limit the cart position for swinging up a pendulum. In this paper we introduce a novel Fuzzy Sliding. Mode (FSM) technique for swinging-up an inverted.
A DECOUPLED FUZZY SLIDING MODE APPROACH TO SWING-UP AND STABILIZE AN INVERTED PENDULUM Mariagrazia Dotoli, Paolo Lino, Biagio Turchiano Dipartimento di Elettrotecnica ed Elettronica Politecnico di Bari Via Re David 200 70125 Bari, Italy
Abstract: In this paper we present a novel decoupled fuzzy sliding mode (DFSM) strategy for swinging-up an inverted pendulum. In the proposed technique the control objective is decomposed into two sub-tasks, i.e., swing-up and stabilization. Accordingly, first a DFSM controller stabilizing the pole is synthesized and optimised via genetic algorithms. Next, a DFSM controller with a piecewise linear sliding manifold is synthesized and optimised, dealing with the task of entering the stabilization zone. Numerical simulations show the effectiveness of the proposed controller for a model encompassing friction as well as a limited control action and a restricted cart travel. Keywords: nonlinear systems, fuzzy control, sliding mode control, decoupled subsystems, genetic algorithms.
1. INTRODUCTION Swinging up an inverted pendulum is a well-known benchmark for the investigation of automatic control methodologies. A pole, hinged to a cart moving on a finite track, is swung-up from its pendant equilibrium and stabilized via a motor motioning the cart. Simultaneously, the cart is placed in an objective position on the track: hence, the system is underactuated. In addition, the control action is limited, since the motor presents a saturation effect. Numerous studies have been developed on the subject in the scientific literature. Wei, et al. (1995) presented a nonlinear technique based on a decomposition of the control law into a sequence of steps, taking into account only the pendulum motion. Wang, et al. (1996) applied to the pendulum subsystem the well known parallel distributed decomposition technique, based on a Takagi-Sugeno fuzzy model of the system. Åström and Furuta (2000) as well as Yoshida (1999) proposed energybased techniques, that work when the track length is unlimited. In such a spirit, recently Chatterjee, et al. (2002) proposed a sliding mode control strategy
using generalized energy control methods in order to limit the cart position for swinging up a pendulum. In this paper we introduce a novel Fuzzy Sliding Mode (FSM) technique for swinging-up an inverted pendulum and controlling the connected cart, while minimizing the cart travel. FSM control techniques are hybrid methodologies combining the effectiveness of sliding mode (SM) approaches in controlling nonlinear systems with the immediacy of fuzzy control algorithms. The SM control theory has been successfully established and employed for over a decade for nonlinear systems, see Slotine and Li (1991) for further reading. FSM controllers (Palm, 1994) approximate the nonlinear input/output map of a SM controller by means of a fuzzy inference mechanism (Jantzen, 1999) applied to a linguistic rule base. This results in a smooth control action, reducing the typical SM chattering, and in a restriction of the fuzzy rule table dimension. However, most of the FSM control methodologies presented in the related literature deal with second order nonlinear systems. Recently Lo and Kuo (1998) proposed a method called Decoupled Fuzzy Sliding Mode (DFSM) that deals with fourth order
systems and is based on the definition of a control surface encompassing two decoupled controllers designed via two corresponding sliding manifolds. Accordingly, in DFSM control two FSM control objectives are simultaneously taken into account. Moreover, Lo and Kuo (1998) apply the proposed method to the stabilization of an inverted pendulum. This paper presents a modified DFSM control strategy for swinging-up an inverted pendulum with restricted cart travel. We decompose the control objective into two sub-tasks, namely, swing-up and stabilization: hence, two target zones are defined in the phase plane. First, a DFSM controller stabilizing the pole is designed; next, a similar DFSM controller is synthesized, dealing with the task of entering the stabilization zone. Moreover, a gain scheduler is introduced in order to select the appropriate controller during operation. In particular, following the approach proposed by some of the authors in (Dotoli, et al., 2001), in this paper the swing-up DFSM regulator is defined using a piecewise linear sliding manifold that is bent towards the far off zones of the pole phase plane, in order to let the pole enter the stabilization zone. In addition, we suggest to optimise the controller via Genetic Algorithms (GAs): the interested reader can refer to Dotoli, et al. (2002) for a general procedure applying GAs to the optimization of FSM controllers. Several numerical simulations show the effectiveness of the proposed method for an inverted pendulum on a cart including friction and a limited control action, while taking into account restrictions on the cart travel.
s(x) = e + λe = x 2 − x 2d + λ( x1 − x1d ) = 0
(2)
where x 2d = x 1d (representing the desired trajectory for the second state variable) and the sliding constant λ is a strictly positive design parameter. It can be shown that at steady state system (1) follows the desired trajectory once s(x)=0, i.e., when the trajectory has reached the sliding line. Hence, in SM control the objective is to design a control law forcing the system onto the sliding surface. Slotine and Li (1991) showed that by selecting the Lyapunov function V=
1 2 s ( x) 2
(3)
the sliding line is attractive if the control action is the following: u = uˆ − Ksign (s(x)b(x)), K > 0
(4)
where ‘sign’ represents the sign function, and uˆ = −b −1 (x)(f (x) − x1d + λe ) .
(5)
Due to the sign function in (4), the SM controller exhibits chattering, i.e., high frequency switching phenomena. These may be avoided introducing a boundary layer of width Φ, i.e., replacing the sign with a saturation function (Slotine and Li, 1991): u = uˆ − Ksat (Φ −1s(x)b(x)), K, Φ > 0 .
(6)
The paper is organized as follows. In section 2 the SM and FSM control techniques are briefly reviewed. In section 3 we outline a fourth-order nonlinear model for the inverted pendulum and summarize the control task. Moreover, in section 4 a concise report on the DFSM control technique is given. In section 5 a DFSM controller for the pendulum stabilization is designed first, then a modified DFSM controller is synthesized in order to swing-up the pendulum. Finally, some conclusions are outlined.
However, using a boundary layer can compromise the tracking accuracy, since the tracking error magnitude directly depends on the boundary layer width. A straightforward method reducing the chattering effect is to fuzzify the sliding surface, i.e., combine fuzzy logic algorithms (Jantzen, 1999) with the SM control methodology by replacing K in (5) or (6) with a fuzzy variable Kfuzz. The distinctive marks of the resulting FSM technique (Palm, 1994) are the small magnitude of chattering and a low computational effort. It can be shown that FSM is an extension of SM control (Palm, 1994).
2. FUZZY SLIDING MODE CONTROL
The FSM controller rule table is determined on the basis of heuristic conditions in the phase plane, so that the overall control surface yields a SM control law with boundary layer (Slotine and Li, 1991). The procedure does not require a complete identification of the plant and keeps the computational effort relatively low. In the following, the rule table of the FSM regulator is as follows (Lo and Kuo, 1998):
In this section we recall the general concepts of SM and FSM control for a second order dynamical system. Consider the following nonlinear second order dynamical system in canonical form: x 1 = x 2 x 2 = f (x) + b(x)u T
(1)
where x=[x1 x2] is the observable state vector, u is the control input and the system output, f(x) and b(x) are nonlinear functions with b(x) of known bound. In addition, consider a given desired trajectory x1d(t) and the tracking error e=x1-x1d. The basic idea of the SM theory is to force the system, after a finite time reaching phase, to a sliding surface (called sliding line for a second-order system) containing the operating point and defined as follows for (1):
R1: If s is Negative Big, then u is Positive Big. R2: If s is Negative Small, then u is Positive Small. R3: If s is Zero, then u is Zero. R4: If s is Positive Small, then u is Negative Small. R5: If s is Positive Big, then u is Negative Big. The corresponding membership functions for s and u (Lo and Kuo, 1998) are triangular with completeness equal to 1 (Jantzen, 1999). If the sup-min compositional rule of inference and the center of area
defuzzification are adopted (Jantzen, 1999), the FSM emulates a SM with boundary layer (see Lo and Kuo, 1998 and Slotine and Li, 1991). 3. INVERTED PENDULUM ON A CART In this section we recall the model of an inverted pendulum (see figure 1). A pole, hinged to a cart moving on a track, is balanced upwards by a horizontal force applied to the cart via a DC motor. The cart is simultaneously motioned to an objective position on the track, which is finite. The system observable state vector is x=[x1 x2 x3 x4]T, including respectively the cart horizontal distance from the track center, the pole angular distance from the upwards equilibrium point and their derivatives. The force motioning the cart may be expressed as F=αu, where u is the input, i.e., the limited motor supply voltage. The system model is (Dotoli, et al., 2001): x 1 = x 3 x 2 = x 4
(7)
x 3 = f 2 (x) + b 2 (x)u x 4 = f1 (x) + b1 (x)u
where f1 (x) =
l cos x 2 (−Tc − µx 24 sin x 2 ) + µg sin x 2 − f p x 4 J + µl sin 2 x 2
b1 (x) =
f 2 (x) =
l cos x 2 α
J + µl sin 2 x 2
, (8a)
,
(8b)
a (−Tc − µx 24 sin x 2 ) + l cos x 2 (µg sin x 2 − f p x 4 ) , J + µl sin 2 x 2
b 2 ( x) =
aα J + µl sin 2 x 2
,
(8c)
(8d)
with l=
Lm p 2(m c + m p )
, a = l2 +
J , µ=(mc+mp)l. mc + mp
Remark 1. The inverted pendulum (7) is an underactuated fourth order system which is not in canonical form. Rather, it includes two second order subsystems, namely, the cart subsystem and the pendulum subsystem, with state vectors respectively [x1 x3]T and [x2 x4]T, coupled by way of the control input u. For each subsystem the state variables of the other one may be regarded as fictitious uncontrollable inputs. Remark 2. The pole dynamics, i.e., the second and fourth equations in (7), are affected by the cart only by way of friction Tc. If we neglect this contribution, the pole subsystem is in the form (1). In other words, the main nonlinearities affect the cart subsystem. Hence, to control both the cart and pole with one input, we can balance the rod first, and then the cart. 4. DECOUPLED FUZZY SLIDING MODE The FSM control technique depicted in Section 2 cannot be applied to a system of the form (7), which is not in canonical form and comprises two coupled subsystems, namely, in the inverted pendulum case, the cart and the pendulum. For such a system Lo and Kuo (1998) proposed the DFSM approach, based on a control surface resulting from two decoupled controllers designed via two corresponding sliding surfaces. In other words, in DFSM control two FSM regulation tasks, addressing the two sub-systems in (7), are taken into account. In the following we briefly review the DFSM technique. Consider a nonlinear fourth order system expressed in the form (7) with b1,2(x) of known bounds. This system includes two second order subsystems in canonical form with states [x1 x3]T and [x2 x4]T, respectively. The basic idea of DFSM control is to design a control law such that the single input u simultaneously controls the two subsystems to accomplish the desired performance. To achieve such a goal, the following sliding surfaces are defined (Lo and Kuo, 1998): s1 (x) = λ1 ⋅ ( x 2 − x 2d − z) + x 4 − x 4d = 0
(9a)
s 2 (x) = λ 2 ⋅ ( x1 − x1d ) + x 3 − x 3d = 0
(9b)
The cart and pole masses are respectively mc and mp, g represents the gravity acceleration, L is half the pole length, J is the cart and pole overall moment of inertia with respect to the system center of mass, fp is the pole rotational friction coefficient and Tc is the horizontal friction acting on the cart, which is a nonlinear function of the cart speed x3. Note that in (7) b1(x) and b2(x) are bounded. Here, the control task is to swing the pole up from its pendant equilibrium and subsequently stabilize it upwards, while the cart is simultaneously controlled to a position on the track. Hence, the system desired trajectory is x1d=x1d(t), x2d=0, x 3d = x 1d ( t ) and x4d=0. There is one control input u, which is bounded: the DC motor saturates.
3
final position
stabilization zone
swing-up zone initial position
2 1
Fig. 1. Pendulum swing-up and stabilization zone.
parameters scheduler cart position
x1
cart velocity
x3
STATE OF THE SISTEM
λ2 +
+
s2
Gs2
CART FLC
Gf2 z
CART FSMC
pendulum angle
x2
pendulum velocity
x4
+
-
λ1 +
+
s1
Gs1
PENDULUM FLC
Gf1
control action
PENDULUM FSMC
Fig. 2. DFSM controller for stabilization and swing-up of an inverted pendulum with restricted cart travel. where z is a value proportional to s2 with the range proper to x2. A comparison of (9a) with (2) shows the meaning of (9a): the control objective in the first subsystem of (7) changes from x2=x2d and x4=x4d to x2=x2d+z and x4=x4d. On the other hand, (9b) has the same meaning of (2) and its control objectives are x1=x1d and x3=x3d. Now, let the control law for (9a) be a FSM emulating a SM with boundary layer (6): u1 = uˆ1 − G f1 sat (s1 (x)b1 (x)G s1 ), Gf1,Gs1>0
(10)
with uˆ1 = −b1−1 (x)(f1 (x) − x 2d + λ1x 4 − λ1x 2d )
(11)
and obvious meaning of the other parameters in (10). So, let the control law for (9b) be a FSM mimicking a SM with boundary layer (Lo and Kuo, 1998): z = sat (s 2 ⋅ G s 2 ) ⋅ G f 2 , 0 < G f 2 < 1
(12)
where Gs2 represents the inverse of the width of the boundary layer to s2, Gf2 transfers s2 to the proper range of x2 and ‘sat’ indicates the saturation function. Notice that in (12) z is a decaying oscillation signal, since Gf2
