Modelling, Filtering and Control for Water Pollution

MODELLING, FILTERING AND CONTROL FOR WATER POLLUTION R. F. Curtain, A. Ichikawa and E. P. Ryan Control Theory Centre, University of Warwick, Coventry CV4 7AL, u.K.

ABSTRACT In this paper we model the concentration z(t,x) of a pollutant at time t and at distance x along a river as the solution of a stochastic partial differential equation of second order. The pollutant is supposed to be discharged at finitely many sites along the river as a compound Poisson process in time. We derive the best linear least squares filter for the profile z(t,x) based on noisy measurements of the concentration at certain measuring stations along the river. A numerical example is presented to illustrate our approach to the filtering problem and a control scheme is briefly discussed.

locations and we take the observations at finitely many sites continuously in time. Earlier models in [2] and [SJ could not model point observations or point inputs and to make this extension we use the recent theory of

[3] .

In §2 we briefly develop the abstract mathematical background from [lJ-[3J and illustrate its use in pollution modelling by a particular river pollution example. In §3 we present computational results for this example. Finally in §4 we discuss extensions of the approach. 2. MATHEMATICAL FORMULATION Our general pollution model will be the following

1. INTRODUCTION

t

The problem we wish to consider is that of obtaining the profile of pollutant in a river, bay or estuary based on continuous measurements of the concentration at certain observation sites. The pollution of a river, bay or estuary has been modelled by a linear partial differential equation with the pollutant dischar~es appearing as a forcing term [SJ, L9] , [10] ,

z(t) = TtZo+ JTt_sB dq(s) (1) o where Tt is a strongly continuous semi-group with generator A on a real separable Hilbert space H, ( ~ ,L,~) is an underlying probability space, K is a real separable Hilbert space, B E ;e(K,H), z E L2 ( n ,~;H) and q is a K-valued stocRastic process given by 00 q(t)= L q. (t)e. (2) i=l l l where {e.} is a complete orthonormal basis l for K and q. (t) are real stochastic processes l with expectation function v.t and joint covariance l function p .. t. The stochastic integral is lJ defined by toot JTt_sB dq(s)= L JTt_sBe i dqi(s) o i=l 0

ell] , [13] , [l4J , [l6J .

For most pollution problems it seems natural to model the pollutant discharges as a Poisson type stochastic input, which gives rise to a special type of stochastic partial differential equation only recently introduced in [1]. The problem of determining the concentration profile is a distributed filtering problem and much work has been done on this for Gaussian white noise (see [4]}. The theory has been extended to allow for Poisson type noise in [2J and this was applied to a water pollution model in [5], which had previous ly been studied in [12J using a finite dimensional approximation. Here we describe a general method for obtaining the concentration profile for rivers, streams or estuaries which can be modelled by a linear partial differential equation. We suppose that the pollutant is discharged at finitely many

t

where JT Be. dq. (s) can be defined -s l l o t for q. a real orthogonal increments proce§s as in Doob [8]. The process (1) is motivated by the stochastic partial differential equation dz(t)= Az(t)dt + Bdq(t); z(O)=zo (3) and for conditions for (1) to be the solution of (3) see [1]. To see how (3) can represent a stochastic partial differential 351

352

R. F. Curtain, A. Ichikawa and E. P. Ryan

equation consider the following model for river pollution: az(t,x)= Da2z(r.~)_ vaz(t,x) at ax 2 ax

In [3J it is shown that (1) remains well-defined for unbounded B on K provided there exists a Banach space X=:>H such that TtE.e(X,H) for t > 0, BE -.t(K,X) and IITtzllH'$ gl (t) Ilzllx for all ZEX where gl E L2 (O,T)

+ t(t,x)

az(t,x)= 0 at x=O, £ ax

M.oreover Z(t) E C(O,T; L 2 ( rI , l.I ;H)), i.e. it is continuous in mean square. For our example above if we choose

(4 )

where z(t,x) is the concentration of a pollutant, D the dispersion coefficient, v the water velocity ~nd is the rate of increase of concentration at (t,x) due to the deposit of wastes. For ~ = 0, (4) can be reformulated on H = L 2 (0, £ ) by

X=(H~+ E (O, £ ))*, the dual of the Sobolev space H~+ E (o, £ ) with E> O arbitrary,

t

.

z = Az ;

then B .E ct(R,X) and Tt E ~(X,H) for t > 0 J with the estimate c 11 Tt Z 11 :0: ~ ~ 11 Z 11 X ' c > 0 L2(0, £ ) t 4 +, E and so (7) is a well defined L 2 (0, £ )valued stochastic process.

z(o)= Zo

where A is the following differential operator on H

a2

a

D- - v ax 2 ax

A

We now suppose that we have noisy measurements of the form

(5)

t

with domain

~(A)={h E H:

y(t)= ah, a h EH; ah=o at x=o, £ } ax ax 2 ax

z(t)= TtZ o + fTt_s dq (s) o

(9)

h(ck)

(6 )

where q(t)= L: q . (t)e., fe. } is an i=l 1 1 1 orthonormal basis for L 2 (0, £ ) and q. (t) are real compound Poiss o n proc§sses. A more realistic model is to suppose th a t the p o llutant discharges occur at finitely many sites bl, ... ,b along the river and that the m discharging occurs according to a compound Poisson process. Formally, this means taking m = L: 0 (x-b. )

q. . (t) j=l J -J which leads us to the integral model m t z(t)= TtZ + L: fTt B. dq . (s) (7) o j=l 0 -s J J where B.= o (x-b . ) is an unbounded

- E

,b j +sJ

IITthl1

Y

$

g2 (t) Ilhll ' H

g2,glg2 E L

2

(0,T).

'1\

J

[b j

for t > o with the e stimate

Moreover, unde r these conditions on Tt and C, we have t t C fTt_sf(s)ds = fCTt_sf(s)ds o 0 for all f E L 2 (0,T;H). k+ In our example if we take Y=H' E (O, £ ) then E t:(H, Y), C E .;e (Y, Rk) and c IITt hll ~ ~ Ilhll Y t 4 ,E H

J

E:

The operator C is linear but unbounded and uncloseable, so (9) needs some further justificatio n and this is done in [3]. It is shown that for Z (t) in (1) t y(t)= f Cz(s)ds + n et) (11) o is we ll defined provide d there exists a Banach space Y dense in H with H =:> ~(C) =:>Y a nd CE '£(Y,Rk ), Tt Ei(H,y)

where

uncloseable operator on H. Alternatively, we c o uld suppose that the discharging occurs over a small but finite region [b . - E , b . + E] and define B . E ~(R,H) byJ J X

+ n et)

,,,here y (t) EO Rk, n (t) is a k-dimensional Wiener process with covariance V and C : H ~ Rk is the following obp servatio: ma ( h .( C )) l Ch = . 0 $ C i $. £ (10)

t

J

'2(s) ds

o

2

and A generates a semigroup Tt on H. In [5] it was assumed that the pollutants were deposited along the length of the river according to a Poisson law , with random amounts of deposit, and the following abstract model was obtained

t (t , x)

f c

We can now pose our general filtering problem for (1) and (11): To find the best least square estimate z(t) of the state z(t) based on the observation yes), 0 s s ~ t, in the class

(8)

elsewhere in which case (7) is well-defined.

t

z (t) =

SK (t , s) dy (s) o

352

+ Z (t)

(12)

353

Modelling, filtering and control

where K(t,·)€L 2 (O,T; ~(Rk,H)) and z(t) is a deterministic function. That is to find K(t,·), z(t). such that E{2} IS minimized for all h€H. This is solved in [2J for bounded Band C and for unbounded Band C the following result is derived in [~ under the foregoing assumptions: The optimal filter is given by K(t,s)= U(t,s)P(s)C* t

(l3)

z(t)= U(t,o)zo+ jU(t,s)Bf(s)ds o where U(t,s) is the unique solution of the integral equation U(t,s)h = Tt_sh t

j u(t,r)p(r)c*v-1CT r-s h dr s

(14 )

Z

E{Z}, f(t)= E q(t)

, and

o 0 P (t) € £(H)n £(H, y)n £(X,H) is the unique solution of the Riccati equation p(t)h = TtPoTth + t

jT _ (BWB*-P(s)c*v-lcp (S))T*t_s h ds o t s (15 ) P is the covariance of z , W is the iRcremental covariance ofOq(t) and pet) is the covariance of the error e (t) =z (t) (t), that is

-z

P = E{(z -2 )o(z - z ) } 00000

(16)

W(t-s)= E{[q(t)-q(S)-E{q(t)-q(s)}]o [q(t)-q(S)-E{q(t)-q(s)}]

f

P (t) = E He (t)-E{e (t))] ° [e (t)-E{e (t) }]f (17 ) and (uow)h = u for all u,w,h E H. We make the cautionary remark that the operators B, C and pet) in (13) (15) must be interpreted carefully for these equations to make sense. For example, Tt B €.;t(K,H), -s CP(s)E~(H,Rk) and so on. If we can choose where to make the observations, it ~s reasonable to ask where the best location is and this problem was examined in [7J, by finding the location which minimized the trace of the covariance of the error. For this problem there is always a best site for observations, although this is not necessarily unique. Returning to our particular example, we observe that glg2 i L 2 (0,T) and so we cannot have point observations and pollutant discharges at points simultaneously. But if we use (8) to define B as a bounded operator we can solve the problem for finitely many point observations.

IFAC S.E.S.- M

3. COMPUTATIONAL RESULTS To illustrate the preceding theory we consider the stochastic model m

dz (t)= Az (t)dt + L 0 (x-b. )dq. (t) i=l ~ ~ (18) €[o,tJ with A and ~CA) defined as in (5). We select the following parameter values: 2 ~ = 32kmi D=2.5km /daYi v=3km/day and assume two pollutant input points (m=2) at b = 0.125~ 4kmi b2=0.25~ = 8km l We model the pollutant discharges by the compound Poisson processes q. (t) , i=1,2, with E qi(t) = Ait~il and~ X

E(qi (t)-Ait~il)2 = Ait~i2

where

Ai (i=1,2) are the Poisson rate parameters with numerical values -1

-1

Al= 0.3 day i A2 = 0.5 day We assume that the jump sizes are uniformly distributed with first and second moments ~l2= 108.33 ~ll= 10 ; ~21= 8

~22=

69.33 these values correspond to jump sizes uniformly distributed over the intervals [5, l5J and [4, l2J respectively. By adopting a finite difference approximation to the differential operator A we reduce (18) to an approximating set of 32 linear ordinary differential equations. We consider the filtering problem over the finite time interval [O,T] with T=7 days. For the observation process we assume a single measuring station Ck=l) located at x=C€[o,~J (referring to (9) we can formally identify the operator C with o(x-c)) with measurement noise covariance V = 4. We seek to determine the location c* which minimizes the trace of the covariance of the error of the estimate, viz. tr p* = tr P(T,c*)= min {tr P(T,c)} CE[O,9,J For different values of c, the approximated state equation (18), Riccati equation (15) and filter equation(12) were integrated using a fourth-order Runge-Kutta method, with initial conditions zo= 2.5 and Po= ~I. Figure 1 depicts the normalised profile tr P(T,c)/ tr p* as a function of measurement location c. As might be expected the optimal location c* = b = 0.259, = 8km 2 coincides with the pollutant discharge point which lies furthest down-

R. F. Curtain, A. Ichikawa and E. P. Ryan

354

stream. In many practical applications this result might not be implementable; in such cases the best strategy is to locate the measuring station at the nearest admissible location downstream from the discharge point. For location c = 9km, i.e. for a single measuring station located lkm downstream from discharge point b 2 , Figure 2 shows the actual and estimated concentration profiles at time t=T (=7 days). Figures 3,4,5 depict the local transient behaviour of concentration and estimate over the interval [O,TJ at points (i) x=xl=c 9km (Figure 3); (ii) x = x 12km (Figure 4); 2 (iii) x = x3 16km (Figure 5). A more detailed presentation of theoretical results and further numerical examples are contained in [17J. 4. EXTENSIONS The abstract theory in [lJ-[3J can be applied to any time-varying linear system or any system with space-varying coefficients and the best linear filter can be obtained in a similar manner. Prediction and smoothing ~roblems can also be solved as in L2J,[3J. In some applications such as 00/ BOO models [9 J, [lOJ , [l1j , [13J , [14J , f15l, it is possible to control the concentration, and [6J suggests a method of attack. Although the separation principle does not hold for non-Gaussian processes, a good suboptimal control design is one based on a feedback law of the filter. This necessitates the solution of a second Riccati equation of type (15). We can allow for control action at discrete points. If there is a delay in observations, as in the DO/BOO case, [13J, [14J, it is natural to take a feedback law of the predictor. REFERENCES 1. R.F. Curtain, Stochastic Evolution Equations with General White Noise Disturbance, to appear in J.Math. Anal. Appl .. 2.

, Estimation Theory for Abstract Evolution Equations Excited by General White Noise Processes, SIAM J. Control & Opt., 14 (1976), pp. 1124-1150.

3.

, Linear Stochastic Control for Distributed Systems with Boundary Control, Boundary Noise and Point Observations, Control Theory Centre Report, No.46, University of Warwick, 1976.

4.

, A survey of Infinite Dimensional Filtering, SIAM Review, 17 (1975), pp. 395-411.

5.

, Infinite Dimensional Estimation Theory Applied to a Water Pollution Problem, Proc.7th IFIP Conference on Optimization Techniques, NiGe, 1975; Lecture Notes in Computer Science, Springer 1976.

6. R.F. Cu~tain and A. Ichikawa, The Separation Principle for Stochastic Evolution Equations, SIAM J. Control & Opt., 15 (1977), to appear. 7. R.F. Curtain, A. Ichikawa and E .P. Ryan, Optimal Location of Point Sensors for Distributed Systems, Control Theory Centre Report, No.50, University of Warwick (1976); Proc. IFIP Working Conference on ~odelling and Identification of Distributed Systems, Rome, 19/6. 8. J.L. Doob, Stochastic Processes, John Wiley, 1953. 9. W. Hullett, Optimal Estuary Aeration; An Application of Distributed Parameter Control Theory, J. Appl. Math. & Opt., 1 (1975), pp. 20-63. 10 R.K. Jain and ~.M. Denn, Short-Term Regulation of BOO Upsets in an Estuary, J. Dynamic Systems, Meas. and Control (ASME), 1976, pp. 30-31. 11 A.J. Koivo and G. Phillips, Optimization of Dissolved Oxygen Concentration in Polluted Streams, IFAC Congress, Boston, Aug. 1975. 12 H. Kwakernaak, Filtering for Systems Excited by Poisson White Noise, Proc. IRIA Int. Symposium on Control Theory, Numerical Methods and Computer Systems Modelling, Rocquencourt, France, June 1974. 13 M.K. Ozgoren, R.W. Longman and C.A. Cooper, Stochastic Optimal Control of Artificial River Aeration, Proc. 1974 JACC, Texas, U.S.A. pp. 235-245. 14 H.J. Perlis and B. Okunseinde, Multiple Kalman Filters in a Distributed Stream ~onitoring System, Proc. 1974 JACC, Texas, U.S.A., pp. 615-623. 15 P.E. Young and B. Beck, The Modelling and Control of Water Quality in a River System, Automatica, 10 (1974), pp. 455-468. 16 N.L. Nemerow, Scientific Stream Pollution Analysis, McGraw-Hill, 1974. 17 R.F. Curtain, A. Ichikawa and E.P. Ryan, Modelling, Filtering and Control for Water Pollution, Control Theory Centre Report, University of Warwick (1977).

Modelling, filtering and control

355

2

tr P(T,c) Figure 1

tr P*

1 0

4

12 16 location c

8

20 (km)

24

28

32

6.5 <>I.

'"

tJI

El

Figure 2

x, -x

z (T ,x)

Eo<

Eo<

-'

-----

N
0.0

24 16 distance x (km)

8

0

Z (T, x) 32

7.0 <>I.

'"El

tJI

Figure 3

.-I

.-I

x x -I-l

z(t,x l )

-I-l

------- . ~ (t,x l )

N
at x = 9km 1

0.0 0

1

2

3

4

time

5

6

7

(days)

6.5 <>I.

~ El

N

Figure 4 N

(t ,x ) 2 -------~ (t,x 2 )

X

, . X,

-I-l

Z

-I-l

N
at x = l2km 2

0.0 1

0

2

4

3 time

5

6

7

(days)

8.0

... -----Figure 5 ~ ~

1---------------------

- - - z et,x 3 )

N
------ 2 (t,x 3 )

O.O+---~--~--~--r_--r_--r_-~

o

1

2

3

4

time (days)

5

6

7