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Concepts

When we talk about model predictive control (MPC), we should know that MPC has other names, e.g., receding horizon control (RHC). What are the differences between the two names? It is usually called MPC in the industrial circle. When we apply the state space paradigm to study MPC with stability guarantee, it is sometimes called RHC for emphasizing the feature of receding-horizon optimization. On the application aspects, besides receding-horizon optimization, MPC has other features, such as model-based prediction and feedback correction. If MPC has no feedback correction, and its prediction is naturally obtained from the model, it is named RHC. Thus, RHC is often used in academic/theoretical research studies.

1.1 PID and Model Predictive Control

It is said that in industry, more than of automatic control loops are utilizing proportional integral derivative (PID). Someone says that this number should be or even . The percentage cannot be very authoritative. PID control strategy is widely used not only in civil industry but also in aerospace, military, and electronic mechanical devices. The use of PID is shown in Figure 1.1. Figure 1.1 has PID, the actuator, the controlled process (controlled device) {plant}, the controlled output {}, sp (the setpoint) of controlled output {}, a plus sign and a minus sign. The measured output feedback. A measurement (meter) block can be added. However, for theoretical research studies, it assumes including the measurement (meter) in the plant.

For many factories, it is optimistic to apply PID for above , since many actuators are manually operated where PID is not operable. The manual operation of actuators is shown in Figure 1.2. Since there are many PIDs, s is added, and both and are vectors.

What is the situation for MPC? According to the statistics, as compared with PID, MPC occupies about 10-15% of automatic control loops in the process control. We should not count based on the upper bounds (PID , MPC ), since other control strategies (different from PID and MPC) misleadingly seem useless. In many factories, 80-90% of actuators are manually operated, i.e., with neither PID nor MPC. Let us concern a modern factory with high-level automation and immense courage to accept advanced control strategies like MPC; according to the statistics in these factories, PID occupies approximately and MPC approximately , while other control strategies are definitely non-mainstream.

Figure 1.1 Control system using PID.

Figure 1.2 Control system using PID+manual.

Figure 1.3 Control system using MPC based on PID.

How does MPC play its role? Sometimes there is misunderstanding. Based on Figure 1.1, MPC acts as in Figure 1.3. In MPC, . MPC lies before PID. is in front of MPC, which is called ss (the steady-state target) of , i.e., sp of MPC. The measurement of is sent to MPC.

In the process control, the controllable input is called manipulated variable (MV); the controlled output is called controlled variable (CV); the measurable disturbance is called disturbance variable (DV), sometimes called the feedforward variable. These names are conventional in the industrial MPC.

Can the "manual" of Figure 1.2 become automatic? If MPC is well-applied and "manual" is also given by MPC, then of MPC includes both and manual, as shown in 1.4. Before applying MPC, "manual" implies that the operator directly operates the valves; PID implies using PID algorithm to manipulate the valve, and the operator has to operate PID sp. By applying MPC, MPC manipulates both valve and PID sp. MPC primarily manipulates PID sp, and secondly directly manipulates some valves.

Figure 1.4 is not a general situation. In practice, some projects cannot utilize MPC on all PID sps (setpoints), i.e., some PID sps are still manually adjusted, as shown in Figure 1.5. Some PID sps are manipulated by MPC, denoted as ; the other PID sps, denoted as , are not manipulated by MPC.

Figure 1.4 Control system manipulating "manual" by MPC.

Figure 1.5 Control system using MPC+PID.

Figure 1.6 Control system using MPC+PID+manual.

Figure 1.5 is still not a general situation. Some "manuals" may not be manipulated by MPC, as shown in Figure 1.6. Applying MPC for all "manuals" represents a high level of control, but all projects are not achievable.

In industry, all of MPC are not actuator positions; some are the actuator positions, and the others are PID sps. There were misunderstandings about this fact.

What does the controlled object of MPC become? It is shown in the dashed box in Figure 1.7. Hence, applying MPC to a real system requires establishing a mathematical model of the object in the dashed box rather than merely building the "plant" model. The model ready for MPC must take PID into account, i.e., the model includes the role of PID. The "manual" in Figure 1.7 becomes DV of MPC, and so is .

Figure 1.7 In the control system using MPC+PID+manual, a controlled object of MPC is in dashed box.

Figure 1.8 Control system using MPC.

Suppose the model in the dashed box is obtained, including the portions for both DV-to-CV and MV-to-CV. In the literature, most researchers are concerned, given , studying

1. (1) how to optimize , i.e., the algorithms, which were dominating in the 1970s and 1980s;
2. (2) whether or not the sequence ( from 0 to ) converges, i.e., stability, which is dominating in the academic theory;
3. (3) whether or not , i.e., the offset-free, which is not mainstream but has some papers.

In the 1990s, there were many mature results in stability. The offset-free is only valid after assuming stability. It seems that the research studies on offset-free has less patterns than on stability.

Let us abbreviate the controlled object of MPC (often referred to as a generalized object in process control), in the dashed box in Figure 1.7, as PLANT wich is capitalized. Then, Figure 1.7 reduces to Figure 1.8. Thus, all three types of studies, mentioned earlier, are for Figure 1.8.

1.2 Two-Layered Model Predictive Control

Are the aforementioned three types of studies closely consistent with state-of-the- art industrial applications? The answer is negative. There are more issues to tackle. When MPC is applied, as in Figure 1.8, it is only called dynamic control (DC) or dynamic tracking, or often, dynamic move calculation in industrial software.

The biggest issue is where comes from. Before using MPC, both valve and PID sp are manually operated. By applying MPC, if is again manually operated, can we have high enough knowledge for well-operating? If we cannot well-operate PID sps, we might not gain big benefits by operating in Figure 1.8. If obtaining is not automatic, it requires a lot of operation experiences and a high-level engineer in order to enhance MPC efficiency. Hence, automating the calculation of is a key to simplifying MPC operation.

Let us take an example, where "PLANT" takes a simple form, i.e., the transfer function model . According to the final-value theorem, we obtain , where is the steady-state gain matrix. In applying PID, for every being controlled, there must be a sp. MPC manipulates not only PID sps, but also some valves. Imagine that the numbers of MVs and CVs may be unequal. When they are unequal, for any , is there satisfying ? By setting a arbitrarily, does MPC necessarily drive CV to ? Obviously not. Looking at the equation with as the unknown, it unnecessarily has a solution for any . Uniqueness of the solution for this equation is rare. In most cases, either there is no solution, or there are infinitely many solutions. For the industrial applications, MPC should automatically calculate not only , but also . The term compatibility refers to, with the given , whether or not there is a solution . The term uniqueness refers to, with the given , whether or not there is a unique solution . In the case of multiple solutions, it should be given the principle to choose .

In industrial operations, it is evident that both and may be related to economy. Any variable related to economy could be involved in an optimization. In MPC, economy is a broad concept; it may include, e.g., increasing money, reducing energy consumption, reducing exhaust gas emission, and reducing pollutants.

Figure 1.8 can be modified. Besides , there is , satisfying . For any , it may fail to find . For any , there is a unique as long as is a real matrix. Since PLANT is , a failure to satisfy brings troubles. The small trouble could be steady-state error (non-offset-free), and the big could be dynamic instability. can easily cause dynamic instability.

In summary, it is important to automatically calculate . MPC in Figure 1.8 is renamed as DC. In order to take out a set of for DC, it needs a so-called steady-state target calculation (SSTC) in front of DC, as shown in Figure...