The Latest Research Progress of Image Dehazing
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摘要: 随着计算机视觉系统的发展及其在军事、交通以及安全监控等领域的发展, 图像去雾已成为计算机视觉的重要研究方向. 在雾、霾之类的恶劣天气下采集的图像会由于大气散射的作用而被严重降质, 使图像颜色偏灰白色, 对比度降低, 物体特征难以辨认, 不仅使视觉效果变差, 图像观赏性降低, 还会影响图像后期的处理, 更会影响各类依赖于光学成像仪器的系统工作, 如卫星遥感系统、航拍系统、室外监控和目标识别系统等. 因此, 需要图像去雾技术来增强或修复, 以改善视觉效果和方便后期处理. 本文归纳总结了两大类图像去雾方法:基于图像增强和基于物理模型的方法, 深入探讨了其中的典型算法和研究成果, 并对这些算法的测试结果进行了定性和定量的分析比较, 最后总结了图像去雾技术目前的研究状况和未来的发展方向.Abstract: With the development of computer vision system and the increasing demand in military, transportation and surveillance applications, image dehazing has been an important researching direction in computer vision. Images acquired in bad weather, such as haze and fog, are seriously degraded by the scatting of the atmosphere, which makes the image color gray, reduces the contrast and makes the object features difficult to identify. The bad weather not only leads to the variation of the visual effect of the image, but also cause the disadvantage of the post processing to the image, as well as inconvenience of all kinds of instruments which rely on optical imaging, such as satellite remote sensing system, aerial photo system, outdoor monitoring system and object identification system. That is the reason why the image need enhancement and restoration for the improvement of the visual effects and convenience of post processing. This paper sums up two kinds of image dehazing methods, which are the methods based on image enhancement and based on the physics model. After that, some algorithms and research results are presented, followed by quantitative and qualitative evaluations of these techniques. Finally, the research progress is summarized and future research directions are suggested.
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Key words:
- Image dehazing /
- image enhancement /
- atmospheric scatting model /
- image processing
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子空间辨识算法由于能对多输入多输出系统采用统一框架建立状态空间模型,在系统辨识和控制工程领域受到广泛关注 [1]. 有一些子空间算法被提出用于开环辨识工业过程,得到一致估计结果 [2]. 但是,由于过程操作安全性和稳定性的需要,许多工业过程限制在闭环条件下进行辨识实验,由于反馈控制作用的影响,使得过程噪声和输入存在相关性,使得传统开环子空间方法产生辨识偏差 [3]. 闭环系统辨识因此在近年来受到很多关注和探讨 [4],一些闭环子空间辨识算法被相继提出 [5]. 这些算法可被归纳为三类,第一类方法 [6-7]采用辅助变量策略消除噪声影响,保证估计结果的一致性;第二类方法 [8]采用最小二乘法对高阶VARX模型(Vector autoregressive with exogenous inputs model)进行计算得到马尔科夫估计参数,由于VARX模型只包括当前时刻的不可测噪声,该噪声和VARX模型的过去时刻输入无关,从而可保证所得参数的一致性;第三类算法 [9]用噪声预估值代替真实值进行计算保证得到一致估计结果.
基于奇偶空间的闭环子空间辨识算法SIMPCAwc [6]采用过去时刻输入输出数据作为辅助变量来消除噪声,以得到无噪声的输入输出数据,然后从无噪声影响的输入输出数据奇偶空间中提取得到扩展可观测矩阵和下三角形Toeplitz矩阵,从而求得系统矩阵,该方法取得较好辨识精度.然而,文献[7]指出当闭环系统设定点输入激励为不相关白噪声序列时,虽然引入辅助变量与噪声不相关,可以有效地消除噪声,但由于该辅助变量和系统设定点输入激励也不相关,导致从无噪声输入输出数据奇偶空间中提取参数可能同时含有过程模型参数信息和控制器参数信息,因而无法对它们进行区分,从而致使过程模型估计出现偏差.针对SIMPCAwc [6]辨识方法存在的问题,本文通过将输入输出数据正交投影到新息数据的正交补空间来消除噪声,以得到新的无噪声数据矩阵,进而从其对应的奇偶空间中提取得到扩展可观测矩阵和下三角形Toeplitz矩阵. 由于新息数据的正交补空间数据和噪声无关,且同时与系统设定输入激励相关,确保本文方法从新的无噪声奇偶空间中提取的参数只包含过程模型参数,有效地保证估计结果的一致无偏性.由于新息数据为不可测量数据,本文通过模型推导,得到和待辨识状态空间模型等价的VARX模型. 在此基础上,采用最小二乘法对VARX模型进行计算以得到新息的一致估计值. 采用新息一致估计值代替真实值,以完成模型参数估计.为了论证说明本文方法的有效性,严格分析和给出了本文算法保证一致估计的条件.
1. 问题描述
本文研究如下线性离散状态空间过程模型:
$S:\left\{ \begin{align} & x(t+1)=Ax(t)+Bu(t)+w(t) \\ & y(t)=Cx(t)+Du(t)+v(t) \\ \end{align} \right.$
(1) 其中,$x(t)\in {{R}^{{{n}_{x}}}}$,$u(t)\in {{R}^{{{n}_{u}}}}$,$y(t)\in {{R}^{{{n}_{y}}}}$分别为系统状态和过程输入和输出. $v(t)\in {{R}^{{{n}_{y}}}}$和$w(t)\in {{R}^{{{n}_{x}}}}$分别为过程测量噪声. A,B,C,D 分别为相应维数的系统矩阵.本文研究系统在闭环工作条件下,利用系统输入和输出观测数据,辨识对象状态空间(亦称子空间)模型.
由于闭环反馈控制作用的影响,使得过程测量噪声和输入存在相关性,若直接通过模型(1)来辨识系统矩阵,很难消除噪声对辨识结果的不利影响.因此,采用卡尔曼滤波原理 [10],将系统模型(1)等价表示为新息形式
${{S}_{I}}:\left\{ \begin{array}{*{35}{l}} \begin{align} & x(t+1)=Ax(t)+Bu(t)+Ke(t) \\ & y(t)=Cx(t)+Du(t)+e(t) \\ \end{align} \\ \end{array} \right.$
(2) 其中,K为卡尔曼滤波增益,新息$e(t)$为零均值白噪声,当$i＜j$时,新息$e(j)$和输入输出$\{u(i),y(i)\}$不相关.
进一步定义$\bar{A}=A-KC$和$\bar{B}=B-KD$,模型(2)可被等价描述为如下预测形式:
${{S}_{P}}:\left\{ \begin{array}{*{35}{l}} \begin{align} & x(t+1)=\bar{A}x(t)+\bar{B}u(t)+Ky(t) \\ & y(t)=Cx(t)+Du(t)+e(t) \\ \end{align} \\ \end{array} \right.$
(3) 其中,假设$\bar{A}$的特征值严格位于单位圆内.
定义过去和将来水平数分别为p和f,过去和将来输入向量分别为$u_p(t)=[u(t-p)^{\textrm T}$ $\cdots$ $u(t-2)^{\textrm T}$ $u(t-1)^{\textrm T}]^{\textrm T}$和$u_f(t)=[u(t)^{\textrm T}$ $\cdots$ $u(t+f-2)^{\textrm T}$ $u(t+f-1)^{\textrm T}]^{\textrm T}$,定义过去和将来输入Hankel 矩阵Up $=$ $[u_p(t)^{\textrm T}$ $\cdots$ $u_p(N)^{\textrm T}]^{\textrm T}$和$U_f=[u_f(t)^{\textrm T}$ $\cdots$ $u_f(N)^{\textrm T}]^{\textrm T}$,输出和新息数据做类似定义.
对式(3)进行迭代可得:
$x(t)={{\bar{A}}^{p}}x(t-p)+{{\bar{L}}_{1}}{{u}_{p}}(t)+{{\bar{L}}_{2}}{{y}_{p}}(t)$
(4) 其中,扩展可观性矩阵分别表示为 $\bar{L}_1=[\bar{A}^{p-1}\bar{B}$ $\cdots$ $\bar{A}\bar{B}$ $\bar{B}]$,$\bar{L}_2=[\bar{A}^{p-1}K$ $\cdots$ $\bar{A}K$ $K]$.初始状态为$x(t$ $-$ $p)$.当p充分大时,$x(t-p)$可被忽略,将式(4)带入式(3})得到等价VARX模型
${{S}_{V}}:y(t)=C{{\bar{L}}_{1}}{{u}_{p}}(t)+C{{\bar{L}}_{2}}{{y}_{p}}(t)+e(t)$
(5) 本文将采用模型(2)和(5)对系统矩阵进行辨识.定义$X_p=[x(t-p)$ $\cdots$ $x(t-p+N-1)]$ 和Xf $=$ $[x(t)$ $\cdots$ $x(t+N-1)]$.通过对式(2)进行迭代可得:
$Y(t)={{\Gamma }_{f}}{{X}_{f}}+{{H}_{f}}{{U}_{f}}+{{G}_{f}}{{E}_{f}}$
(6) 其中,扩展可观测矩阵为$\Gamma_f=[C^{\textrm T}$ $\cdots$ $(CA^{f-1})^{\textrm T}]^{\textrm T}$.下三角形Toeplitz矩阵分别为
$\begin{align} & {{H}_{f}}=\left[ \begin{matrix} D & \cdots & \cdots & 0 \\ CB & \cdots & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ C{{A}^{f-2}} & \cdots & CB & D \\ \end{matrix} \right] \\ & {{G}_{f}}=\left[ \begin{matrix} 0 & \cdots & \cdots & 0 \\ C & \cdots & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ C{{A}^{f-2}} & \cdots & C & 0 \\ \end{matrix} \right] \\ \end{align}$
2. 闭环子空间辨识算法和一致性分析
2.1 闭环子空间辨识算法
在式(6)的基础上,通过同时计算扩展可观测矩阵$\Gamma_f$和下三角形Toeplitz矩阵Hf实现对系统矩阵的辨识.首先将输入数据移至式(6)的左侧,得到:
$[I-{{H}_{f}}]{{W}_{f}}={{\Gamma }_{f}}{{X}_{f}}+{{G}_{f}}{{E}_{f}}$
(7) 其中,$W_f=[Y_f^{\textrm T}U_f^{\textrm T}]^{\textrm T}$,对Wp做同样定义.
为求解式(7)得到$\Gamma_f$和Hf的估计值,需要消除未知状态和新息的影响.通过在式(7)的左右侧同时引入$\Gamma_f$的正交补向量$\Gamma_f^{\bot}$,由于$(\Gamma_f^{\bot})^{\textrm T}\Gamma_f=0$
$\left[ {{(\Gamma _{f}^{\bot })}^{\text{T}}}-{{(\Gamma _{f}^{\bot })}^{\text{T}}}{{H}_{f}} \right]{{W}_{f}}={{(\Gamma _{f}^{\bot })}^{\text{T}}}{{G}_{f}}{{E}_{f}}$
(8) 通过将式(8)正交投影到Ef的正交补空间来消除新息噪声,
$\left[ {{(\Gamma _{f}^{\bot })}^{\text{T}}}-{{(\Gamma _{f}^{\bot })}^{\text{T}}}{{H}_{f}} \right]{{W}_{f}}\Pi _{{{E}_{f}}}^{\bot }=0$
(9) 其中,$\Pi_{E_f}^{\bot}=I_N-E_f^{\textrm T}(E_fE_f^{\textrm T})^{-1}E_f$是Ef的正交补.由于$\lim_{N \to \infty }E_f\Pi_{{E}_f}^{\bot}=0$和$\lim_{N \to \infty }R_f\Pi_{{E}_f}^{\bot}=R_f$ $\neq$ $0$.根据文献[7]结论可知,无噪声数据块$W_f\Pi_{E_f}^{\bot}$的奇偶空间只包含过程模型参数信息,不会包括控制器模型参数信息.因此,通过$W_f\Pi_{E_f}^{\bot}$的奇偶空间可得到过程模型参数的一致估计值.
由于新息Ef未知,不能进行模型参数估计.这里利用等价辅助模型(5)计算Ef的估计值$\hat{E}_f$,再采用估计值代替真实值进行后续计算.定义新的过去输入输出Hankel 矩阵$U_p(t,N+f)=[u_p(t)$ $\cdots$ $u_p(N+f)]$,$Y_p(t,N+f)=[y_p(t)$ $\cdots$ $y_p(N+f)]$,新息数据做同样定义.定义$W_p(t,N+f)=$ $[U_p^{\textrm T}(t,$ $N+f)$ $Y_p^{\textrm T}(t,N+f)]^{\textrm T}$ 和$Y(t,N+f)=$ $[y(t)$ $\cdots$ $y(N+f)]$.同时定义 $\theta= [C\bar{L}_1$ $C\bar{L}_2]$.采用最小二乘法对式(5)进行计算,可得:
$\hat{\theta }=Y(t,N+f)W_{p}^{\dagger }(t,N+f)$
(10) 其中,
$\begin{align} & W_{p}^{\dagger }(t,N+f)= \\ & W_{p}^{\text{T}}(t,N+f){{[{{W}_{p}}(t,N+f)W_{p}^{\text{T}}(t,N+f)]}^{-1}} \\ \end{align}$
由于估计值$\hat{\theta}$和真实值的误差为
$\Delta \theta =\hat{\theta }-\theta =E(t,N+f)W_{p}^{\dagger }(t,N+f)$
(11) 其中,
$E(t,N+f)=[e(t)\cdots (N+f)]$
由于$\lim_{N \to \infty }E(t,N+f)W_p^{\textrm T}(t,N+f)=0$,若$\lim_{N \to \infty }W_p(t,N+f)W_p^{\textrm T}(t,N+f)>0$ (可作为默认成立条件),则$\hat{\theta}$是一致估计值.
将一致估计值$\hat{\theta}$带入估计值$\hat{Y}(t,N+f)$,可以得到一致估计值$\hat{E}(t,N+f)$ (证明见 第2.2节).
$\begin{align} & \hat{E}(t,N+f)=Y(t,N+f)-\hat{Y}(t,N+f)= \\ & Y(t,N+f)-\hat{\theta }{{W}_{p}}(t,N+f)= \\ & Y(t,N+f)[{{I}_{N+f}}-W_{p}^{\dagger }(t,N+f){{W}_{p}}(t,N+f)]= \\ & Y(t,N+f)\Pi _{{{W}_{p}}(t,N+f)}^{\bot } \\ \end{align}$
(12) 其中,
$\begin{align} & \Pi _{{{W}_{p}}(t,N+f)}^{\bot }={{I}_{N+f}}-W_{p}^{\text{T}}(t,N+f)\times \\ & {{\left[ {{W}_{p}}(t,N+f)W_{p}^{\text{T}}(t,N+f) \right]}^{-1}}{{W}_{p}}(t,N+f) \\ \end{align}$
定义
$\Pi _{{{W}_{p}}(t,N+f)}^{\bot }=[{{\beta }_{1}}\cdots {{\beta }_{N+f}}]$
(13) 则未来噪声Hankel矩阵的一致估计值可通过以下方式重构得到:
${{{\hat{E}}}_{f}}=Y(t,N+f)\left[ \begin{matrix} {{\beta }_{1}} & \cdots & {{\beta }_{N}} \\ {{\beta }_{2}} & \cdots & {{\beta }_{N+1}} \\ \vdots & \ddots & \vdots \\ {{\beta }_{f}} & \cdots & {{\beta }_{f+N}} \\ \end{matrix} \right]$
(14) 进一步将Ef用其一致估计值代替,可得一致估计值$W_f\Pi_{\hat{E}_f}^{\bot}$.通过SVD分解得到:
$\begin{align} & {{W}_{f}}\Pi _{{{{\hat{E}}}_{f}}}^{\bot }= \\ & \left[ {{{\hat{U}}}_{1}}\hat{U}_{1}^{\bot } \right]\left[ \begin{matrix} {{{\hat{\Sigma }}}_{1}} & 0 \\ {{{\hat{\Sigma }}}_{2}} & 0 \\ \end{matrix} \right]\left[ \begin{matrix} \begin{matrix} \hat{V}_{1}^{\text{T}} \\ {{(\hat{V}_{1}^{\bot })}^{\text{T}}} \\ \end{matrix} \\ \end{matrix} \right]={{{\hat{U}}}_{1}}{{{\hat{\Sigma }}}_{1}}\hat{V}_{1}^{\text{T}} \\ \end{align}$
(15) 其中,$\hat{U}_1$是$W_f\Pi_{\hat{E}_f}^{\bot}$的前$n_x+fn_u$个特征向量,则$[(\Gamma_f^{\bot})^{\textrm T}-(\Gamma_f^{\bot})^{\textrm T}H_f]$的一致估计值如下(证明见第2.2节):
$\left[ {{(\hat{\Gamma }_{f}^{\bot })}^{\text{T}}}-{{(\hat{\Gamma }_{f}^{\bot })}^{\text{T}}}{{{\hat{H}}}_{f}} \right]={{(\hat{U}_{1}^{\bot })}^{\text{T}}}$
(16) 定义$\hat{U}_1^{\bot}=[P_1^{\textrm T}P_2^{\textrm T}]^{\textrm T}$,其中,$P_1=\hat{U}_1^{\bot}(1:fn_y$,$:)$,$P_2=\hat{U}_1^{\bot}(1+fn_y:end,:)$ (Matlab表示).可得:
$\hat{\Gamma }_{f}^{\bot }={{P}_{1}}$
(17) $-{{(\hat{\Gamma }_{f}^{\bot })}^{\text{T}}}{{{\hat{H}}}_{f}}=P_{2}^{\text{T}}$
(18) 由于$(\Gamma_f^{\bot})^{\bot}=\Gamma_f$,根据式(17),可得:
${{{\hat{\Gamma }}}_{f}}={{(\hat{\Gamma }_{f}^{\bot })}^{\bot }}={{I}_{f{{n}_{y}}}}-{{P}_{1}}{{(P_{1}^{\text{T}}{{P}_{1}})}^{-1}}P_{1}^{\text{T}}$
(19) 为了得到Hf的估计值,做如下定义:
$-P_{1}^{\text{T}}=[{{\phi }_{1}}\cdots {{\phi }_{f}}]$
(20) $P_{2}^{\text{T}}=[{{\varphi }_{1}}\cdots {{\varphi }_{f}}]$
(21) 其中,${{\phi }_{i}}\in {{R}^{(i{{n}_{y}}-{{n}_{x}})\times {{n}_{y}}}},{{\varphi }_{i}}\in {{R}^{(i{{n}_{y}}-{{n}_{x}})\times {{n}_{u}}}}$.
将式(20)和式(21)带入式(18),可得:
$\left[ \begin{matrix} {{\phi }_{1}} & \cdots & {{\phi }_{f-1}} & {{\phi }_{f}} \\ {{\phi }_{2}} & \cdots & {{\phi }_{f}} & 0 \\ \vdots & \vdots & \ddots & \vdots \\ {{\phi }_{f}} & 0 & \cdots & 0 \\ \end{matrix} \right]{{H}_{f1}}=\left[ \begin{matrix} {{\varphi }_{1}} \\ \phi {{i}_{2}} \\ \vdots \\ {{\phi }_{f}} \\ \end{matrix} \right]$
(22) 其中,$H_{f1}=[D^{\textrm T}(CB)^{\textrm T}\cdots(CA^{f-2}B)^{\textrm T}]$.
采用最小二乘法可得$H_{f1}$的一致估计如下:
${{{\hat{H}}}_{f1}}={{\left[ \begin{matrix} {{\phi }_{1}} & \cdots & {{\phi }_{f-1}} & {{\phi }_{f}} \\ {{\phi }_{2}} & \cdots & {{\phi }_{f}} & 0 \\ \vdots & \vdots & \ddots & \vdots \\ {{\phi }_{f}} & 0 & \cdots & 0 \\ \end{matrix} \right]}^{\dagger }}\left[ \begin{matrix} {{\varphi }_{1}} \\ \phi {{i}_{2}} \\ \vdots \\ {{\phi }_{f}} \\ \end{matrix} \right]$
(23) 系统矩阵估计值$\hat{A}$和$\hat{C}$可直接从$\hat{\Gamma}$中提取,即
$\hat{C}=\hat{\Gamma }(1:{{n}_{y}},1:{{n}_{x}})$
(24) $\hat{A}={{{\hat{\Gamma }}}^{\dagger }}(1+{{n}_{y}}:f{{n}_{y}},1:{{n}_{x}})\hat{\Gamma }(1:(f-1){{n}_{y}},1:{{n}_{x}})$
(25) 由于
${{H}_{f1}}=\left[ \begin{matrix} {{I}_{{{n}_{y}}}} & 0 \\ 0 & \Gamma (1:(f-1){{n}_{y}},1:{{n}_{x}}) \\ \end{matrix} \right]\left[ \begin{matrix} \begin{matrix} D \\ B \\ \end{matrix} \\ \end{matrix} \right]$
(26) 系统矩阵估计值$\hat{B}$和$\hat{D}$可采用最小二乘法从$\hat{H}_{f1}$中计算得到:
$\left[ \begin{matrix} \begin{matrix} {\hat{D}} \\ {\hat{B}} \\ \end{matrix} \\ \end{matrix} \right]={{\left[ \begin{matrix} {{I}_{{{n}_{y}}}} & 0 \\ 0 & \hat{\Gamma }(1:(f-1){{n}_{y}},1:{{n}_{x}}) \\ \end{matrix} \right]}^{\dagger }}{{{\hat{H}}}_{f1}}$
(27) 本文基于新息估计和正交投影的闭环子空间辨识方法(Closed-loop subspace identification method using innovation estimation and orthogonal projection,CSIMIEOP)可总结如下:
步骤 1. 由式(10)求解$\hat{\theta}$,再由式(12)计算$\hat{E}(t,N+f)$,采用式(14)构造$\hat{E}_f$.
步骤 2. 通过SVD分解对式(15)进行计算.
步骤 3. 通过式(19})和式(23)计算估计值$\hat{\Gamma}$和$\hat{H}_{f1}$.
步骤 4. 通过式(24),式(25)和式(27)求解系统矩阵$\hat{A}$,$\hat{B}$,$\hat{C}$ 和 $\hat{D}$.
2.2 闭环一致条件分析
由第2.1节闭环子空间辨识算法可知,本文采用新息估计和正交投影消除噪声,得到一致估计结果. 辨识结果是否一致取决于$\hat{E}(t,N+f)$ 和$[(\hat{\Gamma}_f^{\bot})^{\textrm T}$ $-$ $(\hat{\Gamma}_f^{\bot})^{\textrm T}\hat{H}_f]$是否一致,本文对它们的一致估计条件进行分析和说明,给出如下定理.
定理 1. 若$$\lim_{N \to \infty }W_p(t,N+f)W_p^{\textrm T}(t,N+f)>0$$ 则$\hat{E}(t,N+f)$ 为一致估计值.
证明. 由式(12)可知,估计值$\hat{E}(t,N+f)$和真实值的误差为
$\begin{align} & \Delta E(t,N+f)=\hat{E}(t,N+f)-E(t,N+f)= \\ & Y(t,N+f)\Pi _{{{W}_{p}}(t,N+f)}^{\bot }-E(t,N+f)= \\ & [Y(t,N+f)-\hat{Y}(t,N+f)]-E(t,N+f)= \\ & [\theta -\hat{\theta }]{{W}_{p}}(t,N+f)= \\ & \Delta \theta {{W}_{p}}(t,N+f)= \\ & E(t,N+f){{W}_{p}}{{(t,N+f)}^{\dagger }}{{W}_{p}}(t,N+f) \\ \end{align}$
(28) 由于
$\underset{N\to \infty }{\mathop{\lim }}\,E(t,N+f)W_{p}^{\text{T}}(t,N+f)=0$
若
$\underset{N\to \infty }{\mathop{\lim }}\,{{W}_{p}}(t,N+f)W_{p}^{\text{T}}(t,N+f)＞0$
则
$\underset{N\to \infty }{\mathop{\lim }}\,\Delta E(t,N+f)=0$
从而可知$\hat{E}(t,N+f)$为一致估计值.
定理 2. 若系统可控可观,且$E_fE_f^{\textrm T}>0$ 和
$\bar{E}\left\{ \left[ \begin{matrix} {{u}_{p}}(t) \\ {{u}_{f}}(t) \\ {{e}_{p}}(t) \\ {{e}_{f}}(t) \\ \end{matrix} \right]{{\left[ \begin{matrix} {{u}_{p}}(t) \\ {{u}_{f}}(t) \\ {{e}_{p}}(t) \\ {{e}_{f}}(t) \\ \end{matrix} \right]}^{\text{T}}} \right\}＞0$
则$[(\hat{\Gamma}_f^{\bot})^{\textrm T} -(\hat{\Gamma}_f^{\bot})^{\textrm T}\hat{H}_f]$为一致估计值.
证明. 令$\Omega =\left[ \begin{matrix} {{\Gamma }_{f}} & {{H}_{f}} \\ 0 & I \\ \end{matrix} \right]$,由式(9)可知:
$\Omega \left[ \begin{matrix} \begin{matrix} {{U}_{f}} \\ {{X}_{f}} \\ \end{matrix} \\ \end{matrix} \right]\left[ \begin{matrix} {{I}_{N}}-E_{f}^{\text{T}}{{({{E}_{f}}E_{f}^{\text{T}})}^{-1}}{{E}_{f}} \\ \end{matrix} \right]$
(29) 由式(29)可得:
$\begin{array}{l} \mathop {\lim }\limits_{N \to 0} {W_f}\Pi _{{{\hat E}_f}}^ \bot W_f^{\rm{T}} = \\ \Omega \left[ {\begin{array}{*{20}{c}} \begin{array}{l} {U_f}\\ {X_f} \end{array} \end{array}} \right]\left[ {{I_N} - E_f^{\rm{T}}{{({E_f}E_f^{\rm{T}})}^{ - 1}}{E_f}} \right]{\left[ {\begin{array}{*{20}{c}} \begin{array}{l} {U_f}\\ {X_f} \end{array} \end{array}} \right]^{\rm{T}}}{\Omega ^{\rm{T}}} = \\ \Omega \left\{ {{R_1} - {R_2}{{({E_f}E_f^{\rm{T}})}^{ - 1}}R_2^{\rm{T}}} \right\}{\Omega ^{\rm{T}}} \end{array}$
(30) 其中,
$\begin{array}{l} {R_1} = \left[ {\begin{array}{*{20}{c}} {{U_f}{X_f}} \end{array}} \right]{\left[ {\begin{array}{*{20}{c}} {{U_f}{X_f}} \end{array}} \right]^{\rm{T}}}\\ = \bar E\left\{ {\left[ {\begin{array}{*{20}{c}} {{u_f}(t)}\\ {x(t)} \end{array}} \right]{{\left[ {\begin{array}{*{20}{c}} {{u_f}(t)}\\ {x(t)} \end{array}} \right]}^{\rm{T}}}} \right\}\\ {R_2} = \left[ {\begin{array}{*{20}{c}} \begin{array}{l} {U_f}\\ {X_f} \end{array} \end{array}} \right]E_f^{\rm{T}} = \bar E\left\{ {\left[ {\begin{array}{*{20}{c}} {{u_f}(t)}\\ {x(t)} \end{array}} \right]e_f^{\rm{T}}(t)} \right\} \end{array}$
若系统为可控可观系统,则$\Omega$为满秩矩阵
${\rm{rank}}(\Omega ) = {n_x} + f{n_u}$
(31) 由于新息为平稳零均值白噪声系列,可知$E_fE_f^{\rm T}$ $>$ $0$,据文献[11]定理2可知,式(31)中第2项的秩可通过如下计算得到:
$\text{rank}\left\{ \bar{E}\left\{ \left[ \begin{matrix} {{u}_{f}}(t) \\ x(t) \\ {{e}_{f}}(t) \\ \end{matrix} \right]{{\left[ \begin{matrix} {{u}_{f}}(t) \\ x(t) \\ {{e}_{f}}(t) \\ \end{matrix} \right]}^{\text{T}}} \right\} \right\}-f{{n}_{y}}$
(32) 进一步将式(4)带入式(32),可得:
$\begin{align} & \bar{E}\left\{ \left[ \begin{matrix} {{u}_{f}}(t) \\ x(t) \\ {{e}_{f}}(t) \\ \end{matrix} \right]{{\left[ \begin{matrix} {{u}_{f}}(t) \\ x(t) \\ {{e}_{f}}(t) \\ \end{matrix} \right]}^{\text{T}}} \right\}= \\ & \Upsilon \bar{E}\left\{ \left[ \begin{matrix} {{u}_{p}}(t) \\ {{u}_{f}}(t) \\ {{e}_{p}}(t) \\ {{e}_{f}}(t) \\ \end{matrix} \right]{{\left[ \begin{matrix} {{u}_{p}}(t) \\ {{u}_{f}}(t) \\ {{e}_{p}}(t) \\ {{e}_{f}}(t) \\ \end{matrix} \right]}^{\text{T}}} \right\}{{\Upsilon }^{\text{T}}} \\ \end{align}$
(33) 其中,$\Upsilon =\left[ \begin{matrix} 0 & {{I}_{p{{n}_{u}}}} & 0 & 0 \\ {{L}_{1}} & 0 & {{L}_{1}} & 0 \\ 0 & 0 & 0 & {{I}_{p{{n}_{y}}}} \\ \end{matrix} \right]$.
由于系统为可控可观测系统,采用文献[11]定理2可知$\Upsilon$为满秩矩阵.同时如果 式(33)第2项为正定满秩矩阵,根据文献[12]给定的秩条件可得:
$\begin{align} & \text{rank}\left\{ \overline{\text{E}}\left\{ \left[ \begin{matrix} {{u}_{f}}(t) \\ x(t) \\ {{e}_{f}}(t) \\ \end{matrix} \right]{{\left[ \begin{matrix} {{u}_{f}}(t) \\ x(t) \\ {{e}_{f}}(t) \\ \end{matrix} \right]}^{\text{T}}} \right\} \right\}= \\ & f({{n}_{u}}+{{n}_{y}})+{{n}_{x}} \\ \end{align}$
(34) 因此,矩阵(32)为满秩矩阵.
$\text{rank}\left( \left[ {{R}_{1}}-{{R}_{2}}{{({{E}_{f}}E_{f}^{\text{T}})}^{-1}}R_{2}^{\text{T}} \right] \right)=f{{n}_{u}}+{{n}_{x}}$
(35) 由式(30)、式(32)和式(35)可知\vskip0.1mm\noindent
$\text{rank}\left( \underset{N\to 0}{\mathop{\lim }}\,{{W}_{f}}\Pi _{{{{\hat{E}}}_{f}}}^{\bot }W_{f}^{\text{T}} \right)=f{{n}_{u}}+{{n}_{x}}$
(36) 则
$\begin{align} & \text{rank}\left( \underset{N\to 0}{\mathop{\lim }}\,{{W}_{f}}\Pi _{{{{\hat{E}}}_{f}}}^{\bot } \right)= \\ & \text{rank}\left( \underset{N\to 0}{\mathop{\lim }}\,{{W}_{f}}\Pi _{{{{\hat{E}}}_{f}}}^{\bot }W_{f}^{\text{T}} \right)=f{{n}_{u}}+{{n}_{x}} \\ \end{align}$
(37) 由以上秩条件可知,$\lim_{N \to \infty }W_f\Pi_{\hat{E}_f}^{\bot}$的非零特征向量个数为$f(n_u+n_y)-n_x-fn_u=fn_y-n_x$.因此$\lim_{N \to \infty }\hat{U}_1^{\bot}$ 是$\lim_{N \to \infty }W_f\Pi_{\hat{E}_f}^{\bot}$的零特征向量.则
$\underset{N\to 0}{\mathop{\lim }}\,\hat{U}_{1}^{\bot }{{W}_{f}}\Pi _{{{{\hat{E}}}_{f}}}^{\bot }=0$
(38) 由于${\rm rank}(\Gamma_f^{\bot})=fn_y-n_x$ [13],因此
$\text{rank}\left( [{{(\Gamma _{f}^{\bot })}^{\text{T}}}-{{(\Gamma _{f}^{\bot })}^{\text{T}}}{{H}_{f}}] \right)=f{{n}_{y}}-{{n}_{x}}$
(39) 同时,由于
$\underset{N\to 0}{\mathop{\lim }}\,\left[ {{(\Gamma _{f}^{\bot })}^{\text{T}}}-{{(\Gamma _{f}^{\bot })}^{\text{T}}}{{H}_{f}} \right]{{W}_{f}}\Pi _{{{{\hat{E}}}_{f}}}^{\bot }=0$
(40) 所以$[(\Gamma_f^{\bot})^{\textrm T}-(\Gamma_f^{\bot})^{\textrm T}H_f]$和$\lim_{N \to \infty }\hat{U}_1^{\bot}$满足
$\begin{align} & rowspace\left( [{{(\Gamma _{f}^{\bot })}^{\text{T}}}-{{(\Gamma _{f}^{\bot })}^{\text{T}}}{{H}_{f}}] \right)= \\ & rowspace\left( \underset{N\to 0}{\mathop{\lim }}\,\hat{U}_{1}^{\bot } \right) \\ \end{align}$
(41) 因此,$[(\hat{\Gamma}_f^{\bot})^{\textrm T} -(\hat{\Gamma}_f^{\bot})^{\textrm T}\hat{H}_f]=(\hat{U}_1^{\bot})^{\textrm T}$,由于$\hat{E}_f$是一致估计值,可知$(\hat{U}_1^{\bot})^{\textrm T}$为一致估计值,则$[(\hat{\Gamma}_f^{\bot})^{\textrm T}$ $-$ $(\hat{\Gamma}_f^{\bot})^{\textrm T}\hat{H}_f]$为一致估计值.
由于$[(\hat{\Gamma}_f^{\bot})^{\textrm T} -(\hat{\Gamma}_f^{\bot})^{\textrm T}\hat{H}_f]$为一致估计值,根据平移变换法求取系统矩阵的一致不变性,可知系统矩阵估计值也为一致估计值.
2.3 闭环一致条件合理性分析
第2.2节定理2的两个假设条件涉及未知新息信息,以上假设是否合理,将直接决定本文方法能否取得一致估计结果.
由于模型(1)可表示为模型(2),则新息的协方差可表示为$\bar{E}[e(t)e^{\textrm T}(t)]=CP^{\textrm T}C+R_3$ (具体表述可参考文献[10]第5章),其中P为半正定矩阵,对于系统噪声有$R_3=\bar{E}[v(t)v^{\textrm T}(t)]>0$,则$\bar{E}[e(t)e^{\textrm T}(t)]$ $>$ $0$,可知$E_fE_f^{\textrm T}>0$.
对于假设2,由于
$\begin{align} & \underset{N\to 0}{\mathop{\lim }}\,{{W}_{p}}(t,N+f)W_{p}^{\text{T}}(t,N+f)= \\ & \bar{E}\left\{ \left[ \begin{matrix} {{u}_{p}}(t) \\ {{y}_{p}}(t) \\ \end{matrix} \right]{{\left[ \begin{matrix} {{u}_{p}}(t) \\ {{y}_{p}}(t) \\ \end{matrix} \right]}^{\text{T}}} \right\} \\ \end{align}$
(42) 进一步,由式(2)可知
$\begin{align} & {{y}_{p}}(t)={{\Gamma }_{p}}[{{L}_{1}}u_{p}^{\text{T}}(t-p)+{{L}_{2}}e_{p}^{\text{T}}(t-p)]+ \\ & {{H}_{p}}{{u}_{p}}(t)+{{{\bar{G}}}_{p}}{{e}_{p}}(t) \\ \end{align}$
(43) 其中,扩展可观性矩阵分别表示为 ${L}_1=[{A}^{p-1}{B}$ $\cdots$ ${A}{B}$ ${B}]$,${L}_2=[{A}^{p-1}K$ $\cdots$ ${A}K$ $K]$,下三角形Toeplitz矩阵为
${{H}_{f}}=\left[ \begin{matrix} I & \cdots & \cdots & 0 \\ C & \cdots & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ C{{A}^{f-2}} & \cdots & C & I \\ \end{matrix} \right]$
将式(43)带入式(42),可得:
$\begin{align} & \bar{E}\left[ \begin{matrix} {{u}_{p}}(t) \\ {{y}_{p}}(t) \\ \end{matrix} \right]{{\left[ \begin{matrix} {{u}_{p}}(t) \\ {{y}_{p}}(t) \\ \end{matrix} \right]}^{\text{T}}}= \\ & \Xi \bar{E}\left[ \begin{matrix} {{u}_{p}}(t-p) \\ {{u}_{p}}(t) \\ {{e}_{p}}(t-p) \\ {{e}_{f}}(t) \\ \end{matrix} \right]{{\left[ \begin{matrix} {{u}_{p}}(t-p) \\ {{u}_{p}}(t) \\ {{e}_{p}}(t-p) \\ {{e}_{f}}(t) \\ \end{matrix} \right]}^{\text{T}}}{{\Xi }^{\text{T}}} \\ \end{align}$
(44) 其中,$\Xi=\left[\begin{array}{cccc} 0&I_{pn_u}&0&0\\Gamma_{P}L_1&H_p&\Gamma_{P}L_2&\bar{G}_p \end{array}\right]$.
若系统可控可观,由文献[11]定理2可知,${\rm rank}\left(\Xi\right) =p(n_u+n_y)$,$\Xi$是满秩矩阵,则式(42)正定的充分必要条件为式(44)中第2项正定,由于变量p和f可取任何值,则假设2成立的充分必要条件为$\lim_{N \to \infty }W_{p+f}W_{p+f}^{\textrm T}>0$.可通过验证$\lim_{N \to \infty }W_{p+f}W_{p+f}^{\textrm T}>0$来确定假设2是否成立.即若$\lim_{N \to \infty }W_{p+f}W_{p+f}^{\textrm T}>0$,则假设2成立.
3. 仿真研究
考虑文献[6]中研究的闭环系统
$y(t)-0.9y(t-1)=u(t-1)+e(t)+0.9e(t-1)$
(45) 其中,反馈控制结构为$u(t)=- 0.6y(t)+r(t)$.过程噪声设置为方差为0.2的白噪声序列.
对系统设定输入激励$r(k)$为单位方差白噪声序列和相关序列两种情况进行研究. 其中相关序列设为$r(t)=(1+0.8q^{-1}+0.6q^{-2})r_0(t)$,$r_0(t)$为单位方差白噪声序列. 过去和未来水平数均设置为10.在不同数据长度$N\in[200,8 000]$的情况下进行1 000次Monte Carlo仿真. 本文方法的辨识结果与SIMPCAwc算法 [6]进行比较,当$r(t)$为单位方差白噪声系列时,系统极点(真实值为0.9)的估计平均值如图 1所示,系统极点的估计标准方差如图 2 所示.当系统设定输入$r(t)$为相关系列时,系统极点的平均值如图 3所示,系统极点的估计标准方差如图 4所示.
图 1 系统设定输入激励为白噪声的系统极点估计平均值Fig. 1 Mean value of estimated poles for white noise setpoint excitation
图 2 系统设定输入激励为白噪声的系统极点估计标准方差Fig. 2 Standard deviation of estimated poles for white noise setpoint excitation
图 3 系统设定输入激励为相关序列的系统极点估计平均值Fig. 3 Mean value of estimated poles for correlated quasi-stationary setpoint excitation
图 4 系统设定输入激励为相关序列的系统极点估计标准方差Fig. 4 Standard deviation of estimated poles for correlated quasi-stationary setpoint excitation从以上结果可以看出,当系统设定输入为单位方差白噪声系列时,SIMPCAwc算法得出有偏估计结果;只有当系统设定输入激励为相关序列时,SIMPCAwc算法才能保证无偏估计结果.本文CSIMIEOP算法对系统设定输入激励为无关序列和相关序列时均可得到一致无偏结果,并且估计精度优于SIMPCAwc算法.
4. 结论
本文提出一种基于新息估计和正交投影的闭环子空间辨识算法,对系统设定点输入激励为白噪声无关序列和相关序列的情况均可得到一致无偏估计结果,并且相对于近期有关文献给出的方法如SIMPCAwc算法 [6],能进一步提高辨识精度.同时,严格分析和证明了本文算法保证一致估计的条件.最后通过仿真实例验证了本文方法的有效性和优越性.
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