slac-pub-6503 - SLACPUB-6503 June 1994 (A) - A Formal...

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SLACPUB-6503 June 1994 (A) - A Formal Approach to the Design of Multibunch Feedback Systems: LQG Controllers H. Hindi, J. Fox, S. Prabhakar, L. Sapozhnikov, G. Oxoby, I. Linscott, and D. Teytelman Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309 Abstr~t We formulate the multibunch feedback problem ~ a stan- dard control-systems design problem and solve it using Linear Quadratic Gaussian (LQG) regulator theory. Use of a specific optimality criterion allows quantitative eval- uation of different controllers and leads to the design of opt imal LQG cent rollers. Computer simulations are used to show that, as compared to the existing Finite Impulse Response (FIR) control, LQG control can provide the same closed-loop damping for less peak power, thus making more effective use of limited kicker power. furthermore, - LQG cotitrol enables us to use more power to provide bet- ter damping wit bout the problem of driving instabilities with higher loop gains. The code for the LQG filters de scribed h= been written for the Quick prototype installed at ALS. 1 INTRODUCTION The problem of designing the controller (filtering alg~ rithm) for the longitudinal feedback system is bmically a regulator pm~le~. The regulator problem has been stud- ied extensively in control theory, and there are many tech- niques available to solve it. This paper will compare the performances of the existing FIR filter-based control tech- nique to that of the LQG %lter-based technique. One-of the distinguishing features of LQG design is the use of a specificoptimality crfierion. This is in centrast to the FIR- based technique, which is based on an adhoc discretetime approximation of a differentiator [1,2]. Intheprocessof dampingsynchrotronoscillations,mea- surementsof the beamphasearetakenat discretetimes andthefeedbackcorrectionsignalsareappliedat discrete times. Therefore,our analysiswill be carriedout using discretetimecontrolformalism.We will also usestate spacenotationin our descriptionof the plantand con- troller[3,4].Givena systemdescribedby stat+spacema- trices {A, B, C, D}, the transfer function is obtained by H(z) = C(zI– A)-lB+D . (1) 2 LINEAR QUADRATIC GAUSSIAN ~GULATOR THEORY -- Figure 1 sh~ws a block diagram of the LQG regulator prob lem. A precise statement oft he problem follows. Given a *Work supported by Department of Energy contract DW AC03-76SFO0515 . w+ P(z) Cx {A, B, C, D} Figure 1. Block diagram of the LQG regulator problem. linear model of the plant P, described by state matrices {A, B, C, D} (D= O), z(k + 1) = Ax(k) + Bu(k) + w(k) y(k) = Cz(k) + v(k) . (2) where z is the plant state (energy and phase), u is the cent rol input (kicker signal), y is the measured regulated output (phase), and w and v are process and measurement noises, respectively, both of which are assumed to be un- correlated white noises with covariance matrices W and V.
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This note was uploaded on 02/04/2012 for the course ECE 222 taught by Professor Goengi during the Spring '11 term at Maryland.

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slac-pub-6503 - SLACPUB-6503 June 1994 (A) - A Formal...

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