EE 44 Lab#4 PID Cotrol Desig of Rigid Bodies Prepared by: Stacy Caso E-mail: scaso@arri.uta.edu Updated: July 9, 1999 This lab demostrates some key cocepts associated with proportioal plus derivative (PD cotrol ad subsequetly the effects of addig itegral actio (PID. The block diagram for the forward path PID cotrol of a rigid body plat is show i Figure 1. Note that frictio is eglected here. This cotrol scheme, actig o plats modeled as rigid bodies, fids broader applicatio i idustry tha ay other. It is employed i such diverse areas as machie tools, automobiles (cruise cotrol, ad spacecraft cotrol (attitude ad gimbal cotrol. r Referece Iput k k i p + + s s PID Cotroller k hw Hardware Gai PLANT Output Figure 1. Rigid Body PID Cotrol Cotrol Block Diagram Before proceedig, check with lab assistat as to which system you are usig for this lab. The three differet systems are Model 10 (Rectiliear Sprig/Mass Cotrol System, Model 05 (Torsioal Cotrol System ad Model 505 (Iverted Pedulum Cotrol System. Net, follow the appropriate desig procedures below for your particular model. 1
1. Rigid Body PID Cotrol for Model 10 (Rectiliear Sprig/Mass Cotrol System The closed loop trasfer fuctio is give as ( ( m( (1.1 r s + ( m( For the first portio of your lab we shall cosider PD cotrol oly ( k i = 0. With this assumptio i mid, oe may epress (1.1 above as: ( ( m( s + k p (1. r s + k m k s + k ( ( Notice that oe may equate the epressio i (1. to by defiig the followig: c hw ζω s + ω = d p = (1. r s + ζω s + ω k p ω m ζ = mω mk p The effects of k p ad k d o the roots of the deomiator (damped secod-order oscillator of Eq. (1. is studied i the work that follows. PD Cotrol Desig: a. From Eq s (1., 1. ad 1.4 desig a PD cotroller (i.e., fid k p ad k d for a system with atural frequecy ω =15 rad/s ad ζ = 0. 707 k p > 0.08 ad k d > 0. 04. Assume k hw = 1800 N/m ad m =.77 kg. (1.4. (Do ot eceed a value of b. Implemet your cotroller by performig a step respose. Set up a trajectory for a 500 cout closed-loop STEP with 000ms duratio (1 rep. Ask your lab assistat to check your setup before proceedig further. c. Now, eecute this trajectory ad plot the commaded positio ad ecoder positio #1. Plot them both o the same vertical ais so that there is o graphical bias. Save your plots for later compariso. Addig Itegral Actio: a. Implemet a PID cotroller with a value of k i = 0. 01 alog with the previous values of k p ad k d foud i (a. above. Be certai that the followig error see i the backgroud widow is withi 0 couts prior to implemetig this cotroller (if ot choose Zero Positio from the Utility Meu. b. Now, eecute a trajectory for a 500 cout closed-loop step of 000ms duratio (1 rep. Plot the commaded positio ad ecoder positio. Plot them both o the same vertical ais so that there is o graphical bias. Save your plots (eport raw data from
the data meu for later compariso. Questios: a. Show all calculatios for k p ad k d. b. Derive the atural frequecy ω ad dampig ratio ζ i Eq. 1.4. c. What is the effect of the system hardware gai k hw, mass m, ad cotrol gais k p ad k d o the atural frequecy ω ad dampig ratio ζ? d. Describe the effects of atural frequecy ω ad dampig ratio ζ o the characteristic roots of Eq. 1.1. Use a root-locus diagram i your aswer to show the effect of chagig ζ from 0 to for ay give ω. e. What is the effect of addig itegral actio?
. Rigid Body PID Cotrol for Model 05 (Torsioal Cotrol System The closed loop trasfer fuctio is give as ( ( J ( r s + ( J ( For the first portio of your lab we shall cosider PD cotrol oly ( k i = 0. With this assumptio i mid, oe may epress (.1 above as: ( ( J ( s + k p (. r s + k J k s + k ( ( Notice that oe may equate the epressio i (. to by defiig the followig: c hw ζω s + ω = d p (.1 = (. r s + ζω s + ω k p ω J ζ = Jω Jk p (.4 The effects of k p ad k d o the roots of the deomiator (damped secod-order oscillator of Eq. (. is studied i the work that follows. PD Cotrol Desig: a. From Eq s (.,. ad.4 desig a PD cotroller (i.e., fid k p ad k d for a system with atural frequecy ω =10 rad/s ad ζ = 0. 707 k p > 0.08 ad k d > 0. 1. Assume k hw = 17.4 N - m/rad ad. (Do ot eceed a value of J = 0.0108 kg - m. b. Implemet your cotroller by performig a step respose. Set up a trajectory for a 500 cout closed-loop STEP with 000ms duratio (1 rep. Ask your lab assistat to check your setup before proceedig further. c. Now, eecute this trajectory ad plot the commaded positio ad ecoder positio. Plot them both o the same vertical ais so that there is o graphical bias. Save your plots for later compariso. Addig Itegral Actio: a. Implemet a PID cotroller with a value of k i = 0. 05 alog with the previous values of k p ad k d foud i (a. above. Be certai that the followig error see i the backgroud widow is withi 0 couts prior to implemetig this cotroller (if ot choose Zero Positio from the Utility Meu. b. Now, eecute a trajectory for a 500 cout closed-loop step of 000ms duratio (1 rep. Plot the commaded positio ad ecoder positio #1. Plot them both o the same vertical ais so that there is o graphical bias. Save your plots (eport raw data 4
from the data meu for later compariso. Questios: a. Show all calculatios for k p ad k d. b. Derive the atural frequecy ω ad dampig ratio ζ i Eq..4. c. What is the effect of the system hardware gai k hw, mass m, ad cotrol gais k p ad k d o the atural frequecy ω ad dampig ratio ζ? d. Describe the effects of atural frequecy ω ad dampig ratio ζ o the characteristic roots of Eq..1. Use a root-locus diagram i your aswer to show the effect of chagig ζ from 0 to for ay give ω. e. What is the effect of addig itegral actio? 5
. Rigid Body PID Cotrol for Model 505 (Iverted Pedulum Cotrol System The closed loop trasfer fuctio is give as ( ( m( r s + ( m( For the first portio of your lab we shall cosider PD cotrol oly ( k i = 0. With this assumptio i mid, oe may epress (.1 above as: ( ( m( s + k p (. r s + k m k s + k ( ( Notice that oe may equate the epressio i (. to by defiig the followig: c hw ζω s + ω = d p (.1 = (. r s + ζω s + ω k p ω m ζ = mω mk p (.4 The effects of k p ad k d o the roots of the deomiator (damped secod-order oscillator of Eq. (. is studied i the work that follows. PD Cotrol Desig: a. From Eq s (.,. ad.4 desig a PD cotroller (i.e., fid k p ad k d for a system with atural frequecy ω =10 rad/s ad ζ = 0. 707 k p > 0.5 ad k d > 0. 01. Assume k hw = 088 N/m ad m = 0.19 kg.. (Do ot eceed a value of b. Adjust the positio of the balace masses to l t = 10 cm beig sure to secure them o the threaded rode by couter rotatig them. Verify that the dout weights are i place ad secure o the slidig rod. c. Implemet your cotroller by performig a step respose. Set up a trajectory for a 1000 cout closed-loop STEP with 1000ms duratio (1 rep. Ask your lab assistat to check your setup before proceedig further. d. Now, eecute this trajectory ad plot the commaded positio ad ecoder positio #. Plot them both o the same vertical ais so that there is o graphical bias. Save your plots (eport raw data from the data meu for later compariso. Addig Itegral Actio: a. Implemet a PID cotroller with a value of k i = 0. alog with the previous values of k p ad d k foud i (a. above. Be certai that the followig error see i the backgroud widow is withi 0 couts prior to implemetig this cotroller (if ot choose Zero Positio from the Utility Meu. Now, eecute a trajectory for a 1000 cout closed- 6
loop step of 1000ms duratio (1 rep. Plot the commaded positio ad ecoder positio. Plot them both o the same vertical ais so that there is o graphical bias. Save your plots for later compariso. Questios: a. Show all calculatios for k p ad k d. b. Derive the atural frequecy ω ad dampig ratio ζ i Eq..4. c. What is the effect of the system hardware gai k hw, mass m, ad cotrol gais k p ad k d o the atural frequecy ω ad dampig ratio ζ? d. Describe the effects of atural frequecy ω ad dampig ratio ζ o the characteristic roots of Eq..1. Use a root-locus diagram i your aswer to show the effect of chagig ζ from 0 to for ay give ω. e. What is the effect of addig itegral actio? 7