Control Theory meets Machine Learning

Control Theory: Introduction

This blog focuses largely on emphasizing the importance of control theory and its application on complex systems through dimensionality reduction, machine learning, and dynamical systems modeling. First, I will cover a primer about control theory and its connectedness to the so-called ‘Data science.’ The very purpose of an effective control system is its ability to manipulate its behaviour for a given engineering objective actively. The study and practice of manipulating dynamical systems (here, dynamical systems could mean anything from biological to financial models)are broadly known as control theory. It is one of the most successful fields combining the area of applied mathematics and practical engineering. 

Control theory is indivisible from data science, as it relies on sensor measurements (data) obtained from control systems to achieve a given objective. In fact, control theory deals with present data, as a successful technique modifies the system's dynamics, thus changing the characteristics of the measured values. Control theory forces the reader to believe in reality, as simplifying assumptions and model approximations are tested.

Control theory has helped renew the modern technological and industrial landscape. Examples abound, including cruise control in automobiles, fly-by-wire autopilots in aircraft, industrial automation, packet routing in the internet, commercial heating ventilation and cooling systems, stabilization of aerial vehicles, and PID temperature and pressure control in modern espresso machines, to name only a few of the many applications. In the future, control will be increasingly applied to high-dimensional, strongly nonlinear, and multilevel problems, such as turbulence, neuroscience, finance, epidemiology, autonomous robots, and self-driving cars. In these future applications, data-driven modelling and control will be vitally important. All this caused the emergence of ‘Big data technologies.’ Below is a wonderful map created by Brian Douglas that illustrates the entire span of what we can expect and learn from control theory.

How Control Theory is related to Machine Learning ?

I will begin with the pivotal building block of the modern machine learning paradigm: the perceptron. This was a hardware structure built in the ’50s by Rosenblatt to mimic the real neural network in our brains. It came out of control theory literature when people tried to identify highly complex and nonlinear dynamical systems.

The classic sigmoid nonlinear activation function, often used in machine/deep learning as a nonlinear activation, came out of control theory literature. Neural networks, artificial neural networks were first used in a supervised learning scenario to control theory. Kurt Hornik was the first to identify that neural networks were universal approximators. Without classical control theory, we could say there would be no back-propagation (invented by Rumelhart & Hinton in the ’80s based on inspiration from control theory); there would probably not be the LSTM(invented by Horchreiter in 1996) which are used in modelling tapped delay lines in memory-based artificial neural networks. These have found immense use in speech recognition, language models, or time-series sequences.

The antagonism between exploration and exploitation in reinforcement learning is known in control engineering as the conflict between identification (or estimation) and control. You could arguably say massive reinforcement learning problems arose out of Control Engineering research. I am attaching a link to Andrew Ng’s Thesis for a perspective.

If you go through the works of recent machine learning scientists, you’ll find control jargon camouflaged into new diction to make their ideas sound new. What they claim back-propagation, for example, is no more than old-fashioned calculus-based differential chain rule technique. Variants of recurrent neural networks are the only NARX models that you would encounter in any system identification literature. So to answer the question, modern machine learning is a derived class of classical/Modern control theory.

Machine learning is basically functional learning and is vulnerable to errors. i.e., no matter how big your training set is and how well your algorithm is, it’s not going to guarantee that the outcome it predicts is correct. It’s based on probability. Control theory, contrastingly, is based on rigorous mathematical proofs. Provided you model a system dynamics correctly ; you can rely on control theory.

The best proposition would be to use both in autonomous systems building.

As an undergraduate student majoring in instrumentation and control engineering, I found a vast overlap between control theory and machine learning. Most systems we deal with are complex control systems that need to be controlled; this can range from business to biological models. I observed that the approach used in solving machine learning problems is similar to that of control theory. The problem statement in an optimal controller design such as linear quadratic regulator (LQR) requires a cost function that has to be minimized under a specific range and conditions. The analogy of solving machine learning problems is akin, except that now we have to deal with a large dataset and build an intelligent control system. A strong grasp of both linear and non-linear control theory provides a better intuition while solving machine learning problems.

Neural Networks which have been the forefront of Deep Learning and had great success recently are similar to control problems. We consider error (supervised learning) and the input for calculating the output. If the output of Neural Network doesn't match the expected output, then the loss is calculated (loss function or cost function may vary across different problems), and the loss is further used to calculate gradients which further updates the parameters of the network with backpropagation. So, this seems like an optimization problem. Almost, all of the problems in modern machine learning and deep learning are optimization problems. Same is the case for control theory. 

Final Verdict

As you read in this blog, many concepts of machine learning have been derived from control theory. Both of the fields have rich history between them. Machine Learning and Control can be considered as an optimization problem. Control Theory provides useful tools for Machine Learning. Conversely, Machine Learning can be used to solve large control problems. PID which is a type of controller used in control theory for solving control problems can also be tuned with the help of machine learning algorithms. We can even replace PID controllers in some cases with deep learning algorithms (like LSTM) to make the control process more faster.

It will not be wrong to say that, 

"Machine Learning is a derived class of modern control theory"