Multivariable Model Predictive Control of Cleanroom Pressure Cascades

Abstract

Cleanrooms are a fundamental setup of every semiconductor and life science factory, representing strictly controlled ambient conditions designed to minimize the air contamination that could affect a product within the production process. A cleanroom pressure cascade is a technological process associated with the ventilation and air treatment in these plants, with the goal of keeping the rooms within strict parameters. From a control systems perspective, it represents an MIMO ensemble of coupled control loops related to the technological process of air dynamics. In this research, we solve the multivariable control problem and present a novel, systematic approach with a conceptual solution for the following: (1) technological process and modeling challenges; (2) system identification and black-box MIMO process modeling; (3) full-blown controller design with model predictive control in its central place. This research was conducted on a real system at the “Fabrika COVID Vakcina” factory in Belgrade, Serbia.

1. Introduction

Cleanrooms, by definition, are literally rooms used in production processes, mostly related to the pharmaceutical, life science, and semiconductor industries [1,2], in which a high standard for air cleanness is demanded [3]. A cleanroom pressure cascade (CRPC) is a set of architecturally arranged cleanrooms next to each other, built in a special technological order [4]. This order is designed on the basis that proper air pressurization must be applied to each room in the cascade via a ventilation system so that pressure differences are established between rooms and parasitic air flow between them is always moving in a predetermined direction, as depicted in Figure 1. Following this demand, the room with a higher cleanness level must be more pressurized in relation to the surrounding space, or, to put it differently, it must be kept at a higher static pressure than the adjacent ones. By establishing an overpressure of typically 5 Pa to 20 Pa [5], the space and the product inside the room are protected from external contamination [5,6,7]. The word “cascade” in this system implies that pressure levels across multiple rooms can take various cascading or waterfall values, depending on the factory design, technology, production purpose, strategy for staff movement, transport of materials, etc. Apart from pressure level, another major factor for cleanrooms being clean is the designed air flow or, to be precise, the demanded volume of clean and filtered air that is supplied to each room [3]. CRPC relies on a complex ventilation system that consists of strong air handling units (AHUs) with variable speed-driven (VSD) fans, followed by a series of terminal variable-air-volume (VAV) boxes for local inlet air flow and standard modulating dampers for pressure control at their exhaust [8].

From a control systems perspective, CRPC stabilization is a problem to simultaneously regulate multiple cleanrooms on the ventilation side, targeting references for the supply air flow and room pressure in a highly disturbed and coupled physical process [10,11]. Typical industrial solutions for such defined CRPC stabilization problems are based on a distributed single-input-single-output (SISO) architecture with proportional–integral–derivative (PID) controllers [12]. The physical properties of the quantities from the given process represent additional problems, as they are very turbulent in essence. Furthermore, the pressure level for the control room is a highly sensitive quantity, bearing in mind that the range of magnitudes for control is a few tens of Pascals [13].

The aim of the research presented here was to properly analyze and model the multivariable nature of the process, define a scenario for a suitable system identification, and design an adequate model predictive control (MPC) controller for the CRPC in a holistic manner. The resulting MIMO controller is intended to regulate coupled quantities more effectively, both from a reference tracking perspective and from a disturbance rejection viewpoint.

The immediate benefits and contributions of the presented control solution are as follows:

• A conceptual approach to the multivariable nature of the problem and the application of known control theories to the specific CRPC domain, which has generally not been addressed before. This holistic conceptualization results in formalized steps for system identification, controller design, and tuning. As a result, those steps are replicable and can be used in a similar environment.
• System integration architecture, proving and presenting how conceptualization can be applied to a real and existing PLC and drive BMS systems in factories, with the help of additional computer systems on top of it.
• Controller architecture that introduces the MPC for a particular room but that also embraces the interdependencies between rooms, reflected through PID-driven duct pressure control. This is achieved by incorporating duct control in the measured disturbances. In this way, the architecture represents a method to cover the whole cascade of multiple rooms with one pair of fans.
• The discovery that one model of the system is suitable, not only in the vicinity of the identified operating point. On the contrary, it is appropriate for a wider range as long as the constraints are adaptable and shifted together with the setpoint variations according to a pre-captured data set.
• An optimal scenario for proper system identification that suits the developed controller and its architecture. This scenario is derived as a response to the limitations imposed by the mechanical design and process physics.
• Results that prove the assumption that the multivariable approach to this process has an advantage in comparison to the distributed SISO PID approach.

This paper is organized as follows. In Section 2, a review of the most recent scientific studies and research efforts regarding the design, operation, and investigation of the performance of cleanrooms, as well as the design and implementation of modern control strategies in building technology and HVAC systems. The details of the proposed control solution for CRPC, including analysis, modeling, identification, and MPC design, are presented in Section 3. Section 4 presents a case study where the complete proposed design procedure is carried out, and its performance is verified on a real plant. Concluding remarks and some important discussions and directions for future work are given in the final Section 5.

2. Related Work

The gap between industrial practice and control system theory is a well-known and properly identified concern in the academic circles [14,15]. Regarding the case of a cleanroom’s control, to the best of the authors’ knowledge, this problem has not been properly covered by academia so far. From the mechanical and technological point of view, cleanrooms are indeed covered in many aspects, but typical areas of study within them are thermal comfort [16,17], pressurization and air volume rates [18,19], air patterns and distribution [20,21], contamination sources and effects [11,22], and strategies for ventilation optimization and efficacy [23,24]. A comprehensive systematic review of cleanroom ventilation and air distribution systems has been presented in industry [2], with many valuable guidelines on technologically related issues, but not on closed-loop control. The same applies to [25], where a scenario for improved control of the pressure in a cleanroom environment is described, and finally in [13], where we can find a comprehensive analysis of a demand-controlled filtration strategy focused on recirculated air volume. That being said, the strategy given within our novel approach should not be compared but instead combined with those pathways, enriching them with an additional control system layer.

Although cleanrooms themselves are not covered by control-system studies, there are works related to the control of HVAC, ventilation systems, and VAV control via PID, MPC, and other strategies [26,27] that are relevant for a wider context of this research.

MPC is a several decades-old concept now [28,29]. However, each and every implementation of MPC is a new cumbersome task for itself [30]. There are two dimensions of development when it comes to MPC: (1) system modeling as a core component of successful MPC and (2) performance index as an optimization question of what MPC needs to achieve. As this research is focused on real-world industrial practice, we are, in addition, challenging the questions related to implementation and system integration. Recent industrial cases for MPC development are proving its practical feasibility, like carbon-capturing [31], cooling-water networks [32], solar plants [33], and many others. In most of those cases, MPC was used as a supervisory level for slow processes, dictating references for PID controllers, where the PID controllers themselves remained on the base layer, directly connected to process values and actuators [34,35]. Less often, as is the situation with this research, MPC is used for the direct control of the process [36,37]. Some works related to the implementation of MPC in the PLC, as industrial-grade devices designed for real-time control and automation tasks, reported partial success [38,39]. Those attempts have gained mostly theoretical value only, since they suggested that PLC is not suitable for MPC implementation [40]. This result was expected, since the necessary CPU processing power and memory for numerical optimization behind the MPC, like linear or quadratic programming, is more demanding than modern PLCs can provide. This brings us to the various system integration questions and issues. Typically, there are two conventional approaches to how the MPC can be implemented into the industrial environment. The first one is the usage of high-level programming languages such as MATLAB, Python, or C/C++, suitable for code development around the dedicated solver, to be compiled and run on a high-level computer. The second approach is the utilization of standard industrial software solutions like AspenTech DMC+/DMC3, Honeywell Profit Controller, Yokogawa SMOC, and IPCOS INCA, tools that are mostly tailored for the process industry, petrochemical production, and oil and gas in the first place [41].

Applying MPC to building technology and HVAC systems has been a very vivid topic for more than a decade now. HVAC technology represents a large-scale multivariable, multicomponent system of systems, as it consists of subcomplex mutually arranged and interdependent processes. By this we mean heating (from various sources and their combinations), refrigeration, thermal storage, distribution (air), distribution (water), thermal consumption, occupancy, air quality, weather factors, etc. Its technological complexity is additionally combined with advancement in building controllers and supervisory systems. As a result, this has opened a wide field to research in the areas of optimization, prediction, and energy savings against demanded comfort conditions [42,43,44,45,46].

In summary, previous works related to cleanrooms are focused primarily on strategic components, describing various specific innovative engineering scenarios and algorithms for overall technological improvements. Despite that CRPC obviously belongs to the class of complex processes that are challenging for conventional control and promising for the application of MPC, previous studies have not actually explored the implementation of closed-loop control.

3. Control Design Methodology

In this section, a process description and appropriate modeling will lead to decomposition of the CRPC stabilization problem. We will pass through the systematic procedure for the design of the room-by-room ensemble of MPC controllers, with adequate treatment of mutual influences as disturbances.

3.1. Process Description

A single cleanroom [47], with two standard control loops, one of which is the air flow at the inlet duct and the other is the pressure inside the cleanroom, represents a sole 2 × 2 multi-input-multi-output (MIMO) system [48,49]. The process diagram is given in Figure 2. A strong physical interconnection exists between controlled flow rate and pressure due to their mutual influence, stemming from the basics of fluid dynamics [50].

3.2. Process Physics and Justification of the Black-Box Approach

Room pressure calculation is given by the simplified Bernoulli’s equation-based formula [51,52], while damper physics is modeled and given by the ASHREA equation [26,53]:

$$\Delta p={\displaystyle \frac{\rho }{2}}{\left({\displaystyle \frac{V_{sup}-V_{exh}}{A · \mathsf{\mu } · 3600}}\right)}^2$$

(1)

dp = room differential pressure, ρ = relative density, V_sup = supply air flow, V_exh = exhaust air flow, A = room leakage area, µ = discharge coefficient

$$m_a=C_{dp} \sqrt{{\rho }_{a·}\Delta P_{dp}}{ A}_{dp}(\theta )$$

(2)

dP = pressure difference across the device, A_dp(Θ) = flow area, C_dp = intrinsic coefficients

As shown in Figure 3, fan physics is most frequently presented in a non-parametric way. In addition, the duct pressure and resistance are difficult to calculate. They change linearly with the distance from the fan, with abrupt drops with every knee, bend, damper, filter, cross-section change, etc. [50], which is visible from Figure 4.

In practice, this type of design is performed through dedicated CAD software (Computer Aided Design), a range of advanced software tools for engineering drawing and modeling with very sophisticated simulation capabilities developed nowadays. But even in that case, the physics of the process is substantially complex to suit the modeling according to the white-box principle [54]. Given models are especially inaccurate within the room pressure range of magnitude, which is very low and sensitive [55]. And finally, these equations are purely static, as they do not capture the process dynamics, which all force us to step away from a first-principle approach.

3.3. Full Cascade Process Flow Diagram

The whole pressure cascade, on the other hand, represents a complex MIMO structure that usually consists of two general AHU fans (one for the supply and one for the exhaust), both controlled via VSD, and dozens of terminal VAV units (2 per room) [52,56] Physical interconnections due to fluid physics are always present, since there is a mutual dependency between room pressure, flow, fans and shared ducts, adjacent spaces, common corridors, etc. [57], as depicted in Figure 5.

At this point, it is understandable that, due to the complexity of the physical process and its multivariable nature, we will continue forward with the black box approach.

3.4. From the Complete System Model to the Single Cleanroom Transfer Matrix

The model of the process, presented in Figure 5, that properly singles out inputs, outputs, and transfers, can be linearly represented in the following way [47]:

$$\left[\begin{array}{c}\left[\begin{array}{c}{dp}_1\\ {fl}_1\end{array}\right]\\ \left[\begin{array}{c}{dp}_2\\ {fl}_2\end{array}\right]\\ ⋮\\ \left[\begin{array}{c}{dp}_n\\ {fl}_n\end{array}\right]\\ p_{ds}\\ p_{de}\end{array}\right]=\left[\begin{array}{ccccc}\left[\begin{array}{cc}{G}_{11}^1& G_{12}^1\\ G_{21}^1& G_{22}^1\end{array}\right]& W_{12}& \dots & W_{1n}& \left[\begin{array}{cc}{D}_{11}^1& D_{12}^1\\ D_{21}^1& D_{22}^1\end{array}\right]\\ W_{21}& \left[\begin{array}{cc}{G}_{11}^2& G_{12}^2\\ G_{21}^2& G_{22}^2\end{array}\right]& \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots \\ W_{n1}& \dots & \dots & \left[\begin{array}{cc}{G}_{11}^n& G_{12}^n\\ G_{21}^n& G_{22}^n\end{array}\right]& \left[\begin{array}{cc}{D}_{11}^n& D_{12}^n\\ D_{21}^n& D_{22}^n\end{array}\right]\\ W_{se1}& \dots & \dots & W_{sen}& D_s\end{array}\right]\left[\begin{array}{c}\left[\begin{array}{c}{u}_{a1}\\ u_{d1}\end{array}\right]\\ \left[\begin{array}{c}{u}_{a2}\\ u_{d2}\end{array}\right]\\ ⋮\\ \left[\begin{array}{c}{u}_{an}\\ u_{dn}\end{array}\right]\\ u_{os}\\ u_{oe}\end{array}\right]$$

(3)

The given model utilizes the system transfer function matrix that embraces all the rooms in the cascade with their direct input-output dependencies, as well as their mutual influences. The left side of the equation represents corresponding room-related pairs dp_i and fl_i, with two additional quantities related to duct pressure, p_ds (static pressure in the supply duct) and p_de (static pressure in the exhaust duct), in accordance with Figure 5. In the transfer function matrix, shaded matrices Gⁱ represent direct transfer from damper control signals to output signals of room i, matrices W_ij and ${\mathit{W}}_{\mathit{s}\mathit{e}\mathit{i}}$ represent room couplings and transfer of disturbances, while D represents transfer from fan VSD control inputs to outputs. Finally, the input vector contains all control inputs described here—u_ai and u_di control pairs for room i, and additionally u_os and u_oe controls for the VSD of the fans.

This general transfer function is of the order (2n + 2) × (2n + 2), and as such, it is highly complex to be regulated by one comprehensive instance of a controller of the same order.

What lies in the core of the process given by Equation (3) is a multivariable subsystem of a particular cleanroom. When we extract the elements from Equation (3), we can form the segregated single-room transfer function:

$$\left[\begin{array}{c}{dp}_i\\ {fl}_i\end{array}\right]=\left[\begin{array}{cc}{G}_{11}^i& G_{12}^i\\ G_{21}^i& G_{22}^i\end{array}\right]\left[\begin{array}{c}{u}_{ai}\\ u_{di}\end{array}\right]+\sum _{j\ne i}{W}_{ij}\left[\begin{array}{c}{u}_{aj}\\ u_{dj}\end{array}\right]+\left[\begin{array}{cc}{D}_{11}^i& D_{12}^i\\ D_{21}^i& D_{22}^i\end{array}\right]\left[\begin{array}{c}{u}_{os}\\ u_{oe}\end{array}\right]$$

(4)

The main diagonal of Equation (3) is a strongly dominant part when it comes to mutual influences. It contains dependencies characterized by the coupling of two major technological quantities, already mentioned in the introduction: (1) air flow regulated by supply damper modulation and (2) room pressure regulated by exhaust damper modulation. Therefore, our simplest solution is focused only on the room core, in which we conceptualize the influence of fans and other disturbances, W and D, as a non-measurable disturbance. We can eliminate them from our deterministic part, leaving only the dominant G part of the equation, thus isolating a 2 × 2 transfer function.

$$\left[\begin{array}{c}{dp}_i\\ {fl}_i\end{array}\right]=\left[\begin{array}{cc}{G}_{11}^i& G_{12}^i\\ G_{21}^i& G_{22}^i\end{array}\right]\left[\begin{array}{c}{u}_{ai}\\ u_{di}\end{array}\right]$$

(5)

However, this simplification is not always justified. We extend the solution and improve the model by again involving the influence of fans and other disturbances. Using knowledge of the discipline and an understanding of the system, we can reflect all the disturbances, interconnections, and dependencies through the separated control loops of the duct channels. At this point, we unveil the main element of our research and its contribution to the field.

We assume that duct quantities p_ds and p_de will be stabilized by outer control loops. With this assumption, all the non-diagonal (W and D) elements of Equation (3) will be absorbed by the given control loops and will be embodied through common control signals u_os* and u_oe* via summarized transfer D*. If we use these signals in the role of measurable disturbances on the input side, we can obtain the extended 2 × 4 model that effectively captures mutual influences and duct disturbances between rooms, formalized in Equation (6).

$$\left[\begin{array}{c}{dp}_i\\ {fl}_i\end{array}\right]=\left[\begin{array}{c}\begin{array}{cccc}{G}_{11}^i& G_{12}^i& D_{11}^{* i}& D_{12}^{* i}\end{array}\\ \begin{array}{cccc}{G}_{21}^i& G_{22}^i& D_{21}^{* i}& D_{22}^{* i}\end{array}\end{array}\right]\left[\begin{array}{c}{u}_{ai}\\ u_d\\ {u_{os}}^{*}\\ {u_{oe}}^{*}\end{array}\right]$$

(6)

3.5. Optimization Problem

In this subsection we formulate the optimization problem that lies in the backbone of the MPC. We will cover two very symptomatic cases that are related to a specific behavior of system modeling in relation to how the model itself is able to accommodate the wider set of operating points.

Out of many optimization formulations for MPC, ours is a straightforward MPC derivation that minimizes the transient error of quantities in relation to steady state, the transient error of control variables, and finally penalizes their rate of change. Although this is one of the typical performance indexes known in the literature, in our case, its magic lies in properly detected constraints, the dampers’ range movements.

These damper ranges are restrained by the following factors: (1) the saturation of the pressure and flow sensors and (2) critical pressure values in the room itself that can lead to physical damage.

We can now formulate the optimization problem:

3.5.1. Case 1—Single Operating Point

Minimize

$$\mathrm{J}={\sum }_{k=0}^m{\left[\begin{array}{c}{dp}_k-{dp}_{kref}\\ {fl}_k-{fl}_{kref}\end{array}\right]}^T·\mathrm{Q}\left[\begin{array}{c}{dp}_k-{dp}_{kref}\\ {fl}_k-{fl}_{kref}\end{array}\right]+{\left[\begin{array}{c}{u}_{ak}-u_{akss}\\ u_{dk}-u_{dkss}\end{array}\right]}^T·\mathrm{R}\left[\begin{array}{c}{u}_{ak}-{ua}_{kss}\\ u_{dk}-u_{dkss}\end{array}\right]+{\displaystyle \frac{d}{dt}}{\left[\begin{array}{c}{u}_{ak}\\ u_{dk}\end{array}\right]}^T·\mathrm{P}·{\displaystyle \frac{d}{dt}}\left[\begin{array}{c}{u}_{ak}\\ u_{dk}\end{array}\right]$$

(7)

Subject to

Equation (6)

min_u_ai ≤ u_ai ≤ max_u_ai

min_u_di ≤ u_di ≤ max_u_di

P, Q, R = $\left[\begin{array}{cc}{p, q, r}_{11} & 0\\ 0& {p, q, r}_{22} \end{array}\right]$ are diagonal weight factors, m is the maximum discrete time for performance index (prediction horizon), ss is the steady state value, u_ai is the control signal of the supply damper, and u_di is the control signal of the exhaust damper.

Case 1 (Figure 6), is a simple situation with fixed constraints that is suitable for the operation in the vicinity of the unique operating point.

3.5.2. Case 2—Wider Range of Operating Points

The solution for the non-linear systems in case they cannot be modeled by one single build in the vicinity of the operational point typically relies on tools like gain-scheduling or similar adaptive strategies.

However, our results suggest that a single unique model can cover a wider range of operating points, but the constraints must be adaptive! This exactly means that the same MPC can operate across that range, as long as we can provide sliding variable constraints as additional inputs. Variable constraints can be processed in many forms, like mathematical functions or lookup tables. The key point is to have constraints’ function allocated as dedicated inputs to the model, and to have their actual value selected based on the 2D reference, as presented in Figure 7.

The optimization problem for Case 2 is now as follows:

Minimize

Equation (7)

Subject to:

Equation (6)

$$‖f_{min}(dp{i}_{ref},fli\_ref)‖\le ‖f(uai,udi)‖\le ‖f_{max}(dp{i}_{ref},fli\_ref)‖$$

$f(uai,udi)$ is the 2D-bounded constraints function for the dependency between exhaust and supply damper motion that is their corresponding control signals, u_ai and u_di, related to the actual setpoint pair for pressure and flow ($dp{i}_{ref}, fli\_ref$).

3.6. Controller Architecture with MPC Solution

In this subsection, we are proposing our controller architecture, which is as follows. We are forming a series of n parallel and independent MPC controllers of the 2 × 4 type for each room (n is the number of rooms in cascade), and on top of that, we are adding 2 outer PID controllers that will maintain a common pressure in the ducts. According to the general room transfer function in Equation (6), two PID outputs are used as measured disturbance inputs for each MPC 2 × 4 from the series.

This architecture is chosen to fulfill the following challenges. Cleanrooms are isolated entities in which flow and pressure, as the core quantities, are strongly coupled. This is the main reason that positioning such a controller in the heart of the problem is naturally justified. Mutual effects and interconnections from other quantities of adjacent rooms are effectively tied together and compensated by the joint input of measured disturbances from the duct or VSD signal reaction. That way, all the physical links between rooms are indirectly compensated by the a priori reaction of the outer-loop PID controllers. Hence, while the central MPC 2 × 2 performs a decoupling function of the common fluid dynamics of the flow and pressure within the room, the outer loops have a function of suppressing mutual influence among neighboring spaces via joint disturbance signals to each local MPC. Figure 8 represents the process flow diagram of the developed structure. The control block diagram that follows the description of the solution just described here is given in Figure 9.

3.7. System Identification

A valid system model is the key to the MPC design. Modeling via system identification is a wide topic and is deeply covered by academia [58,59]. However, as with any other distinct industrial case and technology, the specifics of both the cleanroom system itself and the MPC design are correlated with the specifics of the successful identification. Our methodical system identification is focused on Equation (6). It consists of several independent estimation steps, to be linearly combined into a complete transfer matrix afterwards.

3.7.1. Preconditions

To execute both modeling and identification given in this research, we define the typical industrial case and frame its limitations.

• Room operating points (pressure and flow) are given upfront by the conceptual factory design, upon very strict technological demands and users’ requirements.
• Based on that, the system must be air-balanced, and the working points of all process parameters must reach their own set values. This is related to duct pressures, all room pressures and room flows, as well as the positions of the actuators related to them.

3.7.2. Non-Uniqueness of the Duct Pressure Setpoint—A New Degree of Freedom

In the air-balanced system of the cleanroom cascade, the operating points for the duct pressures produced by AHU fans are not unique. The system can have an infinite number of stable operating points within the specific range. In theoretical terms we encounter an additional degree of freedom for finding the proper and necessary operating points, which are the duct pressures exactly.

To additionally depict this phenomenon, we need to choose and establish a fixed static pressure in the ducts using VSD fans. This is a self-evident precondition for the flow and dampers’ work if we refer to the process flow diagram from Figure 8. Chosen values can be anywhere in the range where mechanical design allows it, typically a few hundred Pa. Naturally, for differently chosen duct references, room actuators will eventually achieve different control values. If we take a supply side, for example, nominal duct value can be achieved with a high pressure of, e.g., 700 Pa, or a lower pressure, e.g., 300 Pa. Higher pressure will lead the input room dampers to shift towards more closed positions in automatic mode, e.g., 5–30%, and lower pressure to wider openings of 70–90%. In other words, any duct pressure value within the mechanically acceptable range will serve its purpose, as local flow controllers will eventually achieve the desired air flow and accordingly stable room pressure.

We can eliminate this excessive DOF by selecting one fixed reference value and making a technological compromise based on the following side requirements. We have to comply with the waterbed effect outlined below:

• Dampers need to have an acceptable range of free movement for control purposes.
• At the same time, they need to be open as wide as possible to avoid unnecessary air friction and aerodynamic obstruction that could increase the energy consumption of the fans.

To adhere to this semi-empirical criterion, we can look for the point that will keep the dampers in the upper third of their full movement. In the plant engineering practice, this is performed during the commissioning phase by trial-and-error attempts until the desired point is fixed and accepted.

3.7.3. Constraint Determination

It was mentioned before that all constraints are technologically imposed, like sensor saturation, as well as the endurance of the room ceilings and walls to overpressure. In summary, all these constraints are reflected through the limitations in the damper movements. These limitations are determined by moving the dampers in manual control mode, in alternating steps between supply and exhaust, over several iterations. Initially we need to establish the desired operating point, after which we move the dampers in small increments (like 1%) as described. These iterations are repeated until any limiting condition is reached.

Case 1 from Section 3.5.1 is performed around the nominal operating point, but Case 2 from Section 3.5.2 does not differ as well. Namely, for Case 2, we perform the same iterative inspection around a sufficient number of operating points within the desired range (case study—9 points), after which we can compose a segmentally linearized lookup table, a piecewise affine interpolation.

3.7.4. Steps and Sequence for Identification

The goal of the system identification is to obtain the transfer function of the extended 2 × 4 model in Equation (6). We can split this equation into three linear segments:

$$\left[\begin{array}{c}{dp}_1\\ {fl}_1\end{array}\right]=\left[\begin{array}{c}\begin{array}{cc}{G}_{11}& G_{12}\end{array}\\ \begin{array}{cc}{G}_{21}& G_{22}\end{array}\end{array}\right]\left[\begin{array}{c}{u}_{a1}\\ u_{d1}\end{array}\right] \left[\begin{array}{c}{dp}_1\\ {fl}_1\end{array}\right]=\left[\begin{array}{c}\begin{array}{c}{D}_{11}\end{array}\\ \begin{array}{c}{D}_{21}\end{array}\end{array}\right]u_{os} \left[\begin{array}{c}{dp}_1\\ {fl}_1\end{array}\right]=\left[\begin{array}{c}\begin{array}{c}{D}_{12}\end{array}\\ \begin{array}{c}{D}_{22}\end{array}\end{array}\right]u_{oe}$$

(8)

Accordingly, we can perform the identification in three steps.

Step 1: The identification is carried out by recording the response to the generated vector of 2 random-sequence signals, u_a1 and u_d1, towards two damper actuators. This way, the 2 × 2 identification is carried out. Step 2: For fixed dampers at the operating point, a 1 × 2 connection pressure is identified from u_os at the input-supply fan signal. Step 3: The process is repeated in the identical way for the exhaust side, thus obtaining the disturbance transfer functions 1 × 2, with u_oe at the input–exhaust fan signal.

Fitting the entire function: Starting from the repeatedly mentioned assumption that the system is linear, this identification method is adopted and closed by aggregating the three components of Equation (8) into the integrated transfer function of a single 2 × 4 cleanroom, Equation (6)

4. Case Study

4.1. Facility

The research presented in this paper was entirely performed on a test bed that was established in the newly built pharmaceutical production facility, Fabrika COVID Vakcina, Belgrade, Serbia, Figure 10. This factory consists of 14 AHU system pairs (supply and exhaust) and therefore 14 separate and independent cleanroom zones. Each zone, with one dedicated AHU pair, contains between 8 and 19 cleanrooms in cascade.

4.2. Basic Control System

The BMS (building management system) of the facility is constructed for the general control of the entire HVAC process inside the factory, including the described ventilation system. The BMS solution is based on a Siemens PXC 100E.D type PLC. A single cleanroom, with one inlet and one outlet duct, was selected for research purposes and was reconstructed mechanically. Both the inlet and the outlet actuators were equipped with fast-acting Siemens GAP 191 type actuators. The actuator type used has a rapid opening characteristic—2 s movement from 0 to 100% (0 deg to 90 deg). This characteristic is making the whole system controllable in a practical sense, as the actuation speed is comparable to the process response. Fans of the dedicated AHU are equipped with Siemens G120P VSDs with an external control signal from the PLC for speed control.

4.3. High-Level Control and System Integration

The PLC is not practical for advanced control beyond PID. For MPC implementation, the PLC was connected to a standard Windows-based laptop PC for high-level data processing and control. Matlab/Simulink R2021a software package was used for data acquisition, system identification, process conditioning, and multivariable control. For communication purposes, the Kepware OPC Server with the Bacnet/IP driver was used over an Ethernet physical layer, as depicted in Figure 11.

4.4. Experiment Process

In this section, we are performing a breakdown of all the stages of our experiments, methods, and results. All the experiments, both for system identification data recording as well as for testing and validating the controllers, were carried out exclusively on the described testbed and on the designated modified cleanroom. No simulations were conducted in this work whatsoever.

Flowcharts and overviews of all the stages are given through the following points and presented in Figure 12:

• Setting up the factory plant and air-balancing the system in automatic mode and reaching all necessary setpoints according to plant requirements. This is the starting point for the tests and modeling around the nominal operating points.
• Detecting constraints according to the scenario described in the methodological section—Section 3. Constraints are detected, firstly around the nominal setpoint (Case 1 of handling constraints, as per Section 3.5.1), and afterwards at various different operating points (Case 2 of handling constraints, as per Section 3.5.2) in a wider range.
• Performing thorough system identification in three steps:
  • Proceeding with 2 × 2 identification using a pseudo-random generator to excite dampers’ control signals, two signals simultaneously, to capture the response over the tested time.
  • Proceeding with 1 × 2 identification of the supply VSD-to-room signal and capturing the response.
  • Proceeding with 1 × 2 identification of the exhaust VSD-to-room signal and capturing the response.
• Composing the model using system identification tools, linearly combining previous results from Equation (8) into Equation (6). Applying the model to the MPC function, including disturbances from VSDs.
• Tuning the MPC parameters by trial-and-error and experimentation.
• Testing the MPC-isolated pressure control vs. PID-isolated pressure control by using a pseudo-random setpoint generator. Other quantities are fixed at this point (flow and both VSDs).
• Testing MPC 2 × 2 pressure and flow control in parallel vs. two independent PID SISO controls by using a pseudo-random 2D setpoint generator with fixed VSDs.
• Testing MPC 2 × 4 response to VSD disturbance vs. two independent PID SISO controls for pressure and flow by using pseudo-random disturbances as variations added to fixed VSD control inputs.

Throughout the following section, we will go deeper into the details of each of the stages and convey the results of highlighted experiments.

4.4.1. Stage 1—System Balancing and Mechanical Limitations

• We observe the whole HVAC system for the 17 cleanrooms in the cascade with a pair of VSD fans. All the quantities of the system are initially controlled by PID controllers from the PLC software (Siemens PXC100, XWorks 5), whose tuning quality is suboptimal but still responsive enough to reach a stable state in the absence of disturbances.
• Upon being started and put into automatic mode, the ventilation system is isolated from external disturbances and is left to reach a steady state. This means that referenced duct pressure is achieved and stabilized, as well as the whole collection of the cleanroom parameters for all the particular pressures and flows.
• After reaching a steady state, both supply and exhaust fans are frozen using manual control at captured operating points. The other cleanrooms, except for the tested one, are kept in automatic PID-driven mode.
• The starting point for this experiment is the precisely designed operation point for the room under test—30 Pa and 1250 m³/h plant requirements for the selected cleanroom.

4.4.2. Stage 2—Constraint Determination

• Constraint detection is initiated by searching for the throttling range for both the supply and exhaust dampers. The limitations that we do not want to breach are a pressure sensor range of 0 to 100 Pa and flow measurement between 0 and 1450 m³/h (0.402 m³/s). The throttling range is detected by easily changing the position of the dampers with a manual signal until the value near the saturation limit is reached. The constraints in the vicinity of the nominal operating point are supply—55–65%, exhaust—45–62%.
• For the variable constraints’ purposes described in Section 3.5.2, we performed the same set of experiments in nine distinctive operative points, three by each dimension (pressure, flow), and formed the following lookup table,

  Table 1. Constraints lookup table [ua-min, ua-max][ud-min, ud-max].

  	5 Pa	20 Pa	35 Pa
  400 m3/h	[19, 38] [17, 29]	[11, 30] [11, 23]	[3, 20] [5, 14]
  800 m3/h	[34, 55] [33, 54]	[31, 52] [24, 37]	[29, 50] [19, 31]
  1200 m3/h	[51, 71] [62, 77]	[49, 70] [53, 64]	[48, 68] [48, 59]

  .

4.4.3. Stage 3—System Identification Data

With reference to Section 3.7.4, we recall that three independent identification activities are to be performed to capture the parameters of our model in the form of Equation (6).

A fundamental idea is that the data sets should contain proper transient responses and correctly embrace the process dynamics, which are necessary and convenient for the correct identification.

Stage 3, Step 1—recording data for G₁₁, G₁₂, G₂₁, G₂₂

• Supply and exhaust dampers’ position signals are excited by random step signals within their corresponding throttling ranges. The time period for each impulse is 60 s, but the excitation shift between them is 30 s. This means that every 30 s, another actuator is excited in alternating order: sup-30s-exh-30s-sup-30s-exh-30s and so forth. Room pressure and flow quantities are recorded as the outputs.
• The collected data set is a trend of 35 min, which is used for modeling the system. Experimental data are presented in

  Table 2. Experimental data for system identification Step 1.

  Parameter	Description	Value or Range
  [us ue]	Excitation: 2D signal, supply and exhaust damper position signals Form: step impulses of pseudo-random amplitude, 2D	[57–64%; 62–67%]
  Texp	Total experiment duration (overall sampling time)	35 min
  Ti	Impulse duration per dedicated actuator signal	25 s
  Tsh	Impulse shift between two excitation signal changes	12.5 s
  Tq	Discretization time	0.5 s
  [dP Fl]	Output: room pressure, supply air flow	trend

  , while recorded trends are given in Figure 13.

This data set contains proper transient responses and correctly embraces the process dynamics, which are necessary and useful for correct identification.

Stage 3, Step 2—recording data for measured disturbances D₁₁, D₁₂

• Supply fan speed was excited by a pseudo-random step sequence in the range of 80–90%, for gaining the corresponding linear model parts. We recorded the response from the supply VSD control signal to the cleanroom parameters in the role of the first measured disturbance. Experimental data are presented in

  Table 3. Experimental data for system identification Step 2.

  Parameter	Description	Value
  uds	Excitation: 1D pseudo-random signal, supply AHU fan speed control signal Form: step impulses of pseudo-random amplitude	85–97%
  Texp	Total experiment duration (overall sampling time)	6 min
  Ti	Impulse duration	12.5 s
  Tq	Discretization time	0.5 s
  [dP Fl]	Output: room pressure, supply air flow	trend

  and recordings are presented in Figure 14.

Stage 3, Step 3—recording data for measured disturbances D₂₁, D₂₂

• We repeated the same principle for the exhaust side. The exhaust fan speed was excited by a pseudo-random sequence in the range of 70–90% for gaining the final linear model parts. Similarly to the previous step, we were supposed to capture the response from the exhaust VSD to cleanroom pressure and flow in the role of the second measured disturbance. Experimental data are presented in

  Table 4. Experimental data for system identification Step 3.

  Parameter	Description	Value
  ude	Excitation: 1D pseudo-random signal, exhaust AHU fan speed control signal Form: step impulses of pseudo-random amplitude	85–95%
  Texp	Total experiment duration (overall sampling time)	6 min
  Ti	Impulse duration	12.5 s
  Tq	Discretization time	0.5 s
  [dP Fl]	Output: room pressure, supply air flow	trend

  , recordings are presented in Figure 15.

4.4.4. Stage 4—The Model

To generate the identified model, we used a standard MATLAB System Identification toolbox for 2D identification from the MATLAB/SIMULINK environment version R2021a. We chose the Discrete Time Function Model Structure (Z-domain) as an output form, since this form is by default acceptable by the MPC function in MATLAB. According to Section 3.7.4, we had linearly combined three sets of parameters to reach the model from Equation (6), according to the scenario that was previously described multiple times.

As was a priori assumed, the resulting model of the process under consideration is a low-order strongly coupled function set between the quantities.

$$\begin{gathered} G_{11}={\displaystyle \frac{-0.3696 z^{-1}+0.3528 z^{-2}}{1-1.818 z^{-1}+0.8236 z^{-2}}}{ G}_{12}={\displaystyle \frac{0.3485 z^{-1}-0.3491 z^{-2}}{1-1.933 z^{-1}+0.9331 z^{-2}}} \\ {G}_{21}={\displaystyle \frac{0.8701z^{-1}-0.644 z^{-2}}{1-0.638 z^{-1}+0.1231z^{-2}}}{ G}_{22}={\displaystyle \frac{5.28z^{-1}-3.908z^{-2}}{1-0.8847 z^{-1}+0.2208 z^{-2}}} \\ {D}_{11}={\displaystyle \frac{0.2614 z^{-1}}{1-1.318 z^{-1}+0.3697 z^{-2}}}{ D}_{12}={\displaystyle \frac{2.001 z^{-1}}{1-0.499 z^{-1}+0.4237 z^{-2}}} \\ {D}_{21}={\displaystyle \frac{-0.3188 z^{-1}}{1-0.05282 z^{-1}+0.8575 z^{-2}}}{ D}_{22}={\displaystyle \frac{2.001 z^{-1}}{1-1.859 z^{-1}+0.8629z^{-2}}} \end{gathered}$$

(9)

4.4.5. Stage 5—MPC Tuning

Tuning the MPC is a heuristic process supported by domain knowledge. The weight coefficients to be tuned are exactly those from the cost function given in Equation (7). Namely, in our presented 2 × 2 or 2 × 4 system, we have the option to penalize the input trajectory (weight factor R), the output trajectory (weight factor Q), and the input rate of change (weight factor P). In sum, it is necessary to tune six parameters.

At this point, we need to introduce another real requirement from pharmaceutical plants. From a technological aspect, it is vital to place substantially stronger emphasis on the pressure quantity and consequently on the exhaust damper movement. In other words, to achieve a perfect pressure response, we will sacrifice the flow quality to some minor extent and tune the controller by being aware of this trade-off. This is the reason why we will notice a negligible steady-state shift in the flow reference tracking.

Before tuning the weights, we set the sample time, prediction horizon, and control horizon. Process understanding can provide us with the upfront initial guess regarding sample time (more in the case study). The algorithm, which is also visually presented in Figure 16, is as follows:

• Apply the scaling factor by using the range of the data set for two inputs and two outputs: supply and exhaust movement range, pressure and flow effective range.
• Set the initial sample time to 1 s, the prediction horizon to 15 samples, and the control horizon to two samples.
• Suppress the harsh oscillations and overshoots by substantially penalizing the input rate of change in gradual trial-and-error steps.
• Create stable and overdamped control by increasing the weights for both outputs, in circles.
• Gradually decrease the weights for the pressure output and for the exhaust damper input rate of change, leaving the supply flow weights fixed with regard to the above-described philosophy in order to relax the suppressed response.
• If necessary, decrease the damper input rate of change.
• By trial end error, repeat Steps 3–6 until an acceptable response is achieved.
• Repeat the experiment by changing the sample time and both horizons from Step 2 and make enough iterations of 3–6 until a proper result is achieved: overshoot limits are 5 Pa for pressure, and stabilization time is within 5 s.

Many intermediary results will not be presented in this article, but the general trend has given us several conclusions. Control horizon did not have much of an influence once the transition was captured, which was anything longer than 8–10 s. We accepted the number of 15 s to be conservative. Shorter T_s could have possibly given us better results, but during these experiments, the controller was too aggressive and hard to tune. The most convenient results were obtained using the following parameters from

Table 5. Accepted MPC parameter after tuning.

Parameter	Description	Values
[p11, p22]	du/dt rate weights	[1.6 1]
[q11, q22]	u weights (control variables)	[0.1 0.3]
[r11, r22]	y weights (manipulated variables)	[0.54 1]
Ts	MPC sample time	1 s
Tp	Prediction horizon	15 s
Tc	Control horizon	2 s

.

4.4.6. Stage 6—MPC vs. PID 1 × 1 Response to Reference Tracking, Pressure Only

In the figures throughout this section, we convey the results of highlighted experiments in order to confirm and validate the whole concept given in this article so far.

We begin the presentation of our reference tracking test with the pressure response only. In this test, the experimental data of which are given in

Table 6. Experimental data for MPC 1 × 1 and PID 1xSISO test.

Parameter	Description	Value
dP setpoint	Pressure setpoint range, 1D pseudo-random signal Form: step impulses of pseudo-random amplitude	8 Pa–55 Pa
Texp [MPC; PID]	Duration of the experiment, overall sampling time	[15 min; 35 min]
Ti	Impulse duration	30 s
Tq	Data acquisition sample time	0.5 s
dP	1D output signal—room pressure	trend

, all the other control signals were fixed in manual mode. First, we performed the test with MPC as shown in Figure 17, then compared it with the same challenge driven by the PID controller in Figure 18.

4.4.7. Stage 7—MPC vs. PID 2 × 2 Response to Ref. Tracking, Pressure, and Flow

In the next experiment, with experimental data presented in

Table 7. Experimental data for MPC 2 × 2 and PID 2xSISO test.

Parameter	Description	Value
[dP setpoint; Fl setpoint]	Pressure reference, flow reference, 2D pseudo-random signal Form: step impulses of pseudo-random amplitude	[8 Pa–50 Pa; 900 m3/h–1300 m3/h]
Texp	Duration of the experiment, overall sampling time	20 min
Ti	Impulse duration	60 s
Tsh	Impulse shift between the dP setpoint and the Fl setpoint	30 s
Tq	Data acquisition sample time	0.5 s
[dP Fl]	Output: room pressure, supply air flow	trend

, we challenged the multivariability and coupling by testing the simultaneous response of both room pressure and flow to 2D reference changes. Figure 19 is dedicated to the 2 × 2 MPC response, and Figure 20 to the response of two independent PID SISO loops.

4.4.8. Stage 8—MPC 2 × 2 and 2 × 4 vs. PID 2 × 2, Response to Disturbances, Pressure and Flow to VSD Signals Response

Finally, we tested the MPC response to induced disturbances in the form of VSD variable signals in the situation with experimental data given in

Table 8. Experimental data for VSD disturbances test.

Parameter	Description	Value
[uds; ude]	Input VSD disturbances, 2D pseudo-random signal, supply fan speed, and exhaust fan speed Form: step impulses of pseudo-random amplitude	[80–95%; 80–95%]
Texp [MPC; PID]	Duration of the experiment, overall sampling time	[20 min; 12 min]
Ti	Impulse duration	90 s
Tsh	Impulse shift between the VSD supply and the VSD exhaust signal	45 s
Tq	Data acquisition sample time	0.5 s
[dP Fl]	Output: room pressure, supply air flow	trend

. Recordings are in Figure 21, while its PID 2xSISO counterpart is in Figure 22.

4.5. Discussion of the Results

As a general conclusion, we have proved that MIMO controllers such as 2 × 2 or 2 × 4 MPC are slightly superior to the simultaneous process of two independent SISO PID controllers. The scenario when pressure is changeable only, like the experiments whose results are shown in Figure 17 and Figure 18, is good and acceptable for both PID and MPC, even though in that case MPC has some non-essential advantages. The benefit of MPC is most pronounced when both flow and pressure vary simultaneously, like the experiments presented in Figure 19 and Figure 20, which is the scenario where the multivariable nature of the MPC comes into the spotlight. We noticed a negligible steady-state shift for the air flow in the experiment from Figure 19. Although this is not a perfect response of the MPC, it comes as a result of tuning compromises and technological demand, which was explained earlier: higher weight factors must be put on the quality of the room pressure. The trade-off between the quality of the flow and the pressure responses is in our hands, meaning that we have the freedom to choose the tuning parameters in that sense that we can improve the pressure response for the price of a slight degradation of flow. The additional benefit of the MPC is the response to fan signal fluctuations. As shown, we have used these signals as input disturbances for MPC, which resulted in an indisputably better response than with PID.

We have performed a quantitative analysis to summarize the findings and to have objective insight into the results obtained, which are given in

Table 9. Comparative results of controllers by typical time-domain performance criteria.

	Avg. Rise Time dP [s]	Avg. Overshoot dP [%]	Integral Square Error dP	Avg. Rise Time Fl [s]	Avg. Overshoot Fl [%]	Integral Square Error Fl
MPC 1 × 1 pressure reference change	7.72	13.42	N/A	N/A	N/A	964.63
PID 1 × SISO pressure reference change	14.23	2.28	N/A	N/A	N/A	1628.6
MPC 2 × 2 pressure and flow reference change	9.02	18.55	55.25	8.65	18.57	1304.9
PID 2 × SISO pressure and flow reference change	13.97	8.38	2230.7	19.53	7.76	3086.0

.

The analysis was performed using the following scenario:

• For each control strategy regarding responses to step changes, the periods between the instances of reference step changes were identified, and the responses of the controlled variables dP and Fl within the identified period were treated as the results of separate experiments.
• For each experiment:

  ○

  Rise-time (interval between reaching 10% and 90% of final response) and overshoot were calculated for each variable’s response to its own reference change as the performance indices for separate control loops.

  ○

  Integral-square error was calculated for each variable’s response to reference change in the other controlled variable, as a quantifier of (undesired) inter-loop coupling.

• For each control strategy, performance indices obtained in experiments were averaged and presented in the table.
  Note: Since the intensity of reference step changes varied between experiments, the integral-square errors of the coupling obtained in each experiment were pondered according to a relative reference step change intensity (absolute intensity divided by the total span of applied reference step changes in all experiments conducted for a control strategy) and then averaged.

The table suggests that rising time is in favor of MPC in all presented cases. The same conclusion relates to integral square error. On the other hand, PID control achieves smaller overshoots. Superior MPC performance in reaction quickness and inter-loop coupling rejection surely outweighs high overshoot drawbacks, especially for the standard CRPC setup where references are fixed and the main goal is disturbance rejection. MPC not only handles inter-loop coupling disturbances better than PID, but it also outperforms PID solutions in response to external disturbances, as can be seen in Figure 21 and Figure 22.

Another important note is the fact that unlike the MPC that is run on the Windows PC, PID controllers are built into the Siemens PX PLC program. Namely, PID block calculations do not suffer from additional process delay caused by communication (0.2 s to 0.5 s). This gives them a slight starting advantage when compared to MPC or any other controller from an external computer that is established through OPC and Ethernet.

Finally, the quality of the PID response is strongly dependent on the quality of the tuning. We are underlining that the PID tuning used here is performed in the best possible suboptimal way known to the authors. We used the Lambda method [60,61] and enough trial-and-error rounds to achieve results that cannot be easily challenged by other tuning methods.

5. Discussion and Conclusions

5.1. Discussion on the Chosen Architecture, Potential for Other Control Solutions

It is possible to imagine a spectrum of different control architectures and solutions, but not without raising the questions of practicality. One of them is a full-blown MPC covering the whole (2n + 2) × (2n + 2) at once. We cannot exclude the possibility that future research can prove the feasibility of such an approach. However, the complex method and limitations of physical systems lead us to the presumption that comprehensive identification and unique modeling of such a system would not be feasible. Authors have already had problems with the seemingly simplified identification of our solution, which was additionally pointed out in the case study. It is difficult to presume that, within the very narrow frame of physical-mechanical limitations, the existence of feasible and smooth black-box identification for the full-blown MPC is probable.

When it comes to other potential methods, like decouplers for example, such solutions could be practical and suggested for future research. The authors have chosen the MPC solution as a standard industrial solution that is well-known to both the academic community and the commercial sphere of industrial automation, but we can freely support and suggest further research within a range of different methods.

If we consider alternative hardware platforms and their strengths and limitations in comparison to the system integration we provided, Arduino and Raspberry Pi offer low cost, accessibility, and, particularly in the case of Raspberry Pi, the adequate CPU power for simplified MPC tasks, as demonstrated in academic studies. The benefits of the above-mentioned low-cost platform would also be in integrated control and IO solution, contrary to our distributed system based on a PLC for real-time control and a PC for advanced MPC computation. The authors would be very curious to see it in future research; however, these platforms lack deterministic real-time performance, industrial-grade I/O, safety features, and required certifications. As such, they are not suitable for validated industrial cleanroom environments. Our architecture, on the other hand, ensures reliable execution, compliance, and robust system integration and is applicable, with modifications, to existing factory automation PLC-SCADA systems.

5.2. Discussion on Stability of the MPC Controller in the Cleanroom Environment

The stability of linear MPC in the presence of constraints is well-established in the literature. Within a plethora of works, we can point to some fundamental moments that are covered in [62,63]. In our particular case, stability concerns are not especially unique in that sense. During the experimental investigation, the notion of stability was not analyzed in depth. However, an a posteriori analysis based on the controller’s performance and the applied weighting matrices suggested the following insights. According to literature and theoretical principles, ensuring the stability of an MPC system requires the following conditions:

• The accuracy of the LTI model;
• The inclusion of a terminal cost in the optimization problem that penalizes asymptotic steady-state deviation;
• Careful selection of the constraints.

Specifically, the adopted model was considered as an LTI system, and it was experimentally confirmed that this assumption holds over a broad operating range, as stated in Section 3.5.2. Although our performance index, Equation (7), does not explicitly include a terminal cost, the use of a sufficiently long prediction horizon, registered in Table 5, effectively imitates its impact. The weighting matrices P, Q, and R used in the same cost function, Equation (7), are diagonal and positive definite, which is consistent with the conditions for proper optimization. These preconditions align with the above-given findings in academic literature that address the stability theory of MPC solutions. Additionally, since our implementation was based on the Matlab MPC Toolbox, we relied on the advanced internal design of its algorithms, which guarantee implicit stability.

In summary, since MPC stability is not always supported by explicit criteria, we demonstrated it through practical results and system responses under various conditions. We observed stable multivariable interactions between controlled variables, as well as stable responses to disturbances transmitted through air channels, particularly those originating from VSD regulators. Under these conditions, the system demonstrated fully stable behavior.

On the margins of our experiments, the system was also exposed to severe disturbances that are typical in practical operations within a pharmaceutical facility, such as door openings and personnel movement that affect the process as sudden and intensive disturbances. Although these results are not included in this manuscript, we note that a high level of stability was also achieved in those cases.

Finally, under the given testing conditions, we did not identify a need for the use of a state estimator. The controller was designed based on output-feedback measurements. That is, all system states are also measurable outputs, and no points in the control structure were found where a state estimator would have provided performance improvements.

5.3. Conclusions

In this work, we have presented the conceptual approach of designing the MPC-based strategy for cleanroom pressure cascades. The approach covers the undetachable links in the concept, from technological process characteristics reflected in the transfer function to specific steps in system identification to constructing the series of room-related MPCs for controlling the whole cascade. We have demonstrated how a control strategy can be developed from a generic multiple-order system to a semi-dependent series of low-order, low-dimension controllers with included interactions.

Although MPC has been known for decades now, there is still a plethora of process-industry cases that provide room for its expansion. In our case, as expected, the MPC has shown some progress in comparison to conventional PID control. In spite of the fact that PID controllers perform acceptably well in the industrial circumstances in which they are implemented, the greatest progress has been achieved precisely in the multivariable conceptualization of the problem. This result was expected, since we had some deductive understanding of how quantities are coupled in the process. The question of cost and benefit arises from this approach. Namely, although MPC is certainly better in relation to results, there are additional practical obstacles in terms of its implementation and system integration. It is necessary to solve numerous problems related to system integration first, and second, identification of the black-box process model is always a challenge. While PID controllers are available within any modern PLC as pre-programmed blocks, MPC, on the other hand, or any advanced control strategies that require more programming and computer resources, can only be implemented through higher-level computer control. Plant availability and technical setups in the process industry are rarely adapted for academic research. High-level control using supervisory computers and running advanced algorithms is still in practical and commercial development. The intention of this article was, among other goals, to add one more step to overcoming the gap between the theoretical approach in control theory and practice within real-life systems and industrial process plants.