Generalized homogenization of linear controller 

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Homogeneity vs linearity in control system design

Quality of any control system is estimated by many quantitative indices (see e.g. [9], [102], [107]), which reflect control precision, energetic effectiveness, robustness of the closed-loop system with respect to disturbances, etc. From mathematical point of view, the design of a « good » control law is a multi-objective optimization problem. The mentioned criteria frequently contradict to each other, e.g. a time optimal feedback control could not be energetically optimal but it may be efficient for disturbance rejection [19]. In practice, an adjustment of a guaranteed (small enough) convergence time can be considered instead of minimum time control problem, and an exact convergence of systems states to a set-point is relaxed to a convergence into a sufficiently small neighborhood of this set-point.
A well-tuned linear controller, such as PID (Proportional-Integral-Differential) algo-rithm, guarantees a good enough control quality in many practical cases [9]. However, the further improvement of control performance using the same linear strategy looks impossible. Being a certain relaxation of linearity, the homogeneity could provide additional tools for improving control quality. In this context, it is worth knowing if there exist some theoretical features of homogeneous systems, which may be useful (in practice) for a design of an advanced control system.
Finite-time and fixed-time stability are a rather interesting theoretical feature of ho-mogeneous systems [7], [79], [55]. For example, if an asymptotically stable system is homogeneous of positive degree at infinity and homogeneous of negative degree at the origin, then its trajectory reaches the origin (a set point) in a fixed time independently of the initial condition [4]. This idea can be illustrated on the simplest scalar example x˙(t) = u(t); t > 0; x(0) = x0.
where x(t) 2 R is the state variable and u(t) 2 R is the control signal. The control aim is to stabilize this system at the origin such that the condition ju(x)j 1 must be fulfilled for jxj 1.
• The classical approach gives the standard linear proportional feedback algorithm ulin(x) = x.
which guarantees asymptotic (in fact, exponential) convergence to the origin of any trajectory of the closed-loop system: jx(t)j = e tjx0j.

Linear geometric homogeneity

As explained in the standard and weighted homogeneity, once the dilation of system is established, many properties of nonlinear system can be studied easily. In order to extend the homogeneous property to more general systems, a more general form of dilation is introduced as follows x ! d(s)x; s 2 R; x 2 Rn (2.15).
To become a dilation, the family of transformations d(s) : Rn ! Rn must satisfy certain restrictions [34], [41]. Definition 2.1.6. A mapping d : R 7! Rn n is called linear dilation in Rn if it satisfies.
• Group property: d(0) = In and d(t + s) = d(t)d(s) = d(s)d(t);8t; s 2 R.
• Continuity property: s ! d(s) is continuous map, i.e. 8t; > 0;9 > 0 : js tj < ) kd(s) d(t)k.
• Limit property: lims! kd(s)xk = 0 and lims!+1 kd (s)xk = +1 uniformly on the unit sphere S := fx 2 Rn : kxk = 1g.

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Canonical homogeneous norm

In this part, we introduce the canonical homogeneous norm in Rn, which is used for the analysis and design of homogeneous control system. Definition 2.1.9. A continuous function p : Rn ! [0;+1) is said to be d-homogeneous norm in Rn if.
• p(u) ! 0 as u ! 0;
• p( d(s)u) = esp(u) > 0 for u 2 Rnnf0g, s 2 R; where d is a dilation.
The functional p may not satisfy triangle inequality p(u +v) p(u)+p(v), so, formally, it is not even a semi-norm. However, many authors (see e.g. [4], [24], [5]) call functions satisfying the above definition by ”homogeneous norm”. We follow this tradition. For example, if the dilation is given by d(s) = diagfer1s; er2s; :::; ernsg, a homogeneous norm p : Rn ! [0;+1) can be defined as follows [4] n 1 Xi ; u = (u1; u2; :::; un)> 2 Rn: p(u) = jui jri =1.
For strictly monotone dilations the so-called canonical homogeneous norm [80] can be introduced by means of a homogeneous projection to the unit sphere, which is unique in the case of monotone dilation due to Theorem 2.1.2. Definition 2.1.10. ([80]) The function k kd : Rnnf0g ! (0;+1) defined as
kxkd = esx ; where sx 2 R : kd( sx)xk = 1; (2.27) is called the canonical homogeneous norm, where d is a strictly monotone dilation. Obviously, kd(s)xkd = eskxkd and kxkd = k xkd for any x 2 Rn and any s 2 R. The homogeneous norm defined by (2.27) was called canonical since it is induced by a canonical norm k k in Rn and kxkd = 1 , kxk = 1.

Table of contents :

Acknowledge
Abstract
List of Figures
List of Tables
Notations
1 Introduction 
1.1 Quadrotor as unmanned aerial vehicle
1.2 Quadrotor system
1.3 State of the art in quadrotor control
1.4 Experiment setup: QDrone of Quanser
1.5 Contribution and outline of thesis
2 Mathematical backgrounds 
2.1 Homogeneity
2.2 Implicit Lyapunov function method
2.3 Linear Matrix Inequalities
3 Generalized homogenization of linear controller 
3.1 Motivating Example
3.2 Homogenization of linear controllers
3.3 An “upgrade » of a linear controller for Quanser QDrone™
3.4 Conclusion
4 Generalized homogenization of Linear Observer 
4.1 Homogeneous State-Estimation of Linear MIMO Systems
4.2 From a linear observer to a homogeneous one
4.3 An “upgrade » of a linear filter for QDrone of QuanserTM
4.4 Conclusion
5 Homogeneous stabilization under constraints 
5.1 Problem statement
5.2 Controller Design with Time and state Constraint
5.3 Simulation results
5.4 Conclusion
Conclusion and perspective
Bibliography 

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