EE-371 LABORATORY CONTROL SYSTEMSSession 8Position Control Design Project P!"se le"# controller $ Root loc%s Design&P%r'oseIn this project you design and implement a position control system with a cascade phase lead controller using root-locus design. The objectives of this project are: To design a phase-lead compensator using root-locus designs such that the output angle tracks a commanded position. To build the compensated servo plant in SIM!I"# and simulate offline to obtain the response to a s$uare wave input and verify the design. To build the %in&on application' and implement and test the system on the real-time hardware(ntro#%ction(hase lead compensators are used $uite e)tensively in control systems' typically when rate feedback is not possible or the high fre$uency noise would prohibit the use of a (* +.,compensator. The compensator contributes a positive phase angle to the plant open loop transfer function causing the root locus to shift towards the left half s-plane. The compensator is used to improve the transient response' increase the speed of response or reduce the settling time. In !ab - the servomotor position control was achieved by means of a (* compensator. In this project a phase lead controller is designed using root-locus method. The root-locus trajectories show the location of the roots of the characteristic e$uation as one or more parameters are changed. If the root locations are not satisfactory' adding additional poles and .eros via a cascade compensator reshapes the root locus. The introduction of a compensator / 0cG s results in the characteristic e$uation, / 0 / 0 / 0 1c pG s G sHs + = %riting/ 0 / 0 / 0c pG s G sHsas / 0 KGs' we have, / 0 1 KGs + =If ,s is the desired location of a closed-loop pole' it must satisfy the above e$uation' which results in the following angle and magnitude criteria:,+1zi pi = product of vectror lengths from finite polesproduct of vector lengths from finite .erosK =The above e$uations can be used for graphical root-locus design. 1) Ser*o 'l"nt +o#elingIn the position control e)periment /!ab 20 the motor-load transfer function with position as output was found to be

3 3/ 0/ 0m ga eqoi eq m geq a eqKKR JsV s B KKs sJ R J= + + /+.,0or/ 0/ 0 / 0o mi ms aV s ss b=+/+.30 %here 3 3' m g eq m gm ma eq eq a eqKK B KKa bR J J R J = = +/+.40+.3 The open-loop block diagram with the s-domain voltage / 0iVs as input and / 0os as output is shown in 5igure +.,./ 0os mmas b +,s/ 0os iV,ig%re 8)1 6pen-loop plant transfer function.The s-domain unit step response is( ),/ 0momass s b s =+The final value of the response is 1lim / 0 lim / 0so ott s s = = . That is' the response is unbounded.1)1 Position controlIn order to control the output position to follow an input command' consider the addition of a phase lead controller 11/ 0/ 0/ 0ccK s zG ss P+=+/ 0mmass b +11/ 0CK s zs p++/ 0is / 0os / 0cG s,ig%re 8)- closed-loop control system with phase lead controller.The phase lead controller adds a pole and a dominant .ero to the open-loop transfer function' i.e. 1 1 z p >. This would shift the loci towards the left-half s-plane' which improves the transient response.-) Pre L".or"tor/ Assign+ent7valuatema' and mb of the plant transfer function as given by /-.40.The servo plant parameters are given in the position control e)periments /!ab 20. If you have performed !ab 2 you have these values. If you have not performed the position control project /!ab 20 you must derive the plant transfer function' verify the above e$uations and include modeling with this project. 6therwise give reference to !ab 2. 8ssuming a simple proportional controllerK ' draw the root locus to scale for +.4( )/ 0mma KKGss s b=+9ive all pertinent characteristics of the loci such as number of asymptotes' breakaway and:or re-entry points. Indicate the loci directions for increasing K by arrows. Specify range of K for closed loop system stability.-)1 Position control #esignsing root-locus graphical method design a phase lead controller to meet the following time-domain specifications: Step response dominant poles damping ratio 1.-1- = Step response dominant poles time constant1.13 =secse graphical method' and select the controller .ero at 1-2 z = . 8pply angle and magnitude criteria to find the compensator parameters and find the compensator dc gain111cKzap=' /see 7)ample ; in Tutorial III