STATE RETENTION OF AN INVERTED PENDULUM

Transcription

1 ISSN:9-69 Potluri Krihna Murthy et al, Int.J.Computer echnology & Application,Vol 7 (,6-67 SAE REENION OF AN INVERED PENDULUM Potluri Krihna Murthy Aitant Profeor Department of Electrical and Electronic Engineering Vignan Nirula Intitute of echnology and Science for Women Pedapalakaluru, Guntur Rameh Kumar Patro Aitant Profeor Department of Electrical and Electronic Engineering Vignan Intitute of Information echnology Viakhapatnam Ajit Kumar Mohanty Aitant Profeor Department of Electrical and Electronic Engineering Vignan Intitute of Information echnology Viakhapatnam Abtract he Cart Inverted Pendulum Sytem (CIPS ha been conidered among the mot claical and difficult problem in the field of control engineering. he Inverted Pendulum i conidered among the typical repreentative of a cla of non-minimal ytem with non-linear dynamic. he aim of thi work i to tabilize the Inverted Pendulum uch that poition of the cart on track i controlled intantly and accurately o that pendulum i alway maintained erected in it upright (inverted poition. Inverted Pendulum i inherently untable i.e. if it i left without a tabilizing controller it will not be able to remain in an upright poition when diturbed. Hence PID controller i deigned and ytematic iterative method for the tate feedback deign by chooing weighting matrice key to Linear Quadratic Regulator (LQR deign i preented auming all the tate to be available at the output. Where not all tate variable are available, a tate etimator i to be deigned. So oberver baed tate feedback controller ha been deigned. Keyword PID controller; LQR controller; Oberver baed tate feed back controller. I. INRODUCION If we remember ever trying to balance a broom-tick on our inde finger or the palm of our hand, we had to contantly adjut the poition of our hand to keep the object upright. An Inverted Pendulum doe baically the ame thing. But in cae of an Inverted Pendulum the motion i retricted to one dimenion only, where a in cae of a broom-tick the hand i free to move in any direction. An Inverted Pendulum ha it ma above the pivoted point, which i mounted on a cart which can be moved horizontally. he pendulum i table while hanging downward, but the inverted pendulum i inherently untable and need to be balanced. In thi cae the ytem ha one input - the force applied to the cart, and two output - poition of the cart and the angle of the pendulum, making it a SIMO Sytem. here are mainly three way of balancing an inverted pendulum, viz. (i by applying a torue at the pivoted point, (ii by moving the cart horizontally, and (iii by ocillating the upport rapidly up and down. hi can be achieved by two controller, (i LQR controller, and (ii Oberver baed tate feedback controller for tabilizing Cart Inverted Pendulum Sytem. II. MAHEMAICAL MODELLING : Fig: Schematic diagram of Inverted Pendulum Adding all the force on the cart in the horizontal direction, we have, M b N F ( Adding all the force on the pendulum in the horizontal direction, we have, m ml θ coθ ml θ inθ N ( Subtituting euation ( in euation (, we have, ( b ml θ coθ ml θ in θ F ( Adding all the force along the vertical direction of the pendulum, Pinθ N coθ mg inθ ml θ m coθ ( Conidering um of the moment about the center of gravity (C.G of the pendulum, Pl in θ Nl coθ I θ (5 Now, from euation ( & (5 ( θ mgl inθ ml coθ (6 he ytem under conideration i a non-linear ytem. For eae of modeling and imulation, we have to take a mall cae approimation uch that the ytem will be a linear one. Let take the linearization point a θ Π IJCA Jan-Feb 6 6

2 ISSN:9-69 Potluri Krihna Murthy et al, Int.J.Computer echnology & Application,Vol 7 (,6-67 ay θ Π φ Where, ϕ i the angle between the pendulum and vertical upward direction. If we chooe ϕ ᴝ. hen co,in dθ θ θ φ, ( d So, after linearization euation (6 become, ( φ mglφ ml (7 And euation ( become, ( b ml φ F (8 Here, F i the mechanical force to be applied on the moving cart ytem. But in real time model we have to give input voltage proportional to the force F. If the input voltage i u, then euation (8 become, ( b ml φ u A. ranfer Function In control theory, function called tranfer function are commonly ued to characterize the input-output relationhip of component or ytem that can be decribed by linear, time-invariant, differential euation. he tranfer function of a linear, time-invariant, differential euation ytem i defined a the ratio of the Laplace tranform of the output (repone function to the Laplace tranform of the input (driving function under the aumption that all initial condition are zero. Conider the linear time-invariant ytem defined by the following differential euation: ( n ( n ( m ( m a y a y an y an y b b... b b m m Where y i the output of the ytem and i the input. he tranfer function of the ytem i the ratio of the Laplace tranformed output to the Laplace tranformed input when all initial condition are zero, or ranfer function at zero initial condition L( outpu G ( L( inpu X ( ml ( g φ( Subtituting euation(in( ( I Ml g ( φ( ml mlφ( U ( From the above euation φ( U ( b( ( b ml ( ml ( mgl ( ( g φ( bmgl Where [( ( ( ml ] In euation (, it i clear that one pole and zero i at origin. hi lead to cancelation of one pole and zero. So reulting euation will be ml φ ( U ( b( ( mgl bmgl ( Here in thi cae the angle from the vertical poition (( i taken a the output and the applied force to the cart (u( i taken a the input function. Again from euation( ml φ( X ( ( mgl (5 ( m ( m Y ( b b... bm bm ( n ( n X ( a y a y... a y a y n n Now putting the value of ϕ( in euation(, By uing the concept of tranfer function, it i poible to repreent ytem dynamic by algebraic euation in. If the highet power of in the denominator of the tranfer function i eual to n, the ytem i called an nth-order ytem. Laplace tranform of euation (7 ( φ ( mglφ( mlx ( (9 Laplace tranform of euation (8 ( X ( bx ( mlφ ( U ( ( Solving euation (9 for X( X ( ( mgl U ( (( ( ml [ b( I mil ] [( mgl] mglb (6 Here the ditance of the cart from the origin i treated a the output function wherea the applied force on the cart i till the input function. B. State-Space Euation: In a tate-pace ytem repreentation, we have a ytem of two euation: an euation i to determine the tate of the ytem, and the other euation i to determine the output of the ytem. We will ue variable y( a the output of the IJCA Jan-Feb 6 6

3 ISSN:9-69 Potluri Krihna Murthy et al, Int.J.Computer echnology & Application,Vol 7 (,6-67 ytem, ( a the tate of the ytem, and u( a the input of the ytem. We ue the notation for the firt derivative of the tate vector of the ytem, a dependent on the current ytem and current input. We can write thee two euation a A( ( B( u( y( C( ( D( u( If the ytem themelve are time-invariant, we can re-write thi a follow: A( Bu( y ( C( Du( he State Euation how the relationhip between the ytem' current tate and it input, and the future tate of the ytem. he Output Euation how the relationhip between the ytem tate and it input, and the output. hee euation how that in a given ytem, the current output i dependent on the current input and the current tate. he future tate i alo dependent on the current tate and the current input. Now, with thi baic knowledge chooing the tate variable of the above conidered Inverted pendulum, φ; φ (7 Now euation( can be written a: ( ml b u (8 And euation(9 can be written a ( ml mgl (9 Putting the value of from euation (9 in euation (8, we have, b( m gl u ( ( Now putting the value of from euation ( in euation (9 we have, mbl mgl( mlu ( here are two output o, y ( φ( t ( ( t y Now contructing the tate pace matri X AX BU YCX, Where, ; X U u( ; ( Y X φ φ φ ( φ φ Referring from euation 7,, and, we have, b( m gl A mbl mgl( ; C B mlu he deign limitation are he ditance covered by the cart from the tarting point i.e. hould be in the range of. meter. he angle θ hould be in the range of. radian. he applied voltage to the DC motor hould remain within the range of.5 to -.5. III. PID CONROLLER DESIGN: Let conider a following unity feedback ytem Fig: Schematic diagram of the unity feedback control ytem he error ignal i ent to the PID controller, and the controller compute both the derivative and integral of the error ignal. he ignal (u jut pat the controller i now eual to the proportional gain ( K time the magnitude of the P error plu the integral gain ( K time the integral of the error I plu the derivative gain ( K time the derivative of the Error. u K e K D P I e( t dt K D de dt ranfer function of angle of the pendulum i φ( U ( ml mgl ( Now, cancelling the common pole i.e., we have, IJCA Jan-Feb 6 6

4 ISSN:9-69 Potluri Krihna Murthy et al, Int.J.Computer echnology & Application,Vol 7 (,6-67 φ ( U ( ml ( mgl b a If we neglect the frictional force and ma of the pendulum compared to ma of the cart, then the tranfer function of poition of the cart i X ( U ( ( b Where Q Q and R R are weighting parameter that penalize the tate and the control effort, repectively. hee matrice are therefore controller tuning parameter. It i crucial that Q mut choen in accordance to the emphaize we want to give the repone of certain tate, or in other word; how we will penalize the tate. Likewie, the choen value ( of R will penalize the control effort u. Hence, in an optimal problem the control ytem eek to maimize the return from the ytem with minimum cot. In a LQR deign, becaue of the uadratic performance inde of the cot function, the ytem ha a mathematical olution that yield an optimal controller u( K( Where u i the control input and K i the gain given a K R B S. It can be hown in fig that S can be found by olving the algebraic Riccati Euation SA A S Q PBR B S Fig: Schematic diagram of the PID controller IV. Linear Quadratic Regulator (LQR Deign: here are different method, or procedure, to control the inverted pendulum. One i the pole placement procedure having the advantage of giving a much clearer linkage between adjuted parameter and the reulting change in controller behavior. However, one diadvantage with thi method i that the placing of the pole at deired location can lead to high gain. In thi ection a linear uadratic regulator (LQR i propoed a a olution. he principle of a LQR controller are given in figure. Here i the tate pace ytem repreented with it matrice A, B, and C with the LQR controller (hown with the ƒ{k. he LQR problem ret upon the following three aumption: All the tate (( are available for feedback, i.e. it can be meaured by enor etc. he ytem a tabilizable which mean that all of it untable mode are controllable. he ytem are detectable having all it untable mode obervable. o check whether the ytem i controllable and obervable, we ue the function obv(a,c and ctrb(a,b and find thi to be true. hi LQR regulator provide an optimal control law for a linear ytem with uadratic performance inde yielding a cot function on the form J ( Q( u ( Ru( dt Fig: Schematic diagram of LQR controller he proce of minimizing the cot function therefore involve to olve thi euation, which will be done with the ue of MALAB function lr. In thi project the parameter in Q wa initially choen according to Bryon Rule to be 5 Q 5 And the control weight of the performance inde R wa et to. Here we can ee that the choen value in Q reult in a relatively large penalty in the tate and. hi mean that if or i large, large value in Q will amplify the effect of nd in the optimization problem. Since the optimization problem are to minimize J, the optimal control u mut force the tate and to be mall. hi value mut be modified during ubeuent iteration to achieve a good repone a poible. By an iterative tudy when changing Q value and running the command Klr(A,B,Q,R K[ ] C. State Etimation: A mentioned for the cae of the LQR controller, all enor for meauring the different tate are aumed to be available. hi i not valid aumption in practice. Avoid of enor IJCA Jan-Feb 6 65

5 ISSN:9-69 Potluri Krihna Murthy et al, Int.J.Computer echnology & Application,Vol 7 (,6-67 mean that all tate (full-order tate oberver, or ome of the tate (reduced order oberver, are not immediately available for ue in any control cheme beyond jut tabilization. hu, an oberver i relied upon to upply accurate etimation of the tate at all inverted pendulum poition. he chematic of the ytem with the oberver i hown in below figure V. CONCLUSIONS: From the above dicuion, it can be concluded that both the control method of conventional controller (LQR and PID can control the cart poition and the pendulum angle for the linearized ytem. he repone characteritic atified the reuirement of the deigned criteria. VI. RESULS: Fig:Schematic of tate pace control uing a oberver gain and K i the LQR gain matri A can be een from figure the oberver tate euation are given by ˆ Aˆ Bu L( y Cˆ yˆ Cˆ Where ˆ i the etimate of the actual tate. Furthermore ˆ ( A LC ˆ Bu Ly hi, in turn, i the governing euation for a full order oberver, having two input u and y and one output, ˆ.Since we already know A,B and u, oberver of thi kind i imple in deign and provide accurate etimation of all the tate around the linearized point. From the above figure we can ee that oberver i implemented by uing a duplicate of the linearized ytem dynamic and adding in a correction term that i imply a gain on the error in the etimate. hu, we will feed back the difference between the meaured and the etimated output and correct the model continuouly. he proportional oberver gain matri, L, can be found by pole placement method by ue of the place command in MALAB. he pole were determined to be ten time fater then the ytem pole. hee were found to be eig ( A B * K [.6,.585,.59,8.559] Which yield the gain matri L When combining the control-law deign with the etimator deign we can get the compenator. Fig: Repone curve for cart poition uing PID controller Fig:Repone curve for pendulum angle uing PID controller IJCA Jan-Feb 6 66

6 ISSN:9-69 Potluri Krihna Murthy et al, Int.J.Computer echnology & Application,Vol 7 (,6-67 VII. REFERENCES []. A journal by Ahmed Nor Karuddin Bin Nair, Univerity Malayia, Modelling and controller deing for an Inverted pendulum ytem,7. []. M.Athan, he Linear Quadratic LQR Problem,Maachuett Intitute of echnology, Maachuett,98. []. A.E Frazhol, he Control of an Inverted pendulu, School of Aeronautic and Atonautic, Purdue Univerity, Indiana,7. [].J.Lam, Control of an Inverted Pendulum, Georgia ech college of computing, Georgia,9. [5]. J. Hapanha, Undergraduate Lecture Note on LQG/LQR Controller Deign, 7 [6]. Balancing an Inverted Pendulum on a Seeaw by Maim V. Subbotin. [7].PID control ytem deign by Karl Johan Atrom, Fig:Repone curve for pendulum angle uing LQR controller combined with tate etimator [8]. J. Lam, Control of an Inverted Pendulum, Georgia ech College of Computing, Georgia, 9 Fig:Repone curve for cart poition uing LQR controller combined with tate etimator IJCA Jan-Feb 6 67