Observer-Based Predictive Control with One Free Control Move for NCSs with Data Loss

Abstract

In this paper, an efficient output feedback predictive control synthesis based on a prespecified observer for networked control systems is presented. The process of random packet loss between the controller and the actuator is described as Markov chain, and a missing data compensation strategy is induced to cope with the poor performance caused by fading links. The provided model predictive control algorithm optimizes an infinite-horizon objective and parameterizes the infinite-horizon control moves into a free control move followed by output feedback. Further, the corresponding constraints about recursive feasibility and stochastic stability are given by utilizing the linear matrix inequality technique. A numerical example is given to illustrate the applicability of the proposed method.

Appendix. Proof of Theorem 3

(i) According to (29), $\hat{x}(k|k)=\hat{x}(k)$, $e(k|k)=e(k)$, $\eta (k|k)=\eta (k)$ and the successful transmission at initial time k, then (27) can be described as

$$\begin{aligned} \begin{aligned}&\hat{x}^{T}(k)S_{1}\hat{x}(k)+u^{T}(k)Ru(k)+ \hat{x}^{T}(k+1)M_{1}\hat{x}(k+1)\\&\quad +u^{T}(k)N_{1}u(k)+ e^{T}(k+1)P_{e}(k)e(k+1)\le \gamma (k) \end{aligned} \end{aligned}$$

the error $e(k+1)=e(k+1|k)=(A_{0}- LC)e(k|k)$, according to the upper bound of e(k), $e^{T}(k+1)P_{e0}e(k+1) \le \xi (k+1)$ and $P_{e}(k)=(\mu (k) \gamma (k)P_{e0})/\xi (k+1)$, then the above inequality is equal to

$$\begin{aligned}&\hat{x}^{T}(k)S_{1}\hat{x}(k)+u^{T}(k)Ru(k)+ \hat{x}^{T}(k+1)M_{1}\hat{x}(k+1)\nonumber \\&\quad +u^{T}(k)N_{1}u(k)\le (1-\mu (k))\gamma (k) \end{aligned}$$

(39)

multiplying (39) by $\gamma ^{-1}$ and substituting $\gamma M_{1}^{-1}=\bar{M}_{1}$, $\gamma N_{1}=\bar{N}_{1}^{-1}$, using Schur complement, (29) is obtained.

(ii)Based on the quadratic Lyapunov function defined in (24), the contractiveness conditions (25) can be written as

$$\begin{aligned}&E\left\{ \eta ^{T}(k+i|k)\varPsi ^{T}diag \left\{ M_{\theta (k+i+1|\theta (k+i+1)},P_{e}(k),\right. \right. \nonumber \\&\quad \left. N_{\theta (k+i+1|k)|\theta (k+i|k)}\right\} \varPsi \eta (k+i|k)-\eta ^{T}(k+i|k)\nonumber \\&\quad \left. \times diag\left\{ M_{\theta (k+i|k)},P_{e}(k),M_{\theta (k+i|k)}\right\} \eta (k+i|k)\right\} \nonumber \\&\quad +E\left\{ \eta ^{T}(k+i|k)S\eta (k+i|k)+u^{T}(k+i|k)R\right. \nonumber \\&\quad \left. \times u(k+i|k) \right\} \le 0 \end{aligned}$$

(40)

where

$$\begin{aligned} \varPsi (k)=\begin{bmatrix} {A+ \theta (k) BF}&LC&{(1- \theta (k))\tau B}\\ 0&A-LC&0\\ \theta (k) F&0&{(1- \theta (k))\tau I} \end{bmatrix} \end{aligned}$$

(41)

If $\theta (k+i|k)=1$, $E_{\theta (k+i|k)=1}diag\{M_{\theta (k+i+1)}$, $P_{e}(k)$, $N_{\theta (k+i+1)}\}\alpha diag\{M_{0}$, $P_{e}$, $N_{0}\}+(1-\alpha )diag\{M_{1}$, $P_{e}$, $N_{1}\}$, the contractiveness condition is satisfied if and only if the following inequality holds

$$\begin{aligned}&\begin{aligned}&(a)~~\varPsi (k)_{\theta (k)=1}^{T}diag\left\{ W_{1},U_{1}, T_{1}\right\} \varPsi (k)_{\theta (k)=1}-diag\{M_{1},\\&\quad P_{e}, N_{1}\}+diag\left\{ S_{1}, 0, 0\right\} +[F~~0~~0]^{T}R[F~~0~~0] \le 0 \end{aligned}\\&\begin{aligned}&(b)~~\alpha diag\left\{ M_{0},P_{e},N_{0}\right\} +(1-\alpha )diag\left\{ M_{1},P_{e},N_{1}\right\} \\&\quad \le diag\left\{ W_{1},U_{1},T_{1}\right\} \end{aligned} \end{aligned}$$

Pre- and post-multiplying (a) by diag$\{\gamma ^{1/2}M_{1}^{-1}$,$\gamma ^{1/2}P_{e}^{-1}$, $\gamma ^{1/2}N_{1}^{-1}\}$, also pre- and post-multiplying (b) by $diag\{W_{1}^{-1}$, $U_{1}^{-1}$, $T_{1}^{-1}\}$, substituting $\gamma \varGamma ^{-1}=\bar{\varGamma }$, where $\varGamma$ takes $\{M_{1}$, $N_{1}$, $P_{e}$, $M_{0}$, $N_{0}$,$W_{1}$, $U_{1}$, $T_{1}$, $W_{0}$, $U_{0}$, $T_{0}\}$ and $Y_{1}=F \bar{M}_{1}$, according to Schur complement, (30) and (31) are obtained. If $\theta (k+i|k)=0$, using above similar procedure, (32) and (33) can be derived.

(iii) if (25) and (27) hold at sampling time k, then

$$\begin{aligned}&\begin{aligned}&\eta ^{T}(k+2|k)diag\left\{ M_{\theta (k)}, Pe(k), N_{\theta (k)}\right\} \eta (k+2|k)\\&\quad +\hat{x}^{T}(k+1|k)S_{1}\hat{x}(k+1|k)+u^{T}(k+1|k)Ru(k+1|k)\\&\quad \le \hat{x}^{T}(k+1|k)M_{\theta (k)}\hat{x}(k+1|k)+e^{T}(k+1|k)Pe(k)\\&\quad \times e(k+1|k)+u^{T}(k|k)N_{\theta (k)}u(k|k) \end{aligned}\\&\begin{aligned}&\hat{x}^{T}(k+1|k)M_{\theta (k)}\hat{x}(k+1|k)+e^{T}(k+1|k)Pe(k)\\&\quad \times e(k+1|k)+u^{T}(k|k)N_{\theta (k)}u(k|k)\le \digamma (k+1) \end{aligned} \end{aligned}$$

where $\digamma (k+1)\!=\!\gamma (k)\!-\!\hat{x}^{T}(k|k)S_{1}\hat{x}(k|k)\!-\!u^{T}(k|k)Ru(k|k)$, since $e(k+1|k)=e(k+1)$ and $\hat{x}(k+1|k)=\hat{x}(k+1)$. By above two inequalities, the following inequality can be obtained.

$$\begin{aligned} \begin{aligned}&e^{T}(k+2|k)Pe(k)e(k+2|k)\le \digamma (k+1)-\hat{x}^{T}(k+1)S_{1}\\&\quad \times \hat{x}(k+1)-u^{T}(k+1|k)(R+N_{\theta (k)})u(k+1|k)\\&\quad -\hat{x}^{T}(k+2|k)M_{\theta (k)}\hat{x}(k+2|k) \end{aligned} \end{aligned}$$

By $e(k+2|k)=e(k+2)=(A_{0}-LC)e(k+1)$ and $\tilde{x}(k+2)=\hat{x}(k+2|k)=(A+\theta (k)B F)\hat{x}(k+1)+ (1-\theta (k)) \tau u(k-1)+LCe(k+1)$. so it has

$$\begin{aligned} \begin{aligned}&e^{T}(k+2)Pe(k)e(k+2)\le \tilde{\xi }(k+2)= \digamma (k+1)\\&\quad -\hat{x}^{T}(k+1)S_{1}\hat{x}(k+1)-u^{T}(k+1|k)(R+N_{\theta (k)})\\&\quad \times u(k+1|k)-\hat{x}^{T}(k+2|k)M_{\theta (k)}\hat{x}(k+2|k) \end{aligned} \end{aligned}$$

where $\tilde{\xi }(k+2)$ follows the above assumption 1, besides, since $Pe(k)= \pi (k)P_{e0}$. Then

$$\begin{aligned} \begin{aligned} e^{T}(k+2)P_{e0}e(k+2)\le \tilde{\xi }(k+2)/ \pi (k) \end{aligned} \end{aligned}$$

from above results and Theorem 2, it gets the upper bound of estimation error at time $k+2$

$$\begin{aligned} \begin{aligned}&e^{T}(k+2)P_{e0}e(k+2)\le \xi (k+2)\\&\quad =min\left\{ \tilde{\xi }(k+2)/\pi (k);\delta \eta (k+1)\right\} \end{aligned} \end{aligned}$$

For any $k>1$, with similar analysis, we get $e^{T}(k+1)P_{e0}e(k+1)\le \xi (k+1)$.