The position control of the ball and beam system using state-disturbance observe-based adaptive fuzzy sliding mode control in presence of matched and mismatched uncertainties
Introduction
In the real world, nonlinear control faces crucial challenges such as severe nonlinearity, the complexity of equations, uncertainties, and external disturbances. Since the ball and beam system (BBS), as one of the most popular physical-balance systems, encounters almost all challenges of designing nonlinear control methods, it has been used in many research institutes as an experimental setup for practical evaluation of these methods [1]. Moreover, the BBS has only one actuator despite having two degrees of freedom (DOF); as a result, it is called an underactuated mechanical system [2], [3]. Some of the DOFs of this system cannot be directly commanded, which highly complicates the design of control algorithms [3]. Therefore, the BBS can be utilized to evaluate similar underactuated systems in different fields such as in robotics, flexible and mobile systems, aeronautical, and underwater systems [2], [4].
In the BBS modeling process using conventional methods such as Newton and Lagrange, some effects may not be considered. These mathematical simplifications and unmodeled dynamics cause uncertainty in the system and complicate the control process. For instance, in [5], the coupling effect of the dynamic equations for two DOFs is neglected and then the BBS is considered as a system with one DOF. Due to the small angular velocity of the beam during the slow motion of the ball, the authors in [6], [7] neglect this variable. To overcome the simplifications, some model-based control methods have been proposed recently for the BBS [1], [8], [9], [10], [11], [12]. The authors in [8], [9], [10] propose a proportional-integral-derivative (PID), optimal PID, and linear quadratic regulator (LQR)-PID strategy to control the position of the ball on the beam. In [11], based on the virtual spring-damper hypothesis, the control scheme of the redundant manipulator is used to control the BBS. A Takagi-Sugeno (T-S) fuzzy modeling scheme and an adaptive dynamic surface control (DSC) have also been proposed to the BBS control [12]. However, due to the complexity of the dynamic equations, the use of non-model-based control methods has also been suggested [6], [10], [13], [14]. In [10] a PID is given as a non-model-based control strategy. A fuzzy logic controller (FLC) in which the type of the membership functions, its parameters, and rules are tuned using the ant colony optimization (ACO) method has been proposed for position control of the BBS [13]. In many cases, to achieve proper efficiency, intelligent control and evolutionary algorithms such as particle swarm optimization (PSO) and genetic algorithm (GA) optimize the parameters of the proposed controller, resulting in a variety of combined methods of the BBS control [10], [15], [16]. In [10] the parameters of the LQR are tuned using the GA, and the gains of the PID are optimized with the help of the PSO and the fuzzy logic method in [15], [16], respectively
In the above-mentioned literature, although researchers have provided appropriate solutions, some of the problems still available are described as follows:
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The effects of the matched and mismatched uncertainties and external disturbances have not been considered.
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Often, challenges such as selecting the control parameters (e.g., the center and width of the Gaussian membership functions, and the weight of each rule) need supervised and offline learning and small training data, and initial condition lead to complicated control laws and difficult practical implementation.
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The multi-loop control algorithm, the combination of controllers, and the use of evolutionary algorithms may increase the computational burden and be time-demanding.
It is important to note that the relative degree of the BBS is not well-defined and, as a vital challenge, it can be concluded that it is not an input–output linearizable system [17]. In many controller design methods, despite the existence of uncertainties in the BBS, this challenge leads to the elimination of a known part (namely the centrifugal acceleration expression) of the system's dynamic equations. By the use of methods such as approximate feedback linearization [18], [19] or linear models [20], the effects of these uncertainties and omitted terms cannot be easily compensated and the efficiency of the BBS is reduced. Among the foregoing strategies, the sliding mode control (SMC), as one of the most popular and powerful methods in terms of theory and simplicity in practical implementation, has been utilized extensively to control nonlinear, multi-input multi-output (MIMO), underactuated, large-scale, stochastic, and infinite-dimension systems, and it can handle uncertainties and external disturbances [21], [22], [23], [24], [25], [26]. Hence, some researchers have used this method to overcome the BBS challenges [27], [28], [29], [30]. Using a simplified and complete model of the BBS, two static and dynamic SMCs are designed for system control in [27]. In the proposed control scheme in [28], two cascaded control structures as outer and inner loops using two SMCs are used to control two DOFs of the BBS. In [29], [30], by combining the decoupled SMC and the feedback linearization technique, and the SMC with the GA-based back-stepping, robust adaptive control approaches are proposed for the BBS. Most of these methods, despite their suitable performance, focus on robust stability. However, in some of them, despite the existing uncertainties, challenges such as the elimination of known parts of the dynamic equations, the unavailability of the upper bound of the external disturbances and uncertainties, and the phenomenon of chattering have not been resolved yet.
Overcoming the problems of the external disturbances and uncertainties and weakening their effect on the efficiency of the system in the SMC is possible to some extent by increasing the switching gain [31]. However, it is worth mentioning that this increase intensifies the chattering phenomenon at the control input and reduces the lifetime of the actuator, despite the reduction efficiency of the system [23], [31]. As a practical alternative method, the disturbance observer-based control method has been proved to be effective in fully compensating the effects of unknown external disturbances and model uncertainties in many real systems, where they can be very compatible with a variety of SMC techniques [32], [33], [34], [35]. Besides, due to the special operating conditions of the system, it is impossible to install sensors for all state variables or it is expensive in many cases in real systems. In this case, the use of state observers is a proper solution [34], [36], [37]. For instance, in [36], a sliding mode observer is introduced to observe the slope of the beam from the ball position measurement in the BBS. It should be noted that there are challenges in many observer-based control approaches, such as unbiased estimation, global stability, insensitivity to matched disturbances, and sensitivity to mismatched disturbances [33], [37], [38], [39]. Furthermore, in many cases, the proposed observer-based controls are designed for specific structures and cannot be generalized for underactuated systems or systems with mismatched uncertainties [34], [35], [38], [39]. Sometimes, the real data is used to estimate mismatched uncertainties in underactuated systems, which is practically impossible to access and therefore cannot be implemented in practice [31]. On the other hand, in some cases, several solutions have been proposed to overcome matched and mismatched uncertainties in underactuated systems. To achieve the boundary of uncertainties and prevent system instability in cases without accurate estimation of uncertainties, simple forms are considered for uncertainties or the switching gain is increased, which increases the amplitude of the control input and may saturate the actuators [32], [33], [40]. That being said, in all the methods introduced previously, the simultaneous ability to overcome matched and mismatched uncertainties in underactuated systems, adjustment of control coefficients, simplicity, low power consumption, and practical implementation is also of particular importance.
This paper proposes an S-D observer-based adaptive fuzzy sliding mode control (AFSMC) to the position control of the ball and beam system (BBS) as an underactuated mechanical system in the presence of matched and mismatched uncertainties. The main contributions of this study include:
- 1)
The proposed S-D observer-based control scheme provides the estimation of the states, matched and mismatched uncertainties, and their derivatives, which leads to the compensation of their destructive effect on the control process. Although the two sensors in the BBS provide good information about the ball position and the beam angle, the high capability and accuracy of the proposed control increase the reliability of the control system in critical situations.
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The proposed control scheme is easily extensible to similar underactuated mechanical systems that are affected by matched and mismatched uncertainties.
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A combination of the SMC and the S-D observer is proposed for the effective control of the BBS, where the mathematical proof guarantees the global asymptotic stability of the closed-loop system in the presence of uncertainties.
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As an important strength compared to the above methods, not only the use of the proposed adaptive fuzzy approximator in the S-D observer-based SMC does not undermines the system stability but also it preserves the uniform global stability of the closed-loop system in the worst condition, when the designer makes a mistake in adjusting the switching gain.
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The use of adaptive fuzzy gain in the S-D observer-based control structure provides some beneficial features, such as preventing the increase of the control input range and eliminating the chattering which leads to a reduction in energy consumption and an increase in lifetime of the actuator.
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The S-D observer-based control scheme has low computational burden, practical implementation capability, and easy determination of the control coefficients without the need for employing trial and error methods.
The rest of the paper is organized as follows. After providing a conceptual presentation of the BBS in Section 2, its mathematical modeling is introduced in Section 3. The SMC design and its features are presented in 4 SMC design, 5 Advantages and disadvantages of the proposed control, respectively. Section 6 introduced a finite-time S-D observer. Then, the S-D observer is combined with the SMC and the AFSMC in 7 S-D Observer-Based SMC design, 8 S-D Observer-Based AFSMC design, respectively. The validation results are given in Sections 9. Finally, Section 10 concludes the study.
Section snippets
Preliminaries
The BBS as shown in Fig. 1 can be considered as an underactuated mechanical system with two degrees of freedom (DOFs) (the ball position and the beam angle) controlled by the motor input voltage. The ball can roll on the beam, where it changes its position along the length of the beam. On the other hand, the end of the beam is connected to a DC servomotor using a lever arm that can rotate by an angle of and change the angle of the beam. The ultimate goal of the control is to design a
Mathematical model of the BBS
The full nonlinear dynamic equations (consisting of the DC servomotor and the mechanical part) of the BBS can be expressed as [42]:where
Additionally, Table 1 lists all of the parameters, as well as their values, used in the mathematical derivation of the model.
To achieve the compact form of the controller, (1) can be rewritten in the state-space form as follows:
SMC design
In the SMC design process of the BBS, the desired path is considered for and the tracking control goal is fixed . First, the error-based state variables can be defined as:
Note that is a constant value, so its time derivative is zero. Now, by combining (7) and (9), one can obtain that:
The sliding surface is considered as follows:where, are
Advantages and disadvantages of the proposed control
Some advantages of the proposed scheme are listed as follows:
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Mathematical proof shows that the proposed control can overcome the uncertainties in the BBS and guarantees the global asymptotic stability of the closed-loop system.
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Adjustment of input coefficients of the proposed control can be easily accomplished, which can shorten the time required to make the position tracking error zero.
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Since in the proposed control structure , is changed to by properly adjusting of . In this
Finite-Time S-D observer
Consider a nonlinear system in the presence of matched and mismatched uncertainties as follows [33], [40]:where, is the state vector, represents the control input, , are smooth nonlinear functions, and denote the mismatched and matched disturbances, respectively, and is the controlled output. Assumption 2 The mismatched uncertainties in (22a) are time differentiable so that have a known Lipschitz constant [33], [40].
A
S-D Observer-Based SMC design
Since the finite-time S-D observer used together with the state variables of the system is capable of estimating the matched and mismatched disturbances, these estimates are then used to design the controller. Therefore, here the estimations of the state variables vector and disturbances are represented by , , and , respectively. Lemma 1 Considering the BBS expressed by (7), using the proposed finite-time S-D observer (23) and the sliding surface (11), all tracking errors (9) of
S-D Observer-Based AFSMC design
According to the previous section, by selecting if the closed-loop system has asymptotic global stability. Fig. 3, without loss of generality, depicts the behavior of the variations of in for different values of . Referring to this figure, it is obvious that:
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If the value of approaches one, the slope of the curve increases and vice versa, and if its value approaches zero, the slope of the curve decreases.
- 2.
If the value of approaches one, the
Implementation of the AFSMC
As noted in Remark 5, if one wishes to begin the control process with improved reliability, a fuzzy approximator can be added to the S-D observer-based SMC structure. That is, the sum of the S-D observer estimation errors is greater than the coefficient, from the beginning of the control process, then is increasing. If by increasing during the control process, the value of is changed and reaches unity, then the coefficient should be multiplied. As a result, makes the controller
Advantages of the fuzzy approximator
Several benefits of the proposed adaptive fuzzy approximator design include:
- 1.
Its structure never alters the stability analysis of the closed-loop system and, as a result, its presence not only does not undermines the system stability but also enhances the stability of the closed-loop system. If the designer makes a mistake in adjusting the value of, the proposed adaptive fuzzy approximator avoids instability and, consequently, in the worst condition, it preserves the uniform global stability of
Results validation
In this section, two sets of simulations and practical implementation are performed to verify the theoretical considerations and show the effectiveness of the proposed S-D observer-based AFSMC. All of the simulation steps together with practical implementation, as a hardware-in-loop technique, are carried out using the MATLAB/Simulink software. In two of the sets, the sampling time is considered . Parameters of the BBS are given in Table 1. The desired path and matched and mismatched
Simulation results
Simulation 1 In the first part of the simulation step, the performance of the proposed SMC in Section 4 (without the S-D observer and the fuzzy approximator) in the presence according to (38a), (38b)) and absence of matched and mismatched uncertainties is shown. The control coefficients are the same for both conditions and are considered to be and . Fig. 6 shows the ball position tracking performance, where in the absence of uncertainties
Experimental results
The proposed S-D observer-based AFSMC is implemented on the BBS setup in the laboratory, as depicted in Fig. 22. The implementation has been carried out using the MATLAB/Simulink software as hardware-in-the-loop (HIL) in the external mode. The Arduino/Genuino Uno board is used to connect the MATLAB/Simulink environment to the BBS setup. Without loss of generality and with minor changes, the second part of Simulation 1 has been practically implemented. The experimental results are validated in
Conclusion
An S-D observer-based AFSMC for a BBS system was proposed in this study. To begin the design process of the tracking control, dynamic equations of the BBS were presented in an error-based state-space form. Then, an SMC was proposed to the position control of the BBS, where the mathematical proof showed the global asymptotic stability of the closed-loop system in the presence of matched and mismatched uncertainties. In the following, a high accurate and finite-time S-D observer was used to
CRediT authorship contribution statement
Saeed Zaare: Visualization, Investigation, Software, Validation, Data curation, Writing - review & editing. Mohammad Reza Soltanpour: Conceptualization, Methodology, Writing - original draft, Supervision.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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ORCID: 0000-0002-4446-5104.