A numerical method for choice of weighting matrices in active controlled structures (p 55 72)

A feedback control system usually implements active and semiactive control of seismically excited structures. The objective of the control system is described by a performance index, including weighting matrix norms. The choice of weighting matrices is usually based on engineering experience. A new procedure for weighting matrix components choice based on the parametric optimization method is developed in this study. It represents a twostep optimization process. In the first step a discretetime control system is synthesized according to a quadratic performance index. In the second step the weighting coefficients are obtained using the results of the first step. Numerical simulation of a typical structure subjected to earthquakes is carried out in order to demonstrate the effectiveness of the proposed method. It shows that applying the proposed technique provides a choice of the weighting matrices and results in enhanced structural behaviour under different earthquakes. Copyright © 2004 John Wiley Sons, Ltd.

A NUMERICAL METHOD FOR CHOICE OF WEIGHTING MATRICES IN ACTIVE CONTROLLED STRUCTURES

G AGRANOVICH 1

, Y RIBAKOV 2,3

* AND B BLOSTOTSKY 2

1 Department of Electric Engineering, Faculty of Engineering, College of Judea and Samaria, Ariel, Israel

2 Department of Civil Engineering, Faculty of Engineering, College of Judea and Samaria, Ariel, Israel

3 Institute for Structural Concrete and Building Materials, University of Leipzig, Germany

SUMMARY

A feedback control system usually implements active and semi-active control of seismically excited structures The objective of the control system is described by a performance index, including weighting matrix norms The choice of weighting matrices is usually based on engineering experience A new procedure for weighting matrix components choice based on the parametric optimization method is developed in this study It represents a two-step optimization process In the first two-step a discrete-time control system is synthesized according to a quadratic performance index In the second step the weighting coefficients are obtained using the results of the first step Numerical simulation of a typical structure subjected to earthquakes is carried out in order to demonstrate the effectiveness of the proposed method It shows that applying the proposed technique provides a choice of the weighting matrices and results in enhanced structural behaviour under different earthquakes Copyright © 2004 John Wiley & Sons, Ltd.

1 INTRODUCTION Active and semi-active damping of seismically excited structures is usually implemented by a

feed-back control system (Housner et al., 1997; Spencer et al., 1999) The optimal control forces are

gen-erally calculated according to the structural behaviour, which is measured during the earthquake and transferred to a computer These forces are further produced by actuators or dampers installed in the structure

Recent feedback control development methods are based on optimal control theories (Antsaklis and

Mitchel, 1997; Doyle et al., 1989) These methods require the following mathematical description of

the problem First of all mathematical models of the structure and of the excitation should be obtained

A performance index for structural behaviour and control rules should then be chosen The perform-ance index is a measure of the control forces and the regulated variables describing the structural behaviour Minimization of this performance index yields an optimal control law

According to well-known modern approaches the performance index has a form of various matrix

norms, such as L2, H2, and H•(Antsaklis and Mitchel, 1997; Doyle et al., 1989; Spencer et al., 1994; Dyke et al., 1995) In most practical optimization problems these indices do not directly describe the

problem, because they have no direct physical sense Indeed, an integral of squared state vector or control forces vector is very similar to energy But actually arguing about energy minimization has again no physical sense, because generally the performance indexes include a sum of such two squares The sum yields a compromise between the required and the dissipated energy However, a reasonable Published online 9 June 2004 in Wiley Interscience (www.interscience.wiley.com) DOI:10.1002/tal.233

Accepted February 2003

* Correspondence to: Dr Ing Yuri Ribakov, Universitat Leipzig, Marschnerstrasse 31, 04109 Leipzig, Germany E-mail: ribakov@wifa.uni-leipzig.de

question is which energy is ‘more important’ and how it affects the structural response to earthquakes Moreover, sometimes an apparent improvement of the performance index leads to a worse structural response An additional criterion is proposed in the current study in order to improve the performance index and to design a control system, providing more efficient control and yielding further decrease

in structural response to earthquakes

Spencer et al (1999) described several direct criteria for structural control of seismically excited

buildings However, the feedback optimal control solutions are known for performance indices in the form of matrix norms and for linear structural models only Hence these optimization problems are commonly employed for structural control optimization

A classical performance index form is an L2one with an infinite upper horizon:

(1)

where y is a vector of structural displacements, velocities and accelerations, u is a control forces vector, and Q and R are symmetrical non-negative definite weighting matrices describing the balance of the structural behaviour and of the control action (Dyke et al., 1995; Norgaard et al., 2000).

In any performance index described by Equation (1) the relative magnitudes of the control forces

(components of u) and of the regulated variables (components of y) should also be taken into account The matrices Q and R usually have a diagonal form and give different weights to components of vectors y and u These weights take into account the different physical nature of the components and different requirements to their values Spencer et al (1999), Dyke et al (1995), Dyke and Spencer (1997), Battaini et al (2000) and others investigated the influence of different weighting coefficients

on the effectiveness of optimum control algorithms applied to earthquake-excited buildings Gener-ally most of the coefficients are equal to zero For example, in Dyke and Spencer (1997) only the top

storey acceleration weight is non-zero, whereas in Battaini et al (2000) only in the two lower storeys

are absolute displacements weights non-zero

Generally the matrices Q and R are assumed based on practical experience in structural seismic

design An algorithm for weighting matrix components choice based on parametrical optimization method is described in this paper

2 DESCRIPTION OF THE PROPOSED METHOD

As mentioned above, generally the weighting matrices Q and R selection (Equation 1) is based on

engineering experience Technical constraints on variables and controls can also be taken into account Usually this choice is made by a ‘trial and error’ method For more qualitative choice of the per-formance index weighting matrices the following parametrical optimization method is proposed The proposed approach is applicable to various weighted performance indices Its application to acceleration LQG control design of seismically excited structural control is considered in this study The LQG approach is an output feedback design method that has been shown to be effective for design

of acceleration feedback control strategies for this class of systems (Spencer et al., 1994; Dyke et al.,

1995, 1996; Battaini et al., 2000).

Let Joptbe a direct criterion for control strategies evaluation, for example one or several of those

described by Spencer et al (1999) Thus two criteria are obtained The first one is J with a known feedback control solution, and the second one is Jopt, for which the feedback control solution is

unknown A ‘compromise’ solution is to use the first criterion (J) as a ‘working’ criterion for the second one (Jopt)

The above-mentioned working criteria contain some weighting parameters Let these parameters be

defined by W In this case the second criterion will be a function of weighting parameters W of the

J u( ) =Ú•y t Qy t T( ) ( ) +u t Ru t t T( ) ( )d

0

first one, i.e Jopt(W ) For example, for the performance index J described in Equation (1) W collects the matrices Q and R or their components Thus the problem is reduced to a choice of W, at which the optimal control according to criterion J provides a minimum value of the criterion Jopt(W ).

This approach enables application of well-developed numerical parametric optimization methods

for solution of the problem described by Nelder and Mead (1965) and Gill et al (1981) According

to the proposed method, the optimal control synthesis problem should be solved at each step The general linear model of the controlled structure according to the proposed optimization method can be described as follows:

(2)

where: x(t) is the state space vector of the system’s continuous part, which includes the vectors of

story displacements and velocities of the structure, and the state vectors of the actuators and the

meas-urement subsystems; u(kt) is the control signal, which is an output signal of a digital controller for sampling times kt (k = 0, 1, 2, ); t is the controller’s sampling period; g (t) is the ground acceler-ation; and the matrices A c , B c and E cdescribe the continuous part of the whole system The control system should be realized in a digital form, hence the differential equations (2) are transformed to an equivalent system of finite-difference equations (based on an equivalent transformation technique described in Antsaklis and Mitchel, 1997) as follows:

(3) where

(4) The output vector contains structural displacements, velocities and accelerations:

(5)

where matrices C, D and F describe the dependence between the output vector and the structure’s state

vector and excitations

The measurement vector

(6)

contains the floor accelerations of the structure Matrices C m , D m and F m describe the parameters of the measurement subsystem

According to the LQG design approach (Dyke et al., 1995, 1996; Battaini et al., 2000) the ground

acceleration g (kt) and the measurement noise v(kt) are taken to be a stationary white noise with

known intensity An infinite horizon performance index (Equation 1) takes in this case the following form:

(7)

The square root form of the index weight matrices Q = Q1Q1and R = R1R1is chosen to avoid the fol-lowing two problems in parametrical optimization application The first problem is positive definite-ness of index weight matrices constraint, which requires application of much more complicated

J E y k Q Q y k u k R R u k

T

k T

ÎÍ

˘

˚˙

ƕ £

t

˙˙

y k m( t) =C mx(kt) +D u k m ( t) +F x k m˙˙g( t) + ( )v kt

y k( t) =Cx k( t) +Du k( t) +Fx k˙˙g( t)

A e A B e A t B t E e E t

c

A t c

x k( t) =Ax k( t) +Bu k( t) +Ex k˙˙g( t)

˙˙

x t( ) =A x t c ( ) +B u k c ( t) +E x t c g( )

parametrical optimization methods with constraints The second one is the big difference in weight coefficient values, impairing a convergence property of the parametrical optimization process The separation principle allows the control and estimation problems to be considered separately, yielding a discrete-time dynamic controller (Stengel, 1986) Hence the optimal control law is obtained

as follows:

(8)

with the same gain matrix as the deterministic LQ 2 - control, where S is the solution of the Riccati

algebraic equation given by

(9)

A Kalman steady-state estimation (kt) of the system state vector is obtained from the filter equation:

(10)

where (kt) is the optimal estimate of the system’s state vector x(kt).

The filter gain matrix L is determined in the following way:

(11) (12)

where Q m and R m are the intensity matrices of the ground acceleration g (kt) and the measurement noise v(kt) white-noise approximations, respectively.

When the performance index (Equation 7) represents a ‘working’ criterion, Equations (8)–(14) yield

an optimal feedback control LQG optimization problem solution of this index, minimized with linear

dynamic constraints (Equations 3, 5 and 6)

Following Spencer et al (1999), each proposed control strategy is evaluated for four historical earth-quake records: (i) El Centro (California, 1940), (ii) Hachinohe (Hachinohe City, 1968), (iii) North-ridge (California, 1994), (iv) Kobe (Hyogo-ken Nanbu, 1995) The appropriate responses have being used to calculate the evaluation criteria The evaluation criteria (Spencer et al., 1999) are divided into

four categories: building responses, building damage, control devices, and control strategy require-ments The first three categories have both peak- and norm-based criteria Small values of the evalu-ation criteria are generally more desirable Depending on the purpose and priorities of designing one

of the criteria proposed in Spencer et al (1999) or their combination, a direct optimization criterion

Joptcan be chosen As a representative example an optimization problem with a desirable minimum peak inter-storey drift ratio over the time history of each earthquake is considered:

(13)

under constraints on a peak forces value generated by all the control devices over the time history of each earthquake:

h

t i i i

1= ( )

Ï Ì Ô

Ó

Ô Ô

¸

˝ Ô

˛

Ô Ô

max max

,

El Centro Hachinohe Northridge Kobe

˙˙

m T m T

m m T

T

T

T

m m T

T

ˆx

x k( t+t) =Ax k( t) +Bu k( t) +L y k[ m( t) -D u k m ( t) -C x k m ( t)]

ˆx

A SA T - -S A SB R R T ( 1T 1+B SB T )- B SA Q Q T + T =

1

1 1 0

u k( t) = -Kx kˆ( t), K=(R R1T 1+B SB T )- B SA T

1

where 0 £ t £ tmaxis the time-history earthquake range, 1 £ i £ Nfloorsis the building floors range, d i (t)

is the inter-storey drift of the above-ground level over the time history of each earthquake, and h iis the height of the associated storey

The direct criterion was chosen in the following form:

(15) where r(J11) is a penalty function, which possesses zero value if inequality (14) is valid and reaches

a high positive value otherwise The direct criteria (Equations 13, 14 and 15) calculation for the linear structure model (Equations 3, 5 and 6) with feedback control (Equations 8 and 10) consists of the fol-lowing main steps:

(i) Weighted matrices Q1and R1value assignment Note that all or some of those matrices’ elements

are parameters of the direct optimization criterion Jopt(W ).

(ii) Feedback control (Equations 8 and 10) parameters K and L calculation using Equations (8), (9),

(11) and (12)

(iii) Controlled structure simulations over the time history of each earthquake and criteria calculation using Equations (13), (14) and (15)

An integral optimization algorithm consists of three main blocks (see Figure 1)

Note that the proposed procedure is very similar to a neural net training (Norgaard et al., 2000) In

a similar way the proposed algorithm is based on real excitation data, which is obtained from the his-torical earthquake records However, in this case instead of comparison with desired output an optimal control is realized It is obvious that for a particular earthquake this control will be optimal if in the criterion (Equations 13 and 14) only this particular earthquake record is treated Optimization of the

Jopt( ) =W J1+ (r J11)

t i i

11=

Ï Ì Ô

Ó

Ô Ô

¸

˝ Ô

˛

Ô Ô

£ max max

El Centro Hachinohe Northridge Kobe

Structural parameters and initial values

of W 0 assignment

Stepwise procedure: W n+1=W n+s( J opt ( W n ))

and minimum W opt =arg Min ( J opt ( W )) search

Optimized system simulation and analysis Figure 1 Parametrical optimization algorithm

criterion (13) yields a control, providing the best structural response to the worst-case earthquake con-ditions In contrast to Equation (13) the following modified criterion is used:

(16)

It provides the best average result , but not the best structural response to each specific earthquake In this case the direct criterion (15) with the above-described penalty function takes the form

(17)

3.1 Description of the structure and preliminary analysis

In order to demonstrate affectivity and to verify the proposed optimization procedure, MATLAB-based optimum searches and simulations were carried out A typical six-storey steel office building (D’Amore and Astanen-Asl, 1995) designed with UBC-73 (see Figure 2) was chosen for the analysis The struc-tural system consists of a premier welded MR steel frame (Figure 2) Steel ASTM A36 was used for all shapes of columns and grids The stiffness coefficients and floor masses of the building are shown

in Table 1

The natural frequencies of the chosen structure are 1·083, 2·92, 4·799, 9·596, 7·93 and 6·478 Hz

An initial damping ratio of 2% was assumed for the first vibration mode of the uncontrolled structure

Jopt( ) =W J1m+ (r J11)

h

m

t i i i

1 = ( )

Ï Ì Ô

Ó

Ô Ô

¸

˝ Ô

˛

Ô Ô

,

El Centro Hachinohe Northridge Kobe

columns beams W24x68

W14x95 400 cm

W24x68

W14x95 400 cm

W24x68

W14x136 400 cm

W24x68

W14x136 400 cm

W24x102

W14x184 400 cm

W24x116

W14x184 520 cm

6 bays × 610 cm

Figure 2 A six-storey structure used for numerical simulation

Peak inter-storey drifts and story accelerations of the uncontrolled structure under the selected earthquakes are given in Tables 2 and 3 These and following numerical results were obtained using SIMULINK software (MathWorks, 1990) simulation of the structure To this end a version of the

‘Benchmark simulation program for seismically excited buildings’ (Spencer et al., 1999) modified by

the authors was used The above-mentioned four earthquake records were considered with single

mag-nitude level (Spencer et al., 1999), which was equal to 1.

Following Spencer et al (1999) and Battaini et al (2000) it was assumed that the noised

accel-erations of all storeys are available and the control actuators are located at each storey of the struc-ture The dynamics of the measuring instruments and of the control actuators was neglected Similar

to Spencer et al (1999), the control force bound Umax in Equation (14) was assumed to be equal to 10,000 N

Table 1 Structural parameters of the six-storey building Floor number Floor mass (10 5 kg) Stiffness coefficient (10 5 kg/m)

Table 2 Peak inter-storey drifts of the uncontrolled structure (cm)

Earthquake record

Table 3 Peak storey accelerations of the uncontrolled structure (m/s 2 )

Earthquake record

3.2 One- and two-parametric optimization

According to the proposed optimization procedure (Figure 1) the vector of optimized parameters W was chosen The weight matrices Q1and R1of the working performance index J (Equation 7) have

been taken in the following diagonal form:

(18) According to Equation (18) the weights of every control force, inter-storey drifts and storey absolute

velocities have been assumed to be equal to 1, q d and q v, respectively It should be mentioned that multiplying the criterion (Equation 7) by a constant does not affect the solution Thus, only relative values of the weight parameters are relevant For this reason the control weights in Equation (18) are assumed to be equal to one Note that in the optimization procedures described, for example, in Spencer

et al (1999), Dyke et al (1995) and Battaini et al (2000) it is assumed that q d = q v = 0, but R1and

Q1are diagonal matrices with prescribed numerical values Hence only one optimization parameter q a

is used (one-parameter optimization procedure) Some results of the one-parameter q aoptimization procedure are presented in Tables 4(a–d) In these and the following tables

R1=I6 6¥ , Q1=diag{q I d 6 6¥ ,q I v 6 6¥ ,q I a 6 6¥}

Table 4(a) Peak inter-storey drifts and storey accelerations of the controlled structure with one-parametric q a

optimization under the El Centro earthquake Optimization over Optimization according to Optimization according to

Storey Drift (cm) Acc (m/s 2 ) Drift (cm) Acc (m/s 2 ) Drift (cm) Acc (m/s 2 )

Table 4(b) Peak inter-storey drifts and storey accelerations of the controlled structure with one-parametric q a

optimization under the Hachinohe earthquake Optimization over Optimization according to Optimization according to

Storey Drift (cm) Acc (m/s 2 ) Drift (cm) Acc (m/s 2 ) Drift (cm) Acc (m/s 2 )

is the total energy required for the control of the structure, where f i (t) is the control force developed

by the ith control device and x i ¢(t) is the velocity in the ith control device during the earthquake.

First an ‘ideal’ optimization has been performed A real earthquake record was used as an input signal After the parameters of the performance index have been obtained, the same earthquake record has been applied in order to validate the efficiency of the obtained parameters The results of this opti-mization are shown in Tables 4(a–d) (columns 2 and 3)

It is obvious that such optimization is unavailable for application, because it requires prior knowl-edge of the future earthquake The ‘ideal’ optimization has been performed for the following two reasons First, it enables comparison of the subsequent results of a real optimization with an ‘ideal’ structural behaviour Secondly, it is possible to show that the values of the optimized parameters essen-tially depend on the earthquake’s record and not only on the earthquake’s peak ground acceleration (PGA)

Columns 4 and 5, and 6 and 7, in Tables 4(a–d) present results of one-parametric optimization over the four chosen earthquakes according to criteria (15) and (17) The weighting coefficient of storey

accelerations q a(Equation 18) was selected as an optimized parameter

P x t f t i i t

t i

f

£ £

d

0

1 6

Table 4(c) Peak inter-storey drifts and storey accelerations of the controlled structure with one-parametric q a

optimization under the Nothridge earthquake Optimization over Optimization according to Optimization according to

Storey Drift (cm) Acc (m/s 2 ) Drift (cm) Acc (m/s 2 ) Drift (cm) Acc (m/s 2 )

Table 4(d) Peak inter-storey drifts and storey accelerations of the controlled structure with one-parametric q a

optimization under the Kobe earthquake Optimization over Kobe Optimization according to Optimization according to

Storey Drift (cm) Acc (m/s 2 ) Drift (cm) Acc (m/s 2 ) Drift (cm) Acc (m/s 2 )

The optimization for each of these two criteria includes about 30–40 steps Analysis of the opti-mization process for the criterion (15) shows that for the order of 10–20 steps the process tends to reduce maximal inter-storey drift under the Kobe earthquake having the highest PGA After that the peak inter-storey drift values for the Northridge and Kobe earthquakes are similar The subsequent optimization steps tend to provide a compromise between minimum values for these two earthquakes The one-parameter optimization is not ideal Nevertheless for both criteria (15) and (17) and for each of the four considered earthquakes it yields a close structural response compared to the ‘ideal’ control (Tables 4a–d) However, for the Northridge earthquake it requires higher control energy com-pared to the ‘ideal’ control

Tables 5(a–d) present the results of a two-parametric optimization The weighting coefficients of

inter-storey drifts q d and storey accelerations q a(Equation 18) were selected as optimized parameters The process of step optimization for each of two criteria (15 and 17) contains about 50–60 steps

and yields the following results: q d= 7.17 ¥ 107

and q a= 101 Applying two-parametric optimization yields a decrease in the inter-storey drifts, compared to the one-parametric one; however, it results in

an essential increase of floor accelerations It should be noted that the addition of a third optimized

parameter q vdoes not yield any significant improvement compared to the two-parametric optimiza-tion results

Table 5(a) Peak inter-storey drifts and storey accelerations of the controlled structure with two-parametric q d,

q aoptimization under the El Centro earthquake Optimization over Optimization according to Optimization according to

Storey Drift (cm) Acc (m/s 2 ) Drift (cm) Acc (m/s 2 ) Drift (cm) Acc (m/s 2 )

Table 5(b) Peak inter-storey drifts and storey accelerations of the controlled structure with two-parametric q d,

q aoptimization under the Hachinohe earthquake Optimization over Optimization according to Optimization according to

Storey Drift (cm) Acc (m/s 2 ) Drift (cm) Acc (m/s 2 ) Drift (cm) Acc (m/s 2 )