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PAGN: Thanuja -- SCAN: Shubha
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Fuzzy Sets and Systems
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Hybrid adaptive fuzzy controllers with application to robotic systems
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Chih-Min Lin∗ , Yi-Jen Mon
Department of Electrical Engineering, Yuan-Ze University, Chung-Li, Tao-Yuan, Taipei, 32026 Taiwan, Republic of China
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Received 10 December 2001; received in revised form 2 September 2002; accepted 15 October 2002
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Abstract
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This paper presents a hybrid adaptive fuzzy control (HAFC) design methodology, where the nonlinear system is controlled by a state feedback controller and an adaptive fuzzy controller. In the adaptive fuzzy controller, an adaptive law is developed to tune a robust gain of the sliding-mode controller (SMC) so as to cope with the uncertainties and model errors. The proposed design method is applied to investigate the position and tracking control of a two-link robot. A comparison between the proportional-derivative control, SMC, adaptive control and the proposed HAFC for a two-link robot is made. The simulation results demonstrate that, by using the HAFC, the system performances is improved and the system also exhibits stability and robustness. c 2002 Published by Elsevier Science B.V.
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Fuzzy logic control (FLC) is a human knowledge-based design methodology which is driven accordingly by fuzzy membership functions and fuzzy rules. The applications of FLC mainly focus on certain systems that are structurally di:cult to model due to their inherent natural nonlinearities and other modeling complexities [10]. However, the conventional FLC still has two main drawbacks: (1) The fuzzy rules are obtained from human trial-and-error work, and (2) it still lacks the systematic or mathematical methodologies to ensure the system stability. In order to avoid the timeconsuming trial-and-error process for the design of the fuzzy rules, several self-tuning processes have been proposed. The tuning processes can be carried out by the following methods: least-square-error
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1. Introduction
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Keywords: Fuzzy control; Sliding-mode control; Adaptive law; Robotic systems
∗
Corresponding author. Tel.: +886-3-4638800x418; fax: +886-3-4639355. E-mail address:
[email protected] (C.-M. Lin).
c 2002 Published by Elsevier Science B.V. 0165-0114/02/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 2 ) 0 0 5 0 4 - 3
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(LSE) [5], neuro-fuzzy [8,9,11,20], self-organizing [12], adaptive fuzzy [13,19], sliding PI-type fuzzy [7] and adaptive rules insertion [18]. For the stability problem, some useful stability and robustness criterions for FLC have been developed based on some special conditions such as canonical state space models, or phase-plane dynamic models [6,17]. Neural networks are a promising new generation of information processing systems that denotes the ability to learn and recall from training data. The approximation of any continuous function using a neural network seems outstanding and eFective. However, a compact set for approximation is not bright to estimate, the number of neurons is not clear to determine, and it is probably time consuming for an on-line learning. Furthermore, it may have the instability problem caused by parameter drift [4]. Sliding-mode control (SMC) is a robust design methodology developed using a systematic scheme based on a sliding surface and Lyapunov’s stability theorem [3,16]. The main advantage of SMC is that the system uncertainties and external disturbances can be accommodated under the invariance characteristics of system sliding conditions. However, the switching control in SMC results in control gain chattering. Furthermore, the control model should have a canonical form so as to derive an equivalent control gain to drive the system trajectory into a sliding surface. Some adaptive SMC design methods have been proposed [15,22], however, these design methods need to choose suitable sliding surface and equivalent control gain. There has been much research involving designs for FLCs based on the SMC scheme, which is referred to as fuzzy SMC [2,14,21]. Robotic system design has been seen to be an important issue in control engineering. Several model-based robotic system design approaches have been proposed in the past decade. In [9], a neurofuzzy controller based on the robotic dynamic behavior was developed; in this design, the robotic performance is upgraded by tuning the fuzzy membership functions based on the robotic dynamic behavior. However, the initial fuzzy membership functions and fuzzy rules must be appropriately chosen oF-line. In [1], a neural-network methodology combined with an H ∞ scheme was used to control uncertain or unknown robotic systems; in this design, the error dynamics are derived and the neuro-network is used to guarantee the H ∞ performance. However, the parameter tuning processes yield a slow response. In [16], a simple proportional-derivative (PD) control approach was utilized for the position control; meanwhile the SMC and adaptive control approaches have been presented for trajectories tracking control. However, their transient responses are not good enough for a robotic system control. So this paper develops an intelligent control method by introducing the fuzzy inference mechanism into a SMC that can work e:ciently for both position and tracking control. In this paper, a design method of hybrid adaptive fuzzy control (HAFC) is proposed, and then it is applied to a nonlinear robotic system control. The proposed HAFC includes a state feedback controller and an adaptive fuzzy controller. A stable feedback gain is designed to guarantee the system stability and desired control performance for the nominal system. In addition, an adaptive law is derived to tune a robust gain of the SMC so as to cope with the system uncertainties and model errors. This adaptive law is derived in the Lyapunov sense such that the system stability can be guaranteed. Comparing with the neural-network controllers, the proposed design method has the advantage of easy computation, system stability and performance robustness. The proposed HAFC design method is applied for the position and tracking control of a two-link robot. A comparison between the PD control, sliding control, adaptive control, and the proposed HAFC is made to illustrate the eFectiveness of the proposed design method.
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2. Hybrid adaptive fuzzy control for a nonlinear system Consider a multi-input multi-output time-invariant nonlinear system
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x(t) ˙ = F(x(t); u(t));
(1a)
which will be abbreviated as
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(1b)
where x ∈R n is a vector of states, and u ∈Rp is a vector of control input. Assume this nonlinear system can be modeled around the nominal state at [x0 u 0 ], and consider the system uncertainty and external disturbance between the real system and the nominal linear model. The linearized model is formulated as
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x˙ = F(x; u);
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u = ufb + urb ;
where the feedback controller ufb is determined on the basis of the nominal model considering the disturbance-free case, while the robust controller u rb compensates for the system uncertainty. The concept diagram of the control system is shown in Fig. 1, and the design procedure is described in the following. The linear feedback control law is designed as ufb = Gx;
(4)
where G ∈Rp×n is the state feedback gain matrix which can be found from standard methods such as pole placement. And the closed-loop dynamic for the disturbance-free system is given by x˙ = A0 x + B0 u = (A0 + B0 G )x;
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(5)
where the closed-loop dynamics matrix
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Ac = A0 + B0 G
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(3)
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(2)
where A0 =9F=9x0 ∈R n×n and B 0 =9F=9u 0 ∈R n×p ; and d ∈R n is referred to as the lumped uncertainty which represents the diFerence between the nonlinear system and the nominal linear system. This uncertainty is assumed to satisfy the matching condition such that d =B 0 n where n ∈Rp and n6 where is a positive constant and · denotes the induced norm. The necessity of coping with the uncertainty in the model calls for a hybrid control input u of the form
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x˙ = A0 x + B0 u + d ;
has eigenvalues speciLed in the left-half s-plane for the systems stability and desired responses. DeLne denote the sliding surface as s = Cx = [s1
s2
···
s p ]T = 0
v t ¿ ts
(7)
where ts is the time at which the sliding motion starts and C ∈Rp×n is set as a constant matrix to correspond the desired sliding-mode responses.
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Adaptive law (19)
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s
kˆrb sgn( s )
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urb x
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Nonlinear Plant
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u fb
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G Fig. 1. HAFC design concept diagram.
In (3), the robust controller is designed as
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where k rb ¿0 is utilized to cope with the system uncertainties and sgn(·) is a sign function. When the sign function is used as a robustness term, the control law in (8) becomes an SMC. However, the control law in (8) is usually discontinuous across the sliding surface. The application of the robust controller needs to know the bound of uncertainty. However, this bound is di:cult to measure in practical applications; thus, in general, the bound of the uncertainty is chosen large enough to ensure robust stability. However, a large k rb will result in substantial chattering of the control eFort. On the other hand, if the bound is chosen too small, the robust stability cannot be guaranteed. To relax the requirement for the bound of uncertainty, an adaptive design method for the robust gain is proposed. ∗ for the existence of sliding-mode condition, i.e. n¡k ∗ . Assume there is an optimal gain k rb rb ∗ However, the optimal gain k rb cannot be obtained exactly because of the unknown of uncertainty. Therefore an adaptive fuzzy control scheme is developed to estimate this optimal gain. The basic idea for constructing the fuzzy rules is that when the system states are far away from the sliding surfaces then a large robust gain is needed and vice versa. Meanwhile, in the proposed fuzzy rules the antecedent parts are predetermined with appropriate membership functions and the consequent parts are deLned to be the singletons and should be self-generated by the adaptive laws developed later. For this, the robust controller in (8) is modiLed to be an adaptive fuzzy controller
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urb = −krb sgn(s);
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urb = −kˆrb sgn(s);
(9)
where kˆ rb ¿0 can be estimated by the following fuzzy inference mechanism: 21
Rulei : If s is Fi
Then kˆrb is ˆi ;
(10)
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Fig. 2. A generalized Gaussian-type membership function.
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(11)
where s is the fuzzy input, and ws , wc and wd are the parameters of the generalized Gaussian-type function. The equivalent membership function is demonstrated as in Fig. 2, in which the parameter wc represents the center value and the parameter wd denotes the reciprocal value of deviation from the center to which the value on the standardized support set has 0.5. The input s is scaled by the parameter ws . The center-of-gravity method is used for the defuzziLcation r i ˆi ˆ k rb = i=1 ; (12) r i=1 i where i is the Lring weight of the ith rule. From (12), kˆ rb can be rewritten as
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Fi (s) = eln(0:5)(ws s−wc ) wd ;
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where ˆi i =1; 2; : : : ; r, are the adjustable singleton control actions and Fi is the label of the fuzzy set characterized by a generalized Gaussian-type fuzzy membership function [20]:
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where ˆ =[ˆ1 ˆ2 · · · ˆr ]T is a parameter vector and w=[w1 w2 · · · wr ]T is a regressive vector with wi deLned as i wi = r
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(14)
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DeLne the parameter error vector as ˜ = ˆ − ∗ ;
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(15)
where the constant vector ∗ is the constant optimal parameter vector so that T
krb∗ = ∗ w;
where is a positive constant. DiFerentiating V with respect to time yields
(17)
1 ˜T ˆ˙
1 ˜T ˆ˙
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= sT C [A0 x + B0 (Gx − kˆrb sgn(s)) + d ] +
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= sT C [A0 x + B0 u + d ] +
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1 T V˙ = sT s˙ + ˜ ˜˙ 1 T = sT C x˙ + ˜ ˆ˙
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Choose the Lyapunov function 1 T 1 ˜T ˜ s s+ ; V = 2
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1 T = sT C [A0 + B0 G ]x − sT CB 0 kˆrb sgn(s) + sT Cd + ˜ ˆ˙ 1 T = sT CAc x − kˆrb sT CB 0 sgn(s) + sT CB 0 n + ˜ ˆ˙ 1 T 6 Ac sT s − kˆrb CB 0 sT sgn(s) + sCB 0 n + ˜ ˆ˙ 1 T = Ac sT s − sCB 0 (kˆrb − n) + ˜ ˆ˙ 1 T = Ac sT s − sCB 0 (kˆrb − krb∗ − n + krb∗ ) + ˜ ˆ˙
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= Ac sT s − sCB 0 (krb∗ − n) − sCB 0 (kˆrb − krb∗ ) +
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1 T T = Ac sT s − sCB 0 (krb∗ − n) − sCB 0 (˜ w) + ˜ ˆ˙ 1 T ˙ T ∗ ˆ − sCB 0 w : = Ac s s − sCB 0 (krb − n) − ˜
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ˆ˙ = sCB 0 w;
(19)
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Fig. 3. An articulated two-link manipulator.
V˙ = Ac sT s − sCB 0 (krb∗ − n)
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3. Controller design for robotic systems
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Since all the eigenvalues of Ac are chosen to be on the left-half s-plane, V will converge exponentially to zero, i.e. V (t) → 0 at t → ∞. This guarantees the closed-loop stability. In summary, the learning algorithm of the consequents of the fuzzy rules in (10) is given in (13) where the regressive vector w is determined from (14) and the adaptive law of the parameter vector is derived in (19).
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The robot considered is a two-link, articulated manipulator as shown in Fig. 3, whose position can be described by a joint angle vector q =[q1 q2 ]T , and whose actuator input is the torque vector =[1 2 ]T applied at the manipulator joints. The dynamics of such a manipulator are strongly nonlinear, and can be written in a general form H (q)qQ + C (q; q) ˙ q˙ + g(q) = ;
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(20)
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then inequality (18) becomes
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where H (q) is the 2 × 2 manipulator inertia matrix, C (q; q) ˙ q˙ is the vector of centripetal and Coriolis torques, and g(q) is the gravitational torque vector. The control problem for such a system is to
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design the control law such that the required actuator inputs to the robot are su:cient to perform the desired task, e.g. follow a desired trajectory. Assume that the manipulator of Fig. 3 is in the horizontal plane (g(q) ≡ 0), then the dynamics can be written explicitly as [16] H11 H12 qQ1 −hq˙2 −h(q˙1 + q˙2 ) 1 q˙1 + = ; (22) H21 H22 qQ2 0 q˙2 2 hq˙1 where
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H12 = H21 = a2 + a3 cos q2 + a4 sin q2 ;
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with a1 = I1 + m1 l2c1 + Ie + me l2ce + me l21 ;
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and the parameters of the robot are shown in Fig. 3 and their values are given as m1 =1, l1 =1, me =2, e =30◦ , I1 =0:12, lc1 =0:5, Ie =0:25, lce =0:6. Eq. (22) can be rewritten as −1 −1 qQ1 H11 H12 −hq˙2 −h(q˙1 + q˙2 ) H11 H12 1 q˙1 =− + ; (25) qQ2 H21 H22 0 q˙2 H21 H22 2 hq˙1
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a4 = me l1 lce sin
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(23)
is assumed to exist. Eq. (25) can be expressed as a state equation
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where
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=
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;
(27)
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B=−
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(28)
;
(29)
q˜ = q − qd
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(30)
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and qd is the desired joint angles vector. Because the state of (26) is q, ˙ it is necessary to make suitable transformation for (26) such that the controlled states q˜ can be induced. DiFerentiating (30) with respect to time and from (26), it is revealed that q˜Q = A(q˜˙ + q˙d ) + Bu − qQd ;
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q˜Q = Aq˜˙ + Bu + (Aq˙d − qQd ):
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(31)
(32)
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where Aij and Bij imply the element values in the index of the ith row and the jth column for the matrices A and B, respectively. The joint angle error is deLned as
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x˙ = A x + B uR + d x1 00 0 0 0 0 00 1 0 0 0 0 1 x2 0 0 0 0 0 0 + ; = + 0 0 A11 A12 x3 0 0 B11 B12 1 d1 17
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where
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0 0 B21 B22 0 1 5:19 × 10−6
0 0 −2:43 × 10−5 −8:41 × 10−6
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(34a)
(34b)
d2
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(35)
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uR = [0 0 1 2 ]
(38)
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and the nominal states are given as q1 =q2 = q˙1 = q˙2 =0. Based on the system dynamic of (33), the proposed HAFC design procedures are described as follows: Step 1: In order to design the ufb of (4), choose the eigenvalues of A + B G such that the state feedback gain G can be obtained by a conventional control technique. In this paper, a “place” command of MATLABTM is utilized as pole placement technique to calculate the state feedback gain matrix G of (4). Here, the eigenvalues are chosen to be at −20; −30; −200, and −200, so that the feedback gain matrix can be calculated as
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Step 2: In order to design the u rb of (9), Lrst choose a constant matrix C for (7) such that the sliding surface s =Cx =0 can be deLned. In this paper, C is chosen as
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In this paper, the Gaussian-type function is utilized as the membership functions of fuzzy system. Five Gaussian-type membership functions are constructed from (11). The parameters are chosen based on human knowledge and through some trials. The centers are set at wc = − 0:5; −0:25; 0; 0:25 and 0.5, respectively. And for each membership functions, ws and wd are set as ws =1 and wd =0:15. The consequent parts characterized by adjustable singletons ˆi are all initialized at zeros and is set as 0.1.
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(41)
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Fig. 4. (a) Angle error q˜1 for position control; (b) angle error q˜2 for position control; (c) control torque 1 for position control; and (d) control torque 2 for position control.
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The performance of position and tracking control from the simulations are presented and compared with the simulation results of Slotine and Li [16]. In [16], the PD control has been utilized for position control; and the adaptive control and sliding control have been utilized for tracking control. In the simulations, mass uncertainties for m1 and me , and model error of A in (26) are considered. (a) position control: The robot, initially at rest at (q1 =0◦ ; q2 =0◦ ), is commanded to move one step to (qd1 =60◦ , qd2 =90◦ ). The performance of the proposed HAFC is compared with the PD
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control [16]. For the case without mass uncertainties, the simulation results are shown in Fig. 4. It can be seen that the HAFC can achieve better control performance. For the case of 50% mass uncertainties for m1 and me , and 25% model error of A in (26), the simulation results are shown in Fig. 5. The results show that the proposed HAFC controller still can cope with the mass uncertainties and model error to achieve better performance. (b) robust tracking control: The robot, initially at rest at (q1 =0◦ ; q2 =0◦ ), is commanded to follow a desired trajectory qd1 (t)=30◦ (1 − cos(2't)) and qd2 (t)=45◦ (1 − cos(2't)). The performance of the proposed HAFC is compared with the sliding control and adaptive control approaches [16]. For the case without mass uncertainties, the simulation results are shown in Fig. 6, which shows that the HAFC approach has smaller tracking errors, and the transient tracking performance is also better than those using the sliding control or adaptive control. Similar to position control, a 50% mass uncertainties for m1 and me and 25% model error is also simulated to illustrate the robust control
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Fig. 5. (a) Angle error q˜1 for position control with 50% mass uncertainties and 25% model error; (b) joint angle error q˜2 for position control with 50% mass uncertainties and 25% model error; (c) control torque 1 for position control with 50% mass uncertainties and 25% model error; and (d) control torque 2 for position control with 50% mass uncertainties and 25% model error.
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performance, as shown in Fig. 7. This simulation also shows that the proposed HAFC can achieve the best control performance with favorable tracking performance.
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A hybrid adaptive fuzzy controller (HAFC) has been demonstrated to be an e:cient control technology for a robotic system. The adaptive fuzzy controller includes a state feedback control and an adaptive fuzzy robust controller to treat the multi-input multi-output robotic system. The advantage of this approach is that the adaptive fuzzy control law can tune the robust gain of the SMC so as to cope with the uncertainties and modeling error of the nonlinear robotic system. Thus the controller can improve the system performance for the nonlinear robotic system subject to system uncertainties. In the simulation example, the controller developed here is applied to a two-link robot. It is shown that, by using the proposed HAFC, the system performance is considerably improved and the system also exhibits stability and robustness.
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Fig. 7. (a) Angle tracking error q˜1 for tracking control with 50% mass uncertainties and 25% model error; (b) angle tracking error q˜2 for tracking control with 50% mass uncertainty and 25% model error; (c) control torque 1 for tracking control with 50% mass uncertainties and 25% model error; and (d) control torque 2 for tracking control with 50% mass uncertainties and 25% model error.
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This work was supported by the National Science Council of the Republic of China under grant NSC-89-2218-E-155-011. References 3 5
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