Multi-objective optimization of actively controlled structures with topological considerations

Abstract

This paper presents a novel approach for the optimum design of actively controlled structures using both cooperative and Stackelberg game theoretic formulations wherein the structural topology is also optimized. Most of the available literature for design of actively controlled structures deals with structures of a predetermined topology. It is recognized that the structural performance can be improved significantly by the optimization of topology. The optimum topology is created herein by minimizing the strain energy. Once the optimum topology is obtained, a simultaneous sizing and control system optimization of the optimum topology is performed which (i) maximizes the energy dissipated by the controller, (ii) minimizes the structural weight, and (iii) minimizes the controller performance index. The design variables include actuator locations, member cross-sectional areas and entries of state and control weighting matrices. The multi-objective design problem is solved as a bi-level Stackelberg game. A computational procedure based on variable updating using response surface methods is developed for exchanging information between the levels. Two numerical examples illustrating the proposed approach are presented wherein topological, structural and control system aspects of the problem are addressed comprehensively.

Keywords

Active control actuator placement cooperative game Stackelberg game topology optimization

1. Introduction

Because of the strong interactions between structural design and control system design in active vibration control, a simultaneous optimization of both systems is necessary to achieve superior overall design. Over the years, a simultaneous optimization of structure and control systems has attracted significant attention. In these works, the structure and control objective functions have been optimized by linking them through constraints related to control performance, structural performance, or sometimes by combining the structure and control objective functions as a single objective function. The combined structure/control optimization has been formulated (Onoda and Haftka, 1987) to minimize the combined total cost of the structure and control system with constraints on the magnitude of the response. Two approaches, combined and sequential integrated, have been proposed (Fonseca and Bainum, 1995) to solve the simultaneous structural/control optimization problem. While these works address both structural and control optimization, the topological optimization aspects of the structure design were not considered.

An optimization of topology is usually considered in the context of structural design. Topology optimization problems are more challenging than sizing optimization problems because members can be added to or removed from the initial structure and therefore the finite element model of the structure and the number of design variables and constraints change from one iteration to the next. A number of approaches such as the ground-structure method (Xu et. al, 2003), integer programming using 0–1 variables (Ohsaki and Katoh, 2005), genetic algorithms (Liu et. al., 1998), and simulated annealing (Dhingra and Bennage, 1995) have been used for solving the topology optimization problem. All of these approaches are based on discretizing the problem domain at a finite number of nodal points and the resulting optimum topologies are dependent on the underlying distribution of nodes.

Some other recently developed alternatives (Huang and Xie 2007; Rong and Liang 2008; Bruggi and Verani, 2011; Eom et al., 2011; Jia et al., 2011) which treat the problem domain as a continuum instead of a finite collection of nodal points include the homogenization method (Bendsoe and Sigmund, 2003), Solid Isotropic Material with Penalization (SIMP), and evolutionary methods for topology optimization. Recently some works (Diaz and Mukherjee, 2006; Xu et al., 2007; Molter et al., 2010; Silveira and Fonseca, 2010) have appeared which address topological and control considerations simultaneously. The solution approach involves first finding the optimum topology followed by optimum actuator placement according to optimum distribution of piezoelectric material. It may be noted that while these works address control considerations, structural issues such as constraints on stresses, frequencies, etc. are not addressed.

The overall design of an efficient structural control system is of interest to both structural and control engineers. Some of the important problem aspects include a minimum weight design, minimizing the control energy required and the optimum placement of actuators for fast damping of vibrations when the structure is subjected to some external disturbance. While many approaches have been proposed for structural and control system optimization, most of the proposed methods essentially solve a single objective optimization problem through a weighted combination of objective functions. If the objective functions have varying degree of importance such that a hierarchy exists, then a scalarization of objectives is not possible and multi-level optimization techniques are needed. The approach presented in this paper uses a bi-level as well as cooperative game theoretic formulation to comprehensively address the topological, structural and control design aspects of the structure-control design problems.

Stackelberg game theory is a technique for solving bi-level optimization problems and is used in this work. In this formulation, one objective acts as the leader and the other objective acts as the follower. The follower solution, or rational reaction set (RRS), depends on the choices made by the leader. Several methods have been proposed for the computation of the RRS such as response surface methods (Lewis and Mistree, 1998), monotonicity analysis (Rao et al., 1997) and sensitivity-based method (Ghotbi and Dhingra, 2012). Due to the nature of design variables for the problem considered herein, a response surface-based approach is used to construct the RRS.

This paper is organized as follows. Section 2 outlines the solution approach used to determine the optimum structural topology. Details on control system design are presented in Section 3. Section 4 discusses different game theoretic approaches for solving multi-objective problems. The solution procedure for a multi-objective, multi-level structural and control optimization problem is presented in Section 5. The integration of topological considerations in the context of structure and control design of actively controlled structures is illustrated using two design examples in Sections 6 and 7. Finally, conclusions and the contribution of this research are discussed in Section 8.

2. Topology optimization

Topology optimization deals with finding the optimum layout of a structure within a specified region when the only known quantities are the applied loads, possible structural supports, and desired volume of the structure. The approach generally is to find optimum density distribution of material in a fixed domain modeled with a fixed finite element mesh, that is, finding the optimum placement of a given isotropic material in space by determining which points of space should be material points and which points should remain void. For a fixed domain, the topology design problem can be formulated as a sizing problem by modifying the stiffness matrix which can be expressed in terms of the density of the material, which is a design variable. The optimization yields a design consisting almost entirely of a region of material or no material.

The numerical approach to topological design adopted herein starts with a region of material meshed into small finite elements. External loads and boundary conditions are defined next. Every element is assumed to consist of a porous material of density ρ to which external loads and boundary conditions are applied. The purpose of optimization is to find optimum density distribution while maintaining a constant volume constraint. Topology optimization is done by creating design variables associated with Young’s modulus and the density of each element in the design space. The design variable value ranges between 0 and 1 where 0 indicates the element has no stiffness or mass and 1 indicates the element has its solid stiffness and mass. A power law interpolation penalizes intermediate densities to obtain nearly 0/1 material distribution.

Mathematically, the topology optimization problem is formulated as follows. The design domain is divided into $N = N_{x} \times N_{y}$ elements where N_x denotes the number of elements along x-axis and N_y denotes the number of elements along y-axis. The optimization problem to minimize the strain energy is formulated as:

{min}_{ρ} C (ρ) = \frac{1}{2} {P} T {d} = {d} T [K] {d} suchthat \frac{V (ρ)}{V_{o}} = f [K] {d} = {P} 0 < ρ_{min} \leq ρ \leq 1

(1)

Here {d} and {P} denote the global nodal displacement and force vectors, respectively,

[K]

is the global stiffness matrix,

V (ρ)

is the structure volume corresponding to density

ρ,

V_o is the volume of initial structure, f is the prescribed volume fraction, and the density

0 \leq ρ_{xy} \leq 1

for each element. Depending on the finite element type selected to model the structural continuum, the entries in the stiffness matrix will change.

As members are added to and removed from a given topology, the strain energy of the structure changes. The change in the strain energy due to the removal of $i th$ element can be computed as:

Δ C - = \frac{1}{2} {P} T {Δ d} = - \frac{1}{2} {P} T [K] - 1 [Δ K] {d}

(2)

= \frac{1}{2} {d i} T [K i] {d i}

(3)

Here ${d i}$ is the element displacement vector containing the entries of ${d}$ , which are related to the $i th$ element and $Δ K = K - - K = - K_{i}$ where $K -$ is the stiffness matrix after the element is removed and K_i is the stiffness of the $i th$ element. Similarly, the change in strain energy due to the addition of $i th$ element is given by:

Δ C + = - \frac{1}{2} {d i} T [K i] {d i}

(4)

In topology optimization, the objective is to minimize the strain energy while satisfying the volume constraint. For a structural continuum where K_i is generally positive definite, the strain energy of the structure is increased when the material is removed and decreased when material is added. The solution approach adopted herein is to start with an initial structure with a fully connected grid meshed into N elements. To minimize the structural strain energy, it is most effective to remove elements with minimum $Δ C -$ value and add elements with minimum $Δ C +$ value. To keep the structural volume constant, the material added should equal the material removed. A gradient-based optimizer is used to solve the topology optimization problem. The material properties are updated using predefined rules resulting in 0–1 solutions that represent a void or solid state for each element. Once the optimum topology is determined, the next step is to formulate and solve the structural and control system design problem.

3. Control system design

The finite element dynamical equations governing the motion of a controlled structural system are given as:

[M] {\overset{··}{x}} + [C_{D}] {\overset{\cdot}{x}} + [K] {x} = [D] {F_{c}}

(5)

where

[M]

[C_{D}]

and

[K]

are

n \times n

mass, damping and stiffness matrices respectively and n is the number of physical coordinates.

[D]

is the

n \times m

applied force distribution matrix which relates the input control force to coordinate system, m is the number of actuators present,

{x}

is a

n \times 1

vector of physical coordinates and

{F_{c}}

m \times 1

control vector. Using the coordinate transformation

{x} = [φ] {y}

, equation (5) can be represented in state-space form as:

{\overset{\cdot}{u}} = [A] {u} + [B] {F_{c}}

(6)

where

{y}

is the vector of modal coordinates,

{u} = [{y}, {\overset{\cdot}{y}}] T

2 n \times 1

state variable vector,

[φ]

n \times n

modal matrix, and the

2 n \times 2 n

plant matrix

[A]

and

2 n \times m

input matrix

[B]

are defined as:

[A] = [0 I - ω_{i}^{2} - 2 ξ_{i} ω_{i}]

(7)

[B] = [0 φ T D]

(8)

where

ξ_{i} and ω_{i}

denote the damping factor and natural frequency of the

i th

mode, respectively.

Linear quadratic regulator (LQR) theory is used to design a controller for the system governed by equation (6). The optimum control force ${F_{c}}$ is selected to minimize the quadratic performance index, J, which is a compromise between minimum control energy and minimum error requirements, and is defined as:

J = \int_{0}^{\infty} ({u} T [Q] {u} + {F_{c}} T [R] {F_{c}}) dt

(9)

where [Q] is a positive semi-definite state weighting matrix and [R] is a positive definite control weighting matrix. The optimum feedback control law is given as

{F_{c}} = - [κ] {u}

where [κ] is the feedback gain matrix defined as

[κ] = [R] - 1 [B] T [P]

, and

[P]

is the solution to the matrix Riccati equation:

[A] T [P] + [P] [A] + [Q] - [P] [B] [R] - 1 [B] T [P] = [0]

(10)

The minimum value of the quadratic performance index is given as:

J * = {u (0)} T [P] {u (0)}

(11)

where

{u (0)}

is the initial state vector. Since the optimum value of

J *

depends on the initial state which is not always known, it can be shown that the expected value of

J *

E (J *)

, over a set of possible initial states

{u (0)}

is equivalent to

trace [P] .

E (J *) = trace [P]

(12)

A minimization of $trace [P]$ is considered as one of the objective functions in this work.

Substituting the optimum feedback control law, ${F_{c}} = - [κ] {u}$ in equation (6) yields:

{\overset{\cdot}{u}} = ([A] - [B] [κ]) {u} = [A_{cl}] {u}

(13)

The eigenvalues of the closed-loop matrix $[A_{cl}]$ are given as:

λ_{i} = α_{i} \pm j β_{i}, i = 1, \dots, n

(14)

where

j = \sqrt{- 1}

and

| λ_{i} | = \sqrt{α_{i}^{2} + β_{i}^{2}}

. The closed-loop damping ratio

ξ_{i}

associated with

λ_{i}

is given as:

ξ_{i} = - \frac{α_{i}}{\sqrt{α_{i}^{2} + β_{i}^{2}}} i = 1, \dots, n

(15)

The damping ratio controls the settling time of vibrations. For a given initial condition ${u (0)}$ , the solution to equation (13) is given as:

{u (t)} = e [A_{cl}] t {u (0)}

(16)

Next, a performance index is developed which will be used to determine optimum locations of the actuators. This performance index permits a simultaneous determination of optimum feedback gains and optimum actuator locations during the solution process.

3.1. Actuator placement

Rewriting the equations for the closed-loop system given by equation (6) by substituting values for $[A]$ , $[B]$ and ${F_{c}}$ yields:

{\overset{\cdot}{u}} = [[0] [I] [- ω_{i}^{2}] [- 2 ξ_{i} ω_{i}]] {u} - [[0] φ T D] [κ] {u}

(17)

The energy dissipated by the controller is given as, E_c

E_{c} = \int_{0}^{\infty} [y \overset{\cdot}{y}] T D_{c} [y \overset{\cdot}{y}] d t

(18)

where

{y, \overset{\cdot}{y}}

is the state vector and D_c is the damping matrix induced by the active controller and is defined as:

D_{c} = [[0] [0] φ T DR - 1 D T φ P_{III} φ T DR - 1 D T φ P_{IV}]

(19)

Here $P_{III}$ and $P_{IV}$ denote the $n \times n$ block partitions of the $2 n \times 2 n$ Ricatti matrix defined below:

P = [P_{I} P_{II} P_{III} P_{IV}]

(20)

It has been shown (Ali et al., 2013) that equation (18) can be rewritten as:

E_{c} = [y (0) \overset{\cdot}{y} (0)] T H [y (0) \overset{\cdot}{y} (0)]

(21)

where H is the unique solution to the matrix Lyapunov equation:

[A_{cl}] T H + H [A_{cl}] = - D_{c}

(22)

From equation (21), it is clear that the energy dissipated depends on the initial state which is not always known. However, if the initial state is assumed to be a random variable distributed uniformly over the surface of a $2 n$ dimensional unit sphere, maximization of expected value of E_c over the set of possible initial states is the same as maximizing trace $[H]$ . Therefore,

ev [E_{c}] = trace [H] .

(23)

For an efficient controller, $trace [H]$ is maximized by treating the actuator locations as design variables. The natural frequencies of the controlled system and time required to damp out the vibrations can be specified by putting constraints on closed-loop eigenvalues and damping ratios. From a structural viewpoint, the designer wants to minimize the weight of the structure by varying the cross-sectional areas of the members and ensuring that the stresses in members due to applied loads do not exceed permissible limits. For the combined structural control system design problem formulation, the designer wants to (i) minimize the structural weight, (ii) maximize the energy dissipated by the controller (equation (23)), and (iii) minimize the value of quadratic performance index (equation (12)). The procedure outlined in the next section describes how both structural and control objectives can be simultaneously optimized using a game theoretic approach.

4. Game theoretic methods for multi-objective optimization

Multi-objective optimization (MOO) problems requiring a simultaneous consideration of two or more conflicting objective functions frequently arise in design. A general MOO problem has the following form:

Min f (X) = [f_{1} (X), f_{2} (X), \dots f_{p} (X)] subjectto {X | X \in R q, g_{i} (X) \leq 0, h_{j} (X) = 0}

(24)

where

f_{1} (X)

, …

f_{p} (X)

are p objective functions,

g_{i} (X)

and

h_{j} (X)

are inequality and equality constraints, and X is the vector of variables. In a MOO problem, it is not possible to find an optimum point where all objective functions are simultaneously minimized. Therefore, the concept of a Pareto-optimal (PO) solution is used in solving a MOO problem. Frequently, the set of PO solution(s) contains more than one solution. Different methods have been used to determine an optimal compromise solution from the set of PO solutions. Game theory is one such approach which helps determine a compromise solution acceptable to all objective functions (players).

4.1. Game theory method

In the game theory method, the MOO problem is viewed as a game where each player corresponds to an objective function being optimized. These players are competing with each other to improve their overall position subject to some constraints.

There are three types of games in the context of engineering design: cooperative game, non-cooperative (Nash) game, and an extensive game. In a cooperative game, the players have knowledge of the strategies chosen by other players and collaborate with each other to find a PO solution. In a non-cooperative game, each player has a set of variables under his control and optimizes his objective function individually. The player does not care how his selection affects the payoff functions of other players. The players bargain with each other to obtain an equilibrium solution, called the Nash solution. Extensive games refer to situations in which the players make the decisions sequentially. Extensive games with two players have been used in engineering design and are called Stackelberg games. There are two groups of players in this game; one called the leader which dominates the other group called the follower. The leader makes its decision first and according to its decision, the follower optimizes its objective function.

Consider two players, 1 and 2, who select strategies x₁ and x₂ where $x_{1} \in X_{1} \subset R n_{1}$ and $x_{2} \in X_{2} \subset R n_{2}$ . Here X₁ and X₂ are the set of all possible strategies each player can select. The objective functions $f_{1} (x_{1}, x_{2})$ and $f_{2} (x_{1}, x_{2})$ represent the cost function for players 1 and 2, respectively. In a Nash game, each player determines its optimum solution based on the choices made by other player(s). This set of solutions for each player is called the RRS. The RRS for players 1 and 2 are defined as follows:

f_{1} (x_{1}^{N}, x_{2}) = min f_{1} (x_{1}, x_{2}) \to x_{1}^{N} (x_{2}) x_{1} \in X_{1}

(25)

f_{2} (x_{1}, x_{2}^{N}) = min f_{2} (x_{1}, x_{2}) \to x_{2}^{N} (x_{1}) x_{2} \in X_{2}

(26)

where

x_{1}^{N}

is the optimum solution of player 1 which varies depending on the strategy x₂ chosen by player 2. The functions

x_{1}^{N} (x_{2})

and

x_{2}^{N} (x_{1})

denote the RRS for players 1 and 2 respectively. The intersection of these two sets, if it exists, is the Nash solution for the non-cooperative game.

In design situations where the objectives have differing importance and a hierarchical structure exists, the problem can be divided into multiple levels. Solving such multi-level problems as a Stackelberg game is discussed next.

4.2. Bi-level Stackelberg game

The Stackelberg game is a special case of a bi-level game where one player dominates the other player. Suppose player A (the leader) dominates player B (the follower). Player A knows the optimum strategy (solution) of player B. When player A chooses a strategy, player B can see the choices made by player A. Player B solves its problem and finds the optimum solution with respect to choices made by player A. Player A can now adjust its strategy based on choices made by player B.

The model of the Stackelberg game when player A is the leader can be written as follows:

minimize : f_{1} (x_{1}, x_{2}) by varying (x_{1}) subjectto : x_{2} \in X_{2}^{N} (x_{1}) .

(27)

On the other hand, when B is the leader, the problem is:

minimize : f_{2} (x_{1}, x_{2}) by varying (x_{2}) subjectto : x_{1} \in X_{1}^{N} (x_{2}) .

(28)

where

X_{1}^{N} (x_{2}), X_{2}^{N} (x_{1})

are the RRS for players 1 and 2 respectively.

To construct the RRS, approximation techniques such as the response surface method (RSM) or sensitivity-based approaches can be used. In this paper, since some of the design variables (actuator locations) are discrete, sensitivity-based approaches for constructing the RRS cannot be applied. Therefore, the RSM method (Myers and Montgomery, 2002) is used to construct the RRS for the players.

Unlike Nash and Stackelberg games where players do not cooperate, in a cooperative game, the players have knowledge of the strategies chosen by other players and collaborate with each other to find a PO solution. It is not uncommon for players to improve their non-cooperative solution by cooperating.

4.3. Cooperative game theory method

Consider a cooperative game with two players. Let $U_{i} (X)$ be the utility (payoff) function associated with each player $i = 1, 2$ such that if strategy X is selected from a set of alternative strategies $S (X \in S)$ , player i will have payoff $U_{i} (X)$ . The two players compromise to select a mutually beneficial strategy such that their payoffs are as high as possible. It is assumed that if the players decide not to cooperate, their payoffs will be $u *$ and $v *$ where $u * = U_{1} (X_{w})$ and $v * = U_{2} (X_{w})$ and X_w is a status quo point $X_{w} \in S$ . The players want to maximize their distance from X_w.

The bargaining model that determines a compromise solution using the bargaining function $B (X)$ defined as:

B (X) = (u - u *) (v - v *) = Π_{i = 1}^{2} [U_{i} (X) - U_{i} (X_{w})]

(29)

for all

X \in S * \subset S

, where

S * = [X | X \in S,

U_{i} (X) - U_{i} (X_{w}) \geq 0]

An optimum compromise solution is now defined as:

B (X opt) = max B (X), X \in S *

(30)

The bargaining function yields a PO solution $X opt$ which maximizes the payoff for each player.

5. Solution procedure

The complete structural control optimization problem involves determination of optimum topology followed by sizing and control optimization of the optimum topology. A determination of optimum topology begins with defining the region occupied by the structure. This region of material is meshed using finite elements. External loads and boundary conditions are next specified with respect to this domain. The purpose of topology optimization is to find optimum density distribution while maintaining a constant volume constraint. The objective is to minimize the strain energy such that the final weight of the structure should not be more than, say 20% of the initial structure. The topology optimization problem is stated as follows:

Minimize C = \frac{1}{2} {P} T {d} subjectto V \leq 0.2 V_{o} 0 \leq ρ \leq 1

(31)

Once the optimum topology is found, the structural and control system design problems are solved. This multi-objective problem is formulated as a bi-level optimization problem and solved using a Stackelberg game theoretic formulation. The objective function of the leader is the maximization of the energy dissipated by the actuators (max trace [H]) with actuator locations as design variables. The objective functions of the followers are (i) minimize the structural weight with cross-sectional areas of the members as design variables, and (ii) minimize the optimum value of LQR quadratic performance index with entries of state and control weighting matrices as design variables. The problem formulation is given below:

Leader : Maximizetrace [H] byvaryingactuatorlocations (X) subjectto g_{i} (X, Y, Z) \leq 0

(32)

Follower 1 : Minimizeweight byvaryingmembercross - sectionalareas (Y) subjectto g_{i} (X, Y, Z) \leq 0

(33)

Follower 2 : Minimizetrace [P] byvaryingentriesin Q and R matrices (Z) subjectto g_{i} (X, Y, Z) \leq 0

(34)

It is assumed that the follower objective functions (minimize weight and minimize trace [P]) cooperate with each other. A bargaining function $B (X)$ is defined between these functions per equation (30). Equations (32)–(34) can now be rewritten as:

Leader : Maximizetrace [H] byvaryingactuatorlocations (X) subjectto g_{i} (X, areas, Q, R) \leq 0

(35)

Follower : Maximize F_{b \arg} = \frac{(f_{1 w} - f_{1}) (f_{2 w} - f_{2})}{(f_{1 w} - f_{1 b}) (f_{2 w} - f_{2 b})} byvarying (areas, Q, R) subjectto g_{i} (X, areas, Q, R) \leq 0

(36)

where

f_{1 w}

f_{1 b}

f_{2 w}

and

f_{2 b}

denote the worst and best values of the two follower objectives, weight and trace [P]. The constraints imposed on the problem include limits on structural stresses, closed-loop damping ratios, and closed-loop eigenvalues. Since the variables of the leader problem are discrete in nature, a genetic algorithm-based approach is used to solve the leader problem. The variables of the follower problem are continuous; therefore a sequential quadratic programming approach is used to solve the follower problem. A complete flowchart of the solution process is shown in Figure 1 and the bi-level Stackelberg and cooperative game between structural and control objectives are shown in Figure 2.

Figure 1.

Steps for solving the integrated topology and control optimization problem.

Figure 2.

Two level Stackelberg and cooperative game.

Two examples are presented next for solving the structural control optimization problem. For both examples, it is shown that an integration of topological considerations lead to final solutions which outperform the fixed topology optima.

6. Design example 1

The first example deals with sizing and control design of a 10 bar truss (fixed topology) followed by topology, sizing and control design for the same problem. The results for this example show that an integration of topological considerations leads to final solutions which outperform fixed topology optima on both structural and control performance measures.

6.1. Sizing optimization of fixed topology

The 10 bar truss shown in Figure 3 is first considered for structural design. The total length of the truss is 720 inches, equally divided between two bays. The width of the truss is 360 inches.

Figure 3.

Ten bar truss with two applied loads.

Two loads, 5000 lb each, are acting at nodes 2 and 4 in the y-direction whereas nodes 5 and 6 are fixed. The Young’s modulus of the members is 10 × 10⁶psi and the weight density of the material is 0.1 lb/in³. A sizing optimization on this structure is performed first to minimize the weight of the structure subject to the constraint that member stresses should not exceed 25,000 psi. The cross-sectional areas of the members are taken as design variables and are constrained to lie between 0.1–20 in². The minimum weight of the structure is found to be 88.38 lb and the corresponding optimum cross-sectional areas are listed in Table 1.

Table 1.

Optimum cross-sectional areas for the 10 bar truss (fixed topology).

		Optimum areas	Optimum areas
Design variables	Starting values	Stress constraints only	Stress and control constraints
x1	1.0	0.1000	0.2511
x2	1.0	0.1000	0.1000
x3	1.0	0.1379	0.1121
x4	1.0	0.3379	0.3281
x5	1.0	0.1000	0.1000
x6	1.0	0.4621	0.4842
x7	1.0	0.1949	0.1585
x8	1.0	0.1000	0.1434
x9	1.0	0.3707	0.4018
x10	1.0	0.1949	0.1638
Weight	419.64	88.38	93.69

Next, a controller is designed for this structure. The structure has eight degrees-of-freedom (d.f.), two at each of the four free nodes. A non-structural mass of 1.29 lb-s²/in is attached at nodes 1 through 4. A load of 5000 lb is applied downwards at nodes 2 and 4. A total of four actuators are present at the four free nodes and they are assumed to be acting along y-direction only. The passive (material) damping is taken to be 1.0 × 10⁻⁵lb-sec/in. The control weighting matrix [R] used in the LQR controller is a 4 × 4 identity matrix and the state weighting matrix [Q] is taken as 1000* [I]. The design constraints imposed on the problem include: i) The stress in each member should not exceed 25,000 psi $(g_{1} - g_{10})$ ; ii) The closed-loop damping ratio corresponding to the first mode should be greater than 0.6 $(g_{11})$ ; iii) a stability margin of five is required corresponding to the second eigenvalue of the closed-loop system matrix $(g_{12})$ . The complete structural control optimization problem is as follows:

Minimizeweight byvarying (x_{1} - x_{10}) subjectto σ_{1 - 10} - 25000 \leq 0 0.6 - ξ_{1} \leq 0 5 - α_{2} \leq 0 0.1 \leq x_{i} \leq 20 i = 1, \dots 10

(37)

where

x_{1} - x_{10}

are the cross-sectional areas of the elements. This optimization problem is solved using sequential quadratic programming. The integrated structure and control optimization problem yields an optimum structural weight of 93.69 lb and the corresponding cross-sectional areas are also given in Table 1.

6.2. Topology optimization

A formulation to determine the optimum topology is established first. The initial problem domain is defined as a rectangular grid of nodal points as shown in Figure 4 The Young’s modulus and material density are taken as E = 10 × 10⁶psi and ρ = 0.1 lb/in³. Top and bottom nodes on the extreme left are fixed while nodes at the center and the bottom right are subjected to two loads of 5000 lb acting in the y-direction. A topology optimization is performed such that the volume of the final structure should not be more than 20% of the initial structure volume.

Figure 4.

Problem domain with supports and points of load application.

The optimum topology is shown in Figure 5.

Figure 5.

Optimum topology for example 1.

6.3. Sizing optimization of optimum topology

The resulting optimum topology is approximated as a six-bar truss as shown in Figure 6. A sizing optimization of this structure is performed next. Keeping everything the same as in case of initial 10 bar truss (Section 6.1), the minimum weight of the structure is found to be 79.2 lb and the optimum cross-sectional areas are listed in Table 2. From the results in Table 2, it can be seen that an optimization of topology leads to a 10% reduction in the optimum weight of the structure.

Figure 6.

Approximated optimum topology for example 1.

Table 2.

Cross-sectional areas for optimum topology formulation (example 1).

		Optimum areas	Optimum areas
Design variables	Starting values	Stress constraints only	Stress and control constraints
x1	1.0	0.400	0.400
x2	1.0	0.283	0.283
x3	1.0	0.200	0.200
x4	1.0	0.400	0.400
x5	1.0	0.283	0.283
x6	1.0	0.283	0.283
Weight	260.73	79.20	79.20

Next, a controller is designed for this six bar structure. Three actuators are present at the three free nodes and they are assumed to be acting along y-direction only. The control weighting matrix [R] is a 3 × 3 identity matrix and the state weighting matrix [Q] is kept at 1000* [I]. The design constraints are kept the same as those in equation (37). The optimum weight of the structure is found to be 79.2 lb and the cross-sectional areas are listed in Table 2. This design has a 15% lower weight than the corresponding design given in Table 1. The optimum weight for the structure-control optimization problem depends on the values chosen for the controller constraints. For the controller performance parameters selected in this example, the results of the structure only and structure-control optimization problems are identical (columns 3 and 4, Table 2). This is a pure coincidence for this example. In general, the optimum results from these two formulations are expected to be different.

It should be noted that the start point for the optimization problem is selected randomly.

Different starting points were considered when obtaining the results reported in Tables 1 and 2 to rule out the possibility of convergence to a local minimum. It was seen that the same optimum result (reported in Tables 1 and 2) is obtained regardless of the starting point selected.

7. Design example 2

Example 1 demonstrated the benefits of integrating topological considerations in the context of structure and control design of actively controlled structures. Example 2 considers a multi-objective problem where topological, structural and control considerations are addressed using a game theory approach.

7.1. Topology optimization

Consider first the problem domain shown in Figure 7 where a structure is required to support two loads of 1000 lbs each acting in the negative y-direction. The top and bottom left nodes are fixed.

Figure 7.

Problem domain with supports and points of load application for example 2.

A candidate topology for this problem based on a 3 × 2 grid (Ohsaki and Katoh 2005) is shown in Figure 8. A sizing optimization of this topology results in an optimum weight of 13.81 lb.

Figure 8.

3 x 2 plane grid.

The topology optimization is performed by considering the 600 × 400 rectangular region of the problem domain and meshing it using 180 × 90 elements as shown in Figure 7. The volume of the optimum topology should not be more than 25% of the initial volume. The optimum topology is shown in Figure 9. A sizing optimization of this structure results in an optimum weight of 12.7 lb. It can be seen that an optimization of topology leads to a 9% reduction in the optimum weight of the structure.

Figure 9.

Optimum topology with 25% volume constraint.

7.2. Structural and control optimization

The multi-objective structural and control optimization of the optimum topology obtained in Section 7.1 is considered next. Both Stackelberg and cooperative game theory approaches are used to solve the MOO problem given by equations (35) and (36).

7.2.1. Single objective optimization

The optimum topology of Figure 9 is approximated as an eight bar truss shown in Figure 10.

Figure 10.

Approximated topology for example 2.

This structure has eight d.f., two d.f. at each of the four free nodes. The Young’s modulus of the members is 10 × 10⁶psi and the weight density of the material is 0.1 lb/in³. A load of 1000 lb is applied downwards at nodes 3 and 4. The [R] and [Q] matrices are 8 × 8 and 16 × 16 diagonal matrices. The single objective function optimization is performed first to find the best and worst values of the follower objective functions which are weight (objective f₁) and trace $[P]$ (objective f₂) with cross-sectional areas of members and diagonal entries of [Q] and [R] as design variables. It is seen that the best and worst values (in equation (36)) for the weight are 14.7 lb and 61.15 lb respectively. Similarly, the best and worst values of trace $[P]$ are found to be 43814 and 4.26 × 10⁶ respectively.

7.2.2. Multi-objective optimization

The bi-level structural control optimization problem is modeled using the Stackelberg game. The three objective functions, as considered in equations (32)–(34), are: (i) maximize trace $[H]$ , (ii) minimize structural weight, and (iii) minimize trace $[P]$ . Player 1 (leader) wishes to maximize trace $[H]$ by varying $x_{1} - x_{8}$ , which are the actuator locations in all elements. Player 2 (follower) maximizes the bargaining function $(F_{b \arg})$ between weight and trace $[P]$ by varying member cross-sectional areas, ${area}_{1} - {area}_{8}$ , and diagonal entries of state weighting matrix, [Q], and input weighting matrix, [R] namely $Q_{99}$ and $R_{11}$ , $R_{22}$ , $R_{33}$ and $R_{66}$ . The other entries of [Q] and [R] matrices are fixed at 0.1 because they always converged to the lower bound when treated as design variables. For this bi-level problem, the RRS of the follower gives the change in the optimum solution of the follower problem when the leader’s variables are varying. Finding the RRS of the follower involves solving the follower’s problem for various combinations of the leader’s design variables, which are discrete 0 or 1, actuator locations. Since there are eight leader design variables with two possibilities either 0 or 1, the follower problem is solved 2⁸= 256 times to construct the RRS. The sequential quadratic programming method is used to solve the follower problem. Once the RRS for the follower is found, it is inserted into the leader problem to find the optimum solution to the leader problem.

The optimization problem is stated as:

Leader : Maximize trace [H] byvarying (x_{1} - x_{8}) subjectto 0.03 - ξ_{1} \leq 0 200 \leq β_{i} \leq 2500 i = 1, \dots 8 σ_{i} - 25000 \leq 0 0.001 \leq {area}_{i} \leq 20 0.1 \leq Q_{jj} \leq 1000 j = 9 0.1 \leq R_{qq} \leq 1000 q = 1, 2, 3, 6

(38)

Follower : Maximize F_{b \arg} = \frac{(f_{1 w} - f_{1}) (f_{2 w} - f_{2})}{(f_{1 w} - f_{1 b}) (f_{2 w} - f_{2 b})} byvarying (areas, Q, R)

(39)

subject to the same constraints in equation (38). Here

f_{1 w}

f_{2 w}

f_{1 b}

and

f_{2 b}

denote the worst and best values of weight and trace [P] as specified in Section 7.2.1.

7.3. Stackelberg solution

For each $(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8})$ combination, an optimum solution for ${area}_{1} - {area}_{8}$ , $Q_{99}, R_{11}, R_{22}, R_{33}, R_{66}$ is obtained. The response surface regression yields the following RRS for the follower.

A_{1} = 0.1206 - 8.4041 \times 10 - 4 (x_{1} + x_{2} + x_{3}) + 0.9979 \times 10 - 4 x_{4} - 8.4041 \times 10 - 4 (x_{5} + x_{6} + x_{7}) - 1.003 \times 10 - 4 x_{8}

A_{2} = 0.0495 - 5.8829 \times 10 - 4 x_{1} - 7.5156 \times 10 - 4 x_{2} - 6.2875 \times 10 - 4 x_{3} - 2.6468 \times 10 - 4 x_{4} - 7.3374 \times 10 - 4 x_{5} - 5.9925 \times 10 - 4 x_{6} - 7.3127 \times 10 - 4 x_{7} - 4.2876 \times 10 - 4 x_{8}

A_{3} = 0.0749 - 1.6555 \times 10 - 5 (x_{1} + x_{2} + x_{3}) + 1.7269 \times 10 - 5 x_{4} - 1.6555 \times 10 - 5 (x_{5} + x_{6} + x_{7} + x_{8})

A_{4} = 0.0355 - 1.0606 \times 10 - 4 x_{1} + 1.9600 \times 10 - 4 x_{2} + 2.7567 \times 10 - 4 x_{3} - 4.4676 \times 10 - 4 x_{4} + 1.6085 \times 10 - 4 x_{5} + 3.2073 \times 10 - 4 x_{6} + 0.7784 \times 10 - 4 x_{7} - 0.6327 \times 10 - 4 x_{8}

A_{5} = 0.0657 - 2.9621 \times 10 - 4 x_{1} - 3.8406 \times 10 - 4 x_{2} - 3.0136 \times 10 - 4 x_{3} - 3.1952 \times 10 - 4 x_{4} + 1.9552 \times 10 - 4 x_{5} - 2.0022 \times 10 - 4 x_{6} - 3.8406 \times 10 - 4 (x_{7} + x_{8})

A_{6} = 0.0362 - 1.6823 \times 10 - 3 x_{1} - 2.2994 \times 10 - 3 x_{2} - 2.1958 \times 10 - 3 x_{3} + 0.4443 \times 10 - 3 x_{4} - 2.6243 \times 10 - 3 x_{5} - 2.2788 \times 10 - 3 x_{6} - 2.3711 \times 10 - 3 x_{7} - 0.7722 \times 10 - 3 x_{8}

A_{7} = 0.1003 + 7.7238 \times 10 - 5 x_{1} - 7.8801 \times 10 - 5 (x_{2} + x_{3} + x_{4} + x_{5} + x_{6} + x_{7} + x_{8})

A_{8} = 0.0473 - 2.5557 \times 10 - 5 (x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6} + x_{7} + x_{8})

Q_{99} = 2.3815 + 2.7677 \times 10 - 1 x_{1} - 7.5077 \times 10 - 1 x_{2} - 4.3118 \times 10 - 1 x_{3} - 2.1125 \times 10 - 1 x_{4} - 8.9931 \times 10 - 1 x_{5} - 9.4868 \times 10 - 1 x_{6} + 5.1807 \times 10 - 1 x_{7} - 9.8433 \times 10 - 1 x_{8}

R_{11} = 0.1275 + 4.2284 \times 10 - 2 x_{1} - 4.2284 \times 10 - 2 x_{2} + 3.0104 \times 10 - 2 x_{3} + 4.1747 \times 10 - 2 x_{4} - 4.2284 \times 10 - 2 (x_{5} + x_{6}) + 4.2284 \times 10 - 2 x_{7} - 4.2284 \times 10 - 2 x_{8}

R_{22} = 0.1092 - 3.6764 \times 10 - 3 x_{1} + 3.6764 \times 10 - 3 x_{2} - 3.6764 \times 10 - 3 (x_{3} + x_{4} + x_{5} + x_{6} + x_{7}) + 3.6764 \times 10 - 3 x_{8}

R_{33} = 0.1077 - 5.1429 \times 10 - 3 (x_{1} + x_{2}) + 5.1429 \times 10 - 3 (x_{3} + x_{4}) - 5.1429 \times 10 - 3 (x_{5} + x_{6}) + 5.1429 \times 10 - 3 x_{7} - 5.1429 \times 10 - 3 x_{8}

R_{66} = 0.5415 - 1.7660 \times 10 - 1 (x_{1} + x_{2} + x_{3}) + 1.7660 \times 10 - 1 x_{4} - 1.7660 \times 10 - 1 x_{5} + 1.7660 \times 10 - 1 x_{6} - 1.7660 \times 10 - 1 (x_{7} + x_{8})

Substituting the RRS of the follower problem into the leader’s problem, the leader’s problem is solved. Since the leader problem variables are discrete, a genetic algorithm-based approach is used to solve the leader’s problem. The optimum solution of the leader’s problem results in an optimum value of trace

[H]

= 5.99 × 10⁶ with actuators located in elements 4, 7 and 8. The optimum values of the cross-sectional areas are given in Table 3. The optimum weight of the structure is 14.92 lb and the optimum value of trace

[P]

is 6.27 × 10⁴.

Table 3.

Optimum results for design example 2.

Element	Actuator	Areas	R*	Q₉₉
1		0.1198	0.1692	1.7
2		0.0481	0.1055
3		0.0749	0.1128
4	X	0.0351	0.3649
5		0.0646
6		0.0334
7	X	0.1
8	X	0.0473
trace [H]	5.99 × 10⁶
Weight	14.92
trace [P]	6.27 × 10⁴
RMSD	0.0895

R* = First, second, third and sixth diagonal entry of R matrix.

RMSD: root mean square displacement.

The dynamic response of the optimum structure is studied by a unit displacement at node 2 in the y-direction at t = 0. The displacement of all free nodes is accounted for in the root mean square displacement (RMSD). The RMSD is defined as the square root of the sum of the squares of displacements at all free nodes in x and y directions. The RMSD should be damped out quickly so that the structure is brought to its equilibrium position in the shortest possible time.

The RMSD error for the optimum design is about 0.0895 inches and is shown in Figure 11.

Figure 11.

Transient response at optimum design.

8. Conclusions

A unique combination of structural and topological optimization combined with optimum controller design is presented in this paper. The problem is formulated as a MOO for the design of actively controlled structures wherein topological, structural and control aspects are optimized using a bi-level game theoretic formulation. In particular, the MOO problem is solved using a combination of a cooperative and a Stackelberg game. The solution approach involves determination of an optimum topology followed by structure and control system optimization of the optimum topology. The proposed method can be applied to problems with conflicting objectives and with mixed discrete-continuous variables. Through two examples, it is shown that given the problem domain, points of load application and structural and control performance constraints, the proposed approach yields optimum structural configuration with an optimized controller which is able to bring the structure to its equilibrium position when subjected to an external disturbance. The main contribution of this research is showing that structures with improved structural and control performance can be obtained when topological considerations are integrated in the design process. The results from the two examples presented clearly show that structures with optimum topology are better when compared with optimum designs resulting from a fixed topology formulation.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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