Controller Design for Discrete-Time Stochastic Processes With Nonquadratic Loss

This article is devoted to the problem of the optimal synthesis on a subspace for the 2 and 3-dimensional case for the linear control system $\dot{x_1} = u, \dot{x}_i=x_{i-1}, i=\overline{2,n}$ with $|u|\le1$. This problem is related to the problem of optimal synthesis into the point, which solution was presented by V. I. Korobov and G. M. Sklyar and is based on the moment min-problem, but its difference with respect to the original problem is that the number of unknown functions is greater than the number of variables, which requires using the methods for the parametric optimization. As in the problem of optimal synthesis into the point, we search for the optimal solution in the form of piecewise function with $u=\pm 1$ and $n-1$ switching points, which is optimal according to Pontryagyn's maximum principle and the theorem on $n$ intervals. In this article we consider approaches and problems related to the finding of the general solution of the optimization problem and solve the cases of $n=2$ and $n=3$. The interest of the obtain results is the fact that unlike the solution for the single endpoint $x_T$ the general solution for the subspace may have less than $n-1$ switching points, or not have switching points at all, depending on the parameters of the subspace. In our work, we give the solution for the problem of synthesis of the two-dimensional system onto a subspace $G: {x_2 = k x_1}$ for all values of $k$ using the moment min-problem and the optimization methods. For the three-dimensional system we consider the problem of synthesis on a plane $x_3 = k_1 x_1 + k_2 x_2$ and obtain the number of the switching points depending on the values of $k_1$ and $k_2$, construct trajectories and present the equations the for optimal time $\Theta$ for different cases.

The problem of optimal control synthesis of complex nonlinear and multidimensional systems oriented on the building of integrated control systems with the adaptation and self-organization properties is of much current interest. The development of optimization methods employed in control problems demonstrated their effectiveness and has led to the construction of algorithms of approximate solution of optimization problems that significantly extends the class of control problems that can be set. In this work we consider a synthesis of optimal control for affine nonlinear system, in which a special emphasis is placed on the degenerate problems of optimal control. Our approach is based on the extension of the method of minimization of generalized work function by properly transforming the initial nondegenerate problem of optimal control into degenerate problem of optimal control synthesis. The search for the solution is based on the suboptimal strategy that includes the construction of minimizing sequences that converge interval-wise to the optimal solution of the initial problem. We obtain some optimality conditions as well as the necessary and sufficient conditions of existence of optimal solution.

The problem of optimal synthesis of a nonlinear system’s control law parameters of satellite attitude control under constraints is considered. The maximum stability degree of a system is considered as an optimality criterion. The reaction wheels’ control moments and angular velocities achievable within the drives’ physical characteristics are considered as constraints. A linear model of the system converted from the original nonlinear model was used to solve the problem. The decomposition method, based on the transition from real time to relative time, is used to separate the tasks of optimality criterion maximization and constraints consideration. A method of synthesizing the optimal form of system transient process according to the maximum stability degree criterion is developed. An analytical method for determining the optimal values of control law parameters is proposed, based on the structural features of the linear differential equations system. A method that takes into account constraints on control moments and angular velocities of reaction wheels, based on transition scale from relative time to real time and its calculation algorithm, ensuring that conditions of all constraints are met, is developed. An example of solving the problem of optimal system synthesis is considered. The results demonstrate the effectiveness and simplicity of the proposed methods and algorithm for targeted selection of a system’s technical characteristics, taking into account constraints on reaction wheels’ maximum angular velocities and control moments values.

We consider production management using such elements of artificial intelligence as human-machine learning and decision-making procedures with fuzzy or qualitative guidelines. These mechanisms include heuristic elements, and use the knowledge of decision-makers and experts, their language, and the rules formulated in this language. The problem of the control mechanism optimal synthesis, with the guidelines of the expert, is set. This mechanism models the functions of people in conditions of uncertainty. The obtained optimal control mechanism includes learning and stimulation procedures. The proved theorem offers a solution to the problem of optimal synthesis of the learning mechanism using the dichotomous classification procedure, supported by an iterative pattern recognition procedure. This iterative procedure uses both a formal recursive algorithm and the results of recognition by an expert on images of emerging situations. As a result, adaptive norms are formed to evaluate the effectiveness and stimulate production. The use of the optimal control mechanism illustrates by the example of the wagon-repair production of a large-scale corporation Russian Railways.

This paper is concerned with the problem of validation in the context of numerical computations in control. We explore the possibility of using computer algebra tools and interval methods to compute solutions which have guarantees on accuracy, e.g. which are not subject to unknown errors due to rounding or approximation. We demonstrate that this is possible for two common norms of a linear system (L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> and L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> ) and H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> -optimal controller synthesis. We further discuss some of the issues involved in achieving a validation property for other problems of controller synthesis.

The aim of this study is to extend the scope and applicability of the level-crossing method to discrete-time stochastic processes and generalize it to enable us to study multiple discrete-time stochastic processes. In previous versions of the level-crossing method, problems with it correspond to the fact that this method had been developed for analyzing a continuous-time process or at most a multiple continuous-time process in an individual manner. However, since all empirical processes are discrete in time, the already-established level-crossing method may not prove adequate for studying empirical processes. Beyond this, due to the fact that most empirical processes are coupled; their individual study could lead to vague results. To achieve the objectives of this study, we first find an analytical expression for the average frequency of crossing a level in a discrete-time process, giving the measure of the time experienced for two consecutive crossings named as the "waiting time." We then introduce the generalized level-crossing method by which the consideration of coupling between the components of a multiple process becomes possible. Finally, we provide an analytic solution when the components of a multiple stochastic process are independent Gaussian white noises. The comparison of the results obtained for coupled and uncoupled processes measures the strength and efficiency of the coupling, justifying our model and analysis. The advantage of the proposed method is its sensitivity to the slightest coupling and shortest correlation length.

System level synthesis is a promising approach that formulates structured optimal controller synthesis problems as convex problems. This work solves the distributed linear-quadratic regulator problem under communication constraints directly in infinite-dimensional space, without the finite-impulse response relaxation common in related work. Our method can also be used to construct optimal distributed Kalman filters with limited information exchange. We combine the distributed Kalman filter with state-feedback control to perform localized LQG control with communication constraints. We provide agent-level implementation details for the resulting output-feedback state-space controller.

The recent spectacular progress in modern nanoelectronics created a big stimulus towards development of SoCs for embedded applications. Unfortunately, it also introduced unusual silicon and system complexity and heterogeneity, which result in many serious SoC development issues. Additional difficult to solve issues are due to very high throughput and low energy demands of many modern embedded applications. These issues cannot be resolved without new more adequate system architecture concepts, as well as, methods and EDA-tools for an adequate system-level design exploration and multi-objective optimal system architecture synthesis. This tutorial discusses the problems of multi-objective optimal architecture synthesis and trade-off exploitation for complex hard real-time embedded heterogeneous multi-processor SoCs, and model-based semi-automatic architecture synthesis methods that enable its effective and efficient solution. It thoroughly discusses the abstract models of the architecture design issue that involve the abstract system behavior models, system platform models and multi-objective decision models, as well as, the construction of the models and their usage for the actual system architecture exploration and multi-objective optimal architecture synthesis. In the role of examples, it uses the system architecture exploration and synthesis methods and the corresponding EDA-tools that we recently developed, and the SoC architectures synthesised with our tools for several real-world designs related to the newest highly demanding wireless communication and multimedia standards.

This paper considers the control synthesis problem and its solution using symbolic regression. Symbolic regression methods, which were previously called genetic programming methods, allow one to use a computer to find not only the parameters of a given regression function but also its structure. Unlike other works on solving the control synthesis problem using symbolic regression, the novelty of this paper is that for the first time this work employs a training dataset to address the problem of general control synthesis. Initially, the optimal control problem is solved from each point in a given set of initial states, resulting in a collection of control functions expressed as functions of time. A reference model is then integrated into the control object model, which generates optimal motion trajectories using the derived optimal control functions. The control synthesis problem is framed as an approximation task for all optimal trajectories, where the control function is sought as a function of the deviation of the object from the specified terminal state. The optimization criterion for solving the synthesis problem is the accuracy of the object’s movement along the optimal trajectory. The paper includes an example of solving the control synthesis problem for a mobile robot using a supervised machine learning method. A relatively new method of symbolic regression, the method of variational complete binary genetic programming, is studied and proposed for the solution of the control synthesis problem.

An algorithm for optimal antenna synthesis using the Method of Auxiliary Sources is proposed. The physical meaning of the analyticity of the wave fields, which underlies the algorithm, as well as the method itself, is discussed. The physical meaning of reactive fields and the role of the singularities of the analytic continuation of a given far field in solving the problem are discussed. The results of numerical experiments are presented, confirming the theoretical ideas.

A stochastic process or random process consists of chronologically ordered random variables {X t ; t ≥ 0}. For simplicity we assume that the process starts at time t = 0 in X 0 = 0. In this chapter, we consider exclusively processes in discrete time, i.e. processes which are observed at equally spaced points of time t = 0, 1, 2,... . Typical examples are daily, monthly or yearly observed economic data as stock prices, rates of unemployment or sales amount.

Two approaches to control system design for constrained systems are studied. The first involves theoretical investigations of constrained model predictive control algorithms. The second involves extensions of robust linear control theory to handle the nonlinear control schemes commonly used in practice for constrained systems. A novel model predictive control algorithm, with attractive functional and numerical characteristics is developed. This algorithm minimizes peak excursions in the controlled outputs and is particularly suited to regulatory control problems common in continuous process systems. Model predictive control concepts are extended to uncertain linear systems. An on-line optimizing control scheme (RMPC) is developed which has as its objective the minimization of worst-case tracking error for an entire family of linear plants. For model uncertainty descriptions which provide plant impulse response coefficients as affine functions of uncertain parameters, it is shown that the required minimax optimization problem can be recast as a single linear program. The discrete time optimal averaging level control problem is formulated and solved. A finite horizon approximation to the problem is introduced and analytical solutions are obtained in important special cases. A model predictive control formulation is introduced which provides optimal flow filtering and integral action. Analysis tools are provided to characterize the trade-off between flow filtering and rapid integral action. A complete theory is developed for the multivariable anti-windup, bumpless transfer (AWBT) problem. The theoretical framework allows the consideration of any linear time invariant (LTI) control system subject to plant input limitations and substitutions. A general AWBT compensation scheme, applicable to multivariable controllers of arbitrary structure and order, is developed. Conditions are derived under which this general AWBT method reduces to any one of several well-known heuristics for AWBT (e.g., PI anti-reset windup and IMC). The design issues which affect AWBT performance are identified and quantitative analysis methods are developed. Sufficient conditions for nonlinear stability of the AWBT compensated system are provided. These results are a generalization of, and are less conservative than, those available in the AWBT literature. The definition of AWBT performance objectives which are independent of controller structure allows the formulation of a general AWBT synthesis problem. This formal synthesiproblem addresses each of the identified performance objectives in a quantitative manner. The synthesis problem is shown to be a special case of a constrained structure controller synthesis (CSCS) problem. A solution method via reduction to static output feedback is presented and the engineering trade-offs available in the AWBT design are discussed.

We consider the problem of optimal reactive synthesis - compute a strategy that satisfies a mission specification in a dynamic environment, and optimizes a performance metric. We incorporate task-critical information, that is only available at runtime, into the strategy synthesis in order to improve performance. Existing approaches to utilising such time-varying information require online re-synthesis, which is not computationally feasible in real-time applications. In this paper, we pre-synthesize a set of strategies corresponding to candidate instantiations (pre-specified representative information scenarios). We then propose a novel switching mechanism to dynamically switch between the strategies at runtime while guaranteeing all safety and liveness goals are met. We also characterize bounds on the performance suboptimality. We demonstrate our approach on two examples - robotic motion planning where the likelihood of the position of the robot's goal is updated in real-time, and an air traffic management problem for urban air mobility.

This paper tackles the problem of characterizing the natural class, or Riccati rule space, of solutions to a specific equation. Despite the significant theoretical and practical implications, there is limited research exploring the application of spectral decomposition of non-self-adjoint differential operators to explicitly solve this nonlinear Riccati equation. Therefore, investigating operator Riccati equations holds potential to validate the dynamic programming method and address the synthesis problem.

Linear dynamic system consisted of interconnected subsystems is considered. The quality of behavior of each subsystem is characterized by quadratic integral criterion. The problem of multicriterial optimal control synthesis in such systems is formulated and solved in the paper. The solution is obtained on the base of principle of proportional damages, which ensures the following properties the control law is obtained in linear form as in monocriterlon optimization problem; the solution is invariant to the sort of metrics which characterized the distance between vector — criteria’s current value and it’s ideal one; optimal values of criteria of quality have relative losses to be inversely proportional to the coefficient of mutual importance comparative to their ideal values. The numerical example was considered: The keywords. Multicriterial control, dynamic system, optimality, Riccatl equation, the principle of proportional damages.