# Frequency Domain Design of a Series Structure of Robust Controllers for Multi-Input Single-Output Systems

Authors: Montserrat Gil-Martínez; Javier Rico-Azagra; Jorge Elso
Journal: Mathematical Problems in Engineering
 (2018)

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## 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(1) a sequential intervention of loops and (2) 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.


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## Body

## 1. Introduction

In process control, it is common to find several manipulated variables regulating a single measurable output (MISO:Multiple Input Single Outputcontrol). 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 thehabituating control by Henson et al. [15] and themidranging control by Allison and Isaksson [17]. It can be seen as a generalization of thevalve position control (VPC) for the process industry presented by Shinskey [1] and Luyben [18]. Skogestad and Postlethwaite [19] labelled it ascascade control. The architecture seeks a sequential loop intervention from the fastest (bottom) to the slowest (top) loop, in order to recover the set-point r at the d-disturbed output y. Besides, the fastest control variable u2 returns to the setpoint ru2 when the slowest loop resets the fastest loop (input resetting control). Plant models p1 and p2 characterise the dynamic behaviour of the output y in response to the manipulated inputs u1 and u2, respectively. Similarly, pd models the response of y to the disturbance input d. The signal v has been added to account for sensor noise.Figure 1
Classical series architecture (midranging).In steady state, several combinations of the manipulated inputs (u1, u2) could achieve the desired output r=y. Hence, ru2 can be chosen according to different criteria, which are usually linked to efficiency, cost savings, and physical limitations. For example, midranging [17, 20] chooses ru2 as the midpoint of the u2 actuator to preserve the largest maneuvering range in the fast loop. Conversely, in the temperature (y) control of a chemical stirred tank reactor (CSTR), Luyben [21] manipulates the coolant flow to a tank jacket (u2) around the maximum removal capacity (ru2) in such a way that the feed-flow rate (u1) is indirectly maximized, and so is the production rate. Other works [6, 15] prefer reducing ru2 as much as possible since u2 represents a more expensive physical variable than u1, 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 (ru2) in the sludge temperature (y) control of a biochemical reactor operated by the air-flow (u2) and the inlet sludge flow (u1). A smaller ru2 reduces direct expenses by cutting the amount of u2. On the other hand, a higher ru2 deliberately pursues a major spending of u2 in order to force a higher u1, which raises the production rate and makes the plant operation more profitable.Most applications and methods in the literature show integral action in both controllersc1 and c2 to obtain zero steady-state tracking error in r-y and ru2-u2. However, the two integrators in series can make the slowest loop c2c1p1 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 (p2) and low (p1) 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 withad-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. c2) to achieve certain performance in the regulation of y, and then, designing the slowest loop (i.e. c1) that takes u2 to the setpoint ru2. 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 ofn controllers allowing to exploit the benefits of using n 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 (c1 if n=2). It will be shown that this is sufficient to achieve the set-points at the output and at the n-1 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. Section2 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 Figure1 can be expressed in terms of the external inputs as follows (either s-Laplace or ω-frequency domains are possible for variables and functions):(1)y=lt1+ltr-v+pd1+ltd+c1p11+ltru2,(2)u2=c21+ltr-v+-pd1+ltd+-c2c1p11+ltru2,where(3)lt=l2+l1=c2p2+c2c1p1.Let us consider: the sensor noisev(t) exclusively contains high frequency components, i.e. v(t=∞)=vss=0, the p1 and p2 plants model self-regulating processes (absence of integrating dynamics), the r and ru2 set-points take constant values, and the d disturbance is step-type. A suitable feedback design demands both ltj0≫1 and |lt(j0)|≫pdj0 at steady-state s=jω=j0. Besides, distinguishing the fast and slow loops requires that c2j0c1j0p1j0≫c2j0p2j0, i.e. l1j0≫l2j0. Thereby, considering (r-v) and d inputs in (1), a zero steady-state error at the output, yss=y(t=∞)=rss needs of ltj0=∞, which must be necessarily provided by c1j0=∞ since l1j0≫l2j0. An integrator in c1 does the job. Furthermore, c2j0=∞ (an integrator in c2) is also needed to neglect the influence of ru2 at yss in (1), which also preserves l1j0≫l2j0. Regarding (2) both controllers also need integral part to achieve u2ss=ru2ss. 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 l1 is the most troublesome since it contains two integrators, which means a phase lag of -180∘ at ω=0. 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 n>2 manipulated inputs. As each ci controller (i=1,…,n) should contribute with one integrator, the most critical loop l1=p1∏i=1nci would have a phase lag of -90∘n at ω=0. Thus, the loops i>2 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.Figure 2
Series architecture forn actuators.In the new architecture, the system output is(4)y=11+ltltr-v+pdd+∑i=1npirui,where(5)lt=∑i=1nliis the total open-loop transfer function, which is made up of individual loops(6)li=pici∗=pi∏j=incj.This branch open-loop function li defines the direct path from the output set-point r to each pi-plant contribution to the output:(7)yi=li1+ltr-v-pdd-∑j≠ipjli1+ltruj+1+lt-lipi1+ltrui.Eventually, each plant is driven by(8)ui=li/pi1+ltr-v-pdd-∑j≠ipjli/pi1+ltruj+1+lt-li1+ltrui.And particularly, the manipulated input ui>1 steady-states can be conveniently trimmed by the set-points rui>1, while the dependent variable u1 steady-state can be freely adapted by the control law to fight persistent disturbances d or any other kind of system unknowns. So there is not a set-point for u1, 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 r sum-point suffices. This avoids unnecessary inverse gains on the controllers. A second and major advantage is that an integral action in c1 makes ltj0=∞, which suffices to achieve yt=∞=rt=∞ (4), and to achieve ui>1(t=∞)=rui>1(t=∞) (8), under the assumption of constant set-points and step-type disturbances. Furthermore, integrators are no longer needed to yield lij0≫li+1j0. In general, c1 must provide the number of integrators that were needed depending on the external inputs, the plants, and the prescribed steady-state tracking errors.Inputsui must be conveniently arranged in the structure considering that ln will be the fastest loop and l1 will be the slowest loop. Feedback controllers ci will determine the specific working frequencies for each loop li 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 (Figure2) to a parallel one where r-(y+v) would be lead to independent branches li (6) whose controllers would be ci∗. Let us note as controllers ci are of lower order than controllers ci∗=∏j=incj, since each individual controller ci does not need to add dynamics of high order that have been added by cj>i. 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 assumepd=1 and dual input i=2 in the series architecture of Figure 2. A sensitivity function S(s) models the desired performance for disturbance rejection y(s)/d(s); Figure 3(a) depicts its magnitude frequency response Sjω. The closed-loop specification S=1/(1+lt) can be straight away expressed in terms of the open-loop function lt, whose magnitude frequency response ltjω is also depicted on Figure 3(a); ωgc represents the gain cross-over frequency, which quantifies the control system bandwidth. Let us remark that the maximum magnitude peak of Sjω also defines a minimum degree of stability since it bounds a distance from lt(jω) to the critical point (stability margins) [29]. Thereby, ltjω in Figure 3(a) represents the robust performance and stability to be achieved by the MISO control.Figure 3
Frequency goals.
(a) (b) (c) (d)However, there are infinite pairs(l1,l2) that build lt and satisfy that l1 prevails over low frequencies and l2 over high frequencies. An arbitrary pair is depicted on Figure 3(a), revealing that the switching frequency ωsw 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 tsw≈4/ωsw that takes u2 to recover its midrange setpoint ru2.Saving feedback (less gain) at each branch implies each loop dominates over its frequency band. Thus,l1jω rolls off to -∞ from ω≥ωsw and l2jω rolls off to -∞ before ω≤ωsw. 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 lt achieves sufficient stability, by means of a large enough distance of lt to the critical point (stability margins), does not necessary imply that both l1 and l2 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 ofωsw, 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 p1 and p2 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 ωsw at the cross-frequency where p1jωsw=p2jωsw 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; ωsw is chosen sufficiently less than ωRHPz 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 p2 should be the only plant to be involved in the regulation task inside a SISO control structure, since p1 will not improve the performance. In fact, p1 will demand a higher control action than p2 would have at the steady-state. In this case, the trick s u1 physically represents a much less expensive actuation than u2, 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 ofl1(jω) and l2(jω) 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 m uncertain parameters in the set of n+1 plant models pi=1,…,n,pd of Figure 2. And take qj as a vector in the set of all their possible values Q∈Rm. Thus, the MISO uncertain system is defined(9)P=Ps,qj:qj∈Q,where P is a 1×(n+1) vector in the uncertain set.Hence, a set of time responses (4) is possible when a d-disturbance happens. A performance model Wd(s) is chosen to limit those responses. According to QFT principles, the robust performance for disturbance rejection can be expressed in the frequency domain s=jω, ω=[0,∞) as(10)Td=yd=pd1+lt≤Wd;∀P.And the robust stability in s=jω, ω=[0,∞) is being expressed in this work by the set(11)Tsi=li/1+∑j≠ilj1+li/1+∑j≠ilj=li1+lt≤Wsi,i=1,…,n;∀P.For a particular ω frequency, each inequality of (11) defines a forbidden region around the critical point -1 that cannot be violated by li=ci∗pi/1+∑j≠ilj. In this way the prescribed degree of stability can be linked to the uncertainty of each plant pi. Particularly, to achieve certain gain margin GMi above pi gain uncertainty, the upper tolerance is chosen as(12)Wsi=1GMi-1,GMi>1.To enforce a phase margin PMi above pi phase uncertainty, the upper tolerance becomes(13)Wsi=12sin⁡PMi/2.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 ωli and ωhi as the low and high frequencies, respectively, that enclose the interval for the i-loop participation. Since i=n numbers the fastest loop and i=1 numbers the slowest loop, thus ωl1=0 and ωhn=∞. A sequential design of the n 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 controllersci=1,…,n are initially set to zero. Then, the design sequence evolves from k=n to k=1. When the lk-loop design takes place at step k, ck is the only unknown.QFT bounds will translate the robust specifications (10) (11) in terms of the nominal open loop lko at discrete frequencies ω. These bounds can be computed by using the commandgenbnds of Terasoft QFT Toolbox [31], which handles specifications in the general form(14)A+BGC+DG≤W.Therefore, specifications (10) (11) must be conveniently rewritten to identify the coefficients A, B, C, and D of (14) at each step k, being G=ck. Firstly, let us group the loops already designed in the sequential procedure as(15)l-k=∑j=k+1nlj,k<n;l-k≡⌀,k=n,and concatenate their controllers as(16)c-k=∏j=k+1ncj,k<n;c-k≡⌀,k=n.Thus, (10) can be rewritten as(17)pd1+l-k+pkc-kck≤Wd;∀P.And the set (11) can be rewritten as two separated formulas:(18)pkc-kck1+l-k+pkc-kck≤Wsk;∀P,which cares for the stability of the k loop and the set(19)lj1+l-k+pkc-kck≤Wsj,j=k+1,…,n;∀P,which 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 A=pd, B=0, C=1+l-k, D=pkc-k, and W=Wd for (17).At thek step and for a discrete frequency ω, there are a performance bound for (17) and a total amount of n-k+1 stability bounds gathering (18) and (19). Let us denote βlk(ω) to the matching of all those bounds at certain frequency ω. A set of discrete frequencies ω∈[0,∞) is chosen.After computing the intersection bounds and taking into account the working frequencies for thek-loop, the nominal open-loop function lko(jω) is shaped fulfilling βlk(ω) only at ωlk≤ω≤ωhk.The example in Section4.2 details thoroughly the sequential procedure for the bound computation and the loop-shaping of controllers.


## 3.1. Multi-Input Participation in the Frequency Domain

For simplicity, let us assumepd=1 and dual input i=2 in the series architecture of Figure 2. A sensitivity function S(s) models the desired performance for disturbance rejection y(s)/d(s); Figure 3(a) depicts its magnitude frequency response Sjω. The closed-loop specification S=1/(1+lt) can be straight away expressed in terms of the open-loop function lt, whose magnitude frequency response ltjω is also depicted on Figure 3(a); ωgc represents the gain cross-over frequency, which quantifies the control system bandwidth. Let us remark that the maximum magnitude peak of Sjω also defines a minimum degree of stability since it bounds a distance from lt(jω) to the critical point (stability margins) [29]. Thereby, ltjω in Figure 3(a) represents the robust performance and stability to be achieved by the MISO control.Figure 3
Frequency goals.
(a) (b) (c) (d)However, there are infinite pairs(l1,l2) that build lt and satisfy that l1 prevails over low frequencies and l2 over high frequencies. An arbitrary pair is depicted on Figure 3(a), revealing that the switching frequency ωsw 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 tsw≈4/ωsw that takes u2 to recover its midrange setpoint ru2.Saving feedback (less gain) at each branch implies each loop dominates over its frequency band. Thus,l1jω rolls off to -∞ from ω≥ωsw and l2jω rolls off to -∞ before ω≤ωsw. 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 lt achieves sufficient stability, by means of a large enough distance of lt to the critical point (stability margins), does not necessary imply that both l1 and l2 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 ofωsw, 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 p1 and p2 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 ωsw at the cross-frequency where p1jωsw=p2jωsw 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; ωsw is chosen sufficiently less than ωRHPz 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 p2 should be the only plant to be involved in the regulation task inside a SISO control structure, since p1 will not improve the performance. In fact, p1 will demand a higher control action than p2 would have at the steady-state. In this case, the trick s u1 physically represents a much less expensive actuation than u2, 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 ofl1(jω) and l2(jω) 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 m uncertain parameters in the set of n+1 plant models pi=1,…,n,pd of Figure 2. And take qj as a vector in the set of all their possible values Q∈Rm. Thus, the MISO uncertain system is defined(9)P=Ps,qj:qj∈Q,where P is a 1×(n+1) vector in the uncertain set.Hence, a set of time responses (4) is possible when a d-disturbance happens. A performance model Wd(s) is chosen to limit those responses. According to QFT principles, the robust performance for disturbance rejection can be expressed in the frequency domain s=jω, ω=[0,∞) as(10)Td=yd=pd1+lt≤Wd;∀P.And the robust stability in s=jω, ω=[0,∞) is being expressed in this work by the set(11)Tsi=li/1+∑j≠ilj1+li/1+∑j≠ilj=li1+lt≤Wsi,i=1,…,n;∀P.For a particular ω frequency, each inequality of (11) defines a forbidden region around the critical point -1 that cannot be violated by li=ci∗pi/1+∑j≠ilj. In this way the prescribed degree of stability can be linked to the uncertainty of each plant pi. Particularly, to achieve certain gain margin GMi above pi gain uncertainty, the upper tolerance is chosen as(12)Wsi=1GMi-1,GMi>1.To enforce a phase margin PMi above pi phase uncertainty, the upper tolerance becomes(13)Wsi=12sin⁡PMi/2.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 ωli and ωhi as the low and high frequencies, respectively, that enclose the interval for the i-loop participation. Since i=n numbers the fastest loop and i=1 numbers the slowest loop, thus ωl1=0 and ωhn=∞. A sequential design of the n 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 controllersci=1,…,n are initially set to zero. Then, the design sequence evolves from k=n to k=1. When the lk-loop design takes place at step k, ck is the only unknown.QFT bounds will translate the robust specifications (10) (11) in terms of the nominal open loop lko at discrete frequencies ω. These bounds can be computed by using the commandgenbnds of Terasoft QFT Toolbox [31], which handles specifications in the general form(14)A+BGC+DG≤W.Therefore, specifications (10) (11) must be conveniently rewritten to identify the coefficients A, B, C, and D of (14) at each step k, being G=ck. Firstly, let us group the loops already designed in the sequential procedure as(15)l-k=∑j=k+1nlj,k<n;l-k≡⌀,k=n,and concatenate their controllers as(16)c-k=∏j=k+1ncj,k<n;c-k≡⌀,k=n.Thus, (10) can be rewritten as(17)pd1+l-k+pkc-kck≤Wd;∀P.And the set (11) can be rewritten as two separated formulas:(18)pkc-kck1+l-k+pkc-kck≤Wsk;∀P,which cares for the stability of the k loop and the set(19)lj1+l-k+pkc-kck≤Wsj,j=k+1,…,n;∀P,which 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 A=pd, B=0, C=1+l-k, D=pkc-k, and W=Wd for (17).At thek step and for a discrete frequency ω, there are a performance bound for (17) and a total amount of n-k+1 stability bounds gathering (18) and (19). Let us denote βlk(ω) to the matching of all those bounds at certain frequency ω. A set of discrete frequencies ω∈[0,∞) is chosen.After computing the intersection bounds and taking into account the working frequencies for thek-loop, the nominal open-loop function lko(jω) is shaped fulfilling βlk(ω) only at ωlk≤ω≤ωhk.The example in Section4.2 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 u2=Fj (the fastest actuation) midranges around its maximum energy removal capacity ru2=rFj, while large reaction temperature excursions y=Tr are compensated with the feed flow-rate u1=F (the slowest actuation). Furthermore, this smart MISO strategy achieves the highest possible production rate: F is maximised since Fj is set to maximum. Its counterpart is feed temperature d=Tin gets a significant impact on dynamic controllability. When the feed is colder than the reactor (Tin<Tr), the immediate effect of increasing the feed flow-rate is a temporary decrease in the reactor temperature; i.e., F-Tr 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 Fj and a quick auxiliary supply provided by a cooling coil Fc. Figure 4 shows the set-up.Figure 4
MISO control of cooled CSTR.Figure5 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 u1=F, the jacket flow u2=Fj, and the coil flow u3=Fc, to the output, the reactor temperature y=Tr, yields plants p1, p2, and p3, respectively. Three disturbance inputs deviate Tr: the feed concentration d1=Cain, the feed temperature d2=Tin, and the inlet coolant temperature d3=Tjin=Tcin, whose contributions are represented by plants pd1, pd2, and pd3, 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 P (9) collects 240 vectors P=[p1,p2,p3,pd1,pd2,pd3], and Figure 5 shows the envelope of their magnitude frequency responses.Figure 5
Plant frequency responses.It is a fact that the gain ofp1 is higher than the gain of both p2 and p3 along the whole frequency band. However p1 participation will be restricted to low frequencies due to its RHP zero (increasing u1=F produces a temporary decrease in y=Tr). The RHP zero that is closer to the origin for the whole p1 uncertainty set is at s=9.7318×10-4. Thus, ω=1×10-4 is chosen as the border frequency for p1 participation. Beyond that frequency, the contribution of p2 and p3 will be able to improve the performance of the temperature regulation. |p2| dominates medium frequencies (ω below 1×10-3) and |p3| dominates high frequencies (ω above 1×10-3). 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 p2 and p3 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. SinceFc is the fastest actuation, rFc is chosen at exactly the midpoint of the coolant capacity of coil cooling system to achieve maximum maneuverability. However, rFj is chosen near the maximum coolant capacity of jacket cooling system in order to F tends to maximum production rate. A reduction of rFj saves Fj but also reduces F.


### 4.2. MISO Robust Control of CSTR

Three performance specifications for robust disturbance rejection are defined following (10) and being pd equal to pd1, pd2, and pd3, respectively. The required performance is that, in the case of a maximum disturbance happens, the reactor temperature deviation must be less than 0.6  K. And this maximum deviation must be reduced to 0.2  K no later than 20  min and fully extinguished in steady state. To ensure these conditions, the performance upper model is(20)Wdjω=489.3jωjω/0.003+12.The performance specification Wd 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 p2 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 of30∘ (13) on each i-loop, three robust stability specifications (11) take as upper tolerance(21)Wsi=1.932.The discrete set ofω-frequencies to compute QFT templates and bounds is(22)Ω=0.1,1,3,5,7,10,20,30,50×10-4.Section3.3 detailed the procedure for QFT bound computation that represented the robust specifications. Eventually, an intersection bound set βli(ω),ω∈Ω, is obtained. Then, the shaping of lio(jω) 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 c3, c2, and c1 is performed as follows; Figure 6 illustrates it. Each row of plots matches a step, which is designated by the loop under design k, and each column of plots depicts bounds βli(ω) and nominal open-loop function lio(jω) of the i-loop; different line colours distinguish the frequencies in (22).k=3
The procedure begins with the design of controllerc3 in the fastest loop l3. As the other controllers are initially taken as zero, thus l1=l2=0 and lt reduces to solely l3. Hence, bounds at Figures 6(a), 6(b), and 6(c) reveal a situation where l3 should assume the whole control task and no bounds appear for li=1,2; see Figures 6(a) and 6(b). Bounds βl3 are computed from (17) (18) with l-3≡c-3≡⌀. However, lo3 shaping must only assume the control task over its working frequencies; i.e., lo3 must only meet those βl3 over ω≥0.001. The design procedure is driven as usual in QFT: loop gain is conveniently adjusted to bounds from ω=0.001 to roll off frequencies; see Figure 6(c). It yields(23)c3s=-1.6s/0.02+12.The negative gain of controller (23) is due to p3 being inverse gain.k=2
It is aimed as the design ofc2 as part of the loop l2. Now only c1 controller is zero, and lt=l3+l2; i.e., both loops should contribute to the regulation task. As l1=0, this loop does not intervene and there are no bounds for it; see Figure 6(d). Taking l-2=l3 and c-2=c3, bounds βl2 are computed from (17) (18) (19); see Figure 6(e). At frequencies where l3o met its bounds, i.e., over ω≥0.001, βl2 bounds become closed forbidden regions, which prevents l2o from being in counterphase with l3o, and consequently spoiling the previously achieved performance or violating robust stability. Due to the residual gain contributed by l3o over ω<0.001, βl2 bounds show dips at certain phases. Then, lo2 loopshaping is performed meeting βl2 over ω≥0.0001. It yields(24)c2s=5.5s/0.0012+1s/0.00018+1s/0.01+12.Owing to the series structurel2 includes c3 (6). As c3 (23) had negative gain, c2 (24) must have positive gain despite p2 having inverse gain. Furthermore, c2 adds zeros-poles at lower frequencies than c3 did, if needed. Thus, the series arrangement lowers the order of controllers. Once c2 is designed, the bounds βl3 can be updated as Figure 6(f) depicts. Comparing it with Figure 6(c), let us note as l3o now meets bounds that were before violated (βl3 over 0.0001≤ω<0.001) since the specifications have already been achieved at these frequencies by l2.k=1
The procedure ends with the design ofc1 in the slowest loop l1. Taking l-1=l3+l2 and c-1=c3c2, bounds βl1 are computed from (17) (18) (19); see Figure 6(g). They reveal the frequencies where l1o is required to achieve the performance; i.e., βl1 bounds depict nonclosed regions over ω<0.0001. The loopshaping of lo1 yields(25)c1s=-0.00015s.Asl1 includes c2c3 (6); c1 just has to append dynamics at frequencies below those contributed by c2 (24). One integrator in c1 suffices for zero steady-state error at the output and at the two midrange actuations. c1 negative gain cancels c3 negative gain (23) in (6) since p1 does not have inverse gain. Bounds βl2 and βl3 can be now updated; see Figures 6(h) and 6(i). Let us note that l2o and l3o now meet their bounds at the whole frequency band. All branches now participate in the regulation task lt=l3+l2+l1 as Table 1 required.Table 1
Frequency band allocation amongli=1,2,3.
ωl1 ωh1=ωl2 ω h 2 = ω l 3 ω h 3 0.00 0.0001 0.001 ∞Figure 6
Bounds and loop-shaping for each loop (column) and design step (row).
(a)
Stepk=3. l1(jω) (b)
Stepk=3. l2(jω) (c)
Stepk=3. l3(jω) (d)
Stepk=2. l1(jω) (e)
Stepk=2. l2(jω) (f)
Stepk=2. l3jω (g)
Stepk=1. l1(jω) (h)
Stepk=1. l2(jω) (i)
Stepk=1. l3(jω)Figure7 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 li saves the amount of feedback in favour of a smaller v-sensor noise amplification at the actuator ui (Horowitz’scost of feedback [26]).Figure 7
Frequency responses.
(a)
Closed-loop disturbance rejection (b)
Closed-loop stability (c)
Open-loop functionsClosed-loop time responses of main system variables are in Figure8. 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 (δCain=+10%Caino at t=2 h), in the feed temperature (δTin=+5 K at t=8 h), and in the coolant temperature (δTjin=-5 K at t=14 h). Several operating points have been tested, which correspond to jacket flow set-points rFj between 50% and 70% of Fj maximum cooling capacity and a coil flow set-point rFc at the midrange point (50%) of Fc maximum cooling capacity. In all cases, when any disturbance happens the reactor temperature Tr deviates from the set-point (rTr=350 K) less than the maximum permitted 0.6 K and less than 0.2 K after 0.33 h. The actuator collaboration is as follows. When a disturbance happens, the coil flow Fc quickly reacts to initially compensate the reactor temperature deviation. Its intervention is progressively reset by the jacket flow Fj, when this takes control of the regulation task. As long as Fc finally returns to the midrange rFc, maximum maneuverability is preserved if another disturbance happened. Finally, F dominates the situation, and this returns Fj to rFj. Let us note that larger rFj involves larger F; i.e., it pursues increasing the production rate. On the other hand, a smaller rFj pursues saving the amount of Fj. 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.Figure 8
Closed-loop time responses.


## 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 u2=Fj (the fastest actuation) midranges around its maximum energy removal capacity ru2=rFj, while large reaction temperature excursions y=Tr are compensated with the feed flow-rate u1=F (the slowest actuation). Furthermore, this smart MISO strategy achieves the highest possible production rate: F is maximised since Fj is set to maximum. Its counterpart is feed temperature d=Tin gets a significant impact on dynamic controllability. When the feed is colder than the reactor (Tin<Tr), the immediate effect of increasing the feed flow-rate is a temporary decrease in the reactor temperature; i.e., F-Tr 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 Fj and a quick auxiliary supply provided by a cooling coil Fc. Figure 4 shows the set-up.Figure 4
MISO control of cooled CSTR.Figure5 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 u1=F, the jacket flow u2=Fj, and the coil flow u3=Fc, to the output, the reactor temperature y=Tr, yields plants p1, p2, and p3, respectively. Three disturbance inputs deviate Tr: the feed concentration d1=Cain, the feed temperature d2=Tin, and the inlet coolant temperature d3=Tjin=Tcin, whose contributions are represented by plants pd1, pd2, and pd3, 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 P (9) collects 240 vectors P=[p1,p2,p3,pd1,pd2,pd3], and Figure 5 shows the envelope of their magnitude frequency responses.Figure 5
Plant frequency responses.It is a fact that the gain ofp1 is higher than the gain of both p2 and p3 along the whole frequency band. However p1 participation will be restricted to low frequencies due to its RHP zero (increasing u1=F produces a temporary decrease in y=Tr). The RHP zero that is closer to the origin for the whole p1 uncertainty set is at s=9.7318×10-4. Thus, ω=1×10-4 is chosen as the border frequency for p1 participation. Beyond that frequency, the contribution of p2 and p3 will be able to improve the performance of the temperature regulation. |p2| dominates medium frequencies (ω below 1×10-3) and |p3| dominates high frequencies (ω above 1×10-3). 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 p2 and p3 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. SinceFc is the fastest actuation, rFc is chosen at exactly the midpoint of the coolant capacity of coil cooling system to achieve maximum maneuverability. However, rFj is chosen near the maximum coolant capacity of jacket cooling system in order to F tends to maximum production rate. A reduction of rFj saves Fj but also reduces F.


## 4.2. MISO Robust Control of CSTR

Three performance specifications for robust disturbance rejection are defined following (10) and being pd equal to pd1, pd2, and pd3, respectively. The required performance is that, in the case of a maximum disturbance happens, the reactor temperature deviation must be less than 0.6  K. And this maximum deviation must be reduced to 0.2  K no later than 20  min and fully extinguished in steady state. To ensure these conditions, the performance upper model is(20)Wdjω=489.3jωjω/0.003+12.The performance specification Wd 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 p2 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 of30∘ (13) on each i-loop, three robust stability specifications (11) take as upper tolerance(21)Wsi=1.932.The discrete set ofω-frequencies to compute QFT templates and bounds is(22)Ω=0.1,1,3,5,7,10,20,30,50×10-4.Section3.3 detailed the procedure for QFT bound computation that represented the robust specifications. Eventually, an intersection bound set βli(ω),ω∈Ω, is obtained. Then, the shaping of lio(jω) 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 c3, c2, and c1 is performed as follows; Figure 6 illustrates it. Each row of plots matches a step, which is designated by the loop under design k, and each column of plots depicts bounds βli(ω) and nominal open-loop function lio(jω) of the i-loop; different line colours distinguish the frequencies in (22).k=3
The procedure begins with the design of controllerc3 in the fastest loop l3. As the other controllers are initially taken as zero, thus l1=l2=0 and lt reduces to solely l3. Hence, bounds at Figures 6(a), 6(b), and 6(c) reveal a situation where l3 should assume the whole control task and no bounds appear for li=1,2; see Figures 6(a) and 6(b). Bounds βl3 are computed from (17) (18) with l-3≡c-3≡⌀. However, lo3 shaping must only assume the control task over its working frequencies; i.e., lo3 must only meet those βl3 over ω≥0.001. The design procedure is driven as usual in QFT: loop gain is conveniently adjusted to bounds from ω=0.001 to roll off frequencies; see Figure 6(c). It yields(23)c3s=-1.6s/0.02+12.The negative gain of controller (23) is due to p3 being inverse gain.k=2
It is aimed as the design ofc2 as part of the loop l2. Now only c1 controller is zero, and lt=l3+l2; i.e., both loops should contribute to the regulation task. As l1=0, this loop does not intervene and there are no bounds for it; see Figure 6(d). Taking l-2=l3 and c-2=c3, bounds βl2 are computed from (17) (18) (19); see Figure 6(e). At frequencies where l3o met its bounds, i.e., over ω≥0.001, βl2 bounds become closed forbidden regions, which prevents l2o from being in counterphase with l3o, and consequently spoiling the previously achieved performance or violating robust stability. Due to the residual gain contributed by l3o over ω<0.001, βl2 bounds show dips at certain phases. Then, lo2 loopshaping is performed meeting βl2 over ω≥0.0001. It yields(24)c2s=5.5s/0.0012+1s/0.00018+1s/0.01+12.Owing to the series structurel2 includes c3 (6). As c3 (23) had negative gain, c2 (24) must have positive gain despite p2 having inverse gain. Furthermore, c2 adds zeros-poles at lower frequencies than c3 did, if needed. Thus, the series arrangement lowers the order of controllers. Once c2 is designed, the bounds βl3 can be updated as Figure 6(f) depicts. Comparing it with Figure 6(c), let us note as l3o now meets bounds that were before violated (βl3 over 0.0001≤ω<0.001) since the specifications have already been achieved at these frequencies by l2.k=1
The procedure ends with the design ofc1 in the slowest loop l1. Taking l-1=l3+l2 and c-1=c3c2, bounds βl1 are computed from (17) (18) (19); see Figure 6(g). They reveal the frequencies where l1o is required to achieve the performance; i.e., βl1 bounds depict nonclosed regions over ω<0.0001. The loopshaping of lo1 yields(25)c1s=-0.00015s.Asl1 includes c2c3 (6); c1 just has to append dynamics at frequencies below those contributed by c2 (24). One integrator in c1 suffices for zero steady-state error at the output and at the two midrange actuations. c1 negative gain cancels c3 negative gain (23) in (6) since p1 does not have inverse gain. Bounds βl2 and βl3 can be now updated; see Figures 6(h) and 6(i). Let us note that l2o and l3o now meet their bounds at the whole frequency band. All branches now participate in the regulation task lt=l3+l2+l1 as Table 1 required.Table 1
Frequency band allocation amongli=1,2,3.
ωl1 ωh1=ωl2 ω h 2 = ω l 3 ω h 3 0.00 0.0001 0.001 ∞Figure 6
Bounds and loop-shaping for each loop (column) and design step (row).
(a)
Stepk=3. l1(jω) (b)
Stepk=3. l2(jω) (c)
Stepk=3. l3(jω) (d)
Stepk=2. l1(jω) (e)
Stepk=2. l2(jω) (f)
Stepk=2. l3jω (g)
Stepk=1. l1(jω) (h)
Stepk=1. l2(jω) (i)
Stepk=1. l3(jω)Figure7 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 li saves the amount of feedback in favour of a smaller v-sensor noise amplification at the actuator ui (Horowitz’scost of feedback [26]).Figure 7
Frequency responses.
(a)
Closed-loop disturbance rejection (b)
Closed-loop stability (c)
Open-loop functionsClosed-loop time responses of main system variables are in Figure8. 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 (δCain=+10%Caino at t=2 h), in the feed temperature (δTin=+5 K at t=8 h), and in the coolant temperature (δTjin=-5 K at t=14 h). Several operating points have been tested, which correspond to jacket flow set-points rFj between 50% and 70% of Fj maximum cooling capacity and a coil flow set-point rFc at the midrange point (50%) of Fc maximum cooling capacity. In all cases, when any disturbance happens the reactor temperature Tr deviates from the set-point (rTr=350 K) less than the maximum permitted 0.6 K and less than 0.2 K after 0.33 h. The actuator collaboration is as follows. When a disturbance happens, the coil flow Fc quickly reacts to initially compensate the reactor temperature deviation. Its intervention is progressively reset by the jacket flow Fj, when this takes control of the regulation task. As long as Fc finally returns to the midrange rFc, maximum maneuverability is preserved if another disturbance happened. Finally, F dominates the situation, and this returns Fj to rFj. Let us note that larger rFj involves larger F; i.e., it pursues increasing the production rate. On the other hand, a smaller rFj pursues saving the amount of Fj. 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.Figure 8
Closed-loop time responses.


## 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 ormidranging 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.



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*Source: 7531260-2018-10-30.xml*