Identification of systems from multirate data

Master'sMASTER OF ENGINEERIN

Maximum Entropy Vector Kernels for MIMO system identification

Recent contributions have framed linear system identification as a nonparametric regularized inverse problem. Relying on $\ell_2$-type regularization which accounts for the stability and smoothness of the impulse response to be estimated, these approaches have been shown to be competitive w.r.t classical parametric methods. In this paper, adopting Maximum Entropy arguments, we derive a new $\ell_2$ penalty deriving from a vector-valued kernel; to do so we exploit the structure of the Hankel matrix, thus controlling at the same time complexity, measured by the McMillan degree, stability and smoothness of the identified models. As a special case we recover the nuclear norm penalty on the squared block Hankel matrix. In contrast with previous literature on reweighted nuclear norm penalties, our kernel is described by a small number of hyper-parameters, which are iteratively updated through marginal likelihood maximization; constraining the structure of the kernel acts as a (hyper)regularizer which helps controlling the effective degrees of freedom of our estimator. To optimize the marginal likelihood we adapt a Scaled Gradient Projection (SGP) algorithm which is proved to be significantly computationally cheaper than other first and second order off-the-shelf optimization methods. The paper also contains an extensive comparison with many state-of-the-art methods on several Monte-Carlo studies, which confirms the effectiveness of our procedure

Modal identification using optimization approach.

In this thesis, the modal identification problem is pursued using two different optimization approaches. The first approach is a deterministic optimization approach that minimizes the output model error in the time domain between a direct solution using the modal model and the measured response. Examples of single-input single-output identification are used to illustrate this method; it has been shown this approach is robust against noise and can be used to fine-tune the modal parameter, especially for the damping. The second approach is based on probabilistic optimization; the objective function is defined as the a posteriori probabilistic density of the parameters given observations/measurements. The conditional probability density is computed using the Bayesian theory of minimum-mean-square-error estimation. Examples of single-output under ambient excitation are simulated to demonstrate this approach. This methodology allows one to obtain not only the estimated parameters in the form of probabilistic mean but also the uncertainties in the form of covariance. The optimization approaches works though the minimization of an objective function which can be calculated from given set of modal/model parameters. Since there is no gradient or Hessian available for the objective functions defined in this thesis, two direct optimization methods: Nelder-Mead simplex and the Genetic Algorithm are adopted to search the minimum of defined objective functions and thus find the structural parameters. (Abstract shortened by UMI.)Dept. of Civil and Environmental Engineering. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis2005 .L52. Source: Masters Abstracts International, Volume: 44-03, page: 1437. Thesis (M.A.Sc.)--University of Windsor (Canada), 2005

Experimental modeling of a web-winding machine: LPV approaches

This chapter presents the identification of a web-winding system as a linear parameter varying (LPV) system with the reel radius as the time-varying parameter. This system is nonlinear, time-varying and input–output unstable. Two identification methods are considered: in the first one, an LPV model is estimated in a single step using a novel approach based on sparse identification and set membership optimality evaluation. In the second one, several local linear time-invariant (LTI) models are identified using classical identification algorithms, and the overall LPV model is constructed as a weighted sum of the local models. The two methods are applied to experimental data measured on a real web-winding machine

Parsimonious cooperative distributed MPC algorithms for offset-free tracking

We propose in this paper novel cooperative distributed MPC algorithms for tracking of piecewise constant setpoints in linear discrete-time systems. The available literature for cooperative tracking requires that each local controller uses the centralized state dynamics while optimizing over its local input sequence. Furthermore, each local controller must consider a centralized target model. The proposed algorithms instead use a suitably augmented local system, which in general has lower dimension compared to the centralized system. The same parsimonious parameterization is exploited to define a target model in which only a subset of the overall steady-state input is the decision variable. Consequently the optimization problems to be solved by each local controller are made simpler. We also present a distributed offset-free MPC algorithm for tracking in the presence of modeling errors and disturbances, and we illustrate the main features and advantages of the proposed methods by means of a multiple evaporator process case study

Non-Parametric Bayesian Methods for Linear System Identification

Recent contributions have tackled the linear system identification problem by means of non-parametric Bayesian methods, which are built on largely adopted machine learning techniques, such as Gaussian Process regression and kernel-based regularized regression. Following the Bayesian paradigm, these procedures treat the impulse response of the system to be estimated as the realization of a Gaussian process. Typically, a Gaussian prior accounting for stability and smoothness of the impulse response is postulated, as a function of some parameters (called hyper-parameters in the Bayesian framework). These are generally estimated by maximizing the so-called marginal likelihood, i.e. the likelihood after the impulse response has been marginalized out. Once the hyper-parameters have been fixed in this way, the final estimator is computed as the conditional expected value of the impulse response w.r.t. the posterior distribution, which coincides with the minimum variance estimator. Assuming that the identification data are corrupted by Gaussian noise, the above-mentioned estimator coincides with the solution of a regularized estimation problem, in which the regularization term is the l2 norm of the impulse response, weighted by the inverse of the prior covariance function (a.k.a. kernel in the machine learning literature). Recent works have shown how such Bayesian approaches are able to jointly perform estimation and model selection, thus overcoming one of the main issues affecting parametric identification procedures, that is complexity selection. While keeping the classical system identification methods (e.g. Prediction Error Methods and subspace algorithms) as a benchmark for numerical comparison, this thesis extends and analyzes some key aspects of the above-mentioned Bayesian procedure. In particular, four main topics are considered. 1. PRIOR DESIGN. Adopting Maximum Entropy arguments, a new type of l2 regularization is derived: the aim is to penalize the rank of the block Hankel matrix built with Markov coefficients, thus controlling the complexity of the identified model, measured by its McMillan degree. By accounting for the coupling between different input-output channels, this new prior results particularly suited when dealing for the identification of MIMO systems To speed up the computational requirements of the estimation algorithm, a tailored version of the Scaled Gradient Projection algorithm is designed to optimize the marginal likelihood. 2. CHARACTERIZATION OF UNCERTAINTY. The confidence sets returned by the non-parametric Bayesian identification algorithm are analyzed and compared with those returned by parametric Prediction Error Methods. The comparison is carried out in the impulse response space, by deriving “particle” versions (i.e. Monte-Carlo approximations) of the standard confidence sets. 3. ONLINE ESTIMATION. The application of the non-parametric Bayesian system identification techniques is extended to an online setting, in which new data become available as time goes. Specifically, two key modifications of the original “batch” procedure are proposed in order to meet the real-time requirements. In addition, the identification of time-varying systems is tackled by introducing a forgetting factor in the estimation criterion and by treating it as a hyper-parameter. 4. POST PROCESSING: MODEL REDUCTION. Non-parametric Bayesian identification procedures estimate the unknown system in terms of its impulse response coefficients, thus returning a model with high (possibly infinite) McMillan degree. A tailored procedure is proposed to reduce such model to a lower degree one, which appears more suitable for filtering and control applications. Different criteria for the selection of the order of the reduced model are evaluated and compared

Fault detection for the Benfield process using a closed-loop subspace re-identification approach

Closed-loop system identification and fault detection and isolation are the two fundamental building blocks of process monitoring. Efficient and accurate process monitoring increases plant availability and utilisation. This dissertation investigates a subspace system identification and fault detection methodology for the Benfield process, used by Sasol, Synfuels in Secunda, South Africa, to remove CO2 from CO2-rich tail gas. Subspace identification methods originated between system theory, geometry and numerical linear algebra which makes it a computationally efficient tool to estimate system parameters. Subspace identification methods are classified as Black-Box identification techniques, where it does not rely on a-priori process information and estimates the process model structure and order automatically. Typical subspace identification algorithms use non-parsimonious model formulation, with extra terms in the model that appear to be non-causal (stochastic noise components). These extra terms are included to conveniently perform subspace projection, but are the cause for inflated variance in the estimates, and partially responsible for the loss of closed-loop identifiably. The subspace identification methodology proposed in this dissertation incorporates two successive LQ decompositions to remove stochastic components and obtain state-space models of the plant respectively. The stability of the identified plant is further guaranteed by using the shift invariant property of the extended observability matrix by appending the shifted extended observability matrix by a block of zeros. It is shown that the spectral radius of the identified system matrices all lies within a unit boundary, when the system matrices are derived from the newly appended extended observability matrix. The proposed subspace identification methodology is validated and verified by re-identifying the Benfield process operating in closed-loop, with an RMPCT controller, using measured closed-loop process data. Models that have been identified from data measured from the Benfield process operating in closed-loop with an RMPCT controller produced validation data fits of 65% and higher. From residual analysis results, it was concluded that the proposed subspace identification method produce models that are accurate in predicting future outputs and represent a wide variety of process inputs. A parametric fault detection methodology is proposed that monitors the estimated system parameters as identified from the subspace identification methodology. The fault detection methodology is based on the monitoring of parameter discrepancies, where sporadic parameter deviations will be detected as faults. Extended Kalman filter theory is implemented to estimate system parameters, instead of system states, as new process data becomes readily available. The extended Kalman filter needs accurate initial parameter estimates and is thus periodically updated by the subspace identification methodology, as a new set of more accurate parameters have been identified. The proposed fault detection methodology is validated and verified by monitoring process behaviour of the Benfield process. Faults that were monitored for, and detected include foaming, flooding and sensor faults. Initial process parameters as identified from the subspace method can be tracked efficiently by using an extended Kalman filter. This enables the fault detection methodology to identify process parameter deviations, with a process parameter deviation sensitivity of 2% or higher. This means that a 2% parameter deviation will be detected which greatly enhances the fault detection efficiency and sensitivity.Dissertation (MEng)--University of Pretoria, 2008.Electrical, Electronic and Computer Engineeringunrestricte

A Framework for the Use of Mobile Sensor Networks in System Identification

System identification (SID, also known as structural identification in this context) is the process of extracting a system’s modal properties from sensor measurements. Typically, a mathematical model is chosen for data fitting and the identification of model parameters yields modal property estimates. Historically, SID has relied on measurements from fixed sensors, which remain at specific locations throughout data collection. The ultimate flaw in fixed sensors is they provide restricted spatial information, which can be addressed by mobile sensors. In this dissertation, a framework is developed for extracting structural modal estimates from data collected by mobile sensors. The current state of mobile sensor networks applications in SHM is developing; research has been diverse, however limited. Reduced setup requirements for mobile sensor networks facilitate data collection, thus enable expedited information updates on a structure’s health and improved emergency response times to natural disasters. This research focuses on using mobile sensor data, i.e., data from sensors simultaneously recording in time, while moving in space, for comprehensive system identification of real structural systems. Mobile sensing data is analyzed from two perspectives, each requires different modeling techniques: an incomplete data perspective and a complete data perspective. In Chapter 2, Structural Identification using Expectation Maximization (STRIDE) is introduced, a novel application of the Expectation Maximization (EM) algorithm and approach for output-only modal identification. Chapter 3 revisits STRIDE for consideration of incomplete datasets, i.e., data matrices containing missing entries. Such instances may occur as a result of failed communications or packet losses in a wireless sensor network or as a result of sensing and sampling methods, e.g., mobile sensing. It is demonstrated that sensor network data containing a significant amount of missing observations can be used to achieve a comprehensive modal identification. Moreover, a successful real-world identification with simulated mobile sensors quantifies the preservation of spatial information, establishing benefits of this type of network, and emphasizing an inquiry for future SHM implementations. In Maximum Likelihood (ML) estimation theory, on which STRIDE is based, the precision of ML point estimates can be measured by the curvature of the likelihood function. Chapter 4 presents closed-form partial derivatives, observed information, and variance expressions for discrete-time stochastic state-space model entities. Confidence intervals are constructed for natural frequencies, damping ratios, and mode shapes using the asymptotic normality property of ML estimators. In anticipation of high-resolution scanning, mobile sensor data is also perceived to belong to a general class of data called dynamic sensor networks (DSNs), which inherently contain spatial discontinuities. Chapter 5 introduces state-space approaches for processing data from sensor networks with time-variant configurations for which a novel truncated physical model (TPM) is proposed. In typical state-space frameworks, a spatially dense observation space on the physical structure dictates a large state variable space, i.e., more total sensing nodes require a more complex dynamic model. The result is an overly complex dynamic model for the structural system. As sensor networks evolve and with increased use of novel sensing techniques in practice, it is desirable to decouple the size of the structural dynamic system from spatial sampling resolution during instrumentation. The TPM is presented as a novel technique to reduce physical state sizes and permit a general class of DSN data, with an emphasis on mobile sensing. Also, the role of basis functions in the approximation of mode shape regression is established. Chapter 6 discusses the identification of the TPM using an adjusted STRIDE methodology

LPV system identification with globally fixed orthonormal basis functions

A global and a local identification approach are developed for approximation of linear parameter-varying (LPV) systems. The utilized model structure is a linear combination of globally fixed (scheduling-independent) orthonormal basis functions (OBFs) with scheduling-parameter dependent weights. Whether the weighting is applied on the input or on the output side of the OBFs, the resulting models have different modeling capabilities. The local identification approach of these structures is based on the interpolation of locally identified LTI models on the scheduling domain where the local models are composed from a fixed set of OBFs. The global approach utilizes a priori chosen functional dependence of the parameter-varying weighting of a fixed set of OBFs to deliver global model estimation from measured I/O data. Selection of the OBFs that guarantee the least worst-case modeling error for the local behaviors in an asymptotic sense, is accomplished through the fuzzy Kolmogorov c-max approach. The proposed methods are analyzed in terms of applicability and consistency of the estimates

Innovative Surveillance and Process Control in Water Resource Recovery Facilities

Water Resource Recovery Facilities (WRRF), previously known as Wastewater Treatment Plants (WWTP), are getting increasingly complex, with the incorporation of sludge processing and resource recovery technologies. Along with maintaining a stringent effluent water quality standard, the focus is gradually shifting towards energy-efficient operations and recovery of resources. The new objectives of the WRRF demand an economically optimal operation of processes that are subjected to extreme variations in flowrate and composition at the influent. The application of online monitoring, process control, and automation in WRRF has already shown a steady increase in the past decade. However, the advanced model-based optimal control strategies, implemented in most process industries, are less common in WRRF. The complex nature of biological processes, the unavailability of simplified process models, and a lack of cost-effective surveillance infrastructure have often hindered the implementation of advanced control strategies in WRRF. The ambition of this research is to implement and validate cost-efficient monitoring alternatives and advanced control strategies for WRRF by fully utilizing the powerful Internet of Things (IoT) and data science tools. The first step towards implementing an advanced control strategy is to ensure the availability of surveillance infrastructure for monitoring nutrient compositions in WRRF processes. In Paper A, a soft sensor, based on Extended Kalman Filter, is developed for estimating water-quality parameters in a Sequential Batch MBBR process using reliable and inexpensive online sensors. The model used in the soft sensor combines the mechanistic understanding of the nutrient removal process with a statistical correlation between nutrient composition and easy-to-measure parameters. Paper B demonstrates the universality of the soft sensor through validation tests conducted in a Continuous Multistage MBBR pilot plant. The drift in soft-sensor estimation caused by a mismatch between the mathematical model and process behavior is studied in Paper B. The robustness of the soft sensor is assessed by observing estimated nutrient composition values for a period of three months. A systematic method to calibrate the measurement model and update model parameters using data from periodic lab measurements are discussed in Paper B. The term SCADA has been ubiquitous while mentioning online monitoring and control strategy deployment in WRRFs. The present digital world of affordable communication hardware, compact single board processors, and high computational power presents several options for remote monitoring and deployment of soft sensors. In Paper C, a cost-effective IoT strategy is developed by using an open-source programming language and inexpensive hardware. The functionalities of the IoT infrastructure are demonstrated by using it to deploy a soft sensor script in the ContinuousMultistage MBBR pilot plant. A cost-comparison between the commercially available alternatives presented in Paper A and the open-source IoT strategy in Paper B and Paper C highlights the benefits of the new monitoring infrastructure. Lack of reliable control models have often been the cause for the poor performance of advanced control strategies, such as Model Predictive Controls (MPC) when implemented to complex biological nutrient removal processes. Paper D attempts to overcome the inadequacies of the linear prediction model by combining a recursive model parameter estimator with the linear MPC. The new MPC variant, called the adaptive MPC (AMPC), reduces the dependency of MPC on the accuracy of its prediction model. The performance of the AMPC is compared with that of a linear MPC, nonlinear MPC, and the traditional proportional-integral cascade control through simulator-based evaluations conducted on the Benchmark Simulator platform(BSM2). The advantages of AMPC compared to its counterparts, in terms of reducing the aeration energy, curtailing the number of effluent ammonia violations, and the use of computational resources, are highlighted in Paper D. The complex interdependencies between different processes in a WRRF pose a significant challenge in defining constant reference points for WRRFs operations. A strategy that decides control outputs based on economic parameters rather than maintaining a fixed reference set-point is introduced in Paper E. The model-based control strategy presented in Paper D is further improved by including economic parameters in the MPC’s objective function. The control strategy known as Economic MPC (EMPC) is implemented for optimal dosage of magnesium hydroxide in a struvite recovery unit installed in a WRRF. A comparative study performed on the BSM2 platform demonstrates a significant improvement in overall profitability for the EMPC when compared to a constant or a feed-forward flow proportional control strategy. The resilience of the EMPC strategy to variations in the market price of struvite is also presented in Paper E. A combination of cost-effective monitoring infrastructure and advanced control strategies using advanced IoTs and data science tools have been documented to overcome some of the critical problems encountered in WRRFs. The overall improvement in process efficiency, reduction in operating costs, an increase in resource recovery, and a substantial reduction in the price of online monitoring infrastructure contribute to the overall aim of transitioning WRRFs to a self-sustaining facility capable of generating value-added products.Water Resource Recovery Facilities (WRRF), tidligere kjent som avløpsrenseanlegg (WWTP), blir stadig mer komplekse ettersom flere prosess steg tillegges anleggene i form av slambehandling og ressursgjenvinningsteknologi. Foruten hovedmålet om å imøtekomme strenge avløpsvannskvalitetskrav, har anleggenes fokus gradvis skiftet mot energieffektiv drift og gjenvinning av ressurser. Slike nye mål krever økonomisk optimal drift av prosesser som er utsatt for ekstreme variasjoner i volum og sammensetning av tilløp. Bruk av online overvåking, prosesskontroll og automatisering i WRRF har jevnt økt det siste tiåret. Likevel er avanserte modellbaserte kontrollstrategier for optimalisering ikke vanlig i WRRF, i motsetning til de fleste prosessindustrier. Komplekse forhold i biologiske prosesser, mangel på tilgang til pålitelige prosessmodeller og mangel på kostnadseffektiv overvåkingsinfrastruktur har ofte hindret implementeringen av avanserte kontrollstrategier i WRRF. Ambisjonen med denne avhandlingen er å implementere og validere kostnadseffektive overvåkingsalternativer og avanserte kontrollstrategier somutnytter kraftige Internet of Things (IoT) og datavitenskapelige verktøy i WRRF sammenheng. Det første steget mot implementering av en avansert kontrollstrategi er å sørge for tilgjengelighet av overvåkingsinfrastruktur for måling av næringsstoffer i WRRF-prosesser. Paper A demonstrerer en virtuell sensor basert på et utvidet Kalman filter, utviklet for å estimere vannkvalitetsparametere i en sekvensiell batch MBBR prosess ved hjelp av pålitelige og rimelige online sensorer. Modellen som brukes i den virtuelle sensoren kombinerer en mekanistisk forståelse av prosessen for fjerning av næringsstoffer fra avløpsvann med et statistisk sammenheng mellom næringsstoffsammensetning i avløpsvann og parametere som er enkle å måle. Paper B demonstrerer det universale bruksaspektet til den virtuelle sensoren gjennom valideringstester utført i et kontinuerlig flertrinns MBBR pilotanlegg. Feilene i sensorens estimering forårsaket av uoverensstemmelse mellom den matematiske modellen og prosesseatferden er undersøkt i Paper B. Robustheten til den virtuelle sensoren ble vurdert ved å observere estimerte næringssammensetningsverdier i en periode på tre måneder. En systematisk metode for å kalibrere målemodellen og oppdatere modellparametere ved hjelp av data fra periodiske laboratoriemålinger er diskutert i Paper B. Begrepet SCADA har alltid vært til stede når online overvåking og kontrollstrategi innen WRRF er nevnt. Den nåværende digitale verdenen med god tilgjengelighet av rimelig kommunikasjonsmaskinvare, kompakte enkeltkortprosessorer og høy beregningskraft presenterer flere muligheter for fjernovervåking og implementering av virtuelle sensorer. Paper C viser til utvikling av en kostnadseffektiv IoT-strategi ved hjelp av et programmeringsspråk med åpen kildekode og rimelig maskinvare. Funksjonalitetene i IoT-infrastruktur demonstreres gjennom implementering av et virtuelt sensorprogram i et kontinuerlig flertrinns MBBR pilotanlegg. En kostnadssammenligning mellom de kommersielt tilgjengelige alternativene som presenteres i Paper A og åpen kildekode-IoT-strategi i Paper B og Paper C fremhever fordelene med den nye overvåkingsinfrastrukturen. Mangel på pålitelige kontrollmodeller har ofte vært årsaken til svake resultater i avanserte kontrollstrategier, som for eksempel Model Predictive Control (MPC) når de implementeres i komplekse biologiske prosesser for fjerning av næringsstoffer. Paper D prøver å løse manglene i MPC ved å kombinere en rekursiv modellparameterestimator med lineær MPC. Den nye MPC-varianten, kalt Adaptiv MPC (AMPC), reduserer MPCs avhengighet av nøyaktigheten i prediksjonsmodellen. Ytelsen til AMPC sammenlignes med ytelsen til en lineær MPC, ikke-lineær MPC og tradisjonell proportionalintegral kaskadekontroll gjennom simulatorbaserte evalueringer utført på Benchmark Simulator plattformen (BSM2). Fordelene med AMPC sammenlignet med de andre kontrollstrategiene er fremhevet i Paper D og demonstreres i sammenheng redusering av energibruk ved lufting i luftebasseng, samt redusering i antall brudd på utslippskrav for ammoniakk og bruk av beregningsressurser. De komplekse avhengighetene mellom forskjellige prosesser i en WRRF utgjør en betydelig utfordring når man skal definere konstante referansepunkter for WRRF under drift. En strategi som bestemmer kontrollsignaler basert på økonomiske parametere i stedet for å opprettholde et fast referansesettpunkt introduseres i Paper E. Den modellbaserte kontrollstrategien fra PaperDforbedres ytterligere ved å inkludere økonomiske parametere iMPCs objektiv funksjon. Denne kontrollstrategien kalles Economic MPC (EMPC) og er implementert for optimal dosering av magnesiumhydroksid i en struvit utvinningsenhet installert i en WRRF. En sammenligningsstudie utført på BSM2 plattformen viste en betydelig forbedring i den totale lønnsomheten ved bruk av EMPC sammenlignet med en konstant eller en flow proportional kontrollstrategi. Robustheten til EMPC-strategien for variasjoner i markedsprisen på struvit er også demonstrert i Paper E. En kombinasjon av kostnadseffektiv overvåkingsinfrastruktur og avanserte kontrollstrategier ved hjelp av avansert IoT og datavitenskapelige verktøy er brukt for å løse flere kritiske utfordringer i WRRF. Den samlede forbedringen i prosesseffektivitet, reduksjon i operasjonskostnader, økt ressursgjenvinning og en betydelig reduksjon i pris for online overvåkningsinfrastruktur bidrar til det overordnede målet om å gå over til bærekraftige WRRF som er i stand til å generere verdiskapende produkter.DOSCON A