1. Introduction
An article [
1] published in Nature last year successfully applied deep reinforcement learning to magnetic control of tokamak plasmas, which causes a sensation. Of course, this achievement requires overcoming gaps in capability and infrastructure through scientific and engineering advances; for example, an informed trade-off between simulation accuracy and computational complexity, a highly data-efficient RL algorithm that scales to high-dimensional problems, but not the least of which is an accurate and numerically robust simulator. Unfortunately, such an authentic simulator may not be available in the design process of any industrial control system [
2], considering cost and efficiency, in addition to ubiquitous uncertainties in models. Model uncertainty is an inevitable aspect of industrial process control [
3].
Generally, model uncertainty may be induced by (1) the neglected nonlinearities, (2) the unmodeled dynamics, (3) the neglected or incorrectly modeled external disturbances, and (4) the inescapable measurement error [
4]. Without process uncertainties, there is no need for feedback [
5]. In contrast, we can design an optimal open-loop control law if a precise mathematical model is available. Uncertainties are a key ingredient in process control, so the robustness of a control system is a fundamental requirement in designing any feedback control system. This property reflects an ability to maintain adequate performance and in particular, stability in the face of uncertainties [
6].
One significant and fundamental challenge in process control is the trade-off between the robustness and the performance of the closed-loop system. In particular, the predominant proportional-integral-derivative (PID) is a compromise in the industry process control [
7], which has limited performance and passable robustness [
8]. It gradually cannot satisfy industrial control demands, because of increasingly difficult control tasks and its tuning dilemma [
9]. Contrarily, model-based systematic control theories provide perfect closed-loop performance. For example, in the case of linear systems with full-state information, the full-state feedback control (FSFC) achieves the desired closed-loop system [
10], and the linear quadratic regulator (LQR) approach gives a useful and quantitative optimization solution [
11]. Additionally, the model predictive control (MPC) algorithm is a powerful framework for addressing constrained optimal control problems [
12,
13]. However, the above model-based optimal control techniques suffer from robustness deficiency to model uncertainty [
2,
14]. They are shaky in the industrial uncertain environment, due to their reliance on the absolute fidelity of the model used for control design.
Adaptive control [
15,
16,
17] and robust control [
18,
19,
20] are two important schools of thought to deal with model uncertainty. The goal of adaptive control is real-time control of uncertain parameter systems through an adaptation algorithm online [
21]. However, the adaptive control has a severe lack of robustness in the presence of unmodeled dynamics [
22].
H∞ control and
μ-synthesis are the mainstays of robust control methods. They minimize norm-based sensitivity functions to deal with various uncertainties, and give simple and systematic state-space solutions [
23]. But a key issue, which precludes the industrial application of robust control, is that mainstays like
H∞ and
μ-synthesis generally require accurate prior assumptions about uncertainty structure and size, but hard to know in real time in industrial situations [
24]. Moreover, a severe compromise in the closed-loop performance is needed as a result of conflicts between the robustness and the performance of the closed-loop system. Such conflicts are inherent to the traditional feedback control structure because of the intimate relationship between robustness and closed-loop performance.
There is, in addition, one notable point to make: from an engineering perspective, probabilistic robustness control [
25,
26,
27,
28] is developed, using random analysis and Monte Carlo trial. This method aims to meet the robustness requirement of industrial process control in probability, thus partly reducing the practice difficulty and conservatism of
H∞ control. However, it does not eliminate the inherent conflict and still is a trade-off of aggressiveness versus robustness.
As previously mentioned, model uncertainties of practical industrial processes can severely compromise the resulting control design. Generally, model-based control is rarely utilized in industrial process control because it only satisfies specified closed-loop performance, but no guarantees on robustness are provided. Robust control sacrifices closed-loop performance to overcome the robustness challenges. Thus, this article explores an effective control scheme that simultaneously guarantees closed-loop performance and robustness.
Statement of Contributions: In this article, we present a generalized conditional feedback (GCF) system for controlling industrial processes with model uncertainty. The proposed GCF scheme is defined by a control problem that leverages a nominal model and an ancillary feedback controller. Theoretical guarantees on the performance robustness of the closed-loop system and its relationship with conditional feedback (CF) are analyzed. An effective practice procedure is also provided. Furthermore, simulation experiments on six typical industrial processes and a physical half-quadrotor system control test are carried out. The main contributions are summarized as follows:
- (1)
A GCF scheme is proposed to control industrial processes with model uncertainty that simultaneously guarantees closed-loop performance and robustness.
- (2)
The effectiveness of the proposed GCF scheme is validated by case studies and a half-quadrotor system control test.
Organization: In
Section 2, the control problem is defined.
Section 3 introduces the basic idea and structure of the proposed GCF scheme, and then theoretical guarantees on the performance robustness of the closed-loop system and its relationship with CF are analyzed. An effective practice procedure is also provided.
Section 4 is dedicated to demonstrating the effectiveness of the GCF scheme through case studies of six processes and three model-based control methods. In addition, a half-quadrotor system control experiment is presented in
Section 5. Finally,
Section 6 concludes this article.
3. Generalized Conditional Feedback
In this work, the NM (5) is leveraged, not only in the simulation design stage (offline) but also in the industrial application stage (online), to design an efficient generalized conditional feedback (GCF) scheme that can simultaneously guarantee closed-loop performance and robustness.
3.1. Control Algorithm
The GCF scheme consists of the virtual domain and the deviation correction part. In the virtual domain, a primary controller is designed to optimize the trajectory of the virtual NM, depicted as,
where
K0 denotes the controller designed in the virtual domain, and the constraint set
Z0 is a tightened version of the original constraint set (3) such that
Z0 ⊆
Z. The tightened constraint is used to ensure performance robustness and is defined in
Section 3.3. Assumption 2 guarantees the performance of the controller
K0.
In the deviation correction part, another ancillary controller is designed to drive the physical process to track the trajectory of the virtual domain, depicted as,
where
K1 denotes the ancillary correction controller, also designed based on the only known NM, and
ts is the specified time scale, such that the deviation correction controller efficiently drives the physical process to track the trajectory of the virtual domain. Assumption 1 guarantees that such an ancillary correction controller
K1 can be designed.
For additional clarity, the architecture of the proposed GCF control scheme is shown in
Figure 1. This diagram highlights the following facts:
- I.
There are two systems being controlled: the NM (5) is virtual, and the controlled process (1) is physical.
- II.
In the virtual domain, the trajectory of the virtual NM is optimized to be the desired state of the physical process.
- III.
The two systems are connected only by the deviation correction controller, which essentially tries to drive the physical process to track the desired trajectory coming from the virtual domain.
Moreover, comprehensive explanations of the GCF scheme are summarized below:
- I.
The controller in the virtual domain can be any feasible control strategy that is capable of optimizing NM to a specified state.
- II.
The deviation correction controller (e.g., PID, PIDD
2 [
31], DDE [
32], ADRC [
33]) should have strong performance robustness to efficiently drive the uncertain process to track the trajectory of the virtual domain.
- III.
The stability and the optimization are unified. The controller in the virtual domain ensures optimization and the ancillary correction controller guarantees stability.
3.2. Conditional Feedback
Conditional feedback (CF) is proposed to enable a decoupled input–output response and disturbance–output response [
34]. A basic configuration for a linear CF system is shown in
Figure 2, where
Gp denotes the controlled process,
GT denotes the tracking controller,
GD denotes the disturbance rejection controller, and
r,
d, and
y are the setpoint, the external disturbances, and the process output, respectively. From the system shown, we learn that the accurate model is needed for CF system design, and the Laplace transform
Y of the output
y is
which shows that the input–output response is completely determined by the tracking controller
GT, and the feedback controller
GD acts solely to reject disturbance.
The block diagram of the proposed GCF scheme is similar to that of CF. In the uncertainty-free case where q = {0}, the optimal controller in the virtual domain of GCF can be designed as K0 = GT Gp−1, such that GCF also permits designing an input–output response and disturbance–output response independently. In the view of improving performance robustness, CF can be regarded as an example of GCF acting for inverse system control.
However, their original intentions are different. GCF aims to simultaneously guarantee closed-loop performance and robustness for industrial processes with model uncertainty, while CF focuses on removing conflict between the input–output response and disturbance–output response under the assumption of no model uncertainty. Moreover, CF was proposed based on the classical transfer function method, but GCF is open to the well-developed modern control theory and booming artificial intelligence trend.
3.3. Closed-loop Performance Robustness
In the context of this work, a property is considered performance robustness if it holds control constraints (3) in the presence of norm-bounded model uncertainty. Recall that the virtual domain optimizes the trajectory of the simulated NM under the constraint
z0 ∈
Z0. The deviation correction controller then drives the physical process to track this trajectory. Unfortunately, perfect tracking is impossible because of model uncertainty. Thus, choosing
Z0 =
Z will not guarantee performance constraint satisfaction for the physical process. In industrial practice, a tightened version of the original constraint (3) is chosen, as represented in
Figure 3.
Of particular interest is how we set
Z0 to guarantee performance constraint satisfaction. Defining the error
δy =
y −
y0 and
δu =
u −
u0, it follows that
A performance error is defined as
where
H(·) is the defined performance error function that denotes the performance variable of the deviation correction controller driving the uncertain process.
Since the considered model is norm-bounded, it follows that
which means the performance error is bounded by the worst-case bound Δ
z. The value of Δ
z depends on the performance robustness of the deviation correction controller and model uncertainty. Then, the constraint set
Z0 of the virtual domain can be designed as
which is a tightened version of
Z. With this tightened performance constraint set, the nominal trajectory optimized by the virtual domain will account for the tracking error and ensure performance constraint satisfaction.
Performance Robustness. Suppose that the deviation correction controller is performance robust. Then, under the proposed control structure (GCF), the uncertain process (1) will robustly satisfy the performance constraints (3). The closed-loop performance robustness of GCF depends on the performance robustness of the deviation correction controller.
The requirement that the deviation correction controller is performance robust is natural and is satisfied by properly choosing and tuning a non-model-based controller.
3.4. Practice Procedure
The practice procedure of the GCF scheme is now summarized as Algorithm 1. In the online portion, it is suggested that the deviation correction controller is used as a “startup” controller to satisfy basic requirements (e.g., stability, safety). Then, GCF takes over to drive the physical process to guarantee performance constraint satisfaction.
Algorithm 1: Practice procedure of the GCF scheme. |
1: system identification (offline) |
get a nominal model |
2: simulation design (offline) |
K0 ← Equation (6) |
K1 ← Equation (7) |
3: practice (online) |
startup control |
u ← K1 (r, y, u) |
u0 ← u |
then |
u0 ← K0 (r, y0, u0) |
u1 ← K1 (y, y0, u0, u1) |
u ← u0 + u1 |
Additionally, when the “startup” controller is applied to the physical process, the virtual domain is in a tracking stage, such that the simulated NM is controlled by
which ensures a reasonable initial condition when the GCF scheme takes control of the physical process.
4. Simulation Illustration
In this section, several model-based control methods are computed as illustrative examples, based on MATLAB R2023a. Please note that PID control tuned by the Skogestad internal model control (SIMC-PID) [
35] is selected as the deviation correction controller in all simulation experiments, expressed as
Six typical industrial processes are depicted in
Table 1 [
27] and three model-based control methods, namely full-state feedback control (FSFC), linear quadratic regulator (LQR), and model prediction control (MPC), are simulated to illustrate the effectiveness of the proposed GCF scheme.
The concern of simulation experiments is the tracking performance robustness. First, norm-bounded model uncertainties of six typical industrial processes in the simulation experiments are assumed in
Table 2. All tuned controller parameters are listed in
Table 3. In particular, the details of the controller design are explained as follows:
- I.
State observers are designed when FSFC and LRQ are applied to uncertain models. The observer estimation speed is selected to be 3~5 times the closed-loop response.
- II.
For processes, G1(s)~G4(s), the state-space models are all expressed as the second controllable canonical form.
- III.
The pole placements and the cost functions are listed in
Table 4.
- IV.
For the time-delay process,
G5(
s), the standard Smith predictor [
36] is used.
Table 4.
Placed poles and cost functions.
Table 4.
Placed poles and cost functions.
Control Methods | Processes | Placed Poles |
---|
Closed-Loop | State Observer |
---|
FSFC | G1(s) | [−1+j, −1−j, −1 −2] | [−2+2j, −2−2j, −2 −4] |
G2(s) | [−7+5j, −7−5j, −10] | [−15+10j, −15−10j, −15] |
LQR | G3(s) | Cost function: | [−8+4j, −8−4j] |
G4(s) | [−10+5j, −10−5j] |
Tracking responses and Monte Carlo trials [
37] are carried out to quantify the performance robustness of original control methods and GCF schemes. The statistical results are summarized in
Table 5. Obviously, the control scheme, designed based on NM, cannot make the uncertain process behave as expected. Nevertheless, the orange part is closer to the red line than the cyanic part, which means the GCF schemes try to buffer the actual response against model uncertainties. Moreover, in the Monte Carlo trials, the performance indices of GCFs are more clustered and closer to zero than those of the original control methods. It also statistically proves the improvement of the performance robustness with the proposed GCF scheme. Consequently, the effectiveness of the proposed GCF scheme is illustrated for model uncertainty.
6. Conclusions
In this article, a GCF scheme is proposed for controlling industrial processes with model uncertainties. Its basic concept and practical implications are elaborated. The approach leverages nominal models and defines an ancillary feedback controller to guarantee closed-loop performance constraints and robustness simultaneously. This scheme is open, such that based on a nominal model, any existing optimal control theory can be designed in the virtual domain, and any robust control algorithm is used as an ancillary feedback controller to drive the physical process to track the trajectory of the virtual domain. The effectiveness of the proposed GCF scheme is validated by numerous case studies and a half-quadrotor system control test.
Future work: There are some additional considerations in terms of both theoretical and practical significance related to this work. First, a further theoretical analysis is necessary. Second, the optimality of an ancillary feedback controller and uncertainty size could be considered. Third, under physical constraints, such as actuator constraints, the limit of the controller in the virtual domain should be considered. Fourth, extensions to reinforcement learning control or digital-twin-enabled smart control are of significant interest.