Robustness control for nonlinear systems based on homogeneous prescribed slidingmode control

Use your smartphone to scan this QR code and download this article ABSTRACT This paper presents a new homogeneous control using dual sliding mode control, and robustness control using linear matrix inequality (LMI) constraints. The controller is applied for the severe disturbance. A sliding surface function, which relates to an exponential function and itself t-norm, is applied to save the energy consumption of the control system. The constraints related LMI are proposed with the matrices and vectors of the systems following the chosen matrices in control the energy for control. Solution of the constraints is also presentedwith new approach to save the time of calculation. In addition, the proof for the proposed controller is also presented by using the candidate Lyapunov function. In the input control function, the t-norm type is embedded to improve its performance in control disturbance. Besides of the t-norm, themodified sliding surface in the input control is also improve the energy for controlling. The combination of these robustness control elements would bring a new view for the design of control. The advantages of the controller are demonstrated via computer simulation for a seat suspension system. A magneto-rheological fluid seat suspension with its random disturbances is used. To prove the flexibility of the controller, the proposed approach is compared with an existing controller. The compared control has the same structure as shown in the proposed model. However, its design has a disadvantage in control the severe disturbance. The comparison between two controls is a clear view of distinct improvement. The results of simulations show that the controller provides better performance and stability of the system. The stability is also analyzed through the variation of the input control and power spectral density related energy consumption.


INTRODUCTION
Nowadays, the development of modern controls is continuously bringing new surprises in our life through its application such as robotic manipulators, transportations, upper and lower limb exoskeleton, etc. In this development, adaptive controller and its modification with other controls such as proportional-integral-derivative, sliding mode control, prescribed performance, etc. were presented in many featured types of research. Firstly, adaptive fuzzy control with prescribed performance was studied 1,2 . The objective of the prescribed performance 1,2 was to set up upper and lower boundaries for the error of the system. The Nussbaum function was used for unknown direction 2 . Hence the combination of both the Nussbaum function and the prescribed performance was improved the responses 2 . The model of adaptive fuzzy controller with sliding mode controller was studied 3,4 . Sliding mode control 3,4 was conventional types such as the integral model 3 and classical model 4 . However, the classical Lyapunov function was embedded in the input control function 4 to improve performance. The Lyapunov function 4 was a special case of the Riccati equation with an assumption of stability based on the removal of matrix and vector system. It is remarked that the type 1 fuzzy model was applied [1][2][3][4] . The neural networks model was also used in adaptive control 5 . The conventional PID model was applied 5 for the design of adaptation laws. This PID model was also used [6][7][8] . It was a similar structure in control 5,6 , especially in adaptation laws. PID was applied in design the PID-like sliding surface of sliding model control 7 . It is noteworthy that PID could be used in two types: adaptation laws 5,6,8 , sliding surface 8 . The adaptive fuzzy control was also presented [9][10][11] . The interval type 2 fuzzy model was applied [8][9][10][11] . This point is different from the others [1][2][3][4]6,7 . The Riccati-like equation was adopted [8][9][10][11] , and this utilization was different from the properties of the system: disturbance, vibration control, consumption of energy. The sliding mode control was utilized as a first step to combine controls in adaptation laws. Besides, optimal control was also used in the design of adaptive control 10  application of sliding mode control was shown [12][13][14][15][16][17][18] . In these applications, the conventional sliding surface was used. Detail models of sliding surface are analyzed as follows. The saturation function of sliding mode control was applied 12 . The sliding surface 12 was obtained by using a PID-like model, and the saturation function was used in the input control. The signum function was also used in input control [13][14][15]17,18 . It is noteworthy that there are two types of function related to the conventional input control of sliding mode control: saturation function and signum function. The disadvantage of these functions is the chattering phenomenon and sensitive to disturbances. However, these functions could be improved by combining with forms of sliding surface as terminal type (or twisting type) 13,15 , classical type 14,18 , and integral type 16 .
From the above analysis, the adaptive control has remained as a potential controller for modern devices. It can be defined with other control types such as sliding mode control 3,4,[7][8][9][10][11][12]17 , optimal control 10 , prescribed performance 1,2,9 , Riccati-like equation 4,[8][9][10][11] , fuzzy model [1][2][3][4][7][8][9][10][11] , and neural networks 6 for improving the performance of the system. However, the model of sliding mode control 3,4,[7][8][9][10][11][12]17,18 is conventional and this can remain a disadvantage when applying for control design. Hence, in this research, a new novel form of homogeneous controller, which preserves the merits of sliding mode control based on prescribed performance and linear matrix inequality, will be developed. The proposed control is to follow the objective as a simple model and its advantage in control severe disturbance. The main contributions are summarized as follows: (1) A new integration of classical sliding surface, prescribed form of sliding surface and linear matrix inequality for homogeneous control is presented.
(2) A new proof for the design of a homogeneous controller with a new sliding surface of sliding mode control is proposed.
(3) The outstanding property of the proposed approach is validated in simulation for a seat suspension system and compared with a controller 19 .
The rest of this article is organized as follows. In section 2, the proposed design is presented. In section 3, the simulation for two controls including the proposed control, the compared control 19 is described. Finally, the conclusion is presented in section 4 with the main results of the above sections.

DESIGN METHOD OF A HOMOGENEOUS PRESCRIBED SLIDING MODE CONTROL
The control is developed based on a SISO (singleinput single-output) nonlinear system as follows: .
Where f ∇ (x ∇ ) ∈ R n and g ∇ (x ∇ ) ∈ R n are two non- The governing system (1) can be defined as follows: where, The tracking function is described as e ∇ (t) = x 1∇ − x d∇ , where x d∇ is the desired result related the system; k i is Hurwitz parameters which relate to the conventional sliding mode control. It is noteworthy that x 1∇ is the similar value as x ∇ in Eq. (2). The prescribed sliding surface is expressed as follows 10 : Where, c s∇ > 0 is a positive constant. The function φ is defined as follows:

Remark 1. The prescribed sliding surface (3) is used
to control the chattering phenomenon and then guarantee the response following desired boundaries. This advanced property is useful for the system under severe disturbance. The progress to derive the sliding surface (3) was presented 10 .
The governing system (1) can be described as follows: . Where, It is remarked that D s is the maximal boundary of the disturbance; C p , D ∇ are the matrix and the vector when deriving the governing equation following state-space form. (5) is also a change of the property of the governing equation (1) from a nonlinear model to a linear model for the steps of the proposed controller. The linearization can be used in this step to find the linear model which belongs to the property of the system. The new main input u ∇ is proposed as follows:

Remark 2. Equation
the matrices Z ∈ R m×n and P ∈ R n×n are chosen from the constraints; G ζ ∈ R n×n is calculated as G ζ = vW + εI n , I n is the unit matrix; v ∈ R, ε ∈ R are two constants. The element of Z 0 , W, K 0 and K ζ are solved from the constraints of linear matrix inequality (LMI in short) method as Figure 1: (6) is a new design with the assumption that the disturbance is approximately zero. The existence of the disturbance through equation (7) with the relations of K 0 , Z 0 and K ζ will guarantee the stability of the system. Remark 4. The constraint equation (7) is to follow the LMI method. In LMI, the diagonal elements are always designed less than or equal to zero. The other elements are going to zero value when analyzing to infinity of time. From Eq. (7), the constraints for the input control term u are derived as: Where, X ∈ R n×n is chosen matrix; δ x ∈ R is a positive chosen constant. Remark 5. The constraints (8)(9)(10) are chosen from the property of Eq. (7) based on the boundary zero. The constraints are solved by the trial-and-error method. Remark 6. The parameter α ∈ R + in Eq. (7) is chosen from the performance output control. It is remarked that the output control relates the prescribed sliding surface, the results of constraints (8)(9)(10).

Proof of Eq.(7).
The equation (7) implies that (−γW + I n ) B P > 0. A proposed smooth function φ (δ d ) is defined as follows: Where, z ∈ R n , z ̸ = {0} , z is a system variable related to the control progress. The boundary δ d can be seen as a variable of the system, δ d ∈ [δ dmin , 1]. The smooth function (11) is used to depict the operation of a system with the feedback signal, disturbance, and other elements related robust system. A derivative of equation (11) is obtained as: From the boundary of δ d ∈ [δ dmin , 1], it can be determined δ max as 0 < δ min ≤ δ max ≤ 1. On the other hand, equation (11) can be described as Figure 2: From the boundary of 0 < δ min ≤ δ max ≤ 1, values of ln δ dmax δ dmin and δ dmax go to 0 and 1, respectively. Hence equation (14) is rewritten as follows as Figure 3: Theorem: The proposed control is designed with the input control u as shown in Eq. (6) and its constraints for improving the global stability of the system as shown in Eqs. (8)(9)(10). These functions are the frame of the controllers which is proposed for the nonlinear system following the homogeneous sliding mode control. Proof: The Lyapunov candidate for the proposed control is described as follows: Derivative form of Eq.(16) is described as: Eq. (5) can be defined as derivative form as: Where, .

SIMULATION RESULTS AND DISCUSSIONS Dynamic Seat Suspension Model
In this work, the model of vehicle suspension system 11 is used for simulation as shown in Figure 5. The main dynamic equations are described as follows.
Eqs.(26,27) can be rewritten as follows: Where The voltage to be applied to the damper is calculated as follows 11 : Two controllers are used for simulation: an existing control 19 , and the proposed controller. The control 19 is also a start-of-art controller for comparison properties of the proposed control in this study. The detail parameters for the two controllers are shown in Tables 1 and 2. The random step wave road is used for simulation as excitation as shown in Figure 6. This signal is a real signal under the raw surface road which was used [9][10][11] . The main input control is calculated for the force damper with its bounded value is 1000 N. An observer 20 is chosen for two controls. Based on the selected parameters and the matrices, the simulation is executed and the results are shown and discussed.

DISCUSSIONS
The main results of the simulation are shown in Figure 7-Figure 15. Results of the proposed control are shown in Figure 7- Figure 11. In Figure 7, the performance of the proposed controller is good, and the initial excitation as shown in Figure 6 is decreased its values after controlling as shown in Figure 7a. Because of good performance at seat position, the vibration of the driver also decreases as shown in Figure 8. It is noteworthy that the performance of the driver is closed-relation with the seat system. Hence, any variations of the system are also directly affect the comfortable state of the driver. In Figure 9, the values of the displacement seat's response always belong to the boundary of the prescribed performance. So the obtained responses of the proposed control is to guarantee the stability of the system. In the proposed control, the prescribed sliding surface is used and its response is shown in Figure 10. In this figure, the sliding surface has changed its value following the excitation and then adjust the input control to optimize the performance. The damping force of the MR damper when using the proposed control is shown in Figure 11. It          is shown that the controlled force is always lower than the setup value 1000 N. This proves that the proposed control is to save the consumption of energy. Results of the compared control are shown in Figures 12, 13 and 14. In Figure 12, the performance of the seat system is not good and the output value is always in the unstable area as shown in Figure 12c. As mentioned above, the performance of the driver is following the seat vibration. Hence, the driver response is also not good as shown in Figure 13. Furthermore, the con-trol energy following the damping force of the compared control also obtains the maximal value 1000 N as shown in Figure 14. This point shows that the compared control uses the maximal value of input control and the obtained results are not good. To conclude the proposed control and the compared control based on the variation of power for control and the frequency, the power spectral density (PSD) graph as shown in Figure 15 shows that the proposed control always guarantees the stability of the system which is better than the compared control. Figure 15 is also show that the proposed control can save the energy of control better than the compared control.

CONCLUSIONS
A new robustness controller was proposed and applied in this study for vibration control of a vehicle seat system. The controller includes two sliding surfaces as conventional and prescribed model. From the proposed surfaces, the main input controller was designed with its constraints related Lyapunov function. The stability of the controller was proved based on Lyapunov candidate and LMI method. After analyzing, the controller is is simulated and compared with an existing controller 19 . The compared results show that the proposed controller obtains good performance under severe disturbances in both stability and saving energy. In future work, the technique of  this research will concentrate on analyzing the affection of a switching controller to a real model, to improve its flexibility in complicated models.