Input-output Equation Research Articles

Realization of nonlinear MIMO system on homogeneous time scales

The paper addresses a state space realization problem of a set of higher order delta-differential input–output equations, defined on a homogeneous time scale. The algebraic framework of differential one-forms is applied to formulate necessary and sufficient solvability conditions. This approach applies the total differential operator to analytic system equations to obtain the infinitesimal system description in terms of one-forms. This representation can be converted into polynomial system description by interpreting the polynomial indeterminate as the delta derivative acting on one-forms. The system description in terms of two matrices over skew polynomial ring is then used to derive explicit formulas for the differentials of state coordinates that significantly simplify the calculations. The formulas are found from the left quotients computed by the left Euclidean division algorithm.

Robust global identifiability theory using potentials—Application to compartmental models

This paper presents a global practical identifiability theory for analyzing and identifying linear and nonlinear compartmental models. The compartmental system is prolonged onto the potential jet space to formulate a set of input–output equations that are integrals in terms of the measured data, which allows for robust identification of parameters without requiring any simulation of the model differential equations. Two classes of linear and non-linear compartmental models are considered. The theory is first applied to analyze the linear nitrous oxide (N2O) uptake model. The fitting accuracy of the identified models from differential jet space and potential jet space identifiability theories is compared with a realistic noise level of 3% which is derived from sensor noise data in the literature. The potential jet space approach gave a match that was well within the coefficient of variation. The differential jet space formulation was unstable and not suitable for parameter identification. The proposed theory is then applied to a nonlinear immunological model for mastitis in cows. In addition, the model formulation is extended to include an iterative method which allows initial conditions to be accurately identified. With up to 10% noise, the potential jet space theory predicts the normalized population concentration infected with pathogens, to within 9% of the true curve.

Propagation of nanofiber-guided light through an array of atoms

We study the propagation of nanofiber-guided light through an array of atomic cesium, taking into account the transitions between the hyperfine levels $6S_{1/2}F=4$ and $6P_{3/2}F'=5$ of the $D_2$ line. We derive the coupled-mode propagation equation, the input-output equation, the scattering matrix, the transfer matrix, and the reflection and transmission coefficients for the guided field in the linear, quasistationary, weak-excitation regime. We show that, when the initial distribution of populations of atomic ground-state sublevels is flat, the quasilinear polarizations along the principal axes $x$ and $y$, which are parallel and perpendicular, respectively, to the radial direction of the atomic position, are not coupled to each other in the linear coherent scattering process. When the guided probe field is quasilinearly polarized along the major principal axis $x$, forward and backward scattering have different characteristics. We find that, when the array period is far from the Bragg resonance, the backward scattering is weak. Under the Bragg resonance, most of the guided probe light can be reflected back in a broad region of field detunings even though there is an irreversible decay channel into radiation modes. When the atom number is large enough, two different band gaps may be formed, whose properties depend on the polarization of the guided probe field.

Setpoint control of networked systems via static output feedback integral controller

This study studies the problem of setpoint output tracking for networked control systems with network-induced delay, packet disorder and packet dropout. Based on the input-output difference equation model, a novel networked predictive control (NPC) method via static output feedback integral controller (SOFIC) is proposed to actively compensate for the round-trip time (RTT) delay resulting from the aforementioned communication constraints. It is analysed that the resulting NPC system can achieve a similar tracking performance as that of the corresponding local control system. Based on the performance analysis, a necessary and sufficient condition for the stability of the closed-loop NPC system is obtained, which is independent of the RTT delay. Both numerical simulations and practical experiments on an internet-based servo motor system illustrate the effectiveness of the proposed method.

An application of screw theory to the kinematic analysis of a Delta-type robot

This paper reports on the kinematics of a translational parallel manipulator whose topology is close to the architecture of the famous Delta robot. The displacement analysis is presented in closed-form solution by applying a new strategy based on the unknown coordinates of a point embedded to the moving platform. The input-output equations of velocity and acceleration of the robot are systematically obtained by resorting to reciprocal-screw theory. The singularities of the mechanism are explained through the input-output equation of velocity. Finally, a numerical example is provided to show the application of the method.

Polynomial accessibility condition for the multi-input multi-output nonlinear control system

The paper presents a computation-oriented necessary and sufficient accessibility condition for the set of nonlinear higher- order input-output differential equations. The condition is presented in terms of the greatest common left divisor of two polynomial matrices, associated with the system of input-output equations. The basic difference from the linear case is that the elements of the polynomial matrices belong to a non-commutative polynomial ring. The condition found provides a basis for finding the accessible representation of the set of input-output equations, which is a suitable starting point for the construction of an observable and accessible state space realization. Moreover, the condition allows us to check the transfer equivalence of two nonlinear systems.

Hierarchical Network Identification of Large-Scale Systems - An Approach Based on Dissipation Equalities

Recently, dynamical systems in science and engineering problems become increasingly larger and too complex. One of the ways to solve the difficulty is to model the systems with a hierarchically networked interconnection of subsystems. In this paper, we consider a hierarchical network system which is an interconnection of dissipative subsystems and identify both subsystems and network structure, called network system, preserving the dissipativity properties. We first estimate the state sequences of subsystems via the dissipation equalities instead of the matrix input-output equation in ordinary subspace identification methods. As a main result, we show an identification method in open-loop manner for subsystems and network systems based on the least-squares method. We also propose an identification method in a closed-loop manner.

Solvability of the Economic Input-Output Equation by Time Irreversibility

This paper reinterprets the economic input-output equation as a description of a realized situation without considering decision making. This paper uses the equation that the self-sufficiency rate is added to the Leontief type, and discusses its solvability. The equation has a unique solution if and only if each part of the relevant society satisfies the space-time openness condition. This condition means that commodities which a part of the relevant society possesses are not all inputted to its inside. Moreover, if the process of input and output is time irreversible, each part of the relevant society satisfies the space-time openness condition. Therefore, the solvability of the equation is guaranteed by time irreversibility. This proposition seems to be relevant to the grandfather paradox which is a type of time paradox.

Money Circulation Equation Considering Time Irreversibility

The essence of money circulation is that money continues to transfer among economic agents eternally. Based on this recognition, this paper shows a money circulation equation that calculates the quantities of expenditure, revenue, and the end money from the quantity of the beginning money. The beginning money consists of the possession at term beginning, production and being transferred from the outside of the relevant society. The end money consists of the possession at term end, disappearance and transferring to the outside of the relevant society. This equation has a unique solution if and only if each part of the relevant society satisfies the space-time openness condition. Moreover, if money is transferred time irreversibly, each part of the relevant society satisfies the space-time openness condition. Hence, the solvability of the equation is guaranteed by time irreversibility. These solvability conditions are similar to those of the economic input-output equation, but the details are different. An equation resembling our money circulation equation was already shown by Mária Augustinovics, a Hungarian economist. This paper examines the commonalities and differences between our equation and hers. This paper provides the basis for some intended papers by the author.

Infrared Data Measurement Modeling and Non-Uniformity Correction (NUC) Algorithm for Infrared Detector

In this paper, infrared data correction algorithm is suggested when the environmental conditions such as air temperature or detector substrate temperature are varying. To correct the infrared data, infrared detector measurement model is described by a parameter-dependent first-order input-output equation and the relations between infrared data and model environmental parameters are investigated first. By determining the parameters which dominate the influence on the infrared data, the coefficients of a model are obtained. By using only the measurements of ambient temperatures and detector substrate temperatures, infrared detector output data corrupted by environmental temperature variations are compensated based on the model instead of using any target temperature source equipments. It is shown through experiments that the proposed algorithm can reduce the influence of environmental temperatures on the infrared data.

Mobility analysis and kinematics of the semi-general 2(3-RPS) series-parallel manipulator

This work reports on the kinematics of a series-parallel manipulator built with two zero-torsion tangential parallel manipulators assembled in series connection. Although this mechanism has been widely studied, there are some topics that must be revised, e.g. the mobility analysis here reported shows that the robot under study is not precisely a six degrees of freedom spatial mechanism as it has been commonly considered. Furthermore, the traditional hexagonal coupler platform is replaced with a three-dimensional platform which yields a mechanism with a more general topology. The forward and inverse displacement analyses of the robot are obtained in semi-closed form solutions based on simple closure equations which are generated upon the coordinates of three points embedded to the moving platform while the input–output equations of velocity and acceleration of the semi-general series-parallel manipulator are easily derived by resorting to reciprocal-screw theory. A case study is included in order to show the application of the method of kinematic analysis.

Extended observer form: simple existence conditions

The paper focuses on the problem of transforming the discrete-time single-input single-output nonlinear state equations into the extended observer form, which, besides the input and output, also depends on a finite number of their past values. The simple necessary and sufficient conditions for the existence of the extended coordinate change and the output transformation, allowing to solve the problem, are formulated in terms of certain partial derivatives, related to the input–output equation, corresponding to the state equations. Moreover, a certain algorithm for transforming the state equations into the observer form is proposed.

Admissibility and nonuniform exponential dichotomy on the half-line

The aim of this paper is to deduce new conditions for the existence of the nonuniform exponential dichotomy of evolution families on the half-line. We consider an evolution family having a nonuniform exponential growth and we associate to it an input–output equation. We prove that the admissibility of the pair (Cb(R+,X),Lp(R+,X)) with respect to this equation implies the existence of a nonuniform exponential dichotomy. We also present an illustrative example which shows that, generally, the converse implication is not valid in the nonuniform case. Finally, we give an application to the case of uniform exponential dichotomy.

A parallel manipulator inspired by the original Stewart platform

This study addresses the kinematics of a new parallel manipulator inspired by the eight-bar linkage proposed as a flight simulator by Stewart almost five decades ago. Due to its partially decoupled topology, the forward displacement analysis of the robot is obtained in a nearly closed-form solution. The input–output equations of velocity and acceleration of the manipulator are systematically derived by resorting to reciprocal-screw theory. Numerical examples are included in the contribution in order to show the application of the method of kinematic analysis. As far as the authors are aware, the topology proposed in this contribution has not been reported in previous works.

A 2(3-RRPS) parallel manipulator inspired by Gough–Stewart platform

SUMMARYThis study addresses the kinematics of a six-degrees-of-freedom parallel manipulator whose moving platform is a regular triangular prism. The moving and fixed platforms are connected to each other by means of two identical parallel manipulators. Simple forward kinematics and reduced singular regions are the main benefits offered by the proposed parallel manipulator. The Input–Output equations of velocity and acceleration are systematically obtained by resorting to reciprocal-screw theory. A case study, which is verified with the aid of commercially available software, is included with the purpose to exemplify the application of the method of kinematic analysis.

Alternative to Ritt’s pseudodivision for finding the input-output equations of multi-output models

Differential algebra approaches to structural identifiability analysis of a dynamic system model in many instances heavily depend upon Ritt’s pseudodivision at an early step in analysis. The pseudodivision algorithm is used to find the characteristic set, of which a subset, the input-output equations, is used for identifiability analysis. A simpler algorithm is proposed for this step, using Gröbner Bases, along with a proof of the method that includes a reduced upper bound on derivative requirements. Efficacy of the new algorithm is illustrated with several biosystem model examples.

Enhancing the originality of the Stewart platform through its kinematic analyses

In this study, an approach to solve the kinematics of the Stewart platform is introduced with the purpose to enhance its originality. Contrary to the Gough platform, the forward displacement analysis of the Stewart platform is based on closure equations that allow to obtain nearly closed-form solutions. These equations lead to decoupled functions which are solved by means of the well-known Sylvester's dialytic method of elimination. The input–output equations of velocity and acceleration as well as the singularities of the robot are obtained systematically by resorting to reciprocal-screw theory. Numerical examples are provided in order to demonstrate the application of the proposed method. Furthermore, the numerical results obtained via screw theory are successfully compared with simulations generated with the aid of commercially available software. Finally, the contribution may be useful to recognize that the flight simulator proposed by Stewart almost five decades ago and the tire testing machine attributed to Gough are in reality robot manipulators with quite different characteristics.

Symbolic Polynomial Tools for Nonlinear Control Systems

AbstractA collection of functions addressing various modelling problems for nonlinear control systems has been developed in Mathematica environment. The mathematical apparatus behind the programs is based on the theory of non-commutative skew polynomial rings. The functions described in paper allow to find the state-space realization of the set of input-output equations, reduce the system equations, compute the transfer matrix of the nonlinear system, check the transfer equivalence of two systems, and finally, the feedforward and feedback compensators can be found such that the compensated systems transfer function coincides with that of the given system.

Realization of discrete-time nonlinear input–output equations: Polynomial approach

The aim of this paper is to solve reduction and realization problems for discrete-time multi-input multi-output nonlinear control systems by applying the theory of non-commutative polynomials. First, the necessary and sufficient reducibility condition is presented in terms of the greatest common left divisor of two polynomial matrices associated with the set of the higher order input–output (i/o) difference equations of the system. The condition also provides a method for system reduction, i.e. for finding the irreducible representation of the set of the i/o equations, being transfer equivalent to the original system representation. Second, to solve the realization problem, a formula is presented for computing the differentials of the state coordinates directly from the polynomial description of the nonlinear system. The polynomial approach addressed in this paper is more direct and requires noticeably less computations than earlier methods represented in terms of subspaces of differential one-forms.

Reducing the Multiplier-Complexity of Massively Parallel Polyphase 2D IIR Broadband Beam Filters

The superior broadband performance of 2D IIR frequency-planar beam filters, relative to conventional 2D FIR true-time-delay beamforming, has recently been reported using computational electromagnetics and real-time emulations on an antenna test range, resulting in significant improvements of bit-error-rates (BERs) in the presence of broadband interference. Further, massively parallel systolic VLSI circuit polyphase architectures have also been reported (Madanayake et al. in Int. J. Circuit Theory Appl. 2010) for the case of the direct-form signal flow graph (SFG) architecture, operating at a maximum throughput of M-(antenna)-frames-per-clock-cycle (MFPCC). The superior broadband performance of 2D IIR frequency-planar beam filters is extended here from the direct-form signal flow graph (SFG) architecture (Madanayake et al. in Int. J. Circuit Theory Appl. 2010) to the novel differential-form SFG architecture in order to reduce overall complexity. The proposed method employs a differential-form polyphase 2D IIR frequency-planar beam SFG, and a corresponding circuit architecture, to implement the required input-output 2D space-time difference equation. The resultant digital hardware has the significant advantage of much-reduced multiplier complexity, relative to the direct-form structure. For example, when look-ahead pipelining is not employed and for polyphase architectures having two, three, and four phases, the corresponding reductions in multiplier complexity are 20%, 28.6% and 33.3%, respectively. A proof-of-concept prototype circuit is designed and implemented on a Xilinx Sx35 FPGA device for the two-phase case, operating at a frame-rate of 132 million linear frames per second on the uniform linear array (ULA), corresponding to 2-frames-per-clock-cycle at a circuit clock frequency of 66 MHz. The circuit is optimized for low critical path delays (CPDs) using look-ahead pipelining of order three. For ultra-wideband (UWB) radio-frequency (RF) implementations, in such fields as radio astronomy, radar and wireless communications, custom VLSI versions of the proposed circuits are required.