State estimator

The last decade has seen the development of considerable interest in state estimators. The focus of these methods is somewhat different than conventional identification, but the two areas have many similarities. That is the justification for a brief discussion of state estimators in this chapter. 

A rigorous discussion of this extensive subject is beyond the scope of this book, and an extensive treatment is also probably not justified at this point in time. There has been a lot of interest by academia and by industry in these methods, but the jury is still out as to their real practical utility in chemical engineering processes with their very high order, nonlinearity, unknown noise properties, and frequent and unmeasurable load disturbances. Several industrial researchers have reported successful applications. But other industrial reports indicate a significant number of failures, or at least a lack of significant improvement in control over more simple conventional methods. 

The books by T. Kailath [Unear Systems, 1980, Prentice-Hall) and H. Kwakernaak and R. Sivan Linear Optimal Control Systems, 1972, Wiley) can be consulted for detailed information. A recent paper by MacGregor, et al. ( State Estimation For Polymerization Reactors, 1986IFAC Symposium, Bournemouth, U.K.) discusses some of the problems of applying state estimators to chemical engineering systems. 

State estimators are used to provide on-line predictions of those variables that describe the system dynamic behavior but cannot be directly measured. For example, suppose we have a chemical reactor and can measure the temperature in the reactor but not the compositions of reactants or products. A state estimator could be used to predict these compositions. 

State estimators are basically just mathematical models of the system that are solved on-line. These models usually assume linear DDEs, but nonlinear equations can be incorporated. The actual measured inputs to the process (manipulated variables) are fed into the model equations, and the model equations are integrated. Then the available measured output variables are compared with the predictions of the model. The differences between the actual measured output variables and the predictions of the model for these same variables are used to change the model estimates through some sort of feedback. As these differences between the predicted and measured variables are driven to zero, the model predictions of all the state variables are changed. 

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Table 3. Production Capacity in the United States, Estimated 1990...

State estimation is the process of extracting a best estimate of a variable from a number of measurements that contain noise. 

This work was extended by Kalman and Buey (1961) who designed a state estimation proeess based upon an optimal minimum varianee filter, generally referred to as a Kalman filter. 

The Kalman filter multivariable state estimation problem... 

A control system that contains a LQ Regulator/Tracking controller together with a Kalman filter state estimator as shown in Figure 9.8 is called a Linear Quadratic Gaussian (LQG) control system. 

Next, Ah and Ad are written in terms of partition functions (see Section 5.2), which are in principle calculable from quantum mechanical results together with experimental vibrational frequencies. The application of this approach to mechanistic problems involves postulating alternative models of the transition state, estimating the appropriate molecular properties of the hypothetical transition state species, and calculating the corresponding k lko values for comparison with experiment.""- " "P... 

Over the years two ML estimation approaches have evolved (a) parameter estimation based an implicit formulation of the objective function and (b) parameter and state estimation or "error in variables" method based on an explicit formulation of the objective function. In the first approach only the parameters are estimated whereas in the second the true values of the state variables as well as the values of the parameters are estimated. In this section, we are concerned with the latter approach. 

Implicit estimation offers the opportunity to avoid the computationally demanding state estimation by formulating a suitable optimality criterion. The penalty one pays is that additional distributional assumptions must be made. Implicit formulation is based on residuals that are implicit functions of the state variables as opposed to the explicit estimation where the residuals are the errors in the state variables. The assumptions that are made are the following ... 

Ali, Z., Ramachandran, S., Davis, J. F., and Bakshi, B., On-line state estimation in intelligent diagnostic decision support systems for large scale process operations, AIChE Mtg., Los Angeles, CA, November (1997). 

In 1948 the use of DDT insecticides for the control of the hornfly so increased the production of milk and meat that the county agents in several states estimated the benefits to be worth 54,000,000. 

Figure 9.3. Concept of using a state estimator to Figure 9.4. A probable model for a state estimator,...

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 next task is to seek a model for the observer. We stay with a single-input single-output system, but the concept can be extended to multiple outputs. The estimate should embody the dynamics of the plant (process). Thus one probable model, as shown in Fig. 9.4, is to assume that the state estimator has the same structure as the plant model, as in Eqs. (9-13) and (9-14), or Fig. 9.1. The estimator also has the identical plant matrices A and B. However, one major difference is the addition of the estimation error, y - y, in the computation of the estimated state x. 

Here, y = Cx has been used in writing the error in the estimation of the output, (y - y). The (/ x 1) observer gain vector Ke does a weighting on how the error affects each estimate. In the next two sections, we will apply the state estimator in (9-32) to a state feedback system, and see how we can formulate the problem such that the error (y - y) can become zero. 

A system making use of the state estimator is shown in Fig. 9.5, where for the moment, changes in the reference is omitted. What we need is the set of equations that describe this regulator system with state estimation. 

The next task is to make use of the full state estimator equations. Before that, we have to remold Eq. (9-46) as if it were a full state problem. This exercise requires some careful bookkeeping of notations. Let s take the first row in Eq. (9-46) and make it to constitute the output equation. Thus we make a slight rearrangement ... 

The next step is to take the full state estimator in Eq. (9-32),... 

Full-order state estimator Reduced-order state estimator... 

After all this fancy mathematics, we need a word of caution. It is extremely dangerous to apply the state estimate as presented in this chapter. Why The first hint is in Eq. (9-32). We have assumed perfect knowledge of the plant matrices. Of course, we rarely do. Furthermore, we have omitted actual terms for disturbances, noises, and errors in measurements. Despite these drawbacks, material in this chapter provides the groundwork to attack serious problems in modem control. 

On-line State Estimation. Optimal Sensor Selection and Control. Realistically, it is very difficult to have on-line measurements of all the major polymer or latex properties of interest, but perhaps one could rely upon one or two of the sensors available for the on-line measurement of a few states (e.g. conversion). In order to estimate some of the other states (e.g. particle diameter averages), Kalman filters or Observers should be used. A number of papers have investigated these state estimation schemes (58,6 , 67). 

A valid mechanistic model can be very useful, not only in that it can appreciably add to our process understanding, but also in that it can be successfully employed in many aspects of emulsion polymerization reactor technology, ranging from latex reactor simulation to on-line state estimation and control. A general model framework has been presented and then it was shown how it can be applied in a few of these areas. The model, being very flexible and readily expandable, was further extended to cover several monomer and comonomer systems, in an effort to illustrate some of its capabilities. 

The mean-field phase diagram in the WSL calculated by Matsen et al. [138] predicts a transition from C to the disordered state via the bcc and the fee array with decreasing /N. This was not followed here. Transitions from the C to S (at 115.7 °C), to the lattice-disordered sphere - where the bcc lattice was distorted by thermal fluctuations - and finally to the disordered state (estimation > 180 °C but not attained in the study) were observed. It was reasoned to consider the lattice-disordered spheres as a fluctuation-induced lattice disordered phase. This enlarges the window for the disordered one and causes the fee phase to disappear. Even if the latter should exist, its observation will be aggravated by its narrow temperature width of about 8 K and its slow formation due to the symmetry changes between fee and bcc spheres. 

Chemical and hydrocarbon plant losses resulting from fires and explosions are substantial, with yearly property losses in the United States estimated at almost 300 million (1997 dollars).1 Additional losses in life and business interruptions are also substantial. To prevent accidents resulting from fires and explosions, engineers must be familiar with... 

Figure 3. Feedback cooling in cavity QED Evolution of the mean atomic effective energy, with no cooling (top curve), cooling based on direct feedback of the photocurrent signal (middle), cooling based on feedback with a simple Gaussian state estimator (bottom). Note the improved cooling efficiency in the second case.

Source SAMHSA (2001). State Estimates of Substance Abuse. Available at http //www. oas.samhsa.gov/... 

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