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On the Dynamics of Nonlinear Process Systems with Input Constraints
Copyright ,0 IFAC Dvnamics and Control of Process Svstems. Corfu . Greece. 1998 . .
On the Dynamics of Nonlinear Process Systems with Input Constraints Navneet Kapoor'
and
Prodromos Daoutidis "
·Department of Chemical Engineering and Materials Science, University of Minnesota , Minneapolis , MN 55455. Current Affiliation: GE Corporate Research & Development, P. O. Box 8 , Schenectady , NY 12301. ·'Department of Chemical Engineering and Materials Science , University of Minnesota , Minneapolis , MN 55455
Abstract. This work deals with the dynamical analysis of nonlinear process systems with input constraints . In particular, a characterization of the "domain of attraction" of an equilibrium point under bounded control is provided and the notion of regions of invariance within these domains of attraction is introduced and characterized. A case study on a two-dimensional Gray and Scott model of an isothermal, auto-catalytic reaction occurring in a CSTR is also carried out .
Copyright © 1998 IFAC Key Words. nonlinear dynamics, input constraints, chemical reactors
1. Introduction
actor dynamics . In the present paper, we focus on the limitations imposed by input constraints on our ability to control a nonlinear system at a desired equilibrium point. Recent research on the dynamical analysis of constrained non linear systems (see (Colonius and Kliemann, 1992) and the references therein) has provided a system-theoretic characterization of "regions of controllability" i.e., regions in statespace where any two points can be reached from each other with the available control action, and has also introduced the notion of "regions of attraction" of such regions of controllability. We consider a broad class of constrained nonlinear systems for which these regions of controllability evolve around the equilibrium points of the unforced system . For such systems, we focus on the characterization and computation of the regions of attraction of the regions of controllability. Subsequently, we introduce and characterize "regions of invariance" within the regions of attraction, such that all trajectories originating from these regions enter the regions of controllability, under any admissible control law . Some preliminary results in this regard were presented in (Kapoor and Daoutidis, 1994) . Finally, we carry out a detailed study of the Gray and Scott model (Gray and Scott , 1986) of an isothermal , auto-catalytic reaction occurring in a CSTR, to illustrate the theoretical concepts and results.
Chemical process systems exhibit a wide variety of nonlinear phenomena, such as complex patterns of steady state multiplicity and oscillations, finite regions of stability etc . The study of the static and dynamic behavior of such systems , either in open loop or under feedback control, has been an area of active research since the late 50s (see e.g. (Aris and Amundson, 1958; Uppal et al., 1974; Balakotaiah and Luss , 1982)), and has also, to some extent , paved the way for the flourishing research activity in nonlinear process control that we have witnessed during the last decade . A feature that complicates further the behavior of process systems is the presence of constraints in the manipulated inputs. Such input constraints impose fundamental limitations on our ability to modify the dynamics of a process at will. In this regard, there has been some research over the last decade on analyzing the global effects of input constraints on closed-loop dynamics under specific control laws. Specifically, the multiplicity of equilibrium points in nonlinear systems with linear controllers subject to saturation has been associated with certain topological features of the steady-state model , and the bifurcation behavior of these features has been analyzed (Chen and Chang , 1985). Moreover, the bounded nonlinear control for a class of two and three state chemical reactors has been addressed in (Alvarez et al., 1991) , where a criterion to preclude critical points induced by saturation was developed based on physical restrictions and the knowledge of re-
159
2. Preliminaries
either disjoint or identical. We will assume that the equilibrium points of Eq.2 lie in the interior of some control sets (a precise Lie-algebraic condition for verifying this is given in (Colonius and Kliemann, 1992; Theorem 6)). Note the following facts regarding control sets with non-void interiors (see (Colonius and Kliemann, 1992) and the references therein):
We will consider nonlinear systems of the form:
f(x)
+ g(x)u(t)
on X
( 1)
where X is a Coo manifold of dimension n < 00 , f , 9 are analytic vector fields, u denotes a manipulated input variable , and u(t) is an input function that is measurable and takes values in U = [Umin, umaxl C IR (any such function will be referred to as admissible) . We will denote by x(t, x o , u) the unique solution of Eq .l at time t, under the input u(t), starting from an initial state x ° EX . We will also consider the (unforced) system :
f(x)
+ g(x)UO
1. As the range of the manipulated input increases in magnitude, control sets evolve around the equilibrium points of the nominal system . 2. Equilibrium points that belong to a continuum 1, as Uo is varied smoothly over U , belong to the closure of the same control set. 3. Any two points within the interior of a control set are reachable from each other.
(2)
The computation of control sets is a very cumbersome task, even for low-dimensional systems.
obtained from the system of Eq.l for a constant value of u, UO E U . We will denote the equilibrium points of the system of Eq.2 by w(UO,x o ) . It will be assumed that U nom = 0 is the nominal value of uO; the system of Eq .2 under the nominal input will be referred to as the nominal system. We will also assume that the system of Eq .1 is locally accessible. Loosely speaking, a system of the form of Eq.1 is locally accessible if the set of attainable points from each X o E X has a nonempty interior . Local accessibility is a rather weak requirement that is satisfied by most real systems in an appropriate region of the state space. On the other hand, controllability of the system of Eq.1 , is a much stronger requirement than accessibility, which is made even more stringent by the presence of constraints.
Corresponding to each control set, there exists a set of initial conditions starting from where the non linear system can be driven to the control set with an admissible input function. This region, referred to as the domain of attraction of a control set, is defined as follows : Definition 2 (Colonius and Kliemann, 1992): The domain of attraction of a con{x E trol set D is defined as A( D) X , R+(x) D i= 0} . It follows trivially that D ~ A(D) . The domain of attraction of a control set with non-void interior is an open set . In what follows, we present a result that relates the domain of attraction of a control set to the regions of stability of the equilibrium points within the control set and can be viewed as a generalization of the technique employed in (Colonius and Kliemann, 1992) . To this end , consider a control set D (with non-void interior) corresponding to a continuum of equilibrium points for a given control range U. Let us denote by Z. (w( u o , x)) , the region of stability of an equilibrium point w( uO, x), UO E U, and by Z;(D), the union of the regions of stability of all the equilibrium points that belong to D . Thus , Z;(D) Z.(w(UO, x)) . It will also
n
3. Control Sets and their Domains of Attraction : A Review and Some Additional Results The following is a precise definition of the concept of regions of controllability in the presence of input constraints, referred to as control sets: Definition 1 (Colonius and Kliemann, 1992): A set D C X is called a control set if:
=
(i) D C cl R+(xo) for all Xo E D, where cl denotes the closure of a set and R+(xo) denotes the positive reachable set from Xo . (ii) for all Xo E D, there exists an admissible input function u s. t. x( t, Xo, u) E D for all
t
~
U
w(uO,x)ED,uOEU be assumed that int(Z;(D)) i= 0. Also, let us denote the boundary of Z; (D) as P( D) and the normal vector to the boundary at each x E P(D) , pointing outwards of Z;(D), as nx(D) , assuming that P(D) is differentiable.
O.
(iii) D is the maximal set (w. r. t. set inclusion) with these properties .
We are now in a position to present the following theorem, the proof of which is presented in the
A control set D is invariant if clD = clR+(xo) for all Xo E D . Note that a non linear system may have more than one control sets. It, however , follows from the maximality of control sets (property (iii) from above) that control sets are
The set of equilibrium points w( uo, x o ), that evolve as is varied smoothly over a control range U, belong to a continuum iffor each f > 0, 38( f) > 0 such that IUl - u21 < 8(f) ~ IW(Ul ' xd - W(U2, x2)1 < f, where Ul, U2 E U . 1
Uo
160
appendix: Theorem 1: Consider th e system of Eq.l. Then , (i) Z;(D) ~ A(D) . (ii) A(D) Z;(D) , if and only if [J(x) g(x)uojT nx(D) ~ 0, vuo E U, "Ix E P(D) .
=
control sets of th e syst e m of Eq . l which contain , in th eir int erior or th eir boundary, all the equilibrium points w( uo , xo) of th e system of Eq. 2, as u o is varied over U , and let A( Dj) denote the corresponding domains of attraction. For an in variant control set Dj , i E {I , ... , p}, let Y (Dd be defined as the largest invariant subset of A( Dj) such that :
+
The mathematical condition in theorem 1( ii) allows us to establish the equivalence between the union of the regions of stability of the equilibrium points within a control set and its domain of attraction. This condition simply ensures that the 'flow' of the system of Eq.l, under any admissible input function , cannot enter the region Z; (D) from initial conditions outside of Z;(D) . For a large number of chemical engineering examples, the verification of this condition can be performed by mere physical arguments (see case study that follows) . Remark 1: As a consequence of theorem 1, the problem of construction of the domain of attraction of a control set reduces to the problem of computing the regions of asymptotic stability of the equilibrium points of autonomous systems , whenever the condition in theorem l(ii) is satisfied . The characterization and computation of such regions has received considerable attention in the dynamical systems literature (e.g . (Chiang et al. ,
Y(D;)
n clA(D
j )
= 0 , i =/; j
(3)
Then , pro vided th ere are no higher-order attractors within Y(D;) \D j , Y(Dj) is th e strict domain of attraction of Dj.
The computation of the strict domains of attraction of control sets follows directly from the computation of the domains of attraction of control sets . Remark 2: For variant control sets , there exist no regions of invariance in the form of strict domains of attraction . However , it is possible to construct invariant sets , around operating points contained within variant control sets . A systematic procedure to construct them and design controllers that guarantee closed-loop stability, for linear and non linear systems, has been developed in (Kapoor and Daoutidis, 1997; Kapoor and Daoutidis , 1998) . Such regions of closed-loop stability are, of course , subsets of the domain of attraction of the variant control set containing the unstable equilibrium point .
1988)).
4. Strict Domains of Attraction of Control Sets The domain of attraction of a control set is not necessarily a region of invariance; portions of, or the full domain of attraction of a control set may also belong to domains of attraction of other control sets. In what follows we will identify, whenever possible, a region of initial conditions within the domain of attraction of a control set starting from where all trajectories , under any admissible input function , will eventually enter the control set and stay there for all times. Such a concept is analogous to the region of stability of an equilibrium point of an unforced dynamical system ; its precise definition is given below . Definition 3: The strict domain of attraction
5 . Case Study : Cubic Auto-catalysis in a CSTR The following two-reaction scheme (Gray and Scott , 1986) describes cubic auto-catalysis:
(4)
where A is the reactant , B is the catalyst, C is some inert product and kl and k z are the reaction rate constants for the two reactions. The mathematical model for this scheme in an isothermal CSTR is described by the following set of equations:
of a control set is th e larg est subset of its domain of attractIOn such that , under any admissible input function , all trajectori es originating within it enter the control set and stay there for all times .
(5)
It is clear that corresponding to a variant control set , there can be no subset of its domain of attraction with the above properties, since there will always be some admissible input function which will steer the system outside of such a subset . The following theorem , the proof of which is in the appendix, establishes the existence and provides a characterization of the strict domain of attraction of an invariant control set . Theorem 2: Let Dj , J = 1, 2, 3,· . . , p denot e the
where Xl , xz , and f3 are the dimensionless concentrations, T is the dimensionless time , B is the dimensionless residence time and I\. is the dimensionless decay-rate constant. For the system of Eq .5, the reciprocal of the dimension less residence time B will be considered as the variable that will be manipulated for control. Since flow rates are the variables that can be most easily manipulated ,
161
this choice is natural. Moreover , driven by practical considerations we will assume that the range in which the parameter B can be varied is bounded.
equilibrium points , respectively , evolve . From fact 2. we have that equilibrium points corresponding to uO E int(u) within a single continuum are reachable from one another. Now, to determine whether equilibrium points within two different continua are reachable from each other , we will assume that there are potentially two control sets Di,i = 2,3 corresponding to wi(uO,x),i = 2, 3 respectively. We will now proceed with the computation of the domains of attraction of the control sets, by employing theorem l. To this end, note that for any uO E U , the region of stability of any stable equilibrium point in W3( uo, x) is bounded by the stable manifold of the corresponding saddle point in W2( uo, x). Fig.2 shows the stable manifolds M' (Umin) and M' ( u max ) , corresponding to W2(U min , x) and W2(U max , x), respectively. It can easily be verified that the stable equilibrium points W3( u max , x) and W3( u min , x) also lie within the control set D3 since they can be reached from all W3( uO, x), uO E int( U) and vice-versa. Thus , in this figure , Z; (D3) is the region above P(D3) M·(u max ) . On the other hand, Z; (D 2 ) is the union of the stable manifolds corresponding to equilibrium points in W2 (uo. x) that lie within D 2 . Note that the union of the stable manifolds of all the equilibrium points in W2(UO, x) is bounded by M'(u max ) and M'(Umin), i.e. , P(D 2 ) = M'(u max ) U M·(Umin) . Since domains of attraction are open sets, if Z; (D 2 ) = A(D 2 ) , then P(D 2 ) et. Z;(D2)'
Now, in order to obtain the isolated continua of equilibrium points for any given control range , we will employ a bifurcation analysis with B as the bifurcation parameter . For the system under consideration, such an analysis has been performed in (Gray and Scott, 1986) and nine different static bifurcation patterns have been reported for different sets of the process parameters /3 and K . All these bifurcation patterns can be categorized as either unique solutions, isolas or mushrooms (see (Gray and Scott , 1986) for a detailed discussion). As can be seen in (Gray and Scott , 1986) , for the range of parameters for which the system of Eq.5 exhibits unique solutions, there is a single continuum of equilibrium points as the manipulated input is varied over the control range. On the other hand, for the range of parameters for which the system exhibits isolas or mushrooms , there may be more that one continuum of equilibrium points. In what follows, we consider one such scenario.
=
=°
In particular , we consider the case when /3 and K = 0.05 . This situation corresponds to no catalyst in the inflow and the bifurcation pattern is an isola. Fig.l shows a schematic representation of the bifurcation diagram (taken from (Gray and Scott, 1986)). The state-space of interest is X
= [0,1]
x [0, 1 ~ BK] and it will be assumed
To establish that indeed Z;(D;) = A(D;) , i = 2,3 , we verified the condition of Theorem l(ii). It was checked that for an initial condition in Z;(D3) \ Z;(D 2 ) , i.e., above M'(Umin), no trajectories corresponding to Umin can enter Z; (D2 ). This holds Vu O E U because at any point (x j , X2) on M'(Umin), X(Xj,X2 , U) > X(Xj,X2,U m in),U # Umin and X2(Xj,X2,U) > X(Xj , X2 , U m in),U # Umin. Likewise, for an initial condition below M' (u max ), no trajectory can enter Z; (D 2 ) and hence Z;(D 3 ) , under all admissible u. It then follows from theorem 1(ii) that A( D 2 ) = Z; (D2 ) and A(D3) Z;(D3) . Thus, there are two isolated control sets corresponding to the equilibrium points Wi(Uo, x), i = 2, 3.
the maximum allowable range of B is [5 , 60], i.e., U =
[~ !].
The nominal value of u corresponds 60' 5 to B = 25. It can be verified that the compact set X is invariant for the system of Eq.5, VB > 0. We initially consider B E [20 , 30]. Setting
~XTj =
°
°
dx~ . t he steady state soand - = we obtalll dT ' lutions of the system of Eq.5. It can be verified that the only equilibrium point in X for which the Lie-algebraic condition of(Colonius and Kliemann , 1992; Theorem 6) does not hold true is (Xj"X2.) = (1 , 0) . Note that this equilibrium point corresponds to no conversion and is a common zero for the vector fields f( x) and g(x) . This equilibrium point, which we will refer to as Wj (U nom , x), then corresponds to a point control set Dj and is, hence , of no interest to us since it has a void interior. Also , note that for u = U nom , there exist two more solutions Wi(U nom , x) , i = 2, 3 which correspond to a saddle point and a stable equilibrium point, respectively, and both of which lie within the interior of some control set . Moreover, as uO is varied in U , two isolated continua of equilibrium points wi(uO,x),i = 2,3 containing saddle and stable
=
Since D2 consists of saddle equilibrium points, it is clear that it is a variant set . The control set D 3 , on the other hand , contains stable equilibrium points and it can be shown to be invariant . Now in order to identify Y(D3), we need to isolate the invariant portion of the domain of attraction , A(D3), that is not shared by the closure of A(D2) . From Fig.2, it is clear that Y (D3) is the region above M' ( Umin). The parameter B was then allowed to assume values in the range [20 , 39]' i.e., U E U = [1/39 , 1/20]. For this range, there still exists an isolated equilib-
162
rium point wduo , x) and two isolated continua of equilibrium points ,,,.:,(UO , x) , i 2, 3. However. as established in (Gray and Scott, 1986), at around B 37, there is a Hopf bifurcation leading to the evolution of a continuum of periodic orbits corresponding to B E [37.39]. The stable manifold corresponding to W2( 1/39 , x) is shown in Fig.3 which folds into a periodic orbit and has a non-void intersection with the continuum of equilibrium points W3( uO, x) . Thus, W2( uO , x) and W3( uO, x) are reachable from each other and hence belong to the same control set. This illustrates that the presence of periodic orbits may result in equilibrium points within two isolated continua to be reachable from each other. On further enlarging the range of B to [10,60]' note that even though the two isolated continuum of equilibrium points Wj( uo, x), i 2,3 merge into a single continuum, the equilibrium point Wl (UO, x) still forms a separate isolated set . Thus, in the case of isolas, even for an unbounded control range , there may be more than one control set.
7. Appendix
=
Proof of Theorem 1
=
(i) Consider any control set D. Then, VX o E Z;(D) , there exists a constant control Uo E U such that limx(t,xo,uO) =w(UO,xo), where 1-00
w(UO,x o) E D . Thus, Vx o E Z;(D) , there exists an admissible control input u such that x(t, xc , u) D:f. 0. Hence, Z;(D) ~ A(D). (ii) Having established that Z;(D) ~ A(D) , through the result i) above, we will now prove that the condition in theorem 2( ii) is necessary and sufficient for having A(D) = Z;(D). Necessity: Assume that A(D) = Z;(D) . This clearly implies that Vx o ~ Z;(D), there does not exist an admissible control u(t) such that x(t,xo,u)nD :f. 0. This in turn implies that Vx o ~ Z;(D) and Vue E U , x(t,xo , uO)nZ;(D) = 0. Since the domain of attraction of a control set with non-void interior is an open set, it follows that Z; (D) is an open set and that P(D) does not belong to the set Z;(D) . Then, V x E P(D) and Vue E U, the vector (f(x) + g(x)UO] does not point into Z;(D), i.e., the inner product of [/( x) + g( x )UO] and the normal vector to the boundary is non-negative (~ 0) . Sufficiency: Assume that Vx E P(D) and Vue E U , [/(x) + g(x)uO]Tnx(D) ~ O. Thus, Vx o ~ int(Z;(D)), there does not exist an admissible control u(t) such that x(t,xo,u(t))nint(Z;(D)) :f. 0. We will now establish that A(D) ~ int(Z;(D)) by contradiction . To this end, let A(D) :J int( Z; (D)). Clearly, the regions of stability of the equilibrium points in D have a non-void intersection with the control set D. Given that int(Z;(D)) :f. 0, it then follows that int(Z;(D)) D :f. 0. This implies that forVx o E A(D)\int(Z;(D)) , there exists an admissible control u(t) such that x(t,xo,u)nint(Z;(D)) :f. 0 which contradicts the assertion above . Hence by contradiction, A(D) ~ int(Z;(D)) . Then, since from i) we have that Z; (D) ~ A( D), it follows that Z;(D) is an open set and that A(D) = Z; (D). 0
n
=
6. Conclusions
In this paper, we investigated the feasibility of controlling a non linear system at a desired equilibrium point in the presence of constraints. In particular, we obtained a characterization of : i) the largest region in state space from where it is possible to steer the non linear system to a desired equilibrium point, and ii) the largest region in state-space from where the nonlinear system, under any control law, may be steered to a desired region of controllability. A detailed case study was conducted on a cubic auto-catalytic reactor and the following key features , among others, were observed :
n
• enlarging the control range typically results in merging of control sets, • if a continuum contains a globally stable equilibrium point (as may be the case for unique solutions), then every point in the corresponding control set is reachable from the entire state space , and • folded manifolds (such as those that fold into periodic orbits) may result in two isolated continua of equilibrium points being reachable from each other .
Proof of Theorem 2
For an invariant control set Dj , by definition
Acknow ledgments
Y (Dj) is an invariant set and Dj ~ Y (Dj). Hence. Vx o E Y(Dj) and any admissible input function u(t), either the closed-loop trajectories enter Dj or end in a limit set outside Dj. In the first case,
Financial support from the National Science Foundation, CTS-9624725 is gratefully acknowledged .
since Dj is an invariant set, all trajectories which enter it stay within it for all times. In the second case, since the limit set outside Dj cannot be one of the equilibrium points of the system of Eq.2, it
163
has to be a higher-order attractor of the system of Eq.l. Hence, the result. 0
1
0+
Aris. R. and ?'-<.R. Amundson (1958). An analysis of chemical reactor stability and control i.ii.iii. Chem . Eng . Sci. 7. 12!. Balakotaiah. V . and D. Luss (1982). Structure of the steady state solutions of lumped parameter chemically reacting systems. Chem. Eng . Sci. 37, 161!.
J J
::f
8. REFERENCES Alvarez , J .. J. Alvarez and R . Suarez (1991). Nonlinear bounded control for a class of continuous agitated tank reactors. Chem . Eng. Sci. 46, 3235 .
J
0.6
x,
I
I
1
j
O'S f
O.4r 0.3 r
0.1
Chen. L. H and H .C. Chang (1985) . Global effects of controller saturation on closed-loop dynamics . Chem. Eng. Sci. 12. 219l.
0
1
MS ( 1/30)
1 0
1
MS ( 1/20)
0.2,
0.2
Chiang , H.D. , M.W. Hirsch and F.F. Wu (1988) . Stability regions of nonlinear autonomous dynamical systems . IEEE Trans. Autom. Contr. 33 , 16.
0.8
0.6
0.4 x\
Figure 2.
Colonius . F. and W. Kliemann (1992). On control sets and feedback for nonlinear systems. Proc. of IFAC .!lion/in. Contr. Sys . Des. Symp. p. 49 . Gray, P. and S.K. Scott (1986). Auto-catalytic reactions in the isothermal, continuous stirred tank reactor.isola and other forms of multiplicity. Chem . Eng. Commun. 46, 385.
0.9 0 .8
Kapoor , N. and P. Daoutidis (1994). Control of nonlinear systems subject to saturation. AIChE Anuual l\Ieeting. Kapoor, N. and P . Daoutidis (1997) . Stabilization of systems with input constraints. Int. J. of Contr. 66 , 653 .
0.7 0.6 Xl
0.5 0.4[
Kapoor, r-;. and P. Daoutidis (1998). Stabilization of nonlinear processes with input constraints. submitted. Comput. chem. Engng.
0.3 0.2
Cppal. A., W.H. Ray and A . B. Poore (1974). On the dynamic behavior of continuous stirred tank reactor. Chem. Eng. Sci . 29, 967.
0.1 0
solid· stlble equilibril
20
e
MS ( 1/39) 0
0.2
0.4
Figure 3.
dashed· unstlble equilibria
o
MS ( 1/20)
60
Figure I.
164
0.6
0.8
On the Dynamics of Nonlinear Process Systems with Input Constraints Navneet Kapoor'
and
Prodromos Daoutidis "
·Department of Chemical Engineering and Materials Science, University of Minnesota , Minneapolis , MN 55455. Current Affiliation: GE Corporate Research & Development, P. O. Box 8 , Schenectady , NY 12301. ·'Department of Chemical Engineering and Materials Science , University of Minnesota , Minneapolis , MN 55455
Abstract. This work deals with the dynamical analysis of nonlinear process systems with input constraints . In particular, a characterization of the "domain of attraction" of an equilibrium point under bounded control is provided and the notion of regions of invariance within these domains of attraction is introduced and characterized. A case study on a two-dimensional Gray and Scott model of an isothermal, auto-catalytic reaction occurring in a CSTR is also carried out .
Copyright © 1998 IFAC Key Words. nonlinear dynamics, input constraints, chemical reactors
1. Introduction
actor dynamics . In the present paper, we focus on the limitations imposed by input constraints on our ability to control a nonlinear system at a desired equilibrium point. Recent research on the dynamical analysis of constrained non linear systems (see (Colonius and Kliemann, 1992) and the references therein) has provided a system-theoretic characterization of "regions of controllability" i.e., regions in statespace where any two points can be reached from each other with the available control action, and has also introduced the notion of "regions of attraction" of such regions of controllability. We consider a broad class of constrained nonlinear systems for which these regions of controllability evolve around the equilibrium points of the unforced system . For such systems, we focus on the characterization and computation of the regions of attraction of the regions of controllability. Subsequently, we introduce and characterize "regions of invariance" within the regions of attraction, such that all trajectories originating from these regions enter the regions of controllability, under any admissible control law . Some preliminary results in this regard were presented in (Kapoor and Daoutidis, 1994) . Finally, we carry out a detailed study of the Gray and Scott model (Gray and Scott , 1986) of an isothermal , auto-catalytic reaction occurring in a CSTR, to illustrate the theoretical concepts and results.
Chemical process systems exhibit a wide variety of nonlinear phenomena, such as complex patterns of steady state multiplicity and oscillations, finite regions of stability etc . The study of the static and dynamic behavior of such systems , either in open loop or under feedback control, has been an area of active research since the late 50s (see e.g. (Aris and Amundson, 1958; Uppal et al., 1974; Balakotaiah and Luss , 1982)), and has also, to some extent , paved the way for the flourishing research activity in nonlinear process control that we have witnessed during the last decade . A feature that complicates further the behavior of process systems is the presence of constraints in the manipulated inputs. Such input constraints impose fundamental limitations on our ability to modify the dynamics of a process at will. In this regard, there has been some research over the last decade on analyzing the global effects of input constraints on closed-loop dynamics under specific control laws. Specifically, the multiplicity of equilibrium points in nonlinear systems with linear controllers subject to saturation has been associated with certain topological features of the steady-state model , and the bifurcation behavior of these features has been analyzed (Chen and Chang , 1985). Moreover, the bounded nonlinear control for a class of two and three state chemical reactors has been addressed in (Alvarez et al., 1991) , where a criterion to preclude critical points induced by saturation was developed based on physical restrictions and the knowledge of re-
159
2. Preliminaries
either disjoint or identical. We will assume that the equilibrium points of Eq.2 lie in the interior of some control sets (a precise Lie-algebraic condition for verifying this is given in (Colonius and Kliemann, 1992; Theorem 6)). Note the following facts regarding control sets with non-void interiors (see (Colonius and Kliemann, 1992) and the references therein):
We will consider nonlinear systems of the form:
f(x)
+ g(x)u(t)
on X
( 1)
where X is a Coo manifold of dimension n < 00 , f , 9 are analytic vector fields, u denotes a manipulated input variable , and u(t) is an input function that is measurable and takes values in U = [Umin, umaxl C IR (any such function will be referred to as admissible) . We will denote by x(t, x o , u) the unique solution of Eq .l at time t, under the input u(t), starting from an initial state x ° EX . We will also consider the (unforced) system :
f(x)
+ g(x)UO
1. As the range of the manipulated input increases in magnitude, control sets evolve around the equilibrium points of the nominal system . 2. Equilibrium points that belong to a continuum 1, as Uo is varied smoothly over U , belong to the closure of the same control set. 3. Any two points within the interior of a control set are reachable from each other.
(2)
The computation of control sets is a very cumbersome task, even for low-dimensional systems.
obtained from the system of Eq.l for a constant value of u, UO E U . We will denote the equilibrium points of the system of Eq.2 by w(UO,x o ) . It will be assumed that U nom = 0 is the nominal value of uO; the system of Eq .2 under the nominal input will be referred to as the nominal system. We will also assume that the system of Eq .1 is locally accessible. Loosely speaking, a system of the form of Eq.1 is locally accessible if the set of attainable points from each X o E X has a nonempty interior . Local accessibility is a rather weak requirement that is satisfied by most real systems in an appropriate region of the state space. On the other hand, controllability of the system of Eq.1 , is a much stronger requirement than accessibility, which is made even more stringent by the presence of constraints.
Corresponding to each control set, there exists a set of initial conditions starting from where the non linear system can be driven to the control set with an admissible input function. This region, referred to as the domain of attraction of a control set, is defined as follows : Definition 2 (Colonius and Kliemann, 1992): The domain of attraction of a con{x E trol set D is defined as A( D) X , R+(x) D i= 0} . It follows trivially that D ~ A(D) . The domain of attraction of a control set with non-void interior is an open set . In what follows, we present a result that relates the domain of attraction of a control set to the regions of stability of the equilibrium points within the control set and can be viewed as a generalization of the technique employed in (Colonius and Kliemann, 1992) . To this end , consider a control set D (with non-void interior) corresponding to a continuum of equilibrium points for a given control range U. Let us denote by Z. (w( u o , x)) , the region of stability of an equilibrium point w( uO, x), UO E U, and by Z;(D), the union of the regions of stability of all the equilibrium points that belong to D . Thus , Z;(D) Z.(w(UO, x)) . It will also
n
3. Control Sets and their Domains of Attraction : A Review and Some Additional Results The following is a precise definition of the concept of regions of controllability in the presence of input constraints, referred to as control sets: Definition 1 (Colonius and Kliemann, 1992): A set D C X is called a control set if:
=
(i) D C cl R+(xo) for all Xo E D, where cl denotes the closure of a set and R+(xo) denotes the positive reachable set from Xo . (ii) for all Xo E D, there exists an admissible input function u s. t. x( t, Xo, u) E D for all
t
~
U
w(uO,x)ED,uOEU be assumed that int(Z;(D)) i= 0. Also, let us denote the boundary of Z; (D) as P( D) and the normal vector to the boundary at each x E P(D) , pointing outwards of Z;(D), as nx(D) , assuming that P(D) is differentiable.
O.
(iii) D is the maximal set (w. r. t. set inclusion) with these properties .
We are now in a position to present the following theorem, the proof of which is presented in the
A control set D is invariant if clD = clR+(xo) for all Xo E D . Note that a non linear system may have more than one control sets. It, however , follows from the maximality of control sets (property (iii) from above) that control sets are
The set of equilibrium points w( uo, x o ), that evolve as is varied smoothly over a control range U, belong to a continuum iffor each f > 0, 38( f) > 0 such that IUl - u21 < 8(f) ~ IW(Ul ' xd - W(U2, x2)1 < f, where Ul, U2 E U . 1
Uo
160
appendix: Theorem 1: Consider th e system of Eq.l. Then , (i) Z;(D) ~ A(D) . (ii) A(D) Z;(D) , if and only if [J(x) g(x)uojT nx(D) ~ 0, vuo E U, "Ix E P(D) .
=
control sets of th e syst e m of Eq . l which contain , in th eir int erior or th eir boundary, all the equilibrium points w( uo , xo) of th e system of Eq. 2, as u o is varied over U , and let A( Dj) denote the corresponding domains of attraction. For an in variant control set Dj , i E {I , ... , p}, let Y (Dd be defined as the largest invariant subset of A( Dj) such that :
+
The mathematical condition in theorem 1( ii) allows us to establish the equivalence between the union of the regions of stability of the equilibrium points within a control set and its domain of attraction. This condition simply ensures that the 'flow' of the system of Eq.l, under any admissible input function , cannot enter the region Z; (D) from initial conditions outside of Z;(D) . For a large number of chemical engineering examples, the verification of this condition can be performed by mere physical arguments (see case study that follows) . Remark 1: As a consequence of theorem 1, the problem of construction of the domain of attraction of a control set reduces to the problem of computing the regions of asymptotic stability of the equilibrium points of autonomous systems , whenever the condition in theorem l(ii) is satisfied . The characterization and computation of such regions has received considerable attention in the dynamical systems literature (e.g . (Chiang et al. ,
Y(D;)
n clA(D
j )
= 0 , i =/; j
(3)
Then , pro vided th ere are no higher-order attractors within Y(D;) \D j , Y(Dj) is th e strict domain of attraction of Dj.
The computation of the strict domains of attraction of control sets follows directly from the computation of the domains of attraction of control sets . Remark 2: For variant control sets , there exist no regions of invariance in the form of strict domains of attraction . However , it is possible to construct invariant sets , around operating points contained within variant control sets . A systematic procedure to construct them and design controllers that guarantee closed-loop stability, for linear and non linear systems, has been developed in (Kapoor and Daoutidis, 1997; Kapoor and Daoutidis , 1998) . Such regions of closed-loop stability are, of course , subsets of the domain of attraction of the variant control set containing the unstable equilibrium point .
1988)).
4. Strict Domains of Attraction of Control Sets The domain of attraction of a control set is not necessarily a region of invariance; portions of, or the full domain of attraction of a control set may also belong to domains of attraction of other control sets. In what follows we will identify, whenever possible, a region of initial conditions within the domain of attraction of a control set starting from where all trajectories , under any admissible input function , will eventually enter the control set and stay there for all times. Such a concept is analogous to the region of stability of an equilibrium point of an unforced dynamical system ; its precise definition is given below . Definition 3: The strict domain of attraction
5 . Case Study : Cubic Auto-catalysis in a CSTR The following two-reaction scheme (Gray and Scott , 1986) describes cubic auto-catalysis:
(4)
where A is the reactant , B is the catalyst, C is some inert product and kl and k z are the reaction rate constants for the two reactions. The mathematical model for this scheme in an isothermal CSTR is described by the following set of equations:
of a control set is th e larg est subset of its domain of attractIOn such that , under any admissible input function , all trajectori es originating within it enter the control set and stay there for all times .
(5)
It is clear that corresponding to a variant control set , there can be no subset of its domain of attraction with the above properties, since there will always be some admissible input function which will steer the system outside of such a subset . The following theorem , the proof of which is in the appendix, establishes the existence and provides a characterization of the strict domain of attraction of an invariant control set . Theorem 2: Let Dj , J = 1, 2, 3,· . . , p denot e the
where Xl , xz , and f3 are the dimensionless concentrations, T is the dimensionless time , B is the dimensionless residence time and I\. is the dimensionless decay-rate constant. For the system of Eq .5, the reciprocal of the dimension less residence time B will be considered as the variable that will be manipulated for control. Since flow rates are the variables that can be most easily manipulated ,
161
this choice is natural. Moreover , driven by practical considerations we will assume that the range in which the parameter B can be varied is bounded.
equilibrium points , respectively , evolve . From fact 2. we have that equilibrium points corresponding to uO E int(u) within a single continuum are reachable from one another. Now, to determine whether equilibrium points within two different continua are reachable from each other , we will assume that there are potentially two control sets Di,i = 2,3 corresponding to wi(uO,x),i = 2, 3 respectively. We will now proceed with the computation of the domains of attraction of the control sets, by employing theorem l. To this end, note that for any uO E U , the region of stability of any stable equilibrium point in W3( uo, x) is bounded by the stable manifold of the corresponding saddle point in W2( uo, x). Fig.2 shows the stable manifolds M' (Umin) and M' ( u max ) , corresponding to W2(U min , x) and W2(U max , x), respectively. It can easily be verified that the stable equilibrium points W3( u max , x) and W3( u min , x) also lie within the control set D3 since they can be reached from all W3( uO, x), uO E int( U) and vice-versa. Thus , in this figure , Z; (D3) is the region above P(D3) M·(u max ) . On the other hand, Z; (D 2 ) is the union of the stable manifolds corresponding to equilibrium points in W2 (uo. x) that lie within D 2 . Note that the union of the stable manifolds of all the equilibrium points in W2(UO, x) is bounded by M'(u max ) and M'(Umin), i.e. , P(D 2 ) = M'(u max ) U M·(Umin) . Since domains of attraction are open sets, if Z; (D 2 ) = A(D 2 ) , then P(D 2 ) et. Z;(D2)'
Now, in order to obtain the isolated continua of equilibrium points for any given control range , we will employ a bifurcation analysis with B as the bifurcation parameter . For the system under consideration, such an analysis has been performed in (Gray and Scott, 1986) and nine different static bifurcation patterns have been reported for different sets of the process parameters /3 and K . All these bifurcation patterns can be categorized as either unique solutions, isolas or mushrooms (see (Gray and Scott , 1986) for a detailed discussion). As can be seen in (Gray and Scott , 1986) , for the range of parameters for which the system of Eq.5 exhibits unique solutions, there is a single continuum of equilibrium points as the manipulated input is varied over the control range. On the other hand, for the range of parameters for which the system exhibits isolas or mushrooms , there may be more that one continuum of equilibrium points. In what follows, we consider one such scenario.
=
=°
In particular , we consider the case when /3 and K = 0.05 . This situation corresponds to no catalyst in the inflow and the bifurcation pattern is an isola. Fig.l shows a schematic representation of the bifurcation diagram (taken from (Gray and Scott, 1986)). The state-space of interest is X
= [0,1]
x [0, 1 ~ BK] and it will be assumed
To establish that indeed Z;(D;) = A(D;) , i = 2,3 , we verified the condition of Theorem l(ii). It was checked that for an initial condition in Z;(D3) \ Z;(D 2 ) , i.e., above M'(Umin), no trajectories corresponding to Umin can enter Z; (D2 ). This holds Vu O E U because at any point (x j , X2) on M'(Umin), X(Xj,X2 , U) > X(Xj,X2,U m in),U # Umin and X2(Xj,X2,U) > X(Xj , X2 , U m in),U # Umin. Likewise, for an initial condition below M' (u max ), no trajectory can enter Z; (D 2 ) and hence Z;(D 3 ) , under all admissible u. It then follows from theorem 1(ii) that A( D 2 ) = Z; (D2 ) and A(D3) Z;(D3) . Thus, there are two isolated control sets corresponding to the equilibrium points Wi(Uo, x), i = 2, 3.
the maximum allowable range of B is [5 , 60], i.e., U =
[~ !].
The nominal value of u corresponds 60' 5 to B = 25. It can be verified that the compact set X is invariant for the system of Eq.5, VB > 0. We initially consider B E [20 , 30]. Setting
~XTj =
°
°
dx~ . t he steady state soand - = we obtalll dT ' lutions of the system of Eq.5. It can be verified that the only equilibrium point in X for which the Lie-algebraic condition of(Colonius and Kliemann , 1992; Theorem 6) does not hold true is (Xj"X2.) = (1 , 0) . Note that this equilibrium point corresponds to no conversion and is a common zero for the vector fields f( x) and g(x) . This equilibrium point, which we will refer to as Wj (U nom , x), then corresponds to a point control set Dj and is, hence , of no interest to us since it has a void interior. Also , note that for u = U nom , there exist two more solutions Wi(U nom , x) , i = 2, 3 which correspond to a saddle point and a stable equilibrium point, respectively, and both of which lie within the interior of some control set . Moreover, as uO is varied in U , two isolated continua of equilibrium points wi(uO,x),i = 2,3 containing saddle and stable
=
Since D2 consists of saddle equilibrium points, it is clear that it is a variant set . The control set D 3 , on the other hand , contains stable equilibrium points and it can be shown to be invariant . Now in order to identify Y(D3), we need to isolate the invariant portion of the domain of attraction , A(D3), that is not shared by the closure of A(D2) . From Fig.2, it is clear that Y (D3) is the region above M' ( Umin). The parameter B was then allowed to assume values in the range [20 , 39]' i.e., U E U = [1/39 , 1/20]. For this range, there still exists an isolated equilib-
162
rium point wduo , x) and two isolated continua of equilibrium points ,,,.:,(UO , x) , i 2, 3. However. as established in (Gray and Scott, 1986), at around B 37, there is a Hopf bifurcation leading to the evolution of a continuum of periodic orbits corresponding to B E [37.39]. The stable manifold corresponding to W2( 1/39 , x) is shown in Fig.3 which folds into a periodic orbit and has a non-void intersection with the continuum of equilibrium points W3( uO, x) . Thus, W2( uO , x) and W3( uO, x) are reachable from each other and hence belong to the same control set. This illustrates that the presence of periodic orbits may result in equilibrium points within two isolated continua to be reachable from each other. On further enlarging the range of B to [10,60]' note that even though the two isolated continuum of equilibrium points Wj( uo, x), i 2,3 merge into a single continuum, the equilibrium point Wl (UO, x) still forms a separate isolated set . Thus, in the case of isolas, even for an unbounded control range , there may be more than one control set.
7. Appendix
=
Proof of Theorem 1
=
(i) Consider any control set D. Then, VX o E Z;(D) , there exists a constant control Uo E U such that limx(t,xo,uO) =w(UO,xo), where 1-00
w(UO,x o) E D . Thus, Vx o E Z;(D) , there exists an admissible control input u such that x(t, xc , u) D:f. 0. Hence, Z;(D) ~ A(D). (ii) Having established that Z;(D) ~ A(D) , through the result i) above, we will now prove that the condition in theorem 2( ii) is necessary and sufficient for having A(D) = Z;(D). Necessity: Assume that A(D) = Z;(D) . This clearly implies that Vx o ~ Z;(D), there does not exist an admissible control u(t) such that x(t,xo,u)nD :f. 0. This in turn implies that Vx o ~ Z;(D) and Vue E U , x(t,xo , uO)nZ;(D) = 0. Since the domain of attraction of a control set with non-void interior is an open set, it follows that Z; (D) is an open set and that P(D) does not belong to the set Z;(D) . Then, V x E P(D) and Vue E U, the vector (f(x) + g(x)UO] does not point into Z;(D), i.e., the inner product of [/( x) + g( x )UO] and the normal vector to the boundary is non-negative (~ 0) . Sufficiency: Assume that Vx E P(D) and Vue E U , [/(x) + g(x)uO]Tnx(D) ~ O. Thus, Vx o ~ int(Z;(D)), there does not exist an admissible control u(t) such that x(t,xo,u(t))nint(Z;(D)) :f. 0. We will now establish that A(D) ~ int(Z;(D)) by contradiction . To this end, let A(D) :J int( Z; (D)). Clearly, the regions of stability of the equilibrium points in D have a non-void intersection with the control set D. Given that int(Z;(D)) :f. 0, it then follows that int(Z;(D)) D :f. 0. This implies that forVx o E A(D)\int(Z;(D)) , there exists an admissible control u(t) such that x(t,xo,u)nint(Z;(D)) :f. 0 which contradicts the assertion above . Hence by contradiction, A(D) ~ int(Z;(D)) . Then, since from i) we have that Z; (D) ~ A( D), it follows that Z;(D) is an open set and that A(D) = Z; (D). 0
n
=
6. Conclusions
In this paper, we investigated the feasibility of controlling a non linear system at a desired equilibrium point in the presence of constraints. In particular, we obtained a characterization of : i) the largest region in state space from where it is possible to steer the non linear system to a desired equilibrium point, and ii) the largest region in state-space from where the nonlinear system, under any control law, may be steered to a desired region of controllability. A detailed case study was conducted on a cubic auto-catalytic reactor and the following key features , among others, were observed :
n
• enlarging the control range typically results in merging of control sets, • if a continuum contains a globally stable equilibrium point (as may be the case for unique solutions), then every point in the corresponding control set is reachable from the entire state space , and • folded manifolds (such as those that fold into periodic orbits) may result in two isolated continua of equilibrium points being reachable from each other .
Proof of Theorem 2
For an invariant control set Dj , by definition
Acknow ledgments
Y (Dj) is an invariant set and Dj ~ Y (Dj). Hence. Vx o E Y(Dj) and any admissible input function u(t), either the closed-loop trajectories enter Dj or end in a limit set outside Dj. In the first case,
Financial support from the National Science Foundation, CTS-9624725 is gratefully acknowledged .
since Dj is an invariant set, all trajectories which enter it stay within it for all times. In the second case, since the limit set outside Dj cannot be one of the equilibrium points of the system of Eq.2, it
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has to be a higher-order attractor of the system of Eq.l. Hence, the result. 0
1
0+
Aris. R. and ?'-<.R. Amundson (1958). An analysis of chemical reactor stability and control i.ii.iii. Chem . Eng . Sci. 7. 12!. Balakotaiah. V . and D. Luss (1982). Structure of the steady state solutions of lumped parameter chemically reacting systems. Chem. Eng . Sci. 37, 161!.
J J
::f
8. REFERENCES Alvarez , J .. J. Alvarez and R . Suarez (1991). Nonlinear bounded control for a class of continuous agitated tank reactors. Chem . Eng. Sci. 46, 3235 .
J
0.6
x,
I
I
1
j
O'S f
O.4r 0.3 r
0.1
Chen. L. H and H .C. Chang (1985) . Global effects of controller saturation on closed-loop dynamics . Chem. Eng. Sci. 12. 219l.
0
1
MS ( 1/30)
1 0
1
MS ( 1/20)
0.2,
0.2
Chiang , H.D. , M.W. Hirsch and F.F. Wu (1988) . Stability regions of nonlinear autonomous dynamical systems . IEEE Trans. Autom. Contr. 33 , 16.
0.8
0.6
0.4 x\
Figure 2.
Colonius . F. and W. Kliemann (1992). On control sets and feedback for nonlinear systems. Proc. of IFAC .!lion/in. Contr. Sys . Des. Symp. p. 49 . Gray, P. and S.K. Scott (1986). Auto-catalytic reactions in the isothermal, continuous stirred tank reactor.isola and other forms of multiplicity. Chem . Eng. Commun. 46, 385.
0.9 0 .8
Kapoor , N. and P. Daoutidis (1994). Control of nonlinear systems subject to saturation. AIChE Anuual l\Ieeting. Kapoor, N. and P . Daoutidis (1997) . Stabilization of systems with input constraints. Int. J. of Contr. 66 , 653 .
0.7 0.6 Xl
0.5 0.4[
Kapoor, r-;. and P. Daoutidis (1998). Stabilization of nonlinear processes with input constraints. submitted. Comput. chem. Engng.
0.3 0.2
Cppal. A., W.H. Ray and A . B. Poore (1974). On the dynamic behavior of continuous stirred tank reactor. Chem. Eng. Sci . 29, 967.
0.1 0
solid· stlble equilibril
20
e
MS ( 1/39) 0
0.2
0.4
Figure 3.
dashed· unstlble equilibria
o
MS ( 1/20)
60
Figure I.
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0.6
0.8