Applications of Fuzzy Sliding Mode Control for a Gyroscope System

Abstract

The study proposed the application of the fuzzy sliding mode for a gyroscope system status control. The state response analysis of the gyroscope system revealed highly nonlinear and chaotic subharmonic motions of 2𝑇 during state formation. The current study discussed the use of tracking control on the sliding mode control and fuzzy sliding mode control of a gyroscope control system. Consequently, the gyroscope system drives from chaotic motion to periodic motion. The numerical simulation results confirm that the proposed controller provides good system stability and convergence without chattering phenomena.

1. Introduction

Since most systems in the natural world are nonlinear, presenting them in a linear manner is challenging. Some systems exhibit chaotic phenomena after a certain number of intervals. The chaos phenomenon has been widely discussed and researched after many areas, including mechanical engineering, electrical engineering, and communications. Its main feature is that different initial conditions can produce large differences in the system response. According to [1–3] certain conditions generate chaotic behavior in a gyroscope system, which results in system instability and design problems. Wang and Yau used differential transformation (DT) method and Runge-Kutta (RK) method to analyze the dynamic behavior of the gyroscope system [1]. They also proposed the use of a sliding mode controller to stabilize the system and to change its state from chaotic-driven motion to periodic motion. Ge and Lee used the Lyapunov stability theory and adaptive control and random optimization method to synchronize two identical gyroscope systems and to track system parameters [2]. Fei et al. based the adaptive sliding mode control on a radial basis function neural network for three-axis gyroscope used in micro electromechanical systems (MEMS) [4]. Chaos control has attracted the attention of many scholars in recent years [5–13]. Yau et al. used particle swam optimization (PSO) algorithm to find close optimal parameters for PID control and used a PID controller for chaos synchronization control [7]. Aghababa et al. proposed the use of sliding mode control technology to solve finite-time chaos synchronization problems by enabling synchronization of two different chaotic systems with unknown parameters [9]. Li and Kumar proposed the application of nonsingular fast terminal sliding mode control technology in satellite attitude synchronization [10].

This study proposes the use of fuzzy sliding mode control technology for state control of a gyroscope system. Formation of chaotic motion in a gyroscope during subharmonic motions of 2𝑇 is discussed, and controller designs that drive the gyroscope system state from chaotic motion to periodic motion are proposed. The framework of this paper is as follows: Section 2 briefly introduces the gyroscope system and its issues; Section 3 introduces the sliding mode control in the gyroscope system design and its applications; Section 4 proposes the fuzzy sliding mode control for a gyroscope system and introduces the application of the Lyapunov stability theory for designing a fuzzy sliding mode controller; and Section 5 presents the results of the numerical simulation approach used to verify the feasibility of the proposed method and states the conclusions of the study.

2. A Gyroscope System

Figure 1 shows the heavy symmetric gyroscope placed on the structure of the vibration mechanism. The Euler angles 𝜃, 𝜓, and 𝜑 refer to the three-dimensional motion of the system, and multiple harmonic motion ∑^𝑛_𝑘=1𝐴_𝑘sin𝜔_𝑘𝑡 describes the base of the vibrations. The status order is 𝑥=𝜃, 𝑦=𝜃̇, and 𝑧=𝜓̇. The system dynamic equations are as follows [1]: ̇̇𝛽𝑥=𝑦,𝑦=−^𝜓−𝛽_𝜑cos𝑥𝛽_𝜑−𝛽_𝜓cos𝑥𝐼²₁⋅sin³𝑥−𝐶𝐼₁𝑦+𝑀_𝑔𝑙𝐼₁𝑀sin𝑥−^𝑔𝐼₁^𝑛_𝑘=1𝐴_𝑘sin𝜔_𝑘𝑡̇⋅sin𝑥,𝑧=−2cos𝑥𝛽sin𝑥𝑦𝑧+^𝜑𝑦𝐼₁,sin𝑥 (1) where 𝐼₁ is the uniform gyroscope moment of inertia axis, 𝑀_𝑔 is the weight of gravity, and 𝑙 is the distance between the center of the gyroscope and point 𝑂. If only the system of the subharmonic motions of 2𝑇 is considered, the system can be rewritten as ̇̇𝛽𝑥=𝑦,𝑦=−^𝜓−𝛽_𝜑cos𝑥𝛽_𝜑−𝛽_𝜓cos𝑥𝐼²₁⋅sin³𝑥−𝐶𝐼₁𝑦+𝑀_𝑔𝑙𝐼₁𝑀sin𝑥−^𝑔𝐼₁𝐴₂sin𝜔₂𝑡̇sin𝑥,𝑧=−2cos𝑥𝛽sin𝑥𝑦𝑧+^𝜑𝑦𝐼₁.sin𝑥 (2)

Figure 1 

A schematic diagram of a heavy symmetric gyroscope [2].

The system parameter settings are 𝛽_𝜓=2, 𝛽_𝜑=5, 𝐼₁=1, 𝑀_𝑔=4, 𝑙=0.25, 𝐶=0.5, and 𝜔₂=2. The analysis in [1] showed that the chaos phenomenon in a gyroscope system results from subharmonic motions of 2𝑇. Based on the dynamic analysis of [1], 𝐴₂=14.21 was selected for the study. Figure 2 shows the numerical simulation results for the phase plane trajectory of the gyroscope [2]. The results indicate that chaos phenomena resulted from two attractors of system trajectories.

Figure 2 

Phase portrait of gyroscope system.

Since the gyroscope system has chaotic motion, driving its state into periodic motion requires the addition of 𝑢₁(𝑡) and 𝑢₂(𝑡) to the second and third items of the control, respectively. The dynamic equation for the system can then be rewritten as ̇̇𝛽𝑥=𝑦,𝑦=−^𝜓−𝛽_𝜑cos𝑥𝛽_𝜑−𝛽_𝜓cos𝑥𝐼²₁⋅sin³𝑥−𝐶𝐼₁𝑦+𝑀_𝑔𝑙𝐼₁𝑀sin𝑥−^𝑔𝐼₁𝐴₂sin𝜔₂𝑡sin𝑥+𝑢₁(𝑡),𝑧̇=−2cos𝑥𝛽sin𝑥𝑦𝑧+^𝜑𝑦𝐼₁sin𝑥+𝑢₂(𝑡). (3)(4)(5)

A controller was designed into the system to solve two consecutive equations for tracking 𝜃_𝑑 and 𝜓_𝑑 in the system, which allows the gyroscope system to exhibit periodic motion. Through the system control input and state response, the tracking error can be defined as 𝐸(𝑡)=𝑒₁𝑒₂𝑒₃^𝑇=𝑥−𝜃_𝑑𝑦−𝜃̇_𝑑𝑧−𝜓̇_𝑑^𝑇. By integrating the system dynamic equations and system errors equations, the error dynamic equation can be computed as 𝑒₁̇=𝑒₂,𝑒₂̇𝛽=−^𝜓−𝛽_𝜑cos𝑒₁+𝜃_𝑑𝛽_𝜑−𝛽_𝜓cos𝑒₁+𝜃_𝑑𝐼²₁⋅sin³𝑒₁+𝜃_𝑑−𝐶𝐼₁𝑒₂+𝜃̇_𝑑+𝑀_𝑔𝑙𝐼₁𝑙−𝐴₂⋅sin𝜔₂𝑡⋅sin𝑒₁+𝜃̇_𝑑−𝜃̈_𝑑+𝑢₁(𝑡),𝑒₃̇𝑒=−2cos₁+𝜃_𝑑sin𝑒₁+𝜃_𝑑⋅𝑒₂+𝜃̇_𝑑⋅𝑒₃+𝜓̇_𝑑+𝛽_𝜑𝑒₂+𝜃̇_𝑑𝐼₁sin𝑒₁+𝜃_𝑑−𝜓̈_𝑑+𝑢₂(𝑡). (6)

Definition 1. Considering the design of the state signal and the gyroscope system in (3)–(5), if 𝑇=𝑇(𝐸(0))>0 exists, then lim_𝑡≥𝑇‖𝐸(𝑡)‖=0. (7) If 𝑡≥𝑇, then the gyroscope state track to the state signal design within a limited time.

This study designed a fuzzy sliding mode controller to enable the output state of the gyroscope system to tracking the design state. Tracking control is achieved by using the controller for gradual convergence of the error dynamics equation to zero.

3. Sliding Mode Control

In accordance with the analysis of chaotic phenomena in a gyroscope system, Wang and Yau proposed the use of sliding mode controller to tracking system status and harmonic signals [1]. To ensure system stability and error convergence to zero, two sliding surfaces were defined: 𝑠_𝑠1(𝑡)=𝑒₂(𝑡)+𝜆₁𝑒₁(𝑡),𝑠_𝑠2(𝑡)=𝑒₃(𝑡)+𝜆₂^𝑡₀𝑒₁(𝜏)𝑑𝜏, (8) where 𝜆₁ and 𝜆₂ are positive constants. When the system acts on the sliding surface, (8) are satisfied as 𝑠_𝑠1(𝑡)=𝑒₂(𝑡)+𝜆₁𝑒₁(𝑡)𝑠=0,_𝑠2(𝑡)=𝑒₃(𝑡)+𝜆₂^𝑡₀𝑒₁(𝜏)𝑑𝜏=0. (9)

Equations (9) can calculate the equivalent control for the system 𝑢_𝑠1eq(𝑡) and 𝑢_𝑠2eq(𝑡). The equivalent control can drive the system dynamic error to maintain the gradual movement of the sliding surface toward the origin. It can be deducted that this ensures that the dynamic system error progressively approaches zero. In order to make the system error status contact the designed sliding surface, the system selects two sliding mode control inputs, which is designed as 𝑢_𝑠1=−𝛾₁𝛼₁sgn𝑠_𝑠1(𝑡),𝛾₁𝑢>0,_𝑠2=−𝛾₂𝛼₂sgn𝑠_𝑠2(𝑡),𝛾₂>0. (10) Based on [1], 𝛼₁ and 𝛼₂ are 𝛼₁=𝛽_𝜓−𝛽_𝜑cos𝑒₁+𝜃_𝑑𝛽_𝜑−𝛽_𝜓cos𝑒₁+𝜃_𝑑𝐼²₁⋅sin³𝑒₁+𝜃_𝑑𝑓+𝐶𝐼₁−𝜆₁𝑒₂+𝑀_𝑔𝐼₁𝑙+𝐴₂+𝐶𝐼₁𝜃̇_𝑑+𝜃̈_𝑑,𝛼₂=−2cos𝑒₁+𝜃_𝑑sin𝑒₁+𝜃_𝑑𝑒₂+𝜃_𝑑𝑒₃+𝜓̇_𝑑+𝜆₂𝑒₃+𝛽_𝜑𝑒₂+𝜃̇_𝑑𝐼₁sin𝑒₁+𝜃_𝑑+𝜓̈_𝑑. (11) To prevent the sgn(•) equation from forming a noncontinuous control signal in the design, which would cause chattering phenomenon in the system, the controller presented in (10) can be modified as 𝑢_𝑠1=−𝛾₁𝛼₁𝑠_𝑠1𝑠_𝑠1+𝛿₁,𝛾₁𝑢>1,_𝑠2=−𝛾₂𝛼₂𝑠_𝑠2𝑠_𝑠2+𝛿₂,𝛾₂>1. (12)(13)𝛿₁ and 𝛿₂ are small constants and are designed to prevent the denominator from becoming zero.

4. Fuzzy Sliding Mode Control

This section describes the design and implementation of a fuzzy sliding mode controller in the gyroscope system. First, the two controller inputs for the gyroscope were designed for the two sliding planes as follows: 𝑠_𝑓1(𝑡)=𝑒₂(𝑡)+^𝑡₀𝑒₂(𝜏)+𝑘₁𝑒₁(𝜏)𝑠𝑑𝜏,_𝑓2(𝑡)=𝑒₃(𝑡)+^𝑡₀𝑘₂𝑒₃(𝜏)+𝑒₁(𝜏)𝑑𝜏, (14)(15) where 𝑘_𝑖∈𝑅, 𝑖=1,2 are positive numbers. Let (13) and (14) be equal to zero and they can easily fine equivalent control as 𝑢_feq1=𝛽_𝜓−𝛽_𝜑cos𝑒₁+𝜃_𝑑𝛽_𝜑−𝛽_𝜓cos𝑒₁+𝜃_𝑑𝐼²₁⋅sin³𝑒₁+𝜃_𝑑+𝐶𝐼₁𝑒₂+𝜃̇_𝑑−𝑀_𝑔𝐼₁𝑙−𝐴₂𝑒sin𝜔𝑡⋅sin₁+𝜃_𝑑+𝜃̈_𝑑−𝑒₂−𝑘₁𝑒₁,𝑢_feq2=2cos𝑒₁+𝜃_𝑑sin𝑒₁+𝜃_𝑑𝑒₂+𝜃̇_𝑑𝑒₃+𝜓̇_𝑑−𝛽_𝜑𝑒₂+𝜃̇_𝑑𝐼₁⋅sin𝑒₁+𝜃_𝑑+𝜓̈_𝑑−𝑘₂𝑒₃−𝑒₁. (16)

When the gyroscope system error states are during the sliding surface design, the controllers in (16) can be used to maintain the system states in sliding surfaces. Two fuzzy sliding mode controllers are used so that the error state quickly comes into contact with the sliding surfaces: 𝑢_𝑓1𝑒=exp²−𝑒₁𝜀₁𝑆Fuz^𝑓1𝜀_𝑠1,𝑢_𝑓2𝑒=exp³𝜀₂𝑆Fuz^𝑓2𝜀_𝑠2, (17) where 𝜀_𝑖∈𝑅, 𝑖=1,2 are strictly positive constants. The 𝜀_𝑠𝑖 is a normalization factor greater than zero. The Fuz(•) is a fuzzy controller. A fuzzy controller was designed using the following 7 IF-THEN control rules: R₁𝑆:if^𝑓𝑖𝜀_𝑠𝑖(•)RisNB,thenFuzisPB;₂𝑆:if^𝑓𝑖𝜀_𝑠𝑖(•)RisNM,thenFuzisPM;₃𝑆:if^𝑓𝑖𝜀_𝑠𝑖(•)RisNS,thenFuzisPS;₄𝑆:if^𝑓𝑖𝜀_𝑠𝑖(•)RisZE,thenFuzisZE;₅𝑆:if^𝑓𝑖𝜀_𝑠𝑖(•)RisPS,thenFuzisNS;₆𝑆:if^𝑓𝑖𝜀_𝑠𝑖(•)RisPM,thenFuzisNM,₇𝑆:if^𝑓𝑖𝜀_𝑠𝑖isPB,thenFuz(•)isNB, (18) where NB, NM, NS, ZE, PS, PM, and PB are colloquial narratives representing negative big, negative medium, negative small, zero, positive small, positive medium, and positive big, respectively. The 𝑆_𝑓𝑖/𝜀_𝑠𝑖 is a triangle-shaped fuzzy number, controller inference corresponds to fuzzy singleton, and the weighted average defuzzification method is used to calculate the controller output. Through addition of fuzzy controller design to control input, 𝑢₁(𝑡) and 𝑢₂(𝑡) of the gyroscope system (4) and (5) can be designed as 𝑢₁(𝑡)=𝑢_feq1(𝑡)+𝑢_𝑓1(𝑡),𝑢₂(𝑡)=𝑢_feq2(𝑡)+𝑢_𝑓2(𝑡). (19) Adjusting the fuzzy controller can drive the gyroscope system state to tracking the design state.

Theorem 2. Considering the error dynamics equations (6) for the gyroscope system and the designed control inputs (19), the gyroscope system asymptotically synchronizes with the designed signal so that the system error converges to zero.

Proof. A Lyapunov function is defined as 𝑉(𝑡)=𝑆²_𝑓12+𝑆²_𝑓22. (20) Differentiating (20) with respect to time yields 𝑉̇=𝑆_𝑓1𝑆̇_𝑓1+𝑆_𝑓2𝑆̇_𝑓2=𝑆_𝑓1𝑒₂̇(𝑡)+𝑒₂(𝑡)+𝑘₁𝑒₁(𝑡)+𝑆_𝑓2𝑒₃̇(𝑡)+𝑘₂𝑒₃(𝑡)+𝑒₁(𝑡)=𝑆_𝑓1−𝛽_𝜓−𝛽_𝜑cos𝑒₁+𝜃_𝑑𝛽_𝜑−𝛽_𝜓cos𝑒₁+𝜃_𝑑𝐼²₁⋅sin³𝑒₁+𝜃_𝑑−𝐶𝐼₁𝑒₂+𝜃̇_𝑑+𝑀_𝑔𝑙𝐼₁𝑙−𝐴₂⋅sin𝜔₂𝑡⋅sin𝑒₁+𝜃̇_𝑑−𝜃̈_𝑑+𝑢₁+𝑒₂+𝑘₁𝑒₁+𝑆_𝑓2−2cos𝑒₁+𝜃_𝑑sin𝑒₁+𝜃_𝑑⋅𝑒₂+𝜃̇_𝑑⋅𝑒₃+𝜓̇_𝑑+𝛽_𝜑𝑒₂+𝜃̇_𝑑𝐼₁sin𝑒₁+𝜃_𝑑−𝜓̈_𝑑+𝑢₂(𝑡)+𝑘₂𝑒₃(𝑡)+𝑒₁(𝑡). (21) Substituting (19) obtains 𝑉̇=𝑆_𝑓1exp𝑒₂−𝑒₁𝜀₁𝑆Fuz^𝑓1𝜀_𝑠1+𝑆_𝑓2𝑒exp³𝜀₂𝑆Fuz^𝑓2𝜀_𝑠2𝑒≤−exp²−𝑒₁𝜀₁𝑆_𝑓1𝑒−exp³𝜀₂𝑆_𝑓2<0. (22)
Since exp((𝑒₂−𝑒₁)/𝜀₁)>0 and exp(𝑒₃/𝜀₂)>0, its guarantee 𝑉̇<0. The states errors of the system are ensured to converge to zero. This completes the proof.

To verify the performance merits of the designed fuzzy sliding controller, integral absolute error (IAE) was used as a performance index. As described by the gyroscope system verification, IAE is expressed as IAE=^∞_𝑡0|𝑒(𝑡)|𝑑𝑡. (23)

5. Simulation Results

This section presents the results of a numerical simulation performed to evaluate the proposed method. The efficiency of the gyroscope system was evaluated using the sliding mode control method introduced by [1] and the proposed fuzzy sliding mode controller. Performance indices were also used to compare its advantages and disadvantages. In the numerical simulation, the gyroscope system parameters were set to 𝛽_𝜑=5, 𝐼₁=1, 𝑀_𝑔=4, 𝑙=0.25, 𝐶=0.5, and 𝜔₂=2. Calculations were made at 1 ms intervals. The initial value of the gyroscope system is set to 𝑥(0)=−0.3, 𝑦(0)=−0.1, and 𝑧(0)=5. The periodic trajectories designed for the gyroscope system were 𝜃_𝑑=0.5⋅sin(8𝑡), 𝜙_𝑑=0.4⋅sin(4𝑡). During the simulation, control signals were added at 𝑡=5sec to the gyroscope system. According to the sliding mode control in [1], the parameters for sliding surfaces (8) are 𝜆₁=𝜆₂=3; the selected parameters for controllers (12) and (13) are 𝛾₁=𝛾₂=2. Figures 3 and 4 show the simulation results. The solid line represents the status of the gyroscope system; the dashed line represents the cyclical trace design in Figure 3. Figure 4 is the control signal of the sliding mode of the gyroscope system. The parameters for the proposed fuzzy sliding mode controller sliding surfaces (14) and (15) are set to 𝑘₁=2, 𝑘₂=20, and 𝑘₃=10. The parameters chosen for controller (17) are 𝜀₁=1.5, and 𝜀₂=2, 𝜀_𝑠1=1, 𝜀_𝑠2=50. Figures 5 and 6 show the simulation results. The solid line represents the status of the gyroscope system; the dashed line represents the periodic trajectory in Figure 5. Figure 6 shows the control signal for the fuzzy sliding mode for the gyroscope system. Figure 7 shows the tracking error signal where, the solid line represents the sliding mode control and the dashed line the fuzzy sliding mode control. Table 1 shows the control system performance. Thus, the performance indicators confirm the satisfactory performance of the proposed gyroscope system.

Table 1 

Performance index of controllers.

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(b)

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Figure 3 

Time response of gyroscope via sliding mode control.

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Figure 4 

Control input using sliding mode control.

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Figure 5 

Time response of gyroscope via fuzzy sliding mode control.

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Figure 6 

Control input using fuzzy sliding mode control.

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Figure 7 

Time response of tracking error states.

6. Conclusions

The study was able to explore the problem from chaotic motion to periodic motion control of the gyroscope system states. The fuzzy sliding mode control was used to drive the system states into the designed states. Introducing the Lyapunov stability theory in the design of the fuzzy sliding mode controller enables the use of the proposed control law to ensure a stable and robust gyroscope system. Simulation results show that the proposed fuzzy sliding mode control can solve gyroscope system tracking control problems and does not produce chattering.

Acknowledgment

This work was supported by the National Science Council of Taiwan, under Contracts NSC 102-2221-E-022-019 and 101-2221-E-022-014.