TUNING RULES FOR FRACTIONAL PIDs 465 3 Then the five parameters of the fractional PID are to be found using the 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 ini- tial guesses and check all results (also because sometimes unfeasible solutions are found). The first set of rules proposed by Ziegler and Nichols apply to systems with an S-shaped unit-step response, such as the one seen in Fig. 1. 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 = K 1+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 K =1).The specifications used were ω cg =0.5rad/s(9) ϕ m =2/3rad≈ 38 o (10) ω h =10rad/s (11) ω l =0.01 rad/s (12) H = −10 dB (13) N = −20 dB (14) 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. 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 [6, 7]. The parameters of the polynomials involved are given in Table 1. This means that P = −0.0048 + 0.2664L +0.4982T +0.0232L 2 − 0.0720T 2 − 0.0348TL (15) and so on. These rules may be used if 0.1 ≤ T ≤ 50 and L ≤ 2 (16) − Mead direct search simplex minimisation method. This derivative-Nelder freemethod is used to minimise the difference between the desired performance Obtained parameters P , I, λ, D, and μ vary regularly with L and T. Using 3 A First Set of S-shaped Response-Based Tuning Rules 4 466 0 0 K time output tangent at inflection point • inflection point L L+T 0 0 time output P cr Fig. 1. Left: S-shaped unit-step response; right: plant output with critical gain Table 1 . Parameters for the first set of tuning rules for S-shaped response plants Parameters to use when 0.1 ≤ T ≤ 5 P I λ D µ 1 −0.0048 0.3254 1.5766 0.0662 0.8736 L 0.2664 0.2478 −0.2098 −0.2528 0.2746 T 0.4982 0.1429 −0.1313 0.1081 0.1489 L 2 0.0232 −0.13300.0713 0.0702 −0.1557 T 2 −0.0720 0.0258 0.0016 0.0328 −0.0250 LT −0.0348 −0.0171 0.0114 0.2202 −0.0323 Parameters to use when 5 ≤ T ≤ 50 P I λ D µ 12.1187 −0.5201 1.0645 1.1421 1.2902 L −3.5207 2.6643 −0.3268 −1.3707 −0.5371 T −0.1563 0.3453 −0.0229 0.0357 −0.0381 L 2 1.5827 −1.0944 0.2018 0.5552 0.2208 T 2 0.0025 0.0002 0.0003 −0.0002 0.0007 LT 0.1824 −0.1054 0.0028 0.2630 −0.0014 It should be noticed that quadratic polynomials were needed to reproduce the way parameters 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. Rules in Table 2 were obtained just in the same way [6, 7], but for the following specifications: ω cg =0.5rad/s (17) ϕ m =1rad≈ 57 o (18) ω h =10rad/s (19) Val´erio and daCosta control. 4 A Second Set of S-shaped Response-Based Tuning Rules TUNING RULES FOR FRACTIONALPIDs 467 5 ω l =0.01 rad/s (20) H = −20 dB (21) N = −20 dB (22) These rules may be applied if 0.1 ≤ T ≤ 50 and L ≤ 0.5 (23) Table 2. Parameters for the second set of tuning rules for S-shaped response plants P I λ D µ 1 −1.0574 0.6014 1.1851 0.8793 0.2778 L 24.5420 0.4025 −0.3464 −15.0846 −2.1522 T 0.3544 0.7921 −0.0492 −0.0771 0.0675 L 2 −46.7325 −0.4508 1.7317 28.0388 2.4387 T 2 −0.0021 0.0018 0.0006 −0.0000 −0.0013 LT −0.3106 −1.2050 0.0380 1.6711 0.0021 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. 1. The period of such oscillations is the critical period P cr , and the gain causing them is the critical gain K cr . finding the rules in section 3, obtained with specifications (9) to (14), it is cr and P cr . The regularity was again translated into formulas (which are no longer polynomial) using a least-squares fit [8]. The parameters involved are given in Table 3. This means that P =0.4139 + 0.0145K cr +0.1584P cr − 0.4384 K cr − 0.0855 P cr (24) and so on. These rules may be used if P cr ≤ 8andK cr P cr ≤ 640 (25) F 5 A seen that parameters P, I, λ, D,and μ obtained vary regularly with K Plants given by (8) have such a behaviour. Reusing the data collected for irst Set of CriticalGain-Based Tuning Rules 6468 Table 3. Parameters for the first set of tuning rules for plants with critical gain and period Parameters to use when 0.1 ≤ T ≤ 5 P I λ D µ 10.41390.7067 1.3240 0.2293 0.8804 K cr 0.0145 0.0101 −0.0081 0.0153 −0.0048 P cr 0.1584 −0.0049 −0.0163 0.09360.0061 1/K cr −0.4384 −0.2951 0.1393 −0.5293 0.0749 1/P cr −0.0855 −0.1001 0.0791 −0.0440 0.0810 Parameters to use when 5 ≤ T ≤ 50 P I λ D µ 1 −1.4405 5.7800 0.4712 1.3190 0.5425 K cr 0.0000 0.0238 −0.0003 −0.0024 −0.0023 P cr 0.4795 0.2783 −0.0029 2.6251 −0.0281 1/K cr 32.2516 −56.2373 7.0519 −138.9333 5.0073 1/P cr 0.6893 −2.5917 0.1355 0.1941 0.2873 Table 4. These rules may be applied if P cr ≤ 2 (26) Table 4 . Parameters for the second set of tuning rules for plants with critical gain and period P I λ D µ 11.0101 10.5528 0.6213 15.7620 1.0101 K cr 0.0024 0.2352 −0.0034 −0.1771 0.0024 P cr −0.8606 −17.0426 0.2257 −23.0396 −0.8606 P 2 cr 0.1991 6.3144 0.1069 8.2724 0.1991 K cr P cr −0.0005 −0.0617 0.0008 0.1987 −0.0005 1/K cr −0.9300 −0.9399 1.1809 −0.8892 −0.9300 1/P cr −0.1609 −1.5547 0.0904 −2.9981 −0.1609 K cr /P cr −0.0009 −0.0687 0.0010 0.0389 −0.0009 P cr /K cr 0.5846 3.4357 −0.81392.8619 0.5846 Val´erio and daCosta Reusing in the same wise the data used in section 4, corresponding to speci- fications (17) to (22), other rules may be got [8] with parameters given in 6 A Second Set of Critical Gain-Based Tuning Rules TUNING RULES FOR FRACTIONALPIDs 4697 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 [8]. They were obtained from controllers devised to achieve specifications (9) to (14) with plants given by G = K s(s + τ 1 )(s + τ 2 ) (27) It is easy to show that such plants have K cr =(τ 1 + τ 2 )τ 1 τ 2 (28) P cr = 2π √ τ 1 τ 2 (29) 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 0.2 ≤ P cr ≤ 5and1≤ K cr ≤ 200 (30) (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 cope with poles at the origin, the rules in this section do not often cope with plants with a delay. Table 5. Parameters for the third set of tuning rules for plants with critical gain and period P I λ D µ 1 −1.6403 −92.5612 0.7381 −8.6771 0.6688 K cr 0.0046 0.0071 −0.0004 −0.06360.0000 P cr −1.6769 −33.0655 −0.1907 −1.0487 0.4765 K cr P cr 0.0002 −0.0020 0.0000 0.0529 −0.0002 1/K cr 0.8615 −1.0680 −0.0167 −2.1166 0.3695 1/P cr 2.9089 133.7959 0.0360 8.4563 −0.4083 K cr /P cr −0.0012 −0.0011 0.0000 0.0113 −0.0001 P cr /K cr −0.7635 −5.6721 0.0792 2.3350 0.0639 log 10 (K cr )0.4049 −0.9487 0.0164 −0.0002 0.1714 log 10 (P cr )12.6948 336.1220 0.463616.6034 −3.6738 Once more the regular variation of parameters P , I, λ, D,and μ with K 7 A Third Set of Critical Gain-Based Tuning Rules 8470 8 Robustness This section presents evidence showing that rules in sections above provide reasonable, robust controllers. Two introductory comments. Firstly, as stated above, rules usually lead to results poorer than those they were devised to result in overshoots around 25%, but it is not hard to find plants with which attempt to reach always the same gain-crossover frequency, or the same phase cr and P cr applied for wide ranges of those parameters and still achieve a controller that stabilises the plant. Rules from the previous sections always aim at fulfilling the same specifications, and that is why their application range is never so broad as that of Ziegler Nichols rules. G and controllers C were as follows: G 1 (s)= K 1+s e −0.1s (31) C 1a (s)=0.4448 + 0.5158 s 1.4277 +0.2045s 1.0202 (32) C 1b (s)=1.2507 + 1.3106 s 1.1230 − 0.2589s 0.1533 (33) C 1c (s)=12.0000 + 60.0000 s +0.6000s (34) G 2 (s)= K 4.3200s 2 +19.1801s +1 ≈ K 1+20s e −0.2s (35) C 2a (s)=0.0880 + 6.5185 s 0.6751 +2.5881s 0.6957 (36) C 2b (s)=6.9928 + 12.4044 s 0.6000 +4.1066s 0.7805 (37) C 2c (s) = 120.0000 + 300.0000 s +12.0000s (38) G 3 (s)= K 1+ √ s e −0.5s ≈ K 1+1.5s e −0.1s (39) C 3a (s)=0.6021 + 0.6187 s 1.3646 +0.3105s 1.0618 (40) C 3b (s)=1.4098 + 1.6486 s 1.1011 − 0.2139s 0.1855 (41) C 3c (s)=18.0000 + 90.0000 s +0.9000s (42) Val´erio and daCosta − Nichols rules: they are expected toachieve. (The same happens with Ziegler − Nichols rules make nothe overshoot is 100% or even more.) Secondly, Ziegler − vary. This adds some flexibility to Ziegler Nichols rules: they can be − , In what concerns S-shaped response-based tuning rules, three plants − Nichols were devised for each. Plantsand with the first tuning rule of Ziegler margin. Actually, these two performance indicators vary widely as L, T , K dered. Controllers obtained with the two tuning rules from sections 3 and 4 (afirst-order one, a second-order one, and a fractional-order one) were consi- TUNING RULES FOR FRACTIONALPIDs 471 9 ilar step-responses, in what concerns apparent delay and characteristic time- for these plants. Transfer functions are as follows: G 4a (s)= K 20s +1 e −0.2s (43) G 4b (s)= 1 s 3 +2.539s 2 +62.15s ≈ K 20s +1 e −0.2s (44) C 4a (s)=0.0109 + 6.1492 s 0.6363 +2.3956s 0.5494 (45) C 4b (s)=0.3835 + 14.7942 s 0.7480 +3.6466s 0.3835 (46) C 4c (s)=0.8271 + 14.3683 s 0.5588 − 1.6866s 1.2328 (47) C 4d (s)=94.6800 + 237.5910 s +9.43250s (48) The nominal value of K is always 1. The approximation in (35) stems from the values of L and T obtained from its step response. The approximation in (39) is derived from the plant’s step response at t =0.92 s. (It might seem more reasonable to base the approximation on the step response at t =0.5s, but this cannot be done, since the response has an infinite derivative at that time instant.) Notice that due to the approximations involved some controllers have negative gains. This will not, however, affect results. for several values of K, the plant’s gain, which is assumed to be known with uncertainty 1 . The corresponding open-loop Bode diagrams and the gains of sensitivity and closed-loop functions (for K = 1) are also given in those figures. The important thing is that for values of K close to 1, the overshoot does not vary significantly when fractional PIDs are used—the only differ- ence is that the response is faster or slower. And this is true in spite of the different plant structures. This is because fractional PIDs attempt to verify specification (7), which the integer PID does not. And verified it is, together with the other conditions (3) to (6), at least to a reasonable degree, as the frequency-response plots show. (Actually, they are never exactly followed— the approximations incurred by the least-squares fit are to a certain extent responsible for this.) A few minor details. In what concerns plant (31), fractional PIDs can 1 Those time-responses involving fractional derivatives and integrals were obtained using Oustaloup’s approximations [4] for the fractional terms.Approximations were conceived for the frequency range [ω l ,ω h ]= 10 −3 , 10 3 rad/s and make use of 7 poles and 7 zeros. In what concerns critical gain-based tuning rules, two plants (having sim- constant) were considered. Controllers obtained with rules from sections 5, 6, − and 7 and with the second tuning rule of Ziegler Nichols were then reckoned − Figures 2 14 give step responses for the plants and controllers above deal with a clearly broader range of values of K. This is likely because 10 472 0 10 20 30 40 50 0 0.5 1 1.5 time / s output K 10 −2 10 −1 10 0 10 1 10 2 −50 0 50 ω / rad ⋅ s −1 gain / dB 10 −2 10 −1 10 0 10 1 10 2 −600 −400 −200 0 phase / º 10 −2 10 −1 10 0 10 1 10 2 −40 −20 0 ω / rad ⋅ s −1 gain / dB 10 −2 10 −1 10 0 10 1 10 2 −80 −60 −40 −20 0 gain / dB (a) (b) (c) Fig. 2. (a) Step response of (31) controlled with (32) when K is 1/32, 1/16, 1/8, 0 10 20 30 40 50 0 0.5 1 1.5 time / s output K 10 −2 10 −1 10 0 10 1 10 2 −50 0 50 ω / rad ⋅ s −1 gain / dB 10 −2 10 −1 10 0 10 1 10 2 −600 −400 −200 0 phase / º 10 −2 10 −1 10 0 10 1 10 2 −40 −20 0 ω / rad ⋅ s −1 gain / dB 10 −2 10 −1 10 0 10 1 10 2 −80 −60 −40 −20 0 gain / dB (a) (b) (c) Fig. 3. (a) Step response of (31) controlled with (33) when K is 1/32, 1/16, 1/8, =1. the specifications the integer PID tries to achieve are different: that is why responses are all faster, at the cost of greater overshoots. Plant (35) is easier to control, since there is no delay, and a wider variation of K is supported by all controllers. The PID performs poorly with plant (39) because it tries to obtain a fast response and thus employs higher gains (and hence the loop becomes unstable if K is larger than 1/32). Integer PID (48) is unable to stabilise (43). Plant (44) seems easier to control: (48) manages it, and so do (45) and (46). 9 Conclusions In this paper tuning rules (inspired by those proposed by Ziegler and Nichols for integer PIDs) are given to tune fractional PIDs. 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 Val´erio and daCosta 1/4, 1/2, 1 (thick line), 2, 4, and 8. (b) Open-loop Bode diagram when K =1. 1/4, 1/2, 1 (thick line), 2, and 4. (b) Open-loop Bode diagram when K (c)Closed-loop function gain (top) and sensitivity function gain (bottom) when K =1. (c)Closed-loop function gain (top) and sensitivity function gain (bottom) when K =1. TUNING RULES FOR FRACTIONAL PIDs 473 0 2 4 6 8 10 0 0.5 1 1.5 time / s output K 10 −2 10 −1 10 0 10 1 10 2 −50 0 50 ω / rad ⋅ s −1 gain / dB 10 −2 10 −1 10 0 10 1 10 2 −600 −400 −200 0 phase / º 10 −2 10 −1 10 0 10 1 10 2 −40 −20 0 ω / rad ⋅ s −1 gain / dB 10 −2 10 −1 10 0 10 1 10 2 −80 −60 −40 −20 0 gain / dB (a) (b) (c) Fig. 4. (a) Step response of (31) controlled with (34) when K is 1/32, 1/16, 1/8, 1/4, function gain (top) and sensitivity function gain (bottom) when K =1. 0 10 20 30 40 50 0 0.5 1 1.5 time / s output K 10 −4 10 −2 10 0 10 2 −50 0 50 100 ω / rad ⋅ s −1 gain / dB 10 −4 10 −2 10 0 10 2 −150 −100 −50 0 phase / º 10 −4 10 −2 10 0 10 2 −80 −60 −40 −20 0 ω / rad ⋅ s −1 gain / dB 10 −4 10 −2 10 0 10 2 −100 −50 0 gain / dB (a) (b) (c) (b) Open-loop Bode diagram when K =1. (c) Closed-loop function gain (top) and sensitivity function gain (bottom) when K =1. 0 10 20 30 40 50 0 0.5 1 1.5 time / s output K 10 −4 10 −2 10 0 10 2 −50 0 50 100 ω / rad ⋅ s −1 gain / dB 10 −4 10 −2 10 0 10 2 −150 −100 −50 0 phase / º 10 −4 10 −2 10 0 10 2 −80 −60 −40 −20 0 ω / rad ⋅ s −1 gain / dB 10 −4 10 −2 10 0 10 2 −100 −50 0 gain / dB (a) (b) (c) Fig. 6. (a) Step response of (35) controlled with (37) when K is 1/32, 1/16, 1/8, 1/4, 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). 1/2,and 1 (thick line). (b) Open-loop Bode diagram when K = 1. (c) Closed-loop 1/4, 1/2,1(thickline),2,4,8,16,and32. 1/2,1(thickline),2,4,8,16,and32. (b) Open-loop Bode diagram when K =1. Fig. 5. (a) Step response of (35) controlled with (36) when K is 1/32, 1/16, 1/8, (c)Closed-loop function gain (top) and sensitivity function gain (bottom) when K =1. 12474 0 10 20 30 40 50 0 0.5 1 1.5 time / s output K 10 −4 10 −2 10 0 10 2 −50 0 50 100 ω / rad ⋅ s −1 gain / dB 10 −4 10 −2 10 0 10 2 −150 −100 −50 0 phase / º 10 −4 10 −2 10 0 10 2 −80 −60 −40 −20 0 ω / rad ⋅ s −1 gain / dB 10 −4 10 −2 10 0 10 2 −100 −50 0 gain / dB (a) (b) (c) Fig. 7. (a) Step response of (35) controlled with (38) when K is 1/32, 1/16, 1/8, 1/4, 0 10 20 30 40 50 0 0.2 0.4 0.6 0.8 1 1.2 1.4 time / s output K 10 −2 10 −1 10 0 10 1 10 2 −20 0 20 40 60 80 ω / rad ⋅ s −1 gain / dB 10 −2 10 −1 10 0 10 1 10 2 −1000 −500 0 phase / º 10 −2 10 −1 10 0 10 1 10 2 −40 −20 0 ω / rad ⋅ s −1 gain / dB 10 −2 10 −1 10 0 10 1 10 2 −80 −60 −40 −20 0 gain / dB (a) (b) (c) Fig. 8. (a) Step response of (39) controlled with (40) when K is 1/32, 1/16, 1/8, 1/4, function gain (top) and sensitivity function gain (bottom) when K =1. 0 10 20 30 40 50 0 0.2 0.4 0.6 0.8 1 1.2 1.4 time / s output K 10 −2 10 −1 10 0 10 1 10 2 −20 0 20 40 60 80 ω / rad ⋅ s −1 gain / dB 10 −2 10 −1 10 0 10 1 10 2 −1000 −500 0 phase / º 10 −2 10 −1 10 0 10 1 10 2 −40 −20 0 ω / rad ⋅ s −1 gain / dB 10 −2 10 −1 10 0 10 1 10 2 −80 −60 −40 −20 0 gain / dB (a) (b) (c) Fig. 9. (a) Step response of (39) controlled with (41) when K is 1/32, 1/16, 1/8, 1/4, function gain (top) and sensitivity function gain (bottom) when K =1. 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. Val´erio and daCosta 1/2, 1 (thick line), and 2. (b) Open-loop Bode diagram when K =1. (c) Closed-loop 1/2, and 1 (thick line). (b) Open-loop Bode diagram when K =1. (c) Closed-loop (c)Closed-loop function gain (top) and sensitivity function gain (bottom) when K =1. 1/2,1(thickline),2,4,8,16,and32. (b) Open-loop Bode diagram when K =1. [...]... called flatness has been introduced in 1992 by M Fliess, J Lévine, Ph Martin, and 493 J Sabatier et al (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, 493–509 © 2007 Springer 494 Melchior, Cugnet, Sabatier, Poty, and Oustaloup P Rouchon [1, 2, 15, 16], then applied to planes and cranes piloting control processing Thus, each flat system... too long without providing tracking accuracy [ 6] Moreover, during this time the task will not take place In the aerospace industry, flexible mode frequencies are well defined, but weakly damped Here, the input shaper technique reduces vibration in path tracking design Input shaping is obtained by convolving desired input with an impulse sequence This generates vibration-reducing shaped command, which... systems and a controllable and linear system is always flat: taking the Brunovsky’s outputs stemming from controllability canonical forms as flat outputs is sufficient 2.2 Continuous linear systems flatness Let the single input–single output (SISO) time-invariant continuous linear system be defined by the following transfer function in [3]: A( s ) Y ( s ) B(s) U ( s) , (7) in which A(s) and B(s) polynomials,... tracking, control, fractional systems, testing bench 1 Introduction To increase the speed of machine tools, lighter materials are used increasing their flexibility Execution times must be optimized without exciting resonance A prefilter is used in industrial path tracking designs, as it is easy to implement and adapt for reducing overshoots This reduces the highfrequency energy of the path planning... entier Hermès, Paris, in French Podlubny I (1999) Fractional Differential Equations Academic Press, San Diego Valério D, da Costa JS (2005) Ziegler-nichols type tuning rules for fractional PID controllers In Proceedings of ASME 2005 Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Long Beach Valério D, da Costa JS (2006) Tuning of fractional PID controllers... apply flatness principle to a fractional system As soon as the path has been obtained by flatness, a new robust path tracking based on CRONE control is presented Firstly, flatness principle definitions used in control’s theory are reminded The fractional systems dynamic inversion is studied A robust path tracking based on CRONE control is presented Finally, simulations on a thermal testing bench model,... using a low-pass filter with trialand-error determined parameters Nevertheless, for classic linear prefilter 477 J Sabatier et al (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, 477–492 © 2007 Springer 478 Melchior, Poty, and Oustaloup approaches, when overshoots are reduced, dynamic performances are also reduced This type of path tracking,... paper concerns the application of flatness principle to fractional systems In path planning, the flatness concept is used when the trajectory is fixed (in space and in time), to determine the controls inputs to apply without having to integrate any differential equations A lot of developments have been made but, in the case of non-integer differential systems (or fractional systems), few developments are... Description of the thermal testing bench The testing bench copying the behaviour of a non-integer derivatives system involves a semi-infinite-dimensional thermal system, namely, an aluminium rod of large dimension (40 cm) (Fig 1): Fig 1 Aluminium bar, heating resistor 0–12 W and measurement slot 500 Melchior, Cugnet, Sabatier, Poty, and Oustaloup As illustrated in Fig 2, the input of this system is a thermal... Nonlinear effects in the thermal system are not considered because its transfer is obtained by identification in a linear form Therefore, it is not a matter of determining if flatness applies to these systems, since it is proved that any controllable linear system is flat, but demonstrating the corresponding flat output can be generalized under a well-known form in state space The originality of our work . In the aerospace industry, flexible mode frequencies are well defined, but weakly damped. Here, the input shaper technique reduces vibration in path tracking design. Input shaping is obtained. of ASME 2005 Design Engineering Technical Confe- rences and Computers and Information in Engineering Conference, Long Beach. 7. Valério D, da Costa JS (2006) Tuning of fractional PID controllers. PREFILTER IN PATH TRACKING DESIGN Abstract A new approach to path tracking using a fractional differentiation prefilter Keywords 1 Introduction To increase the speed of machine tools, lighter