State Space Model Identification

The matrices in Eqn. (25.2) can be effectively identified using subspace methods. If the sequence of yQi), xQi) and u k) were known, matrices C and D could be computed from Eqa (25.2b) using a least-squares method, with e being the residual. Subsequently, Eqn (25.2a) would form another least squares calculation yielding matrices A, B and K. In other words, when the states x are known, we can use linear least squares to compute the model matrices. 

The states in Eqn (25.2) are now being formed as linear combinations of the -step ahead predicted outputs k= 1, 2,. ..). The literature on state space identification has shown how the states can be estimated directly from the process data by certain projections. (Verhaegen, 1994 van Overschee and de Moor, 1996 Ljung and McKelvey, 1996). The MATLAB function n4sid (Numerical Algorithms for Subspace State Space System Identification) uses subspace methods to identify state space models (Matlab 2000, van Overschee and de Moor, 1996) via singular value decomposition and estimates the state x directly from the data. 

Ljung and McKelvey (1996) propose a procedure where first an ARX model is developed which is used fory-step ahead predictions while u is constant and then the states x are formed by multiphcation of the predictor vector by a state space basis. Subsequently, matrices A, 5, C, D and K are calculated using a least squares approach. 

The Wiener model contains a non-linear relationship. There are various methods to approximate this relationship, such as Tchebychev polynomials (BCreyszig, 1999) or neural 

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M Verhaegen and P Dewilde. Subspace model identification. Part I The output error state space model identification class of algorithms. Int. J. Control 56 1187-1210, 1992. 

The PBL reactor considered in the present study is a typical batch process and the open-loop test is inadequate to identify the process. We employed a closed-loop subspace identification method. This method identifies the linear state-space model using high order ARX model. To apply the linear system identification method to the PBL reactor, we first divide a single batch into several sections according to the injection time of initiators, changes of the reactant temperature and changes of the setpoint profile, etc. Each section is assumed to be linear. The initial state values for each section should be computed in advance. The linear state models obtained for each section were evaluated through numerical simulations. 

As mentioned above, the backbone of the controller is the identified LTI part of Wiener model and the inverse of static nonlinear part just plays the role of converting the original output and reference of process to their linear counterpart. By doing so, the designed controller will try to make the linear counterpart of output follow that of reference. What should be advanced is, therefore, to obtain the linear input/output data-based prediction model, which is obtained by subspace identification. Let us consider the following state space model that can describe a general linear time invariant system ... 

Fig. 1 illustrates the identification result, i.e., validation of identified model. The 4-level pseudo random signal is introduced to obtain the excited output signal which contains the sufficient information on process dynamics. With these exciting and excited data, L and Lu as well as state space model are oalcidated and on the basis of these matrices the modified output prediction model is constructed according to Eq. (8). To both mathematical model assum as plimt and identified model another 4-level pseudo random signal is introduced and then the corresponding outputs fiom both are compared as shown in Fig. 1. Based on the identified model, we design the controller and investigate its performance under the demand on changes in the set-points for the conversion and M . The sampling time, prediction and...

For effective control of crystallizers, multivariable controllers are required. In order to design such controllers, a model in state space representation is required. Therefore the population balance has to be transformed into a set of ordinary differential equations. Two transformation methods were reported in the literature. However, the first method is limited to MSNPR crystallizers with simple size dependent growth rate kinetics whereas the other method results in very high orders of the state space model which causes problems in the control system design. Therefore system identification, which can also be applied directly on experimental data without the intermediate step of calculating the kinetic parameters, is proposed. 

Several identification methods result in a state space model, eithejp by direct identification in the state space structure or by identjLfication in a structure that can be transformed into a state space model. In system identification, discrete-time models are used. The discrete-time state-space model is given by... 

To determine the state space model with system Identification, responses of the nonlinear model to positive and negative steps on the Inputs as depicted in Figure 4 were used. Amplitudes were 20 kW for P,, . 4 1/s for and. 035 1/s for Q. The sample interval for the discrete-time model was chosen to be 18 minutes. The software described In ( 2 ) was used for the estimation of the ARX model, the singular value analysis and the estimation of the approximate... 

The method of lines and system identification are not restricted in their applicability. System identification is preferred because the order of the resulting state space model is significantly lower. Another advantage of system Identification is that it can directly be applied on experimental data without complicated analysis to determine the kinetic parameters. Furthermore, no model assumptions are required with respect to the form of the kinetic expressions, attrition, agglomeration, the occurence of growth rate dispersion, etc. 

The success of MPC is based on a number of factors. First, the technique requires neither state space models (and Riccati equations) nor transfer matrix models (and spectral factorization techniques) but utilizes the step or impulse response as a simple and intuitive process description. This nonpara-metric process description allows time delays and complex dynamics to be represented with equal ease. No advanced knowledge of modeling and identification techniques is necessary. Instead of the observer or state estimator of classic optimal control theory, a model of the process is employed directly in the algorithm to predict the future process outputs. 

To include the information about process d3mamics in the models, the data matrix can be augmented with lagged values of data vectors, or model identification techniques such as subspace state-space modeling can be used (Section 4.5). Negiz and Cinar [209] have proposed the use of state variables developed with canonical variates based realization to implement SPM to multivariable continuous processes. Another approach is based on the use of Kalman filter residuals [326]. MSPM with dynamic process models is discussed in Section 5.3. The last section (Section 5.4) of the chapter gives a brief survey of other approaches proposed for MSPM. 

To include the information about process dynamics in the models, the data matrix can be augmented with lagged values of data vectors, or model identification techniques such as subspace state-space modeling can be used (Section 5.3). Other approaches proposed for MSPM are summarized in Section 5.4). 

For all these reasons, the cutset based analysis needs to be enriched by the determination and assessment of event sequences within the cutset. This evolution justifies the use of state space models to capture the system dynamic and the impact of component s failures and reparation on the system state. Some approaches have been recently developed to determine the critical sequences of events and some basic properties such as minimality and consistency has been proposed for dynamic reparable system (Bouissou Bon 2003, Chaux et al. 2013). However, these approaches are based on deterministic language theory and focus on the identification of a set of events sequences but present some limitations and divergences for... 

Verhaegen, M. (1994) Identification of the deterministic part of MIMO state space models. Auto-matica, 30, 61-74. 

T2601.m direct identification of 10th order state space model... 

The EKF has by far been the most extensively used identification algorithm, for the case of nonlinear systems, over the past 30 years, and has been applied for a number of civil engineering applications, such as structural damage identification, parameter identification of inelastic structures, and so forth. It is based on the propagation of a Gaussian random variable (GRV) through the first-order linearization of the state-space model of the system. Despite... 

Closed-loop identification has been addressed extensively in a linear stochastic control setting (Astrom and Wittenmark, 1989). Good discussions of early results from a stochastic control viewpoint are presented by Box (1976) and Gustavsson et al (1977). Landau and Karimi (1997) provide an evaluation of recursive algorithms for closed-loop identification. Van den Hof and Schrama (1994), Gevers (1993), and Bayard and Mettler (1992) review research on new criteria for closed-loop identification of state space or input-output models for control purposes. 

An approach to the model identification problem is taken by de Wolf et al. (1989) that involves determining an input-output, black-box model that has a state-space structure. The advantage of this approach over a physical-principles model is that no model assumptions are required. The disadvantages are that the estimated parameters have no physical meaning, and parameter estimation may need to be frequently repeated as operating conditions vary. 

Based on the non-linear plant model, a linear dynamic model is derived, either as a set of transfer functions (identification method), or as a state-space description. The last alternative is offered in advanced packages as ASPEN Dynamics . 

Dynamic controllability analysis. Based on the non-linear plant model, a linear dynamic model is derived, either as a set of transfer functions (identification method), or as a state-space description (matrices A, B, C. D). The last alternative is offered in some advanced packages, as Aspen Dynamics , but the applicability to very large problems should be verified. Then a standard controllability analysis versus frequency can be performed. The main steps are ... 

A system identification method is considered parametric if a mathematical dynamic model (often formulated in state-space) is realized in a first step and the dynamic properties of the system estimated from the realized model in the second step. Nonparametric system identification methods directly estimate the dynamic parameters of a system from transformation of data, e.g., Fourier transform or power-spectral density estimation. Time-domain identification methods estimate the dynamic parameters of a system by directly using the measured response time histories, while frequency-domain methods use the Fourier transformation or power-spectral density estimation of the measured time histories. There is also a class of time-frequency methods such as the short-time Fourier transform and the wavelet transform. These methods are commonly used for identification of time-varying systems in which the dynamic properties are time-variant Linear system identification methods are based mi the assumption that the system behaves linearly and... 

B(A) is the probability of observing the system in state A, and B(B) is the probability of observing state B. In this model, the space is divided exactly into A and B. The dividing hyper-surface between the two is employed in Transition State Theory for rate calculations [19]. The identification of the dividing surface, which is usually assumed to depend on coordinates only, is a non-trivial task. Moreover, in principle, the dividing surface is a function of the whole phase space - coordinates and velocities, and therefore the exact calculation of it can be even more complex. Nevertheless, it is a crucial ingredient of the IVansition State Theory and variants of it. 

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