概述
线性二次型规划LQR中二次型期望等于迹
在吴恩达LQG问题的笔记中，有公式为：
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mathbb{E}[w_t^TPhi_{t+1}w_t]=Tr(Sigma_tPhi_{t+1}) quad with quad w_tsimmathcal{N}(0,Sigma_t)
E[wtTΦt+1wt]=Tr(ΣtΦt+1)withwt∼N(0,Σt)
证明如下：
- 引入等式： T r ( x x T A ) = T r ( x T A x ) = x T A x Tr(xx^TA)=Tr(x^TAx)=x^TAx Tr(xxTA)=Tr(xTAx)=xTAx
- 由上述等式
(1) E [ w t T Φ t + 1 w t ] = E [ T r ( w t w t T Φ t + 1 ) ] = T r ( E [ w t w t T Φ t + 1 ] ) = T r ( E [ w t w t T ] Φ t + 1 ) begin{aligned} mathbb{E}[w_t^TPhi_{t+1}w_t]&=mathbb{E}[Tr(w_tw_t^TPhi_{t+1})]\ &=Tr(mathbb{E}[w_tw_t^TPhi_{t+1}])\ &=Tr(mathbb{E}[w_tw_t^T]Phi_{t+1})tag{1} end{aligned} E[wtTΦt+1wt]=E[Tr(wtwtTΦt+1)]=Tr(E[wtwtTΦt+1])=Tr(E[wtwtT]Φt+1)(1)
其中 w t = [ w t 1 , w t 2 , ⋯   , w t n ] T w_t=[w_{t_1},w_{t_2},cdots,w_{t_n}]^T wt=[wt1,wt2,⋯,wtn]T，且 E [ w t 1 ] = 0 mathbb{E}[w_{t_1}]=0 E[wt1]=0
(2) w t w t T = [ w t 1 w t 1 w t 1 w t 2 ⋯ w t 1 w t n w t 2 w t 1 w t 2 w t 2 ⋯ w t 2 w t n ⋮ ⋮ ⋱ ⋮ w t n w t 1 w t n w t 2 ⋯ w t n w t n ] begin{aligned} w_tw_t^T=left[ begin{array}{cccc} w_{t_1}w_{t_1} & w_{t_1}w_{t_2} & cdots & w_{t_1}w_{t_n} \ w_{t_2}w_{t_1} & w_{t_2}w_{t_2} & cdots & w_{t_2}w_{t_n} \ vdots & vdots & ddots & vdots \ w_{t_n}w_{t_1} & w_{t_n}w_{t_2} & cdots & w_{t_n}w_{t_n} end{array} right]tag{2} end{aligned} wtwtT=⎣⎢⎢⎢⎡wt1wt1wt2wt1⋮wtnwt1wt1wt2wt2wt2⋮wtnwt2⋯⋯⋱⋯wt1wtnwt2wtn⋮wtnwtn⎦⎥⎥⎥⎤(2)
所以有
(3) E [ w t w t T ] = [ E [ w t i w t j ] ] n × n = [ E [ w t i w t j ] − E [ w t i ] E [ w t j ] ] n × n = [ E [ ( w t i − E [ w t i ] ) ( w t j − E [ w t j ] ) ] ] n × n = [ c o v ( w t i , w t j ) ] n × n = Σ t begin{aligned} mathbb{E}[w_tw_t^T]&=[mathbb{E}[w_{t_i}w_{t_j}]]_{ntimes n}\ &=[mathbb{E}[w_{t_i}w_{t_j}]-mathbb{E}[w_{t_i}]mathbb{E}[w_{t_j}]]_{ntimes n}\ &=[mathbb{E}[(w_{t_i}-mathbb{E}[w_{t_i}])(w_{t_j}-mathbb{E}[w_{t_j}])]]_{ntimes n}\ &=[cov(w_{t_i}, w_{t_j})]_{ntimes n}\ &=Sigma_ttag{3} end{aligned} E[wtwtT]=[E[wtiwtj]]n×n=[E[wtiwtj]−E[wti]E[wtj]]n×n=[E[(wti−E[wti])(wtj−E[wtj])]]n×n=[cov(wti, wtj)]n×n=Σt(3) - 将公式(3)带入公式(1)得到 E [ w t T Φ t + 1 w t ] = T r ( Σ t Φ t + 1 ) quadmathbb{E}[w_t^TPhi_{t+1}w_t]=Tr(Sigma_tPhi_{t+1}) E[wtTΦt+1wt]=Tr(ΣtΦt+1)
最后
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