Slide# Ketter Hall, North Campus, Buffalo, NY Fax: Tel: 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: H 2 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: H 2 and H  Control  Method 7: Sliding Mode Control  Method 8: Fuzzy Logic Control

Slide# 9 Structure Equations Structure’s Equation Control Force Effective Equation with Control

Slide# 10 Structure Equations Structure’s Equation Denote Variables as:

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: H 2 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: H 2 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: m i =345.6 ton k i =3,404x10 5 kN/m  1 =  2 =2%  I =5.79, 17.18, 27.98, 37.82, 46.38, 53.36, rad/s m d =29.63 ton (=2%W 1 ) c d = 25 ton /m/sec (=7.3%) k d = 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: H 2 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: H 2 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