The Attainable Region and Pontryagin's Maximum Principle

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Ind. Eng. Chem. Res. 1999, 38, 652-659

The Attainable Region and Pontryagin’s Maximum Principle Craig McGregor, David Glasser, and Diane Hildebrandt* School of Process and Materials Engineering, University of the Witwatersrand, Johannesburg, WITS 2050, South Africa

Attainable region analysis has been used to solve a large number of previously unsolved optimization problems. This paper examines its relationship to Pontryagin’s maximum principle and highlights the similarities and differences between the methods. It is shown that not all problems that can be solved by attainable region analysis are readily formulated as maximum principle problems. The fundamental process of mixing seems to be the main reason for this difference and the consequences of this are highlighted. The class of problems that can be formulated as maximum principle problems are then examined and the relationship between the two methods highlighted. Here, the maximum principle gives rise to a set of results that are very useful for finding an attainable region. In fact, from these results and the experience gained by solving AR problems, postulates on the nature of the boundary of the attainable region are proposed. Previous to this work the construction of the attainable region required a trial and error approach and the region thus generated was tested using the necessary conditions. These postulates should allow a more constructive approach to finding the attainable region boundary. Introduction Over the past decade the attainable region method has been used to solve a large variety of optimization problems.1-10 Some of these problems had not been or could not be solved by the classical methods. It is a therefore instructive to compare the AR method with these other methods. In this paper the attainable region methods will be compared with Pontryagin’s maximum principle.11 The aim is to highlight the relationship between the methods and show the advantages and disadvantages of the two approaches. An attempt is made to infer results from Pontryagin’s maximum principle that may prove useful in applying attainable region analysis. It is assumed that the reader has some familiarity with the details of the two methods, although the main features of the two methods are described. Assume that a state space that is used to describe the problem to be addressed has been defined. This state space is defined by a set of variables that entirely describes the state of the system after the values of any free parameters that may also be included in the problem specification have been chosen. The Attainable Region Method The attainable region method is essentially geometric in nature and focuses on finding the region of all attainable outcomes for the specific system. To do this, the fundamental processes that are needed to describe the system being considered must first be identified. A fundamental processes is any physical phenomenon such as reaction, mixing, heat transfer, and mass transfer over which there is control. This control is in the sense that the fundamental process can be used as, and when, required. Clearly sufficient variables are required for the state space to fully describe the fundamental processes. At each point in the state space * To whom correspondence should be addressed. E-mail: [email protected].

a vector or range of vectors can be defined for each fundamental process. If there are no free parameters, then there is a single vector for each fundamental process. If the process has free parameters, then there is a range of vectors, possibly with an upper and lower bound when, for instance, the free parameters are bounded by upper and lower values. The AR can then be defined as the set of values in the state space that can be reached from the given feed or feeds (or initial condition or conditions) using the fundamental processes as defined above. The AR method then focuses on finding the boundary of this region. First, there is a set of necessary conditions that can be used to check any proposed AR to see if it is at least acceptable. At the present time, as there is not a full set of sufficient conditions, there is no guarantee that a region that satisfies these conditions is the AR, only that it is a candidate (denoted ARc). These necessary conditions are as follows: (1) No process vector on the boundary of the AR can point outward except at points where the boundary is itself a constraint. (2) For those systems where mixing is a process the AR must be convex and connected. (3) In steady-state process synthesis problems no stationary point that includes mixing as a fundamental process can exist in the complement of the AR. While these are a powerful set of conditions, they give little help in constructing the candidate attainable region. Where the processes considered are only reaction and mixing, there is a strong set of results about the structure of the boundary:10 (1) The N-1-dimensional hypersurfaces that make up the boundary of the AR in the N-dimensional state space are the union of trajectories where either reaction or mixing occurs on their own. (2) The N-2-dimensional hypercurve that occurs at the intersections of these hypersurfaces can be either smooth or nonsmooth. If the intersection is smooth, there is a unique tangent plane and an extra set of

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conditions can be derived to describe this hypercurve. If the intersection is nonsmooth, there is no unique tangent plane and this is a hypercurve of reaction only. There have been a number of other conjectures and postulates about what the boundary may look like in general, and for specific examples these results have been tested using the necessary conditions. Other than the necessary conditions and these results that are only for the processes of reaction and mixing,10 there are no results that give a firm guide as to the nature of the boundary of the AR. To try to relate these results to those of Pontryagin’s maximum principle, the above discussion will be placed in a form that is amenable to analysis using both methods. In doing this the generality of the discussion may need to be limited as the two methods do not always appear to apply to the same class of problems. However, rather than getting bogged down with these difficulties at present, the discussion will be limited to the cases where there is the required commonality. The Problem Consider a system where a number of fundamental processes p1, p2, ..., pM, may occur. The dynamics of these processes will depend on the state of the system, any values of free parameters that can be chosen, and possibly some other achievable state within the system (as for example with mixing). The state of the system is denoted by a vector c in RN, termed the characteristic vector. The characteristic vector lists all N variables that describe the dynamics of the processes, as well as all the variables that make up the objective function that is to be optimized. The control variables within the system are denoted by a vector u in RK. Some processes, such as mixing, may depend on some other possible state of the system: this state is denoted c* and is a member of the same space as c. The system considered has dynamics described by a net process vector p:

dc dt

M

) p(c,u,c*) )

Rjpj(c,u,c*) ∑ j)1

(1)

The operating policy for process j is denoted Rj and may be constrained to lie between the bounds 0 e Rmin j e Rj e Rmax . A control variable, uk of u may have j bounds umin e uk e umax k k . The variable t is an arbitrary integration parameter. The initial state of the system at t ) 0 can be chosen from a set of possible initial states c01, c02, .... Even though eq 1 appears very general, by writing it in this form there are limitations placed on the system. The AR method allows for stationary points of this equation. For instance, there is a locus of points in a system with reaction and mixing where dc/dt ) 0 for different values of R. It turns out that this locus of points represents the outcomes for CSTR reactors with different residence times. To exacerbate the problem, this locus can have multiple branches, unlike the system described by the differential equation. The CSTR is known to play an important part in the AR solution and so leaving this out will affect the results. At the present time there does not seem to be an easy way to incorporate this set of stationary points into a formulation that can be handled by the maximum principle. So, the

limitations of this formulation must be accepted so that the analysis can proceed. Attainable region theory distinguishes between policy variables, Rj, and control variables, uk. Pontryagin’s maximum principle treats the policy variables as additional control variables that happen to be linear. In attainable region theory the policy variables scale the fundamental process vectors and so relate to when and to what degree the fundamental processes occur within the physical system. Hence, the policy variables help with the procedure of interpreting the attainable region. The aim is to determine the set of all possible states that can be achieved in the system while satisfying all the constraints placed upon the system. This set is termed the attainable region (AR). As has already been noted, all that is required to characterize the attainable region is to describe the boundary of the region, denoted ∂AR. Note that the boundary of the AR contains all the extremes in state that can be achieved in the system. That is, the largest values for objective functions that are a linear combination of the state variables will lie on the boundary of the AR. Pontryagin’s Maximum Principle Pontryagin’s maximum principle tries to solve the same problem stated above, but instead of looking for all possible outcomes in this instance looks for a single state with free end time θ that optimizes an objective function S, where S ) s‚c. This single state is equivalent to a single point in the attainable region boundary. Hence, the equivalent maximum principle problem is the optimization of S(c) subject to eq 1, a free end time θ, fixed initial condition c0, with bounded variables u, R, and c* as optimization variables. First, define a set of adjoint variables, n, such that the Hamiltonian, H, is given by

H ) n‚p(c,u,c*)

(2)

Also, assume at this stage that c* ) c*(t), a known function of t, or a constant. According to the maximum principle the optimum value of the Hamiltonian is a constant and gives the rate of change of the objective function S at θ, the final value of t. Consider the problem where θ is not specified but can be chosen so as to optimize S. In this case a stationary point in the objective function is needed and so the Hamiltonian is zero. In general, the Hamiltonian is not zero, but the attainable region method that is being discussed falls into the time-optimal class of problems that require that H ) 0. Subsequently, it is assumed that the Hamiltonian is always zero. The change in the adjoint variables with t is given by Pontryagin’s maximum principle as

dn dt

)-

∂H ∂c

)-

() ∂p

T

(c,u,c*)n )

∂c

M

-

Rj ∑ j)1

( ) ∂pj

T

(c,u,c*)n (3)

∂c

If the objective function is optimized at some t ) θ, then the adjoint variables at θ are given by the maximum principle as n(θ) ) s. Solving the optimization using Pontryagin’s maximum principle requires that a two-point boundary value problem be solved. The initial state of the system is known; that is, c ) c0; and the final state of the adjoint variables is known, that is n(θ) ) s. So, finding the optimum control policy requires

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the initial value n0 of the adjoint variables. This value is not entirely arbitrary as the condition on the Hamiltonian must still be satisfied; that is,

H ) n0‚p(c0,u,c*) ) 0

(4)

For a system with more than one fundamental process vector or with free parameters different values for all Rj or uk could lead to different adjoint vectors that satisfy eq 4. Hence, there may be an infinite set of possible initial adjoint vectors. Equation 1 that describes the dynamic behavior of the system and eq 3 that gives the change in the adjoint vector can be integrated after making a guess for n0 that satisfies eq 4. During the integration the policy and control variables are selected to ensure that the Hamiltonian remains zero. This gives necessary conditions that help select the values for u and all Rj controls. The result of the integration is a trajectory in the space defined by the characteristic vector c. If at some point c(θ) on this trajectory the adjoint vector n ) s, then the problem has been solved and the value of θ that optimizes S has been found. This is not usually the case and a new acceptable initial state is used for the integration and a new trajectory is found. This procedure can, in principle, be repeated until the correct values for the initial adjoints are found. Note, however, that every point on all of these trajectories found using this trial and error approach is a solution to a different optimization problem. That is, if at t the state of the system is c(t) and the adjoint vector is n(t) ) s′, then this solution optimizes the objective function S′ ) s′‚c. So, if the search is not restricted to finding the initial value n0 that optimizes S but instead searches for all n0 that optimize all possible objective functions S′, then this will determine the set of all possible extremes in the state of the system. This is done by trying all possible unit adjoint vectors n0 that satisfy eq 4 and by integrating until t tends to infinity. Every point that lies on these resulting trajectories is included in a set of possible output states. These extremes in state must be those states that lie in the boundary of the attainable region ∂AR, as ∂AR is itself the set of all extreme states. Consequently the boundary of the AR is made up of all possible trajectories that satisfy Pontryagin’s maximum principle. This implies that boundary points of the attainable region can only be reached via other boundary points and not from interior points. So, finding the boundary of the attainable region requires that one solve all possible Pontryagin’s maximum principle problems with linear objective functions. The attainable region boundary ∂AR is formed by the set of trajectories that result from solving each optimization. Any particular objective function (whether linear or nonlinear) will then be optimized at the point where a hypersurface of constant value of the objective function just touches ∂AR or at an interior point of the AR where the objective function takes on an extreme value. Results Result 1. For the problem formulated above all states c in ∂AR are solutions to Pontryagin optimizations; they must all satisfy the necessary conditions for optimality given by the maximum principle. First, an attainable region analysis falls into the time-optimal class of problems that requires H ) 0. Hence, for all possible

states c ∈ ∂AR the definition of the Hamiltonian in eq 2 requires

n‚p(c,u,c*) ) 0

(5)

This implies that the vector of adjoint variables, n, is normal to the net process vector p. Since the net process vector p is by definition of eq 1 tangent to the boundary of the attainable region ∂AR the vector n is normal to the boundary. Result 1: The vector of adjoint variables, n, is normal to the boundary of the attainable region, ∂AR. Result 2. Consider again the result given by the maximum principle that S is optimized when n ) s. This implies geometrically that the hyperplane with a constant value of S is tangent to ∂AR. This result has long been known for attainable regions. However, the objective function S need not be linear. Then, provided the objective function does not take on its extreme in the AR interior, it will be optimized when the hypersurface of a constant value of S just touches the boundary; that is, the hypersurface is tangent to ∂AR. This is the same result as for Pontryagin’s maximum principle where if the objective function S is not a linear function of c, then n(θ) ) [∂S/∂c](c). Result 2: For the class of problems considered here where the objective function does not take on its extreme value in the AR interior and where the boundary is smooth, the contour of the objective function with an extreme value is tangent to ∂AR at the optimum. Result 3. Each point along a Pontryagin maximum principle trajectory is itself the solution to another optimization problem, so it must be a boundary point in ∂AR. Result 3: All points in ∂AR, except limit points, can only be reached via other points in ∂AR. Result 4. The change in the normal along a trajectory is then given by the adjoint equations; see eq 3. Result 4: The change in the normal to the attainable region boundary along a trajectory in ∂AR is given by the following expression:

dn dt

)-

() ∂p ∂c

T

(c,u,c*)n ) -

M

∑ j)1

( )

Rj

∂pj

T

(c,u,c*)n

(6)

∂c

Result 5. Consider now a control variable uk that is a member of u. Pontryagin’s maximum principle requires for optimality that the choice of uk result in an extreme in operation. Hence, it is necessary that uk is either at a bound or the control is intermediate and φSC k ) ∂H/∂uk ) 0. Result 5: It is necessary that either a free parameter is at a bound or the vector ∂p/∂uk is tangent to the boundary ∂AR for all points on the boundary; that is

or uk ) umax or uk ) umin k k ∂p (c,u,c*) ) 0 φSC k ) n‚ ∂uk

∀ k ∈ 1...K (7)

Also, a switch in control between bounds or to an intermediate value requires that φSC k ) 0. Result 6. Equation 7 gives first-order conditions that may not contain an explicit expression for u(t). However, the second-order conditions obtained by differentiating eq 7 with respect to t also need to be satisfied. Now a smooth change in a control variable may be possible over a finite interval of t so as to have an intermediate value

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of the control variable, that is umin < uk < umax k k . This SC ) dφ requires that the necessary condition φIC k k /dt ) 0 be satisfied over the interval

in which case duk/dt ) 0 from eq 7. Combining these expressions gives n‚(∂pm/∂uk) (duk/dt) for all k. Now

SC ∂φSC k dc ∂φk dn IC φk ) (c,u,c*)‚ + (c,u,c*)‚ + ∂c dt ∂n dt SC ∂φk du (c,u,c*) )0 ∀ k ) 1...K (8) ∂u dt

n‚

Substituting eqs 1 and 3 gives the simplification of eq 9 below. Result 6: The change in free parameters u1 to uK along the length of a trajectory in ∂AR with intermediate control is given by solving the K simultaneous equations evaluated at (c,u,c*), given by

φIC k

∂2p ∂p ∂p ∂2p du ) n‚ p - n‚ + n‚ )0 ∂c∂uk ∂c ∂uk du∂uk dt ∀ k ) 1...K (9)

When there is only a single control, then eq 9 can be simplified to the equation derived using a geometric approach:12

∂p ∂p ∂2p n‚ - n‚ p ∂c ∂u ∂c∂u du ) dt ∂2p n‚ 2 ∂u

(10)

Result 7. Similarly, the maximum principle requires for each process m that the associated operating policy Rm result in an extreme in operation, so either Rm is at a bound or it is necessary that φSP m ) ∂H/∂Rm ) 0. Result 7: It is necessary that either an operating policy is at one of its bounds or the process vector pm is tangent to the boundary ∂AR for all points on the boundary, that is:

or Rm ) Rmax or Rm ) Rmin m m φSP m ) n‚pm(c,u,c*) ) 0

∀ m ∈ 1...M (11)

Also, a switch in operating policy between bounds or to an intermediate value requires φSP m ) 0. Result 8. Now a smooth change in operating policy may be possible over a finite interval of t so as to have an intermediate value of the policy variable (Rmin m < Rm IP SP < Rmax m ). This requires that φm ) dφm /dt ) 0 over the interval

φIP m )

dpm dn dφSP d(n‚pm) ) ) n‚ + ‚p dt dt dt dt m

(12)

Substituting eq 6 for the change of the normal with t and the chain rule on process vector m, (dpm/dt) ) (∂pm/ ∂c) (dc/dt) + (∂pm/∂u) (du/dt) ) (∂pm/∂c)p + (∂pm/∂u) (du/ dt), gives

φIP m

( ) (

)

∂pm du ∂pm ∂p p + - Tn ‚pm + n‚ ) n‚ ∂c ∂c ∂u dt

(13)

Consider a trajectory section where Rm is intermediate so that φSP m ) 0. For each control k the variable uk is either intermediate, in which case n‚(∂pm/∂uk) ) 0 by differentiation of φSP m in eq 11 by uk, or uk is at a bound,

∂pm du ∂pm1 du1 ∂pm1 du2 ) n1 + n1 + ... + ∂u dt ∂u1 dt ∂u2 dt ∂pm2 du1 ∂pm2 du2 + n2 + ... (14) n2 ∂u1 dt ∂u2 dt ∂pm du1 ∂pm du2 + n‚ + ... ) 0 ∂u1 dt ∂u2 dt

) n‚

Result 8: For an intermediate policy of process m it is necessary that

(

φIP m ) n‚

∂pm (c,u,c*)p(c,u,c*) ∂c ∂p (c,u,c*)pm(c,u,c*) ) 0 (15) ∂c

)

Result 9. Continuously satisfying this condition requires the correct choice of the policy Rm for process m. This requires that the following necessary condition be satisfied: IP ∂φIP dφIP m m dc ∂φm dn ) (c,u,c*) + (c,u,c*) + dt ∂c dt ∂n dt ∂φIP m du (c,u,c*) ) 0 (16) ∂u dt

Substituting eq 1 and eq 3, then rearranging, gives the condition given by eq 17 below. Result 9: The operating policies for a trajectory in ∂AR that are intermediate are determined by satisfying simultaneously the necessary conditions, given for each process m that has an intermediate policy, as

(

)

(

)

∂p ∂pm ∂p ∂p ∂ ∂pm p - pm p - n‚ p - pm + n‚ ∂c ∂c ∂c ∂c ∂c ∂c ∂ ∂pm du ∂p n‚ p - pm ) 0 (17) ∂u ∂c ∂c dt

(

)

Result 10. Consider now the choice of the point c* on which some processes, such as mixing, may depend. The point c* is some point within the AR that is chosen so as to extend the boundary of the AR as much as possible. In this respect c* is very similar to the control u. If c* is considered a vector of control variables, then Pontryagin’s maximum principle requires that / or c* ) cbound

∂p (c,u,c*)n ) 0 ∂c*

(18)

Since by definition c* is any achievable output, then c/bound is an element of the bounds on the achievable outputs, which is just the boundary of the AR; the first condition is satisfied by c* ∈ ∂AR. For the second condition the normal in this equation only has meaning on the boundary. The first condition already requires c* to be on the boundary; thus, the second condition does not seem to add any new information.

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Result 10: The possible output c* must be chosen from points that lie in the boundary of the attainable region; that is, c* ∈ ∂AR. The set of points that satisfies this necessary condition is infinite as there are an infinite number of points in ∂AR, so Pontryagin’s maximum principle does not give sufficient information to choose c*. It does, however, provide a bound on the possible choices. Now in AR problems there is usually an effort made to choose the set of variables of c to obey linear mixing laws. This is done because it helps to simplify the solution of many problems. In particular, for steadystate synthesis it makes the AR convex, a very important and powerful result. If the mixing is indeed linear, then

v(c,c*) ) c* - c

(19)

The optimization of the choice of c* is achieved in the same way as that for the policy variables, which are also linear. Thus, there are exactly equivalent results to those of eqs 11 and 15 and these can be written as

φSP v ) n‚v(c,c*) ) 0 φVM ) n‚

∂p - v) ) 0 (dc* dt ∂c

(20) (21)

Result 11. When the fundamental process is mixing, the geometric approach to the attainable region does give necessary conditions for the choice of c*. For the mixing process, v(c,c*) ) c* - c, the geometric condition requires that the mixing trajectory (a straight line) surface meets the process surfaces at both ends smoothly. It can be readily seen that this is true. If it were not, the mixing line could be moved a little further out, so extending ∂AR, until this condition is satisfied. This result requires n(c)‚v(c,c*) ) 0 and n(c*)‚v(c,c*) ) 0. The geometric result also requires that c* is in the boundary of the AR. This first condition is just the switching policy condition, φSP v ) 0, given by eq 11, for the process of mixing a stream of composition c* into a system described by c. This second condition is a new result not obvious from Pontryagin’s maximum principle. Result 11: For mixing trajectories to exist in ∂AR it is necessary that c and c* are elements of ∂AR and that

φSP v * ) n(c*)‚v(c,c*) ) 0

(22)

This condition can also be obtained from eq 11 when considering the converse situation where a stream of composition c is mixed with the system described by c*. Hence, there is symmetry in the conditions that must be satisfied. At this stage no fundamental processes, except mixing, are known to depend on c* and so the general form for eq 22 is not known. It is likely that a similar necessary condition can be derived from geometrical considerations for any other process that depends on c*. Discussion These results only apply to the class of problems that are covered and within the limitations already discussed. Although the results are limited, they help to interpret Pontryagin’s maximum principle in new ways, but also help to throw light on the attainable region method.

The fact that the adjoint variables are normal to the surface of the AR gives a geometric insight into the meaning of the adjoint variables for Pontryagin’s maximum principle. This suggests why they play such an important role in finding the optimum operating conditions. From the AR point of view they give a method for calculating the normal along a path. Also, there is the strong result that boundary points can only be achieved via other boundary points. This results from the fact that each point on a trajectory optimized using Pontryagin’s maximum principle is itself a solution to an optimization problem, albeit, one with a different objective function. This is a result that Horn13 alluded to and that had previously been surmised from attainable region examples. It is also interesting to note that Pontryagin’s maximum principle in its present form does not seem to be able to handle mixing. Although result 11 requires that mixing should occur between points that lie in the boundary it gives no indication of how to choose these mixing points. Furthermore, the process of mixing is rather a peculiar process in terms of Pontryagin’s maximum principle since it allows in effect composite paths. The optimum sometimes requires several parallel paths (depending on the dimensions of the space) that must be mixed together at various points. In terms of Pontryagin’s maximum principle this would require that a number of parallel subproblems be solved. In addition, one would need to be aware that combining these parallel subproblems together could in some way better solve the overall problem. It is not at all clear how such a problem should be formulated using Pontryagin’s maximum principle. The superstructure approach14,15 used to solve these problems tries to incorporate this by allowing for complex interactions between the elements. The stationary points of eq 1 are also not handled using Pontryagin’s maximum principle. However, they play an important role in optimizing systems via attainable region analysis.10 The other big difference between attainable region theory and Pontryagin’s maximum principle is the fact that in the former a distinction is made between the variables that describe the operating policies and those that are free parameters. In the latter they are all lumped together, although allowance is made for choosing variables that are linear. It is convenient to continue the discussion by making certain postulates about the nature of the attainable region boundary and seeing how the results obtained using Pontryagin’s maximum principle confirm these within the limitations already discussed. Structure of the Attainable Region Boundary A number of postulates are now made concerning the structure of the attainable region boundary for the class of systems being discussed. The postulates are based on the nature of the boundary for steady-state reactor synthesis10 and the necessary conditions for optimality. No formal proof is given. However, the extensions arise logically out of the necessary conditions. This section is not supposed to be a thorough proof of the postulates, but to give some idea of how the structure of the AR boundary arises out of the necessary conditions and some of the properties of Pontryagin’s maximum principle.

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Postulate 1: The boundary of the attainable region consists of different surfaces, termed process surfaces, that are the union of process trajectories where all of the operating policies are the same and at either their lower or Rj ) Rmax ∀ j ) or upper bound; that is, Rj ) Rmin j j 1...M. This postulate has been proved10 for the steady-state process synthesis of systems with reaction and mixing, that is, for

dc ) p(c,c*) ) r(c) + Rvv(c,c*) dt

Rv g 0

(23)

In this case the lower bound of Rv for mixing is zero and its upper bound infinite. Hence, the postulate states for this case that the boundary of the attainable region is made up of surfaces comprising the individual processes of reaction and mixing, respectively, exactly as shown by Feinberg and Hildebrandt.10 According to postulate 1 the boundary of the attainable region consists of process surfaces that are the union of trajectories, and that the operating policies are at the same lower or upper bounds along each of these trajectories. The fact that the boundary consists of trajectories is because the dynamics of the system is described by the differential equation of eq 1 and that boundary points can only be reached via other boundary points and not from interior points, as discussed in result 3. Continuity arguments require that the set of bounds is the same for all trajectories on the surface. Postulate 2: The junctions between the various process surfaces, termed intersectors, represent either a change or switch in one or more of the operating policies, or the termination of two different process trajectories that intersect. The evidence accumulated on intersectors is large. Numerous problems solved using attainable region analysis over the years have found the various types of intersectors. When a particular type of intersector was found for the first time, there was usually no preconceived notion that the particular type of intersector was possible, and could be found in the boundary of the attainable region for the system. The particular intersector was the only structure that would give a region that satisfied the necessary conditions. Hence, the use of the necessary conditions, over time, has helped to build up a picture of what the boundary of an attainable region should look like. In particular, Feinberg and Hildebrandt10 consider the nature of the intersectors in systems where only the fundamental processes of reaction and mixing occur. Because of the rather special form of eq 23, the types of intersectors are rather limited and consist only of stationary points (which are not considered here) and the outputs of a differential sidestream reactor, that is, eq 23 with intermediate values of Rv. Consider now a hypersurface in ∂AR that is constructed out of neighboring trajectories that satisfy Pontryagin’s maximum principle. Pontryagin’s maximum principle shows that a switch between bounds of an operating policy along a trajectory only occurs when the necessary condition φSP m ) 0 is satisfied. As the trajectory is a hypercurve, this necessary condition ensures that the switch can only happen at a point along the trajectory. The neighboring trajectories of the hypersurface will have similar switching points. Hence, these switching points form a locus of points that must

lie in a subspace, namely a hypercurve in ∂AR. In the case of intermediate control there is a trajectory that follows this locus; the trajectory, however, still remains in the subspace. This is exactly what was found for the DSR,10 described by eq 23, where only the processes of reaction and mixing are considered. Note further that the maximum principle usually predicts that at these lines of intersection the two hypersurfaces meet smoothly. This is because the equation describing the system, eq 1, is a differential equation. While the operating policies and the controls can have jumps, these only appear in the right-hand side of the differential equation so that the state variables are continuous. Furthermore, the same applies to the adjoints as given by eq 6. Hence, there is usually a unique unit normal that is defined at each point. This implies that not only are the states in ∂AR continuous but also that the different process surfaces meet each other smoothly. The exception to this is for trajectories that arise from points that have no unique unit normal or when Pontryagin’s maximum principle does not apply. When there is no unique unit normal defined for a point, then along any trajectory that arises from this point the adjoints need not be uniquely defined. Hence, there may be no unique unit normal along such a trajectory and the intersections between hypersurfaces need not be smooth. The initial states (or feed points) and points associated with a state constraint very typically have no unique unit normal. In addition, Pontryagin’s maximum principle does not apply to eq 23 as Rv f ∞, as a differential equation can no longer be written. This corresponds to the case where instantaneous mixing is allowed where the intersection between reaction and mixing hypersurfaces need not be smooth.10 When the intersection was not smooth, then the line of intersections was itself the union of reaction only processes. Intersector Postulates Some postulates are now made regarding the nature of intersectors. This is done in the light of previous experience with various examples as well as the results that have been derived here. Smooth Trajectory Intersectors. Postulate 3: A smooth intersector that is the union of trajectories exists where one or more of the operating policies are intermediate in value (between the upper and lower bounds, < Rj < Rmax ). Rmin j j It is clear that because the R’s are constrained to be positive, in order for a trajectory to exist the orientation of the process vectors must be such that a vector sum can be obtained. Such a trajectory is the DSR10 given by eq 23. If a net vector is not possible, then no such trajectory can exist. Smooth Locus Intersectors. Postulate 4: A smooth intersector that is not the union of trajectories is either (1) the union of stationary points that include mixing, (2) the union of points where one or more operating policies switch bounds, (3) the union of points where two different trajectory surfaces intersect and terminate. These are the three cases that remain for smooth intersectors after removing the trajectory case of postulate 3. Postulate 4(1) is not covered by Pontryagin’s maximum principle as it corresponds to the points where the time derivative of eq 1 is zero. This corresponds to the products of a CSTR for systems with

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reaction and mixing. Postulate 4(2) corresponds to the case in Pontryagin’s maximum principle where there is bang-bang control, namely, the switch from one bound to another for a policy variable. Postulate 4(3) allows for the fact that two different parallel paths may intersect smoothly such that both trajectories enter the interior of the attainable region. Such an intersector would seem to be highly unlikely. Nonsmooth Trajectory Intersectors. Postulate 5: A trajectory that forms a nonsmooth intersector either originates at a point with no unique unit normal and has policies fixed at either bounds or has intermediate control of the policy variables so as to continuously satisfy a constraint. Nonsmooth Locus Intersectors. Postulate 6: A nonsmooth intersector that is not the union of trajectories is either the union of points where two different trajectories intersect and terminate or the union of points where a process surface terminates because of a constraint and where a new process surface is initiated. These postulates allow for edges on ∂AR that can arise from points with no unique unit normal or because of system constraints. These edges can themselves be trajectories or loci of points.

Result 12: Points on a trajectory in the attainable region of a steady-state system represent possible states within a flow unit. Result 13. The stationary points of the differential equation description of the system are the solutions to the following equation: M

Rjpj(c,u,c*) ) 0 ∑ j)1

When one of the fundamental processes is mixing, that is, v(c,c*) ) c* - c, and where Rv is the mixing policy, then this equation can be rewritten as

Rv(c* - c) +

dt

M

Rjpj(c,u,c*) ∑ j)1

(26)

Consider the mass and energy balances written for a steady-state well-stirred unit, where the state c of the unit is uniform because of the stirring, and that has a feed of state c0. These balances result in the following algebraic equation describing the unit:

∑Rjpj(c,u,c*) ) Rv(c - c0)

(27)

j*v

The previous section discussed the various types of intersectors. In most problems a number of intersectors will exist in the boundary of the AR. The structure of the boundary is interpreted as a process layout by following various trajectories from one process surface to another via the intersectors, so tracing the sequence of fundamental processes used. Although it may not be entirely clear yet how the interpretation of the boundary is achieved, the easiest way to understand the procedure is to look at problems that have been solved.1-7,9,12 A distinction is made between trajectory surfaces, representing flow units, and stationary points, representing stirred units, when interpreting the vector processes as processing units. This has been stated up to now without justification, and so here these conclusions are formalized. Result 12. Consider the differential mass and energy balances written for a flow unit where there is perfect radial mixing and no axial mixing and where the fundamental processes p1, p2, ..., pM occur in the proportions given by the operating policies R1, R2, ..., RM. The change in the state of the system c along the axial length of the unit l is given by the following differential equation:

) p(c,u,c*) )

∑Rjpj(c,u,c*) ) 0

j*v

Interpreting the Boundary

dc

(25)

(24)

This equation is identical to the differential description of the system given by eq 1 except that dl replaces dt. However, l and t are just arbitrary integration parameters because the state of the system is completely characterized by c and l and t are consequently interchangeable. What this means practically is that if the diameter of the flow unit is changed, the same product can be produced, but in a unit of different length; the volume, which characterizes the system, remains fixed. Equations 1 and 24 are equivalent and so any point that lies on the trajectory resulting from the integration of eq 1 can be interpreted as a possible state in a flow unit.

Hence, stationary points that include mixing as a fundamental process represent the possible states in steady-state systems of a stirred unit. Notice from eq 27 where Rv ) 0 that the net process vector, p(c,u,c*) ) ∑Rjpj(c,u,c*), for the trajectory that leaves the stationary point, is collinear with the mixing vector at the point. Result 13: Stationary points in the attainable region of a steady-state system that include the fundamental process of mixing represent possible states of a stirred unit. Result 14. It is obvious that eq 1 represents the dynamic behavior of a dynamic system, where the arbitrary integration parameter t represents the time elapsed in the system. Hence, any point on a trajectory is a possible product of a batch process. The operating sequence of the process is given by the choice of the control variables u and the operating policies R1, R2, ..., RM with time. Result 14: Points on a trajectory in the attainable region of a dynamic system represent possible states that can be achieved by a batch process. The most important point to realize is that once the boundary has been determined its structure can be interpreted as a process layout for steady-state systems, or process operating sequences for dynamic systems. Conclusion For a certain limited class of problems comparisons have been made between attainable region analysis and Pontryagin’s maximum principle. The results have helped to throw light on each of the methods and to highlight some of the difficulties associated with each. Attainable region theory appears to be able to handle some classes of problems that appear to be difficult to solve using Pontryagin’s maximum principle. Attainable region analysis has been used to find answers to some previously unsolved problems and seems to represent a new optimization method. The paper ends with a set of postulates about the nature of the attainable region boundary. These are based on previous results for more

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limited situations and in some cases are justified by Pontryagin’s maximum principle. These postulates should be very helpful in constructing attainable regions, but as in all cases the result will need to be checked using the necessary conditions. Acknowledgment This work was started while David Glasser was on sabbatical at Princeton University. He would like to thank the U.S. government for a Fullbright Fellowship and Princeton University for a position. In particular, he would like to thank Roy Jackson for valuable discussions relating to the content of this paper. Nomenclature c ) characteristic vector c* ) attainable state H ) Hamiltonian K ) number of free parameters l ) axial position variable along flow unit M ) number of fundamental processes N ) dimension of characteristic vector n ) vector of adjoints, normal to boundary p ) net process vector pj ) vector for fundamental process j S ) objective function s ) objective function vector S′ ) arbitrary objective function s′ ) arbitrary objective function vector t ) arbitrary integration parameter, time u ) vector of free parameters uk ) free parameter k ) lower bound for free parameter k umin k ukmax ) upper bound for free parameter k v ) mixing vector Greek Letters Rj ) operating policy for process j ) upper bound for policy j Rmax j Rmin ) lower bound for policy j j φ ) condition for smoothness θ ) final time in time-optimal problem

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Received for review June 15, 1998 Revised manuscript received November 23, 1998 Accepted November 28, 1998 IE980380L