Computationally efficient controller design for smart base-isolation systems: A reduced-order modal LQG approach

Abstract

Smart base isolation (SBI) is an advanced seismic protection technology for structures. LQG method is a mainstream choice for use in such systems. However, its engineering application is hindered by two critical bottlenecks: the excessively high controller order and the difficulty in determining optimal weighting matrices. For instance, an 8-story 3D base-isolated building alone requires handling 54 × 54 system matrices, and balancing multiple control objectives to obtain optimal control gains remains a formidable challenge. To address these issues, this study adopts a Hankel singular values (HSV)-based index formulated in modal coordinates. The index has an approximate closed-form expression, which takes the solutions of Riccati equations (and the derived controller/estimator gains) as variables and bears explicit physical significance in terms of modal reduction and pole shifting ratios. Leveraging these advantages, the index enables flexible manipulation of both control performance and controller order reduction. The method was applied to the aforementioned benchmark building equipped with MR dampers. The implementation procedure involves three core steps: first, determining optimal weighting matrices via pole allocation; second, calculating the HSV-based order index and deriving the reduced-order controller; third, assigning target damping forces to MR dampers based on real-time structural seismic responses. Further analysis is conducted on seismic performance under different estimator weighting combinations. Results demonstrate that, the method is efficient in that, the LQG order can be reduced from 27 modes to 18 or even 12 modes without sacrificing control efficacy; and optimal weightings can be obtained with minimum trials; the reduced- order controller effectively suppresses large isolation displacements dominated by lower modes without disturbing higher modes, achieving superior seismic control performance compared with passive control and nodal control strategies. Additionally, a 3-5 ratio between estimators and controllers is identified as the optimal matching criterion for enhanced control effects.

Keywords

controller order reduction modal control LQG-balanced model LQG weighting determination base isolated building

Introduction

State-space control strategies are prone to yielding controllers with excessively high orders. Specifically, the linear quadratic (LQ) control methodology, in its basic form, generates a controller whose order is identical to that of the controlled plant. Further modifications—such as frequency-shaped loop transfer recovery, integral control, and other measures aimed at tailoring the frequency-domain characteristics of the controller—will further elevate the controller order. High-order controllers typically incorporate faster poles, which necessitate higher sampling rates in practical applications. This not only increases the cost of analog-to-digital (A/D) and digital-to-analog (D/A) conversion modules but also imposes heavier computational burdens on the control system. In digital control scenarios, faster sampling rates also demand more high-performance processors, leading directly to a substantial rise in overall system costs. For these reasons, controller order reduction is highly desirable whenever feasible in engineering practice. In fact, controller order reduction techniques have been widely implemented in a variety of practical applications, including optimal sensor and controller placement in structural systems (Casciati and Faravelli, 2014; Lu et al., 2015).

Controller order reduction can be achieved via three distinct approaches: (1) Preliminary plant order reduction: A reduced-order approximation of the plant (in terms of degrees of freedom, DOFs) is first derived prior to controller design. For instance, classical control frameworks facilitate the synthesis of non-optimal, low-order controllers: proportional control restricts the controller to a zero-order structure, whereas a lead compensator constrains it to a first-order form (Burl, 1999). (2) Order-constrained controller design: The controller order is explicitly constrained during the design phase. (3) Post-design controller order reduction: An approximate reduced-order controller is constructed subsequent to the synthesis of the full-order controller. For multi-input multi-output (MIMO) systems integrated with modern control techniques such as linear quadratic Gaussian (LQG) control, order reduction is typically implemented after full-order controller design; controller approximation—including the methodology adopted in this study—is recognized as a more favorable approach.

The optimal Hankel singular norm method (Anderson, 1986) and balanced truncation technique (Mustafa and Glover, 1991) have been extensively employed for model reduction. Hankel singular values (HSVs), defined as $γ = \sqrt{λ (W_{c} W_{o})}$ , correspond to the square roots of the eigenvalues $(λ [.])$ of the product of the system’s controllability Gramian ( $W_{c}$ ) and observability Grammian ( $W_{o}$ ) of a system. These values are instrumental in preserving near-optimal feedback control performance. Notably, HSVs remain invariant under modal transformations, whereas Gramians do not; thus, indices derived from HSVs exhibit superior stability. Balanced realizations align the controller and estimator along identical directions in the vector space, enabling the identification of control gains with equivalent significance. These two techniques have revolutionized the paradigm of model reduction, as they provide a priori bounds on frequency response errors and guarantee the stability of the resulting reduced-order models (Kumar et al., 2012). For example, Benner et al. (2016) proposed a balanced truncation method for compensator design, targeting vortex-induced vibrations of the von Kármán type based on a reduced-order plant model. Balanced reduction eliminates weakly controllable and observable system modes by internally balancing the controllability and observability Gramians (Moore, 1981). Zhou et al. (1995) developed a balanced controller order reduction method that simultaneously accounts for closed-loop stability and performance requirements.

A dynamic system can be represented in either nodal or modal coordinates. Modal representation is generally preferred in structural analysis and testing owing to its compactness, simplicity, and clear physical interpretability. Many critical structural characteristics can only be explicitly revealed in the modal domain. Modal control methodologies not only facilitate the elucidation of inherent structural properties and provide mechanistic insights into control law design, but also simplify the overall controller synthesis process. To date, a wealth of procedures and algorithms pertaining to modal control and pole assignment have been reported in the literature (Canciello and Cavallo, 2017; Han, 2020; Lu, 2001). Another salient advantage of modal coordinates is that, under certain conditions, modal control can yield a diagonal-dominant system representation, which substantially simplifies both the solution of control problems and the model reduction process.

Previous studies have demonstrated that the LQG approach constitutes the most natural framework for balancing closed-loop systems (Jonckheer and Silverman, 1983). Nunnari et al. (1994) addressed LQG controller order reduction by formulating closed-loop stability as a structured perturbation problem. However, most existing LQG reduction strategies either focus on developing novel algorithms for solving Riccati equations (Möckel et al., 2011) or leverage the separation principle to simplify the computation of reduced-order estimator gains or controller gains (Leiter and Geller, 2019).

The approximate LQG-balancing reduction technique was initially proposed by Jonckheer and Silverman (1983) and later extended by Opdenacker and Jonckheer (1985). This method incorporates the plant model into the LQG control loop from the outset and circumvents the need to solve the two traditional Riccati equations associated with the Kalman filter. Gawronski further refined this approximate framework within the modal coordinate system, establishing an index to determine the retained system order based on modal significance and enabling direct manipulation of control effects via adjustment of controller weighting matrices (Gawronski, 1998). His work represents an elegant integration of the merits of preceding studies; yet, this approach has seen relatively limited practical application to date.

The integration of control techniques into base isolation systems is widely acknowledged as an effective strategy to enhance the performance of passive base isolation schemes. LQG control is naturally among the most convenient techniques for application in smart base isolation research. The ASCE Committee on Structural Control developed a smart base isolation benchmark problem to enable the international research community to evaluate competing control strategies, encompassing control devices, algorithms, and sensor configurations. Narasimhan et al. (2006) formulated this benchmark problem based on an existing eight-story building located in Los Angeles, California. The building features an L-shaped floor plan with setbacks above the fifth floor; its superstructure is supported by a base slab, with isolators installed between drop panels and the underlying footings. The integrated model of the superstructure (24 DOFs) and isolation system (3 DOFs) thus possesses a total of 27 DOFs. The corresponding state-space model doubles the number of DOFs, resulting in a 54-DOF system for full-state feedback controller design. As a representative complex structure, this benchmark model serves as an ideal platform to validate Gawronski’s method, assess its advantages and effectiveness, and compare the seismic performance improvements achieved by this reduced-order LQG controller against those of alternative control strategies.

The benchmark smart base isolation problem

The nominal isolation system of the benchmark problem comprises 92 isolation bearings, with the controller deployment locations aligned with the positions of these bearings. In this study, magnetorheological (MR) dampers are adopted as the control devices. Based on the optimal placement investigation conducted by Wang and Dyke (2021), the isolation bearings and MR dampers are configured at 10 preselected locations (with dual-directional arrangements in both the x- and y-axes), resulting in a total of 20 MR dampers integrated into the system. The layout of the isolation system and controller configuration is illustrated in Figure 1.

Figure 1.

Elevation view, controller Installation and bearing arrangement of the Benchmark building.

In this study, isolators are specified as linear rubber bearings, with designated stiffness of $k_{b} = 2119$ kN/m. This stiffness configuration yields fundamental isolation periods of T₁ = 2.28 sec (x-), T₂ = 2.18 sec (y-), and T₃ = 1.86 sec (r-). The damping coefficient of the bearings is set to $c_{b} = 241.44$ kN.sec/m, corresponding to damping ratios of $ξ_{1} = 14.7 %$ (x-direction), $ξ_{2} = 14.9 %$ (y-direction), and $ξ_{3} = 17.9 %$ (r-direction) for the first three isolation modes, respectively.

The closed-loop control system and the measurement output y are subsequently described by equations (1) and (2):

\dot{x} = Ax + Bu + Ew = (A - BK) x + Ew

(1)

y = C_{y} x + D_{y} u + v

(2)

where x denotes the state vector comprising displacements and velocities; w represents the general disturbance input (e.g., seismic excitation in this study),

u

is the control force vector generated by the MR damper, and v signifies the measurement noise vector. The system dynamics can be characterized by the matrix set

(A, B, C_{y}, D_{y})

. Specifically,

A

is the system state matrix, and

A_{i} = [\begin{array}{c} 0 & 0 \\ - ω_{i}^{2} & - 2 ζ_{i} ω_{i} \end{array}]

;

ζ_{i}

and

ω_{i}

denote the damping ratio and the natural frequency of the i^th mode, respectively, and

B

is the input influential matrix that quantifies the coupling effect between the control forces and the system states.

For control-oriented investigations, the modal representation that is based on Moore realization $(A_{mb}, B_{mb}, C_{mb}, D_{mb})$ is generally preferred, in which each modal submatrix $A_{mb, i}$ consists exclusively of real-valued elements, which is achieved by decoupling the real and imaginary components of the system eigenvalues. This real-valued modal realization is derived via two successive modal transformations: $x = T_{m} x_{m}$ and $x_{m} = T_{mb} x_{mb}$ . These transformations convert the original state matrix $A$ into a block-diagonal form, namely $A_{mb} = diag (A_{mb, i})$ . The first transformation matrix $T_{m}$ is applied to convert $A_{i}$ into the complex modal form of $A_{mi}$ , and the first transformation matrix $T_{mb}$ is applied to convert $A_{mi}$ into the real-valued block-diagonal submatrix $A_{mbi}$ , as shown in equations (3a) and (3b):

T_{m} = [\begin{array}{c} 1 & 1 \\ - ζ_{i} ω_{i} + j ω_{d, i} & - ζ_{i} ω_{i} - j ω_{d, i} \end{array}], A_{mi} = [\begin{array}{c} - ζ_{i} ω_{i} + j ω_{d, i} & 0 \\ 0 & - ζ_{i} ω_{i} + j ω_{d, i} \end{array}]

(3a)

T_{mb} = \frac{1}{2} [\begin{array}{c} 1 & - j \\ 1 & j \end{array}], A_{mbi} = [\begin{array}{c} - ζ_{i} ω_{i} & ω_{i} \\ - ω_{i} & - ζ_{i} ω_{i} \end{array}]

(3b)

where

j

denotes the imaginary unit. It should be noted that this modal representation is valid for lightly damped structural systems, where the damped natural frequency

ω_{d, i}

satisfies

ω_{d, i} ≅ ω_{i}

The benchmark problem definition (Narasimhan et al., 2006) has established a comprehensive set of evaluation criteria, where all indices are normalized against the corresponding response quantities of the passively isolated system. In this study, three peak response indices and three root mean square (RMS) indices are selected to assess the control performance (detailed in Section 6.4), as summarized in Table 1. For a complete elaboration on the evaluation criteria, refer to (Narasimhan et al., 2006).

Table 1.

Performance indices used in this study.

Peak base drift, J₃	RMS base drift, J₇
Peak 1st-story drift, J₄	RMS 1st-story drift, J₁₁
Peak roof acceleration, J₅	RMS roof acceleration, J₈

Modal LQG control and the approximate weighting formula

Full-state feedback control is a control strategy where the control force $u$ is proportional to the system states, expressed as: $u = - Kx$ , where $K$ denotes the controller gain matrix. However, in practical engineering scenarios, direct measurement of all system states is typically unfeasible. Consequently, the control force is generally derived using estimated states rather than true states, i.e., $u = - K \hat{x}$ , where $\hat{x}$ represents the state estimates obtained from a Kalman estimator (see equation (4)):

\dot{\hat{x}} = A \hat{x} + Bu + Ew + L (y - \hat{y}) = (A - L C_{y}) \hat{x} + (B - L D_{y}) u + Ew + Ly - Lv

(4)

where

L

denotes the estimator gain, whose role is to minimize the estimation error. The combined implementation of

u = - K \hat{x}

(state feedback) and the Kalman estimator is governed by the separation principle. This principle is pivotal as it enables the independent design of the state feedback controller and the estimator, thereby laying the foundation for controller order reduction.

The core objective of the LQG control strategy is to determine an optimal control input u that minimizes the cost function $J$ , which is formulated based on the regulated output $C_{Z} x + D_{Z} u$ as follows:

J = \lim \frac{1}{τ \to \infty τ} E [\int_{0}^{τ} {{(C_{Z} x + D_{Z} u)}^{T} Q (C_{Z} x + D_{Z} u) + u^{T} Ru} dt]

(5)

Where

E [.]

denotes the mathematical expectation operator, Q and R are positive semi-definite and positive definite weighting matrices, respectively, and the superscript T represents the matrix transpose operation.

The solution of $K$ and $L$ hinges on solving for the corresponding Riccati equations to obtain $S_{c}$ and $S_{e}$ :

K = B^{T} S_{c}, L = S_{e} C_{y}^{T}

(6)

When implemented in nodal coordinates, this solution procedure entails intricate computations. Moreover, the weighting matrices Q and R are implicitly contained in the Riccati solutions $S_{c}$ and $S_{e}$ . Their calibration typically relies on an empirical trial-and-error process. However, relevant studies have demonstrated that, approximate analytical formulas can be derived for Q and R when balanced LQG controllers are constructed using Moore’s realization, which yields diagonally dominant state matrices for both the closed-loop plant (i.e., the controlled structure, denoted as (Plant_CL) and the state estimator. These are expressed as:

A - BK = A - B B^{T} S_{c} ≅ A - d i a g (B B^{T}) S_{c} (Plant_CL)

A - L C_{y} = A - S_{e} C_{y}^{T} C_{y} ≅ A - S_{e} d i a g (C_{y}^{T} C_{y}) (Estimator)

(7)

This diagonal dominance property is particularly instrumental for model order reduction, as it allows the direct truncation of states corresponding to negligible diagonal elements without significant loss of control performance.

Other key system properties, such as the Hankel singular values (HSVs), also exhibit a block-diagonally dominant distribution under this modal framework. By adopting this modal representation, it becomes significantly easier to truncate the weakly controllable and observable system modes while preserving the critical input–output characteristics of the original full-order system. Moreover, the diagonal entries of the associated Gramians can be derived via a closed-form approximate formula, as expressed in equation (8):

w_{c, i} = \frac{{‖ B_{m b, i} ‖}_{2}^{2}}{4 ζ_{i} ω_{i}}, w_{c, o} = \frac{{‖ C_{m b, i} ‖}_{2}^{2}}{4 ζ_{i} ω_{i}}, a n d γ_{i} = \sqrt{λ_{\max} (w_{c, i} w_{o, i})} = \frac{{‖ B_{m b, i} ‖}_{2} {‖ C_{m b, i} ‖}_{2}}{4 ζ_{i} ω_{i}}

(8)

where

{‖ . ‖}_{2}

represents the 2-norm,

λ_{\max}

(.) denotes the maximum eigenvalue, and the subscript i corresponds the i^th mode.

Furthermore, drawing on the definition of the cost function in equation (5), by defining $\tilde{Q} = C_{z}^{T} Q C_{z}$ , $\tilde{H} = E_{z} Q E_{z}^{T}$ , it can be established that the weighting matrices are correlated with the HSVs through theorems (Gawronski, 1994):

Theorem 1

Consider diagonal weight matrices $\tilde{Q} = d i a g (q_{i} I_{2})$ and $\tilde{H} = d i a g (h_{i} I_{2})$ , there exist upper bounds ${0 < q}_{i} < q_{o, i}$ , 0< $q_{i} < q_{o, i}$ , $i = 1, \dots, n,$ such that the solutions to the Riccati equations take the block diagonal forms ${\bar{S}}_{c} ≅ diag ({\bar{s}}_{c, i} I_{2})$ , ${\bar{S}}_{e} ≅ diag ({\bar{s}}_{e, i} I_{2})$ . They satisfy the following approximate relationships:

{\bar{S}}_{c, i} ≅ \frac{β_{c, i} - 1}{2 γ_{i}^{2}}, {\bar{s}}_{e, i} ≅ \frac{β_{e, i} - 1}{2 γ_{i}^{2}}, a n d β_{c, i}^{2} = \frac{1 + 2 q_{i} γ_{i}^{2}}{ζ_{i} ω_{i}}, β_{e, i}^{2} = \frac{1 + 2 h_{i} γ_{i}^{2}}{ζ_{i} ω_{i}}

(9)

Here, $q_{o, i}$ and $h_{o, i}$ represent damping authority limit. Specifically, these values correspond to the weighting thresholds at which the i^th resonant peak of the plant transfer function becomes flattened (Gawronski, 1994).

Theorem 2

for the i^th pair, the closed-loop poles for the controller and estimator are shifted along the real axis relative to their open-loop counterparts by factors of $β_{c, i}$ and $β_{e, i}$ , respectively, while the imaginary parts and other poles remain unchanged. i.e.:

Re (λ_{cc, i}) j Im (λ_{cc, i}) ≅ β_{c, i} Re (λ_{co, i}) j Im (λ_{co, i})

Re (λ_{eo, i}) j Im (λ_{eo, i}) ≅ β_{e, i} Re (λ_{eo, i}) j Im (λ_{eo, i})

(10)

where the subscript notations are defined as follows: o = open-loop, e = estimator, the first c = controller, and the second c = closed-loop.

The relative errors of diagonal blocks ${\bar{s}}_{c, i}$ and ${\bar{s}}_{e, i}$ , denoted as $ε_{c, i}$ and $ε_{e, i}$ , satisfy the conditions $ε_{c, i} / {\bar{s}}_{c, i} ≪ 1$ and $ε_{e, i} / {\bar{s}}_{e, i} ≪ 1$ .

Controllers designed under these constraints are classified as low-authority controllers, which only moderately modify the inherent dynamic properties of the system. By sacrificing the undesirable components of structural responses, such controllers yield smaller gain matrices and thus enable the adoption of more compact control devices.

Based on Moore’s representation, there exists a specific modal representation where $S_{c}$ and $S_{e}$ are approximately equal and diagonal dominant. The transformation matrix ${\bar{T}}_{i}$ , $(i = 1, \dots, n)$ defined via the coordinate transformation $x_{mb} = \bar{T} \bar{x}$ , assumes a block-diagonal structure with:

{\bar{S}}_{c} ≅ diag ({\bar{s}}_{c, i} I_{2}), {\bar{S}}_{e} ≅ diag ({\bar{s}}_{e, i} I_{2}), {\bar{S}}_{c} = {\bar{S}}_{e} = \bar{M} = diag (μ_{1}, μ_{2}, \dots, μ_{N}), μ_{1} \geq μ_{2} \geq \dots \geq μ_{N}

(11)

{\bar{T}}_{i} = {({\bar{s}}_{e, i} / {\bar{s}}_{c, i})}^{1 / 4}, a n d μ_{i} = \sqrt{{\bar{s}}_{c, i} {\bar{s}}_{e, i}}

(12)

where

Σ_{lqg} = diag (μ_{i})

, and

μ_{i}, (i = 1, \dots, n)

is called the LQG characteristic (or singular) value (Gawronski, 1994).

This representation is denoted as LQG-balanced system $(\bar{A}, \bar{B}, {\bar{C}}_{y}, {\bar{D}}_{y})$ . Subsequently, equation (5) are transformed into LQG-balanced forms, yielding the balanced feedback gains $\bar{K}$ and $\bar{L}$ . LQG control based on this balanced representation greatly facilitates the computation complexity while retaining explicit physical interpretations of the control parameters.

Semi-active control by MR damper

The control force f generated by the MR damper is numerically simulated using the Bouc-Wen model. This model correlates the damper force with the command input $u$ (i.e., the desired control force) via the following linear relationships:

α = α_{a} + α_{b} u {and c}_{0} = c_{0 a} + c_{0 b} u

(13)

where

α

denotes the hysteretic parameter and

c_{0}

represents the viscous damping coefficient of the MR damper;

α_{a}

α_{b}

c_{0 a}

, and

c_{0 b}

are model-specific calibration coefficients.

To account for the dynamic response characteristics of the MR fluid, a first-order filter is incorporated to relate the command voltage $v$ applied to the control circuit to the desired control force $u$ , as expressed by:

\dot{u} = - η (u - v)

(14)

Detailed formulations of the Bouc-Wen model and the calibration values of all parameters in equations (13) and (14). can be found in references (Dyke et al., 1996; Wang and Dyke, 2021). These parameter settings are specifically calibrated to ensure that the MR damper achieves a maximum output force capacity of 1000 kN.

However, for semi-active systems, the actual control force vector $f$ exerted by the MR damper defers from the ideal command force $u$ generated by the controller, i.e., $f \neq Kk$ . Instead, the real-time measured value of $f$ must be fed back to adjust the subsequent control actions. The updated command force $u$ and the state estimation equations are then given by

u = L^{- 1} {- KL {\begin{array}{c} y \\ f_{meas} \end{array}}}, \dot{\hat{x}} = (A - L C_{y}) \hat{x} + [\begin{array}{l} L & B - L D_{y} \end{array}] {\begin{array}{c} y \\ f_{meas} \end{array}}

(15)

where

L {.}

is the Laplace transform operator, and

f_{m e a s}

is the measured force.

The clipped-optimal control algorithm based on acceleration feedback, proposed by Dyke et al. (1996), was adopted to generate the command signal for the i^th MR damper.

In this study, weightings (corresponding to regulated outputs) are configured to prioritize three key structural responses: corner base drifts, corner base accelerations, and corner top-floor accelerations. The sensor noise is modeled as band-limited white noise, with an RMS magnitude equivalent to 3% of the RMS response of the passively isolated system for each measured quantity. For feedback signal acquisition, accelerometers are installed at corners 2 and 4 from the 3^rd to the 8^th floor, while displacement sensors are placed at corners 2 and 4 of the isolation layer.

To comprehensively evaluate the control performance, four control scenarios are simulated for comparative analysis: the proposed semi-active control, the passive control (without MR dampers), the passive-on (MR dampers energized with a constant 10 V), and the passive-off (MR dampers de-energized with 0 V).

Reduction index and reduction strategy

The order-reduction index is defined as the product of Hankel singular value $γ_{i}$ and LQG characteristic value $μ_{i}$ , and a matrix of the reduction indices matrix is constructed as

σ_{i} = γ_{i}^{2} μ_{i}, where σ_{i} = \frac{\sqrt{(β_{c, i} - 1) (β_{e, i} - 1)}}{2}

(16)

The scaling factors $β_{c, i}$ and $β_{e, i}$ are correlated with mobility of pole pairs shifting toward the left half of the complex plane (equation (10)). If a given pole pair can be shifted with minimal weighting adjustments, the corresponding system states are deemed highly controllable and observable. In contrast, stationary pole pairs correspond to a HSV of $σ_{i} = 0$ ; for mobile pole pairs, $σ_{i} > 0$ . Consequently, $σ_{i}$ serves as a robust indicator of the relative importance of system states and is thus adopted as the core order-reduction index in this study.

By sorting the values of $σ_{i}$ in descending order, the system states can be partitioned into retained states (denoted by the subscript r) and truncated states (denoted by the subscript t). The full-order estimator model is thereby decomposed as follows:

{\begin{cases} {\dot{\hat{\bar{x}}}}_{r} \\ {\dot{\hat{\bar{x}}}}_{t} \end{cases}} = [\begin{array}{c} {\bar{A}}_{r} - {\bar{L}}_{r} {\bar{C}}_{y, r} & 0 \\ 0 & {\bar{A}}_{t} - {\bar{L}}_{t} {\bar{C}}_{y, t} \end{array}] {\begin{cases} {\hat{\bar{x}}}_{r} \\ {\hat{\bar{x}}}_{t} \end{cases}} + [\begin{array}{l} \begin{array}{l} {\bar{L}}_{r} & {\bar{B}}_{r} - \end{array} {\bar{L}}_{r} {\bar{D}}_{y, r} \\ \begin{array}{l} {\bar{L}}_{t} & {\bar{B}}_{t} - \end{array} {\bar{L}}_{t} {\bar{D}}_{y, t} \end{array}] {\begin{array}{c} \bar{y} \\ f_{meas} \end{array}}

(17)

The reduced estimator is then derived by discarding the truncated states:

{\dot{\hat{\bar{x}}}}_{r} = ({\bar{A}}_{r} - {\bar{L}}_{r} {\bar{C}}_{y, r}) {\hat{\bar{x}}}_{r} + (\begin{array}{l} {\bar{L}}_{r} & {\bar{B}}_{r} - \end{array} {\bar{L}}_{r} {\bar{D}}_{y, r}) {\begin{array}{c} \bar{y} \\ f_{meas} \end{array}}

(18)

The control force constitutes the output of the controller. For the full-order system, the control force is calculated as $\bar{K} \hat{\bar{x}}$ . Partitioning $\bar{K}$ as $[\begin{array}{l} {\bar{K}}_{r} & {\bar{K}}_{t} \end{array}]$ , the order-reduced control force is obtained as:

f = {\bar{K}}_{r} {\hat{x}}_{r}

(19)

The implementation procedure of the approximate reduction method is summarized as follows: first, the weighting-dependent factors $β_{c, i}$ and $β_{e, i}$ are selected; second, the ${\bar{S}}_{c}$ & ${\bar{S}}_{e}$ and corresponding $\bar{K}$ & $\bar{L}$ ,are computed; finally, the reduced-order control force is determined using equation (19). The overall control framework is implemented in Simulink, with the corresponding simulation model illustrated in Figure 2:

Figure 2.

Simulink program of the structural control.

Control effect analyses

Pole shifting & order index with weighting

Controller design inherently involves a trade-off between achieving favorable transient response performance and ensuring a sufficiently low bandwidth—this low bandwidth is critical to preventing sensor noise from significantly interfering with actuator operation (Wang and Dyke, 2021). For base isolation systems, a key strategy to balance these two objectives is to assign larger $β_{c . i}$ values to lower-order modes and smaller $β_{c, i}$ values to the higher modes. Meanwhile, the estimator poles are configured to be 2–6 times faster than the controller poles, ensuring that the decay rate of estimator errors is significantly higher than that of the desired system dynamics.

This section begins with Case 1, where the weighting-dependent scaling factors are specified as follows:

• For $β_{c, i}$ s: the first six modes (first two modes in x-, y- and r-directions, respectively) are assigned 4.8, 4.6, 3.1, 1.4, 1.4, 1.02; all remaining higher modes are assigned a uniform value of 1.1.

• For $β_{e, i}$ s: the first three modes are assigned 5.2, 4.8, and 3.0; all other modes are assigned a uniform value of 2.

With this set of scaling factors, the real-part versus imaginary-part distributions of the eigenvalues (i.e., poles) of the closed-loop plant and the estimator are illustrated in Figure 3:

Figure 3.

Pole locations with selected weightings, Case 1.

Figure 3(a) illustrates that the first three pole pairs of the uncontrolled (open-loop) structure are distributed within the range of −1∼−1.5 on the real axis. The full-order Moore balanced control (LQG-CL) shifts these pole pairs leftward to the range of −1.5∼−4, while leaving the other higher-order poles unchanged. In contrast, the approximate algorithm further shifts the first three pole pairs to the left (to the range of −1.5∼−4) and moves the higher-order poles closer to the real axis. It follows that the full-order method yields low-authority controllers, where the damping effect primarily acts on the real-axis components of the poles corresponding to each mode. However, the approximate method partially compromises this property.

Overall, this set of weighting factors effectively suppresses the first three modes, which are the dominant contributors to seismic-induced structural displacements. Meanwhile, the higher-order modes are not significantly perturbed, thereby avoiding any substantial increase in structural accelerations. Although the approximate method exhibits minor discrepancies relative to the exact LQG-CL method, it still achieves satisfactory control performance for the lower (isolation) modes.

Figure 3(b) reveals that the first three pole pairs of the estimator are located even further to the left on the complex plane (−5∼−15), which indicates a more rapid decay rate of estimator noise.

Figure 4 presents the Bode magnitude plots describing the transfer functions from the controller at bearing 3 to the base drift and roof acceleration responses at corner 1, corresponding to the weighting scheme adopted for Case 1.

Figure 4.

Singular values of the uncontrolled (open loop) and controlled systems, Case 1.

Figure 4(a) illustrates that the peak magnitudes of the base drift responses corresponding to the first several modes are reduced. For instance, the peak magnitude of the 1^st mode decreases from −91.2 dB to −81.5 dB and shifts leftward in frequency (from 2.7 rad/s to 1.29 rad/s). The discrepancy between the LQG-balanced and Moore-balanced methods is negligible. Similar trends are observed for the acceleration responses in Figure 4(b).

The controllability of the system and the feasibility of the control strategy can also be evaluated using the LQG characteristic values $μ_{i}$ and the order-reduction index $σ_{i}$ (see Figure 5).

Figure 5.

LQG characteristic values μ and order reduction indices σ, Case 1.

Figure 5 demonstrates that higher controllability is indeed associated with the three isolation modes: both $μ_{i}$ and $σ_{i}$ reach their peak values in these modes, which underpins the feasibility of pole shifting. Additionally, the approximate closed-form method (the red circled lines) exhibits good agreement with the exact singular value decomposition (SVD) method (the blue dotted lines), indicating sufficient accuracy for practical applications. Based on the distributions of $μ_{i}$ and $σ_{i}$ , reducing the controller order to 12–18 modes is deemed appropriate for validating the effectiveness of the proposed method.

Validation by earthquake control performance

In civil engineering, near-field earthquakes—characterized by abundant low-frequency components—are typically the most adverse to structures. This is particularly true for base-isolated buildings, which possess long fundamental natural periods that are prone to resonance with low-frequency seismic waves. To evaluate the seismic performance of the reduced-order LQG algorithm, six near-field earthquake records were selected in this study: Newhall, Sylmar, Rinaldi, Kobe, Ji-ji, and Erzincan (numbered 1 to 6 sequentially). Each earthquake record includes both fault-normal (FN) and fault-parallel (FP) directional components, with their time-history curves presented in Figure 6(a).

Figure 6.

Earthquake records used for time history analysis.

A simple white noise excitation $w (t)$ is insufficient to adequately inform the controller of the frequency-domain characteristics of seismic motions. Thus, a Kanai-Tajimi shaping filter, expressed in equation (20), is adopted to replicate the frequency content of earthquake ground motions and generate a filtered non-white noise input:

G_{KT} (S) = \frac{2_{ζ g} ω_{g} s + ω_{g}^{2}}{s^{2} {+ 2}_{ζ g} ω_{g} s + ω_{g}^{2}}

(20)

With

ω_{g} = 1.5

rad/s, and

ζ_{g} = 0.2

, the filter is capable of covering the envelop of frequency response curves of the earthquake (Figure 6(b)).

Subsequently, the dynamic characteristics of the noise shaping filter are incorporated into the original structural model as an additional degree of freedom (Wang and Dyke, 2007). All seismic response analyses in this study are conducted based on this augmented structural model.

Not to lose generality, the maximum inter-story responses of the structure equipped with the full-order controller and the 18-mode reduced-order controller (designed based on the Case 1 weighting scheme) under one representative earthquake excitation—the Newhall earthquake—are illustrated in Figure 7.

Figure 7.

Max. Response comparison to the Newhall earthquake, Case 1.

It can be observed that the “passive on” case yields the most pronounced reduction in base drift, yet at the cost of notably amplified inter-story drifts in the upper floors and elevated accelerations. By contrast, the dynamic responses of the “passive-off” case exhibit negligible differences from those of the pure “passive” case. Compared with these passive control configurations, the “semi-active” case demonstrates distinct smart control advantages: it effectively mitigates the inter-story drifts of the lower floors while avoiding substantial increases in accelerations.

In addition, both the acceleration and inter-story drift responses indicate that the 18-mode reduced-order controller achieves performance levels comparable to those of the full-order controller. The primary discrepancies between the two controllers are concentrated at the base level: the 18-mode controller induces a greater reduction in base drift but causes a slight increase in base acceleration along the x-direction, whereas the opposite trend is observed along the y-direction.

Figure 8 presents the maximum base drift, roof acceleration, and first-floor inter-story drift responses obtained from the full-order controller and the 18-mode reduced-order controller, respectively. The horizontal axis of these plots denotes the selected earthquake excitations, numbered sequentially as 1 to 7 corresponding to the Newhall, Sylmar, El Centro, Rinaldi, Kobe, Ji-ji, and Erzincan earthquakes (e.g., the Newhall earthquake is designated as #1).

Figure 8.

Max. Response comparison to seven earthquakes, Case 1.

Evidently, the dynamic responses of the “passive” and the “passive-off” systems are nearly identical; the “passive-on” system is the most effective in suppressing base drift but at the cost of amplified accelerations and inter-story drifts within the superstructure. In contrast, the semi-active control system demonstrates outstanding adaptability to diverse earthquake excitations: it achieves a comparable level of base drift reduction to the “passive-on” system while mitigating the undesirable acceleration amplification effect. Furthermore, across all selected earthquake cases, the discrepancies between the 18-mode reduced-order controller and the full-order controller are negligible. These results collectively verify that the proposed reduced-order algorithm is both efficient and robust in enhancing the seismic performance of base-isolated structures.

Parameter analysis: Other sets of weightings

If the estimator gain $L$ is excessively small, the sensor noise can be deemed negligible, but the estimator response will exhibit a sluggish dynamic response. Consequently, the estimation error will be unable to offset the interference induced by the process noise $w$ ; Conversely, a large $L$ value will accelerate the estimator’s response speed, however, this comes at the cost of amplified estimation errors caused by sensor noise. It follows that different estimator weighting schemes will yield distinct control force outputs.

To investigate the effects of estimator gains on control performance, Case 2 is proposed, where the controller scaling factors $β_{c, i}$ are fixed and $β_{e, i}$ are varied. Specifically, Case 2 adopts larger $β_{e . i}$ values for lower modes and smaller values for higher modes compared with Case 1. The values of $β_{e, i}$ for the first nine modes are specified as: 7.0, 4.5, 2.5, 2.5, 2.2, 1.8, 2.1, 1.9, 1.7. The distribution of estimator poles corresponding to Case 2 is illustrated in Figure 9(a), and the LQG characteristic values for this case are presented in Figure 9(b).

Figure 9.

Estimator & LQG characteristic value, Case 2.

A comparison of Figure 9(a) and Figure 3(a) reveals that the first several pole pairs are shifted even further toward the left half of the complex plane under Case 2. Similarly, a comparison of Figure 9(b) and Figure 5 demonstrates that the order-reduction indices for Case 2 are substantially larger in the low-order modes and considerably smaller in the high-order modes. This distribution characteristic enables the truncation of more high-order modes while still maintaining favorable control performance. Based on this observation, a reduction to 12–15 modes is deemed adequate for practical applications.

Figure 10(b) presents the structural dynamic responses obtained with the 12-mode reduced-order controller.

Figure 10.

Maximum structural responses using the 12-mode controller, Case 2.

A comparison of Figure 10(b) and Figure 7 reveals that the 12-mode reduced-order controller derived from this weighting scheme yields even superior performance to both the full-order and 18-mode controllers in Case 1: it achieves a more significant reduction in base drift, while the inter-story accelerations are lower than those of the 18-mode controller in Case 1 and remain close to the levels attained by the full-order controller. These results demonstrate that, for the identical set of controller scaling factors ( $β_{c, i}$ ), the estimator weighting scheme adopted in this case outperforms that in Case 1.

Subsequently, a set of smaller $β_{c, i}$ values was investigated, with the first six modes assigned values of 4.0, 3.5, 2.0, 2.0, 1.5, 1.2, and and all remaining modes assigned a uniform value of 1.05. This parameter configuration is designated as Case 3.

To evaluate the control performance of Case 3 and the 18-mode controller from Case 1, step response analyses were conducted. Figure 11 presents the x-direction displacement of corner #1 in response to the input applied to MR damper #1. In both subplots of this figure, the upper curve corresponds to the closed-loop plant (Plant_cl), while the lower curve corresponds to the reduced-order estimator (Est_rdc). The state-space matrices for both systems are given in equation (7).

Figure 11.

Step responses of the close-loop plant (upper) & the estimator (lower), Case 3.

Figure 11 demonstrates that, for the controlled structure, smaller controller weighting factors lead to a longer settling time (2.21 s in Case 3 versus 1.38 s in Case 2). Nevertheless, the response in Case 3 reaches its peak earlier, indicating that the control action is activated and functions more promptly. For the estimators, the settling times of the two cases are comparable, yet Case 3 exhibits noticeably smaller oscillation amplitudes. These results collectively indicate that Case 3 represents a more optimal controller configuration, as it achieves superior control performance with lower control effort. Notably, the optimal weighting parameters can be determined efficiently through only a limited number of trials, with step response analysis serving as a reliable evaluation tool.

The singular value distributions and maximum inter-story response profiles of Case 3, obtained with the 12-mode reduced-order controller, are presented in Figure 12.

Figure 12.

Responses with case 3 weightings and 12-mode controller, Case 3.

These results further validate the aforementioned conclusions: compared with Case 1 (which adopts larger weighting factors), Case 3 achieves more effective suppression of the fundamental modes (with a magnitude of −89.3 dB versus −91.2 dB in Figure 4(a)), while inducing a smaller frequency shift (1.96 Hz versus 1.13 Hz). These simulation results verify that the proposed order-reduction strategy features high operational explicitness and design efficiency (seeFigure 5 and Figure 9(b)).

Comparison with traditional nodal approach using a full-order controller

While the approximate modal approach with a reduced-order controller has been proven effective for seismic control in comparison with the passive, passive-on, and passive-off cases, its performance relative to alternative control methods remains a topic of interest to researchers. To address this gap, this section presents a comparative analysis between two control schemes: the traditional LQG control based on nodal-coordinate equations with an accurate full-order controller, and the modal-based LQG control (Case 2) with a 12-mode reduced-order controller.

Controller design in the nodal-coordinate framework is inherently non-intuitive, and the optimal weighting parameters are typically determined via a trial-and-error process. Based on the authors’ previous research (Wang and Dyke, 2007), the optimal weighting values were identified as: $q_{b a s e - d r i f t} = 4.462 \times 10^{8}$ , $q_{story - drift} = 4.462 \times 10^{6}$ , and $q_{accelertaion} = 1.145 \times 10^{9}$ . These weighting parameters were applied to the same regulated response positions specified in Section 4.

Normalized response indices, defined relative to the uncontrolled structural responses, are summarized in Table 1. Among these indices, J₃, J₄, J₇, J₁₁ represent displacement-based metrics, while J₅ and J₈ denote acceleration-based metrics. The calculated values of these indices for the six selected earthquake excitations are presented in Table 2. The term “error” is defined as: Error = (Nodal-based results - Modal-based results)/Nodal-based results.

Table 2.

Performance index comparison of the two methods.

Earthquake	Approach	Peak response indices			RMS response indices
Earthquake	Approach	J₃ (base)	J₄ (1^st)	J₅ (roof)	J₇ (base)	J₁₁ (1^st)	J₈ (roof)
Newhall	Nodal full order	1.4942	1.2436	0.766	1.1568	0.7644	0.6988
	Modal reduced order	1.4826	1.3155	0.7344	1.0469	1.1982	0.6879
	Error	0.78%	−5.47%	4.30%	10.50%	−36.20%	1.58%
Sylmar	Nodal full order	1.1384	1.0669	0.7859	0.8829	0.6562	0.5904
	Modal reduced order	1.2789	1.1328	0.7745	0.7516	0.8902	0.5436
	Error	−10.99%	−5.82%	1.47%	17.47%	−26.29%	8.61%
Rinaldi	Nodal full order	1.2382	0.7498	0.8333	1.0482	0.4729	0.6503
	Modal reduced order	1.328	0.7906	0.8209	1.0165	1.0751	0.7168
	Error	−6.76%	−5.16%	1.51%	3.12%	−56.01%	−9.28%
Kobe	Nodal full order	1.4592	0.7478	0.7511	1.1794	0.5227	0.638
	Modal reduced order	1.632	0.7898	0.7908	1.1287	1.2415	0.671
	Error	−10.59%	−5.32%	−5.02%	4.49%	−57.90%	−4.92%
Ji-ji	Nodal full order	1.739	0.8001	0.8624	0.9967	0.6981	0.7597
	Modal reduced order	1.7614	0.8665	0.89	0.7872	1.0464	0.7716
	Error	−1.27%	−7.66%	−3.10%	26.61%	−33.29%	−1.54%
Erzincan	Nodal full order	1.2372	0.6537	0.6457	0.8643	0.5172	0.6043
	Modal reduced order	1.3512	0.6594	0.5934	0.938	0.9353	0.5734
	Error	−8.44%	−0.86%	8.81%	−7.86%	−44.70%	5.39%

It can be observed that both control methods are capable of simultaneously reducing the roof acceleration (J₅ < 1, J₈ < 1) and first-floor inter-story drift (J₄ < 1, J₁₁ < 1) in all but a few cases, albeit at the expense of a slight increase in base displacement (J₁ > 1, J₇ > 1). The modal-based reduced-order controller performs comparably to the nodal-coordinate full-order controller: the discrepancies in their peak response indices for the upper stories (the first two columns of data) are mostly within 5%, and for the base drift indices, deviations exceeding 10% are observed in only 2 out of the 6 selected earthquake cases. In terms of the root-mean-square (RMS) response indices, the deviations between the two methods are acceptable for base drifts and roof accelerations, yet they are remarkably large for the first-floor inter-story drifts. This phenomenon may be attributed to the fact that the relative displacements between the first floor and the base undergo more drastic fluctuations under the reduced-order controller, and RMS values are inherently cumulative metrics that amplify such variations.

Conclusions

This study uses a modal-based approach to reduce the order of the LQG controller for the 27-mode (54-state) benchmark smart base isolation (SBI) building. The order reduction index is constructed by the Hankel singular values (HSV) derived from the modal-and-balanced representation of the system. The index enables pole allocation of the controlled system with explicit physical interpretations and approximate closed-form expressions, thereby simplifying the weighting matrix optimization process. The method was validated on the benchmark building equipped with MR dampers. Analyses of closed-loop poles and settling times revealed that, optimal weightings could be determined through only a few trials; these weightings prioritize the more controllable isolation-dominated lower modes, with negligible interference to higher modes. Earthquake time history simulations under six near-field ground motions were conducted to compare the 12-mode reduced controller with full 27-mode controller. Results confirms that, both controllers effectively reduce the inter-story drifts without amplifying superstructure accelerations; the reduced 12-mode controller achieves comparable control performance to the full-order nodal counterpart, with peak index errors being around 5% for most ground motions. Additionally, sensitivity analyses of estimator weightings indicate that, shifting the estimator poles farther to the left of the real axis than the controller poles is a necessary condition for ensuring satisfactory control effects. In summary, the proposed method enables LQG controller order reduction via straightforward calculations, offering high efficiency and practicality for seismic control of large-scale complex structures.

Footnotes

ORCID iD

Yumei Wang

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research described in this paper was financially supported by the Natural Science Foundation of China, No. 51578517.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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