Observed state variables

The eontrol is implemented using observed state variables... 

If the differenee between the aetual and observed state variables is... 

Comparing the system shown in Figure 8.12 with the original PD eontroller given in Example 5.10, the state feedbaek system may be eonsidered to be a PD eontroller where the proportional term uses measured output variables and the derivative term uses observed state variables. 

In the case of one-dimensional problems, there is only one observable state variable q, i.e. the axial strain a and the generalized force Q corresponds to the axial stress o. From eq. (6) we have a onedimensional non-linear thermo-visco-elastic equation (8) as follows ... 

Fiypothesis unable to explain the values of the state variables observed... 

Motivated by the qualitative observations made above, a set of internal state variables deseribing the internal strueture of the material will be intro-dueed ab initio, denoted eolleetively by k. Their physieal meaning or preeise properties need not be established at this point, and they may inelude sealar, veetor, or tensor quantities. The following eonstitutive assumptions are now made ... 

Equations (8.124) requires that all state variables must be measured. In praetiee this may not happen for a number of reasons ineluding eost, or that the state may not physieally be measurable. Under these eonditions it beeomes neeessary, if full state feedbaek is required, to observe, or estimate the state variables. 

A full-order state observer estimates all of the system state variables. If, however, some of the state variables are measured, it may only be neeessary to estimate a few of them. This is referred to as a redueed-order state observer. All observers use some form of mathematieal model to produee an estimate x of the aetual state veetor x. Figure 8.8 shows a simple arrangement of a full-order state observer. 

A full-order state observer estimates all state variables, irrespeetive of whether they are being measured. In praetiee, it would appear logieal to use a eombination of measured states from y = Cx and observed states (for those state variables that are either not being measured, or not being measured with suffieient aeeuraey). 

The for-end loop in examp88.m that employs equation (8.76), while appearing very simple, is in faet very powerful sinee it ean be used to simulate the time response of any size of multivariable system to any number and manner of inputs. If A and B are time-varying, then A(r) and B(r) should be ealeulated eaeh time around the loop. The author has used this teehnique to simulate the time response of a 14 state-variable, 6 input time-varying system. Example 8.10 shows the ease in whieh the eontrollability and observability matriees M and N ean be ealeulated using c t r b and ob s v and their rank eheeked. 

C is the mxn observation matrix which indicates the state variables (or linear combinations of state variables) that are measured experimentally. 

The state variables are the minimal set of dependent variables that are needed in order to describe fully the state of the system. The output vector represents normally a subset of the state variables or combinations of them that are measured. For example, if we consider the dynamics of a distillation column, in order to describe the condition of the column at any point in time we need to know the prevailing temperature and concentrations at each tray (the state variables). On the other hand, typically very few variables are measured, e.g., the concentration at the top and bottom of the column, the temperature in a few trays and in some occasions the concentrations at a particular tray where a side stream is taken. In other words, for this case the observation matrix C will have zeros everywhere except in very few locations where there will be 1 s indicating which state variables are being measured. 

The only drawback in using this method is that any numerical errors introduced in the estimation of the time derivatives of the state variables have a direct effect on the estimated parameter values. Furthermore, by this approach we can not readily calculate confidence intervals for the unknown parameters. This method is the standard procedure used by the General Algebraic Modeling System (GAMS) for the estimation of parameters in ODE models when all state variables are observed. 

In the above ODEs, X] and x2 represent the biomass and substrate concentration in the chemostat, cF is the substrate concentration in the feed stream (g/L) and D is the dilution factor (h 1) defined as the feed flowrate over the volume of the liquid phase in the chemostat. It is assumed that both state variables, xt and x2 are observed. 

The fifteen layers of constant permeability and porosity were taken as the reservoir zones for which these parameters would be estimated. The reservoir pressure is a state variable and hence in this case the relationship between the output vector (observed variables) and the state variables is of the form y(t,)=Cx(t,). 

Similar findings were observed for the gas-oil ratio or the bottom hole pressure of each well which is also a state variable when the well production rate is capacity restricted (Tan and Kalogerakis. 1991). 

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. 

In order to apply the concepts of modern control theory to this problem it is necessary to linearize Equations 1-9 about some steady state. This steady state is found by setting the time derivatives to zero and solving the resulting system of non-linear algebraic equations, given a set of inputs Q, I., and Min In the vicinity of the chosen steady state, the solution thus obtained is unique. No attempts have been made to determine possible state multiplicities at other operating conditions. Table II lists inputs, state variables, and outputs at steady state. This particular steady state was actually observed by fialsetia (8). 

State variable feedback generally has no effect upon the controllability of a system, but can result in a loss of observability (.9). Since it is intended to measure state variables directly in the present case, observability is not germane. 

For obvious reasons, we need to introduce surface contributions in the thermodynamic framework. Typically, in interface thermodynamics, the area in the system, e.g. the area of an air-water interface, is a state variable that can be adjusted by the observer while keeping the intensive variables (such as the temperature, pressure and chemical potentials) fixed. The unique feature in selfassembling systems is that the observer cannot adjust the area of a membrane in the same way, unless the membrane is put in a frame. Systems that have self-assembly characteristics are conveniently handled in a setting of thermodynamics of small systems, developed by Hill [12], and applied to surfactant self-assembly by Hall and Pethica [13]. In this approach, it is not necessary to make assumptions about the structure of the aggregates in order to define exactly the equilibrium conditions. However, for the present purpose, it is convenient to take the bilayer as an example. 

The molecular modelling of systems consisting of many molecules is the field of statistical mechanics, sometimes called statistical thermodynamics [28,29], Basically, the idea is to go from a molecular model to partition functions, and then, from these, to predict thermodynamic observables and dynamic and structural quantities. As in classical thermodynamics, in statistical mechanics it is essential to define which state variables are fixed and which quantities are allowed to fluctuate, i.e. it is essential to specify the macroscopic boundary conditions. In the present context, there are a few types of molecular systems of interest, which are linked to so-called ensembles. 

A single experiment consists of the measurement of each of the g observed variables for a given set of state variables (dependent, independent). Now if the independent state variables are error-free (explicit models), the optimization need only be performed in the parameter space, which is usually small. 

If all the state variables are not measured, an observer should be implemented. In the Figure 14, the jacket temperature is assumed as not measured, but it can be easily estimated by the rest of inputs and outputs and based on the separation principle, the observer and the control can be calculated independently. In this structure, the observer block will provide the missing output, the integrators block will integrate the concentration and temperature errors and, these three variables, together with the directly measured, will input the state feedback (static) control law, K. Details about the design of these blocks can be found in the cited references. 

The centralized control can be approached using different techniques pole-placement, optimal control and loop decoupling. When the whole state is not accessible, a motivation to introduce a state observer is discussed. A detailed example when all state variables are accessible, i.e. when the state observer it is not necessary, has been explained. It is important to remark that the previously cited techniques are not widely used in CSTR control. This is due to the fact that these procedures require non-intuitive matrix tuning and computations, which are not familiar in the process industry. Nevertheless, for complex processes, these procedures can be the only solution to the control problem, when a limited set of sensors are available. 

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 ... 

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