Piecewise Constant Systems Research Articles

A Novel Polarized Skylight Navigation Model for Bionic Navigation With Marginalized Unscented Kalman Filter

Bionic navigation is an essential technology in GPS-denied environment for vehicle navigation. This paper combined strapdown inertial navigation system, polarized skylight sensors and odometer to design a new navigation model. In particular, a novel measurement model is developed based on polarized skylight. This model primarily utilizes the cross product to compute the error between the measured polarization vector and the theoretical polarization vector. It differs from the error computed by strapdown inertial navigation system in design vector measurement model. It deals with the problem of the directional ambiguity of polarization vector difference directly. Considering the measured polarization vector error due to computation and multiple scattering from atmospheric molecules, the states are augmented with the measured vector error to form a partially nonlinear, bionic integrated navigation model. To reduce the computation burden, marginalised unscented Kalman filter is presented to estimate the unknown states. The observability analysis method of piece-wise constant systems is applied to compute the observability matrix and singular value. Simulation results show that linear maneuver and angular maneuver can improve the degree of the observability of attitude and measured polarized vector error. Finally, experiments further verify the effectiveness of proposed models and method.

Sensitivity to Cumulative Perturbations for a Class of Piecewise Constant Hybrid Systems

We consider a class of continuous-time hybrid dynamical systems that correspond to subgradient flows of a piecewise linear and convex potential function with finitely many pieces, and which includes the fluid-level dynamics of the Max-Weight scheduling policy as a special case. We study the effect of an external disturbance/perturbation on the state trajectory, and establish that the magnitude of this effect can be bounded by a constant multiple of the integral of the perturbation.

A composite pseudospectral method for solving multi-delay optimal control problems involving piecewise constant delay functions

An adaptive numerical method for solving multi-delay optimal control problems with piecewise constant delay functions is introduced. The proposed method is based on composite pseudospectral method using the well-known Legendre–Gauss–Lobatto points. In this approach, the main problem converts to a mathematical optimization problem whose solution is much more easier than the original one. The necessary conditions of optimality associated to nonlinear piecewise constant delay systems are derived. The method is easy to implement and provides very accurate results.

Analysis of Linear Piecewise Constant Delay Systems Using a Hybrid Numerical Scheme

An efficient computational technique for solving linear delay differential equations with a piecewise constant delay function is presented. The new approach is based on a hybrid of block-pulse functions and Legendre polynomials. A key feature of the proposed framework is the excellent representation of smooth and especially piecewise smooth functions. The operational matrices of delay, derivative, and product corresponding to the mentioned hybrid functions are implemented to transform the original problem into a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the proposed numerical scheme.

Numerical Solution of Piecewise Constant Delay Systems Based on a Hybrid Framework

An efficient numerical scheme for solving delay differential equations with a piecewise constant delay function is developed in this paper. The proposed approach is based on a hybrid of block-pulse functions and Taylor’s polynomials. The operational matrix of delay corresponding to the proposed hybrid functions is introduced. The sparsity of this matrix significantly reduces the computation time and memory requirement. The operational matrices of integration, delay, and product are employed to transform the problem under consideration into a system of algebraic equations. It is shown that the developed approach is also applicable to a special class of nonlinear piecewise constant delay differential equations. Several numerical experiments are examined to verify the validity and applicability of the presented technique.

A decidable class of planar linear hybrid systems

In this paper, we show the decidability of a new subclass of linear hybrid automata. These automata are planar, that is, consist of two state variables, monotonic along some direction in the plane and have identity resets, that is, the values of the continuous variables are not reset to a different value on a discrete transition. Our proof uses a combination of a tree construction capturing the edge-to-edge reachability and a finite bisimulation construction. Our class strictly contains the class of two dimensional piecewise constant derivative systems for which decidability of the reachability problem is known.

Analysis of Multi-delay and Piecewise Constant Delay Systems by Hybrid Functions Approximation

In this paper, a new numerical method for solving multi-delay and piecewise constant delay systems is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrices of integration, product and delay are given. These matrices are then utilized to reduce the solution of multi-delay systems and the piecewise constant delay systems to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Controllability and Observability Criteria for Linear Piecewise Constant Impulsive Systems

Impulsive differential systems are an important class of mathematical models for many practical systems in physics, chemistry, biology, engineering, and information science that exhibit impulsive dynamical behaviors due to abrupt changes at certain instants during the dynamical processes. This paper studies the controllability and observability of linear piecewise constant impulsive systems. Necessary and sufficient criteria for reachability and controllability are established, respectively. It is proved that the reachability is equivalent to the controllability under some mild conditions. Then, necessary and sufficient criteria for observability and determinability of such systems are established, respectively. It is also proved that the observability is equivalent to the determinability under some mild conditions. Our criteria are of the geometric type, and they can be transformed into algebraic type conveniently. Finally, a numerical example is given to illustrate the utility of our criteria.

Temporal and differential stabilizability and detectability of piecewise constant rank systems

SUMMARYIn a past note we drew attention to the fact that time‐varying continuous‐time linear systems may be temporarily uncontrollable and unreconstructable and that this is vital knowledge to both control engineers and system scientists. Describing and detecting the temporal loss of controllability and reconstructability require considering piecewise constant rank (PCR) systems and the differential Kalman decomposition. In this note for conventional as well as PCR systems measures of temporal and differential stabilizability and detectability are developed. These measures indicate to what extent the temporal loss of controllability and reconstructability may lead to temporal instability of the closed‐loop system when designing a static state or dynamic output feedback controller. It is indicated how to compute the measures from the system matrices. The importance of our developments for control system design is illustrated through three numerical examples concerning LQ and LQG perturbation feedback control of a non‐linear system about an optimal control and state trajectory. Copyright © 2011 John Wiley & Sons, Ltd.

Solution of piecewise constant delay systems using hybrid of block‐pulse and Chebyshev polynomials

AbstractAn efficient numerical method for finding the solution of piecewise constant delay systems is presented. The method is based upon hybrid of block‐pulse functions and Chebyshev polynomials. The operational matrix of delay is constructed. The properties of hybrid functions together with the operational matrices of integration, delay and product are used to convert the delay differential equation to an algebraic equation, thus greatly simplifying the problem. The method is easy to implement and yields very accurate results. The validity and applicability of the proposed method are demonstrated by considering different types of delay systems. Copyright © 2010 John Wiley & Sons, Ltd.

Temporal Linear System Structure

Piecewise constant rank systems and the differential Kalman decomposition are introduced in this note. Together these enable the detection of temporal uncontrollability/unreconstructability of linear continuous-time systems. These temporal properties are not detected by any of the four conventional Kalman decompositions. Moreover piecewise constant rank systems admit the state dimension to be time-variable. As demonstrated in this note linear continuous-time systems with variable state dimensions enable the well rounded realization theory suggested already by Kalman. The differential Kalman decomposition introduced in this note is associated with differential controllability and differential reconstructability. The system structure obtained from the differential Kalman decomposition may be interpreted as the temporal linear system structure. This note reveals that the difference between controllability and reachability as well as reconstructability and observability is entirely due to changes of the temporal linear system structure. Also, this note reveals how the differential Kalman decomposition relates to the conventional ones.

Controllability and observability of a class of linear impulsive systems

This paper studies the controllability and observability of a class of linear piecewise constant impulsive systems. Necessary and sufficient criteria for reachability and controllability are established, respectively. It is proved that the reachability is equivalent to the controllability under some mild conditions. Then, necessary and sufficient criteria for observability and determinability of such systems are established, respectively. It is also proved that the observability is equivalent to the determinability under some mild conditions. Our criteria are of geometric type, they can be transformed into algebraic type conveniently. Finally, a numerical example is given to illustrate the utility of our criteria.

Achilles and the Tortoise climbing up the hyper-arithmetical hierarchy

In this paper, we characterize the computational power of dynamical systems with piecewise constant derivatives (PCD) considered as computational machines working on a continuous real space with a continuous real time: we prove that piecewise constant derivative systems recognize precisely the languages of the ω k th (respectively ( ω k + 1)th) level of the hyper-arithmetical hierarchy in dimension d = 2 k + 3 (respectively d = 2 k + 4), k ⩾ 0. Hence we prove that the reachability problem for PCD systems of dimension d = 2 k + 3 (resp. d = 2 k + 4), k ⩾ 1, is hyper-arithmetical and is ∑ ω k -complete (resp. ∑ ω k -complete).

Observability analysis of piece-wise constant systems. I. Theory

For pt.II see ibid., vol.28, no.4, p.1068-75, Oct. 1992. A method for analyzing the observability of time-varying linear systems which can be modeled as piece-wise constant systems (PWCS) is presented. An observability matrix for such systems is developed for continuous and discrete time representations. A stripped observability matrix (SOM) is introduced which simplifies the analysis in cases where the use of this matrix is legitimate. The observability analysis is presented as a step-by-step procedure.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Observability analysis of piece-wise constant systems. II. Application to inertial navigation in-flight alignment (military applications)

For pt.I see ibid., vol.28, no.4, p.1056-67, Oct. 1992. The method of analyzing the observability of time-varying linear systems as piecewise constant systems (PWCS) is applied to the analysis of in-flight alignment (IFA) of inertial navigation systems (INS) whose estimability is known to be enhanced by maneuvers. The validity of this approach to the analysis of IFA is proven. The analysis lays the theoretical background to, and clearly demonstrates the observability enhancement of, IFA. The analytic conclusions are confirmed by covariance simulations. Although INS IFA was handled to various degrees in the past, a comprehensive control theoretic approach to the problem is introduced. The analysis yields practical conclusions and a procedure previously unknown. >

General hybrid orthogonal functions and some potential applications in systems and control

A framework for a system of general hybrid orthogonal functions (GHOF) is introduced in the paper. The system of GHOF is shown to be more relevant to many practical situations than the conventional systems of orthogonal functions. The proposed framework of definition combines the natural features of discontinuity of the class of piecewise-constant systems and the inherent characteristics of continuity and differentiability of the systems of continuous systems of polynomial and sinusoidal functions, leading to a general, flexible and piecewise-continuous (and differentiable) system of GHOF. The powerfulness of the proposed system is demonstrated in the representation of functions with discontinuities, solution of the state equation with discontinuous inputs and prediction of the limit cycle of a highly nonlinear van der Pol&apos;s oscillator. The results amply demonstrate the value of the GHOF as a potential basis for function expansion in a large class of real-world problems.

Analysis of piecewise constant delay systems via block-pulse functions

The solution of linear piecewise constant delay systems has been derived by using the block pulse series expansion method. A matrix called the delay operational matrix is introduced first to manipulate the time delay. Then the solution is obtained through using the matrix and the properties of block pulse functions.

Square-root algorithms for parallel processing in optimal estimation

We give explicit algorithms in square-root form that allow measurements for the standard state estimation problem to be processed in a highly parallel fashion with little communication between processors. After this preliminary processing, blocks of measurements may be incorporated into state estimates with essentially the same computation as usually accompanies the incorporation of a single measurement. This formulation also leads to square-root doubling formulae for calculating the steady-state error-covariance matrix of constant models, and an extension of the class of problems for which Chandrasekhar-type algorithms offer computational reductions to include piecewise constant systems with arbitrary initial conditions.