Dynamics-Based Piecewise Constant Control of Omni-Vehicle

In this paper, we consider the dynamics of a mobile vehicle moving under control on a perfectly rough horizontal plane. The vehicle consists of a horizontal platform and three omni-wheels that can rotate independently. An omni-wheel has freely rotating rollers on its rim [1]. We use its simplest model: an omni-wheel on a perfectly rough plane is modelled as a rigid disk with a constraint that its contact point velocity directed perpendicular to the disk's plane. The vehicle is controlled by three direct current motors in wheels' axes. Two terms model torques generated by motors: the rst one is proportional to the voltage, the second one is proportional to the value of the angular velocity of a wheel (counter-electromotive force). We study constant voltage dynamics and boundary-value problems for arbitrary initial and nal mass center coordinates, course angles and their derivatives using a piecewise constant control with one switching point. This problem is reduced to a system of algebraic equations for some specific (symmetric) vehicle model. We numerically model the system and analyze the possibility of optimization. For another vehicle configuration, we get the solution as numerical parametric continuation starting from the solution for the symmetric vehicle.

INVESTIGATION OF PARAMETERS OF THE PROBLEM OF SPLINE APPROXIMATION OF NOISY DATA BY NUMERICAL METHODS OF OPTIMAL CONTROL

Currently, the problems of distortion of measurement data by noise and the appearance of un-certainties in quality criteria have caused increased interest in research in the field of spline approx-imation. At the same time, existing methods of minimizing empirical risk, assuming that the noise is a uniform distribution with heavier tails than Gaussian, limit the scope of application of these studies. The problem of estimating noise-distorted data is usually based on solving an optimi-zation problem with a function containing uncertainty arising from the problem of finding optimal parameters. In this regard, the estimation of distorted noise cannot be solved by classical methods. Aim. This study is aimed at solving and analyzing the problem of spline approximation of data under uncertainty conditions based on the parametrization of control and the gradient projec-tion algorithm. Methods. The study of the problem of spline approximation of noisy data is carried out by the method of approximation of the piecewise constant control function. In this case, para-metrization of the control is possible only for a finite number of break points of the first kind. In the framework of the experimental study, the gradient projection algorithm is used for the numerical solution of the spline approximation problem. The proposed methods are used to study the parameters of the problem of spline approximation of data under conditions of uncertain-ty. Results. The numerical study of the control parametrization approach and the gradient projec-tion algorithm is based on the developed software and algorithmic tool for solving the problem of the spline approximation model under uncertainty. To evaluate the noise-distorted data, numerical experiments were conducted to study the model parameters and it was found that increasing the value of the parameter α leads to an increase in accuracy, but a loss of smoothness. In addition, the analysis showed that the considered distribution laws did not change the accuracy and convergence rate of the algorithm. Conclusion. The proposed approach for solving the problem of spline approx-imation under uncertainty conditions allows us to determine the problems of distortion of measure-ment data by noise and the appearance of uncertainties in the quality criteria. The study of the model parameters showed that the constructed system is stable to the error of the initial approxima-tion, and the distribution laws do not significantly affect the accuracy and convergence of the gra-dient projection method.

Stabilization of a multiconnected controlled continuous discrete system with non-overlapped decompositions with respect to part of variables

This paper considers a multi-connected controllable system with non-overlapping decompositions. Given that most of the control laws are implemented on digital controllers, the control of the system is implemented as a piecewise-constant function. Multiconnectivity of the system, in turn, makes it impossible to use centralized control. Every isolated subsystem must work stably, and intersystem connections can have a destabilizing effect. In this case, piecewise-constant control is constructed as two-level, i.e. in the form of a sum of local and global control. Local control stabilizes the equilibrium positions of individual linear subsystems. Global control acts on intersystem connections. Conditions are obtained under which local control stabilizes linear subsystems, and the equilibrium position of the original multi-connected system is asymptotically stable in part of variables.

A fixed point algorithm for improving fidelity of quantum gates

This work considers the problem of quantum gate generation for controllable quantum systems with drift. It is assumed that an approximate solution called seed is pre-computed by some known algorithm. This work presents a method, called Fixed-Point Algorithm (FPA) that is able to improve arbitrarily the fidelity of the given seed. When the infidelity of the seed is small enough and the approximate solution is attractive in the context of a tracking control problem (that is verified with probability one, in some sense), the Banach Fixed Point Theorem allows to prove the exponential convergence of the FPA. Even when the FPA does not converge, several iterated applications of the FPA may produce the desired fidelity. The FPA produces only small corrections in the control pulses and preserves the original bandwidth of the seed. The computational effort of each step of the FPA corresponds to the one of the numerical integration of a stabilized closed loop system. A piecewise-constant and a smooth numerical implementations are developed. Several numerical experiments with a N-qubit system illustrates the effectiveness of the method in several different applications including the conversion of piecewise-constant control pulses into smooth ones and the reduction of their bandwidth.

Optimal Stabilization Problem for the Quasilinear System with Controllable Parameters

The main concern of this paper is the problem of optimal stabilization of a quasilinear stochastic system with controllable parameters. Systems of this type are described by linear stochastic differential equations with multiplicative noises whose matrices, in general case, are nonlinear functions of control. The performance criterion is a modification of the classic quadratic performance cost. The goal is to minimize the criterion on the set of admissible control processes. This formulation of the problem is interesting because it allows us to study a wide range of optimization problems of linear systems with multiplicative perturbations, including: optimization of design parameters of the system, the problem of optimal stabilization under constraints on the gain matrix of the linear regulator in the form of inequalities, the problem of optimal stabilization of linear stochastic systems under information constraints. The main result of this paper is the necessary conditions for the optimal vector in the problem of stabilization of a quasilinear stochastic system with controllable parameters.The numerical gradient-type procedure for synthesis of the optimal stabilizing vector is also proposed. In addition, using obtained results we construct the algorithm for synthesis of a suboptimal time-dependent control. The result of the proposed algorithm is piecewise constant control, which gives the value of the criterion is guaranteed not worse than for the optimal stabilizing vector. This algorithm is relatively simple and one may use it for calculations in real time. The obtained results are applied to the problem of optimal stabilization under information constraints, in which the necessary optimality conditions are also obtained and the gradient-type procedure for the synthesis of the optimal control is proposed. The use of the obtained results is demonstrated by a model example.

Economic Model Predictive Control of Sampled-Data Linear Systems With Piecewise Constant Control

The paper studies an economic model predictive control (EMPC) problem for sampled-data linear systems with system constraints. The cost function consists of an economic part and a regulatory part, and a new EMPC algorithm with piecewise constant control is designed. Iterative feasibility of the designed optimization problem and input-to-state stability (ISS) of the closed-loop system are proved. In particular, we show that the closed-loop system is input-to-state stable with respect to the supremum norm of the economic cost, and the system state is ultimately bounded within a bound determined by the economic cost. Through thorough simulations, the effectiveness of the designed algorithm is verified and the tradeoff between control and economic performance is demonstrated.

Reachability in Infinite-Dimensional Unital Open Quantum Systems with Switchable GKS–Lindblad Generators

In quantum systems theory one of the fundamental problems boils down to: given an initial state, which final states can be reached by the dynamic system in question. Here we consider infinite-dimensional open quantum dynamical systems following a unital Kossakowski–Lindblad master equation extended by controls. More precisely, their time evolution shall be governed by an inevitable potentially unbounded Hamiltonian drift term H0, finitely many bounded control Hamiltonians Hj allowing for (at least) piecewise constant control amplitudes [Formula: see text] plus a bang-bang (i.e., on-off) switchable noise term ГV in Kossakowski–Lindblad form. Generalizing standard majorization results from finite to infinite dimensions, we show that such bilinear quantum control systems allow to approximately reach any target state majorized by the initial one as up to now it only has been known in finite dimensional analogues. The proof of the result is currently limited to the bounded control Hamiltonians Hj and for noise terms ГV with compact normal V.