Model Predictive Control
Description
Understand the practical side of controlling industrial processes
Model Predictive Control (MPC) is a method for controlling a process according to given parameters, derived in many cases from empirical models. It has been widely applied in industrial units to increase revenue and promoting sustainability. Systematic overviews of this subject, however, are rare, and few draw on direct experience in industrial settings.
Assuming basic knowledge of the relevant mathematical and algebraic modeling techniques, the book's title combines foundational theories of MPC with a thorough sense of its practical applications in an industrial context. The result is a presentation uniquely suited to rapid incorporation in an industrial workplace.
Model Predictive Control readers will also find:
* Two-part organization to balance theory and applications
* Selection of topics directly driven by industrial demand
* An author with decades of experience in both teaching and industrial practice
This book is ideal for industrial control engineers and researchers looking to understand MPC technology, as well as advanced undergraduate and graduate students studying predictive control and related subjects.
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Persons
Baocang Ding, PhD, teaches MPC to both undergraduate and graduate students in the School of Automation, Chongqing University of Posts and Telecommunications, China. His research interests include model predictive control, control of power network, process control, and control software development.
Yuanqing Yang, PhD, teaches MPC to both undergraduate and graduate students in the School of Automation, Chongqing University of Posts and Telecommunications, China. His research interests include model predictive control, fuzzy control, networked control, and distributed control systems.
Content
About the Authors xi
Preface xiii
Acronyms xv
Introduction xvii
1 Concepts 1
1.1 PID and Model Predictive Control 1
1.2 Two-Layered Model Predictive Control 4
1.3 Hierarchical Model Predictive Control 7
2 Parameter Estimation and Output Prediction 11
2.1 Test Signal for Model Identification 11
2.1.1 Step Test 11
2.1.2 White Noise 11
2.1.3 Pseudo-Random Binary Sequence 13
2.1.4 Generalized Binary Noise 14
2.2 Step Response Model Identification 15
2.2.1 Model 15
2.2.2 Data Processing 17
2.2.2.1 Marking or Interpolation of Bad Data 17
2.2.2.2 Smoothing Data 18
2.2.3 Model Identification 19
2.2.3.1 Case Grouping 19
2.2.3.2 Cased Data Preparation for Stable Dependent Variables 19
2.2.3.3 Cased Data Preparation for Integral Dependent Variables 21
2.2.3.4 Least Square Solution to Parameter Regression 22
2.2.3.5 Least Square Solution by SVD Decomposition 24
2.2.3.6 Filtering Pulse Response Coefficients 24
2.2.4 Numerical Example 27
2.3 Prediction Based on Step Response Model and Kalman Filter 30
2.3.1 Steady-State Kalman Filter and Predictor 31
2.3.2 Steady-State Kalman Filter and Predictor Based on Step Response Model 32
2.3.2.1 Open-Loop Prediction of Stable CV 33
2.3.2.2 Open-Loop Prediction of Integral CV 36
3 Steady-State Target Calculation 39
3.1 RTO and External Target 39
3.2 Economic Optimization and Target Tracking Problem 40
3.2.1 Economic Optimization 41
3.2.1.1 Optimization Problem 41
3.2.1.2 Minimum-Move Problem 42
3.2.2 Target Tracking Problem 46
3.3 Judging Feasibility and Adjusting Soft Constraint 46
3.3.1 Weight Method 47
3.3.1.1 An Illustrative Example 47
3.3.1.2 Weight Method 50
3.3.2 Priority-Rank Method 51
3.3.2.1 Ascending-Number Method 52
3.3.2.2 Descending-Number Method 52
3.3.3 Compromise Between Adjusting Soft Constraints and Economic Optimization 55
4 Two-Layered DMC for Stable Processes 57
4.1 Open-Loop Prediction Module 59
4.2 Steady-State Target Calculation Module 61
4.2.1 Hard and Soft Constraints 61
4.2.2 Priority Rank of Soft Constraints 63
4.2.3 Feasibility Stage 64
4.2.4 Economic Stage 66
4.3 Dynamic Calculation Module 67
4.4 Numerical Example 70
5 Two-Layered DMC for Stable and Integral Processes 73
5.1 Open-Loop Prediction Module 74
5.2 Steady-State Target Calculation Module 77
5.2.1 Hard and Soft Constraints 78
5.2.2 Priority Rank of Soft Constraints 80
5.2.3 Feasibility Stage 81
5.2.4 Economic Stage 83
5.3 Dynamic Calculation Module 85
5.4 Numerical Example 87
6 Two-Layered DMC for State-Space Model 95
6.1 Artificial Disturbance Model 95
6.1.1 Basic Model 96
6.1.2 Controlled Variable as Additional State 97
6.1.3 Manipulated Variable as Additional State 98
6.1.4 Kalman Filter 100
6.2 Open-Loop Prediction Module 103
6.3 Steady-State Target Calculation Module 104
6.3.1 Constraints on Steady-State Perturbation Increment 104
6.3.2 Feasibility Stage 106
6.3.3 Economic Stage Without Soft Constraint 107
6.4 Dynamic Calculation Module 108
6.5 Numerical Example 110
7 Offset-Free, Nonlinearity and Variable Structure in Two-Layered MPC 115
7.1 State Space Steady-State Target Calculation with Target Tracking 115
7.1.1 Case all External Targets Having Equal Importance 117
7.1.2 Case CV External Target Being More Important Than MV External Target 117
7.2 QP-Based Dynamic Control and Offset-Free 119
7.3 Static Nonlinear Transformation 125
7.3.1 Principle of Nonlinear Transformation 125
7.3.2 Usual Nonlinear Transformations 127
7.3.2.1 Nonlinear Transformation of Valve Output 127
7.3.2.2 Piecewise Linear Transformation 128
7.4 Two-Layered MPC with Varying Degree of Freedom 129
7.4.1 Numerical Example Without Varying Structure 130
7.4.2 Numerical Example with Varying Number of Manipulated Variables 131
7.5 Numerical Example with Output Collinearity 135
8 Two-Step Model Predictive Control for Hammerstein Model 141
8.1 Two-Step State Feedback MPC 142
8.2 Stability of Two-Step State Feedback MPC 144
8.3 Region of Attraction for Two-Step MPC: Semi-Global Stability 147
8.3.1 System Matrix Having No Eigenvalue Outside of Unit Circle 147
8.3.2 System Matrix Having Eigenvalues Outside of Unit Circle 149
8.3.3 Numerical Example 150
8.4 Two-Step Output Feedback Model Predictive Control 153
8.5 Generalized Predictive Control: Basics 159
8.5.1 Output Prediction 159
8.5.2 Receding Horizon Optimization 161
8.5.3 Dead-Beat Property of Generalized Predictive Control 164
8.5.4 On-line Identification and Feedback Correction 167
8.6 Two-Step Generalized Predictive Control 167
8.6.1 Unconstrained Algorithm 168
8.6.2 Algorithm with Input Saturation 168
8.6.3 Stability Results Based on Popov's Theorem 170
8.7 Region of Attraction for Two-Step Generalized Predictive Control 173
8.7.1 State Space Description 173
8.7.2 Stability with Region of Attraction 174
8.7.3 Computation of Region of Attraction 175
8.7.4 Numerical Example 177
9 Heuristic Model Predictive Control for LPV Model 179
9.1 A Heuristic Approach Based on Open-Loop Optimization 180
9.2 Open-Loop MPC for Unmeasurable State 186
10 Robust Model Predictive Control 195
10.1 A Cornerstone Method 195
10.1.1 KBM Formula 195
10.1.2 KBM Controller 197
10.1.3 Example: Generalizing to Networked Control 199
10.1.3.1 Closed-Loop Model for Double-Sided, Finite-Bounded, Arbitrary Packet Loss 199
10.1.3.2 MPC for Double-Sided, Arbitrary Packet Loss 200
10.1.3.3 Solution of MPC for Double-Sided Packet Loss 201
10.2 Invariant Set Trap 204
10.3 Prediction Horizon: Zero or One 211
10.3.1 One Over Zero 211
10.3.2 One: Generalizing to Networked Control 214
10.3.2.1 Algorithm 215
10.3.2.2 A Numerical Example 218
10.4 Variant Feedback MPC 219
10.5 About Optimality 226
10.5.1 Constrained Linear Time-Varying Quadratic Regulation with Near-Optimal Solution 227
10.5.1.1 Solving KBM Controller 228
10.5.1.2 Solving Problem Without Terminal Cost 229
10.5.1.3 Solving Problem with Terminal Cost 230
10.5.1.4 Overall Algorithm and Analysis 230
10.5.1.5 Numerical Example 231
10.5.2 Alternatives with Nominal Performance Cost 232
10.5.2.1 Problem Formulation 232
10.5.2.2 Robust MPC Based on Partial Feedback Control 233
10.5.2.3 Introducing Vertex Control Moves 235
10.5.2.4 Numerical Example 236
10.5.3 More Discussions 236
11 Output Feedback Robust Model Predictive Control 239
11.1 Model and Controller Descriptions 245
11.1.1 Controller for LPV Model 247
11.1.2 Controller for Quasi-LPV Model 248
11.2 Characterization of Stability and Optimality 249
11.2.1 Review of Quadratic Boundedness 249
11.2.2 Stability Condition 251
11.2.3 Optimality Condition 252
11.2.4 A Paradox for State Convergence 254
11.3 General Optimization Problem 255
11.3.1 Handling Physical Constraints 255
11.3.2 Current Augmented State 256
11.3.3 Some Usual Transformations 258
11.3.4 Handling Double Convex Combinations 259
11.4 Solutions to Output Feedback MPC 260
11.4.1 Full Online Method for LPV 261
11.4.2 Partial Online Method for LPV 262
11.4.3 Relaxed Variables in Optimization Problem 264
11.4.4 Alternative Forms Based on Congruence Transformation 265
11.4.5 Description of Bound on True State 271
References 273
Index 279
1
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) how to optimize , i.e., the algorithms, which were dominating in the 1970s and 1980s;
- (2) whether or not the sequence ( from 0 to ) converges, i.e., stability, which is dominating in the academic theory;
- (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...
