Structural Optimal Control with a Tuned MassDamper during an Earthquake

Kenneth Maples

Prof. Weiqing Gu, Advisor

Prof. Anthony Bright (Engineering), Reader

December 12, 2005

Department of Mathematics

Abstract

In this thesis I develop a mathematical model for a four story apartmentbuilding in a plane during a deterministic earthquake loading. The move-ment of the walls of the building is modeled with a bilinear hysteretic re-action force. A tuned mass damper system attached the the top floor of thestructure controls the movement of the structure during the earthquake.The control law is developed using optimal control theory with an objec-tive functional that is quadratic in state and control. The existence of thecontrol is demonstrated, while the actual control function is found numer-ically using the Miser3 package for Matlab.

Contents

Abstract iii

Acknowledgments vii

1 Introduction 11.1 Model Derivation . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Paper Organization . . . . . . . . . . . . . . . . . . . . . . . . 5

2 A Mathematical Model of a Building During an Earthquake 72.1 The Equations of Motion . . . . . . . . . . . . . . . . . . . . . 7

3 Optimal Control of the Model 113.1 Quadratic Controls . . . . . . . . . . . . . . . . . . . . . . . . 11

4 n Floors, Multiple Controls 154.1 Mathematical Development . . . . . . . . . . . . . . . . . . . 16

5 Continuous Model 195.1 Mathematical Development . . . . . . . . . . . . . . . . . . . 19

6 State Constraints and Non-standard Objective Functionals 216.1 Linear Controls . . . . . . . . . . . . . . . . . . . . . . . . . . 216.2 State Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 22

7 Numerical Results 257.1 ocf.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.2 ocg0.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.3 ocphi.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.4 ocxzero.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

vi Contents

7.5 param.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.6 ocdfdu.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.7 ocdfdx.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.8 ocdg0du.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.9 ocdg0dx.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.10 ocdpdx.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

8 Discussion and Future Work 318.1 Next Semester Research Plans . . . . . . . . . . . . . . . . . . 31

Bibliography 33

Acknowledgments

My acknowledgments will be written at the end of the second semester torecognize the academic, technical, and psychological support of my advi-sors, friends, and family.

Chapter 1

Introduction

Suppose we are asked to develop a system for minimizing the damagethat a building receives during an earthquake. The first, and most obviousmethod for minimizing earthquake damage is passive systems. Naturally,modern construction techniques allow for buildings to resist all but thestrongest earthquakes without severe damage to the structural integrity.However, the problem that an earthquake provides is not a simple ques-tion that can be answered with vibration dampers are creative constructiontechniques.

When an earthquake hits a building, it causes vibrational energy to en-ter the structure which creates waves that travel up and down the building,shaking the structure from side to side. The energy stored in these waves istransferred into heat by the friction inside the walls and by the resistance ofthe air surrounding the building. However, excessive vibrations can causetears in the wall material which can cause structural failure in the building,triggering the collapse of the building and the deaths of its inhabitants. Pas-sive devices limit the risks of building failure by redirecting the energy ofthese waves into specific points where the resonant frequency of the build-ing is damped and contained; these sections will not cause the structuralfailure of the building when they absorb large amounts of kinetic energy.

However, we as building designers can surpass the effects of passivedesign by applying active control techniques to counteract the earthquake.By storing energy in the building before the earthquake hits, an activemechanism can release energy into the building that will counteract the en-ergy of the earthquake and dampen the movement of the structure. In thisway, we can begin to dissipate the energy of the earthquake at a slow ratebefore the earthquake hits, which allows larger earthquakes to be handled

2 Introduction

Figure 1.1: A four story apartment building

by the building.Correctly applying active control techniques to “cancel” the earthquake

is not an obvious task. Improperly designed controllers could instead worsenthe effects of the earthquake by either amplifying the earthquake signal orby creating whiplash effects, where the two waves cancel eachother in away that results in a large peak at a certain point in the building. Thiswhiplash effect would be catastrophic to the structure of the building, be-cause a single point failure of the walls of the structure is sufficient to resultin complete descruction.

With this in mind, how can we decide on an appropriate control thatwill minimize the effects of the earthquake? In this thesis, we experimentwith an application of optimal control theory to the design of a four-storyapartment building, as seen in Figure 1 to determine the minimizing con-trol.

1.1 Model Derivation

Before we can apply optimal control theory to the problem, we need todevelop a mathematical model that we can use for our mathematical ex-periments. There are several different alternatives with varying degrees ofaccuracy and mathematical detail available to us.

The best accuracy that we can get is using a finite element method thatapproximates the building at varying levels of detail as given by the build-ing blueprints. In this case, an accurate model of the building including

Model Derivation 3

Figure 1.2: The same apartment building separated into lumped masses foreach floor.

architectural details and floor layout must be considered. However, thissort of model is beyond the scope of optimal control theory, as solving theoptimization problem that this model would generate requires more com-puting power than is available to the author. In addition, results from themodel would likely be inapplicable to structures other than the one cap-tured by the finite elements.

Another approach is to model the structure as a continuous model withinfinite degrees of freedom. In this case, we can consider the function ofthe displacement u of the building in the x direction (perpendicular to thebuilding) as a function of height and time; that is, u(y, t) : R×R → R.With this model, we can write a partial differential equation (PDE) from thereaction forces between each differential chunk of building mass and eitheruse analytic or numerical techniques to solve for the building position.

This technique has potential, especially if the mass of the building ischaracterized with a mass distribution function ρ(y). Although this modelis not the primary research direction of this paper, some introductory tech-niques are presented in Chapter 5. In the second semester, the full PDE ofthe system will be derived, and if tractable the solution solved.

The final method use, which is the typical method used by structuralcontrol engineers in industry, is a discretized model as shown in Figure 1.1.In this model, each floor of the building is considered as a lumped massthat is connected to adjacent floors by a restoration force that depends onthe relative position and velocity of each floor. The model used for thebulk of this paper was described by Sone et. al. (6). In this model, the

4 Introduction

position and velocity of each floor (along with the state information of thecontrol structure) are recorded in a state vector x with finite dimension.This state vector completely describes the state of the system at any timet, which in a deterministic model can be used to solve for the future stateat any time t1 > t. Given well-defined conditions on the rate of change ofthe state vector, we are guaranteed local existence of a solution x(t) of thestate variable in terms of the input control and earthquake; further, givenbounded equations of motion the state over any finite time interval [t0, t1]can be solved.

1.2 Background

We will use some ideas developed in dynamical series and optimal controltheory. First, some definitions:

We need Pontryagin’s Minimum Principle:

Theorem 1.1 (Pontryagin’s Minimimum Principle)For the control ~u = (u1, ..., um)′ belonging to the admissible control set U andrelated trajectory ~x = (x1, ..., xn)′ that satisfies

xi = gi(~x,~u, t) (state equations)xi(a) = ci (initial conditions)

but with free end conditions, to minimize the performance criterion

J = ψ(~x, t)|ba +∫ b

af (~x,~u, t) dt

it is necessary that a vector~λ = ~λ(t) exist such that

λi = −∂H∂xi

(adjoint equations)

λi(b) = ψxi [~x(b), b] (adjoint final conditions)

where the Hamiltonian

H = f +n

∑i=1

λigi

for all t, a ≤ t ≤ b, and all ~v ∈ U,

H[~λ(t),~x(t),~v] ≥ H[~λ(t),~x(t),~u(t)]. (minimum condition) (1.1)

Paper Organization 5

Finally, we need to establish the existence of an optimal control usingthe theorem below. The proof is given in Fleming and Rishel (2), while thetheorem is applied in Section 3.

Theorem 1.2 (Existence of an Optimal Control) Given the objective functionalJ(~v) =

∫ t f0 F(~x)dt, where U = {~v, piecewise continuous | 0 ≤ v1(t), ..., vm(t) ≤

1 ∀t ∈ [0, t]} subject to system ~x = ~f (t,~x,~v) with ~x(0) = ~x0 then there exists anoptimal control~v∗ such that minvi(t)∈[0,1] J(~v) = J(~v∗) if the following conditionsare met:

1. The class of all initial conditions with a control~v(t) in the admissible controlset along with each state equation being satisfied is not empty.

2. The admissible control set U is closed and convex.

3. Each right hand side of the system ~x = ~f (t,~x,~v) is continuous, is boundedabove by a sum of the bounded control and the state, and can be written as alinear function of vi with coefficients depending on time and the state.

4. The integrand of J(~v) is convex on U and is bounded below by −c2 + c1~v2

with c1 > 0.

1.3 Paper Organization

This chapter introduced the background and reasoning behind the prob-lem description, and offered some insight into why some choices weremade about the model construction. Chapter 2 delves further into the dis-cretized model and culminates in the presentation of the complete equa-tions of motion of the structure and the TMD. Chapter 3 demonstrates theexistence of an optimal control input given an objective functional that isquadratic in the state and control variables. In the second semester, I willtry to find an analytic expression for the control input if possible. The nextchapter discusses how the model can be extended to include structures ofdifferent construction, including variable numbers of stories and controlinputs. Chapter 5 explores the alternative model discussed above, wherethe movement of the building is simulated using a PDE rather than a sys-tem of ODEs. Our goal for next semester is to construct the full PDE forthe building and use harmonic analysis to solve for the building positionin time.

The last three chapters explore nonconventional objective functionalsfor the discretized system, applying state constraints to the system so that

6 Introduction

a controller that maintains the system in some valid operation region ischosen. Then, the numerical results are presented along with a discussionof results completed and work planned in the second semester.

Chapter 2

A Mathematical Model of aBuilding During an Earthquake

Our problem concerns a four story apartment building. Suppose that thestructural engineers have asked us to design an optimal control scheme tominimize the damage from an earthquake.

2.1 The Equations of Motion

First, we need to develop a model to accurately portray the dynamics of thebuilding while an earthquake is happening. Following the development inSone et. al. (6), we decide to discretize the building so that each floorof the structure is represented as a lumped mass with mass mi that hasa single degree of freedom; that is, the floor can oscillate horizontally inthe x direction. The position and velocity of each floor are given by xi, vi,respectively.

2.1.1 Constitutive Law

We need a constitutive law that governs the reaction forces between adja-cent floors. Obviously, floors that are not directly connected together can-not have any constitutive forces. Because position and velocity of each floorare necessary and sufficient to describe the motion of each floor, the consti-tutive law is given by

Fk(xk, vk, xk+1, vk+1);

where Fk is the reaction force between floors k and k + 1. At this stage, wedo not rule out the possibility of memory in Fk.

8 A Mathematical Model of a Building During an Earthquake

Figure 2.1: An example bilinear hysteretic force. Note that when the dis-placement is increased, the component would settle at a new equilibrium.

In order to simplify analysis, we assume that the constitutive law givenabove can be separated into two forces, given by

Fk = Sk(xk, xk+1) + Dk(vk, vk+1)

where Sk and Dk are called the spring and damper forces, respectively.

Spring Forces

To model Sk for each floor, the model from (6) and the discussion from (7)are considered. Therefore, in order to maintain physicality of the model,the spring force is represented as a bilinear hysteretic reaction force. Arepresentation of such a force is shown in Figure 2.1.1.

Damper Force

No sources suggest anything besides a simple linear force between the ve-locities. Therefore,

Dk(vk, vk+1) = −ck(vk − vk+1)

2.1.2 Tuned Mass Damper

Additionally, the building has a tuned mass damper (TMD) attached to theroot from which we can apply our optimal control. The TMD is assumed

The Equations of Motion 9

to be fastened with spring constant ka and damping constant ca, while thedevice has mass ma. The position and velocity of the TMD are in the samedirection as the floors, and are given by xa and va, respectively. The TMDalso has a lever ration λ that increases force applied to the building com-pared to the movement of the damping mass.

2.1.3 Matrix Equations

From these assumptions, we can write the equations of motion for the sys-tem as a system of ordinary differential equations. Let ~x be the state of thesystem, excluding the memory of the hysteretic reaction forces in Sk; ~x isgiven by

~x =[x1 x2 x3 x4 xa v1 v2 v3 v4 va

]T .

The state is subject to the equation

~x′(t) = A~x(t) + h(~x(t)) + Bu(t)= f (t,~x(t), u(t)), 0 ≤ t ≤ t f

where u(t) ∈ U = {u(t) piecewise continuous |∀t ∈ [0, t f ], 0 ≤ u(t) ≤ 1}is the control input to the system. A is a 10× 10 matrix given by,

A =[

0 I−M−1K −M−1C

]where

M−1 =

1/m1 0

1/m21/m3

1/m40 1

,

K =

0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 ka

λ2

(1

m4+ 1

ma

)

,

10 A Mathematical Model of a Building During an Earthquake

and

C =

(c1 + c2) −c2 0 0 0−c2 (c2 + c3) −c3 0 0

0 −c3 (c3 + c4) −c4 00 0 −c4 c4 − ca

λ

0 0 c4λm4

− c4λm4

caλ2

(1

m4+ 1

ma

)

.

h : R10 → R10 is a function that encapsulates the non-linear effects in thesystem; it is described by the equation,

h(~x(t)) =

S1(x1, x2)

S2(x2, x3)− S1(x1, x2)S3(x3, x4)− S2(x2, x3)

−S3(x3, x4)~0

where S(·) is the bilinear hysteretic function. B is a 10× 1 matrix given by,

B =[

~0M−1F

]where

F =[0 0 0 1/λ −1/λ2

(1

m4+ 1

ma

)]T.

The full equation for the state can therefore be represented as

x1x2x3x4xav1v2v3v4va

′

= A

x1x2x3x4xav1v2v3v4va

+~h

x1x2x3x4xav1v2v3v4va

+

000000001

m4λ

−1/λ2(

1m4

+ 1ma

)

u (2.1)

Chapter 3

Optimal Control of the Model

3.1 Quadratic Controls

The objective functional for the system is given by the equation,

J(u) =∫ t f

0L(~x, u, t) dt =

∫ t f

0

(~xT(t)Q~x + ru2(t)

)dt,

where t f is the final time, Q ∈ D(n, n) is the penalty matrix for the statevariables, and r ∈ R is the penalty for the control.

3.1.1 Existence of an Optimal Control

Theorem 3.1 Given the objective functional

J(u) =∫ t f

0

(~xT(t)Q~x + ru2(t)

)dt,

where u(t) ∈ U = {u(t) piecewise continuous |∀t ∈ [0, t f ], 0 ≤ u(t) ≤ 1},Q ∈ R10×10, r ∈ R subject to the system in Equation 2.1 with ~x(0) = ~x0, thereexists an optimal control u∗ ∈ U such that minu∈U J(u) = J(u∗).

We will use Theorem 1.2 [Existence of an Optimal Control] from Sec-tion 1.2. For the first condition, we know that for a given u ∈ U and ini-tial conditions given above, there exists a unique solution to the system inEquation 2.1. Given that the state equations above are bounded for finitestate ~x and time t0, we know that the state at any future t is well defined.

Because U is closed and convex from its definition, the second conditionis satisfied.

12 Optimal Control of the Model

For the third condition, we need to show that the right hand side ofEquation 2.1 is continuous. Obviously, the only term that has any chanceof discontinuities is h(~x(t)). We know that given bounded equations ofmotion for the state ~x(t), the state variables must be continuous in time.Therefore, because the hysteretic reaction force Sk(xk, xk+1) has no disconti-nuities for continuous xk(t) and xk+1(t), this term must also be continuous.Additionally, we need to show that the right hand side is bounded abovegiven a final time t f . Therefore, we can define the supersolution of ~x as X,given by,

~X′(t) = A~X(t) + h(~X(t)) + Bu(t)

where h replaces the hysteretic reaction force F with the linear functionF(δx) = k1δx. The magnitude of the right hand side of this supersolutionis always greater than or equal to the original system, which implies thatif this solution is bounded, the original system is bounded. However, be-cause the supersolution involves only constants it must have a finite upperbound.

Additionally, knowing that the solution is bounded, we can combinethe linear terms from A~X(t) and h(~X(t)) to be A~X(t); therefore,

|~f (t, ~X, u)| ≤ |A~X(t)|+ |Bu(t)|≤ |A||~X(t)|+ |B||u(t)|= C1|~X(t)|+ C2|u(t)|

with C1 the norm of A and C2 of B.For the final condition, we need to show that L is convex; that is, that

L(~x, (1− p)u + pv, t) ≤ (1− p)L(~x, u, t) + pL(~x, v, t).

Therefore,

L(~x, (1− p)u + pv, t) = ~xT(t)Q~x + r[(1− p)u + pv]2

= ~xT(t)Q~x + r[(1− p)2u2 + 2(1− p)puv + p2v2]

while

(1− p)L(~x, u, t) + pL(~x, v, t) = (1− p)[~xT(t)Q~x + ru2] + p[~xT(t)Q~x + rv2]

= ~xT(t)Q~x + r[(1− p)u2 + pv2];

Quadratic Controls 13

So, the difference between these two expressions is,

L(~x, (1− p)u + pv, t)− [(1− p)L(~x, u, t) + pL(~x, v, t)]

= r[(1− p)2u2 + 2(1− p)puv + p2v2]− r[(1− p)u2 + pv2]

= r[(1− 2p + p2)u2 + 2(1− p)puv + p2v2 − (1− p)u2 − pv2]

= r(p2 − p)(u− v)2

< 0

for 0 ≤ p ≤ 1, for all u, v ∈ U . So,

L(~x, (1− p)u + pv, t) ≤ (1− p)L(~x, u, t) + pL(~x, v, t)

as was to be shown.�

Chapter 4

n Floors, Multiple Controls

The previous chapters described the mathematical model and optimal con-trol techniques we can use to control a four story structure. Representingthe structure as a group of lumped mass elements, with one mass corre-sponding to each floor, the equations of motion for the building was foundby summing the forces on each lumped mass. In particular, the internal re-action forces between the floors, manifested as hysteretic spring and lineardamper forces, were modeled to derive a complete set of ODEs describingthe equations of motion of the different masses in the building. Then, an-other lumped mass was attached to the top floor of the building throughidealized spring and damper connections; this mass was called a TMD andprovided the control input into the structure for our optimal control exper-iments.

By moving the large mass on the top floor of the structure, we can re-duce the effects of an earthquake on the structural integrity of the building.However, there are two key assumptions we made that affect the applica-bility of this research. First, assuming that the structure has exactly 4 storiesis inaccurate and possibly misleading in the effects of the control input forstructures with many more or fewer stories. Although there is literaturethat discusses optimal control applied to a single storied building, whenthe structure is very tall (in excess of 10 stories) optimal control is seldomdiscusses. It is not intuitively clear that the general control design for astructure with many more stories would be the same for one with 4 stories;even the placement of the control input might be superior in another floorcloser to the ground.

Likewise, for very tall buildings, a single control input might be insuf-ficient for regulating the building during an earthquake. Therefore, the

16 n Floors, Multiple Controls

model should be extended to handle the application of TMDs on multi-ple stories at the same time, with multiple control inputs in the objectivefunctional. Although the additional (considerable) cost of adding anothercontrol input must be considered in the design of the building, the pres-ence of another active control might be necessary for sufficient structuralresponse to the earthquake.

4.1 Mathematical Development

To extend the results of the previous chapters, we need to find a new math-ematical representation for the control system. Suppose that we developthe equations of motion for an n story building without TMDs; it would bewritten in the form

~x′(t) = A~x(t) + h(~x(t)) (4.1)

where A is a 2n × 2n matrix with constant, real components and h(·) issimilar to the function defined in Section 2.1.3.

First, we can extend the equation above to include a multiple numberof controls by extending the matrix A so that the relative motions of thecontrol on floor j will cause a corresponding reaction force on floor j. Givenm controls, this will produce a new set of equations of motion given by

~y′(t) = A~y(t) + h(~y(t)) + B~u(t) (4.2)

where A is a (2n + 2m)× (2n + 2m) real, constant matrix and y is a vectorwith length 2n + 2m.

The equations developed above are sufficient to determine the optimalcontrol for a system with specified placement of control inputs. However,these developments are not enough to tell us how to find the best storiesto place the TMDs. Naturally, this optimization problem is a good candi-date for further optimal control. We can enumerate the different configu-rations by q, where q has a definite number of TMDs in specified locations.Therefore, there will be a pair of matrices for each configuration Aq andBq, where Aq is a (2n + 2mq) × (2n + 2mq) real constant matrix and Bq isa (2n + 2mq)× mq real constant matrix. We can then combine the differentequations of motion:

vec(~y′q(t)) = (I ⊗ Aq)vec(~yq(t)) + (I ⊗ Bq)vec(~uq(t)) (4.3)

Mathematical Development 17

Then, we can define the objective functional as normal:

J =∫ t f

0vec(~yq)TQvec(~yq) + vec(~u)Rvec(~u) dt

We claim that by minimizing this integral, each of the different optimalcontrols for every system configuration must be minimized, as the formu-lation of Equation 4.3 separated the motion of the different systems. Notethat we can distribute the integral of J through the quadratic forms so thatwe can construct a vector r of the minimal values of the objective functionalfor each system. Therefore, finding the index of the minimal value wouldindicate the best system selection.

Chapter 5

Continuous Model

In the above chapters, the discretized model was developed where eachfloor was represented with a lumped mass. To characterize the currentstate of the system, the position and velocity were recorded in the statevector ~x which allowed the dynamics of the system to be completely de-scribed by the vector. The advantage of this technique is that it allows theprevious state of the system to be ignored, as all future states of the systemare entirely described by the current state vector.

However, the discretization of the building involved a significant ap-proximation of the building motion. Although discretization is the mostcommon method used in structural control and simulation, as explained inChapter 1, continuous models still allow a finer-grained analysis of systembehavior. As described in the introduction, a continuous model that ap-proximates the structure with the mass of the building evenly distributedis an unphysical approximation. But, by applying a mass density functionρ(y) to the system, a partial differential equation (PDE) can be derived thatdescribes the movement of the building in space and time.

5.1 Mathematical Development

Separating the mass of the building vertically into chunks dy, we need tofind the reaction forces between the chunk at vertical position y and y + dy.Obviously, the reaction forces used in the discretized model need to beadapted for use with the differentials. Loosely, this can be accomplishedby comparing the values of u(y, t) and u(y + dy, t), as well as the pairut(y, t) and ut(y + dy, t) corresponding to the spring and damper forces,respectively. For the damper force, we would want the force applied to the

20 Continuous Model

segment dy to be

Fd = c limdy→0

ut(y + dy, t)− ut(y, t)dy

= cuty(y, t)

while the spring force needs to account for hysteretic effects.Once the forces on each segment dy have been found, we can find the

acceleration of the chunk by Newton’s second law of motion, that dF =dm · a, so that the acceleration of the chunk is the forces divided by themass density function ρ(y).

Chapter 6

State Constraints andNon-standard ObjectiveFunctionals

The analysis above for the existence and characterization of the optimalcontrol of the structure used a quadratic objective functional. A quadraticobjective function uses positive definite quadratic forms of the control andstate variables so that excessive control effort and state movement are pe-nalized significantly more than small movements. In this way, a smoothcontrol can be found that balances the necessary displacements of each ofthe variables.

However, the physical relevance of a quadratic control is hard to re-alize in practice. The chief advantage of quadratic objectives is in theirmathematical simplicity; the existence of optimal controls is nearly guar-anteed, while the analysis is simple and tractable with the well-understoodquadratic forms. However, engineers and applied scientists have a hardertime accepting the validity of a quadratic objective functional in the designand application of optimal control theory. In brief, quadratic controls aren’treally what we want to minimize.

6.1 Linear Controls

The application of linear control usually suggests the use of bang-bang typesolutions. These solutions are well suited to regulator-type problems, sincethey can be physically realized using relays and other simple electroniccircuitry. However, linear controls offer some problems for mathematical

22 State Constraints and Non-standard Objective Functionals

analysis in the case where bang-bang controls are not the optimal solutionto the problem.

Using Pontryagin’s minimization principle given in the introduction,we can find the switching point of an objective functional given in the form

J =∫ t f

0Q~x + R~u dt

by finding conditions on the adjoint variables λi when they cross certainthresholds. However, if the adjoint variables stay at the switching points,then the system is called singular and an optimal control must be found bydifferent means. In general, this means a manual numerical search of allthe possible controls in the switching region.

If the existence of a bang-bang type solution could be demonstrated forthe model given in Chapter 2, then linear controls would offer a highly-intuitive objective functional that generates a control that is easy to imple-ment in actual buildings. However, singular controls have questionableapplicability, due to their extreme sensitivity to system parameters, and aretherefore far less desireable.

6.2 State Constraints

Instead, we can consider the situation of the problem and find what char-acteristics of the system must be maintained regardless of their analyticcomplexity. These techniques almost universally rely on numerical solu-tions to the optimization problems, which are problematic for designingcontrollers for the actual devices. However, these techniques may showgeneral forms for the optimal controls that would minimize the desiredcharacteristics and can lend evidence that either the quadratic control sug-gested is indeed applicable to the problem at hand, or show that a differentobjective functional is necessary.

In this thesis, of paramount importance is maintaining minimal damageto the structural integrity of the building. This can be managed by placingstate constraints on the movements of the building so that controls mustbe found that do not allow the displacements of two adjacent floors to dif-fer more than some constant r, which is specified before each simulation.Because for some values of r there are no feasible controls available, de-termining the smallest value of r for which an optimal control exists wouldsolve the equivalent problem of minimizing the maximum oscillation of the

State Constraints 23

building during the earthquake. This corresponding control would there-fore be the control we want to apply to the structure, and we would tocompare it to the value suggested by the quadratic control for relevanceand efficiency.

Chapter 7

Numerical Results

Currently, the numerical simulations are failing (with an infinite f(t,x,u)).Here are the files for the current numerical work:

7.1 ocf.m

function f = ocf(t,x,u,z,upar)

% Return the value of the right hand side of the state equations% as a column vector.

% State vector: x = [x1 x2 x3 x4 xa dx1 dx2 dx3 dx4 dxa]% State space equation: dx = A x + B u + G z%% A = [ 0 I ]% [ (-M^{-1}Kn) (-M^{-1}C) ]%% G is currently not included.%% Parameters: m1 m2 m3 m4 ma k1 k2 k3 k4 ka c1 c2 c3 c4 ca lambda

param;

Minv = diag([1/m1 1/m2 1/m3 1/m4 1]);

Kn = [(k1+k2) -k2 0 0 0; ...

26 Numerical Results

-k2 (k2+k3) -k3 0 0; ...0 -k3 (k3-k4) -k4 0; ...0 0 -k4 k4 0; ...0 0 0 0 (ka/lambda^2)*(1/m4+1/ma)];

C = [(c1+c2) -c2 0 0 0; ...-c2 (c2+c3) -c3 0 0; ...0 -c3 (c3+c4) -c4 0; ...0 0 -c4 c4 (-ca/lambda); ...0 0 (c4/lambda/m4) -(c4/lambda/m4) (ca/lambda^2)*(1/m4+1/ma)];

A = [zeros(5) eye(5); (-Minv*Kn) (-Minv*C)]; % 10 x 10

B = [zeros(5,1); 0; 0; 0; 1/lambda; -1/lambda^2*(1/m4+1/ma)]; % 10 x 1

f = A * x + B * u;

7.2 ocg0.m

function g0 = ocg0 (t,x,u,z,upar,rhoab,kabs)

% Return the value of the objective function integrand.

% Even weights on state and controlQ = eye(10);R = eye(1);

g0 = x’*Q*x + u’*R*u;

7.3 ocphi.m

7.4 ocxzero.m

7.5 param.m

% Unless otherwise noted, values taken from Sone et. al.

ocdfdu.m 27

zeta = 0.02; % ’Modal damping’m1 = 2e4;m2 = 2e4;m3 = 2e4;m4 = 2e4;ma = 100; % About 0.005 mi (’Mass ratio’?)c1 = 4*pi*zeta*2*m1; % Derived from canonical form of spr-damp-massc2 = 4*pi*zeta*2*m2;c3 = 4*pi*zeta*2*m3;c4 = 4*pi*zeta*2*m4;ca = 4*pi*zeta*2*ma;k1 = 2e7;k2 = 2e7;k3 = 2e7;k4 = 2e7;ka = 2e7; % Guessedlambda = 4;

7.6 ocdfdu.m

function dfdu = ocdfdu(t,x,u,z,upar)

param;

B = [zeros(5,1); 0; 0; 0; 1/lambda; -1/lambda^2*(1/m4+1/ma)]; % 10 x 1

dfdu = B;

7.7 ocdfdx.m

function dfdx = ocdfdx(t,x,u,z,upar)

% Return the gradient wrt state of each rhs of the state equations as% a matrix: [df_1/dx_1, ..., df_1/dx_ns; df_2/dx_1, ..., df_2/dx_ns;% ..., df_ns/dx_1, ..., df_ns/dx_ns].

28 Numerical Results

param;

Minv = diag([1/m1 1/m2 1/m3 1/m4 1]);

Kn = [(k1+k2) -k2 0 0 0; ...-k2 (k2+k3) -k3 0 0; ...0 -k3 (k3-k4) -k4 0; ...0 0 -k4 k4 0; ...0 0 0 0 (ka/lambda^2)*(1/m4+1/ma)];

C = [(c1+c2) -c2 0 0 0; ...-c2 (c2+c3) -c3 0 0; ...0 -c3 (c3+c4) -c4 0; ...0 0 -c4 c4 (-ca/lambda); ...0 0 (c4/lambda/m4) -(c4/lambda/m4) (ca/lambda^2)*(1/m4+1/ma)];

A = [zeros(5) eye(5); (-Minv*Kn) (-Minv*C)]; % 10 x 10

dfdx = A;

7.8 ocdg0du.m

function dg0du = ocdg0du(t,x,u,z,upar,rhoab,kabs)

% Return the gradient wrt control of the objective integrand in dg0du.

R = eye(1);

dg0du = 2*R*u;

7.9 ocdg0dx.m

function dg0dx = ocdg0dx(t,x,u,z,upar,rhoab,kabs)

% Return the gradient of objective integrand wrt states in array dg0dx.

ocdpdx.m 29

Q = eye(10);

dg0dx = 2*x’*Q;

7.10 ocdpdx.m

function dpdx = ocdpdx(ig,taut,x,z,upar,rhoab,kabs)

% Return the gradient wrt state of the ig-th phi function as a row vector dpdx.

dpdx = zeros(1,10);

Chapter 8

Discussion and Future Work

This semester began with a review of optimal control theory. During thesummer, I worked with profs. Gu and de Pillis on a tumor modeling projectthat used optimal control theory to find optimal solutions for treatment ofmalignant tumors. Using this background, I researched further into opti-mal control and found sources related to structural control and optimiza-tion. The model presented in (6) was checked and explained in Chapter 2,along with discussion and descriptions of the terms. Then, using the pro-cedure developed during the summer and applying theorems from Flem-ing and Rishel (2), Professor Gu and I proved the existence of the optimalcontrol when the objective functional for quadratic objectives in state andcontrol. Demonstrating the existence of the objective control was the chiefresult of the semester.

Additionally, preliminary work was completed with the numerical por-tion of the thesis. Using the Miser3 Optimal Control Toolkit for Matlab,a numerical simulation was constructed of the model in Chapter 7. How-ever, currently the optimal control cannot be found. The main goal over thenext month is to solve the numerical problem and extend the code so thatnumerical results can be obtained for the multiple floor, multiple controlmodel without major modifications.

8.1 Next Semester Research Plans

For next semester, providing a general control scheme for an n dimensionalbuilding with m different controls, as well as suggestions for the optimalplacement of the m controls, is the main objective of the thesis. Finding op-timal locations for control placement would be a significant result in struc-

32 Discussion and Future Work

tural control theory.As described in the remaining chapters, if time is available I would like

to try to implement the continuous model described in the introductionand Chapter 5; this model would allow for simulation of unconventionalbuilding design and might provide more insight into the behavior of thediscretized model.

Finally, I would like to experiment with state constraints as discussed inChapter 6 The addition of state constraints on the relative displacement andvelocities of the floor would suggest controls that allow minimal damageto the building, but instead prevent catastrophic failure of the structure.These state constraints might suggest controls that minimize the maximumoscillation.

Bibliography

[1] Fabio Casciati and Timothy Yao. Comparison of strategies for the activecontrol of civil structures. In First World Conference on Structural Control,volume 1, pages WA1–3–WA1–12, 1994.

This source introduces a basic sources and the state of theart in active control as of 1994. Different control strategiesare introduced, including state feedback linearization usingLQ, H2, and H∞ controls. The text questions the usefulnessof state feedback linearization as a control scheme, and sug-gests that predictive control, full nonlinear optimal regula-tor techniques, and sliding mode control are stronger controlmethods. The text suggests that both nonlinear optimal con-trol and sliding mode control are promising but are not gen-eral enough for wide applicability. The text also introducesa few control techniques such as neural network control andfuzzy control, which are unrelated to my thesis topic.

[2] Wendell H. Fleming and Raymond W. Rishel. Deterministic and Stochas-tic Optimal Control. Springer-Verlag, 1975.

Fleming and Rishel introduce the basic theorems and theorynecessary to prove the existence of an optimal control. Theybegin with definitions and the analysis of a general optimiza-tion problem, and then move into more complicated deter-ministic and stochastic optimal control theory. For my thesis,I am concerned with the chapter describing the existence anduniqueness theorems for an optimal control for different opti-mal control problems. The main theorem is given in the intro-duction of this thesis without proof, although one is suppliedin Fleming and Rishel.

34 Bibliography

[3] R. E. Kalman. Contributions to the theory of optimal control. Bol. Soc.Mat. Mex., 4:102–119, 1960.

This is the classic paper that details the state of the art in con-trol theory as of 1960. Kalman introduces the theory so that itis applicable to a general control theory problem with an ar-bitrary objective functional; however, most of the techniqueshe gives later are limited to situations with quadratic termsin the functional. Kalman introduces the Riccati equation andits application to find the coefficients for feedback in a linearcontrol system.

[4] J. J. Levin. On the matrix riccati equation. Proceedings Am. Math. Soc.,10:519–524, 1959.

Levin introduces several general results about the Riccatiequation in English.

[5] William T. Reid. A matrix differential equation of riccati type. Am. J.Math., 68:237–246, 1946.

Reid finds solutions to the Riccati differential equation andapplies the results to differential equations and the calculusof variations (and so, to control theory).

[6] Akira Sone, Shizuo Yamamoto, and Arata Masuda. Sliding mode con-trol for building using tuned mass damper with pendulum and levermechanism during strong earthquake. In Second World Conference onStructural Control, pages 531–540, 1998.

Sone et. al. form the foundation of the research performedin this thesis. They introduce the model used in this paper,which is a 4-story apartment building with a tuned massdamper on the top floor. They develop the model introducedin the first section of this thesis, which they use to simulatethe building during the earthquake and develop a slidingmode controller to minimze the oscillations.

[7] Ravi Shanker Thyagarajan. Modeling and Analysis of Hysteretic Struc-tural Behavior. PhD thesis, California Institute of Technology, Pasadena,California, 1989.

Thyagarajan introduces the different types of structural hys-teresis that are used in models for structural analysis and

Bibliography 35

controls. He differentiates between state space representa-tions and representations with memory. He also demon-strates that the more complicated models do not necessar-ily behave closer to their physical counterparts. Thyagarajanalso introduces the bilinear hysteretic model used in this the-sis.

[8] Dennis G. Zill and Michael R. Cullen. Differential Equations withBoundary-Value Problems. Brooks/Cole, 2001.

Zill and Cullen describe basic ordinary differential equationstheory and provide methods for solving systems of ordinarydifferential equations. In particular, they describe the methodof variation of parameters.