Why Parameter Doubt Is An Inadequate Modelling Strategy

We alive inwards a basis of uncertainty. One strategy used inwards economic science is to contain the notion of parameter uncertainty: nosotros receive got the right model, but the parameters receive got some random variation from a baseline value. This strategy is highly inadequate, as well as has been rejected past times robust command theory. The belief that nosotros receive got the right model was an underlying premise of optimal command theory, as well as the weakness of this premise inwards practise explains why optimal command theory was largely abandoned inwards controls engineering. (Interestingly enough, it persists inwards Dynamic Stochastic General Equilibrium (DSGE) models).

In this article, I give an illustration of an abject failure of parameter doubtfulness as a notion of model uncertainty.

Why Care About This Example?

The illustration I give hither is deliberately simple: it is a model amongst only a unmarried parameter. It is non a recognisable economical model. However, it is slow for the reader to experiment amongst this model to validate the failure of parameter uncertainty.

Optimal command failed inwards practise as a outcome of the full general principles illustrated past times this example. To move clear, this is non a model arrangement that caused them difficulty inwards particular. I volition provide to optimal command afterwards I depict the example.

One could hold off at the simplicity of the illustration as well as combat that modern economists are far besides sophisticated mathematically to brand such an error. However, such arguments band hollow when nosotros consider that the optimal command engineers roughshod into a similar trap.

Firstly, the optimal command engineers were to a greater extent than sophisticated mathematically than modern economists. They developed the mathematics that DSGE macro modellers right away use. H5N1 major driver inwards the evolution of optimal command theory was the path-planning required to larn a manned mission to the Luna inwards the 1960s: they were literally rocket scientists.

Secondly, they were working on engineering systems. Macro economists constantly complain that they cannot do experiments on their systems of written report (except past times using DSGE models, of course!). The optimal command engineers had the luxury of doing most whatever tests they wanted on the physical systems they studied to create upward one's heed the dynamics.

Even amongst these advantages, optimal command all the same failed as a pattern technique (outside of said path-planning problems).

The Example

(This article uses a chip of mathematics. Equation-averse readers may move able to skip most of them. The figures were generated past times running simulations that were computed using my sfc_models equation solver. The code is given below, as well as is available inwards the evolution branch of the projection on GitHub.) 


We receive got an extremely uncomplicated system, as well as nosotros receive got a baseline model for it, denoted $P_0$. (We work $P$ to correspond "plant," which is the traditional command systems advert for such a system, as well as it technically is an operator from a discrete-time unmarried input to unmarried output.)

1. We know that the steady-state gain from the input $u$ to the output $x$ is 1. That is, if nosotros receive got a constant input of $k \in R$, the output converges to $k$.
2. We believe that the model lies inside a flat of models defined past times a parameter $a$. The baseline model has $a=.05$, as well as that $a \in [0.005, 0.09]$.
3. The input $u$ is existence ready past times the controller of the system, as well as thus has access to the data.

The parameterised model is defined by: $x[k] = (1-a) x[k-1] + a u[k-1], \forall k > 0, x(0) = 1.$

(Following electrical engineering tradition, I announce the discrete fourth dimension index amongst $[k]$.)

We assume that nosotros receive got access to a nifty bargain of historical information for this system. Once nosotros receive got validated that the steady nation gain is equal to one, the resulting linear arrangement has to receive got the to a higher house shape (or move related to it past times a linear scaling of $x$).

For some reason, nosotros receive got argue to move sure that the parameter $a$ lies inwards the interval $[0.005, .09].$ We tin run simulations of the model amongst 3 parameter values (the baseline $a=0.05$, addition the 2 endpoints of the interval) to larn the fake answer of a stair rising of the input $u$ from 1 to 2. (Figure below.)

We tin as well as so compare the output of the truthful arrangement (in red) to these simulations. We come across that the truthful stair answer is quite roughly the baseline model. (An eagle-eyed reader powerfulness spot the job here, but this would move hard if I buried the truthful arrangement answer amongst random noise.)

As nosotros tin see, the arrangement marches at a leisurely stair from 1 to 2, next the alter inwards the input, However, 1 could imagine that this wearisome adjustment would move seen as suboptimal, as well as nosotros tin as well as so speed the answer up.

We desire the output $x$ to runway a reference signal $r$. We define the tracking mistake $e[k]$ as $x[k[ - r[k]$. We would similar the tracking mistake to receive got the dynamics: $e[k] = \frac{1}{4} e[k-1].$

We tin attain this past times setting $u$ so that it cancels out the existing dynamics for the baseline model, as well as strength $x[k]$ to emulate the to a higher house behaviour. This is achieved past times setting a command law:

$u[k] = \frac{-0.7 x[h] + 0.75 r[k]}{.05}.$

If nosotros copy the unopen loop responses for our baseline system, as well as the systems at the extremes of the parameter set, nosotros come across that demeanour is relatively acceptable for all 3 models.

However, if nosotros apply the command police to the actual arrangement model, nosotros destination upward amongst unstable oscillatory demeanour (in red).

In other words, although parameter doubtfulness covered the opened upward loop demeanour nicely, actual unopen loop demeanour was goose egg roughly what was implied past times the extremes of the parameter set.

The argue for this failure is quite familiar to anyone amongst a background inwards command engineering -- or taking showers -- a lag betwixt input as well as output. In this case, the model is perturbed from the master copy model past times adding 2 lags to the input signal. This is plenty to brand the resulting arrangement unstable. The same lawsuit is felt if 1 is impatient inwards setting a shower temperature. If y'all move along setting the dial temperature based on the electrical flow H2O temperature, y'all volition end  upward repeatedly overshooting betwixt besides hot as well as besides cold. You necessitate to give fourth dimension to permit the H2O to period of time through the piping to come across whether the temperature dial needs farther adjustment. (An automatic shower temperature command is a touchstone command systems engineering project.)

In summary, systems tin deport inwards ways quite different than predicted alone past times parameter uncertainty. Missing dynamics tin move fatal.

So What?

The simplicity of this illustration powerfulness brand some readers impatient. "That's a lag. We know how to contain them into our estimation strategy past times adding parameters."

Not really. Optimal command engineering did non neglect because they did non receive got plenty parameters. I had to work a fairly drastic lag to destabilise the arrangement only because the underlying arrangement (a low-pass filter) is remarkably stable. If the arrangement had oscillatory dynamics of its own, much to a greater extent than subtle perturbations to the model would attain destabilisation.

The optimal command strategy failed because it assumed that the model was known. The methodology was:

1. Take assumed model of the plant.
2. Calculate the optimal trajectory (using some objective function).
3. Force the arrangement trajectory to follow the optimal trajectory past times cancelling out the assumed constitute dynamics.

This physical care for was equivalent to determining the inverse of the mathematical operator of the plant, as well as using that inverse to calculate the target dynamics. Unfortunately, cancelling out an operator amongst an inverse creates severe numerical instability unless the matching is perfect. (Your arrangement matrices destination upward existence ill-conditioned.) This numerical instability made optimal command laws useless inwards practice.

For those interested inwards the history of optimal control, I desire to underline that I do non recollect that they were misled past times parameter uncertainty. When I did my doctorate inwards the early on 1990s, optimal command theory was only studied as a historical curiosity; nobody actually cared what their exact idea processes were. Based on my hazy retentiveness of the literature (and the only optimal command textbook I own), at that topographic point was no formal notion of model uncertainty. Instead, the explanation for mismatches betwixt models as well as reality was explained as follows.

1. Our mensuration of outputs was corrupted past times random noise.
2. Model dynamics included additive random disturbances.

These issues were dealt amongst past times using the Kalman filter to guess the nation despite the noise, as well as the assumed stability of the arrangement would grip (finite energy) external disturbances. (This should audio familiar -- this is just the strategy that DSGE modellers inherited.) As tin move seen, this is inadequate. Incorrect dynamics tin imply a consistent strength driving the actual output away from the theoretical trajectory, a possibility that is non included inwards random disturbance models.

I am unaware of optimal command theory that dealt amongst parameter uncertainty. However, it seems probable that engineers would similar receive got done basic "sensitivity analyses" where they varied parameters. They would receive got done so without the assistance of whatever formal theory to remove them, as well as would probable receive got ended upward amongst an approach similar to what I sketched out to a higher house (what happens if a parameter is close its assumed limit?).

In whatever event, robust command -- sometimes called $H_\infty$ command -- abandoned the approach that nosotros know the constitute model amongst certainty. We all the same receive got a baseline model, but nosotros desire to ensure that nosotros stabilise non only it, but a "cloud" of models that are "close" to its behaviour. There are formal definitions of these concepts, but readers would necessitate to move familiar amongst frequency domain analysis to follow them.

Real World Implications?

In the modern era, nosotros are unlikely to come across disasters similar to those created past times the application of optimal command inwards engineering. People are aware of the number of variable lags, as well as so policy is unlikely to move as aggressive as they were inwards engineering. Furthermore, modern mainstream applications of optimal command to policy is inwards the domain of involvement rates. Fortunately, involvement charge per unit of measurement changes receive got negligible lawsuit on existent economical variables, as well as so policymakers cannot do a lot of damage.

Instead, the applications of the notion of model doubtfulness would move to a greater extent than analytical. We powerfulness commence to inquire ourselves: although nosotros managed to fit some model to historical data, is that fit spurious? How large is the flat of models would yield an every bit expert fit? We only necessitate to move along inwards heed that varying a parameter is a much weaker status that having unknown model dynamics.

Code

(c) Brian Romanchuk 2017