A problem of pursuit with a capture of a maneuvering evader is considered. The notions of a pursuer’s robust capture control and its capture zone are introduced. Based on solutions of auxiliary zero-sum differential games, two versions of the pursuer’s robust capture control are designed and the corresponding capture zones are constructed. An illustrative example of a pursuit-evasion engagement between two flying vehicles is presented.

We consider the general type zero-sum finite horizon linear-quadratic differential game. First, the set of terminal-value problems for three differential equations, associated with the considered game by the solvability conditions, is analyzed. These differential equations are: the game-theoretic Riccati matrix equation, the linear vector equation and the scalar trivial equation. The terminal-value problem for the game-theoretic Riccati matrix differential equation may not have the solution in the entire time-interval of the game’s duration. Based on the artificial parameter method, a sufficient condition (in the terms of the game’s data) for the existence of the solution to this terminal-value problem in the entire time-interval of the game’s duration is presented. Approximate solutions of the terminal-value problems, associated with the considered game, are derived. Using these approximate solutions, an approximation of the value of this game is obtained. Suboptimal state-feedback controls of the players and an approximate-saddle point in this game are derived. The theoretical results are illustrated by the approximate solution of the problem of a pursuit-evasion engagement between two flying vehicles.

The problem of a variable-speed unicycle guidance to the stationary target is considered. The vehicle should be guided to the origin while minimizing the energy loss due to the induced drag. The problem is formulated as a nonlinear optimal control problem with a known velocity profile and, consequently, a known drag coefficient profile. Since no analytical solution is available, a numerical parameter continuation procedure is employed. The parameter continuation algorithm comprises the parameterization of both the system dynamics and the cost by a single parameter. The system dynamics equation is parameterized in such a way that if the parameter is equal to zero, the dynamics equation is that of a quadratic-kinematic approximation of the original system, whereas if the parameter is equal to one, it is the original nonlinear one. In the cost parametrization, the values of zero and one correspond to the cost with constant and variable speed, respectively. A numerical algorithm based on Davidenko's equation is derived. By an extensive simulation, it is shown that the proposed method converges from a wider set of initial and terminal conditions than the classical parameter continuation method and a state-of-the-art alternative solver. This improvement is due to both the dynamics/cost parameterization and choosing an initial guess from the quadratic-kinematic approximation.

An infinite-horizon H infinity linear-quadratic control problem is considered. This problem has the following features: (i) the control cost in the cost functional has a positive small coefficient (small parameter), meaning that the control cost is much smaller than the state cost; (ii) the current cost of the fast state variable in the cost functional is a non-zero positive semi-definite quadratic form. These features require developing a significantly novel approach to asymptotic analysis of the matrix Riccati algebraic equation appearing in the solvability conditions of the considered H infinity problem. Using this solution, an asymptotic analysis of the H infinity problem is carried out. This analysis yields parameter-free solvability conditions for this problem and a simplified controller solving this problem. An example illustrating the theoretical results is presented.

A bi-objective control problem for a linear system in the presence of a disturbance is considered. The first cost functional is a generalized tracking error defined as a Lebesgue-Stieltjes discrepancy integral comprising both continuous and discrete discrepancies. The second cost is the control effort. The relaxed Pareto control, which guarantees a balance between two costs, is defined for an unknown disturbance and controls from L_2 -bounded sets. The solution is constructed based on the auxiliary generalized linear-quadratic differential game formulated in open-loop controls. It is shown that by a proper choice of the cost penalty coefficients, the game-optimal control strategy solves the bi-objective control problem. As a by-product, a novel solvability condition for the game in open-loop controls is derived. Illustrative examples are presented.

A finite-horizon zero-sum linear-quadratic differential game is considered. Its features are: (i) the control cost of the minimizing player in the game's cost functional is much smaller than the control cost of the maximizing player and the state cost; (ii) the cost of the fast state variable in the integrand of the cost functional is a positive semi-definite (but non-zero) quadratic form. These features require developing a significantly novel approach to asymptotic analysis of the matrix Riccati differential equation associated with the considered game. Using this analysis, an asymptotic solution of the game is derived. An illustrative example is presented.

A two-player finite horizon linear-quadratic Stackelberg differential game is considered. The feature of this game is that the control cost of a follower in the cost functionals of both players is small, which means that the game under consideration is a cheap control game. The open-loop solution of this game is studied. Using the game's solvability conditions, obtaining such a game's solution is reduced to the solution of a proper boundary-value problem. Due to the smallness of the follower's control cost, this boundary-value problem is singularly perturbed. The asymptotic behaviour of the solution to this problem is analysed. Based on this analysis, the asymptotic behaviour of the open-loop optimal players' controls and the optimal values of the cost functionals is studied. Using these results, asymptotically suboptimal players' controls are designed. An illustrative example of a supply chain problem with a small control cost of a retailer is presented.

A finite-horizon zero-sum linear-quadratic differential game with non-homogeneous dynamics is considered. The key feature of this game is as follows. The cost of the control of the minimizing player (the minimizer) in the game’s cost functional is much smaller than the cost of the control of the maximizing player (the maximizer) and the cost of the state variable. This smallness is due to a positive small multiplier (a small parameter) for the quadratic form of the minimizer’s control in the integrand of the cost functional. Two cases of the game’s cost functional are studied: (i) the current state cost in the integrand of the cost functional is a positive definite quadratic form; (ii) the current state cost in the integrand of the cost functional is a positive semi-definite (but non-zero) quadratic form. The latter case has not yet been considered in the literature devoted to the analysis of cheap control differential games. For each of the aforementioned cases, an asymptotic approximation (by the small parameter) of the solution to the considered game is derived. It is established that the property of the aforementioned state cost (positive definiteness/positive semi-definiteness) has an essential effect on the asymptotic analysis and solution of the differential equations (Riccati-type, linear, and trivial), appearing in the solvability conditions of the considered game. The cases (i) and (ii) require considerably different approaches to the derivation of the asymptotic solutions to these differential equations. Moreover, the case (ii) requires developing a significantly novel approach. The asymptotic solutions of the aforementioned differential equations considerably differ from each other in cases (i) and (ii). This difference yields essentially different asymptotic solutions (saddle point and value) of the considered game in these cases, meaning it is of crucial importance to distinguish cases (i) and (ii) in the study of various theoretical and real-life cheap control zero-sum linear-quadratic differential games. The asymptotic solutions of the considered game in cases (i) and (ii) are compared with each other. An academic illustrative example is presented.

The dynamical parameter continuation method is considered. It is implemented for the planar non-linear guidance problem with the stationary target. In this approach, the system dynamics is parameterized in such a way that if the parameter is equal to zero, the dynamics is the quadratic-kinematic approximation of the original system, whereas if the parameter is equal to one, it is the original one. We propose an improved formalization of the method that allows convenient handling of the boundary conditions, and derive a numerical procedure. It is shown that the proposed method converges from the wider set of initial and terminal conditions than the classical parameter continuation method.

A zero-sum finite horizon linear-quadratic differential game is considered. First, the terminal-value problem for the game-theoretic Riccati matrix differential equation, associated with the considered game by the solvability conditions, is analysed. This problem may not have the solution in the entire time-interval of the game's duration. Using the method of auxiliary parameter, a sufficient condition (in the terms of the game's data) for the existence of the solution to the aforementioned terminal-value problem in the entire time-interval of the game's duration is derived. An approximate analytical solution to this problem also is obtained as a partial sum of some infinite series of functions arising in the method of auxiliary parameter. Based on this approximate solution, suboptimal state-feedback controls of the players are formally designed and justified. It is shown that the pair of these controls constitutes an approximate-saddle point in the considered game. The theoretical results of the article are illustrated by their application to approximate solution of the problem of pursuit-evasion engagement between two flying vehicles.

A finite-horizon optimal control problem for a nonlinear unicycle with constant linear velocity is considered. The cost functional consists of the squared norm of a final position and the integral penalty term for the control effort, i.e., both the miss distance and the control are soft-constrained. A finite horizon formulation arises, for instance, in coordinated guidance attack against a stationary target, in which all interceptors have to arrive at the target at the same time. The soft constraint on terminal position allows for tradeoff between the miss distance and control effort. Semi-analytical solution is derived by representing the squared norm as a maximum of a quadratic form and by changing the order of maximization and minimization. The inner minimization problem becomes a problem of calculus of variations, which Euler-Lagrange equation writes as a nonlinear pendulum equation. Based on the solution of this equation, a numerical scheme for constructing the suboptimal control is developed. As a by-product of the approach, the posterior control bounds are obtained.

We consider a two-player finite horizon linear–quadratic Stackelberg differential game. For this game, we study the case where the control cost of a leader in the cost functionals of both players is small, which means that the game under consideration is a cheap control game. We look for open-loop optimal players’ controls of this game. Using the game’s solvability conditions, the obtaining such controls is reduced to the solution to a proper boundary-value problem. Due to the smallness of the leader’s control cost, this boundary-value problem is singularly perturbed. Asymptotic behavior of the solution to this problem is analyzed. Based on this analysis, the asymptotic behavior of the open-loop optimal players’ controls and the optimal values of the cost functionals is studied. Using these results, asymptotically suboptimal players’ controls are designed. An illustrative example is presented.

In this article, the problem of precisely tracking a nominal trajectory by an under-actuated multirotor platform is formulated as a game against bounded external disturbances, measurement noise, and initial condition uncertainty. The nominal trajectory is composed of desired position and yaw angle, which is known to be a differentially flat output for the multirotor state dynamics. Using this property, a linearized error dynamics is obtained in the vicinity of the nominal trajectory. Using the linearized error dynamics, a linear-quadratic differential game approach is used to propose an integrated estimation and control method for trajectory tracking. A salient feature of this approach is that the drag coefficients are estimated online from the information of the tracking error. Moreover, there are only six parameters to be tuned for both estimation and control loops. The intuitive rationale behind the tuning of these parameters is also discussed. Simulations as well as experimental validations are presented to demonstrate the performance and applicability of the controllers.

In this paper, the problem of robust trajectory tracking by a multicopter platform is formulated as a linear-quadratic differential game against unknown external disturbances (nature). The desired trajectory is composed of position and yaw constraint, which is known to be a differentially flat output for multicopter state dynamics. Using this property, a linearized error dynamics is obtained in the vicinity of the nominal states and the nominal control inputs. To avoid singularities, the linearized error dynamics is derived using quaternion representation. Based on it, an optimal saddle point strategy for trajectory tracking is presented, where controller gains are calculated \textit{apriori} by numerically solving a differential Ricatti equation. A salient feature of this control design is that both the position and the attitude control loops are integrated, and there are only two tuning parameters — one corresponding to the tradeoff between control effort and tracking accuracy and the other corresponding to robustness. Simulations as well as experimental validations are presented to demonstrate the performance and applicability of the controller.

Polytomous Rasch model (PRM) is a general probabilistic measurement model widely used in psychometrics, social science and educational measurement. It describes the probability of certain ordinal response of an object under test as a function of its ability, given, so called thresholds, characterizing the specific test item. The model was also adapted to business and industry applications. In contrast to the behavior of the median PRM outcome value, monotonically increasing as the ability increases, the ordinal variation behavior, as shown in the article, can be very diverse and it is rather determined by the mutual position of the threshold values of the model. The article studies ordinal variation of the response vs. ability for different thresholds locations arrangements and different amounts of ordered response categories. It is shown under what circumstances this function becomes multimodal. If several objects are involved in the test, attention is paid to the possibility of the total variation decomposition into intra and inter components. Considering the intra object variation helps to avoid overestimation of the real variation between the tested objects as it is demonstrated by illustrative example.

We consider the problem of time-sampling optimization for a Sta-tistical Process Control (SPC). The aim of this optimization is to minimize the expected loss, caused by a delay in the detection of an undesirable pro-cess change. The expected loss is chosen as a cubic polynomial function of this delay. Such a form of the expected loss is justified by some real-life prob-lems. The SPC optimization problem is modeled by a nonlinear calculus of variations problem where the functional is minimized by a proper choice of the sampling time-interval. Theoretical results are illustrated by several academic and real-life examples. In the previous works of the authors, the SPC optimization problem was solved for linear, pure quadratic and quadratic polynomial criteria.

In this work, a finite-horizon zero-sum linear-quadratic differential game, modeling a pursuit-evasion problem, was considered. In the game’s cost function, the cost of the control of the minimizing player (the minimizer/the pursuer) was much smaller than the cost of the control of the maximizing player (the maximizer/the evader) and the cost of the state variable. This smallness was expressed by a positive small multiplier (a small parameter) of the square of the L2-norm of the minimizer’s control in the cost function. Parameter-free sufficient conditions for the existence of the game’s solution (the players’ optimal state-feedback controls and the game value), valid for all sufficiently small values of the parameter, were presented. The boundedness (with respect to the small parameter) of the time realizations of the optimal state-feedback controls along the corresponding game’s trajectory was established. The best achievable game value from the minimizer’s viewpoint was derived. A relation between solutions of the original cheap control game and the game that was obtained from the original one by replacing the small minimizer’s control cost with zero, was established. An illustrative real-life example is presented.

Previous results of the authors on robust controllability of linear systems are extended to the case of non-scalar controls and target linear manifold in ℝn of a dimension other than n−1. Basic concepts, such as the robust transferring strategy and the robust controllability set, are revisited. Novel robust controllability conditions are established. Numerical results for a three-dimensional interception problem with ellipsoidal control constraints are presented.

Two inverse ill-posed problems are considered. The first problem is an input restoration of a linear system. The second one is a restoration of time-dependent coefficients of a linear ordinary differential equation. Both problems are reformulated as auxiliary optimal control problems with regularizing cost functional. For the coefficients restoration problem, two control models are proposed. In the first model, the control coefficients are approximated by the output and the estimates of its derivatives. This model yields an approximating linear-quadratic optimal control problem having a known explicit solution. The derivatives are also obtained as auxiliary linear-quadratic tracking controls. The second control model is accurate and leads to a bilinear-quadratic optimal control problem. The latter is tackled in two ways: by an iterative procedure and by a feedback linearization. Simulation results show that a bilinear model provides more accurate coefficients estimates.