TUNING-RULES FOR FRACTIONAL PID CONTROLLERS

TUNING-RULES FOR FRACTIONAL PID CONTROLLERS Duarte Valério, José Sá da Costa Technical Univ. of Lisbon Instituto Superior Técnico Department of Mechanical Engineering GCAR Av. Rovisco Pais, 49- Lisboa, Portugal Abstract: This paper presents several tuning rules for fractional PID controllers, similar to the first and the second sets of tuning rules proposed by Ziegler and Nichols for integer PIDs. Fractional PIDs so tuned perform better than integer PIDs; in particular, step-responses have roughly constant overshoots even when the gain of the plant varies. eywords: Fractional PIDs, tuning rules. INTRODUCTION PID (proportional integral derivative) controllers are well-known and widely used because they are simple, effective, robust, and easily tuned. An important contribution for this last characteristic was the development of several tuning rules for tuning the parameters of such controllers from some simple response of the plant. The data required by a tuning rule would not suffice to find a model of the plant, but is expected to suffice to find a reasonable controller. Such rules are the only choice when there is really no model for the plant and no way to get it. Even when we do have a model, if our control specifications are not too difficult to attain, a rule may be all that is needed, saving the time and the effort required by an analytical method. Rules have their problems, namely providing controllers that are hardly optimal according to any criteria and that hence might be better tuned (and sometimes have to be better tuned to meet specifications), but since they often (though not always) work and Partially supported by Fundação para a Ciência e a Tecnologia, grant SFRH/BPD/2636/24, funded by POCI 2, POS C, FSE and MCTES. are simple their usefulness is unquestionable (as their widespread use attests). Fractional PID controllers are variations of usual PID controllers C(s) = P + I + Ds () s where the (first-order) integral and the (firstorder) derivative of () are replaced by fractional derivatives like this: C(s) = P + I s λ + Dsµ (2) (In principle, both λ and µ should be positive so that we still have an integration and a differentiation.) Fractional PIDs have been increasingly used over the last years (Podlubny, 999). There are several analytical ways to tune them (Vinagre, 2; Caponetto et al., 22; Caponetto et al., 24). This paper is concerned about how to tune them using tuning rules. It is organised as follows. Section 2 describes an analytical method that lies behind the development of the rules. Sections 3 to 7 describe tuning rules similar to those proposed by Ziegler and

Nichols for (integer) PIDs 2. Section 8 addresses the question of how fractional PIDs can be implemented. Section 9 gives some simple examples and section concludes the paper. 2. TUNING BY MINIMISATION tangent at inflection point inflection point In this tuning method, presented by (Monje et al., 24), we begin by devising a desirable behaviour for our controlled system, described by five specifications (five, because the parameters to be tuned are five): () The open-loop is to have some specified crossover frequency ω cg : C (ω cg ) G (ω cg ) = db (3) (2) The phase margin ϕ m is to have some specified value: π + ϕ m = arg [C (ω cg ) G (ω cg )] (4) (3) To reject high-frequency noise, the closed loop transfer function must have a small magnitude at high frequencies; hence, at some specified frequency ω h, its magnitude is to be less than some specified gain H: C (ω h ) G (ω h ) + C (ω h ) G (ω h ) < H (5) (4) To reject disturbances and closely follow references, the sensitivity function must have a small magnitude at low frequencies; hence, at some specified frequency ω l, its magnitude is to be less than some specified gain N: + C (ω l ) G (ω l ) < N (6) (5) To be robust when gain variations of the plant occur, the phase of the open-loop transfer function is to be (at least roughly) constant around the gain-crossover frequency: d dω arg [C (ω) G (ω)] = (7) ω=ωcg Then the five parameters of the fractional PID are to be chosen using the Nelder-Mead direct search simplex minimisation method. This derivativefree method is used to minimise the difference between the desired performance specified as above and the performance achieved by the controller. Of course this allows for local minima to be found: so it is always good to use several initial guesses and check all results (also because sometimes unfeasible solutions are found). 2 Rules in sections 3 and 4 have already been presented in (Valério and Sá da Costa, 25b; Valério and Sá da Costa, 26). Those in sections 5, 6 and 7 are novel. L time Fig.. S-shaped unit-step response 3. A FIRST SET OF S-SHAPED RESPONSE BASED TUNING RULES The first set of rules proposed by Ziegler and Nichols apply to systems with an S-shaped unitstep response, such as the one seen in Fig.. From the response an apparent delay L and a characteristic time-constant T may be determined (graphically, for instance). A simple plant with such a response is G = L+T + st e Ls (8) Tuning by minimisation was applied to some scores of plants with transfer functions given by (8), for several values of L and T (and with = ). The specifications used were ω cg =.5 rad/s (9) ϕ m = 2/3 rad 38 o () ω h = rad/s () ω l =. rad/s (2) H = db (3) N = 2 db (4) Matlab s implementation of the simplex search in function fmincon was used; (3) was considered the function to minimise, and (4) to (7) accounted for as constraints. Obtained parameters P, I, λ, D and µ vary regularly with L and T. Using a least-squares fit, it was possible to adjust a polynomial to the data, allowing (approximate) values for the parameters to be found from a simple algebraic calculation. The parameters of the polynomials involved are given in Table. This means that P =.48 +.2664L +.4982T +.232L 2.72T 2.348T L (5) and so on. These rules may be used if. T 5 and L 2 (6) It should be noticed that quadratic polynomials were needed to reproduce the way parameters

Table. Parameters for the first set of tuning rules for S-shaped response plants Parameters to use when. T 5 Parameters to use when 5 T 5 P I λ D µ P I λ D µ.48.3254.5766.662.8736 2.87.52.645.42.292 L.2664.2478.298.2528.2746 3.527 2.6643.3268.377.537 T.4982.429.33.8.489.563.3453.229.357.38 L 2.232.33.73.72.557.5827.944.28.5552.228 T 2.72.258.6.328.25.25.2.3.2.7 LT.348.7.4.222.323.824.54.28.263.4 change with reasonable accuracy. So these rules are clearly more complicated than those proposed by Ziegler and Nichols (upon which they are inspired), wherein no quadratic terms appear. 4. A SECOND SET OF S-SHAPED RESPONSE BASED TUNING RULES Rules in Table 2 were obtained just in the same way, but for the following specifications: These rules may be applied if ω cg =.5 rad/s (7) ϕ m = rad 57 o (8) ω h = rad/s (9) ω l =. rad/s (2) H = 2 db (2) N = 2 db (22). T 5 and L.5 (23) 5. A FIRST SET OF CRITICAL GAIN BASED TUNING RULES The second set of rules proposed by Ziegler and Nichols apply to systems that, inserted into a feedback control-loop with proportional gain, show, for a particular gain, sustained oscillations, that is, oscillations that do not decrease or increase with time, as shown in Fig. 2. The period of such oscillations is the critical period P cr, and the gain causing them is the critical gain cr. Plants given by (8) have such a behaviour. Re-using the data collected for finding the rules in section 3, obtained with specifications (9) to (4), it is seen that parameters P, I, λ, D and µ obtained vary regularly with cr and P cr. The regularity was again translated into formulas (which are no longer polynomial) using a least-squares fit. The parameters are given in Table 3. This means that P =.439 +.45 cr +.584P cr.4384 cr.855 P cr (24) and so on. These rules may be used if P cr 8 and cr P cr 64 (25) P cr Fig. 2. Plant with critical gain control time 6. A SECOND SET OF CRITICAL GAIN BASED TUNING RULES Re-using in the same wise the data used in section 4, corresponding to specifications (7) to (22), other rules may be got with parameters given in Table 4. These rules may be applied if P cr 2 (26) 7. A THIRD SET OF CRITICAL GAIN BASED TUNING RULES Unfortunately, rules in the two previous sections do not often work properly for plants with a pole at the origin. The following rules address such plants. They were obtained from controllers devised to achieve specifications (9) to (4) with plants given by G = s(s + τ )(s + τ 2 ) It is easy to show that such plants have (27) cr = (τ + τ 2 )τ τ 2 (28) P cr = 2π τ τ 2 (29) Once more the regular variation of parameters P, I, λ, D and µ with cr and P cr was translated into rules using a least-squares fit. The parameters are those given in Table 5 and may be used if.2 P cr 5 and cr 2 (3) (though the performance be somewhat poor near the borders of the range above). But, if rules above (devised for plants with a delay) did not often

Table 2. Parameters for the second set of tuning rules for S-shaped response plants P I λ D µ.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 Table 3. Parameters for the first set of tuning rules for plants with critical gain and period Parameters to use when crp cr 64 Parameters to use when 64 crp cr 64 P I λ D µ P I λ D µ.439.767.324.2293.884.445 5.78.472.39.5425 cr.45..8.53.48..238.3.24.23 P cr.584.49.63.936.6.4795.2783.29 2.625.28 / cr.4384.295.393.5293.749 32.256 56.2373 7.59 38.9333 5.73 /P cr.855..79.44.8.6893 2.597.355.94.2873 Table 4. Parameters for the second set of tuning rules for plants with critical gain and period P I λ D µ..5528.623 5.762. cr.24.2352.34.77.24 P cr.866 7.426.2257 23.396.866 Pcr 2.99 6.344.69 8.2724.99 crp cr.5.67.8.987.5 / cr.93.9399.89.8892.93 /P cr.69.5547.94 2.998.69 cr/p cr.9.687..389.9 P cr/ cr.5846 3.4357.839 2.869.5846 Table 5. Parameters for the third set of tuning rules for plants with critical gain and period P I λ D µ.643 92.562.738 8.677.6688 cr.46.7.4.636. P cr.6769 33.655.97.487.4765 crp cr.2.2..529.2 / cr.865.68.67 2.66.3695 /P cr 2.989 33.7959.36 8.4563.483 cr/p cr.2...3. P cr/ cr.7635 5.672.792 2.335.639 log ( cr).449.9487.64.2.74 log (P cr) 2.6948 336.22.4636 6.634 3.6738 cope with poles at the origin, the rules in this section do not often cope with plants with a delay. 8. IMPLEMENTATION For implementation purposes, fractional PID controllers are usually converted into integer continuous transfer functions or into discrete transfer functions. This is done replacing each fractional derivative with a suitable approximation. There are many ways of finding integer or discrete transfer functions that approximate a fractional derivative. In what follows one of the most popular integer ones will be considered. On digital approximations of fractional derivatives, see for instance (Valério and Sá da Costa, 25a). Oustaloup s continuous approximation (Oustaloup, 99) consists of a transfer function with poles and zeros recursively placed: s ν = k N n= + s ω z,n + s ω p,n, ν > (3) The approximation is to be valid in a pre-defined frequency range [ω a ; ω b ] (the performance being poor, however, near ω a and ω b ). Gain k in (3) is adjusted so that the approximation shall have unit gain at rad/s. The number of poles and zeros N is chosen beforehand (low values resulting in simpler approximations but also causing the appearance of a ripple in both gain and phase behaviours). Frequencies of poles and zeros are given by

ω z, = ω a η (32) ω p,n = ω z,n α, n =... N (33) ω z,n+ = ω p,n η, n =... N (34) α = (ω b /ω a ) ν N (35) η = (ω b /ω a ) ν N (36) Whatever the approximation used, it is usual, whenever ν >, to make s ν = s n s δ, n + δ = ν n Z δ [; ] (37) and then approximate s δ only. 9. ROBUSTNESS Evidence showing that rules in sections 3 and 4 provide reasonable, robust controllers has been presented in (Valério and Sá da Costa, 25b; Valério and Sá da Costa, 26). Here, similar examples are shown for critical gain based rules. The plant considered is G (s) = 2s + e.2s (38) for several values of. Controllers obtained with rules from sections 5 and 6 are, respectively, C (s) =.9 + 6.492 s.6363 + 2.3956s.5494 (39) C 2 (s) =.3835 + 4.7942 s.748 + 3.6466s.3835 (4) A plant with one pole at the origin and with a similar step-response, in what concerns apparent delay and characteristic time-constant, is G 2 (s) = s 3 + 2.539s 2 + 62.5s (4) The controller obtained with rules from section 7 is C 3 (s) =.827 + 4.3683 s.5588.6866s.2328 (42) Simulations shown in Fig. 3, Fig. 4 and Fig. 5 were obtained using Oustaloup s approximations for the fractional derivatives of (39), (4) and (42). In this particular case, ω a = 3 rad/s (43) ω b = 3 rad/s (44) N = 7 (45) Notice that for values of close to the overshoot does not vary significantly the only difference is that the response is faster or slower. Also notice that specifications (9) to (4) or (7) to (22) are roughly followed, even though not exactly followed this is because of the approximations involved in the process of finding the parameters. An integer PID tuned with the second set of rules by Ziegler and Nichols is unable to stabilise (38). Plant (4) seems easier to control: the PID manages it, and so do (39) and (4). But only fractional PIDs achieve overshoots more or less constant in face of variations of.. CONCLUSIONS In this paper tuning rules (inspired by those proposed by Ziegler and Nichols for integer PIDs) are given to tune fractional PIDs. Two different sets of fixed performance specifications are used; other rules may be similarly obtained for other sets. Such specifications are roughly followed and are more stringent than those aimed at by the rules of Ziegler and Nichols. Though developed for plants with particular forms, the rules presented can usually be applied to other plants with different transfer functions, as long as they have S-shaped unit-step responses or a critical gain control. Fractional PIDs so tuned perform better than rule-tuned PIDs. This may seem trivial, for we now have five parameters to tune (while PIDs have but three), and the actual implementation requires several poles and zeros (while PIDs have but one invariable pole and two zeros). But the new structure might be so poor that it would not improve the simpler one it was trying to upgrade; this is not, however, the case, for fractional PIDs perform fine and with greater robustness. Additionally, examples given show tuning rules to be an effective way to tune the five parameters required. Of course, better results might be got with an analytical tuning method for integer PIDs; but what we compare here is the performance with tuning rules. These reasonably (though not exactly) follow the specifications from which they were built (through tuning by minimisation). One might wonder, since the final implementation has plenty of zeros and poles, why these could not be chosen on their own right, for instance adjusting them to minimise some suitable criteria. Of course they could: but such a minimisation is hard to accomplish. By treating all those zeros and poles as approximations of a fractional controller, it is possible to tune them easily and with good performances, as seen above, and to obtain a understandable mathematical formulation of the dynamic behaviour obtained. So this seems to be a promising approach to fractional control. Future work is possible and desirable, to further explore other means of tuning this type of controller.

.5 2.5 phase / º 5 5 2 2 5 2 4 6 2 2 2 2 2 3 4 5 time / s 2 2 4 2 2 Fig. 3. Left: Step response of (38) controlled with (39) when is /6, /8, /4, /2, (thick line), 2, 4 and 8; centre: open-loop Bode diagram when = ; right: sensitivity function gain (top) and closed-loop gain (bottom) when =.5 2.5 phase / º 5 5 2 2 5 2 4 6 2 2 2 2 2 3 4 5 time / s 2 2 4 2 2 Fig. 4. Left: Step response of (38) controlled with (4) when is /32, /6, /8, /4, /2, (thick line), 2, 4 and 8; centre: open-loop Bode diagram when = ; right: sensitivity function gain (top) and closed-loop gain (bottom) when =.5.5 2 3 4 5 time / s 5 5 2 2 phase / º 2 3 2 2 2 2 4 6 2 2 2 4 6 2 8 2 2 Fig. 5. Left: Step response of (4) controlled with (42) when is /8, /4, /2, (thick line), 2, 4, 8 and 6; centre: open-loop Bode diagram when = ; right: sensitivity function gain (top) and closed-loop gain (bottom) when = REFERENCES Caponetto, R., L. Fortuna and D. Porto (22). Parameter tuning of a non integer order PID controller. In: Electronic proceedings of the 5th International Symposium on Mathematical Theory of Networks and Systems. Caponetto, Ricardo, Luigi Fortuna and Domenico Porto (24). A new tuning strategy for a non integer order PID controller. In: First IFAC Workshop on Fractional Differentiation and its Applications. Bordeaux. Monje, C. A., B. M. Vinagre, Y. Q. Chen, V. Feliu, P. Lanusse and J. Sabatier (24). Proposals for fractional PI λ D µ tuning. In: First IFAC Workshop on Fractional Differentiation and its Applications. Bordeaux. Oustaloup, Alain (99). La commande CRONE : commande robuste d ordre non entier. Hermès. Paris. In French. Podlubny, Igor (999). Fractional differential equations. Academic Press. San Diego. Valério, Duarte and José Sá da Costa (25a). Time-domain implementation of fractional order controllers. IEE Proceedings Control Theory & Applications. Accepted for publication. Valério, Duarte and José Sá da Costa (25b). Ziegler-nichols type tuning rules for fractional PID controllers. In: Proceedings of ASME 25 Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Long Beach. Valério, Duarte and José Sá da Costa (26). Tuning of fractional PID controllers with ziegler-nichols type rules. Signal Processing. Accepted for publication. Vinagre, Blas (2). Modelado y control de sistemas dinámicos caracterizados por ecuaciones íntegro-diferenciales de orden fraccional. PhD thesis. Universidad Nacional de Educación a Distancia. Madrid. In Spanish.