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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

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Issues in Active Control

Control Logic - Algorithms Control Considerations - Stability, etc. Control Implementations - Force Generation Physical Implementations Full Scale Implementations

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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

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Control Considerations

Stability - Liapunov Considerations Controlability Observability Spill-Over

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Control Implementations

Force Generation Time Delay Robustness

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Block Diagram of Control

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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

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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

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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)

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State Space Equation

0

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Quadratic Performance Index

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Optimized Solution

• Solutions of the above leads to basic equations to determine control forces u(t):

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Closed Loop (Feed Back Loop)

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Closed Loop Control

• Ricatti Equation

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Closed Loop Control

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Closed Loop Control

For time independent P :

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Output Control

Computed output Cz(t)

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Open-Closed Loop (Feed Back-Feed Forward)

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Open-Closed Loop Control

Open Loop Control

• Closed Loop Control Excitation

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Open Loop (Feed Forward)

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Open Loop Control

The solution needs the information of loading history and its derivative. Can’t work for earthquake type loading

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Active Tendon System

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Example of Control

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Example’s Parameters

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Example’s Equations

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Optimization Parameters

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SDOF System on Shaking Table

at University at Buffalo (SUNY)

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Active Tendon System

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Frequency Response Function

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Effect of Weighting Matrices

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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

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Method 2: Poles Assignment

Desired eigenvalue for matrix: A+BG

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Determining Gain Matrix

Determinant equation:

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Determining Gain Matrix

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Example of Poles Assignment

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Example of Poles Assignment

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Example of Poles Assignment

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Example of Poles Assignment

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Example of Poles Assignment

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Example of Poles Assignment

Displacements at the two floors:

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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

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Method 3: Instantaneous Optimum

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Method 3: Instantaneous Optimum

Modal formulation:

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Method 3: Instantaneous Optimum

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Method 3: Instantaneous Optimum

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Instantaneous Closed Loop

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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

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Example of Instantaneous Control

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Example of Instantaneous Control

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Example of Instantaneous Control

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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

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Method 4: Independent Modal Space Control (IMSC)

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Method 4: Independent Modal Space Control (IMSC)

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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

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Method 5: Bounded State Control

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Method 5: Bounded State Control

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Method 5: Bounded State Control

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Method 5: Bounded State Control

Prucz, Soong and Reinhorn, 1983

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Method 5: Bounded State Control

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Method 5: Bounded State Control