Homework_4.1
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2. Using MATLAB, we can obtain the unit step response of the closed‐loop system for different values of K.Observe the diagram we can find what as follow: 1) When K ,it yields unstable responses; 2) When K 4.8 ,it yields persistent oscillation; 3) When 1.5385 K 4.8 ,it yields underdamped responses; 4) When K = ,it yields the critical damped response; 5) When K ,it yields overdamped responses; The step responses different values of K(from 0.5 to 4.8,namely,from overdamped to underdamped) show as in Fig. 1.It looks like the responses of a second-order system.
Fig. 1 The Unit Step Response for Different Values of 0＜K≤4.8
However,when K ( the responses are unstable),it looks quite different form a second-order system.The response oscillates periodically with amplitude more and more greater as in Fig. 2.
By inspection of the signal‐flow graph we write the derivatives of the state variables:
x1 x3 x4
x2 x2
Kx4 K d
x2 5 x1 2 x2 x3 3x4
1
By inspection from the graph, we write the output equation:
Solution: 1. Signal‐flow graphs:
d
K
1/s x1 5 -1
1/s pa 1 -2 -1
1/s x3 1
1/s a -3
where
1 dpa 5 dt x2 pa Actual wheel position x1 d a dt x4 a Actual bearing angle x3
2
Fig. 2 The Unit Step Response for K＞4.8
3
Homework 4.1(corrected version)
Student ID 3009202046 Name Liu Qi（刘琪 liú qí） A HelpMate® transport robot is used to deliver food, drugs,laboratory materials and other goods in a hospital settling.Given the block diagram of the robot’s bearing angle (azimuth) control system: 1. Represent the system in State Space, where the actualwheel position and the actual bearing angle are among thestate variables. 2. Using MATLAB, obtain the unit step response of the closed‐loop system for different values of K that yield responses from overdamped to underdamped to unstable.
y x4 a
Rewriting equations in vector-matrix form
ቤተ መጻሕፍቲ ባይዱ
0 1 5 2 x 0 1 0 0
0 K K 0 0 0 x d 0 0 0 1 3 0
y 0 0 0 1 x
Fig. 1 The Unit Step Response for Different Values of 0＜K≤4.8
However,when K ( the responses are unstable),it looks quite different form a second-order system.The response oscillates periodically with amplitude more and more greater as in Fig. 2.
By inspection of the signal‐flow graph we write the derivatives of the state variables:
x1 x3 x4
x2 x2
Kx4 K d
x2 5 x1 2 x2 x3 3x4
1
By inspection from the graph, we write the output equation:
Solution: 1. Signal‐flow graphs:
d
K
1/s x1 5 -1
1/s pa 1 -2 -1
1/s x3 1
1/s a -3
where
1 dpa 5 dt x2 pa Actual wheel position x1 d a dt x4 a Actual bearing angle x3
2
Fig. 2 The Unit Step Response for K＞4.8
3
Homework 4.1(corrected version)
Student ID 3009202046 Name Liu Qi（刘琪 liú qí） A HelpMate® transport robot is used to deliver food, drugs,laboratory materials and other goods in a hospital settling.Given the block diagram of the robot’s bearing angle (azimuth) control system: 1. Represent the system in State Space, where the actualwheel position and the actual bearing angle are among thestate variables. 2. Using MATLAB, obtain the unit step response of the closed‐loop system for different values of K that yield responses from overdamped to underdamped to unstable.
y x4 a
Rewriting equations in vector-matrix form
ቤተ መጻሕፍቲ ባይዱ
0 1 5 2 x 0 1 0 0
0 K K 0 0 0 x d 0 0 0 1 3 0
y 0 0 0 1 x
