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EN
This paper deals with the problem of implementing adaptive radar tracking filters based on continuous-time models of target motion and on discrete-time models of measurement process. The particular difficulties addressed include: nonlinear and non-stationary target movement models with uncertain parameters, and low data rate due to a rotating radar antenna. The proposed tracking filter relies basically on the continuous-discrete variant of the extended Kalman filter (EKF), the probabilistic data association (PDA) technique and the interacting multiplemodel (IMM) state estimation scheme. Numerical properties of the algorithm are discussed and a software implementation is developed using the open-source BLAS library. Several design concepts are combined to assure numerical stability, convergence and efficiency of the estimator.
EN
Reduced-order Kalman filters yield an optimal state estimate for linear dynamical systems, where parts of the output are not corrupted by noise. The design of such filters can either be carried out in the time domain or in the frequency domain. Different from the full-order case where all measurements are noisy, the design equations of the reduced-order filter are not regular. This is due to the rank deficient measurement covariance matrix and it can cause problems when using standard software for the solution of the Riccati equations in the time domain. In the frequency domain the spectral factorization of the non-regular polynomial matrix equation does not cause problems. However, the known proof of optimality of the factorization result also requires a regular measurement covariance matrix. This paper presents regular (reduced-order) design equations for reduced-order Kalman filters in the time and in the frequency domains for linear continuous-time systems. They allow to use standard software for the design of the filter, to formulate the conditions for the stability of the filter and they also prove that the existing frequency domain solutions obtained by spectral factorization of a non-regular polynomial matrix equation are indeed optimal.
EN
The problem of asymptotic stability of continuous-discrete linear systems is considered. Simple necessary conditions and two computer methods for investigation of asymptotic stability of the second Fornasini-Marchesini type model are given. The first method requires computation of the eigenvalue-loci of complex matrices, the second method requires computation of determinants of some matrices. Effectiveness of the methods is demonstrated on numerical example.
EN
The paper considers the stability problem of linear time-invariant continuous-time systems of fractional order, standard and positive, described by the state space model. Review of previous results is given and some new methods for stability checking are presented. Considerations are illustrated by numerical examples and results of computer simulations.
5
Content available remote On parametric Hurwitz stability margin of real polynomials
EN
The paper deals with the problem of determining Hurwitz stability of a ball of polynomials defined by a weighted lp norm in the coefficient space where p is an arbitrary positive integer including infinity. The solution of the case when the weights are supposed to be the same for coefficient being above and below its nominal value corresponding to symmetric ball has been given by Tsypkin and Polyak. However, sometimes it seems to be useful to have a possibility to consider these weights as different, resulting in the asymmetric ball. This is, for example, the situation where the weights express our level of confidence that the real value of a coefficient lies in some interval. Such approach is used if the value of a coefficient is estimated by an expert. Solution of the problem is based on frequency domain plot in the complex plane and on applying the Zero Exclusion Theorem. The main idea consists in separation of the original problem into four subproblems and using an appropriate coordinate transformation which makes the value set independent of frequency. This transformation makes it possible to move the relative value set into the origin of the complex plane and to easily formulate the necessary and sufficient condition of Hurwitz stability of asymmetric ball of polynomials with prescribed radius or determine the maximum radius preserving stability. The whole graphical procedure consists of four plots instead of one, needed in the symmetric case.
EN
The positive realization problem for singular continuous-time linear single-input single-output systems with delays in state and in inputs is addressed. The notion of canonical forms of matrices are extended for singular linear systems with delays. Necessary and sufficient conditions for positivity of the singular continuous-time systems with delays and sufficient conditions for the existence of a positive singular realization are established. A procedure for computation of a positive singular realization of a given transfer function is proposed and illustrated by a numerical example.
EN
This paper summarizes and discusses the results of an extensive simulation based exercise in quest of dependable approaches to identification of continuous-time models from sampled input/output data of continuous-time dynamical systems. Two well-established approaches are considered together with several related techniques of parameter estimation. One is an indirect approach in which well-established discrete-time techniques are applied to first estimate a discrete-time model for the original continuous-time system and the model is then transformed into a continuous-time version. The other is a direct approach in which a continuous-time model is estimated straightaway using well-known continuous-time methods. On the surface, the choice between the two approaches may seem trivial but this paper underlines the need to establish dependability of any approach in terms of certain criteria. The results of extensive simulations clearly show that the direct approach is more dependable than the indirect route.
EN
In MIMO LTI continuous-time systems S(A, B, C) the classical notion of the Smith zeros does not characterize fully the output-zeroing problem. In order to analyze the question we extend this notion by treating multivariable zeros (called further the invariant zeros) as the triples (complex number, nonzero state-zero direction, input-zero direction). Nothing is assumed about the relationship of the number of inputs to the number of outputs nor about the normal rank of the underlying system matrix. The treatment is strictly connected with the output zeroing problem and in that spirit the zeros can be easily interpreted even in the degenerate case (i.e., when any complex number is such zero). A simple sufficient and necessary condition of nondegeneracy is presented. The condition decomposes the class of all systems S(A, B, C) such that and into two disjoint subclasses: of nondegenerate and degenerate systems. In nondegenerate systems, the Smith zeros and the invariant zeros are exactly the same objects which are determined as the roots of the so-called zero polynomial. The degree of this polynomial equals the dimension of the maximal (A, B)-invariant subspace contained in KerC, while the zero dynamics are independent of control vector. In degenerate systems the zero polynomial determines merely the Smith zeros, while the set of the invariant zeros equals the whole complex plane. The dimension of the maximal (A, B)-invariant subspace contained in KerC is strictly larger than the degree of the zero polynomial, whereas the zero dynamics essentially depend upon control vector.
9
Content available remote Interactive-learning Control of Linear Continuous-time Systems with Disturbances
EN
An algorithm of interactive-learning control is presented for multivariable linear continuous-time systems with disturbances. Proposed algorithm enables ones to calculate input equence such that the tracking error is as small as acceptable, also in the case of uncertain plant model. Necessary and sufficient conditions are formulated for the convergence of the learning control system.
EN
This paper refers to application of the Schauder's fixed point theorem together with linear controllability results in getting the sufficient controllability conditions for various kinds of controllability and for different types of nonlinear control systems. The following nonlinear control systems are considered : finite-dimensional systems, systems with delays in control or in the state variables, and infinite-dimensional systems. The paper presents the review of results existing in the literature which show how Schauder's fixed-point theorem can be practically used to solve several controllability problems for different types of nonlinear control systems.
11
Content available remote Analytical Design of Stable Continuous-Time Generalised Predictive Control
EN
With a recently renewed interest in the continuous-time approach to control system design the continuous-time generalised predictive control (CGPC) is also worth considering. The main objective of this presentation is the development of an analytical perspective that results in explicit design procedures for stable control of both minimum-phase and non-minimum-phase SISO systems. The basic project idea is founded on a set of closed-loop prototype characteristics with definite time-domain specifications.
EN
The continuous-time generalised predictive control (CGPC) is considered in the context of control of continuous-time systems having a transportation delay. It is shown that the basic CGPC design strategy can be given in a form which facilitates a clear discussion of relevant design consequences concerning stability issues. The main results that follow incorporate several solutions to the delay-plant control design problem and a verification of the proposed algorithms in terms of the closed-loop stability.
EN
In the process of designing controllers for linear multivariable plants specially effective are algebraic methods which require from the transfer matrices of both, the plant and the controller to be presented in coprime fractional form with factorization carried on with respect to the ring of exponentially-stable, proper real-rational functions. The main objective of the paper is to show that this form of representation with simultaneous parametrization of all linear controllers that provide internal stability of the closed-loop system can be achieved in the simplest and most natural way by analysing the system shown in Fig. 3 - the so-called basic structure. Problems of choosing the parameter to meet some important design specifications, viz. a robust asymptotic tracking of the reference signal with disturbance and noise rejection are also considered and illustrated by two representative examples covering the area of continuous- and discrete-time systems.
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