*Series EditorMaurice Charbit*

First published 2016 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Ltd

27-37 St George’s Road

London SW19 4EU

UK

www.iste.co.uk

John Wiley & Sons, Inc.

111 River Street

Hoboken, NJ 07030

USA

www.wiley.com

© ISTE Ltd 2016

The rights of Smain Femmam to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2016957652

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN 978-1-78630-127-7

This preface gives an overview of signals and systems, generalities and notions of process control and regulation.

The main objective of these two volumes is the analysis and the study of linear, time-invariant, causal signals and deterministic systems of finite dimensions. We will focus our efforts on defining a set of tools useful to automatic control and signal processing, after which we will discuss methods for the representation of linear dynamic systems for the analysis of their behavior. Finally, the identification and the synthesis of control laws will be addressed for the purpose of stabilization and regulation in systems control. Chapter 6 of the other volume [FEM 16] will be dedicated to the use of the Nao robot for a specific application; in this case, home care service.

Whether in the analog or digital field the study of the characteristic properties of signals and systems, the knowledge of mathematical tools, processing and analysis methods are constantly evolving and of late are increasingly significant. The reason is that the current state of technology, in particular of electronics and computer science, makes possible the implementation of very advanced processing systems, which are effective and increasingly less expensive in spite of their complexity. Aims and requirements generally depend on applications. Figure 1 presents the connections between the various disciplines, the scientific and technological resources for their operation with the aim of processing signals or automatic control for the operation or the development of current applications^{1}.

In all areas of physics, for study, analysis and understanding of natural phenomena, a stage for modeling and for the study of the structure of the physical process is necessary. This has led to the development of techniques for modeling, representation and analysis of systems using a fairly general terminology. This terminology is difficult to introduce in a clear manner but the concepts which it relies upon will be defined in detail in the following chapters.

Signal processing concerns the various operations carried out on analog or digital physical quantities with the purpose of analyzing, interpreting and extracting information. These operations are illustrated in Figure 2.

The mastery and the implementation of signal processing techniques require the knowledge of a number of theoretical tools. The objective of this book is to establish the basic concepts of the theoretical study and clarify common processing methods.

A physical process is divided into several components or parts forming what is called a system. This is the case, for example, of an engine that consists of an amplifier, power supply, an electromagnetic part and a position and/or speed sensor. The input to the system is the voltage applied to the amplifier and the output is either the position or the speed of the rotation of the motor shaft.

Among the objectives of the control engineer is the modeling, behavior analysis and the regulation or control of a system, aiming for the dynamic optimization of its behavior. The operation of the system or control is designed to ensure that the variables or system outputs follow a desired trajectory (curve over time in general) or have dynamics defined by the specifications document. For temperature regulation of a speaker to a reference value, one of the following diagrams can be used. Details of vehicle operation is shown in Figure 3.

The objective of automatic control is to design control and operation systems that are able to assign to a dynamic process (physical, chemical physical, biological, economical, etc.) a behavior defined in advance by the operator based on the requirements specifications. For example, we can consider speed regulation of a car that gives the process (the car) a previously determined speed, regardless of the disturbances that may occur (variation of the slope, etc.). Other examples include a radar antenna alignment system for the monitoring of the trajectory of an airplane or a satellite, and an air conditioner designed to stabilize the temperature at a constant value fixed in advance.

A process can be defined by establishing relationships between input and output quantities (this will be formally defined in different ways throughout this book. It is represented in Figure 4.

In the example of the car, the output is the speed and inputs may be the position of the accelerator pedal, the slope of the road and/or any other physical quantities that have an influence on the speed (the output of the system). Inputs consist of variables that can be manipulated (position of the pedal) and upon which no action is possible (the slope of the road). The latter are called disturbance inputs, they may be measurable or not accessible, random or deterministic. The variables that can be manipulated can be used as control inputs.

In order to maintain the constant speed of a vehicle, a mathematical model of the process must be developed, in which the vehicle speed is linked to the position of the accelerator pedal, and then by inverting this model, the necessary input to obtain a specified speed can be derived. As a result, if the system output is not taken into consideration, an open-loop control is carried out (see Figure P.5).

This diagram shows that the control system does not account for disturbances; it cannot function properly. For example, if the vehicle is confronted with different slopes, the slope is considered a disturbance input for the process. It will be required that the model take the slope into account and thus a measurement system of the slope (which would result in a compensation of the measurable disturbances).

To improve the behavior, a control system can be defined that calculates, based on the desired speed obtained speed difference, the necessary action on the pedal to regulate (stabilize) the speed at a value specified by the operator. We will thus obtain an automatic control system of the speed of the vehicle. A sensor measuring the speed obtained is necessary. This system automatically performs what the driver does: it compares the target speed with the actual speed (displayed by the dashboard (sensor)) and acts upon the accelerator to reduce the speed difference to zero. The result will then be a *control system* or *loop system* or servo (servo system). The block diagram (functional) of the principle of a servo system is shown in Figure 7:

- – y
^{d}: setpoint is an electrical quantity that represents the desired value of the output of the system; - –
*ε*: error signal between the setpoint and the actual output of the system; - –
*u*: control signal generated by the control system; - –
*y*: is a physical quantity that represents the system output.

The physical quantity *y* is measured with a sensor that translates it into an electrical quantity. This electric quantity is compared to the setpoint using a comparator.

One of the great advantages of a looped system compared to an open-loop system lies in the fact that the loop system automatically rejects disturbances. In control systems, when the setpoint (reference) is constant, it is referred to *control* (for example oven temperature control and speed control of a motor); in the case where the reference is not constant, this is referred to as *tracking* (for example target tracking by an antenna).

An additional input (measure of the slope), with respect to the equipment existing in the previous example, would complete this diagram with an *anticipation* about the effect of the disturbance (due to the slope variation).

A system is said to be controlled when there is a loop between the output and the input or when the variable to be adjusted is the setpoint input of the system. For example, for the heating system of a house or of an enclosure, the input is the temperature setpoint and the output is the temperature in the enclosure. An open-loop heating system is a system that does not show any loopback, taking into account the effective temperature of the enclosure. Thus, it is sensitive to external shocks, a rise in the external temperature would cause an excess of heating.

*Examples:*

The examples are as follows: temperature system control of an oven, fluid flow servo control, speed system control of the trajectory of a vehicle. When the desired path is reduced to a point, this is referred to as regulation and not as system control because the aim here is to stabilize the output of the system at a point. A control system can be qualified by its degree of stability, accuracy, response speed, sensitivity to disturbances acting on the system, robustness with respect to disturbances on measures and errors or variations of the characteristic parameters of the system. The accuracy of a control system can be characterized by the maximum amplitude of the position error.

In the definition of a control system, we will write transfer functions as follows: *H*(*p*) transfer of the system to be controlled, *p* is the Laplace operator; *R*(*p*) transfer of the sensor or measurement unit, *C*(*p*) transfer of the corrector or controller. The setpoint is *ω*(*t*) and the output to be controlled *y*(*t*). The direct chain consists of *C*(*p*) and *H*(*p*). Block *R*(*p*) constitutes the feedback chain. *e*(*t*) is the difference between output and setpoint also called control error or trajectory tracking error. In order to simplify study, we consider a unity feedback scheme in which *R*(*p*) = 1.

In general, transfers *H*(*p*) and *R*(*p*) are known, estimated or can be obtained and the goal is the determination of a corrector *C*(*p*) that can satisfy the required performances for the closed-loop system (transfer from *w* to *y*).

Several types of systems can be distinguished:

- – continuous systems for which all measured quantities are continuous;
- – discrete systems for which all measured quantities are only measured at very specific times (discontinuous or discrete); these are referred to as sampled data or digital systems;
- – linear systems (they can be described by linear differential equations);
- – nonlinear systems (described by nonlinear differential equation). It is possible, often in the first approximation, to linearize nonlinear systems based on an operation point (equilibrium), considering small variations around this point;
- – time-invariant systems (described by differential equations with constant coefficients) and time-variable systems (described by differential equations with time-variable coefficients).

In this book, we consider time-invariant linear, continuous and sampled-data systems.

NOTATION 1.– *Consider a continuous r-input system denoted u and m outputs y*,

[P.1]

[P.2]

*A minimal state representation of this system will be written as:*

[P.3]

[P.4]

*Its transfer function*

[P.5]

*is denoted as:*

[P.6]

*A is stable: the eigenvalues of A have a real part* < 0.

*G*(*p*) *is stable: the poles of G*(*p*) *are in Re*(*p*) < 0.

*G*(*p*) *is instable: the poles of G*(*p*) *are in Re*(*p*) > 0.

(*p*) = *G*(–*p*)

*G*(*p*) *is an eigen transfer function if G*(∞) *is finite*.

*G*(*p*) *is a strictly eigen transfer function if G*(∞) = 0.

*A ^{T} is the transpose of matrix*

[P.7]

[P.8]

*A ^{–T} is the inverse transpose matrix of A*.

*A* is the conjugate transpose matrix of A* = [*a _{ij}*]

*λ _{i}*(

*σ _{i}*(

(*A*) *and* (*A*) *are the minimal and maximal singular values of A*.

*Diag*(*a _{i}*)

*C _{−} is the set of complex numbers with a negative real part*.

*C _{+} is the set of complex numbers with a positive real part*.

*C ^{n} is the set of complex vectors with elements in C*.

*C ^{nxm} is the set of complex matrices of dimensions* (

*< x, y > is the scalar product of x and y*.

*h u is the convolution product of h*(*t*) *and u*(*t*).

* is the Fourier transform operator*.

*θ _{h} convolution operator by h*(

Λ* _{Hg}* =

* Direct sum of two spaces*.

* Orthogonal space at H*_{2};

with the set *H*_{2}={*H*(*p*) analytic matrix function in *Re*(*p*) > 0}

Π_{1} is the orthogonal projection on *H*_{2} and Π_{2} is the orthogonal projection on .

NOTATION 2.– (*x*, *r*) = *Globe of radius r, centered in x of the space; example* (0,1)*L*_{2} = *L*_{2} *refers to the Globe of unit radius of the space L*_{2}. *L*_{2} *is the set of square-integrable functions*.

In the literature, a real rational transfer function refers to a rational transfer function with real coefficients.

In view of these considerations and following previous and present works, the purpose of this section is the perception for the joint understanding of space and surroundings of an autonomous, cognitive and sociable personal robot. This Nao robot will typically act according to a planned action scheme, ensuring their viability, the consistency of current models and the presence of entities opposed to its initial plan, while interacting with the physical world through perception. The selected approach, which is clearly aimed at integrating perceptual functions on robotic platforms, relies on probabilistic modeling to consider multiple and uncertain percepts. At the sensory level, these percepts will be mainly originating from the vision embedded in the robot. These perceptual functions are to be derived from autonomy, cognition and sociability capabilities outlined for our Nao robot.

An increasingly more worrying situation due to the aging of the population is the increase in the number of elderly people living alone, far away from their loved ones, sometimes physically or mentally vulnerable. Studies recommend that, at the social level as well as at the economic level, people stay at home, eventually resorting to help (cleaning, care, gardening, etc.), to preserve as much as possible the references and the social network of the person. However, with the cost of home care being relatively burdensome, assistance times are limited and are not always enough to fight solitude. In addition, when the elderly person presents risks of accidents (fall, stroke, etc.), or when undergoing medical treatment, monitoring solely based on domestic help has become risky, or even insufficient (admission to hospital then being preferable).

To fight more effectively against isolation and the dependence of elderly people, the presented system aims to implement a prototype for an automated and robotic system for home care. Without claiming that it is capable of recreating or replacing real human contact, the proposed system, based on the integration of several technologies, will create a permanent, friendly and reassuring, presence at home. Due to the use of modern means of audio/video communication (speech recognition, speech synthesis, etc.), by the mobile humanoid Nao robot from Aldebaran Robotics and that of fixed cameras allowing the environment to be perceived, the objective is to provide a set of services that makes it possible to assist senior citizens in their daily tasks (engaging with relatives, remote medical consultation, looking for objects, medication reminders, etc.), entertain them (games, physical exercises, friendly presence, etc.) and detect dangerous or abnormal situations (falls, prolonged inactivity, unusual behavior, etc.) to, in the end, alert the doctor and relatives.