Problems In Control Theory Research Articles

Multi agent through serret-frenet system

Multi agent system present in our daily live. The Multi agent system, starting from a few agents to tens and even millions of agents. Control of multi-agent is one of the most important problem in control theory. Many methods are used to control of multi agent, one of the method is optimal control. Approach with optimum control too many kinds. In this paper, the authors use the Serret-Frenet system. In mathematical models of multi-agent systems written, multi-Serret-Frenet dynamics systems are used. Some agents move with the initial formation from the starting position, towards the final position that is popularized. The movement of maintaining the formation and minimizing the cost of control is translated into the functional cost model. From the dynamic system model and functional model of cost is designed multi-agent control system with the optimum control approach.

ESTIMATION OF MEASUREMENT ERRORS IN A SINGLE-CHANNEL PLATFORM ANGULAR POSITIONING CONTROL SYSTEM BASED ON AN ANGULAR VELOCITY SENSOR

This work is devoted to the important problem of the contemporary automatic control theory – creation of integrated technologies of design of ACS (Automatic Control Systems). Such technologies must provide complex solution of the controlled object’s state evaluation and synthesis of the control law. The main practical result would be ACS accuracy increase. The new concept of ACS design for realization of integrated technologies conditions the urgency to confirm its efficacy’s by applied results in creation of concrete working systems. Research in this direction of automatic control theory is at its initial stage, at which it is important to evaluate possibilities of new technologies on simple ACS’ functioning models. The author presents a model of a one-channel attitude control system for a platform based on an angular velocity sensor providing instantaneous posterior estimation of platform’s rotation angle error. Values of the disturbance torque, acting on the platform, and angular velocity measurement errors on the considered observation interval are set as random numbers that stay constant on this interval. The author presents the analysis of control and state evaluation for the object of control, implemented by the synthesized system for two versions of the object: in the absence and in the presence of its own damping torque. The author demonstrates a modification of adjuster’s structure to achieve accuracy increase for the platform’s parameters evaluation.

Formulation and Solution of a Generalized Chebyshev Problem: First Part

This study is a continuation of the article “Formulation and solution of a generalized Chebyshev problem: First Part,” in which a generalized Chebyshev problem was formulated and two theories of motion for nonholonomic systems with higher order constraints were presented for its solution. These theories were used to study the motion of the Earth’s satellite when fixing the magnitude of its acceleration (this was equivalent to imposing a third-order linear nonholonomic constraint). In this article, the second theory, based on the application of the generalized Gauss principle, is used to solve one of the most important problems of control theory: finding the optimal control force that translates a mechanical system with a finite number of degrees of freedom from one phase state to another in the specified time period. The application of the theory is demonstrated by solving the model problem of controlling the horizontal motion of a cart bearing the axes of s mathematical pendulums. Initially, the problem is solved by applying the Pontryagin maximum principle, which minimizes the functional of the square of the desired horizontal control force, which transfers the mechanical system from a motionless state to a new motionless state in the specified time period with the cart’s horizontal displacement S (the vibration damping problem is considered). This approach is called the first method for solving the control problem. It is shown that a (2s + 4)-order linear nonholonomic constraint is continuously fulfilled. This suggests applying the second theory of motion for nonholonomic systems with higher order constraints to the same problem (see the previous article), the theory developed at the Department of Theoretical and Applied Mechanics of the Faculty of Mathematics and Mechanics of St. Petersburg State University. This approach is called the second method for solving the problem. Calculations are performed for the case s = 2 and show that the results of both methods are practically identical for a short motion of the system but they differ substantially for a long motion. This is due to fact that the control determined using the first method contains harmonics with the system’s natural frequencies, which tends to bring the system into resonance. For a short motion this is not noticeable, and for a long motion there are large fluctuations in the system. In contrast, when the second method is used, the control is polynomial in time, which provides a relatively smooth motion of the system. In addition, in order to eliminate the jumps in the control force at the beginning and end of the motion, it is proposed to solve the generalized boundary value problem. Some special cases occurring when using the second method for solving the boundary value problem are discussed.

Optimal control on SU(2) Lie group with stability analysis

Many configuration spaces of mechanical and non-mechanical problems in physical world involve matrix Lie groups. These Lie groups provide a mathematically rich formulation for studying a variety of control theory problems. While control systems have been specialized to different matrix Lie groups, study of optimal control of these systems by minimizing the input cost function has been a tempting research area. Here we take a left invariant driftless control system on the Lie group SU(2) and analyse controllability and optimal control of the system. Stability of the resulting dynamics from optimal control analysis is elaborated. Then numerical integration and some related properties are discussed via two unconventional integrators.

Qualitative Results in the Bombieri Problem for Conformal Mappings

Bombieri’s numbers σmn characterize a behavior of the coefficient body for the class S of all holomorphic and univalent functions f in the unit disk normalized by f(z) = z + a2z2 + …. The number σmn is the limit of ratio for Re(n − an) and Re (m − am) as f tends to the Koebe function K(z) = z(1 − z)−2. It is showed in the paper that Bombieri’s conjecture about explicit values of σmn implies a sliding regime in an associated control theory problem generated by the Loewner differential equation. We develop also an asymptotical approach in verification of necessary criteria for Bombieri’s conjecture.

An active learning framework for set inversion

Set inversion is a classical problem in control theory that has many important applications in various fields of science and engineering. The state-of-the-art method for solving this problem, Set Inverter Via Interval Analysis (SIVIA), usually does not work well in high dimensions and often fails to recover sets with complicated structures. In this work, we propose a new approach to the problem of set inversion, which employs techniques from machine learning to resolve these issues. Our algorithm can handle problems in high dimensions and achieve the same level of accuracy with fewer data points compared to SIVIA. We illustrate the performance of our method in various simulation studies and apply it to investigate the dynamics of the 17th-century plague in Eyam village, England.

Block Decoupling of Linear Systems by Static-State Feedback

A solution is presented to a long-standing open problem of linear control theory, the block decoupling by static-state feedback. The problem was introduced in 1970 and all past solutions were obtained under restrictive assumptions on system, block structure, and decoupling feedback. The present formulation avoids any restrictive hypotheses. The key notion is that of the block interactor. The existence of decoupling feedback is conditioned by system invariants with respect to the permissible transformations. A block decoupling algorithm is presented, which permits to determine the sizes of the smallest diagonal blocks attainable.

A holistic optimization approach for inverted cart-pendulum control tuning

The inverted cart-pendulum (ICP) is a nonlinear underactuated system, which dynamics are representative of many applications. Therefore, the development of ICP control laws is important since these laws are suitable to other systems. Indeed, many nonlinear control strategies have emerged from the control of the ICP. For these reasons, the ICP remains a canonical and fundamental benchmark problem in control theory and robotics that is of interest to the scientific community. Till now, the trial-and-error method is still widely applied for ICP controller tuning as well as the sequential tuning referring to tune the swing-up controller and thereafter, the stabilization controller. Therefore, the aim of this paper is to automate and facilitate the ICP control in one step. Thus, this paper proposes to holistically optimize ICP controllers. The holistic optimization is performed by a simplified Ant Colony Optimization method with a constrained Nelder–Mead algorithm (ACO-NM). Holistic optimization refers to a simultaneous tuning of the swing-up, stabilization and switching mode parameters. A new cost function is designed to minimize swing-up time, achieve high stabilization performance and consider system constraints. The holistic approach optimizes four controller structures, which include controllers that have never been tuned by a specific method besides by the trial-and-error method. Simulation results on a ICP nonlinear model show that ACO-NM in the holistic approach is effective compared to other algorithms. In addition, contrary to the majority of work on the subject, all the optimized controllers are validated experimentally. The simulation and experimental results obtained confirm that the holistic approach is an efficient optimization tool and specifically responds to the need of optimization technique for the potential-well controller structure and for the Q [diagonal of the matrix and the full matrix] in the linear–quadratic regulator (LQR) technique. Moreover, ICP experimental response analysis demonstrates that using the full Q provides greater experimental stabilization performance than using its diagonal terms in the LQR technique.

Data-Driven Minimum-Energy Controls for Linear Systems

In this letter, we study the problem of computing minimum-energy controls for linear systems from experimental data. The design of open-loop minimum-energy control inputs to steer a linear system between two different states in finite time is a classic problem in control theory, whose solution can be computed in closed form using the system matrices and its controllability Gramian. Yet, the computation of these inputs is known to be ill-conditioned, especially when the system is large, the control horizon long, and the system model uncertain. Due to these limitations, open-loop minimum-energy controls and the associated state trajectories have remained primarily of theoretical value. Surprisingly, in this letter, we show that open-loop minimum-energy controls can be learned exactly from experimental data, with a finite number of control experiments over the same time horizon, without knowledge or estimation of the system model, and with an algorithm that is significantly more reliable than the direct model-based computation. These findings promote a new philosophy of controlling large, uncertain, linear systems where data is abundantly available.

An Estimate for the Hausdorff Distance between a Set and Its Convex Hull in Euclidean Spaces of Small Dimension

We derive estimates for the Hausdorff distance between sets and their convex hulls in finite-dimensional Euclidean spaces with the standard inner product and the corresponding norm. In the first part of the paper, we consider estimates for α-sets. By an α-set we mean an arbitrary compact set for which the parameter characterizing the degree of nonconvexity and computed in a certain way equals α. In most cases, the parameter α is the maximum possible angle under which the projections to this set of points not belonging to the set are visible from these points. Note that α-sets were introduced by Ushakov for the classification of nonconvex sets according to the degree of their nonconvexity; α-sets are used for the description of wavefronts and for the solution of other problems in control theory. We consider α-sets only in a two-dimensional space. It is proved that, if α is small, then the corresponding α-sets are close to convex sets in the Hausdorff metric. This allows us to neglect their nonconvexity and consider such sets convex if it is known that the parameter α is small. The Shapley-Folkman theorem is often applied in the same way. In the second part of the paper, we present an improvement of the estimate from the Shapley-Folkman theorem. The original Shapley-Folkman theorem states that the Minkowski sum of a large number of sets is close in the Hausdorff metric to the convex hull of this sum with respect to the value of the Chebyshev radius of the sum. We consider a particular case when the sum consists of identical terms; i.e., we add some set M to itself. For this case, we derive an improved estimate, which is essential for sets in spaces of small dimension. In addition, as in Starr’s known corollary, the new estimate admits the following improvement: the Chebyshev radius R(M) on the right-hand side can be replaced by the inner radius r(M) of the set M. However, as the dimension of the space grows, the new estimate tends asymptotically to the estimate following immediately from the Shapley-Folkman theorem.

On the linear static output feedback problem: The annihilating polynomial approach

One of the fundamental open problems in control theory is that of the stabilization of a linear time invariant dynamical system through static output feedback. We are given a linear dynamical system defined throughw.=Aw+Buy=Cw. The problem is to find, if it exists, a feedback u=Ky such that the matrix A+BKC has all its eigenvalues in the complex left half plane and, if such a feedback does not exist, to prove that it does not. Substantial progress has not been made on the computational aspect of the solution to this problem.In this paper we consider instead ‘which annihilating polynomials can a matrix of the form A+BKC possess?’.We give a simple solution to this problem when the system has either a single input or a single output. For the multi-input - multi-output case, we use these ideas to characterize the annihilating polynomials when K has rank one, and suggest possible computational solutions for general K. We also present some numerical evidence for the plausibility of this approach for the general case as well as for the problem of shifting the eigenvalues of the system.

Optimal control of Sturm-Liouville type evolution differential inclusions with endpoint constraints

The present paper studies a new class of problems of optimal control theory with linear second order self-adjoint Sturm-Liouville type differential operators and with functional and non-functional endpoint constraints. Sufficient conditions of optimality, containing both the second order Euler-Lagrange and Hamiltonian type inclusions are derived. The presence of functional constraints generates a special second order transversality inclusions and complementary slackness conditions peculiar to inequality constraints; this approach and results make a bridge between optimal control problem with Sturm-Liouville type differential differential inclusions and constrained mathematical programming problems in finite-dimensional spaces.The idea for obtaining optimality conditions is based on applying locally-adjoint mappings to Sturm-Liouville type set-valued mappings. The result generalizes to the problem with a second order non-self-adjoint differential operator. Furthermore, practical applications of these results are demonstrated by optimization of some semilinear optimal control problems for which the Pontryagin maximum condition is obtained. A numerical example is given to illustrate the feasibility and effectiveness of the theoretic results obtained.

Optimal control of a linear system subject to partially specified input noise

SummaryOne of the most basic problems in control theory is that of controlling a discrete‐time linear system subject to uncertain noise with the objective of minimizing the expectation of a quadratic cost. If one assumes the noise to be white, then solving this problem is relatively straightforward. However, white noise is arguably unrealistic: noise is not necessarily independent, and one does not always precisely know its expectation. We first recall the optimal control policy without assuming independence and show that, in this case, computing the optimal control inputs becomes infeasible. In the next step, we assume only the knowledge of lower and upper bounds on the conditional expectation of the noise and prove that this approach leads to tight lower and upper bounds on the optimal control inputs. The analytical expressions that determine these bounds are strikingly similar to the usual expressions for the case of white noise.

Pontryagin maximum principle and Stokes theorem

We present a new geometric unfolding of a prototype problem of optimal control theory, the Mayer problem. This approach is crucially based on the Stokes Theorem and yields to a necessary and sufficient condition that characterizes the optimal solutions, from which the classical Pontryagin Maximum Principle is derived in a new insightful way. It also suggests generalizations in diverse directions of such famous principle.

A Monte Carlo method for backward stochastic differential equations with Hermite martingales

Abstract Backward stochastic differential equations (BSDEs) appear in many problems in stochastic optimal control theory, mathematical finance, insurance and economics. This work deals with the numerical approximation of the class of Markovian BSDEs where the terminal condition is a functional of a Brownian motion. Using Hermite martingales, we show that the problem of solving a BSDE is identical to solving a countable infinite-dimensional system of ordinary differential equations (ODEs). The family of ODEs belongs to the class of stiff ODEs, where the associated functional is one-sided Lipschitz. On this basis, we derive a numerical scheme and provide numerical applications.

Modeling of inverted pendulum system with gravitational search algorithm optimized controller

Abstract The inverted pendulum (IP) is a common classical control theory problem. Many controllers are used to keep the pendulum rod upraise via the control the position of the cart. In this paper, Modelling of IP system is carried out and then verifies the model through comparison with other models in literature. The viscous damping effect on the IP system has been also considered in this work. The PID controller is used to control the IP and is tuned by using both Gravitational Search Algorithm (GSA) and Genetic Algorism (GA). Examination of GSA efficiency has been demonstrated, and then comparison with Genetic Algorism (GA) in the tuning of PID controller is presented. The results proved the superiority of the controller which is tuned by using GSA (PID-GSA). Finally, the optimized controller (PID-GSA) performance is checked through carrying out the robustness test.

The Solution of an Optimal Control Problem for a Third-Order Equation in the Presence of Distributed and Starting Controls

There are a large number of scientific articles and monographs devoted to various problems of control theory in systems described by partial differential equations, which are quite well stated in monographs [1-6] and in many scientific articles. However, in all these works, problems are considered that are related mainly to equations of classical types studied in courses of mathematical physics. Recently, papers that solve various problems obtained from the practice and associated with fourth-order partial differential equations have appeared. Along with it, there are numerous applied problems described by third-order equations or the so called equations of variable type. Despite the fact that the initial-boundary value problems for such equations are studied well, there are only a few works [12-18] in which control problems in such systems are investigated. In this paper, the problem of the quadratic functional minimum for the initial-boundary value problem for a third-order linear equation is studied. First, using the Fourier method, a representation of the solution of the initial-boundary value problem is obtained, and then the optimal control problem is reduced to a conditional extremum problem and its solution is obtained in the form of a series.

Optimal control of evolution differential inclusions with polynomial linear differential operators

In this paper we have introduced a new class of problems of optimal control theory with differential inclusions described by polynomial linear differential operators. Consequently, there arises a rather complicated problem with simultaneous determination of the polynomial linear differential operators with variable coefficients and a Mayer functional depending on high order derivatives. The sufficient conditions, containing both the Euler-Lagrange and Hamiltonian type inclusions and transversality conditions are derived. Formulation of the transversality conditions at the endpoints of the considered time interval plays a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions. The main idea of the proof of optimality conditions of Mayer problem for differential inclusions with polynomial linear differential operators is the use of locally-adjoint mappings. The method is demonstrated in detail as an example for the semilinear optimal control problem and the Weierstrass-Pontryagin maximum principle is obtained. Then the optimality conditions are derived for second order convex differential inclusions with convex endpoint constraints.

New Versions of the Hermite Bieler Theorem in Stability Contexts

The Hermite-Bieler theorem played key roles in several control theory problems including the proof of Kharitonov’s theorem and derivations of elementary proofs of the Routh’s algorithm for determining the Hurwitz stability of a real polynomial. In the present work, we explore the stability of complex continuous-time systems of differential equations. Using the theory of positive paraodd functions, we obtain Hermite-Bieler like conditions for the Routh-Hurwitz stability of such systems. We also look at the problem of stability of discrete-time systems of difference equations. By using suitable conformal mappings, we also establish Hermite-Bieler like conditions for the Schur-Cohn stability of these systems. In both cases, the conditions are necessary as well as sufficient.

Estimation of proximity of controls synthesized on basis of maximum principle and ADAR method

Introduction. A special case of synthesizing the same electromechanical control system by the Pontryagin maximum principle and by the synergetic synthesis method is considered. The task was to solve the synthesis problem of the time optimal electromechanical position control system; herewith the travel resistance modulus linearly depended on the output coordinate of the system. This approach to the selection of the synthesis problem was because the synthesis of time optimal systems is one of the most widespread problems, and it is solved by increasing the efficiency of the existing control systems.Materials and Methods. Synthesis of the time optimal linear control system based on the maximum principle is a widely accepted problem in the modern control theory. However, the procedure of synergistic synthesis does not have such formalization. This being the case, the paper suggests an approach that brings together these two methods, which, in our opinion, will increase the efficiency of the synergistic synthesis method through adding some features of the synthesis methodology for optimal systems.Research Results. The paper formulates two key concepts. The first one is as follows: the application of the maximum principle for an object of the DC motor class when synthesizing the positioning algorithm under the conditions of linear loading functionally dependent on the engine rotation angle allows the time optimal system to be optimized. The second concept states that synthesis of a control system based on the synergistic approach enables to obtain a system close to optimal (quasioptimal), but after modifying the synergetic synthesis method itself. A hypothesis is formulated on the possible connection between the introduced (when implementing the procedure of state space extension in the synergetic synthesis method) time constants with the optimal switching time of control defined in the maximum method.Discussion and Conclusions. The synthesis through the maximum control technique and the ADAR method is performed. In virtue of the comparison of efficiency of these methods, a hypothesis is put forward on the possible compatibility of the studied methods.