ENGINEERING BACKGROUN D F(t) spring (const , k ) Y(t)^H— mas s EL danrping devic e (dashpot) FIGURE 1.4 . If initially y(0) — 0 = dy(0)/dt, the n the Laplace transform y(*) = i F(s) Ms 2 + (i)s + k' EXAMPLE 3 . A typical mode l fo r a linea r tim e invarian t syste m i s a box which has inputs i, outputs #, and internal states x. Th e box is defined by linear maps A, JB, C, D which relate states to inputs and outputs by dx(t)/dt = Ax(t) + Bi(t), x{0) = 0, 0{t) = Cx(t) + £i(0- In Examples 1 and 2 the systems of differential equation s can be written in this first orde r form . Afte r Laplac e transforming , th e i -+ 0 transfer functio n i s gotten from sx(s) - Ax(s) = Bi{s), 0(s) = Cx(s) + Di(s) and is T{s) = D + C(sI-A)~ 1 B. EXAMPLE 4. A BOEIN G 74 7 (BEFOR E TH E CONTRO L SYSTEM I S ADDED). 2 x,y,z positio n coordinates q pitc h rate u,(3,w velocit y coordinates r ya w rate p rol l rate ^ x/u rudsUr \ L aileron, $^ W ZLT a FIGURE 1.5 . Definitio n of aircraft coordinates . 2 Franklin, Feedback control of dynamic systems, Addison-Wesley , Reading , Mass. , 1986, p. 475, Figure 7.29 . Reprinte d wit h permission .

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