The Robotics WEBook

An online textbook about robots and other mechatronic systems

Systems & control

This Chapter introduces what is commonly denoted as systems and control, and summarizes its major paradigms. The following activities are part of every project that needs to design a (complex) controlled system: modelling, analysis, control, signal processing, and identification.

All systems and control paradigms share a similar intellectual goal: to use as few different concepts and terms as possible to express as many possible applications and properties as possible. Often, practitioners of a particular framework are particularly proud of the (subjective) “elegance” with which their beloved framework reaches this trade-off.

All systems and control paradigms share a similar practical goal: to design a controller for a given system. Or better still, (to be allowed) to design the whole system from scratch, including the controller) that gives “the best” trade-off between the conflicting aspects of the “holy grail” of performance, stability, and robustness.

Modelling

Definition: making an abstraction of the real world, with a level of detail adapted to the goal of the system, and represented as a set of mathematical descriptions.

Topics to be described: input-output description; state and state-space description; linear, affine and nonlinear models; block diagram; time-varying dynamic system; system modelling by interconnecting elementary components; (power) ports for physical energy exchange; Bond Graphs, Kirchoff's laws; constitutive relationship; iconic representations; multi-domain model; lumped parameter (finite-dimensional) and distributed/continuous parameters (infinite-dimensional) system; Euler-Lagrange description, Hamiltonian description, Legendre transformation; causality; decoupling of (acausal) modelling by interconnection; discretization of continuous systems; canonical description; …

Analysis

Definition: when a model has been constructed, the properties of that model that are relevant to the control task can be calculated and evaluated.

Topics to be described: order of a system; transfer function, pole, zero; observability, controllability; overshoot; 2nd order system with damping and natural frequency, break frequency, bandwidth, steady state, transient behaviour, time constant, rise time, z-transform frequency-domain characteristics (resonance, phase/amplitude plots, Nyquist and Bode diagram, decibel, decade); conservation law, Casimir function, structural invariance; invariance with respect to transformations in space, time, coordinates; (non)holonomic constraints, scleronomic and rheonomic constraints; virtual motion and work; d'Alembert, Jourdain, Maupertuis, Gauss principles of least constraint; determination of (physical and computational) causality;

Control

Definition: the real-world system is interfaced by “actuators” (to change the energy content of the system), and “sensors” (to collect information about parameters in the mathematical model), and a computer program calculates the “optimal” actuator outputs to bring the system parameters to a desired state, along a desired dynamic trajectory.

One always designs a controller with a given set of systems in mind, so the choice of models used in the controller (and the identification!) depends on this (often implicitly given) set of systems.

Basic trade-offs in control: feedback versus feedforward; stability versus performance versus robustness.

Topics to be described: feedback (closed loop) and feedforward (open loop), stability and robustness, first and second order systems, pole placement, PID, Luenberger observers, passivity, performance, energy shaping, damping injection, sliding mode, gain scheduling, tuning, cascaded loops, model-based and adaptive control, pole placement, (non)collocated control, …

Signal processing

Definition: the raw sensor signals undergo a sequence of mathematical transformations, in order to reduce noise, to increase the relevant parts of the signal, to transform to the time domain, etc.

filtering, smoothing, prediction; ARMAX; low-pass, band-pass, high-pass; correlation, convolution, covariance; pattern matching: outlier detection, peak detection, periodicity detection, jump detection, Principal Component Analysis; adaptive filter; harmonic filtering; …

Identification

Definition: information about the relevant model parameters can often not derived directly from the (processed) sensor signals, so that information about the “hidden” model parameters must be derived from the “measured” parameters, via the mathematical relationships contained in the system model.

Note that developing models is one thing, finding sufficiently accurate values for the model parameters for a given real system is another thing.

Topics to be described: order of a system, persistent excitation, optimal experiment design, least-squares identification, …

System design

Definition: the integration (modelling, analysis, synthesis) of all above-mentioned activities. The physical system to be controlled is not an unchangeable given fact, but can be adapted in order to make control and identification easier, more robust, or more performant.

Bird's eye view: lumped-parameter control is “solved” (= mature systematic approach, major agreement on what works and what doesn't, …), distributed-parameter systems are still too computationally expensive for on-line use, and the research challenges lie in the real integration of finite- and infinite-dimensional systems in the same control design approach. Control is still stuck in “ODEs”, while modelling has matured also in “PDEs”.

Control frameworks and paradigms

A control framework, or a control paradigm is the set of models, thought patterns, techniques, practices, beliefs, systematic procedures, terminology, notations, symbols, implicit assumptions and contexts, values, performance criteria, elementary model building blocks, … shared by a community of scientists and engineers in their modelling, analysis and design of control systems. So, a paradigm is a subjective, collective, cognitive but often unconscious view shared by a group of humans, about how the world works and can/should be controlled.

The following paragraphs summarize the major assumptions of the most popular paradigms in systems and control. None of the paradigms is better than all the others in all possible circumstances of for all possible applications. Getting to know the different paradigms is always helpful, even for people that keep working within the same paradigm: the more one understands of the other paradigms, the better one realizes where the tacit assumptions of one's own paradigm lie, and in what way they limit one's work. Anyway, discussions between “believers” of two different paradigms tend to be difficult, not in the least because of the different terminology and the different value system.