EECE 460 Homework, Spring 2007

F. Mrad EECE 460 Spring 2007 Homework # 9 Solution (Exercises 12.3, 8, 11 & 14) Beiz-3- = Referring to Equation (12-18), the state-feedback gain matrix K can be be given by Klee vIle 4B Ati tm (A) = APY Kp A* +a. Atos I The values of 1, 9, and &3 are determined from the desired characteristic eguation: hea BK) = (St24+ja)(st2-ja)(st10) = 5S? 4 /¥s*+ S85 + 200 = Stolys* + West Oy Thus, WH/t, Wn =be, Ws = 200 Then a7 afF of @ ele foo eA) (| +141 e a 1] +80) 0 0 142% po F. Mrad EECE 460 Spring 2007 =|-5 sp 9 —~F =—¢3 WF Since : a) f { [e ae atela| er mw f =-f/ 64 we have the desired state-feedback gain matrix K as follows: oo! i 179) FF Ke[o@ ry]} / ft ma -3 9 F f -l 66 “7 <-'3 7 aAP7aeF Os FT ¢. Os f FES =[o o sll oesse 00120 -00120 OleEes —@.8/28 ee f2e 47% SS Ff X/-3 «59 7 -F -€3 “#9 (9B 3996 190.269 = ZA 7E3/ ale OIF) r7026F — apye/g | SEES 23.783/ SSE f 2 ¢P SF =[2r7¢3/ $578/ 2.48/97) F. Mrad EECE 460 Spring 2007 B-12-8. From Figure 12-49 we obtain us &; Cr-%1) —ReX —A32X3 = - Ke +r K=D#, Ra Rs | Noting that the rank of . ef e = ios = -é male ag geln|e 1 = is three, arbitrary pole placement is possible, The characteristic equation for this system is s -=/ é |sZ-al= oo s =f |= staés*+ ss WA A o Ss Sté =e tas s+ 48 +43 =9 a =f,» 4,=5, @z,=0 Since the state equation for the system is already in the controllable cano- nical form, we have T = is The desired characteristic equation is (se atjel(st2je(stiD =S+/4s*+ 605 + 200 Sst rr s*+ obese As F. Mrad EECE 460 Spring 2007 from which we obtain Mele, “2=b?, of =200 K=([%-43) -a@e ot)-4,]T7 = [200-0 be-5 (4-6 ]T =[200 s5 #] The state equation for the designed system is Z=Axt Be = Axt BC-Kx+kr) ™~ = QED E+ BAY Since . oO +f of fo oa f 6 A-BK =|0 0 7 \-|o|[20e ss e]=|e o 7 _“ @~s -é} [/ -200 -é0 =H we have x é i © IIx, re X=! 0 of \ixelt] apr (1) x,| |-200 -00 -/¢||% 200 The output equation is ay ; yal’ @ allx, (2) 43 The unit-step response of the desiqned system can be obtained from Equations (1) and (2) by substituting r = 1(t) and finding y(t). A MATLAB program to obtain the unit-step response curve [y(t) versus t curve] is given on the next Page. : ‘ F. Mrad EECE 460 Spring 2007 >> A=[0 10:00 1: -200 -60 -14] A= o 1 0 0 0 1 -200 -60 -14 >> B=[0: 0: 200] B= 0 0 200 >> C=[100] c= 1 0 0 >> step (A.B.C.D) >> grid F. Mrad EECE 460 Spring 2007 B-12-11. A full-order state observer for the given systemis designed by use of MATLAB. The MATLAB program used for the design of the state observer is given below. fe ***" Design of full-order state observer ***** A=[0 1 00 0 1-5 4 O; C=[1 0 oO; L=[-10 -10' -15]; Ke =acker(A\C\Ly ‘Warning: Pole locations are more than 10% in error. Ke= 35 394 1285 Referring to Equation (12-60), the full-order state observer is given by Zo (4~ QQ Et Bat kes = g| [-35 7 off%| |e oS aa -/{2%6 -6§ 4] %, ed fare F. Mrad EECE 460 Spring 2007 B-12-14. We shall use the MATLAB approach to solve this problem. The first MATLAB program given in the next page determines the state feedback gain matrix BR and the observer gain matrix K, The observer to be designed is a full-order erver. % ***** Determination of K and Ke ***** A=[0 1 0;0 0 1,6 -I1 -6h; B=[001; C=[l 0 of Infly -li 3k K's acker(ABJ) The state feedback gain matrix K and the observer gain matrix Ke thus obtained are as follows: Kk=(# ¢ 7] 12 Ke =| 25> ~F2 The second MATLAB program given below determines the transfer fimetion of the observer controller. % Obtaining transfer function of observer controller — full-order observer Aw[0 1 0,0 0 1-6 -11 6; B= (0,01); |c-fi 0 of K=f4 1 I; Ke = [12,25,-72], AA = A-RetC-BPk; BB = Ke, cC=K; DD=0; [num, den] = ss2tf{AA,BB,CC,DD) 0 1.0000 119.0000 618.0000 én= 1.0000 19,0000 121.0000 257.0000 F. Mrad EECE 460 Spring 2007 The transfer function of the observer controiler is ee) S*tHPs + é/8 —Yts) ~ 33+ 9574/2434 257 The transfer function of the given system in state space form is é sty gs*+ffaté A block diagram of the designed system is shown below. Gi) = sees t éf8 aot (Ps (2/3 +2357 / Se4gsteeis té Notice that the designed system is of sixth order.