Box-Jenkins identification revisited—Part III : Multivariable systems

Part I of this series of three papers handles the identification of single input single output Box-Jenkins models on arbitrary frequency grids in an open and closed loop setting. Part II discusses the computational aspects and illustrates the theory on simulations and a real life problem. This paper extends the results of Parts I and II to multiple input multiple output systems. Contrary to the classical time domain approach, the presented technique does not require symbolic calculus for multiple output polynomial Box-Jenkins models.     

Closed loop experiment design for linear time invariant dynamical systems via LMIs

All stationary experimental conditions corresponding to a discrete-time linear time-invariant causal internally stable closed loop with real rational system and feedback controller are characterized using the Youla-Kucera parametrization. Finite dimensional parametrizations of the input spectrum and the Youla-Kucera parameter allow a wide range of closed loop experiment design problems, based on the asymptotic (in the sample size) covariance matrix for the estimated parameters, to be recast as computationally tractable convex optimization problems such as semi-definite programs. In particular, for Box-Jenkins models, a finite dimensional parametrization is provided which is able to generate all possible asymptotic covariance matrices. As a special case, the very common situation of a fixed controller during the identification experiment can be handled and optimal reference signal spectra can be computed subject to closed loop signal constraints. Finally, a brief numerical comparison with closed loop experiment design based on a high model order variance expression is presented.     

Box-Jenkins identification revisited—Part II: Applications

Part I of this series of two papers handles the identification of discrete-time and continuous-time Box-Jenkins models on arbitrary frequency grids in an open and closed loop setting. Part II (i) discusses the practical calculation of the estimators developed in Part I, (ii) illustrates some theoretical results via simulation, and (iii) applies the modeling technique to a real life problem.     

Frequency domain maximum likelihood estimation of linear dynamic errors-in-variables models

This paper studies the linear dynamic errors-in-variables problem for filtered white noise excitations. First, a frequency domain Gaussian maximum likelihood (ML) estimator is constructed that can handle discrete-time as well as continuous-time models on (a) part(s) of the unit circle or imaginary axis. Next, the ML estimates are calculated via a computationally simple and numerically stable Gauss-Newton minimization scheme. Finally, the Cramér-Rao lower bound is derived.     

Direct computation of infimum in discrete-time H∞-optimization using measurement feedback

A direct and non-iterative method for the computation of the infimum for a class of discrete-time H_∞ optimal control problem is considered in this paper. The problem formulation is fairly general and does not place any restrictions on any direct feedthrough terms of the given systems. The method is applicable to systems where (i) the transfer function from the disturbance input to the measurement output is free of unit circle invariant zeros and left invertible, and (ii) the transfer function from the control input to the controlled output of the given system is free of the unit circle invariant zeros and right invertible.     

Frequency-domain identification of continuous-time ARMA models from sampled data

The subject of this paper is the direct identification of continuous-time autoregressive moving average (CARMA) models. The topic is viewed from the frequency domain perspective which then turns the reconstruction of the continuous-time power spectral density (CT-PSD) into a key issue. The first part of the paper therefore concerns the approximate estimation of the CT-PSD from uniformly sampled data under the assumption that the model has a certain relative degree. The approach has its point of origin in the frequency domain Whittle likelihood estimator. The discrete- or continuous-time spectral densities are estimated from equidistant samples of the output. For low sampling rates the discrete-time spectral density is modeled directly by its continuous-time spectral density using the Poisson summation formula. In the case of rapid sampling the continuous-time spectral density is estimated directly by modifying its discrete-time counterpart.     

Stabilization of discrete-time linear systems by delay independent truncated predictor feedback

For a discrete-time linear system with input delay, the predictor feedback law is the product of a feedback gain matrix with the predicted state at a future time instant ahead of the current time instant by the amount of the delay, which is the sum of the zero input solution and the zero state solution of the system. The zero state solution is a finite summation that involves past input, requiring considerable memory in the digital implementation of the predictor feedback law. The truncated predictor feedback, which results from discarding the finite summation part of the predictor feedback law, reduces implementation complexity. The delay independent truncated predictor feedback law further discards the delay dependent transition matrix in the truncated predictor feedback law and is thus robust to unknown delays. It is known that such a delay independent truncated predictor feedback law stabilizes a discrete-time linear system with all its poles at $z=1$ or inside the unit circle no matter how large the delay is. In this paper, we first construct an example to show that the delay independent truncated predictor feedback law cannot compensate too large a delay if the open loop system has poles on the unit circle at $z\neq 1$. Then, a delay bound is provided for the stabilizability of a general linear system by the delay independent truncated predictor feedback.     

Discrete time neural network synthesis using input and outputactivation functions

A new very fast algorithm for synthesis of a new structure of discrete-time neural networks (NN) is proposed. For this purpose the following concepts are employed: (i) combination of input and output activation functions, (ii) input time-varying signal distribution, (iii) time-discrete domain synthesis and (iv) one-step learning iteration approach. The problem of input-output mappings of time-varying vectors is solved. Simulation results based on the synthesis of a new structure of feedforward NN of an universal logical unit are presented. The proposed NN synthesis procedure is useful for applications to identification and control of nonlinear, very fast, dynamical systems. In this sense a feedforward NN for an adaptive nonlinear robot control is designed. Finally, a new algorithm for the direct inverse modeling of input/output nonquadratic systems is discussed.     

Time and frequency domain identification and analysis of a gas turbine engine

The nonlinear autoregressive moving average model with eXogenous inputs (NARMAX) approach is used to analyse the dynamics of a gas turbine engine. The fuel flow-shaft speed relationship is analysed by identifying both time and frequency domain models of the system. The frequency domain analysis is studied by mapping the discrete-time NARMAX models into the generalised frequency response functions (GFRFs) to reveal the nonlinear coupling between the various input spectral components and the energy transfer mechanisms in the system. A continuous-time nonlinear differential equation model is also estimated using the GFRFs.     

Block pulse function approach for the identification of Hammerstein model non-linear systems

Two recursive algorithms based on block pulse functions are presented for identifying continuous Hammerstein models of non-linear systems with (i) a state space model and (ii) an input–output model. Since the continuous non-linear systems are transformed approximately into the corresponding difference equations via block pulse functions, these recursive estimation algorithms can easily be obtained using a derivation similar to that of the discrete-time models expressed by difference equations. Both algorithms derived here are simple and straightforward, and can easily be implemented on-line. As discussed in this paper, these algorithms can also be extended to the identification of certain continuous non-linear systems with a feedback loop or with time delays. The illustrative examples show that these recursive algorithms give satisfactory results for the identification problems of certain continuous non-linear systems.     

Variance-error quantification for identified poles and zeros

This paper deals with quantification of noise induced errors in identified discrete-time models of causal linear time-invariant systems, where the model error is described by the asymptotic (in data length) variance of the estimated poles and zeros. The main conclusion is that there is a fundamental difference in the accuracy of the estimates depending on whether the zeros and poles lie inside or outside the unit circle. As the model order goes to infinity, the asymptotic variance approaches a finite limit for estimates of zeros and poles having magnitude larger than one, but for zeros and poles strictly inside the unit circle the asymptotic variance grows exponentially with the model order. We analyze how the variance of poles and zeros is affected by model order, model structure and input excitation. We treat general black-box model structures including ARMAX and Box–Jenkins models.     

State space realization of bilinear continuous-time input–output equations

This paper studies the realizability property of continuous-time bilinear input–output (i/o) equations in the classical state space form. Constraints on the parameters of the bilinear i/o model are suggested that lead to realizable models. The paper proves that the 2nd order bilinear i/o differential equation, unlike the discrete-time case, is always realizable in the classical state space form. The complete list of 3rd and 4th order realizable i/o bilinear models is given and two subclasses of realizable i/o bilinear systems are suggested. Our conditions rely basically upon the property that certain combinations of coefficients of the i/o equations are zero or not zero. We provide explicit state equations for all realizable 2nd and 3rd order bilinear i/o equations, and for one realizable subclass of bilinear i/o equations of arbitrary order.     

Statistical modelling of communities and ecosystems using the LAMDA software tool

Understanding species interactions is critical to discovering community dynamics. Recently, statistical methods for estimating species interaction strengths from time series data have been developed based on multivariate auto-regressive first-order, or MAR(1), models. However, the complex coding required presents a substantial barrier for most ecologists. We have developed LAMBDA, a software program that allows users to easily fit MAR(1) models to multi-species time series data. The LAMBDA package covers: data input and transformation, selection of the interactions to include via a search algorithm and model selection, estimation of interaction parameters via conditional least squares (CLS) regression or two different maximum-likelihood (ML) algorithms, estimation of confidence intervals via bootstrapping, and computation of community stability properties using the estimated model. We describe performance tests on the variability of estimates, computation speed, and CLS versus ML estimation using simulated data.     

Linear quadratic regulation for linear time-varying systems with multiple input delays

This paper studies the classic linear quadratic regulation (LQR) problem for both continuous-time and discrete-time systems with multiple input delays. For discrete-time systems, the LQR problem for systems with single input delay has been studied in existing literature, whereas a solution to the multiple input delay case is not known to our knowledge. For continuous-time systems with multiple input delays, the LQR problem has been tackled via an infinite dimensional system theory approach and a frequency/time domain approach. The objective of the present paper is to give an explicit solution to the LQR problem via a simple and intuitive approach. The main contributions of the paper include a fundamental result of duality between the LQR problem for systems with multiple input delays and a smoothing problem for an associated backward stochastic system. The duality allows us to obtain a solution to the LQR problem via standard projection in linear space. The LQR controller is simply constructed by the solution of one backward Riccati difference (for the discrete-time case) or differential (for the continuous-time case) equation of the same order as the plant (ignoring the delays).     

A new approach to digital PID controller design

In this note, we present a new approach to the problem of designing a digital proportional-integral-derivative (PID) controller for a given but arbitrary linear time invariant plant. By using the Tchebyshev representation of a discrete-time transfer function and some new results on root counting with respect to the unit circle, we show how the digital PID stabilizing gains can be determined by solving sets of linear inequalities in two unknowns for a fixed value of the third parameter. By sweeping or gridding over this parameter, the entire set of stabilizing gains can be recovered. The precise admissible range of this parameter can be predetermined. This solution is attractive because it answers the question of whether there exists a stabilizing solution or not and in case stabilization is possible the entire set of gains is determined constructively. Using this characterization of the stabilizing set we present solutions to two design problems: 1) maximally deadbeat design, where we determine for the given plant, the smallest circle within the unit circle wherein the closed loop system characteristic roots may be placed by PID control and 2) maximal delay tolerance, where we determine, for the given plant the maximal-loop delay that can be tolerated under PID control. In each case, the set of controllers attaining the specifications is calculated. Illustrative examples are included.     

Minimal realisation of bilinear and quadratic input–output difference equations in state-space form

This article studies the realisability property of discrete-time bilinear and quadratic input–output (i/o) equations in the classical state-space form. Constraints on the parameters of the i/o model are suggested that lead to realisable models. Using new formulae for computing basis vectors of certain vector spaces of differential one-forms, we present in this article the complete list of the third- and fourth-order realisable i/o bilinear models, and a new realisable subclass of an arbitrary order is suggested. Moreover, we provide the sufficient conditions of the second- and third-order realisable i/o quadratic models, respectively. All the developed theory and algorithms are illustrated by means of several examples.     

Identification of linear systems using polynomial kernels in the frequency domain

In prior work we presented an identification algorithm using polynomials in the time domain. In this article, we extend this algorithm to include polynomials in the frequency domain. A polynomial is used to represent the imaginary part of the Fourier transform of the impulse response. The Hilbert transform relationship is used to compute the real part of the frequency response and hence the complete process model. The polynomial parameters are computed based on the computationally efficient linear least square method. The order of the polynomial is estimated based on residue decrement. Simulated and experimental results show the effectiveness of this method, particularly for short input/output data sequence with high signal to noise ratio. The frequency domain polynomial model complements the time domain methods since it can provide a good estimate of the time to steady state for time domain FIR (finite impulse response) models. Confidence limits in time or frequency domain can be computed using this approach. Noise rejection properties of the algorithm are illustrated using data from both simulated and real processes.     

Semiglobal stabilization of linear discrete-time systems subject toinput saturation, via linear feedback-an ARE-based approach

We revisit the problem of semiglobal stabilization of linear discrete-time systems subject to input saturation and give an algebraic Riccati equation (ARE)-based approach to the proof of a fact established earlier (Lin and Saberi, 1995), i.e. a linear discrete-time system subject to input saturation is semiglobally stabilizable via linear feedback as long as the linear system in the absence of the saturation is stabilizable and detectable and all its open-loop poles are located inside or on the unit circle. Moreover, we drastically relax the requirements on the characteristic of the saturation elements as imposed in the earlier work     

General optimal attenuation of harmonic disturbance with unknown frequencies

We present algorithms for optimal harmonic disturbance attenuation in standard discrete-time control structure, based on a parametrisation of (marginally) stabilising controllers. The Frobenius norm and the spectral norm of the closed-loop transfer matrix at the disturbance frequencies are minimised. If there is only one frequency of the disturbance, the controller has an observer–based form, which we obtain by solving a static output feedback (SOF) stabilisation control problem. Although the SOF stabilisation problem is hard, the generical case of nonsquare matrix G ₂₂ is solved by linear algebra methods. Numerical simulation results are presented. As a corollary, we transform the control problem with unit circle invariant zeros into a ?_∞ control problem without such zeros. The elimination of the unit circle invariant zeros is based on the fact that matrix Y(zI???A?+?BF)^?1 is stable, where (Y,?F) with Y?≥?0 is a solution of a discrete-time algebraic Riccati system.     

Distributed predictive control: A non-cooperative algorithm with neighbor-to-neighbor communication for linear systems

This paper presents a novel Distributed Predictive Control (DPC) algorithm for linear discrete-time systems. This method enjoys the following properties: (i) state and input constraints can be considered; (ii) under mild assumptions, convergence of the closed loop control system is proved; (iii) it is not necessary for each subsystem to know the dynamical models of the other subsystems; (iv) the transmission of information is limited, in that each subsystem only needs the reference trajectories of the state variables of its neighbors. A simulation example is reported to illustrate the main characteristics and performance of the algorithm.