Matrix Fraction Description Research Articles

An experimentally validated model for a multivariate drying industrial process

Modeling of industrial production processes is becoming increasingly challenging due to their large number of variables. These variables often are highly correlated and introduce nonlinear and complex features. In this paper, we are interested to model an industrial dicalcium phosphate (DCP) drying process. Attempting to solve this issue is motivated by the need to improve several production conditions, such as minimizing the consumption of natural gas and reducing the pollution rate. In fact, applying an advanced control approach requires a dynamic model for the monitored plant. This work proposes a multivariable mathematical model for the DCP dryer within the Tunisian Chemical Group factory. A steady-state model has been reproduced using Aspen Plus software tool to implement the different functionalities of the system as well as involved reactions. Indeed, since the main operation in the drying process is the combustion reaction of the liquified petroleum gas (LPG) in the furnace that produce the necessary heat to reach a target value of temperature at the dryer outlet, we focus on determining a dynamic model for the furnace. To do so, we have proposed two approaches. The first is based on the Aspen dynamic tool. The second is based on the left matrix fraction description (LMFD) identification approach. The obtained results have been successfully validated using real measurements.

Wafer Stage Modal Model Identification by Modified Vector Fitting Method

Vector fitting is modified in this study for the parametric identification of a model with an undamped rigid body mode in the frequency domain. The modal model of a six-degree-of-freedom (6DOF) wafer stage is identified using the modified vector fitting method. The modal model is typically fitted from the matrix fraction description (MFD) form of the transfer function, which increases the cost function. The vector fitting in frequency domain provides an approach to fitting the modal model directly from frequency response functions(FRFs) via a partial rational basis function, the poles of which can be obtained by pole relocation technology. The traditional vector fitting method is not applicable in identifying systems containing rigid body modes. The method is reformulated here and effectively applied for wafer stage identification. The accuracy of the fitted model is validated by control loop simulation.

Study of modal parameter estimation of time-varying mechanical system in time-frequency domain based on output-only method

With increasing mobile instrument inspection systems like mobile inspection cameras applied in modern industrial automation lines, their time-varying modal parameters need to be thoroughly studied since they are operating under various random vibration excitations which are mostly unpredictable and unrepeatable. This paper investigates modal parameter identification of time-varying mobile mechanical systems through a poly-reference expansion based on output-only method in time-frequency domain. The poly-reference parametric model is presented through a time-frequency-domain right matrix fraction description. Based on the poly-reference parametric model, a poly-reference least square estimator and a poly-reference maximum likelihood estimator of modal parameters of time-varying systems are developed. For the case of only one batch of measurements, a novel approach to estimate the covariance matrix for the poly-reference maximum likelihood estimator by non-repeatable measurement is also proposed in this paper. A numerical example demonstrates the advantageous performance of the proposed estimators, such as the estimation of the close-coupled modes. Finally, a laboratory experiment validates these proposed estimators via a comparative way.

Model specification and selection for multivariate time series

Three major difficulties are identified with an established echelon form approach (see Hannan (1987)) to specifying a Vector Autoregressive Moving Average, VARMA, model for an observed time series. A family of state space representations, valid for each integer, h, is introduced, and collectively referred to as multistep state space representations. This family includes as its special case, with h=0, a state space representation introduced earlier by Akaike (1974), and, with h=1, that introduced by Cooper and Wood (1982). Appropriate generalizations of the notions of minimality, McMillan degree, left matrix fraction description and Kronecker indices, as applicable individually to each member of this family, are presented. The reverse echelon form and state space representation corresponding to the Kronecker indices for each h are derived, and the former illustrated with three examples of standard VARMA processes. The question of how the presence of zero constraints on the coefficients of a reverse echelon form may be detected solely from an inspection of the Kronecker indices is examined. A canonical correlation procedure proposed originally by Akaike (1976) for h=0 is considered for estimating the Kronecker indices with each h. The efficacy of the estimation procedure is investigated by a simulation study. A procedure is suggested for implementing the new approach introduced in this paper with an observed time series, and three different applications of this approach are outlined. This approach is also related to some of its alternatives, including the Kronecker invariants of Poskitt (1992) and the scalar component approach of Tiao and Tsay (1989).

A New control structure for tert-amyl methyl ether production using reactive distillation

The aim of this paper is to introduce new control structures in both centralized and decentralized scheme for tert-amyl methyl ether (TAME) production unit using reactive distillation which developed as a benchmark by Luyben (2005). Steady state and dynamic simulation was performed based on the case study data using Aspen® simulator. Dynamic simulation shows that control performance of the plant has oscillatory behavior using conventional PI controllers. Hence, a proper modification on the process flow sheet was carried out in order to remove oscillatory behavior of the process. Although, the new structure has better dynamic performance than the previous one using PI controller, it is not capable of eliminating the oscillations completely. Non-minimal state space with proportional integral plus controller (NMSS–PIP) controller was applied to dynamic model of the modified process. In this way, the system identification was carried out using refined instrumental variable (RIV) method and state matrices was obtained as left matrix fraction description (LMFD). The control performance of the PI and PIP controller for the modified structure revealed that the NMSS–PIP, completely eliminates the process oscillations while it has smoother responses with better settling time than the PI configuration. Furthermore, comparison of the ITAE criterion between main case and modified process indicates that the ITAE for PI and PIP control decreased by about 20% and 88%, respectively.

Robust block roots assignment in linear discrete-time sliding mode control for a class of multivariable system: gas turbine power plant application

This paper deals with the development of a new extended robust control algorithm based on the assignment of robust block roots to extended linear discrete-time sliding mode control; matrix fraction description (MFD) theory is used to improve the performance of the proposed control algorithm and make it more robust. To validate the proposed control algorithm, an application on a gas turbine system (GE MS5001P), which is used in electrical power generation plants, is presented in this paper. An autoregressive model with exogenous excitation is obtained for the studied system, based on the left MFD multivariable least-squares estimator, using real data obtained on-site in order to obtain a minimum-order model with optimum validation criteria. The main objective of the implementation of the proposed control algorithm within the presented application is to ensure stability in the dynamic behaviour of the gas turbine system. Furthermore, a comparison with some classical and modern controllers has been made to prove the effectiveness and robustness of the proposed control algorithm.

Robust Kalman filter‐based least squares identification of a multivariable system

A novel direct identification using the residual model of Kalman filter (KF) is proposed for multiple-input and multiple-output Box–Jenkins model of the system formed of the signal and disturbance models using the residual model relating the input and output of the system without any a priori knowledge of the statistics of the disturbance and measurement noise corrupting the output. To avoid a non-linear optimisation, the auto-regressive and moving average (MA) residual model is approximated by a high-order MA model, so that the unknown parameters of the KF enter the residual model linearly. A key property of the KF is established, namely that the transfer matrix of the signal model is the matrix fraction description (MFD) model relating the residual and the system input and output. A two-stage identification method is developed here. In stage 1, a high-order KF of the system is identified using the robust, and computationally efficient least-squares method to capture completely both the signal and disturbance models. In stage 2, the KF for the signal is derived using the balanced model reduction technique. The signal model is derived using the key MFD property. The performance of the proposed scheme is successfully evaluated on both simulated and physical systems.

Unimodular equivalence and similarity for linear systems

ABSTRACTThe problem of finding the mapping between unimoduar transformations relating two minimal matrix fraction descriptions (MFDs) of a transfer function, and the similarity transformations relating the respective minimal state-space representations is considered. It is shown that the problem is equivalent to finding the relation of MFDs of the input-state transfer functions of the two systems. This relation turns out to be an equivalence relation involving the unimodular and the similarity matrices relating the MFDs and the state-space systems, respectively. A canonical form for MFDs under this equivalence relation is obtained and it is shown that it leads to a canonical state-space representation, via a realisation procedure.

On λ-matrices and their applications in MIMO control systems design

In the present paper, we have introduced a new control design algorithms based on the theory of matrix polynomials. The first procedure is called block decoupling control which is based on the spectral factors of the denominator of the right matrix fraction description (RMFD), the advantages of this control are the non-interacted behaviour, simplicity in control design and low order controller is obtained due to the cancellation property of the proposed algorithm. The second control algorithm is the whole set of latent-structure assignment via the approaches of block root placement. Of course, the procedure has been developed even if the system is not block transformable, a process done with the aid of conversion between state space and matrix fraction description. The last method defined as a MIMO PID controller design via the placement of block roots with the help of Diophantine equation resolution, the latter systematic procedure retains both regulation and tracking objectives with small gains and minimum error.

Unimodular Transformations and Canonical Forms for Singular Systems

The relationship between the unimodular matrices relating coprime and column reduced matrix fraction descriptions (MFD) of a nonproper transfer function, and the restricted system equivalence (r.s.e.) transformations relating the corresponding generalised state space realisations is considered. It is shown that the r.s.e and unimodular transformations can be directly obtained from each other by inspection. The r.s.e. transformations leading to the canonical form are derived from the unimodular transformations leading to the echelon canonical form of the composite matrix of the MFD of the system.

On lambda-matrices and their applications in MIMO control systems design

In the present paper, we have introduced a new control design algorithms based on the theory of matrix polynomials. The first procedure is called block decoupling control which is based on the spectral factors of the denominator of the right matrix fraction description (RMFD), the advantages of this control are the non-interacted behaviour, simplicity in control design and low order controller is obtained due to the cancellation property of the proposed algorithm. The second control algorithm is the whole set of latent-structure assignment via the approaches of block root placement. Of course, the procedure has been developed even if the system is not block transformable, a process done with the aid of conversion between state space and matrix fraction description. The last method defined as a MIMO PID controller design via the placement of block roots with the help of Diophantine equation resolution, the latter systematic procedure retains both regulation and tracking objectives with small gains and minimum error.

New developments for matrix fraction descriptions: A fully-parametrised approach

This article aims at giving a new answer for the challenging problem of the parametrisation of multi-input multi-output matrix fraction descriptions. In order to reach this goal, new parametrisations of matrix fraction descriptions, called fully-parametrised left matrix fraction descriptions (F-LMFD), are first introduced. Their structural properties as well as their suitability for multi-input multi-output model description are more precisely analysed. As any over-parametrised model description, the F-LMFD cannot describe a transfer function uniquely. The structure of the space of equivalent F-LMFD is then investigated through the determination of its basis. The study carried out in this article is the prelude to a computational improvement of the identification of matrix fraction descriptions with gradient-based optimisation methods.

Characterizations of rectangular (para)-unitary rational functions

We here present three characterizations of not necessarily causal, rational functions which are (co)-isometric on the unit circle: (i) through the realization matrix of Schur stable systems, (ii) the Blaschke-Potapov product, which is then employed to introduce an easy-to-use description of all these functions with dimensions and McMillan degree as parameters, (iii) through the (not necessarily reducible) Matrix Fraction Description (MFD). In cases (ii) and (iii) the poles of the rational functions involved may be anywhere in the complex plane, but the unit circle (including both zero and infinity). A special attention is devoted to exploring the gap between the square and rectangular cases.

Development and Analysis of Linear Model Representations of the Quad-Rotor System

A Quad-rotor, which is an airplane with four rotors, is considered a complex and challenging control problem. Analysis and design of its control systems is usually started by careful consideration of its mathematical models description. This paper presents a detailed mathematical model for a Vertical Takeoff and Landing (VTOL) type Unmanned Aerial Vehicle (UAV) known as the quad rotor. The presented paper focus on the nonlinear modeling of an experimental type quad-rotor. The first objective of this paper aims to develop a nonlinear dynamical model based on Newton-Euler approach. Then, a linear model is obtained around an optimized hovering point. The state space, transfer function matrix and matrix fraction description models are derived. Finally, a linear controller is designed to achieve a stable compensated system around a hovering point. The methodology applied in this paper takes the ST-450 experimental quad-rotor as a case study.

New developments for experimental modal analysis of aircraft structures

This article presents an identification algorithm dedicated to the modal analysis of aircraft structures during flight-tests. More specifically, this algorithm was designed to process short duration tests carried out with multi-input excitations. The identification problem is solved in the frequency domain and the limit effects are considered so as to avoid transient effects with short data sequences. To minimise the effects of the noise, a non-linear gradient-based optimisation method is used. Its performance is improved by the use of an appropriate over-parametrised matrix fraction descriptions. Because the cost function to be minimised is non-convex, this method is however sensitive to the initialisation. For this reason, an iterative instrumental variable method is used to find an initial estimate. This one gives a value of the cost-function sufficiently close to its global minimum so as to ensure a fast convergence of the optimisation. Thus, the algorithm presented in this article is a combination of two iterative methods that gives accurate mode estimations even with high level of noise, as shown on an illustrative example.

Frequency-domain identification of aircraft structural modes from short-duration flight tests

This article presents identification algorithms dedicated to the modal analysis of civil aircraft structures during in-flight flutter tests. This particular operational framework implies several specifications for the identification procedure. To comply with these requirements, the identification problem is formulated in the frequency domain as an output-error problem. Iterative identification methods based on structured matrix fraction descriptions are used to solve this problem and to identify a continuous-time model. These iterative methods are specifically designed to deal with experiments where short-duration tests with multiple-input excitations are used. These algorithms are first discussed and then evaluated through a simulation example illustrative of the in-flight modal analysis of a civil aircraft. Based on these evaluation results, an efficient iterative algorithm is suggested and applied to real flight-test data measured on board a military aircraft.

Characterization and Identification of Matrix Fraction Descriptions for LTI Systems

This paper shows that the matrix fraction description, given by a pair $\{A(z)~B(z)\}$ of matrix polynomials of $z$, for a linear time-invariant system may not be unique even if $A(z)$ is monic, $A(z)$ and $B(z)$ have no common left factor, and the matrix coefficients corresponding to the highest-order terms of $A(z)$ and $B(z)$ are full row rank. The orders of all possible matrix fraction descriptions (MFDs) of a given system are completely characterized. Testing criteria for determining whether a matrix pair is an MFD of the system are derived, which involve rank tests of certain Toeplitz matrices derived from either the impulse response or output correlation functions of the system. A decision procedure is devised that generates sequentially all MFDs for a given system. Identification algorithms are introduced that estimate all MFDs of a given system from its input-output data or output data only. The results are then extended to cover ARMAX systems.

Trajectory planning and tracking control for multiple-input multiple-output, linear time-invariant, sampled data systems in continuous time

In applications, usually sampled data controllers are employed. If the sampling time is sufficiently small, the sampled data structure may be neglected and the system is treated quasi-continuously. If the sampling time is longer, the system is treated in discrete time neglecting the behaviour in between the sample instances and the z-transform is used. This article combines these two approaches. The plant is driven by an input which consists of a sequence of values and a function forming the actuating variable. By use of the modified z-transform, the plant is modelled by a parametric transfer function matrix, whose additional parameter discloses the behaviour between sampling instances. Thus, the output signal is calculated not only at the points of sampling. A right co-prime matrix-fraction description is derived. Building on that description, basis variables are defined in the z-domain. The corresponding basis sequences can be chosen arbitrarily and with them the input sequences and the output functions are fixed and can be calculated without solving a differential or difference equation. This mathematical fact is applied to plan trajectories in continuous time. Hence, the entire output trajectory in continuous time can be taken into account. A tracking controller may be added to ensure that the disturbed system complies with the plan.

Unknown inputs functional observers designs for delay descriptor systems

New time and frequency domain designs of unknown inputs functional observers (UIFO) for a class of descriptor systems with a constant time delay are investigated in this paper. The unknown inputs are present in both the state and the measurement equations. The time procedure design is based on the unbiasedness of the estimation error and the Lyapunov Krasovskii stability theory. After given the existence condition of such observers, the gain implemented in the functional observer with internal delay design is obtained in terms of linear matrix inequalities. The particular cases, where the observer is independent of internal delay and where it is independent of delay are also investigated. A design algorithm that summarises the different steps of the UIFO design is proposed. Then, the frequency procedure design is derived from time domain results by applying the factorisation approach, where we define some useful matrix fraction descriptions. Note that the order of this unknown input observers is equal to the dimension of the vector to be estimated. A numerical example is given to illustrate the effectiveness of the presented approach.

A revisit to inverse optimality of linear systems

In this article, we revisit the problem of inverse optimality for linear systems. By applying certain explicit formulae for coprime matrix fraction descriptions (CMFD) of linear systems, we propose a necessary and sufficient condition for a given stable state feedback law to be optimal for some quadratic performance index. Compared to existing results in the literature, the proposed condition is simpler to check and interpret. Moreover, it reduces the redundancy in the solutions of the associated algebraic Riccati equation (ARE). As a direct application of the proposed results, we consider the problem of inverse optimality of observer-based state feedback. To be specific, for the case where the state is not fully known, we consider the inverse optimality problem of an observer-based state feedback for the closed-loop system augmented by an observer. More precisely, it is shown that the observer-based state feedback is inverse optimal for the closed-loop system with some general forms of cost functions, only if the original state feedback is inverse optimal for the original system with certain cost functions, irrespective of the choice of the observer. This coincides with existing results in the literature. Some other applications of the proposed results are also discussed. We also illustrate the proposed results through an example.