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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 1964IFAC Prague 1967 First Identification SymposiumGeneralized predictive control Automatica 1987A memorable semester as Douglas Holder Visiting FellowOxford in 1988Control is much more than algorithm design; diagnostics,fault detection and reconfiguration are also of primesignificance.
K. J. Åström MVC and MPC
Introduction
Minimum Variance Control
Inspired by practice
Åström 1966(IBM J R&D 1967)
Model structure MISO
Explicit disturbancemodeling
Minimize variance
Identification
Self-tuning
Harris index
Model Predictive Control
Inspired by practice
Richalet 1976(Automatica 1978)
Cutler DMC(ACC 1980)
Model structureMIMO-FIR
Reference trajectory
Captures saturation
Widely used in industry
What can we learn?
K. J. Åström MVC and MPC
Outline
Introduction
The IBM-Billerud Project
Modeling
Minimum Variance Control
Adaptation
Reflections
K. J. Åström MVC and MPC
The Scene of 1960
Servomechanism theory 1945
IFAC 1956 (50 year jubilee in 2006)
Widespread education and industrialuse of control
The First IFAC World CongressMoscow 1960Exciting new ideas
Dynamic Programming Bellman 1957Maximum Principle Pontryagin 1961Kalman Filtering ASME 1960
Exciting new developmentThe 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 fasterthan the mathematical analysis on which it is based.Formal mathematics is frequently inadequate for numerousproblems, a mechanical solution offers the most promise.
Herman Goldstine 1962: When things change by twoorders of magnitude it is revolution not evolution.
Gordon Moore 1965: The number of transistors per squareinch on integrated circuits has doubled approximatelyevery 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
BackgroundIBM and Computer ControlBillerud and Tryggve Bergek
GoalsBillerud: Exploit computer control to improve quality andprofit!IBM: Gain experience in computer control, recover prestigeand find a suitable computer architecture!
ScheduleStart April 1963Computer Installed December 1964System identification and on-line control March 1965Full operation September 196640 many-ears effort in about 3 years
K. J. Åström MVC and MPC
Goals and Tasks
GoalsWhat can be achieved by computer control?Find an architecture of a process control computer!
PhilosophyCram as much as possible into the system!
TasksProduction PlanningProduction SupervisionProcess ControlQuality ControlReporting
Later 1969Millwide 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 installationComputer achitecture for process control
IBM 1800, IBM 360
Methodology
Method for identification of stochastic models
Basic theory, consistency, efficiency, persistent excitation
Minimum variance control
What we misssed
Project was well documented in IBM reports and a fewpapers but we should have written a book
K. J. Åström MVC and MPC
Outline
Introduction
The IBM-Billerud Project
Modeling
Minimum Variance Control
Adaptation
Reflections
K. J. Åström MVC and MPC
Process Modeling
Process understanding and modifications (mixing tanks)
Physical modeling
Logging difficulties
Drastic change in attitude when computer was installedGood support from management Kai Kinberg:
“This is a show-case project! Don’t hesitate to dosomething new if you believe that you can pull it off andfinish it on time.”
The beginning of system identification
Wasted a lot of time on historical data
Big struggle to do real plant experiments
Identifications 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 ofclassical control
Stochastic control theory is a natural formulation ofindustrial regulation problems
State space models for process dynamics anddisturbances
Physical models may give dynamics
Process data necessary to model disturbances
Can we find something similar to frequency response forstate space systems?
K. J. Åström MVC and MPC
Typical Fluctuations
First measurement of fluctuations 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 filtering, quadratic control, separation theorem
Process model
dx = Axdt+ Budt+ dvdy= Cxdt+ de
Controller
dx̂ = Ax̂ + Bu+ K (dy− Cx̂dt)u = L(xm − x̂) + u f f
A natural approach for regulation of industrial processes.
K. J. Åström MVC and MPC
Model Structures
Process model
dx = Axdt+ Budt+ dvdy= Cxdt+ de
Much redundancy z = Tx + noise model. The innovationrepresentation reduces redundancy of stochastics and filtergains appear explicitely in the model
dx = Axdt+ Budt+ Kdε= (A− KC)xdt+ Budt+ Kdy
dy= Cxdt+ dεCanonical form for MISO system removes remainingredundancy, discretization gives (C filter 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 (Identification)
The Likelihood function (Bayes rule)
p(Y t,θ ) = p(y(t)�Y t−1,θ ) = ⋅ ⋅ ⋅ = −12N∑1
ε2(t)σ 2
− N2 log 2πσ 2
θ = (a1, . . . , an, b1, . . . , bn, c1, . . . , cn, ε(1), .., )Ay(t) = Bu(t) + Ce(t) Cε(t) = Ay(t) − Bu(t)
ε = one step ahead prediction errorEfficient computations
�J�ak
=N∑1
ε(t)�ε(t)�akC�ε(t)�ak
= qky(t)
Estimate has nice properties Åström and Bohlin 1965Good match identification and control. Prediction error isminimized 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 loopexperiments
Model validation
20 min for two-passcompilation of Fortranprogram!
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
Reflections
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 fluctuations 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 modelAy(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
True minimum variance control V = E 1T∫ T0 y2(t)dt
K. J. Åström MVC and MPC
Properties of Minimum Variance Control
The output is a moving average
y = Fe, deg F ≤ deg A− deg B+.
Easy to validate!
Interpretation for B− = 1 (all process zeros canceled), y isa moving average of degree npz = deg A− deg B. It isequal to the error in predictiong the output npz step ahead.Closed loop characteristic polynomial is
B+Czdeg A−deg B+ = B+Czdeg A−deg B+deg B−.
The sampling period an important design variable!
Sampled zeros depend on sampling period. For a stablesystem all zeros are stable for sufficiently long samplingperiods.
K. J. Åström MVC and MPC
Performance (B− = 1) and Sampling PeriodPlot prediction error as a function of prediction horizon Tp
Tp
σ 2pe
Td Td + Ts
Td is the time delay and Ts is the sampling period. DecreasingTs reduces the variance but decreases the response time.
K. J. Åström MVC and MPC
Performance and Robustness
Td
0
1
1 2 3 5 5
Strong similarity between all controller for systems withtime delays, minimum variance, moving average and Smithpredictor.
It is dangerous to be greedy!
Rule of thumb: no more than 1-4 samples per dead timemotivated by simulation.
K. J. Åström MVC and MPC
Robustness Analysis
Consider a system with time delay Td design for a closed looptime constantTcl. The main system functions are:
Gt(s) = e−sTd1+ sTcl
Gs(s) = 1− Gcl(s) = 1− e−sTd1+ sTcl
G�(s) = e−sTd1+ sTcl − e−sTd
Sensitivity and complementary sensitivity functions are alwaysless than 2! So things look good!
BUT Look at the delay margins!
K. J. Åström MVC and MPC
Nyquist Plots for Smith Predictors Tcl = 1Td = 1 Td = 2
Td = 4 Td = 8
Re LRe L
Re LRe L
Im LIm L
Im LIm L
−1
K. J. Åström MVC and MPC
Another Robustness Result
A simple digital controller for systems with monotone stepresponse (design based on the model y(k+ 1) = bu(k))
uk = k(ysp− yk) + uk−1, k < 2�(∞)
Ts
Stable if �(Ts) > �(∞)2 kjå: Automatica 16 1980, pp 313–315.
K. J. Åström MVC and MPC
Summary
Regulation can be doneeffectively by minimumvariance control
Easy to validate
Sampling period is thedesign variable!
Robustness dependscritically on the samplingperiod
The Harris Index andrelated criteria
OK to assess but why notadapt?
K. J. Åström MVC and MPC
Outline
Introduction
The IBM-Billerud Project
Modeling
Minimum Variance Control
Adaptation
Reflections
K. J. Åström MVC and MPC
Drawbacks with System Identification
Experiment planning requires prior knowledge
Process perturbations required
Time consuming
Requires competence
Adaptation is an alternative
K. J. Åström MVC and MPC
The Self-tuning Regulator
Process model: Ay(t) = Bu(t− k) + Bf f u f f (t) + Ce(t)Select sampling period and time delay k, rules for stablesystems
Estimate parameters in the model
y(t+ k) = Sy(t) + Ru(t) + R f f u f f (t)If estimate converge
ry(τ ) = 0,τ = k, k+ 1, ⋅ ⋅ ⋅ k+ deg(S)ryu(τ ) = 0,τ = k, k+ 1, ⋅ ⋅ ⋅ k+ deg(R)
If degrees sufficiently large ry(τ ) = 0,∀τ ≥ kConvergence conditions
KJÅ+BW Automatica 9(1973),185-199
K. J. Åström MVC and MPC
Convergence Analysis
IEEE Trans AC-22 (1977) 551–575
Markov processes and differential equations
dx = f (x)dt+ �(x)dw, �p�t = −
�p�x
(� f p�x
)+ 12
�2�x2�
2 f = 0
Lennarts idea
θ t+1 = θ t + γ tϕ e,dθdτ = f (θ ) = Eϕ e
Convergence of recursive algorithms and STR (Ay=Bu+Ce)
Jan Holst: ODE locally stable if ReC(zk) > 0 for B(zk) = 0K. J. Åström MVC and MPC
Paper Machine Control
K. J. Åström MVC and MPC
Industrial Applications
A number of applications inspecial areas
Paper machine control
Ship steering
Rolling mills
Semiconductormanufacturing
Tuning of feedforward verysuccessful
The Novatune
Process diagnostics Harrisand similar indices
K. J. Åström MVC and MPC
Tuning and Adaptation
CategoriesAutomatic TuningGain SchedulingAdaptive feedbackAdaptive feedfoward
ProductsTuning toolsPID controllersTool boxesSpecial purposesystems built intoinstruments
Process dynamics
Varying Constant
Use a controller withvarying parameters
Use a controller withconstant parameters
Unpredictable variations
Predictable variations
Use an adaptive controller Use gain scheduling
Åström Hägglund Advanced PID Control, 2004
K. J. Åström MVC and MPC
Relay Auto-tuning
ProcessΣ
−1
PID
y u y sp
What happens when relay feedback is applied to a system withdynamics? Think about a thermostat?
0 5 10 15 20 25 30
−1
−0.5
0
0.5
1
y
tK. J. Åström MVC and MPC
The Excitation Signal
Relay feedback automatically generates an excitationsignal with good frequency content!The transient is also useful
K. J. Åström MVC and MPC
Temperature Control of Distillation Column
K. J. Åström MVC and MPC
Commercial Auto-Tuners
Easy to useOne-button tuningSemi-automaticgeneration of gainschedulesAdaptation offeedback andfeedforward gains
RobustMany versions
Stand aloneDCS systems
Large numbers
Excellent industrialexperience
K. J. Åström MVC and MPC
Properties of Relay Auto-tuning
Safe for stable systemsClose to industrial practice
Compare manual Ziegler-Nichols tuningEasy to explain
Little prior information. Relay amplitudeOne-button tuningAutomatic generation of test signal
Automatically injects much energy at ω 180 without forknowing ω 180 apriori
Good for pre-tuning of adaptive algorithmsGood industrial experience for more than 25 years. Basicpatents are running out.
K. J. Åström MVC and MPC
Outline
Introduction
The IBM-Billerud Project
Modeling
Minimum Variance Control
Adaptation
Reflections
K. J. Åström MVC and MPC
Interaction with Industry
Contact with real problems is very healthy for research inengineering
Both MVC and MPC emerged in this way
Applied industrial projects can inspire research, providedthat they have enlightened management
New problems may appear
Challenges with publications; importance of good Editors
Necessary to look deeper and to fill in the gaps, even if ittakes a lot of effort and a lot of time - a long range view isnecessary to get real insight
Useful for a project to exchange people between academiaand industry
The Oxford model, the SupAero model, the Lund model
K. J. Åström MVC and MPC
The Knowledge Gap
Richalet Automatica 1963: MPC requires technical staffwith training in:
modeling, identification, digital control,...
The Novatune experience
Projects 73-74
Bengtsson Cold rolling 79
ASEA Innovation 81
30 persons 50M
Transfer to ASEA MasterRelay auto-tuning Hägglund kjå 1981
One button tuning
Can relay auto-tuning be useful for MPC modeling?
K. J. Åström MVC and MPC