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Robust tracking controller design for pneumatic servo system
Int. J. Engng Sci. Vol. 35, No. 10/11, pp. 905-920, 1997
Pergamon
ROBUST
~) 1997 Elsevier Science Limited. All rights reserved Printed in Great Britain 0020-7225/97 $17.00+ 0.00 PII: S0020-7225(97)00037-2
TRACKING
CONTROLLER DESIGN SERVO SYSTEM
FOR
PNEUMATIC
JUNBO SONG* and YOSHIHISA ISHIDA Department of Electronics and Communication, Faculty of Science and Technology, Meiji University, 1-1-1, Higashi-mita, Tama-ku, Kawasaki, 214, Japan (Communicated by S. Minagawa) Abstraet--A new control scheme is introduced for a pneumatic servo system which contain unknown parameters, possible nonlinear uncertainties and additive bounded disturbance. The proposed control scheme is essential to achieve robust stability and good performance for the pneumatic servo system. The design of controller requires only measured input and output signals of system. Compared with other robust control schemes, this scheme not only guarantees a robust stability, but also provides a control signal which is continuous and globally, uniformly bounded. Although the output tracking error cannot be guaranteed to converge to zero, it is able to converge globally and exponentially to a residue set whose radius is an arbitrary positive constant ~. The results of simulation examples and applying to a practical pneumatic servo system prove the effectiveness of this control scheme. ~) 1997 Elsevier Science Ltd.
1. INTRODUCTION Robust feedback controllers are the most common control schemes used for the control of a plant with nonlinear uncertain dynamics. Their popularity is due to the simplicity of their structures as well as the familiarity of designers with feedback control theory. In most robust feedback control schemes presented in Refs [1-5], the authors either use state feedback to place the poles sufficiently far from zero in the left-half-plane, or insert a feedback compensator in the closed loop feedback control system to cancel the effects of system uncertainties and guarantee the stability of the closed loop system. However, for most of these schemes, the transient error response of the closed loop system cannot be specified directly in the design. Therefore, the design of a high-quality tracking controller which can guarantee good system performance is still a challenging topic in the servo control field [6-10]. It is worth mentioning that other nonlinear controllers such as high-gain control and sliding mode control arc: also based on Lyapunov designs and have similar robustness properties except that high-gain control may require excessively large control effect, and sliding mode control is inherently discontinuous [11, 12]. In most nonlinear robust control work, the control objective is state tracking, which may or may not include output tracking as a special case. Asymptotic output tracking results have been reported in the robust control literatures; however, they are usually based on Lyapunov's direct method. In particular, asymptotic stability for output tracking of linear systems is shown in Refs [13, 14], but the stability analysis imposes some restrictions, such that the disturbance must be constant and that the reference signal must be composed of a function that can be generated from a united step function by either integration or differentiation. Our earlier work has proposed a robust sliding mode control scheme. Although it not only guarantees robust stability under bounded additive nonlinear uncertainties or disturbances but also provides a continuous control for pneumatic servo system, full state feedback has to be used in the design of controller [15]. However, in many cases, the measurement of the states of pneumatic servo system is very difficult. In order to overcome this shortage, a new control scheme will be introduced in this paper. * Author to whom all correspondence should be addressed. 905
906
J, SONG and Y. ISHIDA
The aim of this paper is to combine the advantages of both robust control and model reference control so as to propose a new design approach for pneumatic servo control. The resultant nonlinear control guarantees robust stability under bounded additive nonlinear uncertainties or disturbances, and requires only measured input and output signals of pneumatic servo system. Compared with other standard methods less feedback information is used in controller design. Although the output tracking error cannot be guaranteed to converge to zero, it is able to converge globally and exponentially to a residue set whose radius is an arbitrary positive constant e, and then it is continuous and globally, uniformly bounded. This paper is organized as follows: in Section 2 we formulate our problem, whose solution is presented in Section 3. The simulation and experimental results of a practical application experimental results are shown in Sections 4 and 5. Finally, in Section 6 we draw some conclusions of this work.
2. PROBLEM STATEMENT Figure 1 shows the analysis model of a pneumatic cylinder. Considering the pneumatic cylinder in a pneumatic servo system, the dynamics of the pneumatic cylinder can be described by the following second-order nonlinear differential equation: d2y M - - ~ + F(~) + G(y) = u(t) + D(t)
(1)
where y is the displacement of piston, u(t) is the internal pressure of pneumatic cylinder, M is load mass, F(p) is friction dynamics, D(t) is external disturbance and G(y) is the force caused by air compression. G(y) can be considered as a nonlinear spring [16]. Usually, the friction dynamics F(.9) can be considered to satisfy the Dahl model proposed in Ref. [17], and could be described as dy F(2) = 7 / + A(p) dt
(2)
where ~7 is damping coefficient and A(~9) is uncertain friction dynamics. According to the frictional theory h (~9) is less than the static frictional force, and hence it is bounded. Substitute equation (2) into equation (1); the equivalent dynamics of the pneumatic cylinder can be rewritten as d2y dy dt ~ + at, 1 - ~ = kl,[u(t ) + d(y,p,t)]
1
(3)
Pro~ional Control ValveA
i
Control Valve B
i
•
G (y) i i i
:
(~//////j//////////~,
iii
u(O
Pneumatic Cylinder Fig. 1. Analysis model of the pneumatic cylinder.
Robust tracking controller design
907
d(y,p,t) = O(t) + A (y) - G(y).
(4)
where kp = 1/M, apl = rl/M and
We assume that the parameters M and r / a r e unknown, and d(y,p,t) denotes any disturbance or uncertainty that can be bounded by a nonlinear function defined explicitly by y,p and time t. Furthermore, we assume that only the control u(t) and the output y(t) are measurable. When the disturbance d(y,p,t) is absent, we have the following transfer function relationship: r'(s)
U(s)
kp ,t=,, =
- Cp(s)
(5)
s(s + apl)
where Gp(s) can be considered as the linear portion of the plant. Following the definitions of ap~ and kp, we have apt >0 and kp>O. The reference model for the plant to be followed can be represented as the following transfer function:
k.,
Gn,(s) = s2 + a.,ls + an,2
(6)
where a.,~, am2 and k., are positive constants, which are chosen such that the reference equation (6) is stable. It is obvious that G.,(s) is not a strictly positive real transfer function. For the use of controller design in the following, we assume that there exists a first-order monic, Hurwitz polynomial F(s) in s plane, it is chosen such that F(s)G.,(s) is a strictly positive real transfer function. We define that F (s) = s + a
(7)
and = r (s)C.,(s).
(8)
If the input of the reference model is r(t), we define that the output of the reference model is y.,(t), and then assume that r(t) and ym(t) are measurable. Equation (3) is quite general for a pneumatic cylinder in pneumatic servo systems, as it subsumes both the linear and nonlinear modes of the pneumatic cylinder. With respect to the model of the controlled plant defined by equations (3) and (5), we make the following assumptions: (1) The sign of the parameter kp is assumed without any loss of generality to be positive. That is kpin <~ kp < kpm a x in any case for the pneumatic cylinder, where kpmin > 0 and kpm a x > 0 are known constants. (2) The parameter of the controlled plant, apl is an unknown constant and there exists known positive constants ap~m and apf Xsuch that rain apl < apl
~
max apl •
(3) The frictional uncertainty term A(~,) is bounded and there exists a known positive constant fa such that
(4) The lumped disturbance and uncertainty term d(y,p,t) is bounded by a known continuous, nonnegative function p(y,t) such as
Id(y,p,t)] <- p(y,t) for (y,t) E R × R
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J. SONG and Y. ISHIDA
Pneumatic cylinder
Equivalent system
I D (t)
[a(y,~,t)
u(t)+ ~ ~
y(t)
L_] F (:) + G(y) Fig. 2. Pneumatic cylinder and its equivalent system.
where p(y,t) is able to be, in general, a time-varying, high-order nonlinear function that is continuous and uniformly bounded with respect to time t and output y. The uncertainty d(y,p,t) is not necessarily continuous. REMARK 2.1. Usually, the damping coefficient r/ of the pneumatic cylinder is constant. Furthermore, the load mass of the pneumatic cylinder is within the design range. According to the definition of parameters apt and kp and assumptions (1) and (2) are satisfied by a practical pneumatic cylinder. REMARK 2.2. Following from the model of friction dynamics described in Ref. [17], we have that the frictional uncertainty term A (p) is always less than the static frictional force. Hence, A (p) is bounded. REMARK 2.3. If assumption (3) is reasonable, assumption (4) can easily be satisfied [18, 19]. The objective of this work is to combine the advantages of both robust control and model reference control so as to propose a new design approach for a pneumatic servo system if the controlled plant can be described by equation (3) and satisfies assumptions (1)-(4). The resulting method requires only the uncertain bounds or bound functions based on the structure of pneumatic cylinder and measured input and output signals (see Fig. 2). It guarantees robust stability under bounded additive nonlinear uncertainties or disturbances, and at the same time is continuous and globally, uniformly bounded. For the convenience of the controller design in the following sections, we make the following definitions. DEFiNrnor~ 2.1. Let f(.) be a continuous function and F(.) be a functional vector. Then O[f(.)] and ®[F(.)] are known, continuous, nonnegative function and functional vector, respectively, which satisfies [f(.)l -< O[f(.)],
II F(.) II - O[F(.)].
DEFINITION 2.2. In the following of this paper, we make 3-~(.) and @ denote inverse Laplace transform and convolution operation, respectively. 3. THE DESIGN OF ROBUST T R A C K I N G C O N T R O L L E R If the lumped disturbance and uncertainty term d(y,y,t) = 0 for all y,p and t, considering the controlled plant and the reference model defined in equations (5) and (6), perfect tracking can be achieved using the control system shown in Fig. 3. In this figure, Wl(t) and w2(t) are auxiliary state signals defined by wl(t) = aw, + u
(9)
aw2 + y
(10)
w2(t) =
where the constant a has the same definition as equation (7). The control law is given by
u(t) = 0Tw(t) where 0(t) = [ 00, 0,, 02, 03] T, w(t) =
[r(t),w, (t),we(t),y(t)] T.
(11)
909
Robust tracking controller design
r(t)
[ I t
F--1~53 *
-q
L+___~_j
÷I
e(t)
[
Fig. 3. Perfect tracking control scheme of a linear model of pneumatic servo system with perfect knowledge.
Since the affection of d(y,y,t) is neglected, the overall transfer function from r(t) to y(t) of the control system shown in Fig. 3 can be described as
V(s) _ Ookpr(s) R(s) s(s + ap,)(r(s) - 0 , ) - k,(02r(,)+ 03)
(12)
If perfect knowledge of the controlled plant is available, using this transfer function, we can solve the perfect tracking problem. To do so, we simply need to guarantee that
k.,s(s + apl)(F(s) - 0,) - k.,kp( OzF(s) + 03) = Ookt,F(s)(s z + a.,,s + a.a). After solving this equation, the desired parameter vector 0* can be given by
km/kp
I:il ,-=/02j
aml -- apl
1
( - ap, ot z - a2,ot + ap,a.,,ot + a.,,ot 2)
(13)
--
0;
1
2 ( a p l o t 4- apl -- a p l a m l -
am2 -
amlot)
In the case of 0--: 0", the control u(t)= (0") T w(t) can guarantee y(t)---~y.,(t) as t---~ for any r(t) under the condition that the lumped disturbance and uncertainty term d(y,p,t)= O. The question now i,; what control should be used if the controlled plant is unknown and d(y,p,t) ~ 0. The answer of this question is the subject of this section. When the parameters of the controlled plant are unknown, and d(y,p,t) ~ 0, control equation (11) can be rewritten as
u(t) = 0Vw(t) = (0*)Vw(t) - 0Tw(t)
(14)
where 0 is a constant vector containing an arbitrary estimate of 0* and 0 = 0 * - 0 is a parameter error vector. Considering the affection of the lumped disturbance and uncertainty term d(y,y,t), since the desired parameter vector 0* satisfies equation (13), under control equation (14), the controlled plant output dynamics can be represented by
{
1
y(t) = ~ -I(G.,(s)) (~ r(t) + 0---~o
i(1___ r(s)
It is obvious that the key to solve the tracking problem for an unknown plant is to design an
910
J. SONG and Y. ISHIDA
additional control to overcome the output caused by parameter error and d(y,p,t). To do that, we propose the following control such as [20]
u(t) = 0Tw(t) + ur(t).
(15)
Input this control to the controlled plant described in equation (3) and consider 00 = k.,/kp, the plant output becomes kp
01
where ur(t) is a portion of control which must be designed to insure the robustness of the overall control. We now define the output tracking error e(t) to be
e(t) = ym(t) - y(t)
(17)
where ym(t)= 3-1(Gin(S))@r(t). Then the dynamics of the output tracking error is described by
e(t) = ,s- l((~,,(s)) (~) kp
- 3-
1(1[ ~
¢3Tfv ( t ) - .~
01 ])
1-
F(s----)-
@ d(y,p,t)
® Ur(t)
]
(18)
where *(t)
1 ) ®w(t)
= .
(19)
\ r(s)
According to the definition of equation (3), in which the controlled plant is unknown, the design of controller will be carried out under the condition of 0 = 0 without any loss of generality. Then the control law equation (15) becomes
u(t) = ur(t)
(20)
Suppose 0 = 0 is an arbitrary estimate of 0", in this case, 0 can be obtained by 0= 0". It follows from equation (13) and assumptions (1) and (2) that the bound vector of 0 can be defined by km/kp in max
aml q- apl
[03]3
k l a n ( a~jaXa + apl max2 if- apl maxaml -t- am2 + a , . i o 0
031J
p
Then, under the condition of O = 0, the dynamics of the output tracking error can be rewritten as
e(t) = 3-'(G,,,(s)) (~) ~
0Tff(t) -- 3 -
® u(t) -- 3 - '
® d(y,j',t)
. (22)
Robust trackingcontroller design
911
THEOREM 3.1. Consider the system given by equation (3) which satisfies assumptions (1)-(4). If the control is chosen to be
u(t) = aar(t) + sign [e(t)].O[ h ,(t)]
(23)
e(t) to2(t) at(t) = 2(le(t)to(t)l + tr)
(24)
where
/1 to(t)= II 0o(l) II 2+ II g,(t)II 2+2~-1/- ~
\ )®p(y,t)
(25)
and 6 > 0 is a constant, then the controlled plant output tracking error (t) is globally, uniformly ultimately bounded. That is, the output tracking error becomes no larger than a constant which can be made arbitrarily small by choosing the design parameters tr small enough. Furthermore, the robust control u(t) is continuous and globally, uniformly bounded. (See Appendix A about the calculation of O[t~r(t)].) PROOF. Select a minimal realization of the transfer function (3,,(s) as a triple {A,B,C}. Then, a realization of the tracking error dynamics equation (22) can be written as
2,.(t)=Ax~(t)+ B 0Txk(t)--,s
®u(t) -- 3 - '
®d(y,p,t)]
e(t) = Cx~(t).
I~., '
(26)
Since G,,(s) is SPR, equation (26) is controllable and observable. Furthermore, it follows the Kalman-Yakubovich lemma in [21], where, given a symmetric positive defined matrix Q, there exists a symmetric positive define matrix P such that
ATp + PA = - Q, PB = C T.
(27)
Choose a Lyapunov function as
V=
knl
kp
x[Px~.
Differentiating V with respect to time, we have
f/= k"'xT(ATp+PA)x"+2xSPB[
_.~-,(_1
ko, ) ke xTeQxe+2x~PB [ 0Te(t) -- 3 - ' ( \_ _F(s) _1L (~ u(t)
_.~-,(_L
k.,
\ r(s) )®d(Y4',t) ] -
.,
_
-
.~-,(_L
I
k,,, x~Qx~ kp
,+
k,,, x~Qx~ ÷ le(t)I[ II Oo(t)II
kp
[
T
kp x, Qxe + 2e(t) 0Tffv(t)
r(,) )®d~y,,,,) ] = - --
+2e t [ <
- ~ - ' ( F - ~ ) ® u(t)
z + II ~(t) II ~ + 2 . ,
®
p~,t)
]
912
J. SONG and Y. ISHIDA
1
1
-2e(t)3-(--~)(~)u(t)=-
-kp - Xe Qx~ + ]e(t)]to(t) - 2e(t) ~ - '
@ u(t)
(28)
Since it follows the definitions of control law [equation (23)] and 0[.], we have
u(t) = o~ar(t)= sign [e(/)].O[ t~ ,(/)] = sign [e(t)][ la&(t) l + O[ ~ .(/)]}
(29)
Substitute equation (29) into equation (28), and equation (28) becomes
9"<- - - x ~ Q x e +
kp
<_
-
_
le(t)la,(t) -
2[e(t)l
~-
®lota~(t)l + O[ a .(t)]
--~
k., xXeaXe + [e(t)lto(t)- 2 [ e ( / ) l { ~ - l ( - ~ s ) ) ( ~ ) [ o t l G ( t ) l + l ~ r ( t ) l ] } _
kp
<_
--
k., xXQx~ + le(t)la,(t) - 2le(t)lar(t) =
m
I%
=e(t)to(t)
-
e2(t)to2(t) <[e(t)to(t)[ + o" -
kit7
k,,
xX~Qx~
k., x, Qxe = le(t)~o(t)l kp le(t)to(t)[ + tr
~r
(30)
Since both Q and P are the symmetric positive define matrixes, select a constant 7 such that
T(xTpx~) <_xX~Qxe. Then, the differentiation of V with respect to time satisfies
T k., P'<--T(x~Px~)7-- + or= T V + tr k,, Therefore, V converges to a residue set exponentially whose radius is a constant e = tr/y and so does the state xe(t). Consequently, the output tracking error e(t) also converges exponentially to this residue set. The convergence rate is decided by 7. This means that we do have some control over the transient response of the system. Because the output tracking error e(t) is continuous and bounded we can conclude that the output y(t) is also continuous and bounded. Furthermore, we examine the auxiliary control G(t), and we have
e( t) to 2(0 e( t) to ( t) 1 la,(t)l = 2(le(t)to(t)l + ~r) [ -- 2(le(t)~o(t)l + o9 Ito(t) < ~ t o
(31)
As with the above analyses, since the output tracking error e(t) is continuous and bounded, we have to(t) is continuous and bounded, and then the auxiliary control a,(t) is also continuous and bounded. In addition, from the definition of O[Ur(t)] (See Appendix A), OUr(t)] can be ensured to be continuous and bounded. Hence, control u(t) is continuous and bounded. REMARK 3.1. From the design of proposed controller, we have that only the input output and some bounds based on the pneumatic cylinder and bound function are used in the design controller. Furthermore, the robust control u(t) is continuous and globally, uniformly bounded. All of these make the proposed control scheme viable for the practical control problems. REMARK 3.2. We note that the tracking error can be made smaller by choosing smaller value of the design parameters tr. If tr is too small, however, the output of the robust controller may change rapidly and has a large magnitude. Since the bounds resulting from equation (21) and bound function p(y,t) are usually conservative, the design parameter tr should not be set too
Robust tracking controller design
913
small during the transient phase. Instead it should be decreased gradually if the output tracking error is larger than required.
4. S I M U L A T I O N
EXAMPLES
In this section, we provide two examples to show the effectiveness of p r o p o s e d control scheme. The first example is a time-invariant nonlinear s y s t e m - - i n this case, the linear portion of the controlled plant is conducted from the pneumatic cylinder with a load mass of M = 30 kg. The second is a controlled plant with a time-variant nonlinear portion. They permit us to illustrate both asymptotic and uniformly ultimately b o u n d e d stability.
(a) The output tracking and the response of plant with P controller (b) The tracking error e(t) (c) The control u(t) (a) 2
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Times
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Fig. 4. Simulation results of the proposed controller ~r a time-invariant nonlinear system
914
J. SONG and Y. ISHIDA
4.1 An example of time-invariant nonlinear system Consider a plant whose linear portion is conducted from a pneumatic cylinder with load mass of M = 30 kg. This can be described by 6.64
Gp(s)],tty,,}=,, - s(s + 0.35) and then the nonlinear portion of the controlled plant is chosen by G(y) =
sin (y) 1 + y 2 , a ( p ) = 1.511 - e x p ( I P l ) ] .
The bounds of controlled plant are assumed as K p in =
1.99, kp "x = 10.0, apl min = 0.11, aplax = 1.1, p(y,t) = 2.0
Furthermore, select an external disturbance as
O(t) = sin (3/) A reference model tracked by the controlled plant is set by Gin(s) =
100 (s + 10) 2
The reference input is a sine function, that is
r(t) = sin (0.5 ¢rt). Following from the above definitions of the controlled plant and the reference model, the proposed controller can be designed as follows: F(s) = s + 1.0,a = 1.0,or = 0.2, ~T = [50.0,21.1,21.3,72.15]. Figure 4 shows the simulation results of the above-mentioned controlled plant with the proposed controller. Figure 4(a) shows the output tracking and the output response yp(t) of plant with controller P. From the illustration of yp(t), we can know that there is a strong nonlinear uncertainty contained in the controlled plant, Fig. 4(b) is the tracking error and then the control u(t) is illustrated in Fig. 5(c). Following these results, we can find that the tracking error is continuous and converges to a small set. In addition, the control u(t) is also continuous and bounded. Obviously, the proposed control is effective for the above-mentioned controlled plant.
4.2 An example of time-variant nonlinear system In the following the simulation will be done for a time-variant nonlinear plant. This is given by
[,,] [ o.o. *2
=
-Isin(t)l,
,.o irx, Ol - 0.35J[,xdt)J +
[o.oo]U(y,t),y(t)=xt(t) 6.64
where U(y,t)= d(t)+f(y,t)+ u(t), and u(t) is the control resulting from the proposed controller. Then the nonlinear uncertain term is chosen by
f(y,t) -
sin (t) sin (y) 1 + y2 + sin (3t).
A reference model and the parameters of controller selected the same as Section 4.1. Simulation results are shown in Fig. 5. In Fig. 5(a), yl,(t) is the output response of plant with the P controller. Following from the illustration in Fig. 5, it is clear that the proposed control is effective for some time-variant nonlinear systems.
Robust
tracking controller design
915
5. A P P L I C A T I O N TO A P R A C T I C A L P N E U M A T I C S E R V O S Y S T E M
Though the control performance is satisfactory in simulation examples using the proposed control scheme, it may not be able to satisfy a practical pneumatic servo system. Hence a practical application is studied here.
5.1 Configurath~n of pneumatic servo system Figure 6 shows the configuration of a pneumatic position servo system. It is primarily composed of a pneumatic cylinder, a 32 bit personal computer with an 80486 CPU and an associated arithmetic coprocessor, two 12-bit D/A converters, two servo power amplifiers of the voltage-current transformation type and two electro-pneumatic proportional valves. The pneumatic cylinder (CAIBQ63-300: SMC Corporation) is a low friction type whose minimum (a) The output tracking and the response ( b ) T h e t r a c k i n g e r r o r e(t) (c) T h e c o n t r o l u(t)
of plant with P controller
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. . . . 0
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15
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Times (sec) Fig. 5. S i m u l a t i o n
results of the proposed
controller
for a time-variant
nonlinear
system
916
J. SONG and Y. ISHIDA
Pneumatic Source Air Dryer D/ACony. Mist Separator
(>
CounterCircuit
I ServoPowerAmp.
l~t~tor Circuit
I PC-9821
,
Regulator
AirTank Magnescal© 5~msor
Regulator ",,_j
Fig. 6. Configuration of the pneumatic servo system.
pressure is 0.01 MPa, inside diameter is 63 mm, and the stroke is 300 mm. A load mass of 10 kg is always connected to the tip of a position rod. The electro-pneumatic proportional valve is a proportional control valve (VEP3120-1: SMC Corporation) with pressure being of 3 port. The position measurement is carried out by magnetic sensor whose resolving power is 10/~m (SR10100A: Sony Corporation). 5.2
Applications o f the proposed control scheme
In this section the test results of the pneumatic servo system with the proposed controller will be illustrated. We suppose that the pneumatic cylinder in the system mentioned satisfies assumptions (1)-(4) and its dynamic motion equation can be described by equation (3). It follows the controller design described in Section 3 and the structure of the pneumatic cylinder; the controller is designed as follows: 1. The bounds of the controlled plant are chosen based on the design parameters of the pneumatic cylinder as max _- 1.1, p(y,t) = 2.0. k~ in = 1.99, kp ax = 10.0, apm~~n = 0.11, ap,
2. The external disturbance loaded in the input of the servo power amplifiers is set by O(t) =
0
0 -< t <-- 13s
0.976V
t-> 13s
3. Reference model tracked by controlled plant is set by 100 C.,(s)
4.
-
(S + 10) 2
The reference input r(t) is chosen by
-
r(t) =
50
0s - t < 5s
20
5s-<
t <
10s
20 100
10s -< t < 15s 15s -< t < 20s
20
20s -< t < 25s
0
25s <- t < 30s
Robust tracking controller design
917
5. The parameters used in controller is designed by F(s) = s + 1.0,c~ = 1.0,o- = 2, ~T = [50.0,21.1,21.3,72.15] 6. The sampling period is 0.002 s Figure 7 shows ~;he experimental results of the pneumatic servo system with the proposed controller in the case of loading mass M = 30 kg, Fig. 7(a) is the output tracking, Fig. 7(b) is the tracking error and Fig. 7(c) is the control voltage loaded in the input of servo amplifiers. The experimental results suggest that the proposed control scheme is not only effective in the case of the external disturbance D(t)= 0, but also effective in the case of D(t)> 0. Furthermore, a nonoscillation output response can be obtained. The experimental results also illustrate that the tracking error is 'bounded and then control u(t) is bounded and continuous. Figure 8 shows the
(a) The output tracking and the reference input r(t) (b) The o u t p u t tracking error of the pneumatic servo system (c) The input voltage of the servo p o w e r amplifier and the external disturbance
D(t)
(a) 120
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,
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. . . .
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==
y(t)
g 1 ym,, o
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Time (Sec)
(b) 10
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....
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,
I,,
,,
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,, -
....
2 D
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i
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i
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i
i
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I
10
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I''
15
I,,
, ,I
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Controlu(t)
-
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External Disturbance D(t}
I
20
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I
25
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'
'
30
Time (Sec)
Fig. 7. Experimental results of ' .
"matic cylinder with the proposed controller in the case of M = 30 kg
918
J. SONG and Y. ISHIDA
results in the case of load mass M = 60 kg using the same controller in the case of M = 30 kg, Similar results are obtained. Since the output response does not change along with the p a r a m e t e r ' s variant, we can consider that the robustness of the proposed control scheme is good for pneumatic servo systems.
6. C O N C L U S I O N S
In this paper, a robust tracking control scheme has been proposed for pneumatic servo systems. The main contribution of this paper is to find a robust controller (Theorem 3.1) for pneumatic servo systems, which can guarantee that the tracking error converges to an arbitrarily (a) The output tracking and the reference input r(t) (b) The output tracking error of the pneumatic servo system (c) The input voltage of the servo power amplifier and the external disturbance D(t)
(a) 120
....
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,
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,
,
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.
,
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r(t)
- -
y(t)
Z v
,
..... ....
,
ym(t)
4O I---
"5 o.
0
-40
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'
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15
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20
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25
Time (Sec)
(b) ~ , , t l , , , , l , , , , l
I0
E
5
w
0
~
-5
. . . .
I , , , , I , , , ,
I--
-10
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'
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v
r~
0 -2 .
-4
.
.
.
.
.
.
.
.
' ' ' '
.
.......... i-i .......... i ................. .............
.
I'
5
'
''
I ' '
10
''
I ' ' ' '
15
I ' '
20
'
'
I
25
'
'
°
'
30
Time (Sec)
Fig. 8. Experimental results of the pneumatic cylinder with the proposed controller in the case of M=60 kg
Robust tracking controller design
919
small residue set: as time t - - - ~ . T h e key of c o n t r o l l e r design is to find a c o n t i n u o u s b o u n d f u n c t i o n only in t e r m s of system output. This can be easily f o u n d for a practical p n e u m a t i c servo system o p e r a t i n g in a certain e n v i r o n m e n t . I n addition, only the i n p u t a n d o u t p u t signals are used in the design of the controller. All of these bestow u p o n the p r o p o s e d c o n t r o l scheme good practical applicability. F u r t h e r m o r e , from the c o n t r o l l e r design process described in Section 3, we find that the c o n t r o l structure is very simple a n d the o p e r a t i o n s of a r i t h m e t i c are n o t difficult to complete. H o w e v e r , in the case that a c o n t i n u o u s b o u n d f u n c t i o n only in terms of system o u t p u t does n o t exist, it is still very difficult to c o m p l e t e r o b u s t tracking control for p n e u m a t i c servo system using the p r o p o s e d c o n t r o l scheme. This will b e c o m e a topic of work in future papers. Acknowledgements--The authors are grateful to Dr Zongli LIN, assistant professor of State University of New York. for providing some valuable references. The authors also thank the SMC Co. Ltd. for supporting this work.
REFERENCES 1. 2. 3. 4. 5. 6. 7.
Lu, W. M., System Control Letters, 1995, 25, 13 Verma, M. S., International Journal of Control Letters, 1995, 48, 897 Jabbari, E, IEEE Transactions of Automation Control, 1990, 35, 954 Schmite Dorf W. E. and Barmach, B. R., Automatics, 1986, 22, 353 Qu, Z., System Control Letters, 1992, 18, 3 301 Tanaka, K., Simizu,A. and Sakata, K., Journal of the Society of Instrument and Control Engineers, 1994, 30, 1069 Spong, M. W., in New Trends and Applications of Distributed Parameter Control Systems (Lecture Notes in Pure and Applied Mathematics), G. Chen, E. B. Lee, W. Littman and L. Markus, Eds. Marcel Dekker, New York, 1990; System Control Letters, 1987, 1119,310. 8. Khorasani, K., IF,EE Transactions on Automation Control, 1990, 35, 10 1145 9. Ghorbel, E, Hung, J. Y. and Spong, M. W., in Proc. 1989 IEEE Int. Conf. Robotics and Automation, Scottsdale, AZ, 1989, p. 15. 10. Grimm, W. M., buernational Journal of Adaptive Control and Signal Processes, 1992, 4, 501 11. Khalil, H. K. and Saberi, A., IEEE Transactions on Automation Control, 1987, 32, 1031 12. Venkataraman, S. T. and Gulati, S., in Proc. Amer. Conf., 1992, p. 891. 13. Schmite Dorf, W. E. and Barmach, B. R., Automatics, 1986, 22, 353 14. Schmite Dorf, W. E. and Barmach, B. R., Journal of Dynamic Systems, Measurement and Control, 1987, 109, 186 15. Song, J. and Ishida, Y., International Journal of Engineering Science, 1997, (in press). 16. Noritsugu, T. and Fukuzono, K., Journal of the Japan Hydraulics and Pneumatics Society, 1994, 25, 6 733 17. Canudas de Wit, C., Olsson, H. and Lischinsky,P., IEEE Transactions on Automation Control, 1995, 40, 3 419 18. Endo, K., Ishida, Y. and Honda, T., Transactions of the lEE of Japan, 1993, 113, 6 408 19. Noritsugu, T. and Takaiwa, M., Journal of the Society oflnstrument and Control Engineers, 1994, 30, 6 676 20. Yeung, K. S. and Chen, Y. P., IEEE Transactions on Automation Control, 1989, 33, 2 200 21. Khalil, H. K., Nonlinear Systems. Macmillan, New York, 1991. (Received 29 January 1997)
APPENDIX According to the definition of at(t) in equation (24), we have h,(t) = c~(t) . de(t) + ~i(t_~)_) dto(t) ae(t) dt cko(t) dt
(AI)
in which the bound of every term can be obtained easily from the definitions in Section 3. They can be described as follows: ~a(t)
1 -~: -2-
2[e(t)lto(t)+ o" ;S 2
le(t)lto(t)+ o. to2(t) ~ _[- ~7(t) ] (I e(t)Ito(t) + o') 2 to2 t): le(t) [to(t) + o"
~7(t) 1 3eZ(t)toz(t)+ 2le(t)lto(t)o. <- 1.5 le(t)lw(t) ~ [ ~7(t)-] ,~(t) <- "-2 (I e(t)]to(t) + ~)z left) Ito(t) + o. '~ t~[--~-~J.
(A3)
920
J. SONG and Y, ISHIDA
From the definition of tracking error dynamics in equation (18), we have
de(t) _ ~_ '(st~,,,(s)) (~)
d,
~
~'r~'(t) - ~ -
(~) u(t)
-f~
,
1
(y,~,t)l
-
From the above expression the bound function of ~(t) can be obtained by d e(t) s(s + a) d t < kjmax.~ - 1 "s2 + a,,,~s + a,,,2
[to(t) +
(A4)
la,(t) l l t ~ [ ~ J .
Identical to the above calculation, the bound function of tb(t) can be calculated by
(AS)
I~o(t)l~_2icv~(t)cv(t)l+23-t( s~ ) @p(Y,t)~-O[i°(t)] where qv(t) = - aVe(t) . w ( t ) .
From equation (A1), we have an(t) J/~"(t) J -< de(t)
~a(,) alto(t) <_el ~(')1.or de(01 + o[ ~(f}l.o[ dto.~Ol-~etaX')]-
de(t) 4- , 0to(t) dt
dt
L 6,e(t)J
[
dt J
L ato(t)J
[_ tit j
Thus, we obtain the bound function of ~r(l) only using the input and output information. From the analysis in Section 3 we can conduct that the bound function of h,(t) is continuous and bounded,
Pergamon
ROBUST
~) 1997 Elsevier Science Limited. All rights reserved Printed in Great Britain 0020-7225/97 $17.00+ 0.00 PII: S0020-7225(97)00037-2
TRACKING
CONTROLLER DESIGN SERVO SYSTEM
FOR
PNEUMATIC
JUNBO SONG* and YOSHIHISA ISHIDA Department of Electronics and Communication, Faculty of Science and Technology, Meiji University, 1-1-1, Higashi-mita, Tama-ku, Kawasaki, 214, Japan (Communicated by S. Minagawa) Abstraet--A new control scheme is introduced for a pneumatic servo system which contain unknown parameters, possible nonlinear uncertainties and additive bounded disturbance. The proposed control scheme is essential to achieve robust stability and good performance for the pneumatic servo system. The design of controller requires only measured input and output signals of system. Compared with other robust control schemes, this scheme not only guarantees a robust stability, but also provides a control signal which is continuous and globally, uniformly bounded. Although the output tracking error cannot be guaranteed to converge to zero, it is able to converge globally and exponentially to a residue set whose radius is an arbitrary positive constant ~. The results of simulation examples and applying to a practical pneumatic servo system prove the effectiveness of this control scheme. ~) 1997 Elsevier Science Ltd.
1. INTRODUCTION Robust feedback controllers are the most common control schemes used for the control of a plant with nonlinear uncertain dynamics. Their popularity is due to the simplicity of their structures as well as the familiarity of designers with feedback control theory. In most robust feedback control schemes presented in Refs [1-5], the authors either use state feedback to place the poles sufficiently far from zero in the left-half-plane, or insert a feedback compensator in the closed loop feedback control system to cancel the effects of system uncertainties and guarantee the stability of the closed loop system. However, for most of these schemes, the transient error response of the closed loop system cannot be specified directly in the design. Therefore, the design of a high-quality tracking controller which can guarantee good system performance is still a challenging topic in the servo control field [6-10]. It is worth mentioning that other nonlinear controllers such as high-gain control and sliding mode control arc: also based on Lyapunov designs and have similar robustness properties except that high-gain control may require excessively large control effect, and sliding mode control is inherently discontinuous [11, 12]. In most nonlinear robust control work, the control objective is state tracking, which may or may not include output tracking as a special case. Asymptotic output tracking results have been reported in the robust control literatures; however, they are usually based on Lyapunov's direct method. In particular, asymptotic stability for output tracking of linear systems is shown in Refs [13, 14], but the stability analysis imposes some restrictions, such that the disturbance must be constant and that the reference signal must be composed of a function that can be generated from a united step function by either integration or differentiation. Our earlier work has proposed a robust sliding mode control scheme. Although it not only guarantees robust stability under bounded additive nonlinear uncertainties or disturbances but also provides a continuous control for pneumatic servo system, full state feedback has to be used in the design of controller [15]. However, in many cases, the measurement of the states of pneumatic servo system is very difficult. In order to overcome this shortage, a new control scheme will be introduced in this paper. * Author to whom all correspondence should be addressed. 905
906
J, SONG and Y. ISHIDA
The aim of this paper is to combine the advantages of both robust control and model reference control so as to propose a new design approach for pneumatic servo control. The resultant nonlinear control guarantees robust stability under bounded additive nonlinear uncertainties or disturbances, and requires only measured input and output signals of pneumatic servo system. Compared with other standard methods less feedback information is used in controller design. Although the output tracking error cannot be guaranteed to converge to zero, it is able to converge globally and exponentially to a residue set whose radius is an arbitrary positive constant e, and then it is continuous and globally, uniformly bounded. This paper is organized as follows: in Section 2 we formulate our problem, whose solution is presented in Section 3. The simulation and experimental results of a practical application experimental results are shown in Sections 4 and 5. Finally, in Section 6 we draw some conclusions of this work.
2. PROBLEM STATEMENT Figure 1 shows the analysis model of a pneumatic cylinder. Considering the pneumatic cylinder in a pneumatic servo system, the dynamics of the pneumatic cylinder can be described by the following second-order nonlinear differential equation: d2y M - - ~ + F(~) + G(y) = u(t) + D(t)
(1)
where y is the displacement of piston, u(t) is the internal pressure of pneumatic cylinder, M is load mass, F(p) is friction dynamics, D(t) is external disturbance and G(y) is the force caused by air compression. G(y) can be considered as a nonlinear spring [16]. Usually, the friction dynamics F(.9) can be considered to satisfy the Dahl model proposed in Ref. [17], and could be described as dy F(2) = 7 / + A(p) dt
(2)
where ~7 is damping coefficient and A(~9) is uncertain friction dynamics. According to the frictional theory h (~9) is less than the static frictional force, and hence it is bounded. Substitute equation (2) into equation (1); the equivalent dynamics of the pneumatic cylinder can be rewritten as d2y dy dt ~ + at, 1 - ~ = kl,[u(t ) + d(y,p,t)]
1
(3)
Pro~ional Control ValveA
i
Control Valve B
i
•
G (y) i i i
:
(~//////j//////////~,
iii
u(O
Pneumatic Cylinder Fig. 1. Analysis model of the pneumatic cylinder.
Robust tracking controller design
907
d(y,p,t) = O(t) + A (y) - G(y).
(4)
where kp = 1/M, apl = rl/M and
We assume that the parameters M and r / a r e unknown, and d(y,p,t) denotes any disturbance or uncertainty that can be bounded by a nonlinear function defined explicitly by y,p and time t. Furthermore, we assume that only the control u(t) and the output y(t) are measurable. When the disturbance d(y,p,t) is absent, we have the following transfer function relationship: r'(s)
U(s)
kp ,t=,, =
- Cp(s)
(5)
s(s + apl)
where Gp(s) can be considered as the linear portion of the plant. Following the definitions of ap~ and kp, we have apt >0 and kp>O. The reference model for the plant to be followed can be represented as the following transfer function:
k.,
Gn,(s) = s2 + a.,ls + an,2
(6)
where a.,~, am2 and k., are positive constants, which are chosen such that the reference equation (6) is stable. It is obvious that G.,(s) is not a strictly positive real transfer function. For the use of controller design in the following, we assume that there exists a first-order monic, Hurwitz polynomial F(s) in s plane, it is chosen such that F(s)G.,(s) is a strictly positive real transfer function. We define that F (s) = s + a
(7)
and = r (s)C.,(s).
(8)
If the input of the reference model is r(t), we define that the output of the reference model is y.,(t), and then assume that r(t) and ym(t) are measurable. Equation (3) is quite general for a pneumatic cylinder in pneumatic servo systems, as it subsumes both the linear and nonlinear modes of the pneumatic cylinder. With respect to the model of the controlled plant defined by equations (3) and (5), we make the following assumptions: (1) The sign of the parameter kp is assumed without any loss of generality to be positive. That is kpin <~ kp < kpm a x in any case for the pneumatic cylinder, where kpmin > 0 and kpm a x > 0 are known constants. (2) The parameter of the controlled plant, apl is an unknown constant and there exists known positive constants ap~m and apf Xsuch that rain apl < apl
~
max apl •
(3) The frictional uncertainty term A(~,) is bounded and there exists a known positive constant fa such that
(4) The lumped disturbance and uncertainty term d(y,p,t) is bounded by a known continuous, nonnegative function p(y,t) such as
Id(y,p,t)] <- p(y,t) for (y,t) E R × R
908
J. SONG and Y. ISHIDA
Pneumatic cylinder
Equivalent system
I D (t)
[a(y,~,t)
u(t)+ ~ ~
y(t)
L_] F (:) + G(y) Fig. 2. Pneumatic cylinder and its equivalent system.
where p(y,t) is able to be, in general, a time-varying, high-order nonlinear function that is continuous and uniformly bounded with respect to time t and output y. The uncertainty d(y,p,t) is not necessarily continuous. REMARK 2.1. Usually, the damping coefficient r/ of the pneumatic cylinder is constant. Furthermore, the load mass of the pneumatic cylinder is within the design range. According to the definition of parameters apt and kp and assumptions (1) and (2) are satisfied by a practical pneumatic cylinder. REMARK 2.2. Following from the model of friction dynamics described in Ref. [17], we have that the frictional uncertainty term A (p) is always less than the static frictional force. Hence, A (p) is bounded. REMARK 2.3. If assumption (3) is reasonable, assumption (4) can easily be satisfied [18, 19]. The objective of this work is to combine the advantages of both robust control and model reference control so as to propose a new design approach for a pneumatic servo system if the controlled plant can be described by equation (3) and satisfies assumptions (1)-(4). The resulting method requires only the uncertain bounds or bound functions based on the structure of pneumatic cylinder and measured input and output signals (see Fig. 2). It guarantees robust stability under bounded additive nonlinear uncertainties or disturbances, and at the same time is continuous and globally, uniformly bounded. For the convenience of the controller design in the following sections, we make the following definitions. DEFiNrnor~ 2.1. Let f(.) be a continuous function and F(.) be a functional vector. Then O[f(.)] and ®[F(.)] are known, continuous, nonnegative function and functional vector, respectively, which satisfies [f(.)l -< O[f(.)],
II F(.) II - O[F(.)].
DEFINITION 2.2. In the following of this paper, we make 3-~(.) and @ denote inverse Laplace transform and convolution operation, respectively. 3. THE DESIGN OF ROBUST T R A C K I N G C O N T R O L L E R If the lumped disturbance and uncertainty term d(y,y,t) = 0 for all y,p and t, considering the controlled plant and the reference model defined in equations (5) and (6), perfect tracking can be achieved using the control system shown in Fig. 3. In this figure, Wl(t) and w2(t) are auxiliary state signals defined by wl(t) = aw, + u
(9)
aw2 + y
(10)
w2(t) =
where the constant a has the same definition as equation (7). The control law is given by
u(t) = 0Tw(t) where 0(t) = [ 00, 0,, 02, 03] T, w(t) =
[r(t),w, (t),we(t),y(t)] T.
(11)
909
Robust tracking controller design
r(t)
[ I t
F--1~53 *
-q
L+___~_j
÷I
e(t)
[
Fig. 3. Perfect tracking control scheme of a linear model of pneumatic servo system with perfect knowledge.
Since the affection of d(y,y,t) is neglected, the overall transfer function from r(t) to y(t) of the control system shown in Fig. 3 can be described as
V(s) _ Ookpr(s) R(s) s(s + ap,)(r(s) - 0 , ) - k,(02r(,)+ 03)
(12)
If perfect knowledge of the controlled plant is available, using this transfer function, we can solve the perfect tracking problem. To do so, we simply need to guarantee that
k.,s(s + apl)(F(s) - 0,) - k.,kp( OzF(s) + 03) = Ookt,F(s)(s z + a.,,s + a.a). After solving this equation, the desired parameter vector 0* can be given by
km/kp
I:il ,-=/02j
aml -- apl
1
( - ap, ot z - a2,ot + ap,a.,,ot + a.,,ot 2)
(13)
--
0;
1
2 ( a p l o t 4- apl -- a p l a m l -
am2 -
amlot)
In the case of 0--: 0", the control u(t)= (0") T w(t) can guarantee y(t)---~y.,(t) as t---~ for any r(t) under the condition that the lumped disturbance and uncertainty term d(y,p,t)= O. The question now i,; what control should be used if the controlled plant is unknown and d(y,p,t) ~ 0. The answer of this question is the subject of this section. When the parameters of the controlled plant are unknown, and d(y,p,t) ~ 0, control equation (11) can be rewritten as
u(t) = 0Vw(t) = (0*)Vw(t) - 0Tw(t)
(14)
where 0 is a constant vector containing an arbitrary estimate of 0* and 0 = 0 * - 0 is a parameter error vector. Considering the affection of the lumped disturbance and uncertainty term d(y,y,t), since the desired parameter vector 0* satisfies equation (13), under control equation (14), the controlled plant output dynamics can be represented by
{
1
y(t) = ~ -I(G.,(s)) (~ r(t) + 0---~o
i(1___ r(s)
It is obvious that the key to solve the tracking problem for an unknown plant is to design an
910
J. SONG and Y. ISHIDA
additional control to overcome the output caused by parameter error and d(y,p,t). To do that, we propose the following control such as [20]
u(t) = 0Tw(t) + ur(t).
(15)
Input this control to the controlled plant described in equation (3) and consider 00 = k.,/kp, the plant output becomes kp
01
where ur(t) is a portion of control which must be designed to insure the robustness of the overall control. We now define the output tracking error e(t) to be
e(t) = ym(t) - y(t)
(17)
where ym(t)= 3-1(Gin(S))@r(t). Then the dynamics of the output tracking error is described by
e(t) = ,s- l((~,,(s)) (~) kp
- 3-
1(1[ ~
¢3Tfv ( t ) - .~
01 ])
1-
F(s----)-
@ d(y,p,t)
® Ur(t)
]
(18)
where *(t)
1 ) ®w(t)
= .
(19)
\ r(s)
According to the definition of equation (3), in which the controlled plant is unknown, the design of controller will be carried out under the condition of 0 = 0 without any loss of generality. Then the control law equation (15) becomes
u(t) = ur(t)
(20)
Suppose 0 = 0 is an arbitrary estimate of 0", in this case, 0 can be obtained by 0= 0". It follows from equation (13) and assumptions (1) and (2) that the bound vector of 0 can be defined by km/kp in max
aml q- apl
[03]3
k l a n ( a~jaXa + apl max2 if- apl maxaml -t- am2 + a , . i o 0
031J
p
Then, under the condition of O = 0, the dynamics of the output tracking error can be rewritten as
e(t) = 3-'(G,,,(s)) (~) ~
0Tff(t) -- 3 -
® u(t) -- 3 - '
® d(y,j',t)
. (22)
Robust trackingcontroller design
911
THEOREM 3.1. Consider the system given by equation (3) which satisfies assumptions (1)-(4). If the control is chosen to be
u(t) = aar(t) + sign [e(t)].O[ h ,(t)]
(23)
e(t) to2(t) at(t) = 2(le(t)to(t)l + tr)
(24)
where
/1 to(t)= II 0o(l) II 2+ II g,(t)II 2+2~-1/- ~
\ )®p(y,t)
(25)
and 6 > 0 is a constant, then the controlled plant output tracking error (t) is globally, uniformly ultimately bounded. That is, the output tracking error becomes no larger than a constant which can be made arbitrarily small by choosing the design parameters tr small enough. Furthermore, the robust control u(t) is continuous and globally, uniformly bounded. (See Appendix A about the calculation of O[t~r(t)].) PROOF. Select a minimal realization of the transfer function (3,,(s) as a triple {A,B,C}. Then, a realization of the tracking error dynamics equation (22) can be written as
2,.(t)=Ax~(t)+ B 0Txk(t)--,s
®u(t) -- 3 - '
®d(y,p,t)]
e(t) = Cx~(t).
I~., '
(26)
Since G,,(s) is SPR, equation (26) is controllable and observable. Furthermore, it follows the Kalman-Yakubovich lemma in [21], where, given a symmetric positive defined matrix Q, there exists a symmetric positive define matrix P such that
ATp + PA = - Q, PB = C T.
(27)
Choose a Lyapunov function as
V=
knl
kp
x[Px~.
Differentiating V with respect to time, we have
f/= k"'xT(ATp+PA)x"+2xSPB[
_.~-,(_1
ko, ) ke xTeQxe+2x~PB [ 0Te(t) -- 3 - ' ( \_ _F(s) _1L (~ u(t)
_.~-,(_L
k.,
\ r(s) )®d(Y4',t) ] -
.,
_
-
.~-,(_L
I
k,,, x~Qx~ kp
,+
k,,, x~Qx~ ÷ le(t)I[ II Oo(t)II
kp
[
T
kp x, Qxe + 2e(t) 0Tffv(t)
r(,) )®d~y,,,,) ] = - --
+2e t [ <
- ~ - ' ( F - ~ ) ® u(t)
z + II ~(t) II ~ + 2 . ,
®
p~,t)
]
912
J. SONG and Y. ISHIDA
1
1
-2e(t)3-(--~)(~)u(t)=-
-kp - Xe Qx~ + ]e(t)]to(t) - 2e(t) ~ - '
@ u(t)
(28)
Since it follows the definitions of control law [equation (23)] and 0[.], we have
u(t) = o~ar(t)= sign [e(/)].O[ t~ ,(/)] = sign [e(t)][ la&(t) l + O[ ~ .(/)]}
(29)
Substitute equation (29) into equation (28), and equation (28) becomes
9"<- - - x ~ Q x e +
kp
<_
-
_
le(t)la,(t) -
2[e(t)l
~-
®lota~(t)l + O[ a .(t)]
--~
k., xXeaXe + [e(t)lto(t)- 2 [ e ( / ) l { ~ - l ( - ~ s ) ) ( ~ ) [ o t l G ( t ) l + l ~ r ( t ) l ] } _
kp
<_
--
k., xXQx~ + le(t)la,(t) - 2le(t)lar(t) =
m
I%
=e(t)to(t)
-
e2(t)to2(t) <[e(t)to(t)[ + o" -
kit7
k,,
xX~Qx~
k., x, Qxe = le(t)~o(t)l kp le(t)to(t)[ + tr
~r
(30)
Since both Q and P are the symmetric positive define matrixes, select a constant 7 such that
T(xTpx~) <_xX~Qxe. Then, the differentiation of V with respect to time satisfies
T k., P'<--T(x~Px~)7-- + or= T V + tr k,, Therefore, V converges to a residue set exponentially whose radius is a constant e = tr/y and so does the state xe(t). Consequently, the output tracking error e(t) also converges exponentially to this residue set. The convergence rate is decided by 7. This means that we do have some control over the transient response of the system. Because the output tracking error e(t) is continuous and bounded we can conclude that the output y(t) is also continuous and bounded. Furthermore, we examine the auxiliary control G(t), and we have
e( t) to 2(0 e( t) to ( t) 1 la,(t)l = 2(le(t)to(t)l + ~r) [ -- 2(le(t)~o(t)l + o9 Ito(t) < ~ t o
(31)
As with the above analyses, since the output tracking error e(t) is continuous and bounded, we have to(t) is continuous and bounded, and then the auxiliary control a,(t) is also continuous and bounded. In addition, from the definition of O[Ur(t)] (See Appendix A), OUr(t)] can be ensured to be continuous and bounded. Hence, control u(t) is continuous and bounded. REMARK 3.1. From the design of proposed controller, we have that only the input output and some bounds based on the pneumatic cylinder and bound function are used in the design controller. Furthermore, the robust control u(t) is continuous and globally, uniformly bounded. All of these make the proposed control scheme viable for the practical control problems. REMARK 3.2. We note that the tracking error can be made smaller by choosing smaller value of the design parameters tr. If tr is too small, however, the output of the robust controller may change rapidly and has a large magnitude. Since the bounds resulting from equation (21) and bound function p(y,t) are usually conservative, the design parameter tr should not be set too
Robust tracking controller design
913
small during the transient phase. Instead it should be decreased gradually if the output tracking error is larger than required.
4. S I M U L A T I O N
EXAMPLES
In this section, we provide two examples to show the effectiveness of p r o p o s e d control scheme. The first example is a time-invariant nonlinear s y s t e m - - i n this case, the linear portion of the controlled plant is conducted from the pneumatic cylinder with a load mass of M = 30 kg. The second is a controlled plant with a time-variant nonlinear portion. They permit us to illustrate both asymptotic and uniformly ultimately b o u n d e d stability.
(a) The output tracking and the response of plant with P controller (b) The tracking error e(t) (c) The control u(t) (a) 2
,
,
,
,
I
....:
,
,
,
,
;.,
I
,
,
~
,
.... y p ( t ) ~./~
:. A
--
I
V -2
'
'
,
-ym(t)
,
,
/
'
~
I
'
'
'
I
5
'
'
'
'
I
10 Times
,
--y(t) ,.,,;~
V '
'
'
'
20
15 (sec)
(b) ,
0.15
,
I
~
,
,
I
,
,
~
,
I
~
,
,
,
0.100.05
-
0.00
-
J<
-0.05
-
I.-.
-0.10-
ILl
£
,",
-0.15
'
'
f%
'
'
t~
1
'
'
D
'
'
/I
1
'
5
'
D
'
10 Times
'
/X
1
'
'
/'l
'
15
n
'
20
(sec)
(e)
0
~ g
-s
~
-ii 0
5
10
Times
15
20
(sec)
Fig. 4. Simulation results of the proposed controller ~r a time-invariant nonlinear system
914
J. SONG and Y. ISHIDA
4.1 An example of time-invariant nonlinear system Consider a plant whose linear portion is conducted from a pneumatic cylinder with load mass of M = 30 kg. This can be described by 6.64
Gp(s)],tty,,}=,, - s(s + 0.35) and then the nonlinear portion of the controlled plant is chosen by G(y) =
sin (y) 1 + y 2 , a ( p ) = 1.511 - e x p ( I P l ) ] .
The bounds of controlled plant are assumed as K p in =
1.99, kp "x = 10.0, apl min = 0.11, aplax = 1.1, p(y,t) = 2.0
Furthermore, select an external disturbance as
O(t) = sin (3/) A reference model tracked by the controlled plant is set by Gin(s) =
100 (s + 10) 2
The reference input is a sine function, that is
r(t) = sin (0.5 ¢rt). Following from the above definitions of the controlled plant and the reference model, the proposed controller can be designed as follows: F(s) = s + 1.0,a = 1.0,or = 0.2, ~T = [50.0,21.1,21.3,72.15]. Figure 4 shows the simulation results of the above-mentioned controlled plant with the proposed controller. Figure 4(a) shows the output tracking and the output response yp(t) of plant with controller P. From the illustration of yp(t), we can know that there is a strong nonlinear uncertainty contained in the controlled plant, Fig. 4(b) is the tracking error and then the control u(t) is illustrated in Fig. 5(c). Following these results, we can find that the tracking error is continuous and converges to a small set. In addition, the control u(t) is also continuous and bounded. Obviously, the proposed control is effective for the above-mentioned controlled plant.
4.2 An example of time-variant nonlinear system In the following the simulation will be done for a time-variant nonlinear plant. This is given by
[,,] [ o.o. *2
=
-Isin(t)l,
,.o irx, Ol - 0.35J[,xdt)J +
[o.oo]U(y,t),y(t)=xt(t) 6.64
where U(y,t)= d(t)+f(y,t)+ u(t), and u(t) is the control resulting from the proposed controller. Then the nonlinear uncertain term is chosen by
f(y,t) -
sin (t) sin (y) 1 + y2 + sin (3t).
A reference model and the parameters of controller selected the same as Section 4.1. Simulation results are shown in Fig. 5. In Fig. 5(a), yl,(t) is the output response of plant with the P controller. Following from the illustration in Fig. 5, it is clear that the proposed control is effective for some time-variant nonlinear systems.
Robust
tracking controller design
915
5. A P P L I C A T I O N TO A P R A C T I C A L P N E U M A T I C S E R V O S Y S T E M
Though the control performance is satisfactory in simulation examples using the proposed control scheme, it may not be able to satisfy a practical pneumatic servo system. Hence a practical application is studied here.
5.1 Configurath~n of pneumatic servo system Figure 6 shows the configuration of a pneumatic position servo system. It is primarily composed of a pneumatic cylinder, a 32 bit personal computer with an 80486 CPU and an associated arithmetic coprocessor, two 12-bit D/A converters, two servo power amplifiers of the voltage-current transformation type and two electro-pneumatic proportional valves. The pneumatic cylinder (CAIBQ63-300: SMC Corporation) is a low friction type whose minimum (a) The output tracking and the response ( b ) T h e t r a c k i n g e r r o r e(t) (c) T h e c o n t r o l u(t)
of plant with P controller
(a) 2
. . . .
I
,
,
,
,
I
,
,
,
....... yp(t)
,-
,
I
.....
ym(t)
,
,
,
--
y (t)
0.~
f"
0 - I
":-
-2
'~S
. . . .
I
v -v
. . . .
I
5
. . . .
10
~
~
/
'
I
. . . .
15
20
Times (sec)
(b) 0.15
,
=
=
n
I
n
l
I
. . . .
=
I
,
,
,
i
. . . .
,
I
,
,
,
i
. . . .
t
0.10-13J
0.05 m
0.00 .0.05
~-
.0.I0
.0.15
. . . . 0
5
10
15
20
Times ( s e e )
(c) 20
i
,
J
n
I
i
t
I
. . . .
I
.
.
.
.
I
. . . .
I
,
,
,
,
lO
°
8
-10 -20
. . . . 0
5
10
I
. . . .
15
20
Times (sec) Fig. 5. S i m u l a t i o n
results of the proposed
controller
for a time-variant
nonlinear
system
916
J. SONG and Y. ISHIDA
Pneumatic Source Air Dryer D/ACony. Mist Separator
(>
CounterCircuit
I ServoPowerAmp.
l~t~tor Circuit
I PC-9821
,
Regulator
AirTank Magnescal© 5~msor
Regulator ",,_j
Fig. 6. Configuration of the pneumatic servo system.
pressure is 0.01 MPa, inside diameter is 63 mm, and the stroke is 300 mm. A load mass of 10 kg is always connected to the tip of a position rod. The electro-pneumatic proportional valve is a proportional control valve (VEP3120-1: SMC Corporation) with pressure being of 3 port. The position measurement is carried out by magnetic sensor whose resolving power is 10/~m (SR10100A: Sony Corporation). 5.2
Applications o f the proposed control scheme
In this section the test results of the pneumatic servo system with the proposed controller will be illustrated. We suppose that the pneumatic cylinder in the system mentioned satisfies assumptions (1)-(4) and its dynamic motion equation can be described by equation (3). It follows the controller design described in Section 3 and the structure of the pneumatic cylinder; the controller is designed as follows: 1. The bounds of the controlled plant are chosen based on the design parameters of the pneumatic cylinder as max _- 1.1, p(y,t) = 2.0. k~ in = 1.99, kp ax = 10.0, apm~~n = 0.11, ap,
2. The external disturbance loaded in the input of the servo power amplifiers is set by O(t) =
0
0 -< t <-- 13s
0.976V
t-> 13s
3. Reference model tracked by controlled plant is set by 100 C.,(s)
4.
-
(S + 10) 2
The reference input r(t) is chosen by
-
r(t) =
50
0s - t < 5s
20
5s-<
t <
10s
20 100
10s -< t < 15s 15s -< t < 20s
20
20s -< t < 25s
0
25s <- t < 30s
Robust tracking controller design
917
5. The parameters used in controller is designed by F(s) = s + 1.0,c~ = 1.0,o- = 2, ~T = [50.0,21.1,21.3,72.15] 6. The sampling period is 0.002 s Figure 7 shows ~;he experimental results of the pneumatic servo system with the proposed controller in the case of loading mass M = 30 kg, Fig. 7(a) is the output tracking, Fig. 7(b) is the tracking error and Fig. 7(c) is the control voltage loaded in the input of servo amplifiers. The experimental results suggest that the proposed control scheme is not only effective in the case of the external disturbance D(t)= 0, but also effective in the case of D(t)> 0. Furthermore, a nonoscillation output response can be obtained. The experimental results also illustrate that the tracking error is 'bounded and then control u(t) is bounded and continuous. Figure 8 shows the
(a) The output tracking and the reference input r(t) (b) The o u t p u t tracking error of the pneumatic servo system (c) The input voltage of the servo p o w e r amplifier and the external disturbance
D(t)
(a) 120
,
,
,
,
I
. . . .
I
. . . .
I ,
i
i
i
I
. . . .
I
. . . .
. . . . . r(t) 80-
--
==
y(t)
g 1 ym,, o
-40
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t
,
5
I
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10
I
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15
I
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20
'
'
'
'
25
30
Time (Sec)
(b) 10
I
I
l
I
l
I
i
i
i
r
'
I
. . . .
i
6
l
. . . .
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I , , , ,
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,
,
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f
-10
r
i
i
i
j
,
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15
10
5
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i
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'
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20
'
25
30
Tim'~ (Sec)
(c) 4
....
I , ,
,
I,,
,,
I,,
,, -
....
2 D
0
~
-2 -4
i
0
I
i
i
i
i
5
i
'
I
10
'~
' '
I''
15
I,,
, ,I
, , , ,
Controlu(t)
-
J'
External Disturbance D(t}
I
20
''
'
'
I
25
'
'
'
'
30
Time (Sec)
Fig. 7. Experimental results of ' .
"matic cylinder with the proposed controller in the case of M = 30 kg
918
J. SONG and Y. ISHIDA
results in the case of load mass M = 60 kg using the same controller in the case of M = 30 kg, Similar results are obtained. Since the output response does not change along with the p a r a m e t e r ' s variant, we can consider that the robustness of the proposed control scheme is good for pneumatic servo systems.
6. C O N C L U S I O N S
In this paper, a robust tracking control scheme has been proposed for pneumatic servo systems. The main contribution of this paper is to find a robust controller (Theorem 3.1) for pneumatic servo systems, which can guarantee that the tracking error converges to an arbitrarily (a) The output tracking and the reference input r(t) (b) The output tracking error of the pneumatic servo system (c) The input voltage of the servo power amplifier and the external disturbance D(t)
(a) 120
....
i
,
,
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,
,
,
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,
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.
,
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1
,
r(t)
- -
y(t)
Z v
,
..... ....
,
ym(t)
4O I---
"5 o.
0
-40
'
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5
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10
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15
'
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20
'
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30
25
Time (Sec)
(b) ~ , , t l , , , , l , , , , l
I0
E
5
w
0
~
-5
. . . .
I , , , , I , , , ,
I--
-10
'
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0
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5
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10
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I
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=
'
15
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'
20
I
~
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'
25
30
Time (Sec)
(c) 4 -....
2
Control u(t) External Disturbance D(t)
v
r~
0 -2 .
-4
.
.
.
.
.
.
.
.
' ' ' '
.
.......... i-i .......... i ................. .............
.
I'
5
'
''
I ' '
10
''
I ' ' ' '
15
I ' '
20
'
'
I
25
'
'
°
'
30
Time (Sec)
Fig. 8. Experimental results of the pneumatic cylinder with the proposed controller in the case of M=60 kg
Robust tracking controller design
919
small residue set: as time t - - - ~ . T h e key of c o n t r o l l e r design is to find a c o n t i n u o u s b o u n d f u n c t i o n only in t e r m s of system output. This can be easily f o u n d for a practical p n e u m a t i c servo system o p e r a t i n g in a certain e n v i r o n m e n t . I n addition, only the i n p u t a n d o u t p u t signals are used in the design of the controller. All of these bestow u p o n the p r o p o s e d c o n t r o l scheme good practical applicability. F u r t h e r m o r e , from the c o n t r o l l e r design process described in Section 3, we find that the c o n t r o l structure is very simple a n d the o p e r a t i o n s of a r i t h m e t i c are n o t difficult to complete. H o w e v e r , in the case that a c o n t i n u o u s b o u n d f u n c t i o n only in terms of system o u t p u t does n o t exist, it is still very difficult to c o m p l e t e r o b u s t tracking control for p n e u m a t i c servo system using the p r o p o s e d c o n t r o l scheme. This will b e c o m e a topic of work in future papers. Acknowledgements--The authors are grateful to Dr Zongli LIN, assistant professor of State University of New York. for providing some valuable references. The authors also thank the SMC Co. Ltd. for supporting this work.
REFERENCES 1. 2. 3. 4. 5. 6. 7.
Lu, W. M., System Control Letters, 1995, 25, 13 Verma, M. S., International Journal of Control Letters, 1995, 48, 897 Jabbari, E, IEEE Transactions of Automation Control, 1990, 35, 954 Schmite Dorf W. E. and Barmach, B. R., Automatics, 1986, 22, 353 Qu, Z., System Control Letters, 1992, 18, 3 301 Tanaka, K., Simizu,A. and Sakata, K., Journal of the Society of Instrument and Control Engineers, 1994, 30, 1069 Spong, M. W., in New Trends and Applications of Distributed Parameter Control Systems (Lecture Notes in Pure and Applied Mathematics), G. Chen, E. B. Lee, W. Littman and L. Markus, Eds. Marcel Dekker, New York, 1990; System Control Letters, 1987, 1119,310. 8. Khorasani, K., IF,EE Transactions on Automation Control, 1990, 35, 10 1145 9. Ghorbel, E, Hung, J. Y. and Spong, M. W., in Proc. 1989 IEEE Int. Conf. Robotics and Automation, Scottsdale, AZ, 1989, p. 15. 10. Grimm, W. M., buernational Journal of Adaptive Control and Signal Processes, 1992, 4, 501 11. Khalil, H. K. and Saberi, A., IEEE Transactions on Automation Control, 1987, 32, 1031 12. Venkataraman, S. T. and Gulati, S., in Proc. Amer. Conf., 1992, p. 891. 13. Schmite Dorf, W. E. and Barmach, B. R., Automatics, 1986, 22, 353 14. Schmite Dorf, W. E. and Barmach, B. R., Journal of Dynamic Systems, Measurement and Control, 1987, 109, 186 15. Song, J. and Ishida, Y., International Journal of Engineering Science, 1997, (in press). 16. Noritsugu, T. and Fukuzono, K., Journal of the Japan Hydraulics and Pneumatics Society, 1994, 25, 6 733 17. Canudas de Wit, C., Olsson, H. and Lischinsky,P., IEEE Transactions on Automation Control, 1995, 40, 3 419 18. Endo, K., Ishida, Y. and Honda, T., Transactions of the lEE of Japan, 1993, 113, 6 408 19. Noritsugu, T. and Takaiwa, M., Journal of the Society oflnstrument and Control Engineers, 1994, 30, 6 676 20. Yeung, K. S. and Chen, Y. P., IEEE Transactions on Automation Control, 1989, 33, 2 200 21. Khalil, H. K., Nonlinear Systems. Macmillan, New York, 1991. (Received 29 January 1997)
APPENDIX According to the definition of at(t) in equation (24), we have h,(t) = c~(t) . de(t) + ~i(t_~)_) dto(t) ae(t) dt cko(t) dt
(AI)
in which the bound of every term can be obtained easily from the definitions in Section 3. They can be described as follows: ~a(t)
1 -~: -2-
2[e(t)lto(t)+ o" ;S 2
le(t)lto(t)+ o. to2(t) ~ _[- ~7(t) ] (I e(t)Ito(t) + o') 2 to2 t): le(t) [to(t) + o"
~7(t) 1 3eZ(t)toz(t)+ 2le(t)lto(t)o. <- 1.5 le(t)lw(t) ~ [ ~7(t)-] ,~(t) <- "-2 (I e(t)]to(t) + ~)z left) Ito(t) + o. '~ t~[--~-~J.
(A3)
920
J. SONG and Y, ISHIDA
From the definition of tracking error dynamics in equation (18), we have
de(t) _ ~_ '(st~,,,(s)) (~)
d,
~
~'r~'(t) - ~ -
(~) u(t)
-f~
,
1
(y,~,t)l
-
From the above expression the bound function of ~(t) can be obtained by d e(t) s(s + a) d t < kjmax.~ - 1 "s2 + a,,,~s + a,,,2
[to(t) +
(A4)
la,(t) l l t ~ [ ~ J .
Identical to the above calculation, the bound function of tb(t) can be calculated by
(AS)
I~o(t)l~_2icv~(t)cv(t)l+23-t( s~ ) @p(Y,t)~-O[i°(t)] where qv(t) = - aVe(t) . w ( t ) .
From equation (A1), we have an(t) J/~"(t) J -< de(t)
~a(,) alto(t) <_el ~(')1.or de(01 + o[ ~(f}l.o[ dto.~Ol-~etaX')]-
de(t) 4- , 0to(t) dt
dt
L 6,e(t)J
[
dt J
L ato(t)J
[_ tit j
Thus, we obtain the bound function of ~r(l) only using the input and output information. From the analysis in Section 3 we can conduct that the bound function of h,(t) is continuous and bounded,