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-5. The approximating difference equation representation for a continuously operating system is x(k + 1) = 0.75x(k) + u(k). It is desired to bring the system state to the target set S defined by 0.0 < x(2) <2.0 with minimum expenditure of control effort; i.e., minimize J- u(0) + u?(1). The allowable state and control values are constrained by 0.0 < x(k) < 6.0 -1.0 su(k) < 1.0.

-5. The approximating difference equation representation for a continuously operating system is x(k + 1) = 0.75x(k) + u(k). It is desired to bring the system state to the target set S defined by 0.0 < x(2) <2.0 with minimum expenditure of control effort; i.e., minimize J- u(0) + u?(1). The allowable state and control values are constrained by 0.0 < x(k) < 6.0 -1.0 su(k) < 1.0.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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3-5. The approximating difference equation representation for a continuously operating system is x(k + 1) = 0,75x(k) + u(k). It is desired to bring the system state to the target set S defined by 0.0 < x(2) < 2.0 with minimum expenditure of control effort; i.e., minimize J= u-(0) + u?(1). The allowable state and control values are constrained by 0.0 <x(k) < 6,o -1.0 <u(k) < 1.0. Quantize the state values into the levels x(k) 0, 2o, 4.0, 6.0 for k = 0, 1, 2 and the control values into the levels u(k) -1.0, -0.5, 0.0, 0.5, 1.0 for k = 0, 1. (a) Find the optimal control value(s) and the minimum cost for each point on the state grid. Use linear interpolation. (b) What is the optimal control sequence {u*(0), u*(1)} if x(0) = 6.0?
Transcribed Image Text:3-5. The approximating difference equation representation for a continuously operating system is x(k + 1) = 0,75x(k) + u(k). It is desired to bring the system state to the target set S defined by 0.0 < x(2) < 2.0 with minimum expenditure of control effort; i.e., minimize J= u-(0) + u?(1). The allowable state and control values are constrained by 0.0 <x(k) < 6,o -1.0 <u(k) < 1.0. Quantize the state values into the levels x(k) 0, 2o, 4.0, 6.0 for k = 0, 1, 2 and the control values into the levels u(k) -1.0, -0.5, 0.0, 0.5, 1.0 for k = 0, 1. (a) Find the optimal control value(s) and the minimum cost for each point on the state grid. Use linear interpolation. (b) What is the optimal control sequence {u*(0), u*(1)} if x(0) = 6.0?
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