The LQR (linear quadratic regulator) control problem subject to singular system constitutes a optimization problem in which one must be find an optimal control that satisfy the singular system and simultaneously to optimize the quadratic objective functional. In this paper we establish a sufficient condition to obtain the optimal control of discounted LQR optimization problem subject to disturbanced singular system where the disturbance is time varying. The considered problem is solved by transforming the discounted LQR control problem subject to disturbanced singular system into the normal LQR control problem. Some available results in literatures of the normal LQR control problem be used to find the sufficient conditions for the existence of the optimal control for discounted LQR control problem subject to disturbanced singular system. The final result of this paper is in the form a method to find the optimal control of discounted LQR optimization problem subject to disturbanced singular system. The result shows that the disturbance is vanish with the passage of time.
The relationship between internal response-based reliability and conditionality is investigated for Gauss-Markov (GM) models with uncorrelated observations. The models with design matrices of full rank and of incomplete rank are taken into consideration. The formulas based on the Singular Value Decomposition (SVD) of the design matrix are derived which clearly indicate that the investigated concepts are independent of each other. The methods are presented of constructing for a given design matrix the matrices equivalent with respect to internal response-based reliability as well as the matrices equivalent with respect to conditionality. To analyze conditionality of GM models, in general being inconsistent systems, a substitute for condition number commonly used in numerical linear algebra is developed, called a pseudo-condition^number. Also on the basis of the SVD a formula for external reliability is proposed, being the 2-norm of a vector of parameter distortions induced by minimal detectable error in a particular observation. For systems with equal nonzero singular values of the design matrix, the formula can be expressed in terms of the index of internal response-based reliability and the pseudo-condition^number. With these measures appearing in explicit form, the formula shows, although only for the above specific systems, the character of the impact of internal response-based reliability and conditionality of the model upon its external reliability. Proofs for complementary properties concerning the pseudo-condition^number and the 2-norm of parameter distortions in systems with minimal constraints are given in the Appendices. Numerical examples are provided to illustrate the theory.
This article is devoted to the official forms in the inflection of chosen toponyms in Poland and the variety of dialectal singular and plural place names. The variety of place names often diverges from the rules of language use, and causes language users problems. The toponyms have peculiar, locally used inflected forms; the outside-linguistic (non-linguistic) factors that are social and local factors, play an important role in the inflection of place names. The local population often uses other forms than those recommended by official sources. I focus my attention on the genitive forms of toponyms because it is mainly here that one can see clear variations in the official and local inflection of place names. The material shows that the singular masculine toponyms have genitive endings: -a (in the official variety), -u (in the local variety), for example Biłgoraj, gen. Biłgoraja, but in the local dialect: biłgoraju. The singular feminine place names have genitive endings: -ej (in the official variety), -y||-i (in the local variety): Brzezowa, gen. Brzezowej, but Brzezowy in the local dialect. The plural toponyms have genitive endings: -ø, -ów, -i (-y), but in the local circulation the ending -ów is dominant and demonstrates a wider expansion in use. For example Brzózki, gen. Brzózek, in the local variety Brzuskuf; Budy, official gen. Bud, but Buduf; Burnie, gen. Burni, in the local dialect: Burniuf. The gathered material reflects a hesitation in the inflection of toponyms, as the linguistic customs and presented dialectal records of forms of genit ives of place names show a significantly diverse approach towards the Polish language.
This paper presents the design of digital controller for longitudinal aircraft model based on the Dynamic Contraction Method. The control task is formulated as a tracking problem of velocity and flight path angle, where decoupled output transients are accomplished in spite of incomplete information about varying parameters of the system and external disturbances. The design of digital controller based on the pseudo-continuous approach is presented, where the digital controller is the result of continuous-time controller discretization. A resulting output feedback controller has a simple form of a combination of low-order linear dynamical systems and a matrix whose entries depend nonlinearly on certain known process variables. Simulation results for an aircraft model confirm theoretical expectations.
This paper presents a robust model free controller (RMFC) for a class of uncertain continuous-time single-input single-output (SISO) minimum-phase nonaffine-in-control systems. Firstly, the existence of an unknown dynamic inversion controller that can achieve control objectives is demonstrated. Afterwards, a fast approximator is designed to estimate as best as possible this dynamic inversion controller. The proposed robust model free controller is an equivalent realization of the designed fast approximator. The perturbation theory and Tikhonov’s theorem are used to analyze the stability of the overall closed-loop system. The performance of the developped controller are verified experimentally in the position control of a pneumatic actuator system.