An observer‐based fault tolerance control system with a static filter and its application to high‐rise buildings

This article proposes a fault tolerance control (FTC) system that composes of a state observer and a static filter, which can eliminate the adverse effects of sensor failures on the active mass damper control system of high‐rise buildings and realize fault detection and isolation. First, the accurate acceleration responses are obtained through a static filter that is used for minimizing the differences between the estimated and actual feedback signals. The key point is transformed into a gain optimization problem solved by a linear matrix inequality approach. Then, a state observer uses the detected and isolated acceleration responses as the feedback signals to estimate the whole states, which are used to calculate the control forces. Finally, a new observer‐based FTC system is accomplished for high‐rise buildings. To verify its effectiveness, the proposed methodology is applied to a numerical example and an experimental system. The results demonstrate that the fault tolerance controller has a good performance and stable control parameters, which provides good potential for structural vibration control of high‐rise buildings.

types of failures restrict the development of the AMD control system. These failures include four varieties, complete failure, fixed deviation, drift deviation, and accuracy decrease.
Various sensors are set up to measure the structural responses that are regarded as feedback signals to an AMD control system. Generally, the state vectors of each floor include its horizontal displacement and velocity responses that are too difficult to be measured directly. 12,13 Therefore, a state observer is important for high-rise buildings with an AMD control system. References [14][15][16][17] showed that state observers were beneficial to linear uncertain systems and nonlinear systems. Compared with the horizontal displacement and velocity responses, the horizontal acceleration signals are easier to be measured. Therefore, the AMD control system based on acceleration feedback is more robust. 18 However, accelerometers are easy to introduce fault signals which lead to a large estimation error and reduce the control performance. It is necessary to research deeply to improve the fault tolerance and robustness of observer-based systems. Such improved process is often based on a fault tolerance control (FTC) technology. [19][20][21][22][23] The principal methods for establishing a FTC system includes an analytical and hardware redundancy method. The hardware redundancy method aims to provide backup hardware for the components that are prone to failure. Disadvantageously, it increases hardware costs and occupies too much available space. In order to be more appropriate in structural vibration control of high-rise buildings, the analytical redundancy method, which improves the system redundancy through the optimal design of its control gains, has attracted attentions.
A FTC system with an analytical redundancy method contains a passive system and an active system. 24,25 The passive system is regarded as a traditional robust system without online fault identification. On the other hand, the active system utilizes a fault detection and isolation (FDI) technology [26][27][28][29] to realize the online fault identification. A sensor fault compensation system was based on a FDI system and a model predictive control strategy in Reference 30. The results show the reliability of the method to detect and isolate the sensor failure and regulate the steam temperature. In Reference 31, multiple high-order sliding-mode observers with FDI problems were proposed for a certain class of nonlinear systems. A fully decentralized approach towards FDI of autonomous sensors was presented for wireless structural health monitoring systems in Reference 32. Then, a robust FDI scheme was proposed for linear discrete-time systems subject to faults, bounded additive disturbances and norm-bounded structured uncertainties in Reference 33. Similarly, a dynamic filter design method was developed and presented in Reference 34 to overcome the failures in sensors. Even with various fault signals, the control performances of a FTC system with the dynamic filter were stable. However, there are difficulties to achieve a fault-tolerance control process in engineering practices through hardware devices. Future efforts would be focused on a static FDI filter shown as a filter gain matrix to achieve the conversion between the measurement and real outputs. To solve the optimal filter gain easily, a linear matrix inequality (LMI) 35 approach has been widely used. For instance, Reference 36 had a study on the robust fault detection filter design problem for linear time invariant systems with unknown inputs and modeling uncertainties. Besides, the optimal solution of a H ∞ model-matching problem was then presented via a LMI formulation. Reference 37 proposed actuator fault diagnosis of neutral delayed systems with multiple time delays using an unknown input observer that was formulated in terms of linear matrix inequalities. In conclusion, the design problem of a static FDI filter can be transformed into a group of nonlinear matrix inequalities, which can be converted into a group of convex and easily solved linear matrix inequalities.
In this article, the state-space equation of an AMD control system with fault signals is derived first. Then based on a LMI approach, a static filter is built to detect and isolate fault signals through minimizing the difference between the estimated system and the actual system. The detected and isolated signals are regarded as the input of the designed state observer to estimate the whole states, which are used to calculate the control forces of the AMD system. Finally, an observer-based FTC system is completed for a numerical frame and an experimental frame. The control effects and the AMD parameters are used as the control indexes to verify the efficiency of the proposed method.

AN AMD CONTROL SYSTEM WITH FAULT SIGNALS
Fault signals are introduced into an AMD control system that are assumed to follow the form described in Reference 38.
Focusing on a high-rise building (n degrees of freedom), the force equilibrium of its AMD control system is where M 0 , C 0 and K 0 are the mass matrix, the damping matrix and the stiffness matrix of a high-rise building with an AMD control system, respectively. u, w and f are the control forces, the external excitations and the fault signals, respectively. B s , B w , and B e are the position matrices of the control forces, the external excitations and the fault signals, respectively. X is the displacement vectors. The matrices M 0 , C 0 , K 0 , B s , B w , and X are expressed as where x si are the relative displacements of the ith floor and x a is the relative displacements of the auxiliary mass. m a , c a , and k a are the mass, the damping, and the stiffness of the auxiliary mass, respectively. m i , k i , and c i are the mass, the interstorey stiffness, and the damping of the ith floor, respectively. The state vectors Z of the system include the displacements and the velocities. The output vectors Y include the displacements, the velocities, and the accelerations. Specifically, the displacements and velocities are used to verify the control effect of an AMD system, and the accelerations are used to observe the whole states. Therefore, Equation (1) is expressed into the state-space equation as where A, B 1 , and B 2 are the state matrix, the excitation matrix, and the control matrix. C, D 1 , and D 2 are the state output matrix, the direct transmission matrices of the control forces and the external excitations, respectively. E and F are the influence matrices of the fault signals on the state equation and the observation equation. These matrices are expressed as The system (3) is reflected in Figure 1. P is its mathematical model, and C o is an original controller. The system is regarded as increasing a new input when a part of sensors fail to work. When the fault signals cannot be timely eliminated, the control forces are negatively impacted.

Design principle
The static filter C S , which is shown as a gain matrix to achieve the conversion between the measurement and real outputs, is intended to estimate the actual states of the systems through a calculation model which is similar to the controlled structure. The difference r between the estimated and actual state vectors is regarded as the target outputs. The static filter shown in Figure 2 is whereẐ andŶ are the estimated values of the state vectors and the output vectors of the system, r is the detected fault signals, and K f is the filter gain matrix of the static filter. Equation (3) minus Equation (5), and then the residue equation of the state vectors is From (7), the matrix Q is the observable matrix of the system (6) (m dimensions). The system (6) is regarded as a single-input system, so the observable matrix is an m-dimensional square matrix. The states of the system can be completely observable when its observable matrix Q satisfies that the rank is m.
The static filter is completed for a control system, indicating that a relatively small error is obtained. The H ∞ norm of the transfer function between the interference inputs w p (w p = [ u w f ] T ) and the target outputs is a given value, and its H 2 norm is minimized. The design process consists of the following two steps. is a given positive scalar which represents the robust stability of the residue system. If and only if there exists a symmetric positive-definite matrix X 1 such that the following inequality holds.
The residue system (6) is stabilized with a H ∞ performance index according to Reference 39. is a positive scalar which represents the control performance of the residue system. If and only if there exists symmetric positive-definite matrices X 2 and Q such that the following inequalities hold.
The residue system (6) is stabilized with a H 2 performance index according to Reference 40.
Owing to the filter gain matrix K f is coupling with the different matrices of X 1 and X 2 , the variables X 1 , X 2 , and Q are nonconvex and difficult to be solved. Therefore, a variable substitution method 41 cannot be used to linearize these constraints. A public Lyapunov matrix is found to handle the problem.
If and only if there exists symmetric positive-definite matrices P and Q such that the above inequalities hold, the system (3) has a H 2 /H ∞ filter. The second inequality of inequalities (12) can be satisfied by the first inequality.
The above inequalities cannot be solved easily due to P and K f are all unknown variables and coupled together. Therefore, a variable substitution method is now used for linearizing the constraint. Let N = PK f , inequalities (13) is min (14) s.t.
Inequalities (14) are changed as linear matrices. The optimal solutions of N and P are obtained by the solver "mincx" in LMI toolbox of Matlab. The optimal filter gain matrix of the static filter is The state-space equation of the static filter is From the above deduction processes, the static filter shown as Equation (16) is designed by ensuring that the H 2 norm of the residue Equation (6) has an upper bound. A state observer based on structural acceleration responses in Reference 42 is used in the article. The acceleration responses, which are detected and isolated through the static filter, are regarded as the inputs of the state observer. Finally, the estimated state vectors are used for calculating the control forces of the system. Therefore, the control feedback strategy of the system (3) is described as where G is a closed-loop feedback gain matrix, andỸ 1 is the estimated state vectors of the observer. Depending on the deduction above, the rebuilding observer-based FTC system is illustrated in Figure 3. Y 2 is the detected and isolated acceleration signals which are used for calculating the estimated state vectors. The symbol inside the dashed box in the figure represents the static filter.
The simulink diagram of an observer-based FTC system

Numerical verification
In this study, a 10-storey frame has been established for the numerical analysis according to Reference 43. Based on Section 3.1, an observer-based FTC system is developed for the 10-storey frame. The accelerometer on the seventh floor is supposed to introduce into two comparatively significant types of artificially added fault signals, including complete failure and accuracy decrease. For instance, (1) Complete failure: a sine wave (the amplitude of 1 m/s 2 , the period of 2 seconds). (2) Accuracy decrease: a white Gaussian noise (the power is 0.1 dBW, the load impedance is 0.1 ). These fault signals are shown in Figure 4. A numerical example of the above 10-storey frame is presented to verify the effectiveness of the observer-based FTC system. Specifically, its performance is verified by comparing with the system without fault signals (No fault signals). Sine wave stands for the system with the sine wave fault signals. White Gaussian noise stands for the system with the white Gaussian noise fault signals. A correlation coefficient r x,y is used for showing the similarity between the estimated and actual states, which is expressed as where x and y are the real and estimated responses of the structure, respectively. Under a 10-year return period wind load, 42 the correlation coefficients between the estimated and actual states of the system with the sine wave fault signals are listed in Table 1. The structural responses and the AMD parameters of different control systems are shown in Figures 5 to 7, and its corresponding control effects and the AMD parameters are listed in Table 2. In this article, the control effect is quantified as the ratio between the dynamic responses of the structures with and without control.
From Tables 1 and 2, as the accelerometer on the seventh floor fails to work, the control effects of the system without fault tolerance are negative. Therefore, the negative effects of the fault signals need to be paid attentions to, and the filter is important for the FTC system. When the sine wave fault signals exist in the system, the correlation coefficients are nearly equivalent to one, meaning that the static filter estimates the structural responses accurately. After isolating the fault signals, the controller reduces the wind-induced vibration responses obviously, and the control effects and the AMD parameters of the system with the fault signals are close to the system without the fault signals. When the sine wave fault

Index
The eighth The ninth The 10th   From Figures 5 to 7, the control system reduces the structural responses effectively when the accelerometer on the seventh floor is in normal working condition. The observer-based FTC system restrains structural responses availably, and its control performance is consistent with that of the system without the fault signals. The system not only detects and isolates the fault signals effectively, but also maintains the stability of the AMD parameters, which are consistent with the system without the fault signals. The same results are achieved when the fault signals are the white Gaussian noise.

EXPERIMENTAL VERIFICATION
The experimental system shown in Figure 8 consists of a four-storey frame with a servo motor installed on the fourth floor. 13 Specifically, the acceleration signals on the second and fourth floors, which are collected by the controller, are used as the feedback signals to calculate the real-time control forces. A servo motor obtains the force signals from an EtherCAT bus system, and then is used for adding these forces to the experimental frame. The structural responses of each floor are used for verifying the control effectiveness. The observer-based FTC system is applied to the experimental system. Assuming that the accelerometer on the second floor fails to work, and the artificially added fault signals are shown in Figure 4. In these figures, a period of 30 seconds is merely given. The original signals, which include the acceleration responses of the second and fourth floors, are processed by the static filter.
Under a sinusoidal excitation load that has a 1 Hz loading frequency and a 45.89 N peak value, the structural responses of different control systems are shown in Figures 9 and 10, and the corresponding control effects and the AMD parameters are listed in Table 3.
From Table 3, when one of the accelerometers fails to work, the system increases the structural responses and has a negative impact on vibration control. Specifically, the structural response control effects of these systems with the fault signals are negative. For the sine wave fault signals, the control effects and the AMD parameters of the FTC system are nearly equal to the system without the fault signals. Comparing the two different systems, the maximum variations in displacements and acceleration control effects are 2.76% and 9.94%, respectively. At the same time, the AMD parameters of the FTC system are reduced by 0.50 N and increased by 1.25 cm, respectively. For white Gaussian noise signals, the From Figures 9 and 10, the observer-based FTC system restrains structural responses availably. Besides, the peak values of different dynamic responses of the system with the fault signals are consistent with that of the system without the fault signals. It means the FTC system detects and isolates the fault signals with different types effectively, and it restrains the structural responses obviously and maintains the AMD parameters in an appropriate range. For the experimental curves, the structural responses disobey the sine law completely. This phenomenon is caused by the interaction between the control system, the experimental frame and the coupling between horizontal and vertical structural vibrations. Due to the acceleration control needs high-frequency control forces that mitigate the high-order modes of the experimental frame, the control effects of the third floor, which have an opposite high-order phase with the fourth floor, is significantly less than those of the second and fourth floors.

CONCLUSIONS
In this article, an observer-based FTC system with a state observer and a static filter is proposed for high-rise buildings. The state observer based on structural accelerations is designed for estimating the state vectors of an AMD control system, and the static filter is presented to mitigate the negative effect of fault signals in accelerometers. A numerical example and an experimental frame are presented to verify the effectiveness of the proposed method. The main conclusions are as follows.
(1) When several accelerometers fail to work, the control system is introduced into new input signals which bring harmful interferences. An AMD control system with fault signals increases structural responses and has a negative impact on vibration control.
(2) A well-designed static filter is used to isolate fault signals and obtain the accurate acceleration responses which are regarded as the feedback signals to calculate the control forces.
(3) The observer-based FTC system restrains structural responses availably, and the system not only detects and isolates the fault signals effectively, but also maintains the stability of the AMD parameters, which are consistent with the system without the fault signals.
(4) The same results are obtained in the experimental system. The control effects and the AMD parameters of the FTC system are consistent with the system without fault signals, indicating that its robustness is improved.
In conclusion, the observer-based FTC system is a suitable robust control method for high-rise buildings. The filter of the observer-based FTC system is presented as the form of a gain matrix, which presents a static characteristics. It is convenient for implementing the FTC system with hardware devices and achieving the conversion between the measurement outputs and the real outputs quickly. On the other hand, a convergence problem may exist in the solution of a local optimal gain since the stability of the filter is not well controlled. In result, it cannot be applied widely to achieve the complex situation of fault-tolerance control in engineering practices. Thus, future efforts could be focused on a dynamic FDI filter which is easy to converge and widely applied to accomplish online fault identification in structural vibration control of high-rise buildings. Moreover, future investigations should also include a designed reduced-order controller to reduce the calculation time of the control system and a robust controller applied to high-rise buildings with parametric uncertainties.