SE-514 (OPTIMAL CONTROL) OPTIMAL CONTROL FOR SINGLE AND DOUBLE INVERTED PENDULUM. DONE BY: Fatai Olalekan ( Ayman Abdallah (973610)

Transcription

1 SE-54 (OPTIAL CONTROL OPTIAL CONTROL FOR SINGLE AND DOUBLE INVERTED PENDULU DONE BY: Fatai Oaekan (363 Ayman Abdaah (9736 PREPARED FOR: Dr. Sami E-Ferik

2 Tabe of contents Abstract... 3 Introduction... 3 Sine inverted penduum... 4 Description of the System... 4 odein of the system... 5 Linearization... 6 State-Space Representation... 6 Controabiity... 7 Observabiity... 8 Simuation (off-ine... 9 Simuation (Rea-Time... 5 Doube inverted penduum... Description of the System... odein of the system... Linearization... 4 State-Space Representation... 4 Controabiity... 6 Observabiity... 7 Simuation (off-ine... 8 Simuation (Rea-Time Concusion REFERENCES... 36

3 Abstract The project aimed at desinin a inear controer for a sine inverted penduum takin it from its stabe position to the inverted upriht position. A sine inverted penduum consists of a poe mounted on a motor driven cart in such a way that the poe can swin freey in vertica pane. The controer achieves the aim by movin the cart the back and forth on a rai of imited enth by appyin the appropriate of force. The force is appied on the drive by means of compensatin the input votae to the power ampifier of the drivin motor. This is done by introducin the kaman ain and varyin it for effective contro. The desined controer is aso used on a non-inear system but with a inear reationship between the input and the kaman ain, in a rea time impementation. Furthermore, a simiar inear controer is desined to contro a doube inverted penduum. In this instance, a doube inverted penduum consists of a two-stae penduum attached to a motor-drivin cart. The controer desined for both sine and doube inverted penduum ave satisfactory resuts as shown in the state trajectories of both systems. Introduction The contro of an inverted penduum is one of the fundamenta probems in contro fied. The process is non-inear and unstabe with one input sina and severa output sinas. The output of the system depends on the system variabes and the deree of freedom of the system. The aim is to baance a penduum verticay on a motor driven cart. The position of the cart is steered to different positions with a position reference sina. A contro stratey used to stabiize the penduum at the upriht inverted position is the inear quadratic reuator (LQR technique. This technique is effective and it is used in the project. The contro of an inverted penduum is of two fods with two different controers. The swin-up controer, which divere the penduum from the stabe position is one and the stabiization controer is the other, which stabiize the penduum in the unstabe inverted position. In the case of usin a contro technique in a inear mode ike LQR, both controers have to be desined individuay. In this project, ony the stabiization controer was desined. However, the Feedback penduum system s swin-up controer was used for the purpose of the rea time impementation. In this project, the equations of motion for the system is derived and inearized based on certain assumptions and reference position. This is foowed by the desin of the controer and simuation of the sine inverted penduum offine usin a ATLAB Simuink inear mode. A rea time impementation of the Feedback system with the inear controer is aso performed and resuts shown. As a comparison, a simiar controer was desined for a doube inverted penduum and simuated in the same manner. 3

4 Sine inverted penduum Description of the System Where mass of the penduum m mass of the cart enth of the penduum ø ane of penduum from vertica cart position coordinate (horizonta ravitation constant, 9.8 m/s F force appied to the cart The fiure above shows an inverted penduum. The aim is to move the cart aon the direction to a desired point with the penduum in a vertica upriht position. The cart is driven by a DC motor, which is controed by a Kaman fiter. The cart s horizonta position ( and the penduum s ane (, are measured and suppied to the contro system. A disturbance force (F, can be suppied can be suppied on top of the penduum. 4

5 odein of the system The cart, on which the penduum is supported, is abe to move back and forth accordin to the dc motor. We are assumin the movement of the cart wi be smooth and surface friction between the cart and rai is assumed to be minima enouh to be neected. Aso the equiibrium position of the cart is the midde of the track and the correspondin equiibrium ane of the penduum is at its upriht vertica position. A mode of the inverted penduum is used as the basis for contro desin of the rea-time system. The dynamic equations and vaues of the theoretica mode are cacuated to be as cose as possibe to the actua process. In order to obtain the inverted penduum s mode, the system s dynamics is anayzed usin the Larane ethod. Our system is a Two Deree of Freedom System (i.e. and Potentia Enery: V = cos ( Kinetic Enery T = [ ] m ( cos ( sin I Larane Equations: d T T V = F dt d T T V = dt So we et. [ m ] [ cos ] = F cos [ ] [ ( ] I sin = 5

6 Linearization These two equations wi be inearized about = π. Assume that = π θ (θ represents a sma ane from the vertica upward direction. Therefore, cos =, sin = θ, F = u (where u represents the input So that the Equations of otion become: θ [ m ] [ ] = u [ ] [ ] I θ θ = ( State-Space Representation A state-space representation of the inverted penduum dynamics system can be derived from the two previousy inearized equations. Usin these parameters of the Penduum- Cart setup. = 9.8m s, =.7m, m =.k, =.k, I =.3k m We et = θ θ = A Bu u θ θ y = y = C θ θ 6

7 Controabiity The system described by the matrices (A,B can said to be controabe if there eists an unconstrained contro u that can transfer any initia state ( to any other desired ocation (t. For the system = A Bu, the system can be determined to be controabe if the determinant of the controabiity matri is nonzero. >> n =.598e8 System is controabe. 7

8 Observabiity Observabiity refers to the abiity to estimate a state variabe. A system is observabe if, and ony if, there eists a finite time T such that the initia state ( can be determined from the observation history y(t iven the contro u(t. For the same system with output y = C, the system is observabe if the rank of the observabiity matri is 4, which is the fu enth of the observabiity matri. >> m = 4 System is observabe. 8

9 Simuation (off-ine The open-oop behavior of the system can be observed by simuatin a step response to the system. And It is observed with a step input, the system is unstabe. Thus, a controer needs to be desined and impemented to improve and stabiize the system. In order to stabiize the inverted penduum system, a state feedback approach is considered Shown in the bock diaram. A fu-state feedback condition is assumed and the feedback ain, K of the system is to be determined. The feedback matri ain can be cacuated by usin the LQR method, which wi provide with the optima controer vaues. (This script Pots the time response of the system and computes the inverted penduum's system feedback parameters 9

10 (Time response pot

11 (atab output

12 >> K = S = e =.e * i i -.37 P = (Bock Diaram of Fu State Feedback System

13 (Simuaink output 3

14 4

15 Simuation (Rea-Time (Simuink bock diaram for inverted penduum (When cickin on the bue contro system 5

16 (Inverted penduum controer (Cart and penduum contro Gains iven by the Feedback 6

17 (For ane feedback the atan(sin(u[],cos(u[] to take care of quadrant (Output usin Feedback Gains with one disturbance 7

18 (Our Cart and penduum contro Gains (Since our mode was inearized around the upvertica we don't need to take care of quadrant 8

19 (Our Output usin our ode and Gains with one disturbance 9

20 (Our Output usin our ode and Gains with more than one disturbances

21 Doube inverted penduum (Here we are oin to foow the same procedures of sine inverted penduum Description of the System Where m ø ø mass of the st penduum mass of the nd penduum mass of the cart enth of the st penduum enth of the nd penduum ane of st penduum from vertica ane of nd penduum with respect to the st penduum cart position coordinate (horizonta ravitation constant, 9.8 m/s F force appied to the cart

22 odein of the system In order to obtain the doube inverted penduum s mode, the system s dynamics is anayzed usin the Larane ethod. Our system is a Three Deree of Freedom System (i.e., and Potentia Enery: cos( ( cos ( = m m m V Kinetic Enery [ ] [ ] ( ( [ ] ( ( [ ] sin( ( s n cos( ( cos ( sin( ( sin cos( ( cos sin ( cos ( sin ( cos ( = m I m I m T i Larane Equations: F = V T T d dt = V T T d dt = V T T d dt

23 ( ] ( ( [ ] F m m m m m m m m m m = sin( ( ( sin cos( cos( cos So we et. ( ( [ ( ( [ ] ( ( [ ] ( [ ] ( ( ( ( [ ] sin sin( sin sin cos cos cos( cos = m m m m m m m I m m m I m m m [ ] ( ( sin( sin cos cos( = m I ( [ ] ( [ ] m m m I m I m m 3

24 Linearization These three equations wi be inearized about = π. Assume that = π θ and = π (θ represents a sma ane from the vertica upward direction. θ Therefore, cos =, sin = θ, cos =, sin = θ, cos( = θθ, sin( = θ θ,,, F = u (where u represents the input = = = So that the Equations of otion become: ( m m m [( m m ] θ ( m = u [( m ( m m ] I ( m m ( m [ I m ( m ] θ [ ( m ( θ θ ( m m θ ] = [ I m ] θ [ m ] [ I m ( m ] θ [ ( m ( θ θ = I m ] θ State-Space Representation A state-space representation of the doube inverted penduum dynamics system can be derived from the two previousy inearized equations. Usin these parameters of the Penduum-Cart setup. 4

25 et We Bu A = θ θ θ θ θ θ θ θ u = C y = θ = θ θ θ y 5

26 Controabiity The system described by the matrices (A,B can said to be controabe if there eists an unconstrained contro u that can transfer any initia state ( to any other desired ocation (t. For the system = A Bu, the system can be determined to be controabe if the determinant of the controabiity matri is nonzero. >> n = 6

27 -.59e9 System is controabe. Observabiity observabiity refers to the abiity to estimate a state variabe. A system is observabe if, and ony if, there eists a finite time T such that the initia state ( can be determined from the observation history y(t iven the contro u(t. For the same system with output y = C, the system is observabe if the rank of the observabiity matri is 4, which is the fu enth of the observabiity matri. >> m = 7

28 6 System is observabe. Simuation (off-ine The open-oop behavior of the system can be observed by simuatin a step response to the system. And It is observed with a step input, the system is unstabe. Thus, a controer needs to be desined and impemented to improve and stabiize the system. In order to stabiize the doube inverted penduum system, a state feedback approach is considered Shown in the bock diaram. A fu-state feedback condition is assumed and the feedback ain, K of the system is to be determined. The feedback matri ain can be cacuated by usin the LQR method, which wi provide with the optima controer vaues. (This script Pots the time response of the system and computes the inverted penduum's system feedback parameters 8

29 (Time response pot 9

30 (atab output 3

31 >> K = S =.e4 * e = i i i i (Bock Diaram of Fu State Feedback System 3

32 (Simuaink output 3

33 33

34 Simuation (Rea-Time Not accompished (equipments not avaiabe 34

35 Concusion From the resuts of the simuations offine of the sine and doube inverted penduums, it can be concuded that the unstabe trajectory of both systems with just a step input to the system, was controed with the use of the kaman ain to stabiize the trajectory. However, the controabiity and observabiity of the system was verified and found satisfactory before a feedback cosed oop system was created from an open oop system. Durin impementation in rea time situation, it was noticed that the cart responds to a sma disturbance appied to the penduum at the stabe upriht position. The cart moves to the opposite direction to the appied force, to compensate for the appied force and maintain the stabe upriht position of the penduum. Finay, it was observed that the desined controers performed satisfactoriy. Both offine and rea time resuts as shown by the pots of the state variabes, convered after the introduction of the controer. 35

36 REFERENCES. Furuta, K., Yamakita,., and Kobayashi, S. Swin-up contro of inverted penduum usin pseudo-state feedback, J. System. Contro En., 99, 6, pp Wickund,., Kristenson, A., and Astrom, K.J. A new stratey fro swinin up and inverted penduum. Proceedins of IFAC th Word Conress, 99, Vo. 9, pp Astrom, K.J., Furuta, K. Swinin up a penduum by enery contro. IFAC 3 th Word Conress, San Francisco, Caifornia, Rubi, J., Rubio, A., and Aveo, A. Swin-up probem for a sef-erectin doube inverted penduum, IEE Proc.-Contro Theory App., Vo. 49, No., arch. 5. CT Eampe: Inverted Penduum odein, 6. Contro in a ATLAB Environment Diita Penduum (33-5-C, Feedback, Enand,. 7. Kawashima, Takeshi. Swin-up and stabiization of inverted penduum usin ony one sidin mode controer with noninear mode observer. 8. Furuta, T.T.K., Suzuki, S.H.S., and Suiki, A. Swin-up contro of inverted penduum by periodic input Proceedins of IFAC 5 th Word Conress, Barceona, Spain,. 36