CALIFORNIA INSTITUTE OF TECHNOLOGY Control and Dynamical Systems

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1 Control and Dynamical Sytem CDS 0 Problem Set #5 Iued: 3 Nov 08 Due: 0 Nov 08 Note: In the upper left hand corner of the econd page of your homework et, pleae put the number of hour that you pent on thi homework et (including reading).. Conider the block diagram for the following econd-order ytem z c r y a b (a) Compute the tranfer function H yr between the input r and the output y. (b) Show that the following tate pace ytem ha the ame tranfer function, with the appropriate choice of parameter: d x = 0 x 0 r x 2 a 2 a x 2 y = b 2 b x dr Give the value of a i, b i, and d that correpond to the tranfer function you computed in (a). (c) Compute the tranfer function H zr between the input r and the output z. (Hint: It i not H zr =.) 2. Conider the following implified equation of motion for a cruie control ytem (thee are a linearization of the equation from Section 3. in Åtröm and Murray): x 2 m dv = cv bτ F hill, where m = 000 kg i the ma of the vehicle, c = 50 N /m i the vicou damping coefficient, b = 25 i the converion factor between engine torque and the force applied to the vehicle.

2 We model the engine uing a imple firt-order equation dτ = a(τ Tu), where a = 0.2 i the lag coefficient and T = 200 i the converion factor between the throttle input and the teady tate torque. The implet controller for thi ytem i a proportional control, u = k p e, where e = (r v) (r i the reference peed). (a) Draw a block diagram for the ytem, with the engine dynamic and the vehicle dynamic in eparate block and repreented by tranfer function. Label the reference input to the cloed loop ytem a r, the diturbance due to the hill a d, and the output a y (= v). (b) (MATLAB) Contruct the tranfer function H er and H yd for the cloed loop ytem and ue MATLAB to generate the tep repone (tep) and frequency repone (bode) for the each. Make ure to ue the tranfer function computation. (c) Conider a more ophiticated control law of the form dx c = r v, u = k pe k i x c. Thi control law contain an integral term, which ue the controller tate x c to integrate the error. Compute the tranfer function for thi control law and redraw your block diagram from part (a) with the default controller replaced by thi one. (d) (MATLAB) Uing the default gain from previou homework (k p = 0.5 and k i = 0.), ue MATLAB to compute the tranfer function from r to y and plot the tep repone and frequency repone for the ytem. 2

3 Control and Dynamical Sytem CDS 0a Problem Set #5 Iued: 3 Nov 08 Due: 0 Nov 08 Note: In the upper left hand corner of the econd page of your homework et, pleae put the number of hour that you pent on thi homework et (including reading).. Conider the block diagram for the following econd-order ytem z c r y a b (a) Compute the tranfer function H yr between the input r and the output y. (b) Show that the following tate pace ytem ha the ame tranfer function, with the appropriate choice of parameter: d x = 0 x 0 r x 2 a 2 a x 2 y = b 2 b x dr Give the value of a i, b i, and d that correpond to the tranfer function you computed in (a). (c) Compute the tranfer function H zr between the input r and the output z. (Hint: It i not H zr =.) 2. Conider the following implified equation of motion for a cruie control ytem (thee are a linearization of the equation from Section 3. in Åtröm and Murray): x 2 m dv = cv bτ F hill, where m = 000 kg i the ma of the vehicle, c = 50 N /m i the vicou damping coefficient, b = 25 i the converion factor between engine torque and the force applied to the vehicle.

4 We model the engine uing a imple firt-order equation dτ = a(τ Tu), where a = 0.2 i the lag coefficient and T = 200 i the converion factor between the throttle input and the teady tate torque. The implet controller for thi ytem i a proportional control, u = k p e, where e = (r v) (r i the reference peed). (a) Draw a block diagram for the ytem, with the engine dynamic and the vehicle dynamic in eparate block and repreented by tranfer function. Label the reference input to the cloed loop ytem a r, the diturbance due to the hill a d, and the output a y (= v). (b) (MATLAB) Contruct the tranfer function H er and H yd for the cloed loop ytem and ue MATLAB to generate the tep repone (tep) and frequency repone (bode) for the each. Make ure to ue the tranfer function computation. (c) Conider a more ophiticated control law of the form dx c = r v, u = k pe k i x c. Thi control law contain an integral term, which ue the controller tate x c to integrate the error. Compute the tranfer function for thi control law and redraw your block diagram from part (a) with the default controller replaced by thi one. (d) (MATLAB) Uing the default gain from previou homework (k p = 0.5 and k i = 0.), ue MATLAB to compute the tranfer function from r to y and plot the tep repone and frequency repone for the ytem. 3. Åtröm and Murray, Exercie (ÅM08, Exercie 8.6) Conider the linear tate pace ytem dx (a) Show that the tranfer function i = Ax Bu, y = Cx. where b =CB, G() = b n b 2 n2 b n n a n a n, b 2 =CAB a CB,..., b n =CA n B a CA n B a n CB and λ() = n a n a n i the characteritic polynomial for A. (b) Compute the tranfer function for a linear ytem in reachable canonical form and how that it matche the tranfer function given above. 4

5 Control and Dynamical Sytem CDS 20 Problem Set #5 Iued: 3 Nov 08 Due: 0 Nov 08 Note: In the upper left hand corner of the econd page of your homework et, pleae put the number of hour that you pent on thi homework et (including reading).. (ÅM08, Exercie 8.6) Conider the linear tate pace ytem dx = Ax Bu, y = Cx. (a) Show that the tranfer function i where G() = b n b 2 n2 b n n a n a n, b =CB, b 2 =CAB a CB,..., b n =CA n B a CA n B a n CB and λ() = n a n a n i the characteritic polynomial for A. (b) Compute the tranfer function for a linear ytem in reachable canonical form and how that it matche the tranfer function given above. 2. Åtröm and Murray, Exercie Åtröm and Murray, Exercie Chooe one of the following problem below: (a) Åtröm and Murray, Exercie 8.2 (b) Åtröm and Murray, Exercie 8.4 (c) Åtröm and Murray, Exercie DFT DFT 2.8