ECSE 4440 Control System Engineering. Project 1. Controller Design of a Second Order System TA

Transcription

1 ECSE 4440 Control Sytem Enineerin Project 1 Controller Dein of a Secon Orer Sytem TA Content 1. Abtract. Introuction 3. Controller Dein for a Sinle Penulum 4. Concluion

2 1. Abtract The uroe of thi roject i to in a controller for a econ orer ytem(the mot common rototye control roblem). A Proortional Interal erivative(pid) will be aote for the ive ytem. The tuy will be aroache analytically an exerimentally.. Introuction Practically, there are few ytem that on t have any control ytem inie. More comlicate become ytem, more ohiticate controller are neee. To in a controller, everal iue houl be coniere e.., moelin, ytem erformance, tunin ain an tability. For a imle econ orer ytem, thoe iue will be tuie in thi roject. The iven inle enulum ytem i a baic econ orer ytem to be controlle. The ytem will be moele a a linear ytem. The ole an zero of the ytem will be foun to tet the tability of the iven ytem. The controller will be eveloe in the orer of roortional(p) controller, roortional-erivative(pd) controller-roortional an roortional-interal-erivative(pid) controller. To meet the iven ecification an tability, the ain will be tune. The controller will be imlemente in continuou time omain(s lane) an in icrete time omain(z lane), reectively. The analytical controller will be verifie by imulation with imulink of Matlab. 3. Controller Dein for A Sinle Penulum 3.1 Linearization(Tak 1) The analyi an control in are far eaier for the linear than for nonlinear moel. Linearization i the roce of finin a linear moe that aroximate a nonlinear one. Linearization roce een on the exanin the nonlinear tate equation in to a Taylor erie. In the iven ynamic equation (1), two non-linear function exit, e.., n( θ. ), in θ. ( I + I + ml ) θ.. + F θ. + F n( θ. ) + ml in θ = τ v m v c = Parameter Name Value m Ma.048 I Link Inertia m I Motor Inertia.e-7 m m l Ditance.051 m N Gear Ratio F Coulomb Friction Coefficient.014 N-m c F Vicou Friction Coefficient.0034 N-m-ec v Torque Contant (1)

3 n(.. θ ) can be et to be zero ince at the equilibrium tate, θ = 0. For in θ, the firt two term(linear term) of the Taylor erie exanion() are ue. in θ = in θ + coθ θ θ.5in θ θ θ + () The the linearize ytem become ( ) ( )... ( I m N I + ml ) θ + Fv θ + ml coθ θ = τ ml in θ where θ = ( θ ) + (3) θ The Lalace tranform of the equation i α + F + ml coθ ( ) Θ( ) = Ι( ) v where α = I mn + I + ml an I() i the Lalace tranform of τ ml in θ. Then, the tranfer function of the ytem i Θ( ) 1 H ( ) = = I( ) α + Fv + ml coθ (5) From the tranfer function, it ha ole at F coθ v Fv ml ± 4 α α α = (6) an no zero. For the ytem to be table, every ole houl be in left han ie of - lane. In the equation (6), co θ > 0 i the conition for the ytem to be table. In π 3π other wor, 0 < θ < or < θ < π. With thoe iven arameter value an π θ =, the ole are at 0 an an for θ = π, ole are at an Therefore the ytem i marinally table or untable by itelf. 3. Imlementation Simulink iaram for the iven ytem an the to level iaram(with PID Controller) are hown in Fiure 1 an Fiure. The arameter are efine in the file eninit.m. To initialize the arameter, the ubytem block(initialization) i in the ytem. (4) 3

4 Fiure 1 Fiure With only the inle enulum ytem, the te reone oe infinity in fiure P Fee Back Controller(tak 3) Fiure 3 4

5 A a baic controller oeration, the controller i imly an amlifier with a contant ain τ = θ θ. Hence the outut of P controller i an a feeback loo, ( ) relate with the inut of the controller by a roortional contant. Ain a P controller to the iven ytem reult in chanin the tranfer function of overall ytem uch a θ = θ + (7) α + F + ml coθ + a + F + ml coθ + v where iturbance = ml in θ. From (7), the location of the ole een on the iven arameter an ole are v. The F coθ v Fv ml + ± 4 α α α = (8) The location of the ole are chanin by varyin. If > ml coθ, the ytem become table. With iven arameter an = 5, the ole are locate at.474 ± 119i for θ = π an π θ = reult i verifie by the imulation( =5, table for both value. uch that the ytem i table. The analytical i=0 an i=0). In fiure 4, the ytem i Fiure PD Fee Back Controller(tak 4) Even thouh the ytem with P controller i table, the outut ha relatively hih eak overhoot an i ocillatin. The ocillation reult from the exceive amount of 5

6 torque an the lack of amin. Ain the erivative of the inut make the ytem critical ame. In the equation (9), the location of ole are F coθ v + Fv + ml + ± 4 α α α = (9) A it i hown in (9), tunin an make it oible for the ytem to meet the iven ecification e.., rie time (90%), ettlin time (%) mall overhoot(le 5%) an teay tate error(le then %). Once the inie of the quare root i neative, the amin een on it manitue. For fixe =. 1, the te reone are hown for variou value in the fiure 5. The lot how increain increain overhoot( =10). For fixe Fiure 5 reult in ecreain rie time( =1 in the fiure 6, the te reone are hown for variou value. =1,10) but 6

7 Fiure 6 In the fiure, ecreain value lea to ecreae the rie time. For the iven ecification, =.1 an = make the ytem meet the ecification very well in Fiure 7. Fiure 7 From the iven tranfer function for the ytem with PD controller (10), teay tate error can be calculate by ettin = 0. θ = θ + α + F + + ml coθ + a + F + + ml coθ + ( v ) ( v ) where = ml in θ (10) Therefore the teay tate error i the function of θ in (11). ml in θ e = (11) ml coθ + 7

8 π With imulation with the arameter θ = ame with.04 calculate by (11)., =1, the error in Fiure 8 i almot 3.5 Wahout Filter Dein(tak 5) Fiure 8 In ractice,. θ i not meaure. In tea of that, one hih a filter which ha one zero at the oriin in feeback loo. Scoe Ste.+i PI control In1 theta thetaot Terminator Sinle theta To Workace Penulum Simulator +3 Gain Tranfer Fcn SubSytem t1 Clock time Fiure 8-1 The te reone of the wahout filter controller i ocillate at tranient art an ha amin. But the filter reuce the teay tate error in Fiure 8-. 8

9 Fiure PD controller in amle ata imlementation(tak 6) So far, the controller ha been imlemente in continuou time omain. The icrete imlementation, however, become more oular by aearin comuter an mall iital microroceor. In continuou omain, a ifferential equation can be aroximate with a ifference equation e.., ( k ) ( θ ( k ) θ ( k 1) )/ t θ. With the aroximation, the PD controller ytem i imlemente in fiure 9. Fiure 9 The imulation reult i well aroximate in amle ata imlementation in fiure 10. 9

10 I n 1 theta thetaot Fiure 10 θ 3.7 PID controller in continuou an amle ata imlementation (tak 7,8) To comenate the teay tate error, the interal controller houl be ae. One obviou effect of the interal control i that it increae the tye of the ytem by one; that i, if the teay-tate error to a iven inut i contant, the interal control reuce it to zero. The tranfer function of the PID controller ytem i + + i = θ α + F + + ml coθ + + a + F + + ml coθ ( v ) ( ) i ( v ) ( + ) + i where = ml in θ (1) then θ θ oe to zero a S 0. In other wor, the interal controller 0 comenate teay tate error. The continuou an amle ata imlemente PID controller are imulate an comare in fiure 11 an fiure 1. Scoe.+i Ste PI control Gain Sinle Penulum Simulator theta To Workace SubSytem t1 Clock time Fiure 11 10

11 The erformance of both controller meet the iven ecification in fiure 1 an fiure 13. Fiure 13 Fiure 14 Parameter Continuou Controller Samle ZOH controller θ = π θ =π / θ = π θ =π / Overhoot(%) t (ec) t r (ec) t (ec) Steay error (%) 1.4e-4 1.5e e e Trackin(tak 9) U to now, the inut ha been a te function. But in real ytem, the inut ten to be a θ t = π in 4πt time varyin function uch a a inuoial function. The iven inut ( ) ( ) 11

12 increae the tranfer function by becaue the Lalace tranform of the inut i 4π. The total ytem become (13). + 16π + + i 4π θ = + 3 α + F + + ml coθ π a 3 + ( ) ( ) v ( Fv + ) + ( ml coθ + ) + i A hown in (13), the ytem i a fifth orer ytem. Firt of all, ominant econ orer ytem nee to be foun which ha the ole mot cloe to imainary axi. Then tunin the ole of the econ ytem to make the ytem meet the ecification. With the ame ain with tak 8, the outut i hown in Fiure 15. i (13) 3.9 Exerimental Fiure Concluion In thi roject, the controller in of a econ orer ytem(a inle enulum) ha been tuie. A a controller, PID controller ha been aote. Mathematical analyi of the tranfer function ha been ue an imulation jutify the analyi. Even thouh, a PID controller i imle, the robut of the controller ha been exerience an jutifie. 1