IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 4, JULY 2005
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Decoupling Control by Hierarchical Fuzzy Sliding-Mode Controller
Chih-Min Lin and Yi-Jen Mon
Abstract—A design method using hierarchical fuzzy slidingmode (HFSM) decoupling control is proposed to achieve system stability and favorable decoupling performance for a class of nonlinear systems. In this approach, the nonlinear system is decoupled into several subsystems and the state response of each subsystem can be designed to be governed by a corresponding sliding surface. Then the whole system is controlled by a hierarchical sliding-mode controller. In this design, an adaptive law is derived based on the Lyapunov function to tune the coupling factor of the hierarchical sliding-mode controller so as to achieve favorable decoupling performance with guaranteed stability. The proposed design method is applied to investigate the decoupling control of a double inverted pendulum system. Simulations are performed and a comparison between the proposed HFSM decoupling control and a conventional fuzzy sliding-mode (FSM) decoupling control is presented to demonstrate the effectiveness of the proposed design method. Index Terms—Adaptive law, decoupli
ng control, fuzzy control, hierarchical sliding-mode control.
I. INTRODUCTION UZZY LOGIC CONTROL (FLC) using linguistic information can model the qualitative aspects of human knowledge. Since FLC is a model free design method and is more insensitive to plant parameter variations and external disturbances, it has been successfully applied to many engineering systems. As for the FLC system, though it is one of the most effective methods using expert knowledge without knowing the exact model of the controlled systems, the major drawback of the FLC system is lack of adequate analysis and design techniques [1], [2]. In recent years, the sliding mode control (SMC) methodology has been widely used for control design problem for a class of nonlinear systems. SMC is an effective approach to the problem of maintaining stability and consistent performance of a controlled system with imprecise modeling [3], [4]. The main advantage of SMC is that the system uncertainties and external disturbances can be handled under the invariance characteristics of system’s sliding condition. However, fundamental problems still exist in the control of complex systems using sliding mode controllers; e.g., control signal chattering is its main disadvantage. Recently, there has been much research on the design of fuzzy logic controllers based on the sliding-mode control scheme, referred to as fuzzy sliding-mode
Manuscript received August 8, 2003. Manuscript received in final form April 12, 2004. Recommended
by Associate Editor D. A. Schoenwald. This work was supported by the National Science Council of the Republic of China under Grant NSC-90-2213-E-155-016. C.-M. Lin is with the Department of Electrical Engineering, Yuan-Ze University, Chung-Li, 320 Taiwan, R.O.C. (e-mail: u.edu.tw). Y.-J. Mon is with the Department of Computer Science and Information Engineering, Chung-Kuo Institute of Technology, Hsinchu, 300 Taiwan, R.O.C. (e-mail: monbuy@ms.ckitc.edu.tw). Digital Object Identifier 10.1109/TCST.2004.843130
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controls (FSMCs) [5], [6]. FSMC, which is an integration of FLC and SMC, provides a simple way to design FLC systematically. This approach retains the positive property of SMC but alleviates the chattering, and the fuzzy control rules can be determined systematically by the reaching condition of SMC. The main advantage of FSMC is that the control system can achieve asymptotic stability. Recently, a fuzzy sliding-mode (FSM) decoupling control design method has been proposed to achieve decoupling performance of a class of nonlinear coupled systems [7]. Although an intermediate variable has been introduced to incorporate the state information of two subsystems, this intermediate variable must be predetermined by time-consuming trial-and-error procedure. From the authors’ simulations, inadequate choice of the intermediate variable will degrade the control perfo
rmance, even causes the instability of the coupled system. In addition, some researchers have proposed time-varying sliding surface design methods for FSMC [8], [9]. By these approaches, the state trajectories can be controlled to achieve fast convergence and good robustness with regard to parameter variations and external disturbances. In this study, a design method of hierarchical fuzzy slidingmode (HFSM) decoupling control is proposed. In this approach, a class of nonlinear coupled systems can be decoupled into several subsystems and a sliding surface is defined for each subsystem. Then, a hierarchical sliding-mode controller is proposed to control the whole system. In this design, an adaptive law is derived to tune the coupling factor of the hierarchical sliding-mode controller so as to achieve favorable decoupling performance. Since the Lyapunov function is used to derive the control law, the control system can be guaranteed to be asymptotically stable. The proposed HFSM design method is then applied for the decoupling control of a double inverted pendulum system. Simulations are performed and a comparison between the proposed HFSM decoupling control and a conventional FSM decoupling control is presented. II. PROBLEM FORMULATION Consider a single-input–multi-output (SIMO) nonlinear coupled system expressed in the following form:
(1) where are the state variables; , and are bounded is the control input; and nominal nonlinear functions; and are the lumped disturbances, which include , and
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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 4, JULY 2005
the system uncertainties and external disturbances; and assume where and they are bounded by 1, 2 are positive constants. The parameters are all , and in the abbreviated as following description. This system can be treated as two subsystems with second-order canonical form including the states and , respectively. The decoupling control tries to design a single input to simultaneously control the and to achieve desired performance. states The hierarchical sliding-mode control is characterized by defining a pair of sliding surfaces as (2) (3) and are positive constants. The time derivatives of where (2) and (3) are obtained as (4) (5) From (1), (4), and (5), it is obtained that (6) (7) In (6), if the lumped disturbance is not existent, i.e., 0, by the condition of sliding mode 0 [4], an equivalent control law can be obtained as (8) However, this control law does not consider the lumped disturbance ; and it only considers the control of the subsubsystem. system while disregards the control of the For achieving favorable decoupling performance, a HFSM control is proposed in the next section. III. HFSM CONTROL Define a hierarchical coupled sliding surface as (9) is a positive constant and is ref
erred to as the optimal where coupling factor. This coupling factor will play an important role for the interactive control between the sliding surface and . The derivative of this hierarchical sliding surface is obtained as (10) should In (9), for different initial conditions, different be appropriately chosen case by case to achieve satisfactory decoupling performance. However, this is a time-consuming trial-and-error process. Thus, to improve the decoupling performance of the coupled system, an adaptive fuzzy tuning method of the coupling factor is proposed bellow. A fuzzy
inference system is employed to tune a coupling factor to estimate . The fuzzy inference system is expressed as If and then (11)
where denotes the th fuzzy rule, is the fuzzy set in the antecedent part associated with the th input variable at the th fuzzy rule characterized by the fuzzy mem; and is the fuzzy set in the conbership functions sequent part characterized by an adjustable singleton . The is given as [10] Gaussian-type membership function of (12) where the parameter represents the center value and the padenotes the reciprocal value of deviation from the rameter center to which the value on the standardized support set has 0.5. The input is scaled by the parameter . Then the fuzzy output can be inferred as (13) is a parameter vector, and is a regressive vector with defined as , where is the inferred is a positive value, the grade of the th fuzzy rule. Since is also restrained to be positive. estimated value By the u
niversal approximation theorem [1], there exists an such that optimal fuzzy logic system (14) where the time-invariant parameter vector is defined as (15) where on , and are compact sets of suitable bounds , and , respectively, and they are defined as , and , where , and are positive constants specified by the designer. The minimum approxima, which is assumed to be tion error is defined as where is a positive constant. Define the bounded by estimation error of the optimal parameter vector (16) Considering the disturbed system in (6), i.e., control input is defined as , the where
(17) is given in (8), and is the variable structural term where to cope with the lumped disturbance and interactive coupling influence. From (9) and (16), a Lyapunov function is defined as (18)
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 4, JULY 2005
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where is a positive constant. From (6), (9), (16), and (17), the Lyapunov stability condition can be derived as follows:
is uniformly continuous. By using also bounded [11]. Then 0, Barbalat’s Lemma [3], it can be obtained that . that is In summary, the control law is given as
(25) where the coupling factor is adjusted by (13) with adapted by (23). This control law reveals that the disturbance rejection is achieved by the first switching control term and the desired deand coupling control is achieved through the coupling factor the information from two subsystems where the optimal coupling factor is governed by the second switching control term. In general, the switching sign function can be replaced by a saturation function in order to reduce chattering of the control signal. The proposed HFSM control law can also be applied for higher order systems, for example a three subsystems sixth-order system:
(19) Choose (20) and (21) where is a sign function. Then (19) becomes
(22) The adaptive law in (21) is unable to guarantee . Therefore, the adaptive law has to be modified by using the projection algorithm [1], such that the parameter vector will remain inside the constraint. The modified adaptive law is given as follows: if or and (26) The sliding surfaces can be defined as , and , where , and are positive constants. The hierarchical sliding surface can be defined as (27) where the optimal coupling factors and are set as positive constants. Then by the manipulations like (19), the control law can be derived as shown in (28) at the bottom of the page and are the estimated coupling factors which where are adapted by the following fuzzy inference systems: (29) (30) (24) Since the right-hand side of (24) is bounded, it is obtained that . Differentiate with respect to time . Si
nce the entire variables on the right-hand side of (6) and (7) are bounded, it implies is with the adaptive laws shown in (31)–(32) at the bottom of the next page, in which and are the positive learning rates; and are the parameter vectors; and are the regressive vectors; where . and Also, and are restrained to be positive. Similarly, this
if Since plies that , that is and
and (23)
, it imare bounded. Let the function , and integrate with respect to time, then it is obtained that
(28)
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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 4, JULY 2005
Fig. 2. Structure of double-inverted pendulum. TABLE I FUZZY INFERENCE RULES Fig. 1.Concept diagram of HFSM control system.
idea can be extended to even higher order systems by introducing the hierarchical sliding surface. The concept diagram of this HFSM control system is depicted in Fig. 1. IV. SIMULATION RESULTS A double inverted pendulum system is simulated and a comparison between the proposed HFSM and the FSM decoupling control is demonstrated. The structure and controlled states of double inverted pendulum are shown in Fig. 2. Its dynamics are described as:
(33) where is the angle of the pole 1 with respect to the vertical axis, is the angular velocity of the pole 1, is the angle of the pole 2 with respect to the vertical axis, is the angular velocity of the pole 2, is the position of the cart, is the velocity of the cart, is the applied force to move the cart and is the disturbance. The system equations are given in [7]. In the simulations, the parameters are chosen as 1 Kg; 0 m; 9.8 m/s ; and 0.05 /s .
For the coupling factor tuning in (11), same membership functions are defined for and . Five Gaussian-type membership functions are constructed from (12) with centers at 1.5 0.5 0 0.5, and 1.5 to represent the fuzzy sets: negative big (NB), negative small (NS), zero (ZE), positive small (PS) and positive big (PB), respectively. For each membership function, and are set as 1 and 0.15. A 25 rules fuzzy inference system is tabulated in Table I for . The same fuzzy inference system is also used for . The initial values of the adjustable singletons for and are all set as 0.1. For the HFSM deco
upling control, the hierarchical sliding surface is defined as in (27), where , and are built with 0.4, and 0.1. The control law
if if if if
or and or and
and (31)
and (32)
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Fig. 3(a). x (angle of pole 1) response diagram of double-inverted pendulum. (b) x (angle of pole 2) response diagram of double-inverted pendulum. (c) x (position of cart) response diagram of double-inverted pendulum. (d) Control input diagram of double-inverted pendulum. (e) Coupling factors n ^ and n of ^ double-inverted pendulum.
is designed as in (28) with the adaptive laws given in (31) and (32), where 0.05 0.5, and 5. The parameters and are chosen as 0.5 and 0.1, respec-
tively. This is because the second pole is more sensitive than . All the paramethe first pole, so that we choose ters , and are specified based

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