Table of Contents

Cover

Title

Preface

1 Control, Servo-mechanisms and System Regulation

1.1. Introduction
1.2. Process control
1.3. Some application exercises
1.4. Some application exercises
1.5. Application 1: stabilization of a rigid robot with pneumatic actuator
1.6. Application 2: temperature control of an oven

2 System Process Control

2.1. Introduction
2.2. Modeling
2.3. Governability, controllability and observability
2.4. State feedback, control by poles placement and stability
2.5. Linear quadratic (LQ) control
2.6. Optimal control (LQ)
2.7. Comprehension and application exercises

3 Actuators: Modeling and Analysis

3.1. Introduction: electric, hydraulic and pneumatic actuators
3.2. Transmission chains, actuators and sensors
3.3. Pneumatic actuators
3.4. Hydraulic actuators
3.5. Application exercises

4 Digital Control and Polynomial Approach

4.1. Introduction to digital control
4.2. PID controller synthesis and its equivalent digital RST
4.3. Digital control by poles placement
4.4. Diophantine, Bézout, greatest common divisor, least common multiple and division
4.5. A few comprehension and application exercises

5 NAO Robot

5.1. Introduction
5.2. Home care project
5.3. Details of the various programs
5.4. Conclusion

6 Application Problems with Solutions

6.1. Exercise 6.1: car suspension
6.2. Exercise 6.2: electromechanical system
6.3. Exercises: identification and state–space representation
6.4. Exercises: observation and control of nonlinear systems

Bibliography

Index

End User License Agreement

List of Tables

1 Control, Servo-mechanisms and System Regulation

Table 1.1. Routh table results
Table 1.2. Routh table results

2 System Process Control

Table 2.1. Table of the values of the integral J
Table 2.2. Criterion minimization comparison
Table 2.3. Characteristics of the control signal for a step setpoint

3 Actuators: Modeling and Analysis

Table 3.1. Analogy between systems
Table 3.2. Thermodynamic and mechanical equations of the actuator

4 Digital Control and Polynomial Approach

Table 4.1. Conventional forms of standard digital controllers obtained using Euler’s transposition

6 Application Problems with Solutions

Table 6.1. Routh table system no. 2 results
Table 6.2. Routh table system no. 4 results
Table 6.3. Routh table system no. 5 results
Table 6.4. Routh table OL system results

List of Illustrations

Preface

Figure 1. Applied scientific and technological resources
Figure 2. Basic concepts of theoretical study and processing methods
Figure 3. Diagram of the model for vehicle operation
Figure 4. Process with multiple input and output quantities
Figure 5. Open-loop process
Figure 6. Open-loop process with disturbances
Figure 7. Functional block diagram of a servo system
Figure 8. Speed regulation of a motor
Figure 9. Temperature regulation of an oven
Figure 10. Control system with correction
Figure 11. Controlled transfer function with feedback
Figure 12. Controlled transfer function without feedback
Figure 13. Control system with closed-loop system controller
Figure 14. System control with correction of a robotized system

1 Control, Servo-mechanisms and System Regulation

Figure 1.1. Schematic diagram of a controlled system with compensation and feedback sequence
Figure 1.2. Schematic diagram of a feedback system R(p) = 1
Figure 1.3. Speed regulation of a motor
Figure 1.4. Temperature regulation of an oven
Figure 1.5. Schematic diagram of the model for vehicle operation
Figure 1.6. Servo system
Figure 1.7. Control diagram
Figure 1.8. Step response
Figure 1.9. Response with damped oscillations
Figure 1.10. Advance phase control circuit
Figure 1.11. Bode plot of the phase advance controller
Figure 1.12. Phase delay controller circuit
Figure 1.13. Bode plot of the phase delay controller
Figure 1.14. Phase advance circuit
Figure 1.15. Step response of the corrected loop system
Figure 1.16. Step response of the controlled system
Figure 1.17. Step response of the controlled system with correction
Figure 1.18. Bode plots of a system
Figure 1.19. Bode plots of a system with correction
Figure 1.20. Bode plot of the phase advance controller
Figure 1.21. Bode plot of the system after correction
Figure 1.22. Bode plot of the system after correction
Figure 1.23. Bode plot with integral compensator
Figure 1.24. Bode plot of the system without PI control
Figure 1.25. Bode plot of the system with control
Figure 1.26. Bode plot of the system with control
Figure 1.27. Outline of the Bode plot of the system without phase advance control
Figure 1.28. Outline of the Bode plot of the system with phase advance control
Figure 1.29. Frequency response of the unknown system
Figure 1.30. Bode plot of
Figure 1.31. Bode plot of
Figure 1.32. Locus of the roots of H₁(p)
Figure 1.33. Locus of the roots of H₂(p)
Figure 1.34. Bode plot of K.H₂(p). For a color version of this figure, see www.iste.co.uk/femmam/signals.zip
Figure 1.35. Nyquist locus for K = 1(H₁)
Figure 1.36. Nyquist loci for H₂(p) * Ki. For a color version of this figure, see www.iste.co.uk/femmam/signals.zip
Figure 1.37. Black plot for K.H₂(p). For a color version of this figure, see www.iste.co.uk/femmam/signals.zip
Figure 1.38. Bode plot of K.H₁(p). For a color version of this figure, see www.iste.co.uk/femmam/signals.zip
Figure 1.39. Nyquist locus for velocity K.H₁(p). For a color version of this figure, see www.iste.co.uk/femmam/signals.zip
Figure 1.40. Nyquist locus for the system position after velocity feedback K_υ = 1 and K = 1
Figure 1.41. Nyquist locus for the system position after velocity feedback K_υ = 10 and K = 1
Figure 1.42. Nyquist locus for the system position after velocity feedback K_υ = 10 and K = 10
Figure 1.43. Black plot for K_υ = 1 and 10 with K = 1, 10 and 100
Figure 1.44. Model describing the behavior of a system
Figure 1.45. Organigram of the modeling stages for the study of an automated system
Figure 1.46. Approximate model for identification

2 System Process Control

Figure 2.1. System with controller to be automated
Figure 2.2. Satellite control example
Figure 2.3. Differential observer
Figure 2.4. System with state feedback
Figure 2.5. Given noisy system
Figure 2.6. New block diagram
Figure 2.7. Given system with controller
Figure 2.8. Servo system using tachymetric feedback
Figure 2.9. Impulse and step response
Figure 2.10. System with phase advance controller
Figure 2.11. Bode plot of the system
Figure 2.12. Bode plot of the system after correction
Figure 2.13. Servo system using tachymetric feedback
Figure 2.14. Bode plot of the system after correction
Figure 2.15. Bode plot of the system after correction
Figure 2.16. Comparative study (step response)
Figure 2.17. Comparative study (control signal)
Figure 2.18. System with digital controller
Figure 2.19. u(t) (α = 8 ; T_e = 0.05 s)

3 Actuators: Modeling and Analysis

Figure 3.1. Characteristic velocity = f(torque) at constant voltages
Figure 3.2. Adherence model
Figure 3.3. Karnopp model
Figure 3.4. Coulomb model
Figure 3.5. Dahl model
Figure 3.6. Generalized Stribeck curve
Figure 3.7. Friction effect in regions 1–3
Figure 3.8. Diagram of the proposed actuator model
Figure 3.9. DC motor 1
Figure 3.10. DC motor 2
Figure 3.11. Servo motor
Figure 3.12. Motor parameters
Figure 3.13. Speed torque characteristic
Figure 3.14. Region of bad commutation
Figure 3.15. Diagram of a motor in variable state (dynamic)
Figure 3.16. Diagram of a motor in variable state
Figure 3.17. Bond graph for a motor
Figure 3.18. Electric actuators with flexibility
Figure 3.19. Diagram of the motor with speed reducer and an arm
Figure 3.20. 2-DOF arm controlled by electropneumatic servovalves
Figure 3.21. Flow stage of the servovalve
Figure 3.22. Friction model curves
Figure 3.23. Friction force with respect to the relative velocity
Figure 3.24. Example of an arm having a single degree of freedom
Figure 3.25. 1-DOF arm with dry frictions
Figure 3.26. Robot model: hydrodynamic stage

4 Digital Control and Polynomial Approach

Figure 4.1. Digital control principle for an analog system (where G(s) is the process, H(s) is the sensor, C(z) is the controller, F(z) is the prefilter, r is the signal setpoint, y is the output signal (to be regulated), y_m is the output measurement, ε is the deviation or control error, u is the command, w(t) is the input disturbance, υ(t) is the measurement noise and δy(t) is the output disturbance)
Figure 4.2. Principle of sampling hold transposition
Figure 4.3. Effect of the sampling period. Holding Te = 0.01s (continuous and zoh)
Figure 4.4. Effect of the sampling period. Holding Te = 0.1s (continuous and zoh)
Figure 4.5. Effect of the sampling period. Holding T_e = 0.5s (continuous and zoh)
Figure 4.6. An image of the stability region by Euler’s transformation toward the rear
Figure 4.7. Representation of the integration obtained Euler’s transformation
Figure 4.8. Effect on an integrator (continuous and Euler’s)
Figure 4.9. Effect on an phase advance integrator (continuous and Euler’s)
Figure 4.10. Effect on a selective filter (continuous and Euler’s)
Figure 4.11. Detrimental effect of rounding in digital controllers (with and without integrator)
Figure 4.12. Decomposed representation of a digital PID
Figure 4.13. Digital PID without setpoint derivation
Figure 4.14. Representation of saturation on control
Figure 4.15. Response of a loop system with a controller containing an integrator
Figure 4.16. Step response and control for a system without saturation
Figure 4.17. Step response of a loop system with a controller for a system with control saturation
Figure 4.18. RST controller structure
Figure 4.19. System with digital loop without controller
Figure 4.20. Effect of the sampling period on the step response of a looped system
Figure 4.21. System to be controlled
Figure 4.22. Roots locus
Figure 4.23. Step response after adding an integrator
Figure 4.24. Evans’s locus after compensation of the dominant pole of the process
Figure 4.25. Impulse response of the loop system after controller adjustment

5 NAO Robot

Figure 5.1. Choregraphe logo
Figure 5.2. Walk action (move to)
Figure 5.3. Telepathe startup and selection of sensors for capture
Figure 5.4. Capture of measures with Telepathe
Figure 5.5. Video capture with Telepathe
Figure 5.6. Home screen in Monitor
Figure 5.7. Example of a situation in NaoSim
Figure 5.8. Connection between Choregraphe and NaoSim to test a program in the virtual world
Figure 5.9. Example of a situation in Webots
Figure 5.10. Compilation of Naoqi-Matlab-SDK
Figure 5.11. Example – Get joint names from Matlab
Figure 5.12. Deprecated function reference
Figure 5.13. Deprecated function reference
Figure 5.14. Deprecated function reference
Figure 5.15. The result returning a cell array with Strings inside
Figure 5.16. Display of the Root
Figure 5.17. Content of the Root
Figure 5.18. Display of the “ChoiceQuestion” block
Figure 5.19. Display of the “ChoiceQuestion2” block
Figure 5.20. Display of the “CallHelp” box
Figure 5.21. Display of the “Choice” box in “CallHelp”
Figure 5.22. Display of the “CallNeighbor2” block
Figure 5.23. Display of the “CallNeighbor” box
Figure 5.24. Display of the “CallNeighbor2” box
Figure 5.25. Display of the box “No answer”
Figure 5.26. Display of the “CallFamily2” box
Figure 5.27. Displaying of the “CallFamily” box
Figure 5.28. “Collision detection”
Figure 5.29. Display of the Root
Figure 5.30. Display of the “Sleptwell Choice”
Figure 5.31. Display of the “Choice” of the dream
Figure 5.32. Display with the box “News”
Figure 5.33. Displaying of the user’s schedule
Figure 5.34. “Root display”
Figure 5.35. Display of the Root
Figure 5.36. Display of the TimeLine
Figure 5.37. Cleansing movements in NaoQi. Let’s scrub ourselves… Let’s brush our teeth… Let’s comb our hair… Let’s get dressed…
Figure 5.38. Nao does gymnastics
Figure 5.39. View of the Root
Figure 5.40. Nao does gymnastics
Figure 5.41. Display of the Root
Figure 5.42. Display of “Choice”
Figure 5.43. Display of “Say”, the call to Marie the nurse
Figure 5.44. Display with “Say2”, the call to Marie the nurse
Figure 5.45. Display of the Root
Figure 5.46. Display of the “MemoryGame” block
Figure 5.47. Display of the “Color+Word” block
Figure 5.48. Display of the “MyTurn” block
Figure 5.49. Display of the “Wrestling Results” block
Figure 5.50. Display of the “Say” block for repetition
Figure 5.51. Display with Root
Figure 5.52. Display of the timeline and virtual Nao for the movement “search for a face”
Figure 5.53. Display of the “Choice” block
Figure 5.54. Display of the Root
Figure 5.55. Display of the “Choice” block for the genre of book to read
Figure 5.56. Display of the “Choice” block of the type of book
Figure 5.57. Display of the Root
Figure 5.58. Display of the “ChoiceMusic” block
Figure 5.59. Display of the “Choice” block in ChoiceMusic
Figure 5.60. Display of the second choice block in “ChoiceMusic”
Figure 5.61. Display of the “RandomMusic” block
Figure 5.62. Display of the “CrySitting” block
Figure 5.63. Nao sits and cries
Figure 5.64. Display of the “Touch Sequence” block
Figure 5.65. Display of the Root
Figure 5.66. Display of the “Multiplication” block
Figure 5.67. Display of the “PoseQuestion” block
Figure 5.68. Displaying of the “Verif” block
Figure 5.69. Display of the “False Result” box
Figure 5.70. Display of the Root
Figure 5.71. Display of the “Macarena” block
Figure 5.72. Display of the “Root”
Figure 5.73. “Random” and “Switch case” boxes
Figure 5.74. Example of a question in the game
Figure 5.75. If the answer is wrong
Figure 5.76. Display of Root
Figure 5.77. Display of the “Fall” box
Figure 5.78. Display of the “Call” block
Figure 5.79. Display of the “seek and walk” block
Figure 5.80. Display of the “Tracker sound/red” block
Figure 5.81. Display of the Root
Figure 5.82. Display of the “CrySitting” block
Figure 5.83. Display of the “Activity” block

6 Application Problems with Solutions

Figure 6.1. Modeling of the suspension of a quarter of car
Figure 6.2. Step responses (suspension)
Figure 6.3. Asymptotic Bode plot
Figure 6.4. Electromechanical diagram of the system
Figure 6.5. Block diagram of the motor
Figure 6.6. Block diagram of the system companion form
Figure 6.7. Block diagram of the system, Jordan form
Figure 6.8. Block diagram of the system cascade form
Figure 6.9. Step response
Figure 6.10. Impulse response
Figure 6.11. Closed-loop system
Figure 6.12. Step system response in CL
Figure 6.13. Step response of
Figure 6.14. K = 1 ⇒ (ξ = 0.5; ω₀ = 1; ε_p = 0; ε_υ = 1) and K = 9 ⇒ (ξ = 1/6; ω₀ = 3; ε_p = 0; ε_υ = 1/9)
Figure 6.15. Bode plot of
Figure 6.16. Nyquist plot of
Figure 6.17. Bode plot of
Figure 6.18.
Figure 6.19. Bode plot of
Figure 6.20. Nyquist plot of
Figure 6.21. Bode plot of
Figure 6.22. Nyquist plot of
Figure 6.23. Bode plot of
Figure 6.24. Nyquist plot of
Figure 6.25. Bode blot of
Figure 6.26. Nyquist plot of
Figure 6.27. Bode plot of
Figure 6.28. Black–Nichols plot of
Figure 6.29. Resonance maximum amplitude M_r according to ξ
Figure 6.30. Modeling of displacements
Figure 6.31. Block diagram of the system
Figure 6.32. Block diagram of the system
Figure 6.33. Block diagram of the second-order system

Preface

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

Introduction

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.

Signals and systems: generalities

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 applications¹.

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.

Notions of process and operation control

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.

**Figure 1**. *Applied scientific and technological resources*

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.

**Figure 2.** *Basic concepts of theoretical study and processing methods*

**Figure 3.** *Diagram of the model for vehicle operation*

**Figure 4.** *Process with multiple input and output quantities*

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).

**Figure 6.** *Open-loop process with 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.

**Figure 7.** *Functional block diagram of a servo system*

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.

**Figure 8.** *Speed regulation of a motor*

**Figure 9.** *Temperature regulation of an oven*

**Figure 10.** *Control system with correction*

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.

**Figure 11.** *Controlled transfer function with feedback*

**Figure 12.** *Controlled transfer function without feedback*

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).

**Figure 13.** *Control system with closed-loop system controller*

**Figure 14.** *System control with correction of a robotized system*

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] (or Hermitian transpose matrix of A).

λ_i(A) is the ith eigenvalue of A.

σ_i(A) is the ith singular value of A.

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

Diag(a_i) is the diagonal matrix whose diagonal elements are the 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.

Cⁿ is the set of complex vectors with elements in C.

C^nxm is the set of complex matrices of dimensions (nxm) with elements in C.

< 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(t), H(p) should denote the Laplace transform of h(t).

Λ_{H_g} = H(p).g(p) is the Laurent operator or multiplication in the frequency domain.

Direct sum of two spaces.

Orthogonal space at H₂;

with the set H₂={H(p) analytic matrix function in Re(p) > 0}

Π₁ is the orthogonal projection on H₂ and Π₂ is the orthogonal projection on .

NOTATION 2.– (x, r) = Globe of radius r, centered in x of the space; example (0,1)L₂ = L₂ refers to the Globe of unit radius of the space L₂. L₂ is the set of square-integrable functions.

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

Nao robot: application for home care

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.

Signals and Control Systems

Application for Home Health Monitoring

Preface

Introduction

Signals and systems: generalities

Notions of process and operation control

Nao robot: application for home care