Cta Exam
Technische Universit¨t Dortmund a Department of Biochemical and Chemical Engineering Chair of Process Dynamics and Operations Prof. Dr. Sebastian Engell
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
Page 2
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
Page 3
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
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
Page 2
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
Page 3
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