literature [1,2,3,4,5,6]. This trend is driven by RDSs’ numerous practical applications, which include various fields such as chemical, nuclear, aerospace [7], etc. To date, fruitful research results have been achieved in the field of RDSs control, such as event-triggered control [1], saturated control [8], boundary control [9, 10], and output feedback control [11]. It is noteworthy that the above works predominantly focused on single-mode RDSs (RDSs with only one mode). Without a doubt there will be a non-negligible fact that some practical systems are frequently influenced by external uncertainties that lead such systems to exhibit multiple modes [12,13,14], which makes single-mode RDSs challenging in describing more general physical processes. In this regard, how to establish accurate mathematical models for some more complicated and realistic multi-mode systems by reaction-diffusion models is a topic worth exploring.

As reported in the existing works [12, 15], switched RDSs with certain switching mechanisms play an important role in describing systems that exhibit multi-mode characteristics and spatial features. Consequently, on the basis of switched RDSs, it is particularly important to explore the switching mechanisms contained in the subsystems. In accordance with this research direction, Refs. [16, 17] made the first attempts to propose dwell-time (DT) and average dwell-time (ADT) switching mechanisms to describe the switching laws obeyed by the subsystems. Subsequently, due to the limitations of DT’s dwell time and ADT’s switching frequency, they are unable to describe the fast switching phenomenon, which is common in some practical systems, such as complex dynamic networks [18] and mass-spring systems [19]. Based on this fact, an extended persistent dwell-time (PDT) switching mechanism is proposed in [20] to make up for the shortcomings of DT and ADT switching mechanisms. More specifically, for the PDT switching mechanism (PDTSM), there exist infinite disjoint intervals (called \(\tau \)-Portion) in which only one switching occurs, whose length is not less than the dwell time \(\tau _{\mathfrak {p}}\), and the portion between two consecutive \(\tau \)-Portion is called T-Portion whose length is no greater than the persistence period \(T_{\mathfrak {p}}\), where the subsystems can arbitrarily switch [21, 22]. Nevertheless, one point to be noted from the references [21, 22] on the PDTSM is that the results are feasible only under switched ordinary differential equation (ODE) systems. It is well-known that since switched RDSs can be applied to describe many complex systems that cannot be captured by switched ODE systems, such as the temperature control systems of semiconductor power chips [12]. Therefore, it is meaningful to extend PDTSM to switched RDSs.

With the remarkable advances in networked control systems [23,24,25,26,27,28,29,30,31], the control implementation of RDSs has been rapidly networked in control areas such as intelligent robot control [32] and chemical catalytic control [1]. On this trend, the design of networked control for RDSs has yielded a wealth of research results. To mention a few, a class of stochastic sampled-data control was studied in [33]; Selivanov and Fridman in [34], and Song et al. in [1] investigated novel design methods for point measurement and pointwise control, respectively, which have outstanding advantages in terms of reliability, low measurement costs, and ease transmission. However, it should be pointed out that the effect of bandwidth limitations is ignored in the above results [1, 33, 34]. Given the data to be transmitted, a fairly novel hysteresis quantizer with the advantages of avoiding controller chattering and improving bandwidth utilization was developed in [35, 36]. Still, despite the advantages of hysteresis quantization, it is an objective fact that quantization has a negative impact on system’s performance [37]. Therefore, it makes sense to make a trade-off between the quantitative effect and the performance of the system. To the best of our knowledge, research results on the quantified control for switched RDSs with PDTSM, especially integrating the above trade-off, are not currently available, which stimulates the research interest of this work. On the other hand, the existing work on stability analysis of switched systems mainly focused on two traditional ones: asymptotic stability [38] and exponential stability [39]. Unfortunately, these traditional stabilities have more rigorous restrictions and may be not always applicable in various dynamical systems, such as stochastic systems [40] and switched systems [41], where the systems’ states remaining bounded within a certain rate of convergence [called exponential input-to-state stability (ISS)] is sufficient but do not necessarily converge to the equilibrium point. Nonetheless, up to now, despite the results have been achieved in the field of switched ODE systems [41, 42], the problem of exponential ISS of switched RDSs remains a huge room for improvement due to the complexity and uniqueness of switched RDSs, which encourages us to make new attempts in the field. Inspired by the above discussion, in terms of hysteresis quantization, this paper proposes a pointwise control method for the switched RDSs with PDTSM. The specific contributions of this article are emphasized as follows. \(\bullet \) Unlike previous studies biased toward the modeling approach of ODE systems [43, 44], RDSs [1, 8], and switched ODE systems [45,46,47,48], this article proposes a more extensive model of switched RDSs that can characterize more complex dynamical systems. Furthermore, this work makes the first step to extend the PDTSM to represent the switching rules possessed by subsystems, which enhances the proposed method’s applicability compared to the DT and ADT switching mechanisms. \(\bullet \) This paper proposes a unified framework for switched RDS’s signal measurement, transmission, and control. Specifically, different from [23, 25, 26], this work develops a combined point measurement and pointwise control approach, which effectively minimizes the quantities of sensors and actuators. Furthermore, in contrast to [35, 36], we propose the hysteresis quantization with a small parameter to balance the quantitative effect and system’s performance while avoiding the controller’s chattering and improving the bandwidth utilization. \(\bullet \) In practice, different from the asymptotic stability [38] and exponential stability [39], since the states of numerous systems are not strictly required to converge to the equilibrium point, this work attempts to formulate a relatively novel exponential ISS criteria for the switched RDSs with PDTSM, in which the systems’ states only need to remain bounded with certain convergence rate.

Notations: int\([\aleph ]\) denotes the maximal integer less than \(\aleph \). Moreover, for the convenience of expression, define \(\phi _t(x,t) = \frac{\partial \phi (x,t)}{\partial t}\), \(\phi _x(x,t) = \frac{\partial \phi (x,t)}{\partial x}\), \(\phi _{xt}(x,t) = \frac{\partial ^2 \phi (x,t)}{\partial x\partial t}\), and \(\phi _{xx}(x,t) = \frac{\partial ^2 \phi (x,t)}{\partial x^2}\). The other notations are similar to [1] and are not repeated here.

Preliminaries and problem statements

System formulation

Consider the switched RDSs with PDTSM as follows:

$$\begin{aligned} \phi _t(x,t)={\Upsilon }_{\theta (t)}\phi _{xx}(x,t)+{\Lambda }_{\theta (t)}(x)\phi (x,t)+g(x)u_{\theta (t)}, \end{aligned}$$
(1)

subject to the initial condition

$$\begin{aligned} \phi (x,0)=\phi _0(x), \end{aligned}$$
(2)

and boundary conditions

$$\begin{aligned} \phi _x\left( l_1,t\right) =0,\phi _x\left( l_2,t\right) =0, \end{aligned}$$
(3)

where \(\phi (x,t)\in {\mathcal {H}}^n\) denotes the systems’ state at the position \(x\in [l_1,l_2]\) and time \(t\in [0,\infty )\). \(\theta (t): [0,\infty )\rightarrow {\mathbb {J}} \triangleq \{1, 2, 3,..., M \}\) is the PDT switching signal, which is assumed to be a piecewise constant function and right-continuous, and M is the amount of subsystems. For convenience, let \(\theta (t)=j\). Therefore, systems (1) is rewritten as

$$\begin{aligned} \phi _t(x,t)={\Upsilon }_j\phi _{xx}(x,t)+{\Lambda }_j(x)\phi (x,t)+g(x)u_{j}, \end{aligned}$$
(4)

where for any \(j\in {\mathbb {J}}\), \({\Upsilon }_j\in {\mathbb {R}}^n\) is a known constant matrix, and \({\Lambda }_j(x)\) is a polynomial function with \({\bar{\Lambda }}_j \triangleq \max _{x\in [l_1,l_2]} {\Lambda }_j(x)\). The interval \([l_1,l_2]\) is divided into L subintervals. \(x_p\) and \(x_{p+1}\) represent the terminal points of the subinterval, \(p\in {\mathbb {P}}\triangleq \{0,1,\ldots ,L-1\}\). \({\Delta }_p\triangleq x_{p+1}-x_p\) represents the length between the two terminal points and \({{\bar{x}}}_p=\frac{x_p+x_{p+1}}{2}\) denotes the midpoint of the subinterval. The control input \(u_j\) is defined as \(u_j \triangleq col[u_{j0}\ u_{j1}\ \ldots u_{j(L-1)}]\), where \(u_{jp}=col[u_{jp1}\ u_{jp2}\ \ldots \ u_{jpn}]\). \(g(x)\triangleq [g_0(x)\ g_1(x)\ \ \ldots g_{L-1}(x)]\) with \(g_p(x)\triangleq \delta (x-{{\bar{x}}}_p)g_p\) and \(g_p\) is a known scalar. Dirac delta function \(\delta (x-{{\bar{x}}}_p)\) satisfies

$$\begin{aligned} \delta (x&-{{\bar{x}}}_p)=\left\{ {\begin{array}{*{20}{c}} 0,&{}x\ne {{\bar{x}}}_p,\\ \infty , &{}x={{\bar{x}}}_p. \end{array}} \right. \end{aligned}$$

With the above analysis, it is obtained that the space x is divided into L subintervals. The measurement of state information and the implementation of control signals are performed only at the midpoint \({{\bar{x}}}_p\), \(p\in {\mathbb {P}}\) of each subinterval, which can largely reduce the number of sensors and actuators compared to the full domain control. Next, the pointwise controller is provided as follows:

$$\begin{aligned} u_{jp}\left( {{\bar{x}}}_p,t\right) =-K_{jp}\phi \left( {{\bar{x}}}_p,t\right) , \end{aligned}$$
(5)

where \(K_{jp}>0\in {\mathbb {R}}^n\) (\(j\in {\mathbb {J}}\), \(p\in {\mathbb {P}}\)) is the controller gain matrix that needs to be determined.

With the above point measurement and pointwise control, we obtain a control signal in the form of (5). However, during the long-distance transmission of the control signal to the actuators, the bandwidth limitation is a non-negligible problem. Therefore, we dedicate ourselves on improving the utilization of network bandwidth in the following.

Hysteresis quantizer

Indeed, the process of signal transmission over long distances is inevitably restricted by the fact that the bandwidth of the transmission channel has a limited capacity. Therefore, the following hysteresis quantizer is utilized [49]:

$$\begin{aligned} Q(u)\mathrm{=}\left\{ \begin{array}{cc} u^\sigma {\text {sgn}}(u), &{} \frac{u^\sigma }{1+\delta _j}\mathrm{<} \, |u| \, \mathrm{\le }\, u^\sigma ,\dot{u}< 0,\text {or} \\ &{}u^\sigma \, \mathrm{< }|u| \, \mathrm{\le }\, \frac{u^\sigma }{1-\delta _j}, \dot{u}> 0 \\ u^\sigma \left( 1+\delta _j\right) {\text {sgn}}(u), &{} u^\sigma \, \mathrm{<} \, |u| \mathrm{\le }\, \frac{u^\sigma }{1-\delta _j}, \dot{u} \mathrm{<}0, \text {or} \\ &{} \frac{u^\sigma }{1-\delta _j} \, \mathrm{<} \, |u| \mathrm{\le }\, \frac{u^\sigma \left( 1+\delta _j\right) }{\left( 1-\delta _j\right) }, \dot{u}\mathrm{>}0 \\ 0, &{} 0 \le |u|\, \mathrm{<} \, \frac{\zeta _j}{1+\delta _j}, \dot{u} \mathrm{<} 0 \text{ or } \\ &{}\frac{\zeta _j}{1+\delta _j} \mathrm{\le } u \le \zeta _j, \dot{u} \mathrm{>}0, \\ Q\left( u(t^{-})\right) ,&{}\dot{u} =0, \end{array}\right. \nonumber \\ \end{aligned}$$
(6)

where \(u^\sigma ={\nu _j}^{(1-\sigma )}\zeta _j\) with \(\sigma \in {\mathbb {N}}^+\). The constants \(\nu _j \in (0,1)\) and \(\zeta _j\) denote the quantizer density and the range of quantizer’s dead-zone, respectively. The mode-dependent parameter \(\delta _j\) satisfies \(\delta _j=\frac{1-\nu _j}{1+\nu _j}\in (0,1)\). The value of quantizer Q(u) is in the set \({\mathbb {Q}}=\{0,\pm u^\sigma , \pm u^\sigma (1+\delta _j)\}\).

Fig. 1
figure 1

Map of Q(u) for \(u>0\)

Full size image

Remark 1

The hysteresis quantizer not only outperforms the logarithmic quantizer in the number of additional levels of quantization, but also effectively avoids the phenomenon of controller chattering caused by the logarithmic quantizer [35]. As shown in Fig. 1, when the value Q(u) of the hysteresis quantizer transitions from one level to another, there exists some dwell time before the new transition level appears, which avoids the controller’s chattering.

Fig. 2
figure 2

Structural diagram of the proposed control scheme

Full size image

To analyze the hysteresis quantizer’s effect more intuitively, the hysteresis quantizer Q(u) is reformulated as follows:

$$\begin{aligned} Q\left( \phi \left( {{\bar{x}}}_p,t\right) \right) =(1-\varrho )\phi \left( {{\bar{x}}}_p,t\right) +\varrho e\left( {{\bar{x}}}_p,t\right) , \end{aligned}$$
(7)

where \(0<\varrho <1\) is a given scalar, \(e({{\bar{x}}}_p,t)\) is a newly defined error variable.

Remark 2

In [35, 36], the hysteresis quantizer needs to satisfy \(Q(\phi ({{\bar{x}}}_p,t))=\phi ({{\bar{x}}}_p,t)-e(\phi ({{\bar{x}}}_p,t))\). In contrast, the hysteresis quantizer in this paper is improved to a linear combination of \(\phi ({{\bar{x}}}_p,t)\) and \(e(\phi ({{\bar{x}}}_p,t))\) depending the parameter \(\varrho \), and by adjusting which can make a balance between the quantitative effect and the system’s performance.

Then, based on [49], the hysteresis quantizer is set as

$$\begin{aligned} \left| Q\left( \phi ({{\bar{x}}}_p,t)\right) -\phi \left( {{\bar{x}}}_p,t\right) \right| \le \delta _j\left| \phi \left( {{\bar{x}}}_p,t\right) \right| +\zeta _j, \end{aligned}$$
(8)

based on which, the control signal (5) is reformulated as

$$\begin{aligned} u_{jp}\left( {{\bar{x}}}_p,t\right) =-K_{jp}Q\left( \phi \left( {{\bar{x}}}_p,t\right) \right) . \end{aligned}$$
(9)

Combining (4), (7) and (9), the closed-loop switched RDSs has the following form:

$$\begin{aligned} \begin{aligned} \phi _t(x,t)=&{\Upsilon }_j\phi _{xx}(x,t)+{\Lambda }_j(x)\phi (x,t)\\&-g_p(x)K_{jp}\left\{ (1-\varrho )\phi ({{\bar{x}}}_p,t)+\varrho e({{\bar{x}}}_p,t)\right\} . \end{aligned} \end{aligned}$$
(10)

To facilitate the subsequent analysis, the PDTSM should be described in detail.

PDTSM

Definition 1

(see [21]) The switching signal \(\theta (t)\) obeys the PDTSM if there exist two positive numbers \(\tau _{\mathfrak {p}}\) and \(T_{\mathfrak {p}}\) such that the following three conditions hold simultaneously:

\(\bullet \) There are infinite and disjoint alternating intervals, i.e., the \(\tau \)-Portion and the T-Portion.

\(\bullet \) Each disjoint \(\tau \)-Portion is not less than \(\tau _{\mathfrak {p}}\) in length with the switching signal \(\theta (t)\) being a constant value in these intervals.

\(\bullet \) The length of each disjoint T-Portion is no longer than \(T_{\mathfrak {p}}\). In the T-Portion, the switching signal \(\theta (t)\) can switch arbitrarily, but the dwell time cannot exceed \(\tau _{\mathfrak {p}}\).

Figure 2 displays the PDT switching process and the variation of the energy function, where \(t_{s_w}\), \(t_{s_w+1}\), \(t_{s_w+2}\), \(\ldots \), \(t_{s_{w+1}+1}\) are the switching moments. Additionally, the energy function of the system in different subsystems \(\vartheta \) and \(\theta \) is represented by \(V_\vartheta \) and \(V_\theta \) at moments \(t_{s_w}\) and \(t_{s_w+1}\), respectively. The whole stage w consists of \(\tau \)-Portion and T-Portion. it is evident from Fig. 2 that the PDT switching signal is composed of numerous stages, and each stage comprises two portions: \(\tau \)-Portion and \(\tau \)-Portion. In the \(\tau \)-Portion, only one subsystem with a running time no less than dwell time \(\tau _{\mathfrak {p}}\) is activated. In the T-Portion, these switchings can be arbitrary. Furthermore, for \(w\in {\mathbb {Z}}^+\) and \(n\in {\mathbb {Z}}^+\), the total running time \({\mathcal {T}}_{RT-w}\) in a T-Portion of the w-th stage satisfies:

$$\begin{aligned} {\mathcal {T}}_{RT-w}=\sum _{n=1}^{\Upsilon \left( t_{s_w+1}, t_{s_{w+1}}\right) }T\left( t_{s_w+n}, t_{s_w+n+1}\right) \le T_{\mathfrak {p}}, \end{aligned}$$

which means that the running time of the \(T-Portion\) in stage w does not exceed \(T_{\mathfrak {p}}\).

Remark 3

The superiority of the PDTSM: DT switching signal demands that the subsystem’s running time is not less than a fixed time constant \(\tau _\textrm{DT}\); the switching times of ADT switching signal should satisfy \(N(t,l)\le \frac{l-t}{\tau _\textrm{ADT}}+N_0\), where l, \(\tau _\textrm{ADT}\) and \(N_0\) are positive numbers, which leads to the switching frequency of ADT switching mechanism is limited. Furthermore, if the T-Portion of the PDTSM is reduced to zero, the PDT signal can be simplified to a DT signal; if the persistence period of the T-Portion and the switching frequency of the PDT signal are reasonably limited, the PDT signal can be decreased to an ADT signal. Therefore, compared with the DT and ADT switching signals, PDTSM might have stronger applicabilities, which are demonstrated by the fact that it not only can describe the switching process that can be depicted by the DT and ADT switching mechanisms, but also can characterize the fast-switching phenomenon that the DT and ADT switching mechanisms can not.

Control objectives: For the investigated closed-loop switch-ed RDSs (10) with PDTSM, the focus of this work is primarily on the design of a hysteresis quantized pointwise controller to enable the closed-loop system to achieve exponential ISS. To further elaborate the proposed control strategy, the system diagram is shown in Fig. 2.

Before deriving the main results, some useful lemmas are provided in the following.

Lemma 1

(see [22]) In the PDTSM, the times of switching in the \(T-Portion\) are limited and do not produce Zeno behavior. Specifically, the switching times C(v, t) in [v, t) with \(v>0\) satisfy the following expression:

$$\begin{aligned} 0\le C(v, t)\le \left( \left[ \frac{t-v}{T_{\mathfrak {p}}+\tau _{\mathfrak {p}}} \right] +1\right) (\text {int}[T_{\mathfrak {p}}\cdot f]+1). \end{aligned}$$

Lemma 2

(see [50]) The average switching frequency \(f_w\) of the PDTSM has the following form:

$$\begin{aligned} f_w=\frac{C\left( t_{s_w+1}, t_{s_{w+1}}\right) }{T_{w}}, r\in {\mathbb {Z}}^+, \end{aligned}$$

where \(1/f_w\), \(T_{w}\) denote the each switching’s average length, the running time of the subsystem in the w-th stage, respectively. \(C(t_{s_w+1}, t_{s_{w+1}})\) represents the switching times in the T-Portion. \(f\triangleq \max \{f_w\}\), \(1/f\triangleq \min \{1/f_w\}\le \max \{1/f_w\}<\tau _{\mathfrak {p}}\).

Main results

In this section, we first perform stability analysis for the target systems (10). Then, the control objectives are completed based on the obtained results.

Stability analysis

Theorem 3

Preset positive scalars \(\alpha \), \({\Delta }_p\), \(g_p\), \(\lambda _j\), \(\delta _j\), \(T_{\mathfrak {p}}\), f, \(\tau _{\mathfrak {p}}\), and \(\gamma >1\), \(0<\varrho <1\), and provided that there exist positive matrices \(K_{jp}\), \(P_j\), \(j, i \in {\mathbb {J}} (j\ne i)\) and \(p\in {\mathbb {P}}\) such that

$$\begin{aligned} {\Xi }_j&=\left[ \begin{array}{ccc} {\Psi }_{1j} &{} {\Psi }_{2j} &{} {\Psi }_{4j} \\ * &{} {\Psi }_{3j} &{} \varrho ^2\lambda _j \\ * &{} * &{} -\varrho ^2\lambda _j I \end{array}\right] <0,\\ \end{aligned}$$

where

$$\begin{aligned} {\Psi }_{1j}= & {} 2 P_j {\bar{\Lambda }}_j+\alpha P_j-\frac{1}{{\Delta }_p} 2(1-\varrho ) P_j g_p K_{j p}+2 \lambda _j \delta _j^2-\varrho ^2\lambda _j,\\ {\Psi }_{2j}= & {} -\frac{1}{{\Delta }_p} P_j g_p K_{j p}+2 \lambda _j \delta _j^2-\varrho ^2\lambda _j,\\ {\Psi }_{3j}= & {} -2 \frac{\pi ^2}{{\Delta }_p^2} P_j {\Upsilon }_j+2 \lambda _j \delta _j^2-\varrho ^2\lambda _j,\\ {\Psi }_{4j}= & {} -\varrho \frac{1}{{\Delta }_p} P_j g_p K_{j p}+\varrho ^2\lambda _j,\\ P_j\le & {} \gamma P_i, \end{aligned}$$

then the closed-loop switched RDSs (10) can achieve exponential ISS for the PDTSM satisfying

$$\begin{aligned} \beta =(T_{\mathfrak {p}}f+1)\ln \gamma -\alpha \left( \frac{1}{f}+\tau _{{\mathfrak {p}}}\right) < 0. \end{aligned}$$

Proof

Step 1. Constructing the following Lyapunov function for stability analysis:

$$\begin{aligned} V_j(t)=\int _{l_1}^{l_2}\phi ^T(x,t)P_j\phi (x,t)\textrm{d}x. \end{aligned}$$
(11)

Differentiating \(V_j(t)\) in (11) with respect to time t and combining (10), the following expression is obtained

$$\begin{aligned} \begin{aligned} \dot{V}_j(t)&=2\int _{l_1}^{l_2}\phi ^T(x,t)P_j\left\{ {\Upsilon }_j\phi _{xx}(x,t)+{\Lambda }_j(x)\phi (x,t)\right. \\&\quad \left. - g_p(x)K_{jp}\left\{ (1-\varrho )\phi ({{\bar{x}}}_p,t)+\varrho e({{\bar{x}}}_p,t)\right\} \right\} \textrm{d}x. \end{aligned}\nonumber \\ \end{aligned}$$
(12)

According to the boundary conditions (3) and the integration by parts, we get

$$\begin{aligned} \begin{aligned}&2\int _{l_1}^{l_2}\phi ^T(x,t)P_j{\Upsilon }_j\phi _{xx}(x,t)\textrm{d}x \\&\quad =-2\int _{l_1}^{l_2}\phi ^T_x(x,t)P_j{\Upsilon }_j\phi _x(x,t)\textrm{d}x. \end{aligned} \end{aligned}$$
(13)

By defining \(m(x,t)=\phi ({{\bar{x}}}_p,t)-\phi (x,t)\) and based on Wirtinger’s inequalities, it follows that

$$\begin{aligned} \begin{aligned}&-2\int _{l_1}^{l_2}\phi ^T_x(x,t)P_j{\Upsilon }_j\phi _x(x,t)d\textrm{d}x\\&\quad \le -2\sum \limits _{p = 0}^{L-1} \frac{\pi ^2}{{\Delta }_p^2}\int _{x_p}^{x_{p+1}}m^T(x,t)P_j{\Upsilon }_j m(x,t)\textrm{d}x. \end{aligned} \end{aligned}$$
(14)

Similar to (14), the following expression is generated

$$\begin{aligned}{} & {} -2\sum \limits _{p = 0}^{L-1}\frac{1}{{\Delta }_p}\int _{x_p}^{x_{p + 1}}\phi ^T(x,t)P_jg_pK_{jp} \nonumber \\{} & {} \qquad \times \left\{ (1-\varrho )\phi \left( {{\bar{x}}}_p,t\right) +\varrho e\left( {{\bar{x}}}_p,t\right) \right\} \textrm{d}x \nonumber \\{} & {} \quad =-2\sum \limits _{p = 0}^{L-1} \frac{1}{{\Delta }_p}\int _{x_p}^{x_{p + 1}}\phi ^T(x,t)P_jg_pK_{jp}\left\{ (1-\varrho )m(x,t)\right. \nonumber \\{} & {} \qquad \left. +(1-\varrho )\phi (x,t)+\varrho e\left( {{\bar{x}}}_p,t\right) \right\} \textrm{d}x. \end{aligned}$$
(15)

Then, along with (7) and integrating (11)–(15), one can derive

$$\begin{aligned} \dot{V}_j(t)+\alpha V_j(t)\le \sum \limits _{p = 0}^{L-1} \int _{x_p}^{x_{p + 1}}\varsigma ^T{\Xi }_j\varsigma \textrm{d}x +2\lambda _j\zeta _j^2, \end{aligned}$$
(16)

where \(\varsigma =col[\phi (x,t),m(x,t),e({{\bar{x}}}_p,t)]\).

Step 2. Designing proper PDT switching signal.

Considering \({\Xi }_j<0\), (16) ensures that

$$\begin{aligned} \dot{V}_j(t)+\alpha V_j(t)\le 2\lambda _j\zeta _j^2. \end{aligned}$$
(17)

Using Lemma 4 in [51] and making use of the fact that \(P_j\le \gamma P_i\) we can get

$$\begin{aligned} V_{\theta \left( t_{s_{w+1}-1}\right) }(t)\le & {} \gamma e^{-\alpha \left( t-t_{s_{w+1}-1}\right) }V_{\theta \left( t_{s_{w+1}- 2}\right) }\left( t_{s_{w+1}-1}\right) \nonumber \\{} & {} +\int _{t_{s_{w+1}-1}}^t e^{-\alpha \left( t-{\mathfrak {v}}\right) } 2{\lambda _j}\zeta _j^2\textrm{d}{\mathfrak {v}}. \end{aligned}$$
(18)

Analogous to (18), the following expression is produced

$$\begin{aligned} V_{\theta \left( t_{s_{w+1}-2}\right) }\left( t_{{s_{w+1}-1}}\right)\le & {} e^{-\alpha \left( t_{s_{w+1}-1}-t_{s_{w+1}- 2}\right) } V_{\theta \left( t_{s_{w+1}-2}\right) }\left( t_{s_{w+1}-2}\right) \nonumber \\{} & {} +\int _{t_{s_{w+1}-2}}^{t_{s_{w+1}-1}}e^{-\alpha \left( t-{\mathfrak {v}}\right) }2\lambda _j\zeta _j^2\textrm{d}{\mathfrak {v}}. \end{aligned}$$
(19)

Substituting (19) into (18), we derive that

$$\begin{aligned} V_{\theta \left( t_{s_{w+1}-1}\right) }(t)\le & {} \gamma e^{-\alpha \left( t-t_{s_{w+1}- 2}\right) }V_{\theta \left( t_{s_{w + 1} - 2}\right) }\left( t_{s_{w + 1} - 2}\right) \nonumber \\{} & {} + \gamma \int _{t_{s_{w + 1} - 2}}^{t_{s_{w + 1} - 1}} e^{- \alpha \left( t - {\mathfrak {v}}\right) } 2\lambda _j\zeta _j^2\textrm{d}{\mathfrak {v}}\nonumber \\{} & {} + \int _{t_{s_{w + 1} - 1}}^t e^{ - \alpha \left( t - {\mathfrak {v}}\right) } 2{\lambda _j}\zeta _j^2\textrm{d}{\mathfrak {v}}. \end{aligned}$$
(20)

Then, iterating (20) from t to \(t_{s_w}\), we can get

$$\begin{aligned} V_{\theta \left( t_{s_{w + 1} - 1}\right) }(t)\le & {} \gamma ^{C\left( t_{s_w},t\right) }e^{-\alpha \left( t - t_w\right) }V_{\theta \left( t_{s_w}\right) }\left( t_{s_w}\right) \nonumber \\{} & {} + \gamma ^{C\left( t_{s_w},t\right) }\int _{t_{s_w}}^t e^{ - \alpha \left( t - {\mathfrak {v}}\right) } 2\lambda _j\zeta _j^2\textrm{d}{\mathfrak {v}}\nonumber \\\le & {} \gamma ^{C\left( t_{s_w},t\right) }e^{ - \alpha \left( t - t_{s_w}\right) }V_{\theta \left( t_{s_w}\right) }\left( t_{s_w}\right) \nonumber \\{} & {} +2\gamma ^{C\left( t_{s_w},t\right) }\alpha ^{-1}\lambda _j\zeta _j^2, \end{aligned}$$
(21)

where \(C(t_{s_w}, t)\) denotes the switching times in \([t_{s_w}, t)\).

Next, define the two variables \(T_w\) and \(\tau ^{\mathfrak {p}}\) in the w stage:

$$\begin{aligned} \begin{aligned} \left\{ {\begin{array}{*{20}{c}} T_w\triangleq t-t_{s_w+ 1},\\ \tau ^{\mathfrak {p}}\triangleq t_{s_w+1}-t_{s_w}. \end{array}} \right. \end{aligned} \end{aligned}$$
(22)

Combining Lemmas 1–2 and (22), the following inequality is given by

$$\begin{aligned} V_{\theta \left( t_{s_{w+1}-1}\right) }(t)\le & {} \gamma ^{T_{\mathfrak {p}}f+1}e^{-\alpha \left( \frac{1}{f}+\tau _{\mathfrak {p}}\right) }V_{\theta \left( t_{s_w}\right) }\left( t_{s_w}\right) \nonumber \\{} & {} +2\gamma ^{T_{\mathfrak {p}}f+1}\alpha ^{-1}\lambda _j\zeta _j^2. \end{aligned}$$
(23)

To simplify the representation, the following definitions are provided

$$\begin{aligned} \gamma ^{T_{\mathfrak {p}}f+1}e^{-\alpha \left( \frac{1}{f}+\tau _{\mathfrak {p}}\right) }\triangleq e^\beta < 1, \end{aligned}$$
(24)

then, by a series of iterations and recursions, the following expressions hold

$$\begin{aligned} V_{\theta \left( t_{s_{w+1}-1}\right) }(t)\le & {} \frac{1-e^{(w+1)\beta }}{1-e^\beta }\gamma ^{T_{\mathfrak {p}}f+1}\alpha ^{- 1}2\lambda _j\zeta _j^2 \nonumber \\{} & {} +e^{w\beta }V_{\theta \left( t_{s_1}\right) }\left( t_{s_1}\right) . \end{aligned}$$
(25)

Define \({\tilde{\lambda }}(P_j)\triangleq \mathop {\min }\nolimits _{j \in {\mathbb {J}}} \{\lambda _{\min }(P_j)\}\). According to (11), we can deduce that

$$\begin{aligned} V_j(t)&\ge {{\tilde{\lambda }}}\left( P_j\right) \int _{l_1}^{l_2}\phi ^T (x,t)\phi (x,t)\textrm{d}x \\&= {\tilde{\lambda }} \left( P_j\right) ||\phi (x,t)||^2, \forall j\in {\mathbb {J}}, \end{aligned}$$

Furthermore, due to \(\theta (t_{{s_{w+1} -1}})\in {\mathbb {J}}\), we can get

$$\begin{aligned} \begin{aligned} V_{\theta \left( t_{s_{w + 1}-1}\right) }(t) \ge {\tilde{\lambda }}(P_j)||\phi (x,t)||^2. \end{aligned} \end{aligned}$$
(26)

Now, combining (25) and (26), we obtain the state of the system satisfies

$$\begin{aligned} ||\phi (x,t)||^2\le & {} {\tilde{\lambda }}^{-1}(P_j)\left\{ (e^\beta )^w\bar{\lambda }(P_j)||\phi (x,0)||^2\right. \nonumber \\{} & {} \left. + \frac{1 -(e^\beta )^{w+1}}{1-e^{2\beta }}\gamma ^{T_{\mathfrak {p}}f+1}\alpha ^{-1}2{\lambda _j}\zeta _j^2\right\} . \end{aligned}$$
(27)

Inspired by [51], it can be concluded that

$$\begin{aligned} \frac{t-t_{s_1}}{T_{\mathfrak {p}}+\tau _{\mathfrak {p}}}+1\le w. \end{aligned}$$
(28)

Therefore, substituting (28) into (27), we get

$$\begin{aligned} ||\phi (x,t)||^2\le & {} {\tilde{\lambda }}^{-1}(P_j)\left\{ e^{\beta (0.25T_{\mathfrak {p}}t+0.5)}\bar{\lambda }(P_j)||v(x,0)||^2\right. \nonumber \\{} & {} \left. +\frac{1}{1-{e^\beta }}\gamma ^{T_{\mathfrak {p}}f+1}\alpha ^{-1}2\lambda _j\zeta _j^2\right\} . \end{aligned}$$
(29)

Based on (11), the following equation can be generated

$$\begin{aligned} {{\tilde{\lambda }}} \left( P_j\right) ||\phi (x,t)||^2\le V_j(t)\le \bar{\lambda }(P_j)||\phi (x,t)||^2. \end{aligned}$$
(30)

where \(\bar{\lambda }(P_j)\triangleq \mathop {\max }\nolimits _{j \in {\mathbb {J}}} \{\lambda _{\min }(P_j)\}\).

By defining \({\hat{\lambda }}(P_j)\triangleq \bar{\lambda }(P_j){{\tilde{\lambda }}}^{-1}(P_j)\) and combining (30)–(31), one obtains

$$\begin{aligned} V_j(t)\le & {} \hat{\lambda }(P_j)\left\{ e^{\beta (0.25T_{\mathfrak {p}}t+0.5)}\hat{\lambda }(P_j)V_j(0)\right. \nonumber \\{} & {} \left. +\frac{1}{1-{e^\beta }}\gamma ^{T_{\mathfrak {p}}f+1}\alpha ^{-1}2\lambda _j\zeta _j^2\right\} . \end{aligned}$$
(31)

Therefore, based on (31) and the definition of exponential ISS in [49], it is concluded that the closed-loop switched RDSs (10) with PDTSM are exponential ISS. The proof is completed. \(\blacksquare \)

Controller design

Notably, Theorem 3 only offers sufficient conditions for the exponential ISS of the closed-loop switched RDSs (10). To derive the controller’s gain parameters, the following Theorem can be drawn by applying the decoupling technique.

To this end, by defining \({\tilde{K}}_{j p}=P_j K_{j p}\), we can get the controller’s gain parameter satisfies

$$\begin{aligned} K_{j p}={\tilde{K}}_{j p} P_j^{-1}, j\in {\mathbb {J}}, p\in {\mathbb {P}}. \end{aligned}$$

Employing (35), the controller gains can be yielded through solving the inequalities in Theorem 4.

Theorem 4

Given positive constants \(\alpha \), f, \({\Delta }_p\), \(g_p\), \(\lambda _j\), \(\delta _j\), \(T_{\mathfrak {p}}\), \(\tau _{\mathfrak {p}}\), and \(\gamma >1\), \(0<\varrho <1\), the system (10) is exponential ISS, if there exist matrices \({\tilde{K}}_{jp}>0\), \(P_j>0\), \( j, i \in {\mathbb {J}} (j\ne i)\) and \(p\in {\mathbb {P}}\), the following inequalities hold:

$$\begin{aligned} \begin{aligned} {{\tilde{\Xi }}}_j&=\left[ \begin{array}{ccc} {{\tilde{\Psi }}}_{1j} &{} {{\tilde{\Psi }}}_{2j} &{} {{\tilde{\Psi }}}_{4j}\\ * &{} {\Psi }_{3j} &{} \varrho ^2\lambda _j\\ * &{} * &{} -\varrho ^2\lambda _j I \end{array}\right]<0,\\ \beta&=\left( T_{\mathfrak {p}}f+1\right) \ln \gamma -\alpha \left( \frac{1}{f}+\tau _{{\mathfrak {p}}}\right) < 0, \end{aligned} \end{aligned}$$

with

$$\begin{aligned} \begin{aligned} {\tilde{\Psi }}_{1j}&= 2 P_j {\bar{\Lambda }}_j+\alpha P_j-\frac{1}{{\Delta }_p} 2 (1-\varrho ) g_p {\tilde{K}}_{j p}\\&\quad +2 \lambda _j \delta _j^2-\varrho ^2 \lambda _j,\\ {{\tilde{\Psi }}}_{2j}&=-\frac{1}{{\Delta }_p} g_p {\tilde{K}}_{j p}+2 \lambda _j \delta _j^2-\varrho ^2\lambda _j,\\ {{\tilde{\Psi }}}_{4j}&=-\frac{1}{{\Delta }_p} g_p {\tilde{K}}_{j p}+\varrho ^2 \lambda _j,\\ P_j&\le \gamma P_i, \end{aligned} \end{aligned}$$

where \({\Psi }_{3j}\) is given in Theorem 3 and the controller’s gain matrix satisfies \(K_{j p}={\tilde{K}}_{j p} P_j^{-1}\).

Chip temperature control

To validate the effectiveness of the proposed control method, an application example i.e., temperature control of power semiconductor chips is given in this section.

Fig. 3
figure 3

Structure diagram of chips

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The considered power semiconductor chip consists of two DMOS-arrays as shown in Fig. 3. Generally, the internal structure of these two DMOS-arrays can be disregarded for simplicity [12]. The epitaxial layer is assumed to be the only heating layer within the DMOS-array. The heat generated from the chips regions \(T_1\) and \(T_2\) propagates downward in its solid-state material. Thus, the temperature system of the power semiconductor chip can be described with the following RDS model:

$$\begin{aligned} \Re _t(x,t)={\Upsilon }\Re _{xx}(x,t)+{\Lambda }(x)\Re (x,t)+g(x)u, \end{aligned}$$
(32)

subject to the initial condition

$$\begin{aligned} \Re (x,0)=\Re _0(x), \end{aligned}$$
(33)

and boundary conditions

$$\begin{aligned} \Re _x(l_1,t)=0,\Re _x(l_2,t)=0, \end{aligned}$$
(34)

where state \(\Re (x,t)\) of the system represents the temperature of the semiconductor power chip. \({\Upsilon }\) and \({\Lambda }(x)\) denote the diffusion coefficient and thermal resistance of the chip with \({\Upsilon }=\varpi /(\rho \cdot c_s)\) and \({\Lambda }(x)={\mathfrak {T}}/(\varpi \cdot a(x))\), respectively and their parameters are detailed in Table 1. Further, according to the thermal resistance formula, we can discover that the thermal resistance value is inversely proportional to the chip’s area.

Table 1 Model parameters
Full size table

Next, we consider that \(T_1\) or \(T_2\) generates heat. Since the heat conductivity \(\varpi \) is related to the physical structure of \(T_1\) and \(T_2\), the changes of \(T_1\) and \(T_2\) result in the changes of \(\varpi \). Furthermore, since the material and shape of the chip are fixed, the variations of \(\rho \), \(c_s\), \({\mathfrak {T}}\) and a(x), which are related to the material or shape of the chip, can be ignored. Therefore, \({\Upsilon }\) and \({\Lambda }(x)\) with the switching characteristics are rewritten as \({\Upsilon }_{\theta (t)}=\varpi _{\theta (t)}/(\rho \cdot c_s)\) and \({\Lambda }_{\theta (t)}(x)={\mathfrak {T}}/(\varpi _{\theta (t)}\cdot a(x))\), respectively. Therefore, system (32) can be re-modeled to the switched RDS as follows

$$\begin{aligned} \Re _t(x,t)={\Upsilon }_{\theta (t)}\Re _{xx}(x,t)+{\Lambda }_{\theta (t)}(x)\Re (x,t)+g(x)u_{\theta (t)}, \end{aligned}$$
(35)

Inspired by [12], the parameters of the system (35) are chosen as follows: The number of system’s modes is chosen as 2, where the change of the modes is shown in Fig. 4. Next, the coefficients of the system corresponding to each mode are configured. When the mode is 1, \({\Upsilon }_1=1\), \({\Lambda }_1(x)=5exp(-0.1x^2)+0.2\); when the mode is 2, \({\Upsilon }_2=2\), \({\Lambda }_2(x)=2exp(-x^2)+1\). Select the initial condition of the system (35) satisfying \(\Re _0(x)=0.1+cos(\pi x)\). The corresponding open-loop system’s state evolution trajectory is shown in Figure. 5, where the state of the system is remarkably divergent.

Fig. 4
figure 4

Diagram of modal change of PDTSM

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Fig. 5
figure 5

Evolution trajectory of open-loop state

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Therefore, based on the above analysis, it is necessary and meaningful to impose control on the investigated switched RDS (35). Initially, the space length \(l_2-l_1=1\) is divided into five intervals spaced by 0.2, then the midpoint of each interval are \({{\bar{x}}}_1=0.1\), \({{\bar{x}}}_2=0.3\), \({{\bar{x}}}_3=0.5\), \({{\bar{x}}}_4=0.7\), \({{\bar{x}}}_5=0.9\). Subsequently, by selecting parameters \(j=2\), \(p=5\), \(\alpha =0.6\), \(f=5\), \(\gamma =1.01\), \(\Delta _p=0.2\), \(g_p=63.5\), \(\lambda _1=0.03\), \(\lambda _2=0.035\), \(T_{\mathfrak {p}}=4\), \(\tau _{\mathfrak {p}}=3\), \(\nu _1=0.002\), \(\nu _2=0.0025\), \(\zeta _1=0.002\), \(\zeta _2=0.0025\), \(\varrho =0.15\), we obtain the controller gains as shown in (36) by solving the linear matrix inequalities in Theorem 4.

Then, the hysteresis quantified controller (9) is applied to the system (35). The evolution trajectories of the system’s state \(\Re (x,t)\) and quantified control input (9) are shown in Figs. 6 and  7, respectively. According to Figs. 6 and  7, it can be concluded that the system (35) is rapidly stabilized.

$$\begin{aligned} {K_{jp}} = \left\{ {\begin{array}{*{20}{c}} {0.1523,}&{}{x = 0.1,p = 1,j = 1,}\\ {0.1483,}&{}{x = 0.3,p = 2,j = 1,}\\ {0.1270,}&{}{x = 0.5,p = 3,j = 1,}\\ {0.1129,}&{}{x = 0.7,p = 4,j = 1,}\\ {0.0655,}&{}{x = 0.9,p = 5,j = 1.} \end{array}} \right. \nonumber \\ {K_{jp}} = \left\{ {\begin{array}{*{20}{c}} {0.2162}&{}{x = 0.1,p = 1,j = 2,}\\ {0.1734,}&{}{x = 0.3,p = 2,j = 2,}\\ {0.1526,}&{}{x = 0.5,p = 3,j = 2,}\\ {0.1268,}&{}{x = 0.7,p = 4,j = 2,}\\ {0.0611,}&{}{x = 0.9,p = 5,j = 2.} \end{array}} \right. \end{aligned}$$
(36)
Fig. 6
figure 6

Evolution trajectory of close-loop state

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Fig. 7
figure 7

Evolution curve of control input

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Fig. 8
figure 8

Evolution curve of control input

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Next, a logarithmic quantizer is utilized to highlight the advantages of the hysteresis quantizer employed in this paper. Specifically, Fig. 8 shows the trajectory of logarithmic quantified control input. It can be observed from Fig. 8 that the logarithmic quantizer causes controller chattering, which results in a longer time to achieve system’s stability.

Conclusions

This work has addressed the stabilization problem of switch-ed RDSs. Specifically, the switchings between operating modes of the considered systems are consistent with the PDTSM, which is more general than the DT and ADT switching mechanisms. Further, by using the networked control framework of the point measurement, pointwise control and hysteresis quantizer with a adjustable parameter, the exponential ISS of the closed-loop switched RDSs is obtained. Finally, the applicability of the derived theoretical results is verified by the temperature control simulations of power semiconductor chips. The future work will expand the proposed methods to design the secure control strategy for the switched RDSs with network attacks. Meanwhile, to further improve the communication efficiency, we will extend the input quantization proposed in this paper to input–output quantization or input-state quantization.