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# Digital Control Systems ME 6403

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This 0 page Class Notes was uploaded by Chloe Reilly on Monday November 2, 2015. The Class Notes belongs to ME 6403 at Georgia Institute of Technology - Main Campus taught by Staff in Fall. Since its upload, it has received 12 views. For similar materials see /class/234259/me-6403-georgia-institute-of-technology-main-campus in Mechanical Engineering at Georgia Institute of Technology - Main Campus.

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Date Created: 11/02/15

Tracking Control Nader Sadegh The George W WoodrufF School of Mechanical Engineering Georgia Institute of Technology Atlanta Georgia 30332 0405 nadersadegh megatechedu Outline 0 Tracking Control Problem Statement 0 Zero Phase Error Tracking ZPET Control 0 Internal Model Principle 0 Repetitive Control Problem Statement Given the asymptotically closed loop system 30 Y 2 De 30 Find a bounded reference input 7 that forces the output sequence to follow a sequence of desired outputs 1 ZPET Controller Factorize the numerator of the closed loop transfer function denoted by to BzBz D027 where Bz is the quotgoodquot part of the numerator polynomial Bz whose zeros are inside the unit circle and may be cancelled The Zero Phase Error Tracking ZPET Tomizuka 1987 controller for is of the form Bz1Dz lB1lQBz Qz ZPET The compensated system z z B ltzB ltz 1 has zero phase since Qej THej T is always a positive real number hence it has zero phase that s why is called a ZPET Example Consider the closed loop plant b 1 2 W z z z3 z z p with p 05 b 163 X 10 4 zl 026 252 364 Qz is set to an approximate ZEPT of z 053z1 364 QC 4642bz 026 lnternal Model Principle Consider the following outer loop feedback around the closed loop plant Ndlzl and suppose that Ydz Ddz BzDz Figure 1 Tracking Control Block Diagram The steady state error of the system approaches zero if 0 The closed loop poles of the overall system are inside the unit circle 0 The polynomial Ddz divides Dcz so that Dcz Ddzqz for some polynomial 2 Repetitive Learning Control In many practical applications such as in machine tool control and robotics the desired trajectory is repetitive A repetitive controller learns the con trol input while tracking a periodic trajectory The transfer function of a repetitive controller RC is of the form Caz Kl l ll N is the task period K is a positive constant less than 1 is an additional compensation to be determined Klgt0 o Advantage If stable it generates the desired input without requiring the exact knowledge of the plant dynamics and input disturbances o Disadvantage Stability is hard to achieve because of high order dy namlcs o must be designed using the plant model Synthesis of Repetitive Controller Closed loop plant Gclosed is denoted by yk Hzrk r is reference input and y is the output Desired output ydk is periodic WU N WU where N is the period Assumption is asymptotically stable and has no zeros on the unit circle Repetitive Control TF K Cz zN l 1 W K gt 0 learning gain is an additional compensator to be specified must have no poles on the unit circle For the control law to be realizable zNQz must be proper Time Domain Control Law If u and y denote the input and output of the plant and yd the desired output then WC N WC K QZydk yk Why does it Work 0 The short answer is Internal Model principal IMP o The controller transfer function Cz is a generator of N periodic functions yd since Ydz is of the form Yd where Pz is a polynomial of degree N 1 0 Note that this also a generalization of the integral controller N 1 Cz Z511 which can only generate constant functions 0 To prove the IMP in this case let s find Closed Loop TF Ys KZQH 1118 Z W 1 KZQch The error TF EYd Es 1Ys zN l Yd3 Yd3 zN 1 KZQHCZ 60 3 1 WUc N ydkl Since ydkN 6k goes to zero provided the closed loop system is asymptotically stable To state the stability criterion let 27139139 expo Theorem The repetitive controller is asymptotically stable if RQeiHeZ gt 0 i 01 N 1 or equivalently the phase angle of QeiHeZ is between i900 K is selected such that the Nyquist diagram of ZNK1HQZ 1 does not encircle 1 of the complex plane For the proof of this theorem which also includes the MIMO systems see the attached paper Sadegh 1995 3 Synthesis of Repetitive Controller Let be of the form 3ZBz ACZ Choose to be the Zero Phase Error Tracking ZPET Tomizuka 1987 controller for 3127 514127 W ZPET Then exam Bz3z1 B1l2 It can be seen that QeZHeZ has zero phase that s why is called a ZPET thus satisfying the stability condition In practice an approximate plant model can be used to find

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