The control system engineering perspective on climate control

The Control Engineer’s View of Global Temperature Control

Control system engineers think of dynamic systems, e.g. an electric vehicle or world climate, in nested block diagrams where every block represents in itself a dynamic system with its inputs, outputs and internal states. The dynamics of an entire system, being much more than the dynamics of every isolated block, becomes often obvious from studying system block diagrams, with all mutual dependencies and in particular with its feedback loops. These loops can be amplifying or attenuating, representing virtuous or vicious cycles, depending on the point of view.

In order to build a representation of a complex system by combining blocks to compound blocks, and then those again to even more complex diagrams, we need to start with the simplest conceivable blocks. The simplest block is a gain, i.e. the output is the input with a factor applied to it, and without any internal state. If the gain is greater than one, the block amplifies the input, if the gain is smaller than one, the block attenuates the input. The next common block is an integrator which has a single state: its internal value which is the accumulation of the input over time. For instance, a water butt collecting rain water as an input can be thought of as an integrator with the water level being the internal state; it accumulates or integrates the incoming water with the level steadily increasing (until being drained).

What has all that to do with climate change? Well, the world is concerned with how this temperature rise can be contained or controlled. As control system engineer (sic!) I ask myself, how could the world climate and the mean temperature to be controlled be represented in a simple way? What would a controller look like? The starting point is to obtain a simple understanding of the temperature rise (system state and output) with respect to fossil fuel consumption as one of its controllable inputs.

Different authors ascribe climate change, the rise of temperature, to green-house gases from burning of fossil fuel. Among others, Tim Berners-Lee states in his book “There is no Planet B”, Appendix “Climate Change Basics”, Point 2, that “the temperature rise corresponds roughly to the total amount of carbon ever burnt”, i.e. the temperature rise is proportional to the accumulated carbon released or burnt. As control engineer I would say that this statement can be rephrased. Climate change with respect to an average temperature can be expressed like:

temperature rise = some gain x integral of carbon burn rate

with some proportional factor, a gain, and the carbon burn or release rate which is the amount of carbon burnt and released as CO2 into the atmosphere per time unit. If I made the simplistic assumption of a linear system for a certain point of operation and for a short period of time, as well as to say that the obtained temperature is the accumulated temperature rise, or in other worlds, the obtained temperature is the integral of the temperature rise, then the previous statement would translate to the following relationship:

temperature = some gain x integral of the integral of carbon burn rate.

If we wanted to control the temperature, how would we go about that? Control design is the task of finding a way to influence the behaviour of a controlled system in a way to obtain the desired output. The simplest control law is proportional control: you counteract proportionally to the observed deviation. Many system can be controlled in that way which is very intuitive. Free markets operate like that, they compensate in proportion to a perceived deviation; they exhibit a self-regulatory property which is ubiquitous and seems to work well in most cases . In our case, however, if we want to control the mean world temperature, and if our influencing variable was the fossil fuel burn rate, how would the control law look like that allows us to stabilise that temperature?

First, we need to understand the system we try to control. In control engineering terms, and without going into the details, the climate system or plant model is in essence a double integrator which, on its own and in addition to its varying gain, shows a constant phase lag of -180 degrees. Phase lag is in large terms equivalent to a frequency dependent delay or latency. If you excite a system, it will respond with some delay which depends on the excitation frequency. For instance, if you push a swing at a low frequency, it will follow your push effort. When you push at higher and higher frequencies, the motion will lag more and more until it is in anti-phase, i.e. the output does exactly the opposite of the input, the output is at -180 degrees phase lag to the input.

How can a system with -180 degrees phase be controlled? Again, without going into the details of control theory, the temperature can never be stabilised with proportional control alone, i.e. with the effort being proportional to the temperature deviation. Proportionally acting market forces cannot stabilise the system. In order to stabilise such as system anticipatory measures are needed, e.g. an additional effort being proportional to the trend, the time derivative of the temperature. In control engineering terms, a lead element is needed, an element that raises the phase and buys some time. Hence anti-climate-change efforts need a component that is proportional to the change of temperature or even proportional to the rate of temperature change.

In more general terms, the double integrator system is similar to systems with two very large time constants, at least within a certain frequency range, i.e. systems that change very slowly like a huge inertia of a large ship. As control engineer it is perhaps save to say that all plant models that have very large time constants are very difficult to control if a fast closed loop behaviour is desired such as a response within a certain time, e.g. quickly turn an oil tanker, or get the temperature down within a certain time frame in our case. The problem is even exacerbated by delays in the line which add a lot of phase lag. Given a simple proportional control scheme, the result is usually an oscillation of some kind.

Most ecological systems expose usually long time constants. With a little systems or control system engineering knowledge, it should be obvious that the only way to control systems with long time constants and delay is providence, i.e. trying to anticipate the behaviour of the plant model and act before it is too late. Leaving matters to free market feedback mechanisms, which are usually only proportional in the counter-action, results in instability. It takes too long to internalise external cost factors, too long until these become proportional drivers of change in a free market.

The bottom line is that it is necessary to make politicians and economists understand that certain natural effects need to be controlled with modelled anticipation. We cannot afford to wait for the consequences and act then proportionally, i.e. make more effort as the effects increase. One needs to do more: some form of command economy on an international scale, agreements to act in advance.

Contributes to the technological foundations for the self-driving revolution at Five, UK. Interested in sustainable economies and renewable energy.

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