Slide# 1
212 Ketter Hall, North Campus, Buffalo, NY 14260
www.civil.buffalo.edu
Fax: 716 645 3733 Tel: 716 645 2114 x 2400
Control of Structural Vibrations
Lecture #7_1
Active Control - Algorithms
Instructor:
Andrei M. Reinhorn P.Eng. D.Sc.
Professor of Structural Engineering
Slide# 2
Slide# 3
Issues in Active Control
Control Logic - Algorithms Control Considerations - Stability, etc. Control Implementations - Force Generation Physical Implementations Full Scale Implementations
Slide# 4
Control Algorithms
Method 1: Optimal Control Method 2: Poles Assignment Method 3: Instantaneous optimum Method 4: Independent Modal Space Control Method 5: Bounded State Control
Method 6: H2 and H Control Method 7: Sliding Mode Control Method 8: Fuzzy Logic Control
Slide# 5
Control Considerations
Stability - Liapunov Considerations Controlability Observability Spill-Over
Slide# 6
Control Implementations
Force Generation Time Delay Robustness
Slide# 7
Block Diagram of Control
Slide# 8
Control Algorithms
Method 1: Optimal Control Method 2: Poles Assignment Method 3: Instantaneous optimum Method 4: Independent Modal Space Control Method 5: Bounded State Control
Method 6: H2 and H Control Method 7: Sliding Mode Control Method 8: Fuzzy Logic Control
Slide# 9
Structure Equations
M x(t) + C x(t) + K x(t) = D u(t) + E f(t)
1 1 1u(t) = K x(t) + C x(t) + E f(t)
Structure’s Equation
Control Force
1 1 1M x(t) + (C-DC ) x(t) + (K-DK ) x(t) = (E + DE ) f(t)
Effective Equation with Control
Slide# 10
Structure Equations
M x(t) + C x(t) + K x(t) = D u(t) + E f(t)Structure’s Equation
-1 -1 -1 -1 x(t) + M C x(t) + M K x(t) = M D u(t) + M E f(t)
1 2z = z
Denote Variables as:
-1 -1 -1 -12 2 1 z (t) + M C z (t) + M K z (t) = M D u(t) + M E f(t)
1 2z = x(t) z = x(t)
Slide# 11
State Space Equation
0
Slide# 12
Quadratic Performance Index
Slide# 13
Optimized Solution
• Solutions of the above leads to basic equations to determine control forces u(t):
Slide# 14
Closed Loop (Feed Back Loop)
Slide# 15
Closed Loop Control
• Ricatti Equation
Slide# 16
Closed Loop Control
Slide# 17
Closed Loop Control
For time independent P :
Slide# 18
Output Control
Computed output Cz(t)
Slide# 19
Open-Closed Loop (Feed Back-Feed Forward)
Slide# 20
Open-Closed Loop Control
Open Loop Control
• Closed Loop Control Excitation
Slide# 21
Open Loop (Feed Forward)
Slide# 22
Open Loop Control
The solution needs the information of loading history and its derivative. Can’t work for earthquake type loading
Slide# 23
Active Tendon System
Slide# 24
Example of Control
Slide# 25
Example’s Parameters
Slide# 26
Example’s Equations
Slide# 27
Optimization Parameters
Slide# 28
SDOF System on Shaking Table
at University at Buffalo (SUNY)
Slide# 29
Active Tendon System
Slide# 30
Frequency Response Function
Slide# 31
Slide# 32
Effect of Weighting Matrices
Slide# 33
Control Algorithms
Method 1: Optimal Control Method 2: Poles Assignment Method 3: Instantaneous optimum Method 4: Independent Modal Space Control Method 5: Bounded State Control
Method 6: H2 and H Control Method 7: Sliding Mode Control Method 8: Fuzzy Logic Control
Slide# 34
Method 2: Poles Assignment
Desired eigenvalue for matrix: A+BG
Slide# 35
Determining Gain Matrix
Determinant equation:
Slide# 36
Determining Gain Matrix
Slide# 37
Example of Poles Assignment
Slide# 38
Example of Poles Assignment
Slide# 39
Example of Poles Assignment
Slide# 40
Example of Poles Assignment
Slide# 41
Example of Poles Assignment
Slide# 42
Example of Poles Assignment
Displacements at the two floors:
Slide# 43
Control Algorithms
Method 1: Optimal Control Method 2: Poles Assignment Method 3: Instantaneous optimum Method 4: Independent Modal Space Control Method 5: Bounded State Control
Method 6: H2 and H Control Method 7: Sliding Mode Control Method 8: Fuzzy Logic Control
Slide# 44
Method 3: Instantaneous Optimum
Slide# 45
Method 3: Instantaneous Optimum
Modal formulation:
Slide# 46
Method 3: Instantaneous Optimum
Slide# 47
Method 3: Instantaneous Optimum
Slide# 48
Instantaneous Closed Loop
Slide# 49
Example of Instantaneous Control
Data:mi=345.6 tonki =3,404x105 kN/m1= 2=2%I=5.79, 17.18, 27.98, 37.82,
46.38, 53.36, 58.53 rad/smd=29.63 ton (=2%W1)cd = 25 ton /m/sec (=7.3%)kd = 957.2 kN/m
Slide# 50
Example of Instantaneous Control
Slide# 51
Example of Instantaneous Control
Slide# 52
Example of Instantaneous Control
Slide# 53
Control Algorithms
Method 1: Optimal Control Method 2: Poles Assignment Method 3: Instantaneous optimum Method 4: Independent Modal Space Control Method 5: Bounded State Control
Method 6: H2 and H Control Method 7: Sliding Mode Control Method 8: Fuzzy Logic Control
Slide# 54
Method 4: Independent Modal Space Control (IMSC)
Slide# 55
Method 4: Independent Modal Space Control (IMSC)
Slide# 56
Control Algorithms
Method 1: Optimal Control Method 2: Poles Assignment Method 3: Instantaneous optimum Method 4: Independent Modal Space Control Method 5: Bounded State Control
Method 6: H2 and H Control Method 7: Sliding Mode Control Method 8: Fuzzy Logic Control
Slide# 57
Method 5: Bounded State Control
Slide# 58
Method 5: Bounded State Control
Slide# 59
Method 5: Bounded State Control
Slide# 60
Method 5: Bounded State Control
Prucz, Soong and Reinhorn, 1983
Slide# 61
Method 5: Bounded State Control
Slide# 62
Method 5: Bounded State Control