Output-feedback control of nonlinear systems using direct
adaptive fuzzy-neural controller
Wei-Yen Wang
a;∗, Yih-Guang Leu
b, Tsu-Tian Lee
caDepartment of Electronic Engineering, Fu-Jen Catholic University, 510, Chung Cheng Rd., Hsin-Chuang,
Taipei 24205, Taiwan
bDepartment of Electronic Engineering, Hwa-Hsia College of Technology, 235 Chung-Ho City, Taipei, Taiwan cDepartment of Electrical and Control Engineering, National Chiao Tung University, 1001 Ta-Hsieh Road,
Hsinchu, Taiwan
Received 22 August 2001; received in revised form 14 March 2002; accepted 4 November 2002
Abstract
In this paper, a direct adaptive fuzzy-neural output-feedback controller (DAFOC) for a class of uncertain nonlinear systems is developed under the constraint that only the system output is available for measurement. An output feedback control law and an update law are derived for on-line tuning the weighting factors of the DAFOC. By using strictly positive-real Lyapunov theory, the stability ofthe closed-loop system compensated by the DAFOC can be veri7ed. Moreover, the proposed overall control scheme guarantees that all signals involved are bounded and the output ofthe closed-loop system asymptotically tracks the desired output trajectory. To demonstrate the e9ectiveness ofthe proposed method, simulation results are illustrated in this paper.
c
2002 Elsevier B.V. All rights reserved.
Keywords: Fuzzy-neural control; Direct adaptive control; Output feedback control; Nonlinear systems
1. Introduction
Adaptive control theory has been an active area ofresearch for at least a quarter ofa century [23,24,28,11,9,10,12,31,29,22,13]. For linear systems, there have been some researches on stability analysis ofadaptive control systems, design ofadaptive observers and adaptive control ofplants, etc., all with satisfactory results [24,28]. There are also researches focusing on robust adaptive control that guarantee signal boundedness in the presence ofmodeling errors and bounded disturbances [11,9,10]. As for nonlinear systems, adaptive control schemes via feedback linearization have been reported
∗Corresponding author. Tel.: +886-2-2903-1111; fax: +886-2-2904-2638.
E-mail address: wayne@ee.fju.edu.tw(W.-Y. Wang).
0165-0114/03/$ - see front matter c 2002 Elsevier B.V. All rights reserved. PII: S0165-0114(02)00519-5
on continuous-time or discrete-time systems [12,31,29,22,13]. The fundamental ideal of feedback linearization is to transform a nonlinear system into a linear one, so that linear control techniques can be employed to acquire desired performance.
Recently, neural networks and fuzzy systems are applied to several control problems with satisfac-tory results. Because both the neural networks [8] and fuzzy systems [38] are universal approximators, many adaptive control schemes for nonlinear systems based on fuzzy systems [35,14,32], or neural networks [25,16,26,20,6,5,21] have been proposed to obtain better control performance. For a class of nonlinear discrete-time systems, adaptive control using neural networks has been proposed in [27] by feedback linearization. Also, a dynamic recurrent neural-network-based adaptive observer for a class ofnonlinear systems has been presented in [15]. In [39], an output feedback controller using multi-layer neural networks has been developed based on a high-gain observer used to estimate the time derivatives ofthe system output. Moreover, applications offuzzy systems incorporated into neural networks in function approximation, decision systems and nonlinear control systems have been pro-posed in [7,37,36,30,18,19,34,2]. In [37,36,30,18,19], the indirect adaptive fuzzy-neural controllers for nonlinear systems have been proposed and in [30,18] the output feedback control laws, which also are tuned by the indirect adaptive methods, provide robust stability for the closed-loop systems. Theoretical justi7cation on the use ofthe direct adaptive fuzzy controllers [34,2,1,3,17,4] using a state feedback approach is valid when all of the system states are available for measurement. In practice, however, the state feedback control does not always hold because system states are not always available. Estimations of states from the system output for output feedback control design of the direct adaptive fuzzy-neural controller is required. Therefore, problem as to how a direct adaptive fuzzy-neural output-feedback controller (DAFOC) is designed remains to be solved. It is therefore the objective ofthis paper to develop a design algorithm ofthe DAFOC for uncertain nonlinear systems under the constraint that only the system output is available for measurement. Particularly, the output feedback control law and the update law can be on-line tuned. Moreover, the overall adaptive scheme guarantees that all signals involved are bounded and the output ofthe closed-loop system will asymptotically tracks the desired output trajectory.
The paper is organized as follows. In Section 2, the problem is formulated and a brief description ofa fuzzy-neural network is presented. Design methodology ofthe DAFOC is included in Section
3. In Section 4, simulation results are demonstrated to show the e9ectiveness and applicability of the proposed method. Conclusions are included in Section 5.
2. Problem formulation and fuzzy-neural network
Consider the nth order nonlinear dynamical system ofthe form x(n)= f(x; ˙x; : : : ; x(n−1)) + g(x; ˙x; : : : ; x(n−1))u + d;
y = x; (1)
where d is the external bounded disturbance, and u ∈ R and y ∈ R are the control input and system output, respectively. We assume that f and g are uncertain functions, and g is, without loss of generality, a strictly positive function. It is also assumed that a solution for (1) exists. In addition, only the system output y is assumed to be measurable. The control objective is to design a DAFOC
such that the system output y follows a given bounded reference signal ym, and all signals involved
are bounded.
First, we convert the tracking problem to a regulation problem. Eq. (1) can be rewritten as ˙x = Ax + B(f(x) + g(x)u + d); y = CTx; (2) where A = 0 1 0 · · · 0 0 0 1 · · · 0 · · · · 0 0 0 · · · 1 0 0 0 0 0 ; B = 0 0 ... 0 1 ; C = 1 0 ... 0 0 ;
and x = [x; ˙x; : : : ; x(n−1)]T= [x1; x2; : : : ; xn]T∈ Rn is a vector ofstates. De7ne the output tracking error
e = ym− y, the reference vector ym= [ym; ˙ym; : : : ; ym(n−1)] and the tracking error vector e = [e; ˙e; : : : ;
e(n−1)]T= [e1; e2; : : : ; en]T.
Based on the certainty equivalence approach, an optimal control law is u∗ = 1
g(x)[−f(x) + y(n)m + KTc ˆe]; (3)
where ˆe = ym− ˆx, ˆe and ˆx denote the estimates of e and x, respectively. Kc= [kncknc−1: : : k1c]T is the
feedback gain vector, chosen such that the characteristic polynomial of A − BKT
c is Hurwitz because
(A; B) is controllable. Since only the system output y is assumed to be measurable, and f(x) and g(x) are assumed to be uncertain, the optimal control law (3) cannot be implemented. Thus, suppose a control input u is
u = uf+ us; (4)
where uf is designed to approximate the optimal control law (3), and the control term us is employed to compensate the external disturbance and the modeling error. From (2), (3) and (4), we have
˙e = Ae − BKTc ˆe + B[g(x)u∗− g(x)uf− g(x)us− d];
e1= CTe: (5)
Thus, the tracking problem has been converted into the regulation problem ofdesigning a state observer for estimating the state vector e in (5) in order to regulate e1 to zero.
In addition, the con7guration ofthe fuzzy-neural network shown in Fig. 1 consists ofa fuzzy system and neural network. The fuzzy system can be divided into two parts: some fuzzy IF-THEN rules and a fuzzy inference engine. The fuzzy inference engine uses the fuzzy IF-THEN rules to perform a mapping from an input linguistic vector e = [e1; e2; : : : ; en] ∈ Rn to an output linguistic variable uf∈ R. The ith fuzzy IF-THEN rule is written as
Ri: If e
Layer I Layer II Layer III Layer IV
...
...
...
...
...
u1 u2 uh...
uf= T (e) Ain Ai2 Ai1 1 2 h e1 e2 en ϕ ϕ ϕ θ ϕFig. 1. Con7guration ofa fuzzy-neural approximator.
where Ai
1; Ai2; : : : ; Ain and Bi are fuzzy sets [35,14]. By using produce inference, center average and
singleton fuzzi7er, the output of the fuzzy-neural network can be expressed as uf= h i=1 Lui[ n j=1 Ai j(ej)] h i=1[ n j=1 Ai j(ej)] = XT’(e); (7) where Ai
j(ej) is the membership function value of the fuzzy variable, h is the total number ofthe IF-THEN rules, Lui is the point at which Bi( Lui) = 1, X = [ Lu1; Lu2; : : : ; Luh]T is an adjustable parameter
vector, and ’ = [’1; ’2; : : : ; ’h]T is a fuzzy basis vector, where ’i is de7ned as
’i(e) = n j=1 Ai j(ej) h i=1[ n j=1A i j(ej)] : (8)
When the inputs are given into the fuzzy-neural network shown in Fig. 1, the truth value ’i(layer III) ofthe antecedent part ofthe ith implication is calculated by (8). Among the commonly used de9uzzi7cation strategies, the output (layer IV) ofthe fuzzy-neural network is expressed as (7). Therefore, the fuzzy logic approximator based on the nerual network can be established [37,19]. Fig. 1shows the con7guration of the fuzzy-neural function approximator. The approximator has four layers. At layer I, input nodes stand for the input linguistic variables e1; e2; : : : ; en. At layer II, nodes represent the values ofthe membership functions. At layer III, nodes are the values ofthe fuzzy basis vector ’. Each node of layer III performs a fuzzy rule. The links between layer III and layer IV are full connected by the weighting factors X = [ Lu1; Lu2; : : : ; Luh]T, i.e., the adjusted parameters. At layer IV, the output stands for the value of uf.
3. Output feedback control design of direct adaptive fuzzy-neural controller
In this section, our primary tasks are to design an observer that estimates the state vector e in (5), to use the fuzzy-neural network to approximate to the optimal control law u∗ in (3) and to
develop the direct adaptive update law to adjust the parameters ofthe fuzzy neural network in order to achieve the control objective.
First, we replace uf in (4) by the output ofthe fuzzy-neural network, XT’( ˆe) in (7), i.e.,
uf( ˆe|X) = XT’( ˆe); (9)
where ˆe denotes the estimate of e.
Next, consider the following observer that estimates the state vector e in (5) ˙ˆe = A ˆe − BKT
c ˆe + B(gv − gus) + Ko(e1− ˆe1);
ˆe1= CTˆe; (10)
where Ko= [k1o; k2o; : : : ; kno]T is the observer gain vector, chosen such that the characteristic polynomial
of A − KoCT is strictly Hurwitz because (C; A) is observable. The control term v is employed to
compensate the external disturbance d and the modeling error. We de7ne the observation errors as ˜e = e − ˆe and ˜e1= e1− ˆe1. Subtracting (10) from (5), we have
˙˜e = (A − KoCT) ˜e + B[gu∗− guf( ˆe|X) − gv − d];
˜e1= CT˜e: (11)
Besides, the output error dynamics of(11) can be given as
˜e1= H(s)[gu∗− guf( ˆe|) − gv − d]; (12)
where s is the Laplace variable, and H(s) = CT(sI − (A − KoCT))−1B is the transfer function of (11).
In order to derive the direct adaptive update law, the following assumption and lemma must be required.
Assumption 1 (Tsakalis and Ioannou [33]). Let e and ˆe belong to compact sets Ue= {e ∈ Rn :
e6me ¡ ∞} and Uˆe= { ˆe ∈ Rn : ˆe6mˆe ¡ ∞}, respectively, where ˆe denotes the estimate
of e and me and mˆe are designed parameters. It is known a prior that the optimal parameter vector
X∗= arg min
X∈MX[supe∈Ue; ˆe∈Uˆe|u∗ − u( ˆe|X)|] lies in some convex region MX= {X ∈ Rn : X6mX},
where the radius mX is constant.
Lemma 1 (Ioannou and Sun [10]). Consider the linear time-invariant system ˙x(t) = Ax(t) + Bu(t); x(0) = x0;
where x ∈ Rn; u(t) ∈ Rm; A ∈ Rn×n; B ∈ Rn×m. Suppose that A is Hurwitz matrix and u(t) ∈ L
2e.
" ∈ [0; "1], where 0¡"1¡2 0,
x(t) 6 0e− 0tx0 +√B 2 0
0− "ut2";
where ut2"= ( t0e−"(t−!)uT(!)u(!) d!)1=2.
According to Assumption 1, (11) can be rewritten as
˙˜e = (A − KoCT) ˜e + B[guf( ˆe|X∗) − guf( ˆe|X) − gv + w − d];
˜e1= CT˜e; (13)
where w = gu∗− guf( ˆe|X∗) is an approximation error. According to (9), (13) can be rewritten as
˙˜e = (A − KoCT) ˜e + B[g ˜XT’( ˆe) − gv + w − d];
˜e1= CT˜e; (14)
where ˜X = X∗− X. Since only the output ˜e1, in (14) is assumed to be measurable, we use the SPR
Lyapunov design approach to analyze the stability of(14) and generate the direct adaptive update law for X. Eq. (14) can be rewritten as
˜e1= H(s)[g ˜XT’( ˆe) − gv + w − d]; (15)
where H(s) = CT(sI − (A − KoCT))−1B is a known stable transfer function. In order to employ the SPR-Lyapunov design approach, (15) can be written as
˜e1= H(s)L(s)[ ˜XT%( ˆe) − vf+ wf]; (16)
where vf = L−1(s)[gv], wf = L−1(s)[w − d + g ˜XT’( ˆe)] − ˜XT%( ˆe), %( ˆe) = L−1(s)[’( ˆe)] and L(s)
is chosen so that L−1(s) is a proper stable transfer function and H(s)L(s) is a proper SPR transfer function. Supposed that L(s) = sm+ b1sm−1+ b2sm−2+ · · · + bm, where m ¡ n, such that H(s)L(s) is a proper SPR transfer function. Then the state–space realization of (16) can be written as
˙˜e = Ac˜e + Bc[ ˜XT%( ˆe) − vf+ wf];
˜e1= CTc ˜e; (17)
where Ac= (A − KoCT) ∈ Rn×n; BTc = [0 0 · · · b1 b2· · · bm] ∈ Rn and CTc= [10 · · · 0] ∈ Rn. For
the purpose ofstability analysis ofthe DAFOC, the following assumptions and lemma must be required.
Lemma 2 (Wang [35] and Leu et al. [18]). Supposed that the update laws are chosen as ˙X =
' ˜e1%( ˆe) if X ¡ mX or (X = mX and ˜e1XT%( ˆe) ¿ 0);
Pr(' ˜e1%( ˆe)) if X = mX and ˜e1XT%( ˆe) ¡ 0;
where the projection operator [35] is given as Pr(' ˜e1%( ˆe)) = ' ˜e1%( ˆe) − '˜e1X
T%( ˆe)
X2 X:
Then X6mX and ˜X62mX.
Assumption 2. The uncertain function g(x) is bounded by
+16 g(x) 6 +2; (19)
where +1 and +2 are positive constants.
Assumption 3. wf is assumed to satisfy
|wf| 6 ,; (20)
where , is a positive constant.
Remark 1. The assumption of |wf|6, is reasonable because ofAssumption 2, the universal
approx-imate theorem and the external bounded disturbance.
On the basis ofthe above discussions, the following theorems can be obtained.
Theorem 1. Consider system (17) that satis7es Assumptions 1–3. Let X be adjusted by the update law (18), and let v be given as
v =
- if˜e1¿ 0;
−- if˜e1¡ 0; (21)
where -¿,=+1. Then ˜e1(t) converges to zero as t → ∞.
Proof. Given in Appendix A.
Theorem 2. Consider the nonlinear system (1) that satis7es Assumptions 1–3. Let the control term us in (4) and (10) be us= v in (21), such that the state observer (10) becomes
˙ˆe = (A − BKT
c) ˆe + Ko˜e1: (22)
Suppose that the control law is
u = uf( ˆe|X) + us (23)
with the update law (18). Then all signals in the closed-loop system are bounded, and e1(t) con-verges to zero as t → ∞.
Plant: y=x xn f(x)+g(x)u+d ) ( 0 K e (ˆ) T f u
The adaptive law
in (18) us= v in (21) or (24) 1 0~ ˆ ) ( ˆ T e c e K BK A e 1 ˆe 1 e 1 ~ e eˆ T C 1 ~ e f u u s u + + + _ _ + y ym = Σ Σ Σ = – + = . . θ θ
Fig. 2. Overall scheme ofthe proposed direct adaptive fuzzy-neural control system.
According to the above theorems, the design algorithm ofthe DAFOC is described as following.
Design algorithm. Step 1: Select the feedback and observer gain vectors Kc; Ko such that the
ma-trices A − BKTc and A − KoCT are Hurwitz matrices, respectively.
Step 2: Choose an appropriate value - in (21) and ' in (18). In order to remedy the control chattering, (21) can be modi7ed as
v =
- if˜e1¿ 0 and| ˜e1| ¿ ;
−- if˜e1¡ 0 and | ˜e1| ¿ ; where is a positive constant;
- ˜e1= if | ˜e1| ¡ :
(24)
Step 3: Solve the state observer in (22), where v in (21) or (24).
Step 4: Construct fuzzy sets for ˆe(t). Then, from (8) compute the fuzzy basis vector ’. Step 5: Obtain the control law (23), and the update law (18).
To summarize, Fig. 2shows the overall scheme of the direct adaptive fuzzy-neural output feedback control system proposed in this paper.
4. The illustrative example
This section presents simulation results ofthe proposed design algorithm to illustrate that stability ofthe closed-loop system is guaranteed, and all signals involved are bounded.
Example. Consider the DuNng forced oscillation system [34] ˙x1= x2;
˙x2= −0:1x2− x13+ 12 cos t + u + d;
y = x1: (25)
It is assumed that the external disturbance d(t) is a square wave having an amplitude ± 1 with a period of2/. The control objective is to control the state x1 ofthe system to track the reference
trajectory ym, under the condition that only the system output y is measurable. The design parameters
are selected as ' = 0:5 × 103 and - = 20. The feedback and observer gain vectors are given as
Kc= [144 24]T and Ko= [60 900]T, respectively. The 7lter L−1(s) is given as L−1(s) = 1=(s + 2).
The following membership functions for ˆej; j = 1; 2 are given as
A1
j( ˆej) = 1=(1 + exp(5 × ( ˆej+ 3))); A2j( ˆej) = exp(−( ˆej+ 2)
2);
A3
j( ˆej) = exp(−( ˆej+ 1)
2); A4
j( ˆej) = exp(−( ˆej)
2);
A5
j( ˆej) = exp(−( ˆej− 1)
2); A6
j( ˆej) = exp(−( ˆej− 2)
2);
A7
j( ˆej) = 1=(1 + exp(−5 × ( ˆej− 3)):
The initial states are chosen to be x1(0) = x2(0) = 3; ˆx1(0) = ˆx2(0) = −1 and ˆe(0) = ym(0) − ˆx(0).
Simulation results are provided for three cases with di9erent reference trajectories, i.e., ym = 0
(Case 1), ym = sin t (Case 2), and ym= 1 − exp(−t=2) (Case 3), respectively. Figs. 3–13 show the
computer simulation results for these three cases. With reference to Figs. 3, 6, and 9, it is observed that the state observer correctly and responsively generates the estimated state ˆx1. Referring to Figs. 4, 7, and 10, it is also observed that the tracking convergence is fast with only a relatively small tracking error by using the control term v in (24). To avoid the chattering e9ect ofcontrol input, the control term v in (24), instead of v in (21), is used for these three cases as shown in Figs. 3–11. Comparing Figs. 4 with 12 (Case 1), we 7nd that tracking performance using v in (21) is slightly better than that using v in (24). However, the chattering e9ect ofcontrol input using v in (21) is much serious than that using v in (24), as clearly demonstrated in Figs. 5 and 13. As shown in Figs. 5, 8, and 11, chattering e9ect ofthe control input for these three cases almost disappears by using the control term v in (24). That is the reason that v in (24), instead of v in (21), is suggested to derive the control law for practical applications.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 x1 1 ˆx Time (sec)
Fig. 3. Trajectories ofthe states x1 and ˆx1 ofCase 1 using v in (24).
0 2 4 6 8 10 12 14 16 18 20 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 x1 Time (sec)
0 2 4 6 8 10 12 14 16 18 20 -30 -20 -10 0 10 20 30 Time (sec) u (t)
Fig. 5. Control input u ofCase 1 using v in (24).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 x1 1 ˆx Time (sec)
0 2 4 6 8 10 12 14 16 18 20 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 x1 Time (sec) ym
Fig. 7. Trajectories ofthe states x1 and ym ofCase 2 using v in (24).
0 2 4 6 8 10 12 14 16 18 20 -30 -20 -10 0 10 20 30 Time (sec) u (t)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 x1 1 ˆx Time (sec)
Fig. 9. Trajectories ofthe states x1 and ˆx1 ofCase 3 using v in (24).
0 2 4 6 8 10 12 14 16 18 20 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 x1 Time (sec) ym
0 2 4 6 8 10 12 14 16 18 20 -30 -20 -10 0 10 20 30 Time (sec) u (t)
Fig. 11. Control input u ofCase 3 using v in (24).
0 2 4 6 8 10 12 14 16 18 20 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 x1 Time (sec)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -30 -20 -10 0 10 20 30 Time (sec) u (t)
Fig. 13. Control input u ofCase 1 using v in (21).
5. Conclusions
The direct adaptive fuzzy-neural output-feedback controller (DAFOC), which can be subject to on-line tuning for nonon-linear systems, has been proposed in this paper. In designing the output feedback control law, no di9erentiation ofsystem outputs is performed in order to avoid the noise ampli7cation associated with numerical di9erentiation, and no knowledge on nonlinearities ofthe nonlinear systems is required. Also, preliminary o9-line tuning ofthe weighting factors ofthe fuzzy-neural controller is no longer required. The overall adaptive scheme guarantees that all signals involved are bounded and the output ofthe closed-loop system asymptotically tracks the desired output trajectory. Moreover, the proposed design algorithm has been successfully applied to control the nonlinear DuNng forced oscillation system to track a reference trajectory. Simulation results have shown that the DAFOC performs good control and achieve desired performance.
Acknowledgements
This work was supported by the National Science Council, Taiwan, under Grant NSC 89-2213-E-031-008.
Appendix A.
Proof of Theorem1. Consider the Lyapunov-like function candidate V = 1
2˜eTP ˜e + 1 2'˜X
where P = PT¿0. Di9erentiating (A.1) with respect to time and inserting (17) in the above equation yield ˙V = 12˜eT(AT cP + PAc) ˜e + ˜eTPBc[ ˜XT% − vf+ wf] + 1' ˙˜X T ˜X: (A.2)
Because H(s)L(s) is SPR, there exists P = PT ¿ 0 such that
ATcP + PAc= −Q;
PBc= Cc; (A.3)
where Q = QT¿ 0. By using (A.3), (A.2) becomes
˙V = −12˜eTQ ˜e + ˜e1[ ˜XT%:− vf+ wf] +1'˙˜X T
˜X: (A.4)
By using Assumptions 2–3, (21) and the fact min(Q) ˜e2¿ min(Q)| ˜e1|2, where min(Q) ¿ 0, we have
˙V 6 −12 min(Q)| ˜e1|2+ ˜e1˜XT% +1'˙˜X T
˜X: (A.5)
Inserting (18) in (A.5) and after some manipulation yields
˙V 6 −12 min(Q)| ˜e1|2: (A.6)
Eqs. (21) and (A.6) only guarantee that ˜e1(t) ∈ L∞ and ˜e(t) ∈ L∞, but do not guarantee the
convergence. Because all variables in the right-hand side of(17) are bounded, ˙˜e1(t) is bounded, i.e., ˙˜e1(t) ∈ L∞. Integrating both side of(A.6) and after some manipulation yields
∞
0 | ˜e1(t)|
2dt 6 V (0) − V (∞)
(1=2) min(Q): (A.7)
Since the right side of(A.7) is bounded, so ˜e1(t) ∈ L2. Using Barbalat’s lemma [28], we have
limt→∞| ˜e1(t)| = 0. This completes the proof. Appendix B.
Proof of Theorem 2. First, from Theorem 1, we have limt→∞| ˜e1(t)| = 0. Next, consider Eq. (14). De7ne Lu = g ˜XT’( ˆe) − gv + w − d. Because A − KoCT is a Hurwitz matrix, and Lu is bounded from Lemma 2 and under Assumptions 1–3, we have
˜e(t) 6 0e− 0t ˜e(0) +√B 2 0
according to Lemma1. Therefore ˜e(t) ∈ L∞. By using the state observer (22), we obtain the dynamic
system
˙ˆe = (A − BKT
c) ˆe + KoCT˜e
ˆe1= CTˆe: (B.2)
Similarly, because A−BKT
c is a Hurwitz matrix and ˜e(t) is bounded, ˆe(t) is bounded. From ˜e = e− ˆe,
it follows that e1; e ∈ L∞ and e1(t) → 0 as t → ∞. From ˆe; e ∈ L∞, it follows that x; ˆx ∈ L∞. The boundedness of y(t) follows that of e1(t) and ym(t). This completes the proof.
References
[1] Y.C. Chang, Adaptive fuzzy-based tracking control for nonlinear SISO systems via VSS and H approaches, IEEE Trans. Fuzzy Systems 9 (2) (2001) 278–292.
[2] B.S. Chen, C.H. Lee, Y.C. Chang, H∞tracking design ofuncertain nonlinear SISO systems: adaptive fuzzy approach,
IEEE Trans. Fuzzy System 4 (1) (1996) 32–43.
[3] Y. Diao, K.M. Passino, Stable fault-tolerant adaptive fuzzy/neural control for a turbine engine, IEEE Trans. Control Systems Technol. 9 (3) (2001) 494–509.
[4] M.J. Er, S.H. Chin, Hybrid adaptive fuzzy controllers of robot manipulators with bounds estimation, IEEE Trans. Ind. Electron. 47 (5) (2000) 1151–1160.
[5] S.S. Ge, C.C. Hang, T.H. Lee, T. Zhang, Stable Adaptive Neural Network Control, Kluwer Academic Publishers, Norwell, USA, 2001.
[6] S.S. Ge, T.H. Lee, C.J. Harris, Adaptive Neural Network Control ofRobotic Manipulators, World Scienti7c, London, December 1998.
[7] S. Horikawa, T. Furuhashi, Y. Uchikawa, On fuzzy modeling using fuzzy neural networks with the back-propagation algorithm, IEEE Trans. Neural Networks 3 (5) (1992) 801–806.
[8] K. Hornik, M. Stinchcombe, H. White, Multilayer feedforward networks are universal approximators, Neural Networks 2 (1989) 359–366.
[9] P.A. Ioannou, A. Datta, Robust adaptive control: a uni7ed approach, Proc. IEEE 79 (12) (1991) 1735–1768. [10] P.A. Ioannou, J. Sun, Robust Adaptive Control, Prentice-Hall, Englewood Cli9s, NJ, 1996.
[11] P.A. Ioannou, K.S. Tsakalis, A robust direct adaptive controller, IEEE Trans. Automat. Control AC-31 (1986) 1033 –1043.
[12] A. Isidori, Nonliear Control System, Springer, New York, 1989.
[13] B. Jakubezyk, Feedback linearization ofdiscrete time system, System Control Lett. 9 (1987) 411–416. [14] M. Jamshidi, N. Vadiee, T.J. Ress, Fuzzy Logic and Control, Prentice-Hall, Englewood Cli9s, NJ, 1993.
[15] Y.K. Kim, F.L. Lewis, C.T. Abdallah, A dynamic recurrent neural-network-based adaptive observer for a class of nonlinear systems, Automatica 33 (8) (1997) 1539–1543.
[16] E.B. Kosmatopoulos, P.A. Ioannou, M.A. Christodoulou, Identi7cation ofnonlinear systems using new dynamic neural network structures, in: Proc. IEEE Conf. Decision and Control, Tucson, AZ, 1992, pp. 20–25.
[17] C.H. Lee, C.C. Teng, Identi7cation and control ofdynamic systems using recurrent fuzzy neural networks, IEEE Trans. Fuzzy Systems 8 (4) (2000) 349–366.
[18] Y.G. Leu, T.T. Lee, W.Y. Wang, Observer-based adaptive fuzzy-neural control for unknown nonlinear dynamical systems, IEEE Trans. System, Man Cybernet. Part B: Cybernet. 29 (5) (1999) 583–591.
[19] Y.G. Leu, W.Y. Wang, T.T. Lee, Robust adaptive fuzzy-neural controller for uncertain nonlinear systems, IEEE Trans. Robotics Automat. 15 (5) (1999) 805–817.
[20] X. Li, S. Yurkovich, Neural network based, discrete adaptive sliding mode control for idle speed regulation in IC engines, J. Dyn. Systems, Meas. Control (ASME) 122 (2) (2000) 269–275.
[21] Y.H. Lim, F.L. Lewis, High-level Feedback Control with Neural Networks, World Scienti7c, Singapore, December 1999.
[22] S. Monaco, D. Normand-Cyrot, Minimum-phase nonlinear discrete-time systems and feedback stabilization, in: Proc. IEEE Conf. Decision and Control, Los Angeles, CA, 1987, pp. 979–986.
[23] R.V. Monopoli, Model reference adaptive control with an augmented error signal, IEEE Trans. Automat. Control AC-19 (1974) 474–484.
[24] K.S. Narendra, A.M. Annaswamy, Stable Adaptive Systems, Prentice-Hall, Englewood Cli9s, NJ, 1989.
[25] M. Polycarpou, P.A. loannou, Modelling, identi7cation and stable adaptive control ofcontinuous-time nonlinear dynamical systems using neural networks, Proc. Amer. Control Conf., Chicago, IL, 1992, pp. 36–40.
[26] C.A. Rovithakis, M.A. Christodoulou, Adaptive control ofunknown plants using dynamical neural networks, IEEE Trans. System, Man Cybernet. 24 (3) (1995) 400–411.
[27] R.M. Sanner, J.J.E. Slontine, Guassian networks for direct adaptive control, IEEE Trans. Neural Networks 1 (6) (1992) 837–863.
[28] S.S. Sastry, M. Bodson, Adaptive Control: Stability, Convergence, and Robustness, Prentice-Hall, Englewood Cli9s, NJ, 1989.
[29] S.S. Sastry, A. Isidori, Adaptive control oflinearizable systems, IEEE Trans. Automat. Control 34 (1989) 1123– 1131.
[30] S. Seshagiri, H.K. Khalil, Output feedback control of nonlinear systems using RBF neural networks, IEEE Trans. Neural Network 11 (1) (2000) 69–79.
[31] J.J.E. Slontine, W. Li, Applied Nonlinear Control, Prentice-Hall, Englewood Cli9s, NJ, 1991.
[32] S. Tong, T. Wang, J.T. Tang, Fuzzy adaptive output tracking control ofnonlinear systems, Fuzzy Sets and Systems 111 (2000) 1169–1182.
[33] K.S. Tsakalis, P.A. Ioannou, Linear Time-Varying Systems, Prentice-Hall, Englewood Cli9s, NJ, 1993.
[34] L.X. Wang, Stable adaptive fuzzy control of nonlinear systems, IEEE Trans. Fuzzy System 1 (2) (1993) 146–155. [35] L.X. Wang, Adaptive Fuzzy Systems and Control: Design and Stability Analysis, Prentice-Hall, Englewood Cli9s,
NJ, 1994.
[36] W.Y. Wang, M.L. Chan, T.T. Lee, C.H. Liu, Adaptive fuzzy control for strict-feedback canonical nonlinear systems with H∞ tracking performance, IEEE Trans. System, Man and Cybernet. Part B 30 (6) (2000) 878–885.
[37] W.Y. Wang, Y.G. Leu, C.C. Hsu, Robust adaptive fuzzy-neural control of nonlinear dynamical systems using generalized projection update law and variable structure controller, IEEE Trans. System, Man Cybernet. Part B 31 (1) (2001) 140–147.
[38] L.X. Wang, J.M. Mendel, Fuzzy basis functions, universal approximation, and orthogonal least squares learning, IEEE Trans. Neural Networks 3 (5) (1992) 807–814.
[39] T. Zhang, S.S. GE, C.C. Hang, Adaptive output feedback control for general nonlinear systems using multilayer neural networks, Proc. 1998 Amer. Control Conf., Philadelphia, PA, pp. 520–524.
