Second-Order Hierarchical Fast Terminal Sliding Model Control for a Class of Underactuated Systems Using Disturbance Observer

A second-order hierarchical fast terminal sliding mode control method based on disturbance observer (DOSHFTSM) is proposed for a class of fourth-order underactuated systems. In the first step, the fourth-order underactuated system is divided into two subsystems, and the integral sliding surface is designed for each subsystem. Then, the first-order fast terminal sliding surface is defined by using the integral sliding surface and its derivatives of each subsystem, and the switching control items of the system are designed according to the first-order fast terminal sliding surface of the subsystem. Secondly, the second-order sliding surface is designed by using the first-order fast terminal sliding surface of each subsystem. On the premise of ensuring the stability of Lyapunov, the switching control term is designed by using the variable coefficient double power reaching law to eliminate the system jitter. Finally, based on the principle of hyperbolic tangent nonlinear tracking differentiator, a hyperbolic tangent nonlinear disturbance observer (TANH-DOC) is designed to estimate the uncertainties and external disturbances of the system and compensate them to the sliding mode controller to improve the robustness of the system. The stability of the system is proved by using Lyapunov principle. The validity of this method is verified by numerical simulation and physical simulation of inverted pendulum system.


Introduction
In recent years, the public pay more and more attention to the improvement of underactuated systems. It usually appears in mechanical systems where the actuator is less than the control degree of freedom. It is widely applied in space robots, underwater robots, structural flexible robots, bridge cranes and other practical systems, and has published many papers on underactuated system control [1][2][3][4][5][6][7][8][9][10][11][12]. In a nutshell, controller design and stability analysis of nonlinear underactuated dynamic systems have always been an important research field.
Sliding mode control embraces premium robustness to system matching uncertainties and modeling errors. It is a kind of nonlinear control method with great significance that is widely used in various kinds of under actuation [13][14][15][16][17].
But, the traditional sliding mode control method makes the system state gradually converge to the equilibrium point on the sliding mode surface, which makes it difficult to achieve convergence in a finite time. So, terminal sliding model (TSM) greatly improves the convergence speed of the system near the equilibrium point by introducing terminal attractors [18][19][20][21]. However, in the traditional TSM, when the system state approaches the equilibrium state, the convergence speed of the nonlinear sliding mode is slower than that of the linear sliding mode. Therefore, a fast terminal sliding mode (FTSM) is proposed, which not only introduces the terminal attractor to make the system state converge in a limited time, but also retains the fast convergence of linear sliding mode when approaching the equilibrium state, thus realizing the fast and accurate convergence of the system state to the equilibrium state [22][23][24][25][26][27]. However, a large control signal will be generated probably in the steady state and there will be chattering when there is strong disturbance in the system. A novel sliding mode controller based on extended disturbance observer is studied for a class of underactuated systems in reference, aiming to cut down the chattering effect in [28]. But small oscillation still exists in the system state when disturbance is added. Document [29] proposes an adaptive hierarchical sliding mode control method based on extended state observer for the practical application of spherical robots. The designed closed-loop control system of the spherical robot possesses robust and adaptive capabilities to overcome the uncertain rolling resistance but the response time of the system is slow and chattering exists. The proposed controller strategy that the integral sliding mode control and the optimal feedback control law is composed in [30]. The main advantages of the proposed approach are ensuring the robustness throughout the whole system response against the uncertainties, decrease the chattering effect and eliminate the reaching phase. But the simulation experiment system is linearized. A unified adaptive second order sliding mode control method is devised. By using the proposed control structure, the upper bounds of uncertainties are not required, the over-estimation of the control gains are avoided, and the chattering of the conventional sliding mode controllers can be attenuated in [31]. But only simple disturbance phenomena are analyzed, and complex disturbance factors are not analyzed.
In this paper, a hierarchical second-order fast terminal sliding mode based on disturbance observer control approach is proposed for a class of underactuated systems. The contributions of this paper are as follows.
1. The sliding mode controller are designed by using the second-order hierarchical fast terminal sliding mode surface and variable coefficient double power reaching law to reduce the chattering of the system. 2. A hyperbolic tangent nonlinear disturbance observer, which is synchronized with it, has been designed to estimate the uncertainty and external disturbance of the system. 3. The stability of sliding surface at all levels is proof.
Meanwhile, the effectiveness of the method is verified by the numerical simulation experiment of inverted pendulum. The rest of this paper is organized as follows. A class of underactuated systems is formulated in Section 2. An effective second-order hierarchical fast terminal sliding mode Controller based on disturbance observer is devised, and the stability of sliding surface at all levels is analyzed in Section 3. Section 4 conduct the simulation and present the results. Finally, some conclusions are given in Section 5.

Problem Formulation
Consider the following dynamic model of a cart-pole system as [32]: denote the nonlinear functions representing system dynamics; ( ) u t indicates the control input; 1 ( ) d t and 2 ( ) d t are bounded external disturbances and system parameter perturbations.
The terms 1 ( , ) f t x and 2 ( , ) f t x can be expressed as: (2) where 1 '( , ) f x t and 2 '( , ) f x t are the known parts of 1 ( , ) f x t are the unknown parts of 1 ( , ) f x t and 2 ( , ) f x t . The dynamic equation of (1) can also be written as: are the terms of bounded external disturbances and system parameter perturbations. Assumption 1. The system perturbations are assumed to be bounded as 1 1 n ≤ δ and 2 2 n ≤ δ , where 1 δ and 2 δ are unknown positive constants.

Main Results
The control objective of the system is to design a robust controller that enables accurate and fast stable even in the presence of model uncertainties and external disturbances. In order to reach the target, a hierarchical second-order fast terminal sliding mode control scheme combining disturbance observer is designed.

Disturbance Observer for Hyperbolic Tangent Nonlinear Function
The tracking differentiator (TD) was first proposed by the researcher Han and others in China in 1994 [33]. It is used in practical engineering problems to extract continuous filtered signals and differential signals from discontinuous or random noise measurement signals.
In this paper, nonlinear tracking differentiator is constructed by hyperbolic tangent nonlinear function as follows [34].
The stability and convergence of system (4) are proved and detailed regulating rules of design parameters are given in [34]. A special example of TANH-TD can be obtained when 1 2 1 2 , m m h h = = , so that it has fewer tuning parameters and will be more convenient for engineering applications. Theorem 1. No loss of generality, we consider the underactuated mechanical system (3). The design of nonlinear disturbance observer based on tracking differentiator, described as: where 2 1 4 2ˆˆ, , , x n x n are the estimates of 2 1 4 2 , , , x n x n , respectively. If 0, 0 T R > > , we get: In other words, 2 → . In addition, we have Proof: When R → ∞ , the following equation can be obtained： That is, the varies in 1 n and 2 n is much faster than x , respectively. Meanwhile, we can clearly get the equation as: Therefore, when we regard 1 x , it is clear that (5), (6) are established according to theorem of [34]. This is the proof of completion. Remark 1. Please note that in the actual system, any control input is limited. In other words, bandwidth and speed are bounded for any control input u . Therefore, it is reasonable to assume that the varies in 1 n and 2 n is much

Second-Order Hierarchical Fast Terminal Sliding Mode
Corresponding to the two groups of the state variables ( 1 2 , x x ) and ( 3 4 , x x ) of two subsystems we construct a pair of suitable integral sliding surfaces as: where 1 λ and 2 λ are positive constants.
According to (9), the following first-level sliding surface is defined where 1 α , 1 β , 2 α , 2 β are positive constants. 1 1 2 2 , , , q p q p are all positive odd numbers and 1 1 2 2 , q p q p < < . Using the equivalent control method the equivalent control law of the subsystems can be obtained as: The under-actuated system has the characteristic of controlling the multiple outputs with less input. Therefore, the total control input includes the equivalent inputs for all subsystems. We define the total control as: 1 2 eq eq sw u u u u = + + (13) where sw u is the switch control part of the sliding controller. According to the first-level sliding surface of all subsystem, defining the second-level sliding surface and double power approaching law with the variable coefficient as: eq sw eq sw eq sw eq sw eq eq t u u a n n b t u u a n n b t u a b t u S a n n b t u a b t u a n n b t u a n n S b t u a b t a b t The switching control law is defined as: ( ) Therefore, the general control law of the system is given as follows.  2) V ɺ is negative definite except for the equilibrium point.

Stability Analysis
3) Real number 0, 0 k > > α and region N M x converge sat balanced zero point with infinite time.
Theorem 2. The under-actuated system (3) is adopted in the control law of (19), and the second-level sliding surface S and S ɺ converge to the following regions in a finite time.   (21) Proof: Substitute (18) into (17) c c c c c c M a n n a n n k S S V S k S S S a n n a n n k S k S M t D a n n a n n . Then it becomes Lemma 1 shows that the system converges with respect to equilibrium zeros in finite time. Since , we can get outside the area 0 V ≤ ɺ . Therefore, the convergence region at this time is (26).
Lemma 1 shows that the system converges with respect to Underactuated Systems Using Disturbance Observer equilibrium zeros in finite time. Since , we can get outside the area 0 V ≤ ɺ . Therefore, the convergence region at this time is (28).
By synthesizing (26) and (28), S can converges to the following regions in finite time.
Substituting the previous equation into (15), we can get: In conclusion, the theorem 2 is proved to be correct. Theorem 3. If the control law of (19) is used for the underactuated system (3) and the sliding surface is designed as (10), (14). The first-layer sliding mode surfaces 1 2 sosm sosm S S ， are also asymptotically stable.
Proof: According to the theorem 2, we can have S L ∞ ∈ and 2 S L ∞ ∈ . By analyzing the first sliding mode surfaces, we can easily get that 1 Owing where t m is the total mass of the cart-pole system which contains the quality of the pole ( p m ) and the mass of the cart ( c m ), 1 x represents the swing angle of the pole, 2 x expresses the swing speed of the pole, 3 x denotes the position of the cart, 4 x indicates the cart velocity. Contrasting control schemes are considered as below: (a) The sliding surface, reaching law and controller of hierarchical sliding mode control method (HSM) based on exponential reaching law are as follows [30] For the simulation implementation, the constant parameters and initial conditions of the cart . The parameters of controllers are provided in Table 1. In order to better analyze the stability of the system, the following performance evaluation indicators are given: (1) Integral of error squared value (ISE) The performance index values of the inverted pendulum system for the above two evaluation methods are shown in Table 2. It can be seen from Table 2 that the performance evaluation index value of the state 1 x and 3 x of the control method used in this paper are smaller than other control methods. Therefore, the system stability of DOSHFTSM is better than that of other control methods.   4 shows the response curves of the system state, the sliding surfaces, and the system phase trajectory, respectively. It can be seen from them that the system state can converge to the equilibrium point at a faster speed under the control of the proposed scheme and can intuitively show that the control process of the proposed method is smooth and jitter-free. Figure 5 show that the control method of the present invention controls the input signal to be smooth, which can reduce the wear of the motor. Figure 6 show that the disturbance observer designed in this paper can estimate the interference value very well.
In what follows, for the robustness analysis, the simulation studies are repeated with different values of the initial conditions, disturbances and uncertainties terms.   At the same time, ( ) 5sin t interference signal is added to the simulation to 5s. Time responses of the system state, the sliding surfaces, system phase trajectory and control input signal are displayed in Figures 7-10, respectively. These simulation results also confirm that the control scheme has good robustness under different conditions. Therefore, the systems can be stably controlled with different values of the initial conditions, disturbances and uncertainties terms, too.       In this paper, the matlab real-time simulation tool and Simulink toolbox are used to test the designed controller on the actual vehicle system. Figure 11 shows the linear motor inverted pendulum system manufactured by Hopemotion Co., Ltd.

Conclusions
A second-order hierarchical fast terminal sliding mode control scheme based on disturbance observer is proposed for a class of fourth-order underactuated mechanical systems. Equivalent control terms and switching control terms of sliding mode controller are designed by using second-order hierarchical fast terminal sliding mode surface and variable coefficient double power reaching law. The tracking differentiator principle is used to design a hyperbolic tangent nonlinear disturbance observer. The uncertainties and external disturbances of the system are estimated, and the sliding mode controller is compensated to improve the robustness of the system. The stability of the system is verified by Lyapunov principle. Finally, the effectiveness of this method is verified by numerical simulation experiment and physical simulation of inverted pendulum. Therefore, the application of this method to the control of underactuated mechanical systems such as multi-stage inverted pendulum, manipulator system, torch system and spherical plate system deserves further study.