Monte Carlo Simulations with Variance Reduction for Reliability in Control of

Structures

Rajdip Nayek

Department of Civil EngineeringIndian Institute of Science

Submitted on: 23th January, 2013

Abstract

This report deals with the application of Bayesian method of model class selection

in structural engineering problems. Models which have more number of adjustable

parameters will _t the data better but they perform poorly for predictions. Toovercome this problem a method of parsimony is presented inside the

algorithmof Bayesian model class selection to penalize more complex models. Problems

ofincreased size demand a more e_cient and less time taking strategy. The reportthis aspect. Three problems of support condition identication of a beam have

been analyzed. The code written has been tried for parallel processingto note the performance improvement.

Acknowledgments

The following report is based on my attempt to apply Bayesian model class selectionin structural engineering problems. For this valuable educational venture I would like

to express my gratitude to our respected professors and my guides for the preparationof this report Prof. C S Manohar and Dr. Debraj Ghosh, without whose advises

and constant help and guidance this venture would not have been a success. I wouldalso like to thank my friends who have helped me to achieve the same target along

with me.Thanking you,

(SUBHAYAN DE)

Control of Structures

For the last thirty years or so, the reduction of structural response caused by dynamic effects has become a subject of intensive research. Many structural control concepts have been evolved for this purpose, and quite a few of them have been implemented in practice. The concept of structural seismic response control originated in the 1950swith Japanese researchers Kobori and Minai (1960).

Yao’s conceptual paper (1972) marked a significant contribution to structural control research in the United States. He proposed an “error-activated structural system whose behavior varies automatically in accordance with unpredictable variations in the loading as well as environmental conditions and thereby produces desirable responses under all possible loading conditions.” His study has accelerated research in structural control and soon it was identified as an attractive technique for effectively reducing the wind induced vibration and protecting the structure against seismic forces. Control of structures involves many areas, in particular, analytical dynamics for efficient derivation of the equations of motion, structural dynamics for modeling and analysis and control theory for design of control systems. Advances in these areas along with the developments in computer technology and material science have stimulated research to intensify the developments in structural control applications.

The field of structural control is very vast and its various aspects are covered extensively in the existing literature. Excellent state-of-the-art reviews of developments in structural control are provided by Yang and Soong (1988), Soong (1988), Housner et al. (1997), Datta (2003) . Several text books giving the theoretical and practical aspects of structural control are also available. A comprehensive review on structural control is also provided. Basic principles of structural control along with the developments and progress made are provided in the book by Soong (1990).The book by Meirovitch (1989) mainly focuses on the theoretical aspects. The book by Franklin Y. Cheng (2008) gives a good review of mathematical formulations and numerical procedures for various control strategies and their implementation in practice.

The emphasis of the current work is to employ a suitable control strategy for structural control and estimate the probability of failure of the system with active/semi-active elements experimentally.

Structural Control Systems

Depending on the control strategy used, structural control systems are mainly classifiedinto four groups: passive, active, semi-active and hybrid control systems.

Passive Control SystemsIn passive control systems, vibration control is achieved by dynamic absorption of

vibrations, by seismic base isolation or by using systems for additional energy dissipation(Chen and Scawthorn 2002). They do not require external power, sensors, actuators orcontrollers and hence maintenance is easy. Control forces are developed at the location ofinstallation of the passive control mechanisms and these forces are provided by the motionof these mechanisms during dynamic interaction. The commonly used passive controlsystems are base isolation systems (Skinner et al. 1975, Su et al. 1989 and Jangid 2002)and the family of tuned dampers consisting of tuned mass dampers and tuned liquiddampers (Housner et al. 1997 and, Soong and Spencer 2002). Although passive control devices provide simple, robust and less expensive control solutions, they have some inherent limitations. The control action is limited and are non adaptive to the changes in external loading conditions. Passive devices give optimal performance for a particular excitation frequency, for which they are designed and hence a system designed to give optimal performance for a particular earthquake may not perform satisfactorily for another earthquake. So, one needs to adopt active and/or semi-active control systems, to get optimal control performance for a wide range of excitation frequencies.

Active Control Systems

The ability of the actively controlled system to adapt themselves to the excitation frequencies makes it attractive, butthe tremendous power requirement, especially for massive structures, restricts itsapplications (Housner et al. 1997). The book by Soong (1990) provides a detailed accountof the theoretical and practical aspects of active control of civil engineering structures. In active control systems, the vibration control is achieved by applying external force (control force) to the structure in a prescribed manner. Active control systems employ afeedback mechanism to measure the response of the structure using physical sensors andprocess these responses to compute the required control forces, using a control algorithm.Thus, the control force is a function of the structural response.

Semi-Active and Hybrid Control Systems

Both of these combine active and passive control mechanisms and thus are both “hybrid” systems in some sense. They both gain the reliability of passive devices and adaptability of active control systems. The difference is the function of the active device of the two systems. The one in hybrid systems applies control force directly to the structure, while the one in semi-active systems adjusts the behavior of the passive device. In other words, a semi-active system has adjustable properties in real time but cannot directly apply energy to the smart structure to control its seismic response. Thus, the capacity of a semi-active system is somewhat limited by its base, a passive device. A hybrid system such as the hybrid damper- actuator bracing control (HDABC) system can gain the capacity of its active control device in addition to its reliability and adaptability.Many civil engineering structures have been installed with semi-active control systems (Spencer and Nagarajaiah, 2003). It has been found to achieve significantly better results than passive control devices and has the potential to achieve the performance of fully active systems (Dyke et al. 1996, 1998, Jansen and Dyke 2000). In particular, the electro-rheological (ER) and magneto-rheological (MR) dampers serve as examples for semi-active systems, and HDABC systems serve as the example for hybrid systems.

Control Algorithms

Control algorithms are used to determine the control force from the measured structural response. They are implemented by means of software in the control computer. Control algorithms yield a control law, the mathematical model of the controller, for the active structural control system. Development and implementation of the control algorithm also called controller design. This section reviews the concept of performanceindexes and determination of feedback gain and control force by classical controlalgorithms, Riccati optimal active control (ROAC) and pole placement

SDRE

State Dependent Riccati Equation (SDRE) method [1 - 4] is a recently emerged nonlinearcontrol system design methodology for direct synthesis of nonlinear feedback controllers. Usinga special form of the system dynamics, this approach permits the designer to employ linearoptimal control methods such as the LQR methodology and the H∞ design technique for thesynthesis of nonlinear control systems.The SDRE design technique requires the dynamic model of the system to be placed in thestate dependent coefficient (SDC) form. The SDC form has the structure:

xdot = A(x)x + B(x)uNote that the SDC form has the same structure as a linear dynamic system, but with the system matrix A and the control influence matrix B being functions of the state variables. The matrices A(x) and B(x) evaluated at all values of the state vector x are assumed to be such that the system dynamics is controllable.

The second ingredient of the SDRE technique is the definition of a quadratic performance index in state dependent form:

The state dependent weighting matrices Q(x) and R(x) can be chosen to realize the desired performance objectives. In order to ensure local stability, the matrix Q(x) is required to be positive semi-definite for all x and the matrix R(x) is required to be positive definite for all x.

Next, a state dependent algebraic Riccati equation:

is formulated and is solved for a positive definite state dependent matrix P(x). The nonlinear state variable feedback control law is then constructed as:

Additional sophistication can be introduced in the SDRE design approach by including state estimators. An excellent overview of the SDRE design technique can be found in Cloutier.

It may be observed that the crucial part of the control law computation is the solution of the

state-dependent Riccati equation. In rare situations, this Riccati equation may be solvable

in closed-form. In most problems, however, this equation will have to be numerically solved

at each sample instant. A flowchart illustrating the steps involved in the computation of the

SDRE control laws is given in following Figure __. At each time step, the state vector obtained

from feedback sensors or estimators are used to compute the SDC matrices, which are then

used to find the state dependent gains. The product of the state dependent gains and the state

vector then yields the control variables.

State Vector x from the dynamical system˸ΛΓξδγβαẆΩ∏√∆≤

Where m, c, k are the system parameters namely mass, damping and stiffness of the system.

ua(t) is the active control force acting on the system and F(t) is the external force with

intensity σ.

(a) Actively controlled SDOF structure

(b) Free body diagram

The Equations of motion can be written in state space representation i.e. as a system of 1st

order differential equations. Let and then in state space the equation becomes

where is the 2×1 vector of states i.e. displacement and velocities of the structure,

is the 2×2 plant matrix, is the 2×1 location matrix and is the 2×1 excitation vector.

; ; ; ;

The Riccati Optimal Control Algorithm (ROAC) determines the control force ua(t) by

minimizing a standard quadratic index, J, given by,

and satisfying the state equation. and are the initial and final time-instants under

consideration. In this case the weighing matrices and are positive definite. Finally we obtain

whereis obtained by solving the algebraic Riccati equation . Matrix can be obtained by

numerical algorithm lqr.m available in MATLAB.

So now the control force can be expressed in the form of . Now substituting this expression in

eqn(1) we get:

Time variant reliability analysis

Traditionally, structural design has relied on deterministic analyses. Nevertheless the

uncertainties in loads and material properties were not completely neglected. The safety

factors were introduced to separate strengths and loads. The subject of structural reliability

started to evolve from the beginning of the 1920s.The study of structural reliability is

concerned with the calculation and prediction of the probability of limit state violation for

engineered structures at any stage during their life. Here it is implied that the limit state is the

requirements of the safety of the structure against collapse, limitations for damage or

deflections etc.

The actions on the structure, as well as material properties and exploitation conditions

very often explicitly depend on time. Thus in general, the basic variables change in time. In

this work dynamical loads are assumed. Such systems can be taken to governed by equations

of the form:

M(theta)yddot +Q(theta,y ydot) =f(t)

initial conditions Pg22(sundar)

Here theta is p*1 vector of system paramenters modelled as random variables with a specified

joint pdf ptheta, t is the time.Here Θ is a p ×1 vector of system parameters modeled as random

variables with a

specified joint pdf pΘ (θ ) , t is the time, and a dot represents derivative with respect

to t . The excitation F ( t ) is a vector of random processes which, in general, could be non-

Gaussian, non- white, and non-stationary. Similarly, the components of Θ could be mutually

dependent and non-Gaussian. It is often taken that F ( t ) and Θ are independent. The

excitation vector F ( t ) , when not white, can be modeled as the output of linear/non-

linear filters driven by white noise excitation.The resulting governing equation can be cast as

an Ito's SDE of the form as

dX ( t ) A

sundar pg 23

The problem of time variant reliability analysis of non-linear dynamical systems

consists of determining the probability that a response metric of interest, denoted by

h

Θ , X ( t ) , t

, stays within a specified safe region in the state space over specified

time duration. Accordingly, the reliability of the dynamical system under

consideration is written as

pg 23 sundar

Here h* is the permissible threshold value. Typically, the problem is tackled by

converting the time dependent problem into a time invariant format by seeking the

probability that the maximum value of response metric over the specified duration

stays within the specified safe limits. Subsequently Eq. (1.33) can be written as

pg 23 sundar

The following features contribute to the sources of difficulty in evaluating the above

probability:

1. The performance metric h

Θ , X ( t ) , t

is often a non-Gaussian random process

with the non-Gaussianity here originating from non-linear dependence of the

metric on system states and (or) the system mechanics being non-linear.

2. The operation of finding maximum of a random process over a time interval is

highly non-linear in nature.

3. The boundary of safe domain could have a complicated geometry in state space

and may not be easy to compute.

A Monte Carlo estimator for evaluating the probability of failure expressed as Eq.

(1.35) is given as

pg 23

where θ i and X i ( t ) , i = 1, 2, , N1 are random samples of Θ and X ( t ) respectively.

In the numerical work, to obtain these samples, one replaces Eq. (1.32) by an

equivalent discretized map (Kloeden and Platen 1992) of the form

xk +1 φ=

0,1, N

=

k ( xk ,Θ , v k ) , k

(1.37)

x0 = x ( 0 )

Here vk , k = 0,1, , N is a sequence of vector Gaussian random variables with

specified mean and covariance properties, and represents the external excitation. See

Annexure 1 for details of two such discretization schemes. Accordingly, the above

estimator for PF is now obtained as

1 N1

i

i

ˆ =

≤0

P

I h* − max h

θ

x

,

(

)

∑

F

k

0≤ k ≤ N

N1 i =1

where

(x )

k

i

(1.38)

is the i th sample obtained using Eq. (1.37). Clearly, the sampling

ˆ would be inversely proportional to the sample size, and

variance associated with P

as has been already noted, this type of estimators would be practically of limited use.

Question on reduction of sampling variance again become crucial and this has been

considered by several authors. Various methodologies have been documented in literature.

1.3.1 Methods based on time invariant reliability analysis

Here one considers the set of random variables given by

(Θ , V ) ,

with

V = ( v1 , v2 , , v N ) , N

= T / ∆ , where T is the time duration under consideration and

∆ is the time step of discretization, and introduce the performance function,

g (Θ , V=

) h* − max h (Θ , xk , vk )

0≤ k ≤ M

(1.39)

The probability of failure is now given

by PF P

=

g (Θ , V ) ≤ 0

. Consequently, methods for time invariant reliability analysis can be used can be used to

determine PF .

Der Kiureghian (2000) offers an approximate method for the solution of random vibration

problems based on discretizing the external excitation in terms of Gaussian random variables

and applying the concepts of FORM and SORM.

The performance function here is defined with respect to a specific time instant rather

than a time duration, i.e., g ( X =

) h* − h

X ( t )

, 0 ≤ t ≤ T , is the

X (τ )

, where h ( pg 28)

response metric of interest, and h* is the threshold value. This representation of

performance function helps provide a geometric interpretation to the reliability

problem in the standard normal space.

The subset simulation be applied to problems of time variant reliability analysis by discretizing

the excitation in time. The problem of first excursion probability of linear dynamical systems

subjected to Gaussian excitation has been considered by Au and Beck (2001b). The idea here

is to characterize the failure domain of the individual response components through the

corresponding design point, and construct an ispdf to tackle the more complicated problem of

obtaining the probability of excursion of one or more components of the random process

X( t ).

The problems encountered in the application of FORM and SORM to time variant reliability

problems is mainly two-fold (see, for example, Schueller 2003, Adhikari 2004, 2005):

1. The computational time increases with an increase in the dimension of the

random variable vector. This is evident directly from the fact that increased

dimensions leads to higher number of evaluations of gradients (FORM) and

components of the Hessian matrix (SORM).

2. The theory of asymptotic analysis, on which FORM and SORM are based, may

not be valid for higher dimensional problems.

Trajectory Splitting method

We consider the equations

dX(t) =A[X(t),t]dt+sigma(X(t),t)dB pg 31

X(0) =Xo

The probability of failure is obtained as

Double and clump method, Russian roulette and splitting method, and distance controlled

Monte Carlo simulations are some of the trajectory splitting based methods that drive the

response process to failure domain by controlling the evolution of sample trajectories suitably.

The idea in these methods are to associate with each trajectory a weight and judge each of the

trajectories by their proximity to the failure region he least favored trajectories are eliminated from

further evolution, and those trajectories which evolve towards the failure region are split and

multiplied thereby enhancing the number of samples moving towards the failure region. .

Pradlwarter et al., (1994), Pradlwarter and Schueller (1997, 1999), and Melnikov

andDekhyaruk (1999) have considered reliability analysis of randomly driven oscillators

and have discussed methods to drive response trajectories in state space towards

failure regions.

Subset Simulation method

Au and Beck (2001a) have proposed a variance reduction scheme styled as subset

simulations. This method is applicable to both time invariant and time variant

reliability problems. The main idea here is to divide the failure region Ψ into M

Pg16 sundar

Ψ1 Ψ 2 Ψ M = Ψ⊃ ⊃⊃

and

so that the failure probability can be expressed as PF = P ( Ψ1 ) ∏ P ( Ψ i +1 | Ψ i ) . Here P

( ) is the probability measure. The simulation

i =1

1, 2, , M and in estimating the

effort is focused on delineating the sets Ψ i , i =

probabilities P ( Ψ1 ) and P ( Ψ i +=

1, 2, , M − 1 . The effectiveness of the

1 | Ψi ) , i

method is due to the fact that each of these probabilities is much larger than PF and

hence can be estimated reliably with fewer samples.

In simulation studies on Markovian systems governed by Ito’s SDE-s of the form

given in Eq. (1.50), an artificial control force can be introduced into the governing

equation so as to achieve reduction in Monte Carlo sampling variance associated with

desired response statistics (Kloeden and Platen 1992, Milstein 1995, Grigoriu 2002,

Oksendal 2003). The process of replacing a given SDE by a modified equation with

artificial controls in this context is called the Girsanov transformation (Girsanov)

1960). The control force and the estimators are formulated so that the estimate

remains unbiased while at the same time striving to reduce the sampling variance.

Consider the modified version of the SDE given in Eq. (1.50)

dX

Here u ( t ) is the additional control force, Γ ( t ) is a scalar correction term introduced

( t ) is the biased state corresponding to

to account for the addition of controls, and X

the modified excitation σ

X (t ) , t

u (t ) + σ

X (t ) , t

dB ( t ) . It can be shown that

(Milstein 1995, Oksendal 2003, Macke and Bucher 2003)

I h* − max h

X ( t )

≤0

0 ≤ t ≤T

)

(

=

Γ (T ) I h* − max h

X ( t )

≤0

0 ≤ t ≤T

)

Γ0

(1.56)

An estimator for the evaluating the expression on the right hand side can be obtained

as

=

P

F

1

N 2Γ0

∑ Γ ( T ) I h

N2

i =1

*

i

− max h

X (t ) , t

≤0

0 ≤t ≤T

(1.57)

i ( t ) are random draws from Eq. (1.55) . It follows that

where X

= P and

P

F

F

( )

will be dependent on N and the yet to be determined control vector u ( t ) .

Var P

2

F

( )

( )

Var P

ˆ . It can be shown

The control u ( t ) is to be selected such that Var P

F

F

( )

= 0 , but its construction

that an ideal control u* ( t ) exists which yields Var P

F

requires the knowledge of

PF , the very quantity being sought to be determined

(Tanaka 1997, and Macke and Bucher 2003).

The way forward would be to seek a sub-optimal control u ( t ) which help to reduce

the sampling variance appreciably and not to seek the ideal situation of obtaining

( )

= 0 . Here a hypothetical deterministic dynamical system

Var P

F

=

dV ( t ) A

V ( t ) , t

dt + σ

V ( t ) , t

u ( t ) dt ; 0 ≤ t ≤ T

V ( 0) = X 0

(1.58)

is considered. The control u ( t ) is determined such that it minimizes the distance

function

given

by

β (τ

q

m

τm

) = ∑ ∫ u ( t ) dt

2

j

subject

to

the

constraint

j =1 0

m

0 with 0 < τ m ≤ T (Tanaka 1997, Macke and Bucher 2003).

h* − h

V (τ )

=

This approach has been studied in the context of first passage failure of randomly

driven oscillators by Macke and Bucher (2003) and Olsen and Naess (2006). Olsen

and Naess (2007) have proposed an approximation to the ideal control force after

determining the probability of failure of an auxiliary linear system (obtained by

linearizing the non-linear terms). Reliability analysis of dynamical systems subjected

to random excitation with a specified power spectrum has been considered by Ogawa

and Tanaka (2009). The excitation here is represented in terms of series of Gaussian

white noise excitations, and control forces are found with respect to each of these

stationary excitation terms. The study by Newton (1994) and Schoenmakers et al.,

(2002) focuses on the problem of selection of the sub-optimal controls in the context

of the Girsanov transformation based methods. Schoenmakers et al., (2002) have

considered finding the control force by obtaining an approximate solution for the

backward Kolmogorov’s equation. Some of the applications of change in measure

based approaches are, in the area of obtaining the response of the governing SDE, and

improving the available filtering techniques (see, for example, Saha and Roy 2007,

Sarkka and Sottinen 2008, Roy et al., 2008, Saha and Roy 2009, Tara et al., 2013a,

b).

The concept of obtaining the sub-optimal control force by solving the optimization

problem outlined above is related to the concepts of critical excitation (Drenick 1970).

Here, the excitation (from a family of excitations) that drives the structural system to

its maximum response is called critical excitation. Drenick (1970, 1972, and 1977)

has discussed the concepts of critical excitation from a structural engineering

perspective. The study by Westermo (1985) focuses on obtaining the critical

excitation for in-elastic systems. Iyengar and Manohar (1987) obtained the critical

power spectral density of the input excitation which maximizes the response. Abbas

and Manohar (2005a, b) have derived the critical power spectral density of the

excitations by employing three different bases: maximizing the probability of failure,

minimizing the reliability index, and, maximizing the response variance. Saha and

Manohar (2005) considered the problem of designing the structures based on the

concepts of inverse FORM and critical excitations. The study of Au (2006, 2009)

focuses on obtaining the properties of sub-critical excitations in the context of a single

degree of freedom (sdof) elasto-plastic oscillator. A comprehensive review of

methods to obtain critical excitation, and their application is available in the book by

Takewaki (2007).