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 自动化学报  2017, Vol. 43 Issue (12): 2091-2099 PDF

1. 同济大学电子与信息工程学院 上海 201804;
2. 苏州科技大学电子与信息工程学院 苏州 215009

Discussions on Existence of Observers and Reduced-order Observer Design for Discrete-time Switched Systems
ZHU Fang-Lai1, CAI Ming1, GUO Sheng-Hui1,2
1. College of Electronics and Information Engineering, Tongji University, Shanghai 201804;
2. College of Electrical Engineering and Automation, Suzhou University of Science and Technology, Suzhou 215009
Manuscript received : June 16, 2016, accepted: November 23, 2016.
Foundation Item: Supported by National Natural Science Foundation of China (61573256), Shanghai Science and Technology Innovation Fund (16111106502), Zhejiang Province High-end Innovative Carrier Project
Corresponding author. ZHU Fang-Lai     Professor at the College of Electronics and Information Engineering, Tongji University. His research interest covers nonlinear robust control, model-based fault detection and isolation. Corresponding author of this paper
Recommended by Associate Editor SUN Xi-Ming
Abstract: The paper discusses the unknown input observer (UIO) design issue for discrete-time switched systems with unknown inputs. First, a Lyapunov-type equivalent representation of observer matching condition for a general discrete-time system is given and proved. Then, based on the stability theory of discrete-time switched system without unknown inputs, a reduced-order observer is developed for a discrete-time switched system with unknown inputs. The reduced-order observer can eliminate the influences of the unknown inputs directly because of the special selection of the observer gain matrix determined by matrix block computation. Meanwhile, an algebraic unknown input reconstruction method is proposed. Finally, a simulation example is given to verify the correctness and effectiveness of the proposed methods.
Key words: Discrete-time system     unknown input estimation     observer matching condition     switched system

1 问题描述

 $$$\left\{ \begin{array}{l} x(k+1) = Ax(k) + Bu(k) + D\eta(k) \\ y(k) = Cx(k) \\ \end{array} \right.$$$ (1)

 $$$\left\{ {\begin{array}{l} \left( {A+LC} \right)^{\rm T}P\left( {A+LC} \right)-P=-Q \\ D^{\rm T}P=GC \\ \end{array}} \right.$$$ (3)

 \begin{align*} &\varepsilon _1 \left( {\left| {x\left( k \right)} \right|} \right)\le V_i \left( {x\left( k \right)} \right)\le \varepsilon _2 \left( {\left| {x\left( k \right)} \right|} \right) \\& \Delta V_i \left( {x\left( k \right)} \right) := V_i \left( {x\left( {k+1} \right)} \right)-\\&\qquad V_i \left( {x\left( k \right)} \right)\le -\alpha V_i \left( {x\left( k \right)} \right) \end{align*}

$\forall \left( {\sigma \left( {k_v } \right)=i, \sigma \left( {k_v -1} \right)=j} \right)\in \ell \times \ell$, $i\ne j$,

 $V_i \left( {x\left( {k_v } \right)} \right)\le \beta V_j \left( {x\left( {k_v } \right)} \right)$

 $\tau _a \ge \tau _a^\ast =-\frac{\ln \beta }{\ln \left( {1-\alpha } \right)}$

 $\left\{ {\begin{array}{l} x\left( {k+1} \right)=\sum\limits_{i=1}^N {\hbar _i \left( {\sigma \left( k \right)} \right)} \left[ {A_i x\left( k \right)+B_i u\left( k \right)+D_i \eta \left( k \right)} \right] \\ y\left( k \right)=Cx\left( k \right) \\ \end{array}} \right.$

 $$$\left\{ {\begin{array}{l} x\left( {k+1} \right)=A_i^ x\left( k \right)+B_i^ u\left( k \right)+D_i^ \eta \left( k \right) \\ y\left( k \right)=Cx\left( k \right) \\ \end{array}} \right.$$$ (6)

 $$$\left\{ {\begin{array}{l} \left( {A_i +L_i C} \right)^{\rm T}P_i \left( {A_i +L_i C} \right)-\left( {1-\gamma _1 } \right)P_i =-Q_i \\ D_i^{\rm T} P_i =G_i C \\ \end{array}} \right.$$$ (7)

 \begin{align*}&A_i =\left[{{\begin{array}{*{20}c} {A_{i, 11} }&{A_{i, 12} } \\ {A_{i, 21} }&{A_{i, 22} } \\ \end{array} }} \right], B_i =\left[{{\begin{array}{*{20}c} {B_{i, 1} } \\ {B_{i, 2} } \\ \end{array} }} \right] \\&D_i =\left[{{\begin{array}{*{20}c} {D_{i, 1} } \\ {D_{i, 2} } \\ \end{array} }} \right], P_i =\left[{{\begin{array}{*{20}c} {P_{i, 11} }&{P_{i, 12} }\\ {P_{i, 12}^{\rm T} }&{P_{i, 22} } \\ \end{array} }} \right]\end{align*}

 \begin{align*} \theta _1 \left( k \right)=&x_1 \left( k \right)=y\left( k \right) \\ \theta _2 \left( k \right)=&\left[{{\begin{array}{*{20}c} {K_{\sigma \left( k \right)} }&{I_{n-p} } \\ \end{array} }} \right]x\left( k \right)= \\& K_{\sigma \left( k \right)} x_1 \left( k \right)+x_2 \left( k \right)=K_{\sigma \left( k \right)} y\left( k \right)+x_2 \left( k \right) \end{align*}

 $\left[{{\begin{array}{*{20}c} {P_{i, 11} }&{P_{i, 12} } \\ {P_{i, 12}^{\rm T} }&{P_{i, 22} } \\ \end{array} }} \right]\left[{{\begin{array}{*{20}c} {D_{i, 1} } \\ {D_{i, 2} } \\ \end{array} }} \right]=\left[{{\begin{array}{*{20}c} {I_p } \\ 0 \\ \end{array} }} \right]G_i$

 \begin{align*} \theta _2& \left( {k+1} \right)=\left[{{\begin{array}{*{20}c} {K_i }&{I_{n-p} } \\ \end{array} }} \right]x\left( {k+1} \right) =\\&\left[{{\begin{array}{*{20}c} {K_i }&{I_{n-p} } \\ \end{array} }} \right]\left[{{\begin{array}{*{20}c} {A_{i, 11} }&{A_{i, 12} } \\ {A_{i, 21} }&{A_{i, 22} } \\ \end{array} }} \right]\times\\&\left[{{\begin{array}{*{20}c} {I_p }&0 \\ {-K_i }&{I_{n-p} }\\ \end{array} }} \right] \left[{{\begin{array}{*{20}c} {\theta _1 \left( k \right)} \\ {\theta _2 \left( k \right)} \\ \end{array} }} \right]+\\&\left[{{\begin{array}{*{20}c} {K_i }&{I_{n-p} } \\ \end{array} }} \right]\left[{{\begin{array}{*{20}c} {B_{i, 1} } \\ {B_{i, 2} } \\ \end{array} }} \right]u\left( k \right) +\\&\left[{{\begin{array}{*{20}c} {K_i }&{I_{n-p} } \\ \end{array} }} \right]\left[{{\begin{array}{*{20}c} {D_{i, 1} } \\ {D_{i, 2} } \\ \end{array} }} \right]\eta \left( k \right) =\\&\left( {K_i A_{i, 12} +A_{i, 22} } \right)\theta _2 \left( k \right) +\\&\left[{\left( {A_{i, 21}-A_{i, 22} K_i } \right)+K_i \left( {A_{i, 11}-A_{i, 12} K_i } \right)} \right]y\left( k \right) + \\& \left( {K_i B_{i, 1} +B_{i, 2} } \right)u\left( k \right)+\left( {K_i D_{i, 1} +D_{i, 2} } \right)\eta \left( k \right) \end{align*}

 $\begin{array}{*{20}{l}} {{\theta _2}{\mkern 1mu} }&{\left( {k + 1} \right) = \left( {{K_i}{A_{i,12}} + {A_{i,22}}} \right){\theta _2}\left( k \right) + }\\ {}&{\left[ {\left( {{A_{i,21}} - {A_{i,22}}{K_i}} \right) + {K_i}\left( {{A_{i,11}} - {A_{i,12}}{K_i}} \right)} \right]y\left( k \right) + }\\ {}&{\left( {{K_i}{B_{i,1}} + {B_{i,2}}} \right)u\left( k \right)} \end{array}$ (9)

 $\theta \left( {k_v } \right)=T_{\sigma \left( {k_v } \right)} x\left( {k_v } \right)=T_{\sigma \left( {k_v } \right)} x\left( {k_v-1} \right)=$ $T_{\sigma \left( {k_v } \right)} T_{\sigma \left( {k_v-1} \right)}^{-1} \theta \left( {k_v-1} \right)$

 \begin{align} \theta _2& \left( {k_v } \right)=\nonumber\\&\left( {K_{\sigma \left( {k_v } \right)} -K_{\sigma \left( {k_v -1} \right)} } \right)\theta _1 \left( {k_v -1} \right)+\theta _2 \left( {k_v -1} \right) =\nonumber\\& \left( {K_{\sigma \left( {k_v } \right)} -K_{\sigma \left( {k_v -1} \right)} } \right)y\left( {k_v -1} \right)+\theta _2 \left( {k_v -1} \right) \end{align} (10)

 \begin{align*} &{\rm rank}\left( {D^{\rm T}PD} \right)={\rm rank}\left( {GCD} \right)\le {\rm rank}\left( {CD} \right)\le \\ &\quad {\rm rank}\left( D \right) \end{align*}

 \begin{align*} &\left( {\bar {A}_{22} +\tilde {L}_{22} \bar {C}_{22} } \right)^{\rm T}\bar {P}_{22} \left( {\bar {A}_{22} +\tilde {L}_{22} \bar {C}_{22} } \right)-\bar {P}_{22} +\\ &\qquad\left( {\bar {A}_{12} +\bar {L}_{12} \bar {C}_{22} } \right)^{\rm T}\left( {\bar {P}_{11} -\bar {P}_{21}^{\rm T} \bar {P}_{22}^{-1} \bar {P}_{21} } \right)\times\\&\qquad\left( {\bar {A}_{12} +\bar {L}_{12} \bar {C}_{22} } \right)=-\bar {Q}_{22} \end{align*}

 \begin{align*} &\left( {\bar {A}_{22} +\tilde {L}_{22} \bar {C}_{22} } \right)^{\rm T}\bar {P}_{22} \left( {\bar {A}_{22} +\tilde {L}_{22} \bar {C}_{22} } \right)-\bar {P}_{22} =\\ &\quad-\bar {Q}_{22} -\left( {\bar {A}_{12} +\bar {L}_{12} \bar {C}_{22} } \right)^{\rm T}\left( {\bar {P}_{11} -\bar {P}_{21}^{\rm T} \bar {P}_{22}^{-1} \bar {P}_{21} } \right)\times\\&\quad\left( {\bar {A}_{12} +\bar {L}_{12} \bar {C}_{22} } \right) \end{align*}

 \begin{align*} -\bar {Q}_{22}& -\left( {\bar {A}_{12} +\bar {L}_{12} \bar {C}_{22} } \right)^{\rm T}\left( {\bar {P}_{11} -\bar {P}_{21}^{\rm T} \bar {P}_{22}^{-1} \bar {P}_{21} } \right)\times\\ &\left( {\bar {A}_{12} +\bar {L}_{12} \bar {C}_{22} } \right)＜0 \end{align*}

 \begin{align*} \left( {\bar {A}_{22} +\tilde {L}_{22} \bar {C}_{22} } \right)^{\rm T}\bar {P}_{22} \left( {\bar {A}_{22} +\tilde {L}_{22} \bar {C}_{22} } \right)-\bar {P}_{22} ＜0 \end{align*}

$\left( {\bar {A}_{22} , \bar {C}_{22} } \right)$是可检测的, 由引理A1可知假设1成立.

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