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Robust controller design for uncertain nonlinear systems via feedback linearization
Pergamon
Chemical Engineering Science, Vol. 50, No. 9, pp. 1429-1439, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0009 2509/95 $9.50 + 0.00
0009-2509(94)00518-4
ROBUST CONTROLLER DESIGN FOR UNCERTAIN NONLINEAR SYSTEMS VIA FEEDBACK LINEARIZATION YI-SHYONG CHOU* and WEI WU Department of Chemical Engineering, National Taiwan Institute of Technology, Taipei 10672, Taiwan, R.O.C. (First received 25 September 1992; accepted in revised form 25 November 1994)
Abstract--The design of robust stabilizing controllers for uncertain nonlinear systems has been investigated. Uncertainties including uncertain parameters and structured uncertainties, caused by the mismatch between the mathematical model and the true process, are considered. Two strategies are proposed; a nonhigh gain feedback design is employed to achieve robust stability of the controlled system with matched uncertainties, and a parameter diffeomorphism is used as a high gain control to robustify the cancellation of unmatched uncertainties. The proposed methodologies are implemented on the structure of internal model control (IMC) for the output regulation of a CSTR.
INTRODUCTION Recent developments in the theory of geometric nonlinear control provide powerful methods for controller design of nonlinear chemical processes. In almost all research works, two main approaches of iinearization of the closed-loop system can be extracted. First, there is feedback linearization (Hoo and Kantor, 1985; Calvet and Arkun, 1988; Alvarez-Gallegos, 1988) which addresses the transformation of a nonlinear system without output through state and input coordinate changes. Assuming that states are available, in these new coordinates the transformed system exhibits linear dynamics. The other approach is the input-output feedback linearization (Kravaris and Chung, 1987; Kravaris and Kantor, 1990; Bequette, 1991; Henson and Seborg, 1991) which addresses the transformation of a nonlinear system with output so that in the new coordinates the transformed system exhibits linear inout-output dynamics. These appoaches, input-state feedback linearization and input-output feedback linearization techniques, require a perfect model of the plant in order to achieve linearization of the closed-loop system. However, for many real systems, there often exist inevitable uncertainties in their constructed models. In addition, there exist uncertain parameters that are not exactly known or are difficult to estimate. Therefore, the design of a robust controller that deals with uncertainties of a nonlinear system is an important subject. To deal with uncertain nonlinear systems, two main approaches have been proposed: adaptive control approach and Lyapunov-based control approach. If the uncertainties appear strictly as uncertain parameters, adaptive control techniques can be used (Sastry and lsidori, 1989; Taylor et al., 1989; Adebekun, 1992). The Lyapunov-based control relies on an explicit con-
rAuthor to whom correspondence should be addressed.
struction of a Lyapunov function based on which a state feedback control is synthesized using the bounds on the uncertainties (Corless and Leitmann, 1981; Barmish et al., 1983; Kravaris and Palanki, 1988). In order to achieve the control objective of stabilization, however, some assumptions, the structure of the uncertainties and the location of the unknown parameters, were introduced and are often referred to as matching conditions. Some of these methods, which impose matching conditions, have serious limitations. Current methodologies regarding robustness analysis of uncertain dynamic systems without the matching conditions have been investigated. Chen and Leitmann (1987) proposed the critical mismatch threshold condition to guarantee overall stability for a class of uncertain systems. Liao et al. (1992) investigated the output tracking problem of a class of uncertain nonlinear systems, which do not satisfy the conventional matching conditions but states and tracking errors remain bounded, and proposed a robust output tracking controller derived via a Lyapunov-based approach. Calvet and Arkun (1989) proposed the design of ~ stabilizing controller for a nonlinear system in the presence of parametric uncertainty with the approach of a Lyapunov function. Moreover, Arkun and Calvet (1992) recently studied stabilization of a class of nonlinear systems under the influence of both disturbances and modeling errors and presented a robust state feedback control design. Chou and Wu (1994a) proposed a parameterized state feedback to deal with disturbance rejection problem of a chemical reactor. In this paper, we study the design of robust stabilizing controllers for a class of uncertain nonlinear systems. Uncertainties including the uncertain parameters and the structured uncertainties are considered. Two strategies are proposed for the cases that satisfy matching conditions (Barmish et al., 1983) and need not satisfy matching conditions, respectively.
1429
1430
YI-SHYONGCHOU and WEI Wu
For the first method, we use the feedforward-like compensator to avoid a high gain control and to guarantee robust stability of the closed-loop system. The second proposed method does not necessarily require the matching conditions. A parameter diffeomorphism is proposed as a high gain control technique to robustify the cancellation of uncertainties for the second method. The internal model control (IMC) (Morari and Zafiriou, 1989) has proven to be a very useful controller design strategy for linear processes. Furthermore, the IMC framework can be implemented after a nonlinear system is transformed to a linear system with output (Calvet and Arkun, 1988)• In this paper the proposed methodologies are implemented on the structure of IMC for the robust control problem of a nonisothermal CSTR system. Finally numerical simulations in this CSTR example demonstrate the promising performance of the overall nonlinear control structure in disturbance rejection and set-point tracking•
linear form: = Az + by,
(5)
y = Cz.
Also -0
1
0
...
O"
-0-
0 :
0 :
1 :
... '.
0
0
0
0
0
...
1
0
0
0
0
..-
0
1
0
...
0
O]z.
~=
y=[1
z+
• v
(6)
Hence the control law is chosen as u = Ol(x, Po, V) =
-- L~yh(x) ~1 v. LoL ~- ~h(x) LoL ~- ~h(x)
(7)
Here we introduce the Lie derivative of h(x) with respect to two different vector fields by LgLi/- t h(x) = (dL~ - ~h(x), 9(x)) '~ 8L~- I h(x)
PROBLEM FORMULATION
Consider a single input single output (SISO) nonlinear system in the presence of uncertainties
(1)
L°/h(x) = h(x) (9)
y = h(x)
Lkyh(x) = LyLky-lh(x) = ( d L k f - x , f ( x ) ) ,
where x e R", ue R, ye R are the state variables, the control input, and the plant output, respectively. f(x, Po), g(x, po), Af(x, po, p) and Ag(x, pop) are smooth vector fields in R ~. pe R ~ is a set of uncertain parameters in t2, where f~ is a known compact set. poe R s is the nominal constant value. Here we consider the class of SISO nonlinear systems which have a uniform relative degree equal to the dimension of the state vector• The system can be transformed into a normal form with no zero dynamics. Although here we restrict ourselves to consider this class of nonlinear systems, the proposed methodologies, in this paper, can be extended to the problem of input--output linearization of nonlinear systems. The nominal system is then given as follows• = f ( x , Po) + 9(x, po)u.
(2)
It has been shown (Isidori, 1989; Nijmeijer and van der Schaft, 1990) that there exists a local coordinate transformation, ( z . z2 . . . . . z . ) r =
(8)
and we can define higher Lie derivatives of h(x) with respect to the vector field f ( x ) by
0t = f ( x , Po) + Af(x, Po, P) + [O(x, po) + Ag(x, po, p)]u
i = 1,2 . . . . . n
T(x, po) T
= (T1, 7"2. . . . .
Our interest in this paper is in the practical case when there are uncertainties in the system in eq. (1). We will consider the problem of designing a robust controller so that the influence of system uncertainties will be attenuated. In other words, the objective is to design a controller based on the nominal system in eq. (2) that will robustly stabilize the actual process in eq. (1) despite the uncertainties h f ( x , po, p) and Ag(x, po, p). In the following sections, our work will consider two cases of model uncertainty: one is the matched uncertainty, another is the unmatched uncertainty. CONTROL DESIGN WITH MATCHINGCONDITIONS (CDMC) There exist smooth function Af*(.) and A9*(*) in R", such that the uncertainties in eq. (1), for all xe R", satisfy Af(x, po, p) = g(x, po)Af*(x, po, p)
(10a)
Ag(x, po, p) = g(x, po)Ag*(x, po, p)
(10b)
Tn) T
= (h(x), Lfh(x) . . . . . L ) - ~h(x)) r
(3)
satisfying LoL~-lh(x)=O,
k >~ 1.
k = 1,2 . . . . . n - 1 (4)
LoL)-lh(x) # 0 such that eq. (2) can be transformed into the following
where A f * ( x , po, p) and Ag*(x, po, p) are in general referred to as the matched uncertainties (Barmish et al., 1983). Usually the matched uncertainties are compensated by the Lyapunov-based controller (Corless and Leitmann, 1981) and by the variable structure control (Slotine and Li, 1991). However, here we will propose a simpler compensator to solve this problem without using a high gain control•
1431
Robust controller design for uncertain nonlinear systems Consider the system in eq. (1) under the matching conditions [eq. (10)]. By some derivations, it can be easily verified that eq. (1) can be transformed into the following form: Z1
7"2 -~- (Loh)Af* + (Lgh)u(1 + A9*)
=
z2 = z3 + (LgLj.h)Af* + (LgLyh)u(1 + A0*)
:
(11)
~. = L"fh+ (LgL"i-lh)Af * + (LoL"i-lh)u(l + AO*). With the aid ofeqs (4) and (7), eq. (11) can be reduced in the following form: zl = Z 2
(ql, q2). Here we suggest that these parameters are left for on-line adjustment. In essence, we can observe the available information, that is, the output responses of the plant-model mismatch ( y - y,) within IMC framework, to justify the selection of these parameters (Chou and Wu, 1994b). The analysis of the robust control system is now carried out using the Lyapunov-type analysis. Let V(z) = zrP~z be a Lyapunov function for the system in eq. (15), where Ps is a symmetric positive definite matrix. Along the trajectories of eq. (15) with a bounded input r~, that is, [ral <~ Ba and Bd t> 0. Then, we obtain (/(z) = ~T P~z + zr Ps~
Z2 = Z3
(12)
= zr(Ar~P~ + P~A~)z + (brpsz + zTp~b)AB + (brp~z + zrp~b)re(1 + A#*).
~. = v(1 + A0*) + ( L o L y , h)Af* - (L"~h)ao*.
Assume that the external control input v can be expressed as the sum of terms describing external input rd, pole placement Kz, and adding compensator v~. The equation for these combinations is (13)
v = re - K z + vs
where K is a positive constant row vector, and v~ is a feedforward-like compensator of the form v~ = rl(x, Po, P) = [ L~ h(x) ] ql - [ L g L ) - ah(x)] q2
(14) where ql and q2 arc on-line tuning parameters which are scalar constants. Notice that this compensator [eq. (14)] is not unique, it may be represented by the other forms. In particular, here it provides a simple structure with two tuning parameters to compensate for the uncertainties in the control input space beforehand. Introducing eq. (14) into eq. (12), we have = A~z + brd(1 + AO*) + bAB
(15)
where
A~ =
0
1
0
-.-
0
0
0
1
...
0
'
:
:
"..
:
0
0
0
...
1
-kx
--
k2
-
ka
....
By assumption, A~ is stable [eq. (16)] so that P~ is positive definite for a positive definite matrix Q such that A r P~ + P~A~ = - Q.
12 ~< -
,imin(Q)llzll z + 112e~banzll
+ 112P,brn(1 + Ao*)zl[
(1) There exists a scalar C 1 function H, :R" x g --, R with the property HalO, q ) ~ O, Hi(z, q ) > 0 for all z =~ 0 and q = (q,, q2) • 7r. Select two constants ~1 and fl, such that [12P~bABzll <~ ~lllzll2 + [3,Hl(z, q).
(21)
(2) There exists a scalar C 1 function Hz:R" x n --* R with the property H2(O, q)v~ O, H2(z, q ) > 0 for all z :# 0 and q = (q~, q2)• n. Select two constants a2 and 32 such that (22)
In view of the above relations, eq. (20) becomes
(~, +
~911zll 2 + fllHl(z,q)
(23)
+ fl2nz(z, q).
and
(20)
where 2mi.(e) denotes the smallest eigenvalue and II ° II is the usual Euclidean norm. Furthermore, we now make the following assumptions.
12 ~ - gmi.(O)llzll 2 +
k.
(19)
Substituting eq. (19) into eq. (18), yields
l[2Psb(1 + Ag*)zll ~< a~llzll 2 + fl2n2(z,q). (16)
(18)
We can obtain l?(z) ~< 0 by suitably selecting matrix Q and parameters (ql, q2). In consequence, the solution z is ultimately bounded with respect to the set
AB = { - KzAg* + L"~h[ql(1 + Ag*) - Ag*] + L g L 7 1 h [ A f * - q2(1 + AO*)]} o T - ' ( z ) .
(17) In the light of above analysis, it can be seen that the influences of all uncertainties on the closed-loop system will be robustly attenuated if this compensator [eq. (14)] can suitably compensate for the nonlinearities in eq. (17). However, this leaves a great deal of flexibility in selecting the adjustable parameters
F:= {z• R " a n d q •
nlV(z) <~ k,/~> 0}.
(24)
The applicability of stability analysis is quantified by a set of somewhat stringent conditions in eqs (21) and (22). However, it provides a sufficient condition to guarantee robust stabilization of the closed-loop system in eq. (15). From the practical point of view, the present approach for achieving the nonlinearity cancellation in uncertain nonlinear systems transformed
1432
YI-SHYONG CHOU and WEI W u
by feedback linearization technique can be easily employed. We adjust the on-line tuning parameters (qt, q2) and set the pole placement gains (kt . . . . . k,) to satisfy the conditions (21) and (22), such that eq. (23) holds.
or
(dTi*,O(x, po)) = 0 ,
i = 1. . . . . n -
I.
(32)
Therefore, the forms of eq. (30) can be reduced to the following results:
(pdTi, f(x, Po)) + (pdT~, AA(x, Po, P, v) ) = PTi+l + pA~q(x, po, p,v),
CONTROL DESIGN IN THE ABSENCE OF MATCHING
i = 1,2 . . . . . n - 1 .
CONDITIONS (CDAMC)
In this section, the proposed design method does not necessarily require structure matching conditions. In order to introduce appropriate derivations we can express the nonlinear system of eq. (1) with the nominal state feedback in eq. (7) as the following form:
(33)
Also
(d T~*,f(x, Po)) + (d Ti*, AA(x, Po, P, v)) = Ti*+x + pA~ii(x, po, p,v) i = 1, 2 . . . . . n - 1.
.¢¢=f(x, Po) + g(x, po)u + Af(x, po, P) + Ag(x, Po, p)~t (x, Po, P, v)
(34)
The above transformations can be established (25)
Ti*+t = pTi+l = (pdTi, f(x, po)) = (dTi*,f(x, po)),
y = h(x).
i = 1,2 . . . . . n - 1
(35)
Let us define and AA (x, Po, P, v) = Af(x, Po, P)
PA.71i(x, Po, P, v) = (dTi*, AA(x, Po, P, v))
+ Ao(x, po, P)Ol(x, po, p,v ).
(26)
= (pdTi, AA(x, Po, p,v)),
i = 1,2 . . . . . n - 1.
Thus, eq. (25) can be rewritten as the following form:
.~ = f(x, po) + #(x, po)u + AA(x, po, p,v),
y = h(x). (27)
The main focus of this robust design problem is therefore how the technique can be employed to robustly cancel the effect of the perturbed term, AA(x, Po, P, v), in eq. (26). Here we will propose a parameter diffeomorphism to deal with this robust design problem. First, consider the following new transformations
~i = T~*(x, po, p ) = p T i ( x , po),
i = 1,2 . . . . . n
(36) Thus, we can construct the input-transformation from the previous algorithms as follows.
]"* = pT',, = (pdT,,,f(x, po)) + u(pdT,,,g(x, po)) + (pdT,, A A ) = v + pAA,. Hence, we select
v = ( p d T . , f ( x , po)) + u(pdT.,g(x, po)) = (dT,*,f(x, po)) + u ( d T * , g ( x , po))
where p is a scalar constant which can be viewed as an adjustable parameter. Then the system in eq. (27) can be mapped onto a quasi-linear system by this transformations [eq. (28)]. This can be easily verified by substituting eq. (28) into eq. (27), we have
v = L ) T * ( x , po, p) + uLgL"r -1T~'(x, po, p). (39) Notice that we must satisfy the following conditions
(pdT.,g(x, po) ) 4:0
(40)
(dT*,g(x, po)) 4: O.
(41)
or
T
Combining ¢qs (31), (35), and (40) and using the Leibnitz type formula (Isidori, 1989; Kravaris and Chung, 1987), we can show that T* should satisfy the set of partial differential equations
+ gj(x, Po) u + AAi(x, Po, P, v)] = pTi+l + pAAi(x, po, p,v) = Ti*+x + pA.4i(x, po, p,v),
i = 1,2 . . . . . n -- 1. (29)
(dT~',ad~g(x, Po)) = 0 ,
i = 1. . . . . n - 2
(dT*, ad}- tg(x, Po)) # 0
Also
i = 1,2 . . . . . n - 1.
(30)
Since the conditions of eq. (28), the left-hand side of eq. (30) does not depend on u, so that i = 1. . . . . n -
(43)
ad°rg(x, po) = g(x, po)
+ ( p d T , AA(x, Po, P, v)) = p T~+1
( p d T , e(x, Po)) = 0 ,
(42)
where the Lie bracket off(x) and g(x) are defined as
(pdTi, f(x, Po)) + u(pdTi, o(x, po))
+ pAAi(x, po, p,v),
(38)
or
(28)
"
(37)
1
(31)
ad}g(x, Po) = [f(x, Po), #(x, Po)] =
Og(x, Po)., dx
"
j ( x , Pol
(44)
-~xOf(x'P°)o(x' Po).
Moreover, the higher order Lie brackets can be de-
1433
Robust controller design for uncertain nonlinear systems We now make the following assumptions.
fined recursively as
ad~o(x, po) = [f(x, po), adkf-t #(x, po)],
k > 1. (45)
Under the conditions of eqs (28), (42) and (43), it can be easily verified that the system [eq. (27)] can be transformed into the following form: ZI = Z2 + LaA 22 = Z3 -}-
T~'
(1) There exists a scalar C ~ function ~b:R" ---,R with the property ~b(0) 4: 0, ~b(~) > 0 for all ~ ~ 0 and two constants y~ and V2 such that I[2A.4Tp~[] ~< y~[l~[[2 + ])2(~(Z) VZ~eR".
(56)
(2) If the control input rd is bounded, there exists a constant Y3, such that
Laa Lf T *
ll2PsbrdEll ~< 73.
(46)
(57)
In view of the above relations, eq. (55) becomes
z, = L} T* + u LoL"r-' T* + Laa L"y- t T*.
W(E) ~< - Am~,(Q)}12ll2 + ";111~112+ 72~b(~) + )'3.
Hence, let us select the inverse control law as the form of v - / 4 7-, u = ~2(x, Po, P, v) L~L}- ~T* (47)
(58) With the aid of - )~mi,(Q)llz[I2 ~< - Ks, for all z ~ R", and K~ > 0, we obtain -
and v = ra -- K~
(48)
where re is an external output regulator, and K e R" are pole assignment feedback gains• Now the overall control law can be expressed as u=
re - ~ i.:- o k1 i + l Lic T *
L.~
Tl*
L 0L}- 1T*
(49)
Therefore, eq. (46) can be rearranged as the form of
z = Ac~ + bre + A.4
(50)
where
A.4 = [Laa T*, LaALyT* ..... LaAL~- 1T,]T o T*(z-') = p[Laa Tx, L a a L f T l ..... LaaL"f- 1TI]T o T(z). (51) In view of eq. (51), once a suitable parameter p or
e(p = e- 1) is selected, say p ~ 0 or e ~ oo , the effect of the perturbed term will be attenuated. However, one of the most important design specifications that a controlled system is required to meet is stability. Therefore, bounds for parameter p or e will be derived to ensure that the controlled system is stable. To prepare for the stability analysis of the system [eq. (50)], we select a Lyapunov function to be employed through this analysis
w(~) = ~rp~.
(52)
To make the derivative of eq. (52), yields
¢(e) = z~p~ + ~P~.
(53)
Substituting eq. (50) into eq. (53), we obtain
IC/(z") = ~,r(ATp~ + P~A¢)~, + (bTrePs'2 + ~'rp~bre) + (AATp~ + ~rp~AA)
(54)
where A~ is stable and P~ is a solution of the Lyapunov equation in eq. (19). Thus, we have the following inequality W(z-') <~ - Am~,(Q)ll~ll2 + 112P~bra~ll + 112A,4rP~ell.
(55)
-1 -Ks Ami,(Q)ll~[I2 = ---~-)~mi,(Q)llzll 2 ~< - - ~ .
(59)
It is now easy to see that with the choice of e~ =
Ks ~'.~ R". ~,llell 2 + ~4~(e) + ~3 In consequence, the solution is 0 ~< e <~ ~s-
(60)
(61)
The main result is a sufficient condition to guarantee stabilization of the system [eq. (50)]. Observing eqs (50) and (51), the effect of the nonlinearities on the closed-loop system [eq. (51)] can be effectively attenuated when a suitable parameter p or e is tuned. It is worthy of note that if the inappropriate parameter p or e is employed in the control law, it may produce the "peaking" phenomena (Marino et al., 1989) and may destroy closed-loop stability of the system. Here certain conditions are required on the bounds of the parameter p or e in order to avoid an interplay between the peaking, which is induced by the high gain control, and the nonlinearities in AA [eq. (51)]. Therefore, the perturbations [eqs (56) and (57)] have to be limited to certain ranges, which is equivalent to the condition in eq. (61) being satisfied. So that the high gain control is permitted and safe. APPLICATION T O A CSTR
To examine the applicability of the proposed methodologies, an example of composition control of a continuous stirred tank reactor (CSTR) will be demonstrated. In this application, we consider a CSTR in which a single first-order exothermic reaction is taking place. The modeling equations are written as follows (Ray, 1981).
vdCA=F(CA~--CA)--at* Vk°exp(-~----T)CA dT VpoCv-~ = poCvF(Ty - T) + V ( - AH)ko ×exp(-~-T)Ca-hA(T-
To). (62)
1434
YI-SHYONG CHOU and WEI WU
The dimensionless equations for this reactor system are given by
and
#,=--x, +Do(l--x,)exp(x2/(l+~))+d,
L~x,
=[-I- b" exp(x2/(1+ x-32"~')l~o//3 x [-x, + bo(l-x,)exp(x2/(l +~))]
~2--x2+BDo(l-x,)exp(x2/(l+~)) -
fl(Xz -
x2,o) +
flu + d2
where B, Da and fl are uncertain parameters, d~ is an inlet feed composition perturbation and d2 is an inlet feed temperature perturbation, which are both unmeasurable disturbances. Consider that the manipulated control variable u is an admissible deviation between the coolant temperature, To, and some reference value for the coolant temperature Too.xt is a dimensionless composition and x2 is a dimensionless temperature. These notations are defined in the Notation section and the detailed formulations are shown in Uppal et al. (1975) and Ray (1981). The system
A f ( x , Po, P) =
(/(
x 1
1+
-x2+/~D,,(1-x0 (67,
xexp(x2/(e+~))-fi(x2-X2,o)].
The feedback linearization of the CSTR system is well-defined as long as LaLIx I # O, that is, xl # 1. The desired operating conditions are (x], x'2,u')= (0.2 1.33,0.42), while the model parameters /1 = 0.3, /~ = 8,/5° = 0.072, ¢# = 20, and X2¢o= 0 are selected. Furthermore, the plant-model mismatch between eqs (63) and (64) can be expressed as the form
I
+~
equations without uncertain parameters and disturbances can be written in the following nominal form: = f ( x , Po) + g (x, po)u
+ [Do(I - 1,)exp(x2/(l + ~))
(63)
and Ag(x, po, p ) =
(64)
[o]
(68)
f l _ fl + d2 "
where
[
]
and CDMC
#(x, po) = [fl 1 .
(65)
Here we assume that Po = (Da,/~, fl) are known nominal parameters. The control objective is to regulate the reactor outlet composition. Available as the controlled output is the composition xl. Based on eq. (7) we obtain the nonlinear control law as the form of u = ( L g L : x l ) - 1 (v - L}xl)
where
(66)
If we consider that only the system parameter fl is uncertain, that is, fl ~ fl, but Da = Do and B =/~, and also that only the inlet feed temperature has a perturbation, that is d2 # 0 but, dl = 0, then the uncertainties in eq. (68) can be reduced to the case of matched uncertainties, A f ( x , Po, P) = #(x, po)Af*(x, Po, P)
= [~-] ( ~ ) ( x 2
- x2co)
(69)
-
-x
fl
Hence, the proposed feedforward-like compensator based on eqs (13) and (14) is shown as v = r~ - kl(Xx - x~) - k 2 [ A ( x , po) - A(x', po)] + o,
Robust controller design for uncertain nonlinear systems and
1435
where (70)
v~ = ( L } x O q l -- ( L g L f x l ) q 2 .
These proposed results will be implemented on the structure of IMC. Thus, we select the rd as an external regulator on top of IMC structure for observing the plant-model mismatch output and for guaranteeing the asymptotic output tracking. Let ~ = Ac~ + bra
and
(71)
am" [Yra= C~
×I--Xl+Da( 1 --xl)exp(x2f(lq-~))]
be the internal model. ~s R" is the state variable of the model and Ym is the output of the model. Then, the IMC control law based on an inverse model is described as Gl: rd =
s" + k,s " - I + ... + kl
(2s + 1)"
6
(72)
where 2 is a scalar constant as a filter parameter and s is Laplace operator. Here 6 represents 6 = Y~vy + y~ where Y~v is a set-point. This overall control scheme is shown in Fig. 1. CDAMC
Suppose that the parameters p = (D,, B,/3) are uncertain parameters, and d~ and d2 are unmeasurable disturbances. In this case, the uncertainties in eq. (68) do not satisfy the matching conditions in eq. (10). According to the previous analysis, we need to define a parameter diffeomorphism, that is T * = pxl,
T ] = p f l ( x , po)
(73)
for p > 0. Following the conditions in eqs (42) and (43), the constraint
Zat, pxl =Pl)a(1--xl)exp(x2/(l d-~)) has to hold in the neighborhood of the operating points x ~ = Ix], x~2] r. Then we can construct a parameterized state feedback control law based on eq. (49) and p = e-~ as follows: u = (LoLiT*)-X[rd - kle-l(xl
- x])
- k2e- l(fl (x, Po) - f l (x~, Po)) - L } T*]
(75)
Based on the IMC structure, here rd is denoted as a regulator [eq. (72)] for the output regulation. The control scheme is shown in Fig. 2. Let us consider the operating points x ~ = [xl, x 2 ] •~ r as initial conditions and for a practical consideration the manipulated input is bounded with lul ~< 10. The system parameters are perturbed by varying operating conditions and aging process, so we set D a =/),(1 + 60, B =/3(1 + 62) and 13 = fi(1 + ~3), where 6i, i = 1, 2, 3 are the percentage of perturbations. The appropriate pole assignment feedback gains (kl, k z ) = (4,4) are selected and fixed, which should satisfy the Lyapunov equation [eq. (19)]. We present the promising performance of the control schemes as described in the CDMC and CDAMC. In essence, we lay emphasis on the merits of the effects of different tuning parameters (q~, q2, e) on the closedloop system by means of numerical simulations including set-point tracking and disturbances rejection.
Disturbance rejection, First, for the CDMC problem, suppose that the system parameter fl has the maximum variation of 63 = 50%. When the disturbance input d2 = 0.5 enters the system [eq. (63)'I, we would like to evaluate the disturbance rejection capability of the designed control system. A comparative
l
~
d
CSTR I System
x
Iz
=lr~Y.,,.
°
[ • ......... , - 2 i Fig. 1. The proposed control scheme implemented on the structure of IMC.
YI-SHYONG
1436
CHOU
and
WU
WEI
System~ CSTR
Ii_
x
--
I~1
Fig. 2. The proposed control scheme implemented on the structure of IMC.
plot illustrating the advantage employed by the proposed method [eq. (70)] with a filter parameter 2 = 1 [eq. (72)] is depicted in Fig. 3. Here, we observe the output response of the plant-model mismatch (y - y,,) [see, Fig. 3(B)] to estimate a set of appropriate parameters that make the mismatch smallest. Through trial and error by several computer simulations, the solid line in Fig. 3(A) and (B) shows the best performance when the parameters (ql, q:) = (3, 3) are used. Second, for the C D A M C problem, suppose that the system parameters p = (D~, B, fl) have the maxi m u m variation of 5~ = 10%, i = 1, 2, 3. Considering that the external disturbances (d~, d2) = (0.I, 0.1) enter the system, we would evaluate the disturbance rejection capability of the designed high gain control. Fig. 4(A) shows that the suitable value of e can effectively
0.o0 020
\ #^, \ f , \
~ 0.24
r
',
- - -
~=1
--
a=tO
\
o 0.~2 o.2o 0,8 Time Fig. 4(A). Response to disturbance rejection for CDAMC with different tuning parameters ~.
2.O 0.24
/
\
/ 0,23
Iqz,qz)=(O,O)
~,
~qt,q~)=( 1,1 )
~ \
/
(q,,q=)=(3,3) . . . . . . . . . (qDq=)=(5,5)
•
0.0
L
~:~ -2.0
o.22 //
C)~ 0.2!
'd
I{
.~ -0.02
'~',
- - -
~=1
.....
C=2
- •..
0,20
,
< ~-2
-~
~=I0
................... -10.0
x. ,..
0.19
-12.0
Time
o.10
Time Fig. 4(B). The corresponding manipulated input u. Fig. 3(A). Response to disturbance rejection for CDMC with different tuning parameters (q~, q2).
f I
0.08 -
\
/
/
I
E I
;:~
-
0.o0 -
"
I, 0.02
/
\
0.04-
-I
-
s=l
~=10
"'- .....................................
/i" 0.02
.... ,.,
.__ ",,
"',. ",-,.
. . . .
.
.
5 ~
.
.
.
'
.
'
.
.
'
1'o
.
.
. . . .
(q,,qz)=(O,O)
o,oo
/ q , , q z ) = ~ 1,1 ) ~q~,qz)=(3,3) (ql,q2)=(5,5)
1'5
. . . .
-0.02
....
20
~ ....
tb . . . . Time
~'5 . . . .
zo
Time
Fig. Fig. 3(B). The corresponding plant-model mismatch y - ym.
4(C). The
corresponding plant-model Y - Ym.
mismatch
Robust controller design for uncertain nonlinear systems attenuate the influences of system perturbations on the system output. Usually this method is better for the disturbance rejection problem, since it uses a high gain control to suppress the plant-model mismatch as shown in Fig. 4(C). The tuning parameter ~ can be adjusted on-line, since adjusting e is equivalent to adjusting the speed of the closed-loop response. Due to its directness and intuitive appeal it is conceivable that e is left for on-line adjustment by the operating personnel. The key point of this method is that the parameter diffeomorphism [eq. (73)] is used as a high gain technique to robustify the cancellation of the nonlinear terms [eq. (68)]. Note that the presented method based on CDAMC does not require the exact knowledge of the structures of uncertainties and disturbances. From the practical control point of view, this method is suitable for the process control problem.
Set-point tracking. Here, for the CDMC problem, we also consider that the system has the maximum variation of 63 = - 50% and the disturbance input is d2 = 0.5. Simulation results of a set-point change from 0.2 to 0.3 are shown in Fig. 5 for the case of only feedback linearization on top of IMC structure with filter parameters 2~ = 0.3, 2, = 1, and 2~ = 1.5. Let us now compare the merits of the proposed control
1437
method with the only feedback linearization on top of IMC structure. Simulation results of the presented method (CDMC) with the parameters (qx, q2) = (3, 3) are shown in Fig. 6. For all values of the filter parameter 2, the performance of the proposed method is far better than the implementation of IMC structure with feedback linearization transformations. We can observe less overshoot and smoother controller moves. Figure 7 depicts the plant-model mismatch. It is evident that the obvious improvement is made by our proposed method with (ql, q2)= (3,3). Notice that in Fig. 7 the (q~, q2) = (0,0) indicates the case of only IMC structure with feedback linearization transformations being used. For the CDAMC problem, suppose that the system parameters p = (D,,B, fl) have the maximum variation of 6~ = - 2 0 % , i - 1,2,3 and the external disturbances are (dx, d E ) = ( - 0.1, -0.1). Simulation results of set-point tracking from 0.2 to 0.3 are shown in Fig. 8. In Fig. 8(A), the solid line shows the performance of the present proposed method with different adjustable parameters being tuned. Notice that these simulation results show the inverse responses. Since the system output profile from the old equilibrium point [0.2, 1.612] falls firstly towards one of the other equilibrium points [0.03, 0.35] when the perturbed terms, including parameter variability and external disturbance, enter the system,
0.34 0.32
..~ 0.30
. ~ 0.30
.~ 0.20
~
o
0
0.26
tI
- - - X=I.5 . . . . . X=l
li :)
. . . .
-
~,
. . . .
~b . . Time
-
~'s
0.20 0.24
~=0.3
. .
0.28
0.22 . . . .
0.20
~o
Fig. 5(A). Response to set-point tracking for only IMC with feedback linearization transformations under different filter parameters 2.
/ /1"/ / " " ' ' ' /
~ ~ ~
- X=l.5 . . . . . X=I ~=0.3
[ ,'/ ,'/ 5
10
15
Time
Fig. 6(A). Response to set-point tracking for the proposed method with (ql, q2) = (3, 3) under different filter parameters L
" • 1o.o
IO.O - - -
- -
~=1.5
-
5.0.
I~,
~
-
~=0.3
o.o
-5.o
....
~ ....
fo . . . . Time
~'s . . . .
Fig. 5(B). The corresponding manipulated input u. CES
50:9-E
X=l.5
. . . . . ~=1
. . . . . ~=I "~
.5.0
.~
0.0
~
-0.0
-
-
~=0.3
Time Fig. 6(B). The corresponding manipulated input u.
YI-SHYONGCHOU and WEI Wo
1438 0.12
•
/
,.
//
x\
0.10 • 0,00
t
/ I
o.o0
s
-0.08
,
% 0.04 -
----
X=
>..
X=l,(ql,q21=(3,3)
t 0.02
l,(qi,q2)=(O,O)
\ \
>'~-0.16
/
\
- \
~=1
- -
c:5
0.00 -0.02
-0.04
-
. . . .
~
. . . .
1'o Time
. . . .
1~
. . . .
Fig. 7. Comparison of the proposed method with the only IMC under the same filter parameter 2 = 1. The dashed line (ql, q2) = (0, 0): the only IMC. The solid line (ql, q2) = (3, 3): the proposed method.
/
/
/
~
/ /
,' ,'
0 0.20
/ /
,'
/
i
/
i
0.24
~: 1 E=2
.....
/
~=5
/
t, ; /
\ / . . . .
~ . . . .
1'o . . . .
Time
t's . . . .
so
Fig. 8(A). Response to set-point tracking for CDAMC with a fixed filter parameter 2 = I and different tuning parametrs
10.0
"~ 0.0 e~
---
~=1
-
~=5
..... -
£=2
0.o
.~
~-
20
. . . .
1'0
. . . .
1'5
. . . .
20
Time Fig. 8(C). The corresponding plant-model mismatch y - y=.
approach of the feedforward-like compensation method and the parameter diffeomorphism are given. The class of nonlinear systems are those which nominally admit input-state linearization. Two categories of uncertainties, the uncertain parameters and the structured uncertainties, are considered. Also two control strategies are proposed; a feedforward-like control design is used to achieve robust stability of the system with matched uncertainties, and a parameter diffeomorphism is presented as a high gain control technique to robustify the cancellation of unmatched uncertainties. In particular, the proposed control law with the approach of parameter diffeomorphism does not require the exact knowledge of the structured uncertainty. A simple parameter p or e can be adjusted on-line to suppress the influences of uncertainties on the closed-loop system. An illustrative example of a nonisothermal CSTR system demonstrates the applicability of the proposed methods implemented on the structure of I M C for disturbance rejection and set-point tracking. Simulation results show that the proposed methodologies are successfully applied to this system and their performances are satisfactory.
4.0
e~ ~
2.0
....
~, . . . .
t'0 . . . . Time
f5 . . . .
20
Fig. 8(B). The corresponding manipulated input u.
then the response upsurges to the desired level xl = 0.3 due to the control action. In Fig. 8(A), the solid line shows the shortest settling time under the tuning parameters E = 5 and 2 = 1. Figure 8(B) shows the corresponding control action and Fig. 8(C) indicates the corresponding plant-model mismatch. CONCLUSIONS The theory and application of feedback linearization of nonlinear systems with uncertainties by the
A CA Cp
CA$o CAr di E
Y,g F h ko K R T
To
NOTATION heat transfer area concentration of species A in reactor heat capacity base value of concentration used to produce dimensionless concentration reactant concentration in the feed disturbance input activation energy vector fields volumetric feed rate heat transfer coefficient specific rate constant adjustable feedback gains ideal gas constant temperature in reactor coolant temperature
Robust controller design for uncertain nonlinear systems base temperature of coolant reactant feed temperature base temperature of reactant feed diffeomorphism manipulated input external input to the inner loop of the control structure reactor volume vector of state variables vector of new state variables
Tco
r~ r~o T, U /)
V X
z, ~'
Greek letters feedback signal of I M C structure 3 AH heat of reaction adjustable parameter filter parameter of I M C structure internal model state variables adjustable parameter P density Po Dimensionless groups E (D=--
RTIo
B
- AHCA~o ~p poCpTfo koe-~V
Da -- _ _
F
hA [3=-Fpo Cp X2c 0 ~
r,o-Tfo T~o
-
F t=t*-V X1
_ CAr° -- CA CAyo
X2 _ T - Tfo q9
rfo dl = CAIo -- CA~ CAio
d2 - T f - Tfo ~o
U=
rfo T~- T~o - - q ~ Tfo
Mathematical symbols the absolute value of a scalar function I.I the Euclidean norm of a vector or matrix I1"11 2mi.(') the minimum eigenvalue of a matrix ¥ for all belongs to E o composition of functions real line, n-dimensional Euclidean space R, R"
1439
REFERENCES Adsbekun, K., 1992, The robust global stabilization of a stirred tank reactor. A.I.Ch.E.J. 38, 651-659. Alvarez-Gallegos, J., 1988, Application of nonlinear system transformations to control design for a chemical reactor. lEE PtD. 135, 90-94. Arkun, Y. and Calvet, J.-P., 1992, Robust stabilization of input-output linearizable systems under uncertainty and disturbances A.I.Ch.E.J. 38, 1145-1156. Barmish, B. R., Corless, M. and Leitmann, G., 1983, A new class of stabilization controller for uncertain dynamical systems. SIAM J. Control Optimization 21, 246-253. Bequette, B. W., 1991, Nonlinear control of chemical processes: a review Ind. Engn# Chem. Res. 30, 1391-1413. Calvet, J.-P. and Arkun, Y., 1988, Feedforward and feedback linearization of nonlinear systems and its implementation using internal model control (IMC). Ind. Engng Chem. Res. 27, 1822-1831. Calvet, J.-P. and Arkun, Y., 1989, Robust control design for uncertain nonlinear systems under feedback linearization. IEEE Conf. Dec. Control 1, 102-106. Chen, Y. H. and Leitmann, G., 1987, Robustness of uncertain systems in the absence of matching assumptions. Int. J. Control 45, 1527-1542. Chou, Y.-S. and Wu, W., 1994a, Disturbance rejection for nonlinear control systems. J. Chin. I.Ch,E. 25, 163-171. Chou, Y.-S. and Wu, W., 1994b, Disturbance compensation for nonlinear processes. Chem. Engng Commun. (in press). Corless, M. and Leitmann, G., 1981, Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems. IEEE Trans. autom. Control 26, 1139-1148. Henson, M. A. and Seborg, D. E. 1991, Critique of exact linearization strategies for process contol. J. Proc. Control 1, 122-139. Hoo, K. A. and Kantor, J., 1985, An exothermic continuous stirred tank reactor is feedback equivalent to a linear systems. Chem. Engng Commun. 37, 1--10. Isidori, A., 1989, Nonlinear Control Systems, 2nd edn. Springer, Berlin. Kravaris, C. and Chung, C. B., 1987, Nonlinear state feedback synthesis by global input-output linearization. A.I.Ch.E.J. 33, 592-603. Kravaris, C. and Kantor, J. C., 1990, Geometric methods for nonlinear process control. Ind. Engng Chem. Res. 29, 2295-2323. Kravaris, C. and Palanki, S., 1988, Robust nonlinear state feedback under structured uncertainty. A.I.Ch.E.J. 34, 1119-1127. Liao, T. L, Fu, L. C. and Hsu, C. F., 1992, Output tracking control of nonlinear systems with mismatched uncertainties. Syst. Control Lett. 18, 39-47. Marino, R. Respondek, W. and van der Schaft, A. J., 1989, Almost disturbance decoupling for single-input single-output nonlinear systems. IEEE Trans. autom. Control 34, 1013-10t7. Morari, M. and Zafiriou, E., 1989, Robust Process Control. Prentice-Hall, Englewood Cliffs, NJ. Nijmeijer, H. and van der Schaft, A. J., 1990, Nonlinear Dynamical Control Systems. Springer, New York. Ray, W. H., 1981, Advanced Process Control. McGraw-Hill, New York. Sastry, S. and Isidori, A., 1989, Adaptive control of linearizable systems. IEEE Trans. autom. Control 34, 1123-1131. Slotine, J.-J. E. and Li, W., 1991, Applied Nonlinear Control. Prentice-Hall, Englewood Cliffs, NJ. Taylor, D. G., Kokotovic, P. V., Marino, R. and Kanellakopoulos, 1989, Adaptive regulation of nonlinear systems with unmodeled dynamic. IEEE Trans. autom. Control 34, 405-412. Uppal, A., Ray, W. H. and Poore, A. B., 1975, On the dynamic behavior of continuous stirred tank reactors. Chem. Engng Sci. 29, 967-977.
Chemical Engineering Science, Vol. 50, No. 9, pp. 1429-1439, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0009 2509/95 $9.50 + 0.00
0009-2509(94)00518-4
ROBUST CONTROLLER DESIGN FOR UNCERTAIN NONLINEAR SYSTEMS VIA FEEDBACK LINEARIZATION YI-SHYONG CHOU* and WEI WU Department of Chemical Engineering, National Taiwan Institute of Technology, Taipei 10672, Taiwan, R.O.C. (First received 25 September 1992; accepted in revised form 25 November 1994)
Abstract--The design of robust stabilizing controllers for uncertain nonlinear systems has been investigated. Uncertainties including uncertain parameters and structured uncertainties, caused by the mismatch between the mathematical model and the true process, are considered. Two strategies are proposed; a nonhigh gain feedback design is employed to achieve robust stability of the controlled system with matched uncertainties, and a parameter diffeomorphism is used as a high gain control to robustify the cancellation of unmatched uncertainties. The proposed methodologies are implemented on the structure of internal model control (IMC) for the output regulation of a CSTR.
INTRODUCTION Recent developments in the theory of geometric nonlinear control provide powerful methods for controller design of nonlinear chemical processes. In almost all research works, two main approaches of iinearization of the closed-loop system can be extracted. First, there is feedback linearization (Hoo and Kantor, 1985; Calvet and Arkun, 1988; Alvarez-Gallegos, 1988) which addresses the transformation of a nonlinear system without output through state and input coordinate changes. Assuming that states are available, in these new coordinates the transformed system exhibits linear dynamics. The other approach is the input-output feedback linearization (Kravaris and Chung, 1987; Kravaris and Kantor, 1990; Bequette, 1991; Henson and Seborg, 1991) which addresses the transformation of a nonlinear system with output so that in the new coordinates the transformed system exhibits linear inout-output dynamics. These appoaches, input-state feedback linearization and input-output feedback linearization techniques, require a perfect model of the plant in order to achieve linearization of the closed-loop system. However, for many real systems, there often exist inevitable uncertainties in their constructed models. In addition, there exist uncertain parameters that are not exactly known or are difficult to estimate. Therefore, the design of a robust controller that deals with uncertainties of a nonlinear system is an important subject. To deal with uncertain nonlinear systems, two main approaches have been proposed: adaptive control approach and Lyapunov-based control approach. If the uncertainties appear strictly as uncertain parameters, adaptive control techniques can be used (Sastry and lsidori, 1989; Taylor et al., 1989; Adebekun, 1992). The Lyapunov-based control relies on an explicit con-
rAuthor to whom correspondence should be addressed.
struction of a Lyapunov function based on which a state feedback control is synthesized using the bounds on the uncertainties (Corless and Leitmann, 1981; Barmish et al., 1983; Kravaris and Palanki, 1988). In order to achieve the control objective of stabilization, however, some assumptions, the structure of the uncertainties and the location of the unknown parameters, were introduced and are often referred to as matching conditions. Some of these methods, which impose matching conditions, have serious limitations. Current methodologies regarding robustness analysis of uncertain dynamic systems without the matching conditions have been investigated. Chen and Leitmann (1987) proposed the critical mismatch threshold condition to guarantee overall stability for a class of uncertain systems. Liao et al. (1992) investigated the output tracking problem of a class of uncertain nonlinear systems, which do not satisfy the conventional matching conditions but states and tracking errors remain bounded, and proposed a robust output tracking controller derived via a Lyapunov-based approach. Calvet and Arkun (1989) proposed the design of ~ stabilizing controller for a nonlinear system in the presence of parametric uncertainty with the approach of a Lyapunov function. Moreover, Arkun and Calvet (1992) recently studied stabilization of a class of nonlinear systems under the influence of both disturbances and modeling errors and presented a robust state feedback control design. Chou and Wu (1994a) proposed a parameterized state feedback to deal with disturbance rejection problem of a chemical reactor. In this paper, we study the design of robust stabilizing controllers for a class of uncertain nonlinear systems. Uncertainties including the uncertain parameters and the structured uncertainties are considered. Two strategies are proposed for the cases that satisfy matching conditions (Barmish et al., 1983) and need not satisfy matching conditions, respectively.
1429
1430
YI-SHYONGCHOU and WEI Wu
For the first method, we use the feedforward-like compensator to avoid a high gain control and to guarantee robust stability of the closed-loop system. The second proposed method does not necessarily require the matching conditions. A parameter diffeomorphism is proposed as a high gain control technique to robustify the cancellation of uncertainties for the second method. The internal model control (IMC) (Morari and Zafiriou, 1989) has proven to be a very useful controller design strategy for linear processes. Furthermore, the IMC framework can be implemented after a nonlinear system is transformed to a linear system with output (Calvet and Arkun, 1988)• In this paper the proposed methodologies are implemented on the structure of IMC for the robust control problem of a nonisothermal CSTR system. Finally numerical simulations in this CSTR example demonstrate the promising performance of the overall nonlinear control structure in disturbance rejection and set-point tracking•
linear form: = Az + by,
(5)
y = Cz.
Also -0
1
0
...
O"
-0-
0 :
0 :
1 :
... '.
0
0
0
0
0
...
1
0
0
0
0
..-
0
1
0
...
0
O]z.
~=
y=[1
z+
• v
(6)
Hence the control law is chosen as u = Ol(x, Po, V) =
-- L~yh(x) ~1 v. LoL ~- ~h(x) LoL ~- ~h(x)
(7)
Here we introduce the Lie derivative of h(x) with respect to two different vector fields by LgLi/- t h(x) = (dL~ - ~h(x), 9(x)) '~ 8L~- I h(x)
PROBLEM FORMULATION
Consider a single input single output (SISO) nonlinear system in the presence of uncertainties
(1)
L°/h(x) = h(x) (9)
y = h(x)
Lkyh(x) = LyLky-lh(x) = ( d L k f - x , f ( x ) ) ,
where x e R", ue R, ye R are the state variables, the control input, and the plant output, respectively. f(x, Po), g(x, po), Af(x, po, p) and Ag(x, pop) are smooth vector fields in R ~. pe R ~ is a set of uncertain parameters in t2, where f~ is a known compact set. poe R s is the nominal constant value. Here we consider the class of SISO nonlinear systems which have a uniform relative degree equal to the dimension of the state vector• The system can be transformed into a normal form with no zero dynamics. Although here we restrict ourselves to consider this class of nonlinear systems, the proposed methodologies, in this paper, can be extended to the problem of input--output linearization of nonlinear systems. The nominal system is then given as follows• = f ( x , Po) + 9(x, po)u.
(2)
It has been shown (Isidori, 1989; Nijmeijer and van der Schaft, 1990) that there exists a local coordinate transformation, ( z . z2 . . . . . z . ) r =
(8)
and we can define higher Lie derivatives of h(x) with respect to the vector field f ( x ) by
0t = f ( x , Po) + Af(x, Po, P) + [O(x, po) + Ag(x, po, p)]u
i = 1,2 . . . . . n
T(x, po) T
= (T1, 7"2. . . . .
Our interest in this paper is in the practical case when there are uncertainties in the system in eq. (1). We will consider the problem of designing a robust controller so that the influence of system uncertainties will be attenuated. In other words, the objective is to design a controller based on the nominal system in eq. (2) that will robustly stabilize the actual process in eq. (1) despite the uncertainties h f ( x , po, p) and Ag(x, po, p). In the following sections, our work will consider two cases of model uncertainty: one is the matched uncertainty, another is the unmatched uncertainty. CONTROL DESIGN WITH MATCHINGCONDITIONS (CDMC) There exist smooth function Af*(.) and A9*(*) in R", such that the uncertainties in eq. (1), for all xe R", satisfy Af(x, po, p) = g(x, po)Af*(x, po, p)
(10a)
Ag(x, po, p) = g(x, po)Ag*(x, po, p)
(10b)
Tn) T
= (h(x), Lfh(x) . . . . . L ) - ~h(x)) r
(3)
satisfying LoL~-lh(x)=O,
k >~ 1.
k = 1,2 . . . . . n - 1 (4)
LoL)-lh(x) # 0 such that eq. (2) can be transformed into the following
where A f * ( x , po, p) and Ag*(x, po, p) are in general referred to as the matched uncertainties (Barmish et al., 1983). Usually the matched uncertainties are compensated by the Lyapunov-based controller (Corless and Leitmann, 1981) and by the variable structure control (Slotine and Li, 1991). However, here we will propose a simpler compensator to solve this problem without using a high gain control•
1431
Robust controller design for uncertain nonlinear systems Consider the system in eq. (1) under the matching conditions [eq. (10)]. By some derivations, it can be easily verified that eq. (1) can be transformed into the following form: Z1
7"2 -~- (Loh)Af* + (Lgh)u(1 + A9*)
=
z2 = z3 + (LgLj.h)Af* + (LgLyh)u(1 + A0*)
:
(11)
~. = L"fh+ (LgL"i-lh)Af * + (LoL"i-lh)u(l + AO*). With the aid ofeqs (4) and (7), eq. (11) can be reduced in the following form: zl = Z 2
(ql, q2). Here we suggest that these parameters are left for on-line adjustment. In essence, we can observe the available information, that is, the output responses of the plant-model mismatch ( y - y,) within IMC framework, to justify the selection of these parameters (Chou and Wu, 1994b). The analysis of the robust control system is now carried out using the Lyapunov-type analysis. Let V(z) = zrP~z be a Lyapunov function for the system in eq. (15), where Ps is a symmetric positive definite matrix. Along the trajectories of eq. (15) with a bounded input r~, that is, [ral <~ Ba and Bd t> 0. Then, we obtain (/(z) = ~T P~z + zr Ps~
Z2 = Z3
(12)
= zr(Ar~P~ + P~A~)z + (brpsz + zTp~b)AB + (brp~z + zrp~b)re(1 + A#*).
~. = v(1 + A0*) + ( L o L y , h)Af* - (L"~h)ao*.
Assume that the external control input v can be expressed as the sum of terms describing external input rd, pole placement Kz, and adding compensator v~. The equation for these combinations is (13)
v = re - K z + vs
where K is a positive constant row vector, and v~ is a feedforward-like compensator of the form v~ = rl(x, Po, P) = [ L~ h(x) ] ql - [ L g L ) - ah(x)] q2
(14) where ql and q2 arc on-line tuning parameters which are scalar constants. Notice that this compensator [eq. (14)] is not unique, it may be represented by the other forms. In particular, here it provides a simple structure with two tuning parameters to compensate for the uncertainties in the control input space beforehand. Introducing eq. (14) into eq. (12), we have = A~z + brd(1 + AO*) + bAB
(15)
where
A~ =
0
1
0
-.-
0
0
0
1
...
0
'
:
:
"..
:
0
0
0
...
1
-kx
--
k2
-
ka
....
By assumption, A~ is stable [eq. (16)] so that P~ is positive definite for a positive definite matrix Q such that A r P~ + P~A~ = - Q.
12 ~< -
,imin(Q)llzll z + 112e~banzll
+ 112P,brn(1 + Ao*)zl[
(1) There exists a scalar C 1 function H, :R" x g --, R with the property HalO, q ) ~ O, Hi(z, q ) > 0 for all z =~ 0 and q = (q,, q2) • 7r. Select two constants ~1 and fl, such that [12P~bABzll <~ ~lllzll2 + [3,Hl(z, q).
(21)
(2) There exists a scalar C 1 function Hz:R" x n --* R with the property H2(O, q)v~ O, H2(z, q ) > 0 for all z :# 0 and q = (q~, q2)• n. Select two constants a2 and 32 such that (22)
In view of the above relations, eq. (20) becomes
(~, +
~911zll 2 + fllHl(z,q)
(23)
+ fl2nz(z, q).
and
(20)
where 2mi.(e) denotes the smallest eigenvalue and II ° II is the usual Euclidean norm. Furthermore, we now make the following assumptions.
12 ~ - gmi.(O)llzll 2 +
k.
(19)
Substituting eq. (19) into eq. (18), yields
l[2Psb(1 + Ag*)zll ~< a~llzll 2 + fl2n2(z,q). (16)
(18)
We can obtain l?(z) ~< 0 by suitably selecting matrix Q and parameters (ql, q2). In consequence, the solution z is ultimately bounded with respect to the set
AB = { - KzAg* + L"~h[ql(1 + Ag*) - Ag*] + L g L 7 1 h [ A f * - q2(1 + AO*)]} o T - ' ( z ) .
(17) In the light of above analysis, it can be seen that the influences of all uncertainties on the closed-loop system will be robustly attenuated if this compensator [eq. (14)] can suitably compensate for the nonlinearities in eq. (17). However, this leaves a great deal of flexibility in selecting the adjustable parameters
F:= {z• R " a n d q •
nlV(z) <~ k,/~> 0}.
(24)
The applicability of stability analysis is quantified by a set of somewhat stringent conditions in eqs (21) and (22). However, it provides a sufficient condition to guarantee robust stabilization of the closed-loop system in eq. (15). From the practical point of view, the present approach for achieving the nonlinearity cancellation in uncertain nonlinear systems transformed
1432
YI-SHYONG CHOU and WEI W u
by feedback linearization technique can be easily employed. We adjust the on-line tuning parameters (qt, q2) and set the pole placement gains (kt . . . . . k,) to satisfy the conditions (21) and (22), such that eq. (23) holds.
or
(dTi*,O(x, po)) = 0 ,
i = 1. . . . . n -
I.
(32)
Therefore, the forms of eq. (30) can be reduced to the following results:
(pdTi, f(x, Po)) + (pdT~, AA(x, Po, P, v) ) = PTi+l + pA~q(x, po, p,v),
CONTROL DESIGN IN THE ABSENCE OF MATCHING
i = 1,2 . . . . . n - 1 .
CONDITIONS (CDAMC)
In this section, the proposed design method does not necessarily require structure matching conditions. In order to introduce appropriate derivations we can express the nonlinear system of eq. (1) with the nominal state feedback in eq. (7) as the following form:
(33)
Also
(d T~*,f(x, Po)) + (d Ti*, AA(x, Po, P, v)) = Ti*+x + pA~ii(x, po, p,v) i = 1, 2 . . . . . n - 1.
.¢¢=f(x, Po) + g(x, po)u + Af(x, po, P) + Ag(x, Po, p)~t (x, Po, P, v)
(34)
The above transformations can be established (25)
Ti*+t = pTi+l = (pdTi, f(x, po)) = (dTi*,f(x, po)),
y = h(x).
i = 1,2 . . . . . n - 1
(35)
Let us define and AA (x, Po, P, v) = Af(x, Po, P)
PA.71i(x, Po, P, v) = (dTi*, AA(x, Po, P, v))
+ Ao(x, po, P)Ol(x, po, p,v ).
(26)
= (pdTi, AA(x, Po, p,v)),
i = 1,2 . . . . . n - 1.
Thus, eq. (25) can be rewritten as the following form:
.~ = f(x, po) + #(x, po)u + AA(x, po, p,v),
y = h(x). (27)
The main focus of this robust design problem is therefore how the technique can be employed to robustly cancel the effect of the perturbed term, AA(x, Po, P, v), in eq. (26). Here we will propose a parameter diffeomorphism to deal with this robust design problem. First, consider the following new transformations
~i = T~*(x, po, p ) = p T i ( x , po),
i = 1,2 . . . . . n
(36) Thus, we can construct the input-transformation from the previous algorithms as follows.
]"* = pT',, = (pdT,,,f(x, po)) + u(pdT,,,g(x, po)) + (pdT,, A A ) = v + pAA,. Hence, we select
v = ( p d T . , f ( x , po)) + u(pdT.,g(x, po)) = (dT,*,f(x, po)) + u ( d T * , g ( x , po))
where p is a scalar constant which can be viewed as an adjustable parameter. Then the system in eq. (27) can be mapped onto a quasi-linear system by this transformations [eq. (28)]. This can be easily verified by substituting eq. (28) into eq. (27), we have
v = L ) T * ( x , po, p) + uLgL"r -1T~'(x, po, p). (39) Notice that we must satisfy the following conditions
(pdT.,g(x, po) ) 4:0
(40)
(dT*,g(x, po)) 4: O.
(41)
or
T
Combining ¢qs (31), (35), and (40) and using the Leibnitz type formula (Isidori, 1989; Kravaris and Chung, 1987), we can show that T* should satisfy the set of partial differential equations
+ gj(x, Po) u + AAi(x, Po, P, v)] = pTi+l + pAAi(x, po, p,v) = Ti*+x + pA.4i(x, po, p,v),
i = 1,2 . . . . . n -- 1. (29)
(dT~',ad~g(x, Po)) = 0 ,
i = 1. . . . . n - 2
(dT*, ad}- tg(x, Po)) # 0
Also
i = 1,2 . . . . . n - 1.
(30)
Since the conditions of eq. (28), the left-hand side of eq. (30) does not depend on u, so that i = 1. . . . . n -
(43)
ad°rg(x, po) = g(x, po)
+ ( p d T , AA(x, Po, P, v)) = p T~+1
( p d T , e(x, Po)) = 0 ,
(42)
where the Lie bracket off(x) and g(x) are defined as
(pdTi, f(x, Po)) + u(pdTi, o(x, po))
+ pAAi(x, po, p,v),
(38)
or
(28)
"
(37)
1
(31)
ad}g(x, Po) = [f(x, Po), #(x, Po)] =
Og(x, Po)., dx
"
j ( x , Pol
(44)
-~xOf(x'P°)o(x' Po).
Moreover, the higher order Lie brackets can be de-
1433
Robust controller design for uncertain nonlinear systems We now make the following assumptions.
fined recursively as
ad~o(x, po) = [f(x, po), adkf-t #(x, po)],
k > 1. (45)
Under the conditions of eqs (28), (42) and (43), it can be easily verified that the system [eq. (27)] can be transformed into the following form: ZI = Z2 + LaA 22 = Z3 -}-
T~'
(1) There exists a scalar C ~ function ~b:R" ---,R with the property ~b(0) 4: 0, ~b(~) > 0 for all ~ ~ 0 and two constants y~ and V2 such that I[2A.4Tp~[] ~< y~[l~[[2 + ])2(~(Z) VZ~eR".
(56)
(2) If the control input rd is bounded, there exists a constant Y3, such that
Laa Lf T *
ll2PsbrdEll ~< 73.
(46)
(57)
In view of the above relations, eq. (55) becomes
z, = L} T* + u LoL"r-' T* + Laa L"y- t T*.
W(E) ~< - Am~,(Q)}12ll2 + ";111~112+ 72~b(~) + )'3.
Hence, let us select the inverse control law as the form of v - / 4 7-, u = ~2(x, Po, P, v) L~L}- ~T* (47)
(58) With the aid of - )~mi,(Q)llz[I2 ~< - Ks, for all z ~ R", and K~ > 0, we obtain -
and v = ra -- K~
(48)
where re is an external output regulator, and K e R" are pole assignment feedback gains• Now the overall control law can be expressed as u=
re - ~ i.:- o k1 i + l Lic T *
L.~
Tl*
L 0L}- 1T*
(49)
Therefore, eq. (46) can be rearranged as the form of
z = Ac~ + bre + A.4
(50)
where
A.4 = [Laa T*, LaALyT* ..... LaAL~- 1T,]T o T*(z-') = p[Laa Tx, L a a L f T l ..... LaaL"f- 1TI]T o T(z). (51) In view of eq. (51), once a suitable parameter p or
e(p = e- 1) is selected, say p ~ 0 or e ~ oo , the effect of the perturbed term will be attenuated. However, one of the most important design specifications that a controlled system is required to meet is stability. Therefore, bounds for parameter p or e will be derived to ensure that the controlled system is stable. To prepare for the stability analysis of the system [eq. (50)], we select a Lyapunov function to be employed through this analysis
w(~) = ~rp~.
(52)
To make the derivative of eq. (52), yields
¢(e) = z~p~ + ~P~.
(53)
Substituting eq. (50) into eq. (53), we obtain
IC/(z") = ~,r(ATp~ + P~A¢)~, + (bTrePs'2 + ~'rp~bre) + (AATp~ + ~rp~AA)
(54)
where A~ is stable and P~ is a solution of the Lyapunov equation in eq. (19). Thus, we have the following inequality W(z-') <~ - Am~,(Q)ll~ll2 + 112P~bra~ll + 112A,4rP~ell.
(55)
-1 -Ks Ami,(Q)ll~[I2 = ---~-)~mi,(Q)llzll 2 ~< - - ~ .
(59)
It is now easy to see that with the choice of e~ =
Ks ~'.~ R". ~,llell 2 + ~4~(e) + ~3 In consequence, the solution is 0 ~< e <~ ~s-
(60)
(61)
The main result is a sufficient condition to guarantee stabilization of the system [eq. (50)]. Observing eqs (50) and (51), the effect of the nonlinearities on the closed-loop system [eq. (51)] can be effectively attenuated when a suitable parameter p or e is tuned. It is worthy of note that if the inappropriate parameter p or e is employed in the control law, it may produce the "peaking" phenomena (Marino et al., 1989) and may destroy closed-loop stability of the system. Here certain conditions are required on the bounds of the parameter p or e in order to avoid an interplay between the peaking, which is induced by the high gain control, and the nonlinearities in AA [eq. (51)]. Therefore, the perturbations [eqs (56) and (57)] have to be limited to certain ranges, which is equivalent to the condition in eq. (61) being satisfied. So that the high gain control is permitted and safe. APPLICATION T O A CSTR
To examine the applicability of the proposed methodologies, an example of composition control of a continuous stirred tank reactor (CSTR) will be demonstrated. In this application, we consider a CSTR in which a single first-order exothermic reaction is taking place. The modeling equations are written as follows (Ray, 1981).
vdCA=F(CA~--CA)--at* Vk°exp(-~----T)CA dT VpoCv-~ = poCvF(Ty - T) + V ( - AH)ko ×exp(-~-T)Ca-hA(T-
To). (62)
1434
YI-SHYONG CHOU and WEI WU
The dimensionless equations for this reactor system are given by
and
#,=--x, +Do(l--x,)exp(x2/(l+~))+d,
L~x,
=[-I- b" exp(x2/(1+ x-32"~')l~o//3 x [-x, + bo(l-x,)exp(x2/(l +~))]
~2--x2+BDo(l-x,)exp(x2/(l+~)) -
fl(Xz -
x2,o) +
flu + d2
where B, Da and fl are uncertain parameters, d~ is an inlet feed composition perturbation and d2 is an inlet feed temperature perturbation, which are both unmeasurable disturbances. Consider that the manipulated control variable u is an admissible deviation between the coolant temperature, To, and some reference value for the coolant temperature Too.xt is a dimensionless composition and x2 is a dimensionless temperature. These notations are defined in the Notation section and the detailed formulations are shown in Uppal et al. (1975) and Ray (1981). The system
A f ( x , Po, P) =
(/(
x 1
1+
-x2+/~D,,(1-x0 (67,
xexp(x2/(e+~))-fi(x2-X2,o)].
The feedback linearization of the CSTR system is well-defined as long as LaLIx I # O, that is, xl # 1. The desired operating conditions are (x], x'2,u')= (0.2 1.33,0.42), while the model parameters /1 = 0.3, /~ = 8,/5° = 0.072, ¢# = 20, and X2¢o= 0 are selected. Furthermore, the plant-model mismatch between eqs (63) and (64) can be expressed as the form
I
+~
equations without uncertain parameters and disturbances can be written in the following nominal form: = f ( x , Po) + g (x, po)u
+ [Do(I - 1,)exp(x2/(l + ~))
(63)
and Ag(x, po, p ) =
(64)
[o]
(68)
f l _ fl + d2 "
where
[
]
and CDMC
#(x, po) = [fl 1 .
(65)
Here we assume that Po = (Da,/~, fl) are known nominal parameters. The control objective is to regulate the reactor outlet composition. Available as the controlled output is the composition xl. Based on eq. (7) we obtain the nonlinear control law as the form of u = ( L g L : x l ) - 1 (v - L}xl)
where
(66)
If we consider that only the system parameter fl is uncertain, that is, fl ~ fl, but Da = Do and B =/~, and also that only the inlet feed temperature has a perturbation, that is d2 # 0 but, dl = 0, then the uncertainties in eq. (68) can be reduced to the case of matched uncertainties, A f ( x , Po, P) = #(x, po)Af*(x, Po, P)
= [~-] ( ~ ) ( x 2
- x2co)
(69)
-
-x
fl
Hence, the proposed feedforward-like compensator based on eqs (13) and (14) is shown as v = r~ - kl(Xx - x~) - k 2 [ A ( x , po) - A(x', po)] + o,
Robust controller design for uncertain nonlinear systems and
1435
where (70)
v~ = ( L } x O q l -- ( L g L f x l ) q 2 .
These proposed results will be implemented on the structure of IMC. Thus, we select the rd as an external regulator on top of IMC structure for observing the plant-model mismatch output and for guaranteeing the asymptotic output tracking. Let ~ = Ac~ + bra
and
(71)
am" [Yra= C~
×I--Xl+Da( 1 --xl)exp(x2f(lq-~))]
be the internal model. ~s R" is the state variable of the model and Ym is the output of the model. Then, the IMC control law based on an inverse model is described as Gl: rd =
s" + k,s " - I + ... + kl
(2s + 1)"
6
(72)
where 2 is a scalar constant as a filter parameter and s is Laplace operator. Here 6 represents 6 = Y~vy + y~ where Y~v is a set-point. This overall control scheme is shown in Fig. 1. CDAMC
Suppose that the parameters p = (D,, B,/3) are uncertain parameters, and d~ and d2 are unmeasurable disturbances. In this case, the uncertainties in eq. (68) do not satisfy the matching conditions in eq. (10). According to the previous analysis, we need to define a parameter diffeomorphism, that is T * = pxl,
T ] = p f l ( x , po)
(73)
for p > 0. Following the conditions in eqs (42) and (43), the constraint
Zat, pxl =Pl)a(1--xl)exp(x2/(l d-~)) has to hold in the neighborhood of the operating points x ~ = Ix], x~2] r. Then we can construct a parameterized state feedback control law based on eq. (49) and p = e-~ as follows: u = (LoLiT*)-X[rd - kle-l(xl
- x])
- k2e- l(fl (x, Po) - f l (x~, Po)) - L } T*]
(75)
Based on the IMC structure, here rd is denoted as a regulator [eq. (72)] for the output regulation. The control scheme is shown in Fig. 2. Let us consider the operating points x ~ = [xl, x 2 ] •~ r as initial conditions and for a practical consideration the manipulated input is bounded with lul ~< 10. The system parameters are perturbed by varying operating conditions and aging process, so we set D a =/),(1 + 60, B =/3(1 + 62) and 13 = fi(1 + ~3), where 6i, i = 1, 2, 3 are the percentage of perturbations. The appropriate pole assignment feedback gains (kl, k z ) = (4,4) are selected and fixed, which should satisfy the Lyapunov equation [eq. (19)]. We present the promising performance of the control schemes as described in the CDMC and CDAMC. In essence, we lay emphasis on the merits of the effects of different tuning parameters (q~, q2, e) on the closedloop system by means of numerical simulations including set-point tracking and disturbances rejection.
Disturbance rejection, First, for the CDMC problem, suppose that the system parameter fl has the maximum variation of 63 = 50%. When the disturbance input d2 = 0.5 enters the system [eq. (63)'I, we would like to evaluate the disturbance rejection capability of the designed control system. A comparative
l
~
d
CSTR I System
x
Iz
=lr~Y.,,.
°
[ • ......... , - 2 i Fig. 1. The proposed control scheme implemented on the structure of IMC.
YI-SHYONG
1436
CHOU
and
WU
WEI
System~ CSTR
Ii_
x
--
I~1
Fig. 2. The proposed control scheme implemented on the structure of IMC.
plot illustrating the advantage employed by the proposed method [eq. (70)] with a filter parameter 2 = 1 [eq. (72)] is depicted in Fig. 3. Here, we observe the output response of the plant-model mismatch (y - y,,) [see, Fig. 3(B)] to estimate a set of appropriate parameters that make the mismatch smallest. Through trial and error by several computer simulations, the solid line in Fig. 3(A) and (B) shows the best performance when the parameters (ql, q:) = (3, 3) are used. Second, for the C D A M C problem, suppose that the system parameters p = (D~, B, fl) have the maxi m u m variation of 5~ = 10%, i = 1, 2, 3. Considering that the external disturbances (d~, d2) = (0.I, 0.1) enter the system, we would evaluate the disturbance rejection capability of the designed high gain control. Fig. 4(A) shows that the suitable value of e can effectively
0.o0 020
\ #^, \ f , \
~ 0.24
r
',
- - -
~=1
--
a=tO
\
o 0.~2 o.2o 0,8 Time Fig. 4(A). Response to disturbance rejection for CDAMC with different tuning parameters ~.
2.O 0.24
/
\
/ 0,23
Iqz,qz)=(O,O)
~,
~qt,q~)=( 1,1 )
~ \
/
(q,,q=)=(3,3) . . . . . . . . . (qDq=)=(5,5)
•
0.0
L
~:~ -2.0
o.22 //
C)~ 0.2!
'd
I{
.~ -0.02
'~',
- - -
~=1
.....
C=2
- •..
0,20
,
< ~-2
-~
~=I0
................... -10.0
x. ,..
0.19
-12.0
Time
o.10
Time Fig. 4(B). The corresponding manipulated input u. Fig. 3(A). Response to disturbance rejection for CDMC with different tuning parameters (q~, q2).
f I
0.08 -
\
/
/
I
E I
;:~
-
0.o0 -
"
I, 0.02
/
\
0.04-
-I
-
s=l
~=10
"'- .....................................
/i" 0.02
.... ,.,
.__ ",,
"',. ",-,.
. . . .
.
.
5 ~
.
.
.
'
.
'
.
.
'
1'o
.
.
. . . .
(q,,qz)=(O,O)
o,oo
/ q , , q z ) = ~ 1,1 ) ~q~,qz)=(3,3) (ql,q2)=(5,5)
1'5
. . . .
-0.02
....
20
~ ....
tb . . . . Time
~'5 . . . .
zo
Time
Fig. Fig. 3(B). The corresponding plant-model mismatch y - ym.
4(C). The
corresponding plant-model Y - Ym.
mismatch
Robust controller design for uncertain nonlinear systems attenuate the influences of system perturbations on the system output. Usually this method is better for the disturbance rejection problem, since it uses a high gain control to suppress the plant-model mismatch as shown in Fig. 4(C). The tuning parameter ~ can be adjusted on-line, since adjusting e is equivalent to adjusting the speed of the closed-loop response. Due to its directness and intuitive appeal it is conceivable that e is left for on-line adjustment by the operating personnel. The key point of this method is that the parameter diffeomorphism [eq. (73)] is used as a high gain technique to robustify the cancellation of the nonlinear terms [eq. (68)]. Note that the presented method based on CDAMC does not require the exact knowledge of the structures of uncertainties and disturbances. From the practical control point of view, this method is suitable for the process control problem.
Set-point tracking. Here, for the CDMC problem, we also consider that the system has the maximum variation of 63 = - 50% and the disturbance input is d2 = 0.5. Simulation results of a set-point change from 0.2 to 0.3 are shown in Fig. 5 for the case of only feedback linearization on top of IMC structure with filter parameters 2~ = 0.3, 2, = 1, and 2~ = 1.5. Let us now compare the merits of the proposed control
1437
method with the only feedback linearization on top of IMC structure. Simulation results of the presented method (CDMC) with the parameters (qx, q2) = (3, 3) are shown in Fig. 6. For all values of the filter parameter 2, the performance of the proposed method is far better than the implementation of IMC structure with feedback linearization transformations. We can observe less overshoot and smoother controller moves. Figure 7 depicts the plant-model mismatch. It is evident that the obvious improvement is made by our proposed method with (ql, q2)= (3,3). Notice that in Fig. 7 the (q~, q2) = (0,0) indicates the case of only IMC structure with feedback linearization transformations being used. For the CDAMC problem, suppose that the system parameters p = (D,,B, fl) have the maximum variation of 6~ = - 2 0 % , i - 1,2,3 and the external disturbances are (dx, d E ) = ( - 0.1, -0.1). Simulation results of set-point tracking from 0.2 to 0.3 are shown in Fig. 8. In Fig. 8(A), the solid line shows the performance of the present proposed method with different adjustable parameters being tuned. Notice that these simulation results show the inverse responses. Since the system output profile from the old equilibrium point [0.2, 1.612] falls firstly towards one of the other equilibrium points [0.03, 0.35] when the perturbed terms, including parameter variability and external disturbance, enter the system,
0.34 0.32
..~ 0.30
. ~ 0.30
.~ 0.20
~
o
0
0.26
tI
- - - X=I.5 . . . . . X=l
li :)
. . . .
-
~,
. . . .
~b . . Time
-
~'s
0.20 0.24
~=0.3
. .
0.28
0.22 . . . .
0.20
~o
Fig. 5(A). Response to set-point tracking for only IMC with feedback linearization transformations under different filter parameters 2.
/ /1"/ / " " ' ' ' /
~ ~ ~
- X=l.5 . . . . . X=I ~=0.3
[ ,'/ ,'/ 5
10
15
Time
Fig. 6(A). Response to set-point tracking for the proposed method with (ql, q2) = (3, 3) under different filter parameters L
" • 1o.o
IO.O - - -
- -
~=1.5
-
5.0.
I~,
~
-
~=0.3
o.o
-5.o
....
~ ....
fo . . . . Time
~'s . . . .
Fig. 5(B). The corresponding manipulated input u. CES
50:9-E
X=l.5
. . . . . ~=1
. . . . . ~=I "~
.5.0
.~
0.0
~
-0.0
-
-
~=0.3
Time Fig. 6(B). The corresponding manipulated input u.
YI-SHYONGCHOU and WEI Wo
1438 0.12
•
/
,.
//
x\
0.10 • 0,00
t
/ I
o.o0
s
-0.08
,
% 0.04 -
----
X=
>..
X=l,(ql,q21=(3,3)
t 0.02
l,(qi,q2)=(O,O)
\ \
>'~-0.16
/
\
- \
~=1
- -
c:5
0.00 -0.02
-0.04
-
. . . .
~
. . . .
1'o Time
. . . .
1~
. . . .
Fig. 7. Comparison of the proposed method with the only IMC under the same filter parameter 2 = 1. The dashed line (ql, q2) = (0, 0): the only IMC. The solid line (ql, q2) = (3, 3): the proposed method.
/
/
/
~
/ /
,' ,'
0 0.20
/ /
,'
/
i
/
i
0.24
~: 1 E=2
.....
/
~=5
/
t, ; /
\ / . . . .
~ . . . .
1'o . . . .
Time
t's . . . .
so
Fig. 8(A). Response to set-point tracking for CDAMC with a fixed filter parameter 2 = I and different tuning parametrs
10.0
"~ 0.0 e~
---
~=1
-
~=5
..... -
£=2
0.o
.~
~-
20
. . . .
1'0
. . . .
1'5
. . . .
20
Time Fig. 8(C). The corresponding plant-model mismatch y - y=.
approach of the feedforward-like compensation method and the parameter diffeomorphism are given. The class of nonlinear systems are those which nominally admit input-state linearization. Two categories of uncertainties, the uncertain parameters and the structured uncertainties, are considered. Also two control strategies are proposed; a feedforward-like control design is used to achieve robust stability of the system with matched uncertainties, and a parameter diffeomorphism is presented as a high gain control technique to robustify the cancellation of unmatched uncertainties. In particular, the proposed control law with the approach of parameter diffeomorphism does not require the exact knowledge of the structured uncertainty. A simple parameter p or e can be adjusted on-line to suppress the influences of uncertainties on the closed-loop system. An illustrative example of a nonisothermal CSTR system demonstrates the applicability of the proposed methods implemented on the structure of I M C for disturbance rejection and set-point tracking. Simulation results show that the proposed methodologies are successfully applied to this system and their performances are satisfactory.
4.0
e~ ~
2.0
....
~, . . . .
t'0 . . . . Time
f5 . . . .
20
Fig. 8(B). The corresponding manipulated input u.
then the response upsurges to the desired level xl = 0.3 due to the control action. In Fig. 8(A), the solid line shows the shortest settling time under the tuning parameters E = 5 and 2 = 1. Figure 8(B) shows the corresponding control action and Fig. 8(C) indicates the corresponding plant-model mismatch. CONCLUSIONS The theory and application of feedback linearization of nonlinear systems with uncertainties by the
A CA Cp
CA$o CAr di E
Y,g F h ko K R T
To
NOTATION heat transfer area concentration of species A in reactor heat capacity base value of concentration used to produce dimensionless concentration reactant concentration in the feed disturbance input activation energy vector fields volumetric feed rate heat transfer coefficient specific rate constant adjustable feedback gains ideal gas constant temperature in reactor coolant temperature
Robust controller design for uncertain nonlinear systems base temperature of coolant reactant feed temperature base temperature of reactant feed diffeomorphism manipulated input external input to the inner loop of the control structure reactor volume vector of state variables vector of new state variables
Tco
r~ r~o T, U /)
V X
z, ~'
Greek letters feedback signal of I M C structure 3 AH heat of reaction adjustable parameter filter parameter of I M C structure internal model state variables adjustable parameter P density Po Dimensionless groups E (D=--
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B
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Da -- _ _
F
hA [3=-Fpo Cp X2c 0 ~
r,o-Tfo T~o
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F t=t*-V X1
_ CAr° -- CA CAyo
X2 _ T - Tfo q9
rfo dl = CAIo -- CA~ CAio
d2 - T f - Tfo ~o
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rfo T~- T~o - - q ~ Tfo
Mathematical symbols the absolute value of a scalar function I.I the Euclidean norm of a vector or matrix I1"11 2mi.(') the minimum eigenvalue of a matrix ¥ for all belongs to E o composition of functions real line, n-dimensional Euclidean space R, R"
1439
REFERENCES Adsbekun, K., 1992, The robust global stabilization of a stirred tank reactor. A.I.Ch.E.J. 38, 651-659. Alvarez-Gallegos, J., 1988, Application of nonlinear system transformations to control design for a chemical reactor. lEE PtD. 135, 90-94. Arkun, Y. and Calvet, J.-P., 1992, Robust stabilization of input-output linearizable systems under uncertainty and disturbances A.I.Ch.E.J. 38, 1145-1156. Barmish, B. R., Corless, M. and Leitmann, G., 1983, A new class of stabilization controller for uncertain dynamical systems. SIAM J. Control Optimization 21, 246-253. Bequette, B. W., 1991, Nonlinear control of chemical processes: a review Ind. Engn# Chem. Res. 30, 1391-1413. Calvet, J.-P. and Arkun, Y., 1988, Feedforward and feedback linearization of nonlinear systems and its implementation using internal model control (IMC). Ind. Engng Chem. Res. 27, 1822-1831. Calvet, J.-P. and Arkun, Y., 1989, Robust control design for uncertain nonlinear systems under feedback linearization. IEEE Conf. Dec. Control 1, 102-106. Chen, Y. H. and Leitmann, G., 1987, Robustness of uncertain systems in the absence of matching assumptions. Int. J. Control 45, 1527-1542. Chou, Y.-S. and Wu, W., 1994a, Disturbance rejection for nonlinear control systems. J. Chin. I.Ch,E. 25, 163-171. Chou, Y.-S. and Wu, W., 1994b, Disturbance compensation for nonlinear processes. Chem. Engng Commun. (in press). Corless, M. and Leitmann, G., 1981, Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems. IEEE Trans. autom. Control 26, 1139-1148. Henson, M. A. and Seborg, D. E. 1991, Critique of exact linearization strategies for process contol. J. Proc. Control 1, 122-139. Hoo, K. A. and Kantor, J., 1985, An exothermic continuous stirred tank reactor is feedback equivalent to a linear systems. Chem. Engng Commun. 37, 1--10. Isidori, A., 1989, Nonlinear Control Systems, 2nd edn. Springer, Berlin. Kravaris, C. and Chung, C. B., 1987, Nonlinear state feedback synthesis by global input-output linearization. A.I.Ch.E.J. 33, 592-603. Kravaris, C. and Kantor, J. C., 1990, Geometric methods for nonlinear process control. Ind. Engng Chem. Res. 29, 2295-2323. Kravaris, C. and Palanki, S., 1988, Robust nonlinear state feedback under structured uncertainty. A.I.Ch.E.J. 34, 1119-1127. Liao, T. L, Fu, L. C. and Hsu, C. F., 1992, Output tracking control of nonlinear systems with mismatched uncertainties. Syst. Control Lett. 18, 39-47. Marino, R. Respondek, W. and van der Schaft, A. J., 1989, Almost disturbance decoupling for single-input single-output nonlinear systems. IEEE Trans. autom. Control 34, 1013-10t7. Morari, M. and Zafiriou, E., 1989, Robust Process Control. Prentice-Hall, Englewood Cliffs, NJ. Nijmeijer, H. and van der Schaft, A. J., 1990, Nonlinear Dynamical Control Systems. Springer, New York. Ray, W. H., 1981, Advanced Process Control. McGraw-Hill, New York. Sastry, S. and Isidori, A., 1989, Adaptive control of linearizable systems. IEEE Trans. autom. Control 34, 1123-1131. Slotine, J.-J. E. and Li, W., 1991, Applied Nonlinear Control. Prentice-Hall, Englewood Cliffs, NJ. Taylor, D. G., Kokotovic, P. V., Marino, R. and Kanellakopoulos, 1989, Adaptive regulation of nonlinear systems with unmodeled dynamic. IEEE Trans. autom. Control 34, 405-412. Uppal, A., Ray, W. H. and Poore, A. B., 1975, On the dynamic behavior of continuous stirred tank reactors. Chem. Engng Sci. 29, 967-977.