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This article extends the optimal covariance steering (CS) problem for linear stochastic systems subject to chance constraints so as to account for an optimal allocation of the risk. Previous works have assumed a uniform risk allocation in order to cast the optimal control problem as a semidefinite program, which can be solved efficiently using standard SDP solvers. An iterative risk allocation (IRA) formalism is used to solve the optimal risk allocation problem for CS using a two-stage approach. The upper-stage of IRA optimizes the risk, which is a convex problem, while the lower-stage optimizes the controller with the new constraints. The process is applied iteratively until the optimal risk allocation that achieves the lowest total cost is obtained. The proposed framework results in solutions that tend to maximize the terminal covariance, while still satisfying the chance constraints, thus leading to less conservative solutions than previous methodologies. In this article, we consider both polyhedral and cone state chance constraints. Finally, we demonstrate the approach to a spacecraft rendezvous problem and compare the results with other competing approaches. [ABSTRACT FROM AUTHOR]

This paper deals with the coordinated control of the multiple high-speed train system taking into account the sampled characteristic of the communication information. Considering the inertial lag of the servo motor/braking unit in practical scenarios, a third-order nonlinear model is adopted to capture the dynamics of a train, based on which the coordinated control problem is formulated. By virtue of the back-stepping linearization technique, the sampled-data communication based train coordination is transformed to the stabilization of a linear multiple-input system subject to periodic time-varying input delay. An event-triggering communication framework is constructed to lower the dependence on the communication resource. Furthermore, distributed low-gain feedback control protocols are established that, besides achieving coordination of the train operation, ensure the trains track the predefined train speed profile while avoiding unnecessary information transmission. Numerical experiments are carried out to verify the theoretical results. [ABSTRACT FROM AUTHOR]

Highlights • Inversion-based feedforward control technique for stable LTI systems is proposed. • Technique enables tracking of long-duration trajectories online in small batches. • It achieves excellent tracking control for minimum and non-minimum phase systems. • It is used for online compensation of vibration-induced errors of a 3D printer. • Significant improvements in printing accuracy at higher printing speeds are achieved. Abstract A limited-preview filtered B-spline (FBS) approach for minimizing errors in tracking a desired trajectory is presented. In the full-preview FBS approach, the feedforward control input to a stable linear system, with or without non-minimum phase zeros, is decomposed into B-spline basis functions with unknown coefficients; the basis functions are forward filtered using the (modeled) dynamics of the system, and their coefficients selected to minimize tracking errors of the entire trajectory in one batch. Instead, this paper proposes the use of a receding horizon to recursively compute unknown coefficients that minimize tracking errors for small batches (subsets) of the trajectory at a time, by exploiting the local property of B-splines. This allows optimal control signals to be determined at much lower computational cost compared to full-preview FBS, thus enabling online implementation on real-time controllers. The adverse effects of limited preview on tracking accuracy, relative to full preview, are analyzed, and limited-preview FBS is shown in numerical examples to preserve the versatility of full-preview FBS in tracking systems irrespective of their zero locations. The practicality and effectiveness of the limited-preview FBS approach are demonstrated by employing it for online feedforward compensation of tracking errors caused by structural vibrations of a stepper-motor-driven 3D printer. Alleviation of vibration-induced surface waviness and layer-to-layer registration errors, without sacrificing print speed, are demonstrated. Graphical abstract Image, graphical abstract [ABSTRACT FROM AUTHOR]

In this paper, phase trajectory length concept is employed to introduce a performance index in order to investigate the oscillation of linear power systems. At first, Phase Trajectory Length (PTL) concept in the state space of a Multi Input Multi Output (MIMO) linear system is defined as the traversed distance from a certain point in the state-space to the equilibrium point of the system. Moreover, lower and upper bounds of the PTL are computed. In order to evaluate and compare the oscillatory nature of the power system, oscillation number is defined as the ratio of the PTL to the radial distance of a certain point to the equilibrium. Based on this criterion, it is demonstrated that shortening the PTL leads to effective oscillation damping for all variables which are linear combinations of the main states of the intended system. It is proven that considering the aforementioned index in controller design leads to satisfy a certain finite boundary for Integral Absolute Error (IAE) of system states. Predicated on expressed features of the PTL, corresponding Hamilton Jacobi Bellman (HJB) equations with Minimum Length Controller (MLC) is represented to design damping controller. In order to reduce the settling time in oscillation damping, the desired time weight is augmented in the calculation of the considered objective function. The proposed index can be employed to tune the controller parameters. In this regard, a numerical algorithm is suggested to design a full state feedback controller as MLC. At the end, the given linear power examples, both in simulation and experimental results, show the benefits of this approach for analyzing and designing the oscillation damping controller in linear power systems. [ABSTRACT FROM AUTHOR]

This paper is concerned with the finite-time tracking for linear and nonlinear systems, with respect to the reference signal (to be tracked by the system state) obeying some uniformity. Before the control design of finite-time tracking, an interesting and basic problem is whether or not such control exists for the system concerned. This paper aims to this basic problem. First, the sufficient and necessary conditions on finite-time tracking are proposed for linear systems. Such conditions are then extended to those for affine nonlinear systems. It should be pointed out that the sufficient and necessary conditions proposed in the paper essentially reveal the relationship between the structure of the linear/nonlinear systems and the existence of the finite-time tracking control. [ABSTRACT FROM PUBLISHER]

This work presents a Model Predictive Controller (MPC) that is able to handle Linear Time-Varying (LTV) plants with Pulse-Width Modulated (PWM) control. The MPC is based on a planner that employs a Pulse-Amplitude Modulated (PAM) or impulsive approximation as a hot-start and then uses explicit linearization around successive PWM solutions for rapidly improving the solution by means of quadratic programming. As an example, the problem of rendezvous of spacecraft for eccentric target orbits is considered. The problem is modeled by the LTV Tschauner–Hempel equations, whose state transition matrix is explicit; this is exploited by the algorithm for rapid convergence. The efficacy of the method is shown in a simulation study. [ABSTRACT FROM AUTHOR]

Analytical expressions for the control functions constructing various families of the optimal trajectories in the linear-quadratic problem and obeying three-point mixed conditions were obtained. Illustrative examples were given. [ABSTRACT FROM AUTHOR]

This article deals with the stabilisation problem for the switched system composed of two planar linear subsystems via characterising the construction of the optimally arranged switching surfaces. To this end, we first explain the stabilisation mechanism in terms of return ratio and define the most stable trajectory by means of minimising the return ratio. Then we identify the most stable trajectory according to the geometric construction that its turning points constitute the subspaces where the fields of two subsystems are linearly dependent. This observation renders a constructive way to determine the optimally arranged switching surfaces and, therefore, enables us to examine the stabilisability in a necessary and sufficient sense. Finally, we use classical examples to demonstrate the theoretical results. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper, the problem of finite-time stability of linear nonautonomous systems with time-varying delays is considered. Using a novel approach based on some techniques developed for linear positive systems, we derive new explicit conditions in terms of matrix inequalities ensuring that the state trajectories of the system do not exceed a certain threshold over a pre-specified finite time interval. These conditions are shown to be relaxed for the Lyapunov asymptotic stability. A numerical example is given to illustrate the effectiveness of the obtained result. [Copyright &y& Elsevier]

Abstract: In the restricted three-body problem if the Jacobi constant is just below the value corresponding to Lagrangian point only a little neck exists around the equilibrium point and capture trajectories are indicated as low-energy. Capture properties depend on the dynamics around these critical points and qualitative results can be obtained using linearized systems. In this paper, to study transit trajectory properties in the restricted three and four-body problem, the Earth–Moon–Sun–Satellite system is considered as example and studied using different models. In the circular restricted three-body problem (Earth–Moon–Satellite), transit, non transit and asymptotic trajectories, are easily identified by using the principal reference frame. Dynamics around Lagrangian point are then studied introducing the Moon eccentricity into the elliptical restricted three-body model. A preferential region for transit orbit is individuated and studied as a function of eigenvalue properties. To introduce the Sun effect, the bi-circular four-body model is considered and dynamics around Lagrangian points studied as a function of angular distance between Earth–Sun and Earth–Moon line. Finally, results obtained in the elliptical three-body model and bi-circular four-body model, are compared with numerical simulations using real Sun–Moon–Earth ephemeris. [Copyright &y& Elsevier]

Abstract: This paper proposes some ultimate bounded stability analysis and stabilization conditions for systems involving actuators with different nonlinear elements, like for instance both saturation and dead-zone or both saturation and stick–slip. Results are based on the use of a convex differential inclusion approach. Indeed, an adequate property allowing to upper-bound some product terms related to the nonlinearity is provided. Thus, constructive conditions associated to convex optimization schemes are developed to determine suitable regions of the state space in which the closed-loop trajectories can be captured. [Copyright &y& Elsevier]

Abstract: Model predictive control (MPC) is an efficient method for the controller design of a large number of processes. However, linear MPC is often inappropriate for controlling nonlinear large-scale systems, while non-linear MPC can be computationally costly. The resulting optimization-based procedure can lead to local minima due to the, non-convexities that non-linear systems can exhibit. To overcome the excessive computational cost of MPC application for large-scale nonlinear systems, model reduction methodology in conjunction with efficient system linearizations have been exploited to enable the efficient application of linear MPC for nonlinear distributed parameter systems (DPS). An off-line model reduction technique, the proper orthogonal decomposition (POD) method, combined with a finite element Galerkin projection is first used to extract accurate non-linear low-order models from the large-scale ones. Trajectory Piecewise-Linear (TPWL) methodologies are subsequently developed to construct a piecewise linear representation of the reduced nonlinear model, both in a static and in a dynamic fashion. Linear MPC, based on quadratic programming, can then be efficiently performed on the resulting low-order, piece-wise affine system. Our combined methodology is readily applicable in combination with advanced MPC methodologies such as multi-parametric MPC (MP-MPC) (). The stabilisation of the oscillatory behaviour of a tubular reactor with recycle is used as an illustrative example to demonstrate our methodology. [Copyright &y& Elsevier]

Abstract: Min–Max MPC (MMMPC) offers the possibility to consider disturbances and uncertainties in the mathematical model used to predict the future trajectory of the system. The explicit consideration of disturbances and uncertainties in order to obtain a more robust control performance complicates the practical implementation of MMMPC due to the high computational burden required to compute the control law. The computational complexity of the optimization problem can be reduced by using approximate solutions or upper bounds of the worst case cost of the objective function. A computationally efficient MMMPC strategy based on such an upper bound was presented in a previous work also published in this journal (see Section ). One of the main drawbacks of that strategy is the lack of a stability guarantee. In this paper it is shown that input-to-state practical stability of the MMMPC strategy can be guaranteed if a certain initial condition and a semi-feedback approach are used. Furthermore, the MMMPC strategy is validated in experiments with a continuous stirred tank reactor in which the temperature of the reactor is controlled. The behavior of the system and the controller are illustrated by means of experimental results. [ABSTRACT FROM AUTHOR]

A relatively optimal control is a stabilizing controller that, without initialization nor feedforwarding and tracking the optimal trajectory, produces the optimal (constrained) behavior for the nominal initial condition of the plant. In a previous work, for discrete-time linear systems, we presented a linear dynamic relatively optimal control. Here we provide a static solution, namely a deadbeat piecewise-affine state-feedback controller based on a suitable partition of the state space into polyhedral sets. The vertices of the polyhedra are the states of the optimal trajectory; hence a bound for the complexity of the controller is known in advance. We also show how to obtain a controller that is not deadbeat by removing the zero terminal constraint while guaranteeing stability. Finally, we compare the proposed static compensator with the dynamic one. [ABSTRACT FROM AUTHOR]

This paper addresses two strategies for the stabilization of continuous-time, switched linear systems. The first one is of open loop nature (trajectory independent) and is based on the determination of a minimum dwell time by means of a family of quadratic Lyapunov functions. The relevant point on dwell time calculation is that the proposed stability condition does not require the Lyapunov function to be uniformly decreasing at every switching time. The second one is of closed loop nature (trajectory dependent) and is designed from the solution of what we call Lyapunov—Metzler inequalities from which the stability condition (including chattering) is expressed. Being nonconvex, a more conservative but simpler-to-solve version of the Lyapunov—Metzler inequalities is provided. The theoretical results are illustrated by means of examples. [ABSTRACT FROM AUTHOR]

We consider well-posed linear systems whose state trajectories satisfy ẋ = Ax + Bu, where u is the input and A is an essentially skew-adjoint and dissipative operator on the Hilbert space X. This means that the domains of A* and A are equal and A* +A = -Q, where Q ≥ 0 is bounded on X. The control operator B is possibly unbounded, but admissible and the observation operator of the system is B*. Such a description fits many wave and beam equations with colocated sensors and actuators, and it has been shown for many particular cases that the feedback u = -κy+v, with κ > 0, stabilizes the system, strongly or even exponentially. Here, y is the output of the system and v is the new input. We show, by means of a counterexample, that if B is sufficiently unbounded, then such a feedback may be unsuitable: the closed-loop semigroup may even grow exponentially. (Our counterexample is a simple regular system with feedthrough operator zero.) However, we prove that if the original system is exactly controllable and observable and if κ is sufficiently small, then the closed-loop system is exponentially stable. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a continuous extension of the Ott, Grehogi and Yorke control method [Ott et al., 1990]. This extension allows the control of chaotic behavior in higher-order dynamical systems. To begin with, studying the so-called Poincaré Sections (PS) enables determining fixed points and the effect of small changes in the control Parameter on the system trajectory. Introduction of a stabilizer parameter in the control law keeps the dynamical properties and maintains them on a cyclic trajectory. The effectiveness of the proposed method is tested by numerical examples on the two typical Chen and Chua chaotic systems. [ABSTRACT FROM AUTHOR]

In this paper the trajectory tracking control problem for a certain class of propagation processes modeled as quasi-linear parameter varying systems is considered. The propagation physical models are generally described by means of partial differential equations (PDEs). However in real world control problems the PDE models are usually converted into ordinary differential equations (ODEs) models adopting numerical and/or physical approximations. In many practical problems it happens that the propagation dynamics are linear, while the boundary conditions are described by nonlinear algebraic equations. A trajectory following control scheme is proposed for this class of systems together with a robust performance analysis based on the concept of quadratic stability with an H∞ norm bound. An LMI based observer synthesis procedure is also proposed to increase the closed loop system performance. [Copyright &y& Elsevier]

Proposes an approach to the investigation of the stability of the effective solutions of an n-criteria linear trajectory problem. Definitions and lemmas; Theorems and proofs; Corollaries; Numerical example.