[試題] 99下 張帆人 隨機控制 期中考
課程名稱︰隨機控制
課程性質︰選修
課程教師︰張帆人
開課學院：電機資訊學院
開課系所︰電機所
考試日期（年月日）︰2011/04/25
考試時限（分鐘）：100分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.Vectors:X～R^n, g～R^m, h～R^m, and s is scalar.
Find (a) δs/δX (b) δg/δX (c) δ(h^T g)/δX (h^T means transpose of h)
(d) If A～R^(m x n), d(A^-1)/dt = ?
2.Consider a system comprised of two sensors, each making a single measurement,
z1=x+v1 ; z2=x+v2 , where x is unknown constant,v1 and v2 are
random, unbiased but correlated measurement errors, that is, E [v1v2] =
ρσ1σ2 where ρ is a correlation coefficient (|ρ| ≦ 1).
(a) Find k1 for optimal estimate x^(x head) = k1v1 + k2v2
(b) Find minimum E[(x~)^2] (Least mean square of error).
3.X=X1+X2.E[X1]=m1,E[X2]=m2.Var[X1]=(σ1)^2,Var[X2]=(σ2)^2.
(a)If X1 and X2 indepedent, find E[X] and Var[X].
(b)If ρ=(E[X1X2]-E[X1]E[X2])/((σ1)*(σ2)), find E[X] and Var[X].
4.Consider the linear system including forcing function inputs:
t
dx/dt = Fx + Gw.
(a)Given x(t0)=x0, show that
t
x(t)=Φ(t,t0)x(t0)+∫Φ(t,τ)L(τ)u(τ)dτ.
t0
(b)Set P(t0)=P0, show that dP/dt=FP+PF^T+GQG
k k+1
5.m_k=(1/k)*Σ x_i,(σ_k)^2=[1/(k-1)]*Σ (x_i-m_k+1)^2
i=1 i=1
(底線表示足碼)
(a) Show that m_k+1=m_k+[1/(k+1)]*(x_k+1-m_k)
(b) Show that (σ_k+1)^2=(1-1/k)*(σ_k)^2+[1/(k+1)]*(x_k+1-m_k)^2
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