Normal Linear Systems Research Articles

Common Noninteracting Control with Simultaneous Common Partial Zeroing with Application to a Tracked UGV

In several MIMO system applications, the deviations of some output performance variables from their nominal values are required to be controlled independently, while the other performance variables are required to remain at their nominal value. This problem, named noninteracting control with simultaneous partial output zeroing, is important in the case of the common design of multi-model systems. To this end, the problem of a common noninteracting control with simultaneous common partial output zeroing is formulated. The present paper aims to develop a solution to the problem of multi-model normal linear time-invariant systems via regular and static measurement output feedback. The present approach follows the method developed for the solution of the common I/O decoupling problem. The main results of the paper are the introduction and the formulation of the problem at hand, the establishment of the necessary and sufficient conditions for its solvability, and the derivation of the respective general solution of the controller matrices. For the resulting closed-loop system, the additional design requirement of approximate command following a simultaneous I/O stabilizability is studied using a composite norm 2 type cost function and a metaheuristic algorithm for the derivation of the free parameters of the controller. The present results are illustrated through a numerical example of a nonlinear process with two operating points. Moreover, all the above results are successfully applied to the two-model description of a robot-tracked UGV, using a common controller feeding back measurements of the motor currents and the orientation of the vehicle.

Bayes Synthesis of Linear Nonstationary Stochastic Systems by Wavelet Canonical Expansions

This article is devoted to analysis and optimization problems of stochastic systems based on wavelet canonical expansions. Basic new results: (i) for general Bayes criteria, a method of synthesized methodological support and a software tool for nonstationary normal (Gaussian) linear observable stochastic systems by Haar wavelet canonical expansions are presented; (ii) a method of synthesis of a linear optimal observable system for criterion of the maximal probability that a signal will not exceed a particular value in absolute magnitude is given. Applications: wavelet model building of essentially nonstationary stochastic processes and parameters calibration.

Collocation method with fractional powers exponential trial functions for singularly perturbed reaction-convection-diffusion equation

We develop a collocation method for solving the singularly perturbed convection-diffusion equation and reaction-diffusion equation of elliptic type partial differential equations. When the singularly perturbed solution is expressed in terms of a set of fractional powers 2D exponential trial functions, and collocate points to satisfy the boundary conditions and the governing equation, we can obtain a small scale normal linear system, which is solved by the Gaussian elimination method or the conjugate gradient method to determine the expansion coefficients. The numerical algorithm with complexity O(n) is very time saving, effective and accurate to find the solutions of highly singularly perturbed reaction-convection-diffusion problems, which are defined in a rectangular domain as well as in an arbitrary domain, and the numerical examples together with an experimental result confirm the assessments.

Analysis and design of complex-valued linear systems

This paper studies a class of complex-valued linear systems whose state evolution dependents on both the state vector and its conjugate. The complex-valued linear system comes from linear dynamical quantum control theory and is also encountered when a normal linear system is controlled by feedback containing both the state vector and its conjugate that can provide more design freedom. By introducing the concept of bimatrix and its properties, the considered system is transformed into an equivalent real-representation system and a non-equivalent complex-lifting system, which are normal linear systems. Some analysis and design problems including solutions, controllability, observability, stability, eigenvalue assignment, stabilisation, linear quadratic regulation, and state observer design are then investigated. Criterion, conditions, and algorithms are provided in terms of the coefficient bimatrices of the original system. The developed approaches are also utilised to investigate the so-called antilinear system which is a special case of the considered complex-valued linear system. The existing results on this system have been improved and some new results are established.

Solutions to linear bimatrix equations with applications to pole assignment of complex-valued linear systems

We study in this paper solutions to several kinds of linear bimatrix equations arising from pole assignment and stability analysis of complex-valued linear systems, which have several potential applications in control theory, particularly, can be used to model second-order linear systems in a very dense manner. These linear bimatrix equations include generalized Sylvester bimatrix equations, Sylvester bimatrix equations, Stein bimatrix equations, and Lyapunov bimatrix equations. Complete and explicit solutions are provided in terms of the bimatrices that are coefficients of the equations/systems. The obtained solutions are then used to solve the full state feedback pole assignment problem for complex-valued linear system. For a special case of complex-valued linear systems, the so-called antilinear system, the solutions are also used to solve the so-called anti-preserving (the closed-loop system is still an antilinear system) and normalization (the closed-loop system is a normal linear system) problems. Second-order linear systems, particularly, the spacecraft rendezvous control system, are used to demonstrate the obtained theoretical results.

Data-based controllability analysis for generalised linear discrete-time system

ABSTRACTIn this paper, we aim to propose a data-based method to testify whether the system is a normal linear system or a generalised linear system and then analyse its controllability via the measured state data and the control input satisfying certain condition. For this purpose, we first describe a linear discrete-time system in a general form and derive a necessary and sufficient condition for the equivalent condition of complete controllability with a normal linear discrete time system. Second, we check whether the system is normal or singular, and then construct the controllability matrices only based on the input and measured state data. Third, the controllability of the corresponding system is investigated thoroughly based on available data without identifying system parameters. Finally, a numerical example and a stock price example are used to show the effectiveness and feasibility of the proposed data-based method.

Simulation the gas simultaneous adsorption over natural and modified zeolite

Adsorption is of great importance. The unique advantage of adsorption over other separation methods is the higher selectivity that can be achieved by adsorbents. In addition, adsorption phenomena play a vital role in many solid-state reactions and biological mechanisms. In this work, the adsorption process of CO 2 on the clinoptilolite (Skorynskoho field, Transcarpathian region, Ukraine) and SO 2 , NO, and CO 2 adsorption on K 2 CO 3 -modified – γ -alumina in a fixed-bed reactor were theoretical studied and simulated by computer-mathematic methods. The developed mathematical model based on the mass balance in gas and solid phase, the experimental saturation capacities, considering the activity of the adsorbent with respect to the gas by variable coefficients. The model presented by a normal linear system of differential equations with variable coefficients, it was solved by Taylor collocation method. The simulation shows that the data obtained by theoretical study are in agreement with data obtained in the simulation. According to Fisher Criterion the mathematic model adequate in 90 % for modified zeolite and in 75 % for natural zeolite, it can be explained by unordered structure of the natural zeolite. It follows that the offered model adequately describes the dynamics simultaneous adsorption of gases over zeolites. Thus even with a large number of simplifications and assumptions, it is possible to construct efficient mathematical model that can be used in exhaust system. The results indicate that, there is great sense to conduct further researches and simulations to reach the industrial level.

Differentiability properties of the minimum time function for normal linear systems

A result by Sussmann [32], implying that the minimum time function T for normal linear systems is analytic out of a locally finite union of analytic submanifolds, is revisited. The original proof relies on classical properties of subanalytic sets. A constructive and simpler proof is given of a part of it: we show that the singular, i.e., the nondifferentiability, set of T is contained in a lower dimensionally rectifiable set which can be identified through suitable properties of the switching function. Our approach extends to large times the strategy initiated by Hájek in [23]. Furthermore, a result on the propagation of singularities of non-Lipschitz type is proved. Finally, we give explicit formulas for the propagation of first and second order partial derivatives of T along optimal trajectories and we prove that normal cones to the epigraph of T at each (x,T(x)) and to the sublevel of T at x have the same dimension.

Sliding Mode Diagnosers for Leakage Faults on Timed Continuous Petri Nets

Abstract: This paper considers the application of sliding mode diagnosers for leakage fault detection in systems modeled by Timed Continuous Petri Nets under infinite server semantics. In this case the Timed Continuous Petri Net describes a Switching Linear System. Two cases are addressed; in the first one all linear systems are distinguishable from each other (normal and faulty linear systems), while in the second one the linear systems representing the normal behavior are distinguishable from each other, and the linear systems representing the faulty behavior are observable. In the first case a single sliding mode diagnoser, based on a supertwisting switched linear system observer is proposed. For the second case, a combination between the previous diagnoser and models for linear systems representing the faulty behavior are used to detect and isolate the fault.

Non-Lipschitz points and the $${\textit{SBV}}$$ SBV regularity of the minimum time function

This paper is devoted to the study of the Hausdorff dimension of the singular set of the minimum time function \(T\) under controllability conditions which do not imply the Lipschitz continuity of \(T\). We consider first the case of normal linear control systems with constant coefficients in \({\mathbb {R}}^N\). We characterize points around which \(T\) is not Lipschitz as those which can be reached from the origin by an optimal trajectory (of the reversed dynamics) with vanishing minimized Hamiltonian. Linearity permits an explicit representation of such set, that we call \(\mathcal {S}\). Furthermore, we show that \(\mathcal {S}\) is countably \(\mathcal {H}^{N-1}\)-rectifiable with positive \(\mathcal {H}^{N-1}\)-measure. Second, we consider a class of control-affine planar nonlinear systems satisfying a second order controllability condition: we characterize the set \(\mathcal {S}\) in a neighborhood of the origin in a similar way and prove the \(\mathcal {H}^1\)-rectifiability of \(\mathcal {S}\) and that \(\mathcal {H}^1(\mathcal {S})>0\). In both cases, \(T\) is known to have epigraph with positive reach, hence to be a locally \(BV\) function (see Colombo et al.: SIAM J Control Optim 44:2285–2299, 2006; Colombo and Nguyen.: Math Control Relat 3: 51–82, 2013). Since the Cantor part of \(DT\) must be concentrated in \(\mathcal {S}\), our analysis yields that \(T\) is locally \(SBV\), i.e., the Cantor part of \(DT\) vanishes. Our results imply also that \(T\) is differentiable outside a \(\mathcal {H}^{N-1}\)-rectifiable set. With small changes, our results are valid also in the case of multiple control input.

Sturmian theory for linear Hamiltonian systems without controllability

In this paper we present Sturmian separation and comparison theorems for linear Hamiltonian systems when no controllability assumption is imposed. This generalizes the traditional results of W. T. Reid for controllable (or normal) linear Hamiltonian systems to the possibly abnormal case. Our new theory is based on several recent results on linear Hamiltonian systems without controllability by W. Kratz, M. Wahrheit, V. Zeidan, and the author regarding the piecewise constant kernel for conjoined bases, the oscillation and eigenvalue theorems, and the Rayleigh principle. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Observer design and output feedback stabilization for linear singular time-delay systems with unknown inputs

The design of a functional observer and reduced-order observer with internal delay for linear singular time-delay systems with unknown inputs is discussed. The sufficient conditions of the existence of observers, which are normal linear time-delay systems, and the corresponding design steps are presented via linear matrix inequality(LMI). Moreover, the observer-based feedback stabilizing controller is obtained. Three examples are given to show the effectiveness of the proposed methods.

Set Invariance Conditions for Singular Linear Systems Subject to Actuator Saturation

In this note, we establish a set of conditions under which an ellipsoid is contractively invariant with respect to a singular linear system under a saturated linear feedback. These conditions can be expressed in terms of linear matrix inequalities, and the largest contractively invariant ellipsoid can be determined by solving an optimization problem with LMI constraints. With the feedback gain viewed as an additional variable, this optimization problem can be readily adapted for the design of feedback gain that results in the largest contractively invariant ellipsoid. Moreover, in the degenerate case where the singular linear system reduces to a regular system, our set invariance conditions reduce to the existing set invariance conditions for normal linear systems.

Some invariants of discrete-time descriptor systems

Many papers analyze the role of controllability and observability indices of normal linear systems in control theory and their use in the study of structural properties. This work introduces these indices for linear discrete-time descriptor systems. Such indices are studied in two different ways, by means of the state-space approach and using its transfer matrix . A canonical form of the state-space system and a special decomposition of the nonproper transfer matrix are considered in order to discuss the reachability and observability indices.

REGIONAL POLE PLACEMENT BY OUTPUT FEEDBACK FOR A CLASS OF DESCRIPTOR SYSTEMS

The problem of regional pole placement by static output feedback is studied for a class of linear descriptor systems that presents two caracteristics found in some physical models: regularity and absence of direct action of control inputs on the algebraic variables. Thus the structure of the mathematical model of this class of systems is exploited. We show, in particular, how some results from the classical theory of normal (differential) linear systems can be used to solve the considered problem. The paper provides a set of necessary and sufficient conditions to characterize the existence of an output feedback that places the finite closed-loop poles in particular regions of the complex plane. A technique based on the use of an orthogonal decomposition and on the solution of two-coupled Lyapunov equations is also proposed. A numerical example illustrates the approach.

Normal Linear Regression Models With Recursive Graphical Markov Structure

A multivariate normal statistical model defined by the Markov properties determined by an acyclic digraph admits a recursive factorization of its likelihood function (LF) into the product of conditional LFs, each factor having the form of a classical multivariate linear regression model (≡WMANOVA model). Here these models are extended in a natural way to normal linear regression models whose LFs continue to admit such recursive factorizations, from which maximum likelihood estimators and likelihood ratio (LR) test statistics can be derived by classical linear methods. The central distribution of the LR test statistic for testing one such multivariate normal linear regression model against another is derived, and the relation of these regression models to block-recursive normal linear systems is established. It is shown how a collection of nonnested dependent normal linear regression models (≡Wseemingly unrelated regressions) can be combined into a single multivariate normal linear regression model by imposing a parsimonious set of graphical Markov (≡Wconditional independence) restrictions.

Measurement stability of time-optimal feedback control of two-input strictly normal linear systems

The concept of “stability with respect to measurement” was introduced by H. Hermes to characterize those feedback control systems which are tolerant of errors in state variable measurement within the feedback loop. This paper considers autonomous linear strictly normal systems with two-dimensional controls and develops necessary and sufficient conditions for local measurement stability of the time-optimal feedback control system.

Local Time-Optimal Feedback Control of Strictly Normal Two-Input Linear Systems

This paper introduces a new technique for the analysis of time-optimal control of linear systems. A family of easily calculated invariants is developed and, for an important class of two-input systems, is shown to provide a complete description of the time-optimal flow near the origin. The two switching surfaces are described analytically and qualitatively in topological terms. The time-optimal feedback function is defined and analyzed with respect to its complexity and sensitivity to errors in state variable measurement. The results lead to the first explicit construction of a local regular synthesis for multi-input systems of arbitrary order.

Generalized-function solutions of differential and functional differential equations

Recently there has been considerable interest in problems concerning the existence of solutions to differential and functional differential equations (FDE) in various spaces of generalized functions. Many important areas in theoretical and mathematical physics, theory of partial differential equations, quantum electrodynamics, operational calculus, and functional analysis use the methods of the distributions theory. Yet for ordinary differential equations (ODE) and especially FDE, research in this direction is insufficiently developed and remains restricted to isolated results for some secondorder equations or special higher-order systems. We note, in particular. papers [l-6] which contain references to previous work. It is weil known that normal linear homogeneous systems of ODE with infinitely smooth coefficients have no generalized-function solutions other than the classical solutions. In contrast to this case, for equations with singularities in the coefficients, new solutions in generalized functions may appear and some classical solutions may disappear. The number m is called the order of the distribution

On Delayed Action System Control Under Equality and Inequality Constraints

A necessary condition is presented in the form of Pontryagin's maximum principle for the problem of optimally controlling a delayed action system ẋ(t)=f(x(t), x(t-h), u(t), u(t-h), t) subject to various functional equality and inequality constraints. It is further shown that the necessary condition becomes also a sufficient condition for normal linear systems under typical convexity assumptions. Then it is discussed that the results can be effectively utilized to solve classes of (i) function-target control problems, (ii) nonzero duration rendezvous problems and (iii) control problems subject to certain functional constraints.