2 DOF adaptive PID control with a parallel feedforward compensator for nonlinear systems

Abstract

In this paper, we propose a design method of an adaptive PID controller based on output feedback for nonlinear systems with a higher order relative degree and disturbances. To realize an adaptive PID control system, we introduce a PFC for a nonlinear system which does not satisfy OFEP (Output Feedback Exponentially Passive) conditions and design an adaptive feedforward input with a structure of RBF (Radial Basis Function) neural networks in order to remove the steady-state bias error from the PFC output. The proposed method can design a robust adaptive PID controller with higher accuracy on tracking control.

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熊本大学学術リポジトリ熊本大学学術リポジトリ
Kumamoto University Repository SystemKumamoto University Repository System
Title2 DO F Adapti ve PID Control w i th a Paral l el
Feedforw ard Com pensator for N onl i near System s
Author(s)Ikuro, M i zum oto; H i roki , Tanaka; Zenta, Iw ai
CitationProceedi ngs of the 2009 IEEE Internati onal
Conference on N etw orki ng, Sensi ng and Control ,
ICN SC 2009…
Issue date2009-03-26
TypeJ ournal Arti cl e
URLhttp: //hdl . handl e. net/2298/16316
Right(c)2009 IEEE. Personal use of thi s m ateri al i s
perm i tted. H ow ever, perm i ssi on to repri nt/republ i sh
t…

Page 2

2 DOF Adaptive PID Control with a Parallel Feedforward
Compensator for Nonlinear Systems
Ikuro Mizumoto* and H. Tanaka
Department of Intelligent Mechanical Systems
Kumamoto University
2-39-1 Kurokami, Kumamoto, 860-8555, Japan
*ikuro@gpo.kumamoto-u.ac.jp
Zenta Iwai
Kumamoto Prefectural College of Technology
4455-1 Haramizu, Kikuyo-mach, Kikuchi-gun
Kumamoto, 869-1102, Japan
iwai@kumamoto-pct.ac.jp
Abstract—In this paper, we propose a design method of
an adaptive PID controller based on output feedback for
nonlinear systems with a higher order relative degree and
disturbances. To realize an adaptive PID control system, we
introduce a PFC for a nonlinear system which does not satisfy
OFEP(Output Feedback Exponentially Passive) conditions and
design an adaptive feedforward input with a structure of RBF
(Radial Basis Fuction) neural networks in order to remove the
steady-state bias error from the PFC output. The proposed
method can design a robust adaptive PID controller with higher
accuracy on tracking control.
I. INTRODUCTION
In the recent decade, much attention has been paid to high
gain output feedback-based adaptive controls for nonlinear
systems due to their simple structure and high robustness
with respect to uncertainties and disturbances [1], [2], [3].
The most typical design condition for designing the output
feedback-based adaptive control is recognized as the output
feedback exponentially passive (OFEP) condition [3]. This
condition is well known as the ASPR condition for linear
systems [4]. Unlike other adaptive methods, under OFEP (or
ASPR) condition, one can easily design an output feedback-
based adaptive controller without a priori information of the
order of the controlled system and without designing a state
observer.
Recently, auto-tuning and an adaptive PID control strate-
gies based on the almost strictly positive real (ASPR) prop-
erty of the controlled system have been proposed [8], [9] for
linear systems. The PID control is one of the most common
control schemes applied to many industrial processes and
mechanical systems. Since the control has played a very
important role in the improvement of production quality,
accuracy and in reducing production costs, a great deal
of attention has been turned to auto-tuning PIDs including
self-tuning schemes and adaptive control strategies [5], [6],
[7], [8], [9], [10] in order to maintain the desired control
performance and stability during operation.
In this paper, we propose an adaptive PID control system
design scheme for nonlinear systems utilizing the output
feedback exponential passivity (OFEP) [3], [11] of the con-
trolled system. The sufficient conditions for the system to
be OFEP are that (1) the system is globally exponential
minimum-phase, (2) the system has a relative degree of 1 and
(3) the nonlinearities of the system satisfy the Lipschitz con-
ditions [3]. Unfortunately, however, most practical systems
do not satisfy the above-mentioned OFEP conditions. In the
proposed method, the introduction of a parallel feedforward
compensator (PFC) is considered to solve both the relative
degree and minimum-phase problems simultaneously [11],
[12]. Further, since it has been indicated that affects from the
introduced PFC result in degradation of the control perfor-
mance in the output tracking control, we consider introducing
a feedforward signal generated by an adaptive RBF NN
(radial basis function neural networks) [13] directly to the
original controlled system in order to maintain the output
tracking control performance. The proposed adaptive PID
controller can maintain stability a better control performance
even if there are some changes of the system properties by
adjusting PID parameters and RBF NN parameters adap-
tively.
II. PROBLEM STATEMENT
Consider the following SISO affine nonlinear systems:
˙ x(t) = f(x) + g(x)u(t) + p(ω)
y(t) = h(x)
(1)
where x(t) ∈ Rnis a state vector, u(t) ∈ R is a control
input, y(t) ∈ R is an output, and, f(x),g(x) : Rn→ Rn,
h(x) : Rn→ R are of sufficiently smooth (e.g. of class
C∞) functions such that f(0) = 0,h(0) = 0. Further, p(ω)
represents the effect of disturbances. We assume that the
disturbance and a reference signal yrwhich the output of the
system is required to track are generated by the following
exosystem:
˙ ω(t) = s(ω)
yr(t) = py(ω)
s(0) = 0 , py(0) = 0,
(2)
where ω(t) ∈ Rqis a state vector of the exosystem. We
assume that the exosystem (2) satisfies the neutral stabil-
ity property [14]. That is, the linear approximation S =
?∂s
Definition 1: (Output
Passive(OFEP))[3] The system (1) with p(0) = 0 is
said to be OFEP if there exists a smooth output feedback:
∂ω
eigenvalues on the imaginary axis.
?
ω=0of the vector field s(ω) at ω = 0 has all its
FeedbackExponentially
u(t) = α(y) + β(y)v(t)
α(y)= 0,β(0) = 0
(3)
Proceedings of the 2009 IEEE International Conference on
Networking, Sensing and Control, Okayama, Japan, March 26-29, 2009
978-1-4244-3492-3/09/$25.00 ©2009 IEEE 261
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such that the resulting closed loop system from v(t) to y(t)
is exponentially passive, that is for the closed loop system,
the following dissipation inequality (DI) is satisfied
˙V (x) =∂V (x)
∂x
(f(x) + g(x)v) ≤ yv − ζ(x)
(4)
with a (C2) positive definite function V and a positive
definite function ζ(x) having the following properties:
δ1||x||2≤ V (x) ≤ δ2||x||2
δ3||x||2≤ ζ(x),
(5)
where δ1∼ δ3are positive constants and ζ(x) is a positive
definite function.
Sufficient conditions for the system (1) to be OFEP have
been provided in [3] such that (C1) the system has a relative
degree of one, (C2) the system be globally exponential
minimum-phase and (C3) the nonlinearities of the system
satisfy the Lipschitz condition.
In this paper, we consider an adaptive PID control system
design based on the OFEP property for nonlinear systems
which do not satisfy OFEP conditions.
We suppose that the nominal part of the system (1) satisfies
the following assumptions.
Assumption 1: The system (1) has a relative degree of
r and there exists a nonsingular transformation z(t) =
[z1,z2,··· ,zn]T= Φ(x) such that the system (1) can be
transformed into the following canonical form:
˙ zi(t) = zi+1(t) + pi(ω)
˙ zr(t) = a(zξ,η) + b(zξ,η)u(t) + pr(ω)
˙ η(t) = q(zξ,η) + pη(ω)
y(t) = z1(t)
(i = 1,2,··· ,r − 1)
(6)
where zξ= [z1,··· ,zr]T, η = [zr+1,··· ,zn]Tand
a(0,0) = Lr
b(zξ,η) = LgLr−1
f
Assumption 2: Nonlinear functions a(zξ,η), q(zξ,η)
and b(zξ,η) are globally Lipschitz with respect to (zξ,η).
Assumption 3: There exist positive constants¯BM and¯b0
such that b(zξ,η) can be evaluated as
fh(0) = 0, q(0,0) = 0
h(x) ?= 0, ∀x ∈ Rn
0 <¯b0≤ b(zξ,η) ≤¯BM
(7)
Assumption 4: The linear approximation of the system (1)
at x(t) = 0 is stabilizable and its zero dynamics has no
eigenvalue on the imaginary axis.
The objective of this paper is to design a PID control
system which has the output y(t) track the reference signal
yr(t) for uncertain nonlinear systems with higher order
relative degrees and/or non-minimum phase properties. That
is, for uncertain nonlinear systems, the goal is to achieve
lim
t→∞|y(t) − yr(t)| ≤ δ
(8)
for any given small positive constant δ.
III. CONTROL SYSTEM DESIGN
It has been clarified that, under Assumption 4 and the as-
sumption that the exosystem (2) satisfies the neutral stability
property, there exists an ideal state x∗(t) and an ideal input
v∗(t) which attain perfect output tracking:
˙ x∗(t) = f(x∗) + g(x∗)v∗(t) + p(ω)
y(t) = yr(t) = h(x∗)
(9)
and they are given by functions of ω as x∗(t) = π(ω) and
v∗(t) = c(ω) [14], [15].
In the following, we will show an adaptive PID control
system design with the adaptively estimated feedforward
input v(t) of the ideal input v∗(t) for uncertain and non-
OFEP systems.
A. Ideal Input Approximation by RBF NN
Consider approximating the ideal control input v∗(t) by
a Radial Basis Function (RBF) neural networks (NN). We
approximate v∗(t) by the form of RBF NN as
vnn(t) = WTS(ω)
(10)
where W = [w1,··· ,wl]T∈ Rlis the weight vector, l
is the number of NN nodes (weight number) and S(ω) =
[s1(ω),··· ,sl(ω)]Tis the radial basis function vector. This
basis function vector S(ω) is generally designed by the
Gaussian functions such as
?−(ω − μi)T(ω − μi)
si(ω) = exp
η2
i
?
,i = 1,2,··· ,l
(11)
where μi= [μi1,··· ,μiq]Tis the center of the receptive
field and ηiis the width of the Gaussian function.
It has been clarified [13] that, for a sufficiently large l and
a compact set Ωω⊂ Rq, there exists an ideal constant weight
vector W∗such that
?
and thus the ideal input v∗(t) can be approximated by
W∗? arg min
W∈Rl
sup
ω∈Ωω
|v∗− WTS(ω)|?
(12)
v∗(t) = W∗TS(ω) + ?(ω)
|?(ω)| ≤ ?∗
(13)
where ?(ω) is an approximation error.
Here we impose the following assumption.
Assumption 5: For a given NN nodes l, there exists an
ideal weight vector W∗that satisfies (12) for all ω ∈ Ωω.
In practice, W∗is the unknown vector so that one can not
design W in (10) as W = W∗. W will be adaptively adjust
in the proposed control system design which will be shown
later.
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B. Realization of OFEP controlled system
In order to guarantee the stability of the designed adaptive
PID control system, we consider utilizing system’s OFEP
property. Unfortunately, however, most practical systems
including those with relative degree of greater than 1 and/or
non-minimum phase are not OFEP, we here consider an
introduction of a parallel feedforward compensator (PFC) to
realize an OFEP augmented controlled system [11], [12].
Consider a nfth order PFC of the form:
˙ xf(t)= Afxf(t) + bfu(t)
yf(t) = cT
fxf(t)
(14)
and introduce this in parallel with the system (1). The
resulting augmented system can be represented by
˙ x(t) = f(x) + g(x)u(t) + p(ω)
˙ xf(t)= Afxf(t) + bfu(t)
ya(t) = y(t) + yf(t) = h(x) + cT
fxf(t).
(15)
Since the augmented system (15) has a relative degree
of 1, there exists a nonsingular transformation [ya,ηa] =
Φa(x,xf) such that the augmented system (15) can be
transformed into the following form [14]:
˙ ya(t) = aa(ya,ηa) + ba(ya,ηa)u(t) + p1(ω)
˙ ηa(t)= qa(ya,ηa) + pηa(ω)
(16)
Here we impose the following assumption on the aug-
mented system (16).
Assumption 6: The zero dynamics of the augmented sys-
tem (16):
˙ ηa(t) = qa(0,ηa)
(17)
is exponentially stable.
Assumption 7: The PFC (14) is strictly positive real
(SPR).
It should be noted that, under Assumption 2, aa(ya,ηa),
qa(ya,ηa) and ba(ya,ηa) are globally Lipschiz and thus
under Assumption 6, the resulting augmented system is
OFEP [3]. That is, we assume that an SPR PFC which
render the resulting augmented system OFEP is known.
Furthermore, under assumption 3, it follows that there exist
positive constants BM and b0such that
0 < b0≤ ba(ya,ηa) ≤ BM
(18)
C. Adaptive PID System Design
Adaptive PID controller with adaptive NN feedforward
input is designed as follows:
u(t) = ue(t) + v(t)
ue(t) = −K(t)TZ(t)
v(t) =ˆ WT(t)S(ω)
(19)
with
K(t) = [kp(t),kd(t),ki(t)]T,
Z(t) = [¯ ea(t),˙¯ ea(t),w(t)]T
˙ w(t) = ¯ ea(t) − δw(t) ,δ > 0
(20)
＋＋
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Fig. 1.Block diagram of the proposed adaptive control system
and
˙kp(t) = γp¯ e2
˙kd(t) = γd¯ ea(t)˙¯ ea(t) − σdkd(t) , γd> 0,σd> 0
˙ki(t) = γi¯ eaw(t) − σiki(t) , γi> 0,σi> 0
˙ˆ W(t) = −ΓS(ω) − σˆ W(t) , Γ > 0,σ > 0
Where
a(t) − σpkp(t) , γp> 0,σp> 0
(21)
¯ ea= ¯ ya(t) − yr(t)
¯ ya(t) = y(t) + ¯ yf(t)
(22)
(23)
and ¯ yf(t) is defined as the output of the PFC with ue(t) as
an input
˙¯ xf(t)= Af¯ xf(t) + bfue(t)
¯ yf(t) = cT
f¯ xf(t).
(24)
ˆ W(t) is an estimated value of the ideal weight vector W∗.
The overall block diagram of the designed control system
is shown in Fig. 1
D. Stability Analysis
Equivalent representation of the designed control system
given in Fig. 1 is given by the following form using the
augmented system representation given in (16)
˙ ya(t) = aa(ya,ηa) + ba(ya,ηa)(ue(t) + v(t)) + p1(ω)
˙ ηa(t) = q(ya,ηa) + pηa(ω)
¯ ya(t) = ya(t) − ˆ y∗
where ˆ y∗
f(t),
(25)
f(t) is a PFC output with v(t) as an input
˙ˆ xf(t) = Afˆ xf(t) + bfv(t)
ˆ y∗
f(t) = cT
fˆ xf(t)
(26)
Further, the ideal state in which the perfect output tracking
y(t) ≡ yr(t) is achieved is also represented as follows using
the augmented system expression.
˙ y∗
˙ η∗
¯ y∗
a(t) = aa(y∗
a(t) = q(y∗
a(t) = yr(t) = y∗
a,η∗
a,η∗
a) + ba(y∗
a) + pηa(ω)
a(t) − y∗
a,η∗
a)v∗(t) + p1(ω)
f(t),
(27)
where y∗
v∗(t) as an input.
f(t) is a PFC output with the ideal feedforward input
˙ x∗
y∗
f(t) = Afx∗
f(t) = cT
f(t) + bfv∗(t)
fx∗
f(t)
(28)
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From (25) and (27), the error system can be represented by
Δ˙ ya= aa(ya,ηa) − aa(y∗
+ba(ya,ηa)(ue(t) + v(t)) − ba(y∗
Δ˙ ηa= qa(ya,ηa) − qa(y∗
¯ ea(t) = ¯ ya(t) − ¯ y∗
where Δya = ya(t) − y∗
Δy∗
Next considering the difference between (26) and (28),
we have the following PFC error system:
a,η∗
a)
a,η∗
a)v∗(t)
a,η∗
a)
(29)
a= Δya− Δy∗
a(t), Δηa= ηa(t) − η∗
f(t).
f,
a(t) and
f= ˆ y∗
f(t) − y∗
Δ˙ xf = AfΔxf+ bfΔv
Δy∗
f= cT
fΔxf,
(30)
where Δxf= ˆ xf(t) − x∗
From Assumption 7, the PFC error system (30) can be
transformed into the following canonical form:
f(t) and Δv = v(t) − v∗(t).
Δ˙ y∗
˙ ηy = Aηyηy+ cηyΔy∗
f= ayΔy∗
f+ cT
yηy+ byΔv
f
(31)
using an appropriate nonsingular transformation [14].
Therefore it follows from (29) and (31) that the error
system can be rewritten as
˙¯ ea = aa(ya,ηa) − aa(y∗
+ba(ya,ηa)(Δue+ Δv − K∗TZ)
+(ba(ya,ηa) − ba(y∗
−ayΔy∗
Δ˙ ηa= qa(ya,ηa) − qa(y∗
˙ ηy = Aηyηy+ cηyΔy∗
a,η∗
a)
a,η∗
a))v∗
f− cT
yηy− byΔv
a,η∗
a)
f
(32)
where
Δue= ue(t) − u∗
e(t) = −KT(t)Z + K∗TZ
= −ΔKTZ
(33)
ΔKT= [Δkp,Δkd,Δki]T
= [kp(t) − k∗
Δv = v(t) − v∗(t) =ˆ WT(t)S(ω) − W∗TS(ω) − ?
= Δˆ WTS(ω) − ?
From Assumption 6, there exist a positive definite
functionU(Δηa) and positive constants κ1 to κ4 such that
[16]
p,kd(t) − k∗
d,ki(t) − k∗
i]T
(34)
∂U(Δηa)
∂ηa
qa(0,Δηa) ≤ −κ1?Δηa?2
∂U(Δηa)
∂ηa
κ4?Δηa?2≤ U(Δηa) ≤ κ3?Δηa?2.
Further since the PFC error system (31) is SPR from
Assumption 7, representing (31) as
????????≤ κ2?Δηa?
(35)
Δ˙¯ xf =¯AfΔ¯ xf+¯bfΔv
Δy∗
f= ¯ cT
fΔ¯ xf
(36)
with
Δ¯ xf = [Δy∗
f,ηT
ay
cηy
y]T
cT
Aηy
¯Af =
?
y
?
,¯bf=
?
by
0
?
¯ cT
f
= [1,0,··· ,0]
there exist positive definite matrices¯Pf =¯PT
¯QT
is satisfied
¯Af¯Pf+¯Pf¯Af= −¯Qf
¯Pf¯bf= ¯ cf.
f> 0,¯Qf =
f> 0 such that the following Kalman-Yakubovich Lemma
(37)
Now, consider a positive definite function V :
V = Ve+ Vη+ δ1Vxf+ Vk
+(BM+ by)Δˆ WTΓ−1Δˆ W
(38)
with Ve= ¯ e2
b0
γpΔkp2+b0
positive constant δ1,μ1.
The time derivative of Ve= ¯ e2
that
˙Ve= 2¯ ea{aa(ya,ηa) − aa(y∗
+Δv − K∗TZ) + (ba(ya,ηa) − ba(y∗
−ayΔy∗
≤ 2¯ ea{La1(|Δya| + ?Δηa?) + ba(ya,ηa)(Δue
+Δv − K∗TZ) + bMv∗− ayΔy∗
−cT
Where La1is a Lipschiz constant such that
a,Vη= μ1U(Δηa),Vxf= Δ¯ xT
γdΔkd2+b0
f¯PfΔ¯ xf,Vk=
iw2+ b0k∗
γiΔki2+ b0k∗
d¯ e2
aand any
acan be evaluated from (32)
a,η∗
a) + ba(ya,ηa)(Δue
a,η∗
a))v∗
f− cT
yηy− byΔv}
f
yηy− byΔv}.
?aa(ya,ηa)−aa(y∗
and bM is a positive constant that satisfies
a,η∗
a)?<La1(|ya− y∗
a|+?ηa− η∗
a?)
0 < ?ba(ya,ηa) − ba(y∗
The time derivative of Vη= μ1U(Δηa) can be evaluated
a,η∗
a)? ≤ bM
as
˙Vη= μ1∂U
∂ηa
{qa(ya,ηa) − qa(y∗
a,η∗
a)}
????(qa(ya,ηa)
≤ μ1∂U
∂ηa
qa(0,Δηa) + μ1
????
∂U
∂ηa
????qa(y∗
−qa(0,Δηa)) + μ1
≤ −μ1κ1?Δηa?2+ μ1κ2?Δηa?Lq(|ya| + ?η∗
+μ1κ2?Δηa?qM
Where Lqis a Lipschiz constant such that
????
∂U
∂ηa
a,η∗
a)
a?)
?qa(ya,ηa) − qa(0,Δηa)? < Lq(|ya| + ?η∗
and qM is a positive constant that satisfies
a?)
0 < ?qa(y∗
a,η∗
a)? ≤ qM.
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Further the time derivative of Vxf= Δ¯ xT
evaluated as
f¯PfΔ¯ xf can be
˙Vxf= (¯AfΔ¯ xf+¯bfΔv)T¯PfΔ¯ xf
+ Δ¯ xT
= −Δ¯ xT
≤ −λQf?Δ¯ xf?2+ 2Δy∗
≤ −1
−(1
f¯Pf(¯AfΔ¯ xf+¯bfΔv)
f¯QfΔ¯ xf+ 2Δy∗
fΔv
fΔv
2λQf|Δy∗
2λQf− ρy)|Δy∗
f|2−1
2λQf?ηy?2+ 2Δy∗
f|2−1
fΔv
≤
2λQf?ηy?2+1
ρy|Δv|2,
(39)
where λQf
eigenvalues of¯Qf and ρyis any positive constant.
Furthermore for Vkwe have
= λmin[¯Qf] is the minimum value of the
˙Vk=2b0
γp
Δkp(γp¯ e2
a− σpkp) +2b0
γd
Δkd(γd¯ ea˙¯ ea− σdkd)
+2b0
γi
Δki(γi¯ eaw − σiki) + 2b0k∗
+2b0k∗
a−2b0σp
γp
−2b0σd
γd
+2b0k∗
iw(¯ ea− δw)
d¯ ea˙¯ ea
= 2b0Δkp¯ e2
Δkpkp+ 2b0Δkd¯ ea˙¯ ea
Δkdkd+ 2b0Δki¯ eaw −2b0σi
iw¯ ea− 2b0k∗
γi
Δkiki
iδw2+ 2b0k∗
d¯ ea˙¯ ea,
(40)
Thus, the time derivative of V is evaluated by
˙V =˙Ve+˙Vη+ δ1˙Vxf+˙Vk
+(BM+ by)(−ΓS(ω)¯ ea− σˆ W)TΓ−1Δˆ W
+(BM+ by)Δˆ WTΓ−1(−ΓS(ω)¯ ea− σˆ W)
≤ 2La1|¯ ea||Δya| + 2La1|¯ ea|?Δηa? − 2b0k∗
+2bM|¯ ea||v∗| + 2ay|Δy∗
−μ1κ1?Δηa?2+ μ1κ2Lq?Δηa?(|¯ ea+ Δy∗
+?η∗
−δ1
+2?∗?Δˆ W??S(ω)? + ?∗2) + 2(BM+ by)?∗| ¯ ea|
−2b0σp
γp
γp
−2b0σd
γd
γd
−2b0σi
γi
γi
−2(BM+ by)σλΓ?Δˆ W?2
+2(BM+ by)σλM
p¯ e2
a
f||¯ ea| + 2|¯ ea|?cy??ηy?
f+ y∗
a|
f|2
a?) + μ1κ2qM?Δηa? − δ1(1
2λQf?ηy?2+δ1
2λQf− ρy)|Δy∗
ρy(?Δˆ W?2?S(ω)?2
|Δkp|2+2b0σp
|Δkd|2+2b0σd
|Δki|2+2b0σi
|Δkp||k∗
p|
|Δkd||k∗
d|
|Δki||k∗
i| − 2b0δ|k∗
i|w2
Γ?Δˆ W??W∗?,
(41)
where, λΓ = λmin[Γ−1] and λM
minimum and the maximum values of the eigenvalue of Γ−1.
Γ
= λmax[Γ−1] are the
Consequently, we have
˙V ≤ −
?
1κ2
4ρ6
2b0k∗
p− 2La1−L2
?
a1
ρ0
−L2
a1
ρ1
− ρ2−a2
y
ρ3
−?cy?2
ρ4
−μ2
−(μ1κ1− ρ1− ρ5− ρ6− ρ7− ρ8− ρ9)?Δηa?2
−
?δ1
−
γd
?
−δ1
+μ2
4ρ5
4ρ8
+b2
γ2
γ2
+δ2
ρwρ2
y
+(BM+ by)2σ2λM2
ρ14
2L2
q
− ρ13
|¯ ea|2
?
δ(1
2λQf− ρy) − ρ0− ρ3−μ2
?
?2b0σd
2(BM+ by)σλΓ− ρ14− ρw
?
1κ2
q
?η∗
0σ2
p
pρ10|k∗
1S2
M
?∗2+δ2
1κ2
4ρ7
2L2
q
?
|Δy∗
?
?
f|2
−
2λQf− ρ4
?ηy?2−
?
?2b0σp
?2b0σi
γp
− ρ10
|Δkp|2
− ρ11
|Δkd|2−
γi
− ρ12
|Δki|2
−
ρyS2
M
?Δˆ W?2− 2b0δ|k∗
a?2+μ2
p|2+b2
ρy?∗2+(BM+ by)2
i|w2+b2
a|2+μ2
d|2+b2
γ2
M|v∗|2
ρ2
1κ2
2
4ρ9
2L2
1κ2
2L2
q
|y∗
q2
M
0σ2
dρ11|k∗
d
0σ2
iρ12|k∗
i
i|2
1
ρ13
?∗2
Γ
?W∗?2
(42)
with any positive constants ρ0 ∼ ρ14,ρw and SM
max?S(ω)?. Then, considering a sufficiently large ideal
feedback gain k∗such that
=
?
2b0k∗
p− 2La1−L2
−?cy?2
a1
ρ0−L2
a1
ρ1− ρ2−
?
a2
ρ3
y
ρ4
−
μ2
1κ2
4ρ6
2L2
q
− ρ13
> γν> 0
(43)
and setting ρ0, ρ1, ρ3∼ ρ12, ρ12, ρy, ρw, δ1, μ1such as
ρ1 = ρ5= ρ6= ρ7= ρ8= ρ9=μ1κ1
ρy =
ρ10=
ρ14= ρw=
2
δ1 =
7
λQf
4, ρ0= ρ3=δ1
b0σp
γp, ρ11=b0σd
(BM+by)σλΓ
16λQf, ρ4=31
γd, ρ12=b0σi
64δ1λQf
γi
16μ1κ2
λQfκ1, μ1=
2L2
q
(BM+by)σλΓκ1λ2
Qf
128κ2
2L2
qS2
M
,
we have
˙V ≤ −γν|¯ ea|2−1
−b0σp
γp
−1
7μ1κ1?Δηa?2−δ1
|Δkp|2−b0σd
γd
2(BM+ by)σλΓ?Δˆ W?2− 2b0δ|k∗
64λQf?Δ¯ xf?2
|Δki|2
|Δkd|2−b0σi
γi
i|w2+ R (44)
265
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Page 7

with
R =
b2
ρ2|v∗|2+7μ1κ2
+b0σp
+2(BM+by)σλM2
?
Finally we have
M
2
4κ1(L2
p|2+b0σd
q?η∗
d|2+b0σi
?W∗?2
a?2+ L2
q|y∗
a|2+ q2
i|2
M)
γp|k∗
γd|k∗
γi|k∗
Γ
λΓ
+
δ2
ρwρ2
1S2
M
y+
64μ1κ2
λ2
Qfκ1
2L2
q
+BM+by
ρ13
?
?∗2.
(45)
˙V ≤ −αV + R
(46)
where
α =min
?
γν,κ1
7κ3,
λmin[¯Qf]
64λmax[¯Pf],σp,σd,σi,σλmin[Γ−1]
2λmax[Γ−1],2δ
?
(47)
Thus we can conclude that all the signals on the obtained
control system are bounded. Next, we analyze the conver-
gence property of the tracking error e(t). From (46), we
have
lim
t→∞¯ e2
a≤R
α
(48)
This means that for a¯δ such that¯δ2> R/α¯δ, we have
lim
t→∞|¯ ea(t)| <¯δ
(49)
and there exists a time T such that |¯ ea(t)| <¯δ,∀t ≥ T.
Further from (20) and (32), it is easy to show that there
exists a time ∀t ≥ T1 > T and positive constants β1, β2
such that |˙¯ ea| < β1¯δ and |w| < β2¯δ, ∀t ≥ T1. Then, for
kp(t), ki(t) and kd(t), we can confirm from (21) that there
exists a time T2> T1such that
|kp(t)| < γp
|kd(t)| < γdβ1
|ki(t)| < γiβ2
¯δ2
σp
¯δ2
σd
¯δ2
σi
for all t ≥ T2. Thus we can conclude that for a given δ, by
setting design parameters appropriately such that¯δ+β∗¯δ3≤
δ is satisfied for¯δ2> R/α, one can attain that
lim
t→∞|e(t)| ≤ δ.
(50)
Note that we can confirm that such design parameters exist
with certainty from (45).
Finally, we have the following theorem.
Theorem 1: Under Assumptions 1 to 7, all the signals
in the resulting control system with control input (19) are
bounded and the error e(t) converges to any given small
region δ with appropriate setting design parameters. Thus
we can attain
lim
t→∞|y(t) − yr(t)| < δ
IV. CONCLUSIONS
In this paper, we proposed an OFEP-based adaptive PID
control with a PFC and an adaptive RBF NN-based feedfor-
ward input for nonlinear systems. In the proposed method,
the stability of the control system can be guaranteed by
OFEP-based adaptive strategy and the output tracking can be
achieved by a feedforward signal generated by the adaptive
RBF NN. The novel idea presented in this paper is that the
adaptive PID control was designed for nonlinear systems
using OFEP properties and the feedforward signal generated
by the RBF NN was introduced directly to the original
controlled system so as to avoid PFC related problems.
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