Identification Of State-space Models Research Articles

A Subspace Approach to the Structural Decomposition and Identification of Ankle Joint Dynamic Stiffness.

The purpose of this paper is to present a structural decomposition subspace (SDSS) method for decomposition of the joint torque to intrinsic, reflexive, and voluntary torques and identification of joint dynamic stiffness. First, it formulates a novel state-space representation for the joint dynamic stiffness modeled by a parallel-cascade structure with a concise parameter set that provides a direct link between the state-space representation matrices and the parallel-cascade parameters. Second, it presents a subspace method for the identification of the new state-space model that involves two steps: 1) the decomposition of the intrinsic and reflex pathways and 2) the identification of an impulse response model of the intrinsic pathway and a Hammerstein model of the reflex pathway. Extensive simulation studies demonstrate that SDSS has significant performance advantages over some other methods. Thus, SDSS was more robust under high noise conditions, converging where others failed; it was more accurate, giving estimates with lower bias and random errors. The method also worked well in practice and yielded high-quality estimates of intrinsic and reflex stiffnesses when applied to experimental data at three muscle activation levels. The simulation and experimental results demonstrate that SDSS accurately decomposes the intrinsic and reflex torques and provides accurate estimates of physiologically meaningful parameters. SDSS will be a valuable tool for studying joint stiffness under functionally important conditions. It has important clinical implications for the diagnosis, assessment, objective quantification, and monitoring of neuromuscular diseases that change the muscle tone.

N2SID: Nuclear norm subspace identification of innovation models

The identification of multivariable state space models in innovation form is solved in a subspace identification framework using convex nuclear norm optimization. The convex optimization approach allows to include constraints on the unknown matrices in the data-equation characterizing subspace identification methods, such as the lower triangular block-Toeplitz of weighting matrices constructed from the Markov parameters of the unknown observer. The classical use of instrumental variables to remove the influence of the innovation term on the data equation in subspace identification is avoided. The avoidance of the instrumental variable projection step has the potential to improve the accuracy of the estimated model predictions, especially for short data length sequences.

OKID via Output Residuals: A Converter from Stochastic to Deterministic System Identification

This paper presents a new approach for the identification of linear state-space models from input–output measurements in the presence of noise. In contrast to the original OKID/ERA algorithm, which works through the observer Markov parameters, the new approach uses the estimated observer output residuals to convert a stochastic identification problem into a virtually deterministic one. A system state-space model and an associated Kalman-filter gain can be then identified. Because the converted problem is virtually deterministic, any deterministic system-identification method can be used. The proposed stochastic-to-deterministic conversion enables these deterministic algorithms to produce the same performance level as the existing stochastic system-identification algorithms.

Blind multivariable ARMA subspace identification

In this paper, we study the deterministic blind identification of multiple channel state-space models having a common unknown input using measured output signals that are perturbed by additive white noise sequences. Different from traditional blind identification problems, the considered system is an autoregressive system rather than an FIR system; hence, the concerned identification problem is more challenging but possibly having a wider scope of application. Two blind identification methods are presented for multi-channel autoregressive systems. A cross-relation identification method is developed by exploiting the mutual references among different channels. It requires at least three channel systems with square and stably invertible transfer matrices. Moreover, a general subspace identification method is developed for which two channel systems are sufficient for the blind identification; however, it requires the additive noises to have identical variances and the transfer matrices having no transmission zeros. Finally, numerical simulations are carried out to demonstrate the performance of the proposed identification algorithms.

Gain Scheduling Dual Mode MPC for a Solar Thermal Power Plant

A nonlinear gain scheduling control strategy is proposed for a concentrated solar thermal power plant. The strategy involves the identification of local linear time-invariant state space models around a family of operating points, the design of corresponding local linear dual mode model-based predictive controllers and the selection of an appropriate scheduling variable to govern the switching. The local models are estimated directly from input-output data using a subspace identification method while taking into account the frequency response of the plant. Input-output data are obtained from a nonlinear simulation model of the plant rather than the plant itself. The effectiveness of the proposed control strategy in terms of tracking and disturbance rejection is evaluated through two different scenarios created in a nonlinear simulation environment.

Dealing with Transients due to Multiple Experiments in Nonlinear System Identification

Although in many static estimation problems the data are collected from repeated experiments, the default underlying (assumption) setting in most of the system identification tasks is that the data are generated from a single experiment. The rationale for this is that more data can be obtained by increasing the measurement time instead of doing more experiments. As the measurement time tends to infinity, under suitable assumptions it is possible to estimate consistently the model parameters. In most of the real life cases, however, increasing the measurement time is either not possible or it does not cover the whole operating range of the system to be identified, hence data from multiple experiments need to be combined. Furthermore, most of the real life systems are nonlinear and a large variety of nonlinear systems can be described by a nonlinear state space model structure. In this paper, a methodology to deal with the transients arising due to concatenating data from multiple experiments during the identification of Polynomial nonlinear state space (PNLSS) models is described. The methodology is validated on data generated from a laboratory experimental set-up.

IDENTIFICATION OF ON-AND OFF-LINE LINEAR STATE SPACE MODELS USING SUBSPACE METHODS

In this paper, subspace identification methods are proposed to analyze the differences between On-And Off-Line Linear State Space Models Using Subspace Methods. There are several ways that can estimate the order of the system. For this paper, Singular Value Decomposition (SVD) is used to estimate the order of the system. Comparing with the others methods, this method only need a limited number of input and output data for the determination of the system matrices. Two methods of the subspace algorithm are used which is N4SID (Numerical algorithm for Subspace State Space System Identification) and MOESP (Multivariable Output-Error State-Space model identification).

Optimal bilinear observers for bilinear state-space models by interaction matrices

This paper formulates optimal bilinear observers for bilinear state-space models. Observers in bilinear form, as opposed to other nonlinear forms, are required to develop an extension of observer/Kalman filter identification for simultaneous identification of a bilinear state-space model and an associated bilinear observer from noisy input–output measurements. The paper establishes the relationship between the bilinear observer gains and the interaction matrices which are used to convert the original bilinear state-space model to a form that simplifies the identification of such a model. Techniques to find the interaction matrices are developed. In the absence of noises, these matrices produce the gains of the fastest converging observer. In the presence of noises, they minimise the state estimation error in the same manner as a standard steady-state Kalman filter. Numerical examples illustrate both the theoretical and computational aspects of the proposed algorithms.

Chaotic Artificial Bee Colony Algorithm for System Identification of a Small-Scale Unmanned Helicopter

The purpose of this paper is devoted to developing a chaotic artificial bee colony algorithm (CABC) for the system identification of a small-scale unmanned helicopter state-space model in hover condition. In order to avoid the premature of traditional artificial bee colony algorithm (ABC), which is stuck in local optimum and can not reach the global optimum, a novel chaotic operator with the characteristics of ergodicity and irregularity was introduced to enhance its performance. With input-output data collected from actual flight experiments, the identification results showed the superiority of CABC over the ABC and the genetic algorithm (GA). Simulations are presented to demonstrate the effectiveness of our proposed algorithm and the accuracy of the identified helicopter model.

Hybrid evolutionary identification of output-error state-space models

A hybrid optimization method for the identification of state-space models is presented in this study. Hybridization is succeeded by combining the advantages of deterministic and stochastic algorithms in a superior scheme that promises faster convergence rate and reliability in the search for the global optimum. The proposed hybrid algorithm is developed by replacing the original stochastic mutation operator of Evolution Strategies (ES) by the Levenberg-Marquardt (LM) quasi-Newton algorithm. This substitution results in a scheme where the entire population cloud is involved in the search for the global optimum, while single individuals are involved in the local search, undertaken by the LM method. The novel hybrid identification framework is assessed through the Monte Carlo analysis of a simulated system and an experimental case study on a shear frame structure. Comparisons to subspace identification, as well as to conventional, self-adaptive ES provide significant indication of superior performance.

Identification of passive state-space models of strongly frequency dependent wave radiation forces

This paper describes a methodology for obtaining passive state space representations of wave radiation forces for floating bodies with zero forward speed oscillating in multiple degrees of freedom. The method is based on fitting rational transfer functions written on pole-residue form to radiation frequency responses calculated using standard boundary element codes. Determination of poles and residues is facilitated by the vector fitting algorithm, originally developed for fitting of frequency responses of electrical networks. Using the poles and residues as parameters in the fitting has advantages over the more common ratio-of-polynomials transfer function formulation when the model order is high, which is typically required when the frequency dependence is strong and wide-banded such as for multibody floating systems. A methodology for perturbing the residues of slightly non-passive systems such that they become passive is also presented. The method is demonstrated successfully for a five-body wave energy converter, an array of circular cylinders and a single cylinder. A discussion and investigation of the truncation of high frequencies is provided for the wave energy converter, and parameter constraints governing the extrapolation of the rational model above the frequencies present in the dataset are suggested.

Recursive State-space Model Identification of Non-uniformly Sampled Systems Using Singular Value Decomposition

In this paper a recursive state-space model identification method is proposed for non-uniformly sampled systems in industrial applications. Two cases for measuring all states and only output(s) of such a system are considered for identification. In the case of state measurement, an identification algorithm based on the singular value decomposition (SVD) is developed to estimate the model parameter matrices by using the least-squares fitting. In the case of output measurement only, another identification algorithm is given by combining the SVD approach with a hierarchical identification strategy. An example is used to demonstrate the effectiveness of the proposed identification method.

State-Space Rotor Aeroelastic Modeling for Real-Time Helicopter Flight Simulation

This paper introduces a new approach for the identification of linear state-space models of dynamical systems of arbitrary complexity. The identification procedure is described and applied for modeling aeroelastic response of helicopter main rotors. With the aim of developing a tool that might be conveniently applied for real-time simulations of helicopter flight dynamics, the state-space model considered is a reduced-order description of loads transmitted to the airframe due to hub motion and blade pitch controls. In order to validate the proposed approach, loads from the state-space, reduced-order model are compared with those predicted by the complete full-state, nonlinear rotor model for prescribed helicopter maneuvers.

Hybrid DE-PEM algorithm for identification of UAV helicopter

Purpose – The purpose of this paper is to develop a hybrid algorithm using differential evolution (DE) and prediction error modeling (PEM) for identification of small-scale autonomous helicopter state-space model. Design/methodology/approach – In this study, flight data were collected and analyzed; MATLAB-based system identification algorithm was developed using DE and PEM; parameterized state-space model parameters were estimated using the developed algorithm and model dynamic analysis. Findings – The proposed hybrid algorithm improves the performance of the PEM algorithm in the identification of an autonomous helicopter model. It gives better results when compared with conventional PEM algorithm inside MATLAB toolboxes. Research limitations/implications – This study is applicable to only linearized state-space model. Practical implications – The identification algorithm is expected to facilitate the required model development for model-based control design for autonomous helicopter development. Original...

State-space model identification and feedback control of unsteady aerodynamic forces

Unsteady aerodynamic models are necessary to accurately simulate forces and develop feedback controllers for wings in agile motion; however, these models are often high dimensional or incompatible with modern control techniques. Recently, reduced-order unsteady aerodynamic models have been developed for a pitching and plunging airfoil by linearizing the discretized Navier–Stokes equation with lift-force output. In this work, we extend these reduced-order models to include multiple inputs (pitch, plunge, and surge) and explicit parameterization by the pitch-axis location, inspired by Theodorsen׳s model. Next, we investigate the naïve application of system identification techniques to input–output data and the resulting pitfalls, such as unstable or inaccurate models. Finally, robust feedback controllers are constructed based on these low-dimensional state-space models for simulations of a rigid flat plate at Reynolds number 100. Various controllers are implemented for models linearized at base angles of attack α0=0°,α0=10°, and α0=20°. The resulting control laws are able to track an aggressive reference lift trajectory while attenuating sensor noise and compensating for strong nonlinearities.

Learning and Understanding System Stability Using Illustrative Dynamic Texture Examples

System stability is a basic concept in courses on dynamic system analysis and control for undergraduate students with computer science backgrounds. Typically, this was taught using a simple simulation example of an inverted pendulum. Unfortunately, many difficult issues arise in the learning and understanding of the concepts of stability, instability, and marginal stability. Instead, the authors use an example from dynamic texture, a topic with which undergraduate students with computer science backgrounds are both familiar and comfortable. This example provides the opportunity for students with no access to mechanical or electronic equipment to learn system stability. In addition, the designed project also allows students to improve their knowledge in several fields, such as discrete-time dynamic system analysis, parameter identification of state-space models, and MATLAB programming. The assessment by students reveals that this project is interesting and useful for their studies.

N4SID and MOESP Algorithms to Highlight the Ill-conditioning into Subspace Identification

In this paper, an analysis for ill conditioning problem in subspace identification method is provided. The subspace identification technique presents a satisfactory robustness in the parameter estimation of process model which performs control. As a first step, the main geometric and mathematical tools used in subspace identification are briefly presented. In the second step, the problem of analyzing ill-conditioning matrices in the subspace identification method is considered. To illustrate this situation, a simulation study of an example is introduced to show the ill-conditioning in subspace identification. Algorithms numerical subspace state space system identification (N4SID) and multivariable output error state space model identification (MOESP) are considered to study, the parameters estimation while using the induction motor model, in simulation (Matlab environment). Finally, we show the inadequacy of the oblique projection and validate the effectiveness of the orthogonal projection approach which is needed in ill-conditioning; a real application dealing with induction motor parameters estimation has been experimented. The obtained results proved that the algorithm based on orthogonal projection MOESP, overcomes the situation of ill-conditioning in the Hankel's block, and thereby improving the estimation of parameters.

Bayesian identification of non-linear state-space models: Part II- Error analysis

In the last two decades, several methods based on sequential Monte Carlo (SMC) and Markov chain Monte Carlo (MCMC) have been proposed for Bayesian identification of stochastic non-linear state-space models (SSMs). It is well known that the performance of these simulation based identification methods depends on the numerical approximations used in their design. We propose the use of posterior Cramér-Rao lower bound (PCRLB) as a mean square error (MSE) bound. Using PCRLB, a systematic procedure is developed to analyse the estimates delivered by Bayesian identification methods in terms of bias, MSE, and efficiency. The efficacy and utility of the proposed approach is illustrated through a numerical example.

State-space Modeling of Large Domain Wave Propagation Systems by Partitioned C-matrices

Reduced-order models represent an enabling technology in the representa- tion of large-scale dynamic systems. This technology often involves identification of linear state-space models with system matrix A, input matrix B, and output matrix C. Our focus is partitioned C-matrices that facilitate creation of reduced-order discrete- time state-space models appropriate for simulation of large-output wave propagation systems. The Cy-partition method, used to generate the partitioned C-matrices, is suitable when the output dimension is orders of magnitude higher than the num- ber of discrete time samples specifying the time duration of interest. The resulting state-space model is characterized by a relatively small C-matrix component relating a small number of "anchored" or basis outputs to the inputs, and a large C-matrix component relating all remaining outputs to the anchored outputs. The partitioned C-matrix and the associated A, B matrices can be identified from input-output data directly using time-domain signals, without the necessity of identifying or computing transfer functions. The resulting models can be used for accurate and rapid prediction of wave-field responses. The theory is general for modeling short-duration dynam- ics and the applications include modeling of vibrations propagating through a large flexible structure (for damage assessment for example).

Bayesian identification of non-linear state-space models: Part I- Input design

We propose an algorithm for designing optimal inputs for on-line Bayesian identification of stochastic non-linear state-space models. The proposed method relies on minimization of the posterior Cramér Rao lower bound derived for the model parameters, with respect to the input sequence. To render the optimization problem computationally tractable, the inputs are parametrized as a multi-dimensional Markov chain in the input space. The proposed approach is illustrated through a simulation example.