Role In Control Theory Research Articles

SOLVING BIPOLAR FULLY FUZZY SYLVESTER MATRIX EQUATIONS WITH NEGATIVE FUZZY NUMBERS

Sylvester matrix equations play a crucial role in control theory for controller design. Bipolar Fully Fuzzy Sylvester Matrix Equations (FFSME), incorporating both positive and negative components, are employed in controller design to address uncertainties that may affect a system’s performance and stability. However, there is not much existing research on combining bipolar fuzzy numbers and FFSME,and most of them mainly deal with positive coefficients. Thus, this paper presents a method that enables solving the negative coefficient of bipolar FFSME in the form of Left-Right (LR) triangular fuzzynumbers using an Associated Bipolar Linear System (ABLS). To obtain the ABLS, bipolar FFSME is transformed into a bipolar Fully Fuzzy Linear System (FFLS) using the Kronecker product and Vecoperator. Subsequently, the solution is derived through the inverse method, and the equation of the ABLS is rearranged as a bipolar fuzzy matrix. Additionally, this paper provides two numerical examples to illustrate the applicability of the constructed method.

Modeling and Control of Ball and Beam Problem with Controllers’ Performance Comparison

Since nonlinear systems are difficult to analyze but play a critical role in control theory, the construction of an appropriate controller is essential. A robust controller should be capable of dealing with abruwpt changes, uncertainty, disturbances, and other nonlinearities in the system. An accurate control system for nonlinear and unstable ball and beam system is essential, moreover it presents a significant challenge to be controlled by engineers and researchers. The ball and beam problem can be solved by using a controller that would monitor the ball position in the system. The ball and beam system is a very popular system due to its semplicity to be understood and the control techniques that can be studied over it. The system has been built in many ways before some easier and some more effective ones. This system in an open-loop setting is inseparably nonlinear and unstable but with the appropriate steps it can stabilize its response. Physical laws are used to conduct the ball and beam system theoretical analysis. System mathematical modeling is presented clearly and simulations together with their analysis will be done by using MATLAB software. It will be necessary to use mathematical modeling, which will include linearization of the model. The performance of each controller will be evaluated and compared. Two different controllers are used in this work.

Asymptotic Behavior of Reachable Setson Small Time Intervals

The geometric structure of small-time reachable sets plays an important role in control theory, in particular, in solving problems of local synthesis. In this paper, we consider the problem of approximate description of reachable sets on small time intervals for control-affine systems with integral quadratic constraints on the control. Using a time substitution, we replace such a set by the reachable set on a unit interval of a control system with a small parameter, which is the length of the time interval for the original system. The constraints on the control are given by a ball of small radius in the Hilbert space $$\mathbb{L}_{2}$$ . Under certain conditions imposed on the controllability Gramian of the linearized system, this reachable set turns out to be convex for sufficiently small values of the parameter. We show that in this case the shape of the reachable set in the state space is asymptotically close to an ellipsoid. The proof of this fact is based on the representation of the reachable set as the image of a Hilbert ball of small radius in $$\mathbb{L}_{2}$$ under a nonlinear mapping to $$\mathbb{R}^{n}$$ . In particular, this asymptotic representation holds for a fairly wide class of second-order nonlinear control systems with integral constraints. We give three examples of systems whose reachable sets demonstrate both the presence of the indicated asymptotic behavior and the absence of the latter if the necessary conditions are not satisfied.

Robustness and perturbations of minimal bases

Polynomial minimal bases of rational vector subspaces are a classical concept that plays an important role in control theory, linear systems theory, and coding theory. It is a common practice to arrange the vectors of any minimal basis as the rows of a polynomial matrix and to call such matrix simply a minimal basis. Very recently, minimal bases, as well as the closely related pairs of dual minimal bases, have been applied to a number of problems that include the solution of general inverse eigenstructure problems for polynomial matrices, the development of new classes of linearizations and ℓ-ifications of polynomial matrices, and backward error analyses of complete polynomial eigenstructure problems solved via a wide class of strong linearizations. These new applications have revealed that although the algebraic properties of minimal bases are rather well understood, their robustness and the behavior of the corresponding dual minimal bases under perturbations have not yet been explored in the literature, as far as we know. Therefore, the main purpose of this paper is to study in detail when a minimal basis M(λ) is robust under perturbations, i.e., when all the polynomial matrices in a neighborhood of M(λ) are minimal bases, and, in this case, how perturbations of M(λ) change its dual minimal bases. In order to study such problems, a new characterization of whether or not a polynomial matrix is a minimal basis in terms of a finite number of rank conditions is introduced and, based on it, we prove that polynomial matrices are generically minimal bases with very specific properties. In addition, some applications of the results of this paper are discussed.

On an asymptotic analysis problem related to the construction of an attainability domain

Problems of constructing and analyzing the properties of attainability domains play an important role in control theory and its applications. In particular, this applies to control under impulse constraints that reflect the energetics of a process. The situation is complicated by the possible instability of the process under variation (in particular, under relaxation) of constraints related to boundary and intermediate conditions. Stability of the problem is also missing, in general, under relaxation of state constraints. In these cases, it is natural to focus on the asymptotic variant of the statement; this is especially expedient when one has to deal with initially asymptotic requirements. In all these cases, it seems expedient to use analogs of J. Warga’s approximate solutions. At the same time, to seek necessary approximate (and, in fact, asymptotic) solutions, it is natural to use generalized modes. For problems with impulse constraints and discontinuity in the coefficients of control actions, such modes lead to phenomena described by products of discontinuous functions and generalized functions even in the class of linear systems. In a large series of his studies, to overcome the arising difficulties, one of the authors used constructions of extension in the class of finitely additive measures. The present paper follows this approach and is ideologically relevant to the engineering problem of controlling the thrust of an engine under conditions of a given program of variation of its orientation; it is postulated that energy resources are completely consumed in a natural (for a number of impulse control problems) mode of short-duration impulses: the set of time instants at which the instantaneous control is different from zero can be embedded in an interval of vanishingly small length. Within these short periods of time, the engine should consume all energy resources while obeying some other constraints (making the sense of moment constraints) to a high degree of accuracy. In addition, one should take into account the possible discontinuity of the functions defining the coefficients of control actions. As a natural analog of the attainability domain, we use an attraction set, whose construction, together with the subsequent study of its main properties, constitutes the goal of the present study.

Singular Velocities of Even-Rank Affine Distributions

Singular curves, which are projections of singular extremals, play a special role in control theory and the theory of distributions. In this paper, we show that singular velocities, tangent to singular curves, determine affine distributions that are simultaneously of an even-rank and co-rank 2. If D is such a distribution, then its singular velocities span a distribution C D (of a rank two times smaller than that of D) that, together with its Lie square, forms the initial distribution, i.e., \(D={C^{D}}+\left [{C^{D}},{C^{D}}\right ]\).We also provide canonical constructions of vector fields that are C D generators, canonical differential forms that are D cogenerators, and some functional invariants related to distribution of the considered type.

Solving differential matrix Riccati equations by a piecewise-linearized method based on diagonal Padé approximants

Differential Matrix Riccati Equations (DMREs) appear in several branches of science such as applied physics and engineering. For example, these equations play a fundamental role in control theory, optimal control, filtering and estimation, decoupling and order reduction, etc. In this paper a new method based on a theorem proved in this paper is described for solving DMREs by a piecewise-linearized approach. This method is applied for developing two block-oriented algorithms based on diagonal Padé approximants. MATLAB versions of the above algorithms are developed, comparing, under equal conditions, accuracy and computational costs with other piecewise-linearized algorithms implemented by the authors. Experimental results show the advantages of solving stiff or non-stiff DMREs by the implemented algorithms.

Adams–Bashforth and Adams–Moulton methods for solving differential Riccati equations

Differential Riccati equations play a fundamental role in control theory, for example, optimal control, filtering and estimation, decoupling and order reduction, etc. In this paper several algorithms for solving differential Riccati equations based on Adams–Bashforth and Adams–Moulton methods are described. The Adams–Bashforth methods allow us explicitly to compute the approximate solution at an instant time from the solutions in previous instants. In each step of Adams–Moulton methods an algebraic matrix Riccati equation (AMRE) is obtained, which is solved by means of Newton’s method. Nine algorithms are considered for solving the AMRE: a Sylvester algorithm, an iterative generalized minimum residual (GMRES) algorithm, a fixed-point algorithm and six combined algorithms. Since the above algorithms have a similar structure, it is possible to design a general and efficient algorithm that uses one algorithm or another depending on the considered differential matrix Riccati equation. MATLAB versions of the above algorithms are developed, comparing precision and computational costs, after numerous tests on five case studies.

A family of BDF algorithms for solving Differential Matrix Riccati Equations using adaptive techniques

Abstract Differential Matrix Riccati Equations play a fundamental role in control theory, for example, in optimal control, filtering and estimation, decoupling and order reduction, etc. One of the most popular codes to solve stiff Differential Matrix Riccati Equations (DMREs) is based on Backward Differentiation Formula (BDF). In previous papers the authors of this paper showed two algorithms for solving DMREs based on an iterative Generalized Minimum RESidual (GMRES) approach and on a Fixed-Point approach. In this paper we present two contributions to improve the above algorithms. Firstly six variants of previous algorithms are carried out by using one of above algorithms in the first step and another algorithm to carry out the other steps until reaching convergence. Numerous tests on four case studies have been done comparing both precision and computational costs of MATLAB implementations of the above algorithms. Experimental results show that in some cases these algorithms improve on the speed and convergence of the original algorithms. Secondly, using the previous experimental results and since all algorithms have a similar structure and there is no best algorithm to solve all problems, two general-purpose adaptive algorithms have been designed for selecting the most appropriate algorithm, which can be chosen using a parameter that indicates the stiffness of the DMRE to be solved.

Solving Differential Matrix Riccati Equations by a piecewise-linearized method based on the conmutant equation

Differential Matrix Riccati Equations play a fundamental role in control theory, for example, in optimal control, filtering and estimation, decoupling and order reduction, etc. In this paper a piecewise-linearized method based on the conmutant equation to solve Differential Matrix Riccati Equations (DMREs) is described. This method is applied to develop two algorithms which solve these equations: one for time-varying DMREs and another for time-invariant DMREs, also MATLAB implementations of the above algorithms are developed. Since MATLAB does not have functions which solve DMREs, two algorithms based on a BDF method are also developed. All implemented algorithms have been compared, under equal conditions, at both precision and computational costs. Experimental results show the advantages of solving non-stiff DMREs and in particular stiff DMREs by the proposed algorithms.

Lagrange multiplier necessary conditions for global optimality for non-convex minimization over a quadratic constraint via S-lemma

In this paper, we present Lagrange multiplier necessary conditions for global optimality that apply to non-convex optimization problems beyond quadratic optimization problems subject to a single quadratic constraint. In particular, we show that our optimality conditions apply to problems where the objective function is the difference of quadratic and convex functions over a quadratic constraint, and to certain class of fractional programming problems. Our necessary conditions become necessary and sufficient conditions for global optimality for quadratic minimization subject to quadratic constraint. As an application, we also obtain global optimality conditions for a class of trust-region problems. Our approach makes use of outer-estimators, and the powerful S-lemma which has played key role in control theory and semidefinite optimization. We discuss numerical examples to illustrate the significance of our optimality conditions.

A GMRES-based BDF method for solving differential Riccati equations

Differential Riccati equations play a fundamental role in control theory, for example, optimal control, filtering and estimation, decoupling and order reduction, etc. The most popular codes to solve stiff differential Riccati equations use backward differentiation formula (BDF) methods. In this paper, a new approach to solve differential Riccati equations by means of a BDF method is described. In each step of these methods an algebraic Riccati equation is obtained, which is solved by means of Newton’s method. In the standard approach, this system is transformed into a Sylvester equation, which could be solved by means of the well-known Bartels–Stewart method. In our code, we obtain a system of linear equations, defined from a Kronecker product of matrices related to coefficient matrices of the differential Riccati equation, that is solved by means of the iterative generalized minimum residual (GMRES) method. We have also implemented an efficient matrix–vector product in order to reduce the computational and storage cost of the GMRES method. The above approach has been applied in the development of an algorithm to solve differential Riccati equations. The accuracy and efficiency of this algorithm has been compared with the BDF algorithm that uses the Bartels–Stewart method. Experimental results show the advantages of the new algorithm.

A less conservative stability test for second-order linear time-varying vector differential equations

System stability and stability bounds play an essential role in control theory. This note is concerned with the exponential stability of a class of second-order linear time-varying vector differential equations with real piecewise continuous coefficient matrices. A less conservative explicit condition for stability of such a system is derived using the matrix measure theory and a more accurate upper bound for the decay exponent of its stable solution is established. Examples are included for illustration.

The Role of Control Theory in Computer Vision and Image Processing

The Role of Control Theory in Computer Vision and Image Processing

Singular Trajectories and Their Role in Control Theory

Linear Systems and the Time Optimal Control Problem. - Exercises. - Optimal Control for Nonlinear Systems. - Exercises. - Geometric Optimal Control. - Exercises. - Singular Trajectories and Feedback Classification. - Exercises. - Controllability, Higher Order Maximum Principle, Legendre-Clebsch and Goh Necessary Optimality Conditions. - Exercises. - The Concept of Conjugate Points in the Time Minimal Control Problem for Singular Trajectories, C0-Optimality. - Time Minimal Control of Chemical Batch Reactors and Singular Trajectories. - Generic Properties of Singular Trajectories. - Exercises. - Singular Trajectories in Sub-Riemannian Geometry. - Exercises. - Micro-Local Resolution of the Singularity near a Singular Trajectory, Lagrangian Manifolds and Symplectic Stratifications. - Exercises. - Numerical Computations. - Conclusion and Perspectives. - Exercises. - References. - Index

On the Spectral Multiplicity of Some Volterra Operators in Sobolev Spaces

=1.Note that the notion of spectral multiplicity plays an important role in control theory [1].Namely, a linear dynamical system is described by the equationdx(t)dt=Ax(t)+Bu(t),t≥ 0, (1)with the operators A∈ [X]andB= B U, B∈ [U,X], whereX, U are Banach spaces (thestate space and the subspace of control, respectively)andx(t)∈ X, u(t)∈ U.The first problem in control theory is to find all spaces U having the following controllabilityproperty: for any state x∈ X,anye>0, and any instant of time t>0, there exists a (smooth)control function u(·)for which the solution of system (1)arrives in thee-neighborhood of x, x−x(t) <e, starting from some fixed initial state x(0)(for instance,x(0)=0).The well-known Kalman theorem states that the system A,B U is controllable if and only ifBU∈ CycA, i.e., if the minimal dimension of the subspace of control in (1)is equal toµ

Solving the Generalized Lyapunov Equation by the Bartels-Stewart Method Using Standard Software Libraries for Linear Algebra Computations

The generalized Lyapunov equation, in its continuous-time and discretetime versions, plays an essential role in control theory, as in the stabilization and optimal control of linear systems. This article focuses on solving the generalized Lyapunov equation by the Bartels-Stewart method. Sequential and block-oriented implementations using standard software libraries for linear algebra computations BLAS and LAPACK are presented. Furthermore, parallel versions on distributed memory computers using the parallel library ScaLAPACK are described.

Obtaining robust stability operationally

Robust stabilization is studied within the context of time-domain feedback control of single-input-single-output distributed parameter systems using approximate models. Using the operational methods of Mikusinski (1983), one can easily obtain strong versions of several famous theorems on robust stabilization. The operational methods of Mikusinski can play a powerful role in control theory. Using these methods, one is able to avoid the technical difficulties of convergence, existence, inversion, etc. associated with the Laplace transform. Unlike similar estimates obtained via the contraction mapping and small gain theorems, one is able to obtain robust stabilization free of restrictions on the size of certain error operators. >

A new method of solving the optimal control problem for a partially observable stochastic Volterra process

A new method is proposed for solving the linear-quadratic problem of optimal control for a partially observable stochastic Volterra process. The method relies on the representation and optimal estimation of optimal control in the form of integrals over the observable process. The integrands are non-stochastic and are defined by some system of integral equations, which may be solved numerically in advance. The optimal control is constructed directly from observations. An example demonstrating the implementation of the method by computer is given. Integral Volterra equations first arose in creep theory and they are the foundation of this theory /1, 2/. They include a fairly large class of equations with a memory /3–5/, which play a central role in control theory and in various applications. The theory of optimal control of Volterra equations is a natural outgrowth of the theory of controllable differentiable equations. Filtering and optimal control theory for stochastic integral equations is rapidly developing. The classical solution of the problem by the “separation principle” /6, 7/ reduces to solving the optimal filtering problem and the optimal control problem under complete information for some subsidiary controllable system. The optimal control of the original problem is obtained /8/ as a linear functional of the (mean-square) optimal estimate of the optimal trajectory of motion, which in its turn is the solution of a system of stochastic integral equations. Thus, in order to construct the optimal control at each instant of time t we need to solve a system of stochastic integral equations in the interval [0, t].

Computer control systems for automotive power trains

The problems of idle speed control and transmission control are reviewed to illustrate how control-theoretic techniques are used in the design of power-train control systems. The structure of current automobile control systems is examined. Possible future system structures and the role of control theory in the development of such systems are discussed. >