Abstract
The regulation of a disturbed output can be improved when several manipulated inputs are available. A popular choice in these cases is the series control scheme, characterized by a sequential intervention of loops and faster loops being reset by slower loops, to keep their control action around convenient values. This paper tackles the problem from the frequency-domain perspective. First, the working frequencies for each loop are determined and closed-loop specifications are defined. Then, Quantitative Feedback Theory (QFT) bounds are computed for each loop, and a sequential loop-shaping of controllers takes place. The obtained controllers are placed in a new series architecture, which unlike the classical series architecture only requires one controller with integral action. The benefits of the method are greater as the number of control inputs grow. A continuous stirred tank reactor (CSTR) is presented as an application example.
1. Introduction
In process control, it is common to find several manipulated variables regulating a single measurable output (MISO: Multiple Input Single Output control). Remarkable examples are distillation columns [1], chemical and biochemical reactors [2–4], heat-exchangers [5], paper-machines [6], and medical control systems [7, 8]. Recent fields of application are the automotive industry [9, 10], consumer-electronics [11, 12], and robotics and unmanned aerial vehicles [13, 14].
The architectures for MISO control can be grouped in two: the parallel [15, 16] and the series disposition of controllers. The latter (see Figure 1) originally appeared in the habituating control by Henson et al. [15] and the midranging control by Allison and Isaksson [17]. It can be seen as a generalization of the valve position control (VPC) for the process industry presented by Shinskey [1] and Luyben [18]. Skogestad and Postlethwaite [19] labelled it as cascade control. The architecture seeks a sequential loop intervention from the fastest (bottom) to the slowest (top) loop, in order to recover the set-point at the -disturbed output . Besides, the fastest control variable returns to the setpoint when the slowest loop resets the fastest loop (input resetting control). Plant models and characterise the dynamic behaviour of the output in response to the manipulated inputs and , respectively. Similarly, models the response of to the disturbance input . The signal has been added to account for sensor noise.
In steady state, several combinations of the manipulated inputs (, ) could achieve the desired output . Hence, can be chosen according to different criteria, which are usually linked to efficiency, cost savings, and physical limitations. For example, midranging [17, 20] chooses as the midpoint of the actuator to preserve the largest maneuvering range in the fast loop. Conversely, in the temperature () control of a chemical stirred tank reactor (CSTR), Luyben [21] manipulates the coolant flow to a tank jacket () around the maximum removal capacity () in such a way that the feed-flow rate () is indirectly maximized, and so is the production rate. Other works [6, 15] prefer reducing as much as possible since represents a more expensive physical variable than , and global plant operating expenses are dominated by steady state control actions. Parallel structures of controllers can equally attain those aims. For instance, Nájera et al. [4] shows several ways to conduct the air flow set-point () in the sludge temperature () control of a biochemical reactor operated by the air-flow () and the inlet sludge flow (). A smaller reduces direct expenses by cutting the amount of . On the other hand, a higher deliberately pursues a major spending of in order to force a higher , which raises the production rate and makes the plant operation more profitable.
Most applications and methods in the literature show integral action in both controllers and to obtain zero steady-state tracking error in and . However, the two integrators in series can make the slowest loop conditionally stable [22]. Moreover, the more manipulated inputs there are, the harder the control design becomes. This is one of the concerns of the present work. A second one has to do with the frequency bands at which each input contributes to the regulation. The location of each input inside the control structure determines which plant works in high () and low () frequencies, respectively, whereas the controllers determine the border between these two ranges. These decisions may not be trivial. As stated before, some considerations are linked to the steady-state and permitted range of the manipulated variables themselves. Others have to do with the best achievable performance according to the plant frequency responses. Most contributions in the literature solve practical examples with ad-hoc decisions and tailor-made design methods, although some general design methods are described in [1, 15, 22–25]. Their common approach is firstly designing the fastest loop (i.e. ) to achieve certain performance in the regulation of , and then, designing the slowest loop (i.e. ) that takes to the setpoint . One important remaining challenge is how to perform the switching-off of the fast loop since global performance or even stability may be compromised [22]. Moreover, no method has been reported on how to distribute the performance amongst more than two branches as far as the authors are aware.
Under these premises, the present work proposes a new series structure of controllers allowing to exploit the benefits of using manipulated inputs while avoiding the inconveniences of having integral action in all controllers. In this work, integrators appear only in the controller that is next to the plant that works at the lowest frequencies ( if ). It will be shown that this is sufficient to achieve the set-points at the output and at the midrange inputs with zero steady-steady error. The present work adopts a frequency domain perspective, showing how to allocate the control load among the different loops. In particular, a robust methodology is proposed in the framework of Quantitative Feedback Theory (QFT) [26–28], which includes plant uncertainty in the controller design process. The new method details how to compute the QFT bounds and how to perform a sequential loop-shaping of controllers.
The paper is organised as follows. Section 2 justifies the new series structure for MISO control. Section 3 describes the multi-input participation from a frequency domain perspective and proposes a frequency domain method to achieve robust stability and performance in the regulation of the disturbed output and midrange inputs. Section 4 illustrates the usefulness of the proposed methodology to improve the controllability of a continuous stirred tank reactor (CSTR). Finally, Section 5 presents the main conclusions.
2. A New Series Architecture
The two controlled variables in Figure 1 can be expressed in terms of the external inputs as follows (either -Laplace or -frequency domains are possible for variables and functions):where
Let us consider: the sensor noise exclusively contains high frequency components, i.e. , the and plants model self-regulating processes (absence of integrating dynamics), the and set-points take constant values, and the disturbance is step-type. A suitable feedback design demands both and at steady-state . Besides, distinguishing the fast and slow loops requires that , i.e. . Thereby, considering and inputs in (1), a zero steady-state error at the output, needs of , which must be necessarily provided by since . An integrator in does the job. Furthermore, (an integrator in is also needed to neglect the influence of at in (1), which also preserves . Regarding (2) both controllers also need integral part to achieve . Beyond their usefulness for steady-state purposes, integrators become a problem for linear stability, in addition to other practical issues such as wind-up phenomena. Regarding the stability, the loop is the most troublesome since it contains two integrators, which means a phase lag of at . Because of that, [22] discusses some examples of conditional stability. Achieving stable loops would become more and more challenging if more manipulated inputs were used. Let us suppose the Figure 1 structure were generalised for manipulated inputs. As each controller () should contribute with one integrator, the most critical loop would have a phase lag of at . Thus, the loops would be conditionally stable and demand lead-lag networks that would increase the order of controllers and make harder the design procedure. The new series arrangement of controllers in Figure 2 overcomes the above problems.
In the new architecture, the system output iswhereis the total open-loop transfer function, which is made up of individual loopsThis branch open-loop function defines the direct path from the output set-point to each -plant contribution to the output:Eventually, each plant is driven byAnd particularly, the manipulated input steady-states can be conveniently trimmed by the set-points , while the dependent variable steady-state can be freely adapted by the control law to fight persistent disturbances or any other kind of system unknowns. So there is not a set-point for , as in classical structures. A first advantage of the new architecture is that negative comparisons of Figure 1 are no longer needed at midrange variable sum-points since the negative feedback at sum-point suffices. This avoids unnecessary inverse gains on the controllers. A second and major advantage is that an integral action in makes , which suffices to achieve (4), and to achieve (8), under the assumption of constant set-points and step-type disturbances. Furthermore, integrators are no longer needed to yield . In general, must provide the number of integrators that were needed depending on the external inputs, the plants, and the prescribed steady-state tracking errors.
Inputs must be conveniently arranged in the structure considering that will be the fastest loop and will be the slowest loop. Feedback controllers will determine the specific working frequencies for each loop in the global regulation task. These frequencies also condition the time at which an upper loop disconnect a lower loop inside the structure. The next section details a frequency domain design procedure for those controllers.
There is an equivalent of the series structure (Figure 2) to a parallel one where would be lead to independent branches (6) whose controllers would be . Let us note as controllers are of lower order than controllers , since each individual controller does not need to add dynamics of high order that have been added by . The counterpart of simpler controllers is a lower flexibility of the structure. Thus, the series structure sets the time sequence of plant intervention, while the controllers could freely allocate it in a parallel structure [16]; furthermore, this can make several plants work in the same frequency band.
3. A Robust Frequency Domain Control Design
3.1. Multi-Input Participation in the Frequency Domain
For simplicity, let us assume and dual input in the series architecture of Figure 2. A sensitivity function models the desired performance for disturbance rejection ; Figure 3(a) depicts its magnitude frequency response . The closed-loop specification can be straight away expressed in terms of the open-loop function , whose magnitude frequency response is also depicted on Figure 3(a); represents the gain cross-over frequency, which quantifies the control system bandwidth. Let us remark that the maximum magnitude peak of also defines a minimum degree of stability since it bounds a distance from to the critical point (stability margins) [29]. Thereby, in Figure 3(a) represents the robust performance and stability to be achieved by the MISO control.
(a)
(b)
(c)
(d)
However, there are infinite pairs that build and satisfy that prevails over low frequencies and over high frequencies. An arbitrary pair is depicted on Figure 3(a), revealing that the switching frequency must be specified by the designer. This frequency is closely related to when the fastest loop is disconnected by the slowest loop, i.e. to the time that takes to recover its midrange setpoint .
Saving feedback (less gain) at each branch implies each loop dominates over its frequency band. Thus, rolls off to from and rolls off to before . An abruptly roll-off would require quite complex controllers (great number of poles and zeros), not only to achieve a sharp gain increment/decrement but also to guarantee system stability (let us remind that according to Bode integrals the magnitude slope is closely related to its phase lag [27]). Besides, the fact that achieves sufficient stability, by means of a large enough distance of to the critical point (stability margins), does not necessary imply that both and show large enough distances to the critical point. Thus, as long as stability margins are chosen in consonance with the uncertainty about plant models, it will be of interest to define sufficient stability margins for both loops, which hampers their disconnection slopes.
Other relevant points are the frequency allocation of plants in the frequency band and the best choice of , which are intimately linked to the application and designer expertise. Let us describe some general examples in Figures 3(b), 3(c), and 3(d). They depict the magnitude frequency responses of plants and that have already been chosen to contribute over low and high frequency bands, respectively. It is important to remark that plants must be conveniently scaled [19] for a fair comparison. Figure 3(b) depicts two minimum phase plants with different frequency characteristics. Thus, choosing at the cross-frequency where will make the most of both plants to achieve the required performance while saving feedback (smaller controller gains). This case is thoroughly exploited inside a parallel structure in [16]. Similarly the case depicted in Figure 3(c) also makes the most of input-output characteristics. In this case, the low-frequency gain superiority and the presence of a RHP zero in one of the plants make it the option to work at low frequencies; is chosen sufficiently less than for an appropriated degree of stability. On the other hand, Figure 3(d) shows an apparent contradiction considering only the plant frequency responses. At a first glance should be the only plant to be involved in the regulation task inside a SISO control structure, since will not improve the performance. In fact, will demand a higher control action than would have at the steady-state. In this case, the trick s physically represents a much less expensive actuation than , which justifies its intervention.
In summary, quantitative and frequency design methods can be of great help to guide the control designer. Particularly, the loop-shaping of and appears to be the wiser approach.
3.2. Robust Control Problem Statement
Plant model uncertainty and unknown disturbances justifies feedback [26, 30]. The explicit consideration of uncertainty in the design makes the control robust. Let us consider uncertain parameters in the set of plant models of Figure 2. And take as a vector in the set of all their possible values . Thus, the MISO uncertain system is definedwhere is a vector in the uncertain set.
Hence, a set of time responses (4) is possible when a -disturbance happens. A performance model is chosen to limit those responses. According to QFT principles, the robust performance for disturbance rejection can be expressed in the frequency domain , asAnd the robust stability in , is being expressed in this work by the setFor a particular frequency, each inequality of (11) defines a forbidden region around the critical point -1 that cannot be violated by . In this way the prescribed degree of stability can be linked to the uncertainty of each plant . Particularly, to achieve certain gain margin above gain uncertainty, the upper tolerance is chosen asTo enforce a phase margin above phase uncertainty, the upper tolerance becomes
Then robust stability (11) is straight forward related to individual loops, while robust performance (10) is a collaborative task among loops. Thus, the robust performance must be distributed among the loops along the frequency band. Let us explicitly define and as the low and high frequencies, respectively, that enclose the interval for the -loop participation. Since numbers the fastest loop and numbers the slowest loop, thus and . A sequential design of the controllers is following proposed to carry out a quantitative allocation of the frequency band among the loops.
3.3. Robust Frequency Domain Design Method
The controllers are initially set to zero. Then, the design sequence evolves from to . When the -loop design takes place at step , is the only unknown.
QFT bounds will translate the robust specifications (10) (11) in terms of the nominal open loop at discrete frequencies . These bounds can be computed by using the command genbnds of Terasoft QFT Toolbox [31], which handles specifications in the general formTherefore, specifications (10) (11) must be conveniently rewritten to identify the coefficients , , , and of (14) at each step , being . Firstly, let us group the loops already designed in the sequential procedure asand concatenate their controllers asThus, (10) can be rewritten asAnd the set (11) can be rewritten as two separated formulas:which cares for the stability of the loop and the setwhich cares for the stability of the loops that have already been designed. Now the control specifications (17)-(19) can be easily identified with the format (14). As an example, let us take , , , , and for (17).
At the step and for a discrete frequency , there are a performance bound for (17) and a total amount of stability bounds gathering (18) and (19). Let us denote to the matching of all those bounds at certain frequency . A set of discrete frequencies is chosen.
After computing the intersection bounds and taking into account the working frequencies for the -loop, the nominal open-loop function is shaped fulfilling only at .
The example in Section 4.2 details thoroughly the sequential procedure for the bound computation and the loop-shaping of controllers.
4. Example: MISO Control of Continuous Stirred-Tank Reactor
The usefulness of the proposed methodology is being illustrated through the control of a Chemical Stirred Tank Reactor (CSTR), which is a recurrent benchmark in the process control literature because of its unquestionable importance in the chemical and materials industry [21].
4.1. Continuous Stirred-Tank Reactor
A coolant flow (usually water), through either a cooling jacket or a cooling coil [32], or both [33], removes the necessary energy to prevent the exothermic and irreversible reaction runaway and to regulate the reactor temperature. Due to the limited heat-removal capacity of the coolant flow, the manipulation of the reactant flow can contribute to temperature control. Thus [21] presents a MISO strategy founded on VPC [1], which follows the classical series architecture in Figure 1. The jacket coolant flow (the fastest actuation) midranges around its maximum energy removal capacity , while large reaction temperature excursions are compensated with the feed flow-rate (the slowest actuation). Furthermore, this smart MISO strategy achieves the highest possible production rate: is maximised since is set to maximum. Its counterpart is feed temperature gets a significant impact on dynamic controllability. When the feed is colder than the reactor (), the immediate effect of increasing the feed flow-rate is a temporary decrease in the reactor temperature; i.e., behaves as a nonminimum phase (NMP) plant. Then, a wise frequency distribution of MISO dynamic controllability is of importance and can be quantitatively accomplished through the robust frequency domain method that this paper develops. Beyond that and to illustrate the ability to deal with more than two manipulated inputs, the cooling capacity is being contributed from two sources: a main supply provided by a cooling jacket and a quick auxiliary supply provided by a cooling coil . Figure 4 shows the set-up.
Figure 5 depicts the magnitude frequency response of the linear input-output relations that intervene. In agreement with the notation in the series control architecture (Figure 2), the contribution from the manipulated inputs, the feed flow , the jacket flow , and the coil flow , to the output, the reactor temperature , yields plants , , and , respectively. Three disturbance inputs deviate : the feed concentration , the feed temperature , and the inlet coolant temperature , whose contributions are represented by plants , , and , respectively. A detailed procedure on how to obtain the plant models is appended at the end of the paper. Several operating points are considered for the inputs, which yield the plant parameter uncertainty in the linearised models. A plant matrix (9) collects 240 vectors , and Figure 5 shows the envelope of their magnitude frequency responses.
It is a fact that the gain of is higher than the gain of both and along the whole frequency band. However participation will be restricted to low frequencies due to its RHP zero (increasing produces a temporary decrease in ). The RHP zero that is closer to the origin for the whole uncertainty set is at . Thus, is chosen as the border frequency for participation. Beyond that frequency, the contribution of and will be able to improve the performance of the temperature regulation. dominates medium frequencies ( below ) and dominates high frequencies ( above ). This agrees with the cooling system dimensioning to the extent that the coil cooling becomes an auxiliary system with a quicker but less powerful response than the jacket cooling. Let us also mention that and have inverse gain, since raising any of the coolant flows makes the reactor temperature drop.
The other relevant point in MISO control is a convenient selection of the midrange setpoints. Since is the fastest actuation, is chosen at exactly the midpoint of the coolant capacity of coil cooling system to achieve maximum maneuverability. However, is chosen near the maximum coolant capacity of jacket cooling system in order to tends to maximum production rate. A reduction of saves but also reduces .
4.2. MISO Robust Control of CSTR
Three performance specifications for robust disturbance rejection are defined following (10) and being equal to , , and , respectively. The required performance is that, in the case of a maximum disturbance happens, the reactor temperature deviation must be less than K. And this maximum deviation must be reduced to K no later than 20 min and fully extinguished in steady state. To ensure these conditions, the performance upper model isThe performance specification has been precisely defined. It could not be achieved neither with a SISO control using the jacket flow-rate as single control variable, nor with a MISO strategy using the cooling jacket as single midrange control variable. A roughly proof of this is the cross-over frequency of the scaled plant in Figure 5. Thus the dual cooling (jacket and coil) becomes necessary, and the MISO control must have three loops (two midrange inputs).
To guarantee minimum phase margins of (13) on each -loop, three robust stability specifications (11) take as upper tolerance
The discrete set of -frequencies to compute QFT templates and bounds is
Section 3.3 detailed the procedure for QFT bound computation that represented the robust specifications. Eventually, an intersection bound set , is obtained. Then, the shaping of is performed. In accordance to comments on plant peculiarities in Section 4.1, Table 1 shows the frequency band distribution among loops to be achieved. The sequential design of controllers , , and is performed as follows; Figure 6 illustrates it. Each row of plots matches a step, which is designated by the loop under design , and each column of plots depicts bounds and nominal open-loop function of the -loop; different line colours distinguish the frequencies in (22).The procedure begins with the design of controller in the fastest loop . As the other controllers are initially taken as zero, thus and reduces to solely . Hence, bounds at Figures 6(a), 6(b), and 6(c) reveal a situation where should assume the whole control task and no bounds appear for ; see Figures 6(a) and 6(b). Bounds are computed from (17) (18) with . However, shaping must only assume the control task over its working frequencies; i.e., must only meet those over . The design procedure is driven as usual in QFT: loop gain is conveniently adjusted to bounds from to roll off frequencies; see Figure 6(c). It yields The negative gain of controller (23) is due to being inverse gain.It is aimed as the design of as part of the loop . Now only controller is zero, and ; i.e., both loops should contribute to the regulation task. As , this loop does not intervene and there are no bounds for it; see Figure 6(d). Taking and , bounds are computed from (17) (18) (19); see Figure 6(e). At frequencies where met its bounds, i.e., over , bounds become closed forbidden regions, which prevents from being in counterphase with , and consequently spoiling the previously achieved performance or violating robust stability. Due to the residual gain contributed by over , bounds show dips at certain phases. Then, loopshaping is performed meeting over . It yields Owing to the series structure includes (6). As (23) had negative gain, (24) must have positive gain despite having inverse gain. Furthermore, adds zeros-poles at lower frequencies than did, if needed. Thus, the series arrangement lowers the order of controllers. Once is designed, the bounds can be updated as Figure 6(f) depicts. Comparing it with Figure 6(c), let us note as now meets bounds that were before violated ( over ) since the specifications have already been achieved at these frequencies by .The procedure ends with the design of in the slowest loop . Taking and , bounds are computed from (17) (18) (19); see Figure 6(g). They reveal the frequencies where is required to achieve the performance; i.e., bounds depict nonclosed regions over . The loopshaping of yields As includes (6); just has to append dynamics at frequencies below those contributed by (24). One integrator in suffices for zero steady-state error at the output and at the two midrange actuations. negative gain cancels negative gain (23) in (6) since does not have inverse gain. Bounds and can be now updated; see Figures 6(h) and 6(i). Let us note that and now meet their bounds at the whole frequency band. All branches now participate in the regulation task as Table 1 required.
(a) Step .
(b) Step .
(c) Step .
(d) Step .
(e) Step .
(f) Step .
(g) Step .
(h) Step .
(i) Step .
Figure 7 proves the fulfilment of control specifications for the whole set of 240 plant cases: (a) what concerns the robust rejection at the output of the three disturbance inputs; (b) what concerns the robust stability of the three loops; (c) what concerns the frequency band allocation among loops. Let us remark the smooth disconnection of loops around their switching frequencies, which preserves stability and avoids higher order of controllers. Beyond its switching frequency a steep gain reduction of saves the amount of feedback in favour of a smaller -sensor noise amplification at the actuator (Horowitz’s cost of feedback [26]).
(a) Closed-loop disturbance rejection
(b) Closed-loop stability
(c) Open-loop functions
Closed-loop time responses of main system variables are in Figure 8. These results have been obtained using the proposed control system in the nonlinear model of CSTR (at this point it is recommended to see the annex for a full understanding of the physical units and experiments). The simulation shows the system behaviour for step-type changes: in the feed concentration ( at h), in the feed temperature ( K at h), and in the coolant temperature ( K at h). Several operating points have been tested, which correspond to jacket flow set-points between 50% and 70% of maximum cooling capacity and a coil flow set-point at the midrange point (50%) of maximum cooling capacity. In all cases, when any disturbance happens the reactor temperature deviates from the set-point ( K) less than the maximum permitted K and less than K after h. The actuator collaboration is as follows. When a disturbance happens, the coil flow quickly reacts to initially compensate the reactor temperature deviation. Its intervention is progressively reset by the jacket flow , when this takes control of the regulation task. As long as finally returns to the midrange , maximum maneuverability is preserved if another disturbance happened. Finally, dominates the situation, and this returns to . Let us note that larger involves larger ; i.e., it pursues increasing the production rate. On the other hand, a smaller pursues saving the amount of . In summary, the use of three manipulated inputs allowed not only improving the performance but also enhancing the controllability. As a matter of fact three variables are controlled, the output and two manipulated inputs.
5. Conclusions
This paper highlighted the relevance of a frequency domain method to design feedback controllers inside a new series architecture that involved several manipulated inputs. They not only intervened in the dynamic regulation of the output but also in returning all but one of the manipulated inputs to conveniently chosen set-points (midrange inputs).
Controllers and loops were arranged to participate in a series fashion when a disturbance deviated the output: the fastest open loop transfer function included a single controller, meanwhile the slowest open loop incorporated the whole set of controllers in cascade. The novelty of the new series architecture was that the controllers were on the feedback path instead of on the direct path to the midrange input set-points, as it happened in classical structures (valve position control or midranging control). In this way, integral action was only needed in the controller next to the slowest actuation to achieve zero steady state error at the regulated variables (output and midrange inputs). Since the stability of each loop was not compromised any more by a high number of integrators, the procedure to design the controllers became easier and the order of controllers was lower, especially when the number of manipulated inputs increased.
Concerning the controller design method, the frequency band was allocated among the loops that had to work together to achieve certain robust performance in disturbance rejection. Robust stability was also of concern. Achieved stability margins were linked to the uncertainty of input-output plant models. Quantitative Feedback Theory was the framework for robust control design. The multi-loop design was accomplished in a sequential way in order to use well-known tools to compute QFT bounds, which represented the robust performance and stability. A method was proposed to shape each loop according to the bounds and the frequency allocation.
As a challenging example the temperature of a chemical stirred tank reactor was regulated manipulating coil and jacket cooling flows, and the reactant flow, despite several conditions on the flow temperatures and the feed concentration tried to deviate the desired temperature in the reactor. Besides, the cooling flows were regulated to certain set-points to achieve maximum maneuverability and to manage the production rate. The system frequency responses under different operating conditions allowed the frequency band allocation among the three loops in a quantitative way. The provided QFT control design method achieved a frequency distribution of the best MISO controllability in order to meet prescribed specifications on robust performance and stability.
Appendix
CSTR Dimensioning and Dynamic Modelling
A dual cooling CSTR (Figure 4) was used in the example of Section 4. This CSTR has been dimensioning mainly following the procedure in [21], which has been conveniently modified to split the cooling capacity into two: the jacket and coil cooling systems. An -parameter distributes their participation: means a coil-full cooling and means a jacket-full cooling.
Process parameters and reaction initial conditions are in Table 2. In the nominal equilibrium the following is adopted: a reactor temperature of , a conversion level of , a cooling distribution of , and a feed flow-rate of . Then, considering mass and energy equations, the cooling CSTR is sized and the nominal equilibrium is computed for the remaining variables. Table 3 collects the most relevant values. Let us remark as the nominal values for the manipulated inputs , and mean 37%, 65%, and 50% of full capacities , , and , respectively.
The system behaviour can be modelled by the following nonlinear ordinary differential equations:(i)Component A balance:(ii)Reaction rate:(iii)Reactor energy balance:(iv)Jacket energy balance:(v)Coil energy balance:
Coil energy balance (A.5) is a simplification to obtain a small-signal linear model later. However, for nonlinear simulations in Figure 8, the coil is divided in segments whose dynamic behaviour yields:andwhere is the temperature at the input of the -segment, the temperature at the output of the -segment, and is the energy exchanged with the reactor along the -segment. The total energy that is absorbed by the segments of the coil is
The nonlinear equations (A.1)-(A.5) are linearised around certain steady-state values of the system variables. It yields the small-signal linear model in (A.9), where the equilibrium of a variable is denoted by an upper bar and the small deviation of variables around equilibrium is denoted with the -symbol. There is supposed to be the same inlet coolant temperature at both cooling systems, i.e., .
An effective control of CSTR should consider several operating points. The set of main process variables at certain equilibrium consists of output, three manipulated inputs, and three disturbance inputs. Then, several equilibria are being considered. Since the reactor temperature is the controlled variable, let us adopt a single equilibrium for it: . The three disturbance inputs are taken as independent variables, whose equilibrium values are estimated asbeing the nominal feed concentration;which represents a variation range of around nominal , andwhich represents a variation range of around nominal . The cooling flow-rates must return to desired set-points after temperature regulation. Therefore, they complete the set of independent variables. The coil flow-rate is the fastest actuation for reactor temperature recovering. Thus the coil-flow set-point is fixed at the middle of its range to preserve maximum maneuverability, which means . The jacket flow-rate set-point can be conveniently moved such thatLet us remark that larger involves larger , which increases the production rate. The combination of all possible values in (A.11), (A.12), (A.13), and (A.14) yields 240 cases. and complete the set of independent variables for each of the 240 operating points. Substituting them in (A.9), the dependent variable can be computed. Eventually, (A.9) models the system dynamic behaviour for 240 operating points (A.10).
Applying the Laplace transform to (A.9), and conveniently rearranging the equations, it yields the transfer functions:being , , , , , and . Coefficients and () depend on process parameters (see Tables 2 and 3) and on the 240 equilibrium values .
Multivariable systems must be conveniently scaled for control design tasks [19]. Here the presence of a single output makes only necessary to scale the ranges of the five inputs. The scaling gains for , , and are , 5, and 5, respectively, according to the smallest distance from , , and to the edge of intervals (A.11), (A.12), and (A.13), respectively. The scaling gain for is according to the smallest distance from any equilibrium value (A.14) to the maximum capacity . In a similar way a scaling gain of for and a scaling gain of for is computed. These scaling gains are applied to plants (A.15). Eventually, the plant matrix is obtained which collects 240 vectors , whose magnitude frequency response is depicted in Figure 5.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors are grateful for the assistance provided by University of La Rioja.