Written Midterm Examination Control Theory & Application Master Program: Automation & Robotics WS 2010/11 9th of December, 2010 Duration: 75 minutes

Task

Achievable Score

Score Achieved

Σ Question 1: a) b) c) d) e) Σ Question 2: Σ Question 3: a) b) Σ Question 4: a) b) c) d) e) f) Σ Question 5: Σ Question 6: a) b) Σ Total

2.5 0.5 0.5 0.5 0.5 0.5 3 1.5 1 0.5 3 0.5 0.5 0.5 0.5 0.5 0.5 4 (0.5x8) 3 1 2 17.0

Task

Achievable Score

Score Achieved

Σ Problem 1: a) b) Σ Problem 2: a) b) c) Σ Problem 3: a) b) Σ Total

9 3 6 6 2 2 2 5 2 3 20

Midterm Examination: Control Theory and Applications, 9.12.2010

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Question 1: Stability (2.5 Points)

State True or False a. A dynamic system can be operated at an unstable equilibrium point, if a stabilizing feedback controller is applied to it. ( ) b. A system is stable according to Lyapunov if the system converges to some equilibrium point for all initial states. ( ) c. According to Lyapunov, the domain of attraction of a nonlinear system with one stable equilibrium point is always the complete state-space. ( ) d. An autonomous linear system having two eigenvalues λ1 = 0, λ2 < 0 will always converge to the origin. ( ) e. An autonomous linear system having two eigenvalues λ1 = λ2 = 0 is always a simply stable system. ( )

Midterm Examination: Control Theory and Applications, 9.12.2010

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Question 2: SVD (3 Points)

A two tank system is shown in the ﬁgure. A constant input ﬂow is fed through valve V1 and then distributed equally to both tanks. The outlet ﬂow of the system goes through valves V2 and V3 . The system has two manipulated variables, which are the opening positions of the valves V2 and V3 . The state variables are the heights of the ﬂuid inside the tanks h1 and h2 .

V1

h1

h2

V2

Tank 1

Tank 2

V3

Figure 1: The two tank system The system has the following properties: 1. It takes less eﬀort for the system in order to increase or decrease the heights of the ﬂuid in both tanks at the same time and this can be done as follows: a. Closing V2 and V3 will increase the heights in both tanks. b. Opening V2 and V3 will decrease the heights in both tanks. 2. It takes more eﬀort for the system in order to increase the height of the ﬂuid in one tank and decrease it in the other at the same time and this can be done as follows: a. Closing V2 and opening V3 will increase the height in tank 1 and decrease it in tank 2. b. Opening V2 and closing V3 will decrease the height in tank 1 and increase it in tank 2. h1 V2 and the input vector is: which represents the valve opening h2 V3 percentages. The SVD (Singular Value Decomposition) is given by: The state vector is:

Midterm Examination: Control Theory and Applications, 9.12.2010 vT i . vT j

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U = ui uj ,

Σ=

100 0 , 0 10

VT =

Hint: The ﬁrst property of the system can be related to the ﬁrst and the larger singular value (i.e 100) and the second property of the system can be related to the last and the smaller singular value (i.e 10)

Task: Choose appropriate values of ui , uj , v i and v j for the above mentioned system that well describe the previous characteristics. • For the output matrix U , choose from the following vectors: T T u1 = 0.7071 −0.7071 , u2 = −1.23 −0.5 , u3 = −0.7071 −0.7071 T

, u4 = 1.23 −0.5

T

• For the input matrix V T , choose from the following vectors: T T v 1 = −0.6638 0.7479 , v 2 = 1.34 −1.34 , v 3 = 1.34 −0.576 T

, v 4 = 0.7479 0.6638

T

Midterm Examination: Control Theory and Applications, 9.12.2010

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Question 3: Linearity (1.5 Points)

Assume we have a linear system f (x) that is stimulated by two diﬀerent inputs x1 and x2 . The responses are shown in ﬁgures 2(a) and 2(b). Now, assume that the system is stimulated by a third input x3 , where...