Decoupling control analysis, design and tuning for multivariable processes

TRUONG NGUYEN LUAN VU VO LAM CHUONG DECOUPLING CONTROL: ANALYSIS, DESIGN, AND TUNING FOR MULTIVARIABLE PROCESSES HCMC - VNU PUBLISHING HOUSE TRUONG NGUYEN LUAN VU VO LAM CHUONG DECOUPLING CONTROL: ANALYSIS, DESIGN, AND TUNING FOR MULTIVARIABLE PROCESSES VNU-HCM PRESS - 2022 ABOUT THE AUTHORS Truong Nguyen Luan Vu is currently an Associate Professor of Mechanical Engineering at Ho Chi Minh City University of Technology and Education, Vietnam He received his B.S degree from Ho Chi Minh City University of Technology, Ho Chi Minh City National University in 2000, and his master’s and Ph.D degrees from Yeungnam University, the Republic of Korea in 2005 and 2009, respectively He has also taught at Yeungnam University for two years in terms of being an International Professor His research interests include multivariable control, fractional control, PID control, process control, automatic control, and control hardware Vo Lam Chuong is a lecturer of Mechatronics Department of Mechanical Engineering at Ho Chi Minh City University of Technology and Education, Vietnam He received his B.S degree from Ho Chi Minh City University of Technology, Ho Chi Minh City in 2002, and his master’s degree with the majority of Cybernetics at the same university in 2005 Currently, he is a Ph.D candidate of Mechanical Engineering of Ho Chi Minh City University of Technology and Education His research interests are multivariable process, fractional control, and nonlinear control PREFACE Recently, the changes of industrial needs and various kinds of advances in engineering and technology had a profound impact on the process control and control industrial processes as well Modern process control should be considered as an efficient integration of real-time information management with respect to traditional control strategies Accordingly, the decoupling control is one of the widely used multivariable control strategies for the industry today, due to the industrial perspective, efficiency, improved productivity, and product quality goals since multiloop control systems satisfy all of those demands for more effective operational strategies and allow more advanced tools to be implemented Over a decade, we have concentrated our research field on decoupling control, multi-loop PID control system, and their tuning We decided to summarize our experience and knowledge on those topics in this book based on a number of our articles published in the ISI journals, and those of other renowned authors This book consists of seven chapters Some of them are proper for undergraduate students, but the remaining parts require profound backgrounds of advanced process control for the monograph levels We hope that the readers can find out the helpful insights and can apply our methodologies for both theoretical and practice fields The authors would like to thank the Ho Chi Minh City University of Technology and Education (HCMUTE) for their financial supports We also express our special thanks to our colleagues at the Faculty of Mechanical Engineering (FME) for their spiritual encouragement and their comments on this book 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 summarised as below: Abbreviations BLT Biggest Log-Modulus Tuning CM Coefficient Matching DLT Decentralized Lambda Tuning DRGA Dynamic Relative Gain Array DTC Dead Time Compensator FOLF First Order Lag Filter FOPI Fractional-Order PI FOPID Fractional-Order PID FOPTD First Order Plus Time Delay IAE Integral Absolute Error IMC Internal Model Control MFD Matrix Fraction Description MIMO Multi-Input Multi-Output MP Minimum Phase NMP Nonminimum Phase NP Nominal Performance NS Nominal Stability P&ID Pipe and Instrumentation Diagram PEM Prediction Error Method PRBS Pseudo Random Binary Signal PSO Particle Swarm Optimization RGA Relative Gain Array RHP Right-Haft Plane RP Robust Performance RS Robust Stability SAT Sequential Auto-Tuning SDSP Simplified Decoupling Smith Predictor SISO Single-Input Single-Output SOPTD Second Order Plus Time Delay SP Smith Predictor SSV Structured Singular Value TITO Two-Input Two-Output Symbols G(s) Process transfer function matrix D(s) Decoupler matrix Q(s) Decoupled matrix C(s), Gc(s) or G~ c (s) Multi-loop controller matrix Gii+ Non-minimum part of process model Gii- Minimum part of process model H(s), T(s) Closed-loop transfer function matrix 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 Kc Proportional gain KI Integral gain KD Derivative gain τ I Integral time τ D Derivative time Greek Characters γ λ µ ( A ) ∏ ρ ( A ) σ ( A ) σ ( A ) Robust level Block diagonal matrix containing the uncertainty RGA matrix of the matrix A Closed-loop time constant SSV of A for the same specific set ∆ Set of possible plants Spectral radius of A Maximum singular value of A Minimum singular value of A Superscripts H Complex conjugate transpose of a matrix T Transpose of a matrix APPENDIXES APPENDIX A: LAPLACE TRANSFORMATION Laplace transforms play a key role in important process control concepts and techniques A.1 Definition The Laplace transform function F(s) of a time function f(t) is: = F ( s ) L= [ f (t )] ∞ − st ∫0 f ( t ) e dt (A.1) where F(s) is the symbol for the Laplace transform, L is the Laplace transform operator, and f(t) is some function of time, t Note: The L operator transforms a time domain function f(t) into an s domain function, F(s) s is a complex variable: s = a + bj A.2 Laplace transform of some typical functions Constant Function: Let f(t) = a (a constant), where a is a real constant ∞ ∫0 ae − st a dt =− e − st s ∞  a a =0 −  −  = (A.2)  s s In process control, the input variable is typically taken as a unit step function to analyze the dynamic response of processes The unit stepfunction is defined as 0 , t <  us (t ) =   (A.3) 1 , t ≥  Figure A.1 Unit step function 167 One can show that the Laplace transformation of the unit step function is given by = U s (s) = L{us (t )} ∞ − st ∞ s (t )e dt ∫0 1e ∫0 u= − st dt ∞  − st  = 1 =− e − [e−∞ − e0 ] = s s  s  , s>0 (A.4) Exponential functions: Let f ( t ) = ce−bt , where b>0 Then = L e−bt    ∞ −bt − st ∞ −( b + s )t = ce e dt c dt ∫ ∫ e 0 ∞ c c − b+ s t =  −e ( )  =   b+s s+b (A.5) Figure A.2 Exponential functions Laplace Transforms of some common functions can be listed in Table A.1 as follows: Table A.1 Laplace Transform of some basic functions 168 A.3 Important properties of laplace transform Property (Linear relationship): Laplace transform is a linear operation: Assume that F(s), G(s) are Laplace transforms of time functions f(t), g(t) {af (t ) ± bg (t )} = aF ( s ) ± bG ( s ) (A.6) Proof: {af (t ) ± bg (t )}= ∞ ∫ [af (t ) ± bg (t )]e − st dt ∞ ∞ 0 = a ∫ f (t )e − st dt ± b ∫ g (t )e − st dt = a { f (t )} ± b {g (t )} = aF ( s ) ± bG ( s ) Property (Differential of time function): Converting the differential operation of a time function into algebraic operation by multiplying transfer variable s and minus initial value Transform of the first derivative of f(t): d    f= (t )  sF ( s ) − f (0)  dt  (A.7) Proof: ∞  ∞ d  d  ∞   ∫0  f (= t )  e − st dt  [ f (t )e-st ] − ∫0 f (t ) e − st dt dt    dt  ∞ = − f (0) − ∫0 f (t )(− s )e − st dt ∞ = s ∫0 f (t )e − st dt − f (0) = sF ( s ) − f (0) When initial condition is f (0) = 0: d    f (t )  = sF ( s )  dt  Property (Integral of time functions): Converting the integral operation of a time function into algebraic operation by dividing transfer variable s at Laplace transform domain 169  {∫ f (τ )dτ } = 1s F (s) (A.8) t Proof: { } t =  ∫ f (τ )dτ ∞ t 0 ∫= ∫ f (τ )dτ e dt ∫ − st ∞ ∞ f (τ ) ∫ e − st dt dτ τ ∞ ∞  −st  dτ ∞ f (τ )e− sτ dτ (A.9) τ = f ( ) ∫0 − s e  s ∫0 =  τ = F (s) s Property (Exponential summation of time function): An exponential summation at time domain is a parallel transference function at Laplace domain {e at f (= t )} F ( s − a ) (A.10) Proof: ∞ {e at f (t )} = ∫ e at f (t )e − st dt = ∫ ∞ t f (t )e − ( s − a )= dt F ( s − a ) A parallel transference at one domain is an operation of exponential summation at the other at time domain or frequency domain Property (Time delay): A time delay is an exponential weighting in Laplace domain { f (t − d )} = e − ds F ( s ) (A.11) Proof: { f (t − d )}= ∫ ∞ ∫ f (t − d )e − st dt= d ∞ f (t − d )e − st dt + ∫ f (t − d )e − st dt d ∞ = ∫ f (v)e − s ( v + d ) dv ∞ = e − sd ∫ f (v)e − sv dv =e 170 − sd F (s) Figure A.3 Time delay function A.4 Inverse laplace transform By definition, the inverse Laplace transform operator, L−1 , converts an s-domain function back to the corresponding time domain function: f ( t ) = L−1  F ( s )  (A.12) Both L and L−1 are linear operators Thus, L  ax ( t ) + by ( t )=  aL  x ( t )  + bL  y ( t )  = aX ( s ) + bY ( s ) (A.13) A.5 Solution of linear ordinary equation (odes) by laplace transforms Procedure: Take the L of both sides of the ODE Rearrange the resulting algebraic equation in the s domain to solve for the L of the output variable, e.g., Y(s) Perform a partial fraction expansion Use the L−1 to find y(t) from the expression for Y(s) Figure A.4 Solution of Linear Ordinary Equation (ODEs) 171 Example A.1 Solve the ODE given by: dy = + 4y dt y ( ) (A.14) = First, take L of both sides, ( sY ( s ) − 1) + 4Y ( s ) = (A.15) s Rearrange, Y (s) = 5s + s ( 5s + ) (A.16) Take L−1 ,  5s +  y ( t ) = L−1   (A.17)  s ( 5s + )  Hence, y (= t ) 0.5 + 0.5e−0.8t (A.18) Noted that: b − b  b −b y ( t ) = L  e −b1t + e −b2t  = b1 − b2  b2 − b1  where b1 = 0.8, b2 = 0, b3 = 0.4 s + b3 (A.19) ( s + b1 )( s + b2 ) A.6 Partial fraction expansions Basic idea: Expand a complex expression for Y(s) into simpler terms, each of which appears in the Laplace Transform table Then you can take the L-1 of both sides of the equation to obtain y(t) F ( s= ) F1 ( s ) + F2 ( s ) +  + Fr ( s ) (A.20) The time-domain function can be obtained as: = f ( t ) L−1 = F ( s )  L−1  F1 ( s ) + F2 ( s ) +  + Fr ( s )  = L−1  F1 ( s )  + L−1  F2 ( s )  +  + L−1  Fr ( s )  172 (A.21) Example A.2 Consider the Laplace-domain solution of a differential equation that yields: Y (s) = s+5 ( s + 1)( s + ) (A.22) Perform a partial fraction expansion (PFE): α1 α s+5 = + ( s + 1)( s + ) s + s + (A.23) where coefficients α1 and α have to be determined To find α1 : Multiply both sides by s + and let s = -1: ∴ α1= s+5 s+4 = s =−1 (A.24) To find α : Multiply both sides by s + and let s = -4: ∴ s+5 s +1 α2 = = − s =−4 (A.25) Therefore, s+5 = − ( s + 1)( s + ) ( s + 1) ( s + ) (A.26) It is given: c L ce−bt  =   s+b (A.27)  4e−t e −4t − = 3  (A.28) Then,  L-1  −  ( s + 1) ( s + ) 173 APPENDIX B: MATRIX OPERATIONS B.1 Matrices Let A be square n x n matrix, which can be expressed as:  a11 a12  a1n  a a22  a2 n  21  = A = = aij  , i, j 1, 2, , n          an1 an  ann  The transpose of a matrix A is a n x n matrix and defined as (B.1)  a11 a T   = A a ji   12 =     a1n a21  an1  a22  an  T =  aij  , = i, j 1, 2, , n (B.2)      a2 n  ann  The determinant of a matrix is a useful operation in solving a set of linear algebraic equation The determinant is defined for a square matrix only and can be computed using the minors of a matrix, det ( A= ) A= n ∑a C i =1 ij ij (B.3) 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 = (B.4) det ( A ) A B.2 Eigenvalues and eigenvectors Eigenvalues appear in the solution of linear system of equations and are often referred to as the solution of the roots of a characteristic equation 174 Consider a matrix operation on a nonzero vector, Ax = λ x (B.5) The eigenvalues λi are the n solutions to n’th-order characteristic equation det ( A - λ I ) = (B.6) The eigenvalue decomposition of a matrix can be expressed as A = XΛX −1 (B.7) where The eigenvalue matrix containing n eigenvalues of A in the diagonal can be given by: λ1 0 λ Λ=    0  0   (B.8)     λn  The matrix X is the eigenvector matrix whose columns correspond to the eigenvector xi associated with the eigenvalue λi X = [ x1 x2  xn ] (B.9) Remarks - The eigenvectors are usually normalized to have unit length, i.e xiH xi = - The largest of the absolute values of the eigenvalues of A is the spectral radius of A ρ ( A )  max λi ( A ) (B.10) i - The sum of the eigenvalues of A is equal to the sum of the diagonal elements of A tr ( A ) = ∑ i λi (B.11) - The product of the eigenvalues of A is equal to the determinant of A det ( A ) = ∏ i λi (B.12) 175 - A and AT have the same eigenvalues (different eigenvectors) - Gershgorin’s theorem The eigenvalues of the n x n matrix A lie in the union of n circles in the complex plane, each with centre aii and radius ri = ∑ j ≠i aij They also lie in the union of n circle, each with centre aii and radius r'i = ∑ j ≠i a ji - The eigenvalues of a Hermitian matrix are real - A Hermitian matrix is positive definitely if and only if all its eigenvalues are positive B.3 Singular value decomposition Let A be an n x n complex (constant) matrix, then the positive square roots of the eigenvalues AHA (where AH means the complex conjugate transpose of A) are called the singular values of A = σi (A) ( ) = λi A H A λi ( AA H ) (B.13) The maximum and minimum singular values of A are denoted by σ ( A ) and σ ( A ) , respectively σ ( A ) = λmax ( A H A ) (B.14) σ ( A ) = λmin ( A H A ) (B.15) The singular values bound the magnitude of the eigenvalues: σ ( A ) ≤ λi ( A ) ≤ σ ( A ) (B.16) B.4 Relative gain array The RGA of a complex non-singular n x n matrix A, denoted RGA(A) or Λ ( A ) is defined as: ( ) RGA ( A ) ≡ Λ ( A )  A ⊗ A -1 T (B.17) where the operation ⊗ denotes element-by-element multiplication (Hadamard or Schur product) Remarks ( ) ( ) Λ A -1 = Λ AT = Λ (A) T The sum of elements in each row (or each column) of RGA is n n That is, ∑ λij = and ∑ λ ji = i =1 176 j =1 B.5 Vector and matrix norms The concept of norm is used to measure the size of a vector, matrix, signal, or system It is denoted by  , that satisfies the following properties: Non-negative: a ≥ Positive: a =0 ⇔ a = Homogeneous: α ⋅ a = α a for any scalar a Triangle inequality a1 + a2 ≤ a1 + a2 (B.18) B.6 Vector norms Consider a vector, a = [ a1 a2  an ] (B.19) T Vector 1-norm This is sometimes referred to the “taxi-cab”norm a  ∑ (B.20) i Vector 2-norm The most common norm is Euclidean norm a2 n ∑a i =1 i (B.21) Vector 2-norm The ∞ -norm gives the largest magnitude in the vector a ∞  max (B.22) i B.7 Matrix norms Consider an m x n matrix A= =  aij  , i 1,= 2, , n j 1, 2, , m A norm on a matrix A is a matrix norm if it satisfies the multiplicative property (also called the consistency condition) A ⋅ B ≤ A ⋅ B (B.23) Sum matrix norm This is the sum of the element magnitudes: A sum = ∑ aij (B.24) i, j The Euclidean norm (or Frobenius matrix norm) for a matrix is analogous to the vector 2-norm, 177 A = F n,m ( ∑ a= A = i , j =1 ) tr A H A (B.25) ij where the trace (tr) operator is the sum of the diagonal elements of the matrix Max element norm This is the largest element magnitude, A = max aij (B.26) max i, j B.8 Vectors, matrices, and norms in Matlab Matlab is a formidable mathematics analysis package, which is a very useful tool for control engineering Over the number of years, Matlab become the industry standard software package for the control system design Table B.1 Matlab commands are frequently used in the control engineering Equation Matlab command a  ∑ norm(a,1) i a2 a ∞ n ∑a i =1 norm(a,2) i  max norm(a,’inf’) i AT =  a ji  det ( A= ) A= = A −1 A = F 178 A’ n ∑a C i =1 ij det(A) ij adj ( A ) CT = det ( A ) A A = n,m ∑ a= i , j =1 ij inv(A) ( tr A H A ) norm(A,’fro’) A A sum = ∑ aij sum(sum(abs(A))) max = max aij max(max(abs(A))) i, j i, j ρ ( A )  max λi ( A ) max(abs(eig(A))) i = σi (A) RGA(A) ( ) = λi A H A λi ( AA H ) svd(A) A.*pinv(A).’ 179 Decoupling control Analysis, design and tuning for multivariable processes Truong Nguyen Luan Vu, Vo Lam Chuong Trường Đại học Sư phạm Kỹ thuật Thành phố Hồ Chí Minh NHÀ XUẤT BẢN ĐẠI HỌC QUỐC GIA THÀNH PHỐ HỒ CHÍ MINH Trụ sở: Phịng 501, Nhà Điều hành ĐHQG-HCM, phường Linh Trung, thành phốThủ Đức, Thành phố Hồ Chí Minh ĐT: 028 62726361 E-mail: vnuhp@vnuhcm.edu.vn Văn phòng đại diện: Tòa nhà K-Trường Đại học Khoa học Xã hội & Nhân văn, số 10-12 Đinh Tiên Hoàng, phường Bến Nghé, Quận 1,Thành phố Hồ Chí Minh ĐT: 028 62726390 Website: www.vnuhcmpress.edu.vn Chịu trách nhiệm xuất nội dung TS ĐỖ VĂN BIÊN Biên tập SIN KẾ DUYÊN Sửa in NHƯ NGỌC Trình bày bìa TRƯỜNG ĐẠI HỌC SƯ PHẠM KỸ THUẬT THÀNH PHỒ HỒ CHÍ MINH Đối tác liên kết TRƯỜNG ĐẠI HỌC SƯ PHẠM KỸ THUẬT THÀNH PHỒ HỒ CHÍ MINH Xuất lần thứ Số lượng in: 250 cuốn, khổ 16 x 24cm Số XNĐKXB: 778-2022/CXBIPH/2-07/ĐHQGTPHCM QĐXB số: 39/QĐ-NXB cấp ngày 18/3/2022 In tại: Cơng ty TNHH In Bao bì Hưng Phú; Địa chỉ: 162A/1, KP1A, phường An Phú, TP Thuận An, Bình Dương Nộp lưu chiểu: Năm 2022 ISBN: 978-604-73-8893-6 Bản quyền tác phẩm bảo hộ Luật Xuất Luật Sở hữu trí tuệ Việt Nam Nghiêm cấm hình thức xuất bản, chụp, phát tán nội dung chưa có đồng ý tác giả Nhà xuất ĐỂ CÓ SÁCH HAY, CẦN CHUNG TAY BẢO VỆ TÁC QUYỀN! NXB ĐHQG-HCM ISBN: 978-604-73-8893-6 786047 388936 ... DECOUPLING DESIGN FOR TYPICALPROCESSES This section analytically develops simplified decoupling for × 2, × 3, and × processes using Eq (2.19) and Eq (2.20) 2.2.1 Simplified decoupling for × processes. ..TRUONG NGUYEN LUAN VU VO LAM CHUONG DECOUPLING CONTROL: ANALYSIS, DESIGN, AND TUNING FOR MULTIVARIABLE PROCESSES VNU-HCM PRESS - 2022 ABOUT THE AUTHORS Truong Nguyen... simplicity, robust performance, analytical form, and effectiveness for simplified decoupling systems The overall procedure for deriving the tuning rules is as follows: For the simplified control system,