A new notion of a realization of transfer matrix of (P;Q; V)-cone-system for discrete-time linear systems is proposed. Necessary and sufficient conditions for the existence of the realizations are established. A procedure is proposed for computation of a realization of a given proper transfer matrix T(z) of (P;Q; V)-cone-system. It is shown that there exists a realization of T(z) of (P;Q; V)-cone-system if and only if there exists a positive realization of T(z) = V T(z)Q!1, where V;Q and P are non-singular matrices generating the cones V;Q and P respectively.

In the paper there has been made an advantage of the non-classical operational calculus to determination of the response of the certain discrete time-systems. The Z-transform is often used to analysis of the stationary discrete time-systems. However, the use of the Z-transform to determination of the response especially of the non-stationary discrete time-systems is doubtful or may cause complications. This method leads to differential equations of n-th order of variable coefficients, whose solutions are very difficult or impossible. The non-classical operational calculus can be used to analysis both of the stationary and non-stationary discrete time-systems. The presented method with the use of the Heaviside operator soon leads to the target without unnecessary differential equations.

In this paper, we show that signal sampling operation can be considered as a kind of all-pass filtering in the time domain, when the Nyquist frequency is larger or equal to the maximal frequency in the spectrum of a signal sampled. We demonstrate that this seemingly obvious observation has wideranging implications. They are discussed here in detail. Furthermore, we discuss also signal shaping effects that occur in the case of signal under-sampling. That is, when the Nyquist frequency is smaller than the maximal frequency in the spectrum of a signal sampled. Further, we explain the mechanism of a specific signal distortion that arises under these circumstances. We call it the signal shaping, not the signal aliasing, because of many reasons discussed throughout this paper. Mainly however because of the fact that the operation behind it, called also the signal shaping here, is not a filtering in a usual sense. And, it is shown that this kind of shaping depends upon the sampling phase. Furthermore, formulated in other words, this operation can be viewed as a one which shapes the signal and performs the low-pass filtering of it at the same time. Also, an interesting relation connecting the Fourier transform of a signal filtered with the use of an ideal low-pass filter having the cut frequency lying in the region of under-sampling with the Fourier transforms of its two under-sampled versions is derived. This relation is presented in the time domain, too.

The global stability of discrete-time nonlinear systems with descriptor positive linear parts and positive scalar feedbacks is addressed. Sufficient conditions for the global stability of standard and fractional nonlinear systems are established. The effectiveness of these conditions is illustrated on numerical examples.

The stability analysis for discrete-time fractional linear systems with delays is presented. The state-space model with a time shift in the difference is considered. Necessary and sufficient conditions for practical stability and for asymptotic stability have been established. The systems with only one matrix occurring in the state equation at a delayed moment have been also considered. In this case analytical conditions for asymptotic stability have been given. Moreover parametric descriptions of the boundary of practical stability and asymptotic stability regions have been presented.

Necessary and sufficient conditions for the pointwise completeness and the pointwise degeneracy of linear discrete-time different fractional order systems are established. It is shown that if the fractional system is pointwise complete in one step (q = 1), then it is also pointwise complete for q = 2, 3…

The positivity and absolute stability of a class of nonlinear continuous-time and discretetime systems are addressed. Necessary and sufficient conditions for the positivity of this class of nonlinear systems are established. Sufficient conditions for the absolute stability of this class of nonlinear systems are also given.

Controlling mechanical systems with position and velocity cascade loops is one of the most effective methods to operate this type of systems. However, when using low-rate sampling electronics, the implementation is not trivial and the resulting performance can be poor. This paper proposes effective tuning rules that only require establishing the bandwidth of the inner velocity loop and an estimation of the inertia of the mechanism. Since discrete-time mechatronic systems can also exhibit unstable behavior, several stability conditions are also derived. By using the proposed methodology, a P-PI control algorithm is developed for a desktop haptic device, obtaining good experimental performance with low sampling-rate electronics.

This paper proposes a design procedure for observer-based controllers of discrete-time switched systems, in the presence of state’s time-delay, nonlinear terms, arbitrary switching signals, and affine parametric uncertainties. The proposed switched observer and the state- feedback controller are designed simultaneously using a set of linear matrix inequalities (LMIs). The stability analysis is performed based on an appropriate Lyapunov–Krasovskii functional with one switched expression, and in the meantime, the sufficient conditions for observer-based stabilization are developed. These conditions are formulated in the form of a feasibility test of a proposed bilinear matrix inequality (BMI) which is a non-convex problem. To make the problem easy to solve, the BMI is transformed into a set of LMIs using the singular value decomposition of output matrices. An important advantage of the proposed method is that the established sufficient conditions depend only on the upper bound of uncertain parameters. Furthermore, in the proposed method, an admissible upper bound for unknown nonlinear terms of the switched system may be calculated using a simple search algorithm. Finally, the efficiency of the proposed controller and the validity of the theoretical results are illustrated through a simulation example.

Consider the semilinear system defined by

x(i+1) = Ax(i) + f(x(i)), i≥ 0

x(0) = x0 ϵ ℜn

and the corresponding output signal y(i)=Cx(i), i ≥ 0, where A is a n x n matrix, C is a p x n matrix and f is a nonlinear function. An initial state x(0) is output admissible with respect to A, f, C and a constraint set Ω in ℜp if the output signal (y(i))i associated to our system satisfies the condition y(i) in Ω, for every integer i ≥ 0. The set of all possible such initial conditions is the maximal output admissible set Γ(Ω). In this paper we will define a new set that characterizes the maximal output set in various systems (controlled and uncontrolled systems). Therefore, we propose an algorithmic approach that permits to verify if such set is finitely determined or not. The case of discrete delayed systems is taken into consideration as well. To illustrate our work, we give various numerical simulations.

Schemes are presented for calculating tuples of solutions of matrix polynomial equations using continued fractions. Despite the fact that the simplest matrix equations were solved in the second half of the 19th century, and the problem of multiplier decomposition was then deeply analysed, many tasks in this area have not yet been solved. Therefore, the construction of computer schemes for calculating the sequences of solutions is proposed in this work. The second-order matrix equations can be solved by a matrix chain function or iterative method. The results of the numerical experiment using the MatLab package for a given number of iterations are presented. A similar calculation is done for a symmetric square matrix equation of the 2nd order. Also, for the discrete (time) Riccati equation, as its analytical solution cannot be performed yet, we propose constructing its own special scheme of development of the solution in the matrix continued fraction. Next, matrix equations of the n-th order, matrix polynomial equations of the order of non-canonical form, and finally, the conditions for the termination of the iterative process in solving matrix equations by branched continued fractions and the criteria of convergence of matrix branching chain fractions to solutions are discussed.

Most construction projects involve subcontracting some work packages. A subcontractor is employed on the basis of their bid as well as according to their availability. A viable schedule must account for resource availability constraints. These resources (e.g. crews, subcontractors) engage in many projects, so they become at the disposal for a new project only in certain periods. One of the key tasks of a planner is thus synchronizing the work of resources between concurrent projects. The paper presents a mathematical model of the problem of selecting subcontractors or general contractor’s crews for a time-constrained project that accounts for the availability of contractors, as well as for the cost of subcontracting works. The proposed mixed integer-binary linear programming model enables the user to perform the time/cost trade-off analysis.

This paper addresses weighted L2 gain performance switching controller design of discrete-time switched linear systems with average dwell time (ADT) scheme. Two kinds of methods, so called linearizing change-of-variables based method and controller variable elimination method, are considered for the output-feedback control with a supervisor enforcing a reset rule at each switching instant are considered respectively. Furthermore, some comparison between these two methods are also given.

The paper addresses the problem of constrained pole placement in discrete-time linear systems. The design conditions are outlined in terms of linear matrix inequalities for the Dstable ellipse region in the complex Z plain. In addition, it is demonstrated that the D-stable circle region formulation is the special case of by this way formulated and solved pole placement problem. The proposed principle is enhanced for discrete-lime linear systems with polytopic uncertainties.