System model u x ˙ = Ax + Bu x y C
Control law −K
Observer ˙ = Ax x ˜ ˜ + Bu + Ly ˜ y ˜ = y − Cx ˜ + −
C Compensator
Figure 1.
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4. Consider the system represented in state variable form x ˙ = Ax+Bu y = Cx+Du where A= 1 2 −5 −10 ,B= −4 1 , C = ( 6 −4 ) and D = ( 0 ).
Verify that the system is observable and controllable. If so, design a full-state feedback law and observer by placing the closed-loop system poles at s1, 2 = −1 ± j and the observer poles at s1,2 = −10.
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3. A system has a matrix differential equation ˙= x 0 1 2 0 x+ b1 b2 u
What values for b1 and b2 are required so that the system is controllable?
˙ = x
x +
u
2. Consider a system with state-space model given below. −3 1 0 1 ˙ = 0 −3 1 x + 0 u x 0 0 −3 1 a1 0 0 y = x 0 1 0 a) Is the system observable? Does it depend on a1? b) Find the range of values of a1 for which the system is observable? c) Is the system controllable? Does it depend on a1?
Design a full-state feedback gain matrix and an observer gain matrix to place the closed-loop system poles at s1,2 = −1.4 ± j 1.4, s3,4 = −2 ± j and the observer poles s1, 2 = −18 ± j 5, s3, 4 = −20. Construct the state variable compensator using Figure 1 as a guide and simulate the closed-loop system using Simulink. Select several values of initial states and initial state estimates in the observer and display the tracking results on an x, y -graph.
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Modern Control Theory and Application - Homework 3 1. A state-space model of a system is given below. Is the model controllable? −2 1 0 0 0 0 −2 0 0 0 0 0 −1 0 0 0 0 0 −5 0 0 0 0 0 −4 0 2 1 1 2
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5. Consider the system in state variable form 0 1 0 0 0 0 1 0 x + ˙ = x 0 0 0 1 −2 −5 −1 −13 y = ( 1 0 0 0 )x+( 0 )u 0 0 u 0 1