The proposed control algorithm is first developed to stabilize a class of second-order nonlinear systems subject to both matched and mismatched disturbances. Four different study cases are performed to verify the effectiveness of the proposed control algorithm.
INTRODUCTION
Background
The finite-time disturbance detector used in this work can be easily found in the literature. The same disturbance observer is employed in [46] to solve the problem of air gap control of a MAGnetic LEViation (MAGLEV) suspension vehicle in the presence of both matched and non-matched disturbances.
Problem statement
Second, a new terminal slip mode surface developed based on the disturbance evaluation results increases the robustness to external disturbances, uncertainties, and unmodeled dynamics of the control system. Once the desired slip mode s=0 is established, the dynamics of the system can be expressed as.
Research Objectives and Outline
In chapter 4, four different case studies are presented to verify the effectiveness of the proposed control algorithm. The first numerical simulation is given to help evaluate and discuss the performance of the proposed control algorithm theoretically.
DISTURBANCE-OBSERVER-BASED TERMINAL SLIDING MODE
Nonlinear Finite-time Disturbance Observer Design
It is noted in these above references that the observer design parameters, i, i(i=1, 2,3), are chosen recursively in such a way that i, i ensures the convergence of the observer with Lipschitz -the constant L given in Assumption 2. The trade-off is as follows: in general, the larger the parameters, the faster the convergence of the disturbance observer, even higher sensitivity to input noise and the sampling step, and greater peaking phenomenon in the control signal if the initial values of system states and observer states are different.
Disturbance Observer-based Chattering-free Full-order Terminal Sliding Mode
The authors also mention that these parameters can be easily changed as it is not very sensitive to their values. Then, according to Proposition 8.1 and its proof in [47], it can be concluded that the origin is a global finite-time stable equilibrium for system (29), which shows that the state x t1( ) of system (1 ) will be in the finite time converged to zero. Note that the proposed control law only uses the sign of the sliding surface instead of its value.
Based on this observation, the same idea proposed in [30] is shown here to avoid the use of an "acceleration" signal. Another option is to reuse the nonlinear disturbance observer structure to meet this need and is presented below. Since the perturbation observer is shown to be finite-time convergent, after the transition period z4 =xˆ2 can be treated as.
However, using a different perturbation observer will increase the computational burden, which may represent a disadvantage in practical applications. In this case, the sliding surface (19) and the control law (20) will be reduced to that of the conventional TSMC.
GENERALIZATION TO THE NTH-ORDER NONLINEAR SYSTEMS
Towards the Problem of Stabilization
Then, a full-order chatter-free TSMC for nth-order systems can be developed based on the disturbance observer as follows. This implies that the system states will arrive at the slip plane =0 in finite time. Then, according to Proposition 8.1 and its proof in [47], it can be concluded that the origin is a global finite-time stable equilibrium for the system (47), indicating that the system state x t1( ) will do so.
Towards the Problem of Tracking Control
First, the nth-order finite-time perturbation observer developed in the previous section is reproduced here for clearer illustration. Whereas, by Theorem 1, the perturbation observer developed in the previous section ensures that z1i =xi, z2i =di, z3i =di ,…, z(n 1)i (ndi1). Then, according to Proposition 8.1 and its proof in [47], it can be concluded that the origin is a globally finite-time stable equilibrium for the system (64), indicating that the state error e t1( ) will eventually converge to zero in finite time.
SIMULATIONS AND EXPERIMENTS
Case Study 1: Numerical Example
Therefore, the TSMC signals in the following figures are suppressed for better visualization of the other two algorithms. This explanation can be observed by comparing the orbital shape of the state x2( )t in Figure 2 with that of the mismatched perturbation dt1(. Reducing observer gains from the ESOSMC may be a solution to alleviate the severity of this phenomenon .
Looking closely at Figure 3, there is an abnormal increase in the proposed control signal. This phenomenon occurs due to the use of an absolute value function in the controller design. The control command signal is smooth except for some abnormal spikes that appear in the transition period, which is caused by the piecewise linearity of the absolute value functions used in the controller design.
It is evident that the system state trajectory of the conventional TSMC and that of the proposed control algorithm are identical before t=5 s and start to differ only after that. In summary, this section uses a numerical example to verify the effectiveness of the proposed control algorithm as it is compared with other controllers.
Case Study 2: Stabilizing an Electro Hydrostatic Actuator System in Simulation
The parameters of the EHA system that will be used for the mathematical modeling are defined in Table 2 below. According to [52], the dynamic model of the EHA system studied here consists of force dynamics and pressure dynamics as follows. The stroke length of the cylinder is 0.3[ ]m , so the operating range of the actuator is set to.
This section compares TSMC and the proposed DOBTSMC to verify the effectiveness of the proposed controller. The position and velocity responses of the EHA system are shown in Figure 11 and Figure 12. This lowers the performance of the disturbance estimation and consequently the performance of the control system as a whole.
In summary, this section partially reflects the effectiveness of the proposed control algorithm in practice as it is employed to stabilize an EHA system, which is subject to both time-varying mismatched and matched disturbances. This simulation study also considers the negative impact of measurement noise and parameter uncertainty to verify the robustness of the proposed control algorithm.
Case Study 3: Voltage Control for Bidirectional DC-DC Converter
The aim is to demonstrate the effectiveness of the proposed control algorithm developed in the previous section. To verify the effectiveness of the chattering illumination method, the simulation study is first performed using the unfiltered term un = −sign( ) and later compared with the results obtained using the low-pass filter-like method as in (95) , where T =50, and =(2+kT +)=5e7. The trajectories of the duty cycle before and after filtering are shown in Figure 24, which illustrates the severe flipping phenomenon of the signum function.
The small value of components (L and CL) appearing in the denominator of a fraction in system modeling and large value control parameters used in controller design lead to the amplification of the chattering effect found in any part of the control system . Whereas Figure 25b shows the effectiveness of the proposed disturbance observer with an estimation error of . Although high accuracy is seen in the observer's performance, it still exhibits a shading phenomenon.
The signals illustrated in Figure 26 and these expressions above explain why the magnitude of the control parameter, =(2+kT +)=5e7 is set to be large to overcome the influence of corresponding disturbances. However, the existence of the multiplier, LC , in the denominator of the virtual control input u (87) forces the virtual control size u to be large.
Case study 4: Experimental Validation on an Electro Hydrostatic Actuator (EHA)
Actuator displacement and system working pressure are recorded by a linear encoder (WTB5-500 MM) and pressure transducers (DS-230), respectively. To verify the effectiveness of the proposed control algorithm, the popular PID controller is selected for comparison purposes. Despite the high nonlinearity of the target system, PID can produce a relatively good control error in the range of .
This can be explained by mechanical failure of the structure and delayed effects in the sensing system. The fact that the defects are only on one side of the external load trajectory may be due to the asymmetric structure of the single-bar cylinder, where the areas of the two chambers differ from each other. The discrepancy in the area of the two chambers explains the difference in the magnitude of the two pressures.
In this section, the effectiveness of the proposed controller in solving the position tracking control problem for the EHA system was evaluated. The proposed controller achieves much higher position tracking accuracy due to both the relatively accurate estimation of mismatched disturbances and the robustness of the sliding mode control.
CONCLUSION AND FUTURE WORK
Summary
However, this approach leads to a disadvantage since the virtual control signal is required to be large, and subsequently large control parameters are chosen to produce such a control signal. Whereas, in the study of position control for an electro-hydrostatic actuator, although it is carried out in an experiment, different working scenarios must be added to the scheme. Also the effect of the asymmetric cylinders on the system performance must be carefully studied.
Future work
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Yu, "Sliding-Mode Control for Systems With Mismatched Onsekerhede via 'n Disturbance Observer," IEEE Transactions on Industrial Electronics, vol. Phadke, "Sliding Mode Control for Mismatched Uncertain Systems Using an Extended Disturbance Observer," IEEE Transactions on Industrial Electronics, vol. Wang, "Versteuringswaarnemer-gebaseerde integrale glymodusbeheer vir stelsels met onooreenstemmende steurings," IEEE Transactions on Industrial Electronics, vol.
Fang, "Extended-State-Observer-Based Chattering Free Sliding Mode Control for Nonlinear Systems With Mismatched Disturbance", IEEE Access, vol. Moon, "Perturbations-Based Observers for Continuous Finite Time Sliding Mode Control Against Matching and Mismatching Perturbations".
