Time variant systems

In Sections 41.2 and 41.3 we applied a recursive procedure to estimate the model parameters of time-invariant systems. After each new measurement, the model parameters were updated. The updating procedure for time-variant systems consists of two steps. In the first step the system state j - 1) at time /), is extrapolated to the state x(y) at time by applying the system equation (eq. (41.15)) in Table 41.10). At time tj a new measurement is carried out and the result is used to... 

For systems that are nearly linear or time-variant, the concept of the impulse (complex frequency) response is still applicable. For weakly non-linear systems the characterization can be extended by including measurements of the non-linearity (noise, distortion, clipping point). For time-variant systems the characterization can be extended by including measurements of the time dependency of the impulse response. Some of the additional measurements incorporate knowledge of the human auditory system which lead to system characterizations that have a direct link to the perceived audio quality (e.g. the perceptually weighted signal to noise ratio). 

Each of these categories can then be combined into more descriptive categories, e.g., a nonlinear, time-variant system or a static, time-invariant discrete system. 

FIGURE 20.41 Model of the atmospheric radio channel as a linear time-variant system with additive noise and additive interference. 

The steady-state optimal Kalman filter can be generalized for time-variant systems or time-invariant systems with non-stationary noise covariance. The time-varying Kalman filter is calculated in two steps, filtering and prediction. For the nonlinear model the state estimate may be relinearized to compensate the inadequacies of the linear model. The resulting filter is referred to the extended Kalman filter. If once a new state estimate is obtained, then a corrected reference state trajectory is determined in the estimation process. In this manner the Filter reduces deviations of the estimated state from the reference state trajectory (Kwon and Wozny, 1999 Vankateswarlu and Avantika, 2001). In the first step the state estimate and its covariance matrix are corrected at time by using new measurement values >[Pg.439]

The algorithm given in Eqs. 12,13,14,15, and 16 distinguishes itself from the classical Kalman filter by including the correlation between the process noise vector W[jt] and the measurement noise vector V[jt], i.e., S 0. In the equations above, the system is assumed to be time invariant. The algorithm can, however, also be applied to time-variant systems, resulting in system matrices A[,t] G[,t] depending on the... 

Problems of time variant system reliability, which involve more than one performance metric, can also be formulated, but their study is far more involved. 

Strictly spealdng Eq. (6.5) describes, in the sense of system theory, a linear signal transfer which is analogue, continuous and time invariant (note that j3 is constrmt). In particular in Section 7.3.6, we have to be concerned with time-variant systems. 

The process in question involved the reaction of two materials, A and B, to produce a product C. The reaction was noncatalytic, homogeneous, and in the gas phase. It took place in a tubular reactor which could not be considered either adiabatic or isothermal. The reactor was divided into four sections, the first three of which were cooled while the fourth was adiabatic. Coking of the reactor tube introduced a time variant in the system, requiring adjustment of operating conditions and eventual shutdown for cleaning. 

A disadvantage of the system characterization approach is that although the characterization is valid for a wide variety of input signals it can only be measured on the basis of knowledge of the system, This leads to system characterizations that are dependent on the type of system that is tested. A serious drawback in the system characterization approach is that it is extremely difficult to characterize systems that show a non-linear and time-variant behavior. 

The big advantage of the perceptual approach is that it is system independent and can be applied to any system, including systems that show a non-linear and time-variant behavior. A disadvantage is that for the characterization of the audio quality of a system one needs a large set of relevant test signals (speech and music signals). 

Time-variant means that the parameters of the system change over time, such as an autoinduction process that increases a drug s hepatic clearance with repeated administration. Time-invariant or stationary parameters do not change over time. It is typically assumed that a drug s pharmacokinetics are stationary over time so that the principle of superposition1 applies. With a static model, the output depends only on the input and does... 

The consideration of time dependent interfacial area is of importance for many experimental investigations. Various models are described The linearised theories for small area changes, which belong to relaxation theories and are outlined in Chapter 6. The flow in the liquid layer adjacent to a time variant surface area is also incorporated into the theoretical models. Systems such as liquid films flowing down an inclined plane or growing bubbles and drops are described quantitatively, assuming laminar and radial flow patterns (MacLeod Radke 1993). 

These issues, positive and negative, are reflected in the available correlations. These correlations are both highly useful and also limited. Some are useful because the inputs are easily measured and adjusted as needed however, correlations are mostly empirical or semi-empirical, which means that they are not widely applicable but, rather, are bioreactor design dependent at best. Hence, geometric similarity is very important. Furthermore, most studies are performed in air-water systems while most industrial processes use much more complicated and time-variant liquids. In other words, the airhft bioreactor correlations have similar problems as those for stirred-tank bioreactors and bubble columns and are due to the fact that they share the problem source bubble-bubble interactions. Bubble-bubble interactions are highly variable and lead to hydrodynamics which, in turn, are difficult to quantify and predict. Hence, the result has been that the airlift bioreactor correlations and models are either system dependent or not adequately constrained. 

Czamecki, A. A. Nowak, A. S. 2008. Time variant reliability profiles for girder bridges. Structura Sa 30 49-64. Eamon, Ch. D. Nowak, A. S. 2004. Effect of secondary element on bridge stractural system reliability considering... 

Only one aspect of the methodology is dealt with in this paper, namely the way in which time variant mechanical reliability for a mechanical component can be integrated into the functional and dysfunctional modeling of a complex (notably mechatronic) system achieved using Petri Nets. The first part describes how the Petri Nets are applied. The second part reexamines the principle of time variant mechanical reliability, and more specifically the PHI2 outcrossing rate that enables analysis of time variant failure... 

In this paper, we have presented an approach that enables us to assess time variant failure probability using the PH 12 method, which we propose integrating into the Petri Nets rehahihly analysis of a mechatronic system. The physical model of the system under consideration (a mechatronic system) allows us to obtain the functional times of the various system components. These times are used in the functional transitions of... 

Andrieu-Renaud C. 2002. Fiabilite mecanique des structures soumises d des phenomenes physiques dependant du temps. PhD thesis, Universite Blaise Pascal. Andrieu-Renaud C., Sudret B., and Lemaire M. 2004. The phi2 method a way to compute time-variant reliability. Reliability Engineering and System Safety, 84, 75-86. Demri A., Charki A., Guerin R, and Christofol H. 2007. Analyses fonctionnelle et dysfonctionnelle d un systeme mecatronique. Qualita. 

ABSTRACT This study addresses the time-variaut reliability assessment in relation to systems exhibiting a non-stationary random process during their operation, such as thermal-hydrauhc passive systems for advanced reactors, relying on natural circulation. The reliability assessment efforts conducted so far don t deal with this specific aspect the dependence upon time is usually ignored, or at most the system unavaUabUity is intended to he assessed per mission time, during which the parameter values, as t-h parameters for instance, are assumed as constant quantities. The paper presents an effort for a consistent approach to model and evaluate the natural circulation passive systems, in terms of time-variant performance parameters, as for instance mass flow-rate and thermal power, to cite any. 

This concept is depicted in Figure 1, with reference to the well known R — S (Resistance - Stress) (Melchers 1999) or load-capacity interference model, within a rehahility physics framework, where the system prohahihty o f failure Pf is evaluated by comparing the distributions of the two quantities, (Burgazzi 2003, Apostolakis et al. 2005). The curves 1 and 2 in Fig. lb represent respectively the system failure prohahihty Pf in case of time variant and time invariant stochastic process. 

It has to be noted that the notion of time-variant reliabUity is not to be confused with the expression of reUabUity corresponding to a defined mission time, or time dependent reliability. In fact it is important to distinguish between the reUabihty of the passive system which refers to its capabiUty to perform the required mission (e.g. decay heat removal) for the designated time period (for instance 72 hours corresponding to the grace period for advanced reactors) while the reli-abdity changing in time corresponds to the conditional probability at time t. 

With reference to our case of thermal-hydraulic passive system, let s consider the characteristic time-variant parameter W t) (its evolution during time will depend on the transient/accident scenario under consideration). The lower hoimd for natural circulation operation is denoted as Wi t), which, according to the failure criterion provided above (see chapter 4), is a fraction (0,8) of the flow-rate in nominal conditions. 

The non-stationary process related to the natural circulation prompts the attention for the time-variant aspect of the reliability analysis of thermal-hydraulic passive systems. 

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