Adaptive fuzzy CMAC control for a class of nonlinear
systems with smooth compensation
T.-F. Wu, P.-S. Tsai, F.-R. Chang and L.-S. Wang
Abstract: Adaptive fuzzy cerebellar model articulation controller (CMAC) schemes are proposed to solve the tracking problem for a class of nonlinear systems. The proposed method provides a simple control architecture that merges CMAC and fuzzy logic, so that the complicated structure and the input space dimension in CMAC can be simplified. Adaptive laws are developed to tune all of the control gains online, thereby accommodating the uncertainty of nonlinear systems without any learning phase. In particular, smooth compensation is adopted to overcome the chattering problem associated with conventional switching compensation. By Lyapunov stability analysis, it is guaranteed that all of the closed-loop signals are bounded and the tracking errors converge exponentially to a residual set whose size can be adjusted by changing the design parameters. Simulation results for its applications to three examples are presented to demonstrate the perform-ance of the proposed methodology.
1 Introduction
Owing to its intrinsic difficulty, interest on the control of nonlinear systems has persisted for many years. Various control methodologies have been developed from the per-spective of system theory and traditional feedback control theory[1, 2]. Notably, these methods mostly depend on a thorough understanding of the controlled system’s dynamics, which makes their application unfavourable for uncertain systems. To deal with uncertainties on dynamical models or disturbances, some techniques in intelligent control have found an application. Examples include neural networks (NNs) using appropriated learning phases
[3 – 5] and fuzzy control by capturing human experiences
[6 – 9].
One subclass of NNs, introduced by Albus[10], called the cerebellar model articulation controller (CMAC), has attracted much attention because of faster learning, better generalisation and simpler computation. In particular, a trained CMAC can approximate nonlinear functions in a generalised lookup-table manner over a domain to any desired accuracy. Numerous researchers have applied it to design the controller of unknown nonlinear systems such as robot manipulators [11] and spacecraft[12]. Recently, various modifications of CMAC have been proposed to enhance the performance. The CMAC with a robust com-pensation achieves H1 tracking performance [13]. The merging of CMAC and the Hamilton – Jacobin – Bellman (HJB) optimisation theory yields an optimal control design[14]. Combining a fuzzy reasoning mechanism, the
resulting fuzzy CMAC (FCMAC) brings about a simple control architecture[3, 12, 15, 16].
Traditionally, the weights in CMAC were trained by an off-line learning phase, so the setting of CMAC may take a long time. The effectiveness of CMAC is limited in treating the problem that requires online tuning. Several studies have suggested the use of the adaptive law to update the CMAC weights online. The tracking perform-ance of CMAC coupled with adaptive laws has been shown by Peng and Woo [3] and Kim and Lewis[14]for the robot manipulators, by Wai et al.[17]for linear piezo-electric ceramic motors and by Lin and Peng [18] for a Chua’s chaotic circuit. In these applications, a compensation is required in the adaptive CMAC to attenuate the error of CMAC approximation. This compensation is usually designed to involve a switching function, which gives rise to chattering on the control signals, and an undesirable phenomenon may be excited in turn[1, 2].
To solve the chattering problem without sacrificing the performance, we propose a modified adaptive FCMAC (AFCMAC) scheme on the basis of previous work [16], for dealing with the tracking problem of a class of nonlinear systems. The AFCMAC approximation is adopted as rough tuning, and the smooth compensation is developed as fine tuning, so that (i) the design methodology is easy to realise, (ii) all of the control gains, including the CMAC weights, can be updated online without a prior learning phase, (iii) chattering can be prevented, and (iv) the tracking performance is guaranteed.
2 Structure of fuzzy CMAC
This section introduces the basic structure of CMAC and its modification called the FCMAC that will be used later in this paper. In general, to achieve the desired accuracy with a FCMAC, a complicated structure and a sufficient number of rules may be constructed, so that the dimension of the underlying system becomes higher [19]. However, for the real-time requirement in physical applications, the computational load increases along with the complexity of CMAC and may cause instability owing to the effect of #The Institution of Engineering and Technology 2006
IEE Proceedings online no. 20050362 doi:10.1049/ip-cta:20050362
Paper first received 30th April 2005 and in revised form 7th January 2006 T.-F. Wu, P.-S. Tsai and F.-R. Chang are with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, Republic of China L.-S. Wang is with the Institute of Applied Mechanics, National Taiwan University, Taipei, Taiwan, Republic of China
time-delay. Some advanced control theories should be integrated with a FCMAC to enhance the system performance if the number of rules is reduced. Consequently, simple controller structures are used in many proposed schemes.
2.1 Basic CMAC design
The physical system to be controlled is assumed to have only one control input and all of the state variables are assumed available. Therefore a single-output CMAC [15] is designed and the output is given by
zCMAC¼FðsÞ ð1Þ
where F: RL!R is a nonlinear function of CMAC input variable s ¼ [s1, . . . , sL]T[ S , RL. To mimic the oper-ation of the cerebellum, the inputs (sensors) are related to the output (response) through an association mechanism with the association memory space A. Any element in A consists of M number of 0s and 1s according to the pattern of the inputs. Mathematically, the relation (1) can be represented by a pair of mappings
G: S ! A; s 7! GðsÞ ¼ aðsÞ [ A ð2Þ
P: A ! R; a 7! PðaÞ ð3Þ
In particular, we may choose the function P which generates the output zCMACas follows
zCMAC¼PðaÞ ¼ aTw ð4Þ
where the vector w denotes the CMAC weight vector. As an example, we consider the case with two input variables s1and s2in the range of [22, 2].Fig. 1shows a possible partition of the input variables of the CMAC, in which both s1 and s2 are divided into four sub-regions such that 16 blocks (m, n), with m, n ¼ 22, 21, 1, 2, are formed. These sub-regions are further grouped into two regions, (A, B) and (a, b), for s1 and s2, respectively, in the first layer. Their combinations Aa, Ab, Ba and Bb are the hypercubes. By shifting the first layer step-by-step on the sub-regions, we obtain the second and third layers such that an association vector can be expressed as aT; [Aa Ab Ba Bb Cc Cd Dc Dd Ee Ef Fe Ff ].
The map G defined in (2) is then given by associating
a set of input variables with the association vector according to the corresponding block. For instance, if [s1, s2]T¼ [0.5, 0.2]T[ S, the corresponding block is (1, 1) and the
associ-ated hypercubes are Aa ¼ 1, Dd ¼ 1 and Ff ¼ 1, which yields aT¼ [1 0 0 0 0 0 0 1 0 0 0 1]. The output z
CMAC can then be obtained by using (4) as shown by the solid-line in Fig. 2. To illustrate the CMAC mapping, the process of determining the output for another input (21.5, 1.2) is shown by the dotted-line in Fig. 2. In this paper, we will only consider the cases of two input variables and the above-described CMAC mapping will be adopted.
2.2 FCMAC design
Owing to possible disturbances on the sensors, the input data, namely s, may not be exact. To accommodate this fuzziness and simplify the input partition, the structure of FCMAC was proposed by Chen et al. [15]. For a two-input problem, a fuzzy system with N fuzzy rules may be designed, each of which in the form of
RðiÞ: IF s1 is F1i and s2 is F2i; THEN z ðiÞ f is a
T iw ð5Þ where i ¼ 1, 2, . . . , N, and the THEN part is extracted from the CMAC. Given the membership function of fuzzy set Fki, k ¼ 1, 2, denoted by mFki, the following defuzzification process is chosen to compute the output zFCMAC
zFCMAC ¼ v1aT1w þ v2aT2w þ þ vNaTNw v1þv2þ þvN ¼ PN i¼1aTiwvi PN i¼1vi ð6Þ where vi¼ Q k¼1 2
mFki(sk). The preceding equation may be re-written compactly as zFCMAC¼hTAw ð7Þ where h ¼ ½h1 h2 hNT; hi¼ vi PN i¼1vi and A ¼ aT 1 aT2 .. . aT N 2 6 6 6 6 4 3 7 7 7 7 5 ð8Þ
Fig. 1 Schematic diagram of fuzzy sets integrated with CMAC
Fig. 2 Mapping of basic CMAC
IEE Proc.-Control Theory Appl., Vol. 153, No. 6, November 2006 648
In (7), the matrix A (determined by CMAC) and the vector h (determined by fuzzy rules) are typically fixed, but the weight vector w is adjustable herein.
For the two-input problem, a set of membership functions may be chosen as shown inFig. 1, in which P (positive) and N (negative) fuzzy sets are imposed on each variable. Accordingly, there are four fuzzy rules with four association vectors, a1, a2, a3, a4, attached to (P, P), (N, P), (N, N ), (P, N ), respectively. It is seen that, for this example, there are 16 association vectors in CMAC, while only four are used in FCMAC. To determine ai in FCMAC, the logical operation ‘OR’ is performed on all possible (in the same region) association vectors in CMAC. For instance, if (s1, s2) is in the class (P, P), there are nine blocks in the region and, by performing ‘OR’ on the
corres-ponding nine association vectors, we obtain
a1
T¼ [1 1 1 1 0 0 0 1 1 1 1 1].
Table 1 shows the
relationship between the fuzzy rules and the CMAC, from which, the matrix A in (7) is given by
A ¼ 1 1 1 1 0 0 0 1 1 1 1 1 1 1 0 0 0 1 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 0 1 1 0 0 1 1 1 1 1 1 2 6 6 6 4 3 7 7 7 5 ð9Þ
Intrinsically, the chosen FCMAC is more like a fuzzy system, with the output function (7) depending on the CMAC structure. Although a fuzzy system has been proven to be a universal approximator [6], it is not easily implemented with a large number of inputs in real-time applications. In this work, we shall adopt the simplest structure of FCMAC comprising two input vari-ables and four fuzzy rules, so that the adaptive control can be integrated easily to achieve basic performance. Moreover, to suppress the approximated error of the FCMAC, a switching compensation is imposed, and a smooth compensation is developed to resolve the arising chattering problem.
3 Adaptive FCMAC controller design
The tracking problem of a class of nonlinear systems is first described. The AFCMAC with different compensation designs is the proposed on the basis of the FCMAC introduced in Section 2.
3.1 Problem description and AFCMAC design
Consider a physical system that can be modelled by the fol-lowing nth-order nonlinear equations[1, 18]
xðnÞ¼f ðx; _x; . . . ; xðn1ÞÞ þgðx; _x; . . . ; xðn1ÞÞu ð10Þ
y ¼ x ð11Þ
where f, g: Rn!R are continuous nonlinear bounded func-tions; u [ R is the system input and y [ R is the system output. Define the state vector x [ Rnas
x ¼ ½x1; x2;. . . ; xnT; ½x; _x; . . . ; xðn1ÞT ð12Þ For x in a certain controllability region Vc[ R
n , it is necessary that g(x) = 0. As g(x) is continuous, we may assume that g(x) . 0 for all x [Vcwithout loss of gener-ality. Furthermore, to implement the adaptive law proposed in this paper, the function g is assumed to be known. The goal here is to design a controller u such that the system output y follows a desired smooth trajectory yd. Letting ~y ; y 2 yd, the aggregate tracking error vector ~y is defined as
~y ¼ ½~y1; ~y2;. . . ; ~ynT; ½~y; _~y; . . . ; ~yðn1ÞT ð13Þ Now, if the system dynamics f is known, we may apply the ideal control law u given by
u¼g1ðxÞ½f ðxÞ þ yðnÞd cT~y ð14Þ where c ; [cn, cn21, . . . , c1]Twith ci, i ¼ 1, 2, . . . , n being positive constants such that the polynomial D(l) ¼lnþ c1ln21þ þcn is Hurwitz. With (14), the error dynamics of the closed-loop system becomes
~yðnÞþc1~yðn1Þþ þcn~y ¼ 0 ð15Þ and it follows that the tracking error ~y exponentially approaches zero. However, the ideal control law (14) cannot be directly applied if the function f is unknown. One may need to find a scheme to approximate f, so that an approximated control law can be used. The AFCMAC provides such a scheme.
In applying the two-input FCMAC with simple structure developed in Section 2.2, the input variables are chosen as s1¼dT~y; s2¼_s1¼dT_~y ð16Þ
Table 1: Relationship between fuzzy rules and CMAC
Continuous input (s1, s2) [Aa, Ab, Ba, Bb, Cc, Cd, Dc, Dd, Ee, Ef, Fe, Ff ]
Class Blocks Hypercubes aiT, i ¼ 1, 2, 3, 4
(P, P) (21, 21) [1 0 0 0 0 0 0 1 1 0 0 0] [1 1 1 1 0 0 0 1 1 1 1 1] (21, 1) [1 0 0 0 0 0 0 1 0 1 0 0] (21, 2) [0 1 0 0 0 0 0 1 0 1 0 0] (1, 21) [1 0 0 0 0 0 0 1 0 0 1 0] (1, 1) [1 0 0 0 0 0 0 1 0 0 0 1] (1, 2) [0 1 0 0 0 0 0 1 0 0 0 1] (2, 21) [0 0 1 0 0 0 0 1 0 0 1 0] (2, 1) [0 0 1 0 0 0 0 1 0 0 0 1] (2, 2) [0 0 0 1 0 0 0 1 0 0 0 1] (N, P) ... ... [1 1 0 0 0 1 0 1 1 1 1 1] (N, N ) ... ... [1 0 0 0 1 1 1 1 1 1 1 1] (P, N ) ... ... [1 0 1 1 0 0 1 1 1 1 1 1]
where d ¼ [d1, d2, . . . , dn] T
is the coefficient vector. The errors of different orders are thus synthesised in one vari-able, which is similar to the concept of the sliding surface
[1, 15]. With the optimal weight vector w that is
assumed to exist, it is desired that the output of FCMAC (7), rewritten as u
FCMAC¼ hTAw, would be close to the ideal control feedback u in (14). Let 1 denote their differ-ence
1 ¼ uuFCMAC ð17Þ
which is assumed to be bounded with a small positive bound D [17]
j1j D ð18Þ
By including u in the system (10) and (11), we obtain yðnÞ¼xðnÞ¼f ðxÞ þ gðxÞ½u uþu
¼f ðxÞ þ gðxÞ u uh þg1ðxÞðf ðxÞ þ yðnÞd cT~yÞi
¼yðnÞd cT~y þ gðxÞ½u u ð19Þ
which implies that
_~y ¼ F~y þ g½u u ð20Þ
where F ¼ 0ðn1Þ1 In1 cT 2 4 3 5; g ¼ 0ðn1Þ1 gðxÞ 2 4 3 5 ð21Þ
Replacing u in (20) by (17), the error equations become _~y ¼ F~y þ g½u u
FCMAC1 ð22Þ
As D(l) is Hurwitz, so then is the n n matrix F. Hence for any n n symmetric positive definite matrix Q, there exists an n n symmetric positive definite matrix P, so that the following Lyapunov matrix equation holds
PF þ FTP ¼ Q ð23Þ
The solution P of (23) can then be used to construct a Lyapunov function V ; (1/2)~yTP~y, with its rate being com-puted as _ V ¼1 2~y TðPF þ FTPÞ~y þ ~yTPgðu u FCMAC1Þ ¼ 1 2~y TQ~y þ eðu u FCMAC1Þ ð24Þ
where e ¼ ~yTPg. Now if wand D are available, we may use the control
u ¼ u
FCMACþur ð25Þ
to steer the system, where uris designed as
ur ¼ D sgnðeÞ ð26Þ
with sgn(.) being the signum function[1, 2]
sgnðeÞ ; 1; e . 0 0; e ¼ 0 1; e , 0 8 < : ð27Þ
It can be then shown that the tracking error (13) will converge exponentially to zero. In this design, the switching mechanism of the compensation ur is devel-oped to accommodate the approximation error using
FCMAC. Nevertheless, the optimal weight and the bound are mostly unknown in applications. When f is uncertain, the optimal weight vector w is conventionally trained by some off-line learning process, which may take a long period of time. Alternatively, the adaptive laws described subsequently can be used to estimate w and D online.
3.2 AFCMAC with switching compensation
As stated earlier, the unknown optimal weights raise a serious problem in the implementation of the FCMAC-based scheme. To solve this problem, the adaptive laws to estimate the optimal weights and the bound are incorporated to yield the AFCMAC. We shall use the following laws to find the estimates ^w and ^D
_^w ¼ g1eATh ð28Þ
_^D ¼g2jej; Dð0Þ . 0^ ð29Þ
where the constantsg1 andg2 are the design parameters. As g(x) is known, the variable e is available and hence the previous laws are well-defined. With these estimates, the AFCMAC control scheme is designed as
u ¼ uAFCMACþuar ð30Þ
where
uAFCMAC¼hTA ^w ð31Þ
uar¼ ^D sgnðeÞ ð32Þ
The ‘Thm. 1’ part of Fig. 3exhibits the nonlinear system subject to the AFCMAC control scheme (28) – (32), whose performance is summarised in the following theorem. Theorem 1: If the design parameters g1 and g2 are both positive, the application of the AFCMAC control scheme (28) – (32) to the nonlinear system (10) and (11) yields a closed-loop system in which
1. all signals are bounded;
2. the tracking error ~y converges asymptotically to zero (i.e. ~y(t)!0 as t ! 1).
Fig. 3 Architecture of the AFCMAC
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Proof: Consider the Lyapunov function candidate[1] V ;1 2~y TP ~y þ 1 2g1w~ Tw þ~ 1 2g2D~ 2 ð33Þ where ~w ; ^w 2 wand ~D ; ^D 2 D. The time derivative of V along the trajectory of the closed-loop system derived from (22) is found as _ V ¼1 2~y TðPF þ FTPÞ~y þ eðu u FCMAC1Þ þ 1 g1 _~wTw þ~ 1 g2 ~ D_~D ¼ 1 2~y TQ~y þ eðu AFCMACuFCMACþuar1Þ þ 1 g1 _~w T ~ w þ 1 g2D~ _~D ð34Þ
where (23) and the control law (30) have been applied. Next, from (28) and (29), and from (31) and (32), it follows that
_ V ¼ 1
2~y
TQ~y þ ehTA ~w ^Djej 1e ehTA ~w þ ð ^D DÞjej
¼ 1 2~y
TQ~y 1e Djej
ð35Þ By invoking the inequality
+1e j1ej Djej ð36Þ
we then obtain
_ V 1
2~y
TQ~y ð37Þ
which shows that V is non-increasing. Therefore V is bounded, that is, V [L1, which implies that all signals in the closed-loop system are bounded (~y, ~w, ~D [L1). Next, as the right-hand side of (22) is bounded, we have _~y [ L1. Integrating and rearranging (37) yields
lim t!1 ðt 0 ~yTðtÞ~yðtÞdt 2 lminðQÞ ½V ð0Þ lim t!1V ðtÞ , 1 ð38Þ where lmin(Q) (.0) denotes the minimum eigenvalue of the matrix Q. Therefore ~y [L2. Now the Barba˘lat lemma [1, 2, 20] can be invoked to conclude that the tracking error ~y converges to zero asymptotically. A Although the AFCMAC control scheme described earlier can be used to track the reference trajectory, the switching compensation (32) may cause chattering in the control input, which, in turn, may lead to undesirable effects [1, 2]. To deal with this problem, a modified AFCMAC scheme is proposed, in which the switching compensation is replaced by a smooth compensation, so that chattering can be eliminated, while keeping the performance satisfactory.
3.3 AFCMAC with smooth compensation
(modified AFCMAC)
Basically, the chattering of control input comes from the discontinuity of the switching function. To prevent chatter-ing, one may replace the switching by other continuous
maps. Owing to its similarity to the switching function such that the performance can be preserved, the saturation function would be a good choice. As a result, the compen-sation uarin (32) of the previous AFCMAC control scheme is modified to
uar¼ ^D satbðeÞ ð39Þ
where satb(.) denotes the saturation function[1, 2] satbðeÞ ; sgnðeÞ if jej . b
e=b otherwise
ð40Þ with the constant b . 0 specifying the boundary layer. It is seen that as b ! 0, the saturation function approaches the switching function. Moreover, to assure the convergence rates of the estimation, the adaptive laws (28) and (29) are replaced by
_^w ¼ s1w ^ g1eATh ð41Þ
_^D ¼ s2D þ^ g2jej; Dð0Þ . 0^ ð42Þ where the constantsg1,g2,s1ands2 are the design par-ameters. The first terms on the right-hand side of (41) and (42) represent thes-modification, increasing the robustness of the adaptive law (28) and (29)[20, 21]. The closed-loop system of the nonlinear system subject to this modified AFCMAC control scheme (30) and (31) and (39) – (42) is depicted by the ‘Thm. 2’ part in Fig. 3. The following theorem summarises the main result of this study.
Theorem 2: Consider the nonlinear system (10) and (11) controlled by the modified AFCMAC scheme (30) and (31) and (39) – (42). If the design parameters (g1,g2, s1,
s2, b) are positive, and the parameters (s1,s2) are chosen such that max(s1, (3/4)s2) ,lmin(Q)/lmax(P), where
lmax(P) (.0) denotes the maximal eigenvalue of the matrix P, then
1. all signals in the closed-loop system are bounded; 2. the tracking error ~y converges exponentially to a residual set that can be made small by adjusting the parameterss1,
s2and b.
Proof: The same Lyapunov function candidate (33) as in Theorem 1 shall be used to perform the analysis. Referring to (34), the time derivative of V along the trajec-tory of the closed-loop system by using the modified laws (39) – (42) becomes _ V ¼ 1 2~y TQ~y ^ De satbðeÞ 1e þ ð ^D DÞjej s1 g1w~ Tw ^ s2 g2D ^~D ð43Þ
Depending on the values of jej, the following two cases are considered separately
Case 1: jej . b. As satb(e) ¼ sgn(e), (43) yields _ V ¼ 1 2~y TQ~y 1e Djej s1 g1w~ Tw ^ s2 g2D ^~D ð44Þ
The application of the inequality (36) leads to _ V 1 2~y TQ~y s1 g1w~ Tðw þ w~ Þ s2 g2Dð ~~ D þ DÞ ð45Þ
As for a, b [ Rn, we have +a . b (1/2)kak2þ(1/2)kbk2, it follows that _ V 1 2~y TQ~y s1 2g1w~ Tw ~ s2 2g2D~ 2 þ s1 2g1kw k2 þ s2 2g2D 2 ð46Þ
Now for any b1. 0, the previous inequality may be rewritten as _ V b1V þb1 1 2~y TP ~y þw~ Tw~ 2g1 þ ~ D2 2g2 ! 1 2~y TQ~y s1 2g1w~ Tw ~ s2 2g2D~ 2 þ s1 2g1kw k2þ s2 2g2D 2 b1V þ ðb1lmaxðPÞ lminðQÞÞ k~yk2 2 þ ðb1s1Þ ~ wTw~ 2g1 þ ðb1s2Þ ~ D2 2g2þa1 ð47Þ
in which the inequalities ~yTP~y lmax(P)k~yk 2
and 2 ~yTQ~y 2lmin(Q)k~yk2have been used, and the constant
a1is defined by a1; s1 2g1kw k2þ s2 2g2D 2. 0 ð48Þ Now, if b1¼min lminðQÞ lmaxðPÞ ;s1;s2 ð49Þ is selected, we have _ V b1V þa1 ð50Þ
which implies that all the signals are bounded according to Ioannou and Kokotovic[21]. Moreover, if the parameters
s1ands2are chosen such that maxðs1;s2Þ, lminðQÞ lmaxðPÞ ð51Þ andb1is selected as b1 ¼minðs1;s2Þ ð52Þ
the inequailty (47) implies that _
V b1V þ ðb1lmaxðPÞ lminðQÞÞ k~yk2
2 þa1 ð53Þ
Now, define the residual set as
G1; ~y : k~yk2, 2a1
lminðQÞ b1lmaxðPÞ
ð54Þ It is seen that outside the residual set, we have
_
V b1V ð55Þ
such that the tracking error ~y converges exponentially[21].
Case 2: jej b. For this case, satb(e) ¼ e/b, and (43) yields _ V ¼ 1 2~y TQ~y D^ bjej 2þ ^Djej 1e Djej s1 g1w~ Tw ^ s2 g2D ^~D 1 2~y TQ~y D^ b jej b 2 2 b 2 4 " # s1 g1w~ Tw^ s2 g2D ^~D ð56Þ
by applying (36). As (jej 2 b/2)20, using similar tech-niques as in Case 1, we find
_ V 1 2~y TQ~y þb 4ð ~D þ DÞ s1 2g1w~ Tw ~ s2 2g2D~ 2 þ s1 2g1 kwk2þ s2 2g2 D2 ð57Þ
Owing to the following inequality b 4ð ~D þ DÞ ¼ 1 4 ffiffiffiffiffi g2 s2 r b ffiffiffiffiffi s2 g2 r ~ D þ ffiffiffiffiffi g2 s2 r b ffiffiffiffiffi s2 g2 r D s2 8g2D~ 2 þ s2 8g2D 2þ g2 4s2 b2 ð58Þ
Equation (57) can be further expressed as _ V 1 2~y TQ~y s 1 ~ wTw~ 2g1 3 4s2 ~ D2 2g2þ s1 2g1kw k2 þ5s2 8g2 D2þ g2 4s2 b2 ð59Þ
It is observed that the inequality (59) is similar to (46), and we may use an analogous method to perform the analysis. In particular, by defining a2; s1 2g1kw k2þ5s2 8g2D 2þ g2 4s2 b2. 0 ð60Þ
choosings1ands2such that maxðs1;34s2Þ, lminðQÞ lmaxðPÞ ð61Þ and selectingb2to be b2¼min s1; 3 4s2 ð62Þ we can prove that all the signals are bounded, and ~y con-verges exponentially to the residual set given by
G2 ; ~y : k~yk2,
2a2
lminðQÞ b2lmaxðPÞ
ð63Þ In order to have the modified AFCMAC perform satisfac-tory, it is further required that for both cases, the residual sets can be made arbitrarily small. This can be achieved by adjusting the parameters (g1, g2,s1, s2, b) such that (a1,a2) are small enough, as kwkand D are fixed. In par-ticular, the residual set G1can be made small and contained in the region f ~y: k~yTPgk bg, which corresponds to the region of Case 2. For that setting, the exponential conver-gence is assured for Case 1. When the errors enter the regime of Case 2, they are driven exponentially to the small G2, which guarantees the performance of the proposed
modified AFCMAC scheme. A
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4 Applications of the modified AFCMAC
To illustrate the performance of the proposed modified AFCMAC, three applications shall be discussed. The first one is on the tracking of an inverted pendulum with friction. The second is on the one-link robotic manipulator[5, 18].
For the third example, the tracking problem for a third-order highly nonlinear system is attacked.
4.1 Example 1: inverted pendulum with friction The inverted pendulum consists of a thin homogeneous rod of mass m and length l, with a load of point mass mL attached to the end, as depicted inFig. 4. Assume that fric-tion torque exists in the joint that can be modelled[22]as
bvð_qÞ ¼ tcsgnð _qÞ þ cv_q; _q = 0 te; _q ¼ 0; jtej,ts tssgnðteÞ; _q ¼ 0; jtej ts 8 < : ð64Þ
where te denotes the external torque, tc is the Coulomb friction, ts is the breakaway torque and cv is the coeffi-cient of viscous friction. Let q denote the joint angle. Euler’s law can be applied to find the equations of
0 10 20 30 40 50 60 -1 -0.5 0 0.5 1 responsedesired a time, sec qd1 q1 0 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1 error zero b time, sec 0 10 20 30 40 50 60 -50 0 50 c time, sec 0 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1 error zero d time, sec 0 10 20 30 40 50 60 -50 0 50 e time, sec 0 50 -5 0 5 0 50 -5 0 5 0 50 -5 0 5 0 50 -5 0 5 0 50 -5 0 5 0 50 -5 0 5 0 50 -5 0 5 0 50 -5 0 5 0 50 -5 0 5 0 50 -5 0 5 0 50 -5 0 5 0 50 -5 0 5 time, sec 0 10 20 30 40 50 60 0 5 10 15 20 time, sec 0 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1 error zero h time, sec tr ac king response , r a d control input, Nm tr ac ki ng error , r a d CMA C w e ightings wh1 wh7 wh10 wh11 wh8 wh5 wh2 wh3 wh6 wh9 wh12 wh4 control input, Nm appro
ximation error bound (Dh)
tr ac king error , r a d tr ac king error , r a d g f
Fig. 5 Numerical results of Example 1
a Tracking response e Control input
b Tracking error f Updated CMAC weights
c Control input g Updated approximation error bound d Tracking error h Tracking error
q = 0 (rad)
motion as €q ¼ 3 ðm þ 3mLÞl2 bvð_qÞ þ 3ðm þ 2mLÞg0 2ðm þ 3mLÞl sin ðqÞ þ 3 ðm þ 3mLÞl2 t ð65Þ y ¼ q ð66Þ
where t is the torque applied to the joint and g0¼ 9.8026 is the gravitational constant. Here the exter-nal torque in the friction model (64) is given by
te¼tþ(m/2 þ mL)lg0 sin(q). As a specific example, it is assumed that m ¼ l ¼ 1, mL¼ 2/3 and tc¼ 0.023, vc¼ 0.001, ts¼ 0.035.
The objective is to drive the inverted pendulum and follow the desired trajectory yd, defined by the reference model
€qd ¼ 5_qd5qdþ5r ð67Þ
yd ¼qd ð68Þ
with qd(0) ¼ 0, q˙d(0) ¼ 0, where the reference input r is a unit periodic rectangular signal with period T ¼ 32 s. Now suppose that in designing the controller, the friction model is unknown, and thus the proposed AFCMAC control schemes are applicable. First, the AFCMAC control scheme discussed in Section 3.2 is applied. By choosing the following design parameters
^ Dð0Þ ¼ 10; g1¼30; g2¼0:05; d ¼ ½1 0:001T; c ¼ ½1 2T and Q ¼ 10 0 0 10 ð69Þ the corresponding F, P andlmin(Q)/lmax(P) can be found to be F ¼ 0 1 1 2 ; P ¼ 15 5 5 5 and lminðQÞ lmaxðPÞ ¼0:5858 ð70Þ
The simulation was then performed with the results shown inFigs. 5aandbfor the tracking response and the tracking error, respectively. It is seen that the tracking error indeed converges asymptotically to zero. However, as shown in
Fig. 5c, the control input is bounded, but suffers from
severe chattering.
To resolve the chattering problem, the modified AFCMAC control scheme is then applied. In addition to selecting the same design parameters as before, we choose
s1 ¼0:5; s2¼0:01 and b ¼ 1 ð71Þ
that meet with the requirement given in Theorem 2. The simulation results show that the response remains rapid and the tracking error (Fig. 5d) converges exponentially to a residual set whose size can be roughly estimated to be +0.0096 rad. It is further seen that the control input (Fig. 5e) is now bounded without chattering because of a smooth compensation. In Figs. 5f and g, the updated weights and the upper bound of approximation error are given, respectively, which indicates that all of the updated control gains are bounded.
To further appreciate the effects of the parameters on the performance, a different setting of parameters is chosen in the modified AFCMAC as
s1¼0:05; s2 ¼0:001 and b ¼ 0:1 ð72Þ From the simulation results, it is seen that the control performance is the same as that obtained previously, except that the residual set of the tracking error is now about +0.0037 rad (Fig. 5h), which is smaller than that in the previous case. This comparison justifies that the residual set can be made smaller by using smaller s1,
s2and b.
Note that a large ^D(0) is chosen here, as the bound on the approximation error is not clear. Through the
s-modification in Theorem 2, the actual bound can be attained presumably, so that the chattering can be attenu-ated. One may argue that with small ^D(0), the chattering phenomenon may not appear by using the adaptive law with Theorem 1. However, as shown in Figs. 6a – d, the chattering for ^D(0) ¼ 0.01, g1¼ 300, g2¼ 2, still exists. On the contrary, if the modified AFCMAC is used (Theorem 2), the chattering can be suppressed by choosing
^
D(0) ¼ 10,s1¼ 0.5,s2¼ 0.001, b ¼ 1, as shown inFigs. 6e, f.
4.2 Example 2: one-link rigid robotic manipulator
To compare the proposed modified AFCMAC control scheme with the methods used by Zhihong et al.[5] and Lin and Peng[18], the same one-link rigid robotic manipu-lator is chosen as
ml2€q þ b_q þ mlgvcosðqÞ ¼ u ð73Þ where l is the link length, m is the mass and q is the angular position with initial conditions q(0) ¼ 20.1 and q˙(0) ¼ 0. Let the state variables be x1¼ q and x2¼ q˙, so that the model (73) can be expressed as
_x1 _x2 ¼ 0 1 0 0 x1 x2 þ 0 1 ðf þ gu þ dÞ ð74Þ
where f ¼ (2b/ml2)x22 (gv/l ) cos(x1) and g ¼ (1/ml2). The parameters are given by m ¼ l ¼ b ¼ gv¼ 1, and it is assumed that d is the external square wave disturbance with magnitude 0.1 and period 2p, which is the same as in Zhihong [5] and Lin and Peng [18]. The reference signal xdis generated by the following model
_xd1 _xd2 ¼ 0 1 16 8 xd1 xd2 þ 0 1 rðtÞ ð75Þ
with initial condition [xd1, xd2]T¼ [0, 0]T, where r(t) is a periodic rectangular signal with a period of 6 s.
The proposed modified AFCMAC control scheme is now used to perform the tracking. The design parameters are selected as ^ Dð0Þ ¼ 30; g1¼30; g2¼0:01; s1¼0:001; s2¼0:0001; b ¼ 0:1; d ¼ ½1 0:001T; c ¼ ½9 6Tand Q ¼ 90 9 9 2 ð76Þ IEE Proc.-Control Theory Appl., Vol. 153, No. 6, November 2006 654
such that F ¼ 0 1 9 6 ; P ¼ 30 5 5 1 and lminðQÞ lmaxðPÞ ¼0:0353 ð77Þ
The simulation was then performed with results shown in Figs. 7a– c. Although the system performances (Figs. 7a– c) are similar to that given by Zhihong [5] and Lin and Peng [18], the tracking error (Fig. 7b) is smaller than that by Zhihong [5], and the choice of the initial bound of approximation error D(0) ¼ 30^ is more reasonable than that ofdˆ(0) ¼ 0.01 in Lin and Peng[18].
4.3 Example 3: third-order nonlinear system To demonstrate that the proposed control schemes can handle a higher order nonlinear system, a third-order one is constructed as
€x_ ¼ 4€x23ej_xj2 sinðxÞ þ u ð78Þ
y ¼ x ð79Þ
where y is the system output, u is the control input. Now, consider the following reference model
€x_d ¼ 6€xd11_xd6xdþr ð80Þ
yd ¼xd ð81Þ
where r is a piecewise constant signal. Suppose that f ¼ 24€x22 3e2jx˙j22 sin(x) is uncertain. We now apply the proposed control scheme.
0 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1 error zero a time, sec 0 10 20 30 40 50 60 -50 0 50 b time, sec 0 10 20 30 40 50 60 -50 0 50 c time, sec 0 10 20 30 40 50 60 -50 0 50 d time, sec 0 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1 error zero e time, sec 0 10 20 30 40 50 60 -50 0 50 f time, sec tr ac king error , r a d control input (u = u + u ), Nm FCMA C r control input (u ), Nm FCMA C control input (u ), Nmr tr ac king error , r a d control input (u = u +u ), Nm FCMA C r
Fig. 6 Numerical results of Example 1
a Tracking error d Control input (ur) b Control input (u ¼ uFCMACþur) e Tracking error
The modified AFCMAC control scheme is first used with the following parameters
^ Dð0Þ ¼ 10; g1¼1:5; g2¼0:5; s1¼0:1; s2¼0:01; b ¼ 1; d ¼ ½3 2 1T; c ¼ ½1 2 3T Q ¼ 10 0 0 0 10 0 0 0 10 2 6 4 3 7 5; F ¼ 0 1 0 0 0 1 1 2 3 2 6 4 3 7 5; P ¼ 23 21 5 21 46 13 5 13 6 2 6 4 3 7 5 and llminðQÞ maxðPÞ ¼0:1617 ð82Þ
Figs. 8a and b show the tracking response and tracking
error, respectively. Obviously, the tracking error converges exponentially to a residual set whose size is about 0.0036. The control input is shown inFig. 8c, which is seen to be bounded without chattering. Simulation was conducted
with some adjusting of the parameters as
s1¼0:01; s2 ¼0:001 and b ¼ 0:5 ð83Þ which also satisfies the requirement in (62). The perform-ances are similar, but the residual set of the tracking error
0 50 100 150 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 response drsired a tr ac king response 0 50 100 150 -0.02 0 0.02 0.04 0.06 0.08 0.1 error zero d tr ac king error 0 50 100 150 -0.02 0 0.02 0.04 0.06 0.08 0.1 error zero b tr ac king error 0 50 100 150 0 1 2 3 4 5 6 c control input time, sec time, sec time, sec time, sec y yd
Fig. 8 Numerical results of Example 3
a Tracking response b Tracking error c Control input d Tracking error 0 2 4 6 8 10 12 -0.1 -0.05 0 0.05 0.1 error zero 0 2 4 6 8 10 12 -20 -15 -10 -5 0 5 10 15 20 0 2 4 6 8 10 12 -0.5 0 0.5 1 1.5 response drsired a b c tr ac king response tr ac king error control input, Nm time, sec time, sec time, sec xd1 x1
Fig. 7 Numerical results of Example 2
a Tracking response b Tracking error c Control input
IEE Proc.-Control Theory Appl., Vol. 153, No. 6, November 2006 656
becomes about 0.00036 (Fig. 8d) that is smaller than that in the previous case. This example shows that the proposed control scheme is applicable to deal with a class of compli-cated, higher-order nonlinear systems.
5 Conclusions
In this study, a modified AFCMAC scheme was developed to solve the tracking problem for a class of nonlinear systems. The proposed method was based on the CMAC technique that was integrated into the THEN part of a fuzzy reasoning mechanism. The resulting architecture, the FCMAC, was simpler than that of the basic CMAC. The FCMAC method was then combined with the adaptive law, so that the entire controller gains or weights could be adjusted online without preliminary off-line learning. To accommodate the approximation error of the control from the ideal control input, two different compensations were considered. Although the AFCMAC control scheme using the switching compensation drives the tracking error to con-verge asymptotically to zero, severe chattering resulted that could cause undesirable effects. Alternatively, the modified AFCMAC scheme with the saturation compensation prevented the chattering and steered the tracking error to converge exponentially to a residual set whose size could be adjusted. The application of the method to three examples demonstrated the effectiveness of the proposed modified AFCMAC scheme, as shown by the corresponding simulation results.
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