ARTICLE IN PRESS JID: JTICE [m5G;August 24, 2017;12:55] Journal of the Taiwan Institute of Chemical Engineers 0 0 0 (2017) 1–15 Contents lists available at ScienceDirect Journal of the Taiwan Institute of Chemical Engineers journal homepage: www.elsevier.com/locate/jtice Optimization approach for the analytical design of an industrial PI controller for the optimal regulatory control of ﬁrst order processes under operational constraints Rodrigue Tchamna, Moonyong Lee∗ School of Chemical Engineering, Yeungnam University, Gyeongsan 712-749, Republic of Korea a r t i c l e i n f o Article history: Received 10 November 2016 Revised 21 March 2017 Accepted 3 August 2017 Available online xxx Keywords: Constrained optimal regulatory control Proportional-integral (PI) controller design Operational constraints Optimization based approach Stable ﬁrst order process a b s t r a c t In this paper, an optimization-based approach for the closed-form design of an industrial proportionalintegral (PI) controller was proposed for the optimal regulatory control of ﬁrst order process under three typical operational constraints. An ingenious parameterization with Lagrangian multiplier method was used to convert the constrained optimal control problem in the time domain to an unconstrained optimization problem to derive an analytical solution for the optimal regulatory control. Three typical operational constraints could be taken into account in the controller design stage, explicitly. The proposed analytical design method required no complicated optimization steps and guaranteed global optimal closedloop performance and stability. The proposed analytical approach also provides useful insights into the optimal controller design and analysis. A practical and facile procedure for designing optimal PI parameters and a feasible constraint set was also proposed. © 2017 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved. 1. Introduction For economic and/or safety reasons, the control and operation of many modern chemical processes are restricted by the stringent operational constraints associated with the process variable, manipulated variable, and its allowable rate of change. For example, if a control system cannot consider the actuator limitation explicitly, the control algorithm may supply a control signal that the actuator cannot achieve. This will lead to actuator saturation, which in turn results in severe degradation of the control performance and even instability of the control loop, even in the case of an open-loop stable process. This tough environment in process control highlights the need to design more precise controllers that can cope with both the control performance and operational constraints in an eﬃcient manner. In order to manage the performance and robustness together in optimal control, the control objective should take into account not only the variations in the controlled variable, but also in the controller output. Optimal control of a process may become even more challenging when constraints need to be considered in the design stage. PID controllers used for industrial processes rarely tackle the constraints in an explicit manner [1]. Optimal control is a remarkable challenge when multiple constraints need to be ∗ Corresponding author. E-mail address: mynlee@yu.ac.kr (M. Lee). considered, i.e., constrained optimal control, in its design. In a popular approach using Pontryagin’s principle or the Hamilton–Jacobi– Bellman equation for a classical optimal control framework [2–6], the optimal controller parameters are obtained by solving the nonlinear constrained optimization numerically. On the other hand, despite the use of complicated optimization packages for solving non-linear optimization, the existing numerical methods neither guarantee a global optimal solution nor provide truthful insights and physical interpretations regarding the complicated relationships and effects among the process parameters and optimal solutions. The limitations of the conventional black-box approach by solving non-linear optimization has prompted research into novel analytical approaches for ﬁnding the optimal controller parameters, but mainly focusing on integral processes [7–12]. Recently, Thu and Lee [13] extended the analytical approach to constrained optimal servo control of ﬁrst order processes and showed the resulting PI controller guarantees the global optimality without the necessity of sophisticated optimization packages. In addition to servo control, regulatory control against disturbances is major consideration in most process control loop. This study extends the analytical design approach for the servo optimal control to the regulatory control. The proposed PI controller provides the globally optimal control performance, whilst satisfying the given operational limits of the process variable, controller output, and its rate of change. A formulation of the optimal regulatory control problem under constraints was ﬁrst provided http://dx.doi.org/10.1016/j.jtice.2017.08.012 1876-1070/© 2017 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Please cite this article as: R. Tchamna, M. Lee, Optimization approach for the analytical design of an industrial PI controller for the optimal regulatory control of ﬁrst order processes under operational constraints, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.08.012 ARTICLE IN PRESS JID: JTICE 2 [m5G;August 24, 2017;12:55] R. Tchamna, M. Lee / Journal of the Taiwan Institute of Chemical Engineers 000 (2017) 1–15 Fig. 1. Block diagram of the feedback control of a stable ﬁrst order process. for a ﬁrst order process that has a clear practical signiﬁcance as a representative basic and starting process dynamics in all model reductions for process control purpose. By applying the Lagrange multiplier method, re-express it in the form of a corresponding unconstrained optimization problem in terms of two new parameters that provide a clear physical meaning and useful graphical interpretation for controller design. The shape and trends of the constraints and objective function mainly depend on the parameterization of the control parameters. Since the graphical analysis essentially depends on this parameterization, the choice of these new parameters should be done such as the graphical analysis can be easily performed. After the parameterization of the system, the analytical design for an optimal PI controller and feasible constraint set was obtained from meticulous graphical examination of the possible location of the global optimum. The proposed optimal controller enhanced the closed-loop performance signiﬁcantly by ensuring a global optimal solution whilst taking care of the critical functioning requirements in an explicit way. The other good point of the proposed design technique is that, it relates the control parameters to the plant parameters in a closed-form/explicit way which is very appealing to control practitioners. To the authors’ best knowledge, this study is the ﬁrst work to develop the analytical PI controller design of optimal regulatory control for the ﬁrst order process under operational constraints. It also offers valuable insights into the optimal performance and its physical understanding, which is not available in black-box based approaches applying the numerical non-linear optimization directly. 2.2. Formulation of the constrained optimal regulatory control problem The goal of the constrained optimal regulatory problem is to minimize the weighted sum of the process variable error, e(t), and the rate of change in the manipulated variable, u (t), for a given step disturbance change, D, subjected to the following three typical operational constraints: maximum allowable limit in (1) the controlled variable, ymax , (2) the rate of change in the manipulated variable, u max , and (3) the manipulated variable, umax . Consequently, the optimal controller parameters is obtained by minimizing the performance index in Eq. (4.1) for a step disturbanceD/s, and subjected to the constraints given in Eqs. (4.2), (4.3), and (4.4). min = ∞ 0 (4.2) u (t ) ≤ u max , (4.3) |u(t )| ≤ umax . (4.4) After some mathematical operations, the above performance index can be transformed to the form expressed in Eqs. (5.1)–(5.4), with ζ and τ c the two new design parameters. (See the Appendix for the derivation details). β 1 1 + 2 2 τc 4 ζ ε β 1 (τc − τ )2 3 2 = ατc ζ + + , τc 4 ζ 2 τ2 Fig. 1 presents a schematic diagram of the system. The Laplace transfer function of the system in Fig. 1 is expressed as follows: τI s + 1 D(s ), ε τc τI s 2 + ε τI s + 1 (1) τ τ 1 = (1 + K Kc ) ε K Kc (3.1) 1 . 1 − ττc (3.2) and ε= The closed-loop damping ratio of the above system is ζ= 1 2 ε τI 1 = τc 2 τ τI . τ − τc τc (5.1) τc g(ζ ) ≤ γg , (5.2) h(ζ , τc ) ≤ γh (5.3) f (ζ , τc ) ≤ γf , (5.4) (2) where τc = ατc3 ζ 2 + subject to and U (s ) = − (4.1) |y(t )| ≤ ymax , min (ζ , τc ) = 2.1. Control system description K τε τc τI s Y (s ) = D (s ) ε τc τI s 2 + ε τI s + 1 ωy (y(t ) − ysp (t ))2 + ωu (u (t ))2 dt subject to 2. Optimal PI controller design ∗ (3.3) where K 2 ωu τ ymax α = 2 ω y D ; β= (D )2 ; γg = ; 2 K D τ u u γh = max ; γ f = max . D D (6) g(ζ ), h(ζ , τ c ), and f(ζ , τ c ) are given in Eqs. (A13), (A38), and (A28.1), (A28.2) in the Appendix, respectively. In order to obtain the coeﬃcients in Eq. (6), the plant parameters, the weighting factors, the constraints and the disturbance size must be known. Please cite this article as: R. Tchamna, M. Lee, Optimization approach for the analytical design of an industrial PI controller for the optimal regulatory control of ﬁrst order processes under operational constraints, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.08.012 ARTICLE IN PRESS JID: JTICE [m5G;August 24, 2017;12:55] R. Tchamna, M. Lee / Journal of the Taiwan Institute of Chemical Engineers 000 (2017) 1–15 In practice, the size of the disturbance is usually unknown and changed. Choosing a maximum value expected would be a practical way for specifying the disturbance size for controller design purpose. Disturbance can also be estimated by using a disturbance observer. For disturbance estimation, interested readers may refer to [14]. As shown in Eqs. (5.2)–(5.4), the three constraints are only function of ζ and τ c . Therefore, a simple graphical examination of the contour of the objective function and the constraints in (ζ , τ c )space allows to ﬁnd the location of global optimal solution without complex numerical optimization process. 2.3. Optimal PI controller design The Lagrangian multiplier [15] can be applied in order to convert the constrained optimization problem in Eqs. (5.1)–(5.4) to an equivalent unconstrained problem as follows: min L(ζ , τc , 1 , 2 , 3 , σ1 , σ2 , σ3 ) = (ζ , τc ) + 1 (γh − h(ζ , τc ) − σ12 ) + 2 γ f − f (ζ , τc ) − σ32 β 1 (τc − τ )2 3 2 = ατc ζ + + τc 4 ζ 2 τ2 +1 (γh − h(ζ , τc ) − σ12 ) + 2 γg − g(ζ )τc − σ22 +3 γ f − f (ζ , τc ) − σ32 +3 γg − g(ζ )τc − σ22 3 the corresponding conditions of the Lagrange multipliers and slack variables as follows: † † Case A (ϖ1 = ϖ2 = ϖ3 = 0): The extreme point, (ζ , τc ), which is located inside the feasible region, is therefore the global optimum. Solving Eqs. (8) and (9) simultaneously, the global optimum can be determined in explicit form as: ζ = † τc† = 1 1 + 2 2 τ 1 ζ† β 4α β 4α 1/2 1/2 (14.1) 1/4 (14.2) Case B (σ 1 = ϖ2 = ϖ3 = 0): The global optimum, symbolized as (ζ ∗h , τc∗h ), is positioned on the constraint, γ h = h(ζ , τ c ). ζ ∗h and τc∗h can be obtained by substituting σ 1 = ϖ2 = ϖ3 = 0 into Eqs. (8)– (10), thus solving the following system of equations: γh − h(ζ , τc ) = 0 (15.1) −1 ∂ ∂ ∂h ∂h − = 0. ∂ζ ∂ τc ∂ τc ∂ζ (15.2) Case C (ϖ1 = σ 2 = ϖ3 = 0): For this case, the global optimum, (ζ ∗g , τc∗g ), is located on the constraint, γ g = τ c g(ζ ). ζ ∗g and τc∗g are obtained by substituting ϖ1 = σ 2 = ϖ3 = 0 into Eqs. (8), (9) and (11), (7) where ϖi and σ i are the Lagrange multiplier and the slack variable, respectively. The necessary conditions for an optimal solution are ∂L ∂ ∂ h ( ζ , τc ) = + 1 − + 2 [−g(ζ )] ∂ τc ∂ τc ∂ τc ∂ f ( ζ , τc ) +3 − = 0, ∂ τc (8) ∂L ∂ ∂ h ( ζ , τc ) ∂ g( ζ ) ∂ f ( ζ , τc ) = − 1 − 2 τc − 3 = 0, ∂ζ ∂ζ ∂ζ ∂ζ ∂ζ and solving the following system of equations: γg − g(ζ )τc = 0 (16.1) ∂ ∂ ∂g g − τc =0 ∂ζ ∂ τc ∂ζ (16.2) Case D (σ 1 = σ 2 = ϖ3 = 0): Using Eqs. (10) and (11), the global gh optimum represented by (ζ gh , τc ) is located on the intersection gh point by γ h = h(ζ , τ c ) andγ g = τ c g(ζ ). ζ gh and τc can be calculated by solving γg − τc g(ζ ) = 0 (9) (17.1) γh − h(ζ , τc ) = 0 (17.2) (ζ ∗ f , τc∗ f ), ∂L = γh − h(ζ , τc ) − σ12 = 0, ∂ 1 (10) ∂L = γg − τc g(ζ ) − σ22 = 0, ∂ 2 Case E (ϖ1 = ϖ2 = σ 3 = 0): The global optimum, is lo∗f cated on the constraint, γ f = f (ζ , τc ). ζ ∗f and τc can be found by solving the following system of equations: (11) γf − f (ζ , τc ) = 0 (18.1) ∂L = γ f − f (ζ , τc ) − σ32 = 0, ∂ 3 (12) ∂ f ∂ ∂ f ∂ − =0 ∂ τc ∂ζ ∂ζ ∂ τc (18.2) ∂L ∂L ∂L = −21 σ1 = 0; = −22 σ2 = 0; = −23 σ3 = 0. ∂ σ1 ∂ σ2 ∂ σ3 (13) The simultaneous solutions of Eqs. (8)–(13) for possible combinations of σ i = 0, σ i = 0, ϖi = 0, and ϖi = 0 are associated with the corresponding optimal cases. Note that instead of introducing the slack variables, Karush–Kuhn–Tucker conditions can be utilized for solving the optimization problem by Eqs. (5.1)–(5.4). Fig. 2 presents seven possible cases for the location of global optimum: the global optimum can be found inside the feasible region (case A), or on the boundary of one constraint (cases B, C, and E), or on the intersection point of two constraints (cases D, F, and G). The global optima of the seven cases can be evaluated by inspecting their geometrical characteristics in (ζ , τc )space as well as Case F (ϖ1 = σ 2 = σ 3 = 0): Using Eqs. (11) and (12), the global gf optimum, (ζ g f , τc ), is on the intersection point created by γ f = f (ζ , τc ) and γ g = τ c g(ζ ). ζ gf and τc are calculated by solving the following equations: gf γf − f (ζ , τc ) = 0 (19.1) γg − τc g(ζ ) = 0 (19.2) Case G (σ 1 = ϖ2 = σ 3 = 0): Using Eqs. (10) and (12), the global fh optimum, (ζ fh , τc ), is located on the intersection point of the fh curved γ f = f (ζ , τc ) and γ h = h(ζ , τ c ). ζ fh and τc are calculated by solving the following set of equations: γ f − f ( ζ , τc ) = 0 (20.1) γh − h(ζ , τc ) = 0 (20.2) Please cite this article as: R. Tchamna, M. Lee, Optimization approach for the analytical design of an industrial PI controller for the optimal regulatory control of ﬁrst order processes under operational constraints, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.08.012 ARTICLE IN PRESS JID: JTICE 4 [m5G;August 24, 2017;12:55] R. Tchamna, M. Lee / Journal of the Taiwan Institute of Chemical Engineers 000 (2017) 1–15 Fig. 2. Typical contours and constraints for the seven possible cases with respect to the location of the global optimum. After the global optimum is determined in (ζ , τc ) space, the optimal PI parameters corresponding to each case can then be calculated using Eqs. (3.1)–(3.3): Kcopt = τ 1 K ε τ opt opt c ; τIopt = 4 ε opt (ζ opt )2 τcopt , (21.1) every condition for discriminating the seven possible global optimum cases was derived. Table 1 lists the results for the conditions and characteristics of the global optima. Fig. 4 also presents the overall procedure for determining the global optimum quickly. where ε opt = 1 opt 1 − τcτ . (21.2) The conditions for the seven possible cases were also evaluated based on a meticulous analysis of the graphical shape of the constraints and contours in (ζ , τ c ) space. The concept of the relative locations between the extreme point and its projections to the constraints were used mainly to develop the conditions to discriminate each case associated with the global optimum. Fig. 3 shows an example of the projection of the extreme point and its notation rule used in this study. The notation, ζ †f , represents the abscissa of the projection of the extreme point on the †h †g constraint curve by f(ζ , τ c ). Similarly, τc and τc indicate the ordinate of the projection of the extreme point on the constraint † †h curves by h(ζ , τ c ) and g(ζ ), respectively. If τc is such that τc ≤ †g † τc ≤ τc , then the global optimum is above the constraint, h(ζ , τ c ), and below the constraint, g(ζ ). Referring to Fig 2, this corresponds to cases A, E or possibly G. In this case, it is apparent from Fig. 2 that if ζ †f < ζ †, it belongs to case A (i.e., the extreme point is the global optimum), otherwise it belongs to either case E or G. ∗f Case E and G can be distinguished simply by comparing τc and fh fh τc , where τc is the ordinate of the intersection of the constraints ∗f fh by f and h. As shown in Fig. 2, if τc > τc , the global optimum case belongs to case E, otherwise case G. Using similar reasoning, 3. Design of a feasible constraint set 3.1. Effect of the constraints on the feasible region Fig. 5 shows the effects of the constraint speciﬁcations on the feasible region in (ζ , τ c ) space. The constraint imposed by Eq. (5.4) lays a vertical line that shifts rightward but bends leftward as umax decreases. The constraint given in Eq. (5.3) shifts up as u max decreases, whereas the constraint in Eq. (5.2) shifts down as ymax decreases. Note that the feasible region exists only when the constraint curve of u max is below that ofymax . As indicated in Fig. 5, the feasible region shrinks as u max and ymax decrease, and will eventually cease to exist if the constraints are lower than certain minimum allowable values. Fig. 6 presents the tangent (ζ t , τct ), where the two constraint curves by u max and ymax touch each other in a single point in (ζ , τ c )space, for the different speciﬁcations of u max and ymax . The tangent consists of the smallest feasible u max for a given ymax (or the smallest feasible ymax for a given u max ). As ymax decreases, the smallest feasible u max that can still yield a feasible region must increase, as expected from the inherent tradeoff between the process variable and the manipulated variable in the control response. As ymax decreases, the controller requires more control actions, which indicates increases in Kc to cope with the tight constraint; thus, τct decreases in (ζ , τ c )space, as shown in Fig. 6. Please cite this article as: R. Tchamna, M. Lee, Optimization approach for the analytical design of an industrial PI controller for the optimal regulatory control of ﬁrst order processes under operational constraints, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.08.012 ARTICLE IN PRESS JID: JTICE [m5G;August 24, 2017;12:55] R. Tchamna, M. Lee / Journal of the Taiwan Institute of Chemical Engineers 000 (2017) 1–15 5 Fig. 3. Projection of the extreme point on the constraints in (ζ , τ c ) space. Table 1 Global optimums of the constrained optimization problem in Eqs. (5.1)–(5.4). Case Constraint speciﬁcation Condition Global optimum Location of global optimum Calculation of global optimum A mild u max mild ymax mild umax ζ †f ≤ ζ † and τc†h ≤ τc† ≤ (ζ † , τc† ) In the interior of the constraint set. ζ † = [ 12 + B tight u max mild ymax mild umax ζ †f ≤ ζ ∗ h ≤ ζ gh and τc† < τc†h (ζ ∗h , τc∗h ) on γ h = h(ζ , τ c ) { ∂ C mild u max tight ymax mild umax ζ †f ≤ ζ ∗ g ≤ ζ gh and τc∗ f ≤ τcg f γ and τc† ≥ g(ζg† ) (ζ ∗g , τc∗g ) on τc = g(ζg ) D tight u max tight ymax mild [ζ ∗ h ≥ ζ gh and τc† < τc† ] or γ [ζ ∗ g > ζ gh and τc† ≥ g(ζg† ) ] (ζ gh , τcgh ) E mild u max mild ymax tight umax ζ † < ζ †f and τcf h ≤ τc∗ f or ζ gh > ζ ∗ g and τcg f ≥ τc∗ f ≥ τcf h (ζ ∗ f , τc∗ f ) F mild u max tight ymax ζ† > ζ γg g( ζ † ) h f ∗g >ζ gh ∗f c and τ >τ gf c on the vertex by γg and γh = h (ζ , τc ) g( ζ ) G tight u max mild ymax tight umax [ζ † < ζ †f and τc∗ f ≤ τcf h ] or [ζ gh ≥ ζ ∗ h ≥ ζ †f and τc† < τc†h ] (ζ f h , τcf h ) 3.2. Discrimination and design of the feasible constraint set In a constrained optimization problem, the existence of the feasible solution region should be checked ﬁrst for a given constraint set. Moreover, it is often required to control a system on the tightest allowable constraint set. On the other hand, the three constraint speciﬁcations cannot be selected randomly or independently due to their interrelation. For this reason, the constraint set must be determined not only by considering the process requirements, but also by satisfying a feasibility condition. 3.2.1. Necessary condition for feasible ymax , u max , and umax ymax and umax should be set to ymax ≥ 0 and umax ≥ |D|because lim y(t ) = 0 and | lim u(t )| = |D|. Furthermore, t→∞ t→∞ γ h − h ( ζ , τc ) = 0 { γg − τc g(ζ ) = 0 γ h − h ( ζ , τc ) = 0 on γ f = f(ζ , τ c ) γ − f (ζ , τc ) = 0 { ∂ f f ∂ ∂ f ∂ on the vertex by γ − f (ζ , τc ) = 0 { f γg − τc g(ζ ) = 0 γ f = f (ζ , τc ) and τc = tight umax ]1/2 ; τc† = ζ1† ( 4βα )1/2 γg − g(ζ )τc = 0 { ∂ ∂ ∂g ∂ζ g − τc ∂ τc ∂ζ = 0 τc = gf c 1/2 ( βα ) ∂ ∂ h −1 ∂ h ∂ζ − ∂ τc [ ∂ τc ] ∂ζ = 0 γ (ζ , τ ) gf 1 2τ 2 ∂ τc ∂ζ − ∂ζ ∂ τc = 0 γg g( ζ ) on the vertex by γ f = f(ζ , τ c ) and γ h = h(ζ , τ c ) { γf − f (ζ , τc ) = 0 γ h − h ( ζ , τc ) = 0 when ζ approaches ∞, Eq. (A38) becomes lim h(ζ , τc ) = ζ →∞ τ − τc = γh . τ τc (22) Let τch (∞ ) be the solution of Eq. (22). τch (∞ ) can be found as τch (∞ ) = γ ττ+1 . Hence, τch (∞ ) is always positive as long as u max h ≥ 0. Therefore, ymax ,u max , and umax should be set at least to satisfy the following: ymax ≥ 0, umax ≥ |D|, umax ≥ 0 (23) 3.2.2. Condition for a feasible (ymax , u max ) For a given ymax , there is a minimum available u max value below which the optimal control problem is not feasible. Let u t max be the tightest/smallest possible value of u max . As stated in the Please cite this article as: R. Tchamna, M. Lee, Optimization approach for the analytical design of an industrial PI controller for the optimal regulatory control of ﬁrst order processes under operational constraints, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.08.012 JID: JTICE 6 ARTICLE IN PRESS [m5G;August 24, 2017;12:55] R. Tchamna, M. Lee / Journal of the Taiwan Institute of Chemical Engineers 000 (2017) 1–15 Fig. 4. Flow chart for ﬁnding the global optimum and PI parameters. Fig. 5. Effect of the constraint speciﬁcations, ymax , umax , and u max , on the feasible region. Please cite this article as: R. Tchamna, M. Lee, Optimization approach for the analytical design of an industrial PI controller for the optimal regulatory control of ﬁrst order processes under operational constraints, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.08.012 ARTICLE IN PRESS JID: JTICE [m5G;August 24, 2017;12:55] R. Tchamna, M. Lee / Journal of the Taiwan Institute of Chemical Engineers 000 (2017) 1–15 7 Fig. 6. Tangent points by the constraint curves of u max and ymax . previous section, u t max can be obtained when ζ = ζ t . ζ t can be calculated by solving the following equation: dh dh(ζ , τc ) = dζ ζ , γg g−1 = 0. dζ (24) umax = h ζ t |D|. h ζ , γg g −1 and dh = γh ζ , γg g−1 =0 dζ (25) (26) (27) Finally, if ymax ≥ ytmax , the constraint set (ymax , u max ) is feasible. Otherwise, it is not feasible. 3.2.3. Feasible umax for a given feasible set (ymax , u max ) For a feasible (ymax ,u max ), it can be inferred intuitively from geometrical analysis of the constraint curves that any positive umax will lead to a feasible region if there is no intersection between the constraint curves by τ c g(ζ ) = γ g and h(ζ , τ c ) = γ h . Furthermore, in the case where the intersection, ζ gh , exists, the constraint, umax , is feasible if ζ gh ≥ ζ gf . To evaluate the existence of an intersection between the two constraint curves, i.e., by τ c g(ζ ) = γ g and h(ζ , τ c ) = γ h , it is important to calculate τ c (∞), which is the value of τ c where the two constraint curves are secant when ζ → ∞. τ c (∞) of each constraint curve can be obtained by solving the following equations: [τc (∞ )g(ζ ) = γg ]ζ →∞ Because (28.2) lim g(ζ ) = 1 and ζ →∞ lim h(ζ , τc ) = (τ − τc )/τ τc = γh , ζ →∞ τcg (∞ ) = γg t , then the constraint set (y Finally, if u max ≥ umax max , u max ) is feasible. Otherwise, it is infeasible. Similarly, for a given u max , there is a minimum available ymax value below which the optimal control problem is not feasible. Let ytmax be the tightest/smallest possible ymax . The values of ytmax and ζ t can be obtained by solving the following system of equations simultaneously: {h[ζ , τc (∞ )] = γh }ζ →∞ τ c (∞) of the two constraint curves are t Once ζ t is obtained, ymax can be derived as follows: t and (28.1) and τch (∞ ) = (29.1) τ . γh τ + 1 (29.2) A vertex ζ gh exists when τc (∞ ) ≥ τch (∞ ). Therefore, g τ γg ≥ . γh τ + 1 (30) Substitution of Eq. (6) for γ g and γ h into Eq. (31) yields ymax |K D| u τ max + 1 ≥ 1. |D| (31) Overall, for a given feasible (ymax ,u max ), umax is feasible if either 1) Eq. (31) is not satisﬁed or 2) Eq. (31) is satisﬁed and ζ gh ≥ ζ gf . Otherwise, umax is not feasible and should be increased until one of the conditions is satisﬁed. Note that if ymax u max |K D| (τ |D| + 1 ) < 1, no vertex point is formed by γ g = τ c g(ζ ) and γ h = h(ζ , τ c ), i.e., case D does not exist. This situation is likely to happen if the speciﬁcation, u max and/or ymax , are set mildly with a large value. In this situation, ζ gh can be considered as a sufﬁciently large value, for an evaluation of the conditions presented in Table 1 and Fig. 4. Fig. 7 presents the overall procedure to test the feasibility of a constraint set (ymax ,u max ,umax ) and design a feasible constraint set. 4. Simulation of the closed-loop performance Consider the following stable ﬁrst order process as G p (s ) = 10 10s + 1 (32) Please cite this article as: R. Tchamna, M. Lee, Optimization approach for the analytical design of an industrial PI controller for the optimal regulatory control of ﬁrst order processes under operational constraints, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.08.012 ARTICLE IN PRESS JID: JTICE 8 [m5G;August 24, 2017;12:55] R. Tchamna, M. Lee / Journal of the Taiwan Institute of Chemical Engineers 000 (2017) 1–15 The proposed PI controllers provide the optimal responses satisfying the ymax , umax , and u max constraints strictly. 5. Discussion 5.1. Effect of the weighting coeﬃcient Fig. 7. Overall procedure for designing and checking a feasible constraint set. Table 2 Constraint speciﬁcations and resulting optimal PI parameters. Example 1 2 3 4 5 6 7 case A B C D E F G Constraint speciﬁcation PI parameter ymax umax u max KC τI 0.70 0.70 0.36 0.285 0.70 0.30 0.70 2.70 2.70 2.70 2.70 1.105 1.20 1.20 2.70 1.11 2.70 2.10 2.70 2.70 1.37 1.32 1.11 1.83 2.10 1.76 2.07 1.37 1.32 1.11 1.43 0.76 2.18 0.92 1.36 The values of the weighting coeﬃcients were selected arbitrarily as ωy = ωu = 0.5. Examples of cases A–G were simulated for various constraint speciﬁcations as listed in Table 2. Figs. 8 and 9 present the responses of y(t), u(t), and u (t) for the seven examples. The weighting coeﬃcient, ω, is a main parameter to regulate the tradeoff between the robustness and performance in optimal control. The weighting factor ωy is used to emphasize on the control performance. As ωy increases, the optimal control results in a tighter control response. While on the other, if the main control objective is to have a smooth control action rather than fast response, then larger values of ωu should be chosen. Fig. 10 presents the effects of the weighting coeﬃcient on the trajectories of the global optimum and the extreme point as the weighting coeﬃcient increases. The process in Eq. (32) was considered with ymax = 0.36, umax = 2.7, u max = 2.7, and ωu = 0.5. When ωy is small, e.g. ωy = 0.05 (or conversely ωu is large enough), the variation of u (t) mainly determines the performance measure of the optimal control. Accordingly, the controller yields a tight response of u (t) and the response is possibly constrained by ymax . In this case, the extreme point is likely to be above the boundary of γ g = g(ζ ) in (ζ , τc ) space, and the global optimum will be on γ g = g(ζ ), i.e., case C. For a larger ωy , the variation of y(t) governs the performance measure more, and the optimal controller yields a tight response of y(t). Accordingly, the extreme point shifts down in (ζ , τc ) space, as predicted from Eq. (14.2), and becomes the global optimal point, i.e., case A. This is physically logical in that as a largerωy is applied, a larger Kc is required to obtain a tight response of the process variable, which corresponds to the lower position of the extreme point in (ζ , τ c ) space. As ωy is increased further to a suﬃciently large value, the optimum is then likely to be on the constraint, τc2 = γh h(ζ ), i.e., case B, where the response is constrained by the u max speciﬁcation, as seen in Fig. 10. Fig. 11 presents the corresponding closed-loop responses of the proposed optimal PI controller for the process examined in Fig. 8. Responses of y(t), u (t), and u(t) for cases A to D. Please cite this article as: R. Tchamna, M. Lee, Optimization approach for the analytical design of an industrial PI controller for the optimal regulatory control of ﬁrst order processes under operational constraints, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.08.012 JID: JTICE ARTICLE IN PRESS [m5G;August 24, 2017;12:55] R. Tchamna, M. Lee / Journal of the Taiwan Institute of Chemical Engineers 000 (2017) 1–15 9 Fig. 9. Responses of y(t), u (t), and u(t) for cases E to G. Fig. 10. Effects of the weighting factor on the trajectories of the extreme point and global optimum in (ζ , τ c ) space. Fig. 10 for different weighting factors of ωy = 0.05, 0.85, and 0.98. The inherent trade-off between the process variable and the manipulated variable can be seen from this ﬁgure. An increase in ωy leads to a faster and tighter control response as increases, and conversely, a decrease in ωy (or an increase in ωu ) makes the manipulated variable and its rate smoother and smaller. For ωy = 0.05, only ymax constrains the optimal control response, i.e., case C. For ωy = 0.85, the process variable departs from the constraint, ymax . None of the three constraints constrains the optimal responses, i.e., case A. Thereafter, the constraint on the controlled variable is released completely as ωy increases to ωy = 0.98. Instead, u max constrains the optimal response, i.e., case B. These features of the optimal responses as the weighting factor varies match perfectly with those indicated in Fig. 10. The proposed optimal controller strictly satisﬁes all the constraints speciﬁcations on ymax , umax , and u max , regardless of the value of the weighting factor. opt Remark 1. For case A or an unconstrained case, the ratio of Kc to τIopt is related only to the weighting factors ωy and ωu as follows: Please cite this article as: R. Tchamna, M. Lee, Optimization approach for the analytical design of an industrial PI controller for the optimal regulatory control of ﬁrst order processes under operational constraints, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.08.012 ARTICLE IN PRESS JID: JTICE 10 [m5G;August 24, 2017;12:55] R. Tchamna, M. Lee / Journal of the Taiwan Institute of Chemical Engineers 000 (2017) 1–15 Fig. 11. Closed-loop responses by the proposed PI controller for various weighting coeﬃcients. Fig. 12. Effect of the process gain on the extreme point and global optimum in (ζ , τ c ) space. Kcopt opt I τ = ωy . ωu the process variable and that of the rate of change of the manipulated variable. (33) Eq. (33) shows and interesting result: the ratio of the optimal PI control parameters for the regulatory control of ﬁrst order systems is totally independent on process parameters. Whatever the values of the plant parameters, the ratio of the optimal PI control parameters only depends on the ratio of the weighing factors of 5.2. Effect of the process parameters The process gain and time constant affect the trend and location of the constraints as well as the location of the global optimum in (ζ , τ c ) space, which results in different behaviors of the optimal controller. Figs. 12 and 13 show how the extreme point, constraint curves, and global optimum vary according to Please cite this article as: R. Tchamna, M. Lee, Optimization approach for the analytical design of an industrial PI controller for the optimal regulatory control of ﬁrst order processes under operational constraints, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.08.012 JID: JTICE ARTICLE IN PRESS R. Tchamna, M. Lee / Journal of the Taiwan Institute of Chemical Engineers 000 (2017) 1–15 [m5G;August 24, 2017;12:55] 11 Fig. 13. Effect of the time constant on the extreme point and global optimum in (ζ , τ c ) space. Fig. 14. Closed-loop responses by the proposed PI controller for various process gains. the time constant and process gain. The process in Eq. (32) was considered again with ymax = 0.36, umax = 2.7, u max = 2.7 , and ωy = ωu = 0.5 but with varying process parameters. As shown in Fig. 12, as the process gain increases, the constraint curve byymax and the extreme point move down in (ζ , τ c )space while both the constraint curves by umax and u max are independent of the process gain. This tendency of the extreme point and each constraint for varying the process gain is matched well with that predicted from each corresponding equation, i.e., Eqs. (14.2), (A13), (A38) , (A28.1) and (A28.2). As a result, in this particular example, as the process gain increases from K = 3–15, the optimal response shifts from the unconstraint case, i.e., case A, through case C constrained by ymax , and ﬁnally to case D constrained by both ymax and u max . On the other hand, as shown in Fig. 13, the extreme point and the constraint curve by ymax shift down in (ζ , τ c ) space, as Please cite this article as: R. Tchamna, M. Lee, Optimization approach for the analytical design of an industrial PI controller for the optimal regulatory control of ﬁrst order processes under operational constraints, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.08.012 JID: JTICE 12 ARTICLE IN PRESS [m5G;August 24, 2017;12:55] R. Tchamna, M. Lee / Journal of the Taiwan Institute of Chemical Engineers 000 (2017) 1–15 Fig. 15. Closed-loop responses by the proposed PI controller for various process time constants. the time constant decreases. The constraint curve by umax rotates slightly leftwards as the time constant decreases, whereas that by u max is insensitive to changes in the time constant. These observations match well with those predicted from their corresponding equations. As a result, as the process time constant is decreased from τ = 20 to 7, the optimal response of this process shows unconstraint case A, then case C, and ﬁnally case D. Figs. 14 and 15 show the corresponding closed loop responses for various process gains and time constants examined in Figs. 12 and 13, respectively. As observed in Fig. 14, the optimal response shifts from the unconstraint case A to case C and ﬁnally to case D as the process gain increases from K = 3–15. The reverse phenomenon can be observed in Fig. 15; an increase in the process time constant makes the system shift from case D to C and then to A. These observations validate what are observed in (ζ , τ c ) space in Figs. 13 and 14. Note that the proposed PI controller always provides the optimal responses with no constraint violations in every case examined. The proposed optimization-based analytical approach offers the following useful insights, which are diﬃcult to obtain using blackbox or numerical approaches, into the optimal design of the stable process with operational constraints: (1) this approach can explain the effects of the process and control parameters on the optimal responses; (2) it can determine if a constraint set is feasible in the constrained optimal problem, and if not, it suggests how to design the operational constraints to make the problem feasible; and (3) it provides the optimal PI parameters of the constrained optimal control problem without the need for complex optimization packages, and guarantees its stability and global optimality. The same analysis presented in this paper can be extended to more general cases such as three term PID controllers, higher order and dead time processes by taking a hybrid method combining analytical and numerical approaches and/or a dead time compensator such as Smith Predictor and IMC. Remark 2. The validity of the results in this paper is under the assumption that the system model is a perfect model. When a process model is not perfect, the responses of the system will possibly violate the constraints. One practical way to cope with the constraint violation problem by the plan-model mismatch would be to use a conservative or tighter constraint value considering these uncertainties. This work was supported by the 2016 Yeungnam University Research Grant. 6. Conclusions y(t ) = A novel analytical design method was developed for the optimal regulatory control of stable ﬁrst order process under operational constraints. The original constrained optimal control problem was formulated in (ζ , τ c ) space using clever parameterization, and then converted to an equivalent unconstrained problem using the classical Lagrangian multiplier method to derive the analytical solutions for the optimal PI parameters. Seven possible cases for the constrained optimal control problem were then analyzed in terms of the location of the global optimum in (ζ , τ c ) space. Acknowledgments Appendix A. Proof of the Performance Index in Eq. (5.1) When a step change in the disturbance, D(s) = D/s is applied to the stable ﬁrst order process presented in Fig. 1, y(t) and u (t) given in Eqs. (1) and (2) are derived as D r 1 r 2 τ I e r 1 t − e r 2 t Kc u (t ) = −D r1 − r2 for r1 = r2 (A1) r r 1 2 τI r1 er1 t − r2 er2 t + er1 t − er2 t for r1 = r2 , r1 − r2 (A2) where r1 and r2 are the roots of the characteristic equation, ε τc τI s2 + ε τI s + 1 = 0. r1 = −1 + xi −1 − xi , r2 = 2 τc 2 τc (A3) Please cite this article as: R. Tchamna, M. Lee, Optimization approach for the analytical design of an industrial PI controller for the optimal regulatory control of ﬁrst order processes under operational constraints, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.08.012 ARTICLE IN PRESS JID: JTICE [m5G;August 24, 2017;12:55] R. Tchamna, M. Lee / Journal of the Taiwan Institute of Chemical Engineers 000 (2017) 1–15 and x= ∀ζ , g(ζ ) > 0; consequently, Eq. (4.2) can be expressed as 1 − ζ2 for 0 < ζ ≤ 1 . ζ ζ2 − 1 = for ζ > 1 ζ (A4) = ωy ωy ∞ ∞ (y(t ) )2 dt + ωu 0 2 2 D τ I r1 r2 1 0 Kc r1 − r2 − + 2 u (t ) dt 2r1 r1 + r2 2 1 − + 2r1 r1 + r2 2r2 2r2 s + 1 τc s+ 1 ε τc τI 2 n τc ; r1 r2 = ωn2 = 1 ε τc τI − cos x τ − 2τ x c t + sin t 2 τc xτ 2 τc (A15) (A5) (A6) ; r1 − r2 = t τ − 2 τc −1 + t for ζ = 1 2τ 2τ τ t c x c τ −2τ x c = −D 1 + exp − − cosh t + sinh t 2 τc 2 τc xτ 2 τc for ζ > 1 From the characteristic equation of the system, along with Eq. (A3), the roots of the characteristic equation satisfy the following set of equations: −1 t 2 τc The steady-state value of u(t) is uss (t → ∞ ) = −D. = s + 2ζ ωn s + ω = 0. r1 + r2 = − 2ωn ζ = = −D 1 + exp − . 2 u(t ) = −D 1 + exp − The characteristic equation of the system is given as 2 (A14) u(t) for a step change in the disturbance can be derived using the inverse Laplace transform of Eq. (2) as follows: for 0 < ζ < 1 r r 2 r r 2r1 r2 1 2 + ωu (−D )2 τI2 − 1 − 2 + r1 − r2 2 2 ( r1 + r2 ) 2 1 1 + ypeak < ymax ⇒ |K D| τc g(ζ ) ≤ ymax . τ Appendix C. Proof of Eq. (5.4) Note that the above notation for xare used all along this Appendix. The optimal control performance index can be derived as follows: = 13 xi τc (A7) (A16) Furthermore, the peak time for the largest peak of u(t) can be obtained from du(t)/dt = 0 as follows: ⎧ 2 ⎪ 2 τc ⎪ −1 2ζ (τc − τ ) ζ − 1 ⎪ tanh for ζ > 1 ⎪ ⎨ x 2 ζ 2 ( τc − τ ) + τ tupeak = . ⎪ 2 ζ ( τc − τ ) 1 − ζ 2 2 τc ⎪ −1 ⎪ tan + kπ , k ∈ Z for ζ < 1 ⎪ ⎩ x 2 ζ 2 ( τc − τ ) + τ By substituting Eq. (A7) into Eq. (A5), can be found as follows: β 1 1 β 1 (τc − τ )2 3 2 = ατc ζ + + = ατc ζ + + . τc 4 ζ 2 ε 2 τc 4 ζ 2 τ2 3 2 (A8) Appendix B. Proof of Eq. (5.2) y(t) for a step change in the disturbance can be obtained using the inverse Laplace transform of Eq. (1) as follows: x t 2DK τc exp − sin t y(t ) = τ x 2 τc 2 τc t DK for 0 < ζ < 1 (A9) Therefore, the steady-state value of y(t) is yss (t → ∞ ) = 0. (A10) Taking the derivative of y(t), the peak time for the largest peak of y(t) can be obtained as 2τc atan(x ) x = 2 τc 2τc atanh(x ) = x tpeak = for 0<ζ <1 for ζ =1 for ζ >1 . where K τ τc g ( ζ ), (A11) (A12) tupeak <0 ⇔ 2ζ (τc −τ )+τ > 0 ⇒ ζ < τ 2 ( τ − τc ) = ε 2 = ζ0 f (A18) To avoid a negative peak time, and also to ensure that the peak time will be the smallest peak time among all the peak time sequences, the peak time must be chosen as follows: if ζ < ζ0 f tupeak = else Therefore, the peak of y(t) is as follows: ypeak = D However, the theoretical values of the peak times do not always correspond to the actual peak time. Since τc = 1+τK Kc ⇒ ττc = 1 + K Kc > 1. This means that τ > τ c , hence τ c − τ < 0. for ζ < 1, the peak time of u(t) must satisfy two conditions: tupeak must be the smallest time among the time series in (A17), and must be a positive value. For k = 0, e.g., the peak time will be negative if the following relation is satisﬁed 2 exp − t for ζ = 1 τ 2τc x t 2DK τc = exp − sinh t for ζ > 1. τ x 2 τc 2 τc = (A17) tupeak = 2 τc 2ζ 2 (τc − τ )x tan−1 x 2 ζ 2 ( τc − τ ) + τ 2 τc 2ζ 2 (τc − τ )x tan−1 x 2 ζ 2 ( τc − τ ) + τ +π . (A19) Thus, the peak time for ζ < 1 can be ⎧ 2 τc 2ζ 2 (τc − τ )x ⎪ tan−1 +π ⎪ 2 ⎪ 2 ζ ( τc − τ ) + τ ⎪ ⎪ x ⎪ ⎨ 2ζ 2 (τc − τ )x tupeak = 2τc tan−1 2 ⎪ x 2 ζ ( τc − τ ) + τ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ( τc − τ ) ⎩ 2 τc 2 τc − τ expressed as for 0 < ζ < ζ0 f and for ζ0 f ≤ ζ < 1 ζ <1 . tan −1 x 2 g( ζ ) = √ exp − for 0 < ζ < 1 for ζ = 1 2 x 1+x = 2 exp(−1 ) , (A13) (A20) for ζ = 1 −1 tanh x 2 = cite this exp − for ζ > 1 Please √ 1 − x2 article as: xR. Tchamna, M. Lee, Optimization approach for the analytical design of an industrial PI controller for the optimal regulatory control of ﬁrst order processes under operational constraints, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.08.012 . ARTICLE IN PRESS JID: JTICE 14 [m5G;August 24, 2017;12:55] R. Tchamna, M. Lee / Journal of the Taiwan Institute of Chemical Engineers 000 (2017) 1–15 For ζ > 1, it should be noted that tupeak > 0⇒ζ > ζ 0f . On the other hand, since lim [tanh(x )] = 1, the following relation x→∞ tanh (x) = λ physically holds only if λ ≤ 1. Hence, x 2ζ (τ − τ )ζ 2 − 1 c tanh t = 2τc upeak 2 ζ 2 ( τc − τ ) + τ 2 ζ ( τc − τ ) ζ 2 − 1 ⇒ ≤1 2 ζ 2 ( τc − τ ) + τ 2 ζ ( τc − τ ) ζ 2 − 1 1 τ =1⇒ζ = = ζ1 f 2 τc ( τ − τc ) 2 ζ 2 ( τc − τ ) + τ τ ζ >1 τ τc ≥ 2 f ( ζ , τc ) = 1 else (A21) (A22) (A23) 2 τc for if ζ 0f and ζ 1f need to be compared. ζ 1f /ζ 0f can be expressed as ζ1 f = ζ0 f and ⎧ 4 −τ τc + τc2 ζ 2 + τ 2 ⎪ ⎪ ⎪ 1+ ⎪ ⎪ τ 2 ⎪ ⎪ ⎪ 2 ⎪ ⎨ exp − 1 tanh−1 2ζ (τc − τ )x x 2 ζ 2 ( τc − τ ) + τ f ( ζ , τc ) = ⎪ for ζ < ζ < ζ ⎪ 0f 1f ⎪ ⎪ ⎪ ⎪ ⎪ 1 for ζ = ζ1 f ⎪ ⎪ ⎩ undeﬁned f or ζ > ζ1 f (A28.2) Hence, ζ1 f > ζ0 f for ζ1 f < ζ0 f for τc < τc > ζ1 f = ζ0 f = 1 for τc = τ with 2 τ (A24) 2 f or ζ ≥ ζ1 f , tanh x tupeak = 1 ⇒ tupeak = ∞. 2 τc (A25) τc > τ upeak = D f (ζ , τc ) where f ( ζ , τc ) = 1 + 4 −τ τc + τ 2 c 2 τ ζ +τ (A29) (A30) u (t) can be derived by taking the derivative of equation (A15) t τ − τ x D c exp − cos t τc 2 τc τ 2 τc 1 − x 2 τ − 2 τc x − sin( t) for 0 < ζ < 1 2 xτ 2 τc t τ − 2τ t D τ −τ u (t ) = − =− exp − c − + τc 2 τc τ 4 τc τ t τ − τ x D c =− exp − cosh t τc 2 τc τ 2 τc 1 + x 2 τ − 2 τc x − sinh( t) for ζ > 1 2 xτ 2 τc c for ζ =1 (A31) The steady-state value of u (t) is (A26) uss (t → ∞ ) = 0. (A27) f or (A32) The peak time of u (t) can be obtained by solving du (t)/dt = 0 as follows: ζ < 1, 2 τc tu peak = 2 1 τ 2 τc ( τ − τc ) Appendix D. Proof of Eq. (5.3) elsei f Therefore, ζ1 f = ; |D| · f (ζ , τc ) ≤ umax 2 tupeak = ∞ τ τc < ⎧ 2 2 ⎪ 2 τc −1 2ζ (τc − τ ) ζ − 1 ⎪ ⎨ tanh for ζ0 f < ζ < ζ1 f x 2 ζ 2 ( τc − τ ) + τ tupeak = ⎪ for ζ = ζ1 f ⎪∞ ⎩ unde f ined for ζ > ζ1 f else 2 τc ζ −1 tupeak = tanh x ζ2 − 1 τ 2 ( τ − τc ) Note that ∀(ζ , τ c ),f(ζ , τ c ) > 0. Therefore, Eq. (4.4) can be converted to 2 Using (A24) and (A25), the peak time for ζ > 1 can be expressed as if ζ0 f = τ Note that ζ 1f can be seen like a cutoff damping ratio. This means that, if ζ > ζ 1f , set ζ = ζ 1f . This implies that x tan −1 4 ζ 2 ( τc − τ ) + τ 4 ζ 2 ( τc − τ ) − 2 τ c + 3 τ 1 − ζ2 ζ + kπ , k ∈ Z 2 1 2ζ 2 (τc − τ )x × exp − tan−1 x 2 ζ 2 ( τc − τ ) + τ (A33) − π x ζ <1 4 −τ τc + τc2 ζ 2 + τ 2 = 1+ τ2 1 2ζ 2 (τc − τ )x × exp − tan−1 for ζ0 f ≤ ζ <1 x 2ζ 2 (τc −τ )+τ 2 τ −τ |τ − 2τc | ( c ) = 1+ exp − for ζ = 1 (A28.1) τ 2 τc − τ For k = 0, for example, the sign of the peak time is given as tu peak > 0 for ζ < ζ01h or ζ > ζ02h < 0 for for 0 < ζ < ζ0 f and with ζ01h = τ ζ01h < ζ < ζ02h 4 ( τ − τc ) ; ζ02h = Considering that (1) tu peak 3 τ − 2 τc . 4 ( τ − τc ) (A34) (A35) must be the smallest positive time value among the above series of times, and (2) for ζ > ζ 01h , the Please cite this article as: R. Tchamna, M. Lee, Optimization approach for the analytical design of an industrial PI controller for the optimal regulatory control of ﬁrst order processes under operational constraints, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.08.012 ARTICLE IN PRESS JID: JTICE R. Tchamna, M. Lee / Journal of the Taiwan Institute of Chemical Engineers 000 (2017) 1–15 theoretical maximum of |u (t)| obtained from du (t)/dt = 0 is always lower than |u (0)|, and tu can be expressed as tu peak = peak 2 τc 4 ζ ( τc + τ ) − τ tan −1 x 4 ζ 2 ( τc + τ ) − 2 τc − 3 τ 2 1−ζ 2 ζ ζ < ζ01h = 0 for ζ ≥ ζ01h for (A36) In fact, it can be shown graphically that for ζ ≥ ζ 01h , the curves of |u (t)| start from the value 1 at t = 0, then decrease and start oscillating with the maximum peaks lower than 1. This means that the mathematical value of the peak of |u (t)|is not the actual peak. The actual peak when ζ ≥ ζ 01h is obtained at the initial time. Therefore, the peak of u (t)can be expressed as upeak = Dh(ζ , τc ) (A37) with h ( ζ , τc ) = 1 2 τ τc ζ 4 τc ( τc − τ ) ζ 2 + τ 2 1 × exp − tan−1 x u ( 0 ) = τ − τc = D τ τc 4ζ 2 (τc −τ )+τ x 2 4ζ (τc −τ )+3τ − 2τc for ζ ≥ ζ01h for ζ < ζ01h (A38) Therefore, Eq. (4.3) can be expressed as |D|h(ζ , τc ) ≤ umax (A39) [m5G;August 24, 2017;12:55] 15 References [1] Chiu T, Christoﬁdes PD. Nonlinear control of particulate processes. AIChE J 1999;45:1279–97. [2] Liou C-T, Chien YS. The effect of nonideal mixing on input multiplicities in a CSTR. Chem Eng Sci 1991;46:2113–16. [3] Rao AS, Chidambaram M. Control of unstable processes with two RHP poles, a zero and time delay. Asia Paciﬁc J Chem Eng 2006;1:63–9. [4] Ali E, Al-humaizi K. Temperature control of ethylene to butene-1 dimerization reactor. Ind Eng Chem Res 20 0 0;39:1320–9. [5] Uma S, Chidambaram M, Rao AS. Enhanced control of unstable cascade processes with time delays using a modiﬁed smith predictor. 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Lee, Optimization approach for the analytical design of an industrial PI controller for the optimal regulatory control of ﬁrst order processes under operational constraints, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.08.012

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