# Robust Feedback Control for Nonminimum Phase, Delayed, or Unstable Systems with Multiple Inputs

Authors: Montserrat Gil-Martínez; Javier Rico-Azagra
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper presents a feedback control solution that achieves robust stability and disturbance rejection in systems with multiple manipulated inputs and a single measurable output. The uncertain plant models may exhibit either nonminimum phase, or delay, or unstable phenomena, which makes it not easy to take full advantage of the frequency response of each plant. In the framework of quantitative feedback theory (QFT), a methodology is proposed to decide the best control bandwidth distribution among inputs and to design the set of parallel controllers with as small as possible gain at each frequency. The temperature regulation in a continuous stirred-tank reactor (CSTR) illustrates the benefits of a quantitative frequency distribution of the dynamic controllability between the jacket flow and the feed flow. The main challenge is that the feed flow exhibits a higher temperature regulation capacity and also produces a temporary decrease in the reactor temperature (nonminimum phase behaviour).


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## Body

## 1. Introduction

The collaboration of multiple inputs at a time to regulate a single measurable output (MISO control) is often considered in process control to increase control authority, performance, or flexibility. How to use the additional degrees of freedom has led to different designations, structures, and design methods in the scientific literature. On the basis of the inputs differing significantly in their dynamic effect on the output and the relative cost of manipulating each one,valve position control [1] and habituating control [2] pursue the participation of each input along different frequency bands. Thus, a chained intervention of gradually slower loops takes place whenever the output is deviated from its set point. This also makes the slowest inputs reset the fastest inputs (input resetting control [3]) to convenient set points (midranging control [4]), which prevents the saturation of actuators and may report economical benefits in different ways. For example, since the cheapest input manages the stationary [2, 5], or since the midrange set points condition a more profitable equilibria of the whole set of manipulated inputs [6, 7]. A serial arrangement of feedback controllers [2, 8] offers a more transparent design methodology in accordance with the previous objectives. Alternatively, a parallel arrangement (Figure 1) is more general and flexible [2]. In fact, it provides the ability of several inputs collaborating along the same frequency band [9], and eventually, the control bandwidth distribution among the inputs as desired [10]. It is also common to separate a master centralised controller from the parallel structure of branch elements [11] (slaves), as in load-sharing applications [12]. This paves the way to much more complex distribution structures [13] and specific algorithms among which the model predictive control and optimization techniques are frequent [2, 4, 9]. In general, it can be regarded as a hierarchical structure where the high-level linear controller faces the single output regulation by yielding a single virtual control input. Then, a low-level control allocation algorithm coordinates the different manipulated inputs such that they together produce the desired virtual control effort, if possible [14]. Nonlinear approaches to control allocation are common in flight control, marine applications, automotive, and other industries where mechatronics prevails [15, 16]. Optimization-based allocation methods are referred in [17]. The relevance of different multi-input control approaches is supported by recent applications in several fields such as process control [7, 18, 19], medical systems [20–22], robotics [23–25], unmanned aerial vehicles [26, 27], automotive industry [28], and electrical power systems [29].Figure 1
Parallel control structure for MISO systems.Nonminimum phase (NMP) behaviours (or delays) are frequent in MISO process control. The design methods in [1, 2, 30] pay special attention to them inside serial arrangements of controllers. Those strategies make the inputs collaborate over different bands of frequency, such that the NMP input handles the low-frequency band and does not limit the achievable performance. However, as long as the inputs cannot collaborate along the same frequencies, a control effort reduction is not possible by using redundant control inputs. On the other hand, authors in [9, 11] present methods for some kind of parallel structure, which allows a potential collaboration of inputs over the same frequencies. Alvarez-Ramirez et al. [9] illustrate how less-expensive control effort can be obtained with the addition of redundant control inputs, but such a cost saving is obtained at the expense of a sluggish closed-loop response if any additional control input is of the NMP nature. Thus, the drawback of the method is that the inputs are forced to collaborate along the same frequency band. Schroeck et al. [11] propose a master-slaves structure and develop a PQ method for control design. Firstly, the slaves are designed bearing in mind the following issues: (a) they condition the zeros of the equivalent slaves-plants avoiding that they are located in the right half-plane (RHP) and may limit the achievable closed-loop performance; (b) they determine the relative magnitude contribution to the output (a separated contribution of the inputs along the frequency band is pursued); (c) the relative phase contribution may be critical if they are out of phase when their magnitude contribution is nearly the same (destructive interference). Secondly, the master achieves the desired performance and stability. The PQ method has been proved in a hard disk drive application [11] and in a turbocharged engine control [31]. Some drawbacks of the method are the inputs eventually collaborate in different frequencies, the slaves are designed open loop with no reference to the closed-loop specifications, which can negatively interfere with the master task, and the control structure and the design method are complex when the number of inputs grows. Finally, any specific method in the scientific literature has dealt with the potential collaboration of unstable systems to the best of the author’s knowledge.The frequency domain appears to be the most meaningful approach to determine the best input participation, under the assumption that the system can be approximated by linear models. And, the parallel structure in Figure1 is being used to freely distribute the frequency band. A proper scaling of all ui input ranges allows fairer comparisons as it considers saturation constraints. Then, the frequency response y/ui,i.e.,pijω, reveals the relative “capacity” of each input; pi plant models can include uncertainty, for example, corresponding to unmodelled dynamics or small-signal linear approximations of nonlinearities. The capacity of each pijω plant is in reference to the closed-loop bandwidth (required performance). Inside the framework of quantitative feedback theory (QFT), the bounds express, at discrete frequencies, the set of closed-loop specifications in terms of the nominal open-loop transfer function, which initially matches the nominal plant [32, 33]. In this way, special bounds [10] depict in a transparent and quantitative way the capacity of each plant of the MISO system to achieve the specifications. A comparison of those bounds helped to distribute the control bandwidth among loops in order to get the set of controllers of minimum gain at each frequency [10]. Several inputs could either collaborate at a certain frequency to reduce the individual control effort or be inhibited to avoid useless fatigue and saturation risk of actuators. However, only magnitude contributions of plants (inputs) were taken into account, which only works well for minimum-phase (MP) plants. This paper develops a method that takes into account the relative phase of the branches under the potential presence of RHP dynamics (zeros or poles) or delays. If branches were out of phase, they would fight each other and the total output magnitude would drop considerably .The paper is organised as follows: Section2 formulates the feedback control problem to be solved inside a QFT framework. Then, it puts forward the challenges to decide the best control bandwidth distribution when output/input relations are not in the minimum phase; in particular, the unstable case is analysed by an example. Finally, the section summarises the methodology to find the set controllers of smallest gain at each frequency. Section 3 applies it for the dual-input control of a continuous stirred-tank reactor (CSTR), where a convenient distribution of feedback improves the system controllability despite the nonminimum phase behaviour of one manipulated input. Conclusions are in Section 4.


## 2. MISO Control to Reduce the Feedback Demand

### 2.1. Statement of the Robust Regulation Problem

Figure1 depicts an architecture of n parallel feedback controllers ci=1,…,ns. Each plant pis models the dynamic response from the manipulated input ui to the single controlled output y. The plant pds determines how a nonmeasurable disturbance d deviates the output y from its set point r. Each actuator contribution in the steady state can be conveniently selected by set points rui for n−1 manipulated inputs at most. At least a plant (i.e., p1) will work in the low-frequency band, and ui=1 will compensate any output deviation in steady state. External input v represents the high-frequency noise associated with the single sensor that measures the output y.Uncertainty of plant models is being explicitly considered in the collaborative control design. Thus, let us take a1×n+1 vector of plant transfer functions P=pi=1,…,n,pd, which can be any element of(1)P=Ps,qj:qj∈Q,where qj is a vector with the m uncertain parameters of plant models and Q is the set in ℝm that is defined by all its possible values.Then, if a certain performance model is desired ford disturbance rejection at the y output, QFT defines the robust specification in the frequency domain s=jω, ω=0,∞ as(2)Tdjω=yjωdjω=pdjω1+ltjω≤Wdjω;∀Pjω,∀ω,where lt=∑i=1nli denotes the total open-loop transfer function that is contributed by branches li=pici. Additionally, occasional changes in rui may be of interest, and robust specifications on y/rui responses can be also taken into account. From a mathematical point of view, these can be treated identically to (2). Thus, they are here being omitted for simplicity.Robust stability ins=jω, ω=0,∞ is being defined by the set(3)Tijω=lijω/1+∑j≠iljjω1+lijω/1+∑j≠iljjω=lijω1+ltjω≤Wsi,i=1,…,n;∀Pjω,∀ω,where the upper tolerance Wsi means the minimum distance from lijω/1+∑j≠iljjω to the critical point −1. It is usually of interest to assure a certain phase margin PMi by taking(4)Wsi=12sinPMi/2,or, if preferred, to choose a certain gain margin GMi by taking(5)Wsi=1GMi−1,GMi>1.Robust stability (3) is straight forward related to individual loops, while robust performance (2) is a collaborative task among loops.Thecicontroller design is being executed by loop-shaping a nominal open-loop loijω=poijωciω on a mod-arg chart. loijω has to meet the QFT bounds βliω, which are computed at discrete frequencies ω∈Ω. These βli-bounds translate the closed-loop robust specifications (2)–(3) in terms of loi, and they can be computed by using the command genbnds of Terasoft QFT Toolbox [34]. It handles specifications in the general form:(6)A+BGC+DG≤W,where G is the ci-controller to be designed and functions A, B, C, D, and W can be easily identified by comparison with (2)–(3).The aim is obtaining the feedback control solutionci=1,…,n that achieves (2)–(3) using as small as possible control gain at each frequency. The challenge is how to distribute the collaborative task over the frequency band among loops despite the fact that plants pi,…,n contain RHP poles, RHP zeros, or delays.


### 2.2. Challenges in the Collaboration of Plants with RHP Dynamics

Let us suppose a system where two inputs can intervene in the output regulation. The uncertainty-free plant models are(7)p1s=1s+1,p2s=1s−1,pds=1.The disturbance rejection model(8)Wds=1.75ss+12,defines the desired y/d behavior (2). For a minimum phase margin of 40∘ at both loops (3), the tolerance choice (4) is(9)Wsi=1,2=1.462.QFT bounds will represent those closed-loop specifications in terms of open-loop transfer functions at the discrete set ofω-frequencies:(10)Ω=0.1,0.3,1,2,3,6,10,20,50rad/s.In order to compare the capacity of each plant in the regulation task, let us firstly solve the problem assuming that only one of the two plants (inputs) participates at a time. For the case ofp1 alone participation, the bounds βl1 in Figure 2(a) show the feedback demand in terms of lt=l1. The initial location of this open-loop function is p1 since c1=1. A possible solution l1=p1c1 that fulfils the bounds is given by(11)c1=7.3s+1ss/252+2×0.6/25s+1.Figure 2
Loop-shaping bounds: (a)βl1, (b) βl2, and (c) βl2′; frequencies in different colours.
(a)
(b)
(c)On the other hand, let us only usep2 in the regulation. As long as p2 is not a minimum phase plant, the QFT method allows two ways of conducting the loop-shaping. The former is depicted in Figure 2(b) where bounds βl2 express the feedback demand in terms of lt=l2=c2p2, which is not minimum phase and initially c2=1 matches p2. Another procedure [35, 36] is shown in Figure 2(c). In this case, bounds βl2′ express the feedback demand in terms of l2′, which initially matches p2′. This plant corresponds to a p2 plant whose RHP pole has been mirrored to the LHP (in this example, p2′is equal to p1). Then, l2′ and p2′ are minimum phase, and the phase shift due to p2 RHP pole has been translated to the bounds βl2′. Let us note as the relative position between plant frequency responses (p2 or p2′) and their respective bounds (βl2 or βl2′) is the same. As expected, both procedures can yield the same solution, for example,(12)c2=7.3s/0.8+1ss/252+2×0.7/25s+1.Placing the open-loop function exactly onto the bounds at each design frequency means using the minimum controller gain to achieve the specifications, which favours a smaller control gain beyond the gain cross-over frequencyωgc [32, 37], where feedback becomes useless or even harmful (e.g., high-frequency unmodeled dynamics or sensor noise amplification that can saturate the control input). Then, the minimum quantity of feedback to achieve the specification at each design frequency is the distance between the plant frequency response and the corresponding bound. To quantify it, Rico-Azagra et al. [10] proposed the use of bounds βcω, which were calculated scaling classical bounds βlω by the nominal plant magnitudepojω, i.e., βcω=βlω/pojω. Performing this on bounds of Figure 2 yields the bounds of Figure 3.Figure 3
Feedback demand bounds: (a)βc1, (b) βc2, and (c) βc2′; frequencies in different colours.
(a)
(b)
(c)At any frequency, several locations onto the bound are possible for the loop-shaping: one for each phase in the horizontal axis. Thus, the design phase becomes important. Let us analyse the lowest frequencyω=0.1 in Figure 2. Either the control solution through p1 (11) or through p2 (12) contains an integrator to achieve zero steady-state error r−y when a d-step happens. The integrator is usually the first element added during the loop-shaping, which produces a phase lag of −90∘. Secondly, the control gain is raised to fulfil the low-frequency bound. After performing these operations in Figures 2(a) and 2(b), the same control element 7.3/s makes l1jω and l2jω to fulfil the lowest frequency bounds, βl10.1 and βl20.1, respectively. Obviously, as long as both plants (7) have the same magnitude at low frequencies, both demand the same control gain. However, the design phase to measure and compare the demanded gain is different: about −90∘ for l1 in Figure 2(a) and about −180∘ for l2 in Figure 2(b). The exact quantities can be measured on Figures 3(a) and 3(b), respectively. Applying the same reasoning for other frequencies may be a challenging task since it is not so obvious which design phases have to be compared between l1 and l2. However, those phases for comparisons become the same on the pictures for l1 and l2′, see Figures 2(a) and 2(c). In summary, the bound height (feedback demand) can be easily compared between bounds in Figures 3(a) and 3(c) looking at the same phase. Then, bearing only this criterion in mind, both plants p1 and p2 could be indistinctly used to solve the regulation problem since both require the same control gain at any frequency.However, the fact that two plants require the same feedback when working by their own does not necessarily imply that a collaboration of both would have that quantity of feedback inside an MISO structure. This is only true for minimum-phase plants: for example, the use of two twin plantsp1 would yield two controllers c1/2. On the contrary, p1 and p2 (or l1 and l2) are completely in counter phase at ω=0, and are only in phase at ω=∞, as shown in Figures 2(a) and 2(b). Then, they can collaborate over the high frequencies but not over the low frequencies to reduce the individual feedback gain since the function lt is the vectorial sum of l1+l2. Any attempt of designing minimum-phase controllers c1 and c2 to make p1 and p2 collaborate at low frequencies would yield controllers of higher gains than c1 (11) and c2 (12). As a general rule of thumb, to build a vector ltjω with vectors l1jω and l2jω that have a smaller magnitude than lt, the phase difference between l1jω and l2jω has to be less than ±90∘. Looking back at Figure 2, the information about the p2 phase that can be read in Figure 2(b) has been translated to the bounds that appear shifted in Figure 2(c). Therefore, the phase shift between true plants p1 and p2 can be read in the phase shift between bounds βc1 and βc2′ in Figure 3; the vertical segment is taken as a reference that marks the critical phase −180∘ on the bounds.Once detected at which frequencies the two plants are not out of phaseω≥1, the amount of feedback that each one would demand becomes of interest. This focuses on the height difference (magnitude in dB) between bounds βc1 and βc2′ in Figure 3 at the design phase Θd that is selected for each frequency ω≥1. Regardless of the bound height difference, a collaboration would always reduce the gain of controllers (let us remind the loops are not out of phase). However, these reductions may not justify the major complexity of performing the design. A magnitude difference of 20 log2 between bounds is a practical guideline to attempt a collaboration, and it is funded on the ideal collaboration of two equal plants, which would reduce the controller gain two times.Applying previous guidelines, Table1 summarises the frequency band distribution between the two loops of the example (7). The next step is the loop-shaping of l1 and l2 that achieves it, for which viewing the real phase of l1 and l2 becomes of great importance. Thus, bounds of the kind of βl2′ in Figure 2(c) are not advisable, but bounds βl2 in Figure 2(b) are preferred. The standard procedure to compute βli bounds and to perform iteratively the loop-shaping of each li is fully described in [10]. The main steps for the present example are illustrated in Appendix A.Table 1
Frequency band distribution between control inputs of system (7).
ω
0.1
0.3
1
2
3
6
10
20
50
p
1
×
×
×
×
×
×
×
p
2
×
×
×
×
×
×
×
×
×The final elements that integrate the MISO control are(13)c1=63.47ss/1.3+1s/0.2+1s/0.5+1s/322+2×0.6/32s+1,(14)c2=7.01s/2.7+1s/1.7+1s/0.23+1ss/0.82+2×0.8/0.8s+1s/322+2×0.6/32s+1.Figure4 compares different control solutions: c1s is the SISO controller (11), c2s is the SISO controller (12), and c1−c2 the MISO control solution (13)–(14). It proves the control gain reduction over the frequencies where the two loops collaborate ω>1. To illustrate the challenges due to RHP dynamics, let us build pe=p1+p2, and design for it a controller(15)ce=4s/0.8+1s+1s2s/352+2×0.7/35s+1,to fulfil the specifications. It has forced both plants to collaborate over the whole frequency band, which has implied a huge gain at frequencies where both plants are out of phase ω<1. Over the frequencies where only one branch works, the MISO control is obviously equal to any SISO control ω<1.Figure 4
Benchmark of controllers for system (7).


### 2.3. Proposed Methodology

#### 2.3.1. Stage 1: Feedback Sharing at Each Frequency of Design

This first stage distributes the control bandwidth among the plants that could contribute to the robust regulation problem (2)–(3) in order to reduce as much as possible the amount of feedback (gain) that is demanded by the controllers at each frequency. These decisions are made by the comparison of control bounds βci′ω,i=1,…,n.β
c
i
′
ω quantifies the necessary feedback cijω at each discrete frequency ω in the event that a single plant pijω worked to achieve the robust control requirements. In order to calculate the said bounds, nominal MP plants poi′ must be employed. For those plants that were not minimum phase, poi′ is obtained from poi by mirroring all RHP dynamics into the LHP and neglecting any possible delay (it fulfils poijω=poi′jω). Nominal plants can be any Pjω of the uncertainty set (1). Then, common QFT bounds on the nominal open-loop function loi′jω are computed, which yields βli′ω. These are computed in the event that only pijω assumes the regulation task. Eventually, βci′ω=βli′ω/poi′jω. Bounds βci′ωshift with respect to−180∘an amount equal to the phase difference between poijω and poi′jω.Theβci′ω bound comparison among loops i=1,…,n focuses at the desired design phase Θd∈−360∘,0∘, which is particularly chosen for each frequency and is unique for all loops (Θd is the phase in which the nominal open loop loi will be modified through the addition of gain, pole, or zero elements during the loop-shaping at Stage 2). The criteria for the control bandwidth distribution are(a)
Criterion 1: m plants m≥2 can collaborate at ω, if the phase differences between the m bounds βck=1,…,m′ are less than ±90∘ and if the heights of the said bounds do not differ from each other more than ±20logm dB at the design phase Θd. Additionally, there is no bound j≠k of lesser height. Consequently, the remaining plants j≠k should be inhibited, reducing their control gain as much as possible at ω during the design stage.(b)
A single plant is in charge of the regulation atω if criterion 1 is not met.(c)
The highestm is sought m≤n since the gain saving on each one of the m branch controllers is proportional to m, i.e., to the number of plants that collaborate.The amount20logm is a practical guideline. It is funded on the ideal collaboration of m equal plants, which would reduce the controller gain m times. The phase shifting between bounds is vital when plants contain RHP dynamics and delays. This argument condition determines if the loops are not out of phase, and therefore, lotjω≈∑lokjω can be built with lokjω≤lotjω.


#### 2.3.2. Stage 2: Design of Controllers

The designs of thei=1,…,n loops are carried out sequentially such that loijω fulfils the specifications (bounds βliω) according to the feedback sharing that was established in Stage 1. The general procedure was provided in [10], and it does not depend on the plants being MP or not. Bounds βli are computed and referenced to the true plant poi being MP or not.


## 2.1. Statement of the Robust Regulation Problem

Figure1 depicts an architecture of n parallel feedback controllers ci=1,…,ns. Each plant pis models the dynamic response from the manipulated input ui to the single controlled output y. The plant pds determines how a nonmeasurable disturbance d deviates the output y from its set point r. Each actuator contribution in the steady state can be conveniently selected by set points rui for n−1 manipulated inputs at most. At least a plant (i.e., p1) will work in the low-frequency band, and ui=1 will compensate any output deviation in steady state. External input v represents the high-frequency noise associated with the single sensor that measures the output y.Uncertainty of plant models is being explicitly considered in the collaborative control design. Thus, let us take a1×n+1 vector of plant transfer functions P=pi=1,…,n,pd, which can be any element of(1)P=Ps,qj:qj∈Q,where qj is a vector with the m uncertain parameters of plant models and Q is the set in ℝm that is defined by all its possible values.Then, if a certain performance model is desired ford disturbance rejection at the y output, QFT defines the robust specification in the frequency domain s=jω, ω=0,∞ as(2)Tdjω=yjωdjω=pdjω1+ltjω≤Wdjω;∀Pjω,∀ω,where lt=∑i=1nli denotes the total open-loop transfer function that is contributed by branches li=pici. Additionally, occasional changes in rui may be of interest, and robust specifications on y/rui responses can be also taken into account. From a mathematical point of view, these can be treated identically to (2). Thus, they are here being omitted for simplicity.Robust stability ins=jω, ω=0,∞ is being defined by the set(3)Tijω=lijω/1+∑j≠iljjω1+lijω/1+∑j≠iljjω=lijω1+ltjω≤Wsi,i=1,…,n;∀Pjω,∀ω,where the upper tolerance Wsi means the minimum distance from lijω/1+∑j≠iljjω to the critical point −1. It is usually of interest to assure a certain phase margin PMi by taking(4)Wsi=12sinPMi/2,or, if preferred, to choose a certain gain margin GMi by taking(5)Wsi=1GMi−1,GMi>1.Robust stability (3) is straight forward related to individual loops, while robust performance (2) is a collaborative task among loops.Thecicontroller design is being executed by loop-shaping a nominal open-loop loijω=poijωciω on a mod-arg chart. loijω has to meet the QFT bounds βliω, which are computed at discrete frequencies ω∈Ω. These βli-bounds translate the closed-loop robust specifications (2)–(3) in terms of loi, and they can be computed by using the command genbnds of Terasoft QFT Toolbox [34]. It handles specifications in the general form:(6)A+BGC+DG≤W,where G is the ci-controller to be designed and functions A, B, C, D, and W can be easily identified by comparison with (2)–(3).The aim is obtaining the feedback control solutionci=1,…,n that achieves (2)–(3) using as small as possible control gain at each frequency. The challenge is how to distribute the collaborative task over the frequency band among loops despite the fact that plants pi,…,n contain RHP poles, RHP zeros, or delays.


## 2.2. Challenges in the Collaboration of Plants with RHP Dynamics

Let us suppose a system where two inputs can intervene in the output regulation. The uncertainty-free plant models are(7)p1s=1s+1,p2s=1s−1,pds=1.The disturbance rejection model(8)Wds=1.75ss+12,defines the desired y/d behavior (2). For a minimum phase margin of 40∘ at both loops (3), the tolerance choice (4) is(9)Wsi=1,2=1.462.QFT bounds will represent those closed-loop specifications in terms of open-loop transfer functions at the discrete set ofω-frequencies:(10)Ω=0.1,0.3,1,2,3,6,10,20,50rad/s.In order to compare the capacity of each plant in the regulation task, let us firstly solve the problem assuming that only one of the two plants (inputs) participates at a time. For the case ofp1 alone participation, the bounds βl1 in Figure 2(a) show the feedback demand in terms of lt=l1. The initial location of this open-loop function is p1 since c1=1. A possible solution l1=p1c1 that fulfils the bounds is given by(11)c1=7.3s+1ss/252+2×0.6/25s+1.Figure 2
Loop-shaping bounds: (a)βl1, (b) βl2, and (c) βl2′; frequencies in different colours.
(a)
(b)
(c)On the other hand, let us only usep2 in the regulation. As long as p2 is not a minimum phase plant, the QFT method allows two ways of conducting the loop-shaping. The former is depicted in Figure 2(b) where bounds βl2 express the feedback demand in terms of lt=l2=c2p2, which is not minimum phase and initially c2=1 matches p2. Another procedure [35, 36] is shown in Figure 2(c). In this case, bounds βl2′ express the feedback demand in terms of l2′, which initially matches p2′. This plant corresponds to a p2 plant whose RHP pole has been mirrored to the LHP (in this example, p2′is equal to p1). Then, l2′ and p2′ are minimum phase, and the phase shift due to p2 RHP pole has been translated to the bounds βl2′. Let us note as the relative position between plant frequency responses (p2 or p2′) and their respective bounds (βl2 or βl2′) is the same. As expected, both procedures can yield the same solution, for example,(12)c2=7.3s/0.8+1ss/252+2×0.7/25s+1.Placing the open-loop function exactly onto the bounds at each design frequency means using the minimum controller gain to achieve the specifications, which favours a smaller control gain beyond the gain cross-over frequencyωgc [32, 37], where feedback becomes useless or even harmful (e.g., high-frequency unmodeled dynamics or sensor noise amplification that can saturate the control input). Then, the minimum quantity of feedback to achieve the specification at each design frequency is the distance between the plant frequency response and the corresponding bound. To quantify it, Rico-Azagra et al. [10] proposed the use of bounds βcω, which were calculated scaling classical bounds βlω by the nominal plant magnitudepojω, i.e., βcω=βlω/pojω. Performing this on bounds of Figure 2 yields the bounds of Figure 3.Figure 3
Feedback demand bounds: (a)βc1, (b) βc2, and (c) βc2′; frequencies in different colours.
(a)
(b)
(c)At any frequency, several locations onto the bound are possible for the loop-shaping: one for each phase in the horizontal axis. Thus, the design phase becomes important. Let us analyse the lowest frequencyω=0.1 in Figure 2. Either the control solution through p1 (11) or through p2 (12) contains an integrator to achieve zero steady-state error r−y when a d-step happens. The integrator is usually the first element added during the loop-shaping, which produces a phase lag of −90∘. Secondly, the control gain is raised to fulfil the low-frequency bound. After performing these operations in Figures 2(a) and 2(b), the same control element 7.3/s makes l1jω and l2jω to fulfil the lowest frequency bounds, βl10.1 and βl20.1, respectively. Obviously, as long as both plants (7) have the same magnitude at low frequencies, both demand the same control gain. However, the design phase to measure and compare the demanded gain is different: about −90∘ for l1 in Figure 2(a) and about −180∘ for l2 in Figure 2(b). The exact quantities can be measured on Figures 3(a) and 3(b), respectively. Applying the same reasoning for other frequencies may be a challenging task since it is not so obvious which design phases have to be compared between l1 and l2. However, those phases for comparisons become the same on the pictures for l1 and l2′, see Figures 2(a) and 2(c). In summary, the bound height (feedback demand) can be easily compared between bounds in Figures 3(a) and 3(c) looking at the same phase. Then, bearing only this criterion in mind, both plants p1 and p2 could be indistinctly used to solve the regulation problem since both require the same control gain at any frequency.However, the fact that two plants require the same feedback when working by their own does not necessarily imply that a collaboration of both would have that quantity of feedback inside an MISO structure. This is only true for minimum-phase plants: for example, the use of two twin plantsp1 would yield two controllers c1/2. On the contrary, p1 and p2 (or l1 and l2) are completely in counter phase at ω=0, and are only in phase at ω=∞, as shown in Figures 2(a) and 2(b). Then, they can collaborate over the high frequencies but not over the low frequencies to reduce the individual feedback gain since the function lt is the vectorial sum of l1+l2. Any attempt of designing minimum-phase controllers c1 and c2 to make p1 and p2 collaborate at low frequencies would yield controllers of higher gains than c1 (11) and c2 (12). As a general rule of thumb, to build a vector ltjω with vectors l1jω and l2jω that have a smaller magnitude than lt, the phase difference between l1jω and l2jω has to be less than ±90∘. Looking back at Figure 2, the information about the p2 phase that can be read in Figure 2(b) has been translated to the bounds that appear shifted in Figure 2(c). Therefore, the phase shift between true plants p1 and p2 can be read in the phase shift between bounds βc1 and βc2′ in Figure 3; the vertical segment is taken as a reference that marks the critical phase −180∘ on the bounds.Once detected at which frequencies the two plants are not out of phaseω≥1, the amount of feedback that each one would demand becomes of interest. This focuses on the height difference (magnitude in dB) between bounds βc1 and βc2′ in Figure 3 at the design phase Θd that is selected for each frequency ω≥1. Regardless of the bound height difference, a collaboration would always reduce the gain of controllers (let us remind the loops are not out of phase). However, these reductions may not justify the major complexity of performing the design. A magnitude difference of 20 log2 between bounds is a practical guideline to attempt a collaboration, and it is funded on the ideal collaboration of two equal plants, which would reduce the controller gain two times.Applying previous guidelines, Table1 summarises the frequency band distribution between the two loops of the example (7). The next step is the loop-shaping of l1 and l2 that achieves it, for which viewing the real phase of l1 and l2 becomes of great importance. Thus, bounds of the kind of βl2′ in Figure 2(c) are not advisable, but bounds βl2 in Figure 2(b) are preferred. The standard procedure to compute βli bounds and to perform iteratively the loop-shaping of each li is fully described in [10]. The main steps for the present example are illustrated in Appendix A.Table 1
Frequency band distribution between control inputs of system (7).
ω
0.1
0.3
1
2
3
6
10
20
50
p
1
×
×
×
×
×
×
×
p
2
×
×
×
×
×
×
×
×
×The final elements that integrate the MISO control are(13)c1=63.47ss/1.3+1s/0.2+1s/0.5+1s/322+2×0.6/32s+1,(14)c2=7.01s/2.7+1s/1.7+1s/0.23+1ss/0.82+2×0.8/0.8s+1s/322+2×0.6/32s+1.Figure4 compares different control solutions: c1s is the SISO controller (11), c2s is the SISO controller (12), and c1−c2 the MISO control solution (13)–(14). It proves the control gain reduction over the frequencies where the two loops collaborate ω>1. To illustrate the challenges due to RHP dynamics, let us build pe=p1+p2, and design for it a controller(15)ce=4s/0.8+1s+1s2s/352+2×0.7/35s+1,to fulfil the specifications. It has forced both plants to collaborate over the whole frequency band, which has implied a huge gain at frequencies where both plants are out of phase ω<1. Over the frequencies where only one branch works, the MISO control is obviously equal to any SISO control ω<1.Figure 4
Benchmark of controllers for system (7).


## 2.3. Proposed Methodology

### 2.3.1. Stage 1: Feedback Sharing at Each Frequency of Design

This first stage distributes the control bandwidth among the plants that could contribute to the robust regulation problem (2)–(3) in order to reduce as much as possible the amount of feedback (gain) that is demanded by the controllers at each frequency. These decisions are made by the comparison of control bounds βci′ω,i=1,…,n.β
c
i
′
ω quantifies the necessary feedback cijω at each discrete frequency ω in the event that a single plant pijω worked to achieve the robust control requirements. In order to calculate the said bounds, nominal MP plants poi′ must be employed. For those plants that were not minimum phase, poi′ is obtained from poi by mirroring all RHP dynamics into the LHP and neglecting any possible delay (it fulfils poijω=poi′jω). Nominal plants can be any Pjω of the uncertainty set (1). Then, common QFT bounds on the nominal open-loop function loi′jω are computed, which yields βli′ω. These are computed in the event that only pijω assumes the regulation task. Eventually, βci′ω=βli′ω/poi′jω. Bounds βci′ωshift with respect to−180∘an amount equal to the phase difference between poijω and poi′jω.Theβci′ω bound comparison among loops i=1,…,n focuses at the desired design phase Θd∈−360∘,0∘, which is particularly chosen for each frequency and is unique for all loops (Θd is the phase in which the nominal open loop loi will be modified through the addition of gain, pole, or zero elements during the loop-shaping at Stage 2). The criteria for the control bandwidth distribution are(a)
Criterion 1: m plants m≥2 can collaborate at ω, if the phase differences between the m bounds βck=1,…,m′ are less than ±90∘ and if the heights of the said bounds do not differ from each other more than ±20logm dB at the design phase Θd. Additionally, there is no bound j≠k of lesser height. Consequently, the remaining plants j≠k should be inhibited, reducing their control gain as much as possible at ω during the design stage.(b)
A single plant is in charge of the regulation atω if criterion 1 is not met.(c)
The highestm is sought m≤n since the gain saving on each one of the m branch controllers is proportional to m, i.e., to the number of plants that collaborate.The amount20logm is a practical guideline. It is funded on the ideal collaboration of m equal plants, which would reduce the controller gain m times. The phase shifting between bounds is vital when plants contain RHP dynamics and delays. This argument condition determines if the loops are not out of phase, and therefore, lotjω≈∑lokjω can be built with lokjω≤lotjω.


### 2.3.2. Stage 2: Design of Controllers

The designs of thei=1,…,n loops are carried out sequentially such that loijω fulfils the specifications (bounds βliω) according to the feedback sharing that was established in Stage 1. The general procedure was provided in [10], and it does not depend on the plants being MP or not. Bounds βli are computed and referenced to the true plant poi being MP or not.


## 2.3.1. Stage 1: Feedback Sharing at Each Frequency of Design

This first stage distributes the control bandwidth among the plants that could contribute to the robust regulation problem (2)–(3) in order to reduce as much as possible the amount of feedback (gain) that is demanded by the controllers at each frequency. These decisions are made by the comparison of control bounds βci′ω,i=1,…,n.β
c
i
′
ω quantifies the necessary feedback cijω at each discrete frequency ω in the event that a single plant pijω worked to achieve the robust control requirements. In order to calculate the said bounds, nominal MP plants poi′ must be employed. For those plants that were not minimum phase, poi′ is obtained from poi by mirroring all RHP dynamics into the LHP and neglecting any possible delay (it fulfils poijω=poi′jω). Nominal plants can be any Pjω of the uncertainty set (1). Then, common QFT bounds on the nominal open-loop function loi′jω are computed, which yields βli′ω. These are computed in the event that only pijω assumes the regulation task. Eventually, βci′ω=βli′ω/poi′jω. Bounds βci′ωshift with respect to−180∘an amount equal to the phase difference between poijω and poi′jω.Theβci′ω bound comparison among loops i=1,…,n focuses at the desired design phase Θd∈−360∘,0∘, which is particularly chosen for each frequency and is unique for all loops (Θd is the phase in which the nominal open loop loi will be modified through the addition of gain, pole, or zero elements during the loop-shaping at Stage 2). The criteria for the control bandwidth distribution are(a)
Criterion 1: m plants m≥2 can collaborate at ω, if the phase differences between the m bounds βck=1,…,m′ are less than ±90∘ and if the heights of the said bounds do not differ from each other more than ±20logm dB at the design phase Θd. Additionally, there is no bound j≠k of lesser height. Consequently, the remaining plants j≠k should be inhibited, reducing their control gain as much as possible at ω during the design stage.(b)
A single plant is in charge of the regulation atω if criterion 1 is not met.(c)
The highestm is sought m≤n since the gain saving on each one of the m branch controllers is proportional to m, i.e., to the number of plants that collaborate.The amount20logm is a practical guideline. It is funded on the ideal collaboration of m equal plants, which would reduce the controller gain m times. The phase shifting between bounds is vital when plants contain RHP dynamics and delays. This argument condition determines if the loops are not out of phase, and therefore, lotjω≈∑lokjω can be built with lokjω≤lotjω.


## 2.3.2. Stage 2: Design of Controllers

The designs of thei=1,…,n loops are carried out sequentially such that loijω fulfils the specifications (bounds βliω) according to the feedback sharing that was established in Stage 1. The general procedure was provided in [10], and it does not depend on the plants being MP or not. Bounds βli are computed and referenced to the true plant poi being MP or not.


## 3. Application: MISO Control of a Continuous Stirred-Tank Reactor

In this section, the benefits of the proposed methodology are shown on a recurrent benchmark of the chemical and materials industry, the continuous stirred-tank reactor (CSTR) of unquestionable importance in the transformation of raw materials into valuable chemicals or products [6, 38, 39].


### 3.1. The Process and the Control Design

The control setup of the MISO plant is depicted in Figure5. A coolant flow (usually water) through a jacket u2=Fj removes the necessary energy to prevent the exothermic and irreversible reaction runaway and to regulate the reactor temperature y=Tr. Due to the limited heat-removal capacity of the coolant flow, the manipulation of the reactant flow u1=F can contribute to the temperature control. The linear dynamic models linked to these manipulated inputs are p1 and p2. The work [6] presented a MISO control strategy inside a series control architecture following the VPC of [40]. In this paper, the parallel architecture of Figure 1 is used. The jacket coolant flow (the fastest actuation) midranges around ru2=rFj, while large reaction temperature excursions are compensated with the feed flow rate (the slowest actuation). Thus, beyond the dual-input collaboration in disturbance compensation, the MISO strategy can achieve the highest possible production rate: F is maximised when Fj set point is set to maximum. Three disturbance inputs are being considered: the feed concentration d1=Cain, the feed temperature d2=Tin, and the inlet coolant temperature d3=Tjin, whose contributions to the output Tr deviation are being represented by linear plants pd1, pd2, and pd3, respectively. Appendix B details the procedure on how to obtain all linear plant models from nonlinear differential equations of dynamic mass and energy balances [6]. Small-signal linearisation is executed about the system operating point(16)qj=Tre,Fe,Fje,Caine,Tine,Tjine,which relates the equilibrium (subscript “e”) of main process variables. A discrete set of 192 possible operating points is being considered (see Appendix B), which yields uncertainty about linear plant models. Thus, there is a plant matrix P (1) that collects 192 vectors Ps;qj=p1,p2,pd1,pd2,pd3. Figure 6 shades the envelopes of magnitude frequency responses of all plant cases and highlights the nominal plants(17)p1o=31.1342−s/0.002224+1s/0.003147+1s/0.003329+1s/0.000326962+2×0.6697/0.00032696s+1,(18)p2o=−8.7745s/0.0008615+1s/0.003329+1s/0.000326962+2×0.6697/0.00032696s+1,(19)pd1o=3.0482s/0.003147+1s/0.003329+1s/0.000326962+2×0.6697/0.00032696s+1,(20)pd2o=1.6408s/0.003147+1s/0.0008615+1s/0.003329+1s/0.000326962+2×0.6697/0.00032696s+1,(21)pd3o=4.5446s/0.0008615+1s/0.003329+1s/0.000326962+2×0.6697/0.00032696s+1,which correspond to the nominal values of equilibrium (16): Tro=350K, Fo=4.377×10−3m3/s, Fjo=11.3×10−3m3/s, Caino=8.01kmol/m3, and Tino=Tjino=294K.Figure 5
MISO control of CSTR.Figure 6
CSTR. Frequency responses of scaled plants.
(a)
(b)Regarding Figure6, the gain of p1 is higher than that of p2 over the whole frequency band; this reveals the feed flow F is the most powerful regulation variable. Its counterpart is the feed temperature Tin which 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., p1 behaves as a nonminimum phase (NMP) plant. In particular, the RHP-plane uncertain zero of p1 at ω∈0.0014,0.0029 makes the p1 participation unsuitable in control tasks over high frequencies. Then, a convenient frequency distribution of MISO dynamic controllability is of importance and can be quantitatively accomplished through the robust frequency domain method that this paper develops. Let us also mention that p2 plant has a negative gain since raising the jacket flow rate Fj makes the reactor temperature Tr to drop.Three performance specifications for robust disturbance rejection are defined following (2), 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 has to be less than 1K; and this maximum deviation has to be reduced to 0.25K no later than 2.5h after and fully extinguished in the steady state (settling time less than 4.0h to stay within a ±5% band). To ensure these conditions, the performance upper model is(22)Wdjω=6796jωjω/0.0004+12.Besides, two robust stability specifications (3) adopt as upper tolerance(23)Wsi=1,2=1.462,to guarantee minimum phase margins of 40∘ on each of the two loops (4). The discrete set of ω-frequencies to compute QFT bounds is(24)Ω=1,2,3,5,8,10,50×10−4rad/s.


#### 3.1.1. Stage 1: Feedback Sharing at Each Frequency of Design

Boundsβci′ are calculated. Figure 7 depicts them at the most illustrative frequencies, which yields the following conclusions according to the criteria in Section 2.3:At low frequenciesω≤3×10−4, bounds βc1′ω and βc2′ω are found phase-shifted less than ±90∘. However, βc1′ω are located over 20log2=6 dB lower than βc2′ω, which make the participation of p2jω useless in the regulation. Thus, the branch of p1jω should assume only the y regulation, and the branch of p2jω should be switched off at ω≤3×10−4.At medium frequencies3×10−4<ω≤8×10−4, bounds βc1′ω and βc2′ω are found phase-shifted less than ±90∘ and present a similar height around Θd=−90∘. Thus, plants p1jω and p2jω can collaborate over this interval of frequencies.At frequenciesω>8×10−4, the phase shifting between bounds βc1′ω and βc2′ω increases beyond ±90∘ such that the collaboration between plants is no longer possible. Thus, p2jω should assume the control tasks while p1jω should be inhibited by making l1jω≈0, which avoids the limitations introduced by the RHP zero of p1.Figure 7
CSTR. Feedback demands in the event of single plant intervention.
(a)
(b)


#### 3.1.2. Stage 2: Loop-Shaping of Controllers

To fulfil the objectives set in Stage 1, the loop-shaping of the two nominal open-loop functions is carried out following a sequential methodology as detailed in [10]. Figure 8 shows the final arrangement of bounds βli=1,2ω at design frequencies (24) and the nominal open-loop transfer functions loi=1,2jω that satisfy them. Let us note that lo1 contains an RHP zero. The resulting controllers are(25)c1=3.2×10−5s/0.0003+1s/0.0084+1ss/0.0017+12s/0.0152+2×0.5/0.015s+1,(26)c2=−0.37s/0.0003+1s/00018+1s/0.001+1s/0.00242+2×0.8/0.0024s+1.Figure 8
CSTR. Loop-shaping of controllers.
(a)
(b)


### 3.2. Results: Analysis and Comparatives

Figure9 proves the fulfilment of control specifications for the whole set of 192 plant cases: Figure 9(a) shows what concerns the robust rejection at the output of the three disturbances and the performance specification (2) and (22), and Figure 9(b) shows what concerns the robust stability of the two loops and the stability specification (3) and (23). Figure 10 illustrates the frequency band allocation between both loops: ltjω matches l1jω over low frequencies, ltjω matches l2jω over high frequencies, and ltjω matches l1jω+l2jω over midfrequencies.Figure 9
MISO control of CSTR. Closed-loop frequency responses.
(a)
(b)Figure 10
MISO control of CSTR. Open-loop frequency responses.Closed-loop time responses of main system variables are shown in Figure11. These results have been obtained using the proposed control system in the nonlinear model of CSTR (at this point it is recommended to see Appendix B for a full understanding of the physical units and experiments). The simulation shows the system behaviour for step-type disturbances in the feed temperature (δTin=5K at t=2h), in the coolant temperature (δTjin=−5K K at t=8h), and in the feed concentration (δCain=5%Caino at t=14h). Measured output Tr incorporates sensor noise, which is generated by a band-limited white noise source vt (noise power: 0.02; sample time = 100). Several set points on the coolant flow rate rFj between 65% and 85% of Fj maximum capacity have been tested. In all cases, the reactor temperature Tr deviates less than the maximum permitted (1 K) and recovers the set point (350 K) in less than 4.0 h after any disturbance happens. The steady-state feed flow F increases when the coolant set point rFj increases. This proves the smart VPC strategy [6] of indirectly maximizing the production rate by using a dual-input control. The collaboration of inputs (plants) in disturbance rejection is as follows: when a disturbance happens, the coolant flow rate Fj quickly reacts to compensate the reactor temperature deviation. Then, as the feed flow F takes control of the situation, Fj is progressively reset. Eventually, Fj returns to rFj after approximately 30 min. F cannot return to a particular set point rF while any disturbance persists. However, rFj can be wisely moved to bring F towards a particular range: a larger rFj involves a larger F, which increases the production rate; on the other hand, a smaller rFj pursues saving the amount of Fj. Both F and Fj move between their margins of operation [0% and 100%], despite the fact perturbations are of maximum amplitude. The evolution of other internal system variables, like Tj and Ca, is also included in the simulations.Figure 11
MISO control of CSTR. Closed-loop time responses.
(a)
(b)
(c)
(d)
(e)
(f)Following, there are argued benefits of MISO control versus SISO control. In this last case, the same robust control specifications are being satisfied with solely a loop intervention. Let us denoteF-SISO control when u1=F is in charge of the regulation tasks, and thus u2=Fj remains at a constant value. In that case, the control design process yields(27)c1SISOs=3.1354×10−5s/0.0012+1s/0.0004+1s/0.00045+1ss/0.003+1s/0.006+1.On the other hard, let us denoteFj-SISO control when u2=Fj is in charge of the regulation task and u1=F remains at a constant value. Thus, the single controller results(28)c2SISOs=−1.2402×10−4s/3.488×10−42+2×0.76/3.488×10−4s+1ss/0.001+1s/0.004+1.Figures12 and 13 depict frequency and time responses, respectively; their first and second columns of plots correspond to F-SISO and Fj-SISO control, respectively. The beam of frequency responses in Figures 12(c)–12(f) corresponds to the 192 cases of plant uncertainty. The time experiments of Figure 13 are run in a nonlinear simulator of CSTR and under the same conditions than the MISO control. In particular, Figure 11(e) details the changes of disturbance inputs, and five cases are being considered: for F-SISO, the coolant flow Fjt takes constant values 65%,70%,75%,80%,and85%, and for Fj-SISO, the feed flow Ft adopts constant values 55%,57.5%,60%,62.5%,and65%. Correlation between time and frequency responses agrees with the superiority of MISO over SISO control as follows.Figure 12
SISO control of CSTR. Frequency responses: (a), (c), and (e)F-SISO; (b), (d), and (f) Fj-SISO.
(a)
(b)
(c)
(d)
(e)
(f)Figure 13
SISO control of CSTR. Closed-loop time responses: (a), (c), and (e)F-SISO; (b), (d), and (f) Fj-SISO.
(a)
(b)
(c)
(d)
(e)
(f)The required performance and stablility cannot be achieved byp1 on its own due to the limited bandwidth imposed by the RHP zero; F-SISO (27) yields a reasonable trade-off solution between disturbance rejection and stability commitments. Figure 12(e) shows how Td1 and Td3 slightly violate Wd over midfrequencies, despite the fact this worsening is hardly noticeable in Figure 13(a) in comparison with Figure 11(a). Additionally, Figure 12(e) shows violations of robust stability Ws, which means a reduction of the phase margin from 40∘ to 35∘. Nevertheless, the major drawback of F-SISO control can be noticed from the comparison of control gains in Figure 12(a). p1-plant, which has an RHP zero at ω∈0.0014,0.0029, has been forced to work beyond ω=0.0008 at the expense of a huge increase of the controller gain c1SISOjω over high frequencies. It produces an inadmissible sensor noise amplification at the control input Ft,see Figure 13(c), which did not happen in MISO control—see Figures 11(c) and 11(d). In that case, a minimum phase p2-plant worked over the high-frequency band, despite apparently being less powerful (less gain) than a nonminimum phase p1-plant, see Figure 6(a). Therefore, c2jω≪c1SISOjω over high frequencies reports important benefits. Additionally, c1jω≈c1SISOjω at low frequencies, which means that the same Ft is needed to manage the steady states at Figure 13(c) and at Figure 11(c) if the noise components are dismissed from Figure 13(c).On the contrary, Figures12(d) and 12(f) check how the minimum phase p2-plant can achieve the specifications on its own, which means the temperature recovery Trt meets the time requirements in Figure 13(b) as it happened in Figure 11(a). The counterpart is that the p2-plant has to do the p1-plant work at ω≤0.0003, despite the fact p2 is less powerful (gain) than p1, see Figure 6(a). It results in higher control gains c2SISOjω≫c1jω over low frequencies and specifically at ω=0. In this way, δFj in each steady state, see Figure 13(f), is larger than δF in each steady state, see Figure 11(c). As an example, after the step change of the feed temperature (δTin=5K) at t=2h, the SISO control (case F=65%) requires a steady-state variation δFj of 14%, which approaches the available maximum of Fj; meanwhile, the MISO control (case rFj=85%) only required a steady-state variation δF of 5%. Additionally, c2jω≈c2SISOjω at high frequencies, which means the same sensor noise amplification at Fj in Figure 13(f) and in Figure 11(d).In summary, the superiority of MISO versus the SISO control of CSTR relies on the following facts. A better performance for reactor temperature control can be achieved without high sensor noise amplification at the manipulated inputs, which prevents actuator saturation and fatigue. One inputF is in charge of the steady state, which resets the fastest input Fj to prescribed values rFj. This is important from two points of view. As the fastest input returns to the set point after a disturbance rejection, it preserves the initial manoeuvring range to quickly react to new disturbances. Additionally, rFj gives places to infinite combinations of manipulated inputs to achieve the same reactor temperature. Some convenient ones are increasing rFj to maximize the production rate F or decreasing rFj to reduce the coolant cost Fj.


## 3.1. The Process and the Control Design

The control setup of the MISO plant is depicted in Figure5. A coolant flow (usually water) through a jacket u2=Fj removes the necessary energy to prevent the exothermic and irreversible reaction runaway and to regulate the reactor temperature y=Tr. Due to the limited heat-removal capacity of the coolant flow, the manipulation of the reactant flow u1=F can contribute to the temperature control. The linear dynamic models linked to these manipulated inputs are p1 and p2. The work [6] presented a MISO control strategy inside a series control architecture following the VPC of [40]. In this paper, the parallel architecture of Figure 1 is used. The jacket coolant flow (the fastest actuation) midranges around ru2=rFj, while large reaction temperature excursions are compensated with the feed flow rate (the slowest actuation). Thus, beyond the dual-input collaboration in disturbance compensation, the MISO strategy can achieve the highest possible production rate: F is maximised when Fj set point is set to maximum. Three disturbance inputs are being considered: the feed concentration d1=Cain, the feed temperature d2=Tin, and the inlet coolant temperature d3=Tjin, whose contributions to the output Tr deviation are being represented by linear plants pd1, pd2, and pd3, respectively. Appendix B details the procedure on how to obtain all linear plant models from nonlinear differential equations of dynamic mass and energy balances [6]. Small-signal linearisation is executed about the system operating point(16)qj=Tre,Fe,Fje,Caine,Tine,Tjine,which relates the equilibrium (subscript “e”) of main process variables. A discrete set of 192 possible operating points is being considered (see Appendix B), which yields uncertainty about linear plant models. Thus, there is a plant matrix P (1) that collects 192 vectors Ps;qj=p1,p2,pd1,pd2,pd3. Figure 6 shades the envelopes of magnitude frequency responses of all plant cases and highlights the nominal plants(17)p1o=31.1342−s/0.002224+1s/0.003147+1s/0.003329+1s/0.000326962+2×0.6697/0.00032696s+1,(18)p2o=−8.7745s/0.0008615+1s/0.003329+1s/0.000326962+2×0.6697/0.00032696s+1,(19)pd1o=3.0482s/0.003147+1s/0.003329+1s/0.000326962+2×0.6697/0.00032696s+1,(20)pd2o=1.6408s/0.003147+1s/0.0008615+1s/0.003329+1s/0.000326962+2×0.6697/0.00032696s+1,(21)pd3o=4.5446s/0.0008615+1s/0.003329+1s/0.000326962+2×0.6697/0.00032696s+1,which correspond to the nominal values of equilibrium (16): Tro=350K, Fo=4.377×10−3m3/s, Fjo=11.3×10−3m3/s, Caino=8.01kmol/m3, and Tino=Tjino=294K.Figure 5
MISO control of CSTR.Figure 6
CSTR. Frequency responses of scaled plants.
(a)
(b)Regarding Figure6, the gain of p1 is higher than that of p2 over the whole frequency band; this reveals the feed flow F is the most powerful regulation variable. Its counterpart is the feed temperature Tin which 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., p1 behaves as a nonminimum phase (NMP) plant. In particular, the RHP-plane uncertain zero of p1 at ω∈0.0014,0.0029 makes the p1 participation unsuitable in control tasks over high frequencies. Then, a convenient frequency distribution of MISO dynamic controllability is of importance and can be quantitatively accomplished through the robust frequency domain method that this paper develops. Let us also mention that p2 plant has a negative gain since raising the jacket flow rate Fj makes the reactor temperature Tr to drop.Three performance specifications for robust disturbance rejection are defined following (2), 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 has to be less than 1K; and this maximum deviation has to be reduced to 0.25K no later than 2.5h after and fully extinguished in the steady state (settling time less than 4.0h to stay within a ±5% band). To ensure these conditions, the performance upper model is(22)Wdjω=6796jωjω/0.0004+12.Besides, two robust stability specifications (3) adopt as upper tolerance(23)Wsi=1,2=1.462,to guarantee minimum phase margins of 40∘ on each of the two loops (4). The discrete set of ω-frequencies to compute QFT bounds is(24)Ω=1,2,3,5,8,10,50×10−4rad/s.


### 3.1.1. Stage 1: Feedback Sharing at Each Frequency of Design

Boundsβci′ are calculated. Figure 7 depicts them at the most illustrative frequencies, which yields the following conclusions according to the criteria in Section 2.3:At low frequenciesω≤3×10−4, bounds βc1′ω and βc2′ω are found phase-shifted less than ±90∘. However, βc1′ω are located over 20log2=6 dB lower than βc2′ω, which make the participation of p2jω useless in the regulation. Thus, the branch of p1jω should assume only the y regulation, and the branch of p2jω should be switched off at ω≤3×10−4.At medium frequencies3×10−4<ω≤8×10−4, bounds βc1′ω and βc2′ω are found phase-shifted less than ±90∘ and present a similar height around Θd=−90∘. Thus, plants p1jω and p2jω can collaborate over this interval of frequencies.At frequenciesω>8×10−4, the phase shifting between bounds βc1′ω and βc2′ω increases beyond ±90∘ such that the collaboration between plants is no longer possible. Thus, p2jω should assume the control tasks while p1jω should be inhibited by making l1jω≈0, which avoids the limitations introduced by the RHP zero of p1.Figure 7
CSTR. Feedback demands in the event of single plant intervention.
(a)
(b)


### 3.1.2. Stage 2: Loop-Shaping of Controllers

To fulfil the objectives set in Stage 1, the loop-shaping of the two nominal open-loop functions is carried out following a sequential methodology as detailed in [10]. Figure 8 shows the final arrangement of bounds βli=1,2ω at design frequencies (24) and the nominal open-loop transfer functions loi=1,2jω that satisfy them. Let us note that lo1 contains an RHP zero. The resulting controllers are(25)c1=3.2×10−5s/0.0003+1s/0.0084+1ss/0.0017+12s/0.0152+2×0.5/0.015s+1,(26)c2=−0.37s/0.0003+1s/00018+1s/0.001+1s/0.00242+2×0.8/0.0024s+1.Figure 8
CSTR. Loop-shaping of controllers.
(a)
(b)


## 3.1.1. Stage 1: Feedback Sharing at Each Frequency of Design

Boundsβci′ are calculated. Figure 7 depicts them at the most illustrative frequencies, which yields the following conclusions according to the criteria in Section 2.3:At low frequenciesω≤3×10−4, bounds βc1′ω and βc2′ω are found phase-shifted less than ±90∘. However, βc1′ω are located over 20log2=6 dB lower than βc2′ω, which make the participation of p2jω useless in the regulation. Thus, the branch of p1jω should assume only the y regulation, and the branch of p2jω should be switched off at ω≤3×10−4.At medium frequencies3×10−4<ω≤8×10−4, bounds βc1′ω and βc2′ω are found phase-shifted less than ±90∘ and present a similar height around Θd=−90∘. Thus, plants p1jω and p2jω can collaborate over this interval of frequencies.At frequenciesω>8×10−4, the phase shifting between bounds βc1′ω and βc2′ω increases beyond ±90∘ such that the collaboration between plants is no longer possible. Thus, p2jω should assume the control tasks while p1jω should be inhibited by making l1jω≈0, which avoids the limitations introduced by the RHP zero of p1.Figure 7
CSTR. Feedback demands in the event of single plant intervention.
(a)
(b)


## 3.1.2. Stage 2: Loop-Shaping of Controllers

To fulfil the objectives set in Stage 1, the loop-shaping of the two nominal open-loop functions is carried out following a sequential methodology as detailed in [10]. Figure 8 shows the final arrangement of bounds βli=1,2ω at design frequencies (24) and the nominal open-loop transfer functions loi=1,2jω that satisfy them. Let us note that lo1 contains an RHP zero. The resulting controllers are(25)c1=3.2×10−5s/0.0003+1s/0.0084+1ss/0.0017+12s/0.0152+2×0.5/0.015s+1,(26)c2=−0.37s/0.0003+1s/00018+1s/0.001+1s/0.00242+2×0.8/0.0024s+1.Figure 8
CSTR. Loop-shaping of controllers.
(a)
(b)


## 3.2. Results: Analysis and Comparatives

Figure9 proves the fulfilment of control specifications for the whole set of 192 plant cases: Figure 9(a) shows what concerns the robust rejection at the output of the three disturbances and the performance specification (2) and (22), and Figure 9(b) shows what concerns the robust stability of the two loops and the stability specification (3) and (23). Figure 10 illustrates the frequency band allocation between both loops: ltjω matches l1jω over low frequencies, ltjω matches l2jω over high frequencies, and ltjω matches l1jω+l2jω over midfrequencies.Figure 9
MISO control of CSTR. Closed-loop frequency responses.
(a)
(b)Figure 10
MISO control of CSTR. Open-loop frequency responses.Closed-loop time responses of main system variables are shown in Figure11. These results have been obtained using the proposed control system in the nonlinear model of CSTR (at this point it is recommended to see Appendix B for a full understanding of the physical units and experiments). The simulation shows the system behaviour for step-type disturbances in the feed temperature (δTin=5K at t=2h), in the coolant temperature (δTjin=−5K K at t=8h), and in the feed concentration (δCain=5%Caino at t=14h). Measured output Tr incorporates sensor noise, which is generated by a band-limited white noise source vt (noise power: 0.02; sample time = 100). Several set points on the coolant flow rate rFj between 65% and 85% of Fj maximum capacity have been tested. In all cases, the reactor temperature Tr deviates less than the maximum permitted (1 K) and recovers the set point (350 K) in less than 4.0 h after any disturbance happens. The steady-state feed flow F increases when the coolant set point rFj increases. This proves the smart VPC strategy [6] of indirectly maximizing the production rate by using a dual-input control. The collaboration of inputs (plants) in disturbance rejection is as follows: when a disturbance happens, the coolant flow rate Fj quickly reacts to compensate the reactor temperature deviation. Then, as the feed flow F takes control of the situation, Fj is progressively reset. Eventually, Fj returns to rFj after approximately 30 min. F cannot return to a particular set point rF while any disturbance persists. However, rFj can be wisely moved to bring F towards a particular range: a larger rFj involves a larger F, which increases the production rate; on the other hand, a smaller rFj pursues saving the amount of Fj. Both F and Fj move between their margins of operation [0% and 100%], despite the fact perturbations are of maximum amplitude. The evolution of other internal system variables, like Tj and Ca, is also included in the simulations.Figure 11
MISO control of CSTR. Closed-loop time responses.
(a)
(b)
(c)
(d)
(e)
(f)Following, there are argued benefits of MISO control versus SISO control. In this last case, the same robust control specifications are being satisfied with solely a loop intervention. Let us denoteF-SISO control when u1=F is in charge of the regulation tasks, and thus u2=Fj remains at a constant value. In that case, the control design process yields(27)c1SISOs=3.1354×10−5s/0.0012+1s/0.0004+1s/0.00045+1ss/0.003+1s/0.006+1.On the other hard, let us denoteFj-SISO control when u2=Fj is in charge of the regulation task and u1=F remains at a constant value. Thus, the single controller results(28)c2SISOs=−1.2402×10−4s/3.488×10−42+2×0.76/3.488×10−4s+1ss/0.001+1s/0.004+1.Figures12 and 13 depict frequency and time responses, respectively; their first and second columns of plots correspond to F-SISO and Fj-SISO control, respectively. The beam of frequency responses in Figures 12(c)–12(f) corresponds to the 192 cases of plant uncertainty. The time experiments of Figure 13 are run in a nonlinear simulator of CSTR and under the same conditions than the MISO control. In particular, Figure 11(e) details the changes of disturbance inputs, and five cases are being considered: for F-SISO, the coolant flow Fjt takes constant values 65%,70%,75%,80%,and85%, and for Fj-SISO, the feed flow Ft adopts constant values 55%,57.5%,60%,62.5%,and65%. Correlation between time and frequency responses agrees with the superiority of MISO over SISO control as follows.Figure 12
SISO control of CSTR. Frequency responses: (a), (c), and (e)F-SISO; (b), (d), and (f) Fj-SISO.
(a)
(b)
(c)
(d)
(e)
(f)Figure 13
SISO control of CSTR. Closed-loop time responses: (a), (c), and (e)F-SISO; (b), (d), and (f) Fj-SISO.
(a)
(b)
(c)
(d)
(e)
(f)The required performance and stablility cannot be achieved byp1 on its own due to the limited bandwidth imposed by the RHP zero; F-SISO (27) yields a reasonable trade-off solution between disturbance rejection and stability commitments. Figure 12(e) shows how Td1 and Td3 slightly violate Wd over midfrequencies, despite the fact this worsening is hardly noticeable in Figure 13(a) in comparison with Figure 11(a). Additionally, Figure 12(e) shows violations of robust stability Ws, which means a reduction of the phase margin from 40∘ to 35∘. Nevertheless, the major drawback of F-SISO control can be noticed from the comparison of control gains in Figure 12(a). p1-plant, which has an RHP zero at ω∈0.0014,0.0029, has been forced to work beyond ω=0.0008 at the expense of a huge increase of the controller gain c1SISOjω over high frequencies. It produces an inadmissible sensor noise amplification at the control input Ft,see Figure 13(c), which did not happen in MISO control—see Figures 11(c) and 11(d). In that case, a minimum phase p2-plant worked over the high-frequency band, despite apparently being less powerful (less gain) than a nonminimum phase p1-plant, see Figure 6(a). Therefore, c2jω≪c1SISOjω over high frequencies reports important benefits. Additionally, c1jω≈c1SISOjω at low frequencies, which means that the same Ft is needed to manage the steady states at Figure 13(c) and at Figure 11(c) if the noise components are dismissed from Figure 13(c).On the contrary, Figures12(d) and 12(f) check how the minimum phase p2-plant can achieve the specifications on its own, which means the temperature recovery Trt meets the time requirements in Figure 13(b) as it happened in Figure 11(a). The counterpart is that the p2-plant has to do the p1-plant work at ω≤0.0003, despite the fact p2 is less powerful (gain) than p1, see Figure 6(a). It results in higher control gains c2SISOjω≫c1jω over low frequencies and specifically at ω=0. In this way, δFj in each steady state, see Figure 13(f), is larger than δF in each steady state, see Figure 11(c). As an example, after the step change of the feed temperature (δTin=5K) at t=2h, the SISO control (case F=65%) requires a steady-state variation δFj of 14%, which approaches the available maximum of Fj; meanwhile, the MISO control (case rFj=85%) only required a steady-state variation δF of 5%. Additionally, c2jω≈c2SISOjω at high frequencies, which means the same sensor noise amplification at Fj in Figure 13(f) and in Figure 11(d).In summary, the superiority of MISO versus the SISO control of CSTR relies on the following facts. A better performance for reactor temperature control can be achieved without high sensor noise amplification at the manipulated inputs, which prevents actuator saturation and fatigue. One inputF is in charge of the steady state, which resets the fastest input Fj to prescribed values rFj. This is important from two points of view. As the fastest input returns to the set point after a disturbance rejection, it preserves the initial manoeuvring range to quickly react to new disturbances. Additionally, rFj gives places to infinite combinations of manipulated inputs to achieve the same reactor temperature. Some convenient ones are increasing rFj to maximize the production rate F or decreasing rFj to reduce the coolant cost Fj.


## 4. Conclusions

This work has presented a control solution where multiple inputs to a system can be used to govern its single measurable output. The method aimed a set of parallel controllers with the smallest possible gain at each frequency to achieve prescribed robust stability and robust performance in disturbance rejection. At a frequency, the gain of a branch controller was reduced when the magnitude of the associated plant contributed significantly to increasing the total magnitude of the vector sum of the branches (each branch included a controller and a plant). However, this was not easily to foresee due to counter-phase effects of RHP dynamics (or delays), due to uncertainties in the plant models, or due to the intervention of a great number of plants (inputs). The quantitative feedback theory (QFT) framework succeeds in the challenge. Then, some QFT control bounds were proposed to quantify the capacity of each plant to achieve the performance at each frequency. The lower the height the control bounds represented, the lower the control magnitude required by the associated plant. Besides, RHP dynamics and delays resulted in a phase shifting of these control bounds. Thus, comparing not only their relative magnitude but also their relative phase, the criteria of plant collaboration or inhibition at certain frequency to yield the control solution of the smallest gain were established. At the end of this first stage, the frequency band was allocated amongst the plants. At a second stage, the loop-shaping of the controllers carried out the predefined regulation sharing. Classical QFT bounds were computed to guide each loop-shaping. To illustrate the method, the MISO control of an unstable system was conducted.As a challenging example, the temperature of a continuous stirred-tank reactor was regulated through the coolant and feed flows pulling together along frequency bands that slightly overlapped. The feed flow capacity dominated the steady state, but from a dynamic point of view, it involved a temporary decrease in the reactor temperature. This nonminimum phase behaviour made useful the contribution of the coolant flow in the high-frequency band to improve the performance. Several feed flow temperatures, cooling flow temperatures, and reactant concentrations were considered. The provided control design method achieved a quantitative frequency distribution of MISO dynamic controllability in order to meet the prescribed robust performance and stability. Besides, a conveniently chosen set point for the coolant flow (midrange variable) governed indirectly the feed flow in the steady state, which conditioned the production rate.



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*Source: 3206864-2020-04-04.xml*