Early History Edmund Lee (1745) Windmill Fantail
Industrial Age James Watt (1788) Centrifugal governor patented for the steam engine
Mathematical Age James Clerk Maxwell (1831-1879) On Governors. Proc. Roy. Soc. 16 (1868) 270-283. Stability Concept Simple Mathematical Models Importance of integral action Linearization
Mathematical Age Stability Concept "It will be seen that the motion of a machine with its governor consist in general of a uniform motion, combined with a disturbance which may be expressed as the sum of several component motions. These components may be of four different kinds: 1. The disturbance may continually increase. 2. It may continually diminish. 3. It may be an oscillation of continually increasing amplitude. 4. It may be an oscillation of continually decreasing amplitude. The first and third cases are evidently inconsistent with the stability of the motion: and the second and fourth alone are admissible in a good governor. This condition is mathematically equivalent to the condition that all the possible roots, and all the possible parts of the impossible roots of a certain equation shall be negative."
Mathematical Age Proportional Action Concept "Most governors depend on the centrifugal force of a piece connected with a shaft of the machine. When the velocity increases, this force increases, and either increases the pressure of the piece against a surface or moves the piece, and so acts on a break or a valve. In one class of regulators of machinery, which we may call moderators, the resistance is increased by a quantity depending on the velocity."
Mathematical Age Integral Action Concept "But if the part acted on by centrifugal force, instead of acting directly on the machine, sets in motion a contrivance which continually increases the resistance as long as the velocity is above its normal value, and reverses its action when the velocity is below that value, the governor will bring the velocity to the same normal value whatever variation (within the working limits of the machine) be made in the driving-power or the resistance."
Mathematical Age 1877 Vishnegradsky – stability 1893 Lyaponov – stability of nonlinear differential equations 1895 Heaviside – transient behavior of systems 1927 Black – negative feedback 1932 Nyquist – design of stable amplifiers 1938 Bode – frequency response (Bell Telephone Laboratories) Time Domain Frequency Domain
Control Implementation Technology Pneumatic single loop controllers (3-15 psig standard) Electronic controllers (4-20 ma standard)
Hardware Age Control Implementation Technology Pneumatic ingenuity 7216 Variable Ratio Regulators are used with nozzle-mix burners to achieve temperature uniformity while using minimum excess air. … A high quality stainless steel spring is used to bias 7216 Regulator air/gas ratio. As air rate is turned down towards low fire, gas rate drops faster, giving increasing percentages of excess air …
Migration to the Computer Age Control Implementation Technology Block logic (PID, summer, selectors, digital points, ratio, cascade, etc. Control logic programs Distributed vs. Centralized Architecture
Computer Age Control Strategy Technology What should be controlled ? Simple variables, measurements Computed combinations of measurements Inferred measurements Laboratory measurements How to control process variables Single loop control Cascade, override, multivariable, feedforward, etc. Tuning, performance considerations Control Objectives Migration from "hold constant" to "find optimum"
Today's Age Control to Maximize Profit Control Objectives Measures of profit Real time changes to the profit function (pricing, demand, etc.) Process Constraints Measureable/predictable, unpredictable Current time or future time Operation at process constraints Changing Targets Maintaining good control when the target is always moving Interface with supply chain management Managing Process Variability Control strategy design
Today's Age Models A model in this context is a mathematical description of a process Linear / Nonlinear Continuous / Discrete Deterministic / Stochastic First principle / Empirical Process Model Inputs: valve positions flow rates energy inputs Outputs: temperature composition properties Parameters
Today's Age Models Example: ( linear, no dynamics, continuous ) Process y = au + b Inputs: u Outputs: y Parameters: a, b
Today's Age Models Example:( linear, dynamic, continuous ) Process Parameters: A, B, C
Today's Age Models Example: ( linear, dynamic, continuous, uncertainty descriptions ) Process Parameters: A, B, C, N(µ, σ)
Coming Age Models Example:( nonlinear, dynamic, continuous, known uncertainty description ) Process Parameters: a, b, c, …
Coming Age Control of Uncertain Processes Characterization of Uncertainty Mean, standard deviation (normal distribution) Other distributions Non stationary distributions Characterization of Risk Determination of the cost of risk Deciding the risk/reward tradeoff point
Coming Age The PSUADE Uncertainty Quantification Project Uncertainty quantification is defined as the identification (Where are the uncertainties?), characterization (What form they are in?), propagation (How they evolve during the simulation?), analysis (What are their impacts?), and reduction of ALL uncertainties in simulation models.
Coming Age Models Current research: Improved, more accurate models System identification methods to develop models Model reduction techniques Mathematical methods to drive models to optimum points in an optimum manner (given a mathematical description of 'optimum'). Optimizers exploit model weaknesses and can find poor, undesirable answers As solution techniques get stronger we need better models
Coming Age Process Uncertainty Models provide a template upon which to overlay process data Uncertainty descriptions coupled with models can suggest "best" answers in a probabilistic sense. Techniques for handling model size, complexity, unknown parameters, probability distributions, initialization, etc. are barriers Deterministic World Probabilistic World
Coming Age Process Model Structure Extracted Data Fit to Model Structure Uncharacterized Process Information
Coming Age Process Inputs Refine Parameterize Analyze Optimize Strategize Invert Filter Add Logic Clamp Limit etc., etc.
Coming Age How should we describe the remainder of the process information ? Other process information/behavior we don't know about … 0100101110100110 … Process information we are not measuring.
Coming Age Measure of risk Probabilistic answers Stochastic/chaotic Random behavior Process Inputs Other process information/behavior we don't know about
Final Thoughts Compute Model Inverse Process Model Desired State Process Output Estimate Applications Biological Systems Diabetes Drug dosing strategies Virus spread Predator/prey dynamics Population demographics Economics Stock market Fiscal policy Supply/demand Financial modeling