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module8 - Module 8: Observer Based Design 49 8 8.1 Module...

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Module 8: Observer Based Design 49 8 Module 8: Observer Based Design 8.1 Overview This module continues the state space approaches initially covered in module 7. The satellite antenna dish problem, illustrated in Figure 8, is again used as an example. θ Figure 8: Satellite antenna dish This module will use this example to study full and reduced order observer designs. The problems at the end will extend this to reference tracking and integral control designs. 8.2 Notes In this case, only the dish angle, θ ( t ), is available for measurement. Again a discrete time implementation will be used so it is actually θ ( k ) that will be measured. The ±rst objective will be to design a full order estimator to estimate θ ( k ) and ˙ θ ( k ). The new open loop plant, P θ ( s ), is illustrated in Figure 9. Again a continuous time wind input ( Td ( t )) simulation is performed with Tc ( t ) = 0. P θ ( s ) ± ± ± θ ( t ) ( t ) ( t ) Figure 9: Open loop plant: Satellite antenna dish
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50 ECE 147b: Matlab Computer Modules An full order estimator is designed. The design conFguration is illustrated if ±igure 10. The estimated discrete states are θ e ( k ) and ˙ θ e ( k ). In the simulation of this estimator, ˙ θ ( k ) is also calculated and displayed. P θ d ( z ) est ± ± ± ± ± ± ± ± ± θ ( k ) Td ( k ) Tc ( k ) ˙ θ e ( k ) θ e ( k ) ±igure 10: Estimator design conFguration The estimator, est , is tested and found to work well. A state feedback design is done and this is connected up to the estimator outputs. The resulting system is illustrated in ±igure 11. P θ d ( z ) est - K ± ± ± ± ± ± ± ± ± θ ( k ) ( k ) ( k )= u ( k ) ˙ θ e ( k ) θ e ( k ) ±igure 11: ±ull order estimator based control design This discrete time control system is now studied using Simulink . The full set up is shown in ±igure 12. The continuous response, θ ( t ), was found to be very close to θ ( k ). The module next considers the problem of designing a reduced order estimator. The reduced order problem is set up such that the reduced order estimator, estr , is a plug in replacement for est in ±igures 11 and 12. However only, ˙ θ ( k ) is estimated. The measurement of θ ( k ) is used directly into - K . ±igure 13 shows the construction of estr in order to achieve this. The notation in the module and the lecture notes are very similar to make referring from one to the other easy.
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Module 8: Observer Based Design 51 est - K P θ ( s ) ZOH ± ± . . . . .. T ± ± ± ± ² ± ³ ± ± θ ( t ) Td ( t ) Tc ( t )= u ( t ) ˙ θ e ( k ) θ e ( k ) u ( k ) θ ( k ) Figure 12: Sampled data system with full order estimator based control design L r A bb - L r A ab z - 1 A ba - L r A aa B b - L r B a ² + ² + ³³ ³ ± ´ ² ± ± ³ ´ ´ ´ ± ± ± x c ( k +1) x c ( k ) ˙ θ e ( k ) θ e ( k ) u ( k ) θ ( k ) Figure 13: Structure of the reduced order estimator: estr
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52 ECE 147b: Matlab Computer Modules 8.3 Module 8: Listing and Graphics clf;axis([1,2,3,4]);axis; echo on %--------------------------------------------------------- % % Dept. of Electrical & Computer Eng.
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This note was uploaded on 04/06/2010 for the course ECE 145 taught by Professor Rodwell during the Spring '07 term at UCSB.

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module8 - Module 8: Observer Based Design 49 8 8.1 Module...

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