There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Time-delay systems: an overview of some recent advances and open problems

Being a motion on a discontinuity set of a dynamic system, sliding mode is used to keep accurately a given constraint and features theoretically-infinite-frequency switching. Standard sliding modes provide for finite-time convergence, precise keeping of the constraint and robustness with respect to internal and external disturbances. Yet the relative degree of the constraint has to be 1 and a dangerous chattering effect is possible. Higher-order sliding modes preserve or generalize the main properties of the standard sliding mode and remove the above restrictions. r-Sliding mode realization provides for up to the rth order of sliding precision with respect to the sampling interval compared with the first order of the standard sliding mode. Such controllers require higher-order real-time derivatives of the outputs to be available. The lacking information is achieved by means of proposed arbitrary-order robust exact differentiators with finite-time convergence. These differentiators feature optimal asymptot...

/pdf/higher-order-sliding-modes-differentiation-and-output-438zdzb0ld.pdf

Higher-order sliding modes, differentiation and output-feedback control

Differential-Difference Equations. By Richard Bellman and Kenneth L. Cooke. Pp. xvi, 465. 1963. (Academic Press)

Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

Linear systems

In this paper we propose a discrete time optimal control law to stabilize the four-rotor rotorcraft in attitude and position. The main objective of this kind of control law is to save energy and therefore increase the effective time in takeoff and hover flight phases for this robotic platforms. The optimal control law is synthesized considering a infinite horizon combined with an exact linearization by applying a nonlinear control law over nonlinear equations describing the robot dynamic model. The control law obtained is simple, easy and better adapted to be programmed in a micro-controller. Both simulation and experimental test and results show a satisfactory UAV behavior.

Discrete optimal control for a quadrotor UAV: Experimental approach

This paper addresses the problem of estimation and tracking of a road using the new tilt-rotor convertible MAV (mini aerial vehicle). For that objective, we consider the following scenario: (i) no previous knowledge of the road, i.e. shape, dimension and color, (ii) loss of information by the sensors is considered and (iii) nonlinear dynamics of the MAV is taken into consideration. Aiming at this goal, two operational regions are defined: road detected and road not detected by the sensors. A switching between the measurements of imaging and inertial sensors enables estimation of the required states in both operational regions. For dealing with both aforementioned cases, a Lyapunov-based switching control for stabilizing the vehicle's position is proposed. Unmodeled dynamics such as friction forces are estimated by means of the proposed controller. The global exponential stability of the position subsystem together with the switching controller is demonstrated exploiting the fact that the individual closed-loop systems are globally exponentially stable and the switching is sufficiently slow, so as to allow the transient effects to dissipate after each switch. The control law is validated on the Quad-plane experimental platform, showing the expected behavior during autonomous navigation.

/pdf/lyapunov-based-switching-control-for-a-road-estimation-and-16so8lxrly.pdf

Lyapunov-Based Switching Control for a Road Estimation and Tracking Applied on a Convertible MAV

In this paper we present the dynamic model and the stabilisation of an original configuration of an small helicopter having eight rotors. This configuration is not yet presented in the literature and in our knowledge is also the first prototype of this kind of helicopters. The equations of motion for the VTOL aircraft are obtained using Euler-Lagrange approach. Due to the configuration of the aircraft the attitude dynamics are practically independent of the translational dynamics corresponding to the lateral displacements (x and y). A nonlinear control strategy based in the Lyapunov analysis is proposed to stabilize this aircraft at hover. The performance of the nonlinear control algorithm is illustrated with real-time experiments.

Modelling and real-time control stabilization of a new VTOL aircraft with eight rotors

In this paper we propose an alternative approach to estimate the UAV translational velocity and position applying an onboard optical flow sensor. This, is a appropriated sensor to be used in a mini UAV due to its lightness and smallness. With this approach we are able to perform real-time autonomous outdoor and indoor flight without using the GPS information, which in some case is blocked, denied or jammed. Additionally, a PD controller is robustly tuned in order to compensate the uncertain velocity measurements provided by the optical flow sensor. The experimental results obtained by using this approach shown the feasibility of the proposed sensor-controller scheme.

Visual odometry for autonomous outdoor flight of a quadrotor UAV

Multivariable linear systems with several delays are considered. Delays are allowed in the state as well as in the input or the output. Passivity and dissipativity of such systems are considered both in the input-output and state space framework in connection with hyperstability. In the input-output framework necessary and sufficient conditions of passivity and hyperstability of delay systems are given in the language of positive real functions. Furthermore, necessary and sufficient conditions for positive realness are derived for a class of meromorphic functions describing delay systems.

https://ieeexplore.ieee.org/iel5/7285/19732/00914236.pdf

Rogelio Lozano

Papers

Discrete optimal control for a quadrotor UAV: Experimental approach

Lyapunov-Based Switching Control for a Road Estimation and Tracking Applied on a Convertible MAV

Modelling and real-time control stabilization of a new VTOL aircraft with eight rotors

Visual odometry for autonomous outdoor flight of a quadrotor UAV

Input-output passive framework for delay systems