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module7 - Module 7: State Feedback 35 7 7.1 Module 7: State...

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Module 7: State Feedback 35 7 Module 7: State Feedback 7.1 Overview State space design methods are introduced in this module. Chapter 6 of the text covers all of the state space design approaches. We will start with full state feedback. This assumes that all state variables can be measured. A satellite tracking dish is used as an example. A more detailed study of this problem is given in Appendix A.2 of the text. Figure 4 illustrates the plant. θ Figure 4: Satellite antenna dish 7.2 Notes This module covers a relatively involved state-feedback problem. The accompa- nying block diagrams are discussed in the notes section. It is often easier to have the equivalent picture in your mind when you are looking at the mathematics. The open loop system is formed in the continuous domain. This is illustrated schematically in Figure 5. A wind disturbance, at input Td ( t ), is modelled as low passed random noise signal. For the open loop simulation Tc ( t ) = 0. This simulation forms the base line for comparison of the closed loop schemes. P ( s ) ± ± ± ± θ ( t ) ( t ) ˙ θ ( t ) ( t ) Figure 5: Open loop plant: Satellite antenna dish Now a zero order hold equivalent is obtained and used for state feedback design.
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36 ECE 147b: Matlab Computer Modules The closed loop, discrete time, state feedback system is illustrated if Figure 6. The discrete time system, Pd ( z ), is a zero order hold equivalent to P ( s ). For the design purposes, we are only interested in the control input, Tc . ( z ) - K ± ± ± ± ² ² ± θ ( k ) ˙ θ ( k ) ( k )= u ( k ) Figure 6: Design con±guration for discrete time state feedback The objective of the state feedback design is to drive the state to zero. Often, the non zero state results from an initial condition. In this case the wind disturbance will drive the states away from zero. The simulation is now much more complex than any we have previously per- formed. The controller, - K , is discrete time. I know it doesn’t have any dynamics, however, we designed it for a discrete time plant, to put the poles at good locations in the unit disk. The wind disturbance signal, Td ( t ), is con- tinuous and drives the states of the continuous plant P ( s ). To simulate this sampled-data (i.e. mixed continuous and discrete) system we use Simulink . The full system is illustrated in Figure 7. - K ZOH P ( s ) ± ± . . .. . . T ² ² ² ² ² ± ± ± ± ( t ) ( t u ( t ) θ ( t ) ˙ θ ( t ) θ ( k ) ˙ θ ( k ) u ( k ) Figure 7: Sampled data feedback system used for simulation This state space representation of this interconnected system is constructed in m7sd.mdl .
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Module 7: State Feedback 37 7.3 Module 7: Listing and Graphics clf;axis([1,2,3,4]);axis; echo on %--------------------------------------------------------- % % Dept. of Electrical & Computer Eng. % University of California, Santa Barbara. % % ECE 147 B: Digital Control Systems % % module 7: State feedback % % ------------------------------------------------- % State feedback design is studied in this module.
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module7 - Module 7: State Feedback 35 7 7.1 Module 7: State...

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