Dynamic output‐feedback event‐triggered H∞ control for singular active seat suspension systems with a human body model

Funding information Australian Research Council, Grant/Award Number: LP160100132; China Scholarship Council, Grant/Award Number: 201608210142; Natural Science Foundation of Heilongjiang Province, Grant/Award Number: YQ2019F004; Natural Science Foundation of China, Grant/Award Number: 62073094; China Postdoctoral Science Foundation, Grant/Award Numbers: 2018M63034, 2018T110275 Abstract The output-feedback event-triggered H∞ controller is designed in this paper for singular active seat suspension systems. The human body is simply separated into two parts (body torso and head) and is considered with the seat suspension system. The accelerations of each part are considered as part of the system states, which makes the system as a singular system. The seat suspension deflection, relative velocity, the accelerations of the seat frame, body torso, and head are defined as the system outputs, which are all measurable in realtime. The output-feedback H∞ controller is designed at first. Then, the event-triggered scheme is designed for the seat suspension system to reduce data transmission. Linear matrix equalities criteria are presented to get the controller and event-triggered scheme and ensure the admissibility of the seat suspension system. Simulation results are shown to illustrate the effectiveness of the considered control method.


INTRODUCTION
Seat suspensions have been a hot research topic because of their important roles to reduce vibrations of drivers and high driving comfort requirements of some vehicles, especially for heavy vehicles like trucks, buses, agricultural vehicles, et al. [1]. Generally, active seat suspensions have the best control performance to reduce vibrations among different types of seat suspensions, and there have been a lot of work to investigate different control methods of active seat suspensions [2]. Some works simplify the seat suspension into a second-order system and use the absolute or relative displacement and velocity of the seat suspension as the controller, which increases the burden of data transmission and leads to time-delay problem [8]. Event-triggered control is one approach to deal with this problem which can evaluate if it is necessary to transmit the current data [9]. Hence, eventtriggered control can reduce the workload of data transmission and deal with the time-delay problem to control performance [10,11]. During the recent years, it is frequently used in many areas [12,13]. An event-triggered H ∞ static output-feedback controller is designed for active vehicle suspension systems with network-induced delays in [14]. Based on event-triggered control, H ∞ control for singular systems with state delay and disturbance is concerned in [15]. In most previous suspension related works, the absolute or relative displacement and velocity signals are considered as system states while the acceleration signals are not considered even though they can be easily measured in real-time by acceleration sensors [16]. Moreover, absolute displacement or velocity signals usually cannot be measured in real-time in practice. In this paper, the human body is simply separated into two parts (head and body torso), which is considered as part of the seat suspension system together with the seat frame. The relative displacements, velocities, and accelerations of each part are defined as system states, which makes the seat suspension system as a singular system [17]. The outputs of this system are chosen as the deflection and relative velocity of the seat frame, the accelerations of the upper seat suspension, body torso, and head, so these signals are easily measured in realtime. The output feedback controller design is only based on the measurable states, which makes the control method more practical. Then the discrete output event-triggered scheme is designed for the seat suspension system, which can avoid the "redundant" data packets and reduce the workload of data transmission.
The main contributions of this paper are listed as follows: • A singular seat suspension system with a human body in the loop is established. Compared with traditional seat suspension systems, the singular seat suspension system also defines the acceleration signals as system states, which makes full use of all measurable signals in real-time. • The output feedback controller of the active singular seat suspension system is designed by only using easily measurable state variables, which makes the control method more practical. • The discrete-time event-triggered scheme is designed for the singular active seat suspension system, which can make the value of the controller only changes when the defined event triggers. Therefore, it can reduce the processor load and data transmission load. • The output-feedback H ∞ controller is designed at first and then the event-triggered scheme is designed based on the obtained controller. Compared with the traditional outputfeedback controller, this two-step design method can minimise the complication of the control system, especially for this complicated seat suspension system, which makes the control process simpler. The rest of this paper is organised as follows: The model of the singular active seat suspension system with a human body in the loop is established in Section 2. The output-feedback H ∞ controller is designed in Section 3. The even-triggered scheme is designed in Section 4. Some simulation results are given in Section 5 and it concludes in Section 6. Figure 1 shows the simplified active seat suspension with human body model. m h , m b , and m s are the masses of human's head, body torso, and seat frame, respectively; k h , k b , k s are their spring stiffness, respectively; c h , c b , c s are their damping coefficients, respectively; u is the active control input; z h , z b , z s and z v are the displacements of human head, body torso, seat frame and the base platform, respectively. The dynamic equations of this model for each part can be derived as follows.

MODEL OF SINGULAR ACTIVE SEAT SUSPENSION SYSTEM WITH A HUMAN BODY
Seat frame: (1) Body torso: Head: Here, we define some state variables as and the state vector as The deflection variable between seat frame and the base platform x 1 can be measured by a laser displacement sensor in realtime and its relative velocity x 2 can be derived by differentiating x 1 . The acceleration variables of the seat frame x 3 , body torso x 6 , and human's head x 9 can be measured by accelerometers in real-time. These five measurable system state variables are chosen as the system state output y, which are used to design the output-feedback controller. The human head's acceleration is closely related to the driving comfort, so x 9 is chosen as the system performance output z. Define the disturbance of the seat suspension system as =z v . Then, the state space equation of the seat active suspension system with a human body can be written as follows: where

THE OUTPUT-FEEDBACK H ∞ CONTROLLER DESIGN
Define the output-feedback controller as u(t ) = Ky(t ). Substituting it into (4), we have To design the controller K , we consider the following equivalent system:Ē̇x ] . (6) is admissible with H ∞ performance index if there exist matrices P > 0 and Q with appropriate dimensions and Z withĒ T Z = 0 satisfying the following conditionsĀ

Theorem 1. The continuous-time singular system
Proof. Firstly, we need to prove the system (6) is regular and impulse-free. Since the system (6) is equivalent to the system (5), the regularity and impulse free are the same for these two systems. When the control input u and disturbance are not considered, according to the system (5), we have Then, we can get det(sE − A) = a 6 s 6 + a 5 s 5 + +a 4 s 4 + +a 3 (5) and (6) are regular and impulse free.
Then, we need to prove the stability of the system (6). Define If the disturbance is not considered, define a Lyapunov function as can be derived aṡ The inequality (7) ensuresV (t ) < 0, so the system (6) is stable. Therefore, when (t ) ≡ 0, the system (6) is admissible. Then, considering the condition (t ) ≠ 0, we define and a state vector as Based on the inequality (11), under zero initial conditions, we can get The inequality (8) ensures J < 0, so the system (6), under zero initial conditions, is admissible with H ∞ performance . Then, the proof has been completed. □ The controller K cannot be solved by Theorem 1 through LMI toolbox because it is coupled with other matrices. To solve this issue, we introduce the following theorem. (6) is admissible with H ∞ performance index if there exist matrices G 11 , G 12 , G 13 , M and L with appropriate dimensions satisfying the following condition:

Theorem 2. The continuous-time singular system
where then the controller gain can be obtained as K = G −1 12 M.
Proof. Define G = [ G 11 LG 12 , where L is a given matrix to set weights for different state variables. Set G = (PĒ + ZQ) T and M = G 12 K . Substituting them to inequalities (7) and (8), we can get inequalities (15) and (16). Based on the proof of Theorem 1, Theorem 2 is conformed. □

EVENT-TRIGGERED SCHEME DESIGN
The output feedback controller is designed in the above section. To reduce the data transmission load of the system, a discrete event-triggered scheme is designed here. Firstly, define the sampling interval of the seat suspension system as h. The output of the system y(t ) is firstly sampled by the sensors at time instants ih (i ∈ N) as y(t k + ih). Then the event-triggered scheme checks whether the output signal should be transmitted or not by the following condition where t 0 , t 1 , … , t k , t k+1 , … k ∈ N are event-triggered time instants, M is the maximum time delay, and Ω > 0, 0 < < 1.
The event-triggered control diagram for the system (4) is shown in Figure 2. The external disturbance z v (t ) influences the plant of the seat suspension system with a human body directly. Some sensors collect the measurable system state signals y(ih) at the sampling time interval h in real-time. Then, the eventdetector estimates whether the data should be transferred or not according to the condition (17). If triggered, the event detector allows the output y(t k ) to transmit to the zero-order holder (ZOH), which holds the data value until the next event is triggered. The event-triggered output data y(t k ) come to H ∞ controller, so u(t ) holds its value during the time intervals [t k , t k+1 ), k ∈ N. Therefore, the event-triggered scheme can decrease the workload of data transmission.
Considering the event-triggered output y(t k ) for ∀t ∈ [t k , t k+1 ), for convenience, we define where t ∈ [t k , t k+1 ) and we have where m and M are the minimum and maximum value of (t ), respectively. Based on the event-triggered scheme, the system (4) can be rewritten as where K is obtained in the above section and e(t − (t )) = y(t k + ih) − y(t k ).
Proof. At first, we need to prove the system (20) to be regular and impulse free. Since rank E = r < n, there must exist two invertible matrices M and N such that ] .
According to (24), we have Φ 11 < 0. Due to Q 1 > 0 and Q 2 > 0, so we have Pre-and post-multiplying (26) by N T and N , respectively, we have where • represents matrix elements which do not need to be considered. Hence, we can see that 1 A T 22 U 22 + 1 U T 22 A 22 < 0, so A 22 is non-singular. Therefore, the singular seat suspension system (6) is regular and impulse free.
To verify the stability of the system (20), assuming (t ) = 0, we design a Lyapunov function as where Now we define a state vector as follows: Calculating the derivative of V (t ) for t ∈ [t k , t k+1 ), we havė wherė (t ). [ Based on the event-triggered condition (17), we can get (35) Combining Equations (30) to (35) and according to Lemma 1, we haveV Based on (24), we haveV (t ) < 0, so the system (20) is stable. Then, when the condition (t ) ≠ 0, we define and a new state vector as Based on the inequality (36), under zero initial conditions, we can get The inequality (25) ensures J < 0, so the system (20), under zero initial conditions, is admissible with H ∞ performance . Then, we complete the proof. □

SIMULATION RESULTS
In this section, some simulation results of the output-feedback event-triggered H ∞ controller are given for the singular active

Bump road response
A bump disturbance is a common road excitation, so it is chosen here to evaluate the performance of the proposed control method for the seat suspension system. A bump road surface is defined as This bump road excitation firstly transfers to a vehicle suspension [19] and then the output is considered as the input disturbance of the seat suspension. To show the effectiveness of the proposed control method, the performance of the outputfeedback controller K with and without event-triggered scheme for the seat suspension system is compared with the uncontrolled seat suspension system. Figure 3 shows the seat deflection and the head's acceleration comparison under the bump road disturbance among the output-feedback controlled seat suspension system with the event-triggered scheme (red solid curve), without event-triggered scheme (blue dashed curve), and the uncontrolled seat suspension system (green dotted curve). Compared with the response of the uncontrolled seat suspension system, here we can see both two controllers can reduce the seat deflection and the head's acceleration effectively without big differences, so the controllers can both improve The output of the controllers, the release instants and release intervals of the seat suspension system with a bump road disturbance driving comfort for drivers. Figure 4 shows the controller comparison between the output-feedback controller with the event-triggered scheme (red solid curve) and without the eventtriggered scheme (blue dotted curve) and the release intervals of the event-triggered scheme. Here, we can see that the values of the event-triggered output-feedback controller only change when the event trigger happens, so it can reduce the workload of data transmission of this seat suspension system.

Sinusoidal vibration response
Set z v = 0.02 sin(1.4 × 2 t ) as the sinusoidal disturbance. Figure 5 shows the seat suspension deflection and the head's acceleration comparison among the output-feedback controlled system with and without the event-triggered scheme and the uncontrolled seat suspension system with a sinusoidal disturbance. From this figure, it is clear to see that both two controllers can greatly reduce both the seat suspension deflection and the head's acceleration with little differences compared with the response of the uncontrolled seat suspension system. Therefore, the proposed control method can largely improve driving comfort for vehicle drivers. Figure 6 shows the output of the two controllers comparison, the release instants, and release intervals of the seat suspension system with a sinusoidal disturbance. From this figure, we can see the values of the output-feedback controller with the event-triggered scheme only change when the event trigger happens, which can greatly reduce the workload of data transmission. Table 2 shows the average data transmission times per second of the output-feedback controller with and without eventtriggered scheme under bump road and sinusoidal disturbance. The seat suspension deflection and the head's acceleration comparison for the seat suspension system with a sinusoidal disturbance

FIGURE 6
The output of the controllers, the release instants and release intervals of the seat suspension system with a sinusoidal disturbance We set the sampling time as h = 0.002 s, so the controller without event-triggered scheme needs 500 times of data transmission per second. However, the controller with the eventtriggered scheme only needs 36.8 and 21.2 times averagely of data transmission for bump road disturbance and sinusoidal disturbance, respectively, which can reduce 92.64% and 95.76% of data transmission without big influence on the control performance to reduce vibration and improve driving comfort.

CONCLUSION
An output-feedback event-triggered H ∞ controller is designed for the singular active seat suspension system with a human body in the loop in this paper. The acceleration signals of upper seat suspension frame, body torso, and head of the human body are considered as parts of the system states, which makes the seat suspension system as a singular system. The outputfeedback controller is designed by using all easily measurable signals and the event-triggered scheme is designed to reduce the workload of data transmission. The LMIs criteria are given to obtain the controller and event-triggered scheme. Some simulation results show the effectiveness of the proposed control method to improve driving comfort and reduce data transmission. Experimental tests will be considered as our future work.