530 S. E. Hamamci and N. Tan / ISA Transactions 45, (2006) 529–543bilizing P, PI, and PID controllers ͓7–10͔. How-ever, beyond stabilization it is important to designcontrollers that guarantee speciﬁed performancemeasures such as gain and phase margins, settlingtime, and overshoot. Although there are some clas-sical formulas that exist which link frequency andtime domain performances such as Ϸ PM/ 100,these approximate formulas do not work for timedelay systems or high order control systems. Itseems that the coefﬁcient diagram method ͑CDM͒ Fig. 1. A standard block diagram of the CDM controlcombined with the methods for the computation of system.stabilizing controllers can be a good candidate fordesigning controllers which attain the time and ts in the same plane, ͑k p , ki͒ plane, is obtained.frequency domain speciﬁcations simultaneously. This graphical representation is called the fre-CDM is a polynomial approach which was devel- quency and time domain performances (FTDP)oped and introduced for a good transient response map. Thus, one can easily obtain a set of PI con-of the control systems by Manabe ͓11͔. The most trollers which attains the desired frequency andimportant properties of this method are the adap- time domain performances using the FTDP map.tation of the polynomial representation for both The paper is organized as follows: The CDM isthe plant and controller, the use of the two-degree- revisited in Section 2. The proposed controller de-of-freedom ͑2DOF͒ control system structure, the sign method is explained in Section 3. Simulationnonexistence ͑or very small͒ of the overshoot in examples are given in Section 4. Section 5 in-the step response of the closed-loop system, the cludes some concluding remarks.determination of the desired settling time and themaximum overshoot at the start and to continue 2. Coefﬁcient diagram methodthe design accordingly, and good robustness of thecontrol system with respect to the parameter The standard block diagram of the CDM controlchanges. system is shown in Fig. 1, where r is the reference In this paper, a new method is proposed for the input, y is the output, d is the disturbance, and u iscomputation of a set of PI controllers which give the control signal. N p͑s͒ and D p͑s͒ are the nu-the prescribed frequency and time domain perfor- merator and denominator polynomials of themance criteria such as gain margin, phase margin, transfer function of the plant and have not anyovershoot, and settling time simultaneously. In this common factors. A͑s͒ is the forward denominatormethod, using the stability boundary locus ap- polynomial while B͑s͒ and F͑s͒ are the feedbackproach of Refs. ͓10,12͔, all the stabilizing values numerator and the reference numerator polynomi-of the parameters of the PI controller are ﬁrst ob- als of the controller transfer function. Since thetained. This stability region is called the global transfer function of the CDM controller has twostability region. In addition to this, all the stabiliz- numerators, the control system resembles to aing PI controllers within the global stability region 2DOF system structure. A͑s͒ and B͑s͒ are de-which provide desired frequency domain perfor- signed as to satisfy the desired transient behaviormance measures are identiﬁed. These subsets of and deﬁned asthe global stability region are called the local sta- pbility region. Then, the settling time and overshoot A ͑ s ͒ = ͚ l is i ,which are very important time domain perfor- i=0mance measures are chosen for time domain qspeciﬁcations. Using the CDM, some explicit for-mulations are obtained between the PI controller B ͑ s ͒ = ͚ k is i ͑1͒ i=0parameters and the time domain speciﬁcations.For this, a method is given for removing errors in the polynomial forms while preﬁlter F͑s͒ is de-due to approximation used for the time delay term. termined as the zero order polynomial and used toFinally, a graphical relation on GM, PM, MO, and provide the steady-state gain. Better performance
S. E. Hamamci and N. Tan / ISA Transactions 45, (2006) 529–543 531Fig. 2. ͑a͒ The effect of the equivalent time constant on the rate of the closed-loop time response. ͑b͒ The effect of the stabilityindex, ␥1 on the time response shape.can be expected when using a 2DOF structure, the target characteristic polynomial of the closed-because it can focus on both tracking the desired loop system. From Eq. ͑4͒, the characteristic poly-reference signal and disturbance rejection. Un- nomial in Eq. ͑3͒ can be expressed in terms of stable pole-zero cancellations and the use of more and ␥i as ͫͭ ͚ ͩ ͟ ͪ ͮ ͬnumbers of integrators are also avoided in imple-mentations with this structure. The output of the n i−1 1CDM control system in Fig. 1 is deﬁned as Ptarget͑s͒ = a0 ͑s͒i + s + 1 . i=2 j=1 ␥i−j j N p͑ s ͒ F ͑ s ͒ A ͑ s ͒ N p͑ s ͒ y= r+ d, ͑2͒ ͑5͒ P͑s͒ P͑s͒where P͑s͒ is the characteristic polynomial of the The equivalent time constant speciﬁes the time re-closed-loop system. This polynomial is a Hurwitz sponse speed ͑settling time, especially͒. The sta-polynomial with real positive coefﬁcients and de- bility indices affect the stability and the shape ofscribed by the time response ͑overshoot, especially͒. For ex- n ample, consider a characteristic polynomial whose P ͑ s ͒ = A ͑ s ͒ D p͑ s ͒ + B ͑ s ͒ N p͑ s ͒ = ͚ a is i, ai Ͼ 0. degree is chosen as n = 2. According to this poly- i=0 nomial, choosing ␥1 = 3 as constants and changing ͑3͒ the equivalent time constant in the interval of ͑1– 4͒, the unit step responses of the control systemIn the literature, it is reported that there are some are shown in Fig. 2͑a͒. It is seen in this ﬁgure thatcertain relations between characteristic polynomial is very effective on the settling time of the unit-coefﬁcients and important time domain measures step response. If the equivalent time constant issuch as settling time and overshoot ͓13–15͔. increased, the settling time is also increased. OnManabe integrated these studies with the basic the contrary, if is reduced, the system time re-principles of his method and determined two im- sponse can be accelerated as desired. When theportant design parameters. These parameters are choice = 1 is considered and ␥1 is changed in theequivalent time constant ͑͒ and stability indices interval of ͑0.5–5͒, the change of the time re-͑␥i͒ which are deﬁned as sponse shapes are shown in Fig. 2͑b͒. This ﬁgure = a1/a0, ␥i = a2/͑ai+1ai−1͒ , indicates that the stability index ␥1 is much effec- i tive on the response shape and stability. If ␥1 is ͑4͒ made bigger, the stability of the control system is i = 1 ϳ ͑n − 1͒, ␥0 = ␥n = ϱ. increased and the overshoot is zero. But ␥1 is of aIt is important that these parameters are speciﬁed the small size, the stability decreases and over-before the design and then used for determining shoot is nonzero in this case.
532 S. E. Hamamci and N. Tan / ISA Transactions 45, (2006) 529–543 lim C͑s͒ = ϱ and lim C f ͑s͒/C͑s͒ = 0. ͑6͒ s→0 s→0 If C͑s͒ includes an integrator then these condi- tions can be satisﬁed. Thus, C͑s͒ can be chosen as k i k ps + k i C͑s͒ = k p + = ͑7͒ s s Fig. 3. A 2DOF control system structure. in the type of the conventional PI element and C f ͑s͒ is an appropriate element satisfying Eq. ͑6͒. For a special case, it is recommended that stan- The proposed design approach consists of a num-dard Manabe values for the time response without ber of distinct steps, which can be summarized asovershoot and with the smallest settling time are follows.used for the CDM design. Stability indices arechosen as ␥i = ͕2.5, 2 , 2 , . . . , 2͖ for i = 1 ϳ ͑n − 1͒n ͑I͒ Frequency response performanceϾ 3 in this form. For a third order system ␥i ͑I a͒ Computation of global stability region= ͕2.7, 2͖ and ␥1 = 3 in a second order system, Consider a 2DOF control system shown in Fig.overshoot is zero. When the standard Manabe val- 3 whereues are chosen, the settling time is about͑2.5– 3͒. The selection of the standard values can N ͑ s ͒ −sbe relaxed according to the various performance G a͑ s ͒ = G ͑ s ͒ e −s = e ͑8͒ D͑s͒requirements. and a PI controller of the form of Eq. ͑7͒. The problem is to compute the global stability region3. Controller design which includes all the parameters of the PI con- troller of Eq. ͑7͒ which stabilize the given system. The proposed method which is used to design a The closed-loop characteristic polynomial P͑s͒ ofPI controller satisfying the required time and fre- the system, i.e., the numerator of 1 + C͑s͒Ga͑s͒,quency domain speciﬁcations consists of two can be written assteps. In the ﬁrst step, a method is proposed to P͑s͒ = sD͑s͒ + ͑k ps + ki͒N͑s͒e−scompute the global and the local stability regionsusing the stability boundary locus approach = ansn + an−1sn−1 + ¯ + a1s + a0 , ͑9͓͒10,12͔. In the second step, the CDM method is where all or some of the coefﬁcients ai, iused to design PI controllers for which the step = 0 , 1 , 2 , . . . , n are the function of k p, ki, and e−sresponses have a required overshoot and an ac- depending on the order of N͑s͒, and D͑s͒ polyno-ceptable settling time. As a result of combining mials. In the parameter space approach, there arethese two steps, the FTDP map is obtained. The three possibilities for a root of a stable polynomialFTDP map, which is a graphical tool, shows the to cross over the imaginary axes ͑to become un-relation between the stabilizing parameters of the stable͒:PI controllers and the chosen frequency and timedomain performance criteria on the same ͑k p , ki͒ ͑a͒ Real Root Boundary: A real root crossesplane. Thus, one can choose a PI controller pro- over the imaginary axis at s = 0. Thus, theviding all of the desired GM, PM, MO, and ts real root boundary can be obtained fromspeciﬁcation values together. substituting s = 0 in P͑s͒ of Eq. ͑9͒ which A general schema of the 2DOF control system is gives a0 = 0.shown in Fig. 3. This representation is the rational ͑b͒ Inﬁnite Root Boundary: A real root crossesequivalence of the control system given in Fig. 1. over the imaginary axis at s = ϱ. Thus, theHere, C͑s͒ = B͑s͒ / A͑s͒ is the main controller and inﬁnite root boundary can be characterizedC f ͑s͒ = F͑s͒ / B͑s͒ is the set point ﬁlter ͓16͔. It can by taking an = 0 from Eq. ͑9͒.be shown that the steady-state error to the unit step ͑c͒ Complex Root Boundary: Eq. ͑9͒ becomeschange and the unit step disturbance become zero unstable at s = j when its roots cross therobustly if imaginary axis which means that the real
S. E. Hamamci and N. Tan / ISA Transactions 45, (2006) 529–543 533 and the imaginary parts of Eq. ͑9͒ become k p͓Ne cos͑͒ + 2No sin͔͑͒ zero simultaneously. Thus, the complex root boundary can be obtained as follows: + ki͓No cos͑͒ − Ne sin͔͑͒ = − De . ͑12b͒ Decomposing the numerator and the denomina-tor polynomials of G͑s͒ in Eq. ͑8͒ into their even Letand odd parts, and substituting s = j , gives Q͑͒ = Ne sin͑͒ − 2No cos͑͒ , N e͑ − 2͒ + j N o͑ − 2͒ R͑͒ = Ne cos͑͒ + No sin͑͒ , ͑13a͒ G͑ j ͒ = . ͑10͒ D e͑ − 2͒ + j D o͑ − 2͒ X ͑ ͒ = 2D o ,For simplicity ͑−2͒ will be dropped in the fol- S͑͒ = Ne cos͑͒ + 2No sin͑͒ ,lowing equations. Thus, the closed-loop character-istic polynomial of Eq. ͑9͒ can be written as ͑13b͒ U͑͒ = No cos͑͒ − Ne sin͑͒, Y ͑͒ =P͑ j͒ = ͓͑kiNe − k p2No͒cos͑͒ − De . + ͑kiNo + k pNe͒sin͑͒ − Do͔ 2 Then, Eqs. ͑12a͒ and ͑12b͒ can be written as + j͓͑kiNo + k pNe͒cos͑͒ − ͑kiNe k pQ ͑ ͒ + k iR ͑ ͒ = X ͑ ͒ , − 2k pNo͒sin͑͒ + De͔ = R P + jI P = 0. k pS ͑ ͒ + k iU ͑ ͒ = Y ͑ ͒ . ͑14͒ ͑11͒ From these equations X͑͒U͑͒ − Y ͑͒R͑͒Then, equating the real and imaginary parts of kp = ,P͑j͒ to zero, two equation are obtained as Q͑͒U͑͒ − R͑͒S͑͒ Y ͑͒Q͑͒ − X͑͒S͑͒ k p͓− 2No cos͑͒ + Ne sin͔͑͒ ki = . ͑15͒ Q͑͒U͑͒ − R͑͒S͑͒ + ki͓Ne cos͑͒ + No sin͔͑͒ = Do , 2 Substituting Eqs. ͑13a͒ and ͑13b͒ into Eq. ͑15͒, the ͑12a͒ PI controller parameters are obtained as ͑2NoDo + NeDe͒cos͑͒ + ͑NoDe − NeDo͒sin͑͒ kp = , ͑16͒ − ͑ N 2 + 2N 2͒ e o 2͑NoDe − NeDo͒cos͑͒ − ͑NeDe + 2NoDo͒sin͑͒ ki = . ͑17͒ − ͑ N 2 + 2N 2͒ e oThe stability boundary locus, l͑k p , ki , ͒ can be the stability boundary locus has been obtainedconstructed in the ͑k p , ki͒ plane using Eqs. ͑16͒ then it is necessary to test whether stabilizing con-and ͑17͒. If at any particular frequency value the trollers exist or not since the stability boundarydenominator of Eqs. ͑16͒ and ͑17͒ Ne + 2N2 = 0,o locus, l͑k p , ki , ͒, the real root and inﬁnite rootthen this value of frequency must not be used. In boundary lines ͑if they are exist͒ may divide thethis case, a discontinuous stability boundary locus parameter plane ͓͑k p , ki͒ plane͔ into stable and un-will be obtained and this will not be a problem for stable regions. It can be found that the line ki = 0 isthe computation of stabilizing controllers. Once the real root boundary line obtained from substi-
534 S. E. Hamamci and N. Tan / ISA Transactions 45, (2006) 529–543tuting = 0 into Eq. ͑9͒ and equating it to zero ͑ N oD e − N eD o͒since a real root of P͑s͒ of Eq. ͑9͒ can crossover tan͑͒ = = f ͑͒ . ͑19͒ N eD e + 2N oD othe imaginary axis at s = 0. Generally, the order ofD͑s͒ is greater than order of N͑s͒ for proper trans- Thus, c is the solution of Eq. ͑19͒ in the intervalfer function. Therefore, there will be no inﬁnite ͑0 , ͒. By plotting tan͑͒ and f͑͒ vs , it canroot boundary line. be seen that c is the smallest value of at which It can be seen that the stability boundary locus is plots of tan͑͒ and f͑͒ intersect with eachdependent on the frequency which varies from 0 other. If there is more than one real value of to ϱ. However, one can consider the frequency which satisﬁes Eq. ͑19͒ then the frequency axisbelow the critical frequency, c, or the ultimate can be divided into a ﬁnite number of intervalsfrequency since the controller operates in this fre- and by testing each interval the stability region canquency range. Thus, the critical frequency can be be computed.used to obtain the stability boundary locus over apossible smaller range of frequency such as ͑I b͒ Computation of local stability regions ͓0 , c͔. Since the phase of G p͑s͒ at s = jc isequal to −180°, one can write Phase and gain margins are two important fre- ͩ ͪ ͩ ͪ quency domain performance measures which are No Do widely used in the classical control theory for the tan−1 − tan−1 − = − Ne De controller design. Consider Fig. 3 with a gain- phase margin tester ͓17͔, Gc͑s͒ = Ae−j, which is ͑18͒ connected in the feed forward path. From Eqs.or ͑16͒ and ͑17͒, ͑2NoDo + NeDe͒cos͑h͒ + ͑NoDe − NeDo͒sin͑h͒ kp = , ͑20͒ − A ͑ N 2 + 2N 2͒ e o 2͑NoDe − NeDo͒cos͑h͒ − ͑NeDe + 2NoDo͒sin͑h͒ ki = , ͑21͒ − A ͑ N 2 + 2N 2͒ e owhere h = + . Here, the gain-phase margin the industry can be mostly described by a FOPDTtester is not an actual part of the system which is model such asincluded in the system in order to obtain the sta-bility regions for prespeciﬁed values of the gain N m͑ s ͒ −s K −s G m͑ s ͒ = e = e , ͑22͒and phase margins. To obtain the stability bound- D m͑ s ͒ Ts + 1ary locus for a given value of gain margin A, oneneeds to set = 0 in Eqs. ͑20͒ and ͑21͒. On the where K is the gain, T is the time constant, and other hand, setting A = 1 in Eqs. ͑20͒ and ͑21͒, one is the dead time. The experimental identiﬁcationcan obtain the stability boundary locus for a given of these models using many techniques is well de-phase margin . Thus, the local stability regions scribed in Ref. ͓4͔. It is necessary to use thefor speciﬁed gain and phase margins can be iden- FOPDT model for the proposed method, since thetiﬁed within the global stability region. CDM requires high order controllers for high or-͑II͒ Time response performance der models. The term e−s which also representsThe CDM technique is used to obtain time domain the time delay in Eq. ͑22͒ is approximated asperformances. However, in this case, it is neces- e−s Ϸ 1 − s using the ﬁrst-order Taylor numeratorsary to use the ﬁrst-order plus dead time ͑FOPDT͒ approximation. The Taylor numerator approxima-model of the plant. The processes encountered in tion is used in order to not affect the denominator
S. E. Hamamci and N. Tan / ISA Transactions 45, (2006) 529–543 535Fig. 4. Time delay approximation for corrected and uncorrected cases: ͑a͒ selection of k p and ki parameters and ͑b͒ the stepresponses for selected k p and ki values.of Eq. ͑22͒. If the Taylor denominator approxima- by Eq. ͑24͒ are replaced in Eq. ͑3͒. Hence, a poly-tion or Pade approximation is used, theirs denomi- nomial depending on the parameters k p and ki isnators lead to a higher order of the approximate obtained. For the FOPDT model transfer function,transfer function of the plant and consequently to the characteristic polynomial of the control systemmore complex resulting controllers. The results is determined asobtained in Section 4 show that the ﬁrst-order Tay-lor numerator approximation is acceptable and P͑s͒ = ͑T − Kk p͒s2 + ͑1 − Kki + Kk p͒s + Kki .gives good results. Thus, for the model transferfunction of Eq. ͑22͒, plant polynomials are ͑25͒ N p͑s͒ = − Ks + K, D p͑s͒ = Ts + 1. ͑23͒ Using Eq. ͑4͒, ␥1 and can be obtained asFor a selection of good controllers, the degrees ofthe controller polynomials in Eq. ͑1͒ get impor- ␥1 = ͑1 + Kk p − Kki͒2/Kki͑T − Kk p͒ ,tance. The most important fact that affects the de-grees is the existence of a disturbing signal and its ͑26a͒type. It is advised that the minimum degree poly-nomials are chosen depending on the type of thedisturbance. In this paper, the controller polynomi- = ͑1 + Kk p − Kki͒/Kki . ͑26b͒als are chosen for the step disturbance signal. Inthis case, the controller polynomials have the For Eqs. ͑26a͒ and ͑26b͒, ki can be found for ␥1forms and separately as A͑s͒ = l1s, B ͑ s ͒ = k 1s + k 0 . ͑24͒ Table 1Thus, the conﬁguration in Fig. 3 is transformed to ␥1 and k values for some overshoot values.a PI-based control system. From Eqs. ͑7͒ and ͑24͒, MOit can be seen that k1 = k p, k0 = ki and l1 = 1. ͑%͒ ␥1 values k values ͑ts Х k͒͑II a͒ Computation of the PI controller parametersfor the time domain performance. 20 0.8 11.2In the design, a feedback controller is chosen by 10 1.4 6the pole-placement technique and then, a set point 5 1.9 5.2ﬁlter is determined so as to match the steady-state 0a 3a 3a 0 5 3.9gain of the closed-loop system. According to this, athe controller polynomials which are determined Standard Manabe values.
536 S. E. Hamamci and N. Tan / ISA Transactions 45, (2006) 529–543Fig. 5. For K = 1, T = 1, and = 2 in Eq. ͑22͒: ͑a͒ ki-k p curves of ␥1 and , ͑b͒ the step responses for = 4 and different valuesof ␥1 selected from Table 1. ki = ͓K͑2 + 2Kk p + ␥1T − ␥1Kk p͒ may not be exactly obtained since an approxima- tion for time delay has been used. Therefore, a − ͱ⌬͔/͑2K22͒ for ␥1 , ͑27a͒ correction process should be implemented. To do this, a second order Taylor numerator approxima-where ⌬ = K2͓͑2 + 2Kk p + ␥1T − ␥1Kk p͒2 tion, e−s Ϸ 1 − s + 0.52s2 is used for the same− 42͑1 + Kk p͒2͔ and controller of Eq. ͑24͒. Repeating the same math- ki = ͑1 + Kk p͒/͑K + K͒ for . ͑27b͒ ematical derivations for and ␥1, it can be seen that Eq. ͑26b͒ remains same but Eq. ͑26a͒ isFrom the global stability region obtained in step changed to the formI a, an interval for k p can be found as ͑k pmin-k pmax͒. Using this interval and Eqs. ͑27a͒ and ␥1 = ͑1 + Kk p − Kki͒2/Kki͑T − Kk p + 0.5K2ki͒ .͑27b͒, ki-k p curves can be plotted. The values of k p ͑28͒and ki parameters at the intersection of two curvesprovide desired vs ␥1 values. At this point one Using the higher order approximations for correc-should be careful that desired -␥1 speciﬁcations tion does not effect the relations for and ␥1. kiFig. 6. For K = 1, T = 1, and = 2 in Eq. ͑22͒: ͑a͒ ki-k p curves of ␥1 and , ͑b͒ the step responses for ␥1 = 3 and different valuesof .
S. E. Hamamci and N. Tan / ISA Transactions 45, (2006) 529–543 537 uncorrected cases are shown in Fig. 4͑a͒. From this ﬁgure, it can be seen that ͑ki, k p͒ values which give the desired speciﬁcations are different. The step responses for PI controllers obtained from corrected and uncorrected ki-k p curves are shown in Fig. 4͑b͒ which indicate that the step response for the uncorrected case have a 3.2% overshoot and settling time of 18.2 s, however, the step re- sponse for the corrected case has no overshoot and a settling time of 11.9 s. These values of the set- tling time have been computed using a tolerance band of 2% for the steady-state value ͓18͔. Thus, the error due to the time delay approximation can be removed by this way. The reference numerator polynomial F͑s͒ which isFig. 7. The global stability region including all of the sta- deﬁned as the preﬁlter element is chosen to bebilizing PI parameters for the given plant in Eq. ͑32͒. F͑s͒ = ͉ P͑s͒/Nm͑s͉͒s=0 = ki . ͑30͒values for this case can be obtained as This way, the value of the error that may occur in ki = ͓K͑2 + 2Kk p + ␥1T − ␥1Kk p͒ the steady-state response of the closed-loop sys- tem is reduced to zero. Finally, the set point ﬁlter − ͱ⌬͔/͓K22͑2 − ␥1͔͒ for ␥1 , ͑29a͒ in Fig. 3 is obtained bywhere ⌬ = K2͓͑2 + 2Kk p + ␥1T − ␥1Kk p͒2 F͑s͒ ki+ 2 ͑2 − ␥1͒͑1 + Kk p͒2͔ and 2 C f ͑s͒ = = . ͑31͒ B ͑ s ͒ k ps + k i ki = ͑1 + Kk p͒/͑K + K͒ for . ͑29b͒ Note that the parameters of C f ͑s͒ depend on the PIThus, the PI controller parameters which nearly parameters directly. Therefore, the designer doesgive the desired and ␥1 values can be obtained. not need the extra calculation for the set point ﬁl-For example, let us choose K = 1, T = 1, and = 2 ter.in Eq. ͑22͒ and set ␥1 = 3 and = 4 for a step re- ͑II b͒ Choice of the key parameter values for thesponse which has no overshoot and approximately desired time domain properties.12 s settling time. ki-k p curves for corrected and One of the most important properties of the pro- Fig. 8. Local stability regions: ͑a͒ for some GM values and ͑b͒ for some PM values.
538 S. E. Hamamci and N. Tan / ISA Transactions 45, (2006) 529–543 Fig. 9. ͑a͒ ki-k p curves for different ␥1 values in Table 1. ͑b͒ The step responses of these ␥1 values.posed method is that the settling time and over- different values of are shown in Fig. 6 whichshoot information of the control system are deter- clearly shows that there is no overshoot and themined at the beginning before starting to design. It settling time values determined in Table 1 are alsois generally aimed that the time response of the met.control system must have no overshoot and de- ͑III͒ Building of the FTDP mapsired settling time. Table 1 shows the relation be- The FTDP map is built by plotting the frequencytween ts and for the desired settling time for the domain stability regions and the time domain per-various overshoot properties. These values of the formance curves in the same ͑k p, ki͒ plane. Thegeneral relation are obtained by normalizing the frequency domain stability regions include thecharacteristic polynomial of Eq. ͑5͒ for n = 2 and global and local stability regions which can be ob- = 1. For a transfer function with K = 1, T = 1, and tained using Eqs. ͑16͒, ͑17͒, ͑20͒, and ͑21͒, and the = 2 in Eq. ͑22͒, the step responses for = 4 and time domain performance curves which corre-different values of ␥1 selected from Table 1 are spond to the various overshoot and settling timeshown in Fig. 5 where it can be seen that the maxi- values can be plotted using Eqs. ͑29a͒ and ͑29b͒.mum overshoot values given in Table 1 are exactly Thus, the designer can choose any points, ͑k p, ki͒met. Similarly, the step responses for ␥1 = 3 and values, from the FTDP map which satisfy the de-Fig. 10. ͑a͒ Four different PI controllers in the GMϾ 2 region and on the ␥1 = 3 curve. ͑b͒ The unit step responses withoutovershoot for the selected PI controllers.
S. E. Hamamci and N. Tan / ISA Transactions 45, (2006) 529–543 539 Table 2 The time domain and frequency domain speciﬁcations cor- respond to selected four points. 1 2 3 4 kp 3.0941 2.5002 2.0601 1.7201 ki 1.1573 0.8216 0.6142 0.4768 MO ͑%͒ 0 0 0 0 ts ͑s͒ 6.2 8.8 10.8 12.7 GM 2.3 2.95 3.6 4.4 PM ͑°͒ 38.8 47.7 54.6 60 ͑2͒ ﬁnding ki = g͑k p , ͒ curves which have dif-Fig. 11. A local stability region with 2.3Ͻ GMϽ 5 and ferent values of settling time using Eq.30° Ͻ PMϽ 60° ͑ﬁlled in gray color͒ and ␥1 = 3 curve ex- ͑29b͒.pressed no overshoot property. • Step 3. Plotting the frequency domain perfor-sired frequency and time domain properties. After mance regions together with the time domainbuilding the FTDP map and choosing the control- performance curves, thus, obtaining the FTDPler parameters, the control system shown in Fig. 3 map and ﬁnding the desired PI controllers fromis simulated using the actual process and the de- this map.signed PI controller. However, for some transferfunctions there are not any ͑k p, ki͒ values whichgive the desired performances because of the hardproperties of the plant. For such situations, it is 4. Simulation examplesnecessary to relax the desired frequency and timedomain speciﬁcations. 4.1. Example 1 In view of the earlier developments, the FTDP- A ﬁrst order process with a time delay is chosenmap design procedure linking the frequency and astime domain performances is summarized as fol-lows for the ease of reference: 1 G a͑ s ͒ = e−s , ͑32͒ 5s + 1• Step 1. Computation of the frequency response performance regions in the ͑k p, ki͒-plane for which has a pole near the imaginary axis. The aim Eq. ͑8͒: is to investigate the PI controller parameters which have a satisfying frequency and time domain ͑1͒ construction of the global stability region speciﬁcations such as gain margin, phase margin, using Eqs. ͑16͒ and ͑17͒, the stability overshoot, and settling time. For this, it will be boundary locus l͑k p , ki , ͒ and the real root made use of the FTDP map explained in Section 3. boundary line ki = 0; The global stability region which is shown in ͑2͒ obtaining the local stability regions corre- Fig. 7 is computed using the procedure given in spond to desired GM and PM values using Section 3I a. The global stability region includes Eqs. ͑20͒ and ͑21͒. all PI controllers which stabilize the given system. However, one needs to choose suitable k p and ki values within this region to obtain the required• Step 2. Computation of the time performance frequency and time domain performances. The lo- curves in the global stability region using cal stability regions for the speciﬁed gain and FOPTD model of actual plant given in Eq. ͑8͒: phase margins can be identiﬁed within the global ͑1͒ obtaining ki = f͑k p , ␥1͒ curves which have stability region using the procedure given in Sec- different values of overshoot using Eq. tion 3I b as shown in Figs. 8͑a͒ and 8͑b͒. If the ͑29a͒; local regions for the GM and PM values are com-
540 S. E. Hamamci and N. Tan / ISA Transactions 45, (2006) 529–543 Fig. 12. ͑a͒ Global and desired local stability regions. ͑b͒ Time domain curves related to the overshoot and settling time.bined, the intersection local regions can be ob- in Fig. 10͑a͒ give unit step responses without antained satisfying both the GM and PM together. overshoot for ␥1 = 3 as shown in Fig. 10͑b͒. Now, it is important that how the PI parameters Eq. ͑29b͒ can be used to obtain the desired set-which provide the desired frequency domain per- tling time. The four intersection points in Fig.formances in the local region will exhibit time do- 10͑a͒ express four different settling time valuesmain performances. Using Eq. ͑29a͒ to get the for ␥1 = 3. The intersection points of ␥1 and overshoot information of the control system for curves give important information: ␥1 representsdifferent ␥1 values in Table 1, Fig. 9͑a͒ is ob- the overshoot, and the settling time obtained fromtained. According to this ␥1 values, the step re- ts = k. Thus, the values of k p and ki correspond tosponses are shown in Fig. 9͑b͒ for k p = 1 and cor- the intersection points that give the values of theresponding ki values which are computed from the overshoot and the settling time. The step responses␥1 curves. From the step responses, one can see for these points are shown in Fig. 10͑b͒. It can bethat the values of the resultant overshoots are seen that the settling time increases when movingequal to the results given in Table 1. For example, from point 1 to point 4 as shown in Fig. 10͑b͒. Itfor ␥1 = 1.4, the expected value of the overshoot has been observed that the values of the settlingfrom Table 1 is about 10% and this value is ex- time and overshoot given in Table 1 are met.actly obtained as shown in Fig. 9͑b͒. Similarly, for The local region with properties of 2.3Ͻ GM␥1 = 3 a response without an overshoot is expected Ͻ 5 and 30° Ͻ PMϽ 60° is shown in Fig. 11.and it can be seen that which is also met. It is vital to point out that all the values of k pand ki obtained over any ␥1 curve do not neces-sarily give a response having its overshoot prop-erty given in Table 1. For satisfactory perfor-mance, the gain margin should be greater than 2,and the phase margin should be between 30° and60° ͓18͔. However, the simulation results showedthat especially the gain margin is very effective onthe overshoot property. The part of any ␥1 curveswithin the stability region for GMϾ 2 gives k pand ki values for which the step responses have theovershoot values given in Table 1. Note that GM= 2 is a boundary value and causes a time responsewith a little overshoot. For example, the PI con-trollers which correspond to points 1, 2, 3, and 4 Fig. 13. FTDP map for example 2.
S. E. Hamamci and N. Tan / ISA Transactions 45, (2006) 529–543 541 Fig. 14. Set point and disturbance responses for ͑a͒ points 1, ͑b͒ 2 and ͑c͒ 3.Four points are selected from this ﬁgure within the neously. However, for some transfer functionsshaded region. The frequency and time domain there is not any ͑k p , ki͒ values which give desiredproperties for each point are given in Table 2. performances. For example, there is not anyThus, it can be seen that all PI controllers for ͑k p , ki͒ value for the example studied earlier whichwhich the gain margin is between 2.3 and 4.4, the simultaneously provide the gain margin to be 5phase margin is between 38.3° and 60°, the set- and a response without the overshoot, because thistling time is between 6.2 and 12.7 s, and with no property is out of the desired region. For such situ-overshoot is between the points of 1–4 on ␥1 = 3 ations, it is necessary to relax the desired fre-curve within the local region. quency and time domain speciﬁcations. From the simulation studies using the FTDP- 4.2. Example 2map method, it has been observed that there aremany ͑k p , ki͒ values which satisfy the proper time Consider a higher order process with a largeand frequency domain performances simulta- time delay as 1 G a͑ s ͒ = e−4s . ͑33͒ ͑s + 1͒͑0.5s + 1͒͑0.25s + 1͒͑0.125s + 1͒The aim is to design PI controllers which make the local stability regions are obtained as shown ingain margin of the control system greater than 2.3 Fig. 12͑a͒. For time domain performances it isand the phase margin between 30° and 60°. Using necessary to use the the FOPDT model which isEqs. ͑16͒, ͑17͒, ͑20͒, and ͑21͒ the global and the
542 S. E. Hamamci and N. Tan / ISA Transactions 45, (2006) 529–543Table 3 ͑iii͒ The FTDP map gives a visual chance toThe time domain and frequency domain speciﬁcations cor- see which parts of the global stability re-respond to the selected three points. gion is important for the required design 1 2 3 speciﬁcations. ͑iv͒ A set of PI controllers which provide pre- kp 0.2463 0.1295 0.1034 deﬁned performance speciﬁcations can be ki 0.1317 0.1194 0.1055 computed from FTDP-map. MO ͑%͒ 5 10 5 ts ͑s͒ 25.2 29.3 31 The given simulation examples clearly show GM 2.6 2.8 3.1 that the results presented are useful for the design PM ͑°͒ 57 58.5 60 of PI controllers. It is known that the mathematical techniques for obtaining the exact correlation between the stepgiven in Ref. ͓19͔ as Gm͑s͒ = e−4.462s / ͑1.521s + 1͒. transient response and frequency response are␥1 and curves which are directly related to the available but they are very laborious and of littletime domain performances as explained earlier can practical value ͓18͔. However, the method pre-be seen from Fig. 12͑b͒. Combining Figs. 12͑a͒ sented in this paper has a potential to show aand 12͑b͒, the FTDP map shown in Fig. 13 is ob- graphical relation between the frequency and timetained. From this ﬁgure, it is clear that it is not domain performances of high order or time delaypossible to obtain a step response without an over- systems. Therefore, the results obtained can be ap-shoot since the ␥1 = 3 curve does not pass through plied for many real applications. The extension ofthe shaded region. Looking at Fig. 13, it is seen the method to the PID controller will be very im-that the shaded region is approximately bounded portant in the control system design.by ␥1 = 1.9 and ␥1 = 0.8 which give a 5% and 20%overshoot as stated in Table 1, respectively. There-fore, all PI controllers in the shaded region give Referencesthe desired time and frequency domain perfor-mances. Three points are selected within this re- ͓1͔ Åström, K. J. and Hägglund, T., The future of PID control. Control Eng. Pract. 9, 1163–1175 ͑2001͒.gion which are indicated by 1, 2, and 3 as shown ͓2͔ Ziegler, J. G. and Nichols, N. B., Optimum settings forin Fig. 13. Figs. 14͑a͒–14͑c͒ show the time re- automatic controllers. Trans. ASME 64, 759–768sponses of the selected three points according to ͑1942͒.the set point changes applied at t = 0 s and the dis- ͓3͔ ÅströmK. J., Hägglund, T., Hang, C. C., and Ho, W. K., Automatic tuning and adaptation for PIDturbance applied at t = 70 s. The time and fre- controllers—A survey. Control Eng. Pract. 1, 699–quency domain speciﬁcations of these points are 714 ͑1993͒.shown in Table 3. ͓4͔ Åström, K. J. and Hägglund, T., PID Controllers: Theory, Design, and Tuning. Instrument Society of5. Conclusions America, Research Triangle Park, NC, 1995. ͓5͔ Zhuang, M. and Atherton, D. P., Automatic tuning of optimum PID controllers. IEE Proc.-D: Control A new method which is simple to use has been Theory Appl. 140, 216–224 ͑1993͒.presented to design PI controllers giving the de- ͓6͔ Ho, W. K., Hang, C. C., and Cao, L. S., Tuning of PIDsired frequency and time domain performances. controllers based on gain and phase margins speciﬁca-One of the most important features of the pro- tions. Automatica 31, 497–502 ͑1995͒.posed method is that the desired frequency and ͓7͔ Datta, A., Ho, M. T., and Bhattacharyya, S. P., Struc- ture and Synthesis of PID Controllers. Springer, Newtime domain speciﬁcations can be speciﬁed before York, 2000.the design. It has been shown that: ͓8͔ Ackermann, J. and Kaesbauer, D., Design of robust PID controllers. Proceedings of the 2001 European ͑i͒ The global and the local stability regions Control Conference, 2001, pp. 522–527. can be computed using the stability ͓9͔ Söylemez, M. T., Munro, N., and Baki, H., Fast cal- boundary locus approach. culation of stabilizing PID controllers. Automatica 39, ͑ii͒ The values of the parameters of PI con- 121–126 ͑2003͒. ͓10͔ Tan, N., Kaya, I., and Atherton, D. P., Computation of troller which achieve desired values of stabilizing PI and PID controllers. Proceedings of the settling time and overshoot can be esti- CCA2003 IEEE Conf. on the Contr. Appl., 2003. mated using the CDM. ͓11͔ Manabe, S., Coefﬁcient Diagram Method. Proceed-
S. E. Hamamci and N. Tan / ISA Transactions 45, (2006) 529–543 543 ings of the 14th IFAC Symposium on Automatic Con- Serdar Ethem Hamamci re- trol in Aerospace, Seoul, 1998. ceived B.Sc. and M.Sc. degrees from Erciyes University, Kayseri,͓12͔ Tan, N., Computation of stabilizing PI and PID con- Turkey, in 1992 and Firat Univer- trollers for processes with time delay. ISA Trans. 44, sity, Elazig, Turkey, in 1997, re- 213–223 ͑2005͒. spectively. He obtained a Ph.D.͓13͔ Naslin, P., Essentials of Optimal Control. Boston degree from Firat University, in 2002. He is currently working as Technical, Cambridge, 1969. a research assistant in the depart-͓14͔ Lipatov, A. and Sokolov, N., Some sufﬁcient condi- ment of electrical and electronics tions for stability and instability of continuous linear engineering at Inonu University, stationary systems. Autom. Remote Control ͑Engl. Malatya, Turkey. His primary area Transl.͒ 39, 1285–1291 ͑1979͒. of research is polynomial control methods ͑especially the coefﬁ-͓15͔ Kim, Y. C., Keel, L. H., and Bhattacharyya, S. P., cient diagram method͒ and their applications. Transient response control via characteristic ratio as- signment. IEEE Trans. Autom. Control 48͑12͒, 2238– 2244 ͑2003͒. Nusret Tan was born in Malatya,͓16͔ Chen, C. T., Analog and Digital Control System De- Turkey, in 1971. He received a B.Sc. degree in electrical and sign: Transfer Function, State-Space and Algebraic electronics engineering from Hac- Methods. Saunders College, New York, 1992. ettepe University, Ankara, Turkey,͓17͔ Argoun, M. B. and Bayoumi, M. M., Robust gain and in 1994. He received a Ph.D. de- phase margins for interval uncertain systems. Proceed- gree in control engineering from ings of the 1993 Canadian Conference on Electrical University of Sussex, Brighton, UK, in 2000. He is currently and Computer Engineering 1993, pp. 73–78. working as an associate professor͓18͔ Ogata, K., Modern Control Engineering. Prentice- in the department of electrical and Hall, Englewood Cliffs, NJ, 1970. electronics engineering at Inonu͓19͔ Mann, G. K. I., Hu, B.-G., and Gosine, R. G., Time- University, Malatya, Turkey. His primary research interest lies in domain based design and analysis of new PID tuning the area of systems and control. rules. IEE Proc.: Control Theory Appl. 148͑3͒, 251– 261 ͑2001͒.