Higher-order continuous-time implicit systems: Consistency and weak consistency, impulse controllability, geometric concepts, and invertibility properties

In this work, necessary conditions for the impulsive time-optimal control of finite-dimensional Lagrangian systems are stated. The conditions are obtained by the application sub-differential calculus techniques to extended-valued lower semi-continuous functionals. The considered functional is a generalized Bolza functional that is evaluated on multiple intervals. Contrary to the approach in literature so far, the instant of possibly impulsive transition is considered as an instant of Lebesgue measure zero. This approach is in comparison to other impulsive necessary conditions consistent with different hybrid system modeling methods in which transitions happen instantaneously. The necessary conditions provide necessary criteria for the determination of optimal transition times and locations.KeywordsImpulsive Optimal ControlImpactive Systemsnon-smooth analysishybridvariational inequalities

In this paper, based on the inverse of block companion matrix pencils associated with matrix polynomials, a first-order state-space realization in descriptor system form for a general high-order linear system is derived. An important feature of the proposed realization is that it allows the high-order linear system to be singular. While in the case that the high-order system is nonsingular, two realizations in normal state-space system forms are directly deduced. With the help of the proposed realization, a generalized Popov Belevitch Hautus (PBH) criterion for high-order systems is then established in terms of the original system matrix polynomials. Examples are studied which well demonstrate the proposed theories.

FSID - A Frequency Weighted MIMO Frequency Domain Identification and Rational Matrix Approximation Method for Python, Julia and Matlab

<abstract><p>We construct a West Nile virus epidemic model that includes the interaction between the bird hosts and mosquito vectors, mosquito life stages (eggs, larvae, adults), and the dynamics of both larvicide and adulticide. We derive the basic reproduction number for the epidemic as the spectral radius of the next generation matrix. We formulate two impulsive optimal control problems which seek to balance the cost of insecticide applications (both the timing and application level) with the benefit of (1) vector control: reducing the number of mosquitoes or (2) disease control: reducing the disease burden. We reformulate these impulsive optimal control problems as nonlinear optimization problems and derive associated necessary conditions for the optimal controls. Numerical simulations are used to address three questions: How does the control and its impact on the system vary with the objective type? Is it beneficial to optimize the treatment timing? How does the control and its impact on the population vary with the type of pesticide used?</p></abstract>

The paper is concerned with a nonlinear optimal impulsive control problem with trajectories of bounded variation. Vector Lebesgue-Stieltjes measures play the part of controls. Studied were state and mixed constraints in the conventional and fast times, as well as joint conditions for trajectory and impulsive control. A method of reduction to the classical problem of optimal control was developed relying on the discontinuous time reparameterization. The original and reduced problems were established to be equivalent.

Many multi-beam energy harvesters appearing in the literature require custom analytical or finite-element models to compute their eigensolutions and piezoelectric coupling effects. This paper discusses the use of the transfer matrix method to derive analytical solutions to beam structures with point-wise discontinuities or with lumped inertias between members or at the tip. In this method, transfer matrices are developed for the beam’s states (deflection, slope, shear force, and bending moment) analogously to the state transition matrix of a linear system. Euler-Bernoulli beam theory is used to derive transfer matrices for the uniform beam segments, and point transfer matrices are derived to handle discontinuities in the structure. The transfer matrix method is shown to be advantageous for analyzing complex structures because the size of the transfer matrix does not grow with increasing number of components in the structure. Furthermore, the same formulation can be used for a wide range of geometries, including arbitrary combinations of beam segments — single- or multi-layered — and lumped inertias. The eigensolution of the transfer matrix is shown to produce the natural frequencies and mode shapes for these structures. Subsequently, the electromechanical coupling effects are incorporated and the base excitation problem is considered. The electromechanical equations of motion are decoupled by mode and shown to be a generalization of existing analytical models. Parametric case studies are provided for beam structures with varying piezoelectric layer coverage.

PROBLEMS OF DISCRETE-TIME PERIODIC REALIZATION OF A PERIODIC COLLECTION OF RATIONAL MATRICES

For an arbitrary rational matrix (RM), a new descriptor realization of its all-pass factorization is presented. The coefficients of the factors are expressed explicitly in the solution of a linear matrix inequality, which is well known in control theory. Using the all-pass factorization, necessary and sufficient conditions for existence of a solution of an equation with RMs where the unknown RM has its poles in the left closed complex half plane and it can be improper are presented, as well as all solutions. As an application of the latter result, the notion of almost disturbance decoupling, i.e., when the infimum of the ${\mathscr {H}}_{\infty}$ norm of the transfer matrix from the disturbance input to the controlled output over the stabilizing controllers is zero, is generalized. The generalization is such that instead of the whole frequency axis, the frequency axis with excluded intervals of small lengths, is considered. The controller consists of output feedback plus disturbance feedforward designed independently. A necessary and sufficient solvability condition is presented, which is expressed in terms of the given data. The three main results are illustrated by an example.

The question of achieving a variety of specific design goals (e.g., arbitrary pole placement, decoupling, and exact model matching) for linear multivariable systems subjected to unknown step disturbances on the entire internal state and output is constructively resolved via a synthesis procedure which simultaneously combines integral feedforward compensation with asymptotic state estimation. The technique relies on the ability to express the desired closed-loop transfer matrix of the compensated system as the product of the open-loop transfer matrix and any proper rational transfer matrix, assuming the given system has no zeros at the origin and a stable closed-loop design is sought. The method employs some recent results involving the differential operator approach to multivariable system analysis and synthesis, and the treatment is confined almost exclusively to the frequency domain.

The well-known Richards' Theorem of the continuous-time filter theory is reformulated in the digital domain in a convenient manner, leading to a simple derivation of cascaded lattice digital filter structures, realizing lossless bounded transfer functions. The theorem is also extended to the matrix case, leading to a derivation of m -input p -output cascaded lattice filter structures with lossless building blocks, that realize an arbitrary p \times m digital Lossless Bounded Real (LBR) transfer matrix. Extensions to the synthesis of arbitrary, stable p \times m transfer matrices in the form of such cascaded lattices is also outlined. The derivation also places in evidence a means of testing the stability of an arbitrary p \times m transfer matrix of a discrete-time linear system.

The Exterior-Matrix Method (EMM) is applied to the in-plane dynamical equations of a shallow elastic catenary. The transfer matrices of the composite system are expanded so that asymptotic analysis can be done for the general case. The main idea is to find the 4-by-4 transfer matrices for each span of the cable system, and the transfer matrices for the joint, which could be an insulated pole. Then, by using the exterior matrices, the 6-by-6 matrices which correspond to the transfer matrices can be found. At first it seems surprising to use 6-by-6 matrices instead of the 4-by-4 transfer matrices, but with the exterior matrices, the determinant of the product need not be taken. This allows one to immediately take approximations of the matrices without worry that the highest-order terms will cancel. As a bonus to using the exterior matrices, an angle for the insulator will be allowed, which generalizes the work done by Simpson (Proc IEE 113(5):870–878, 1966). The practical use of this method is illustrated by finding the normal frequencies of a symmetrical 3-span system.

Aim This study investigated whether time-estimation task exposure influences the severity of Attention Deficit Hyperactivity Disorder (ADHD), as well as theta band activity in the dorsolateral prefrontal cortex and ventrolateral prefrontal cortex. Material and methods Twenty-two patients with ADHD participated in a crossover experiment with a visual time-estimation task under control conditions (without exposure to time estimation tasks) and experimental (thirty days exposure to time-estimation tasks) in association with electroencephalographic analysis of theta band. Results ADHD patients with thirty days of time-estimation task exposure presented a worse performance of the time-estimation task, as revealed by the measurements of the absolute error and relative error (p ≤ 0.05). However, our findings show the improvement of self-reported symptoms of attention, impulsivity, and emotional control in patients after the time-estimation task exposure (p = 0.0001). Moreover, the theta band oscillations in the right dorsolateral prefrontal cortex and in the ventrolateral prefrontal increased with thirty days of time-estimation task exposure (p ≤ 0.05). Conclusion We propose that the decrease in EEG theta power may indicate an efficient accumulation of temporal pulses, which could be responsible for the improvement in the patient cognitive aspects as demonstrated by the current study. Time-estimation task improves ADHD cognitive symptoms, with a substantial increase in cortical areas activity related to attention and memory, suggesting its use as a tool for cognitive timing function management and non-invasive therapeutic aid in ADHD.

Collective action against collective disadvantage is an important socio-psychological phenomenon that represents a powerful pathway to social change. One key conclusion from the psychological literature on collective action is that a strong subjective sense of social identity is almost a necessity for disadvantaged group members to become motivated to undertake collective action against collective disadvantage. We propose, however, that even those individuals who may not identify strongly with their group might engage in collective action when they are in a crowd. This is because one core feature of the crowd, the physical co-presence of in-group members, conveys a psychological cue to in-group support for collective action against collective disadvantage. This is particularly relevant for lower identifiers, because, unlike higher identifiers, they do not view themselves as similar to other group members and thus do not expect in-group support. As a consequence, expectations of in-group support and therefore interest in collective action should increase for lower identifiers when they are in a crowd. We tested this idea in a psychological experiment in which higher and lower identifiers with a disadvantaged group were randomly assigned to a physical co-presence or control condition. Results showed indeed that the physical co-presence of in-group members increased only lower identifiers' expectations of in-group support and their interest in collective action against collective disadvantage. We discuss the theoretical and practical implications of these results.

A novel method for factorizing the transfer function matrix of a multivariable system described by the triple (A, B, C) is presented. The factorization consists of a product of two rational function matrices, one of which is the inverse of a rational function matrix. Various properties of the factorization are given and the factorization procedure is illustrated by an example.

For an implicitly defined discrete system, a new implicit form of the Kalman filter is developed and an efficient numerical approach for its implementation is given. Unlike the traditional explicit approach, the implicit filter can be applied to ill-conditioned systems. It is shown that when the system matrices of an implicit system are sparse, the implicit Kalman filter requires significantly less computer time and storage to implement than the traditional explicit Kalman filter. Simulation results are presented to illustrate and substantiate the theoretical developments.