Resilient active seismic response control of structural systems

Abstract

This paper proposes a dissipative resilient observer and controller (DROC) design for a network controlled system (NCS) that handles faults, implementation errors, or cyberattacks that can be modeled as bounded controller or observer gain perturbations. It presents linear matrix inequality (LMI) conditions for the robust stability of the system in the presence of bounded perturbations in the observer and controller. Furthermore, a new LMI-based time-delay control (TDC) algorithm that mitigates the effects of perturbations due to time-delays in the NCS is introduced. The robust methodology is applied to active control of a scaled model of a structural system equipped with an active mass driver system. The results demonstrate that the proposed methodology is robust and ensures stable system response due to various types of earthquake base excitations.

Keywords

networked control system control of structural systems time-delay compensation dissipativity dissipative LMI resilient observer and controller design

Introduction

An actively controlled structural system, such as building and bridge, can be viewed as a complex networked control system (NCS) that includes a communication network through which sensor and control signals are transmitted via wireless and/or wired means within a close the control loop. The communication system in NCS is prone to random time-delays, packet drops, and furthermore NCS may be susceptible to cyberattacks. Hence, NCS must be carefully designed to mitigate the effects of time-delay and packet drops, and to ensure resilient system against various forms of disturbances. Additional sources of time-delay can be identified in a control system due to computational time and hardware (sensor data acquisition, actuator, etc.). While the total end-to-end time-delay within the control loop is typically in the order of the system-sampling rate, larger random time-delays may occur. However, delays, even orders of magnitude smaller than the sampling rate may cause instability.

The terms “fragile” and “non-resilient” are used interchangeably for a controller. Such controllers are subject to significant degradation in the system response due to implementation errors, failure, or cyberattacks. A resilient control system is “one that maintains state awareness and an accepted level of operational normalcy in response to disturbances, including threats of an unexpected and malicious nature” [Reiger et al., 2009]. Robust resilient observer and/or controller design for discrete-time systems and cyber-physical systems (CPS) compensate for bounded errors in the observer gain that may result in an unacceptable system response [Duan and Yu, 2013; Fadali and Yaz, 2018, Feng et al., 2015; Keel and Bhattacharyya, 1997; Yaz et al.,2005]. Yaz et al. [2005] introduced a resilient non-fragile observer design for discrete-time systems. They used linear matrix inequalities (LMI) to design observers with guaranteed stability and performance indices. Feng et al. [2015] designed resilient full-order controllers and observers that are robust with respect to perturbations in both controller and observer gains. They selected the locations of the closed-loop eigenvalues subject to $H_{2}$ norm bounds. Li and Yongheng [2009] introduced non-fragile L₂–L_∞ control for a class of singular time-delay systems via an observer-based controller with additive gain variations and derived a delay-dependent stability condition using LMI. They derived sufficient conditions for the existence of a non-fragile observer and an observer-based L₂–L_∞ stabilization controller for all allowable time-varying delays and controller gain variations with a prescribed L₂–L_∞ performance index.

Fadali and Yaz [2018] presented an observer design that is robust with respect to implementation errors and cyber attacks in the form of perturbations in the observer gain. They derived LMIs that guarantee QSR dissipativity and observer stability in terms of the nominal observer gain. Yan et al. [2017] proposed a hybrid framework to detect a data injection attack and to ensure plant operation. Their method utilizes a mechanism of observer-based detection and a passivity balance defense framework. The administrator prevents attacks by separating user data from bad data, thus ensuring plant operation even after the attack has been detected. The defense mechanism is obtained by a passivation transformation M-matrix design. The design guarantees passivity for the hybrid automaton under attack and the system operates with desired performance even if the system is under attack. The passivity-based design does not require the intense computation required for Kalman filter, LQG controller, or linear-optimization-based game theory methods. Torfi [2018] proposed a resilient controller design for linear time invariant (LTI) systems subject to probabilistic attacks on the sensors and the actuators. The probabilistic attack and the time interval over which each sensor/actuator subject to attack are modeled using Bernoulli random variables.

If the controller and observer are part of a NCS, the network dynamics influences the behavior of the control system. The network introduces a delay between the controlled system and the observer, as well as a delay between the controller and the actuator. Limited network capacity can cause the time-delays to vary randomly and can result in dropping packets of measurement data delivered to the observer or control input data delivered to the actuator via the communication network. The packet delay or loss can lead to signification degradation of the time response, or to instability. Hence, it is important to consider the effect of delays while designing a resilient observer and controller for NCS.

Mathiyalagan et al. [2015] modeled network delays in NCSs as a set of switches between the observer and the controller. A resilient observer for a NCS utilizing the switched system was designed and the system behavior with delay, without delay, and with packet drops was examined. They proposed an exponential dissipative discrete-time switched control with time-varying delays by using the average dwell time approach and a Lyapunov–Krasovskii function. Moreover, using LMIs, the authors designed an observer-based state feedback controller that guarantees the exponential stability of the NCS and strict QSR dissipativity for all network delays and packet losses. Zhang et al. [2008] proposed an observer design for linear system with delayed measurements. They used a functional-based transformation to transform the system with delayed measurements into a system without delay and designed a full-order observer and a reduced-order observer for the non-delayed system. Naghshtabrizi and Hespanha [2005] provided a design procedure of a linear observer-based output-feedback controller for a remote linear plant. The controller design uses the Lyapunov–Krasovskii theorem and a linear cone complementarity algorithm. The controller stabilizes the plant in the presence of delay, sampling, and dropout effects in the measurement and actuation channels.

Time-delay compensation has been investigated in various structural control implementations (e.g. Liu et al. [2016], Lezgy-Nazargah et al. [2020]). Jang et al. [2013] proposed a time-delay control (TDC) algorithm for vibration mitigation in a building structure that estimates system dynamics and disturbances by utilizing immediate past response history. The design is robust to unknown dynamics and disturbances. Agrawal and Yang [2000] evaluated five different commonly used delay compensation methods in civil engineering control applications. Their results showed that the phase shift method is not reliable when the time-delay is large or when the level of active damping is high. Furthermore, the recursive response method and the state-augmented compensated method give more reliable results in terms of stability and control performance. The Schmitt trigger method guaranteed stability in the presence of short time-delays without performance distortion. In general, Pade approximation methods guarantee stability for large or small delays with small performance distortion. Chen and Cai [2009] proposed an optimal discrete TDC algorithm based on an augmented system derived by transforming the equations of motion of the TDC system into a difference equation. The control forces were linearly related to the time-delay states and the state feedback gain was designed by minimizing a specified performance index. This approach ensured stability and mitigated the effects of time-delay.

Time-delays, implementation errors, faults, or cyberattacks can be modeled as a state-dependent or output-dependent term that is equivalent to a gain perturbation for the controller or the observer [Feng et al., 2015; Fattahi and Afshar, 2018]. The present study proposes a dissipative resilient observer and controller (DROC) design that is resilient in the face of bounded perturbations in controller or observer gain due to implementation errors, faults, or cyberattacks. Linear matrix inequality (LMI) conditions for the robust stability of the system with bounded perturbations in the observer and controller gains are provided. Moreover, a delay compensation algorithm is designed to mitigate the adverse effects of time-delays. For the random time-delay compensation algorithm, it is assumed that sensor measurements are time-stamped so that network delay is known as measurements are received. Because the sampling period is short relative to the network dynamics, it is also assumed that the controller-to-actuator delay is approximately equal to the sensor-to-controller delay. An application is demonstrated through simulations on actively controlled model structure presented by Spencer et al. [1998]. Other sources of external disturbances that may occur during the implementation of a plant control in engineering structures are not considered in this paper but can be accounted for in the error dynamics using our methodology, as shown in Fadali and Yaz [2018].

The main significant contributions of this paper are: a) the design of a dissipative resilient NCS controller and observer to handle bounded perturbations, b) the lower computational load of the proposed design in comparison to other methodologies, such as Kalman filters, LQG controllers or linear-optimization, which promotes its practical implementation, and c) an effective random or deterministic time-delay compensation method that applies to both observer and controller, and ensures resilient NCS by means of a computationally inexpensive algorithm, as demonstrated on a benchmark experimental model.

The paper starts with the derivation of general NCS model design with resilient dissipative observer and controller using LMIs. Next, a delay compensation algorithm is proposed to eliminate the adverse effects of time-delay on the system response. The structural control application that utilizes an Active Mass Driver (AMD) system [Spencer et al., 1998] is described and the numerical simulation of the proposed designs is demonstrated. The simulation results and evaluation of both time and frequency domain performance of the designs are discussed and conclusions are presented.

Dissipative resilient observer and controller (DROC) design

Observer design

The state-space model of a linear time invariant continuous-time plant is given as

\dot{x} (t) = A_{a} x (t) + B_{a} u (t)

(1)

y = Cx (t) + Du (t)

(2)

where

x (t) \in ℛ^{n}

is the state vector,

u (t) \in ℛ^{p}

is the control input,

y (t) \in ℛ^{p}

is the measurement vector,

A_{a}

is the state matrix,

B_{a}

is the input matrix,

C

is the output matrix, and

D

is the feedforward matrix. For digital control, the continuous-time model can be discretized using hold equivalence to obtain

x (k + 1) = Ax (k) + Bu (k)

(3)

y (k) = Cx (k) + Du (k)

(4)

in which

A = e^{A_{a} T_{s}}, B = \int_{0}^{T_{s}} e^{A_{a} τ} B_{a} dτ

, and

T_{s}

is the sampling period.

Since it is practically not possible to measure all of the state variables, an observer is used to provide estimates of the state variables. The nominal observer dynamics is given by

\hat{x} (k + 1) = A \hat{x} (k) + Bu (k) + L [y (k) - C \hat{x} (k)]

(5)

where

\hat{x}

is the state estimate and

L \in ℛ^{n \times p}

is the observer gain matrix.

If the system suffers from a fault or cyberattack, an output-dependent perturbation can occur in the term $Ly (k)$ . Assuming a linear approximation of the perturbation, the perturbation takes the form $Δ Ly (k)$ . Implementation errors can also result in an observer gain perturbation. Thus, the nominal observer gain matrix is in practice be replaced by the perturbed matrix $\tilde{L} = (L + Δ L)$ due to computational errors, faults, or malicious attacks, where $Δ L$ is a bounded perturbation subject to

Δ L^{T} Δ L \leq γ I_{p}

(6)

Subtracting the observer equation from the equation of the plant gives the error dynamics. The augmented dynamics including the observer error dynamics and the dynamics for the system with full state feedback control is

[\begin{matrix} \hat{x} (k + 1) \\ e (k + 1) \end{matrix}] = [\begin{matrix} A - BK & - BK \\ 0 & A - \tilde{L} C \end{matrix}] [\begin{matrix} \hat{x} (k) \\ e (k) \end{matrix}]

(7)

where

e (k) = x (k) - \hat{x} (k)

is the estimation error vector,

L

is the observer gain and K is the feedback controller gain (see Figure 1). Equation (7) is the well-known separation principle [Fadali and Visioli, 2020] that allows the design of the controller and the observer separately. Substituting for the output in the observer dynamics yields

\hat{x} (k + 1) = \tilde{A} \hat{x} (k) + (B - \tilde{L} D) u (k) + \tilde{L} y (k)

(8)

where

\tilde{A} = (A - \tilde{L} C)

is the perturbed observer matrix. The estimation error dynamics can be written in terms of the observer matrix as

e (k + 1) = \tilde{A} e (k)

(9)

Figure 1.

Block diagram of the closed-loop NCS.

The performance of the system depends on the error and the dependence is encapsulated in a performance output defined as

z (k) = C_{z} e (k)

(10)

where

z (k) \in ℛ^{p}

. The solution of equation (10) in terms of

e (k)

becomes

e (k) = \tilde{A} e_{0}

(11)

where

e_{0}

is the initial value of the error vector. To characterize the behavior of the error dynamics, we use the concept of dissipativity [Willems, 1972]. A dynamic system is said to be dissipative with respect to the supply rate

w (u (k), z (k))

if a nonnegative storage function

V : R^{n} \to R

exists for all

u (k) \in R^{p}

such that

V (k + 1) - V (k) \leq w (u (k), z (k))

(12)

A dissipative system is called QSR dissipative if the supply rate function is the quadratic [Hill and Moylan, 1976; Marquez, 2003; Moylan and Hill, 1978]

w (u (k), z (k)) = z {(k)}^{T} Qz (k) + z {(k)}^{T} Su (k) + u {(k)}^{T} Ru (k)

(13)

with Q, S, and R matrices with appropriate dimensions and with symmetric matrices Q and R The choice of matrices determines the type of passivity for the system.

Letting $V (k) = e {(k)}^{T} Pe (k)$ where $P > 0$ and for $u (k) = 0$ , equation (12) becomes

e {(k + 1)}^{T} Pe (k + 1) - e {(k)}^{T} Pe (k) \leq z {(k)}^{T} Qz (k)

(14)

Substituting energy and performance outputs in equation (14) gives the asymptotic stability condition for the perturbed observer dynamics

P + C_{z}^{T} Q C_{z} - {\tilde{A}}^{T} P \tilde{A} \geq 0

(15)

where

Q = - δ I_{p}, δ > 0.

Assuming that the pair

(\tilde{A}, C)

is observable for all perturbations, the observer is asymptotically stable if and only if the observer matrix satisfies the Lyapunov stability condition of equation (15) [Fadali and Visioli, 2020]. To provide a robust observer, the QSR dissipative design of [Fadali and Yaz, 2018] can be used to obtain the following LMI

[\begin{matrix} P + C_{z}^{T} Q C_{z} - γ C^{T} C & {(PA - YC)}^{T} & 0 \\ PA - YC & P & P \\ 0 & P & I_{n} \end{matrix}] \geq 0

(16)

with

γ

> 0. After solving the LMI for p and Y, the observer gain L can be determined from

L = P^{- 1} Y

(17)

The results of this section are summarized in the following theorem [Fadali and Yaz, 2018].

Theorem 1

For the system of (1), the observer of (5) subject to the gain perturbations of (6) is robustly asymptotically stable if there exists a solution $(P, Y), P > 0,$ for the LMI (16). The robust observer gain is given by (17).

Controller design

Consider the discrete-time system of equations (3) and (4) with the state feedback control

u (k) = - Kx (k)

(18)

The closed-loop system is given by

x (k + 1) = (A - BK) x (k) = A_{c} x (k)

(19)

where

A_{c} = A - BK

(20)

Based on the separation principle of (7), the controller can be designed using (19) and the state can replace the state in the calculation of the state feedback.

If the objective is to drive the system to the origin, the error is given by $e (k) = x (k)$ and the tracking error dynamics becomes

e (k + 1) = A_{c} e (k)

(21)

Due to implementation errors, faults or cyberattacks, a perturbation can occur in the controller. Wang and Soh [2000], Fattahi and Afshar [2018] represented such faults and cyberattacks as additive perturbation classes in the feedback controller gain matrix. For example, Wang and Soh [2000] proposed two classes of gain perturbations in terms of solutions to algebraic Riccati equations for state feedback control designs. Accordingly, one of the class gain perturbations, $Δ K$ , was defined as an additive form.

Fattahi and Afshar [2018] assumed that attackers may cause gain fluctuations in the network, and modeled the controller gain as the additive form of perturbation. Similarly, this study assumes that, computational errors, actuator faults, or malicious attacks can be modeled as a perturbation $Δ K$ of the nominal gain matrix $K$ . The perturbation $Δ K$ changes the closed-loop state matrix to

\tilde{A} = A_{c} - B Δ K

(22)

The error dynamic equation (21) changes to

e (k + 1) = \tilde{A} e (k)

(23)

The controller gain error $Δ K$ is subject to the perturbation bound

Δ K^{T} Δ K \leq γ_{c} I_{n}

(24)

The closed-loop dynamics become

x (k + 1) = \tilde{A} x (k)

(25)

To design a controller to converge to the desired steady state, the quadratic error energy function

V_{k} = e_{k}^{T} P e_{k}, P > 0

(26)

Must satisfy the dissipativity condition [Willems, 1972]

V (k + 1) - V (k) \leq z {(k)}^{T} Qz (k)

(27)

where

z (k)

is the performance output of the system defined in equation (22) Substituting for

z (k)

gives the inequality

{[C_{z} e_{k}]}^{T} Q [C_{z} e_{k}] \geq e_{k}^{T} [{\tilde{A}}^{T} P \tilde{A} - P] e_{k}

(28)

The inequality is equivalent to the LMI

P + C_{z}^{T} Q C_{z} - {\tilde{A}}^{T} P \tilde{A} \geq 0

(29)

Therefore, substituting $\tilde{A} = A_{c} - B Δ K$ in equation (29) yields

P + C_{z}^{T} Q C_{z} - {(A_{c} - B Δ K)}^{T} P (A_{c} - B Δ K) \geq 0

(30)

which can be expanded as

P + C_{z}^{T} Q C_{z} - A_{c}^{T} P A_{c} + A_{c}^{T} PB Δ K + Δ K^{T} B^{T} P A_{c} - Δ K^{T} B^{T} PB Δ K \geq 0

(31)

Equation (31) can be manipulated by adding and subtracting terms to include the matrix $A_{c}^{T} PB Δ K + Δ K^{T} B^{T} P A_{c}$ and the equation can be rewritten as

P + C_{z}^{T} Q C_{z} - A_{c}^{T} (P + I_{n}) A_{c} + {(A_{c} + PB Δ K)}^{T} (A_{c} + PB Δ K) - Δ K^{T} B^{T} (P + P^{2}) B Δ K \geq 0

(32)

This simplifies the sufficient condition (32) to

P + C_{z}^{T} Q C_{z} - A_{c}^{T} (P + I_{n}) A_{c} - Δ K^{T} B^{T} (P + P^{2}) B Δ K \geq 0

(33)

Applying Lemma-1 (see the Appendix) to equation (33) gives

P + C_{z}^{T} Q C_{z} - A_{c}^{T} (P + I_{n}) A_{c} - Δ K^{T} B^{T} (P + P^{2}) B Δ K \geq 0 \Leftrightarrow [\begin{matrix} P + C_{z}^{T} Q C_{z} & {(A_{c} - B Δ K)}^{T} P \\ P (A_{c} - B Δ K) & P \end{matrix}] \geq 0

(34)

Then applying Lemma-2 (see the Appendix) results in

[\begin{matrix} 0 & - Δ K^{T} B^{T} P \\ - PB Δ K & 0 \end{matrix}] \leq [\begin{matrix} ϵ Δ K^{T} Δ K & 0 \\ 0 & ϵ^{- 1} PB B^{T} P \end{matrix}] \leq [\begin{matrix} ϵ γ I_{n} & 0 \\ 0 & ϵ^{- 1} PB B^{T} P \end{matrix}]

(35)

A sufficient condition for dissipativity is obtained as

[\begin{matrix} P + C_{z}^{T} Q C_{z} - ϵ γ I_{n} & {(A - BK)}^{T} P \\ P (A - BK) & P - ϵ^{- 1} PB B^{T} P \end{matrix}] \geq 0

(36)

Using Lemma-1 gives the inequality

P - ϵ^{- 1} PB B^{T} P \geq 0 \Leftrightarrow [\begin{matrix} P & PB \\ B^{T} P & ϵ I_{m} \end{matrix}] \geq 0

(37)

Combining the two LMIs of equations (36) and (37) gives the sufficient dissipativity condition

[\begin{matrix} P + C_{z}^{T} Q C_{z} - ϵ γ I_{n} & A^{T} P - K^{T} B^{T} P & 0 \\ PA - PBK & P & PB \\ 0 & B^{T} P & ϵ I_{m} \end{matrix}] \geq 0

(38)

where

Y = PBK

After solving the LMI (38) for $P, Y, ϵ$ , the resilient controller gain K can be calculated using

K = B^{#} P^{- 1} Y

(39)

where

B^{#}

denotes the pseudoinverse.

Substituting for the control in the observer equation gives the controller–observer dynamics as in (7)

\hat{x} (k + 1) = \tilde{A} \hat{x} (k) + L y (k)

(40)

where

\tilde{A} = A - LC - BK + LDK, y (k) = Cx (k)

L

is the observer gain, and

K

is the controller gain.

The results of this section are summarized in the following theorem.

Theorem 2

For the system of (1), the control of (18) subject to the gain perturbations of (24) is robustly asymptotically stable if there exists a solution $(P, Y, ϵ), P > 0,$ for the LMI (38). The robust controller gain is given by (39).

Time-Delay Compensation (TDC) Algorithm for NCS

In the NCS system of Figure 1, it is assumed that the data acquisition system samples sensor measurements of the plant output and sends them through a communication channel in the form of data packets that are input to the observer. The observer computes state estimates and the controller uses them to provide observer state feedback. The control values are sent to the actuator and plant as data packets through the communication network. The data packets sent through the communication network experience random time-delay that must be accounted for if the NCS is to function properly. Although the controller-to-actuator and sensor-to-observer/controller network delay may differ, the two delays are approximately equal if the sampling period is sufficiently small relative to network dynamics [Al-Hammouri et al., 2006]. Therefore, the controller-to-actuator time-delay is assumed to be equal to the sensor-to-observer/controller time-delay.

If the data delivered to the observer is time-stamped [Nilsson, 1998], the delay $T_{d} (k)$ is known and can be used to provide timely control to the actuator by correcting for the time-delay. For simplicity, $T_{d} (k) \leq T_{s}$ is assumed so that the control remains constant at each step regardless of the delay. However, as it is demonstrated in a subsequent section, the control with time-delay correction performs satisfactorily even if the time-delay exceeds $T_{s}$ for short durations because the change in the control action from one sampling period to the next is relatively small. The process can be generalized mathematically but the generalized equations are not considered for simplicity.

Because of the sensor-to-observer time-delay, the observer receives the measurement $y (k T_{s} - T_{d} (k))$ at time $k T_{s}$ .The objective is to predict the future state at the time that the control command reaches the actuator using the mathematical model of the plant. The state estimate required is $\hat{x} (k T_{s} + T_{d} (k))$ , where $k T_{s} + T_{d} (k)$ is the time at which the control command reaches the controller. The desired state estimate $\hat{x} ((k + 1) T_{s} + T_{d} (k))$ is calculated as

\hat{x} ((k + 1) T_{s} + T_{d} (k)) = (A_{d} - B_{d} K) \hat{x} (k + 1) T_{s} - T_{d} (k T_{s}))

(41)

in which

A_{d} = e^{A (2 T_{d} (k))}, B_{d} = \int_{0}^{(2 T_{d} (k))} e^{A ((2 T_{d} (k)) λ} B d λ

and A is the state matrix, B is the input matrix of the continuous-time plant. The corresponding control input is

u (((k + 1) T_{s} + T_{d} (k)) = - K \hat{x} ((k + 1) T_{s} + T_{d} (k))

(42)

The controller and observer designs are based on LMIs as in earlier work presented by Fadali and Yaz [2018]. However, the perturbation model used in the present study and the use of LMI design to mitigate the effects of the perturbation are new contributions. The proposed approach may be of interest in many control applications, such as the control of structural systems. The approach has the following advantages and critical limitations:

1. The design handles any type of delay, random, or deterministic, provided that it is less than a specified maximum allowable delay. Although the algorithm is not guaranteed to succeed if the network delay exceeds the maximum allowable delay, as demonstrated through a numerical case study, the control with time-delay compensation performs satisfactorily even if the time-delay exceeds the maximum allowable delay for short durations.

2. Because observer dynamics are used to predict the states, the computational cost of the proposed delay compensation method is low and is suitable for real-time applications

3. The present approach assumes that delays for sensor-to-controller and controller-to-actuator are the same for the proposed delay compensation design. For systems where network dynamics change faster than the system dynamics, this assumption is not valid.

Numerical case studies: structure with an active mass driver

The implementation and performance of the proposed DROC and time-delay compensation (TDC) algorithm are demonstrated through numerical response simulations of a benchmark control problem [Dyke, 2011] introduced by Spencer et al. [1998]. The benchmark structure was designed as a scaled model of a prototype building [Chung et al., 1989], and subjected to uniaxial base excitations for various control performance evaluations. The 158 cm tall, steel model structure and its 2D idealization are shown in Figure 2. The total mass was 309 kg, which included structural frame (77 kg), added similitude masses (75.6 kg per floor), and 5.2 kg AMD moving mass. The dominant frequencies of vibration for the three structural modes were 5.81, 17.68, and 28.53 Hz. For control purposes, a simple implementation of an Active Mass Driver (AMD) was placed on the third floor of the structure. The AMD consists of a single hydraulic actuator with steel masses attached to the ends of the piston rod (Figure 2). The servo-actuated hydraulic cylinder has a 3.8 cm diameter and a 30.5 cm stroke. The readers are referred to Spencer et al. [1998] for complete details.

Figure 2.

The benchmark structure with AMD [Spencer et al., 1998].

The numerical model was subjected to seven ground motion excitations summarized in Table 1. The ground motion records were both time (1:5) and amplitude (7:2) scaled conforming to the similitude requirements. However, due to relatively high seismic demand imposed due to some of the ground motions, the amplitude scale factors listed in the table were used to ensure controlled responses remain below predefined physical constraints. Acceleration, velocity, and displacement response histories of two selected ground motions are shown in Figure 3. For all of the TDC simulations, the following four cases are considered: 1) uncontrolled, 2) controlled with no time-delay, 3) controlled with random time-delay history

(max (T_{d} (t)) \leq T_{s} / 2)

, and 4) controlled with four different random time-delay history

(max (T_{d} (t)) \leq 2 T_{s} / 3, T_{s}, 3 T_{s} / 2, a n d 2 T_{s})

and with time-delay compensation.

Table 1.

Ground motion records.

Eq Id	Record name	$M_{w}$	Recording Station	Dur. (s)	PGA (g)	PGV (cm/s)	PGD (cm)	SF_amp
E01	Northridge, 1994	6.7	Beverly Hills-Mul	23.95	0.62	40.7	8.56	1.75
E02	Chi-Chi, Taiwan, 1999	7.6	TCU045	89.98	0.51	40.0	14.3	1.75
E03	Imperial Valley, 1979	6.5	El Centro Array 7	36.80	0.46	109.3	44.7	1.00
E04	Erzincan, Turkey, 1992	6.7	Erzincan	21.30	0.52	95.5	27.7	1.00
E05	Chi-Chi, Taiwan, 1999	7.6	TCU102	89.98	0.30	112.5	89.2	1.00
E06	Imperial Valley, 1940	7.0	El Centro Array 9	53.70	0.23	31.3	18.4	3.50
E07	Takochi-oki, Japan, 1968	7.9 (M_s)	Hachinohe	36.00	0.23	18.3	7.0	3.50

Figure 3.

Scaled acceleration, velocity, and displacement records of sample earthquake ground motions.

As noted earlier, the proposed LMI-based time-delay compensation is designed to ensure that NCS system will perform satisfactorily for time-delays less than or equal to the sampling period (i.e., $max (T_{d}) \leq T_{s}$ ). For practical applications in structural control, random network delays are typically limited to and/or smaller than delays in other communication networks. On the other hand, subsequent simulation results show that the proposed resilient control system can tolerate larger time-delays. To demonstrate this aspect, a threshold limit was placed on the maximum random time-delay such that when $T_{d} (t)$ exceeds the sampling rate, $T_{s}$ , respective value for $T_{d}$ in equations (41) and (42) was capped to $T_{s} .$ Furthermore, a minimum time-delay, $\min (T_{d}) = T_{s} / 4$ was imposed. A sample random time-delay history is shown in Figure 4 for one of the simulations where the actual imposed random time-delay through the network is specified within $T_{s} / 4 \leq T_{d} (t) \leq 2 T_{s}$ . However, it was limited to the sampling rate during state estimation and computation of control input. In addition, the performance of the proposed LMI-based control design (DROC) is demonstrated subject to perturbations in the controller gain.

Figure 4.

Sample random normalized time-delay history.

Evaluation Model and Control Designs

Spencer et al. [1998] introduced the evaluation model for the AMD system with a sample control design as follows. Consider the continuous time state-space model of the AMD Linear Time Invariant (LTI) system

\dot{x} (t) = Ax (t) + Bu (t) + E (t)

(43)

y (t) = C_{y} x (t) + D_{y} u (t) + F_{g} \ddot{x} (t) + v

(44)

z (t) = C_{z} x (t) + D_{z} u (t) + F_{z} {\ddot{x}}_{g} (t)

(45)

where

A, B, E, C_{y}, D_{y}, F, C_{z}, D_{z}, F_{z}

are matrices of appropriate dimensions,

x (t) \in ℛ^{28}

is the state vector,

u (t) \in ℛ^{p}

is the control input,

z (t) = {[x_{1}, x_{2}, x_{3}, x_{am}, {\dot{x}}_{1}, {\dot{x}}_{2}, {\dot{x}}_{3}, {\dot{x}}_{am}, {\ddot{x}}_{1}, {\ddot{x}}_{2}, {\ddot{x}}_{3}, {\ddot{x}}_{am}]}^{T}

is the response vector to be regulated,

y (t) = {[x_{m}, {\ddot{x}}_{1}, {\ddot{x}}_{2}, {\ddot{x}}_{3}, {\ddot{x}}_{am}, {\ddot{x}}_{g}]}^{T}

is the measurement vector, and

{\ddot{x}}_{1}, {\ddot{x}}_{2}, {\ddot{x}}_{3}

are the absolute accelerations of floor 1,2,3 respectively,

{\ddot{x}}_{am}

is the absolute acceleration of AMD, and

v

is the measurement noise vector.

To simplify controller design, the following reduced-order model is used

\dot{x} (t) = A_{r} x (t) + B_{r} u (t) + E_{r} (t)

(46)

y (t) = C_{yr} x (t) + D_{yr} u (t) + F_{yr} {\ddot{x}}_{g} (t) + v_{r}

(47)

where

x_{r} (t) \in ℛ^{10}

is the state vector,

u (t) \in ℛ^{p}

is the control input,

y_{r} (t) = {[{\ddot{x}}_{1}, {\ddot{x}}_{2}, {\ddot{x}}_{3}, {\ddot{x}}_{am}]}^{T}

is the measurement vector, and

A_{r}, B_{r}, E_{r}, C_{yr}, D_{yr}, F_{yr}

are matrices of appropriate dimensions. In the following numerical simulations, the measurement noise is assumed to be Gaussian white noise.

The evaluation model was verified using an experimentally recorded response and was adopted for subsequent performance evaluations with various control designs. A sample Linear Quadratic Gaussian (LQG) design was provided by Spencer et al. [1998]. LQG control yields an optimal output-feedback law by minimizing the expected value of a quadratic cost function with linear systems and additive white Gaussian noise input. The AMD system response due to the LQG controller is used as the basis for comparison with the proposed DROC and TDC simulations. In addition to direct comparison of response histories for various simulation cases, performance evaluation is presented using the measures summarized in Table 2.

Table 2.

Performance measures.

Criterion	Performance measure
Floor displacement	$J_{1} = m a x [\frac{\| d_{1} (t) \|}{x_{3 o}}, \frac{\| d_{2} (t) \|}{x_{3 o}}, \frac{\| d_{3} (t) \|}{x_{3 o}}]$	(48)
Absolute floor acceleration	$J_{2} = m a x [\frac{\| {\ddot{x}}_{1} (t) \|}{{\ddot{x}}_{3 o}}, \frac{\| {\ddot{x}}_{2} (t) \|}{{\ddot{x}}_{3 o}}, \frac{\| {\ddot{x}}_{3} \|}{{\ddot{x}}_{3 o}}]$	(49)
AMD displacement	$J_{3} = m a x [\frac{\| x_{m} (t) \|}{x_{3 o}}]$	(50)
AMD velocity	$J_{4} = m a x [\frac{\| {\dot{x}}_{m} (t) \|}{{\dot{x}}_{3 o}}]$	(51)
AMD acceleration	$J_{5} = m a x [\frac{{\ddot{x}}_{a m} (t)}{{\ddot{x}}_{a 3 o}}]$	(52)

In Equations (48)–(52), $d_{i}$ and ${\ddot{x}}_{ai}$ are interstory displacement and absolute acceleration responses at the $i^{t h}$ level, $x_{3 o}$ , ${\dot{x}}_{3 o}$ , and ${\ddot{x}}_{3 o}$ are the maximum displacement, velocity, and absolute acceleration responses measured at the third level of the uncontrolled structure, $x_{m}$ , ${\dot{x}}_{m}$ , and ${\ddot{x}}_{am}$ are the displacement, velocity, and absolute acceleration responses of the AMD mass, respectively. For the present study, LMI-based DROC was designed and simulated using a Simulink [Mathworks, 2012] model for its numerical implementation. For all the numerical simulations sensor and controller sampling, $T_{s} = 100 H z$ was assumed. However, a fixed integration time step of 1000 Hz and fifth order Dormand–Prince solver was used. Finally, the model includes all other implementation constraints as presented in Spencer et al. [1998]. Displacement and acceleration responses measured at the third floor level of the model structure are shown in Figure 5 for uncontrolled, LQG, and the proposed LMI-based designs for two selected ground motions E02 (far-field long duration) and E04 (near-fault short duration) (Table 1). Clearly, both designs result in comparable responses.

Figure 5.

Response history comparison of displacements and absolute accelerations due to (a) E02 and (b) E04.

Verification of DORC design

The proposed method was applied to the state-space model described in the previous section. The discrete resilient observer and controller were designed based on discretized reduced-order system dynamics given by equations 46 and 47, based on the model of the structure together with a zero-order hold (ZOH) and sampler with $T_{s} = 0.01 s$ . The following design parameters were used: $Q = - 10^{- 4} I_{4}, γ = 10^{- 6}$ . Using the MATLAB LMI Toolbox, equation (16) is solved for p and Y, and the resilient observer gain L can be calculated using equation (17) as

L = 10^{- 3} [\begin{matrix} - 0.1897 & - 0.0541 & 0.1022 & - 0.0621 \\ - 0.1579 & - 0.0676 & 0.1141 & 0.0698 \\ 0.0579 & - 0.1158 & 0.0023 & 0.0015 \\ 0.1545 & - 0.1475 & 0.0871 & 0.0530 \\ - 0.4099 & - 0.4233 & - 0.6606 & - 0.3975 \\ 0.4521 & 0.4772 & 0.6737 & 0.4049 \\ 0.0013 & 0.0036 & 0.0021 & 0.0013 \\ - 0.0134 & - 0.0186 & - 0.0473 & - 0.0283 \\ 0.0013 & 0.0014 & 0.0043 & 0.0026 \\ - 0.0440 & - 0.0585 & - 0.1065 & - 0.0640 \end{matrix}]

A−LC is stable with all its eigenvalues inside the unit circle as shown in Figure 6.

Figure 6.

Eigenvalues for A-LC and A-BK

The discrete resilient controller is designed by using the proposed method. The same discretized reduced-order system model is used to design the controller. The parameters of the design for the resilient controller are $Q = - 10, R = 0$ , (strictly output passivity [Marquez, 2003]) where $γ = 10$ . Utilizing the resilient controller design method and using the LMI Toolbox of MATLAB, the parameters $P, Y, ϵ$ are calculated using equation (38). Therefore, the resilient controller gain $K$ is obtained using equation (39) as

K = [18.5591, - 34.0284, 9.083, 36.9855, - 2.821, - 9.2406, 1.7306, - 2.6447, - 0.5315, - 7.7258]

The nominal closed-loop state matrix $A - B K$ is stable with all its eigenvalues inside the unit circle as shown in Figure 6. The performance of DROC design is demonstrated by imposing various levels of perturbations on the controller gain, K as

K_{p} = K + Δ K w h e r e Δ K = α_{p} K

(53)

For the present study, the perturbation factor α_p was varied from 1.2 to 4.0 to assess the effect of a range of perturbations in the controller gain on system performance.

Figure 7 through 10 demonstrate seismic response comparisons for various unperturbed and perturbed systems for two of the ground motions; a relatively long duration, far-field (E02) and short duration, near-fault (E04). Response history comparisons are presented for displacement $(x_{m})$ and acceleration $({\ddot{x}}_{a m})$ of AMD mass, third floor relative displacement $(d_{3})$ , and absolute acceleration $({\ddot{x}}_{3})$ . In general, the proposed LMI-based DROC consistently result in stable and acceptable seismic response for both ground motions for a wide range of perturbations. Note that the perturbations may be interpreted as faults in the control system and hardware, implementation errors, or cyberattacks. Higher perturbation factors imply larger consequential deviations from normal operating parameters and control implementation. Clearly, the propagation of the state estimation error compounds and becomes more critical for excitations of longer duration, leading to instability even for relatively small perturbations (Figures 7 and 8). In contrast, the proposed LMI-based DROC design ensures stability for both short and long duration excitations and for large perturbations.

Figure 7.

Response comparison—Ground motion: E02, $α_{p}$ = 1.4.

Figure 8.

Response comparison—Ground motion: E02, $α_{p} = 1.8$ .

Figure 9.

Response comparison—Ground motion: E04, $α_{p} = 1.4$ .

Figure 10.

Response comparison—Ground motion: E04, $α_{p} = 4.0$ .

Figure 11.

Performance index comparison for various time-delay cases.

Verification of TDC in terms of time and frequency domain responses

Time-delay is inevitable in practical applications of active as well as semi-active control of structural systems. In a complete closed-loop system, various sources of time-delay can be identified due to network, computational time, and hardware (sensor data acquisition, actuator, etc.). While the total end-to-end time-delay within the control loop may typically be in the order of the system-sampling rate, larger random time-delays may be expected. The proposed LMI-based time-delay compensation (TDC) method addresses random time-delays up to the effective sampling period, and ensures stable response. However, its performance does not degrade due to larger random time-delays.

Additional numerical simulations were conducted to demonstrate the efficacy of the proposed LMI-based (TDC) approach. For this purpose, the same set of ground motions (Table 1) were used with the control cases listed in Table 3. The performance evaluation is presented using the performance measures,

J_{i}

summarized in Table 2. The value for a particular performance index in Table 3 was calculated as the maximum value of all corresponding values for the ground motions considered. Figure 11 demonstrates that smaller perforance indices, particularly for

J_{1}

and

J_{2}

, indicate more desirable response reductions due to AMD relative to uncontrolled response. Large

J_{2}

values evaluated for all controlled responses are primarily due to system’s response to E02 ground motion, which is a near-fault type record with a large velocity-pulse as shown in Figure 3. While the AMD may not be a feasible active control approach in case of near-fault ground motions, this aspect is outside the scope of the present study. On the other hand, relative performance of the proposed TDC approach is demonstrated. The potential adverse effects of time-delay on the seismic response of actively controlled model are shown in Figure 12 for two ground motions. In these simulations, the maximum random time-delay was assumed to be half of the sampling period. The robustness of the proposed LMI-based TDC approach is evident in Figure 13 and Figure 14 as the structural system remains stable and controller performance is maintained even when large random time-delays occur between the controller, sensors and actuator.

Table 3.

Maximum performance indices.

Control case	J ₁	J ₂	J ₃	J ₄	J ₅
LQG	0.566	1.129	0.836	1.542	5.283
LMI	0.550	1.005	0.683	0.916	2.077
LMI delay (max (T_d) = 1/2T_s)	2.002	20.12	1.473	4.466	35.22
LMI delay comp (max (T_d) = 2/3T_s)	0.544	1.024	0.697	0.905	3.505
LMI delay comp (max (T_d) = T_s)	0.547	1.072	0.689	1.111	4.730
LMI delay comp (max (T_d) = 3/2T_s)	0.550	1.295	0.698	1.236	6.185
LMI delay comp (max (T_d) = 2T_s)	0.553	1.366	0.741	1.410	8.178

Figure 12.

Effect of time-delay (Td(t) ≤ Ts/2) on the controlled response due to (a) E02 and (b) E04.

Figure 13.

Seismic response due to E02 ground motion with time-delay control.

Figure 14.

Seismic response due to E04 with time-delay control.

Finally, examples of frequency response functions (FRFs) and corresponding phase angles between the base excitations and third floor absolute floor acceleration response are shown in Figure 15 for two of the ground motions, namely E01 and E04. The figures present comparisons for cases with no time-delay $T_{d} (t) = 0$ , uncompensated time-delay $(max (T_{d} (t)) \leq T_{s})$ , and cases with compensated time-delay $(max (T_{d} (t)) \leq T_{s})$ and $(max (T_{d} (t)) \leq 2 T_{s})$ ). The apparent peaks in FRFs correspond to the structural modes of vibration. The presence of time-delay lead to significant amplification of FRF magnitude over a wide range of frequencies, particularly beyond the first structural mode of vibration. Furthermore, distortion of phase response due to random time-delay is evident in these figures. The undesired response amplification at high frequencies results in high acceleration response and—in this particular case—requires large control forces that would lead to actuator saturation implied by J₂ and J₅ performance indices. Hence, the efficiency of the proposed TDC in achieving desired response control in terms of both FRF magnitude and phase is demonstrated in Figure 15.

Figure 15.

Frequency response comparisons (a) E01 and (b) E04.

Conclusions

This paper proposes a resilient control system design for networked controlled structural systems. The controller design includes a resilient observer and controller designed using dissipative LMI. The design can tolerate errors in implementation, cyberattacks, and inevitable time-delays within the control loop. The NCS controller design is applied to a state-space dynamic model of a benchmark three-story structure with an active mass driver system using the SIMULINK MATLAB toolbox. The model was subjected to a series of historic far-field and near-fault ground motions. Simulation results indicate that the proposed NCS ensures resilient and stable structural response as demonstrated in both time and frequency domains. Furthermore, the network time-delays in the order of twice the sampling rate can be tolerated without significantly affecting the maximum response of the structural and control system. For further studies, the proposed approach can be extended to nonlinear models.

ORCID iD

Gökhan Pekcan https://orcid.org/0000-0002-9745-1603

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Appendix

Lemma-1:The Schur Complement [Boyd et al., 1994]

For matrices $Q = Q^{T}, S = S^{T},$ then for any compatible matrix $R$

[\begin{matrix} Q & R \\ R^{T} & S \end{matrix}] > 0 \Leftrightarrow S > 0, Q - R S^{- 1} R^{T} > 0

\Leftrightarrow Q > 0, S - R^{T} Q^{- 1} R > 0

Lemma-2 [Fadali and Yaz, 2018]

\exists α > 0, [\begin{matrix} α Q_{1}^{T} Q_{1} & \pm Q_{1}^{T} Q_{2} \\ \pm Q_{2}^{T} Q_{1} & α^{- 1} Q_{2}^{T} Q_{2} \end{matrix}] \geq 0

In addition, if

Q_{1}^{T} Q_{1} < γ Q_{4},

then

\exists α > 0, [\begin{matrix} αγ Q_{4} & \pm Q_{1}^{T} Q_{2} \\ \pm Q_{2}^{T} Q_{1} & α^{- 1} Q_{2}^{T} Q_{2} \end{matrix}] \geq 0

Proof

{[\begin{matrix} x \\ y \end{matrix}]}^{T} [\begin{matrix} α Q_{1}^{T} Q_{1} & \pm Q_{1}^{T} Q_{2} \\ \pm Q_{2}^{T} Q_{1} & α^{- 1} Q_{2}^{T} Q_{2} \end{matrix}] [\begin{matrix} x \\ y \end{matrix}]

= α x^{T} Q_{1}^{T} Q_{1} x + α^{- 1} Q_{2}^{T} Q_{2} y \pm 2 x^{T} Q_{1}^{T} Q_{2} y

= {(\sqrt{α} Q_{1} x \pm Q_{2} \frac{y}{\sqrt{α}})}^{T} (\sqrt{α} Q_{1} x \pm Q_{2} \frac{y}{\sqrt{α}}) \geq 0

Q_{1}^{T} Q_{1} < γ Q_{4},

then

αγ x^{T} Q_{4} x + α^{- 1} Q_{2}^{T} Q_{2} y \pm 2 x^{T} Q_{1}^{T} Q_{2} y

\geq α x^{T} Q_{1}^{T} Q_{1} x + α^{- 1} Q_{2}^{T} Q_{2} y \pm 2 x^{T} Q_{1}^{T} Q_{2} y \geq 0

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