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Optimal measurement control strategies for naturalresource systemsARTEMIS PAPAKYRIAZIS aa Department of Economics , California Polytechnic State University , San Luis Obispo,California, 93407, U.S.APublished online: 30 May 2007.

To cite this article: ARTEMIS PAPAKYRIAZIS (1988) Optimal measurement control strategies for natural resource systems,International Journal of Systems Science, 19:1, 195-224, DOI: 10.1080/00207728808967598

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Optimal measurement control strategies for natural resource systems

ARTEMIS PAPAKYRlAZISf

An explicit treatment of the uncertainty in the state of environmental quality of a natural resource system can provide a quantitative basis for information gathering decisions. This paper presents the formulation of a class of optimal measurement control problems in the context of natural resource systems. In particular, the design of optimal polluting level measurement control to be used for the optimal estimation or prediction of the state of a stochastic environment system is considered. Filtering theory, an extension of bayesian analysis to dynamic systems, is used to formulate a dynamic pollution level measurement optimization problem in which 'the state of the plant' is given by the elements of the variance-covariance matrix of pollution level estimates and the control is related to the determination of the locations of sampling sites and to finding the sequencing of measurements in time. The problem statement is in such a form so that the deterministic as well as the stochastic principle of Pontryagin can be applied to obtain the necessary conditions. The analysis shows that the formulation of the problem considered in this paper is sufficiently realistic and equally valid for the theory of the measurement control of any renewable or non-renewable resource system.

1. Introduction The objectives of environmental quality control are quantified in environmental

quality standards and the enforcement of environmental standards is the feed back mechanism of environmental quality control. However, to achieve effective environ- mental control the planner must know the state of the environmental system.

In a world of certainty, the necessary information could be obtained from data collection programs in'a relatively straightforward manner. Unfortunately, we do not live in such a world of certainty, and environmental policy makers must recognize explicitly that natural resource systems in general and environmental systems in particular are fraught with uncertainty.

While the environmental externalities controI problem under specific types of uncertainty has received some attention in the last few years (see, for example, Bawa 1975, Yohe 1976 and Roberts and Spence 1976, it is usually assumed that the state of the environmental system (i.e. pollution levels) is known with certainty, which violates reality. In the real world, policy makers must recognize explicitly that the environ- mental externality dynamics, including the evolution of pollution concent rat ions over time and the impact of control actions and random disturbances upon that evolution, involves uncertainty. Furthermore, the environmental system state (pollution levels) is not directly accessible to the policy maker and hence a costly observation (measure- ment) system must be employed.

Perhaps more important in the economic literature of environmental externalities, it is usually assumed that uncertainty is, as it were, a fixed element of the environment within which the decision maker operates, and that it cannot, thus, be affected by any

Received 24 July 1986. t Department of Economics, California Polytechnic State University, San Luis Obispo,

California 93407, U.S.A.

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196 A. Papakyriazis

human activity such as information gathering. However, in actual environmental quality management problems, (measurement) uncertainty is not entirely exogenous to the environmental decision maker and he can respond in a more active way in order to overcome it. In particular, choices of the number of observations as well as the type and precision of the measurement system are also control variables available to the policy maker. Therefore, the precision of the information gathering system along with conventional environmental measures (such as taxes) are control variables available to the environmental control agencies and under the assumption of efficiency, there is no reason to expect the minimization of uncertainty and infor- mation gathering costs to be less important than the gains obtained from internalization of external ties. Yet, most existing economic models make no provision for evaluating such measurement considerations.

This paper is the sequel to a previous paper (Papakyriazis 1984), which analysed both control agency and firm behaviour under standards-taxes, dynamic stochastic dispersion relationships, probabilistic environmental standards, uncertainty about the prices in preference function parameters and uncertainty in the measurement of pollution levels. Exploring the implications of the above elements of uncertainty, in the context of a stochastic intertemporal optimization problem, the study indicated that the optimal control sets are two-way separable in the sense that the optimal conditions may be stated in terms of two problems; the first, a multiperiod stochastic control problem, is concerned with joint optimal emission and price information gathering controls; and the second, a non-linear deterministic problem, is concerned with optimal pollution measurement controls. Thus the problems of optimally designing monitoring programs in the context of natural resource systems in general, and environmental systems in particular, is a topic of direct relevance. The purpose of this paper is to extend the design of measurement control theory in the context of environmental resource models and hence help remedy deficiencies in the existing theory of environmental externalities. More precisely, this paper uses the Kalman filtering technique (Kafman L960), to formulate the pollution level measurement control problem as a problem in the efficient use of limited resources.

The layout of the paper is as follows: In 4 2 we present the formal structure of the model. The multiperiod pollution level measurement control problem and its solution for the case of linear process dynamics and the measurement equation is obtained in $ 3 . Section 4 deals with the formulation and solution of the optimal non-linear environmental quality measurement problem, and Section 5 concludes.

2. Models 2.1. Firm

Public control of environmental externalities involves an attempt to influence the amount of residuals or waste products discharged into the environment by firms and households. For the sake of simplicity, we shall assume that all residuals are generated in production and no residuals are generated in consumption. Moreover, we shall assume that each firm produces only one commodity. Within a particular airshed or watershed, then, each firm produces its product with a set of variable (flow) factors and stock factor 'capital'. Hence, each firm is subject to both internal and external adjustment costs in the purchase and employment of the capital factor (see, for example, Nerlove 1972).

There are J firms in the region, each with a production function

q,t =ht(L,jt, Klj,) j = 1,2, ..., J; for all t

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Optimal measurement control strategies

where qjt is the rate of output produced by firm j at time t,

is the set of variable factors employed by firm j at time t, I l l , is the rate of purchase of the capital used by the jth firm at time t, and K l j r is the capital stock of firm j at time t. (For a rigorous mathematical treatment of the theory of the firm, see Kogiku 1971.) The investment variable, l I j , , in the production function of the jth firm in ( I ) represents the internal costs of adjustment concerning the capital factor of production (see Lucas 1967). Finally fj,( ) is well behaved with

s = l , 2 ,..., S; j = l , 2 ,..., J for a11 t .

Jointly with qj r , I different residuals denoted by

are produced. These residuals will, in general, be a function of three factors:

(i) the amount of output produced, which is given by L,,,, Klj, and IIjt;

(ii) the technology used, which is also given by LIj,, Klj, and I l l , ; and

(iii) the set of variable factors, L2,, = (L2( , , ... LZsj,) = { L ~ ~ , ~ Jf=(Sl + and capital K,,,, used for waste treatment by the jth firm at time t .

We can then write the residuals generating function as follows:

for all t. We assume that each gij,( ), i = 1,2, ..., 1, is twice continuously differentiable and

convex.

2.2. Dispersion relationships and measurement of pollution levels 2.2.1. Measure of dispersion. The quantity of pollutants actually being consumed by the public is typically related to the quantity produced by a myriad of weather and topographical random variables.

This observation uncovers a source of significant uncertainty which has been thus far ignored in the economic literature of environmental externalities. In this subsec- tion, we develop a discrete dynamic system equation model which describes dispersion functions summarizing relationships between pollution levels at each of the receptor locations throughout the airshed and emission rates at each of the emission sources.

In general, a distinction has to be made between cases in which the current level of pollution is onfy a function of the current level of polluting activities (flow dispersion

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198 A. Papakyriazis

measures), and cases where it depends only on the history of past levels of pollution so that pollutant concentrations accumulate over time, thus constituting an ever increasing environmental threat (stock dispersion measures). The stock and flow dispersion functions are, however, extreme polar cases. In reality, the current level of pollution concentrations depends on both the current level of emission rates and the historic levels of polluting activities. Moreover, dispersion functions depend on climate conditions, geography and chemical reactions and they involve four main faces: transport, dilution, depletion and reaction.

Expressed somewhat more formally, the dispersion functions describing the determination of the I pollution levels at each of the K receptor locations at time t may be written as follows:

Here P,, the state of the system at time t, is a column vector of dimension N = (K x I); that is, there is one state (pollution,levei) for each receptor point and pollution type existing at that point. G, is a (J x I)-dimensional vector of emission rates from the J emission sources at time t and 2, is an N-dimensional vector whose components measure environmental conditions. To be sure, although the level of waste emissions a t time r can be somewhat controlled, we have little if any control over the exogenous variables describing the vector try which for all practical purposes can be considered to be random.

Assuming F( ) is linear, we may write the set of relationships (3 a) as follows:

P,=U),P,,, + B,G,+e, for all t ( 3 b)

where

the typical element of P,, is the amount of pollution i a t the receptor location k at time t measured in termsofconcentration, {gil,; i = l y 2, ..., the typical element of G,, is the amount of the ith emission rate of the jth source at time I , expressed as a weight per unit time basis, (/IkI,; i = I , 2, .. ., I ; k = 1,2, ..., K):. , , the typical element of B,, is the diffusion coefficient which converts the ith emission rate of source j into the average concentration of the it h pollutant at receptor location k at time t (see, for example, Gustafson and Kortanek 1972), and {&; i = 1,2, ..., I):=, , the typical element of O,, represents the proportion of the ( t - 1) period's ith pollution level at receptor location k that is not dispersed by the time period t begins at the kth location. (The diffusion coefficient f l , , , i , is functionally related to average wind velocity and direction and the location of emitter j ois-d-vis the receptor location k, among other things.) Finally, e, is an n-dimensional random vector which is assumed to be normally distributed with mean zero and (known) positive semidefinite variance-covariance 46,,.; that is,

et - N O ; R,,) If the parameters

are assumed to be time invariant, it is possible to add the following dynamic model for the vector 0,

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Optimal measurement control strategies 199

If, however, the parameters are time-varying, one can postulate a more general model of the form

where @, is an ( [ N ~ + (K x I)] x [N2 + (K x I)]) transition matrix and S, is an ( [ N 2 + (K x I)] x 1) vector of serially independent random variables with zero mean and covariance Q,,; that is, we have: E(S,) = 0 and E(S, Si.) = Q,,S,,4. The vector of random variables S, is added to compensate for any random variation in 8, (see, for example, Mendel 1973). Examples of the general transition equations (4 b) are the multivariable random walk model

and the first-order Gauss-Markov model

where +,, is an ([N' + (K x I ) ] x EN2 + (K x I ) ] ) diagonal matrix with elements 4 ,,,. lying in the region ( - 1 < 4,,,* < I), n' = I , 2, ..., (N2 + [ K x a).

In order to complete the specification of our model, we assume that: . .

(i) Po and 8, (the initial amount of pollution levels and parameter values, respectively) are intertemporally normally distributed with known means, {Po, 8,) and variance-covariance {cp0; Eea) respectively; and

(ii) the vectors Po and 8, are independent.

2.2.2. Measurement and monitoring of pollution concentrations. Since precise measure- ment of pollution levels is essential for:

(i) the establishment of quantitative relationships between pollution levels and emission rates; and

(ii) the assessment of the effects of ambient concentrations on man and his environment (see, for example, Lave and Seskin 1977), it becomes clear that measurement of environmental quality (pollutant concentrations) is perhaps the single most important task in the evaluation and control of environmental externalities.

The practical problem of measuring pollution levels, however, is one of the most vexing problems faced by those responsible for designing optimal environmental policy. Since it is possible to locate monitors only at a small number of receptor locations, an ambient concentrations measuring system must be designed to help estimating pollution levels at off-sampling site receptor locations. Hence, both precision of measurements as well as measurement costs are becoming control variables and the optimal monitoring activity becomes a problem in the efficient use of limited resources.

Fortunately, the problems of monitoring environmental systems are not unique. In particular, control engineers have been faced with similar problems. Design and control of space flight vehicles, which are described by stochastic dynamics and observed with noisy incomplete measurement systems, are highly successful. Esti- mation theory (as discussed by Jazwinski 1970, for example) provides the basis for analysis in space flight programs.

Although, in general, estimation theory deals with statistical phenomena, it also includes the dynamic behaviour (or model) of systems and it is particularly useful

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when the system characterization (model) is multidimensional. The power of estimation theory is based on this unique combination of model and observed data. In this study, prior knowledge of the dispersion measure between pollution levels and emission rates as well as meteorological factors are summarized in the stochastic- dynamic models (3) and (4). Then, at times when data on the environmental system (i.e. pollution levels) are collected, a 'filtering' procedure will be used to give an improved estimate of the pollution levels. To obtain the minimum error estimate of pollution levels, the Kalman filter will be used (Kalman 1960, Schweppe 1973). While the Kalman filtering technique of estimation theory is generally considered to be a data analysis technique with suitable and reasonably realistic approximations, it can be extended to provide capabilities of control in environmental systems as well as in the design of (optimal) data collection programs. In fact, the processes of analysis, control and collection are combined in this study, providing a potentially powerful tool in the study of complex environmental systems.

Denoting the measured values of the different pollution levels at K M receptor locations a t time t by Y,, the data collection processes (measurement equations) can be modelled in the following form:

where V, is an

the number of H,(ut) and D,(

M (=[(y 1.) x K,])-dimensional measurement vector with I, as 9 = l

' pollutants measured at monitoring station q (I, < 1). The matrices u,) are of dimensions (M x N) and (M x M*) respectively with D,(u,)

of full rank for all u, and they are, in general, either linear or non-linear functions of the measurement control vector u,, which in turn determines measurement locations in the airshed, (measurement) instrument specifications, frequency of sampling, etc. Finally V, is an (M* x 1) gaussian measurement noise vector with zero mean and variance-cova riance

E(V, V,!) = IL,6,,. ( 6 )

It will be assumed that the two random vectors e, and V, are uncorrelated with each other for all t , and uncorrelated with the initial state vector Po (Kalman 1960). Also, i t is assumed that:

are all mutually independent. Several special cases of measurement equations (5) are of interest. Consider first

the case in which the measurement equations (5) take the form: Y, = H, P, + u,V,, where Y,, P, and V, are as in (5), H, is a known matrix, but u,, which is a scalar, denotes the measurement policy variable. The parameter u, represents the accuracy of observations at time t . If we consider the situation in which p,, denotes the number of (independent and identically distributed) observations taken at time t at each monitoring stat ion and for all kinds of pollutants and then average them to obtain the observations:

Y, = [ ( p i ' Yll) . . . ( p i Per YMf)I1

then u, is related to the number of samples, A,, by the relation: u, = ( ~ , ) - l ' ' . It is noted that the variance-covariance of the measurement noise term becomes: E(u: V, V;) = u:IIv, = p~ Rv,, which, according to the above interpretation of p,,

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Optimal measurement control strategies 20 1

indicates that the measurement variance-covariance matrix is inversely proportional to the number of samples taken at time I . Note in particular that p,, = 0 implies no observations whatsoever are made at time t , and p,, = m implies perfect observation of states at time t. Finally, in the context of our discussion, u, might be interpreted as the precision of the pollution levels measurement instruments. In view of this interpretation, the measurement equations (5) take the form: Y, = HIP, + u,V, where u, = {avm1)~ = 1 and E(V,V;) = R,, = I. It is 'noted that the variance-covariance of (u,V,) becomes:

o l = diag (a:,, . . . a$_*)

Of course, the mixed case in which both the number of samples, taken as well as the precision of the measurement instruments can be chosen at time t might be considered.

A second special case of measurement equations (5) is: Y, = u,(H, P,) + V,, where u, is a continuous scalar defined in the region of positive numbers. The situation represented by this set of measurement equations can be interpreted more or less as a signal- to-noise-ratio measurement control; hence, when u, is large (because of the precise instrumentation in the measurement process, perhaps), the signal ( H, P,) is boosted with respect to the noise V,.

Another special case of measurement equations (5) can be written as follows: Y, = u,(H,P, + V,), where u, E (0, 1). This equation is of the measure-no-measure type. If u, = 1, a measurement of fixed quality (specified perhaps by the variance- covariance matrix Rv, and/or the elements of the measurement matrix H,) is obtained; if u, = 0, no measurement is taken.

A fourth special case of equations (5) is:

F where {u,, E (0, I)):=, and the additional constraint u,, = 1 is imposed. This

f = 1 . represents a situation in which the measurement control selects one of F monitoring locations (configurations) available at time t (including perhaps the zero point implying no measurement at all).

(It is important to realize that according to our assumptions each measurement point-configuration-can provide a group of noisy measurements at certain receptor locations. For example, suppose that we deal with a basic grid system where five potential monitoring locations have been selected for the measurement of one type of pollution in an airshed for period t . Then, measurement point one may take measurements at monitoring stations one, two and five, say, while measurement point-configuration-two may yield measurements at all five stations and measure- ment configuration three may yield a measurement only at station one. In such cases the measurement equations (5) take the following form:

where the dimension of (Y,) = dim (Y,,) when uJ, = 1. A generalization of this case may be obtained if we allow the selection at each

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period of time and the best combination of measurement configurations, as opposed to the best measurement point-configuration. In this case, of course, the measure-

- ment equations are:

F but the additional constraint 2 u,, = 1 is not imposed, so that more than one

f = l

measurement configuration can take the value of one. This case is, however, mathematically more complex to handle.)

A final special case of the general measurement equations (5) is Y, = u, P, + V,, where the measurement (observation) matrix itself is controlled; that is, u, = H, and thus u, is an (M x N) matrix. This case, in the context of an environmental control - problem, is motivated by the need to choose certain locational variables so that precise estimates of the pollution contributions of each source are obtained.

2.3. Probabilistic environmental standards and policy costs

To evaluate alternative policies, an objective function must be specified. On efficiency grounds, this function should reflect the costs of controlling environmental externalities, the damages resulting from these externalities and policy costs. While some progress in deriving damage cost functions has been made (see, for example, Maler and Wyzga 1976), comprehensive results are not yet available. The placement of damage cost functions that are ill-defined and imprecise in the objective function of the environmental control problem leaves no practical means of filling gaps in our knowledge of managing uncertainty. Hence, in this paper we operate with a more realistic specification, namely in a set of environmental quality constraints.

If the pollution levels are stochastic, then the environmental control agency can only realistically forrpulate environmental quality standards in probabilistic terms. Such constraints take the following form:

where Pr stands for probability,

is the ith pollution level at receptor location k at time t,

is the highest limit on the amount of pollutant i at receptor k at time t and a,, is the prescribed probability of violating the nth environmental constraint at time c. (Under the Clean Air Act, for example, air pollution control programs in the United States are required to achieve air quality standards which are not to be exceeded, on the average, more than once a year. For standards with one-hour averaging periods, this is equivalent to obtaining a probability of exceeding the standard of: Pr ([Pollution level], >, [standard],) < ( 118760) = 0-00683%.)

Alternatively, the environmental policy constraints (7) can be expressed in the form:

SO that E(Pn,), a,, and [V(Pn,)]'12 are not independent but are related by (8).

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Optimal measurement control strategies 203

(A(a, , ) [~(u, , ) ] '~~ is determined from the distribution form of P,,, or from a non- parametric relationship such as Chebyshev's inequality-see, for example, Feller 1968.) We shall define P,, and A*(u,,) to be

so that (8) becomes

( ~ , , + A * ( a . , ) [ V ( P . , ) 1 1 1 2 ) 2 0 , n = l , 2 ...., N ( 10)

(For example, if a,, = 0.05, -A(1 - 0.05) = - 1.645 so that A*(0.05) = 1.645.) In this model, the decision-maker has to decide (or specify) two things for each pollutant at each receptor location before actually selecting the optimal environmental control strategy, see the Appendix. They are the environmental standard P,,, and the value of risk level a,,. (It is very helpful for the decision maker if he can estimate the implications of various values of a, on the optimum objective function. It is possible to do this by using the evaluators in the environmental control problem:

Minimize [(costs of controlling externalities) + (policy costs) J

Subject to Pr (P , , 2 P,,,) < a,,, n = 1,2, ..., N

In solving this environmental control problem, we shall obtain the lagrangian multiplier which corresponds to the probability of violating the nth standard at period t, a,,, for all n = 1,2, ..., N. Alternatively, we can obtain the value of lagrangian multipliers by solving the dual of the environmental control problem.)

To evaluate alternative policies, we must have some definition of policy costs. However, to develop a definition of policy costs which is sufficiently encompassing is a hard task. Do policy costs, for example, include all the costs of designing and implementing a new institution (such as the Environmental Protection Agency-EPA) aimed at controlling environmental externalities, or only the costs of implementing alternative environmental control policies within an already rigidly mandated institutional structure? The answer to this question, to a large extent at least, is arbitrary and depends on the particular analyst. Our purpose in this study will be to provide a limited definition of policy costs. In particular, we define policy costs as the costs of measurement, enforcement and administration so that a predetermined environmental constraint is obtained; that is, we have:

Measurement costs are encountered for obtaining knowledge on every aspect of the environmental quality program, including knowledge of emissions, pollution levels, damage and establishment of environmental standards. In this study, we shall consider only the (measurement) costs of obtaining emissions and pollution levels, and presume that all other costs of measurement are zero. (Quite arbitrarily, they do not include the collection of information costs about output and input prices which are included in the value of other goods functionals of the environmental control problem.) A general cost-of-measurement equation can be specified as follows:

where C,,(u,) is a well defined (linear or non-linear) function of u, indicating the costs of measuring ambient concentrations at time t borne by the pollution control agency. The functional form of CAt( ) can be chosen depending on the application.

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204 A. Papakyriazis

Enforcement costs, in general, include the costs of legal settlements or court determination of the amount of emissions, taxing or subsidy mechanisms, administra- tive costs and firm expenses associated with maintaining records, etc. In order not to complicate the discussion unduly, we shall assume in this paper that any enforcement costs may be lumped together with the monitoring costs. Finally, administrative costs are assumed to be overhead costs associated with the measurement and enforcement efforts. In this context, then, the administrative costs may be lumped together with the other two policy costs.

3. Linear pollution level measurement control problem 3.1. Problem statement

As shown by Papakyriazis (1984) in the case of known transfer function parameters, the pollution level information collection (measurement) control variables are separable from regulatory, price information gathering and emission rate measurement (monitoring) controls. Hence, the (linear process dynamics and measurement equation) multiperiod pollution level measurement control problem the control sequence (u,):. , which maximizes

subject to:

(i) ~ ( P t ) = ( @ r v ~ P t - l l @ i +na,)-(@tvlpr-l]@;+%,)

H:(U,) [H,(ut)(@lvcP,-, I@; + n,)H;(ut) + ot(ur)Q, o : (uo l -

x H,(u,)(@tVCf',- 1 I@: + no,)

V(P,) = 2, (known) (13)

t = 1,2, . . . , T are assumed to be known (15)

where v(P,) is the variance-covariance matrix of pollution level estimates at time t, and p = 1/(1 + o), w being the appropriate discount rate. This is a (non-linear) deterministic control problem where the elements of the variance-covariance matrix ~ ( p , ) play the role of states and the elements of vector u, act as the controls.

Consider now the pollution level measurement control problem which results when the environmental system process dynamics and/or the measurement equations are non-linear. In particular, consider the class of non-linear time-varying environ- mental systems and the class of non-linear time-varying measurements described by

Y, = F , ( P , , u,, t ) + Dt(u,)Vt, t = 1, 2, ..., T (18)

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Optimal measurement control strategies 205

where e, - N ( 0 ; fie,), Vt - N ( 0 ; q), Po - N ( F ~ ; rpo) and {e,, V,, Po) are mutually independent. (The zero-mean, whiteness, and mutual independence assumption on

' the environmental system and measurement error terms is not as restrictive as it may seem. Standard techniques are available to transform the non-zero mean, coloured error, correlated process dynamics and measurement error problem into the form of the problem stated above-see, for example, Sage and Melsa 1971.)'

The problem of estimating P, from observations up to time t in the context of (17) and (18), known as the non-linear filtering problem, is an important field of estimation and control (see Jazwinski 1970). Many different approaches have been taken, among which perhaps the extended Kalman filter (EKF) is the best known one. It is based on linearization of the state and measurement equations at each step and on the use of linear estimation theory (the Kalman filter). More specifically, for the system ( 17) and ( 18) the basic extended Kalman filter equations have the following form:

f?,t = E(P,IY') = ~P~(f 'r- , I t - l , G t , 0 + (C(~FP~C I/~P,-l)lP,.,-P1-,.-, I x [V(P,-1)IC(dFP1[ l /aPt-l) I~.- , - ,-,,,- 11'+%,)

x (CaFJ 31aPtl IP.- F~,~P,-..-,. ~ . . t ; 1 ) '

x {(I?F",( ) / ~ P l l l , - ~ ~ , , ~ ,-,,

x (C(dFP,( )/dP,-,)l,-,-~ a - , , #-,I x C~ (~ ' , - , ) IC (~FP .C~ l / ~ P , - l ) l , ~ , - ~ ,-,,,., I)+ R.,)

([aFul( )/aptl ~ P , = F ~ , ( P . ~ ~ ~ ~ ~ ~ , G ~ , ~ ~ ~ '

+ D,(~t)Rv,D:(ut))-'

( y f - [(FutL . 1) ~ P ~ - F ~ , ( P ~ - ~ ~ ,-,, G . , t ) l )

E(P0) = Po ~ ( f ' f ) = V(P,IYf) =(C(aFp1C. l / ~ P t - , ) l ~ ~ ~ ~ - ~ ~ ~ ~ ~ . ~ ~ l

x Cv(Pt-l)l[(aF,C - 1 / ~ P * - 1 ) l , ~ , - ~ , ~ ~ ~ , ~ , 3 ' +4) - ( [ ( ~ F P ~ [ l / ~ ~ t - l ~ l ~ . ~ ~ - ~ , ~ , ~ , ~ ~ l c ~ ~ ~ , - ~ ~ l

x C(~FP.C l / ~ P ~ - l ) l ~ ~ ~ ~ - ~ , ~ ~ ~ , ~ , l ' + ~ * ) x (CaF,( ) / ~ P ~ I I P ~ - F ~ , ~ P ~ - , ~ - ~ . G ~ , ~ ~ ) ~

{ (CaFut ( ' ) / a p t ~ ~ ~ f = F p , ( ~ ~ - l M - l , G r , , ) )

([(dF,C ]/apt-1) t P , - l - ~ l

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The solution of equation (20) requires the solution from the equation

Hence, although ~ ( p , ) = V(P, I V') does not depend on Y,, it depends on E( P, I Y' - I ) ,

which in turn depends on Pt - ,I,- = E(P,-, 1 Yr- ' ), and, of course, on the vector of pollution level measurement controls u,.

Again, assuming that the desired goal is the discounted monitoring cost minimiz- ation, the non-linear concentration optimization problem can be stated as follows. Choose the control sequence {u, I:=, > 0 to maximize ( 12) subject to (20) and ( 14). This, strictly speaking, is a non-linear stochastic control problem, since V(P,) = E(P, I Y t ) as given by (20) is a priori a stochastic process.

3.2. Linear pollution level measurement control problem 3.2.1. Optimal location problem. A method for the design of optimal sampling locations in a particular airshed will be suggested in this subsection. (Note that in this subsection only it is assumed that monitoring instrument precision as well as sequencing of pollution level measurements have already been determined. This assumption will -be relaxed in the next subsection.) More specifically, suppose that attention is limited to lo admissible monitoring configurations which have been chosen by the environmental management decision maker so that the airshed under consideration is adequately covered and that, at each period of time t, the environ- mental manager is able to choose only one out of the c0 possible measurement configurations. (It is important to realize that according to our assumptions here each pollution level measurement configuration can provide a group of noisy measure- ments at certain receptor locations. For example, suppose that we deal with a basic grid system where five potential pollution level monitoring Iocations have been selected as indicated in the diagram. Then, pollution level measurement configuration No. 1 may take measurements a t monitoring stations (1,3 and 51, while configuration No. 2 may yield measurements at all five stations and measurement configuration No. 3 may yield measurement only at station one.

Limiting our attention to c0 possible measurement configurations may be viewed as a two-part assumption: first, that monitoring must be restricted to a given region in the airshed (a matter of necessity); and secondly, that within that region they must fall at only to locations (a matter of convenience). It should be clear, however, that actual determination of the number of measurement configurations and their exact specifi-

East ( Miles)

Grid system for pollution level monitoring locations (monitoring station location denoted by 9.

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Optimal measurement control strategies 207

cation (number of monitoring stations, variables measured, instrument specifications, etc.), is not a trivial task and depends on many factors, including prior knowledge of the pollution levels in the airshed.)

Mathematically, we can express this as follows:

i 1, if the jth measurement configuration is chosen at time t "it = f nit = 1

0, otherwise j = 1

(21) I

Since (Ht(ut) Dt(u,)) = njtIijt{ / I) we can write the system of measure- I

ment equations (9, the present value of pollution level measurement costs (12), the environ~ental constraints ( 14), and the variance-covariance matrix of the pollution level estimates given by the Kalman filter as follows

( pt n ( = (Pollution level monitoring costs) t = 1

x (@*vcPt- ,I@; +a,) (25)

where use has been made of the fact that n;, = n,, and

n j , n j~ t (~ , t~@tv rP , - I]@; + Q,lHi, + = 0

since either nj, or nj., is zero, for all j Zj' . Note that V(Pt) in (25) requires the inversion of

instead of the measurement covariance Rvf. However, when ( @ , v [ P , - ~ ]a; +ad and Rvb are positive definite, using the matnx inversion lemma (Sorenson 1966), one can obtain from (25) the following

where use has been made of the fact nj, = nj, and

nit nit = (H;,fi;,l Hit) = 0

since either nj, or nl., is zero, for all j # j'.

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208 A. Papakyriazis

The problem of optimal location can now be formulated in a manner which suggests the use of Pontryagin's maximum principle. In particular, the multiperiod location problem considered in this subsection can be stated as follows: determine the control sequence {nj,; j = 1,2, ..., c,}:. , which maximizes

(see (23)) subject to:

(ii)

(iii)

1, if the jth measurement configuration is chosen at time t Ca

nit = 1 nit = 1 0, otherwise j = 1

(see (21)).

GI, nv,,, + t , Bty Hjt; j = 1,2, Co

t = 1, 2, . . . , T are known (27) The monitoring station location problem stated above is a deterministic (discrete

time) optimal control problem where the elements [ ~ ( p , ) ] ~ ~ . , {n, n' = 1,2, ..., N ) , of the variance-covariance matrix of pollution level estimates V(Bt) play the role of the state variables of a dynamical system whose equation of motion is governed by the matrix variance-covariance difference equation (26), the njt play the role of control variables, and the objective function and the static constraints depend upon the values of the control and state variables, njl, ([v(Pt)],), for j = 1,2, ..., Ca, n = l ,2 , ..., N , t = l , 2 ,..., T

If, instead of satisfying the deterministic equivalent of the probabilistic environ- mental constraints (24), the environmental management decision maker wishes to maximize the upper bound of the nth pollution level at time t (i.e. environmental standard P,,,, n = l ,2, ..., N) below which he can be sure that the nth pollutant concentration at time t, P,,, will fall with probability (1 - a,,), the optimal location problem can be reformulated as follows: determine the control sequence (njt; j = 1,2, ..., 1,):. , which maximizes

N { ( t = l [ j = l t j t { ~ ( ~ n t ) +~*(ant ) (var n- 1

subject to (26), (21) and (27).

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Optimal measurement control strategies 209

We shall use the discrete maximum principle of Pontryagin to derive a set of optimality (necessary) conditions. Before we apply the maximum principle, however, it is necessary to convert the state variable inequality constraints (24) to equality constraints (see, for example, McGill 1965). Towards this end, we may define a new variable by

where h(Kt + A*[ant] [(var ~p,,,))~] I t ' ) is a modified Heaviside step function defined such that

h(pnt + A * c ~ ~ , I [(var [ p R t I) j~ lJ2 ) 7

Kn z 0; n = 1, 2, ..., N J and where the initial condition is

It is clear then that v,, is a direct measure of the penetration of the state variable T

inequality constraint oO, = (oO, - v0(, - ,,). We shall require that the final value of r = l

oo, is zero: v O T = O (3 1)

which will impose the restriction that we do not violate the inequality constraint (24). We now define the real-valued function H, called the modified discounted

hamiltonian, as follows:

where {$,,; t = 1,2, ..., T) is the costate at time t corresponding to the state variable u,, and {Y,; t = 1, 2, ..., T} is the costate matrix (whose nn'th element is the costate which corresponds to the @l(pt)],. state variable; (n, n' = 1,2, ..., N) corresponding to the covariance matrix ( V C ~ ~ ] ) . Alternatively, the modified hamiltonian can be written as

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The properties of the trace function can now be utilized to obtain:

The necessary conditions for optimality of the monitoring station location problem can now be stated. In particular, if {n;; j = 1,2, ..., 1,) is the optimal control sequence, and if (v[P,]*), ( v & ) and (Y:) , ($&) are the optimal trajectories of the states and costates respectively, then the following conditions hold:

(i) Maximization of hamiltonian The inequality

must hold at each t = 1, 2, .. ., T and for all nj, E {0, 1).

(ii) Canonical equations

{v(Pt)* -V(Pf- I)*) = [ ( d H m d t [ l/dyt)I*I

= ( @ f v f ~ , - , ~ @ : + ~ e , - v ~ P , - l ~ ) - (@rv~Pt- ,~a;+n,) Dow

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(iii) Boundary conditions

The solution of the above set of necessary conditions establishes the optimal trajectory and optimal control for the monitoring station location problem. Needless to say, this will not be aneasy task because of the occurrence of a non-linear two-point boundary-value problem (TPBVP). The difficulties of solving the two-point boundary-value problems have led to a search for computational methods of a different kind which hopefully circumvent the problems associated with TPBVP solutions. Perhaps the commonest technique used for this purpose is the gradient method or method of steepest descent (see, for example, Bryson and Ho 1969). The gradient method consists of searching for an optimum by making each new iteration move in the direction of the gradient, which points locally in the direction of the maximum increase (or decrease) of the objective function. Standard gradient techniques, however, cannot be used for the location problem under consideration because of the constraints on the control variables {n,; j = 1,2, ..., [,}:= ,. However, of course, other computational techniques could be used to obtain numerical solutions for the optimal monitoring station location problem. The essence of one such technique is now discussed.

It is clear that we cannot require (28), (30) and (3 1) in a computational algorithm because there is no way to ensure that the constraint will not be violated. So we add the final values of the Vo as penalty functions such that the new objective function becomes

where K O is a suitably chosen constant and

The computation technique can be outlined as follows.

Step 1 . An initial guess is made for the values of nj, (say n i for all j = l ,2 , . . ., 5,; t = 1,2 ,..., T).

Step 2. Equation (37) is solved numerically forward in time using the initial condition (40) to give a trajectory v ( P , ) O , t = I , 2, ..., T

Step 3. Using the value of V(PT)O, the terminal value of t,boI, t,bgr, can be found from

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212 A. Papakyriazis

S t e p 4. The trajectories of PP and $8, can be found by solving numerically (38) and (39), backwards in time, starting at the known conditions YO, = 0 and

Step 5. The modified discounted hamiltonian (34) may now be used to determine new values for njt(say njr for all j = 1 ,2 ,..., lo, t = 1,2 ,..., T).

Step 6 . Steps (1)-(5) are repeated until a suitable criterion (based upon the change of the objective function with successive iterations, perhaps) is satisfied.

3.2.2. Optimal frequency and scheduling problem. The problem of optimal frequency and scheduling of pollution level measurements can be considered in the context of the following model

P , = @ , P , - , + B , G , + e ,

where H, is known, and

{urn,; m = 1,2, ... , M) can be either discrete or continuous scalars defined in the region of positive numbers. (In this subsection we assume that both the instrumental precision and the monitoring station locations have already been selected and the only variables that need to be selected are the frequency and scheduIing of measurements.) Of particular interest are the positive scalar measurement parameters {u,,; m = l ,2 , ..., M), which represent the accuracy of the observations at time t. I l one considers the case in which during the measurement period t (e-g. day, week, month, year, etc.) pCmt independent observations of a particular pollutant at a particular monitoring station are obtained and then averaged to obtain the observation:

then

is related to the number of samples, {pc,,; m = 1,2, ..., M), by the following relation:

Dt(ut) = diag b I , , u,,, . . . , u,,) = diag p;:I2 . . . p$i2) (47)

(If the measurement period is one day and p,,, = 12, for example, this would imply that observations of the mth state are taken every hour during the tth day.) It is noted that the variance-covariance matrix of the measurement noise becomes

E{( Dt L-~llVf) (Dt [utIVt)') = E{(~ctVt) (PC1 Vt)') = (~clfi"P;*)

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Opt irnal measurement control strategies 213

where p,, = diag (&:I2 . .. Equation (48), according to the above interpret- ation of {&,; m = l ,2, . .., M), indicates that the measurement variance-covariance matrix is inversely related to the number of samples taken at time t. Note in particular that (A,,,# = 0; m = L,2, . . . , M) implies that no observations whatsoever are made at time t, and (I,,= a ; m = 1, 2, ..., M) implies perfect observation of the states measured at time t. It is further to be noted that the above interpretations imply that the pollution level measurement costs in period r take the following form

Given the above definitions and assumptions, the frequency and scheduling control problem can be stated as follows:

subject to:

( 1 0 )

maximize: (- pt c ~ ~ [ ~ ~ ~ ] ) ( b ; m - ~ . Z . . . , M ) : . , t = l

(1 b) {V(Pt)1-' = {(@,vCpt- I]@; +%,I- ' + (H;CrO%.~~,3-' HI)}

{V(P,)] - = - (known) (51 b)

( 4) ; m = 1,2, ..., M ) ; , are positive integers ( 54)

Strictly speaking, the frequency and scheduling problem stated above is an integer (discrete time) non-linear deterministic control problem, since the pc,, m = 1,2, ..., M, 1,2, ..., 7j are required to take integer values only. However, in practice little is lost by treating the p,, as continuous in solving the above frequency and scheduling control problem and then rounding off. Hence, the monitoring problem considered in this subsection is a standard (discrete time) deterministic control problem. (In the control literature, such a problem is known as the fixed-time, free- end-point control problem-see, for example, A thans and Falb 1966.)

Applying Pontryagin's maximum principle to the non-linear measurement control

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214 A. Papakyriazis

problem defined in (49) -(54), one defines the scalar modified discounted hamiltonian by

~ c m d t ( f ~ ( ' t ) l , ocO~, v t l , $cOt, vfl~ pt)

= { - ~ ~ C ( C , , l l r c r l ) + t r I ~ ~ r ( @ , ~ C P t - l l @ ; + ~ , ) - ( V C P , - , I ) -(@,vrfj,-,l@; +fi,)(H:r~,(@,vrP,-,i@; +aa,) x H; + (~ctQv,CI;t)l - H')(@lVC',- , I@; + Q,,)1

+ ( $ o c r C ( p n t + A * f ~ n r I Cvar ( P n r [ ~ c t l ) ] ' I2 ) '

x h ( K + A * C a n t l Cvar (@.t~r,1)3112)1)1) where

and ((I,,,; (I,,,.,; t = 1,2, ..., T) are the costates corresponding to the state variables u,,, and {v[P,]).~ n = n' = 1,2, ..., N, n = n', respectively. If p: is the number of samples set which rhaximizes (50), and v Z t , {v[P~])&, are the corresponding optimal states, then there exists corresponding costates I(/%, and I),*,,., such that the following necessary conditions are satisfied:

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for all { p C m , ; m = 1,2, ..., MI bO.

( v ~ P t ~ * - v C P f - 11') = (LaHcmdt( 1.1 =([@,v(pf- l )@:+n,-~(Pt- , ) l -r@tv(P,-,)@:

+ n.llH:CHt(@,Vl?f- I@: + n,)W + (r,,fiV,r2l-'

~ , ~ @ t ~ ( f ' l - l ) @ : + Qe11)I*) (61)

( V;CI~ - ~ , *0 r - 1 ) = ( CaHodt ( 9 l a $ O c t l 1 * = + A*(am,)(var ~Pn t (~c t ) l ) '"1'

x h(P;, + A*Ca.tl Cvar (~nt[~ct])1~"1)1* (62)

(PYX -@:-I) = (CdHc,( )/a(VCP,-l1)11*) (63)

(~ll/csot - $%t- 1) = 0

V(P,) =fPb (given)

The necessary conditions specified by (59) -(65) form a two-point boundary-value problem (TPBVP) whose solution yields the desired optimum sample size values {k.,; m = 1,2, ..., M}:, ,. The numerical solution of such TPVBPs is a significant problem in its own right, especially in view of the non-negative constraint in the controls, {pcmt 20; m = 1,2, ..., M I L , , and the 'state' inequality constraints (52). Many techniques are available for determining such solutions (see Bryson and Ho 1969). In the remainder of this subsection we outline one such technique, the gradient method, as it applies to the measurement control problem under consideration. (For a description of various gradient methods, see Fletcher and Powell 1963, Sage 1968, Polack and Ribiere 1969, Perry 1976, and computer programs which embody several of these techniques, namely MINOS due to Murtagh and Saunders 1977, and LSGRG due to Mantell and Lasdon 1977.)

Again, as in the previous subsection, we cannot require (56) and (58) in the gradient procedure because we cannot be sure that the constraint will not be violated. Thus we add the final values of the uc, as penalty functions to obtain the following new objective function

where K: is a suitably chosen constant and

For convenience, the v are usually selected as square terms, although this is not necessary. The gradient computation then proceeds in the usual way. In particular, the main features of this computation can be indicated in the following steps.

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216 A. Papakyriazis

S t e p 1 A nominal set of non-negative control variables for the zero iteration is chosen, say

{ P : ~ ; m = 1, 2, ..., MI:= 2 0 Step 2

The control variables and the initial conditions (40) are used to solve the system (51) numerically forward in time. That is, p:o and ep0 are used in (51) to calculate V(Pl) . Then V(P,) and p:l are used to calculate V(P,), etc. Finally, at the terminal time 7; v(P,) is obtained and is used in turn in the terminal condition

P ' ~ : O T ~ ~ ~ / ~ [ ~ C O T ~ ) = $COT

to determine I/&,.

S t e p 3 The costate equations (63) and (64) are solved backward in time numerically

from period T to period 1 to obtain {YL, $Po,) through {Y:o, $$ ). At this point, all the first-order conditions are satisfied except the optimality conditions (dH,,,[ ]/dpc,,) = 0, m = 1,2, ..., M, t = 1,2, ..., T and even these may be satisfied. To check this, (dH,,,, [ . ]lap,,) is obtained for all time periods using the nominal control {P :~ , ; rn = 1,2, . . ., M):. , >, 0 and the state and costates calculated lrom them in the manner described above. Since usually not all the elements in (dHc,,,[ a ]/dc(,,), m = 1,2, ..., M, are sufficiently close to zero for all time periods, the problem is to move the controls in such a direction that the optimality conditions are more likely to be met on the next iteration. Towards this goal, we obtain the set control for the next iteration

where K, is the distance to move in the gradient direction. If some non-negativity constraints {P:~:., 2 0; m = 1,2, ..., M) are violated, we substitute the zero constrained boundary values for those time periods which have binding constraints.

Step 4 Once the new measurement controls have been obtained from (68) , the process is

repeated again starting with the system of state equations (51). The procedure continues until the optimality conditions are satisfied to the desired accuracy.

4. Non-linear pollution level measurement control problem In this section, the pollution Ievel measurement problem that results when the

environmental system model and/or the pollution measurement model are non-linear is formulated. A special case of the general non-linear pollution level measurement control problem, which is of particular interest for the study of linear systems with unknown dynamic parameters, is when the environmental system is described by equations

and

In particular, consider the following environmental system model:

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Assuming that the

are time invariant, we can adjoint the known parameter vector 8, to the environmental state PI, by defining a new state vector POI, to obtain:

We are consequently faced with a non-linear environmental system model and a linear pollution level measurement model; that is, we have

In addition, instead of considering the measurement control problem where monitor- ing costs are minimized subject to probabilistic constraints, the more general problem where the performance index of the environmental control agency is the weighted sum of a scalar-valued function of the pollution level variance-covariance matrix and ineasurement costs will be considered. (In general, the environmental control agency wants to consider a sequence of pollution level measurement controls {u,):=, . so as to minimize both the uncertainty in the estimate of pollution levels and the measurement costs. This may be done in three ways. First, the control agency may choose to minimize measurement costs subject to keeping uncertainty below a certain level-see 4 3. t and 6 3.2 above. Secondly, it may minimize the remaining uncertainty, subject to some upper bound on measurement costs. Finally, the environmental control agency may minimize the weighted sum of the remaining uncertainty and the measurement costs. Since, by appropriate manipulation of the weights of the remaining uncertainty and measurement costs, solutions can be obtained which are equivalent to the first and second approaches above (the weights taking on the role of lagrangian multipliers), only the third approach will be considered in this section.)

Since the variance-covariance matrix of pollution levels depends on the measure- ment controls, u,, it is reasonable to require that the estimation error covariance matrix is minimized. Obviously, the smaller the estimation error covariance the better are the estimates. However, as can be seen from (20), the variance-covariance of pollution level estimates is a matrix; and hence the selection of a suitable norm for optimization becomes very important. One may consider the following norms:

where tr ( ) = trace ( ), det ( ) = determinant ( ), and 9, is a weighting matrix.

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2 18 A. Papakyriazis

Of course, for optimization purposes, one may use any norm one desires. The approach taken in this section is to minimize tr (.~,D/(P,)]). This seems particularly appropriate in the context of our measurement problem since tr (9, CV(P,)]) is the weighted sum of the variances in the error of the state estimates. The weighting 9, in tr (9,[V(Pt)]) may result from a desire to estimate certain combinations of states accurately. (The pollution level measurement optimization problem in the case of linear systems with unknown (transfer) parameters can be formulated so that it is possible to design the measurement system with the objective of providing good estimates of the (unknown) model parameters 8 rather than-or in addition to-the pollution-levels, P. It is achieved through the weighting matrix, d in tr (G[V(P)]). Since the uncertain (unknown) parameters 0 can be included as states in the model, they can be weighted heavily in the objective function by use of the 9 matrix. This is a useful strategy when the measurement system is designed to collect data for the first time from a particular environmental system, and insufficient prior data exists for model specification. The first round of data collection is aimed a t model parameter estimation. After improving the parameter estimates, subsequent data collection efforts are then directed a t state-pollution level-estimation.) Suppose one is not interested in the estimates of the states (pollution levels) themselves, but desires to estimate the vector Zd, (damage functions) defined by Z,, = A,P, accurately. Each element of Zd, is a linear combination of the elements of P,. Note that A,, is time varying to indicate that the emphasis on estimation accuracy may change over the measuring period. For this case, the optimum linear filtered estimate t,, is given by

where P, is the optimal estimate of P,. Then the current error covariance for t,, denoted ~ ( t , , ) , is given by

Hence, the minimum of the sum of the error variances of estimates of Id, is obtained by minimizing

where 9, = (A&, A,,). The non-linear pollution level measurement control problem considered in this

section can now be stated as follows: determine the optimal measurement controls {u,)T-, such that the following objective function

is minimized for allowable u,, subject to the state equality constraint of (20), where A,, and A,, indicate appropriate weights. It is apparent that the solution of (20) requires the solution from (19).

A remark regarding the above non-linear measurement optimization is in order here. I t relates to the form of the objective function J, . In particular, J , is a function of the approximate error covariance matrix V(P,) given by (20); V(P,) is dependent upon E(P,l Yf- '), which in turn depends on the actual observations Y-'. Thus, v(P,) is a stochastic process a priori. This means that the objective function as written in

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Optimal measurement control strategies 2 19

equation (72) is not, strictly speaking, meaningful for the non-linear pollution level measurement control problem. At least this is true if we wish to perform the optimization a priori, that is, prior to taking the observations. (Of course, it makes absolutely no sense in the context of our discussion to perform the optimization after taking the observations. After all, the whole purpose of the optimization problem is to be able to take better observations. Hence, we must perform the optimization either a priori or in real time-sequentially.) Two ways around this problem are the optimal open-loop feedback (OLF) pollution level measurement control and the stochastic maximum principle (SMP) established by Kushner and Schweppe (1964).

4.1. Optimal open-loop feedback (OLF) pollution level measurement control One way around the problem of v(P,), and thus of J,, being stochastic is to

approximate the non-linear pollution level measurement control problem in real time, that is at the beginning of the measurement process, the environment control agency calculates a series of measurement controls, (up):=, , utilizing the prior guesses about Po (i.e. Po and Ep0). (Only the first of these measurement controls, uy, will actually be implemented, of course, but it is necessary to make tentative plans as to what will be done later in the measurement process in order to make an optimal choice of what is to be done in the first period.) As the pollution level observations of the first period, Y, , become available, the control agency decision maker estimates P , ,, , and hence P, ,, , and recalculates the optimum values for the remaining measurement controls, {u: ):, , . Thus, as the pollution level measurement process progresses, more and more measurements become available and the initial states may be replaced by better estimates, which in turn are used to update the design of the remainder of the pollution level measurement process.

Formally, the OLF pollution level measurement control procedure can be described as follows. Denote the current period by t. Let us assume that the pollution level measurement sequence {u: , u: , . . . , u: ) has been applied and the corresponding pollution level observation sequence (Y,)Z=, is obtained. We want to choose the future pollution level measurement sequence {u:+, , ..., u$J based on the information up to time t . Let us denote the optimization at this point as follows:

J,*,-, = Minimize: (J, = J,[(u;, u;, ..., u,*; u,+,, ..., u,; (u,+I, -.., ~ r )

{Pk = BkJ;i:; { Pj = kj):=t)]) (73)

subject to:

(a)

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A. Papakyriazis

(b) V( f',) = 2 , (given); V(PT) = cP, (unrestricted) (75)

where P, = E(P,JYt) is the estimate of P based on the pollution level observation (Yt = (Y, , Y,, ..., Y,)}. The minimization carried out by the above problem yields the following sequence of pollution level measurement controls

The optimal OLF pollution level measurement control sequence is obtained from (77) by choosing those measurement controls for which t = t'; that is,

{u: = (u,,,; t = 1, 2, ..., T ) ) (78)

4.2. Pollution level measurement problem: stochastic maximum principle and stochastic approximation solution

An alternative approach to the non-linear pollution level measurement control problem is to consider the following more suitable objective function:

and then apply the stochastic minimum principle (see Kushner and Schweppe 1964) to carry out the optimization. More precisely, the pollution level measurement control problem can be stated as follows: determine the measurement control sequence {u,}; , 2 0, so that (79) is minimized, subject to the variance-covariance (state) equality constraint (20). The indicated expectation in (79) are, with respect to the stochastic processes of the pollution level measurement problem, e, and V,. Of course, neither e, nor V, appear explicitly in the formulation of the problem, but the variables of the measurement control problem under consideration are functions of these stochastic processes. (Y, is a function of P,, which in turn depends on e,, and V,; p,., is a function of Y,; V(Pt) is a function of f',~,.)

Applying the stochastic maximum principle, we proceed to define the hamiltonian

The necessary conditions for optimality of the pollution level non-linear measurement problem can now be stated as follows: If (u:; t = 1,2, ..., T ) is the optimal measurement control sequence, and if (v[P,J*) and $: are the optimal state and costate trajectories, respectively, then the following conditions are satisfied:

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Optimal measurement control strategies 22 1

assuming an interior solution exists (i.e. u, > 0), or

The above conditions define a two-point boundary-value problem which may be solved by the use of standard computational techniques (see 4 3.2). In cases in which the equations from the stochastic maximum principle are impractical to solve, the following stochastic approximation method (see Sage and Melsa 197 1) may be used.

(a) Choose an initial sequence of measurement controls {ui; t = 1, 2, . . . , T). ( b ) From the known joint probability density of e, and V,, select a typical

sequence {ef ; t = 1, 2, . . . , T). (c) Solve the state equations dynamic system

CV(P,)Ii = F"(DI( Pt- 1 )li, 4 , ef, Vl, 1)

V(P,) = f, (given)

forward in time from 1 to T

(d) Solve the adjoint equation

(PYt - y t - 1 ) ' = -(affL*Cm I / ~ D / ( P t - I ) ] ' )

backward in time witb the terminal condition Y , = 0. (e) Determine a new control iteration from the stochastic approximation

aigorithm

where ( KU)' must satisfy certain stochastic approximation requirements (see, for example, Sage and Melsa 1971, Chap. 5).

(f) Repeat the computations starting with Step b.

The above stochastic approximation technique is iterative in nature and hopefully converges to the optimum measurement controls, u,.

5. Conclusion This paper is concerned with the optimal choice of pollution level measurement

controls. In particular, the development of optimization methods to aid in the selection of monitoring locations and frequencies of sampling for both linear and non- linear system and measurement models are considered.

A first-order conditional mean estimator is assumed. The linear problem is formulated as the minimization of monitoring costs subject to contraints on the pollution level accuracy and to the equality constraint of the error variance- covariance matrix difference equation. The necessary conditions for optimization result from an application of the deterministic maximum principle and the solution obtained is the optimum solution, except for numerical problems associated with the

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222 A. Papakyriazis

solution of two-point boundary-value problems. In addition, in the linear case the solution is deterministic and is completely specified a priori.

The non-linear pllution level measurement problem on the other hand is formulated as the minimization of an objective function which expresses the desired result in terms of the estimation error covariance and pollution level measurement costs. The optimization is subject to constraints on the measurement controls and to the equality constraint of the state variance-covariance matrix difference equation. The necessary conditions for optimization result from an application of the stochastic maximum principle of Pontryagin or the OLF control law. When the measurement system is non-linear in the states, however, the solution of the measurement problem is not deterministic and thus no a priori specification is possible. In this case, sequential computations are required because the two-point boundary-value problem, which depends on exact pollution level observations, is required to be solved sequentially.

The solution to the pollution level measurement problem presented in this paper provides a means of giving concise consideration to the varied and sometimes conflicting factors involved in determining the optimal allocation of measurement resources. Under the criterion of efficiency, there is no reason to expect the minimization of measurement costs and state uncertainty to be less important from the gains obtained from internalization of environmental externalities. Unfortunately, pollution level measurement considerations have thus far been ignored in the economic literature of environmental externalities.

ACKNOWLEDGMENT The author wishes to thank K. C. Kogiku, Adam Rose, Robert Beaver and

Payagio t is Papa k yriazis for helpful comments and suggestions.

Appendix In the formulation given in 5 2.3, equation (lo), the satisfaction of the probabilistic

constraints in (7) is considered as the occurrence of events (standards satisfaction) with their respective marginal probabilities (1 -ant), where n = 1, 2, ..., N. Alterna- tively it may be required that (7) be replaced by the following type of joint probabilistic constraint:

0 c a t < 1, where St = {(P,,, ..., P,,): P,, 2 P,, , , ..., P,, 3 P,,,)

and P, = (P,, , P,,, .. . , P,,) is an N-dimensional random vector, the components of which need not be independent random variables. Obviousiy, if the P,, components are independent random variables, the above constraint is equivalent to

Since Pt is an N-dimensional normal random vector with mean E(P,) and variance-covariance matrix E[(P, - ECP,]) (P, - ECP,])'] = Z,,, where IZPtl > 0 then there exists an orthogonal matrix A, such that

(AiEplA,) = diag (A,, , . . . , A,,) = 0,

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Optimal measurement control strategies 223

Without loss of generality, ( D; ' I 2 A&, At D; 'I2) = I,,,. Let P,, = (D; /2Ai P,), then E( P,,) = (D; 'I2 A: + ECP,]) and

EL( Pd, - ECPd,l)(Pd, - ECPd,I)'l = ECD; 112A;(Pt - ECP,l)(Pt - ECPtI)

x (P, - E[P,])'A,D; 'I2] = [D; ' I 2 ~ ; E ~ , ~ + D;'I2] = I

Hence, P,, - N(E[P,,], I) and

PA = (P,, - [D; '"A, E(P,)]) - N ( 0 , I, N )

so that

Formerly then the constraint Pr (P, 2 P,,) < a, is equivalent to

where P,, = ( D; ' I 2 A; P,,) and P, = (P ,,,, P ,,,, . . . , P,). However, in making the last environmental constraint operational, insurmount able computational difficulties arise from the process of evaluating the function of multivariate normal probability integrals. In fact, presentIy sufficiently fast numerical anIytica1 procedures and tables are available only for the bivariate normal distribution function in the literature, and very l i t t le or no experience exists in handling similar computations for N > 2. Alternatively, the joint probabilistic constraint Pr (P, 2 P,) ,< a, can be reformulated as follows:

( i ) K,, = (P,,, - ECP,,]), n = 1, 2, ..., N; and

(ii) [a ... j Pr(P,, . .PN,)d~l , . . .dPN,hat, where Pn,=Pn,-E(Pn,)', n = #* #dl

1,2 ,..., N.

Again, insurmountable computational difficulties arise from the process of evaluatine the second constraint.

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