Keywords

1 Introduction

Manufacturing process real-time optimization (MPRTO) [1] is a concept, that even if it can be considered as well-defined, the emerging Industry 4.0 Key Enabling Technologies have helped in its reappearance into spotlight. Also, it is a fact that the documentation of the respective challenges and aspects that need to be addressed has started some years ago (in 1995) [2], linking the manufacturing process control requirements to the implementation (i.e. communication protocols) and the application itself (manufacturing process addressed).

Furthermore, it can be easily found that it is closely related to classical control theory; if one considers a dynamic (linear) system S of state x, its exact controllability or complete controllability [3] can be defined as follows: “The system S is controllable on an interval [t0, t1] if ∀ x0, x1 ∈ Rn, ∃ controllable function u ∈ L2([t0, t1]: Rm) such that the corresponding state space satisfying x(t0) = 0 also satisfies x(t1) = x1”. This statement means that we can define the behavior of the system, at least for the second time point.

An additional concept that is highly useful, is that of structural controllability [4, 5], implying that it is the physical structure of the dynamic system, and not its occasional parameter that define the controllability. It is worth noting that these two concepts cannot directly be applied in the manufacturing process area, as this will be shown below.

After having defined the concept of control, a wide area of branches can be defined, such as static controllers, PID controllers, adaptive controllers [6] in the case where the system is unknown, empirical controllers [7] that bypass theoretical calibration, even robust controllers [8] that address the case of systems with uncertainties, either structural or functional. Furthermore, there are secondary types, like the Fuzzy Control which can extend other concepts (i.e. the PID concept [9]).

Modelling plays an important role in the application of this. As shown in literature [8, 10], the uncertainties or the non-linearities can affect the model as well as the controller that will have to be chosen. Also, the manufacturing processes modelling is not straight-forward, as the criteria (i.e. quality) can affect the choice of the I/O model, requiring the definition of the so-called Intermediate Variables [10]. These primarily constitute the first challenges come across in the MPRTO procedure.

Another useful concept can be the stochastic control (i.e. through polytopes) [11], that must not be confused with statistical control. Optimal control [12], next, can be used to address the various extra (manufacturing) criteria, such as energy efficiency, while the practicality and the generality can be addressed through some intelligent control approach, such as neural modelling, or even neural control [13, 14].

Multiple-Input-Multiple-Output (MIMO) control from the classical control theory [15] can be also highly useful in this attempt to address manufacturing process control. This consecutive integration of definitions could be summarized by the depiction of Fig. 1. This is a (non-unique) approach, aiming at describing the digital-twin (DT) based manufacturing process control.

Fig. 1.
figure 1

Framework integrating control theory into DT-driven MPRTO.

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Moreover, additional issues, such as security/safety and certification [16] and cloud services could be added in the requirements, leading to the depiction of Fig. 2. These are side-challenges that may appear in the procedure. Additionally, digital twins, being the result of extensive integration [17, 18] are in any case umbrellas of various technologies [19], exploiting to a large extent Artificial Intelligence techniques, among others, so that they are able to handle the data management. The complexity management, thus, may also hinder the modelling and the controlling procedures.

Thus, after reviewing the potential challenges set by the above mentioned technologies, the practicality of control as real-time optimization should be investigated. This paper introduces some concepts expected to be highly useful as a future reference for digital twin design frameworks (due to small execution time). In addition, the MPRTO will be benefited from the integration of control theory techniques (Figs. 1 and 2).

Fig. 2.
figure 2

Additional requirements for digital twin driven MPRTO.

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2 Method for Integrating Manufacturing Process Control in DTs

2.1 Input-Output Approach: Adopting a Meta-modelling Framework

Modelling for application of control theory requires the existence of a dynamic system. This is also the case with manufacturing process control, except that the I/O definition is not always straight-forward; metamodelling can be of high importance, since the knowledge about the process can help defining the Process Parameters, the Performance Indicators, as well as Key Variables that are implicated in optimizing a manufacturing process [10]. As there is quite some variability in process models, this is a expected to be a good policy, as it reduces complexity. Thus, as indicated in Fig. 3, the application and the criteria define the Input Key and Output Key variables. However, this may not be enough. The reference values have to be defined. This may have a two-fold implication in terms of objectives: either to perform tracking on a signal, or for the output to have specific features (i.e. overshoot for quality or time scale for process time, etc.). The latter renders the procedure a rather complicated one.

It is crucial for a digital twin to be able to manipulate such information, thus making tacit knowledge on aspects quality or any other manufacturing attribute relevant. In addition, the interpretation of such knowledge into information on the selection of Input Key and Output Key Variables is essential to the whole procedure. Setting the criteria also requires the existence of a specific database, while the communication in real-time needs to handle the monitoring signals as well as the control signals. Finally, the extraction of knowledge to assist explicability is an extra criterion, however, this exceeds the purposes of the current work. It is noted, in any case, that linearization is the primary route to control. Generic non-linear approaches exceed the purposes of this work.

Fig. 3.
figure 3

An Input-Output model of a closed loop MPRTO.

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2.2 Theoretical Definitions Facilitating DT-Driven Process Control

The first type of control that would need to be discussed under the concept of manufacturing processes control, would be the Direct Process Control. This is particularly for the case of operation in manufacturing. In this case, under classical control theory, given an integro-differential operator T, an output key variable (signal) y and an input key variable u, without loss of generality, the following systemic description stands.

$$ Ty = x $$
(1)

It is noted that the operator T can include either a first-order temporal derivative (e.g. in the linear area of laser processes) or a second-order temporal derivative (e.g. in the elastic region in the case of mechanical processes).

In any case, direct control, given a desired output y0, would be to select an input x0 = Ty0. However, this is not always applicable, since it implies the use of fields x and y as functions of space. An alternative would be what could be defined as Boundary Direct Control, taking into account the boundary conditions of the problem, e.g. in the case of laser incidence on metals, which is modeled as flux [7]. A mixed type could also be defined, given the fact that in some cases, the hybrid triggering of the field through both types may be needed. An example would be Beer-Lambert penetration of photons due to material-laser combination (i.e. CO2 of 10.6um wavelength vs. SiO2/Al2O3 material [20]) or due to scattering in powder [21].

Additional requirements could be added on these types of control, leading to Restricted Control, extending the Empirical Control that has been aforementioned. In this case, the key input variable may be subject to extra requirements, such as the positivity, and since formulation may not be applicable or the feasibility of the optimization problem may not be always the case, Heuristic Restricted control may be needed. An example for this case is also given in Sect. 3.1 below.

An additional definition that would be highly relevant in the case of process design is given herein. Given the structural controllability presented in the introduction, it is quite normal to assume that the system structure cannot always originate from a systemic approach, due to the distributed (in space) character of the processes.

Subsequently, in lumped approaches, such as in the case of mechatronics, the structural controllability is of high relevance. To this end, we are looking for the minimum type of system invasions (Δ1, Δ2,) so that the final system (A, B, C, D) is (state) controllable (Eq. 2, given that the ρ operator denotes the rank of a matrix and that n is the dimension of square matrix A).

(2)

In addition, it would be highly useful to connect this to graphs annotation, so that the network interventions are also visualized. This could be defined as invasive control.

3 Numerical Results and Discussion

3.1 Application of DT-Driven Control Integrating New Concepts

The current application is related to enforcing direct process control in a simple process. The term simple implies the consideration of a linear area (i.e. Equation 4), while a specific spatio-temporal temperature profile has to be tracked (Eq. 3 describing heating up). There is a first-order temporal derivative, implying an adoption of thermal processes. Figure 4 depicts the outcome of the direct control application, as well as a potential experimental configuration; a metal sheet (blue surface) heated by a distribution of heating/cooling elements (yellow surfaces).

$$ \nabla^2 T + Q = k\frac{\partial T}{{\partial t}} $$
(3)
$$ T = 500e^{ - x^2 } \left( {0.2y + 1} \right)\frac{1}{t^2 + 1} $$
(4)
Fig. 4.
figure 4

Application of direct manufacturing process control: a corresponding experimental setup (planar surfaces), the desired spatial profile (left) and the flux required (right) at a given time.

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Next, it would be quite interesting to also apply empirical control. In the case below, the temporal profile of Eq. 5 needs to be tracked for the model of Eq. 6 (an ordinary differential equation). The experimental configuration corresponding to this case could be 1D motion control of a machine tool. The requirement is that the input needs to be pulsed (restriction of the motor). After some trial and error (manually), this is achieved, as shown in Fig. 5. The oscillation is imposed by the dynamics of the equation, which is a crucial limitation of control.

$$T=3t{e}^{-t}$$
(5)
(6)
Fig. 5.
figure 5

Empirical direct control: Control performance (left) and input signal (right).

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In addition, one could apply also restricted control. This would address machine-related restrictions, on top of aforementioned process-related ones, such as the very dynamics discussed above. For the model of Eq. 6, the control has to fulfil the approximate criterion of Eq. 7, where input has been assumed to be positive, as an extra restriction (\({q}_{n}^{2}\)>0), using a specific set of pulses, denoted in Eq. 8 and 9 (duration of pulse is td and H being the Heaviside function). Results are shown in Fig. 6.

(7)
$$p(t,{t}_{0})=\left(H\left(t-{t}_{0}\right)-H\left(t-{t}_{0}-{t}_{d}\right)\right){Sin}^{0.1}(2\pi \frac{t-t0}{2{t}_{d}})$$
(8)
$${u}_{c}\left(t\right)={\sum}_{n=0}^{{N}_{d}}\left|{q}_{n+1}\right|p(t,(n+1){t}_{d})$$
(9)
Fig. 6.
figure 6

Empirical restricted control applied.

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3.2 Repercussions on the Digital Twins’ Workflows

Empirical models and thus empirical manufacturing process control (or equivalently MPRTO based on closed loop schemas) have been proved to be inherently useful for many cases of manufacturing process control, however, the digital twin architecture that is adopted needs to be able to support all this. Thus, it can be claimed that human-in-the-loop optimization of manufacturing processes modelling and control or AI-based digital twins’ workflows are of high importance towards achieving such operations. To this end, and to address robust manufacturing, it would be crucial to integrate monitoring capabilities towards identifying the actual thermal model of the part manufactured. This implies taking into account uncertainties in modelling, like temperature dependencies, or spatial variations due to material characteristics. This could be addressed by profiling the surficial temperature distribution at different part planes, and extracting the internal status of the part; implying further integration of AI in the digital twin.

4 Conclusions and Future Outlook

Digital twins are quite hard to be implemented, with control theory interpreted into MPRTO being an integral part of their set of operations. As such, the empirical use of process control needs take into account several aspects. At the same time, the application of empirical process control is promising for a variety of applications, especially in cases of low digital maturity. This can be integrated either in the form of trial and error (automatically or using a human-in-the-loop scenario), or through a specific optimization procedure.

In addition, in the case of designing, invasive control can be a useful definition. This would need, in any case, further elaboration, before suggesting a framework that is implementable. With respect to future outlook, specific data driven models should be elaborated for testing theoretically and practically the use of manufacturing process control, seemingly requiring a long series of tests, for a variety of processes and configurations, as well as AI-based monitoring techniques elaboration, as aforementioned.