About one problem of optimal control synthesis

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.

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.

Considered the problem of flying over restricted areas by an unmanned aerial vehicle (UAV), which have various shapes and restrictions, set on the basis of the international airspace classification system for aviation in accordance with the Chicago Convention and the recommended principles for the formation of forbidden zones, rules for creating a flight route along forbidden zones and actions in case of border violations of restricted areas. The problem of analytical synthesis of the control acceleration of an unmanned aerial vehicle (UAV) is solved during its flight along a route passing along the boundaries of the forbidden zone of a given shape, along a given trajectory, which consists of subsequent segments located at the same height relative to the earth’s surface, in a given coordinate system. The optimal control synthesis problem is solved as an analytical definition of the optimal control of a linear non-stationary system based on the quadratic quality functional. A mathematical model of UAV motion in the horizontal plane is proposed, in the form of a system of ordinary differential equations in the Cauchy form. A law for measuring the control acceleration of the UAV’s center of mass is obtained on the basis of specifying the minimized quality functional and the corresponding constraints, which is a feature of the considered method of solving the problem. The proposed quality functional takes into account the parameters of coordinates and speed of the UAV, which correspond to the given points in the airspace, which characterize the necessary trajectory for flying around the restricted area. The derived mathematical dependences make it possible to implement them on board a UAV and minimize energy costs when guiding a UAV moving through specified points in space. Computer modeling of the derived analytical results, mathematical dependencies representing the optimal trajectory of the UAV flight along the boundaries of the forbidden zone, as well as the corresponding processes of changing the control acceleration and speed of the UAV movement was carried out, which made it possible to draw conclusions about the efficiency of the proposed method and the feasibility of its further use as a basis. for the initial stage of the synthesis of the UAV control system.

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.

Theoretical physics seems to be in a kind of schizophrenic state. Many phenomena in the observable macroscopic world obey nonlinear evolution equations, whereas the microscopic world is governed by quantum mechanics, a fundamental theory that is supposedly linear. In order to combine these two worlds in a common formalism, at least one of them must sacrifice one of its dogmas. I claim that linearity in quantum mechanics is not as essential as it apparently seems since quantum mechanics can be reformulated in terms of nonlinear Riccati equations. In a first step, it will be shown where complex Riccati equations appear in time-dependent quantum mechanics and how they can be treated and compared with similar space-dependent Riccati equations in supersymmetric quantum mechanics. Furthermore, the time-independent Schrödinger equation can also be rewritten as a complex Riccati equation. Finally, it will be shown that (real and complex) Riccati equations also appear in many other fields of physics, like statistical thermodynamics and cosmology.

In this paper we investigate the optimal controller synthesis problem, so that the system under the controller can reach a specified target set while satisfying given constraints. Existing model predictive control (MPC) methods learn from a set of discrete states visited by previous (sub-)optimized trajectories and thus result in computationally expensive mixed-integer nonlinear optimization. In this paper a novel MPC method is proposed based on reach-avoid analysis to solve the controller synthesis problem iteratively. The reach-avoid analysis is concerned with computing a reach-avoid set which is a set of initial states such that the system can reach the target set successfully. It not only provides terminal constraints, which ensure feasibility of MPC, but also expands discrete states in existing methods into a continuous set (i.e., reach-avoid sets) and thus leads to nonlinear optimization which is more computationally tractable online due to the absence of integer variables. Finally, we evaluate the proposed method and make comparisons with state-of-the-art ones based on several examples.

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.

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.

The problem of optimal controller synthesis for discrete-time stochastic nonlinear processes is examined. When the objective for design is specified as the expected value of a nonsymmetric, nonquadratic loss functional, it is necessary to relate the closed-loop process dynamics to the stationary probability density function (pdf). Because explicit relationships are available for only a few special cases, a general solution to the problem is not possible. In this work a technique for developing approximations of the optimal control law by application of a dynamics–pdf approximation is presented. Numerical simulations illustrate application of the proposed technique.

In this thesis, a number of search techniques are proposed as a solution for program and discrete controller synthesis (DCS). Classic synthesis techniques facilitate exhaus- tive search, while genetic programming has recently proven the potential of generic search techniques. But is genetic programming the right search technique for the synthesis prob- lem? In this thesis we challenge this belief and argue in favor of simulated annealing, a different class of general search techniques. We show that, in hindsight, the success of genetic programming has drawn from what is arguably a hybrid between simulated annealing and genetic programming, and compare the fitness of classic genetic program- ming, the hybrid form, and pure simulated annealing. Our experimental evaluation suggests that pure simulated annealing offers better results for automated programming than techniques based on genetic programming. Discrete Controller Synthesis (DCS) and Program Synthesis have similar goals: they are automated techniques to infer a control strategy and an implementation, respectively, that is correct by construction. We also investigate the application of the search tech- niques that we have been used for program synthesis for the computation of deterministic strategies solving symbolic Discrete Controller Synthesis (DCS) problems, where a model of the system under control is given along with desired objective behaviours. We experi- mentally confirm that relative performance results are similar to program synthesis, and give a complexity analysis of our simulated annealing algorithm for symbolic DCS. From the performance results we obtain, we draw the conclusion that simulated annealing, when combined with efficient model-checking techniques, is worth further investigating to solve symbolic DCS problems. A tool is designed to explore the parameter space of different synthesis techniques. Besides using it to synthesise a discrete control strategies for reactive systems (controller synthesis) and for protocol adapters for the coordination of different threads (software synthesis), we can also use it to study the influence of turning various screws in the syn- thesis process. For simulated annealing, PranCS allows the user to define the behaviour of the cooling schedule. For genetic programming, the user can select the population size.

An identification control is a control of an object which mathematical model is unknown. For such objects initially the problem of identification is solved, and then for the resulting object model the problem of control synthesis is solved. As a result of control synthesis we obtain a multi-dimensional function that describes the dependence of control on the object state. After the implementation of this function in the control unit of the real object, it achieves the goal of control with optimal value of the quality criteria. To solve the identification and synthesis problems we use the method of network operator, which with the help of evolutionary algorithms finds the structure and parameters of the functions describing the model and functional dependence of control from the object state. In this paper, an example of using a network operator to solve the problem of identification control synthesis for a multilink robot is given.

The object of this paper is to present new approximation schemes for the nonlinear matrix Riccati differential equation. They are obtained from any one-step or multistep method applied to the original linear quadratic control problem. They lead to the same type of schemes. However only one matrix inversion is required at each discretization node even if the Riccati equation is a nonlinear equation. This important computational advantage is obtained without altering the original nodal asymptotic convergence rate. It is proved that this rate is the same as the one of the initially chosen scheme.

The purpose of this work is to propose and demonstrate a way to explicitly transform polynomial ODE systems to linear ODE systems. With the help of an additional first integral, the one-dimensional Riccati equation is transformed to a linear system of three ODEs with variable coefficients. Solving the system, we can find a solution to the original Riccati equation in the general form or only to the Cauchy problem. The Riccati equation is one of the most interesting nonlinear first order differential equations. It is proved that there is no general solution of the Riccati equation in the form of quadratures; however, if at least one particular solution is known, then its general solution is also found. Thus, it is enough only to find a particular solution of the linear system of ODEs. The applied transformation method is a special case of the method described in our work [Zaytsev M. L., Akkerman V. B. (2020) On the identification of solutions to Riccati equation and the other polynomial systems of ODEs // preprint, Research Gate. DOI: 10.13140 / RG.2.2.26980.60807]. This method uses algebraic transformations and transition to new unknowns consisting of products of the original unknowns. The number of new unknowns becomes less than the number of equations. For the multidimensional Riccati equations, we do not present the corresponding linear system of ODEs because of the large number of linear equations obtained (more than 100). However, we present the first integral with which this can be done. In this paper, we also propose a method for finding the first integral, which can be used to reduce a search for the solution of any polynomial systems of ODEs to a search of solutions to linear systems of ODEs. In particular, if the coefficients in these equations are constant, then the solution is found explicitly.

The Riccati equation with variable coefficients expansion algorithm to find more exact solutions of nonlinear differential equations

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.