Question

Problem 1 A certain dynamic process is governed by the following differential equations: 121 +22+ u where xo = const is an exogenous variable. Assuming ro is a measured quantity, a control law is required, which 5, and . Places the closed loop poles at s =-8 . Forces T2 to zero in the steady state. Problem 3 Suppose that the exogenous i is of c2, i.e, nput zo is white noise with spectral density v, and that the only measuremert where w is white noise with spectral density w = 1. (a) Give the differential equations equations that govern the Kalman filter that produc mum) estimates o of the state variables 1, 2 (b) Using the steady state variance equation, find a relationship between the Kalman filter not depend on the spectral density v s least-squares (opti- gains that dos

Answer 1

01) To2 and such that lose oop pole stsen is located at S-8135 22 become zeru assted state) Xj hsxihing above closed loop system in malar form, e 2 2-92 GI-2 え] 3 ,@hc re λ, ad2_ are d问gonal elemen卜 pole main Determ]のaatcios eet (oop 지.dz= prodacトイeigen valyen hie have つ 89-39 -2+92 = 52

3 A135 3 01 st3 s+3-5 Now Xt (s) (-Go +5+13)名15) 大21s) - ! rm as im er r) (-@fSH3)스 lwn SX 3 Hence tne centel vanable shoudd bet the femm

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