Tuning of fractional PID controllers with Ziegler Nichols-type rules

Transcription

1 Signal Processing 86 (26) Tuning of fractional PID controllers with Ziegler Nichols-type rules Duarte Vale rio,, Jose Sá da Costa Department of Mechanical Engineering, Instituto Superior Técnico, Technical University of Lisbon, GCAR, Av. Rovisco Pais, 49- Lisboa, Portugal Received 2 April 25; received in revised form 29 October 25; accepted 6 December 25 Available online 9 March 26 Abstract In this paper two sets of tuning rules for fractional PIDs are presented. These rules are quadratic and require the same plant time response data used by the first Ziegler Nichols tuning rule for (usual, integer) PIDs. Hence no model for the plant to control is needed only an S-shaped step response is. Even if a model is known rules quickly provide a starting point for fine tuning. Results compare well with those obtained with rule-tuned integer PIDs. r 26 Elsevier B.V. All rights reserved. eywords: Fractional controllers; PID; Ziegler Nichols tuning rules. Introduction The of PID controllers (proportional derivative integrative controllers) is a linear combination of the input, the derivative of the input and the integral of the input. They are widely used and enjoy significant popularity, because they are simple, effective and robust. One of the reasons why this is so is the existence of tuning rules for finding suitable parameters for PIDs, rules that do not require any model of the plant to control. All that is needed to apply such Corresponding author. Tel.: addresses: dvalerio@dem.ist.utl.pt (D. Vale rio), sadacosta@dem.ist.utl.pt (J.S. da Costa). URLs: ist.utl.pt/pessoal/sc/. Partially supported by Fundac-ão para a Ciência e a Tecnologia, grant SFRH/BPD/2636/24, funded by POCI 2, POS_C, FSE and MCTES. rules is to have a certain time response of the plant. Examples of such sets of rules are those due to Ziegler and Nichols, Cohen and Coon, and the appa Tau rules []. Actually, rule-tuned PIDs often perform in a nonoptimal way. But even though further fine-tuning be possible and sometimes necessary, rules provide a good starting point. Their usefulness is obvious when no model of the plant is available, and thus no analytic means of tuning a controller exists, but rules may also be used when a model is known. Fractional PIDs are generalisations of PIDs: their is a linear combination of the input, a fractional derivative of the input and a fractional integral of the input [2]. Fractional PIDs are also known as PI l D m controllers, where l and m are the integration and differentiation orders; if both values are, the result is a usual PID (henceforth called integer PID as opposed to a fractional PID) /$ - see front matter r 26 Elsevier B.V. All rights reserved. doi:.6/j.sigpro

2 2772 D. Valério, J.S. da Costa / Signal Processing 86 (26) Even though fractional PIDs have been increasingly used over the last years, methods proposed to tune them always require a model of the plant to control [3,4]. (An exception is [5], but the proposed method is far from the simplicity of tuning rules for integer PIDs.) This paper addresses this issue proposing two sets of tuning rules for fractional PIDs. Proposed rules bear similarities to the first rule proposed by Ziegler and Nichols for integer PIDs, and make use of the same plant time response data. The paper is organised as follows. Section 2 sums up the fundamentals of fractional calculus needed to understand fractional PIDs. Then, in Section 3, two analytical methods for tuning fractional PIDs when a plant model is available are addressed; these are used as basis for deriving the tuning rules given in Section 4. Section 5 gives some examples of application and Section 6 draws some conclusions. 2. Fractional order systems 2.. Definitions Fractional calculus is a generalisation of ordinary calculus. The main idea is to develop a functional operator D, associated to an order n not restricted to integer numbers, that generalises the usual notions of derivatives (for a positive n) and integrals (for a negative n). Just as there are several alternative definitions of (usual, integer) integrals (due to Riemann, Lebesgue, Steltjes, etc.), so there are several alternative definitions of fractional derivatives that are not exactly equivalent. The most usual definition is due to Riemann and Liouville and generalises two equalities easily proved for integer orders: c D n x f ðxþ ¼ Z x c ðx tþ n f ðtþ dt; n 2 N, () ðn Þ! D n D m f ðxþ ¼D nþm f ðxþ; m 2 Z _ n; m 2 N. (2) The full definition of D becomes c Dn x f 8 ðxþ R x ðx xþ n c f ðxþ dx if no; Gð nþ >< ¼ f ðxþ if n ¼ ; D n ½ c Dx n n f ðxþš if n4; >: n ¼ minfk 2 N : k4ng: ð3þ It is worth noticing that, when n is positive but noninteger, operator D still needs integration limits c and x; in other words, D is a local operator for natural values of n (usual derivatives) only. The Laplace transform of D follows rules rather similar to the usual ones: L½ D n x 8 f ðxþš s n FðsÞ >< if np; ¼ >: s n FðsÞ Pn s k Dn k x f ðþ if n onon 2 N: k¼ ð4þ This means that, if zero initial conditions are assumed, systems with a dynamic behaviour described by differential equations involving fractional derivatives give rise to transfer functions with fractional powers of s. Even though n may assume both rational and irrational values in (4), the names fractional calculus and fractional order systems are commonly used for purely historical reasons. Some authors replace fractional with non-integer or generalised, however. Thorough expositions of these subjects may be found in [2,6,7] Integer order approximations The most usual way of making use, both in simulations and hardware implementations, of transfer functions involving fractional powers of s is to approximate them with usual (integer order) transfer functions with a similar behaviour. So as to perfectly mimic a fractional transfer function, an integer transfer function would have to include an infinite number of poles and zeroes. Nevertheless, it is possible to obtain reasonable approximations with a finite number of zeroes and poles. One of the best-known approximations is due to Manabe and Oustaloup and makes use of a recursive distribution of poles and zeroes. The approximating transfer function is given by [8] s n k YN n¼ þðs=o z;n Þ ; n4. (5) þðs=o p;n Þ The approximation is valid in the frequency range ½o l ; o h Š. Gain k is adjusted so that both sides of (5) shall have unit gain at rad/s. The number of poles and zeroes N is chosen beforehand, and the good performance of the approximation strongly depends

3 D. Valério, J.S. da Costa / Signal Processing 86 (26) thereon: low values result in simpler approximations, but also cause the appearance of a ripple in both gain and phase behaviours; such a ripple may be practically eliminated increasing N, but the approximation will be computationally heavier. Frequencies of poles and zeroes in (5) are given by p ffiffi o z; ¼ o l Z, ð6þ o p;n ¼ o z;n a; n ¼...N, ð7þ o z;nþ ¼ o p;n Z; n ¼...N, ð8þ a ¼ðo h =o l Þ n=n ð9þ Z ¼ðo h =o l Þ ð nþ=n. ðþ The case no may be dealt with inverting (5). But if jj4 n approximations become unsatisfactory; for that reason (among others), it is usual to split fractional powers of s like this: s n ¼ s n s d ; n ¼ n þ d ^ n 2 Z ^ d 2½; Š. () In this manner only the latter term has to be approximated. If a discrete transfer function approximation is sought, the above approximation (or any other alternative approximation) may be discretised using any usual method (Tustin, Simpson, etc.). But there are also formulas that directly provide discrete approximations. None shall be needed in what follows. See, for instance [9] for more on this subject. 3. Analytical tuning methods The transfer function of a fractional PID is given by CðsÞ ¼P þ I s l þ Dsm. (2) In this section, two methods published in the literature for analytically tuning the five parameters of such controllers are given. 3.. Internal model control The internal model control methodology may, in some cases, be used to obtain PID or fractional PID controllers. It makes use of the control scheme of Fig.. In that control loop, G is the plant to control, G is an inverse of G (or at least a plant as close as possible to the inverse of G), G is a model of G and F is some judiciously chosen filter. If G were exact, the error e would be equal to disturbance d. If, additionally, G were the exact inverse of G and F + - e F G* G d + + Fig.. Block diagram for internal model control. +- e were unity, control would be perfect. Since no models are perfect, e will not be exactly the disturbance. That is also exactly why F exists and is usually a low-pass filter: to reduce the influence of high-frequency modelling errors. It also helps ensuring that product FG is realisable. The interconnections of Fig. are equivalent to those of Fig. 2 if the controller C is given by FG C ¼ FG G. (3) Controller C is not, in the general case, a PID or a fractional PID, but in some cases it will, if G ¼ þ s m T e Ls. (4) Firstly, let F ¼, ð5þ þ st F G ¼ þ sm T, ð6þ G ¼ þ s m ð slþ. ð7þ T Note that the delay of G was neglected in G but not in G, where an approximation consisting of a truncated McLaurin series has been used. Then (3) becomes C ¼ =ðt F þ LÞ þ T=ðT F þ LÞ s s m. (8) This can be viewed as a fractional PID controller with the proportional part equal to zero. G C G + + Fig. 2. Block diagram equivalent to that of Fig.. d

4 2774 D. Valério, J.S. da Costa / Signal Processing 86 (26) Secondly, let F ¼, ð9þ G ¼ þ sm T, ð2þ G ¼ þ s m T þ sl. ð2þ Then (3) becomes C ¼ þ =L þ T=L s s m þ T sm. (22) If one of the two integral parts is neglectable, (22) will be a fractional PID controller. (The price to pay for neglecting a term is some possible slight deterioration in performance.) Finally, if a Pade approximation with one pole and one zero is used in G, F ¼, ð23þ G ¼ þ sm T, ð24þ G ¼ þ s m T sl=2 þ sl=2, ð25þ then (3) becomes C ¼ 2 þ =L s þ T=L s s m þ T 2 sm. (26) Again, (26) will be a fractional PID if one of the two integral parts is neglectable. Obviously, should m 2 Z, Eqs. (8), (22) and (26) become usual PIDs Tuning by minimisation Monje et al. [4] proposed that fractional PIDs be tuned by requiring them to satisfy the following five conditions (C being the controller and G the plant): () The gain-crossover frequency o cg is to have some specified value: jcðjo cg ÞGðjo cg Þj ¼ db. (27) (2) The phase margin j m is to have some specified value: p þ j m ¼ arg½cðjo cg ÞGðjo cg ÞŠ. (28) (3) So as to reject high-frequency noise, the closedloop transfer function must have a small magnitude at high frequencies; thus it is required that at some specified frequency o h its magnitude be less than some specified gain: Cðjo h ÞGðjo h Þ þ Cðjo h ÞGðjo h ÞoH. (29) (4) So as to reject disturbances and closely follow references, the sensitivity function must have a small magnitude at low frequencies; thus it is required that at some specified frequency o l its magnitude be less than some specified gain: þ Cðjo l ÞGðjo l ÞoN. (3) (5) So as to be robust in face of gain variations of the plant, the phase of the open-loop transfer function must be (at least roughly) constant around the gain-crossover frequency: d do arg½cðjoþgðjoþš ¼. (3) o¼ocg Conditions are five because five are the parameters to tune. To satisfy them all the authors proposed the use of numerical optimisation algorithms, namely of Nelder Mead s simplex method as implemented in Matlab s function fmincon (the condition in (27) is assumed as the condition to minimise; conditions in (28) (3) are assumed as constraints). This is effective but allows local minima to be obtained. In practice most solutions found with this optimisation method are good enough, but they strongly depend on initial estimates of the parameters provided. Some may be discarded because they are unfeasible or lead to unstable loops, but in many cases it is possible to find more than one acceptable fractional PID. In other cases, only wellchosen initial estimates of the parameters allow finding a solution. 4. Tuning rules The first Ziegler Nichols rule for tuning an integer PID assumes the plant to have an S-shaped unit-step response, as that of Fig. 3, wherel is an apparent delay and T may be interpreted as a time constant, such as the one resulting from a pole. The method cannot be applied if the unit-step response is shaped otherwise. The simplest plant with such a response is GðsÞ ¼ þ st e Ls. (32)

5 D. Valério, J.S. da Costa / Signal Processing 86 (26) L tangent at inflection point inflection point time Fig. 3. S-shaped unit-step response. The minimisation tuning method presented above in Section 3.2 was applied to plants given by (32) for several values of L and T,with ¼. The parameters of fractional PIDs thus obtained vary in a regular manner with L and T. Using the least-squares method, it is possible to translate that regularity into polynomial formulas to find acceptable values of the parameters from the values of L and T. Theseare given in the two following subsections. 2 Two comments. Firstly, to implement the minimisation tuning method the last condition was verified numerically, evaluating argument in (3) at two frequencies, equal to o cg =:22 and :22 o cg (this corresponds to 2 of a decade for each side of o cg ). It is of course possible to evaluate the argument at other frequencies around o cg ; actually, the larger the interval where the argument is constant (or nearly so) the better, and thus using more than two points might ensure that. However, it was verified that such stronger requirements are so difficult to meet that they often prevent a solution from being found. Secondly, least-squares method-adjusted formulas cannot exactly reproduce every change in parameters. This means that fractional PIDs tuned with the rules presented below never behave as well as those tuned analytically (including those they are based upon), neither are they so robust. In other words, conditions in Eqs. (27) (3) will only approximately be verified. 2 These rules were already presented in []. L+T 4.. First set of rules A first set of rules is given in Table. This is to be read as P ¼ :48 þ :2664L þ :4982T þ :232L 2 :72T 2 :348TL ð33þ and so on. They may be used if :ptp5 and Lp2 (34) and were designed for the following specifications: o cg ¼ :5 rad=s, ð35þ j m ¼ 2=3 rad 38, ð36þ o h ¼ rad=s, ð37þ o l ¼ : rad=s, ð38þ H ¼ db, ð39þ N ¼ 2 db. ð4þ Recall that specifications are only approximately verified Second set of rules A second set of rules is given in Table 2. This second set of rules may be applied if :ptp5 and Lp:5. (4) Only one set of parameters is needed in this case because the range of values of L with which these rules cope with is more reduced. They were designed for the following specifications: o cg ¼ :5 rad=s, ð42þ Table Parameters for the first set of tuning rules P I l D m Parameters to use when :ptp5 :48 :3254 :5766 :662 :8736 L :2664 :2478 :298 :2528 :2746 T :4982 :429 :33 :8 :489 L 2 :232 :33 :73 :72 :557 T 2 :72 :258 :6 :328 :25 LT :348 :7 :4 :222 :323 Parameters to use when 5pTp5 2:87 :52 :645 :42 :292 L 3:527 2:6643 :3268 :377 :537 T :563 :3453 :229 :357 :38 L 2 :5827 :944 :28 :5552 :228 T 2 :25 :2 :3 :2 :7 LT :824 :54 :28 :263 :4

6 2776 D. Valério, J.S. da Costa / Signal Processing 86 (26) Table 2 Parameters for the second set of tuning rules P I l D m :574 :64 :85 :8793 :2778 L 24:542 :425 :3464 5:846 2:522 T :3544 :792 :492 :77 :675 L 2 46:7325 :458 :737 28:388 2:4387 T 2 :2 :8 :6 : :3 LT :36 :25 :38 :67 :2.5.5 j m ¼ rad 57, ð43þ o h ¼ rad=s, ð44þ o l ¼ : rad=s, ð45þ H ¼ 2 db, ð46þ N ¼ 2 db. ð47þ Of course, sets of rules other than those two above might have been found in the same way as these were for different sets of specifications. 5. Examples In this section the rules from Section 4 are applied to three different plants. The performance of the results is then asserted and compared to the performance obtained with integer PIDs tuned with the first Ziegler Nichols rule. Two comments. Firstly, as stated above, rules usually lead to results poorer than those they were devised to achieve. (The same happens with Ziegler Nichols rules: they are expected to result in an overshoot around 25%, but it is not hard to find plants with which the overshoot is % or even more.) Secondly, Ziegler Nichols rules make no attempt to reach always the same gain-crossover frequency, or the same phase margin. Actually, these two performance indicators vary widely as L and T vary. This adds some flexibility to Ziegler Nichols rules: they can be applied for wide ranges of L and T and still achieve a controller that stabilises the plant. Rules from the previous section always aim at fulfilling the same specifications, and that is why their application range is never so broad as that of Ziegler Nichols rules. Bode diagrams presented are exact; all timeresponses involving fractional derivatives and integrals were obtained with simulations making use of Oustaloup s approximation described in the subsection dealing with approximations, with (b) o l ¼ 3 rad=s, o h ¼ 3 rad=s, N ¼ 7. ω / rad s - ω / rad s - Fig. 4. Step response of (5) controlled with (52) when is 32, 6, 8, 4, 2, (thick line), 2, 4 and 8. (b) Open-loop Bode diagram when ¼. Closed-loop function gain (top) and sensitivity function gain (bottom) when ¼. ð48þ ð49þ ð5þ

7 D. Valério, J.S. da Costa / Signal Processing 86 (26) ω / rad s (b) -2-2 (b) ω / rad s ω / rad s Fig. 5. Step response of (5) controlled with (53) when is 32, 6, 8, 4, 2, (thick line). (b) Open-loop Bode diagram when ¼. Closed-loop function gain (top) and sensitivity function gain (bottom) when ¼ ω / rad s Fig. 6. Step response of (5) controlled with (54) when is 32, 6, 8, 4, 2 and (thick line). (b) Open-loop Bode diagram when ¼. Closed-loop function gain (top) and sensitivity function gain (bottom) when ¼.

8 2778 D. Valério, J.S. da Costa / Signal Processing 86 (26) First-order plant with delay The plant considered was GðsÞ ¼ þ s e :s. (5) The nominal value of is. Controllers obtained with the two tuning rules from the previous section and with the first Ziegler Nichols rule are C ðsþ ¼:4448 þ :558 s :4277 þ :245s:22, ð52þ C 2 ðsþ ¼:257 þ :36 s :23 :2589s:533, ð53þ C ZN ðsþ ¼2: þ 6: þ :6 s. ð54þ s (Note that due to the approximations involved one of the gains is negative. This will not, however, affect results.) Corresponding step responses are given in Figs These show what happens for several values of, the plant s gain, which is assumed to be known with uncertainty. It should be noticed that fractional PIDs can deal with a clearly broader range of values of. This is likely because the specifications the integer PID tries to achieve are different: that is why responses are all faster, at the cost of greater overshoots. More important is that the overshoot is fairly constant with fractional PIDs at least for those values closer to. This is because fractional PIDs attempt to verify (3), which the integer PID does not. Data on these responses are summed up in Table 3. (In this and the following tables, the rise time is reckoned according to the 9% rule and the settling time is reckoned according to the 5% rule.) The corresponding open-loop Bode diagrams and the gains of sensitivity and closed-loop functions are also given in those figures. They show that the desired conditions given by Eqs. (27) (3) are reasonably though not exactly followed. The approximations incurred by the least-squares fit are responsible for the conditions being only approximately verified Second-order plant The plant considered was GðsÞ ¼ 4:32s 2 þ 9:8s þ þ 2s e :2s (55) with a nominal value of of. The approximation stems from the values of L and T obtained from its step response. Controllers obtained with the two rules given above and with the first Ziegler Nichols rule are C ðsþ ¼:88 þ 6:585 s :675 þ 2:588s:6957, ð56þ C 2 ðsþ ¼6:9928 þ 2:444 s :6 þ 4:66s :785, ð57þ C ZN ðsþ ¼2: þ 3: þ 2: s. ð58þ s The step responses obtained (together with openloop Bode diagrams and sensitivity and closed-loop functions gains) are given in Figs Table 4 presents data on the step responses. This time, since Table 3 Data on step responses of Figs. 4 6 Controller of Eq. (52) Controller of Eq. (53) Controller of Eq. (54) Rise time Overshoot Settling time Rise time Overshoot Settling time Rise time Overshoot Settling time (s) (%) (s) (s) (%) (s) (s) (%) (s)

9 D. Valério, J.S. da Costa / Signal Processing 86 (26) there is no delay, the plant is easier to control and a wider variation of is supported by all controllers. But fractional PIDs still achieve an overshoot that is more constant in spite of the plant having a structure different from that used to derive the rules ω / rad s ω / rad s (b) (b) ω / rad s ω / rad s Fig. 7. Step response of (55) controlled with (56) when is 32, 6, 8, 4, 2, (thick line), 2, 4, 8, 6 and 32. (b) Open-loop Bode diagram when ¼. Closed-loop function gain (top) and sensitivity function gain (bottom) when ¼. Fig. 8. Step response of (55) controlled with (57) when is 32, 6, 8, 4, 2, (thick line), 2, 4, 8, 6 and 32. (b) Open-loop Bode diagram when ¼. Closed-loop function gain (top) and sensitivity function gain (bottom) when ¼.

10 278 D. Valério, J.S. da Costa / Signal Processing 86 (26) Fractional-order plant with delay (b) ω / rad s - ω / rad s - Fig. 9. Step response of (55) controlled with (58) when is 32, 6, 8, 4, 2, (thick line), 2, 4, 8, 6 and 32. (b) Open-loop Bode diagram when ¼. Closed-loop function gain (top) and sensitivity function gain (bottom) when ¼ The plant considered was GðsÞ ¼ p þ ffiffi e :5s s þ :5s e :s (59) with a nominal value of of. The approximation is derived from the plant s step response at t ¼ :92 s. (It might seem more reasonable to base the approximation on the step response at t ¼ :5s, but this cannot be done, since the response has an infinite derivative at that time instant.) Controllers obtained with the two rules given above and with the first Ziegler Nichols rule are C ðsþ ¼:62 þ :687 s :3646 þ :35s:68, C 2 ðsþ ¼:498 þ :6486 s : :239s:855, C ZN ðsþ ¼8: þ 9: þ :9 s. s ð6þ ð6þ ð62þ The step responses obtained (together with openloop Bode diagrams and sensitivity and closed-loop functions gains) are given in Figs. 2. Table 5 presents data on the step responses. The PID performs poorly because it tries to obtain a fast response and thus employs higher gains (and hence the loop becomes unstable if is larger than 32 ), but that is not what is most relevant. The most relevant result here is that fractional PIDs still achieve practically constant overshoots, since, in spite of the different plant structure, the conditions they were expected to verify are still verified to a reasonable degree, as the frequency response plots show. In this case it is possible to find IMC-tuned fractional PIDs to compare results. Using the parameters of plant (59) ( ¼, T ¼, L ¼ :5 and m ¼ :5; these are not the parameters of the approximation used with the tuning rules), Eqs. (22) and (26) yield the two following transfer functions: C IMC ¼ þ 2 s þ 2 s þ =2 s=2, C IMC ¼ 2 þ 2 s þ 2 s þ =2 2 s=2. ð63þ ð64þ In none of the two cases is clear which of the two integral terms is better to discard. By trying both possibilities, it is found out that it is better to keep

11 Table 4 Data on step responses of Figs. 7 9 Controller of Eq. (52) Controller of Eq. (53) Controller of Eq. (54) D. Valério, J.S. da Costa / Signal Processing 86 (26) Rise time Overshoot Settling time Rise time Overshoot Settling time Rise time Overshoot Settling time (s) (%) (s) (s) (%) (s) (s) (%) (s) the terms with the first derivative: C IMC2 ¼ þ 2 s þ s=2, C IMC ¼ 2 þ 2 s þ 2 s=2. ð65þ ð66þ Step responses obtained are shown in Fig. 3 and compare well with those of Figs. and. It is seen that rule-tuned fractional PIDs perform nearly as well as those found with IMC. 6. Comments and conclusions In this paper, two analytical methods (among others published in the literature) for tuning the parameters of fractional PIDs were reviewed. The optimisation method of [4] was then used for developing two sets of tuning rules similar to those of the first set of Ziegler Nichols rules. These new tuning rules make use of two parameters (L and T) of the unit-step response of the plant, which should be S-shaped: otherwise they cannot be applied. The most obvious difference is that the new rules are clearly more complicated than those of Ziegler Nichols: they have to be quadratic (approximations of lower order were tried but proved unsatisfactory). And the broader the application range of the rules is to be, the more complicated they become: the first rule needs two tables of parameters, while the second, good for a narrower interval of values of L only, needs only one. The usefulness of these rules is that of all sets of rules: they may be applied even if no model of the plant is available, provided a suitable time response is; they may be used as a departing point for finetuning (this is, for instance relevant if the optimisation tuning method is used, since its results depend significantly from the initial estimate provided); they are easier and faster to apply than analytic methods. Their drawbacks are also those of all sets of rules: their performance is often inferior to the one sought, fine-tuning being often needed; they perform worse than controllers tuned analytically; they cannot be applied to all types of plants, but only to those with a particular sort of time response. Fractional PIDs tuned with these new rules compare well with integer PIDs tuned according to the first Ziegler Nichols rule, even though the comparison is made difficult because Ziegler Nichols rules achieve different specifications for different values of T and L while the new rules attempt to always keep a uniform result. The advantage fractional PIDs provide is a roughly constant overshoot when the gain of the plant undergoes variations. (It is of course likely that carefully tuned integer PIDs perform better than rule-tuned fractional PIDs just as carefully tuned fractional PIDs are likely to perform better than rule-tuned integer PIDs.) It is surely possible to improve these tuning rules. Rules similar to the second Ziegler Nichols rule (making use of a closed-loop response of the plant) are certainly possible, and are currently being

12 2782 D. Valério, J.S. da Costa / Signal Processing 86 (26) ω / rad s ω / rad s (b) (b) ω / rad s - Fig.. Step response of (59) controlled with (6) when is 32, 6, 8, 4, 2, (thick line) and 2. (b) Open-loop Bode diagram when ¼. Closed-loop function gain (top) and sensitivity function gain (bottom) when ¼ ω / rad s - Fig.. Step response of (59) controlled with (6) when is 32, 6, 8, 4, 2 and (thick line). (b) Open-loop Bode diagram when ¼. Closed-loop function gain (top) and sensitivity function gain (bottom) when ¼. 2 2

13 D. Valério, J.S. da Costa / Signal Processing 86 (26) (b) ω / rad s - ω / rad s - Fig. 2. Step response of (59) controlled with (62) when is 32. (b) Open-loop Bode diagram when ¼. Closed-loop function gain (top) and sensitivity function gain (bottom) when ¼. developed. Rules specific for non-minimum phase plants (with which Ziegler Nichols rules do not properly deal) may also be of interest Table 5 Data on step responses of Fig. 2. Controller of Eq. (52) Controller of Eq. (53) Controller of Eq. (54) Rise time Overshoot Settling time Rise time Overshoot Settling time Rise time Overshoot Settling time (s) (%) (s) (s) (%) (s) (s) (%) (s) s 3 3. s

14 D. Valério, J.S. da Costa / Signal Processing 86 (26) References (b) Fig. 3. Step response of (59) controlled with (65) when is 32, 6, 8, 4 and 2. (b) Step response of (59) controlled with (66) when is 32, 6, 8, 4, 2 and (thick line). [] T. Ha gglund,. Åstro m, Automatic tuning of PID controllers, in: W.S. Levine (Ed.), The Control Handbook, CRC Press, Boca Raton, FL, 996, pp [2] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations to Methods of their Solution and Some of their Applications, Academic Press, San Diego, 999. [3] R. Caponetto, L. Fortuna, D. Porto, Parameter tuning of a non integer order PID controller, in: Fifteenth International Symposium on Mathematical Theory of Networks and Systems, Notre Dame, 22. [4] C.A. Monje, B.M. Vinagre, Y.Q. Chen, V. Feliu, P. Lanusse, J. Sabatier, Proposals for fractional PI l D m tuning, in: Fractional Differentiation and its Applications, Bordeaux, 24. [5] Y.Q. Chen,.L. Moore, B.M. Vinagre, I. Podlubny, Robust PID controller autotuning with a phase shaper, in: Fractional Differentiation and its Applications, Bordeaux, 24. [6].S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 993. [7] S.G. Samko, A.A. ilbas, O.I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach, Yverdon, 993. [8] A. Oustaloup, La commande CRONE: commande robuste d ordre non entier, Hermès, Paris, 99. [9] B.M. Vinagre, I. Podlubny, A. Hernández, V. Feliu, Some approximations of fractional order operators used in control theory and applications, Fractional Calculus & Appl. Anal. 3 (2) [] D. Valério, J. Sa da Costa, Ziegler Nichols type tuning rules for PID controllers, in: Fifth ASME International Conference on Multibody Systems, Nonlinear Dynamics and Control, Long Beach, 25.