Global optimal actuator placement for adaptive structures: New formulation and benchmarking

Abstract

Civil structures are often overdesigned to meet safety and functionality criteria under rare, strong events. Adaptive structures, however, can modify their response through sensing and actuation to satisfy design criteria more efficiently with better material utilization, which results in lower resource consumption and associated environmental impacts. Adaptation is performed through actuators integrated into the structural layout. Several methods exist for optimal actuator placement to control displacements and internal force flow. In discrete systems like trusses and frames, actuator placement is typically a binary assignment. Most existing methods use bilevel and heuristic formulations, leading to suboptimal solutions without proving global optimality. This paper introduces a Mixed Integer Programming (MIP) method that produces global optimum solutions by optimizing both actuator placement and commands. Two objective functions are used: minimizing the number of actuators and minimizing control energy. The optimization considers structural and serviceability limits and control feasibility. An extensive benchmark compares the new formulation’s global optima with solutions from greedy, stochastic, and heuristic methods. Results show that the new method consistently produces higher-quality solutions than all other methods benchmarked in this study.

1. Introduction

1.1. Previous work

Building construction and operation are responsible for a substantial share of global energy consumption, greenhouse gas emissions, and waste generation (United Nations Environment Programme (IEA), 2022). The construction sector is a major consumer of mined raw materials leading to the depletion of resources once believed unlimited (e.g. sand, metals) (United Nations Environment Programme, 2014). The material used for construction causes 9% of the annual global CO₂ emissions, and overall the building industry contributes to more than 34% of the global energy demand (United Nations Environment Programme (IEA), 2022). Addressing these challenges requires a change in the way civil structures are designed. Civil structures must satisfy safety and serviceability criteria under strong and rare events and therefore are typically overdesigned for most of their service life. Instead, structures could be adaptive in response to strong but infrequent loads and other environmental actions (Preumont and Kazuto, 2008; Soong and Spencer Jr, 2000; Utku, 1998; Wani et al., 2022). A key aspect that allows adaptive structures to be resource-efficient is the ability to reach a compromise between the stiffness provided through geometry and mass (i.e. element sizing) and the extent of active compensation of stress and displacement responses caused by external actions. Active response mitigation against strong but infrequent loading events enables material, embodied carbon and energy minimization, especially for stiffness-governed configurations such as tall and slender buildings (Blandini et al., 2022; Reksowardojo and Senatore, 2023; Senatore et al., 2018), long-span bridges (Reksowardojo et al., 2024a; Zeller et al., 2023) and floor systems (Nitzlader et al., 2022; Reksowardojo et al., 2024b). In these cases, it is much more efficient to compensate the response under loading actively when needed than embodying additional material, which is a cost penalty for most of the service life of the structure since typical design loads are caused by strong events that occur infrequently (Senatore et al., 2019). Response mitigation under loading can also be employed to increase durability and extend the service by reducing the stress response below relevant fatigue initiation thresholds which is particularly relevant for bridge structures (Canny et al., 2024).

For discrete structural assemblies such as trusses and frames, adaptation can be performed by linear actuators that are integrated into the structural layout. The action of linear actuators, for example, length changes, has been employed to control the response of truss and frame configurations, including high-rise and bridge structures (Abdel-Rohman and Leipholz 1983; Geiger et al., 2020; Rhode-Barbarigos et al., 2010; Rodellar et al., 2002; Wagner et al., 2020). Several actuator placement methods have been formulated to maximize the efficacy of controlling displacements and the internal force flow (Böhm et al., 2019; Lu et al., 1992; Ramesh et al., 1991; Senatore and Reksowardojo, 2020; Steffen et al., 2022). In most previous methods for discrete assemblies, actuator placement has been implemented as a binary (0–1) assignment to element sites. The binary decision variable is typically relaxed by evaluating the effect of a unitary action of candidate actuators on displacement and internal forces, which is collated in sensitivity (i.e. influence) matrices (Du Pasquier and Shea, 2022; Senatore et al., 2019; Steffen et al., 2020). Other approaches including stochastic methods (Du Pasquier and Shea, 2021; Furuya and Haftka, 1995; Yan and Yam, 2002), bilevel approaches (e.g. Stackelberg Game) (Ali et al., 2015; Marker and Bleicher, 2022; Wagner et al., 2018; Zeller et al., 2022) and other sequential formulations (Manguri et al., 2023; Rashid et al., 2022; Saeed et al., 2023; Wani and Tantray, 2022) have been implemented. Typically, these methods produce actuator placements with optimality degrading as the problem dimension increases and constraints are tightened (e.g. displacement limits).

All-In-One (AIO) structure-control optimization methods have been formulated in (Senatore and Wang, 2024; Wang and Senatore, 2020) that can produce global least-weight and minimum energy adaptive structures. In these formulations, variables related to the structural layout (e.g. sizing, topology) and actuator placement are optimized simultaneously. For this reason, using such methods it is not possible to evaluate the effect of certain objective functions and constraints only on the actuator positions and to benchmark solutions with other actuator placement methods. This is particularly important for retrofitting applications; in which cases the structural parameters must be kept constant since the structure is already built. Methods for the optimal retrofitting of multi-story structures that minimize the number of actuators and the control energy that produce global optima have not yet been formulated.

Previous studies (Virgili et al., 2023) do not include benchmarks between solutions produced by sequential and AIO methods. Importantly, no systematic investigation has been carried out in the literature to compare the performance of diverse actuator placement methods against global benchmarks.

1.2. New contribution

The new contributions of this work are:

• A method based on Mixed-Integer Programming (MIP) that can optimize simultaneously actuator positions and commands, hereafter referred to as “All-In-One” (AIO) actuator placement. Two objective functions are formulated: (O1) minimize the number of actuators, and (O2) minimize the control energy, both under quasi-static loads. Optimization constraints include structural, serviceability and control feasibility.

• New objective and constraint functions specific to the actuator placement problem are provided. The formulation is linear for O1 and quadratic for O2, and therefore in both cases, it can be solved to a global optimum using state-of-the-art branch and bound algorithms.

• A proof of the problem convexity to minimize the control energy (O2), which contains bilinear constraint functions.

• Rigorous comparison of the performance of diverse methods based on sequential approaches including heuristics, greedy and stochastic strategies against the global benchmarks obtained with the AIO approach.

• Benchmarks are carried out in terms of two key performance indicators, namely the number of actuators and the control energy required to satisfy target control objectives that include stress, displacement and actuator force limits.

Section 2 provides the formulation for the structural and control system model. Section 3 outlines different optimal control strategies under equivalent static representations of dynamic loads (e.g. wind). In Section 4, performance metrics are defined to evaluate the quality of the solutions. In Section 4.3.3, a benchmark is carried out between the new AIO formulation and several sequential methods. Case studies on multi-story building structures are employed to evaluate the influence of different input parameters including the number of structural elements and control constraints.

2. System modeling

Pin-jointed structures are considered with the assumption of small deformations and linear elastic material. Under static loads, the system equilibrium is

$$\mathit{K}\mathit{d}=\mathit{P}$$

(1)

where $\mathit{K}\in {\mathbb{R}}^{n^{\mathrm{dof}}\times n^{\mathrm{dof}}}$ is the stiffness matrix, $\mathit{d}\in {\mathbb{R}}^{n^{\mathrm{dof}}}$ the nodal displacement, $\mathit{P}\in {\mathbb{R}}^{n^{\mathrm{dof}}}$ the external forces and $n^{\mathrm{dof}}$ the number of degrees of freedom. Actuation is implemented through the action of “force-serial” liner actuators (Böhm et al., 2020) that are placed in the truss elements to modify the internal forces and reduce displacements through controlled length changes. Force-serial refers to the housing element and actuator taking the same force since they are in series as shown in Figure 1. The external loads are distinguished into control ${\mathit{B}}^u\mathit{u}$ and disturbance forces ${\mathit{B}}^z\mathit{z}$

$$\mathit{P}={\mathit{B}}^u\mathit{u}+{\mathit{B}}^z\mathit{z}$$

(2)

Figure 1.

Uncontrolled state: (a) displacements, (b) internal forces; all elements have identical cross-section $\mathrm{\alpha }$ .

The control input matrix ${\mathit{B}}^u\in {\mathbb{R}}^{n^{\mathrm{dof}}\times n^{\mathrm{el}}}$ maps the control commands $\mathit{u}\in {\mathbb{R}}^{n^{\mathrm{el}}}$ , in this case the actuator length changes $\mathbf{\Delta }\mathit{L}\in {\mathbb{R}}^{n^{\mathrm{el}}}$ , onto equivalent external loads. Assuming there are $n^{\mathrm{act}}<n^{\mathrm{el}}$ actuators, $\mathbf{\Delta }\mathit{L}$ is non-zero only for the entries corresponding to the active elements. The disturbance matrix (or primary input matrix) ${\mathit{B}}^z\in {\mathbb{R}}^{n^{\mathrm{dof}}\times n^{\mathrm{dof}}}$ models external disturbances $\mathit{z}\in {\mathbb{R}}^{n^{\mathrm{dof}}}$ (e.g. external loads, temperature changes). The actuation force is in series with the force taken by the housing element, and thus the stiffness of the coupled system is

$$\frac{1}{k}=\frac{1}{k^{\mathrm{el}}}+\frac{1}{k^{\mathrm{act}}}$$

(3)

For simplicity, since the actuator stiffness $k^{\mathrm{act}}$ is typically much greater than the element stiffness $k^{\mathrm{el}}$ , the assembly (housing element + actuator) stiffness can be approximated as

$$k=k^{\mathrm{el}}=\frac{E\alpha }{l_{\mathrm{eff}}}$$

(4)

For short truss elements and large cross-sections, the actuator stiffness might affect the serial stiffness and therefore should be considered.

Consider the truss structure in Figure 1, which is equipped with one actuator to reduce the horizontal displacement response under load P , as shown in Figure 2. The constitutive law of the element-actuator assembly is

$$\mathrm{\Delta }{l}_i=\mathrm{\Delta }{L}_i+\frac{F_i}{k_i^{\mathrm{el}}}$$

(5)

Figure 2.

Controlled state with complete compensation of horizontal displacement $\mathrm{\Delta }d=-d_0$ : (a) displacements, (b) internal forces; all elements have identical cross-sections $\mathrm{\alpha }$ .

$\mathrm{\Delta }{l}_i$ is the total deformation which is the sum of the actuator length change $\mathrm{\Delta }{L}_i$ and the elastic extension (or contraction) caused by the element axial force $F_i$ , $k_i^{\mathrm{el}}$ is the element stiffness. For clarity, the actuator length change $\mathrm{\Delta }{L}_i$ is inelastic. The total extension of the element i can be computed from the nodal displacements

$$\mathrm{\Delta }{l}_i={\mathit{A}}_i^T\mathit{d}$$

(6)

where $\mathit{d}\in {\mathbb{R}}^{n^{\mathrm{dof}}}$ is the vector of nodal displacements and $\mathit{A}\in {\mathbb{R}}^{n^{\mathrm{dof}}\times n^{\mathrm{el}}}$ is the equilibrium matrix containing the element direction cosines. The virtual work of the conservative external forces $\mathit{P}$ is

$$\delta W_{\mathrm{ext}}={\mathit{P}}^T\delta \mathit{d}$$

(7)

The virtual work due to internal forces $\mathit{F}\in {\mathbb{R}}^{n^{\mathrm{el}}}$ is

$$\delta W_{\mathrm{int}}={\mathit{F}}^T\delta \mathbf{\Delta }\mathit{l}$$

(8)

Expressing $\mathbf{\Delta }l$ from equation (6), that is compatibility between nodal displacements and element deformations, and applying virtual work, the system equilibrium is

$$\delta W={\mathit{F}}^T{\mathit{A}}^T\delta \mathit{d}={\mathit{P}}^T\delta \mathit{d}$$

(9)

Expressing $\mathit{F}$ from equation (5) as a function of the elastic deformation and replacing in equation (9), the virtual work of the internal forces can be rewritten as

$$\delta W={\left((\mathbf{\Delta }\mathit{l}-\mathbf{\Delta }\mathit{L}){\mathit{K}}^{\mathrm{el}}\right)}^T{\mathit{A}}^T\delta \mathit{d}$$

$$\delta W={\left(\mathbf{\Delta }\mathit{l}{\mathit{K}}^{\mathrm{el}}\right)}^T{\mathit{A}}^T\mathbf{\delta }\mathit{d}+{(-{\mathbf{\Delta }\mathit{L}\mathbf{K}}^{\mathbf{el}})}^{\mathbf{T}}{\mathit{A}}^{\mathbf{T}}\delta d$$

$$\delta W=(\mathit{A}{\mathit{K}}^{\mathrm{el}}\mathbf{\Delta }\mathit{l}-\mathit{A}{\mathit{K}}^{\mathrm{el}}\mathbf{\Delta }\mathit{L})\delta \mathit{d}$$

(10)

where ${\mathit{K}}^{\mathrm{el}}\in {\mathbb{R}}^{n^{\mathrm{el}}\times n^{\mathrm{el}}}$ is an unconnected stiffness matrix (diagonal for pin-jointed structures) containing the elements’ axial stiffness. Isolating the contribution due to the inelastic length change $\mathbf{\Delta }\mathit{L}$ , the equivalent actuation load is considered an external load and is expressed as

$${\mathit{P}}^{\mathrm{act}}=\mathit{A}{\mathit{K}}^{\mathrm{el}}\mathbf{\Delta }\mathit{L}$$

(11)

The equivalent actuation load consists of a pair of self-equilibrated forces applied by the active element to the connecting nodes, producing an inelastic displacement $\mathbf{\Delta }\mathit{L}$ equivalent to a thermal deformation or a lack of fit (Senatore et al., 2019). For clarity, the equivalent actuation load is not the required control force, but a fictitious external load that is convenient for modeling the effect of actuator length changes (inelastic).

The controlled system is described as

$$\mathit{K}\mathit{d}={\mathit{B}}^z\mathit{z}+\mathit{A}{\mathit{K}}^{\mathrm{el}}\mathbf{\Delta }\mathit{L}$$

(12)

The control input matrix can thus be written as

$${\mathit{B}}^u=\mathit{A}{\mathit{K}}^{\mathrm{el}}$$

(13)

The change of displacement caused by $\mathbf{\Delta }\mathit{L}$ is

$${\mathit{d}}^{\mathrm{act}}={\mathit{K}}^{-1}\mathit{A}{\mathit{K}}^{\mathrm{el}}\mathbf{\Delta }\mathit{L}$$

(14)

The internal element force change $\mathbf{\Delta }F\in {\mathbb{R}}^{n^{\mathrm{el}}}$ caused by actuation is computed from equations (5) and (6) as

$$\mathbf{\Delta }{F}^{\mathrm{act}}={\mathit{K}}^{\mathrm{el}}({\mathit{A}}^T{\mathit{d}}^{\mathrm{act}}-\mathbf{\Delta }\mathit{L})$$

(15)

In equation (13), the element force is obtained by subtracting from the total deformation ${\mathit{A}}^T{\mathit{d}}^{\mathrm{act}}$ the inelastic contribution caused by the actuator length change $\mathbf{\Delta }\mathit{L}$ . For clarity, since in this formulation, the actuator inputs are always length changes, the notation u is replaced with $\mathbf{\Delta }\mathit{L}$ hereafter.

Figures 1(b) and 2(b) show the variation of the internal forces for a negative length change of the actuator positioned in the diagonal element #3 to achieve a complete compensation of the horizontal displacement in the upper nodes (see Figure 2(a)). The effect of actuation to reduce the displacement response can be typically controlled to obtain an overall homogenization by increasing the utilization of elements that are not critically stressed.

For simplicity to achieve a formulation that can produce global optima, the effect of geometric nonlinear has not been considered. The assumption of small strains and displacements has been experimentally validated on several large-scale prototype structures including a 6m span simply supported bridge (Senatore et al., 2017) and a 36 m tall 12-story adaptive tower (Dakova et al., 2024). Geometric non-linearity was accounted for a new method to design structures that adapt to loading through large shape reconfigurations, that is, shape morphing (Reksowardojo et al., 2022; Reksowardojo and Senatore, 2023), which is not considered in this work.

Dynamic effects are only considered in the definition of static-equivalent loading conditions since the focus is on determining effective positions to place linear actuators and not on closed-loop dynamics. For slender and lightweight structures with low vibration periods (T₁ > 1 s) and with the principal vibration mode shapes similar to the deformed shape under static loading, controlling the steady-state response is usually the dominant design requirement for the control system. This is the case for slender cantilever structures under lateral loads as the one considered in the case studies in Section 4.3.3. The actuator performance in compensating for static displacements is representative of the upper bound (i.e. largest amplitude) of the dynamic response. In the case of seismic design, equivalent static loads can be determined from the response spectrum, which, for structures meeting regularity criteria, for example, simple geometry with uniform mass and stiffness distribution, is adequate to model the effect of seismic loads (European Committee for Standardization (CEN), 2005).

3. Constrained optimal control

Several strategies have been proposed to achieve optimal response control under static load cases. A formulation proposed in (Wagner et al., 2018), uses a Moore-Penrose pseudo-inverse matrix (Penrose, 1955) to determine the control inputs that yield the minimum L ₂ -norm of the displacements for the controlled nodes. In (Böhm et al., 2019) the same approach is proposed to achieve stress homogenization in structural elements. However, these methods do not account for specific load cases nor structural and actuation constraints. In (Senatore and Reksowardojo, 2020), control strategies are formulated based on constrained minimization to obtain optimal control inputs for different targets: (C1) simultaneous displacement and element force control to satisfy strength and serviceability limits; (C2) displacement control without causing a change in element forces (Impotent eigenstrain); (C3) element force without causing a change in displacements (Nilpotent eigenstrain); (C4) displacement control through actuation energy minimization subject to stress and stability constraints.

Following (Senatore and Reksowardojo, 2020), in this work, the control problem is formulated through the minimization of the discrepancy between a correction target to satisfy serviceability limits $\mathbf{\Delta }{\mathit{d}}_j^{\mathrm{SLS}}\in {\mathbb{R}}^{n^{\mathrm{cdof}}}$ and the compensation achieved through actuation ${\mathit{S}}^d\mathbf{\Delta }\mathit{L}$ subject to constraints on the element forces and actuation length changes. The optimal control problem is

$$\begin{array}{cc}\begin{array}{c}\underset{\mathrm{\Delta }\mathbf{L}}{min}\Vert {\mathbf{S}}^d\mathrm{\Delta }{\mathbf{L}}_j-\mathbf{\Delta }{\mathbf{d}}_j^{\mathrm{SLS}}{\Vert }_2\hfill \\ \multicolumn{1}{c}{s.t.}\\ \multicolumn{1}{c}{\underset{¯}{\mathbf{F}}<{\mathbf{F}}_j^0+{\mathbf{S}}^F\mathrm{\Delta }{\mathbf{L}}_j<\overline{\mathbf{F}}}\\ \multicolumn{1}{c}{\underset{¯}{\mathrm{\Delta }\mathbf{L}}\le \mathrm{\Delta }{\mathbf{L}}_j\le \overline{\mathrm{\Delta }\mathbf{L}}}\end{array}\hfill & \forall j\in [1,n^{\mathrm{LC}}]\hfill \end{array}$$

(16)

where ${\mathit{S}}^d\in {\mathbb{R}}^{n^{\mathrm{cdof}}\times n^{\mathrm{act}}}$ and ${\mathit{S}}^F\in {\mathbb{R}}^{n^{\mathrm{el}}\times n^{\mathrm{act}}}$ are the displacement and force actuation sensitivity (i.e. influence) matrices. In this case, $\mathbf{\Delta }\mathit{L}\in {\mathbb{R}}^{n^{\mathrm{act}}}$ is reduced to the entries corresponding to the active elements. ${\mathit{F}}_j^0\in {\mathbb{R}}^{n^{\mathrm{el}}}$ is the internal force in the uncontrolled state. The lower and upper force limits $\{\overline{\mathit{F}},\overline{\mathit{F}}\}$ are determined so that the element stress $\sigma$ is kept below resistance ${\mathbf{\sigma }}_{\mathit{rd}}\in {\mathbb{R}}^{n^{\mathrm{el}}\times 1}$ and buckling ${\sigma }_{\mathrm{blk}}\in {\mathbb{R}}^{n^{\mathrm{el}}}$ limits. For the active elements, the internal force must be lower than the maximum actuator force capacity $\{{\overline{\mathit{F}}}^{\mathrm{act}},{\overline{\mathit{F}}}^{\mathrm{act}}\}$ (i.e. the maximum force that an actuator can apply)

$${\overline{\mathit{\sigma }}}_{\mathrm{rd}}\mathit{\alpha }\le \mathit{F}\le {\overline{\mathbf{\sigma }}}_{\mathbf{rd}}\mathit{\alpha }$$

$${\overline{\mathit{\sigma }}}_{\mathrm{blk}}\mathit{\alpha }\le \mathit{F}$$

$${\overline{\mathit{F}}}^{\mathrm{act}}<{\mathit{F}}^{\mathrm{act}}<{\overline{\mathit{F}}}^{\mathrm{act}}$$

(17)

where $\mathit{\alpha }\in {\mathbb{R}}^{{\mathbf{n}}^{\mathbf{el}}}$ is the cross-section area. $\{\underset{¯}{\mathbf{\Delta }\mathit{L}},\overline{\mathbf{\Delta }\mathit{L}}\}$ are the lower and upper actuator length change limits. The actuation sensitivity matrices collate the effect on nodal displacements ${\mathit{S}}^d$ and element forces ${\mathit{S}}^F$ of a unitary length change of each element in turn. In (Senatore and Reksowardojo, 2020) the derivation of the actuation sensitivity matrices is formulated through the Integrated Force Method. Alternatively, with the Direct Stiffness Method using equations (14) and (15) and setting $\mathbf{\Delta }\mathit{L}$ to an identity matrix, they can be expressed as

$${\mathit{S}}^d={\mathit{K}}^{-1}\mathit{A}{\mathit{K}}^{\mathrm{act}}$$

$${\mathit{S}}^F={\mathit{K}}^{\mathrm{el}}{\mathit{A}}^T{\mathit{S}}^d-{\mathit{K}}^{\mathrm{act}}$$

(18)

where ${\mathit{K}}^{\mathrm{act}}\in {\mathbb{R}}^{n^{\mathrm{el}}\times n^{\mathrm{act}}}$ is the unconnected element stiffness matrix reduced column-wise to the entries corresponding to the active elements. ${\mathit{S}}^d$ is usually premultiplied by an output matrix $\mathit{C}\in {\mathbb{R}}^{n^{\mathrm{cdof}}\times n^{\mathrm{dof}}}$ to select the controlled degrees of freedom.

4. Actuator layout optimization

The determination of an efficient actuator set is typically a combinatorial optimization problem that involves selecting a subset from a finite set of possible placements. When all elements are candidate sites for actuators, the number of placements increases exponentially with the number of structural elements

$$\sum _{n^{\mathrm{act}}=1}^{n^{\mathrm{el}}}\left(\frac{n^{\mathrm{el}}}{n^{\mathrm{act}}!(n^{\mathrm{el}}-n^{\mathrm{act}})!}\right)\simeq 2^{n^{\mathrm{el}}}$$

(19)

Computation of all possible sets is not feasible, optimal solutions have been obtained through methods involving integer relaxation (Senatore et al., 2019) through efficacy ranking, mixed-integer programming formulations (Senatore and Wang, 2024; Wang and Senatore, 2021), and metaheuristics (Furuya and Haftka, 1995; Zeller et al., 2022).

Minimizing the number of actuators in control system design is important due to cost efficiency, control strategy simplicity, reduced risks of failure, and low maintenance and installation efforts. On the other hand, employing a higher number of actuators allows smoother and more precise control over a wider range of load events. Similarly, limiting the operational energy of the system is important to ensure control feasibility, reduce operational costs, enhance system reliability, and reduce adverse environmental impacts. These two objective functions are inversely correlated: reducing the number of actuators generally requires more energy for the system to satisfy the control targets.

4.1. Performance metrics

Given a truss structure, the assignment of an actuator to an element position $i$ is expressed through a binary vector denoted as $\mathbb{A}\mathbb{C}\mathbb{T}\in {\{0,1\}}^{n^{\mathrm{el}}}$

$$\mathbb{A}\mathbb{C}{\mathbb{T}}_i=\{\begin{array}{c}1\mathrm{if}\mathrm{actuator}\mathrm{assigned}\mathrm{to}\mathrm{position}\mathrm{i}\hfill \\ 0\mathrm{if}\mathrm{actuator}\mathrm{is}\mathrm{not}\mathrm{assigned}\mathrm{to}\mathrm{position}\mathrm{i}\hfill \end{array}$$

(20)

If the i^th entry of $\mathbb{A}\mathbb{C}\mathbb{T}$ is 1, the i^th element houses an actuator and 0 otherwise. To evaluate the optimality of an actuator placement for structural control several metrics have been proposed among with the number of actuators $n^{\mathrm{act}}$ required to satisfy a control target. It is preferable to keep the number of actuators as low as possible due to practical requirements that include cost-effectiveness, reliability, and control system simplicity. Limits on the actuation forces are also important to ensure implementation feasibility concerning actuation capacity and power density. When selecting an actuator layout, it is important to exclude those locations where very high forces are required. Additionally, assessing the maximum actuation force required to control the response is a useful parameter to evaluate control feasibility.

The actuation energy for control is a key metric for adaptive structures (Reksowardojo et al., 2022; Senatore et al., 2011). Assuming a linear elastic force-displacement relationship and small deformations, an estimate for the actuation energy (or control energy) $E_{\mathrm{ij}}^{\mathrm{act}}$ of actuator i under load case j is

$$E_{\mathrm{ij}}^{\mathrm{act}}=F_{\mathrm{ij}}^0\mathrm{\Delta }{L}_{\mathrm{ij}}+\frac{1}{2}\mathrm{\Delta }{F}_{\mathrm{ij}}\mathrm{\Delta }{L}_{\mathrm{ij}},\forall i\in \{1,2,..n^{\mathrm{act}}\}$$

(21)

where ${\mathit{F}}^0$ are the element forces in the uncontrolled state (i.e. before actuation). It is assumed that the external load is applied first, the response is monitored and then controlled via actuation. Figure 3 gives an illustration of the control energy for a single actuator. To compute a conservative estimate, all potential energy gains are neglected. Energy gains might occur when the actuator force and length change have the same sign, for example, when performing an extension (positive) under a tension force (positive) or contraction (negative) under a compression force (negative).

Figure 3.

Control energy $E_{\mathrm{ij}}^{\mathrm{act}}$ for a force-serial linear actuator (Senatore et al., 2019).

The energy parameter $E^{\mathrm{act}}{\left(\mathbb{A}\mathbb{C}\mathbb{T}\right)}_j$ computed for the entire actuator set can be normalized with respect to an upper bound ${\overline{E}}^{\mathrm{act}}$ and lower bound ${\overline{E}}_j^{\mathrm{act}}$ . The normalized value ${E^{\mathrm{act}||}}_j{\left(\mathbb{A}\mathbb{C}\mathbb{T}\right)}_j$ of the operational energy for a single load case j is expressed as

$$E^{\mathrm{act}||}{\left(\mathbb{A}\mathbb{C}\mathbb{T}\right)}_j=\frac{E^{\mathrm{act}}{\left(\mathbb{A}\mathbb{C}\mathbb{T}\right)}_j-{\overline{E}}_j^{\mathrm{act}}}{{\overline{E}}^{\mathrm{act}}-{\overline{E}}_j^{\mathrm{act}}}$$

(22)

The operational energy lower bound ${\overline{E}}_j^{\mathrm{act}}$ for a specific load case can be determined considering the following quadratic minimization problem

$$\begin{array}{c}\underset{\mathrm{\Delta }\mathbf{L}}{min}\Vert {\underset{¯}{E}}_j^{\mathrm{act}}{\Vert }_2\hfill \\ s.t.\hfill \\ {\mathbf{S}}^d\mathrm{\Delta }{\mathbf{L}}_j-\mathbf{\Delta }{\mathbf{d}}_j^{\mathrm{SLS}}=0\forall j\in [1,n^{\mathrm{LC}}]\hfill \\ \underset{¯}{\mathbf{F}}<{\mathbf{F}}_j^0+{\mathbf{S}}^F\mathrm{\Delta }{\mathbf{L}}_j<\overline{\mathbf{F}}\hfill \\ \underset{¯}{\mathrm{\Delta }\mathbf{L}}\le \mathrm{\Delta }{\mathbf{L}}_j\le \overline{\mathrm{\Delta }\mathbf{L}}\hfill \end{array}$$

(23)

In equation (23) all elements are considered active, hence no placement integer variable is necessary. The only variable is the control command vector $\mathbf{\Delta }\mathit{L}$ . Compared to the problem stated in equation (16), in equation (23) the control targets are equality constraints while the objective function is the minimization of the control energy. Note that a layout where all elements are active requires the minimum control energy since the controller has no limitations in choosing which actuator to operate within the given constraints. Considering a specific control system and power plant, the control energy upper bound ${\overline{E}}^{\mathrm{act}}$ can be set as the product of the maximum available power by the period of the first vibration mode $T$ . The work frequency of the actuators is assumed to be equal to the structure natural frequency, which is likely to dominate the response for most building scenarios. ${\overline{E}}^{\mathrm{act}}$ can be considered as a control energy threshold above which the system will not be able to operate. The energy parameter ${E^{\mathrm{act}||}}_j$ takes value in the range [0,1]. A value of ${E^{\mathrm{act}||}}_j$ greater than one indicates an unfeasible solution. For multiple load cases, the global operational energy parameter $E^{||}$ is given by

$$E^{\mathrm{act}||}\left(\mathbb{A}\mathbb{C}\mathbb{T}\right)=\frac{\sum _{j=1}^{\mathrm{LC}}{E}^{\mathrm{act}||}{\left(\mathbb{A}\mathbb{C}\mathbb{T}\right)}_j}{n^{\mathrm{LC}}}$$

(24)

Two metrics are adopted to assess the ability of a selected actuator set to satisfy the target control outputs: one is based on the compensability Gramian formulation in (Wagner et al., 2018), and the other on an efficacy measure formulated in (Senatore et al., 2019). The compensability quantifies the ability of a set of actuators to compensate for the effects of a generic disturbance on displacement or internal forces. Denote ${\mathit{d}}_j^c\in {\mathbb{R}}^{n^{\mathrm{cdof}}}$ the system output vector (i.e. controlled displacements) for the controlled response under load case j, the output error ${\mathit{e}}_j\in {\mathbb{R}}^{n^{\mathrm{cdof}}}$ . is given with respect to the serviceability limit state $d_j^{\mathrm{SLS}}\in {\mathbb{R}}^{n^{\mathrm{cdof}}}$ , which is the target output

$${\mathit{e}}_j=({\mathit{d}}_j^{\mathrm{SLS}}-{\mathit{d}}_j^c)$$

(25)

The compensability Gramian ${\tilde{\mathit{W}}}_j\in {\mathbb{R}}^{n^{\mathrm{cdof}}\times n^{\mathrm{cdof}}}$ for load case $j$ is obtained as a symmetric matrix of inner products $〈e_{\mathrm{ji}}{e}_{\mathrm{jk}}^T〉\forall i,k$

$${\mathit{e}}_j{\mathit{e}}_j^T={\tilde{\mathit{W}}}_j$$

(26)

${\tilde{\mathit{W}}}_j$ quantifies the efficacy of an actuator set to achieve minimum output error. The trace of ${\tilde{\mathit{W}}}_j$ , denoted as $J^{||}{\left(\mathbb{A}\mathbb{C}\mathbb{T}\right)}_j=\mathrm{tr}{\left({\tilde{\mathit{W}}}_j\right)}_{\mathbb{A}\mathbb{C}\mathbb{T}}$ , is proportional to the average squared output error norm and can be employed as a measure of response compensability. The compensability metric is normalized with respect to the uncontrolled system $J{\left(0\right)}_j$

$$J^{||}{\left(\mathbb{A}\mathbb{C}\mathbb{T}\right)}_j=\frac{J{\left(0\right)}_j-J{\left(\mathbb{A}\mathbb{C}\mathbb{T}\right)}_j}{J{\left(0\right)}_j}$$

(27)

$J^{||}{\left(\mathbb{A}\mathbb{C}\mathbb{T}\right)}_j$ takes value in the range [0,1], $J^{||}{\left(\mathbb{A}\mathbb{C}\mathbb{T}\right)}_j$  = 1 when the control error is zero. $J^{||}{\left(\mathbb{A}\mathbb{C}\mathbb{T}\right)}_j$ is further averaged over the number of load cases $n^{\mathrm{LC}}$

$$J^{||}\left(\mathbb{A}\mathbb{C}\mathbb{T}\right)=\frac{\sum _j^{n^{\mathrm{LC}}}{J}^{||}{\left(\mathbb{A}\mathbb{C}\mathbb{T}\right)}_j}{n^{\mathrm{LC}}}$$

(28)

The efficacy of an actuator set $\mathrm{eff}{\left(\mathbb{A}\mathbb{C}\mathbb{T}\right)}_j$ is quantified by evaluating the contribution towards a target displacement and force compensation for a given control input $\mathbf{\Delta }{\mathit{L}}_{\mathbf{j}}$ under load case $j$ (Senatore et al., 2019). When the control objective is displacement compensation, the efficacy is the ratio between the displacement change $\mathbf{\Delta }{\mathit{d}}_{\mathbf{j}}$ caused by $\mathbf{\Delta }{\mathit{L}}_{\mathbf{j}}$ and the desired output correction $\mathbf{\Delta }{\mathit{d}}_j^{\mathrm{SLS}}$

$$\mathrm{eff}{\left(\mathbb{A}\mathbb{C}\mathbb{T}\right)}_j=\frac{\sum (\left({\mathit{S}}^d\mathrm{\Delta }{L}_j\right)\odot \frac{1}{{\mathbf{\Delta }d}_j^{\mathrm{SLS}}})}{n^{\mathrm{cdof}}}$$

(29)

Alternatively, if the control objective is force compensation

$$\mathrm{eff}{\left(\mathbb{A}\mathbb{C}\mathbb{T}\right)}_j=\frac{\sum (\left({\mathit{S}}^F\mathrm{\Delta }{L}_j\right)\odot \frac{1}{\mathbf{\Delta }{\mathit{F}}_j})}{n^{\mathrm{el}}}$$

(30)

The symbol ⊙ denotes the Hadamard product (i.e. elementwise). The actuator set efficacy is then averaged for all load cases

$$\mathrm{Eff}\left(\mathbb{A}\mathbb{C}\mathbb{T}\right)=\frac{{\sum }_{j=1}^{\mathrm{LC}}\mathrm{eff}{\left(\mathbb{A}\mathbb{C}\mathbb{T}\right)}_j}{n^{\mathrm{LC}}}$$

(31)

The efficacy metric provides an average relative error for the controlled displacements, while the Compensability Gramian trace correlates with the average squared output error norm. This implies that using the efficacy metric will penalize even minor violations of the control targets compared to the Gramian-based metric, which instead gives a bias to absolute target errors, such as large displacements at the free end of a cantilever compared to those at the constrained end. In the context of the application of this metric for actuation placement, it is assumed that the uncontrolled state always violates serviceability conditions and that the controlled state should not overshoot the target displacements. This typically guarantees a smoother shape in the controlled state.

4.2. Sequential actuator placement formulation

Most previous methods (e.g. Senatore et al., 2019; Wagner et al., 2018; Zeller et al., 2022) have adopted sequential approaches to formulate the actuator placement problem, which can be categorized under Stackelberg Game optimization (Ali et al., 2015; Marker and Bleicher, 2022; Wagner et al., 2018; Zeller et al., 2022). This consists of a sequential or bi-level process with two players that are referred to as the leader and follower. The process is represented diagrammatically in the flowchart in Figure 4. The same approach can also be thought of as a nested optimization. In the outer or “leader” the actuation binary vector is the optimization variable. A candidate actuator placement is therefore set in the lead process. Subsequently, in the nested or “follower” process, optimal control inputs are obtained for each candidate actuator layout. The process repeats iteratively to minimize a cost function. The cost functions considered in this work are the minimization of the actuator number and control energy. Depending on the formulation, these can be minimized directly (e.g. global search) or indirectly (local search). The optimal control inputs are determined by solving the constrained optimization problem stated in equation (16).

Figure 4.

Sequential formulation flowchart.

4.2.1. Local search through integer relaxation (IR) and efficacy measure $\mathrm{Eff}\left(\mathbb{A}\mathbb{C}\mathbb{T}\right)$

An actuator placement method based on the efficacy measure outlined in Section 4.1 and constrained optimization is given in (Senatore et al., 2019). The number of actuators $n^{\mathrm{act}}$ is set as the number of controlled degrees of freedom $n^{\mathrm{cdof}}$ plus the static indeterminacy $r$

$$n^{\mathrm{act}}=n^{\mathrm{cdof}}+r$$

(32)

This is the number of releases required to transform a statically indeterminate structure into a controlled mechanism (Senatore et al., 2019). However, a lower number of actuators can often be employed to satisfy displacement and stress limits, as also shown in (Senatore and Wang, 2024; Wang and Senatore, 2020). The actuator placement is performed by relaxing the integrality condition of the assignment variable adopting a sequential approach.

First, all elements are considered active, that is, $\mathbf{\Delta }\mathit{L}\in {\mathbb{R}}^{{\mathbf{n}}^{\mathbf{el}}}$ . The actuator commands are determined by minimizing the difference between the uncontrolled state and a target displacement state $\mathbf{\Delta }{\mathit{d}}_{\mathbf{j}}^{\mathbf{SLS}}$ subject to a force redirection that satisfies stress and buckling limits, which is the problem formulated in equation (16). Subsequently, the efficacy $\mathrm{Eff}\left(\mathbb{A}\mathbb{C}\mathbb{T}\right)$ of each actuator ${\mathrm{eff}}_{\mathrm{ij}}^{\mathrm{\Delta }L}$ is quantified by evaluating the effect of the corresponding $\mathrm{\Delta }{L}_{\mathrm{ij}}$ using equations (29) and (30). Note that in this case, since the actuator layout is not known, the actuator command vector $\mathbf{\Delta }\mathit{L}$ contains only the i^th value corresponding to the position of the actuator under evaluation, and all other elements are set to zero (Senatore et al., 2019). The objective function (minimizing the number of actuators) is considered indirectly by selecting the $n^{\mathrm{act}}$ elements with the highest efficacy.

4.2.2. Local search through Greedy selection and compensability measure $J^{||}\left(\mathbb{A}\mathbb{C}\mathbb{T}\right)$

Following (Wagner et al., 2018), a Greedy algorithm is implemented to solve the actuator placement problem. The performance metric adopted for the greedy search is the normalized compensability $J^{||}\left(\mathbb{A}\mathbb{C}\mathbb{T}\right)$ . Starting from a set that includes all elements (i.e. all elements are considered active), $J^{||}\left(\mathbb{A}\mathbb{C}\mathbb{T}\right)$ is computed iteratively by removing each actuator in turn $s=\{1...n^{\mathrm{el}}-1\}$ . The actuator removal that causes the least deterioration is evaluated, and the corresponding actuator index is removed from the set. The Greedy algorithm is also implemented in the reverse direction, that is, starting from an empty set and adding sequentially the actuator that gives the greatest improvement to $J^{||}\left(\mathbb{A}\mathbb{C}\mathbb{T}\right)$ . The objective function (e.g. minimizing the number of actuators) is considered indirectly by choosing the actuator layout that satisfies the target output using the minimum number of actuators.

Greedy algorithms, although simple to implement, typically produce sub-optimal solutions because the overall problem structure is not considered and there is no means of backtracking. When starting from a configuration where all elements are considered active, early-stage selection might not be effective because the performance metric $J^{||}(\mathbb{A}\mathbb{C}\mathbb{T})$ changes marginally between adjacent actuator placements that differ by only 1 element. This might result in low-quality solutions.

4.2.3. Global search

A critical limitation of approaches based on local search is that the actuator positions and the number of actuators are procedural variables that are determined sequentially. This causes a disconnection between the objective function minimization and constraint satisfaction. For example, solutions with a low number of actuators might be obtained with a significant deterioration of the constraint residuals. To overcome these limitations, global search approaches are implemented which provide a more robust strategy by exploring the search space more extensively. With these strategies, it is possible to consider the actuator positions as explicit design variables, which enables the incorporation of the number of actuators in the objective function as in equation (33). The objective also includes the compensability metric to obtain actuator placements that can satisfy the control target with small residuals.

$$\varphi \left(\mathbb{A}\mathbb{C}\mathbb{T}\right)=\nu \frac{n^{\mathrm{act}}}{n^{\mathrm{el}}}+\eta (1-J^{||}\left(\mathbb{A}\mathbb{C}\mathbb{T}\right)),\nu +\eta =1$$

(33)

Although the search is global, the optimization involves a nested process (the follower) to evaluate constraint satisfaction, the output of which must be considered in the objective function to steer the search towards feasible regions. Two methods are implemented for global search: a Genetic Algorithm (GA) and a Surrogate Algorithm (SA). In both cases, the objective function is given in equation (33) while the actuator commands are determined by solving the problem stated in equation (16). The GA employed in this work is the algorithm implemented in Matlab (Chipperfield and Fleming, 1995). Regarding SA, the objective function in equation (33) is evaluated for a set of candidate actuator layouts (i.e. sample points), which are used to make a surrogate through radial basis function (RBF) interpolation. The minimum of the surrogate function is employed as a new sample point to improve the interpolation. When the distance of the new minimum is within a certain tolerance to a previously obtained minimum, it means that the surrogate approximation is accurate, and a local minimum has been found. Subsequently, the search starts again using new sample points that are generated to explore a different region of the solution space. The surrogate model is employed to drive the search for the global optimum, allowing for efficient exploration of the solution space and identification of good regions where high-quality solutions are likely to be obtained (Mueller, 2014; Wang and Shoemaker, 2014).

4.3. All-In-One (AIO) actuator placement formulation

An All-In-One (AIO) formulation refers to solving a multi-variate and/or multi-objective optimization problem by obtaining all variables simultaneously. This typically requires an in-depth knowledge of the problem and a certain creativity in modeling all necessary equations simultaneously within a single problem statement. AIO formulations typically produce high-quality solutions because they can model interdependencies between variables such as the case of actuator positions and control commands.

4.3.1. Actuator number minimization

In this work, since the aim is to compare actuator placement methods, the structural parameters are pre-determined through a separate optimization process. The actuator placement problem is stated as

$$\underset{\mathbb{A}\mathbb{C}\mathbb{T},\mathbf{\Delta }\mathit{L},{\mathit{F}}^{\mathit{act}}}{\min }\sum \mathbb{A}\mathbb{C}{\mathbb{T}}_i$$

$$s.t.$$

$$\mathrm{\Delta }{\overline{\mathit{d}}}^{\mathrm{SLS}}\le {\mathit{S}}^d\mathbf{\Delta }\mathit{L}\le \mathbf{\Delta }{\overline{\mathit{d}}}^{\mathbf{SLS}}$$

$$\mathbb{A}\mathbb{C}\mathbb{T}\underset{¯}{\mathbf{\Delta }\mathit{L}}\le \mathbf{\Delta }\mathit{L}\le \mathbb{A}\mathbb{C}\mathbb{T}\overline{\mathbf{\Delta }\mathit{L}}$$

$$(1-\mathbb{A}\mathbb{C}\mathbb{T})\overline{\mathit{F}}\le {\mathit{F}}^0+{\mathit{S}}^F\mathbf{\Delta }\mathit{L}-{\mathit{F}}^{\mathbf{act}}\le (1-\mathbb{A}\mathbb{C}\mathbb{T})\overline{\mathit{F}}$$

$$\mathbb{A}\mathbb{C}\mathbb{T}{\underset{¯}{\mathit{F}}}^{\mathrm{act}}\le {\mathit{F}}^{\mathrm{act}}\le \mathbb{A}\mathbb{C}\mathbb{T}{\overline{\mathit{F}}}^{\mathrm{act}}$$

$${\underset{¯}{\mathit{F}}}^{\mathrm{act}}\le \mathbb{A}\mathbb{C}\mathbb{T}{\mathit{F}}_0\le {\overline{\mathit{F}}}^{\mathrm{act}}$$

(34)

The optimization objective is the minimization of the number of actuators

$$\underset{\mathbb{A}\mathbb{C}\mathbb{T},\mathbf{\Delta }\mathit{L},{\mathit{F}}^{\mathbf{act}}}{\min }\sum \mathbb{A}\mathbb{C}{\mathbb{T}}_i$$

(35)

The design variable is the actuator assignment $\mathbb{A}\mathbb{C}\mathbb{T}$ while state variables are the actuator length changes $\mathbf{\Delta }\mathit{L}$ and forces ${\mathit{F}}^{\mathrm{act}}$ . Displacement limits are enforced through the inequality constraints:

$$\mathbf{\Delta }{\underset{¯}{\mathit{d}}}^{\mathrm{SLS}}\le {\mathit{S}}^d\mathbf{\Delta }\mathit{L}\le \mathbf{\Delta }{\overline{\mathit{d}}}^{\mathbf{SLS}}$$

(36)

where $\mathbf{\Delta }{\underset{¯}{\mathit{d}}}^{\mathrm{SLS}}$ and $\mathbf{\Delta }{\overline{\mathit{d}}}^{\mathrm{SLS}}$ are lower and upper bounds of the target displacement corrections. The actuator inputs $\mathbf{\Delta }\mathit{L}$ are limited by lower and upper bounds { $\underset{¯}{\mathbf{\Delta }\mathit{L}}$ , $\overline{\mathbf{\Delta }\mathit{L}}$ } and set to zero for elements that do not house an actuator (i.e. $\mathbb{A}\mathbb{C}{\mathbb{T}}_i=0$ ) using the assignment variable

$$\mathbb{A}\mathbb{C}\mathbb{T}\underset{¯}{\mathbf{\Delta }\mathit{L}}\le \mathbf{\Delta }\mathit{L}\le \mathbb{A}\mathbb{C}\mathbb{T}\overline{\mathbf{\Delta }\mathit{L}}$$

(37)

The actuator force variable ${\mathit{F}}^{\mathrm{act}}$ is employed to set appropriate limits for the element housing actuators and those that do not. The element force in the controlled state, which is expressed as a function of the actuator command ${\mathit{F}}^0+{\mathit{S}}^F\mathbf{\Delta }\mathit{L}$ , is correlated to the actuator force as

$$(1-\mathbb{A}\mathbb{C}\mathbb{T})\mathit{F}\le {\overline{\mathit{F}}}^0+{\mathit{S}}^F\mathbf{\Delta }\mathit{L}-{\mathit{F}}^{\mathbf{act}}\le (1-\mathbb{A}\mathbb{C}\mathbb{T})\overline{\mathit{F}}$$

(38)

In this process $F^0$ , the uncontrolled force state is constant and can be precomputed since all structural parameters are obtained through a separate process. When $\mathbb{A}\mathbb{C}{\mathbb{T}}_i=0$ , equation (38) constrains the element forces in the controlled state within lower and upper bounds $\{\overline{\mathit{F}},\overline{\mathit{F}}\}$ to satisfy stress and buckling conditions. Instead, when element i houses an actuator ( $\mathbb{A}\mathbb{C}{\mathbb{T}}_i=1$ ), equation (38) reduces to the following equality

$$F_i^0+\mathit{S}{\mathbf{\Delta }}_i\mathit{L}-{\mathbf{F}}_{\mathbf{i}}^{\mathbf{act}}=0$$

(39)

so that the element force $F_i^0+{\mathit{S}}_i^F\mathbf{\Delta }\mathit{L}$ and the actuator force $F_i^{\mathrm{act}}$ are identical since the actuator is assumed to be placed in series with the element. The actuation force variables ${\mathit{F}}^{\mathrm{act}}$ are constrained within lower and upper bounds $\{{\overline{\mathit{F}}}^{\mathrm{act}},{\overline{\mathit{F}}}^{\mathrm{act}}\}$ , which account for the force capacity limits according to equation (17)

$$\mathbb{A}\mathbb{C}\mathbb{T}{\overline{\mathit{F}}}^{\mathrm{act}}\le {\mathit{F}}^{\mathrm{act}}\le \mathbb{A}\mathbb{C}\mathbb{T}{\overline{\mathit{F}}}^{\mathrm{act}}$$

(40)

Actuator placement is avoided for elements that are subjected to a force in the uncontrolled state $F_i^0$ exceeding the actuator capacity, which could cause the response control to be not feasible

$${\overline{\mathit{F}}}^{\mathrm{act}}\le \mathbb{A}\mathbb{C}\mathbb{T}{\mathit{F}}^0\le {\overline{\mathit{F}}}^{\mathrm{act}}$$

(41)

Note that this constraint can be enforced when the structural parameters are known, that is, the structural sizing and topology are obtained through a separate process.

4.3.2. Energy minimization

The formulation given in equation (34) does not consider the control energy directly, which can be computed after a solution is obtained. To benchmark the quality of the solutions obtained through equation (34) concerning the control effort, an additional formulation is given in equation (42). The objective function is the minimization of the control energy that is stated in equation (21)

$$\underset{\mathbb{A}\mathbb{C}\mathbb{T},\mathrm{\Delta }\mathbf{L},{\mathit{F}}^{\mathrm{act}}}{\min }\sum _j\sum _i-(E_{\mathrm{ij}}^0+E_{\mathrm{ij}}^{\mathrm{\Delta }F})$$

$$s.t.$$

$$\mathbf{\Delta }{\overline{\mathit{d}}}^{\mathrm{SLS}}\le {\mathit{S}}^d\mathbf{\Delta }\mathit{L}\le \mathbf{\Delta }{\overline{\mathbf{d}}}^{\mathbf{SLS}}$$

$$\mathbb{A}\mathbb{C}\mathbb{T}\underset{¯}{\mathbf{\Delta }\mathit{L}}\le \mathbf{\Delta }\mathit{L}\le \mathbb{A}\mathbb{C}\mathbb{T}\overline{\mathbf{\Delta }\mathit{L}}$$

$$(1-\mathbb{A}\mathbb{C}\mathbb{T})\underset{¯}{\mathit{F}}\le {\mathit{F}}^0+{\mathit{S}}^F\mathbf{\Delta }\mathit{L}-{\mathit{F}}^{\mathbf{act}}\le (1-\mathbb{A}\mathbb{C}\mathbb{T})\overline{\mathbf{F}}$$

$${\underset{¯}{\mathit{F}}}^{\mathrm{act}}\le \mathbb{A}\mathbb{C}\mathbb{T}{\mathit{F}}^{\mathrm{act}}\le {\overline{\mathit{F}}}^{\mathrm{act}}$$

$${\underset{¯}{\mathit{F}}}^{\mathrm{act}}\le \mathbb{A}\mathbb{C}\mathbb{T}{\mathit{F}}^0\le {\overline{\mathit{F}}}^{\mathrm{act}}$$

$${\mathit{E}}^0\le {\mathit{F}}^0\mathbf{\Delta }L$$

$${\mathit{E}}^{\mathrm{\Delta }F}\le \frac{1}{2}({\mathit{F}}^{\mathrm{act}}-{\mathit{F}}^0)\mathbf{\Delta }L$$

$${\mathit{E}}^0\le 0$$

$${\mathit{E}}^{\mathrm{\Delta }F}\le 0$$

$$\sum \mathbb{A}\mathbb{C}\mathbb{T}\le {\overline{n}}^{\mathrm{act}}$$

(42)

Referring to equation (21) for the control energy, only negative terms are accounted for to avoid considering energy gains. To eliminate discontinuities caused by absolute terms, two auxiliary variables are introduced ${\mathit{E}}^0$ and ${\mathit{E}}^{\mathrm{\Delta }F}$ . ${\mathit{E}}^0$ is employed to model the linear term ${\mathit{F}}^0\mathbf{\Delta }\mathbf{L}$ . ${\mathit{E}}^{\mathrm{\Delta }F}$ is a bilinear term due to the element force variation $\mathbf{\Delta }F$ , which can be expressed as a function of the actuator force ${\mathit{F}}^{\mathrm{act}}$ and the forces in the uncontrolled state ${\mathit{F}}^0$ , that is, $\mathbf{\Delta }F={\mathit{F}}^{\mathrm{act}}-{\mathit{F}}^0$ . The auxiliary variables ${\mathit{E}}^0$ and ${\mathit{E}}^{\mathrm{\Delta }F}$ are therefore related to their corresponding physical values by the following constraints

$${\mathit{E}}^0\le {\mathit{F}}^0\mathbf{\Delta }L$$

$${\mathit{E}}^{\mathrm{\Delta }F}\le \frac{1}{2}({\mathit{F}}^{\mathrm{act}}-{\mathit{F}}^0)\mathbf{\Delta }L$$

(43)

which will be satisfied with equality only when the solution of minimum energy is obtained. To avoid potential energy gains (positive work with the adopted sign convention), the variables ${\mathit{E}}^0$ and ${\mathit{E}}^{\mathrm{\Delta }F}$ must always be negative and at most zero when the actuator length changes and forces have an equal sign

$${\mathit{E}}^0\le 0$$

$${\mathit{E}}^{\mathrm{\Delta }F}\le 0$$

(44)

Limits on the number of actuators are formulated as

$$\sum \mathbb{A}\mathbb{C}{\mathbb{T}}_i\le {\overline{n}}^{\mathrm{act}}$$

(45)

This way it is possible to evaluate solutions with the minimum number of actuators as well as actuator placements that require the least energy given a certain number of actuators that can be placed.

Since the actuator force is proportional to the length change via the force influence matrix ${\mathit{F}}^{\mathrm{act}}~{\mathit{S}}^F\mathrm{\Delta }L$ , the formulation is quadratic. The term is ${\mathit{E}}^{\mathrm{\Delta }F}\le \frac{1}{2}({\mathit{F}}^{\mathrm{act}}-{\mathit{F}}^0)\mathbf{\Delta }\mathit{L}$ in equation (43) is the only bilinear constraint. The function ${\mathit{E}}^{\mathrm{\Delta }F}=\frac{1}{2}({\mathit{F}}^{\mathrm{act}}-{\mathit{F}}^0)\mathbf{\Delta }\mathit{L}$ is non-convex since its Hessian matrix H is not positive semi-definite for all points of its domain:

$$\begin{array}{c}\multicolumn{1}{c}{H=\left[\begin{array}{cc}\frac{\partial {\mathit{E}}^{\mathrm{\Delta }F}}{\partial {{\mathit{F}}^{\mathrm{act}}}^2}\hfill & \frac{\partial {\mathit{E}}^{\mathrm{\Delta }F}}{\partial {\mathit{F}}^{\mathrm{act}}\partial \mathbf{\Delta }\mathit{L}}\hfill \\ \multicolumn{1}{c}{\frac{\partial {\mathit{E}}^{\mathrm{\Delta }F}}{\partial \mathbf{\Delta }\mathit{L}\partial {\mathit{F}}^{\mathrm{act}}}}& \frac{\partial {\mathit{E}}^{\mathrm{\Delta }F}}{\partial \mathbf{\Delta }{\mathit{L}}^2}\hfill \end{array}\right]=\left[\begin{array}{cc}0\hfill & \frac{1}{2}\hfill \\ \multicolumn{1}{c}{\frac{1}{2}}& 0\hfill \end{array}\right]}\end{array}$$

(46)

The eigenvalues of H are in fact ${\lambda }_1=\frac{1}{2}$ and ${\lambda }_2=-\frac{1}{2}$ . However, a sign condition on ${\mathit{E}}^{\mathrm{\Delta }F}$ isolates the part of the domain where the function is either convex or concave. Figure 5 illustrates a non-convex bilinear function (a) whose domain is restricted to a convex region (b). In other words, the condition on ${\mathit{E}}^{\mathrm{\Delta }F}\le 0$ in equation (44) guarantees the convexity of the solution domain therefore the problem can be solved to a global optimum using state-of-the-art branch and bound solvers (Pia et al., 2017).

Figure 5.

(a) Non-convex bilinear function, (b) convex region of a non-convex bilinear function.

The computational complexity of MIP problems is typically exponential in the worst case due to the integrality of the variables. For the problems investigated in this work, the computational complexity is $O\left(2^{n^{\mathrm{el}}}\right)$ exponential with the number of structural elements. For MIQP convex problem, such as that given in equation (42), the continuous part of the problem can be solved in polynomial time. Instead for nonconvex problems also the continuous part becomes NP-hard. Typically, most of the computational time is spent verifying the global optimality of the best upper-bound solution. When the problem is nonconvex, this might become very hard requiring computational expensive strategies implementing cutting-plane methods, introducing auxiliary variables and additional constraints. This increases the size of the problem, leading to a larger search space and requiring significantly larger computational resources and time to solve.

4.3.3. AIO actuator placement problem solution

The AIO formulations stated in equations (34) and (42) have been successfully solved using the branch and bound algorithm implemented in Gurobi (Gurobi Optimization LLC, 2020). A branch and bound algorithm systematically explores the solution space in the form of a search tree by branching on discrete variables and solving linear or nonlinear programming problems to obtain feasible solutions (Bertsimas and Tsitsiklis, 1998). The solution of the continuous relaxation is referred to as lower Bound (LB), when it is integer feasible, it is referred to as an upper bound UB. The mixed-integer programming gap (MIP-gap) is defined as

$$\mathrm{MIP}\mathrm{-}\mathrm{gap}=\frac{\mathrm{UB}-\mathrm{LB}}{\mathrm{UB}}$$

(47)

When the gap is zero (i.e. UB = LB), it provides proof of the solution’s global optimality. For high-dimensional problems, the optimality of an integral solution might not be proved. MIP problems are known to be exponentially complex in the worst case, therefore, especially for problems with many integer variables, it is not always possible to prove global optimality. However, the MIP-gap can still be used to evaluate the solution quality.

The AIO formulation stated in equation (34) has also been solved using a mixed-integer Genetic Algorithm (GA) and Surrogate Algorithm (SA). Regarding the AIO-GA, different from sequential approaches, additional terms are added to the objective function to penalize unfeasible solutions that violate the constraints. The binary variable integrality is forced through heuristics that involve rounding feasible points to integers (Deb, 2000; Deep et al., 2009). Similarly, regarding the AIO-SA, the surrogate function is generated using sample points that at first satisfy constraints containing continuous variables only, and subsequently round the integer variables using a heuristic to preserve the constraint feasibility (Mueller, 2014; Wang and Shoemaker, 2014).

5. Case studies

The actuator placement methods presented in Section 4 are compared using numerical examples varying the number of structural elements, load cases and geometric dimensions. All case studies consider slender high-rise multi-story buildings modeled as pin-jointed trusses, which are shown in Figure 6.

Figure 6.

Dimensions and boundary conditions; controlled nodes are indicated with circles.

5.1. Geometry and loading

Three planar (2D) configurations with an increasing number of elements under two load cases and one spatial (3D) configuration under four load cases are considered. The structural and control parameters are given in Table 1. All elements are made of structural steel S355 and have cylindrical hollow sections with a wall thickness set to 10% of the external diameter. The minimum diameter is 100 mm.

Table 1.

Structural and control parameters.

Parameter	Case 1	Case 2	Case 3	Case 4
Structural system parameters
Dimensions	2D	2D	2D	3D
Number of elements	40	75	135	144
Number of free degrees of freedom	32	64	96	96
Degrees of static indeterminacy	8	11	39	48
Module height $h^m$ (m)	15	8	8	8
Floor height $h^f$ (m)	3.5	4	4	4
Total height (m)	120
Width (m)	10
Slenderness ratio	12
Load cases	2	2	2	4
First mode period T1 (s)	1.58	1.54	1.51	2.81
Control system parameters
Number of controlled degrees of freedom	8	15	15	10
Actuation force limit (kN)	5000	4500	3500	8000
Maximum actuator stroke	30 (mm)

The self-weight (SW) is determined from the cross-section dimensions, and the dead load (DL) is set to 4,92 kN/m² (500 kg/m²). The imposed live load (LL) is set to 2.94 kN/m² (300 kg/m²). A wind load (WL) is also applied. The wind loads are computed according to EN 1991-1-4 considering a reference velocity v = 30 m/s, using the parameters given in Table 2. For the 2D cases, an out-of-plane dimension equal to the width of the structure is considered to determine the load influence areas. The floor height is set to 3.5 m for case study #1 and 4.0 m for case study #2, #3, and #4. Gravitational loads (SW, DL, LL) are applied according to the number of floors contained in each module, which are 4 for case study #1 and 2 for case study #2, #3, and #4. A prevailing wind direction is considered for case study #4 (3D) as indicated in Table 2.

Table 2.

Wind load parameters.

Wind load parameters	Case study 1–3	Case study 4
Basic wind velocity vb	30 m/s
Terrain category	II
Directional factor cdir,	1	1.00/0.93/0.74/0.72
Seasonal factors cseason	1
Structural factors cscd	1

The design load combinations are given in Table 3.

Table 3.

Load cases and combinations.

Limit state	Load case	Load combinations
ULS	LC0 LC1, LC2…	${\gamma }_{G1}$ (SW+DL)+ ${\gamma }_Q(\mathrm{LL})+{\gamma }_Q({\Psi }_{0,2}$ WL1…) ${\gamma }_{G2}$ (SW+DL)+ ${\Psi }_{0,1}{\gamma }_Q(\mathrm{LL})+{\gamma }_Q(\mathrm{WL}1\dots )$
SLS	LC1, LC2…	SW+DL+ ${\Psi }_{0,1}$ LL+(WL1…)
Partial factors: ${\gamma }_{G1}$  = 1.35; ${\gamma }_{G2}$  = 1.05; ${\gamma }_Q$  = 1.5; Combination factors ${\Psi }_{0,1}$  = 0.7; ${\Psi }_{0,2}$  = 0.6

For the 3D case (#4), a rigid diaphragm is employed to model each floor. The diaphragm assumption allows reducing the horizontal degrees of freedom to two translational and one rotational degree of freedom for each floor. To simplify the analysis further, the floor elements are omitted by applying internal translational and rotational constraints on the corresponding nodal degrees of freedom with respect to a floor master node. This model simplification results in a reduction of the free degrees of freedom from 198 to 96. The controlled degrees of freedom are indicated with circles in Figure 6. Thanks to the rigid diagram assumption, only one controlled degree of freedom per floor is sufficient for the 3D case study.

5.2. Structural design

The elements of the adaptive structures are sized to satisfy ultimate limit states (ULS), while serviceability limit states (SLS) do not have any influence on element sizing as they are satisfied through active control. The maximum stress under the ULS combination is limited to 80% of the yield stress. The reader is referred to (Senatore et al., 2019) for a detailed description of the sizing optimization process. To limit optimization complexity, global stability is not considered. The displacement limits are set to H^m/500 where H^m is the height of each module with respect to the ground. Figure 7 displays the deformed shapes under a single load case of the passive structures. For the passive case, the element cross-sections are sized to satisfy ULS and SLS requirements. Figure 8 illustrates the deformed shapes of the adaptive solutions in the uncontrolled state and gives mass savings compared to the passive solutions. Figure 9 illustrates the controlled displacement state.

Figure 7.

Passive systems: structural mass (primary structure only) and deformed shapes under a wind-load characteristic combination: (1) mass = 687 ton, (2) mass = 631 ton, (3) mass = 571 ton, (4) mass = 815 ton.

Figure 8.

Adaptive systems: structural mass (primary structure only), mass savings, uncontrolled state under wind load characteristic combination; active elements are indicated with purple lines: (1) mass = 293 ton (−57%), (2) mass = 289 ton (−54%), (3) mass = 266 ton (−53%), (4) mass = 440 ton (−46%).

Figure 9.

Adaptive systems: controlled state under wind load characteristic combination, actuator layouts obtained through the AIO formulation: (1) case study 1 (MIP-gap = 0.0%), (2) case study 2 (MIP-gap = 0.0%), (3) case study 3 (MIP-gap = 5.0%), (4) case study 4 (MIP-gap = 12.5%).

5.3. Actuator placement

The actuator placement is performed by implementing each method described in Section 4. The Integer Relaxation strategy IR is implemented as in (Senatore et al., 2019). The Stackelberg Game strategies are implemented using:

• (a) Greedy-culling load-independent algorithm SG (a) as described in (Wagner et al., 2018)

• (b) Greedy-culling load-dependent algorithm SG (b)

• (c) Greedy-additive load-dependent algorithm SG (c)

• (d) Genetic algorithm SG (d)

• (e) Surrogate algorithm SG (e)

Greedy algorithms use the Compensability Gramian-based metric defined in equation (28) while Genetic and Surrogate algorithms adopt the objective function defined in equation (33) (see Section 4.2.3), where the weight coefficients are set to $\nu =0.5,\eta =0.5$ and when the control tolerance is above the prescribed bound, $\nu =1$ to penalize the solution.

The AIO formulation for the minimization of the actuator number in equation (34) is implemented using:

• (a) Mixed-Integer Linear Programming AIO-MILP

• (b) Mixed-Integer Genetic algorithm AIO-MIGA

• (c) Mixed-Integer Surrogate algorithm AIO-MISA

The AIO formulation for the minimization of the control energy in equation (42) is implemented as a Mixed-Integer Quadratic Programming AIO-MIQP. The solvers employed for each formulation are indicated in Table 4.

Table 4.

Optimization solvers.

Strategy	Method	Solver 1
Integer relaxation (IR)	(a) Single actuator efficacy evaluation.	Matlab lsqlin
Stackelberg game (SG)	(a) Load independent Greedy-Cull (b) Load dependent Greedy-Cull (c) Load dependent Greedy-Add	Matlab lsqlin
	(d) Genetic	Matlab GA and Matlab lsqlin
	(e) Surrogate	Matlab surrogate-opt
AIO-MILP 2	(a) Mixed -integer linear programming	Gurobi
	(b) Mixed-integer genetic	Matlab GA
	(c) Mixed-integer surrogate	Matlab surrogate-opt
AIO-MIQP 3	(d) Mixed -integer quadratic programming	Gurobi

1

All optimization settings are kept at default unless otherwise specified.

2

Actuator number minimization.

3

Control energy minimization.

Referring to the problems stated in equations (16), (34), and (42), a tighter constraint on control feasibility concerning element stress and actuation forces is applied by reducing the tolerance on inequality conditions to 1e-5. Instead, since displacement control is not a safety-critical condition, the related tolerance is relaxed by allowing the controlled node position to be in the range expressed in equation (48)

$$\mathbf{\Delta }{d}_i^{\mathrm{SLS}}-\frac{h^m}{1000}\le {\mathit{S}}_i^d\mathrm{\Delta }L\le \mathbf{\Delta }{d}_i^{\mathrm{SLS}}+\frac{h^m}{1000}$$

(48)

where $h^m$ is the height of the module (15 m for case #1 and 8 m for case #2, #3 and #4). This tolerance is set to limit the interstory drifts to $h^f/500$ . For the problem stated in equation (16), since the displacement correction target is part of the objective function, this condition is set through the optimality tolerance.

For the methods IR, SG(a), SG(b), SG(c), the number of actuators $n^{\mathrm{act}}$ is not an explicit variable and therefore must be pre-assigned or obtained through compensability and efficacy evaluation. To benchmark the solutions of these methods with the AIO solutions, $n^{\mathrm{act}}$ is set to that obtained by AIO-MILP. This way, it is possible to appreciate the solution quality concerning all other metrics (e.g. control energy and actuation forces, etc.) which otherwise could be maximized by increasing arbitrarily the number of actuators. In addition, the number of actuators obtained from AIO-MILP is set as the actuator number upper limit for the energy minimization formulation AIO-MIQP.

Table 5 gives optimization metrics for all placement methods for case study #2 which is taken as an illustrative case. Due to the tight tolerance limits on internal stresses and actuator length changes, the solutions provide feasible control inputs that guarantee structural integrity at the expense of the displacement target attainment. The performance of each method is evaluated considering (a) the number of actuators, (b) the maximum actuator force (c) the control energy (d) the absolute displacement error compensation, (e) the control efficacy, and (f) the computational time. Sequential and heuristic approaches, often produce feasible solutions only if constraints on the maximum actuation forces are relaxed to some extent.

Table 5.

Performance metrics for case study #2.

Strategy	Method	Number of actuators	Max actuation force (4500 kN)	Control energy (MJ)	Control target error compensation (%)	Control efficacy (%)	Time (s)
Integer Relaxation (IR)	(a) Single actuator efficacy evaluation	15	5593 kN	0.82	95	85	0.1
Stackelberg Game (SG)	(a) Load independent Greedy-Cull	15	4078 kN	0.70	90	73	3.1
	(b) Load dependent Greedy- Cull	15	4888 kN	0.68	99	72	21.8
	(c) Load dependent Greedy-Add	15	6297 kN	0.85	100	85	16.0
	(d) Genetic	17	4624 kN	0.65	100	99	133.0
	(e) Surrogate	16	4117 kN	0.72	100	99	786.9
All in One (AIO)	(a) Mixed-integer linear programming	15	4072 kN	0.57	100	98	14.8
	(b) Mixed-integer genetic	15	4048 kN	0.59	100	75	793.8
	(c) Mixed-integer surrogate	22	4007 kN	0.47	100	75	1879.3
	(a) Mixed-integer quadratic programming	15	4081 kN	0.50	100	98	1046.5

For methods implementing Genetic algorithms (SG (d), AIO (b)), the convergence criterion is the objective function stationarity over 10 generations with a tolerance of 1e-4, the maximum number of generations is set to $50n^{\mathrm{el}}$ . For methods implementing a Surrogate algorithm (SG (e), AIO (c)), the optimization stops when the maximum number of function evaluations is reached, that is also set to $50n^{\mathrm{el}}$ . In case studies #3 and #4, for all AIO formulations, the computation time limit is set to 24 and 30 h respectively.

Table 6 gives upper and lower bounds for the MILP solutions for each case study. Table 7 gives upper and lower bounds for the MIQP solutions for each case study. Figure 10 shows the control energy savings obtained through AIO-MIQP (control energy minimization) with respect to AIO-MILP (actuator number minimization).

Table 6.

Results for actuator number minimization, AIO-MILP.

Case study	Upper Bound (UB) ( $n^{\mathrm{act}}$ )	Lower Bound (LB) ( $n^{\mathrm{act}}$ )	Number of elements $n^{\mathrm{el}}$	MIP gap (%)	Best UB solution time (s)	Total time (s)
1	10	10	45	0.0	0.97	1.15
2	15	15	75	0.0	8.1	14.8
3	20	19	135	5.0	1459	3090
4	32	28	144	12.5	11,265	36,022

Table 7.

Results for control energy minimization, AIO-MIQP.

Case study	Upper Bound (UB) ( $E^{\mathrm{act}}$ ) (MJ)	Lower Bound (LB) ( $E^{\mathrm{act}}$ ) (MJ)	Number of elements $n^{\mathrm{el}}$	MIP gap (%)	Best UB solution time (s)	Total time (s)
1	0.841	0.841	40	0.0	3.9	8.2
2	0.503	0.503	75	0.0	682.4	1046.5
3	0.380	0.365	135	3.9	9145	21613
4	0.711	0.550	144	23.1	38,316	108,000

Figure 10.

Control energy saving obtained through AIO-MIQP (control energy minimization) with respect to AIO-MILP (actuator number minimization).

For case studies #1 to #3, a Pareto analysis has been performed progressively increasing the number of actuators in formulation (42) until actuation force limitations prevent further actuator assignments. Figure 11 shows the Pareto fronts control energy vs number of actuators. For clarity, the value of the control energy optimization (AIO-MIQP) are also given in Table 8. Generally, the required control energy decreases asymptotically as the number of actuators increases. This is expected since the minimum control energy is obtained when all elements are active (see Section 4.1). The discrepancy in the control energy magnitude between the three cases can be attributed to the varying stiffness levels, as the degree of redundancy increases from case #1 to case #3. This study gives a rigorous confirmation, since the solutions are global optima, of the trend that was already observed in (Senatore and Reksowardojo, 2020). Note that for case studies #2 and #3, when the maximum actuator number is below 15 and 20, respectively, it is necessary to increase the maximum actuator force to obtain feasible solutions, hence the corresponding Pareto curves are indicated with a dashed pattern.

Figure 11.

Pareto fronts control energy vs number of actuators obtained using the AIO-MIQP formulation.

Table 8.

Control energy vs number of actuators obtained using the AIO-MIQP formulation, actuation force capacity increased by 50% (see Table 1) for solutions indicated by shaded cells.

5.4. Benchmark

For a rigorous benchmark, for each case study k, all metrics $X_m$ are standardized by centering around the mean value ${\mu }_k$ and scaling to a standard deviation of 1

$$X_m^{|}=\psi \frac{X_m-{\mu }_k}{{\sigma }_k}.$$

(49)

${\sigma }_k$ is the standard deviation of the original distribution, $\psi$ is a sign unit coefficient that is employed to indicate the solution quality depending on the considered metric. For example, $\psi =1$ for metrics related to control target error compensation and efficacy because they are maximized; $\psi =-1$ for all other parameters including the actuator number since they are minimized. Standardization provides a distribution of values that is less sensitive to outliers. To achieve a consistent range of values, all standardized metrics are also normalized:

$$X_m^{||}=\frac{X_m^{|}-{X^{|}}_{\min }}{X^{|}{{|}_{\min }}_{\max },}$$

(50)

where ${X^{|}}_{\max }$ and ${X^{|}}_{\min }$ are the maximum and minimum standardized values across all case studies. $X_m^{||}$ takes values in the range [0,1]. Figure 12 shows standardized results through radar charts for each placement strategy according to six performance metrics as described in Section 5.3. Each case study is represented by a different polygonal area providing a visual ranking of each placement method. The further out the edge of the polygon, the better the performance. Note that for the methods IR, SG (a), SG (b), SG (c) the number of actuators is pre-assigned to match that obtained from the AIO-MILP. Referring to the radial charts, the AIO formulations produce the highest quality solutions in terms of the number of actuators, target error, control efficacy and energy. The sequential formulations require a lower computation time but they produce lower-quality solutions. Trade-offs between control energy and the number of actuators can be observed, as solutions with a higher number of actuators generally require less control energy.

Figure 12.

Performance metrics, standardized and normalized across all case studies.

To offer a more concise overview of the performance of each placement method, results are averaged across all case studies and metrics. The standard deviation of the optimization performance for each method across the case studies gives an indication of the “reliability” of each method, that is, with what variability each method performs better or worse with respect to others. The computational time parameter is considered separately. The radar charts of the three global parameters are shown in Figure 13. The method SG (a) has the worst global performance as indicated by the mean performance chart. The performance is consistently low across multiple case studies as indicated by a low standard deviation. On the other hand, AIO-MILP and AIO-MIQP have the best global performance across multiple case studies consistently (i.e. high mean performance with low standard deviation). However, the computational effort for AIO-MIQP is among the highest.

Figure 13.

Mean performance, standard deviation, mean normalized process time.

Additional insights are given in Figure 14 showing the number of actuators for each method and case study that is required to satisfy control constraints within the assigned tolerance. For clarity, control constrain tolerances are satisfied for methods in which $n^{\mathrm{act}}$ is an explicit design variable (AIO and SG with global search). For the methods IR, SG(a), SG(b), SG(c), it is often necessary to increase the number of actuators to fully satisfy the tolerance on the control target (displacement), since it is implemented as a least square minimization that often produces solutions with relatively large residuals.

Figure 14.

Number of actuators for each method and case study to satisfy displacement constraints within the assigned tolerance.

6. Discussion

Actuator layouts produced by methods that consider specific load cases have a significantly higher performance compared to methods that are independent of the disturbance SG (a). This is expected because the placement is determined to perform best for a smaller subset of cases. Generally, the efficacy measure Eff provides a well-weighted assessment of the control error. Since compensability Gramian-related metrics are based on a non-normalized error, they tend to overestimate the importance of response states that require larger compensation while underestimating response states that require small compensation. A direct assignment of the actuator positions through the integer relaxation approach IR (a) considering the control efficacy contribution of each actuator yields satisfactory layouts with a very small computational effort.

Stackelberg Game strategies that implement greedy selections as lead player tend to converge rapidly. However, the solution quality is significantly lower compared to the global search approaches such as Genetic Algorithm SG (d) and Surrogate algorithm SG (e). Different than greedy culling (SG (a), SG (b)), additive greedy algorithms SG (c) can recover from early choices later in the process producing better quality solutions. However, solutions often include redundant actuators, since early assignments are irreversible. Stackelber Game strategies (SG (d), SG (e)) implementing global search techniques as lead player outperform local optimization approaches. Genetic and Surrogate algorithms are able to minimize cost functions that include multiple parameters without stability issues in the selection process, and accounting for different control requirements (e.g. penalizing actuation forces over control energy or the number of actuators over control efficacy).

AIO solutions implemented through metaheuristic approaches (AIO-MIGA, AIO-MISA) are generally better compared to those obtained using sequential formulations. For larger problems, however, the performance gap becomes negligible and the computational cost of an AIO formulation increases substantially, while for the sequential strategy remains within the same order of magnitude. The AIO formulation implemented through MILP (AIO-MILP) provides globally optimum solutions. However, the computational cost is significant. For relatively small problems (45–90 integer variables) this can be achieved within a reasonable time, for larger problems most of the computational effort (60% or more of the processing time) is used to verify global optimality, without significant improvement of the best upper bound solution. Generally, even when global optimality is not verified due to time limitations and the MIP-Gap is relatively low (<20%–30%), the solutions generated through MIP have a higher quality compared to other methods. Time limitations may not allow to prove the global optimality of the solution, however, this does not exclude that the solution is a global optimum.

The computational time required by AIO-MILP and AIO-MIQP can be up to four orders of magnitude longer than that of an Integral Relaxation (IR) or Greedy algorithms (see Table 5). However, the maximum time needed for the best upper bound is approximately 10 h (see case study #4 in Table 7) which is acceptable in most design practices. AIO-MILP and AIO-MIQP formulations were solved using Gurobi v10 running on an AMD Threadripper Pro 5995wx using 32 cores in parallel.

The solutions obtained through the formulation stated in (34) (AIO-MILP) have the minimum number of actuators to ensure the feasibility of control under given constraints. However, this can be further improved with the formulation given in (42) minimizing the control energy. When ${\overline{n}}^{\mathrm{act}}$ is set as the global minimum obtained from (34) any solution with $\sum \mathbb{A}\mathbb{C}{\mathbb{T}}_i<{\overline{n}}^{\mathrm{act}}$ is infeasible therefore the number of possible combinations is significantly reduced from the exponential term $2^{n^{\mathrm{el}}}$ to the factorial term given by the binomial coefficient $\left(\begin{array}{c}{n}^{\mathrm{el}}\hfill \\ \multicolumn{1}{c}{n^{\mathrm{act}}}\end{array}\right)$ at the cost of introducing a bilinear term in the constraints (see 4.3).

7. Conclusions

Depending on the control objective and requirements, different metrics and constraints can be considered for actuator placement in adaptive structures. Actuator layouts produced by considering only displacement or force compensability separately are often not practical due to power and actuation force constraints. Since the design of a civil structure is typically carried out under the assumption of well-defined load cases, it is convenient that the actuator placement follows the same approach. This work has shown that actuator placement considering specific load cases, produces better-quality solutions in terms of control efficacy and feasibility.

Genetic and Surrogate algorithms perform best in sequential approaches using the actuator positions as design variables of the lead player and control inputs obtained in the follower player using constrained least square optimization. The AIO-MILP and AIO-MIQP formulations produce actuator layouts with the lowest number of actuators and the smallest control energy, respectively, required to satisfy structural integrity, serviceability, and control feasibility. For problems with many variables, due to computational limits, it might not be possible to verify global optimality. However, results have shown that, generally, the AIO upper-bound solutions (MIP gap < 30%) perform significantly better compared to the solutions produced by any other strategy tested in this work.

It has been proved numerically that the control energy decreases asymptotically as the number of actuators increases, which confirms the results obtained in previous work. However, the study carried out in this work offers rigorous numerical proof since the solutions considered in the Pareto analysis are global optima.

The formulations presented in this work can be extended to frame and shell structures. The main limitation is the number of variables that would be required for structures comprising beams and shells, which is significantly higher than for trusses, therefore model reduction techniques would be required. Future work will develop extension to include explicit consideration of dynamic actions (e.g. earthquakes) and control objectives will be formulated to include closed-loop performance parameters for vibration damping.