MVC and MPC K. J. strm Department of Automatic Control, Lund - PowerPoint PPT Presentation

MVC and MPC K. J. Åström Department of Automatic Control, Lund University K. J. Åström MVC and MPC

Congratulations to a Stellar Career! Points of tangency , IFAC Teddington 1964 IFAC Prague 1967 First Identi fi cation Symposium Generalized predictive control Automatica 1987 A memorable semester as Douglas Holder Visiting Fellow Oxford in 1988 Control is much more than algorithm design; diagnostics, fault detection and recon fi guration are also of prime signi fi cance. K. J. Åström MVC and MPC

Introduction Minimum Variance Control Model Predictive Control Inspired by practice Inspired by practice Åström 1966 Richalet 1976 (IBM J R&D 1967) (Automatica 1978) Model structure MISO Cutler DMC (ACC 1980) Explicit disturbance modeling Model structure MIMO-FIR Minimize variance Reference trajectory Identi fi cation Captures saturation Self-tuning Widely used in industry Harris index What can we learn? K. J. Åström MVC and MPC

Outline Introduction The IBM-Billerud Project Modeling Minimum Variance Control Adaptation Re fl ections K. J. Åström MVC and MPC

The Scene of 1960 Servomechanism theory 1945 IFAC 1956 (50 year jubilee in 2006) Widespread education and industrial use of control The First IFAC World Congress Moscow 1960 Exciting new ideas Dynamic Programming Bellman 1957 Maximum Principle Pontryagin 1961 Kalman Filtering ASME 1960 Exciting new development The space race (Sputnik 1957) Computer Control Port Arthur 1959 IBM and Nordic Laboratory 1961 K. J. Åström MVC and MPC

The Role of Computing Vannevar Bush 1927. Engineering can proceed no faster than the mathematical analysis on which it is based. Formal mathematics is frequently inadequate for numerous problems, a mechanical solution offers the most promise. Herman Goldstine 1962: When things change by two orders of magnitude it is revolution not evolution. Gordon Moore 1965: The number of transistors per square inch on integrated circuits has doubled approximately every 12 months. Moore+Goldstine: A revolution every 10 year! Unfortunately software does keep up with hardware Roughly 10 years between MVC and MPC K. J. Åström MVC and MPC

The Billerud-IBM Project Background IBM and Computer Control Billerud and Tryggve Bergek Goals Billerud: Exploit computer control to improve quality and pro fi t! IBM: Gain experience in computer control, recover prestige and fi nd a suitable computer architecture! Schedule Start April 1963 Computer Installed December 1964 System identi fi cation and on-line control March 1965 Full operation September 1966 40 many-ears effort in about 3 years K. J. Åström MVC and MPC

Goals and Tasks Goals What can be achieved by computer control? Find an architecture of a process control computer! Philosophy Cram as much as possible into the system! Tasks Production Planning Production Supervision Process Control Quality Control Reporting Later 1969 Millwide control K. J. Åström MVC and MPC

Computer Resources IBM 1720 (special version of 1620 decimal architecture) Core Memory 40k words (decimal digits) Disk 2 M decimal digits 80 Analog Inputs 22 Pulse Counts 100 Digital Inputs 45 Analog Outputs (Pulse width) 14 Digital Outputs Fastest sampling rate 3.6 s One hardware interrupt (special engineering) Home brew operating system K. J. Åström MVC and MPC

The Billerud Plant K. J. Åström MVC and MPC

Summary Industrial A successful installation Computer achitecture for process control IBM 1800, IBM 360 Methodology Method for identi fi cation of stochastic models Basic theory, consistency, ef fi ciency, persistent excitation Minimum variance control What we misssed Project was well documented in IBM reports and a few papers but we should have written a book K. J. Åström MVC and MPC

Outline Introduction The IBM-Billerud Project Modeling Minimum Variance Control Adaptation Re fl ections K. J. Åström MVC and MPC

Process Modeling Process understanding and modi fi cations (mixing tanks) Physical modeling Logging dif fi culties Drastic change in attitude when computer was installed Good support from management Kai Kinberg: “This is a show-case project! Don’t hesitate to do something new if you believe that you can pull it off and fi nish it on time.” The beginning of system identi fi cation Wasted a lot of time on historical data Big struggle to do real plant experiments Identi fi cations requires a great range of skills K. J. Åström MVC and MPC

Basis Weight and Moisture Control Two important loops Triangular coupling MISO works K. J. Åström MVC and MPC

Modeling for Control Modeling by frequency response key for success of classical control Stochastic control theory is a natural formulation of industrial regulation problems State space models for process dynamics and disturbances Physical models may give dynamics Process data necessary to model disturbances Can we fi nd something similar to frequency response for state space systems? K. J. Åström MVC and MPC

Typical Fluctuations First measurement of fl uctuations in basis weight 1963 Availability of sensor crucial! A lot of effort to obtain this curve! K. J. Åström MVC and MPC

Stochastic Control Theory Kalman fi ltering, quadratic control, separation theorem Process model dx = Axdt + Budt + dv dy = Cxdt + de Controller d ˆ x = A ˆ x + Bu + K ( dy − C ˆ xdt ) u = L ( x m − ˆ x ) + u f f A natural approach for regulation of industrial processes. K. J. Åström MVC and MPC

Model Structures Process model dx = Axdt + Budt + dv dy = Cxdt + de Much redundancy z = Tx + noise model. The innovation representation reduces redundancy of stochastics and fi lter gains appear explicitely in the model dx = Axdt + Budt + K d ǫ = ( A − K C ) xdt + Budt + K dy dy = Cxdt + d ǫ Canonical form for MISO system removes remaining redundancy, discretization gives ( C fi lter dynamcis) A ( q − 1 ) y ( t ) = B ( q − 1 ) u ( t ) + C ( q − 1 ) e ( t ) K. J. Åström MVC and MPC

Modeling from Data (Identi fi cation) The Likelihood function (Bayes rule) N ǫ 2 ( t ) p ( Y t , θ ) = p ( y ( t )� Y t − 1 , θ ) = ⋅ ⋅ ⋅ = − 1 − N � 2 log 2 πσ 2 σ 2 2 1 θ = ( a 1 , . . ., a n , b 1 , . . . , b n , c 1 , . . ., c n , ǫ ( 1 ) , .., ) Ay ( t ) = Bu ( t ) + Ce ( t ) C ǫ ( t ) = Ay ( t ) − Bu ( t ) ǫ = one step ahead prediction error Ef fi cient computations N � J ǫ ( t )� ǫ ( t ) C � ǫ ( t ) � = = q k y ( t ) � a k � a k � a k 1 Estimate has nice properties Åström and Bohlin 1965 Good match identi fi cation and control. Prediction error is minimized in both cases! Cleaned up by Lennart Ljung ... K. J. Åström MVC and MPC

Practical Issues Sampling period To perturb or not to perturb Open or closed loop experiments Model validation 20 min for two-pass compilation of Fortran program! Control design Skills and experiences K. J. Åström MVC and MPC

Results K. J. Åström MVC and MPC

Outline Introduction The IBM-Billerud Project Modeling Minimum Variance Control Adaptation Re fl ections K. J. Åström MVC and MPC

Control Conventional PI(D) at lower level Simple digital control for non-critical loops Limited computational capacities Time delay dynamics stochastic fl uctuations domimating Mild coupling basis weight and moisture control Minimum variance control and moving average control Robustness performance trade-offs K. J. Åström MVC and MPC

Minimum Variance (Moving Average Control) Process model Ay ( t ) = Bu ( t ) + Ce ( t ) Factor B = B + B − , solve (minimum G -degree solution) AF + B − G = C Cy = AFy + B − Gy = F ( Bu + Ce )+ B − Gy = CFe + B − ( B + Fu + Gy ) Control law and output are given by B + Fu ( t ) = − Gy ( t ) , y ( t ) = Fe ( t ) where deg F ≥ pole excess of B / A � T True minimum variance control V = E 1 0 y 2 ( t ) dt T K. J. Åström MVC and MPC

Properties of Minimum Variance Control The output is a moving average deg F ≤ deg A − deg B + . y = Fe , Easy to validate! Interpretation for B − = 1 (all process zeros canceled), y is a moving average of degree n pz = deg A − deg B . It is equal to the error in predictiong the output n pz step ahead. Closed loop characteristic polynomial is B + Cz deg A − deg B + = B + Cz deg A − deg B + deg B − . The sampling period an important design variable! Sampled zeros depend on sampling period. For a stable system all zeros are stable for suf fi ciently long sampling periods. K. J. Åström MVC and MPC

Performance ( B − = 1 ) and Sampling Period Plot prediction error as a function of prediction horizon T p σ 2 pe T p T d + T s T d T d is the time delay and T s is the sampling period. Decreasing T s reduces the variance but decreases the response time. K. J. Åström MVC and MPC

Performance and Robustness 1 5 5 0 1 2 3 T d Strong similarity between all controller for systems with time delays, minimum variance, moving average and Smith predictor. It is dangerous to be greedy! Rule of thumb: no more than 1-4 samples per dead time motivated by simulation. K. J. Åström MVC and MPC