Uncertainty Versus Randomness – A Command Theorist’S View
I volition Federal Reserve notation that this article represents my views, as well as non that of whatever my ex-colleagues inward academia. Since I am questioning the usefulness of some approaches to command systems, it goes without proverb that some academics associated alongside those approaches would disagree alongside my assessment. I should also Federal Reserve notation that I receive got been removed from command systems for most ii decades, as well as so it may move that my views are out of date.
Survey Of Different Modelling Approaches
I do non wish to laissez passer on a total history of command systems engineering, but I volition summarise it every bit beingness an expanse of applied mathematics that looks the mathematical problems associated alongside getting technology scientific discipline systems to human activeness inward a desired fashion. Within this article, I volition utilization the extended illustration of the pattern of an aircraft flying command system. Mechanical as well as aeronautical engineers volition pattern the plane, but it is necessary to develop the mathematical algorithms that map the inputs from the airplane pilot to the agency of controlling the flying of the aircraft – ailerons, rudders, thrust vectoring, etc.
The total nonlinear model of an aircraft is extremely hard to bargain with. So what has been typically done inward exercise is to create a issue of linearisations of the nonlinear model, each based roughly unlike operating points of the aircraft. Control rules are developed for each of these linearisations, as well as so the rules are stuck together to create a nonlinear command dominion that covers the total hit of operations. (I cite this utilization of linearisations only to betoken out that the way inward which economists fountain betwixt nonlinear economical models as well as linearisations is rather painful to watch.)
Let us bring 1 linear model, state for score flight. There are a issue of ways to approach the problem. (Note that I receive got an additional modelling philosophy inward the appendix.)
Deterministic
H5N1 model is created which has nation variables (aircraft seat as well as velocity). H5N1 controller is designed that translates airplane pilot command inputs to actions that touching on the nation variables. Pilot actions tin thence command the trajectory of nation variables.
Deterministic models are what are taught to undergraduates, as well as what are typically used inward practice. They appear to receive got obvious drawbacks – what happens if the airplane hits turbulence? What happens if the linearisation is incorrect? Design techniques were developed to brand controllers based on deterministic models robust. But alongside the advent of digital computers, some mathematically inclined command systems engineers wanted to utilization optimisation techniques to brand “optimal command rules”. These controllers were a disaster when implemented, every bit I discussed inward an before article. Economists receive got embraced optimisation throughout their models, fifty-fifty though they were rejected past times engineers based on their practical experience.
Robust Deterministic
Robust Deterministic Control Theory (often referred to every bit “H-∞” theory, because “H-infinity” sounds cool) was the initial response of the theoretical wing of command systems to these problems. To interpret this viewpoint into the damage used past times economists, this is an approach that incorporates uncertainty. The incertitude takes ii forms:
- Although nosotros e'er operate alongside deterministic models, nosotros do non actually know the “true” deterministic model. We assume that the “true” model is “close” to our base of operations illustration linear model. “Close” is defined inward damage of the mathematical properties of an unknown perturbation to the model. In our example, this agency that the parameters of our linear model may move off, or nosotros may fifty-fifty move missing some dynamics exclusively (e.g., the style of the air frame to flex). This agency that the “true” nation variables may include novel variables that are non included inward the laid upward of nation variables of our base of operations illustration model. (Technically, the dimension of the nation vector may change, every bit good every bit the model dynamics.)
- We also assume that the arrangement is striking alongside unknown somewhat arbitrary disturbances. In our example, these would move air pockets. But nosotros receive got to move realistic most what nosotros tin bargain with. No command dominion could move on a airplane that has been flown into the side of the mount on its master copy desired flying plan. We capture this mathematically past times insisting that the disturbance has finite energy, as well as shaping its frequency characteristics.
Even though this framework has a really realistic amount of uncertainty, randomness, inward the shape of probability theory, does non appear.
Stochastic Models
In stochastic (random) command theory, the deterministic base of operations illustration model is augmented past times ii forms of randomness:
- The development of the nation variables include random disturbances. This agency that the differential equations plough into messy stochastic differential equations, alongside Brownian motions as well as so forth. However, the nation variables rest the same every bit the base of operations illustration model.
- The arrangement is also striking alongside random impulses as well as measuring errors. As inward the deterministic case, things similar air pockets would autumn nether this category.
Although at that spot appear to move meaning differences betwixt these ii philosophies, they goal upward giving the same answers for linear systems. From the betoken of sentiment of the publish-or-perish environs inward academia, this is brilliant. What this agency is that in 1 lawsuit the deterministic robust command results are derived, it is possible to re-derive the exact same affair inside a stochastic framework.
Unfortunately for the stochastic approach, the correspondence breaks downwards in 1 lawsuit nonlinearities are introduced. It is at this betoken the shortfalls of the stochastic approach are apparent – the approach assumes that the model is correct, at that spot is but some randomness involved. This nevertheless assumes besides much most the validity of the model; nosotros tin receive got dynamics that are non modelled which are perverse. For example, the aircraft frame may resonate at a exceptional frequency. This could move really bad for a command dominion that causes oscillations at that frequency, as well as this exactly non a random effect.
Randomness Is Necessary In Some Contexts
Although I am non a huge fan of stochastic modelling frameworks, they are necessary inward some cases. For example, pick pricing makes piffling sense inside a deterministic model, fifty-fifty 1 alongside model uncertainty. Within engineering, the related champaign of communication systems relies heavily on probability. If communication systems were designed alongside the same conservative pattern philosophies of command engineers, jail cellphone phones would weight 10 pounds. But pattern conservatism has its place; people are to a greater extent than ofttimes than non tolerant of some static on their telephone calls, but a airplane landing 100 metres before the rails is to a greater extent than ofttimes than non not viewed every bit acceptable.
How This Relates To Economic Models
This digression into the philosophy of command systems pattern is applicable to economical modelling. Academic economists receive got similar a shot embraced stochastic mathematics, as well as they receive got layered their theory inward unreadable stochastic jargon. They presumably sentiment this approach every bit beingness mathematically sophisticated. However, this mathematical sophistication does non interpret automatically into modelling sophistication.
Using random variables to stand upward inward for incertitude creates the supposition that the model as well as its parameters are correct. But the mathematical complexity introduced past times using stochastic concepts makes it hard to sympathize the models as well as the model dynamics.
I prefer the approach taken inward the Stock-Flow Consistent models text of Godley as well as Lavoie. Deterministic models are developed as well as solved, allowing the users to run into what the actual nonlinear dynamics of the models are. Since the models are tied together past times accounting frameworks, nosotros know the dynamics are internally coherent. Although it is non the direction they receive got taken their theory, 1 could at to the lowest degree promise to rigourously comprise model incertitude inside their models. Even though the models volition inherently move weaker than models of technology scientific discipline systems, it is possible to run into what forms of incertitude are the most unsafe past times trial-and-error mathematical experimentation.
This comes upward inward my modelling of financial dynamics. Rather than bury my give-and-take of the authorities budget constraint nether layers of stochastic sludge, I prefer to hash out the constraint inward damage of an unknown deterministic trajectory. This is non beingness simplistic, rather it is next a to a greater extent than realistic modelling formalism. I volition illustrate this inward an upcoming article.
Appendix – Game Theoretic Approach
I skipped over some other way of deriving the same “robust” linear controls, which is using a game-theoretic framework. This framework is characterised by:
- The model is assumed to move correct, as well as at that spot are no “random” shocks.
- The arrangement is striking past times perturbations generated past times a malevolent “opponent”; the command police is the solution to a mathematical game inward which the influence of the opponent is minimised.
In damage of my flying command example, this framework says that model is perfectly known as well as at that spot are no “random” disturbances similar air pockets, but the aircraft is plagued past times a gremlin that is attempting to crash the aircraft. The objective of the controller is to counteract the gremlin so that the aircraft follows its desired trajectory.
Although this framework appears really different, it is possible to generate the same command laws for a linear arrangement every bit the other approaches. Once again, this is useful for academics, every bit it possible to re-derive existing results inward a novel framework, padding the publication count on resumés.
I would fence that this was non a skilful philosophy for modelling existent globe systems. The approach has no concept of uncertainty, rather y'all are certain that your model is correct, as well as y'all are certain that y'all are facing an opponent that volition e'er select the worst possible strategy to destabilise your system.
The fact that 1 tin derive the same controllers for linear systems is purely an artefact of linear systems – y'all tin derive the same command laws inward a huge issue of ways.* But this equivalence breaks downwards when y'all striking nonlinearities. And the philosophy of reasoning inward damage of an optimisation (the command police is the optimal police for the least-bad outcome) is dangerous. The malevolent opponent volition e'er exploit pathological properties of the mathematical arrangement inward companionship to come upward up alongside the most unsafe disturbance. This agency that the solution volition move dominated past times the most extreme business office of your mathematical model’s behaviour, which is in all probability the expanse of the model where its check to reality is the worst.
It almost goes without proverb that this was the approach that was embraced past times some economists when they wanted to apply some robustness concepts to economical theory. Economists tend to move attracted to game theory, as well as so this was a natural fit. The fact that this approach misses the entire objective of robust command – dealing alongside model incertitude – seems to non receive got been noticed.
Since this business office of the literature is almost unreadable as well as makes piffling sense from a modelling perspective, I am inward no hurry to covert it inward detail.
Footnote:
* The solution for the "optimal robust controller" is given past times an algebraic Riccati equation. You but laid upward the terminal solution to move the same Riccati equation, as well as so y'all operate backwards to decide your initial modelling framework.
See Also:
- The text past times Godley as well as Lavoie on stock-flow consistent (SFC) models - Monetary Economics: An Integrated Approach to Credit, Money, Income, Production as well as Wealth (affiliate link).
- My article on optimal command theory, inward which I explicate why it was abandoned.
No comments