Inherent Dynamic Operability of Processes: General Definitions and

Ind. Eng. Chem. Res. 2002, 41, 421-432

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Inherent Dynamic Operability of Processes: General Definitions and Analysis of SISO Cases Derya Uztu 1 rk and Christos Georgakis* Chemical Process Modeling and Control Research Center, and Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015

An optimization-based approach is presented for assessing and enhancing the dynamic operability characteristics of processes. A dynamic operability measure is used that captures the dynamic performance based on how quickly a system can react to changes in the nominal operating conditions. The measure is evaluated using an open-loop optimal control formulation and, therefore, provides an upper bound on the achievable control performance of the process. Some key operating spaces are defined representing the ranges of desired outputs, expected disturbances, and available inputs. A dynamic operability index, referred to as the dOI, is defined based on this measure and the operating spaces, quantifying the dynamic performance of the process over the entire ranges of the desired outputs and anticipated disturbances. The dOI gives a quantitative figure of the extent to which the desired dynamic performance requirements can be achieved with the available input ranges. The measure is further used in an optimization framework to calculate the ranges of the inputs that are required to satisfy the desired dynamic performance requirements. In this enhancement study, the aim is to provide insights into the identification of limitations on the dynamic performance caused by the constraints on the inputs. The proposed approaches are conceptualized using a generic formulation in the case of linear systems. Their application is demonstrated using SISO example problems. 1. Introduction In the past, process design and control system design have been mostly carried out as two sequential and isolated tasks. The structure of the plant and the size of the equipment were selected with the objective of minimizing capital and operating costs. Operational issues and dynamic characteristics were only considered during control structure selection. Although operability concerns were quite well recognized by the design engineer, these concerns were rarely taken into account as an integral part of the objective in the process of making design decisions. This shortcoming can be partly attributed to the understanding of operability being qualitative at best. Also, at that time, a measure that could successfully quantify operability characteristics of a design was not available. Another factor was that, in the past, the integration of process units was not an important issue possibly because less attention was paid to heat and material integration, which increase process complexity and reduce the operability characteristics. Therefore, the majority of the early tools for evaluating the performance of processes focused on helping with the design of the control system. In the past 20 years, environmental regulations and global competition have pushed the chemical processing industries to operate their plants at their physical and economical limits. This has created significant pressure toward the exploration of better design practices. In contemporary process design practice, design engineers are expected to produce process designs that are capable of operating reliably and cost efficiently under * Corresponding author. Address: CPMC Research Center, D-311 Iacocca Hall, 111 Research Drive, Lehigh University, Bethlehem, PA 18015, USA. Phone: (610) 758-5432. Fax: (610) 758-5297. E-mail: [email protected].

all circumstances. Increasing customer expectations and aggressive global competition have made the job of process designers more difficult than ever before. Today’s processing plants need to exhibit more flexibility and integrity to satisfy the variety of products and tight quality constraints. All of these factors combine to increase the risk of designing a plant that is not easy to operate. In such a scenario, it is imperative that every prospective design be tested for operability. It is emphasized in the literature that, during operation, the performance of a control system in responding to disturbances and uncertainties is determined not only by the controller, but also by the process itself. Thus, the ability of the system to cope with these changes can only be guaranteed if the control requirements are considered during the early stages of the design where a simple design modification might often convert a difficult control problem into one that does not require special attention. Motivated by these factors, recent efforts have been directed toward the development of approaches whereby performance limitations can be identified and evaluated at the design phase. This new effort is referred to as the integration of design and control. The need for addressing process operability issues during the design stage is certainly not a new discovery. The effect of process design on achievable control performance has been discussed in the literature for over five decades. A 1943 paper by Ziegler and Nichols1 is cited by various authors as one of the milestones of this discussion. Morari and Perkins2 quote the following remark from this paper, emphasizing the importance of design for operability: “In the application of automatic controllers, it is important to realize that controller and process form a unit; credit or discredit for results obtained are attributable to one as much as the other. A poor controller is often able to perform acceptably on

10.1021/ie0101792 CCC: $22.00 © 2002 American Chemical Society Published on Web 10/20/2001

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a process which is easily controlled. The finest controller made, when applied to a miserably designed process, may not deliver the desired performance. True, on badly designed processes, advanced controllers are able to eke out better results than older models, but on these processes there is a definite end point which can be approached by instrumentation and it falls short of perfection.” Although the need for integrated design was noted quite early, significant progress has been made only during the past 20 years. During the early and mid 1980s, considerable work was done in identifying and understanding the fundamental limitations on control performance. Using the internal model control (IMC) framework, Morari3 established the relationship between the invertibility of the transfer function matrix of a system and its achievable control performance. The argument is that perfect control can be achieved by manipulating the inputs based on the inverse of the process. Therefore, one can infer that plants that are easier to invert should be easier to control. Morari3 identified four fundamental factors that prevent the inversion of the process and, thus, deviate the control objective from the perfect one: (1) right-half-plane (RHP) zeros, (2) time delays, (3) constraints on the input variables, and (4) model uncertainty. Various tools and measures have been developed for assessing the extent of these limitations for both SISO and MIMO linear(ized) systems. Morari and co-workers studied the effect of these fundamental limitations in a series of papers (see, e.g., Holt and Morari4,5 and Skogestad and Morari6,7). Psarris and Floudas8,9 investigated the effect of time delays and transmission zeros on the dynamic operability of MIMO systems. Condition number, disturbance condition number, relative gain array, block relative gain, and closed-loop disturbance gain are just a few of the tools used in the literature in an attempt to address the operability problem. One major shortcoming in using these tools is that, whereas all of the dynamic performance specifications are defined in the time domain, these tools are mostly formulated in the frequency domain. An accurate representation of these specifications in the frequency domain is not always possible. Another problem arises when the results of these tools are interpreted. The question of what is considered too large a value of the condition number is not clear and, therefore, can be misleading. In most cases, the interpretation of these tools requires experience, and more than one technique is usually needed to draw a satisfactory conclusion. A further problem is that each indicator considers only one performance limitation at a time, which makes it impossible to understand the combined effect of these limitations on the operability of a process. A comprehensive list and detailed discussion of such tools can be found in the review articles by Morari,10 Morari and Perkins,2 and Perkins and Walsh.11 Over the past decade, advances in computer technology and optimization methods have enabled the use of optimization more extensively as a tool for assessing the achievable control performance of candidate process designs. Several optimization-based tools have been proposed to address this problem. Perkins and Walsh11 make the following comment about the use of optimization as quoted in Chenery:12 “First, it has been shown that it is possible to devise absolute controllability tests by this means, that is tests which if failed imply no real

plant based on the tested partial design can meet the performance requirements. Second, it is possible to include the actual performance requirements (if known!) into the test itself.” One way of classifying optimization-based techniques is in terms of the type of optimization set of controllers used in the assessment of the dynamic performance:12 (1) realistic controllers or (2) ideal controllers. When a realistic set of controllers is used in the assessment, the analysis gives a conservative bound on the achievable control performance of the process. If the process along with the selected set of controllers can satisfy the performance requirements, then this type of performance will be guaranteed in practice. However, if the requirements cannot be met by the specific process and controller set examined, there is still the possibility that another controller exists and gives the desired performance. The set of controllers can be selected to include only PI controllers (e.g., Mohideen et al.13) or to cover the entire set of linear time-invariant controllers (e.g., Swartz14 and Chenery12). Typically, a broader set of realizable controllers gives a more realistic bound, but the problem can be computationally expensive. If the set of controllers is too restricted, one runs the risk of identifying the performance limitations of the controllers rather than the inherent limitations of the process. Techniques using ideal controllers constitute the second category (e.g., Carvallo,15 Cao,16 and Chenery12). They provide an upper bound for the achievable control performance of the process. That is, if the ideal controller fails to satisfy the performance requirements, no feedback controller can satisfy them in practice. However, even if the requirements are met by the ideal controller, it does not necessarily mean that a realizable controller exists that will give the same performance. A thorough critique of these optimization-based approaches can be found in Chenery.12 The assessment of achievable control performance using optimization-based techniques is a promising field in need of more attention. These techniques offer a flexible formulation and give results that can be interpreted rather easily. Because the analysis is performed in the time domain, performance specifications and limitations on control performance, particularly input constraints and uncertainty, are easily incorporated into the formulation. However, the question of whether the bounds they produce are realistic, is always valid, as it is practically impossible to solve the problem for the whole set of realizable controllers or to design an ideal controller that is realizable. In this article, a new optimization-based approach is proposed for assessing and enhancing the dynamic operability characteristics of processes. The proposed approach is based on an ideal controller formulation, and as discussed above, it represents an upper bound for the achievable control performance of a process. In the assessment problem, dynamic operability is quantified on the basis of how quickly a system can respond to set-point changes and disturbances. A dynamic operability index is defined to evaluate the dynamic performance of a system over the entire ranges of desired outputs and expected disturbances. In the enhancement problem, the ranges of inputs needed for a system to satisfy the dynamic performance requirements are calculated. This provides a tool that can be effectively used to identify bottlenecks due to input availability and to gain insight on how to modify the ranges of the inputs

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to make the process operable. The proposed approaches are conceptualized using a generic formulation in the case of linear systems. Their application is demonstrated using SISO example problems. 2. Dynamic Operability of Processes The operating characteristics of processing plants have been rigorously studied in the literature over the past two decades. Several terms, such as flexibility, controllability, resiliency, and operability, have been used to refer to different characteristics of process designs. However, these terms have often been used interchangeably to define the same or similar aspects (e.g., see Hashimoto and Zafiriou17 for a qualitative description of these terms). Because there is no general agreement on which terms best define which characteristics, it is customary to adopt an existing definition in accordance with the proposed performance indicator. Here, the following, rather generic, definition of operability is adopted from Vinson18 and used throughout this paper. Definition 1. A process is operable if the available set of inputs is capable of satisfying the desired steadystate and dynamic performance requirements defined at the design stage, in the presence of the set of anticipated disturbances, without violating any process constraints. 2.1. Dynamic Operability Measure. Quantitative measures of dynamic performance are needed to perform an effective dynamic operability analysis. Rise time, settling time, overshoot, and integral square error are well-known examples of such measures. Here, a measure similar to settling time is used to quantify the dynamic performance. The following is the definition of this dynamic operability measure. Definition 2. Dynamic operability is measured as the shortest time required for a system to settle to the desired set point after a set-point change and/or disturbance occurrence. This operability measure is based on the idea that the time spent away from the desired set point is linked to potential losses due to off-specification products. An estimate of this operability measure can be obtained using different types of controllers. However, a performance measure independent of the controller and capable of assessing the inherent limitations of the process is desirable. A minimum-time optimal controller suits these demands very well. The use of such an openloop optimal controller is based on the implied assumption that a feedback controller exists that will deliver a dynamic performance that is close to the one calculated using the minimum-time optimal controller. This point will be further discussed in the next section with respect to an example problem. For multivariable systems, the shortest response time is associated with the time required for the slowest output to settle down after a set-point change or a disturbance. One should note that the slowest output might not be the same for all set points and disturbances. Furthermore, if some outputs are considered to be less important than others, a set of primary outputs can be defined and dynamic operability can be measured in terms of only these outputs. A similar setting is used in Subramanian19 et al. to assess the dynamic operability of CSTR systems in which some of the outputs

are controlled at a set point whereas others are controlled within a set interval. Carvallo15 used the same operability measure within a minimum-time optimal control framework to develop a worst-case performance index, which is defined as the minimum time necessary to overcome the worst setpoint change and/or disturbance that can enter a system. Bahri et al.20 used the response time as a performance measure in an integrated design strategy in which they incorporated the operability aspects into the design problem formulation. They applied an upper bound on the response time to ensure a rapidly responding process design and then varied the bound to observe how the plant design and economical objective were affected. Jaisathaporn21 used the time spent in producing off-specification products as a measure of lack of operability. He studied the load rejection and switchability characteristics of CSTR designs in which the performance was evaluated using PI controllers. 2.2. Minimum-Time Optimal Control Problem. The minimum-time optimal control problem for generic discrete-time linear systems can be formulated as in the following optimization problem

t/f (ysp, d) ) min tf

(1)

s.t. xk+1 ) Φxk + Γuuk + Γdd

(2)

˜ + Σ3d + Σ4 e 0 Σ1x˜ + Σ2u

(3)

Σ5xtf + Σ6utf + Σ7d + Σ8 ) 0

(4)

uk

x0, u0, ysp, d given where t/f (ysp, d) represents the magnitude of the minimum time necessary to respond to a set-point change and/or a disturbance. Equation 2 represents the discretetime linear model of the system, where xk is the nx-dimensional state vector; uk is the nu-dimensional input vector at time k; and k ) 0, 1, ..., tf. Note that, in this discrete-time formulation, tf is equal to the number of samples, and its equivalent value in time units can be obtained by multiplying it by the sampling time, T. Φ, Γu, and Γd represent the system matrices of the discrete-time state-space formulation of the dynamics of the process. The constraints in eq 3 represent the process constraints, including the bounds on the magnitudes and the rate of change of the input variables, ˜ ) [uT0 uT1 ‚‚‚ utTf ]T are the where x˜ ) [ xT0 xT1 ‚‚‚ xtTf ]T and u stacked state and input vectors, respectively. Additional constraints in eq 4 are the final-time constraints, which are typically set (see eqs 5 and 6 below) to ensure that the system reaches or returns to the set point, ysp, at the final time tf and stays there afterward, i.e., xk+1 ) xk for all k g tf.

Cxtf - ysp ) 0

(5)

(Φ - I)xtf + Γuutf + Γdd ) 0

(6)

The Σj’s, in eqs 3 and 4, represent the constantcoefficient matrices forming the corresponding constraints. In the operability framework presented here, set points and disturbances entering the process are

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assumed to be in step form. For this reason, ysp and d appear in the formulation as time-independent variables. For the optimal control problem in eqs 1-4 to have a solution, it is necessary that the set-point changes be tractable and that the disturbances rejectable at steady state. In other words, the constraints in eqs 2-4 must have at least one feasible solution at steady state; otherwise, the optimal control problem has no solution. The solution of the minimum-time control problem can be computationally intensive. Several different numerical techniques exist for solving this optimal control problem. For the linear discrete-time formulation considered here, a commonly used technique is the one proposed by Zadeh and Whalen.22 In this technique, the optimal control problem is partitioned into a set of linear programming (LP) problems. For a given final time tf, one solves a LP problem whose objective is to minimize a norm of the final-time conditions in eq 4. The minimum time is the smallest tf for which eq 4 is satisfied. It is with this technique that the example problems of this paper are solved. The details of this LP formulation are given in the Appendix. Also, one can find a detailed discussion of different LP formulations for solving minimum-time control problems in Gutman.23 2.3. Dynamic Operating Spaces. The definition of the dynamic operating spaces follows the work of Vinson and Georgakis24,25 on the steady-state operability analysis of processing plants. In their work, Vinson and Georgakis24,25 proposed a performance index, called the operability index (OI), that measures the ability of a process design to reach the desired outputs and reject the anticipated disturbances at steady state. Using this approach, candidate process designs can be ranked according to their OIs, and inoperable designs can be eliminated. The proposed approach can also be used to identify potential design changes by calculating the ranges of inputs that will guarantee steady-state operability. Subramanian and Georgakis26,27 extended this approach to the case of nonlinear systems and analyzed the operability characteristics of chemical reactors and reacting systems. Subramanian et al.19 proposed an optimization-based approach that further extends the original OI formulation to include nonsquare systems. A brief review of the steady-state operability analysis is given below before dynamic operating spaces are introduced. The first definition is that of the available input space (AIS), which is defined as the set of values that the input variables can take. The mathematical definition of the AIS is given by

ranges of the disturbances that are expected to enter the process constitute the expected disturbance space (EDS).

AIS ) {u|Ru e Λuu e βu}

dDOpS ) {(tf, ysp, d) | tf e tfd(ysp, d), ∀ysp ∈ DOS, ∀d ∈ EDS} (12)

(7)

where Λu is a constant-coefficient matrix corresponding to the physical constraints on the inputs. For most processes, available input values range from a minimum value, often equal to zero, to a maximum value, often a multiple of the nominal value. The desired operating window of the process outputs forms the desired output space (DOS).

DOS ) {y|Ry e Λyy e βy}

(8)

Here, Λy is a constant-coefficient matrix that typically corresponds to different product grades, transition regions, and physical constraints on the outputs. The

EDS ) {d|Rd e Λdd e βd}

(9)

The EDS includes all disturbances that can be captured with the steady-state model of the process, including uncertainties in the model parameters such as heat of reaction, heat-transfer coefficients, kinetic constants, etc. The set of input values required for the system to reach all outputs in the DOS forms the servo desired input space (DIS). The servo DIS can be calculated by mapping the DOS onto the input space using the inverse of the steady-state model. A similar DIS calculation can be performed in the regulatory case to obtain the set of input values necessary to reject all disturbances in the EDS. The main objective is to be able to reach all of the desired outputs and, at the same time, reject all of the expected disturbances. The overall DIS is calculated as the set-theoretic union of the input spaces required for the servo and regulatory tasks. The operability index is defined on the basis of the AIS and the overall DIS and gives a quantitative figure of how much of the DIS is covered by the AIS. The system is said to be steadystate operable, i.e., OI ) 1.0, if the AIS is large enough to cover the entire DIS. The concept of steady-state operability is necessary but not sufficient for the system to be completely operable. It must be complemented by dynamic operability. Dynamic operating spaces are the extensions of steady-state operating spaces to the dynamic case. They include process constraints and specifications that need to be considered during transient operation. The first operating space, called the dynamic available input space (dAIS), is the dynamic equivalent of the AIS and is defined as the set of values that the input variables and their combination can take. A typical dAIS includes the constraints on the magnitudes and the rate of change of the inputs as given in eqs 10 and 11 (u0 is given).

j ∀k ∈ [1, tf] u e uk e u

(10)

∆u e uk - uk-1 e ∆u ∀k ∈ [1, tf]

(11)

The dynamic desired operating space (dDOpS) is defined as the space formed by the combination of the DOS, EDS, and desired response times. The mathematical definition of the dDOpS is

where tdf (ysp, d) represents the desired dynamic performance, or the maximum allowable response time, in tracking a set-point change, ysp, in the DOS and/or recovering from a disturbance, d, in the EDS. The dDOpS can easily be formulated for addressing the servo and regulatory problems separately by setting the variables that are not considered (disturbance or set point, respectively) to their nominal values. Note that, even though a lower bound for the response time tf is not specified, the response time has an implicit lower bound of zero.


The dynamic achievable operating space (dAOpS) is defined as the operating space representing the dynamic performance that can be achieved by the system for a given choice of the dAIS, DOS, and EDS. Mathematically, the dAOpS can be defined as

dAOpS ) {(tf, ysp, d) | tf g t/f (ysp, d), ∀ysp ∈ DOS, ∀d ∈ EDS, u ∈ dAIS} (13) The lower bound, t/f (ysp, d), for the response time is obtained from the minimum-time optimal control calculation. This also establishes an upper bound for the achievable control performance of the system. Because the response of a stable system to any (ysp, d) can take an infinite time to reach the desired steady-state when u is changed to the corresponding steady-state value, the dAOpS is not bounded by an upper value of the response time. Note that, if the system is not steadystate operable for some values of (ysp, d) then a solution to the minimum-time control problem, or t/f (ysp, d), does not exist at these values. In the following section, a dynamic operability index is presented for the operability assessment problem. The index exploits the concept of operating spaces to quantify the dynamic performance over the entire operating ranges. 3. Assessment of Dynamic Operability 3.1. Dynamic Operability Index. A comparison of the desired and achievable dynamic operating spaces (dDOpS and dAOpS, respectively) is used to define an index for dynamic operability. This index is referred to as the dynamic operability index (dOI) and is defined as follows: Definition 3. The dynamic operability index is defined as the ratio of the size of the operating space that can be achieved within a desired response time tdf (ysp, d) to that of the desired operating space, given the ranges of the available inputs in the dAIS. To mathematically define the operability index, two additional operating spaces are introduced. The first operating space, referred to as S1, is created by the union of the set points in the DOS and the disturbances in the EDS

S1 ) {(ysp, d) | ∀ysp ∈ DOS, ∀d ∈ EDS}

(14)

S1 basically represents the combined ranges of changes in operating conditions that are desired to be overcome with the available inputs. The second operating space, referred to as S2, represents the ranges of set points and disturbances that can be achieved with the minimumtime optimal controller within tdf (ysp, d) or less. S2 is obtained by projecting the intersection of the dDOpS and dAOpS onto S1 and defined as

S2 ) {(ysp, d) | t/f (ysp, d) e tdf (ysp, d), ∀ysp ∈ DOS, ∀d ∈ EDS} (15) This intersection is not difficult to perform in the onedimensional examples we consider later. In higher dimensions, the intersection could possibly be achieved by the use of Delaunay triangulation in a manner similar to that of Vinson.18

The dOI is mathematically defined on the basis of these operating spaces as

dOI )

µ(S2) µ(S1)

(16)

where µ represents a function calculating the size of the corresponding space. The dOI can take values varying between 0 and 1, representing the worst and the best performances, respectively. Because the achievable response times are computed using an idealized controller, the dOI represents an upper bound for the achievable control performance of the process. That is, if the minimum-time controller fails to satisfy the performance requirements in the dDOpS (i.e., dOI < 1), then no realizable controller can satisfy them in practice. However, even if the performance requirements are met by the optimal controller (i.e., dOI ) 1), it does not necessarily mean that a feedback controller exists that will give the same exact performance. Note that the dOI can be calculated as a servo or regulatory dynamic operability index by selecting the appropriate dDOpS. In the following subsections, illustrative examples are given to discuss the application of this assessment technique. The example systems are selected as SISO processes for the sake of graphical presentation of the defined concepts. In these examples, minimum-time optimal control problems are solved numerically as described in the Appendix using GAMS with CPLEX as the LP solver. 3.2. Illustrative Example. The first example, taken from Uztu¨rk and Georgakis,28 discusses the dynamic operability of a first-order system

y(s) )

Kpe- τds u(s) τs + 1

(17)

where τ, Kp, and τd represent the time constant, process gain, and time delay, respectively. Because the LP formulation for solving the optimal control problem is in discrete time, the system is first discretized and then transformed into the state-space formulation. In each of the processes studied below, a sampling time of 0.1 is used. The desired operating window for the process output is specified as in the following DOS

DOS ) {y| -1.0 e y e 1.0}

(18)

It is desired that the system reaches all operating points in the DOS within 1.0 time unit (i.e., tdf ) 1.0), which forms the following dDOpS

dDOpS ) {(tf, ysp) | tf e 1.0, ∀ysp ∈ DOS} (19) Note that, in the discrete-time formulation presented in the previous section, tf is defined to be equal to the number of samples. However, in the examples discussed here, tf is referred to in terms of time units for the ease of presentation. Consider three first-order systems with process gains of Kp ) 1.0, 2.0, and 4.0; an open-loop time constant of τ ) 1; and a time delay of τd ) 0. Also, assume that the nominal operating point for each system is at y0 ) 0. The dAIS for these systems is specified as

dAIS ) {uk| -1.0 e uk e 1.0 ∀k}

(20)

426


Figure 1. Comparison of the dynamic operability of first-order processes with different process gains, Kp.

The t/f values for each one of these systems and for all ysp in the DOS are calculated using the LP formulation given in the Appendix. The resulting performance bounds are depicted in Figure 1. The region above each curve corresponds to the region of achievable dynamic performance, i.e., the dAOpS. The curves represent the bounding values of the response times within which the set points can be reached. For example, if the system with the gain of 1.0 starts at the nominal point y0 ) 0, it can reach a new set point, say, equal to 0.8, at times greater than or equal to 1.65 (i.e., t/f ) 1.65). The horizontal line at tf ) 1.0 represents the bounding value of the desired response times as defined in the dDOpS with tdf . The dOI for the system with the gain of 1.0 is calculated to be 0.63, whereas the other two systems are dynamically operable, i.e., dOI ) 1.0. These index values can easily be obtained as the fractions of the setpoint range that can be achieved within the desired response time of 1.0 time units. These results show that, as the process gain increases, the minimum time becomes smaller, and the region of achievable dynamic performance, or the dAOpS, becomes larger. Hence, the dynamic operability of a system with higher gain is superior to that of one with smaller gain for the entire range of the output. This is an expected behavior as the process gain amplifies the effect that a certain change in the input variable has on the output variable. The next example examines the performance of a system when it initially operates at different nominal operating points. Consider the system in eq 17 with Kp ) 2.0, τ ) 1.0, and τd ) 0. The dAOpS for this system is calculated for the different initial operating points y0 ) - 0.6, 0, and 0.4. The resulting performance bounds are depicted in Figure 2. The bounding curves show that the dynamic performance of a system depends on its initial operating point. For instance, the system cannot be driven from 0 to -0.6 as fast as it can be driven from -0.6 to 0. This behavior arises because, in the second direction, the magnitude of the maximum forcing power is much higher than it is in the first direction. In the first direction, the magnitude of the available maximum input is 1, whereas in the reverse direction, it is 1.6. Finally, the limitations caused by time delay and input constraints on the dynamic performance are discussed using two first-order systems. Both systems are selected to have a time constant of τ ) 1, a process gain of Kp ) 1.0, and a nominal operating point of y0 ) 0. The first system (A) has a time delay of τd ) 0.3 and its input is bounded with the constraint |u| e 2, whereas

Figure 2. Comparison of the dynamic operability of first-order processes with different nominal operating points.

Figure 3. Comparison of the effect of time delay, τd, versus the effect of input constraints on the dynamic operability of a firstorder process.

the second system has no time delay and its input is bounded with the constraint |u| e 1. The performance bounds for these systems, calculated using the minimumtime optimal controller, are shown in Figure 3. The aim in this example is to show that, although time delay is a performance limitation, it might not be the only factor determining the dynamic performance of the system. It can be seen from Figure 3 that system A has a better overall performance than system B. Specifically, the dynamic operability of system A is equal to 1.0 for a desired response time of tdf ) 1.0 time units (dashed line). The corresponding dynamic operability of system B is calculated to be 0.63. The performance limitations caused by the smaller input ranges in system B are more restrictive than limitations due to the time delay in system A. System A has a faster response if the new set point is in either one of the two intervals (-1, - 0.4) and (0.4, 1). The negative effect of dead time is more pronounced at smaller magnitudes of set-point changes. Naturally, the system with dead time can reach no other set point within a time that is smaller than its dead time. This simple example clearly demonstrates that dynamic performance limitations caused by open-loop dynamics and input constraints should not be evaluated one at a time but must be considered simultaneously. Also, it shows that these limitations can be effectively captured in the calculations of the dAOpS and dOI. 3.3. Rate-of-Change Constraints on Inputs. The effect of input rate-of-change constraints on dynamic operability has rarely been examined in the literature. In practice, most of the manipulated variables are


Figure 4. Comparison of the dynamic operability of a SISO process when different rate-of-change constraints are imposed on the input.

Figure 5. Servo dynamic operability of a SISO process (disturbance is kept at its nominal value).

bounded by constraints on both their magnitude and their rate of change. In this subsection, an example problem is studied that analyzes the effect of these constraints on the dynamic performance. A third-order system with the following transfer function is considered

y(s) )

6 u(s) (2s + 1)(4s + 1)(6s + 1)

(21)

and a DOS of the same form as in eq 18 is used. A sampling time of 0.2 is used for discretizing the system. The dynamic operability of this system is examined for different dAIS specifications.

dAIS ) {uk| -0.2 e uk e 0.2, -δu e uk+1 - uk e δu, ∀k} (22) Three cases are studied where the rate-of-change constraints are selected as (1) δu f ∞ (i.e., no rate-of-change constraints), (2) δu ) 0.03, and (3) δu ) 0.01. The dAOpS for each of these cases is calculated using the LP formulation, and the results are depicted in Figure 4. The lowest curve in Figure 4 represents the system with no constraints on the rate of change of the input. It can be seen that, when the bounds on the rate-ofchange constraints are tightened, the system reacts to set-point changes more slowly, and the dynamic performance deteriorates. For instance, if the desired dynamic performance in the dDOpS is specified to be, say, tdf ) 15, then the system with no rate-of-change constraints has a dOI of 0.75, whereas the system with δu ) 0.01 has a dOI of only 0.54. This comparative study clearly shows that the rate-of-change constraints can considerably degrade the dynamic performance of the system and that this degradation can be effectively captured by the dOI. 3.4. Servo and Regulatory Operability. In the previous examples, only servo operability problems were discussed. In this example, both servo and regulatory operability problems are studied for illustration purposes. Consider the following SISO system

y(s) )

4.5 u(s) + (s + 1)(5s + 1)(25s + 1) 5.5 d(s) (23) (s + 1)(11s + 1)

Figure 6. Regulatory dynamic operability of a SISO process (set point is kept at its nominal value).

with a DOS and EDS of the forms

DOS ) {y| -5.0 e y e 5.0}

(24)

EDS ) {d| -1.0 e d e 1.0}

(25)

The available ranges of the inputs are specified as in the following dAIS

dAIS ) {uk| -3.0 e uk e 3.0, -δu e uk+1 - uk e δu, ∀k} (26) where two cases are studied, namely, (1) δu f ∞ and (2) δu ) 0.01. Here, servo and regulatory problems are studied separately for the sake of simplicity in the presentation, but one can easily study the cases in which a disturbance enters at the same time a set-point change occurs. The nominal operating points are selected to be at the origin for both problems, i.e., y ) 0 and d ) 0. A sampling time of 1.0 is used for discretizing the system. In the regulatory problem, it is assumed that the disturbance model is given, and the disturbance is considered known. Naturally, this is the most ideal case. Figures 5 and 6 show the performance bounds for the servo and regulatory problems, respectively. For this particular system, when there are no rate-of-change constraints, the figures show that the regulatory tasks are more demanding than the servo ones, that is, they have a larger minimum time. However, when the rateof-change constraints are present, the tasks become equally demanding.

428


3.5. Feedback versus Optimal Controller. The main advantage of using optimal controllers in assessing dynamic performance is that they give a measure of operability that is independent of the feedback controller to be used and, therefore, that reflects the inherent performance limitations of the process design. Such an operability measure constitutes an upper bound for the achievable control performance of the process. In other words, any feedback controller will give a performance with a larger response time than the optimal controller. Furthermore, the operability characteristics captured by the optimal controller should be representative of the best performance that one can achieve with feedback controllers. In this context, an example problem is solved that compares the performance of different feedback controllers with the minimum-time optimal controller calculations. The primary aim here is not to compare the dynamic performance of feedback controllers themselves but to determine whether a feedback controller exists that has dynamic characteristics that can come close to those predicted by the open-loop optimal one. The comparison here is based on an example case, yet, it is implied that these results are applicable to a much wider class of linear dynamic systems, possibly to all linear constrained dynamic systems. However, a mathematical proof of such a conjecture is not offered at the present time. One should also note that the performance characteristics for the feedback controllers depend on their tuning and that different tunings might give numerical results that are slightly different from the ones presented here. However, it is not expected that changes in tuning will affect the relative performance of the families of the controllers examined here. In addition to an assessment of relative performance among classes of controllers, the calculations to be presented in the following will serve as a demonstration of the existence of at least one feedback controller that closely approximates the characteristics of the optimal controller. This will add extra validity to the optimal controller calculations. Consider the third-order system given in eq 21 with a DOS of the same form as in eq 18 and a dAIS as in eq 22. The rate-of-change constraints are selected as (1) δu f ∞ and (2) δu ) 0.03. The three different types of feedback controllers used in this example are PI, IMC, and MPC. The response times, tf, for these feedback controllers are calculated using the same tolerance as was used in calculating the minimum-time controller. The response time is recorded when the system reaches and stays in the (10-3 vicinity of the set point. The resulting performance bounds are depicted in Figures 7 and 8. It can be seen from these figures that the response times of the feedback controllers are larger than those of the optimal controller, as expected. The performance of the PI and IMC controllers is quite inferior to that of the optimal one, but the MPC controller can provide a performance that is very close to that of the optimal controller. Consequently, for this particular example, the performance bound calculated by the optimal controller is quite realistic. This is true both when rate-of-change constraints are imposed (Figure 7) and when they are not (Figure 8). The inevitable relative comparison of the closed-loop performance of the different controllers can also be rationalized as follows. The MPC controller explicitly accounts for the process dynamics and the input constraints, the IMC accounts only for the process dynamics, and the PI does

Figure 7. Comparison of the minimum-time optimal controller with different feedback controllers (no rate-of-change constraints on the input).

Figure 8. Comparison of the minimum-time optimal controller with different feedback controllers when rate-of-change constraints are imposed on the input.

not explicitly account for either. If tuned in a similar manner, these controllers would respond in the order presented in these figures. Apparently, one can make the MPC respond more slowly than PI or IMC by choosing to drastically penalize the control variable moves. However, this is not the point here. 4. Enhancement of Dynamic Operability The assessment technique presented in the previous section determines whether a candidate process design can achieve the desired dynamic performance or ranks alternative designs based on their dOIs. However, if the performance specifications cannot be met by a present design, it is desirable to generate design changes that will improve the performance of the process. Constraints on the input variables constitute one of the major sources of performance limitations. Accordingly, the enhancement problem is formulated here to identify the ranges of inputs that will guarantee the desired dynamic performance for the entire ranges of set points and disturbances, i.e., a dOI ) 1.0. 4.1. Problem Formulation. The enhancement problem involves the calculation of the smallest necessary ranges of the inputs, referred to as the bounding box, with which the process can achieve the desired dynamic performance requirements as defined in the dDOpS for all ysp in the DOS and all d in the EDS. The calculation


of this bounding box is formulated in two steps. In the first step, the smallest bounding box of inputs is calculated for a given (ysp, d). In the second, the set of (ysp, d) that would require the largest input bounds is identified for each input variable separately. These formulations are presented and discussed below. The first step, from now on referred to as P1, can be formulated as in the following optimization problem in the case of linear discrete-time systems

P1:

J(u j *) ) min J(u j) uk,u j

(27)


(28)

Cxtf - ysp ) 0

(29)


(30)

j |uk| e u

(31)

|uk+1 - uk| e ∆u

(32)

x0, u0, ysp, d, tf given where the objective function, J, typically represents the cost associated with demanding larger ranges on the inputs and is selected as an appropriate norm of u j. Equation 28 represents the discrete-time linear model of the system. The constraints in eqs 29 and 30 represent the final-time specifications. Constraints on the inputs and the rates of change of the inputs, defined in the dAIS, are given in eqs 31 and 32, respectively. Here, the min operator iterates over the possible choices of the input trajectory u searching for the smallest input ranges that will satisfy the final-time constraints in eqs 29 and 30 for a given (ysp, d) and a fixed final time of tdf . The resulting bounding box is represented by the vector u j *. Note that P1 is an optimal control problem in which the final time is fixed (tf ) tdf ), as opposed to the minimum-time optimal control problem considered earlier, which is a free final-time control problem. Therefore, it is computationally less involved. As discussed above, the idea in the enhancement problem is to identify the ranges of inputs for which the process is dynamically operable for the entire operating ranges. To achieve this objective, a search over all ysp ∈ DOS and all d ∈ EDS has to be made, which would give the largest bounding box in P1. However, the most demanding (worst) set of (ysp, d) might not be the same for each input. As a result, the bounding value of each input should be calculated separately. This problem, referred to as P2, is formulated as in the following optimization problem

P2:

j *j} u j /j ) max {u ysp∈DOS d∈EDS

(33)

where u j *j is the jth element of the nu-dimensional bounding-box vector u j * calculated using P1. Here, the max operator searches over ysp and d looking for the largest bound of input uj out of the input bounds calculated through P1. This bound is referred to as u j /j . Note that problem P1 is nested in P2, which makes the latter quite complex to solve. On the other hand, one

would need to solve P2 once for each input, uj, to obtain the bounding values of all inputs. The enhancement problem can also be demonstrated graphically using a process with two inputs. Figure 9 depicts three bounding boxes, u j *(yisp, di), for such a process. Each of these boxes corresponds to the solution of a P1 problem for a fixed value of (ysp, d). For instance, u j *(y1sp, d1) represents the bounding box of inputs required for the process to overcome (y1sp, d1) within the response time tdf as defined in the dDOpS. Of interest is the calculation of the union of all bounding boxes for all set points, ysp, in the DOS and all disturbances, d, in the EDS. This is depicted in Figure 9 with the boldface rectangle and is the solution of the problem P2 defined in eq 33. Note that the calculation of the bounding box also reveals the most demanding set points and disturbances that require the largest bounds on the inputs. The solution of this optimization problem can be computationally intensive, especially in the case of MIMO nonlinear systems. However, for linear systems, the most demanding sets of (ysp, d) would be at the vertices of the DOS and EDS. Therefore, one can enumerate these sets, solve as many P1 problems as the number of vertices, and determine the bounding box by inspecting the results. This would eliminate the need for solving the second problem, P2, in an explicit manner, which simplifies the enhancement problem significantly. Yet, as the number of vertices, 2m, grows exponentially with m, where m is the total number disturbances and set-point controlled outputs, the number of optimization problems one needs to solve might become overwhelming for large systems. Next, an example problem is solved that discusses the application of the enhancement analysis. Then, the example of the rate-of-change constraints on inputs presented in the previous section is revisited for further analysis of the effect of these constraints. 4.2. Illustrative Example. The following example, taken from Uztu¨rk and Georgakis,28 aims at using both the steady-state and dynamic operability analyses to demonstrate how they can be employed to extract information related to the bounding values of the inputs that will guarantee the desired performance. Consider the first-order system in eq 17 with the system parameters Kp ) 2.0, τ ) 1.0, and τd ) 0. First, the steadystate operability of the process is examined where a DOS of the same form as in eq 18 is used. In the initial design, the input bounds were selected as in the following AIS

AIS ) {u| -0.4 e u e 0.4}

(34)

Note that this initial design is not steady-state operable because the ranges of the input, as specified in the AIS, are not large enough to cover the entire ranges of the output given in the DOS. Input bounds that will guarantee steady-state operability can be calculated by constructing the DIS. For this purpose, the DOS is mapped onto the input space using the inverse of the process gain, Kp. For Kp ) 2.0, the resulting DIS is given by

DIS ) {u| -0.5 e u e 0.5}

(35)

Steady-state operability is guaranteed once the input ranges are modified as specified in the DIS.

430


Figure 9. Graphical representation of problems P1 and P2 defined in the enhancement formulation.

Figure 10. Comparison of the dynamic operability of a first-order process before and after the enhancement study is applied.

Next, the dynamic operability of the process is studied. In the dynamic case, following the steady-state analysis, the ranges of the input are selected as given in the DIS. Accordingly, the dAIS is defined as

dAIS ) {uk| -0.5 e uk e 0.5, ∀k}

(36)

It is desired that the system reach all outputs in the DOS within 1.2 time units when the initial operating point is at y0 ) 0, which results in the following dDOpS

dDOpS ) {(tf, ysp)|tf e 1.2, ∀ysp ∈ DOS}

(37)

The achievable performance bounds for this system are calculated as in the assessment problem, and the resulting dAOpS is depicted in Figure 10. The area under the horizontal dashed line represents the dDOpS. It can be seen that the process cannot achieve the dynamic performance specifications given in the dDOpS for set points that are larger than 0.7 and smaller than -0.7. In other words, the system is not dynamically operable, and it has a dOI ) 0.7. Input ranges that will guarantee dynamic operability are computed using the optimization formulations P1 and P2. The cost function in P1 is selected to be J ) u j . Here, P2 can be eliminated because it is known a priori that the most demanding set-point change is at the vertices of the DOS, i.e., at ysp ) 1.0 or -1.0. Because of symmetry with respect to the nominal operating point y0 ) 0, only one of these boundary points is needed. The optimization problem P1 is then solved for ysp ) 1.0 and tf ) 1.2, and the

Figure 11. Relationship between the desired dynamic perforj , calculated using the mance, tdf , and the required input bound, u enhancement formulation. The dynamic operability of a SISO process is compared when different rate-of-change constraints are imposed on the input.

ranges of the input are found to be -0.7155 e uk e 0.7155. Note that these ranges are larger than the ones found using the steady-state operability analysis. Using these input ranges, the performance bounds are recalculated, and the dAOpS is depicted in Figure 10. One can see that now the process can reach all set points in the DOS within the desired response time of 1.2, and it has a dOI ) 1. This simple example shows how the enhancement analysis can be used to evaluate the ranges of the inputs that are required to satisfy the dynamic performance specifications given in the dDOpS. 4.3. Rate-of-Change Constraints on Inputs. In this subsection, the effect of rate-of-change constraints on the achievable control performance of a system is examined using the enhancement framework. Consider the third-order system given in eq 21 with a DOS as in eq 18 and a dAIS as in eq 22. Three cases are studied where the rate-of-change constraints are selected as (1) δu f ∞ (i.e., no rate-of-change constraints), (2) δu ) 0.10, and (3) δu ) 0.03. In this study, the ranges of the input that will guarantee dynamic performance are computed using the optimization formulations P1 and P2 for different values of tdf . Once again, note that the worst set point is at the vertices of DOS, and thus, P2 is eliminated. The resulting u j values are depicted in Figure 11. Each curve represents the minimum desired range of the input variable necessary for the dynamic system to have the desired dynamic performance for a given rate-of-change constraint. The area to the right of each curve represents the region in which the system is operable. The horizontal line, labeled steady state, represents the magnitude of the input required to track all set-point changes in the DOS at steady state, that is, without worrying about how slowly these changes can be achieved. Note that, as tdf increases, the input bounds converge to this steady-state value and the effect of rate-of-change constraints become less important. When there are no rate-of-change constraints, the minimum response time, depicted in Figure 11, approaches zero as the input bound goes to infinity. On the other hand, the bounding curves in the same figure show that, when the rate-of-change constraints are present, there exists a critical point after which the performance of the system cannot be further improved


by increasing the ranges of the input. For instance, in the case in which δu is equal to 0.10, the critical point is observed at tdf ) 9.2 and u j ) 1.05. In this example, the system cannot react faster than 9.2 time units to the worst set-point change in the DOS even if the input bound is larger than 1.05. This is because, at the critical point and thereafter, the rate constraints are saturated throughout the control trajectory. An increase in the magnitude of the input bound does not affect the saturation of the rate constraints, and an improvement in the performance cannot be achieved.

The enhancement problem involves the calculation of the input ranges that will guarantee a desired dynamic operability. As in the assessment problem, these input bounds are optimistic. In other words, a feedback controller might require larger input bounds to achieve the same performance. The enhancement analysis can be used effectively to identify bottlenecks due to input availability and to gain insight into how to modify the ranges of the inputs to make the process operable.

5. Conclusions

The financial support of the industrial sponsors of the Chemical Process Modeling and Control Research Center at Lehigh University and P. C. Rossin College of Engineering and Applied Science through the Milestone Fellowship is greatly appreciated.

A new technique is presented for dynamic operability analysis of process designs. The study is based on the solution of an open-loop optimal control scheme and, therefore, represents an upper bound for the achievable control performance of a process. A dynamic operability measure is introduced for capturing the dynamic performance in response to set-point changes and disturbances. The measure is used in defining a dynamic operability index, dOI, which quantifies the dynamic performance over the entire operating ranges. The study is performed in two directions: (1) assessment and (2) enhancement. The assessment problem involves the calculation of the achievable dynamic operating space for the given input ranges, which is then used to calculate the operability index. Because the analysis is based on an open-loop optimal controller, the results are optimistic and should be treated in the following way. If the operability assessment of a system indicates that the system is not dynamically operable, then no realizable controller can achieve the desired performance requirements. However, if the system can achieve the desired performance, it does not necessarily mean that there exists a realizable controller that can give the same performance. The latter observation suggests a followup problem that needs to be examined, namely, the design of a control system that will provide, in closedloop form, an operability that is as close as possible to the inherent operability of the process calculated by the optimal control problem. As demonstrated here, a MPC controller comes very close to the inherent dynamic operability characteristics calculated using the optimal controller. The MPC controller is faster than the IMC controller, which, in turn, is faster than the PI controller. This is not surprising as the MPC controller explicitly accounts for the process dynamics and the input constraints, the IMC controller accounts for only the process dynamics, and the PI does not explicitly account for either. The application of the assessment problem is presented using SISO systems in which the desired and achievable operating spaces can be displayed graphically. The examples show that the proposed analysis can be used effectively to assess the dynamic characteristics of processes. The operability index, dOI, gives a quantitative measure of the achievable control performance of a process under ideal circumstances. The dOI is quite easy to interpret and is independent of variable scaling. The performance loss due to the presence of rate-ofchange constraints is also examined using an example problem. This study shows that the effect of these constraints can be quite significant when fast operation is required.

Acknowledgment

Appendix. Solution of the Minimum-Time Control Problem using a LP Formulation Consider the following form of the minimum-time optimal control problem defined in eqs 1-4

min tf

(38)


(39)

Cxtf - ysp ) 0

(40)


(41)

j u e uk e u

(42)

∆u e uk+1 - uk e ∆u

(43)

uk

x0, u0, ysp, d given Methods based on the LP formulation are commonly employed in solving this minimum-time control problem. In these methods, the free-final-time optimal control problem in eqs 38-43 is converted into a sequence of fixed-final-time control problems. The fixedfinal-time subproblem is set to minimize the final-time constraints in eqs 40 and 41 as in the following22

min ||Cxtf - ysp||p + ||(Φ - I)xtf + Γuutf + Γdd||p uk (44) s.t. xk+1 ) Φxk + Γuuk + Γdd

(45)

u e uk e u j

(46)

∆u e uk+1 - uk e ∆u

(47)

x0, u0, ysp, d, tf given The minimum time is obtained by iterating over tf, and it is the smallest tf for which the objective function in eq 44 is less than a tolerance. All examples in this work are solved using a tolerance of (10-3. In this LP formulation, ||‚||p is typically selected to be the l1 or l∞ norm. The latter is used in this paper, in which case

432


the objective function is rewritten as in following linear form with additional constraints29

min (z1 + z2)

(48)

-z1 e (Cxtf - ysp)j e z1

(49)

-z2 e ((Φ - I)xtf + Γuutf + Γdd)l e z2

(50)

uk

where z1 and z2 are positive scalars, j ) 1, 2, ... ny, and l ) 1, 2, ... nx. Literature Cited (1) Ziegler, J. G.; Nichols, N. B. Process lags in automatic control circuits. Trans. ASME 1943, 65, 433-444. (2) Morari, M.; Perkins, J. Design for Operations. In Foundations of Computer Aided Process Design; CACHE: Austin, TX, 1994. (3) Morari, M. Design of resilient processing plants 3: A general framework for the assessment of dynamic resilience. AIChE J. 1983, 38, 1881-1891. (4) Holt, B. R.; Morari, M. Design of resilient processing plants 5: The effect of dead time on dynamic resilience. Chem. Eng. Sci. 1985, 40, 1229-1237. (5) Holt, B. R.; Morari, M. Design of resilient processing plants 6: The effect of right-half-plane zeros on dynamic resilience. Chem. Eng. Sci. 1985, 40, 59-74. (6) Skogestad, S.; Morari, M. Design of resilient processing plants 9: The effect of model uncertainty on dynamic resilience. Chem. Eng. Sci. 1987, 42, 1765-1780. (7) Skogestad, S.; Morari, M. Effect of disturbance directions on closed-loop performance. Ind. Eng. Chem. Res. 1987, 26, 20292035. (8) Psarris, P.; Floudas, C. A. Dynamic operability of MIMO systems with time delays and transmission zeroes 1: Assessment. Chem. Eng. Sci. 1991, 46, 2691-2707. (9) Psarris, P.; Floudas, C. A. Dynamic operability of MIMO systems with time delays and transmission zeroes 2: Enhancement. Chem. Eng. Sci. 1991, 46, 2709-2728. (10) Morari, M. Effect of Design on the Controllability of Chemical Plants. In Interactions Between Process Design and Process Control; Perkins, J. D., Ed.; Pergamon Press: Elmsford, NY, 1992. (11) Perkins, J. D.; Walsh, S. P. K. Optimization as a tool for design/control integration. Comput. Chem. Eng. 1996, 20, 315323. (12) Chenery, S. D. Process Controllability Analysis using Linear and Nonlinear Optimization. Ph.D. Dissertation, University of London, London, U.K., 1997.

(13) Mohideen, M. J.; Perkins, J. D.; Pistikopoulos, E. N. Optimal design of dynamic systems under uncertainity. AIChE J. 1996, 42, 2251-2272. (14) Swartz, C. L. E. A computational framework for dynamic operability assessment. Comput. Chem. Eng. 1996, 20, 365-371. (15) Carvallo, F. D. Design of Chemical Processes for Controllability. Ph.D. Dissertation, Carnegie Mellon University, Pittsburgh, PA, 1989. (16) Cao, Y. Control Structure Selection for Chemical Processes using Input-Output Controllability Analysis. Ph.D. Dissertation, University of Exeter, Exeter, U.K., 1995. (17) Hashimoto, I.; Zafiriou, E. Design for Operations and Control. AIChE Symp. Ser. 1995, 91 (304), 103-104. (18) Vinson, D. R. A New Measure of Process Operability for the Improved Steady-State Design of Chemical Processes. Ph.D. Dissertation, Lehigh University, Bethlehem, PA, 2000. (19) Subramanian, S.; Uztu¨rk, D.; Georgakis, C. An optmization-based approach for the operability analysis of continuously stirred tank reactors. Ind. Eng. Chem. Res. 40, 4238-4252. (20) Bahri, P. A.; Bandoni, J. A.; Romagnoli, J. A. Operability assessment in chemical plants. Comput. Chem. Eng. 1996, 20, 787-792. (21) Jaisathaporn, P. Trade-Offs Between Load Rejection and Switchability. Master’s Thesis, Lehigh University, Bethlehem, PA, 2000. (22) Zadeh, L. A.; Whalen, B. H. On optimal control and linear programming. IEEE Trans. Autom. Control 1962, 7, 45-46. (23) Gutman, P. O. Controllers for Bilinear and Constrained Systems. Ph.D. Dissertation, Lund Institute of Technology, Lund, Sweden, 1982. (24) Vinson, D. R.; Georgakis, C. A New Measure of Process Output Controllability. In 5th IFAC Symposium on Dynamics and Control of Process Systems; Georgakis, C., Ed.; Elsevier: Amsterdam, 1998. (25) Vinson, D. R.; Georgakis, C. A new measure of process output controllability. J. Process Control 2000, 10, 185-194. (26) Subramanian, S.; Georgakis, C. Steady-state operabilty characteristics of reactors. Comput. Chem. Eng. 2000, 24, 15631568. (27) Subramanian, S.; Georgakis, C. Steady-state operability characteristics of idealized reactors. Chem. Eng. Sci. 2001, 56, 5111-5130. (28) Uztu¨rk, D.; Georgakis, C. An Optimal Control Perspective on the Inherent Dynamic Operability of Processes. Presented at the Annual AIChE Meeting, Miami, FL, Nov 15-19, 1998; Paper 217a. (29) Bertsimas, D.; Tsitsiklis, J. N. Introduction to Linear Optimization; Athena Scientific: Cambridge, MA, 1997.

Received for review February 26, 2001 Revised manuscript received June 29, 2001 Accepted June 29, 2001 IE0101792

Inherent Dynamic Operability of Processes: General Definitions and

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