08.Performance of control systems
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Exercises for Chapter 8 |
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Step response 1 |
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The material in this chapter has focused upon the unity gain feedback loop and its relation to the solution of Problem 6.41 concerning design for input/output systems. In the next two problems, you will investigate a few aspects of performance for Problem 6.48 where static state feedback is considered, and for Problem 6.53 where static output feedback is considered. Recall that the block diagram representation for static state feedback is as in Figure 8.5, and that the block diagram representation for static output feedback is as in
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D
Figure 8.5 Block diagram for static state feedback
358 8 Performance of control systems 22/10/2004
Figure 8.6. You may also wish to refer to Theorems 6.49 and 6.54 concerning the form of
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D
Figure 8.6 Block diagram for static output feedback
the transfer function under static state feedback and static output feedback.
E8.7 In this exercise we consider a controllable SISO linear system Σ = (A, b, ct, 01) and a state feedback vector f. You may suppose that (A, b) is in controller canonical form, and that f Ss(Σ).
(a)Compute the transfer function from the reference rˆ(s) to the error rˆ(s) −yˆ(s) for the closed-loop system of Figure 8.5.
(b)Determine the system type for the closed-loop system of Figure 8.5. Note that the system type will depend on the relationship between the state feedback vector f and the system Σ.
Now we will consider a concrete example of the above situation by taking
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(c)What are the possible values for the system type for the closed-loop system in this case?
(d)For what values of f0 and f1 does the system type achieve its maximum possible value?
(e)Let f be a state feedback vector from part (d), i.e., so that the system type of the closed-loop system is maximal. Plot the step response of the closed-loop system. What is the steady-state error?
(f) Let f be a state feedback vector that is not of the type which answers part (d), i.e., so that the system type of the closed-loop system is not maximal. Plot the step response of the closed-loop system. What is the steady-state error?
E8.8 In this exercise we consider a controllable SISO linear system Σ = (A, b, ct, 01) and an output feedback constant F . You may suppose that (A, b) is in controller canonical form.
(a)Compute the transfer function from the reference rˆ(s) to the error rˆ(s) −yˆ(s) for the closed-loop system of Figure 8.6.
Exercises for Chapter 8 |
359 |
360 |
8 Performance of control systems |
22/10/2004 |
(b)Determine the system type for the closed-loop system of Figure 8.6. Note that the system type will depend on the relationship between the output feedback constant F and the system Σ.
Now we will consider a concrete example of the above situation by taking
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(c)What are the possible values for the system type for the closed-loop system in this case?
(d)For what values of F does the system type achieve its maximum possible value?
(e)Let F be an output feedback constant from part (d), i.e., so that the system type of the closed-loop system is maximal. Plot the step response of the closed-loop system. What is the steady-state error?
(f)Let F be a output feedback constant that is not of the type which answers part (d), i.e., so that the system type of the closed-loop system is not maximal. Plot the step response of the closed-loop system. What is the steady-state error?