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Looking deeper into the devices – what do we discover with control theory?

Control theory is a branch of science that tries to find an algorithm to control a time-varying object so that it meets certain conditions. Such an object may be a heater, an electrical circuit, a robot or an autonomous car. The simplest algorithm is “on-off”, which is used to control e. g. pumps. I rely on the fact that if the water level is too low, more water is poured in. When the level is right, such a pump is switched off.

However, such control may be too aggressive for the system and may cause damage. An alternative is proportional control (P), which for pumps would mean that the greater the difference between the current and the target water level, the faster the water level rises. As a result, the pump slowly slows down as the difference approaches zero.

The P control is part of the most popular control algorithm in the industry, the PID controller [1]. Different versions of it are used for temperature control, robot arms, hydraulic presses. PID is a popular algorithm because it is simple to design, implement, perform mathematical analysis, or even combine with other algorithms such as gravity compensation [2].

Unfortunately, if the controlled object is too complicated, problems arise. For example, to control the drone smoothly, 6 controllers from the PID family are used at the same time [3,4]. In addition, the PID controller does not take into account limitations (e. g. maximum speed) or optimality of the solution. Predictive control (MPC) is a much better algorithm. It has three components:

The cost function tells you how good a solution is and can take into account any factors such as mission time or fuel consumption. A system of equations and inequalities that contains information about restrictions imposed by the designer (e. g. speed limit) and an object model. The model contains information about the physics of the object, which allows you to predict the future.

Solver that is able to solve the system of equations and find possible and allowed trajectories of the system. Then choose the most optimal among them. Such solvers are already optimized in terms of speed and can be found ready-made solutions in different programming languages such as Python (SciPy) [5] or Matlab (optimization toolbox) [6].

MPC [7] is able to control even such complex systems as human-like robots from Boston Dynamics [8, 9] or autonomous car control [10]. Unfortunately, MPC also has disadvantages. The main disadvantage is that the larger the equation layout and prediction window, the more time the computer needs to find a solution. 

Drone control in practice

In the case of a drone, a delay of 0. 1s may already be too long and cause a dangerous flight. In addition, solving the system of equations with the drone model is not easy, since the system contains angles, and thus also functions of sine and cosine. In order to quickly find a solution, the position is approximated and assumed to be small angles [11], for which a cosine of one and a sine of the angle is assumed.

Due to such simplifications and random factors (e. g. wind), predictions contain an error, which may cause system constraints to be temporarily violated. Robust MPC takes this difference into account and for the trajectory to be taken into account, a group of similar-real trajectories must be allowed. The most popular implantation is Tube MPC [12]. Unfortunately, in order to use TMPC in practice, additional simplifications are often required, which causes the algorithm to sacrifice optimality for security and speed of calculations.

Research is currently underway on how to accelerate robust MPC. One of the interesting and fresh ideas is to use neural networks to replicate the algorithm. Thus, the calculation time of the robust MPC is only important during the training of the neural network and not during operation of the machine. Such neural network was already used to control the drone [13,14] or the KUKU robot arm [15].

It is worth noting that control theory, as a scientific discipline, significantly impacts various aspects of the Polish market, primarily the industrial sector.

  1. “Advanced PID Control”; K.J. Åström, T. Hägglund
  2. “Low-Cost Automation for Gravity Compensation of Robotic Arm”; W. Montalvo, J. Escobar-Naranjo, C A. Garcia, M V. Garcia.
  3. https://nl.mathworks.com/videos/drone-simulation-and-control-part-1-setting-up-the-control-problem-1539323440930.html
  4. “Successive Loop Closure Based Controller Design for an Autonomous Quadrotor Vehicle”; A.M. Singh, D.J. Lee, D. Hong, K.T. Chong
  5. https://docs.scipy.org/doc/scipy/reference/optimize.html
  6. https://www.mathworks.com/products/optimization.html
  7. “Model Predictive Control: Theory, Computation, and Design”; J.B. Rawlings, D.Q. Mayne, M.M. Diehl
  8. https://www.google.com/search?client=firefox-b-d&q=boston+dynamics+youtube#fpstate=ive&vld=cid:de355194,vid:-e1_QhJ1EhQ
  9. https://www.bostondynamics.com/resources/blog/picking-momentum
  10. “Comfort-Oriented Design of Model Predictive Control in Assisted and Autonomous Driving”, S.Luciani, A. Bonfitto, N. Amati, A. Tonoli
  11. “A SIMPLE TUBE CONTROLLER FOR EFFICIENT ROBUST MODEL PREDICTIVE CONTROL OF CONSTRAINED LINEAR DISCRETE TIME SYSTEMS SUBJECT TO BOUNDED DISTURBANCES”: S.V. Raković, D.Q. Mayne
  12. “Trajectory Tracking and Stabilization of a Quadrotor UsingModel Predictive Control of Laguerre Functions”; M. Chipofya, D.J. Lee, K.T. Chong
  13. https://www.youtube.com/watch?v=28zQFktJIqg
  14. “Demonstration-Efficient Guided Policy Search via Imitation of Robust Tube MPC”; A. Tagliabue, D. Kim, M. Everett, J.P. How
  15. “Safe and Fast Tracking on a Robot Manipulator: Robust MPC and Neural Network Control”; J. Nubert, J. Köhler, V. Berenz, F. Allgöwer, S. Trimpe
Filip Surma
Bio:
I received my Master’s with distinction in Robotics from the University of Birmingham, UK, in 2020. I am currently a PhD candidate in the Mathematical Decision-Making group at the Department of Control and Operations, TU Delft.
Written by:

Filip Surma, PHD student

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