Robust model predictive control for polytopic uncertain systems via a high-rate network with the FlexRay protocol

System models

The system with polytopic uncertainties is given as follows:

(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 $u(t_k)\in {\mathbb{R}}^{n_u}$, $y(t_k)\in {\mathbb{R}}^{n_y}$ and $x(t_k)\in {\mathbb{R}}^{n_x}$ represent the control input, measure output and state, respectively. $A(t_k)$, $B(t_k)$ and $C(t_k)$ are certain matrices which belong to a convex hull given by

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

The following hard constraints are taken into account so as to correspond with the engineering:

(3) $$\{\begin{array}{cc}\underset{w}{max}|{[u(t_k)]}_w|\le \overline{u},\hfill & w\in \{1,2,...,n_u\}\hfill \\ \underset{j}{max}|{[x(t_k)]}_j|\le \overline{x},\hfill & j\in \{1,2,...,n_x\}.\hfill \end{array}$$

Communication protocols

In Fig. 1, the data are transmitted from sensors to controllers through a limited bandwidth shared communication network. In order to reduce the transmission burden, a hybrid scheduling strategy FRP is utilized to orchestrate the data transmission order. More specifically, only one sensor node has the right to access the shared network at each time instant, while others are transmitted according to corresponding strategy.

The structure of FlexRay is shown in Fig. 2. It is composed of four parts, including static segment, dynamic segment, symbol window and network idle time (NIT). The part of static segment is composed of equal length time slots and transmits time-triggered periodic data. The deterministic time of message transmission can be ensured in the static segment, which is suitable for scenarios that need to transmit data at fixed time intervals. The dynamic segment uses flexible time division multiple access technology and can transmit event-driven signals that are transmitted only when needed, so it is able to support asynchronous processes. Thus, the flexible allocation of bandwidth can be achieved by the data volume requirement of periodic transmission and the event-driven amount in the network. In addition, the time slot in the dynamic segment is generally smaller than the time slot in the static segment, because the static segment needs enough time slots to ensure that the signal is transmitted successfully, while the signal in the dynamic segment is transmitted only when it is needed. The symbol window is a time slot to transmit data for network management purposes, when there is no symbol data to be transmitted, the symbol window can be configured to be of length 0, that is, no symbol data is sent. NIT is a period without communication so as to clock synchronization, it can provide a time buffer against possible communication delays or exceptions. The lengths of symbol window and NIT are quite small compared to the lengths of static and dynamic segments, therefore the time lengths of the former will be ignored.

In this article, the RRP and the TODP are implemented in the static and dynamic scheduling parts, respectively, and assume that there are N packets that should be sent from sensors to controller though the communication channel. The N packets are divided into two sets, where the first part $h\ge 1$ nodes belong to the set ${\mathrm{\aleph }}_1\stackrel{\mathrm{\Delta }}{=}\{1,...,h\}$ and the other nodes belong to the set ${\mathrm{\aleph }}_2\stackrel{\mathrm{\Delta }}{=}\{h+1,...,N\}$. Taking varying real-time demands of sensor nodes into account and without loss of generality, in the set ${\mathrm{\aleph }}_1$ we use the RRP and the nodes belonging to set ${\mathrm{\aleph }}_2$ are scheduled by the TODP. Further elaboration regarding the process of identifying the chosen transmission nodes for the RRP and the TODP is provided in the following descriptions:

(1) RRP:

(4) $$\rho (i)=\mathrm{m}\mathrm{o}\mathrm{d}(i-1,h)+1$$where $\rho (i)\in {\mathrm{\aleph }}_1$ is an element of ${\mathrm{\aleph }}_1$ and the $\rho$th node has the right to access the communication channel at time $i$.

(2) TODP:

(5) $$\varsigma (i)=\arg \underset{m=h+1,...N}{max}\{{\tilde{y}}_m^T(i){\chi }_m(i){\tilde{y}}_m(i)\}$$where “arg” refers to the minimum value of $m$ that maximizing the quadratic, $\varsigma (i)\in {\mathrm{\aleph }}_2$ denotes the node has the token at time instant $i$ according to the rule of TODP, ${\tilde{y}}_m(i)\stackrel{\mathrm{\Delta }}{=}{y}_m(i)-{\overline{y}}_m^{\ast }(i)$ represents the difference between the signal $y_m(i)$ at time $i$ and the last transmitted signal ${\overline{y}}_m^{\ast }(i)$ by node m before time $i$. ${\chi }_m(i)$( $m\in {\mathrm{\aleph }}_2$) are the weighted matrices.

$Remark1.$ The static segment of FRP uses time division multiple access technology, and the communication cycle is divided into dynamic segment and static segment time slots, and the frame ID is used to transmit the corresponding data. The static slot is fixed and is used to transmit periodic data. Each frame ID corresponds to a time slot and the communication node sends the message when the corresponding frame ID is assigned. The static slot frame ID is fixed and determined by the system designer in the initial stage, which is suitable for transmitting periodic and high reliability data. The smaller the frame ID of the dynamic segment, the higher the priority of the transmission data, which is suitable for the transmission of data that needs to obtain priority through competition, and the real-time requirement of the data is relatively low.

Suppose there are $s$ measurement signals that are stacked into N packets and transmitted through the communication network. The FRP is utilized to orchestrate the order of the N network nodes. The measurement outputs of static segment and dynamic are given by

(6) $$\{\begin{array}{c}{y}^{[1]}\stackrel{\mathrm{\Delta }}{=}{[y_1^T(\cdot ),y_2^T(\cdot ),....,y_h^T(\cdot )]}^T\hfill \\ y^{[2]}\stackrel{\mathrm{\Delta }}{=}{[y_{h+1}^T(\cdot ),y_{h+2}^T(\cdot ),....,y_N^T(\cdot )]}^T\hfill \end{array}$$where $y_i(\cdot )\in {\mathbb{R}}^{s_i}$ with ${\mathrm{\Sigma }}_{i=1}^N{s}_i=s$. The scheduling protocol of the first $h$ measurement outputs are RRP while the remaining $N-h$ outputs are scheduled by TODP.

$Remark2.$ High-rate communication network is taken into account in this article so as to save communication resources and the order of sensors is determined by FRP. Before further discussion, the assumption is made for the high-rate communication network as follows:

$Assumption1.$ The transmission signal communication period in the high-rate communication channels is defined as $p_c=p/(h+1)$ with $p_c$ as a constant and the first signal transmitted when the signal is sampled at 0th, that is to say $t_0=T_0$.

The measurement though the share communication network is defined as $\overline{y}(t_k)\stackrel{\mathrm{\Delta }}{=}[({\overline{y}}^{[1]}(t_k){)}^T,({\overline{y}}^{[2]}(t_k){)}^T{]}^T$, where ${\overline{y}}^{[1]}(t_k)\stackrel{\mathrm{\Delta }}{=}[{\overline{y}}_1^T(t_k),{\overline{y}}_2^T(t_k),...{\overline{y}}_h^T(t_k){]}^T$ and ${\overline{y}}^{[2]}(t_k)\stackrel{\mathrm{\Delta }}{=}[{\overline{y}}_{h+1}^T(t_k),{\overline{y}}_{h+2}^T(t_k),...{\overline{y}}_N^T(t_k){]}^T$. ${\overline{y}}_i(t_k)$ $i$ is the $i$th (i = 1,2,…N) element of the $\overline{y}(t_k)$.

Then, in terms of Assumption 1, when the transmission occurs at time interval $(t_{k-1},t_k]$, the transmission instants can be expressed as $t_{k}p-h{p}_c,t_{k}p-(h-1)p_c,...,t_{k}p-p_c,t_{k}p$, for the purpose of simplifying the expression, denote $t_k^i\stackrel{\mathrm{\Delta }}{=}{t}_k-i{p}_c(i=1,2,...,h)$. Next, how to combine the FRP with high-rate will be described. The RRP is used to schedule the transmission date at instant $t_{k}p-i{p}_c(i=1,2,...,h)$ and the TODP is utilized to determine the node which most needed transmission at time $t_{k}p$.

Next, according to the periodic feature of RRP and the characteristic of TODP, the zero-input strategy and the zero-order holder are adopted, respectively. The corresponding formulations are illustrated as follows:

(7) $$\{\begin{array}{c}{\overline{y}}^{[1]}(t_k)={[y_1^T(t_k^h)y_2^T(t_k^{h-1})...y_h^T(t_k^1)]}^T\hfill \\ {\overline{y}}^{[2]}(t_k)={\gamma }_{\varsigma (t_k)}{y}^{[2]}(t_k)+(I_{N_2}-{\gamma }_{\varsigma (t_k)}{\overline{y}}^{[2]}(t_{k-1})\hfill \end{array}$$where $N_2\stackrel{\mathrm{\Delta }}{=}{\mathrm{\Sigma }}_{i=h+1}^N{s}_i$, ${\gamma }_{\varsigma (t_k)}\stackrel{\mathrm{\Delta }}{=}$ diag $\delta (h+1-\varsigma (t_k))I_{s_{h+1}},...,\delta (N-\varsigma (t_k))I_{s_N}$. In more detail, it can be rewritten as

(8) $$\{\begin{array}{c}{\overline{y}}^{[1]}(t_k)=y^{[1]}(t_{k-1})\hfill \\ {\overline{y}}^{[2]}(t_k)={\gamma }_{\varsigma (t_k)}{y}^{[2]}(t_k)+(I_{N_2}-{\gamma }_{\varsigma (t_k)}{\overline{y}}^{[2]}(t_{k-1})\hfill \\ ={\gamma }_{\varsigma (t_k)}({\overline{c}}^{[2]}(t_k)x(t_k)+(I_{N_2}-{\gamma }_{\varsigma (t_k)}{\overline{y}}^{[2]}(t_{k-1})\hfill \end{array}$$where

$$\begin{array}{rl}& {\overline{c}}^{[1]}(t_k)\stackrel{\mathrm{\Delta }}{=}[c_1^T(t_k)c_2^T(t_k)c_3^T(t_k)....c_h^T(t_k){]}^T,\\ & {\overline{c}}^{[2]}(t_k)\stackrel{\mathrm{\Delta }}{=}[c_{h+1}^T(t_k)c_{h+2}^T(t_k)c_{h+3}^T(t_k)....c_N^T(t_k){]}^T.\end{array}$$

According to Formula (8), $\overline{y}(t_k)$ is defined as

(9) $$\overline{y}(t_k)=I_1{y}^{[1]}(t_{k-1})+I_2{\gamma }_{\varsigma (t_k)}{\overline{c}}^{[2]}(t_k)x(t_k)+I_2(I_{N_2}-{\gamma }_{\varsigma (t_k)}{\overline{y}}^{[2]}(t_{k-1})$$where $I_1\stackrel{\mathrm{\Delta }}{=}[I_{N_1\times N_1},0_{(s-N_1)\times N_1}^T{]}^T$, $N_1\stackrel{\mathrm{\Delta }}{=}{\mathrm{\Sigma }}_{i=1}^h{s}_i$, $I_2\stackrel{\mathrm{\Delta }}{=}[0_{(s-N_2)\times N_2}^T,I_{N_2\times N_2}{]}^T$.

$Remark3.$ In this article, the assumption should highlight that the communication period $p_c$ is independent of the number of sensors. Because only one node has the transmission right when the communication channel transmits the signal in the high-rate network. In addition, without loss of generality, the communication period $p_c=p/(h+1)$ and $h$ are prescribed in terms of engineering practice application.

Problem of interests

Generally speaking, the states of the practice system are difficult to obtain directly. Hence, the following controller with static output feedback strategy in the high-rate communication network is designed for the system Eq. (1) under the FRP.

(10) $$u(t_k)=F_{\iota (t_k)}(t_k)\overline{y}(t_k),$$

$$\iota (t_k)=\{\begin{array}{cc}\varrho (t_k),\hfill & \iota \in {\mathrm{\aleph }}_1\hfill \\ \varsigma (t_k),\hfill & \iota \in {\mathrm{\aleph }}_2\hfill \end{array}$$where $F_{\iota (t_k)}$ is the feedback gain of the controller to be designed.

According to the control law Eq. (10), the form of closed-loop system can be rewritten as follows:

(11) $$\eta (t_{k+n+1}|t_k)={\overline{R}}_{\iota (t_{k+n}|t_k)}(t_k)\eta (t_{k+n}|t_k),$$where

(12) $$\eta (t_{k+n}|t_k)={\left[x{(t_{k+n}|t_k)}^T{y}^{[1]}{(t_{k+n-1}|t_k)}^T{\overline{y}}^{[2]}{(t_{k+n-1}|t_k)}^T\right]}^T,$$

$$\begin{array}{r}{\overline{R}}_{\iota (t_{k+n}|t_k)}(t_k)=\left[\begin{array}{ccc}\mathcal{Z}(t_k)\hfill & \tilde{B}(t_k)I_1\hfill & \tilde{B}(t_k)I_2{\overline{\gamma }}_{\iota (t_{k+n}|t_k)}\hfill \\ I_1{\overline{c}}^{[1]}(t_k)\hfill & 0\hfill & 0\hfill \\ {\gamma }_{\iota (t_{k+n}|t_k)}{\overline{c}}^{[2]}(t_k)\hfill & 0\hfill & {\overline{\gamma }}_{\iota (t_{k+n}|t_k)}\hfill \end{array}\right],\end{array}$$

${\overline{\gamma }}_{\iota (t_{k+n}|t_k)}=I_{N2}-{\gamma }_{\iota (t_{k+n}|t_k)}$, $\mathcal{Z}(t_k)=A(t_k)+B(t_k)F_{\iota (t_{k+n}|t_k)}{\gamma }_{\iota (t_{k+n}|t_k)}{\overline{c}}^{[2]}(t_k)I_2$,

$\tilde{B}(t_k)=B(t_k)F_{\iota (t_{k+n}|t_k)}$,

moreover, denote the initial state $\eta (0)=[x^T(0)(y^{[1]}(-1){)}^T({\overline{y}}^{[2]}(-1){)}^T{]}^T$.

The following “min-max” problem is put forward over infinite horizon for system Eq. (1) with polytopic uncertainties to design the controllers:

(13) $$\underset{F_{\iota (t_{k+n}|t_k)},}{min}\underset{(A(t_k),B(t_k),C(t_k))\in \mathrm{\Gamma }}{max}{J}_{\mathrm{\infty }}(t_k),$$where $J_{\mathrm{\infty }}(t_k)$ is defined by

(14) $$J_{\mathrm{\infty }}\left(t_k\right)\stackrel{\mathrm{\Delta }}{=}\sum _{n=0}^{\mathrm{\infty }}[{\eta }^T(tk+n)\mathcal{Q}\eta (t_k+n)+u^T(tk+n)Ru(t_k+n)]$$where $\mathcal{Q}=diag\{{\mathcal{Q}}_1,{\mathcal{Q}}_2,{\mathcal{Q}}_3\}$ and R being prescribed symmetric and positive weighting matrices.

By virtue of the min-max problem Eq. (13), OP1 is illustrated to design the controllers:

$$\mathbf{O}\mathbf{P}\mathbf{1}.\underset{F_{\iota (t_{k+n}|t_k)},}{min}\underset{(A(t_k),B(t_k),C(t_k))\in \mathrm{\Gamma }}{max}{J}_{\mathrm{\infty }}(t_k),$$

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

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

$$\eta (t_{k+n}|t_k)\in \phi (P_{\iota (t_{k+n}|t_k)},3\tau ),$$where $\phi$ is terminal constraint set and is defined by

(15) $$\phi \stackrel{\mathrm{\Delta }}{=}\{\eta (t_{k+n}|t_k)|{\eta }^T(t_{k+n}|t_k)P_{\iota (t_{k+n}|t_k)}\eta (t_{k+n}|t_k)\le 3\tau \}$$representing a quadratic function positive definite matrix with appropriate dimensions which can improve the performance of a system with polytopic uncertainties. It is noticeable that only the first component $u(t_k)$ of the predicted inputs set $\{u(t_k),u(t_{k+1}),u(t_{k+2}),...\}$ is valid for plant at each time instant.

In this article, we try to ensure the asymptotic stability of the system Eq. (1) in the framework of OFRMPC strategy and take both a high-rate network and FRP Eq. (8) into consideration. More precisely, in order to find the parameter matrices $F_{\iota (t_k)}$, an auxiliary optimization problem OP1 is put forward. Furthermore, the stability of the closed-loop system is ensured under the auxiliary optimization when the following requirements to all the admissible parameters are satisfied simultaneously:

• R1: A sub-optimization solution can be obtained from the auxiliary optimization problem which is utilized to represent the OP1.

• R2: The closed-loop system is stable through the obtained parameter matrices $F_{\iota (t_k)}$.

MPC under FRP without hard constraints

In this section, some sufficient conditions are established to guarantee the performance of unconstrained systems by virtue of the quadratic function approach. Then, we will get the controllers in the framework of static OFRMPC strategy. First, sufficient conditions are given to satisfy the terminal constraint set condition in OP1, i.e., $\eta (t_{k+n}|t_k)\in \phi (P_{\iota (t_{k+n}|t_k)},3\tau )$. Afterwards, an auxiliary optimization problem is put forward so as to seek the sub-optimal solution for the unconstrained systems. Furthermore, the unavailable state $x(t_k)$ of the established auxiliary optimization problem can be solved by inequality analysis technique. Then, considering the recursive solvability, another auxiliary optimization problem is constructed and the closed-loop system is stabilized by the sufficient conditions which are obtained from solving the online auxiliary optimization problem.