PALM: FISHERY REGULATION VIA CONTROL THEORY 



specifying the components of the weighting ma- 

 trices. A common procedure for doing this is to 

 choose the components by the rule: 



Qn=- 



1 



Im 



R = 



1 



where .ri^ is the maximum desired magnitude of 

 the deviation .ri of the population A'', and )(„, is the 

 maximum desired magnitude of the deviation u of 

 the fishing rate./'. The components Q22 and Q33 are 

 chosen in a similar manner. Here we assume the 

 maxima are specified to be: 





5% deviation from A^^^ = 0.008 

 1% deviation from W^^^ = 0.008 

 50% deviation from/^q = 0.0025. 



Thus: 



Iw 



3m 



= 1.6 X 104 = 0.1 



Assuming that the variation .(•2 in the food density 

 is not of direct interest, we set Q22 = 0- Since ./ 

 from Equation (13) depends only on the relative 

 magnitudes of the weighting factors, we can 

 choose these factors to be: 



R = 1 

 Qii = Q33 = 0.1 

 Q22 = 0. 



For this three-variable model, the symmetric Ric- 

 cati matrix [P] has nine elements, three of which 

 are redundant. Computer solution of the six 

 coupled differential equations resulting from 

 Equation (A-9) and use of Equations (A-7) and 

 (A-8) yield the following feedback control func- 

 tion: 



u = -[K]x = 0.149.ri + 0.00027.r2 + 0.026.r3. (14) 



Use of Equation (14) and the definitions of x^, x^, x^, 

 and u. gives the optimal fishing effort as a feed- 

 back function of the system variables: 



/=/eq + 0.149 (iV-iV,q) 



+ 0.00027 {E - E,^)+ 0.026 fPT - W^^). 



Substitution of the equilibrium values gives: 



/= -0.045 + 0.149iV + 0.00027£' + 0.026^^.(15) 



Substitution of u from Equation (14) into 

 Equation (12) gives the set of linearized equations 

 describing the behavior of the model under feed- 

 back control. The matrix [A] to be used in Equa- 

 tion (A-5) becomes 



[A] = 



-0.119 -0.00027 0.534 



-5.73 -0.12 -0.81 



0.14 0.02 -2.02 



The roots of Equation (A-5) are: .'^■ = -2.09, -0.096 

 ± 0.17j, where j = \fA. The dominant time con- 

 stant is the negative reciprocal of the least negative 

 real part, and here is equal to 1/0.096 = 10.4 time 

 units. In a similar way the dominant time constant 

 for the system without feedback is 45.5 time units. 

 Thus the feedback control given by Equation (15) 

 reduces the effects of disturbances in one-fourth 

 the time. These linearized results have been 

 verified by simulation of the original nonlinear 

 model. Other simulations are discussed bv Palm 

 (1975). 



Before concluding this example, we note from 

 Equation (15) that ./' is a function of all three 

 variables. This is due to the coupling between the 

 three equations. Also, although the choice of the 

 weighting factors is somewhat arbitrary, this 

 should not obscure the fact that the Linear- 

 Quadratic Optimal Control Problem provides a 

 systematic method for determining the feedback 

 gain matrix [A^. A systematic approach is needed 

 because the number of components of [K] becomes 

 so large for multivariable problems that a trial- 

 and-error approach is prohibitive. As long as [Q] 

 and [R] are chosen to be positive-definite, the 

 resulting [K\ will stabilize the system. Various 

 choices of [Q] and [/?] merely affect the time con- 

 stants and form of response (oscillatory vs. non- 

 oscillatory return to equilibrium). This is the main 

 advantage of this technique. 



With this model the effects of mesh size regula- 

 tion can be studied by using Wh as an additional 

 control variable. Also, the food supply rate a is 

 another possible control variable if the model is 

 used to analyze fish farming. The linear-quadratic 

 control technique could be used in both cases. 



CONCLUSION 



In this introductory paper we have presented 

 only the deterministic case of the Linear-Quadrat- 



835 



