Optimization Based Approach for Constrained Optimal Servo Control of Integrating Process Using Industrial PI Controller

Optimization Based Approach for Constrained Optimal Servo Control of Integrating Process. Using Industrial PI Controller Nguyen Viet Ha, Moonyong Lee*. Abstract—Constrained optimal control can be remarkably. challenging to control. In this paper, optimization based approach for constrained optimal control of the integrating process is investigated through inventory control from using of the PI controller in the servo problem. The Lagrangian multiplier method is applied to find the PI parameters by solving the constrained optimal problem. The developed method explicitly deals with the important control constraints as well as minimizes the optimal performance measure. The method can be applied for any general integrating process.. Keywords—Constrained optimal control, Integrating process, Inventory control, PI controller tuning, Servo control. . I. INTRODUCTION Some processes where the streams are comprised of gases, liquids, powers, slurries and melts do not naturally settle out at a steady state operating level, which refer as the non self-regulating or more commonly, as integrating processes. In process industry, many of the level, temperature, pressure, pH and other loops have the characteristics of the integrating processes [1]. In practice, a liquid level control system with a pump attached to the outflow line contains the integrating process and can be represented as a typical integrating model [2]. Another example of an integrating process is pressure control for a vapor system. In modern control, the integrating process also appears in the some applications such as the space telescope control system, the lightweight robotic arm and pilot crane control system, etc., [3]. In most research, the disturbance change referred to as the regulator problem has been studied [4], [5]. However, in some cases such as the reactor liquid level in chemical process [6], the Hubble telescope pointing controls and the control system of the disk drive head reader [3], both the regulator and servo problems need to be concerned.. ( )spY s c. I. K sτ. Dk s. pk s. cK. ( )U s ( )Y s. ( )D s. +. − + +. +. + ( )spY s. c. I. K sτ. Dk s. pk s. cK. ( )U s ( )Y s. ( )D s. +. − + +. +. +. In this paper, an analytical design method for PI controller is developed for constrained optimal control of integrating process. The constrained optimal control associated with the three constraints is formulated and then converted into a simple form with two independent variables by using a proper variable transformation.. Manuscript received December 30, 2008. This work was supported by Yeungnam University. 2008. * Corresponding author Nguyen Viet Ha and Moonyong Lee are with the School of Chemical Engineering and Technology, Yeungnam University, 214-1 Dae-dong, Gyeongsan , 712-749, South Korea. (Tel/Fax: 053-810-3241/811-3262; Email: mynlee@yu.ac.kr ). The Lagrangian multiplier method is utilized to handle the constrained optimization and then the PI parameters are found from the analysis of the global optimal condition. Liquid level control system is used throughout the case study section as a typical example for the constrained optimal servo control of the integrating process. The proposed method has explicitly brought out the way to find the optimal PI parameters in inventory control.. II. CONTROL SYSTEM DESCRIPTION . The control system presented in Fig. 1 is simply described as:. ( ) ( ) ( )pD kk. Y s D s U s s s. = + (1). To avoid the proportional kick for a servo problem, a modified PI controller is widely accepted as a practical solution. . ( )( ) ( ) ( ) ( )cc s I. K pK Y s Y s Y ssτ. = + −U s (2). where and cK Iτ denote the dimensionless proportional gain and the integral time constant, respectively. The closed-loop transfer functions for the level control system are. 2 2. 1 (s)= ( ) ( ). 1 1 D H I. sp H I I H I I. k s Y D s. s s s s Y s. τ τ τ τ τ τ τ τ. + + + + +. (3). H 2 2. H H. 1. 1 1. ( ) ( ) ( ) ( )D c H I c sp. I I I I. k K s K s U s D s Y s. s s s s +. = − + + + +. τ τ τ τ τ τ τ τ τ. (4). where H 1. p ck K −. =τ (5). Fig.1. Block diagram of the closed loop feedback control of. the integrating process . Proceedings of the International MultiConference of Engineers and Computer Scientists 2009 Vol II IMECS 2009, March 18 - 20, 2009, Hong Kong. ISBN: 978-988-17012-7-5 IMECS 2009. mailto:mynlee@yu.ac.kr. the damping factor of the closed-loop transfer functions is . 2 1. exp( 1) 1. 2. 1 tanh exp 1. 2. for. x x for. x. −. − = =. ⎛ ⎞− = − >⎜ ⎟. ⎝ ⎠. ζ. ζ. H. 1 2. Iτζ τ. = (6) . III. FORMULATION OF CONSTRAINED OPTIMAL CONTROL where. 21 0 1x for. ζ ζ. ζ −. = < ≤ The control objective in the optimal control is to minimize the rate of change of the manipulated variable, , and the controlled variable, , whereas operating under the following constraints:. ' ( )u t ( )y t. 2. 1 1. x for= > −. ζ ζ ζ. Applying the Lagrangian multiplier with slack variables converts the constrained optimization problem in (8a)-(8d) into an unconstrained equivalent problem with an augmented objective function, as follows:. (1) The maximum allowable controlled variable, ; maxy (2) The maximum allowable rate of change of the manipulated variable, ; 'maxu 1 2 3 1 2 3. 2 2 1 14. 2 2 2 2 3. min ( , , , , , , , ) 1. (1 4 ) ( ( ) ). ( ( ) ) ( ( ). H. H h. g H f. L. h. g f. τ ζ ϖ ϖ ϖ σ σ σ. 2. 3 ). ϖ τ γ ζ σ. ϖ γ ζ σ ϖ τ γ ζ σ. Η 3 Η. = ατ + ζ + β + − − τ ζ. + − − + − −. (14) (3) The maximum allowable manipulated variable, . maxu Therefore, the constrained optimal control problem can be defined as finding the controller parameters that minimize the performance measure in (7a) for a given , subject to. the constraints in (7b)-(7d): /spy s∆. where iϖ is the Lagrange multiplier ( 0iϖ ≤ ), and iσ is a slack variable. . ( ) ( )22 'min ( ) ( )y uy t dt u t dtδ ∞ ∞. 0 0 Φ = ω +ω∫ ∫ (7a) . The necessary condition for an optimum solution of Eq. (14) is:. subject to max( )y t y≤ (7b). '( ) 'maxu t u≤ (7c) 2 1 34 4. 3 (1 4 ) 2 0H. H H. L β α ζ ϖ τ ϖ. τ τ ζ ∂. = + − + + = ∂. (15) max( )u t u≤ (7d). ' ' ' 1 2 33 5. 4 8 ( ) ( ) ( ) 0(16)H L K. H. L h g k. β ατ ζ ϖ γ ζ ϖ ζ ϖ γ ζ. ζ τ ζ ∂. = − − − − = ∂. 2 2 h 1. 1. ( ) 0H L. hτ γ ζ σ ϖ ∂. = − − = ∂. (17). where ; is a weighting factor. ( ) ( ) ( )spy t y t y tδ = − ω After some mathematical manipulations, the above. constrained optimal control problem can be expressed in terms of Hτ and ζ as follows: . 2 H 3 4. 1 min ( , ) (1 4 )H. H. τ ζ ατ ζ β τ ζ. Φ = + + (8a) 2 2. 2. ( ) 0 (18)g L. g ∂. = − − = ∂. γ ζ σ ϖ. 2 f 3. 3. ( ) 0H L. fτ γ ζ σ ϖ ∂. = − − = ∂. (19) subject to ( )g gγ ζ≥ (8b). 2 H h . ( )hτ γ ζ≥ (8c). 1 1 1. 2 0 L. ϖ σ σ. ∂ = − =. ∂ ; 2 2. 2. 2 0 L. ϖ σ σ ∂. = − = ∂. ; H f . ( )fτ γ ζ≥ (8d). where 2( ) 2. y spyα ∆. ω = ;. 2. 32 spu. p. y k. β ω ⎛ ⎞∆. = ⎜⎜ ⎝ ⎠. ⎟⎟ (9). 3 3 3. 2 L. ϖ σ σ ∂. 0= − = ∂. (20). max g. sp. y y. γ = ∆. ; ' max. h sp. p. y k u. γ ∆. = ; max. sp f. p. y k u. γ ∆. = (10) . ( )g ζ , ( )h ζ and ( )f ζ are given by. ( ) 1 exp 0 1. 1. g x. for. ⎛ ⎞= + − < <⎜ ⎟ ⎝ ⎠. =. π ζ ζ. ζ 1. for. ≥. (11) . The solutions of (15)-(20) for 0ϖ = , 0σ = , 0ϖ ≠ , and 0σ ≠ are associated with the possible optimum cases. Fig. 2. shows seven possible instances of a global optimum with the contour of the objective function and the constraints in (8a)-(8d). . 2. 1 ( ) 0. 4 h ζ. ζ = for ζ > (12). 2 11 tan ( ) exp 0 1. 2 x x. f x. −⎛ ⎞+ = − <⎜ ⎟. ⎝ ⎠ ζ ζfor < (13). Notice that the constraint by (8b) is vertical line in the ( , )Hζ τ space and the constraints in (8c) and (8d) bring the curves with a similar shape. It is clear that, the feasible region given by constraints , and with any positive value is always available and unbounded. The global optimum of. maxy 'maxu maxu. ( , )Hζ τ for each possible case is evaluated as follows:. Proceedings of the International MultiConference of Engineers and Computer Scientists 2009 Vol II IMECS 2009, March 18 - 20, 2009, Hong Kong. ISBN: 978-988-17012-7-5 IMECS 2009. Fig.2. Typical contour and constraints with seven possible cases of optimum location. Case A ( ): The global optimum,. denoted by 1 2 3 0ϖ ϖ ϖ= = =. † †( , )Hζ τ , is in the interior of the feasible region.. This case occurs when [ † †max( ( ) , (H h fh f≥τ γ ζ γ † ))ζ ]. and [ †minζ ζ≤ ]. The global optimum is calculated by using (15) and (16) as follows: . . 1 4† 4. H. β τ. α =. ⎛ ⎞ ⎜ ⎟ ⎝ ⎠. ; † 1 2. ζ = (21). It should be noted that the optimal damping coefficient †ζ is independent of the process dynamics and weighting. factor. Case B ( 1 2 3 0σ ϖ ϖ= = = ): The global optimum, denoted. by ( ,h hH )ζ τ ∗ ∗ , is located on the constraint 2 ( )H h hτ γ ζ= .. This case occurs when the following conditions are satisfied: † † † and max( ( ) , ( ))H h fh f<τ γ ζ γ ζ minmax( , ). h hf∗ >ζ ζ ζ. Then, from (15)-(17), the following equation is given.. 2 2 5. 2 '. 2 3 4. 4 8. ( ) ( ). 3 (1 4( ) ( )( ). 2 ( ) ( ) 2 ( ). h h h. h. h h. h h h h. h. h h h. ∗. ∗ ∗. ∗ ∗. ∗ ∗ ∗. − −. + − =. β αζ. γ ζ ζ. β α ζ ζ. γ ζ ζ ζ ). 0 (22). hζ ∗ can be calculated by using any simple root finding method from (22). . Finally, hHτ ∗ can be accordingly found by using (17) as:. ( ) 1 2( )hH hhτ γ ζ. ∗ ∗= h (23). Case C ( 1 2 3 0ϖ σ ϖ= = = ): The global optimum, denoted. by * *( , )g gHζ τ , is on the constraint g( )g ζ γ= . This case. happens when the conditions [ †minζ ζ> ] and [ max( , )g gh gfH H. ∗ > Hτ τ τ ] are satisfied. . The minimum allowable damping factor, minζ , in the feasible region is calculated from (18). gζ ∗ is equivalent to. minζ . g. Hτ ∗ is obtained from (15) as follows:. min gζ ζ∗ = ;. 1. 4 2 4 min min. 3 ( ). (1 4 ) g. H. β τ. α ζ ζ ∗ =. + (24). Case D ( 1 2 3 0σ σ ϖ= = = ): The global optimum,. denoted by ( ),gh ghHζ τ , is located on the vertex point formed by 2 h ( )H hτ γ ζ= and g( )g ζ γ= . This case occurs either. when [ † † †max( ( ) , ( )) H h f. h f≥τ γ ζ γ ζ †min ], [ ], and. [ max( , )gh g H H. ∗> gf H. τ τ τ ] or when [ max( , )gh hf ∗> hζ ζ ζ ] and. [ † †max( ( ) , ( )) H h f. h f< †τ γ ζ γ ζ ].. ζ ζ>. The optimum point ghζ and ghHτ are evaluated as follows:. Proceedings of the International MultiConference of Engineers and Computer Scientists 2009 Vol II IMECS 2009, March 18 - 20, 2009, Hong Kong. ISBN: 978-988-17012-7-5 IMECS 2009. min ghζ ζ= ; ( ). 1 2. h min( ) gh H hτ γ ζ= (25). Case E ( ): The global optimum, denoted. by 1 2 3 0ϖ ϖ σ= = =. * *( , )f fHζ τ , is on the constraint ( )H f fτ γ ζ= . This case. happens when [ † †max( ( ) , ( )) H h f. h f< †τ γ ζ γ ζ ] and. [ *min f hfζ ζ ζ< < ].. Once the global optimum is obtained in terms of ζ and Hτ , the corresponding optimal PI parameters can be directly calculated using (5) and (6):. 1 c opt. p H. K k τ. − = ; 24( )opt optI Hτ ζ τ= (30). The PI controller designed by the proposed method gives the optimal responses whereas strictly satisfying all the given three constraint specifications. From (15), (16) and (19). fζ ∗ is evaluated by using (26): The overall procedure for a quick finding of the global. optimum and PI parameters is presented in Fig. 3. 3 3 5. 2 4 4 4. 4 8 ( ). ( )( ). 3 ( (1 4( ) ) '( )). ( )( ). f f f f f. f. f f ff f. f. f f. f f. ∗ ∗. ∗ ∗. ∗ ∗. ∗ ∗. − −. − + =. β αγ ζ ζ. γ ζ ζ. β α ζ γ ζ. γ ζ ζ 0. (26) . IV. CASE STUDY The most common integrating process is the liquid level system. In this section, a servo problem for the liquid level control system shown in Fig.4 is considered to demonstrate the performance of the proposed method. In this particular integrating process, Dk , , , , and Y s . correspond to. pk ( )D s ( )U s ( ). 1. A ,. 1 A −. , , , and( ) i. Q s ( ) o. Q s ( )H s ,. respectively. . f Hτ ∗ is then found by using (19) as follows: . (fH ) f. f fτ γ ζ ∗ ∗= (27). Case F ( ): The global optimum, denoted. by 1 2 3 0ϖ σ σ= = =. ( , )gf gfHζ τ , is located on the vertex point formed by ( )H f fτ γ ζ= and ( ) gg ζ γ= . This case happens either. when [ † † †max( ( ) , ( ))H h fh f≥τ γ ζ γ ζ †. min ], [ ], and. [ max( ,gf gH H ∗> ghHτ τ τ ] or when [. * f gf hfζ ζ ζ< < ] and. [ † †max( ( ) , ( )) H h f. h f< †τ γ ζ γ ζ ]. . ζ ζ>. ) where Q is maximum allowable rate of change of outlet flow rate, m. max'o 3/min2; is maximum allowable of. outlet flow rate, m maxoQ. 3/min; is maximum allowable level deviation from set-point, m.. maxH. gfζ is equivalent to minζ . gf Hτ is calculated as follows: . In the level control, the modified PI controller is expressed in (31):. min gfζ ζ= ; (28) min(. gf H f fτ γ ζ= ). ). Case G ( ): The global optimum, denoted. by 1 2 3 0σ ϖ σ= ==. ( ,hf hfHζ τ , is located on the vertex point formed by ( )H f fτ γ ζ= and. 2 ( )H h hτ γ ζ= . This case occurs when. [ † †max( ( ) , ( )) H h f. h f< †τ γ ζ γ ζ ], [ h < fγ γ ], and. [ * min. max( , )h hf ∗< < fζ ζ ζ ζ ].. (( ) ( ) ( ) ( )setLo L I. K )K H s H s H s sτ. = + −Q s (31). where maxo cL. span. Q K K. H ν=. ∆. The closed-loop transfer functions for the level control system are:. 2 H. 1 (s)= ( ). 1 set. I I. H s sτ τ τ+ +. H s (32) hfζ and hfHτ are calculated as follows: . H 2. H 1 ( ) ( )setLo. I I. K s Q s H s. s s = −. + + τ. τ τ τ (33) f h( ) ( ). hf hff hγ ζ γ ζ= ; (29) ( )hfH f hffτ γ ζ=. Note that when h < fγ γ , the vertex point formed by the. two constraints 2 ( )H h hτ γ ζ= and ( )H f fτ γ ζ= exists.. The derivative of the constraint 2 ( )H h hτ γ ζ= and the constraint are always negative with any value. of ζ . Furthermore, at the vertex point, the derivative of the constraint 2 ( )H h hτ γ ζ= is always greater than the derivative of the constraint ( )H f fτ γ ζ= , which means. the constraint ( )H f fτ γ ζ= are always upper than the. constraint 2 ( )H h hτ γ ζ= for min hfζ ζ ζ< < . . where VH L c. A K K. τ τ = = ;. V max max. ( )span T o o. H A V Q Qν ν. τ ∆. = =. ( )H f fτ γ ζ= . Fig.4. Schematic of a level control loop. Proceedings of the International MultiConference of Engineers and Computer Scientists 2009 Vol II IMECS 2009, March 18 - 20, 2009, Hong Kong. ISBN: 978-988-17012-7-5 IMECS 2009. † † † H h fif max( h( ) , f ( ))τ ≥ γ ζ γ ζ. † minif ζ ≤ ζ. † † H( , )ζ τ. g *g H( , ). ∗ζ τ. Global optimum. h fif γ < γ hf. minif ζ ≤ ζ *f hf. minif ζ < ζ < ζ f *f. H( , ) ∗ζ τ. gf gf H( , )ζ τ. f minif. ∗ζ ≥ ζ. h hfif ∗ζ ≤ ζ hf hfH( , )ζ τ. *h *h H( , )ζ τ. h minif. ∗ζ ≤ ζ gh ghH( , )ζ τ. *h *h H( , )ζ τ. N. Y Y. Y. Y. Y. N. N. N. Y. Y. Y. N. opt opt H( , )ζ τ. c opt p H. 1K k. − =. τ opt 2 opt. I H4( )τ = ζ τ;. gh g gf H H Hif max( , ). ∗τ > τ τ. g gh gf H H Hif max( , ) ∗τ > τ τ. gf g gh H H Hif max( , ). ∗τ > τ τ. gh gh H( , )ζ τ. gf gf H( , )ζ τ. N. Y. Y. Y N. N. † † † H h fif max( h( ) , f ( ))τ ≥ γ ζ γ ζ. † minif ζ ≤ ζ. † † H( , )ζ τ. g *g H( , ). ∗ζ τ. Global optimum. h fif γ < γ hf. minif ζ ≤ ζ *f hf. minif ζ < ζ < ζ f *f. H( , ) ∗ζ τ. gf gf H( , )ζ τ. f minif. ∗ζ ≥ ζ. h hfif ∗ζ ≤ ζ hf hfH( , )ζ τ. *h *h H( , )ζ τ. h minif. ∗ζ ≤ ζ gh ghH( , )ζ τ. *h *h H( , )ζ τ. N. Y Y. Y. Y. Y. N. N. N. Y. Y. Y. N. opt opt H( , )ζ τ. c opt p H. 1K k. − =. τ opt 2 opt. I H4( )τ = ζ τ;. gh g gf H H Hif max( , ). ∗τ > τ τ. g gh gf H H Hif max( , ) ∗τ > τ τ. gf g gh H H Hif max( , ). ∗τ > τ τ. gh gh H( , )ζ τ. gf gf H( , )ζ τ. N. Y. Y. Y N. N. Fig.3. Procedure to find a global optimum and PI parameters . The PI parameters were found based on the minimum of the sum of the rate of change of the outlet flow rate and the level deviation.. 2 ' 02 ' 2. max max. (1 min ( )] ( )] (34). ( )o t dt Q t dt. H Q ∞ ∞. 0 0. ω − ω) Φ = [ Η + [∫ ∫δ 2 . where 2 max. y H ω. ω = ,. ' 2 max. (1 ( )u oQ. ω − ω). = , . ( ) ( ) ( ) sett H t H tδΗ = −. The controller should satisfy the following conditions: (1) The maximum allowable level deviation; (2) The maximum allowable rate of change of the outlet flow (3) The maximum allowable outlet flow Therefore, the three constraints corresponding to (7b)-(7d) are max( )H t H≤ (35). ' ( ) 'oQ t Q≤ omax (36). max( )o oQ t Q≤ (37). The parameters α , β , hγ , gγ , and fγ given in (9) and (10) are then: . 2. max. ( ) 2. setH H. α ∆ω. = ; 2. max. 1 32 '. set. o. A H Q. ω β. ⎛− ∆⎛ ⎞= ⎜⎜ ⎟ ⎝ ⎠ ⎝ ⎠. ⎞ ⎟ (38). max' h. set. o. A H Q. γ ∆= ; maxg set H H. γ = ∆. ; max. f. set. o. A H Q. γ ∆= ( 39). Suppose that the liquid level of a drum with a cross section. area of 1 and a working volume 2m spanA H∆ of 2 is controlled by the PI controller. The initial steady- state level is 50% and the nominal flow rates of the inlet and outlet are both 1 /min. The maximum outflow is 5m. 3m. 3m maxovQ 3/min.. The expected maximum set point change in the level spH∆ is 0.6m. In Table.1, seven examples are considered for illustrating seven corresponding to cases A-G. . Table.1. Constraints for illustrative examples. example case. max'oQ (m3/min2) . maxoQ (m3/min). maxH (m). ω . 1 A 3.0 4.0 0.72 0.5 2 B 1.5 1.5 0.72 0.7 3 C 3 4.0 0.612 0.3 4 D 1.5 4.0 0.609 0.5 5 E 5 0.5 0.72 0.5 6 F 3 0.5 0.603 0.4 7 G 3 0.6 0.72 0.7 . Proceedings of the International MultiConference of Engineers and Computer Scientists 2009 Vol II IMECS 2009, March 18 - 20, 2009, Hong Kong. ISBN: 978-988-17012-7-5 IMECS 2009. Fig.5. Response of level deviation, rate of change of outlet. flow, and outlet flow rate for Examples (1-4). . Fig.6. Response of level deviation, rate of change of outlet. flow, and outlet flow rate for Examples (5-7). . Figs. 5 and 6 compare the responses of the level deviation, the rate of change of the outlet flow rate, and the outlet flow rate for several cases.. A set point step change of 0.6m was introduced at t=1. As seen in the figures, the PI controller designed by the proposed. method gives the optimal responses whereas strictly satisfying the given , , and specifications. maxH max'oQ maxoQ. V. CONCLUSION A novel PI controller design method for constrained optimal servo control of the integrating process system is proposed by solving the constrained optimization problem to deal with the important three specifications in the optimal control as well as minimize the optimal performance measure. A simple constrained problem with two independent variables is obtained from a proper variable transformation. The Lagrangian multiplier method is applied to find the analytical optimal solution. The performance of the proposed method has been demonstrated through the example of typical liquid level system. . REFERENCES. [1] H. Wade, Basic and Advanced Regulatory Control: System Design and. Application 2nd Edition, ISA, pp. 149-170, 2004.. [2] D.E Seborg, T.F. Egard, Process Dynamics and Control, 2nd Edition,. WILEY, 2004, pp. 268-276. . [3] R.C. Dorf and R.H. Bishop, Modern Control Systems, 10th Edition,. Pearson Educational International, 2005, pp. 683-692.. [4] J. Shin, J. Lee, S. Park, K. Koo, and M. Lee, “Analytical design of a. proportional-integral controller for constrained optimal regulatory control of inventory loop”, Cont.Eng.g Prac.e, vol.16, no.11, pp. 1391-1397,Nov, 2008.. [5] M. Lee and J. Shin, “Constrained Optimal Control of Liquid Level Loop Using a Conventional PI controller”, Chem.Eng. Comm., 196, 1-17, (2009). [6] K. Wu, C. Yu, and Y. Cheung, “A two degree of freedom level control”, J.. of Proc.s Cont.l, vol.11, no.3,pp. 311-319,June,2001.. [7] K. MacDonald, T. McAvoy, and A. Tits, “Optimal averaging level. control”, AIChE J., vol.32, no.1, pp. 75-86, Jan,1986.. [8] R. Rice and D.J. Cooper, “Improve control of liquid level loops”,. AIChE100 CEP, pp. 54-61, June, 2008.. [9] T. Cheung and W. Luyben, “Liquid-level control in single tanks and. cascades of tanks with P-only and PI feedback controllers”, Ind. Eng. Chem. Fund., vol.1, no.1, pp.15-21, 1979.. [10] D.E. Rivera, M. Morari, and S. Skogestad, “Internal model control. 4.. PID controller design”, Ind. Eng. Chem. Proc. Des. Dev., vol.25, no.1, pp. 252-265, Jan, 1986.. Proceedings of the International MultiConference of Engineers and Computer Scientists 2009 Vol II IMECS 2009, March 18 - 20, 2009, Hong Kong. ISBN: 978-988-17012-7-5 IMECS 2009. INTRODUCTION CONTROL SYSTEM DESCRIPTION FORMULATION OF CONSTRAINED OPTIMAL CONTROL CASE STUDY CONCLUSION

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