State variable unmeasured

The linear time invariant system in Eqs. (9-1) and (9-2) is completely observable if every initial state x(0) can be determined from the output y(t) over a finite time interval. The concept of observability is useful because in a given system, all not of the state variables are accessible for direct measurement. We will need to estimate the unmeasurable state variables from the output in order to construct the control signal. 

The pole placement design predicates on the feedback of all the state variables x (Fig. 9.1). Under many circumstances, this may not be true. We have to estimate unmeasureable state variables or signals that are too noisy to be measured accurately. One approach to work around this problem is to estimate the state vector with a model. The algorithm that performs this estimation is called the state observer or the state estimator. The estimated state X is then used as the feedback signal in a control system (Fig. 9.3). A full-order state observer estimates all the states even when some of them are measured. A reduced-order observer does the smart thing and skip these measurable states. 

The internal structure of an observer is based on the model of the considered system. Of course, the model can be extremely simple or reduced to a simple algebraic relationship binding available measurements. However, when the model is of the dynamical type, the value of a variable is no longer influenced uniquely by the inputs at the considered moment but also by the former values of the inputs as well as by other system variables. These phenomena are then described by differential equations. Since these models carry information on the interactions between the inputs and the state variables, they are used to estimate unmeasured variables from the readily available measurements. 

Do not be confused with the unknown inputs observers theory in which the goal is to estimate state variables of a system subjected to unmeasured inputs. Here, the goal is precisely to estimate these unknown inputs and not only certain state variables. 

Given that the nonlinearities f x t),t) are unknown, the asymptotic observer is designed in such a way that it enables us the reconstruction of the unmeasured states from the measured ones, whatever the unknown nonlinearities are. This can be done by hnding a suitable linear combination of the state variables w t) = Nx t) with N G 3i ", such that ... 

Proof. Let e+ = x)( —xi and = x —xf be the observation errors associated to (21) which are related to the unmeasured state variables for the upper and lower bounds, respectively. For simplicity, the notation e is sused here to represent any of the errors e+ or since their d3mamics have the same mathematical structure. Then, it is straightforward to verify that ... 

In order to tackle the problem of uncertainties in the available model, nonlinear robust and adaptive strategies have been developed, while, in the absence of full state measurements, output-feedback control schemes can be adopted, where the unmeasurable state variables can be estimated by resorting to state observers. The development of model-based nonlinear strategies has been fostered by the development of efficient experimental identification methods for nonlinear models and by significantly improved capabilities of computer-control hardware and software. 

The optimal filtering problem (the Kalman-Bucy filter) can be solved independent of the optimal control for the LQP and provides a means for estimating unmeasured state variables which may be corrupted by process and instrument noise. 

Figure 4.29 shows a block diagram of a reactor with manipulated inputs U. other measured inputs W, and unknowm or unmeasured inputs N. We may assume that this reactor is more complicated than a simple plug-flow reactor or a CSTR. It may be more along the lines of the fluidized catalytic cracker that we showed in Fig. 4.4. The reactor can be described by a set of nonlinear differential equations as we have previously demonstrated. This results in a set of dynamic state variables X The state vector is often of high dimension and we normally only measure a subset of all the states. Y is the vector of all measurements made on the system.

Intensive—independent of quantity of material extensive—dependent of quantity of material. Measurable—temperature, pressure unmeasurable—internal energy, enthalpy. State variable—difference in value between two states depends only on the states path variable—difference in two states depends on trajectory reaching in the final state. 

Furthermore, the deviations, of he unmeasured state variables should be small, since any large deviations would imply that large variations would be needed in the parameters, a , of the second subsystem of equation (24.75). Accordingly we shall need to keep the mean-squared values of x small ... 

A way of improving the performance of state estimators in the presence of modeling errors and unmeasurable disturbances is to use adaptive estimation schemes. This involves the simultaneous estimation of state variables and model parameters. One approach is to assume a model for each parameter (usually random walk type models are used) and then to estimate the parameters together with the state variables by using a state estimator such as an EKF or non-linear high gain observer [139,154,155]. 

We now aim to relate the two path-dependent quantities, q and lu, to state functions like U and 5, for two reasons. First, such relationships give ways of getting fundamental but unmeasurable quantities U and S from measurable quantities q and w. Second, there is an increase in predictive power w henever q and w depend only on state variables and not on process variables. The First Law of thermodynamics gives such a relationship. 

Steady-state process variables are related by mass and energy conservation laws. Although, for reasons of cost, convenience, or technical feasibility, not every variable is measured, some of them can be estimated using other measurements through balance calculations. Unmeasured variable estimation depends on the structure of the process flowsheet and on the instrument placement. Typically, there is an incomplete set of instruments thus, unmeasured variables are divided into determinable or estimable and indeterminable or inestimable. An unmeasured variable is determinable, or estimable, if its value can be calculated using measurements. Measurements are classified into redundant and nonredundant. A measurement is redundant if it remains determinable when the observation is deleted. 

Consequently, the following can be stated. The structural pair (Aj A2) is completely solvable with respect to the unmeasured variables, if the following two conditions are satisfied ... 

These two conditions stated for determinability correspond to those for the existence of an output set, given by Steward (1962). The first condition warrants that the number of equations is at least equal to the number of unmeasured variables, while the second condition of accessibility takes into account the existence of a subset of equations containing fewer variables than equations. We have shown that if either of the above two conditions is not satisfied, the structural pair (Aj A2) admits a decomposition analogous to that given in the previous section. Thus the same results are still valid when only the structural aspects are considered. A graphical interpretation of these two conditions is instructive. 

Using a classification algorithm we can determine the measured variables that are overmeasured, that is, the measurements that may also be obtained from mathematical relationships using other measured variables. In certain cases we are not interested in all of them, but rather in some that for some reason (control, optimization, reliability) are required to be known with good accuracy. On the other hand, there are unmeasured variables that are also required and whose intervals are composed of over measured parameters. Then we can state the following problem Select the set of measured variables that are to be corrected in order to improve the accuracy of the required measured and unmeasured process variables. 

Consider a system that after the classification has all the unmeasured variables determinable. Suppose also that the system under study has some overmeasured variables. Then we want to select which of the overmeasured variables need not be measured, while preserving the condition of determinability for the unmeasured variables. That is, we want to minimize the number of measurements in such a way that all the unmeasured variables are determinable. This problem can be stated as... 

In some cases, we do not want all the variables to be determinable only those that are required. Consequently, we must identify which of the measurable variables have to be measured. Let p be the set of variables that for various reasons should be known correctly p may be composed of measured and unmeasured variables. Sometimes we are not interested in the whole system being determinable, so we want to select which of the process variables have to be measured to have complete determinability of the variables in set p. This problem can be stated as follows Select the necessary measurements for the subset of required variables to be determinable. 

An alternative decomposition can be performed using a Q-R factorization of matrix A2 to decouple the unmeasured variables from the measured ones (Sanchez and Romagnoli, 1996). Let us state the following theorem (Dahlquist and Bjork, 1974). 

A subsystem comprising seven units is shown in Fig. 4. In this case we have eight measured process variables and seven unmeasured ones. Matrices Ai and A2 are stated as follows ... 

The unmeasured variables are then eliminated using, for example, the orthogonal factorization procedure discussed before. Once the subset of equations containing only measured variables has been identified, the problem stated by Swartz (1989) is resolved ... 

An estimator (or more specifically an optimal state estimator ) in this usage is an algorithm for obtaining approximate values of process variables which cannot be directly measured. It does this by using knowledge of the system and measurement dynamics, assumed statistics of measurement noise, and initial condition information to deduce a minimum error state estimate. The basic algorithm is usually some version of the Kalman filter.14 In extremely simple terms, a stochastic process model is compared to known process measurements, the difference is minimized in a least-squares sense, and then the model values are used for unmeasurable quantities. Estimators have been tested on a variety of processes, including mycelial fermentation and fed-batch penicillin production,13 and baker s yeast fermentation.15 The... 

At a practical level, the load variables are classified either as major or minor, and the effort is directed at developing a relationship which incorporates the major load variables, the manipulated variables, and the controlled variable. Such a relationship is termed the steady-state model of the process. Minor load variables are usually very slow to materialize and are hard to measure. Minor load variables are easily handled by a feedback loop. The purpose of the feedback loop Is to trim the forward calculation to compensate for the minor or unmeasured load variations. 

That means that some set of L (= rankB) columns of B(z), not comprising the j-th column constitutes a basis of ImB(z) the L column vectors are thus linearly independent. By standard arguments, one concludes that the equality (8.5.45) holds true also in some neighbourhood of point z. We thus can state that the variable yj is not observable. It is not even locally observable, because the condition (8.5.43) is necessary even for local observability. The statement (8.5.45) thus disqualifies the j-th unmeasured variable we cannot expect that with an arbitrary measured (and adjusted) x, the value of yj will be determined. It can happen that the condition (8.5.43) is fulfilled at certain particular values of z, and even that the yj-value is uniquely determined by some it, see the example 4 in Section 8.1, Fig. 8-2. But such case is exceptional, due to some coincidence. It is left to the reader s taste, if he then will call the variable unobservable or perhaps observable at some x. 

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