Design of Optimal Feedback for Structural Control

194 Pages·2021·20.487 MB·English

by  Ido Halperin, Grigory Agranovich, Yuri Ribakov

Design of
Optimal Feedback
for Structural Control
Ido Halperin
Department of Civil Engineering
Ariel University, Israel
Grigory Agranovich
Department of Electrical and Electronic Engineering
Ariel University, Israel
Yuri Ribakov
Department of Civil Engineering
Ariel University, Israel
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A SCIENCE PUBLISHERS BOOK
First edition published 2021
by CRC Press
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Library of Congress Cataloging-in-Publication Data
Names: Halperin, Ido, 1979- author. | Agranovich, Grigory, 1948- author. |
Ribakov, Yuri, author.
Title: Design of optimal feedback for structural control / Ido Halperin,
Grigory Agranovich, Yuri Ribakov.
Description: First edition. | Boca Raton : CRC Press : Taylor & Francis
Group, 2021. | Includes bibliographical references and index.
Identifiers: LCCN 2020046707 | ISBN 9780367354121 (hardcover)
Subjects: LCSH: Structural control (Engineering) | Structural optimization.
| Feedback control systems. | Control theory.
Classification: LCC TA654.9 .H35 2021 | DDC 624.1/71--dc23
LC record available at https://lccn.loc.gov/2020046707
ISBN: 9780367354121 (hbk)
ISBN: 9780367767006 (pbk)
ISBN: 9780429346330 (ebk)
Typeset in Times New Roman
by Radiant Productions
In the memory of Prof. Jacob Gluck and
Prof. Vadim F. Krotov
Preface
Structural control is a promising approach aimed at the suppression of unwanted
dynamic phenomena in structures. It proposes the use of approaches and tools
from control theory for analysis and manipulation of structures’ dynamic
behavior, with emphasis on suppression of seismic and wind responses. This
book addresses problems in optimal structural control. It gathers some of our
recent contributions to methods and techniques for solving optimal control
problems that rise in structural control problems. Namely, it deals with solving
optimal control design problems that are related to passive and semi-active
controlled structures. The formulated problems consider constraints and
excitations that are common in structural control. Optimal control theory is used
in order to solve these structural control design problems in a rigorous manner,
and based on a firm theoretical background. This monograph begins with a
discussion on models that are commonly used for civil structures and control
actuators. Modern theoretical concepts, such as dissipativity and passivity
of dynamic systems, are discussed in the context of the addressed problems.
Optimal control theory and suitable successive methods are over-viewed.
Novel solutions for optimal passive and semi-active control design problems
are derived, based on firm theoretical foundations. These results are verified by
numerical simulations of typical civil structures subjected to different types of
dynamic excitations.
The books is suitable for researchers and graduate students with background
in civil engineering, structural control, control theory and basic knowledge in
optimal control.
We would like to accentuate here two famous researchers whose contributions
had significant influence on the results that are presented in this monograph—
Prof. Jacob Gluck (1934–2001) and Prof. Vadim F. Krotov (1932–2015).
Prof. Gluck was an Israel civil engineering researcher. He was an expert in
Preface < v
seismic design as well as pioneer in structural computer-aided-design and
structural control. In addition to being a famous researcher, he was a wonderful
teacher and a pleasant person who devoted his life to advancing the field of civil
engineering in Israel out of sincere concern for the lives and safety of people.
Prof. Krotov was a honored Russian researcher, whose work focused on theory
of nonlinear controlled dynamic systems and variational analysis in physics.
His ideas have granted two main contributions to the theory of optimal control:
a new theoretical approach to sufficient optimality conditions, and an iterative
computational technique for deriving a sequence of monotonically improving
control laws. This monograph is inspired by Prof. Gluck’s vision and Prof.
Krotov’s bright ideas.
Contents
Preface iv
List of Symbols viii
1. Introduction 1
2. Dynamic Models of Structures 5
2.1 Plant Models 5
2.2 Viscous and Semi-active Dampers 10
2.3 Dissipative Systems 15
3. Optimal Control 25
3.1 Lagrange’s Multipliers 27
3.2 Pontryagin’s Minimum Principle 28
3.3 Karush-Kuhn-Tucker Necessary Conditions 30
3.4 Krotov’s Sufficient Conditions 34
4. Optimal Control: Successive Solution Methods 39
4.1 Steepest Descent 39
4.2 Parametric Optimization: Newton’s Method 41
4.3 Krotov’s Method—Successive Global Improvements 42
of Control
5. Control Using Viscous Dampers 47
5.1 Optimal Control by Viscous Dampers: Free Vibration 48
5.2 Optimal Control by Viscous Dampers: White Noise Excitation 57
Contents < vii
6. Semi-Active Control 69
6.1 Semi-active Control Constraints 70
6.2 Constrained LQR 74
6.3 Constrained Bilinear Biquadratic Regulator 86
6.4 Constrained Bilinear Quadratic Regulator 121
7. Dampers’ Configuration 155
7.1 Efficient Damper’s Configuration
156
Bibliography 171
Index 181
List of Symbols
(v)N A vector of order N with n The plant order
i i=1
elements v n The number of control inputs
i u
v A column vector of (v)N n The number of control forces
i i=1 w
V A matrix n The number of dynamic
z
(V) Element ij of matrix V DOFs
i j
diag(v) A diagonal matrix with the g State ground acceleration
vector v in its main diagonal input vector, state equation
sign The customary sign function excitation input trajectory
¡ → {–1,0,1} Q State cost
A State equation dynamic R, r Control input cost
i
matrix γ Shear strain
B State equation control input γ DOFs ground acceleration
matrix input vector
C Output equation state t Time variable
observation matrix
f State derivatives vector
C Damping matrix function
d
D Output equation input fc Constraints equation
observation matrix
h Output equation vector
eA The exponential matrix of A function
Ψ DOF’s control forces input H Hamiltonian function
matrix
p Costate/Lagrange multipliers
G Shear modulus vector function
Φ State transition matrix J Performance index
K Stiffness matrix l , l Terminal and integrand cost
f
M Mass matrix functionals
List of Symbols < ix
τ Shear stress f The jacobian matrix of
x
μ KKT multiplier trajectory f with relation to x
x State trajectory U The set of admissible control
trajectories
y Output trajectory
H The set of admissible state
u control input trajectory
trajectories
uˆ Feedback form for u
q Improving function
w Control forces trajectory
CBQR Constrained bilinear
z DOF displacements
quadratic regulator
trajectory
CBBR Constrained bilinear
ż, z¨ The first and second full
biquadratic regulator
derivative of trajectory z
DOF Dynamic degrees of freedom
z , ż MRD elongation, elongation
d d KKT Karush-Kuhn-Tucker
rate and state vector
LQ Linear quadratic
z¨ Ground acceleration
g
¡, ¡n×m The set of real scalars, the LQR Linear quadratic regulator
set of real n × m matrices LTI Linear time invariant
M– 1 The inverse of a matrix M MR Magnetorheological
M T The transpose of a matrix M MRD Magnetorheological damper
e , e The gradient (row vector) MRF Moment resisting frame
g gg
and the hessian matrix of a VD Viscous dampers
function e