Multi loop pid controller analysis, design, and tuning for multivariable processes

TRUONG NGUYEN LUAN VU MULTI LOOP PID CONTROLLER ANALYSIS, DESIGN, AND TUNING FOR MULTIVARIABLE PROCESSES VIETNAM NATIONAL UNIVERSITY – HO CHI MINH CITY PRESS HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY[.]

TRUONG NGUYEN LUAN VU MULTI-LOOP PID CONTROLLER: ANALYSIS, DESIGN, AND TUNING FOR MULTIVARIABLE PROCESSES VIETNAM NATIONAL UNIVERSITY – HO CHI MINH CITY PRESS HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY AND EDUCATION TRUONG NGUYEN LUAN VU MULTI-LOOP PID CONTROLLER: ANALYSIS, DESIGN, AND TUNING FOR MULTIVARIABLE PROCESSES Vietnam National University Ho Chi Minh City Press - 2018 PREFACE In recent years, multi-loop proportional-integral/proportionalintegral-derivative (PI/PID) controllers are one of the most widely used industrial controllers The reason for this popularity is due to the industrial perspective, efficiency, improved productivity, and product quality goals since multi-loop control systems satisfy all of those demands for more effective operational strategies and allow more advanced tools to be implemented In addition, they are also easily understood by their sound theoretical foundation, required fewer parameters to tune than multivariable control systems that help maximize the economic benefits, and easily obtained the loop failure tolerance that is very important for practical applications Therefore, many design methods of multi-loop PI/PID controllers have been suggested in the process of control literature over the years Among them, the wellknown IMC-PID approach is often utilized because of its powerful framework and effectiveness for the design and implementation It attracts the control engineers and academics because the tradeoff between the robustness and performance are directly related to one design parameter as the time constant of IMC filter or the closed-loop time constant, equivalently This is a very useful characteristic of the IMCPID approach, which guarantees multi-loop control systems to be tuned satisfying the robust stability condition Furthermore, the direct synthesis for the multi-loop PI/PID control system that has the similar features of the IMC-PID approach is also considered by many authors In this textbook, the novel analytical methods based on the generalized multi-loop IMC-PID approach and direct synthesis are proposed different states of the robust design of multi-loop proportionalintegral/proportional-integral-derivative (PI/PID) controllers The proposed methods are aimed at achieving the desired closed-loop responses for multivariable processes with multiple time delays It is known that the interactions between input/output variables are very common phenomena encountering in the multi-loop feedback control loops, which can be a serious obstacle to achieve a good performance of the multi-loop control system To overcome this impediment, several techniques are suggested for taking the interaction effects fully into account such as weighted sum Mp criterion, multi-loop Ms criterion, etc The robust stability and performance are also two important issues that are considered in this book Basically, the gap between the actual system and its model may be a cause for the closed-loop instability The control system is robust if it performs satisfactorily under the various possible uncertainties Therefore, many multi-loop control systems are designed by considering the robust performance and stability analysis Several simulation studies are considered to compare and analyze many well-known tuning methods Notation and Nomenclature It is attempted to define the notation used for equations in the text However, the most important nomenclature used for the multi-loop feedback control are summaried as below: Abbreviations BLT Biggest Log-Modulus Tuning DLT Decentralized Lambda Tuning IMC Internal Model Control IAE Integral Absolute Error MIMO Multi-Input Multi-Output MP Minimum Phase NMP Nonminimum Phase NP Nominal Performance NS Nominal Stability RGA Relative Gain Array RHP Right-Haft Plane RP Robust Performance RS Robust Stability SAT Sequential Auto-Tuning SISO Single-Input Single-Output SSV Structured Singular Value Symbols E(s) Continuous error signal G(s) Process transfer function matrix Gii+ Non-minimum part of process model Gii- Minimum part of process model ~ G c (s) Multi-loop controller Gc0 Integral term of the multi-loop controller G c1 Proportional term of the multi-loop controller Gc Derivative term of the multi-loop controller Kc Proportional gain H(s) Closed-loop transfer function matrix Lcm A multivariable closed-loop log modulus r(s) Continuous set-point (set-point vector) R(s) The desired closed-loop responses u(s) Manipulated variable vector WI (s) (Wo(s)) Input (output) weights for the MIMO case y(s) Control variable vector Greek Characters   s Robust level Block diagonal matrix containing the uncertainty for the continuous (discrete) case   A RGA matrix of the matrix A  Closed-loop time constant   A SSV of A for same specific set   Set of possible plants   A Spectral radius of A   A Maximum singular value of A   A Minimum singular value of A Superscripts H Conplex conjugate transpose of a matrix T Transpose of a matrix Special Notation  For all  Such that CONTENTS PREFACE NOTATION AND NOMENCLATURE CHAPTER FUNDAMENTALS OF MATHEMATICAL ANALYSIS CHAPTER TRADITIONAL MULTI-LOOP PID TUNING METHODS 15 CHAPTER MULTI-LOOP PID CONTROLLER DESIGN FOR ENHANCED DISTURBANCE REJECTION IN MULTIVARIABLE PROCESSES 37 CHAPTER DESIGN OF MULTI-LOOP PID CONTROLLERS BASED ON THE GENERALIZED IMC-PID METHOD WITH MP CRITERION 55 CHAPTER IMC-PID CONTROLLER FOR IDEAL DECOUPLING EMBEDDED IN MULTIVARIABLE SMITH PREDICTOR CONTROL 73 CHAPTER MULTI-LOOP PI CONTROLLER DESIGN BASED ON DIRECT SYNTHESIS 85 CHAPTER ANALYTICAL DESIGN OF MULTI-LOOP PI CONTROLLERS FOR INTERACTIVE MULTIVARIABLE PROCESSES 105 CHAPTER DESIGN METHOD OF SIMPLIFIED DECOUPLING FOR MULTIVARIABLE PROCESSES 131 CHAPTER INDEPENDENT DESIGN OF MULTI-LOOP PID CONTROLLERS 161 APPENDIX 189 REFERENCES 191 Chapter FUNDAMENTALS OF MATHEMATICAL ANALYSIS MATRIX OPERATIONS AND NORMS The vector, matrix, and norm theories are very important background materials, which should ideally be studied to have a clear understanding of this book It should be appreciated the difference between various norms of vector, matrices and how these norms can be utilized to measure performance 1.1 Matrices Let A be square n x n matrix, which can be expressed as:  a11 a A   21    an1 a1n  a2 n    a  , i, j  1, 2,   ij   ann  a12 a22 an ,n (1.1) The transpose of a matrix A is a n x n matrix and defined as:  a11 a T A   a ji    12    a1n a21 a22 a2 n an1  an  T   aij  , i, j  1, 2,   ann  ,n (1.2) The determinant of a matrix is a useful operation in solving a set of linear algebraic equations The determinant is defined as a square matrix only and can be computed by using the minors of a matrix, n det  A   A   aij Cij (1.3) i 1 where Cij denotes the cofactor of aij If det(A) = 0, the matrix is then called singular matrix Singularity implies that it is linearly dependent Using the concept of minors, the rank of matrix can be defined as the order of the highest non-vanishing minor of matrix The rank is a measure of the number of independent columns or rows For a square matrix, rank deficiency also implies singularity By definition, the inverse of a non-singular matrix A, denoted A-1, satisfies A-1A = AA-1 = I, and is defined as: A 1  adj  A  CT  det  A  A (1.4) 1.2 EIGENVALUES AND EIGENVECTORS Eigenvalues appear in the solution of linear system of equations and are often referred to the solution of the roots of a characteristic equation Consider a matrix operation on a nonzero vector, Ax   x (1.5) The eigenvalues i are the n solutions to n’th-order characteristic equation det  A -  I   (1.6) The eigenvalue decomposition of a matrix can be expressed as A  XX1 (1.7) where The eigenvalue matrix containing n eigenvalues of A in the diagonal can be given by: 1 0     0 0    n  (1.8) The matrix X is the eigenvector matrix whose columns correspond to the eigenvector xi associated with the eigenvalue i X   x1 x2 xn  (1.9) Remarks - The eigenvectors are usually normalized to have unit length, i.e x xi  H i 10 ...HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY AND EDUCATION TRUONG NGUYEN LUAN VU MULTI- LOOP PID CONTROLLER: ANALYSIS, DESIGN, AND TUNING FOR MULTIVARIABLE PROCESSES Vietnam National University Ho... CHAPTER MULTI- LOOP PID CONTROLLER DESIGN FOR ENHANCED DISTURBANCE REJECTION IN MULTIVARIABLE PROCESSES 37 CHAPTER DESIGN OF MULTI- LOOP PID CONTROLLERS BASED ON THE GENERALIZED IMC -PID METHOD... model ~ G c (s) Multi- loop controller Gc0 Integral term of the multi- loop controller G c1 Proportional term of the multi- loop controller Gc Derivative term of the multi- loop controller Kc Proportional