Robust MPC for polytopic uncertain systems via a high-rate network with the round-robin scheduling

System models

Let us consider the discrete-time linear polytopic uncertain systems:

(1) $$\{\begin{array}{c}x(t_{k+1})=A(t_k)x(t_k)+B(t_k)u(t_k)\hfill \\ y(t_k)=C(t_k)x(t_k)\hfill \end{array}$$where $t_k(k\ge 0)$ represents the $(k+1)$th sampling instant, $x(t_k)\in {\mathbb{R}}^{n_x}$, $u(t_k)\in {\mathbb{R}}^{n_u}$ and $y(t_k)\in {\mathbb{R}}^{n_y}$ are the state, the control input and the measurement output, respectively. $A(t_k)$, $B(t_k)$ and $C(t_k)$ with appropriate dimensions are unknown matrices, which belong to a polytope given by

(2) $$\mathrm{\Delta }:=\{\mathrm{\Theta }|\mathrm{\Theta }=\sum _{l=1}^L{\lambda }_l{\mathrm{\Theta }}^{(l)},\sum _{l=1}^L{\lambda }_l=1,0\le {\lambda }_l\le 1\},$$where $\mathrm{\Theta }:=(A(t_k),B(t_k),C(t_k))\in \mathrm{\Delta }$, and matrices ${\mathrm{\Theta }}^{(l)}$ are previously known and defined as ${\mathrm{\Theta }}^{(l)}:=(A^{(l)},B^{(l)},C^{(l)}),l=1,2,\dots ,L$, which are the vertices in the convex hull $\mathrm{\Delta }$.

In engineering systems, consider the actual applicability of the strategy that we propose, the constraints on the inputs and states are given by

(3) $$\{\begin{array}{c}\underset{c}{max}|{[u(t_k)]}_c|\le \overline{u},c\in \{1,2,\dots ,n_u\}\\ \underset{d}{max}|{[x(t_k)]}_d|\le \overline{x},d\in \{1,2,\dots ,n_x\}\end{array}$$where $[.{]}_j$ denotes the $j$th element of a vector, $j\in \{c,d\}$. $\overline{u}>0$, $\overline{x}>0$ are known scalars.

Communication schedulings

In Fig. 1, the data are transmitted by the sensors to the controller via a shared network with high-rate communication channels. For the sake of reducing the network transmission burden and avoiding congestion, a famous RR scheduling is used to arrange the data transmission sequence. To be more specific, only one sensor node with the permission has the access to the shared network, while others maintain the previous information by zero-order-holders (ZOHs). In particular, under the RR scheduling, data are transmitted periodically in cyclic order according to a predetermined scheduling principle.

Let ${\hat{y}}_i(t_k)$ mean the measurement obtained from sensor $i$. The update of ${\hat{y}}_i(t_k)$ under the RR scheduling can be denoted by

(4) $${\hat{y}}_i(t_k)=\{\begin{array}{cc}{y}_i(t_k),\hfill & ifmod(t_{k-i},m)=0\hfill \\ {\hat{y}}_i(t_{k-1}),\hfill & otherwise\hfill \end{array}$$

The combination of ${\hat{y}}_i(t_k)$ $(i=1,2,3,...,m)$ means the information obtained from the controller at the time instant $t_k$, ${\hat{y}}_i(t_k)$ is defined by $\hat{y}(t_k)\stackrel{\mathrm{\Delta }}{=}[{\hat{y}}_1^T(t_k),{\hat{y}}_2^T(t_k),\dots ,{\hat{y}}_m^T(t_k){]}^T$.

$Remark1.$ In this article, the measurement outputs of the sensors are transmitted to the remote controller via the high-rate communication channels. Because a large number of sensors are deployed, the data collision occur when sensors transmit the measurement signals at the same time. In order to prevent data conflict, the RR scheduling is used to determine the transmission sequence of sensors in high-rate communication channels. Before the further discussion, the following assumption on the high-rate communication channels is made.

$Assumption1.$ The time interval between two adjacent transmissions in the high-rate communication channels is $t_h=t_s/d$ with $d\ge 1$ being a positive integer. The sampling period of the sensors is denoted as $t_s\stackrel{\mathrm{\Delta }}{=}{t}_{k+1}-t_k$. In addition, the first transmission occurs at the initial sampling instant $t_0$.

Denoting the $(k^{\mathrm{\prime }}+1)$th $(k^{\mathrm{\prime }}\ge 0)$ transmission time instant as $T_k^{\mathrm{\prime }}$, we have $T_{k^{\mathrm{\prime }}+1}=T_{k^{\mathrm{\prime }}}+t_h$. In addition, it is easy to know the measurement output sent by sensor $i$ at time instant $T_{k^{\mathrm{\prime }}}$ is

(5) $$y_i(T_{k^{\mathrm{\prime }}})=y_i(t_k),T_{k^{\mathrm{\prime }}}\in [t_k,t_{k+1}).$$

Now, we are going to explain the scheduling effect of RR scheduling in high-rate communication channels in detail. Let $\check{y_i}(T_{k^{\mathrm{\prime }}})$ be the actual received measurement output from sensor $i$ at time instant $T_{k^{\mathrm{\prime }}}$. From the knowledge of the RR scheduling, we have

(6) $${\check{y}}_i(T_{k^{\mathrm{\prime }}})=\{\begin{array}{c}{y}_i(T_{k^{\mathrm{\prime }}}),if\xi (T_{k^{\mathrm{\prime }}})=i;\hfill \\ {\check{y}}_i(T_{k^{\mathrm{\prime }}-1}),otherwise\hfill \end{array}$$where $\xi (T_{k^{\mathrm{\prime }}})\stackrel{\mathrm{\Delta }}{=}mod(k^{\mathrm{\prime }}-1,n)+1$ represents the sensor which is given the transmission access at time instant $T_{k^{\mathrm{\prime }}}$. Moreover, we set $\check{y_i}(T_{k^{\mathrm{\prime }}})=0$ for any $k^{\mathrm{\prime }}<0$. Denote

$$\check{y}(T_{k^{\mathrm{\prime }}})\stackrel{\mathrm{\Delta }}{=}col\{\check{y_1}(T_{k^{\mathrm{\prime }}}),\check{y_2}(T_{k^{\mathrm{\prime }}}),\dots ,\check{y_n}(T_{k^{\mathrm{\prime }}})\},$$

$$y(t_k)\stackrel{\mathrm{\Delta }}{=}col\{y_1(t_k),y_2(t_k),\dots ,y_n(t_k)\},$$

$${\mathrm{\Psi }}_{\xi (T_{k^{\mathrm{\prime }}})}\stackrel{\mathrm{\Delta }}{=}diag\{\delta (\xi (T_{k^{\mathrm{\prime }}}),1)I,\delta (\xi (T_{k^{\mathrm{\prime }}}),2)I,\dots ,\delta (\xi (T_{k^{\mathrm{\prime }}}),n)I\}$$

with $\delta$ (.,.) being the Kronecker delta function. From Eqs. (5) and (6), one has

(7) $$\check{y}(T_{k^{\mathrm{\prime }}})={\mathrm{\Psi }}_{\xi (T_{k^{\mathrm{\prime }}})}y(t_k)+(I-{\mathrm{\Psi }}_{\xi (T_{k^{\mathrm{\prime }}}))}\check{y}(T_{k^{\mathrm{\prime }}-1}),T_{k^{\mathrm{\prime }}}\in [t_k,t_{k+1}).$$

From Assumption 1, we know that there are $d$ transmissions occurring in the time interval $(t_{k-1},t_k]$. Denote the time instant when the $\beta$th transmission in the time interval $(t_{k-1},t_k]$ occurs as ${\overline{t}}_k^{\beta }\stackrel{\mathrm{\Delta }}{=}{t}_{k-1}+\beta t_h(\beta =1,2,...,d)$. From Eq. (7), we have

$$\check{y}(t_k)={\mathrm{\Psi }}_{\xi (t_k)}y(t_k)+(I-{\mathrm{\Psi }}_{\xi (t_k)})\check{y}({\overline{t}}_k^{d-1}),$$

$$\check{y}({\overline{t}}_k^{d-1})={\mathrm{\Psi }}_{\xi ({\overline{t}}_k^{d-1})}y(t_{k-1})+(I-{\mathrm{\Psi }}_{\xi ({\overline{t}}_k^{d-1})})\check{y}({\overline{t}}_k^{d-2}),$$

$$⋮$$

$$\check{y}({\overline{t}}_k^1)={\mathrm{\Psi }}_{\xi ({\overline{t}}_k^1)}y(t_{k-1})+(I-{\mathrm{\Psi }}_{\xi ({\overline{t}}_k^1)})\check{y}(t_{k-1}).$$

By iterating the above equations, we further obtain

(8) $$\check{y}(t_k)={\mathrm{\Psi }}_{\xi (t_k)}y(t_k)+{\varphi }_{\zeta ({\overline{t}}_k^d)}^{1}y(t_{k-1})+{\varphi }_{\zeta ({\overline{t}}_k^d)}^2\check{y}(t_{k-1})$$where

$$\zeta ({\overline{t}}_k^d)\stackrel{\mathrm{\Delta }}{=}(\xi ({\overline{t}}_k^1),\xi ({\overline{t}}_k^2),\dots ,\xi ({\overline{t}}_k^d)),$$

$${\varphi }_{\zeta ({\overline{t}}_k^d)}^1\stackrel{\mathrm{\Delta }}{=}{\mathrm{\Sigma }}_{i=1}^{d-1}({\mathrm{\Pi }}_{j=0}^{i-1}(I-{\mathrm{\Psi }}_{\xi ({\overline{t}}_k^{d-j})})){\mathrm{\Psi }}_{\xi ({\overline{t}}_k^{d-i})},$$

$${\varphi }_{\zeta ({\overline{t}}_k^d)}^2\stackrel{\mathrm{\Delta }}{=}{\mathrm{\Pi }}_{i=0}^{d-1}(I-{\mathrm{\Psi }}_{\xi ({\overline{t}}_k^{d-i})}).$$

$Remark2.$ It can be seen that the matrix $X_{\zeta ({\overline{t}}_k^d)}(X$ stands for ${\varphi }^1,{\varphi }^2,{\varphi }^3,\overline{\mathrm{\Psi }}$ and ${\overline{\mathrm{\Psi }}}^i)$ is dependent on $\xi ({\overline{t}}_k^1),\xi ({\overline{t}}_k^2),\dots ,\xi ({\overline{t}}_k^d)$ and therefore $X_{\zeta ({\overline{t}}_k^d)}$ should be written as $X_{\xi ({\overline{t}}_k^1),\xi ({\overline{t}}_k^2),\dots ,\xi ({\overline{t}}_k^d)}$. However, for convenience of presentation, we introduce a new notation $\zeta ({\overline{t}}_k^d)$ to represent $\xi ({\overline{t}}_k^1),\xi ({\overline{t}}_k^2),\dots ,\xi ({\overline{t}}_k^d)$, and $X_{\xi ({\overline{t}}_k^1),\xi ({\overline{t}}_k^2),\dots ,\xi ({\overline{t}}_k^d)}$ is represented by $X_{\zeta ({\overline{t}}_k^d)}$.

Problem of interests

Because it is difficult to get the states of the system in actual practice, for the underlying system (Eq. (1)), the following static output-feedback controller based on MPC is designed under the RR scheduling in the high-rate communication channel:

(9) $$u(t_k)=F_{\xi (t_k)}(t_k)\check{y}(t_k),$$

$F_{\xi (t_k)}(t_k)$ is the feedback gain to be designed by optimization.

On the basis of the control law (Eq. (9)), we can adapt the closed-loop system which is on the prediction horizon as the following:

(10) $$\chi (t_{k+1})={\overline{A}}_{\xi (t_{k+n}|k)}(t_k)\chi (t_k),$$where

$$\chi (t_k)\stackrel{\mathrm{\Delta }}{=}col\{x(t_k),y(t_{k-1}),\check{y}(t_{k-1})\},$$

$${\overline{A}}_{\xi (t_{k+n}|k)}(t_k)=\left[\begin{array}{ccc}A(t_k)+B(t_k)F_{\xi (t_k)}{\mathrm{\Psi }}_{\xi (t_k)}C(t_k)& B(t_k)F_{\xi (t_k)}{\mathrm{\Phi }}_{\zeta ({\overline{t}}_k^d)}^1& B(t_k)F_{\xi (t_k)}{\mathrm{\Phi }}_{\zeta ({\overline{t}}_k^d)}^2\\ C(t_k)& 0& 0\\ {\mathrm{\Psi }}_{\xi (t_k)}C(t_k)& {\mathrm{\Phi }}_{\zeta ({\overline{t}}_k^d)}^1& {\mathrm{\Phi }}_{\zeta ({\overline{t}}_k^d)}^2\end{array}\right]$$

The initial state is obtained by $\chi (0)=col\{x(0)y(-1)\check{y}(-1)\}$.

For the system (Eq. (10)) with polyhedral parameter uncertainties, the “min-max” problem over an infinite time horizon is used to design the controllers:

(11) $$\underset{F_{\xi (t_{k+n}|k)}}{min}\underset{(A(t_k),B(t_k),C(t_k))\in \mathrm{\Delta }}{max}{J}_{\mathrm{\infty }}(t_k),$$

we define the objective function $J_{\mathrm{\infty }}(t_k)$ by

(12) $$J_{\mathrm{\infty }}(t_k)\stackrel{\mathrm{\Delta }}{=}\sum _{n=0}^{\mathrm{\infty }}\left[{\chi }^T(t_k)Q\chi (t_k)+u^T(t_k)Ru(t_k)\right].$$

$Q_1,Q_2,Q_3$ and R denote the symmetric and positive definite weighting matrices, and $Q=diag\{Q_1,Q_2,Q_3\}$.

Based on the min-max problem (Eq. (12)), for the aim of designing the controllers in the framework of out-feedback MPC, we present the online optimization problem:

$$\mathrm{O}\mathrm{p}1:\underset{F_{\xi (t_{k+n}|k)}}{min}\underset{(A(t_k),B(t_k),C(t_k))\in \mathrm{\Delta }}{max}{J}_{\mathrm{\infty }}(t_k),$$

$$\mathrm{s}.\mathrm{t}.\underset{c}{max}|[u(t_k){]}_c|\le \overline{u},$$

$$\mathrm{s}.\mathrm{t}.\underset{d}{max}|[x(t_k){]}_d|\le \overline{x},$$

$$\chi (t_k)\in \mathrm{{\rm Y}}(P_{\xi (t_{k+n}|k)},3\rho ),$$where $\mathrm{{\rm Y}}$ is what is called a terminal constraint set, we define it by

(13) $$\mathrm{{\rm Y}}\stackrel{\mathrm{\Delta }}{=}\{\chi (t_k)|\chi {}^T(t_k)P_{\xi (t_{k+n}|k)}\chi (t_k)\le 3\rho \}$$

$P_{\xi (t_{k+n}|k)}$ denotes a positive definite matrix of aquadratic function. We will present more details in the next section. Note that only the first component $u(t_k)$ of predicted inputs $\{u(t_k),u(t_{k+1}),u(t_{k+2}),...\}$ will be worked on the plant at each time instant.

In this article, the static output-feedback controllers (Eq. (9)) are designed within the framework of RMPC to make the system (Eq. (1)) asymptotically stable under RR scheduling (Eq. (4)) in high-rate communication channels. To be more specific, an auxiliary optimization problem Op1 is provided to find the required parameter matrices $F_{\xi (t_k)}$, so that we can guarantee the stability of the closed-loop system. In order to reach this goal, we need to satisfy the following requirements for any admissible parameter $k$ at the same time:

$R1.$ an auxiliary optimization issue is supplied to denote the problem Op1, so that we can get the suboptimal solution;

$R2.$ according to the parameter matrices $F_{\xi (t_k)}$ that we obtain, the closed-loop system under the RR scheduling (Eq. (4)) in high-rate communication channels is asymptotically stable.

MPC under RR scheduling without hard constraints

This section will provide several sufficient conditions for the systems without constraints to ensure the desired performance through the quadratic function method. Accordingly, we obtain the static output-feedback controllers under the RMPC framework. To be exact, firstly, sufficient conditions are given to meet the condition of the terminal constraint set in Op1, i.e., $\chi (t_k)\in \mathrm{{\rm Y}}(P_{\xi (t_{k+n}|k)},3\rho )$. After that, an auxiliary optimization problem is proposed to find the suboptimal solution for the unconstrained system. In addition, the inequation analysis technology is used to deal with the unavailable state $x(t_k)$ problem of the auxiliary problem, we provide another auxiliary problem for the solvability. In the end, by solving this kind of online auxiliary optimization issue, we obtain sufficient conditions to guarantee the stability of the closed-loop system.

Terminal constraint set

Before the main results are developed, we first present the significant definition.

$Definition1.$ For system (Eq. (1)) under the control law (Eq. (9)), the set $\mathrm{{\rm Y}}$ is a robust positive invariant (RPI) set if $\chi (t_k)\in \mathrm{{\rm Y}}$ implies $\chi (t_{k+1})\in \mathrm{{\rm Y}}$.

Based on the online optimization problem Op1, we need to satisfy the following two conditions such that the set $\mathrm{{\rm Y}}(P_{\xi (t_{k+n}|k)},3\rho )$ is the terminal constraint set for Op1:

C1: we define a quadratic function by

(14) $$V(\chi (t_{k+n}|k))\stackrel{\mathrm{\Delta }}{=}{\chi }^T(t_{k+n}|k)P_{\xi (t_{k+n}|k)}\chi (t_{k+n}|k),$$

such that

$$V=V(\chi (t_{k+n+1}|k))-V(\chi (t_{k+n}|k))$$

(15) $$V\le -\chi {}^T(t_{k+n}|k)\}Q\chi (t_{k+n}|k)\}-u^T(t_{k+n}|k)Ru(t_{k+n}|k).$$

C2: the set $\mathrm{{\rm Y}}(P_{\xi (t_{k+n}|k)},3\rho )$ is an RPI set.

Next, we will discuss the above conditions. For the sake of simplicity, we define $\xi (t_{k+n}|k)=r$, $\xi (t_{k+n+1}|k)=t$.

The following theorem is shown to ensure the condition C1 of terminal constraint set.

$Lemma1.$ Let us give symmetric positive definite matrices $Q_1$, $Q_2$, $Q_3$ and R. For the system (Eq. (10)) which is controlled by Eq. (9), if there is a positive scalar $\rho >0$, symmetric and positive definite matrices ${\tilde{Q}}_{ir}$, ${\tilde{Q}}_{it}(i=1,2,3)$ and matrices $Y_r$, for any $(r,t)\in \mathbb{S}\times \mathbb{S}$, $l=1,2,...,L$, such that the following conditions are established:

(16) $$\left[\begin{array}{cccc}{\hat{Q}}_r^l& \ast & \ast & \ast \\ {\hat{A}}_r^l& {\tilde{Q}}_t& \ast & \ast \\ \sqrt{R}\overline{M}& 0& \rho I& \ast \\ \mathrm{\Xi }& 0& 0& \overline{\rho }I\end{array}\right]\ge 0,$$where

$$\overline{\rho }I=\left[\begin{array}{ccc}\rho I& 0& 0\\ 0& \rho I& 0\\ 0& 0& \rho I\end{array}\right],$$

$$\overline{M}=F_r\left[\begin{array}{ccc}{\mathrm{\Psi }}_{\xi (t_k)}{C}^l{T}^{l}S& {\mathrm{\Phi }}_{\zeta ({\overline{t}}_k^d)}^1{S}_{11}& {\mathrm{\Phi }}_{\zeta ({\overline{t}}_k^d)}^2{S}_{11}\end{array}\right],$$

$${\hat{Q}}_r^l=\left[\begin{array}{ccc}{\overline{Q}}_{1r}& 0& 0\\ 0& {\overline{Q}}_{2r}& 0\\ 0& 0& {\overline{Q}}_{3r}\end{array}\right],$$

$${\hat{A}}_r^l=\left[\begin{array}{ccc}{A}^l{T}^{l}S+B^l\mathrm{\Psi }{Y}_r& B^l{\mathrm{\Phi }}^1{Y}_r& B^l{\mathrm{\Phi }}^2{Y}_r\\ C^l{T}^{l}S& 0& 0\\ \mathrm{\Psi }{C}^l{T}^{l}S& {\mathrm{\Phi }}^1{S}_{11}& {\mathrm{\Phi }}^2{S}_{11}\end{array}\right],$$

$$\mathrm{\Xi }=\left[\begin{array}{ccc}\sqrt{Q_1}{T}^{l}S& 0& 0\\ 0& \sqrt{Q_2}{S}_{11}& 0\\ 0& 0& \sqrt{Q_3}{S}_{11}\end{array}\right],$$

$${\overline{Q}}_{1r}=(T^{l}S{)}^T+T^{l}S-{\tilde{Q}}_{1r},$$

$${\overline{Q}}_{2r}=(S_{11}{)}^T+S_{11}-{\tilde{Q}}_{2r},$$

$${\overline{Q}}_{3r}=(S_{11}{)}^T+S_{11}-{\tilde{Q}}_{3r},$$

$$Y_r=F_r{S}_{11},$$

and we have Eq. (15) with $V(t_{k+n})$ by Eq. (14). In addition, the related output-feedback gains in the control law (Eq. (9)) are given by

(17) $$F_r=Y_r{S_{11}}^{-1}.$$

Proof: We can choose the token-dependent quadratic function that is defined by Eq. (14), i.e.,

(18) $$V(\chi (t_{k+n}|k))={\chi }^T(t_{k+n}|k)P_r\chi (t_{k+n}|k),$$where $P_r=diag\{P_{ir}\}(i=1,2,3)$ is the symmetric positive definite matrix that we wil design.

Calculate the difference of Eq. (14) along the trajectory of the system (Eq. (10)) yields.

$$\mathrm{\Delta }V(\chi (t_{k+n}|k))=V(\chi (t_{k+n+1}|k))-V(\chi (t_{k+n}|k))$$

$$={\chi }^T(t_{k+n+1}|k)P_t\chi (t_{k+n+1}|k)-{\chi }^T(t_{k+n}|k)P_r\chi (t_{k+n}|k)$$

(19) $$={\chi }^T(t_{k+n}|k)({\overline{A}}_r^T(t_k)P_t{\overline{A}}_r(t_k)-P_r)\chi (t_{k+n}|k)$$

We denote $T^l=[(C^l{)}^T(C^l(C^l{)}^T{)}^{-1}(C^l{)}^{\mathrm{\perp }}]$, in which $(C^l{)}^{\mathrm{\perp }}$ represents the orthogonal basis of the null space for $C^l$, and we introduce a free matrix:

(20) $$S=\left[\begin{array}{cc}{S}_{11}& 0\\ 0& S_{22}\end{array}\right],$$

$S_{11}$ is the arbitrary diagonal matrix, $S_{22}$ is the arbitrary matrix with appropriate dimension.

Substitute the following conditions $(i=2,3)$

$$\begin{gathered} T^{l}S+(T^{l}S{)}^T-{\tilde{Q}}_{1r}-(T^{l}S{)}^T{\tilde{Q}}_{1r}^{-1}{T}^{l}S \\ =-({\tilde{Q}}_{1r}-T^{l}S){\tilde{Q}}_{1r}^{-1}({\tilde{Q}}_{1r}-T^{l}S{)}^T\le 0, \end{gathered}$$

$$\begin{gathered} S_{11}+(S_{11}{)}^T-{\tilde{Q}}_{ir}-(S_{11}{)}^T{\tilde{Q}}_{ir}^{-1}{S}_{11} \\ =-({\tilde{Q}}_{ir}-S_{11}){\tilde{Q}}_{ir}^{-1}({\tilde{Q}}_{ir}-S_{11}{)}^T\le 0 \end{gathered}$$

into Eq. (16), we will get

(21) $$\left[\begin{array}{cccc}{\check{Q}}_r^l& \ast & \ast & \ast \\ {\hat{A}}_r^l& {\tilde{Q}}_t& \ast & \ast \\ \sqrt{R}\overline{M}& 0& \rho I& \ast \\ \mathrm{\Xi }& 0& 0& \overline{\rho }I\end{array}\right]\ge 0,$$where

$${\check{Q}}_r^l=\left[\begin{array}{ccc}{(T^{l}S)}^T{\tilde{Q}}_{1r}^{-1}(T^{l}S)& 0& 0\\ 0& {(S_{11})}^T{\tilde{Q}}_{2r}^{-1}(S_{11})& 0\\ 0& 0& {(S_{11})}^T{\tilde{Q}}_{3r}^{-1}(S_{11})\end{array}\right].$$

If pre- and post-multiplying the following inequalities with diag $\{(T^{l}S{)}^T,(S_{11}{)}^T,(S_{11}{)}^T,\underset{7}{\underbrace{I,...,I}}\}$ and its transpose, Eq. (21) will be obtained:

(22) $$\left[\begin{array}{cccc}{\tilde{Q}}_r^{-1}& \ast & \ast & \ast \\ {\overline{A}}_r^l& {\tilde{Q}}_t& \ast & \ast \\ \sqrt{R}M& 0& \rho I& \ast \\ \sqrt{Q}& 0& 0& \overline{\rho }I\end{array}\right]\ge 0,$$where $Q=diag\{Q_1,Q_2,Q_3\}$,

$${\tilde{Q}}_r^{-1}=\left[\begin{array}{ccc}{\tilde{Q}}_{1r}^{-1}& 0& 0\\ 0& {\tilde{Q}}_{2r}^{-1}& 0\\ 0& 0& {\tilde{Q}}_{3r}^{-1}\end{array}\right],$$

$${\overline{A}}_r^l=\left[\begin{array}{ccc}{A}^l+B^l{F}_r\mathrm{\Psi }{C}^l& B^l{F}_r{\mathrm{\Phi }}^1& B^l{F}_r{\mathrm{\Phi }}^2\\ C^l& 0& 0\\ \mathrm{\Psi }{C}^l& {\mathrm{\Phi }}^1& {\mathrm{\Phi }}^2\end{array}\right],$$

$$M=F_r\left[\begin{array}{ccc}{\mathrm{\Psi }}_{\xi (t_k)}{C}^l& {\mathrm{\Phi }}_{\zeta ({\overline{t}}_k^d)}^1& {\mathrm{\Phi }}_{\zeta ({\overline{t}}_k^d)}^2\end{array}\right].$$

Because the system (Eq. (10)) is polytopic uncertain, i.e., inequality Eq. (22) is affine in $\mathrm{\Omega }$, Eq. (22) means that

(23) $$\left[\begin{array}{cccc}{\tilde{Q}}_r^{-1}& \ast & \ast & \ast \\ {\overline{A}}_r(k)& {\tilde{Q}}_t& \ast & \ast \\ \sqrt{R}M& 0& \rho I& \ast \\ \sqrt{Q}& 0& 0& \overline{\rho }I\end{array}\right]\ge 0.$$

According to the Schur Complement, we can deduce from Eq. (23) that

(24) $${\tilde{Q}}_r^{-1}-{\overline{A}}_r^T(t_k){\tilde{Q}}_t^{-1}{\overline{A}}_r(t_k)-M^T{\rho }^{-1}RM-{\rho }^{-1}Q\ge 0.$$

Multiply both sides of Eq. (24) with $\rho >0$ and define $P_j=\rho {\tilde{Q}}_j^{-1}(j\in \{r,t\})$, we can get

(25) $$P_r-{\overline{A}}_r^T(t_k)P_t{\overline{A}}_r(t_k)-M^{T}RM-Q\ge 0.$$

Pre- and post-multiplying Eq. (25) with ${\chi }^T(t_k)$ and its transpose means

(26) $${\chi }^T(t_k)P_r\chi (t_k)-{\nu }_r(t_k)-{\omega }_r-{\chi }^T(t_k)Q\chi (t_k)\ge 0.$$where

$${\nu }_r(t_k)=({\overline{A}}_r(t_k)\chi (t_k){)}^T{P}_t({\overline{A}}_r(t_k)\chi (t_k)),$$

$${\omega }_r=(M\chi (t_k){)}^{T}R(M\chi (t_k)).$$

It is noted that we have Eqs. (9), (10) and (19), then the condition (Eq. (15)) can be ensured by Eq. (26). So the proof is complete.

$Remark4.$ Despite the utilization of RR scheduling in the high-rate communication channel can avoid data conflict and network congestion, many side effects may happen because of the change of data transmission sequence. To deal with such a barrier, we employ the token-dependent Lyapunnov-like method. So the effect of the underlying RR scheduling can be reflected well in the design of the controller in this article. In Zou, Wang & Gao (2016a), Song et al. (2019), Zou et al. (2017b), to choose the token-dependent Lyapunov function for systems with various communication schedulings, a technical approach is proposed.

$Remark5.$ Of course, our proposed approach has some limitations. First, the RR protocol is a static protocol, only a part of the information can be updated at every time instant. So the amount of data in the system at each time will be relatively small, and the accuracy will also be affected. Second, the value of d cannot be too large in high-rate communication channels, which can be improved and optimized additionally.

Next, we will study the condition C2 of the terminal constraint set. Namely, all we have to do is to obtain the sufficient conditions, which can satisfy that the set $\mathrm{{\rm Y}}(P_r,3\rho )$ is an RPI set.

Based on Definition 1, we need to satisfy the following two requirements to ensure the RPI set $\mathrm{{\rm Y}}(P_r,3\rho )$:

$R1.$ at the time instant $n=0$, the initial state belongs to the set $\mathrm{{\rm Y}}$, i.e.,

$$\chi {}^T(t_k){\tilde{Q}}_r^{-1}\chi (t_k)\le 3;$$

$R2.$ the future states $\chi (t_{k+n}|k),n>0$ are part of the set $\mathrm{{\rm Y}}$ too.

In the following content, we address above requirements one by one.

According to the Schur Complement, R1 holds if and only if

(27) $$\left[\begin{array}{cc}3& \ast \\ \chi (t_k)& {\tilde{Q}}_r\end{array}\right]\ge 0.$$

In addition, on account of Lemma 1, it is easy to see from Eq. (27) that

(28) $$V(\chi (t_{k+n+1}))\le V(\chi (t_{k+n}))\le ...\le V(\chi (t_k))\le 3\rho$$

which means that all the future states $\chi (t_{k+n}|k)$ are part of the set $\mathrm{{\rm Y}}$ so long as $\mathrm{{\rm Y}}$ includes the initial state $\chi (t_k)$. Thus we can also guarantee that the set $\mathrm{{\rm Y}}$ is an RPI set.

Up to now, by conditions Eqs. (16) and (27), the terminal constraint set can be ensured. That is to say, condition $\chi (t_{k+n}|k)\in \mathrm{{\rm Y}}(P_r,3\rho )$ of OP1 is satisfied.

Auxiliary optimization problems

For the system without constraints in this article, we discuss how to deal with the Op1.

Op1 is an optimization problem which includes parameter uncertainties over an infinite time horizon, it is very difficult to handle it directly. On the contrary, to find a sub-optimal solution, a certain auxiliary optimization issue will be proposed. We try to give such an auxiliary problem next.

We have Eq. (15) if the condition (Eq. (16)) holds. This means that $\chi (\mathrm{\infty }|k)=0$ and $V(\mathrm{\infty })=0$. Sum up both sides of Eq. (15) from $n=0$ to $n=\mathrm{\infty }$ and use Eq. (11) yields

(29) $$J_{\mathrm{\infty }}(t_k)\le V(t_k)=\chi {}^T(t_k)P_r\chi (t_k)\le 3\rho ,$$

which means

(30) $$\underset{(A(t_k),B(t_k),C(t_k))\in \mathrm{\Delta }}{max}{J}_{\mathrm{\infty }}(t_k)\le 3\rho .$$

An upper bound of the objective function of Op1 is given.

On the basis of the above analysis, we are ready to present the following auxiliary optimization problem for the system without constraints:

$$\mathrm{O}\mathrm{p}2:\underset{{\tilde{Q}}_{ir},{\tilde{Q}}_{it}(i=1,2,3),(r,t)\in \mathbb{S},F_r}{min}3\rho ,$$s.t. (Eqs. (16) and (27)).

The condition (Eq. (27)) is unable to be checked online because of the immeasurable state $x(t_k)$. We will handle the issue of unavailable states in constraint (Eq. (27)) next. Before continuing to work hard, the following significant assumption is showed.

$Assumption2.$ According to the initial state of the system (Eq. (1)), a known set is showed:

(31) $$x(0)\in \{x(t_k)|x^T(t_k)S^{-1}x(t_k)\le 1\},$$

matrix $S>0$ can be predefied from practical experience.

$Lemma2.$ The system (Eq. (1)) is considered that it is controlled by Eq. (9), if there exist symmetric positive definite matrices ${\hat{Q}}_{jr}(j=1,2,3)$, for the preseted Assumption 2, such that

(32) $$\left[\begin{array}{ccc}2& \ast & \ast \\ y(t_{k-1})& {\tilde{Q}}_{2r}& \ast \\ \check{y}(t_{k-1})& 0& {\tilde{Q}}_{3r}\end{array}\right]\ge 0,$$

(33) $$\left[\begin{array}{cc}{\tilde{Q}}_r& \ast \\ {\overline{A}}_r^l{\tilde{Q}}_r& {\hat{Q}}_r\end{array}\right]\ge 0,$$

(34) $${\tilde{Q}}_{1r}\ge {\hat{Q}}_{1r},{\tilde{Q}}_{1r}\ge S,{\tilde{Q}}_{2r}\ge {\hat{Q}}_{2r},{\tilde{Q}}_{3r}\ge {\hat{Q}}_{3r},$$

(35) $$\left[\begin{array}{ccc}{\tilde{Q}}_r& \ast & \ast \\ [0,I,0]{\overline{A}}_r^l{\tilde{Q}}_r& 2/3{\hat{Q}}_{2r}& \ast \\ [0,0,I]{\overline{A}}_r^l{\tilde{Q}}_r& 0& 2/3{\hat{Q}}_{3r}\end{array}\right]\ge 0.$$

hold, where ${\hat{Q}}_r\stackrel{\mathrm{\Delta }}{=}\{{\hat{Q}}_{1r},{\hat{Q}}_{2r},{\hat{Q}}_{3r}\}$ and ${\overline{A}}^l$ mean the vertices of ${\overline{A}}_{\mathrm{\zeta }({\mathit{T}\mathit{k}}^{\mathrm{\prime }}\mathit{+}\mathit{n}\mathit{)}}(k)$, $l=1,2,3,...,L,$ thus we can always guarantee condition (Eq. (27)).

Proof: By taking the similar to Ding et al. (2019), we can obtain the above lemma, which is thus omitted. So the proof is completed.

According to Schur Complement and Lemma 1, we can transform Eqs. (33) and (35) into the following conditions respectively $(l=1,2,3,\dots ,L)$:

(36) $$\left[\begin{array}{cc}{\hat{Q}}_r^l& \ast \\ {\hat{A}}_r^l& {\hat{Q}}_r\end{array}\right]\ge 0,$$

(37) $$\left[\begin{array}{ccc}{\hat{Q}}_r^l& \ast & \ast \\ {\text{𝒳}}_r& 2/3{\hat{Q}}_{2r}& \ast \\ {\text{𝒴}}_r& 0& 2/3{\hat{Q}}_{3r}\end{array}\right]\ge 0,$$where

$${\text{𝒳}}_r=\left[\begin{array}{ccc}{C}^l{T}^{l}S& 0& 0\end{array}\right],$$

$${\text{𝒴}}_r=\left[\begin{array}{ccc}\mathrm{\Psi }{C}^l{T}^{l}S& {\mathrm{\Phi }}^1{S}_{11}& {\mathrm{\Phi }}^2{S}_{11}\end{array}\right].$$

For the solvability, we can transform the issue Op2 into the following approximate optimization based on Lemma 2 and Assumption 2:

$$\mathrm{O}\mathrm{p}3:\underset{{\tilde{Q}}_{ir}>0,{\tilde{Q}}_{it}>0,{\hat{Q}}_{ir}>0(i=1,2,3),(r,t)\in \mathbb{S}\times \mathbb{S},F_r}{min}3\rho ,$$s.t. Eqs. (16), (31), (32) and Eqs. (34), (36), (37).

Feasibility and stability

We will make the feasibility of the proposed issues clear. And to move forward a single step, we will show the stability of the system (Eq. (1)), which is controlled by Eq. (9).

$Theorem1.$ Let us give the symmetric positive definite matrices $Q_1,Q_2,Q_3$ and R. We take the system (Eq. (1)) controlled by Eq. (9) into account. If a feasible solution for the optimization issue Op3 at the initial time instant $t_k$ exists, then at any future time instant $t>t_k$, the corresponding feasible solution also exists. Beside, the closed-loop system is asymptotically stable and Eq. (9) determines feedback gains.

Proof: (1) $Feasibility$. At the initial time instant $t_k$, let us assume that the optimization issue Op3 is feasible. For the time instant $t_{k+n},n\ge 1$ in the future, our need is proving that the issue Op3 is feasible too. It is not hard to see that only the condition (Eq. (16)) depends on the states, but other conditions are feasible at the time instant $t>t_k$ in the future so long as they are feasible at the time instant $t_k$. To this extent, we only need to prove that the condition (Eq. (16)) is feasible at the future time instant. Namely, the feasibility of the condition (Eq. (27)) in Op2 need to be proved. From Eqs. (1) and (10), we can get the following relations:

(38) $$\chi (t_{k+n}|k+n)=\chi (t_{k+n}),n\ge 1$$

(39) $$\chi (t_{k+1}|k)={\overline{A}}_{\xi (t_k)}(t_k)\chi (t_k|t_k),$$

(40) $$\chi (t_{k+1})={\overline{A}}_{\xi (t_k)}(t_k)\chi (t_k)forsome{\overline{A}}_{\xi (t_k)}(t_k)\in \mathrm{\Delta }$$

Based on the property of the RPI set, we have

(41) $${\chi }^T(t_{k+1}|k)Q_t^{-1}\chi (t_{k+1}|k)<{\chi }^T(t_k)Q_r^{-1}\chi (t_k)<3.$$

Using Eqs. (39) and (40), we can get the condition from Eq. (41):

(42) $${\chi }^T(t_{k+1})Q_t^{-1}\chi (t_{k+1})<3.$$

This indicates that the condition (Eq. (27)) is feasible at the time instant $t_{k+1}$. In addition, this process can continue at any time $t_{k+2},t_{k+3},...$ in the future.