Michał Niezabitowski

Silesian University of Technology · Faculty of Automatic Control, Electronics and Computer Science

We consider a linear discrete time-invariant control system with linear time-varying state feedback. Our main goal is to determine what types of asymptotic dynamics are possible for this system. We introduce and investigate certain discrete-time analogs for notions of global attainability, global Lyapunov reducibility, and global assignability of L...

In this paper,we present a theorem about stability of nonlinear fractional difference equation with Riemann-Liouvile difference operator. The result is a version of classical theorem on linear approximation and to derive them,we prove the variation of constants formula for nabla Riemann- Liouville fractional difference equations.We also present som...

We prove that the spectra (sets of values) of the upper and lower Sergeev frequencies of zeros of a linear difference equation of order higher than one are Suslin sets of the interval [0,1]. Moreover, we prove the inverse theorem for upper frequencies of zeros of second-order equations under the additional assumption that 0 belongs to the spectrum.

We investigate a problem of proportional local assignability of the Lyapunov spectrum for discrete time-varying linear systems. We propose the concept of non-multiple proportional local assignability and discuss the relations of this notion and splitness of the corresponding free system. We prove that, if a given open-loop system is uniformly compl...

We investigate the behavior of the Lyapunov spectrum of a linear time-varying discrete system under the action of small perturbations in order to obtain some verifiable conditions for stability and openness of the Lyapunov spectrum. To this end we introduce the concepts of broken away solutions and splitted systems. The main results obtained are a...

We consider a linear discrete time-varying input-output system. Our goal is to study the problem of local assignability of the Lyapunov spectrum by static output feedback control. To this end we introduce the notion of uniform consistency for discrete-time linear systems which is the extension of the notion of uniform complete controllability to in...

Semi-linear discrete-time control systems with periodic coefficients are considered. The problem of uniform global asymptotic stabilization of the zero equilibrium of the closed-loop system by state feedback is studied. It is assumed that the free dynamic system has a Lyapunov stable zero equilibrium. The method for constructing a damping control i...

We study the problem of arbitrary eigenvalue spectrum assignment for a linear discrete-time control system with delays defined by a linear difference equation of the nth order with multiple inputs and multiple outputs via static output feedback with delays. Necessary and sufficient conditions are obtained for arbitrary eigenvalue spectrum assignmen...

We investigate the behavior of the Lyapunov spectrum of a linear discrete-time system under the action of small perturbations in order to obtain some verifiable conditions for stability and openness of the Lyapunov spectrum. To this end we introduce the concepts of broken away solutions and splitted systems. The main results obtained are a necessar...

A generalized Nicholson blowflies model with harvesting or immigration and random effect is considered. We discuss the existence of positive global solution and provide the estimate on the lower bound of the Lyapunov exponent. Moreover, we show that the nontrivial equilibrium solution is mean square exponential stability and stable in probability....

This paper is concerned with a non-linear stochastic delay differential system with delay-dependent impulsive perturbations. In this work, the size of the jump is defined as a general non-linear delay-dependent state variable and the solution of the impulsive stochastic delay differential system corresponding to the system without impulsive perturb...

In the chapter two the most important properties of fractional order dynamical systems, namely, controllability and stability are presented. At the beginning the basic notations and the fundamental definitions are recalled. The first part of the chapter is devoted to controllability and contains the formulation of the problem, main hypotheses and t...

In this paper, we establish some criteria for boundedness, stability properties, and separation of solutions of autonomous nonlinear nabla Riemann‐Liouville scalar fractional difference equations. To derive these results, we prove the variation of constants formula for nabla Riemann‐Liouville fractional difference equations.

We formulate fractional difference equations of Riemann–Liouville and Caputo type in a functional analytical framework. Main results are existence of solutions on Hilbert space-valued weighted sequence spaces and a condition for stability of linear fractional difference equations. Using a functional calculus, we relate the fractional sum to fractio...

We discuss relations between the four formulations of the problem of assignability of the Lyapunov spectrum for discrete linear time-varying systems by a time-varying feedback. For two of them: global assignability and proportional local assignability, we have already [2, 3, 4] obtained sufficient conditions in terms of uniform complete controllabi...

We consider a discrete nonlinear control time-varying system x(k + 1) = f(k, x(k), u(k)), k ∈ ℕ, x ∈ ℝⁿ, u ∈ ℝr. A control process of this system is a pair (x(k), u(k))k ∈ N consisting of a control (u(k))k∈N and some solution (x(k))k∈N of the system with this control. We assume that the control process is defined for all k ∈ N. We have obtained suf...

In this paper, the variation of constant formulas for Grünwald-Letnikov-type fractional difference equations are established. As an application, we prove a stability result for solutions of a class of autonomous scalar fractional difference equations.

In this paper, we establish a lower bound on the separation between two distinct solutions of a scalar Riemann-Liouville fractional differential equation. As a consequence, we show that the Lyapunov exponent of an arbitrary non-trivial solution of a bounded linear scalar Riemann-Liouville fractional differential equation is always non-negative.

In this paper we present necessary and sufficient conditions for two given functions to be the Lyapunov and the upper Bohl exponent of a certain discrete linear system with diagonal coefficients. The obtained conditions have a form of easily verifiable algebraic conditions.

We consider a local version of the pole assignment problem for linear discrete time-varying systems. Our aim is to obtain sufficient conditions to place the Lyapunov spectrum of the closed-loop system in an arbitrary position within some neighborhood of the Lyapunov spectrum of the free system using an appropriate time-varying linear feedback. More...

In the paper, properties of the upper Bohl exponents and senior upper general exponent of discrete linear time-varying systems are investigated. The relation of these exponents to uniform exponential stability is discussed. Moreover, an example of system, which is not uniformly exponentially stable but each trajectory tends uniformly and exponentia...

In this paper we investigate the problem of assignability of the so-called regularity coefficients and central exponents of discrete linear time-varying systems. The main result presents a possibility of assignability of Lyapunov, Perron, Grobman regularity coefficients and central exponents by a linear time-varying feedback under the assumptions o...

We formulate fractional difference equations of Riemann-Liouville and Caputo type in a functional analytical framework. Main results are existence of solutions on Hilbert space-valued weighted sequence spaces and a condition for stability of linear fractional difference equations. Using a functional calculus, we relate the fractional sum to fractio...

In this paper, we establish variation of constant formulas for both Caputo and RiemannLiouville fractional difference equations. The main technique is the Z -transform. As an application, we prove a lower bound on the separation between two different solutions of a class of nonlinear scalar fractional difference equations.

In this paper we investigate the problem of influence of parametric perturbations on the Lyapunov spectrum of the discrete linear time-varying system. The main result of the paper is that for any two sequences of positive real numbers and any rate of convergence there exist a discrete linear time-varying system and a perturbation tending to zero wi...

In this paper we consider regularity coefficients of a discrete linear time‐varying systems. The main result presents a complete description of the relations between Lyapunov, Perron and Grobman regularity coefficients of mutually adjoint systems.

We consider a version of the pole assignment problem for linear discrete time-varying systems with linear state feedback. Our aim is to prove that all the systems from the closure (in the topology of pointwise convergence) of all shifts of the original system have assignable Lyapunov spectrum if and only if the original system is uniformly complete...

In this paper, we present some results for existence of global solutions and attractivity for mulidimensional fractional differential equations involving Riemann-Liouville derivative. First, by using a Bielecki type norm and Banach fixed point theorem, we prove a Picard-Lindel\"of type theorem on the existence and uniqueness of solutions. Then, app...

In this paper, we present some results for existence of global solutions and attractivity for mulidimensional fractional differential equations involving Riemann-Liouville derivative. First, by using a Bielecki type norm and Banach fixed point theorem, we prove a Picard-Lindel\"of type theorem on the existence and uniqueness of solutions. Then, app...

In this paper an application of the fractional calculus to path control is studied. The integer-order derivative and integral are replaced with the fractional-order ones in order to solve the inverse kinematics problem. As an approximation of the fractional differentiator the Al-Alaoui operator with power series expansion (PSE) is used. The propose...

In this paper we give the necessary and sufficient conditions for a function f to be an upper Bohl function of the diagonal discrete linear time-varying systems.

This paper presents a short survey on the recent result in the area of exact controllability of selected types of second order semilinear infinite dimensional dynamical systems. In particular, it shows the current state of knowledge in the area of trajectory controllability, boundry controllability and impulsive controllability, all for determinist...

In this paper we consider the influence of parametric perturbations on the maximal Lyapunov exponent of the discrete time-varying system. The main result states that the exact upper movability boundaries of the greatest Lyapunov exponent under arbitrary small perturbations is the central exponent.

In this paper we investigate properties of the Perron exponents of diagonal discrete linear time-varying systems. We give necessary and sufficient conditions for a function / to be the Perron function of the considered systems.

In this paper we give the necessary and sufficient conditions for a function f to be a lower Bohl function of a diagonal discrete linear time-varying systems.

For discrete linear time-varying systems with bounded system matrices we discuss the pole assignment problem utilizing linear state feedback. It is shown that uniform complete controllability is sufficient for the Lyapunov exponents being arbitrarily assignable by choosing a suitable feedback.

The following article presents recent results of controllability problem of dynamical systems in infinite-dimensional space. Generally speaking, we describe selected controllability problems of fractional order systems, including approximate controllability of fractional impulsive partial neutral integrodifferential inclusions with infinite delay i...

This book collects papers from the 8th Conference on Non-Integer Order Calculus and Its Applications that have been held on September 20-21, 2016 in Zakopane, Poland. The preceding two conferences were held in Szczecin, Poland in 2015, and in Opole, Poland, in 2014. This conference provides a platform for academic exchange on the theory and applica...

In the article unconstrained controllability problem of positive discrete-time switched fractional order systems is addressed. A solution of discrete-time switched fractional order systems is presented. Additionally, a transition matrix of considered dynamical systems is given. A sufficient condition for unconstrained controllability in a given num...

In this paper an application of the fractional calculus to path control is studied. The integer-order derivative and integral are replaced with the fractional-order ones in order to solve the inverse kinematics problem. The proposed algorithm is a modification of the existing one. In order to maintain the accuracy and to lower the memory requiremen...

The article concerns the output controllability of discrete-time linear switched systems. The primary objective is to find a control signal which performs system from any initial state to the given output value in a finite time interval regardless of the switching signal. In addition, the search control signal is performed on the assumption that th...

The main objective of this article is to present the state of the art concerning approximate controllability of dynamic systems in infinite-dimensional spaces. The presented investigation focuses on obtaining sufficient conditions for approximate controllability of various types of dynamic systems using Schauder’s fixed-point theorem. We describe t...

This paper outlines development process of unmanned multirotor aerial vehicle HF-4X, which consists of design and manufacturing semi-autonomous UAV dedicated for indoor flight, which would be capable of stable and controllable mission flight. A micro air vehicle was designed to participate in the International Micro Air Vehicle Conference and Fligh...

The main purpose of the paper is to discuss controllability problems for fractional discrete linear time-varying infinite - dimensional systems. The necessary and sufficient condition for the exact and approximate controllability of the system are proposed and proved.

Singular trajectories are rarely used because of issues during realization. A method of planning trajectories for given set of points in task space with use of graphs and neural networks is presented. In every desired point the inverse kinematics problem is solved in order to derive all possible solutions. A graph of solutions is made. The shortest...

Using the coefficients matrices we obtained upper and lower exact boundaries of mobility of Lyapunov, Perron, upper and lower Bohl exponents for linear difference systems under small perturbations of the coefficients matrices.

This paper presents the way of modeling thrust generated by a quadrotor flying unit. The highly non-linear equations describing rotor thrust are derived with use of momentum theory and approximated with use of generalized polynomial. The article includes the model formulation and the example of approximation. The polynomial approximation ensures th...

Brain-Computer Interfaces (BCIs) are systems capable of capturing and interpreting the consent changes in the activity of brain (e.g. intention of limb movement, attention focus on specific frequency or symbol) and translating them into sets of instructions, which can be used for the control of a computer. The most popular hardware solutions in BCI...

In the paper we present a complete description of relations between general exponents of the discrete linear time-varying system. A necessary and sufficient condition for four real numbers, guaranting that there exists a system such that the numbers are its general exponents, is presented in the form of inequalities between the numbers.

In this paper a simple method of orientational path planning for commonly used 6R manipulators is presented. The chosen criterion is to minimize changes of values of manipulator's joints. For a given set of locations a proper orientation of end-effector has to be computed. The three Euler angles (α, β, γ) have been replaced with one angle (δ). In o...

In the paper we examine the influence of parametric perturbations on the Lyapunov exponents of the linear discrete time-varying system. We show on an example that even the perturbations tend to zero exponentially they may change the values of Lyapunov exponents. However, the main result of the paper shows that if the perturbations tends to zero exp...

The transmission of electric fields from a primary bioelectric source through biological tissue towards measurement sensors is known as the volume conduction. This phenomenon is widely exploited in many biosensors used for the measurement of bioelectromagnetism. One example of such sensors is an electroencephalograph (EEG) used for the recording an...

It is well known that eye movement tracking may reveal information about human intentions. Therefore, it seems that it would be easy to use gaze pointing as a replacement for other traditional human computer interaction modalities like e.g. mouse or trackball, especially when there are more and more affordable eye trackers available. However, it oc...

The topic of this paper concerns a way of robotic manipulator control with use of hand gestures. The previously developed hand landmarks detection and localization algorithm is now translated to work with robotic arm manipulator. The proposed algorithm is used for hand gesture recognition and continuous feature points tracking. The points coordinat...

In this paper the problem of joint state and parameter estimation of quadrotor is addressed. It was assumed that quadrotor is equipped with Inertial Measurement Unit (IMU) and GPS module. Based on this instrumentation it was possible to estimate complete state vector describing the movement of rigid body, along with parameters related to its propul...

Performing complex activities that accompany piloting an airplane requires a lot of concentration, coordination and reflex from the pilots. Additionally, the responsibility that lies on a person that controls any kind of flying platform can lead to the increased amount of pressure and stress. Considering all of the aforementioned conditions, it is...

The main aim of this project was to design and construct the laboratory setup to perform testing of the ABS car systems. In this paper a realistic car brake system for a one wheel with suspension is used. Development and implementation of ABS architecture for integrated real-time, embedded control software is described. At the same time, it include...

In the work we present the stability criteria for difference linear equations obtained by the freezing method. The results are presented for discrete version of damped oscillator equation. We also report stability criteria for another type of equations obtained by this method. Our results are illustrated by numerical examples.

The main aim of this article is to review the existing state of art concerning the complete controllability of semilinear dynamical systems. The study focus on obtaining the sufficient conditions for the complete controllability for various systems using the Banach fixedpoint theorem. We describe the results for stochastic semilinear functional int...

Growing interest in hand gesture recognition system, bring new possibilities in developing new ways of intuitive control. Controlling the everyday surrounding can be much easier with use of natural body language. This article presents the review of hand gestures systems basing on color images, focusing on hand feature points detection. Additionally...

We investigate properties of partial exponents (in particular, the Lyapunov and Perron exponents) of discrete time-varying linear systems. In the set of all increasing sequences of natural numbers, we define an equivalence relation with the property that sequences in the same equivalence class have the same partial exponent. We also define certain...

The joint spectral radius of a set of matrices is a generalization of the concept of spectral radius of a matrix. Such notation has many applications in the computer science, and more generally in applied mathematics. It has been used, for example in graph theory, control theory, capacity of codes, continuity of wavelets, overlap-free words, tracka...

Robot manipulator teaching is a time consuming procedure where qualified operator programs the execution path. In this paper we introduce and discuss the improvement of traditional teaching method with application of hand gesture recognition system. The paper presents the most common robot programming and hand gesture recognition issues and present...

In this paper the controllability problem for discrete-time linear switched systems is considered. The main goal is to find a control signal that steers any initial state to a given final state independently of the switching signal. In the paper, it is assumed that there are some constraints posed on the switching signal. Moreover, we present a nec...

Development of a reliable high-performance multirotor unmanned aerial vehicle (UAV) requires an accurate and practical model of the vehicle dynamics. This paper describes the process and results of the dynamic modeling of an unmanned aerial platform known as quadrotor. To model a vehicle dynamics, elementary physical and aerodynamical principles ha...

The article outlines the process of developing a mathematical model of the human arm. The model arm consists of two joints connected by the rotational link. Each of them is represented as a truncated cone prism which changes its shape during motion. The shape change instant depends on the two-state coordinates being the angular displacements q1 and...

Baseline in signals is a relatively complicated problem in analysis of signals obtained in various analytical techniques such as chromatography and spectroscopy. In this article there are presented results of tests on four algorithms for a baseline estimation in chromatographic signals. Two of them are based on a polynomial fitting in a region of d...

In this paper we present three concepts of stability for discrete linear inclusions, i.e. uniform power equistability, power equistability and selectable stability. The main result states that first two concepts are equivalent. Moreover they are equivalent to the fact that a numerical quantity called generalized spectral radius is less than one. Fi...

The Lyapunov, Perron and Bohl exponents are the most important numerical characteristics of dynamic systems used in control theory. Properties of the first one are well investigated. Properties of the Perron and Bohl exponents are much less described in the literature, for example the questions about the structure of the sets of values of these qua...

In the paper we present a complete description of relations between Lyapunov coefficients of irregularity of mutually adjoint linear discrete time-varying systems.

In this paper we address the problem of water basins surface measurements. This problem is important in regions where coal extraction took place because of it significant impact on the landscape. We propose method based on usage of quadrotor equipped with non-metrical camera. Presented concept include discussion about quadrotor path planning, image...

The presented research work considers stability criteria of second-order differential equation. The secondorder discrete-time oscillator equation is obtained from discretization of second order continuous-time equation using the forward difference operator. The stability criteria are drawn with freezing method and are presented in the terms of the...

For the discrete linear time-varying systems we present basic facts and definitions concerning the Lyapunov transformation, kinematic similarity and reducibility in the context of stability and Lyapunov exponents theory. Moreover, the paper contains the original result giving the necessary and sufficient conditions for the reducibility of a system...

In this paper we consider the finite-dimensional dynamical control system described by scalar semilinear ordinary differential state equation with variable delay. The semilinear state equation contains both pure linear part and nonlinear perturbation. We extend the concept of controllability on trajectory controllability for systems with point dela...

Modern approach to the FOREX currency exchange market requires support from the computer algorithms to manage huge volumes of the transactions and to find opportunities in a vast number of currency pairs traded daily. There are many well known techniques used by market participants on both FOREX and stock-exchange markets (i.e. fundamental and tech...

With the advent of portable and high density microelectronic devices, the minimization of power consumption in CMOS VLSI circuits is becoming a critical concern. An embedded system is a combination of electronic hardware and software and sometimes additional parts designed to perform a dedicated function. In many cases system (microprocessor) must...

For the discrete linear time-varying systems the necessary and sufficient conditions to being kinematically similar to a system with block-triangular or constant coefficients are presented. In particular, conditions for the system to be kinematically similar to a system with identity matrices coefficients are provided.

The Bohl exponents, similarly as the Lyapunov exponents, are one of the most important numerical characteristics of dynamical systems used in control theory. By contrast with properties of the Lyapunov characteristics, properties of the Bohl exponents are much less investigated. In this paper we present some new properties of the upper Bohl exponen...

In this paper we consider the controllability problem for discrete-time linear switched systems. The problem consists of finding a control signal that steers any initial condition to a given final state regardless of the switching signal. In the paper a necessary and sufficient conditions for this type of controllability are presented. Moreover, we...