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output linearization
Computerschem. EngngVol.21, No. 8, pp. 891-903. 1997
Pergamon PII: S0098.1354(96)00307-9
CopyrightO 1997ElsevierScienceLtd Printedin GreatBritain.All rightsreserved 0098-1354/97S17.00+o.0o
Controller synthesis for time-varying systems by input/output linearization Srinivas Palanki a* and Costas Kravaris b a Department of Chemical Engineering, Florida A & M University/Florida State University, 2525 Pottsdamer Street, Tallahassee, FL 32310-6046, U.S.A.
b Department of Chemical Engineering, The University of Michigan, 2200 Hayward Street, Ann Arbor, MI 48109, U.S.A.
(Received 1 September 1994; revised 23 May 1996) Abstract Recently, there have been major developments in the nonlinear systems theory for the regulation of nonlinear timeinvariant systems. In this paper this methodology is extended to nonlinear time-varying systems. First, the relevance of modeling systems in chemical engineering with time-varying parameters is shown. Then, modified Lie derivatives are defined to account for the explicit time dependence of the system model on time. A time-dependent invertible coordinate transformation is derived to quantify the zero dynamics of the system. State feedback laws are synthesized, which provide a linear time-invariant input-output response. It is observed that the internal stability of this state feedback is not analogous to the results of the time-invariant case. A dynamic output feedback controller for the time-varying system is synthesized. This methodology is used to derive feedback laws for the regulation of linear time-varying systems. Finally, the proposed methodology is illustrated by a simulation example. © 1997 Elsevier Science Ltd
Introduction Recently, there have been major developments in the nonlinear systems theory literature using techniques from differential geometry (see Isidori, 1989). Brockett (1972) provided the first glimpse of the potential of differential geometric methods for designing nonlinear controllers. Some of the most important results in the area include solutions of problems of reachability (Lobry, 1970; Krener, 1974), invertibility (Hirschorn, 1981a, b), input-state linearization (Jakubcsyk and Respondek, 1982; Su, 1982; Hunt etal., 1983)and input/ output linearization (Singh and Rugh, 1972; Claude et al., 1983; Isidori and Ruberti, 1984). These developments have been motivated by the geometric approach of linear systems theory (Wonham, 1979). Applications of these results have appeared in the chemical engineering literature (Hoo and Kantor, 1986; Kravaris and Chung, 1987). Hoo and Kantor (1986) developed a global feedback transformation using the results from Hunt et al. (1983), which linearized the state
response in an unstable biological reactor. A linear controller was computed using linear state feedback control. Using the input-output linearization approach, Kravaris and Chung (1987) developed the globally linearizing controller (GLC), where a nonlinear controller was designed by first using a state feedback to make the input-output relationship linear and then using an external linear controller around the input-output linear system. This two-step procedure was demonstrated numerically and experimentally in Soroush and Kravaris (1992). This approach was utilized to synthesize a dynamic output feedback controllers for minimum-phase nonlinear processes in Daoutidis and Kravaris (1992). All these results in the literature have focused on nonlinear time-invariant systems. Control of nonlinear time-varying systems has received little attention in the literature. However, complex systems are often described by low order time-varying models. This motivates the problem of designing controllers for nonlinear time-varying systems of the following form: Jcf ~x,t) + g(x,Ou
* To whom correspondenceshould be addressed.
y=h(x,t)
891
(1)
892
S. PALANKI and C. KRAVARIS
wherefand g are C ~ vector fields in R n × R, h is a C ~ scalar field on R n × R, and ue'~. We restrict ourselves to single-input-single-output (SISO) systems in this paper. However, the methodology can be extended easily to a system with multiple inputs and outputs.The basic question that we address in this paper is: do the results developed for time-varying systems differ from those of time invariant systems? l f so, what are the differences? The controller design methodology for time-varying systems will be developed in the lines of input-output linearization of nonlinear time-invariant systems. The differences and similarities between the two results will be identified clearly. It will be shown that, in the timevarying case, the definition of relative order has to be modified. Furthermore, the calculation of an invertible transformation and internal stability analysis differ from the time-invariant case. The results obtained from this analysis will be simplified to obtain explicit formulae for the regulation of linear time-varying systems. These results are expected to be useful in practical situations where complex systems are represented by low-order time-varying models as shown in the next section. Models with time-varying parameters A real system, placed in its real environment, could represent a very complex situation. Processes such as biological processes are characterized by a large number of individual reactions (of the order of 1000 reactions). A detailed mathematical model which accounts for each reaction of the system becomes too complex for practical applications such as bioreactor design and process control (Joshi and Palsson, 1988). A balance between physiological reality and simplicity calls for reduction in the complexity of the mathematical description to develop a tractable model that is of practical utility. For this reason, most complex chemical processes are described by simplified models by ignoring or lumping together some of the system dynamics. The parameters of these simplified models are constant under the assumption of constancy of certain environmental conditions such as pH, temperature, dissolved oxygen, etc. The model parameters change with time if the environmental conditions change with time. For instance, Chu (1987) considered the effect of temperature and pH on the parameters of a simple model for the production of cephalosporin C by glucose. The utilization of glucose was modeled as: dS dt
m_
1 Iz~SX Ys (Ks+S)
where #~, Ks and Y, are the parameters of the model and are constant for a given pH and temperature. By conducting experiments at various temperatures and pH, Chu found that these parameters were strong functions of temperature and pH. Using Pontryagin's Minimum Principle, the optimal pH and temperature profiles to maximize the yield of cephalosporin C were calculated. When the fermenter is operated at this pH and tern-
perature profile, the parameters /z m, K~ and Y~ become time-varying,Model parameters can become time-varying by model reduction in some specific situations. For instance, consider the following second order reaction in a batch reactor catalyzed by a catalyst E: A . - , B. The catalyst is subject to a first-order decay. We can write a mathematical model for this process as follows: dCA = -- a.kA.C2A
dt da - - = - a'kE dt where a is the activity of the catalyst. This set of two equations can be reduced to one equation by integrating the balance for the catalyst activity, a, and substituting in the balance for species A. This results in: dCa = _ kA.(exp( _ kEt)).C2A. dt Note that the right-hand side of the above equation depends explicitly on time, whereas the original system of two equations is time-invariant. Furthermore, one can consider the term kA.(exp( - kEt)) as a lumped parameter k(t) which is time-varying. Thus, model reduction by eliminating the equation for catalyst activity leads to a time-varying parameter in the balance for species A. Matsumura et al. (1981) used this concept to develop a model for cephalosporin C with a time-varying parameter to account for the effect of an enzyme whose activity decreased with time. Similarly, Staniskis and Levisauskas (1984) described the production of a-amylase in a batch fermentation by a nonlinear model with timevarying parameters. Thus, we observe that simple models developed for process control applications could consist of timevarying parameters. The time-varying parameters contain information about the process dynamics. This information could be used in controller design to provide a more effective control. This provides the motivation for developing a nonlinear time-varying controller methodology for systems described by equation (1). Mathematical preliminaries
Differential geometry provides the necessary nonlinear analysis tools for nonlinear controller synthesis. The so-called Lie algebra is a nonlinear analog of matrix algebra and provides a compact way of representing a sequence of additions, multiplications and differentiations. The standard Lie derivative for a time-invariant system: ~=y~x) + g(x)u y=h(x) is defined as:
(2)
Controller synthesis for time-varying systems
L~h(x)=(x) .
Oh
= ,~, f ~x~ (x).
(3)
Based on this definition, higher-order Lie derivatives can be defined as:
L~h(x)=(x); k= 1,2 .....
893
to the ( r - l)th time derivative results in the generation of successive modified Lie derivatives of the system• Furthermore, it is clear that the relative order of the timevarying system is the smallest time derivative of the output with respect to time which depends explicitly on the input.
Zero dynamics
However, this notation is inadequate for time-varying systems and modifications need to be made to this notation to explicitly account for the time variability of the model. To account for time variability of the model, modified Lie derivatives are defined as follows:
L °h(x,t)=h(x,t)
In this section, we compute the zero dynamics of the time-varying system• These will be used later on to study the stability of the closed-loop system under input/output linearizing state feedback.
Invertible transformations Consider the time-varying coordinate transformation fromR" × R t o R " × R o f t h e form:
Oh Z~h(x,t)=(x,t)+ ~ (x,t)
(, = 01(t,x,,x2,...,x,)
~,=02(t.v,:2...~.) ~'.= 0,(t,x,,xv...,x,)
0 L~h(x,t) L 2h(x't)=(x't)+ Ot
(7)
where O,. 02 ..... O. are scalar fields on R" × R. Lemma 1. The transformation (7) is invertible if and
only if the row vectors of the Jacobian matrix: L~h(x,t)=(x,t)+ O L2h(x,t)
(4)
L~h(x,t)=(x,t)+ 0 L~-lh(x,t).
00,
00,
O0, -
Oxl
Ox2
Ox.
ao2
o02
a~
OxL
Ox:
Ox. (8)
Relative order Definition 1. For a system of the form (1), the relative order of the output y with respect to the manipulated input u is the smallest integer r for which
LgL~- Ih(x,t)#O
ao.
ao.
oo.
OxI
Oxz
Ox,
(5)
for all t • [to, o0]. If such an integer does not exist, we say that the system has an infinite relative order. Successive differentiations of the output with respect to time yield the following expressions:
y=h(x,t) dy =L~h(x,t) dt
are linearly independent. Proof. See Appendix A. Lemma 2. Consider the system represented by equation (1) with relative order r. The scalar fields h(x,t),L~h(x,t),L~(x,t),...,L~-'h(x,t) are linearly independent. Proof. See Appendix B. An immediate consequence of Lemma 2 is that the ( r - 1 ) × n matrix:
dt 2
Oh Ox~
Oh Ox.
dr-iy
a~h
aLh
dtr- i =Z ~- Jh(x,t)
ax~
Oxn
bL~-2h
aL~-2h
Ox~
c)xn
d2~Y =L~h(x,t)
dry =L ~(x,t)+ (L~L ~- lh(x,t))u. dtr
(6)
The reason for modifying the standard Lie derivative notation is obvious from the above expressions; successive differentiation of the output with respect to time up
has rank ( r - 1).
(9)
894
S. PALANKIand C. KRAVARIS
We can always rotate the indices of the state variables x~ to x. such that:
Oh Ox._,+ ~
Oh Ox._ j
Ox._.÷,
Ox._,
Under the coordinate transformation (11 ), system (l) can be represented in the following form:
(I =F,((,t)
(1o)
oL[-2h Ox._.÷,
L_,=F,-,((,O L-,+, =~o-,+~
oL[-2h Ox._,
Also, by rotation, we can assure that the nth component of the vector g(x,t) is non-zero. Byrnes and Isidori (1985) derived an invertible coordinate transformation for time invariant nonlinear systems which provides a minimal order realization of the inverse of the nonlinear system. This result can be extended easily to time-varying systems as shown by the theorem below. Note that, in this case, the coordinate transformation is time-varying. The existence of a finite relative order for all t~[to, oo] ensures that the coordinate transformation is well behaved.
(13)
L= ~,t)+G((,Ou y=(.-,.~
where:
Theorem 1. Consider the system represented by equation (1) with relative order r. If r is strictly less than n, it is always possible to find ( n - r ) functions tl(x,t),t2(x,t) ..... t, - r(x,t) such that the mapping: 8t.. F i= < d t f f > + --'
t,(x,t)
Ot
i= 1,2 ..... n - r
t._,(x,t) (= T(x,t)=
h(x,t)
L~h(x,O
(11)
¢~--L~x,t) L[-2h(x.O LV 'h(x,O has a Jacobian matrix which is non-singular and therefore qualifies as an invertible coordinate transformation. Furthermore, it is always possible to choose h(x,t), [~(x,t) ..... tn_~(x,t) such that: Ltti(x,t) =O for all 1 <-i<-(n - r) and for all x. (12) Proof. See Appendix C.
G=l~i,~- '(x,t).
Corollary 1. Consider a system of the form (1) with relative order r which has been transform~l into the form (13). Then the dynamic system:
Controller synthesis for time-varying systems
895
which is minimum phase and has finite relative order, r. It is desirable to find a state feedback of the form:
dy d2y dr-ty ~=F~ t,z~..... Zn-~y, dt ' dt'- " ' " dt ~ [ J
u= A(x,t)+ p.(x,t)v
(18)
where v is an external input, so that the closed-loop system:
• // dy d2y d ' - I y "~ z,= F,.~ t,ZD..... Zn-~Y, -~ , - ~ ..... dt,_l
)
.t = [f(x,t) + g(x,t),~(x,t) ] + g(x,t)l.*(x,t) ]v
(14)
~,_~=F._r(t,z, ..... z,_~y, dy d2y d~-IY) dt ' dt" ..... dt'-t
y = h(x,t)
(I 9)
1. is time invariant 2. has linear input/output dynamics.
d'Y - ~ ( t , Z , ..... Z,_~y, dy d2y d~-'y ) dt ~ dt ' d t 2 ..... d t ' - t
The result is summarized in the following theorem. Theorem 2. The state feedback law:
U=
dy d2y d r-ty ) G t,z~..... Z,-r,Y, d t ' ---~ dt ..... dt r-~
v-
is a minimal order realization of the inverse of the nonlinear system (1).The notion of forced and unforced dynamics are determined by considering the first n - r equations of the above normal form. This definition is completely analogous to the time-invariant case (Kravails, 1988); however, the zero dynamics will be explicit functions of time. This will have implications in deriving local stability results as will be shown in next section. Forced zero dynamics. The forced zero dynamics of the system (I) are given by: zl = Fl(t,zt ..... z.-r,U~ ..... U,) ~2= F2(t,Zl ..... Z. . . . U i..... Ur)
(15)
~°_,=F._,(t,z~ ..... Zn-r,U, ..... UO Unforced zero dynamics. The unforced zero dynamics of the system (1) are given by: z l =Fl(t,zl ..... z,_~,O ..... O) ~2=Fz(t, Zj ..... Zn_.O ..... 0)
(16)
~n-r=F,-,(t,Z~ ..... Z,-r,O ..... O)
State feedback
Consider a nonlinear time-varying system of the form:
fl~LgZ[-'h(x,t)
(20)
induces linear, time invariant closed-loop input/output dynamics described by:
flk dky k=O -d~ =V.
(21)
Proof. The proof is an immediate consequence of the definition of relative order. Note that the relative order is a function of both x and t. As in the case of time invariant systems, it is possible to find singular points in the phase plane where LsL~h(x,t)=O for all k>0. At these points, there will be loss of controllability and care must be taken to avoid these points during controller design. Furthermore, since the relative order is time-dependent, it is possible to find a time interval (instead of a singular point), where LsL~h(x,t)=O for all k>0. In this time interval, the system fails to have a well-defined relative order. In such cases, one has to resort to approximate linearization techniques such as extended linearization or pseudolinearization. Wang and Rugh (1989) and Reboulet and Champetier (1984) showed that a nonlinear system which fails to have a well-defined relative order could be approximated by a linear system or a family of linear systems that are input-output linearizable. Hauser et al • (1992) provided a methodology for constructing an approximate nonlinear system that is input--output linearizable for systems with ill-defined relative order. These approaches can be used for the time-interval over which the relative order of the time-varying system becomes infinite. Linear controller design with state feedback
± =fix,t) + g(x,t)u y=h(x,t)
u=
E l~,Lkh(x,t)
k=O
(17)
We saw in the previous section that under the state feedback law, equation (20), the time-varying nonlinear
896
S. PALANKI a n d C. KRAVARIS
system (1) is transformed into a time-invariant linear input-output system (21). Note that the parameters ~ , fit ..... /~r are completely arbitrary, which means that the v-y system can have arbitrarily placed poles and, therefore, prespecified bounded-input/bounded-output (BIBO) stability characteristics. An external controller with integral action such as a PI controller:
(22) can then be used for offsetless tracking of the desired trajectory yd(t). The resulting structure is called the GLC structure. Thus, the design procedure using the GLC structure is extremely simple and can be summarized as follows: 1. Compute the linearizing state feedback, equation (20) from the system model. 2. Select flo, fit, /~ ..... 3~ so that the poles of the transformed system are stable and "fast". 3. Tune a PI controller (or any appropriate linear controller) for the linear time-invariant v--y system.
lim ~i(t)=0 i= 1,2..... n - r. t---*ee
(24)
Under an appropriate choice of fl~, the output y and its successive time derivatives will get arbitrarily close to zero in finite time. Thus, for practical purposes, the internal stability will depend on the stability characteristics of the unforced zero dynamics of the open-loop system. Local stability results can be obtained by examining the stability of the linearized system. For the timeinvariant case, it is sufficient to check for local asymptotic stability of the unforced zero dynamics to ensure local asymptotic stability of the closed-loop system (Kravaris, 1988). However, as shown below, this result does not translate to the time-varying case. We first review the definition of iinearization of a timevarying system. Consider the nonlinear time-varying system:
~= F(t,z).
(25)
Suppose F(t,0)=0 t->0 and F(t,.) is continuously differentiable. Define:
Internal stability Theorem 3. Consider the system (I) under the state feedback (20) and assume that flo, BJ, 1~..... ~r have been chosen such that the roots of the polynomial ~,s ~ + ... + 3 ~ s + ~ are in the open left-half plane. The unforced closed loop system will be internally stable if the zero dynamics of the open-loop system are stable. Proof. Under the state feedback transformation (20),
(26)
F*(t,z)=F(t,z) - A(t)z.
(27)
and
the system (1) is transformed into the following form: ~',=F.((.t)
A(t)=I OF(t'z) ]f=o
If the following relation holds:
lim s,u~a
llF*(t,z)ll
=0
(28)
L-,=F.-,(~,t) then the system: (23)
~=A(Oz
: ° = - N :n-''
'
3,-.(.
N
y=~'._,+j Since the roots of the polynomial/3~sr+ ,.. + fl~s+flo are in the open left-half plane, it is clear that the modes (._r+l to ~'. will be stable. Thus, closed-loop stability will depend on the forced zero dynamics of the openloop system. The closed-loop system will be internally uniformly asymptotically stable if the forced dynamics are stable in the sense that for any set of initial conditions g'(0) and any exponentially decaying inputs
U~,U~..... Ur:
(29)
is the linearized system of the nonlinear system (25) around the equilibrium point 0 (Vidyasagar, 1978).If a linearized system for the time-varying"nonlinear system exists, then local stability results can be obtained using Lyapunov's indirect method which states (Vidyasagar, 1978): If the equilibrium point 0 of the linearized system (29) is uniformly asymptotically stable over [0:o], then the equilibrium point of the nonlinear system (25) is also
uniformly asymptotically stable. Consider the system represented by (23) with an equilibrium point at 0, Assuming that relation (28) holds, the linearized system can be represented as:
~ffiA(t)~
(3O)
Controller synthesis for time-varying systems
897
where
A(t) =
OFi
OFt
aF~ a(,
aF~ a¢,
aF2 a¢.
aF~_,
aFo_,
~F,-r
(31)
a~ 0 0
0
0
0
0
0 0
1
&
BF In the case of time-invariant systems, local stability of the unforced zero dynamics guarantees the local stability of the closed loop system. However, in the time-varying linearized system represented by equation (30), the first (n - r) rows of the matrix A(t) are explicit functions of time. Because of this time dependence, local stability of the subsystem which represents the unforced zero dynamics of the lineaxized system does not guarantee that the linearized system represented by equation (30) is stable (as in the case of time-invariant systems). This is a direct consequence of the time-varying nature of the original nonlinear system. Thus, to obtain local stability results, it is not sufficient to check the stability of the zero dynamics; one has to check the stability of the linearized system represented by equation (30).
~=0
C'(+ fl-~(y~p- y) - ~o ~Lkh(w't) %
u=
=
where: -0
0
1
0
0
1
0 0
A*=
(33) 1 0
)'2 %
-- "YI
%
-0
Output feedback controller synthesis via GLC In many practical situations, on-line measurements of all the states are not available. In a typical situation, only the output is measured on-line and this motivates the problem of synthesizing nonlinear output feedback control laws for accurate tracking of set-points and effective rejection of disturbances. This synthesis problem can be tackled conveniently within the GLC framework, if the input/output linearizing feedback is combined with an appropriate state observer (Daoutidis and Kravaris, 1992).
Theorem 4. Consider a nonlinear system of the form (1), with relative order r. Then, the dynamic system: ~=A *¢+ b*(y~ - y)
¢v=J(w,t)+g(w,t)
C*¢+ ~' (y~p- y) - k~o 13,L~h(w,t) % B,LsL[-Ih(w,t)
(32)
/3z~L ~- ' h(w,t)
-
0
b* ~
•
(34)
1
%
C*= B0
/3,
3'r
t ..... I f i r _ l - / 3 r % ~
(35)
represents the (n+r)th order state space realization of a dynamic output feedback controller which induces the closed-loop input/output behavior: dky Y+ ~ ~A =Y~-
(36)
898
S. PALANKI and C. KRAVARIS
Proof. The proof is along the lines of Daoutidis and Kravaris (1992) and is given in Appendix D.
c~Ck-,(t)
Ck(t)= - -
Ot
+
Ck-,(t).A(t).
Linear time-varying systems It can be easily seen that the modified Lie derivatives The development of a rigorous "first principles model" often is not practical for describing complex systems. In this situation, one resorts to the development of empirical models directly from experimental data. When the system states can be measured or estimated frequently, a linear time-varying system is employed for system identification. For instance, Harmon et al • (1987) expressed the process dynamics of a fermentation by a linear time-varying parameter model. This motivates the development of control strategies for regulating processes described by linear time-varying models. The problem of calculating the inverse of linear timevarying systems has been studied in the literature and the applicability of a system inverse either implicitly or explicitly to numerous control problems has been demonstrated. Freund (1971) derived a full order process inverse for a linear time-varying system. Silverman (1968) showed that a reduced order process inverse of order ( n - r ) could be constructed by a time-varying coordinate transformation for a linear time-varying system. However, this method involves considerable matrix manipulations (Bell and Tsang, 1990). In this section, we use the results of the previous section to provide a simpler and more transparent procedure for the regulation of linear time-varying systems. Consider a SISO linear time-varying system described by:
are:
L °h(x,t)=Co(t)x L~h(x,t)=C,(t)x L~h(x,t)=C2(t)x L 3h(x,t)=C3(t)x
L~h(x,t)=Ck(t)x.
Applying Definition 1, we immediately see that for a system of the form (37), the relative order r of output y with respect to the manipulated input u is the smallest integer for which:
C,_ ,(t)B(t)~O.
(41)
The state feedback [equation (20)] is of the form:
u=
v - t~o ~,Ck(t)x " fl, C,_ ,(t)B(O
(42)
The output feedback controller becomes:
±=A(t)x+B(t)u y=C(t)x
(40)
~=A *~+ fl*(Ysp - Y) (37)
where A(t), B(t) and C(t) are matrices of dimensions n X n, n x 1 and n x 1, respectively. This is a special case of (1) for:
C*~+ fir (y,p - y) - ~ Ck(t)w 'Yr
fv=A(t)w+B(t)
tffiO
B,C,- ,(t)w
f(x,t)=A(t)x C*~+ ~ tYsv - Y) - ~0 Ck(t)w
g(x,t)=B(t) h(x,t)=C(t)x.
u=
(38)
#rCr- , (OW
(43)
where A*, b* and C* are defined by equations (33), (34) and (35)•
Define:
CoCt)=c(t) Zero dynamics of linear time-varying systems
c,(t)= -8C(t) - ~ +C(t).A(t)
C2(t)=
Consider a linear time-varying system of the form (37) with relative order r and linearly independent scalar fields C0(t)x, C,(t)x . . . . . C,-l(t)x. Assume that the components of the vector B are as follows:
OCl(t) at +Cl(t).A(t)
B(t)= [bt(O b2(t) ... b,(t)] r C3(t)=
OC2(t) +C2(t)'A(O Ot
(39)
(44)
where b.(t)@0. Then it can be easily verified that the transformation:
Controller synthesis for time-varying systems
899
e2= &r
OSt140. 0
The unsteady state mass balances are given by (see Notation section for explanation of symbols):
d dt
XI [I[ x2
=
&=x,-,-
SX” n
u
(48) (49)
Due to the time-varying nature of e,, the term Da, is a time-varying parameter. The system is of the form with:
5.-,+*=W)x
L-,=c,-,(0x 5”= crwx
x= :
XI [I [
1 --x, - Da+, +Du&
Du,x, -x2 -Da&-Dug;
x2 ;flx,t)=
Da& -x3
x3
is invertible and transforms equation (37) to the form represented by equation (23).
g(x,r)=
Illustrative example
Reaction mechanisms which involve successive reaction steps are common in many areas of chemistry. The system: A==B+C
1
0
y=x,
L - ,+I = G(e
HI (1) 1 [I +
Da.9 : - x,
X3
(45)
0
1 -x, -Da,x,+Da&
Da,x,-x2-Da&-Da.&
(6)
is a well-studied system in the presence of bifunctional catalysts (see, for instance, Froment and Bischoff, 1979). For instance, the isomerization of paraffin using platinum-silica-alumina catalyst is represented by the above reaction scheme. In this case, the platinum catalyzes the first reaction and the silica-alumina catalyzes the second reaction and neither catalyzes the reaction of the other component (Eley et al., 1962). Similarly, the conversion of penicillin N to deacetoxycephalosporin C to deacetylcephalosporin C in C. acremonium is catalyzed by the bifunctional enzyme expandaselhydroxylase (Skatrud er al., 1989). Due to the presence of separate reaction sites on the catalyst, the decay kinetics of these sites is usually different.In this example, we study a regulation problem in a system represented by the above kinetics where the reaction sites catalyzing the second reaction undergo substantial decay, whereas the reaction sites catalyzing the first reaction are stable in the time scale of operation. We assume that the rates of reaction for the scheme represented by equation (46) are given by:
It
can
be
easily
seen
;
0 1 0
that
.
(50)
=0
and
L&h(x,r)=2Da+~~#O. Thus r=2. Furthermore: E;h(x,r)=Du~& -x3 ~;h(~,t)=2Da~~(Da,x, -xz-Du& - (Dug; -x,)+x;
(51) - Da&)
%.
(52)
XI 1
The transformation:
T=
x3
(53)
[ Da& - xj
will transform the system into the form (13). Let xla, x2,, x,,, and u, be the operating point. Define: 5,=x, -.% 4s=x3
l3=Da3(4
-x:,1
-x3,
-
(~3
-x3,)
(54)
Then, it can be easily shown that the forced zero dynamics are given by
r,=k,e,C, -kze,Ci r,=k,e,Ci
(47)
f;=-&-Da,&+
‘2(12+6;).
(55)
3
where e, and es represent the activity of the two reaction sites on the bifunctiottal catalyst. The activity of the first site is constant whereas the activity of the second site follows second order decay kinetics from O-40 h due to sintering (Fogler, 1986). Thus:
Clearly, the above equation is stable for exponentially decaying inputs & and & and so the system (48) has stable zero dynamics. The performance of the dynamic output feedback
900
S. PALANKI a n d C. KRAVARIS
(equation 32) was evaluated by numerical simulations. The control objective was to regulate the output at y=0.500. The adjustable parameters were chosen as t o = l , fl~=3,~=2 and f12=y2=l, in order to obtain a critically damped response. The following controller designs were compared. 1. Dynamic control law obtained for time-varying system. 2. Dynamic control law obtained by assuming that the system is time-invariant and applying to time-varying system. 3. Dynamic control law obtained by assuming that the system is time-invariant and but updating the timevarying parameters online. Fig. 1 shows the simulated closed-loop response under the assumption that the model is perfect. This case is compared with the case where the time-varying parameter Da3 is assumed to be constant in the state feedback law. It is observed that the constant parameter control law is unable to bring the system to set point till t=40, after which the catalyst activity e 2 is virtually constant. This is because, in the time interval from t=0 to t=40, the parameter Da3 decreases continuously and a controller with constant parameters would require additional integral action to drive the system to set-point. Robustness analysis of the dynamic control law was performed through numerical simulations. It was assumed that there is an error of about 10% in estimating the time-varying parameter Da3 in the dynamic control law. Fig. 2 shows the time profile of the time varying parameter Da3 as well as the estimated profile used in the control law. Fig. 3 shows the simulated closed-loop response under this error in estimating the time-varying parameter. It is observed that an error of about 10% in the estimation of the parameter Da 3 leads to an error of only 0.6% in tracking the set-point. Note that the dynamic output feedback is a function of the time-varying parameter Da3 and its time derivative (see equation (52)). If the system states are sampled "fast enough",
0.55
ODa 3
the t e r m
at
will be small and can be
-
Exact Error of + 10% Error of .10 %
1.0
.., O
Ix. x, I ~
.
~
.
.
0.8 ~u .~
0.6
~
.4
["
0.2
........
0
5
10
15
20
25
30
35
40
45
Time Fig. 2. Time-v~u,'ying parameter Da 3.
0.55
-
-
L) 0.50
: "
0.45 o e~ o 0.40 ..~
f
=
0.35
o L)
0.30
......
. . . . . . Error of -10% Exact Linearization - - Error of + 10%
0.25
0.20 0
I
I
I
I
I
I
I
I
I
5
10
15
20
25
30
35
40
45
Time
Fig. 3. Robustness analysis.
0551 0.50
•--
..
,.~ 0.45 O o
0.40
'~
0,35 I"4
~--
Sampling Interval = 5
r..) 0.30
0.50
0.25
L) 0.45 o
0.20
0
..o 0.40
5
10
15
20
25
30
35
40
45
Time
0.35
Fig. 4. Comparison with adaptive control.
Exact Linearization - - -- With Constant Parameters . . . . . . Set Point
0.30 0.25 0.20 0
I
I
I
I
I
I
I
I
I
5
I0
15
20
25
30
35
40
45
Time Fig. I. Closed loop response for perfectmodel.
ignored. In this situation,the feedback is equivalent to deriving a feedback law by assuming constant parameters and updating this feedback law by new estimates of the time-varying parameter (as is done in adaptive control algorithms). We studied the effect of sampling time through numerical simulations. The results are shown in Fig. 4.
Controller synthesis for time-varying systems Assuming the same values of the adjustable parameters /3o, fl~, ~ , % and ~2 as in Fig. 1, we simulated the controller performance for the case where the timevarying parameter in the dynamic feedback law is assumed to be constant but updated with new estimates at certain sampling instants. In this simulation, we assume that at each sampling instant, we get a time averaged estimate of the time-varying parameter Da3 of the previous sampling interval. It is observed from Fig. 4 that, when the sampling is "fast enough" (sampling interval=l), the "adaptive" feedback law drives the system very close to the set-point. When the sampling interval is increased (sampling interval=5, 10), the system exhibits damped oscillations.
Acknowledgements Financial support by NSF Research Grant BCS-8912627 is gratefully acknowledged.
Nomenclature CA= concentration of species A in reactor CAF= feed concentration of species A CB= concentration of species B in reactor Cc = concentration of species C in reactor F= volumetric feed rate NBF= molar feed rate of species B k~= first-order rate constant k2, k3= second-order rate constants t* = time t= t*F/V xl = CA/CAF x2= Ca/CaF x~= Cc/C~F Dal= kleiV/F Da~= k2eiCg~/F Da3= k3e2Ca~/F
References Bell, D. J. and Tsang, W. (1990) Invertibility and decoupling of linear time-varying systems. Intl J. Contr. 52, 4 983 Brockett, R. W. (1972) System theory on group manifolds and coset spaces. SIAM J. Contr. 10, 265 Byrnes, C. I. and A. Isidori, Global Feedback Stabilization of Nonlinear Systems. Proc. 24th IEEE CDC, Ft Lauderdale, FL, p. 1031 (1985). Chu, W. B., Modeling, optimization and computer control of the cephalosporin C fermentation process. Ph.D. Thesis, Rutgers, The State University of New Jersey, New Brunswick (1987). Claude, D., Fliess, M. and Isidori, A. (1983) Immersion, direct et par bouclage, d'un systeme non lineaire dans un lineaire. C.R. Acad. Sci. Paris 296, 237 Daoutidis, E and Kravaris, C. (1992) Dynamic output feedback control of minimum-phase nonlinear processes. Chem. Engng Sci. 4% 4 837 Eiey, D. D., E W. Selwood and E B. Weisz, Advances in Catalysis, Vol. 13. Academic Press, New York (1962). Fogler, H. S., Elements of Chemical Reaction Engineering. Prentice-Hall, Englewood Cliffs, NJ (1986). Freund, E. (197 l) Design of time-variable muitivariable systems by decoupling and by the inverse. IEEE Trans. Automat. Contr. 16, 183
901
Froment, G. F. and K. B. Bischoff, Chemical Reactor Analysis and Design. Wiley, New York (1979). Harmon, J., Svoronos, S. A. and Lyberatos, G. (1987) Adaptive steady state optimization of biomass productivity in continuous fermenters. Biotechnol. Bioengng 30, 335 Hauser, J., Sastry, S. and Kokotovic, P. (1992) Nonlinear control via approximate input-output linearization: the ball and beam example. IEEE Trans. Automat. Contr. 37, 392 Hirschorn, R. M. (1981) Output tracking in multivariable nonlinear systems. IEEE Trans. Automat. Contr. 26, 593 Hirschorn, R. M. (1981) (A-B) Invariant distributions and disturbance decoupling of nonlinear systems. SIAM J. Contr. Optimiz. 19, 1 Hoo, K. A. and Kantor, J. C. (1986) Linear feedback equivalence and control of an unstable biological reactor. Chem. Engng Commun. 46, 385 Hunt, L. R., Su, R. and Meyer, G. (1983) Global transformations of nonlinear systems. IEEE Trans. Automat. Contr. AC-28, 24 Isidori, A., Nonlinear Control Systems, 2nd edition. Springer-Verlag (1989). Isidori, A. and Ruberti, A. (1984) On the synthesis of linear input-output responses for nonlinear systems. Syst. Contr. Lett. 4, 17 Jakubcsyk, B. and Respondek, W. (1982) On linearization of control systems. Bull, Acad. Polon. Sci. Ser. Sci. Math. 28, 60 Joshi, A. and Palsson, B. (1988) Escherichia coli growth dynamics: a three-pool biochemically based description. Biotech. Bioengng 31, 3 102 Kravaris, C. (1988) Input/output linearization: a nonlinear analog of placing poles at process zeros. AIChE J. 28, 935-945. Kravaris, C. and Chung, C. B. (1987) Nonlinear state feedback synthesis by global input output linearization. AIChE J. 33, 4 592 Krener, A. J. (1974) A generalization of Chow's theorem and the bang-bang theorem to nonlinear systems. SIAM J. Contr. 12, 43 Lobry, C. (1970) Controllabilit6 des systemes non lineaires. SIAM J. Contr. g, 573 Matsumura, M., Imanaka, T., Yoshida, T. and Taguchi, H. (1981) Modeling of cephalosporin C production and its application to fed-batch culture. J. Ferment. Technol. 59, 115 Reboulet, C. and Champetier, C. (1984) A new method for linearizing nonlinear systems: the pseudolinearization. Intl J. Contr. 40, 631 Siiverman, L. M. (1968) Properties and application of inverse systems. IEEE Trans. Automat. Contr. 13, 436 Singh, S. N. and Rugh, W. J. (1972) Decoupling in a class of nonlinear systems by state variable feedback. ASME Trans. J. Dyn. Sys. Meas. Contr. 94, 323 Skatrud, P. L., Tietz, A. J., Ingolia, T. D., Cantwell, C. A., Fisher, D. L., Chapman, J. L. and Queener, S. w. (1989) Use of recombinant dna to improve production of cephalosporin C Cephalosporium acremonium. BioTechnology 7, 477 Soroush, M, and Kravaris, C. (1992) Nonlinear control of a batch polymerization reactor: an experimental study. AIChE J. 38, 9 1429
902
S. PALANKIand C. KRAVARIS
Staniskis, J. and Levisauskas, D. (1984) An adaptive control algorithm for fed-batch culture. Biotechnol. Bioengng 26, 419 Su, R. (1982) On the linear equivalents of nonlinear systems. Syst. Contr. Lett. 2, 48 Vidyasagar, M., Nonlinear Systems Analysis. PrenticeHall, Englewood Cliffs, NJ (1978). Wang, J. and Rugh, W. J. 0989) On the pseudolinearization problem for nonlinear systems. Syst. Contr. Lett. 12, 161 Wonham, W. M., Linear multivariable control: a geometric approach. In Lecture Notes on Control and Information Science, 2nd edition, Vol. 10. SpringerVerlag (1979).
H(X)=h(x,t) L~H(X)=L~h(x,t) LkH(X)=L~h(x,t) (62)
L V IH(X) =L ~- 'h(x,t) and the relative order of the prolonged system is the same as the relative order of the original system.It can be shown easily (Isidori, 1989) that given ~b, a real valued function andfand g vector fields, all defined in R *+1,for any choice of s, k, r >0:
r
g(x)>= i
Appendix A
-I
,(r)i Lf
.
Proof of Lemma 1 Define x0= t and consider the "prolonged" vectors:
= 0
(=[(o (~ "" L]t
=(
X= [xo xl "'"x.] T
(56)
where (o=Oo=xo. Then, using (56), we can represent (7) and (o= Oo in compact form as: (= 0(X).
0
o
002 OXo
00, __ Oxt
00~ Ox.
005 CgXo
a~ Ox~
00,. axe, 50.
dH(X) 7 dL~H(X) I | [G(X) ad~G(X).., ad~F- 'G(X)] =
i
/ dL~-'n(X)]
0
]
o (58)
(65)
- I)'- '-JLGL~ - 'H(X)~O.
The abov conditions, all together, show that the matrix:
(57)
Using the inverse function theorem, the above transformation is invertible if and only if:
I'
(64)
and for all i,j such that i+ j= r - I:
0=[0o 0~ ... 0,] ~
det ~00(X) ]J -=d e t
(63)
Using the above expression and the definition of relative order, we have for all i,j such that i+j<-r - 2:
!
* * ,
i
J(66)
has rank r and thus the row vectors dH(X),dL~H(X),...,dL~F-tH(X) are linearly independent in R"+l. Using (62), we conclude that the scalar fields h(x,t),L~h(x,t),Lfh(x,t),...,L~-Ih(x,t) are linearly independent in R"×R.
00, Oxo
OX~
~Xn
This implies that the row vectors of (8) are linearly independent.
Appendix C Proof of Theorem 1
Appendix B
Consider the vector g. We can find functions fi,t2,...,t._ ~ in (x,t) such that:
Proof of Lemma 2
Ot~ Ott Ott ~xlgt+ ~x2g2+"'+ ~x g,=O
Define Xo=t and define the following "prolonged" vectors: X=[x0 x] r
Ot2 + Ot2
G(X)= [0 g(X,Xo)]T
Ot2
ox_g, ~g~+...+~g.=
F(X)=[1 J(x,ro)] T
0 (67)
(59)
and the "prolonged" scalar:
H(X)=h(x,xo).
(60) c~tn_ i
Then, the "prolonged" system can be written as:
X= F(X) + G(X)u y=n(x). It can be seen easily that:
~t. _ i
c3t._ i
Oxj- gl + ~-x2 g:+...+ - ax. - g:--O To prove that: (61)
dim( span {dh ,dt:..... dt. _ i} +span ldh,dL~h,...,dL~- Ih l )=n.
(68)
Controller synthesis for time-varying systems
903
We prove this by contradiction. Suppose that:
dim( span {dt l,dt,"..... dt, _ i} + span{dh,dL~h,...,dL~- 2h })
vmC'~+ ~ (Ysp- Y)(69) Then, it can be easily verified that the system:
~=A*/~+b*(y,p-y) (70)
This implies that
(span I dt l,dh ..... dr,_ i })" y(x,t) = 0 (span{dh,dL~h,...,dL~-'h }).~x,t)=0.
(71 ) (72)
From equations (67) and (71), we get:
y(x,t)=span{g}
aL~h
Oh
oL~h
v=C';~+ fl' (Ysp-Y) Yr represents a minimal state-space realization of: • I 9'k ~d% ~E /=
Oh
~0/~ d~k (Y~P- y)
oL~h -0 (74)
v - ~o/g,L~h(w,t)
fv=[(w,t)+g(w,t)
fl,LsL~f- ih(w,t ) r
In other words: (75)
This is a contradiction by definition of relative order. Thus, equation (68) is true. Since {dh,L~r-lh,...,L~-th} has dimension r, we can choose ( n - r ) vectors {t,, t: .... t._r} such that the vectors {tt,t,,...,t,_,dh.L~-'h,...,L~-~h} are linearly independent and thus qualify as an invertible transformation.
Proof of Theorem 4 Define the auxiliary variable:
-
v - kE=ol~,L~h(w,t) u= ~,L~L~-'h(w,t)
aL~-Ih + dL[-thg,+...+ oL~-'h g,=0 ax--~ g' Ox~ Ox,=0.
(78)
with input, the error ( y ~ - y ) and the output, the auxiliary variable v. which can be interpreted as a choice of the linear controller in the GLC structure. The other component of the controller becomes:
g' + -Tffx g'+"" + -Tffx. g"-
Appendix D
(77)
(73)
Substituting in equation (72), we get: Oh
(76)
(79)
which is the input--output linearizing state feedback law with the states reconstructed through an open-loop observer. When w(0)=x(0), it easily follows that w(t)=x(t). Then (79) induces exactly the behavior: dkY ,=0 l~, d 7 = v.
(80)
Combining (21) and (79), we easily obtain the desired closed loop input-output dynamics: • dky y+ k_~,y ~ =y~
(81)
Pergamon PII: S0098.1354(96)00307-9
CopyrightO 1997ElsevierScienceLtd Printedin GreatBritain.All rightsreserved 0098-1354/97S17.00+o.0o
Controller synthesis for time-varying systems by input/output linearization Srinivas Palanki a* and Costas Kravaris b a Department of Chemical Engineering, Florida A & M University/Florida State University, 2525 Pottsdamer Street, Tallahassee, FL 32310-6046, U.S.A.
b Department of Chemical Engineering, The University of Michigan, 2200 Hayward Street, Ann Arbor, MI 48109, U.S.A.
(Received 1 September 1994; revised 23 May 1996) Abstract Recently, there have been major developments in the nonlinear systems theory for the regulation of nonlinear timeinvariant systems. In this paper this methodology is extended to nonlinear time-varying systems. First, the relevance of modeling systems in chemical engineering with time-varying parameters is shown. Then, modified Lie derivatives are defined to account for the explicit time dependence of the system model on time. A time-dependent invertible coordinate transformation is derived to quantify the zero dynamics of the system. State feedback laws are synthesized, which provide a linear time-invariant input-output response. It is observed that the internal stability of this state feedback is not analogous to the results of the time-invariant case. A dynamic output feedback controller for the time-varying system is synthesized. This methodology is used to derive feedback laws for the regulation of linear time-varying systems. Finally, the proposed methodology is illustrated by a simulation example. © 1997 Elsevier Science Ltd
Introduction Recently, there have been major developments in the nonlinear systems theory literature using techniques from differential geometry (see Isidori, 1989). Brockett (1972) provided the first glimpse of the potential of differential geometric methods for designing nonlinear controllers. Some of the most important results in the area include solutions of problems of reachability (Lobry, 1970; Krener, 1974), invertibility (Hirschorn, 1981a, b), input-state linearization (Jakubcsyk and Respondek, 1982; Su, 1982; Hunt etal., 1983)and input/ output linearization (Singh and Rugh, 1972; Claude et al., 1983; Isidori and Ruberti, 1984). These developments have been motivated by the geometric approach of linear systems theory (Wonham, 1979). Applications of these results have appeared in the chemical engineering literature (Hoo and Kantor, 1986; Kravaris and Chung, 1987). Hoo and Kantor (1986) developed a global feedback transformation using the results from Hunt et al. (1983), which linearized the state
response in an unstable biological reactor. A linear controller was computed using linear state feedback control. Using the input-output linearization approach, Kravaris and Chung (1987) developed the globally linearizing controller (GLC), where a nonlinear controller was designed by first using a state feedback to make the input-output relationship linear and then using an external linear controller around the input-output linear system. This two-step procedure was demonstrated numerically and experimentally in Soroush and Kravaris (1992). This approach was utilized to synthesize a dynamic output feedback controllers for minimum-phase nonlinear processes in Daoutidis and Kravaris (1992). All these results in the literature have focused on nonlinear time-invariant systems. Control of nonlinear time-varying systems has received little attention in the literature. However, complex systems are often described by low order time-varying models. This motivates the problem of designing controllers for nonlinear time-varying systems of the following form: Jcf ~x,t) + g(x,Ou
* To whom correspondenceshould be addressed.
y=h(x,t)
891
(1)
892
S. PALANKI and C. KRAVARIS
wherefand g are C ~ vector fields in R n × R, h is a C ~ scalar field on R n × R, and ue'~. We restrict ourselves to single-input-single-output (SISO) systems in this paper. However, the methodology can be extended easily to a system with multiple inputs and outputs.The basic question that we address in this paper is: do the results developed for time-varying systems differ from those of time invariant systems? l f so, what are the differences? The controller design methodology for time-varying systems will be developed in the lines of input-output linearization of nonlinear time-invariant systems. The differences and similarities between the two results will be identified clearly. It will be shown that, in the timevarying case, the definition of relative order has to be modified. Furthermore, the calculation of an invertible transformation and internal stability analysis differ from the time-invariant case. The results obtained from this analysis will be simplified to obtain explicit formulae for the regulation of linear time-varying systems. These results are expected to be useful in practical situations where complex systems are represented by low-order time-varying models as shown in the next section. Models with time-varying parameters A real system, placed in its real environment, could represent a very complex situation. Processes such as biological processes are characterized by a large number of individual reactions (of the order of 1000 reactions). A detailed mathematical model which accounts for each reaction of the system becomes too complex for practical applications such as bioreactor design and process control (Joshi and Palsson, 1988). A balance between physiological reality and simplicity calls for reduction in the complexity of the mathematical description to develop a tractable model that is of practical utility. For this reason, most complex chemical processes are described by simplified models by ignoring or lumping together some of the system dynamics. The parameters of these simplified models are constant under the assumption of constancy of certain environmental conditions such as pH, temperature, dissolved oxygen, etc. The model parameters change with time if the environmental conditions change with time. For instance, Chu (1987) considered the effect of temperature and pH on the parameters of a simple model for the production of cephalosporin C by glucose. The utilization of glucose was modeled as: dS dt
m_
1 Iz~SX Ys (Ks+S)
where #~, Ks and Y, are the parameters of the model and are constant for a given pH and temperature. By conducting experiments at various temperatures and pH, Chu found that these parameters were strong functions of temperature and pH. Using Pontryagin's Minimum Principle, the optimal pH and temperature profiles to maximize the yield of cephalosporin C were calculated. When the fermenter is operated at this pH and tern-
perature profile, the parameters /z m, K~ and Y~ become time-varying,Model parameters can become time-varying by model reduction in some specific situations. For instance, consider the following second order reaction in a batch reactor catalyzed by a catalyst E: A . - , B. The catalyst is subject to a first-order decay. We can write a mathematical model for this process as follows: dCA = -- a.kA.C2A
dt da - - = - a'kE dt where a is the activity of the catalyst. This set of two equations can be reduced to one equation by integrating the balance for the catalyst activity, a, and substituting in the balance for species A. This results in: dCa = _ kA.(exp( _ kEt)).C2A. dt Note that the right-hand side of the above equation depends explicitly on time, whereas the original system of two equations is time-invariant. Furthermore, one can consider the term kA.(exp( - kEt)) as a lumped parameter k(t) which is time-varying. Thus, model reduction by eliminating the equation for catalyst activity leads to a time-varying parameter in the balance for species A. Matsumura et al. (1981) used this concept to develop a model for cephalosporin C with a time-varying parameter to account for the effect of an enzyme whose activity decreased with time. Similarly, Staniskis and Levisauskas (1984) described the production of a-amylase in a batch fermentation by a nonlinear model with timevarying parameters. Thus, we observe that simple models developed for process control applications could consist of timevarying parameters. The time-varying parameters contain information about the process dynamics. This information could be used in controller design to provide a more effective control. This provides the motivation for developing a nonlinear time-varying controller methodology for systems described by equation (1). Mathematical preliminaries
Differential geometry provides the necessary nonlinear analysis tools for nonlinear controller synthesis. The so-called Lie algebra is a nonlinear analog of matrix algebra and provides a compact way of representing a sequence of additions, multiplications and differentiations. The standard Lie derivative for a time-invariant system: ~=y~x) + g(x)u y=h(x) is defined as:
(2)
Controller synthesis for time-varying systems
L~h(x)=
Oh
= ,~, f ~x~ (x).
(3)
Based on this definition, higher-order Lie derivatives can be defined as:
L~h(x)=
893
to the ( r - l)th time derivative results in the generation of successive modified Lie derivatives of the system• Furthermore, it is clear that the relative order of the timevarying system is the smallest time derivative of the output with respect to time which depends explicitly on the input.
Zero dynamics
However, this notation is inadequate for time-varying systems and modifications need to be made to this notation to explicitly account for the time variability of the model. To account for time variability of the model, modified Lie derivatives are defined as follows:
L °h(x,t)=h(x,t)
In this section, we compute the zero dynamics of the time-varying system• These will be used later on to study the stability of the closed-loop system under input/output linearizing state feedback.
Invertible transformations Consider the time-varying coordinate transformation fromR" × R t o R " × R o f t h e form:
Oh Z~h(x,t)=
(, = 01(t,x,,x2,...,x,)
~,=02(t.v,:2...~.) ~'.= 0,(t,x,,xv...,x,)
0 L~h(x,t) L 2h(x't)=
(7)
where O,. 02 ..... O. are scalar fields on R" × R. Lemma 1. The transformation (7) is invertible if and
only if the row vectors of the Jacobian matrix: L~h(x,t)=
(4)
L~h(x,t)=
00,
00,
O0, -
Oxl
Ox2
Ox.
ao2
o02
a~
OxL
Ox:
Ox. (8)
Relative order Definition 1. For a system of the form (1), the relative order of the output y with respect to the manipulated input u is the smallest integer r for which
LgL~- Ih(x,t)#O
ao.
ao.
oo.
OxI
Oxz
Ox,
(5)
for all t • [to, o0]. If such an integer does not exist, we say that the system has an infinite relative order. Successive differentiations of the output with respect to time yield the following expressions:
y=h(x,t) dy =L~h(x,t) dt
are linearly independent. Proof. See Appendix A. Lemma 2. Consider the system represented by equation (1) with relative order r. The scalar fields h(x,t),L~h(x,t),L~(x,t),...,L~-'h(x,t) are linearly independent. Proof. See Appendix B. An immediate consequence of Lemma 2 is that the ( r - 1 ) × n matrix:
dt 2
Oh Ox~
Oh Ox.
dr-iy
a~h
aLh
dtr- i =Z ~- Jh(x,t)
ax~
Oxn
bL~-2h
aL~-2h
Ox~
c)xn
d2~Y =L~h(x,t)
dry =L ~(x,t)+ (L~L ~- lh(x,t))u. dtr
(6)
The reason for modifying the standard Lie derivative notation is obvious from the above expressions; successive differentiation of the output with respect to time up
has rank ( r - 1).
(9)
894
S. PALANKIand C. KRAVARIS
We can always rotate the indices of the state variables x~ to x. such that:
Oh Ox._,+ ~
Oh Ox._ j
Ox._.÷,
Ox._,
Under the coordinate transformation (11 ), system (l) can be represented in the following form:
(I =F,((,t)
(1o)
oL[-2h Ox._.÷,
L_,=F,-,((,O L-,+, =~o-,+~
oL[-2h Ox._,
Also, by rotation, we can assure that the nth component of the vector g(x,t) is non-zero. Byrnes and Isidori (1985) derived an invertible coordinate transformation for time invariant nonlinear systems which provides a minimal order realization of the inverse of the nonlinear system. This result can be extended easily to time-varying systems as shown by the theorem below. Note that, in this case, the coordinate transformation is time-varying. The existence of a finite relative order for all t~[to, oo] ensures that the coordinate transformation is well behaved.
(13)
L= ~,t)+G((,Ou y=(.-,.~
where:
Theorem 1. Consider the system represented by equation (1) with relative order r. If r is strictly less than n, it is always possible to find ( n - r ) functions tl(x,t),t2(x,t) ..... t, - r(x,t) such that the mapping: 8t.. F i= < d t f f > + --'
t,(x,t)
Ot
i= 1,2 ..... n - r
t._,(x,t) (= T(x,t)=
h(x,t)
L~h(x,O
(11)
¢~--L~x,t) L[-2h(x.O LV 'h(x,O has a Jacobian matrix which is non-singular and therefore qualifies as an invertible coordinate transformation. Furthermore, it is always possible to choose h(x,t), [~(x,t) ..... tn_~(x,t) such that: Ltti(x,t) =O for all 1 <-i<-(n - r) and for all x. (12) Proof. See Appendix C.
G=l~i,~- '(x,t).
Corollary 1. Consider a system of the form (1) with relative order r which has been transform~l into the form (13). Then the dynamic system:
Controller synthesis for time-varying systems
895
which is minimum phase and has finite relative order, r. It is desirable to find a state feedback of the form:
dy d2y dr-ty ~=F~ t,z~..... Zn-~y, dt ' dt'- " ' " dt ~ [ J
u= A(x,t)+ p.(x,t)v
(18)
where v is an external input, so that the closed-loop system:
• // dy d2y d ' - I y "~ z,= F,.~ t,ZD..... Zn-~Y, -~ , - ~ ..... dt,_l
)
.t = [f(x,t) + g(x,t),~(x,t) ] + g(x,t)l.*(x,t) ]v
(14)
~,_~=F._r(t,z, ..... z,_~y, dy d2y d~-IY) dt ' dt" ..... dt'-t
y = h(x,t)
(I 9)
1. is time invariant 2. has linear input/output dynamics.
d'Y - ~ ( t , Z , ..... Z,_~y, dy d2y d~-'y ) dt ~ dt ' d t 2 ..... d t ' - t
The result is summarized in the following theorem. Theorem 2. The state feedback law:
U=
dy d2y d r-ty ) G t,z~..... Z,-r,Y, d t ' ---~ dt ..... dt r-~
v-
is a minimal order realization of the inverse of the nonlinear system (1).The notion of forced and unforced dynamics are determined by considering the first n - r equations of the above normal form. This definition is completely analogous to the time-invariant case (Kravails, 1988); however, the zero dynamics will be explicit functions of time. This will have implications in deriving local stability results as will be shown in next section. Forced zero dynamics. The forced zero dynamics of the system (I) are given by: zl = Fl(t,zt ..... z.-r,U~ ..... U,) ~2= F2(t,Zl ..... Z. . . . U i..... Ur)
(15)
~°_,=F._,(t,z~ ..... Zn-r,U, ..... UO Unforced zero dynamics. The unforced zero dynamics of the system (1) are given by: z l =Fl(t,zl ..... z,_~,O ..... O) ~2=Fz(t, Zj ..... Zn_.O ..... 0)
(16)
~n-r=F,-,(t,Z~ ..... Z,-r,O ..... O)
State feedback
Consider a nonlinear time-varying system of the form:
fl~LgZ[-'h(x,t)
(20)
induces linear, time invariant closed-loop input/output dynamics described by:
flk dky k=O -d~ =V.
(21)
Proof. The proof is an immediate consequence of the definition of relative order. Note that the relative order is a function of both x and t. As in the case of time invariant systems, it is possible to find singular points in the phase plane where LsL~h(x,t)=O for all k>0. At these points, there will be loss of controllability and care must be taken to avoid these points during controller design. Furthermore, since the relative order is time-dependent, it is possible to find a time interval (instead of a singular point), where LsL~h(x,t)=O for all k>0. In this time interval, the system fails to have a well-defined relative order. In such cases, one has to resort to approximate linearization techniques such as extended linearization or pseudolinearization. Wang and Rugh (1989) and Reboulet and Champetier (1984) showed that a nonlinear system which fails to have a well-defined relative order could be approximated by a linear system or a family of linear systems that are input-output linearizable. Hauser et al • (1992) provided a methodology for constructing an approximate nonlinear system that is input--output linearizable for systems with ill-defined relative order. These approaches can be used for the time-interval over which the relative order of the time-varying system becomes infinite. Linear controller design with state feedback
± =fix,t) + g(x,t)u y=h(x,t)
u=
E l~,Lkh(x,t)
k=O
(17)
We saw in the previous section that under the state feedback law, equation (20), the time-varying nonlinear
896
S. PALANKI a n d C. KRAVARIS
system (1) is transformed into a time-invariant linear input-output system (21). Note that the parameters ~ , fit ..... /~r are completely arbitrary, which means that the v-y system can have arbitrarily placed poles and, therefore, prespecified bounded-input/bounded-output (BIBO) stability characteristics. An external controller with integral action such as a PI controller:
(22) can then be used for offsetless tracking of the desired trajectory yd(t). The resulting structure is called the GLC structure. Thus, the design procedure using the GLC structure is extremely simple and can be summarized as follows: 1. Compute the linearizing state feedback, equation (20) from the system model. 2. Select flo, fit, /~ ..... 3~ so that the poles of the transformed system are stable and "fast". 3. Tune a PI controller (or any appropriate linear controller) for the linear time-invariant v--y system.
lim ~i(t)=0 i= 1,2..... n - r. t---*ee
(24)
Under an appropriate choice of fl~, the output y and its successive time derivatives will get arbitrarily close to zero in finite time. Thus, for practical purposes, the internal stability will depend on the stability characteristics of the unforced zero dynamics of the open-loop system. Local stability results can be obtained by examining the stability of the linearized system. For the timeinvariant case, it is sufficient to check for local asymptotic stability of the unforced zero dynamics to ensure local asymptotic stability of the closed-loop system (Kravaris, 1988). However, as shown below, this result does not translate to the time-varying case. We first review the definition of iinearization of a timevarying system. Consider the nonlinear time-varying system:
~= F(t,z).
(25)
Suppose F(t,0)=0 t->0 and F(t,.) is continuously differentiable. Define:
Internal stability Theorem 3. Consider the system (I) under the state feedback (20) and assume that flo, BJ, 1~..... ~r have been chosen such that the roots of the polynomial ~,s ~ + ... + 3 ~ s + ~ are in the open left-half plane. The unforced closed loop system will be internally stable if the zero dynamics of the open-loop system are stable. Proof. Under the state feedback transformation (20),
(26)
F*(t,z)=F(t,z) - A(t)z.
(27)
and
the system (1) is transformed into the following form: ~',=F.((.t)
A(t)=I OF(t'z) ]f=o
If the following relation holds:
lim s,u~a
llF*(t,z)ll
=0
(28)
L-,=F.-,(~,t) then the system: (23)
~=A(Oz
: ° = - N :n-''
'
3,-.(.
N
y=~'._,+j Since the roots of the polynomial/3~sr+ ,.. + fl~s+flo are in the open left-half plane, it is clear that the modes (._r+l to ~'. will be stable. Thus, closed-loop stability will depend on the forced zero dynamics of the openloop system. The closed-loop system will be internally uniformly asymptotically stable if the forced dynamics are stable in the sense that for any set of initial conditions g'(0) and any exponentially decaying inputs
U~,U~..... Ur:
(29)
is the linearized system of the nonlinear system (25) around the equilibrium point 0 (Vidyasagar, 1978).If a linearized system for the time-varying"nonlinear system exists, then local stability results can be obtained using Lyapunov's indirect method which states (Vidyasagar, 1978): If the equilibrium point 0 of the linearized system (29) is uniformly asymptotically stable over [0:o], then the equilibrium point of the nonlinear system (25) is also
uniformly asymptotically stable. Consider the system represented by (23) with an equilibrium point at 0, Assuming that relation (28) holds, the linearized system can be represented as:
~ffiA(t)~
(3O)
Controller synthesis for time-varying systems
897
where
A(t) =
OFi
OFt
aF~ a(,
aF~ a¢,
aF2 a¢.
aF~_,
aFo_,
~F,-r
(31)
a~ 0 0
0
0
0
0
0 0
1
&
BF In the case of time-invariant systems, local stability of the unforced zero dynamics guarantees the local stability of the closed loop system. However, in the time-varying linearized system represented by equation (30), the first (n - r) rows of the matrix A(t) are explicit functions of time. Because of this time dependence, local stability of the subsystem which represents the unforced zero dynamics of the lineaxized system does not guarantee that the linearized system represented by equation (30) is stable (as in the case of time-invariant systems). This is a direct consequence of the time-varying nature of the original nonlinear system. Thus, to obtain local stability results, it is not sufficient to check the stability of the zero dynamics; one has to check the stability of the linearized system represented by equation (30).
~=0
C'(+ fl-~(y~p- y) - ~o ~Lkh(w't) %
u=
=
where: -0
0
1
0
0
1
0 0
A*=
(33) 1 0
)'2 %
-- "YI
%
-0
Output feedback controller synthesis via GLC In many practical situations, on-line measurements of all the states are not available. In a typical situation, only the output is measured on-line and this motivates the problem of synthesizing nonlinear output feedback control laws for accurate tracking of set-points and effective rejection of disturbances. This synthesis problem can be tackled conveniently within the GLC framework, if the input/output linearizing feedback is combined with an appropriate state observer (Daoutidis and Kravaris, 1992).
Theorem 4. Consider a nonlinear system of the form (1), with relative order r. Then, the dynamic system: ~=A *¢+ b*(y~ - y)
¢v=J(w,t)+g(w,t)
C*¢+ ~' (y~p- y) - k~o 13,L~h(w,t) % B,LsL[-Ih(w,t)
(32)
/3z~L ~- ' h(w,t)
-
0
b* ~
•
(34)
1
%
C*= B0
/3,
3'r
t ..... I f i r _ l - / 3 r % ~
(35)
represents the (n+r)th order state space realization of a dynamic output feedback controller which induces the closed-loop input/output behavior: dky Y+ ~ ~A =Y~-
(36)
898
S. PALANKI and C. KRAVARIS
Proof. The proof is along the lines of Daoutidis and Kravaris (1992) and is given in Appendix D.
c~Ck-,(t)
Ck(t)= - -
Ot
+
Ck-,(t).A(t).
Linear time-varying systems It can be easily seen that the modified Lie derivatives The development of a rigorous "first principles model" often is not practical for describing complex systems. In this situation, one resorts to the development of empirical models directly from experimental data. When the system states can be measured or estimated frequently, a linear time-varying system is employed for system identification. For instance, Harmon et al • (1987) expressed the process dynamics of a fermentation by a linear time-varying parameter model. This motivates the development of control strategies for regulating processes described by linear time-varying models. The problem of calculating the inverse of linear timevarying systems has been studied in the literature and the applicability of a system inverse either implicitly or explicitly to numerous control problems has been demonstrated. Freund (1971) derived a full order process inverse for a linear time-varying system. Silverman (1968) showed that a reduced order process inverse of order ( n - r ) could be constructed by a time-varying coordinate transformation for a linear time-varying system. However, this method involves considerable matrix manipulations (Bell and Tsang, 1990). In this section, we use the results of the previous section to provide a simpler and more transparent procedure for the regulation of linear time-varying systems. Consider a SISO linear time-varying system described by:
are:
L °h(x,t)=Co(t)x L~h(x,t)=C,(t)x L~h(x,t)=C2(t)x L 3h(x,t)=C3(t)x
L~h(x,t)=Ck(t)x.
Applying Definition 1, we immediately see that for a system of the form (37), the relative order r of output y with respect to the manipulated input u is the smallest integer for which:
C,_ ,(t)B(t)~O.
(41)
The state feedback [equation (20)] is of the form:
u=
v - t~o ~,Ck(t)x " fl, C,_ ,(t)B(O
(42)
The output feedback controller becomes:
±=A(t)x+B(t)u y=C(t)x
(40)
~=A *~+ fl*(Ysp - Y) (37)
where A(t), B(t) and C(t) are matrices of dimensions n X n, n x 1 and n x 1, respectively. This is a special case of (1) for:
C*~+ fir (y,p - y) - ~ Ck(t)w 'Yr
fv=A(t)w+B(t)
tffiO
B,C,- ,(t)w
f(x,t)=A(t)x C*~+ ~ tYsv - Y) - ~0 Ck(t)w
g(x,t)=B(t) h(x,t)=C(t)x.
u=
(38)
#rCr- , (OW
(43)
where A*, b* and C* are defined by equations (33), (34) and (35)•
Define:
CoCt)=c(t) Zero dynamics of linear time-varying systems
c,(t)= -8C(t) - ~ +C(t).A(t)
C2(t)=
Consider a linear time-varying system of the form (37) with relative order r and linearly independent scalar fields C0(t)x, C,(t)x . . . . . C,-l(t)x. Assume that the components of the vector B are as follows:
OCl(t) at +Cl(t).A(t)
B(t)= [bt(O b2(t) ... b,(t)] r C3(t)=
OC2(t) +C2(t)'A(O Ot
(39)
(44)
where b.(t)@0. Then it can be easily verified that the transformation:
Controller synthesis for time-varying systems
899
e2= &r
OSt140. 0
The unsteady state mass balances are given by (see Notation section for explanation of symbols):
d dt
XI [I[ x2
=
&=x,-,-
SX” n
u
(48) (49)
Due to the time-varying nature of e,, the term Da, is a time-varying parameter. The system is of the form with:
5.-,+*=W)x
L-,=c,-,(0x 5”= crwx
x= :
XI [I [
1 --x, - Da+, +Du&
Du,x, -x2 -Da&-Dug;
x2 ;flx,t)=
Da& -x3
x3
is invertible and transforms equation (37) to the form represented by equation (23).
g(x,r)=
Illustrative example
Reaction mechanisms which involve successive reaction steps are common in many areas of chemistry. The system: A==B+C
1
0
y=x,
L - ,+I = G(e
HI (1) 1 [I +
Da.9 : - x,
X3
(45)
0
1 -x, -Da,x,+Da&
Da,x,-x2-Da&-Da.&
(6)
is a well-studied system in the presence of bifunctional catalysts (see, for instance, Froment and Bischoff, 1979). For instance, the isomerization of paraffin using platinum-silica-alumina catalyst is represented by the above reaction scheme. In this case, the platinum catalyzes the first reaction and the silica-alumina catalyzes the second reaction and neither catalyzes the reaction of the other component (Eley et al., 1962). Similarly, the conversion of penicillin N to deacetoxycephalosporin C to deacetylcephalosporin C in C. acremonium is catalyzed by the bifunctional enzyme expandaselhydroxylase (Skatrud er al., 1989). Due to the presence of separate reaction sites on the catalyst, the decay kinetics of these sites is usually different.In this example, we study a regulation problem in a system represented by the above kinetics where the reaction sites catalyzing the second reaction undergo substantial decay, whereas the reaction sites catalyzing the first reaction are stable in the time scale of operation. We assume that the rates of reaction for the scheme represented by equation (46) are given by:
It
can
be
easily
seen
;
0 1 0
that
.
(50)
=0
and
L&h(x,r)=2Da+~~#O. Thus r=2. Furthermore: E;h(x,r)=Du~& -x3 ~;h(~,t)=2Da~~(Da,x, -xz-Du& - (Dug; -x,)+x;
(51) - Da&)
%.
(52)
XI 1
The transformation:
T=
x3
(53)
[ Da& - xj
will transform the system into the form (13). Let xla, x2,, x,,, and u, be the operating point. Define: 5,=x, -.% 4s=x3
l3=Da3(4
-x:,1
-x3,
-
(~3
-x3,)
(54)
Then, it can be easily shown that the forced zero dynamics are given by
r,=k,e,C, -kze,Ci r,=k,e,Ci
(47)
f;=-&-Da,&+
‘2(12+6;).
(55)
3
where e, and es represent the activity of the two reaction sites on the bifunctiottal catalyst. The activity of the first site is constant whereas the activity of the second site follows second order decay kinetics from O-40 h due to sintering (Fogler, 1986). Thus:
Clearly, the above equation is stable for exponentially decaying inputs & and & and so the system (48) has stable zero dynamics. The performance of the dynamic output feedback
900
S. PALANKI a n d C. KRAVARIS
(equation 32) was evaluated by numerical simulations. The control objective was to regulate the output at y=0.500. The adjustable parameters were chosen as t o = l , fl~=3,~=2 and f12=y2=l, in order to obtain a critically damped response. The following controller designs were compared. 1. Dynamic control law obtained for time-varying system. 2. Dynamic control law obtained by assuming that the system is time-invariant and applying to time-varying system. 3. Dynamic control law obtained by assuming that the system is time-invariant and but updating the timevarying parameters online. Fig. 1 shows the simulated closed-loop response under the assumption that the model is perfect. This case is compared with the case where the time-varying parameter Da3 is assumed to be constant in the state feedback law. It is observed that the constant parameter control law is unable to bring the system to set point till t=40, after which the catalyst activity e 2 is virtually constant. This is because, in the time interval from t=0 to t=40, the parameter Da3 decreases continuously and a controller with constant parameters would require additional integral action to drive the system to set-point. Robustness analysis of the dynamic control law was performed through numerical simulations. It was assumed that there is an error of about 10% in estimating the time-varying parameter Da3 in the dynamic control law. Fig. 2 shows the time profile of the time varying parameter Da3 as well as the estimated profile used in the control law. Fig. 3 shows the simulated closed-loop response under this error in estimating the time-varying parameter. It is observed that an error of about 10% in the estimation of the parameter Da 3 leads to an error of only 0.6% in tracking the set-point. Note that the dynamic output feedback is a function of the time-varying parameter Da3 and its time derivative (see equation (52)). If the system states are sampled "fast enough",
0.55
ODa 3
the t e r m
at
will be small and can be
-
Exact Error of + 10% Error of .10 %
1.0
.., O
Ix. x, I ~
.
~
.
.
0.8 ~u .~
0.6
~
.4
["
0.2
........
0
5
10
15
20
25
30
35
40
45
Time Fig. 2. Time-v~u,'ying parameter Da 3.
0.55
-
-
L) 0.50
: "
0.45 o e~ o 0.40 ..~
f
=
0.35
o L)
0.30
......
. . . . . . Error of -10% Exact Linearization - - Error of + 10%
0.25
0.20 0
I
I
I
I
I
I
I
I
I
5
10
15
20
25
30
35
40
45
Time
Fig. 3. Robustness analysis.
0551 0.50
•--
..
,.~ 0.45 O o
0.40
'~
0,35 I"4
~--
Sampling Interval = 5
r..) 0.30
0.50
0.25
L) 0.45 o
0.20
0
..o 0.40
5
10
15
20
25
30
35
40
45
Time
0.35
Fig. 4. Comparison with adaptive control.
Exact Linearization - - -- With Constant Parameters . . . . . . Set Point
0.30 0.25 0.20 0
I
I
I
I
I
I
I
I
I
5
I0
15
20
25
30
35
40
45
Time Fig. I. Closed loop response for perfectmodel.
ignored. In this situation,the feedback is equivalent to deriving a feedback law by assuming constant parameters and updating this feedback law by new estimates of the time-varying parameter (as is done in adaptive control algorithms). We studied the effect of sampling time through numerical simulations. The results are shown in Fig. 4.
Controller synthesis for time-varying systems Assuming the same values of the adjustable parameters /3o, fl~, ~ , % and ~2 as in Fig. 1, we simulated the controller performance for the case where the timevarying parameter in the dynamic feedback law is assumed to be constant but updated with new estimates at certain sampling instants. In this simulation, we assume that at each sampling instant, we get a time averaged estimate of the time-varying parameter Da3 of the previous sampling interval. It is observed from Fig. 4 that, when the sampling is "fast enough" (sampling interval=l), the "adaptive" feedback law drives the system very close to the set-point. When the sampling interval is increased (sampling interval=5, 10), the system exhibits damped oscillations.
Acknowledgements Financial support by NSF Research Grant BCS-8912627 is gratefully acknowledged.
Nomenclature CA= concentration of species A in reactor CAF= feed concentration of species A CB= concentration of species B in reactor Cc = concentration of species C in reactor F= volumetric feed rate NBF= molar feed rate of species B k~= first-order rate constant k2, k3= second-order rate constants t* = time t= t*F/V xl = CA/CAF x2= Ca/CaF x~= Cc/C~F Dal= kleiV/F Da~= k2eiCg~/F Da3= k3e2Ca~/F
References Bell, D. J. and Tsang, W. (1990) Invertibility and decoupling of linear time-varying systems. Intl J. Contr. 52, 4 983 Brockett, R. W. (1972) System theory on group manifolds and coset spaces. SIAM J. Contr. 10, 265 Byrnes, C. I. and A. Isidori, Global Feedback Stabilization of Nonlinear Systems. Proc. 24th IEEE CDC, Ft Lauderdale, FL, p. 1031 (1985). Chu, W. B., Modeling, optimization and computer control of the cephalosporin C fermentation process. Ph.D. Thesis, Rutgers, The State University of New Jersey, New Brunswick (1987). Claude, D., Fliess, M. and Isidori, A. (1983) Immersion, direct et par bouclage, d'un systeme non lineaire dans un lineaire. C.R. Acad. Sci. Paris 296, 237 Daoutidis, E and Kravaris, C. (1992) Dynamic output feedback control of minimum-phase nonlinear processes. Chem. Engng Sci. 4% 4 837 Eiey, D. D., E W. Selwood and E B. Weisz, Advances in Catalysis, Vol. 13. Academic Press, New York (1962). Fogler, H. S., Elements of Chemical Reaction Engineering. Prentice-Hall, Englewood Cliffs, NJ (1986). Freund, E. (197 l) Design of time-variable muitivariable systems by decoupling and by the inverse. IEEE Trans. Automat. Contr. 16, 183
901
Froment, G. F. and K. B. Bischoff, Chemical Reactor Analysis and Design. Wiley, New York (1979). Harmon, J., Svoronos, S. A. and Lyberatos, G. (1987) Adaptive steady state optimization of biomass productivity in continuous fermenters. Biotechnol. Bioengng 30, 335 Hauser, J., Sastry, S. and Kokotovic, P. (1992) Nonlinear control via approximate input-output linearization: the ball and beam example. IEEE Trans. Automat. Contr. 37, 392 Hirschorn, R. M. (1981) Output tracking in multivariable nonlinear systems. IEEE Trans. Automat. Contr. 26, 593 Hirschorn, R. M. (1981) (A-B) Invariant distributions and disturbance decoupling of nonlinear systems. SIAM J. Contr. Optimiz. 19, 1 Hoo, K. A. and Kantor, J. C. (1986) Linear feedback equivalence and control of an unstable biological reactor. Chem. Engng Commun. 46, 385 Hunt, L. R., Su, R. and Meyer, G. (1983) Global transformations of nonlinear systems. IEEE Trans. Automat. Contr. AC-28, 24 Isidori, A., Nonlinear Control Systems, 2nd edition. Springer-Verlag (1989). Isidori, A. and Ruberti, A. (1984) On the synthesis of linear input-output responses for nonlinear systems. Syst. Contr. Lett. 4, 17 Jakubcsyk, B. and Respondek, W. (1982) On linearization of control systems. Bull, Acad. Polon. Sci. Ser. Sci. Math. 28, 60 Joshi, A. and Palsson, B. (1988) Escherichia coli growth dynamics: a three-pool biochemically based description. Biotech. Bioengng 31, 3 102 Kravaris, C. (1988) Input/output linearization: a nonlinear analog of placing poles at process zeros. AIChE J. 28, 935-945. Kravaris, C. and Chung, C. B. (1987) Nonlinear state feedback synthesis by global input output linearization. AIChE J. 33, 4 592 Krener, A. J. (1974) A generalization of Chow's theorem and the bang-bang theorem to nonlinear systems. SIAM J. Contr. 12, 43 Lobry, C. (1970) Controllabilit6 des systemes non lineaires. SIAM J. Contr. g, 573 Matsumura, M., Imanaka, T., Yoshida, T. and Taguchi, H. (1981) Modeling of cephalosporin C production and its application to fed-batch culture. J. Ferment. Technol. 59, 115 Reboulet, C. and Champetier, C. (1984) A new method for linearizing nonlinear systems: the pseudolinearization. Intl J. Contr. 40, 631 Siiverman, L. M. (1968) Properties and application of inverse systems. IEEE Trans. Automat. Contr. 13, 436 Singh, S. N. and Rugh, W. J. (1972) Decoupling in a class of nonlinear systems by state variable feedback. ASME Trans. J. Dyn. Sys. Meas. Contr. 94, 323 Skatrud, P. L., Tietz, A. J., Ingolia, T. D., Cantwell, C. A., Fisher, D. L., Chapman, J. L. and Queener, S. w. (1989) Use of recombinant dna to improve production of cephalosporin C Cephalosporium acremonium. BioTechnology 7, 477 Soroush, M, and Kravaris, C. (1992) Nonlinear control of a batch polymerization reactor: an experimental study. AIChE J. 38, 9 1429
902
S. PALANKIand C. KRAVARIS
Staniskis, J. and Levisauskas, D. (1984) An adaptive control algorithm for fed-batch culture. Biotechnol. Bioengng 26, 419 Su, R. (1982) On the linear equivalents of nonlinear systems. Syst. Contr. Lett. 2, 48 Vidyasagar, M., Nonlinear Systems Analysis. PrenticeHall, Englewood Cliffs, NJ (1978). Wang, J. and Rugh, W. J. 0989) On the pseudolinearization problem for nonlinear systems. Syst. Contr. Lett. 12, 161 Wonham, W. M., Linear multivariable control: a geometric approach. In Lecture Notes on Control and Information Science, 2nd edition, Vol. 10. SpringerVerlag (1979).
H(X)=h(x,t) L~H(X)=L~h(x,t) LkH(X)=L~h(x,t) (62)
L V IH(X) =L ~- 'h(x,t) and the relative order of the prolonged system is the same as the relative order of the original system.It can be shown easily (Isidori, 1989) that given ~b, a real valued function andfand g vector fields, all defined in R *+1,for any choice of s, k, r >0:
r
g(x)>= i
Appendix A
-I
,(r)i Lf
Proof of Lemma 1 Define x0= t and consider the "prolonged" vectors:
(=[(o (~ "" L]t
X= [xo xl "'"x.] T
(56)
where (o=Oo=xo. Then, using (56), we can represent (7) and (o= Oo in compact form as: (= 0(X).
0
o
002 OXo
00, __ Oxt
00~ Ox.
005 CgXo
a~ Ox~
00,. axe, 50.
dH(X) 7 dL~H(X) I | [G(X) ad~G(X).., ad~F- 'G(X)] =
i
/ dL~-'n(X)]
0
o (58)
(65)
- I)'- '-JLGL~ - 'H(X)~O.
The abov conditions, all together, show that the matrix:
(57)
Using the inverse function theorem, the above transformation is invertible if and only if:
I'
(64)
and for all i,j such that i+ j= r - I:
0=[0o 0~ ... 0,] ~
det ~00(X) ]J -=d e t
(63)
Using the above expression and the definition of relative order, we have for all i,j such that i+j<-r - 2:
!
* * ,
i
J(66)
has rank r and thus the row vectors dH(X),dL~H(X),...,dL~F-tH(X) are linearly independent in R"+l. Using (62), we conclude that the scalar fields h(x,t),L~h(x,t),Lfh(x,t),...,L~-Ih(x,t) are linearly independent in R"×R.
00, Oxo
OX~
~Xn
This implies that the row vectors of (8) are linearly independent.
Appendix C Proof of Theorem 1
Appendix B
Consider the vector g. We can find functions fi,t2,...,t._ ~ in (x,t) such that:
Proof of Lemma 2
Ot~ Ott Ott ~xlgt+ ~x2g2+"'+ ~x g,=O
Define Xo=t and define the following "prolonged" vectors: X=[x0 x] r
Ot2 + Ot2
G(X)= [0 g(X,Xo)]T
Ot2
ox_g, ~g~+...+~g.=
F(X)=[1 J(x,ro)] T
0 (67)
(59)
and the "prolonged" scalar:
H(X)=h(x,xo).
(60) c~tn_ i
Then, the "prolonged" system can be written as:
X= F(X) + G(X)u y=n(x). It can be seen easily that:
~t. _ i
c3t._ i
Oxj- gl + ~-x2 g:+...+ - ax. - g:--O To prove that: (61)
dim( span {dh ,dt:..... dt. _ i} +span ldh,dL~h,...,dL~- Ih l )=n.
(68)
Controller synthesis for time-varying systems
903
We prove this by contradiction. Suppose that:
dim( span {dt l,dt,"..... dt, _ i} + span{dh,dL~h,...,dL~- 2h })
vmC'~+ ~ (Ysp- Y)(69) Then, it can be easily verified that the system:
~=A*/~+b*(y,p-y) (70)
This implies that
(span I dt l,dh ..... dr,_ i })" y(x,t) = 0 (span{dh,dL~h,...,dL~-'h }).~x,t)=0.
(71 ) (72)
From equations (67) and (71), we get:
y(x,t)=span{g}
aL~h
Oh
oL~h
v=C';~+ fl' (Ysp-Y) Yr represents a minimal state-space realization of: • I 9'k ~d% ~E /=
Oh
~0/~ d~k (Y~P- y)
oL~h -0 (74)
v - ~o/g,L~h(w,t)
fv=[(w,t)+g(w,t)
fl,LsL~f- ih(w,t ) r
In other words: (75)
This is a contradiction by definition of relative order. Thus, equation (68) is true. Since {dh,L~r-lh,...,L~-th} has dimension r, we can choose ( n - r ) vectors {t,, t: .... t._r} such that the vectors {tt,t,,...,t,_,dh.L~-'h,...,L~-~h} are linearly independent and thus qualify as an invertible transformation.
Proof of Theorem 4 Define the auxiliary variable:
-
v - kE=ol~,L~h(w,t) u= ~,L~L~-'h(w,t)
aL~-Ih + dL[-thg,+...+ oL~-'h g,=0 ax--~ g' Ox~ Ox,
(78)
with input, the error ( y ~ - y ) and the output, the auxiliary variable v. which can be interpreted as a choice of the linear controller in the GLC structure. The other component of the controller becomes:
g' + -Tffx g'+"" + -Tffx. g"-
Appendix D
(77)
(73)
Substituting in equation (72), we get: Oh
(76)
(79)
which is the input--output linearizing state feedback law with the states reconstructed through an open-loop observer. When w(0)=x(0), it easily follows that w(t)=x(t). Then (79) induces exactly the behavior: dkY ,=0 l~, d 7 = v.
(80)
Combining (21) and (79), we easily obtain the desired closed loop input-output dynamics: • dky y+ k_~,y ~ =y~
(81)