## Monday, May 5, 2014

### Uncertainty Versus Randomness – A Control Theorist’s View

Within economics, there is a somewhat obscure debate regarding the distinction between randomness and uncertainty. This debate was started by Keynes, and is still referred to by Post-Keynesian economists. I believe that this debate is important for the strategy of model building, but I think of it in terms of my experience in the theory of control systems. The distinction between randomness and uncertainty has an operational difference in the frameworks for mathematical models. Once one looks at a concrete example, the distinction is a lot easier to understand.

I will note that this article represents my views, and not that of any my ex-colleagues in academia. Since I am questioning the usefulness of some approaches to control systems, it goes without saying that some academics associated with those approaches would disagree with my assessment. I should also note that I have been removed from control systems for about two decades, and so it may be that my views are out of date.

### Survey Of Different Modelling Approaches

I do not want to give a full history of control systems engineering, but I will summarise it as being an area of applied mathematics that looks the mathematical problems associated with getting engineering systems to act in a desired fashion. Within this article, I will use the extended example of the design of an aircraft flight control system. Mechanical and aeronautical engineers will design the plane, but it is necessary to develop the mathematical algorithms that map the inputs from the pilot to the means of controlling the flight of the aircraft – ailerons, rudders, thrust vectoring, etc.

The full nonlinear model of an aircraft is extremely difficult to deal with. So what has been typically done in practice is to create a number of linearisations of the nonlinear model, each based around different operating points of the aircraft. Control rules are developed for each of these linearisations, and then the rules are stuck together to create a nonlinear control rule that covers the full range of operations. (I mention this role of linearisations only to point out that the way in which economists jump between nonlinear economic models and linearisations is rather painful to watch.)

Let us take one linear model, say for level flight. There are a number of ways to approach the problem. (Note that I have an additional modelling philosophy in the appendix.)

Deterministic

A model is created which has state variables (aircraft position and velocity). A controller is designed that translates pilot control inputs to actions that affect the state variables. Pilot actions can thus control the trajectory of state variables.

Deterministic models are what are taught to undergraduates, and what are typically used in practice. They appear to have obvious drawbacks – what happens if the plane hits turbulence? What happens if the linearisation is incorrect? Design techniques were developed to make controllers based on deterministic models robust. But with the advent of digital computers, some mathematically inclined control systems engineers wanted to use optimisation techniques to make “optimal control rules”. These controllers were a disaster when implemented, as I discussed in an earlier article. Economists have embraced optimisation throughout their models, even though they were rejected by engineers based on their practical experience.

Robust Deterministic

Robust Deterministic Control Theory (often referred to as “H-∞” theory, because “H-infinity” sounds cool) was the initial response of the theoretical wing of control systems to these problems. To translate this viewpoint into the terms used by economists, this is an approach that incorporates uncertainty. The uncertainty takes two forms:

1. Although we always work with deterministic models, we do not really know the “true” deterministic model. We assume that the “true” model is “close” to our base case linear model. “Close” is defined in terms of the mathematical properties of an unknown perturbation to the model. In our example, this means that the parameters of our linear model may be off, or we may even be missing some dynamics entirely (e.g., the tendency of the air frame to flex). This means that the “true” state variables may include new variables that are not included in the set of state variables of our base case model. (Technically, the dimension of the state vector may change, as well as the model dynamics.)
2. We also assume that the system is hit with unknown somewhat arbitrary disturbances. In our example, these would be air pockets. But we have to be realistic about what we can deal with. No control rule could keep a plane that has been flown into the side of the mountain on its original desired flight plan. We capture this mathematically by insisting that the disturbance has finite energy, and shaping its frequency characteristics.

Even though this framework has a very realistic amount of uncertainty, randomness, in the form of probability theory, does not appear.

Stochastic Models

In stochastic (random) control theory, the deterministic base case model is augmented by two forms of randomness:

1. The evolution of the state variables include random disturbances. This means that the differential equations turn into messy stochastic differential equations, with Brownian motions and so forth. However, the state variables remain the same as the base case model.
2. The system is also hit with random impulses and measurement errors. As in the deterministic case, things like air pockets would fall under this category.

Although there appear to be significant differences between these two philosophies, they end up giving the same answers for linear systems. From the point of view of the publish-or-perish environment in academia, this is brilliant. What this means is that once the deterministic robust control results are derived, it is possible to re-derive the exact same thing within a stochastic framework.

Unfortunately for the stochastic approach, the correspondence breaks down once nonlinearities are introduced. It is at this point the shortfalls of the stochastic approach are apparent – the approach assumes that the model is correct, there is just some randomness involved. This still assumes too much about the validity of the model; we can have dynamics that are not modelled which are perverse. For example, the aircraft frame may resonate at a particular frequency. This could be very bad for a control rule that causes oscillations at that frequency, and this precisely not a random effect.

### Randomness Is Necessary In Some Contexts

Although I am not a huge fan of stochastic modelling frameworks, they are necessary in some cases. For example, option pricing makes little sense within a deterministic model, even one with model uncertainty. Within engineering, the related field of communication systems relies heavily on probability. If communication systems were designed with the same conservative design philosophies of control engineers, cell phones would weight ten pounds. But design conservatism has its place; people are generally tolerant of some static on their phone calls, but a plane landing 100 metres before the runway is generally not viewed as acceptable.

### How This Relates To Economic Models

This digression into the philosophy of control systems design is applicable to economic modelling. Academic economists have now embraced stochastic mathematics, and they have layered their theory in unreadable stochastic jargon. They presumably view this approach as being mathematically sophisticated. However, this mathematical sophistication does not translate automatically into modelling sophistication.

Using random variables to stand in for uncertainty creates the assumption that the model and its parameters are correct. But the mathematical complexity introduced by using stochastic concepts makes it difficult to understand the models and the model dynamics.

I prefer the approach taken in the Stock-Flow Consistent models text of Godley and Lavoie. Deterministic models are developed and solved, allowing the users to see what the actual nonlinear dynamics of the models are. Since the models are tied together by accounting frameworks, we know the dynamics are internally coherent. Although it is not the direction they have taken their theory, one could at least hope to rigourously incorporate model uncertainty within their models. Even though the models will inherently be weaker than models of engineering systems, it is possible to see what forms of uncertainty are the most dangerous by trial-and-error mathematical experimentation.

This comes up in my modelling of fiscal dynamics. Rather than bury my discussion of the government budget constraint under layers of stochastic sludge, I prefer to discuss the constraint in terms of an unknown deterministic trajectory. This is not being simplistic, rather it is following a more realistic modelling formalism. I will illustrate this in an upcoming article.

### Appendix – Game Theoretic Approach

I skipped over another way of deriving the same “robust” linear controls, which is using a game-theoretic framework. This framework is characterised by:

1. The model is assumed to be correct, and there are no “random” shocks.
2. The system is hit by perturbations generated by a malevolent “opponent”; the control law is the solution to a mathematical game in which the influence of the opponent is minimised.

In terms of my flight control example, this framework says that model is perfectly known and there are no “random” disturbances like air pockets, but the aircraft is plagued by 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 very different, it is possible to generate the same control laws for a linear system as the other approaches. Once again, this is useful for academics, as it possible to re-derive existing results in a new framework, padding the publication count on resumés.

I would argue that this was not a good philosophy for modelling real world systems. The approach has no concept of uncertainty, rather you are certain that your model is correct, and you are certain that you are facing an opponent that will always choose the worst possible strategy to destabilise your system.

The fact that one can derive the same controllers for linear systems is purely an artefact of linear systems – you can derive the same control laws in a huge number of ways.* But this equivalence breaks down when you hit nonlinearities. And the philosophy of reasoning in terms of an optimisation (the control law is the optimal law for the least-bad outcome) is dangerous. The malevolent opponent will always exploit pathological properties of the mathematical system in order to come up with the most dangerous disturbance. This means that the solution will be dominated by the most extreme part of your mathematical model’s behaviour, which is probably the area of the model where its fit to reality is the worst.

It almost goes without saying that this was the approach that was embraced by some economists when they wanted to apply some robustness concepts to economic theory. Economists tend to be attracted to game theory, and so this was a natural fit. The fact that this approach misses the entire objective of robust control – dealing with model uncertainty – seems to not have been noticed.

Since this part of the literature is almost unreadable and makes little sense from a modelling perspective, I am in no hurry to cover it in detail.

Footnote:

* The solution for the "optimal robust controller" is given by an algebraic Riccati equation. You just set the final solution to be the same Riccati equation, and then you work backwards to determine your initial modelling framework.  (c) Brian Romanchuk 2014