9 CHAPTER 7 FRACTIONAL ORDER SYSTEMS WITH FRACTIONAL ORDER CONTROLLERS 7. FRACTIONAL ORDER SYSTEMS Fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. This is the main advantage of fractional derivatives in comparison with classical integer-order models, in which such effects are in fact neglected. The advantages of fractional derivatives become apparent in modeling mechanical and electrical properties of real materials, as well as in the description of rheological properties of rocks, and in many other fields. The mathematical modeling and simulation of systems and processes, based on the description of their properties in terms of fractional derivatives, naturally leads to differential equations of fractional order and to the necessity to solve such equations. The idea of fractional derivatives and integrals seems to be quite a strange topic, very hard to explain, due to the fact that, unlike commonly used differential operators, it is not related to some important geometrical meaning, such as the trend of functions or their convexity. For this reason, this mathematical tool could be judged far from reality. But many physical phenomena have intrinsic fractional order description and so fractional order calculus is necessary in order to explain them. Fractional order differential equations accumulate the entire information of the function in weighted form.
2 The transfer function of a single input, single output (SISO) system, with the output and the input related by a linear, time-invariant differential equation, is the ratio of the Laplace transforms of the output and of the input, when all initial conditions are zero. This definition also stands when the differential equation involves fractional orders. In general the LTI systems can be classified as integer order systems and fractional order systems. Fractional order systems can further be classified as commensurate order or non commensurate order systems. In commensurate order systems the fractional powers are integer multiples of a fractional number. In non commensurate order systems there is no such generalization occurs. function is given by In case of commensurate order systems the continuous transfer G( s) ( m ) b s m b s L b s b 2 m m ( n ) a s n a s L a s a n n (7.) where b, b 2, b m+ are numerator coefficients a, a 2, a n+ are denominator coefficients is a fractional number function is given by In case of non commensurate order systems the continuous transfer G( s) b s m m b s m - m L b s b a s n n a s n - n L a s a (7.2)
2 where b m, b m-, b are numerator coefficients a n, a n-, a are denominator coefficients m, m-,... are numerator fractional powers n, n-, are denominator fractional powers. For integer order systems once the maximum order of the system to be identified is chosen, the parameters of the model can be optimized directly. For fractional order systems, the identification requires (a) the choice of the number of fractional operators, (b) the fractional powers of the operator, and (c) the coefficients of the operators. But to identify the form of transfer function itself, with order and coefficient, an approach from experimental data of frequency domain analysis should be the starting point, to identify unknown systems. 7.. Stability analysis of LTI fractional order systems It is known from the theory of stability that an LTI (linear timeinvariant) system is stable if the roots of the characteristic polynomial are negative or have negative real parts if they are complex conjugate. It means that they are located on the left half of the complex plane. In the fractionalorder LTI case, the stability is different from the integer one. Interesting notion is that a stable fractional system may have roots in the right half of complex plane. Figure 7. shows the stable and unstable regions of fractional order systems.
22 Figure 7. Stability of LTI FOS with order <q< It has been shown that system is stable if the following condition is satisfied arg(eig(a)) > q (7.3) where < q < and eig(a) represents the eigenvalues of matrix A. Matignon s stability theorem says the fractional transfer function G(s) = Z(s)/P(s) (7.4) is stable if and only if the following condition is satisfied in s-plane: arg(s) > q (7.5) where s = s q. When s = is a single root of P(s), the system cannot be stable. For q =, this is the classical theorem of pole location in the complex plane, no pole is in the closed right half plane of the first Riemann sheet.
23 Generally, consider the following commensurate fractional order system in the form: D q = f( ) (7.6) where < q < and R n. The equilibrium points of system are calculated via solving the following equation f( ) = (7.7) The equilibrium points are asymptotically stable if all the eigen values j, (j =,2,...,n) of the Jacobian matrix J = f/ (7.8) evaluated at the equilibrium, satisfy the condition arg(eig(j)) = arg( j ) > q (7.9) where j =,2,...,n. Apart from many advantages fractional order systems also have some disadvantages which are summarized below. Fractional order differential equations accumulate the entire information of the function in weighted form. Fractionally differentiated state variables must be known as long as the system has been operated. This is known as initialization function. For integer order systems, the initialization function is constant and for fractional order systems it is time varying. Integer order system set of state along with system equations is sufficient to predict the response.
24 The fractional dynamic variables do not represent the state of the system. Fractional dynamics require history of states or sufficient number of points by short-memory principle, for initialization function computation. The above memory effect requires large memory. The evolving developments to reduce this requirement in form of power series expansion and continued fraction expansion of the generating function in digital domain are an ongoing process. 7.2 MODELING OF FRACTIONAL ORDER SYSTEM Because of the closeness of fractional description to the real system, the same spherical tank liquid level system is modeled as a fractional order system. Two different approaches are tried in this research work. In both the approaches a non commensurate model of fractional order system is selected. 7.2. First Method In this approach the dead time was not considered first. A fitness function is used to model the spherical tank as a non commensurate model of fractional order system. The fitness function is defined as f F = t i C ( t ) - P(t) 2 (7.) m
25 Where, k C (s) = (7.) m s is the rationalized integer order transfer function model of spherical tank and l P(s) = (7.2) a 2 a a 2 s s s is the fractional order transfer function model of the same spherical tank. The selected non commensurate fractional order transfer function model (Podlubny 999) is G(s) ans n n a s... a s a s n (7.3) where k is an arbitrary real number, n > n- >. > >, a k is an arbitrary constant, Here n is selected as 2. So the resultant FOTF is G(s) = 2 a s a s a s 2 (7.4) As said previously for modeling the spherical tank as a FOS, first the dead time was not considered; only the remaining first order transfer function of FOPDT was approximated as a fractional order system using
26 curve fitting method and then the resultant FOTF was combined with the dead time. An operating point of 2cm was selected. Step input was applied to the FOPDT model and the obtained response was taken as C m (t). Then the step input of same magnitude is applied to the selected fractional order model system (FOS) and obtained response was taken as P(t). The coefficients and fractional powers of FOS were adjusted until the least square error between C m (t) and P(t) was near zero. The curve fit of step responses of FOS and IOS is shown in Figure 7.2. The parameters of IOS and FOS at the operating point of 2cm are listed in Table 7. and Table 7.2 respectively..35 Curve Fitting Method.3.25.2 FOS FOPDT.5..5.2.4.6.8.2.4.6.8 2 2.2 Time(sec) x 4 Figure 7.2 Step responses of FOS and IOS at the operating point of 2cm Table 7. Parameters of IOS at the operating point of 2cm K d.32 9.74 8
27 Table 7.2 Parameters of FOS at the operating point of 2cm a 2 a a 2 3.5.5.5.4 point of 2 cm is The resultant fractional order transfer function at the operating P (s) = e 8s.5.4 3.5s.5s (7.5) 7.2.2 Second Method Alternatively the dead time of FOPDT model was approximated by Pade s approximation method. The resultant transfer function took the following form. C m (s) = k s 2 2 d d s s (7.6) Then the same least square method was applied to obtain the non commensurate FOTF model. The modeling parameters a, a, a 2,, and 2 of FOS were adjusted until the least square error between the step response of FOPDT model and the step response of fractional order system was almost zero. The step responses of IOS (C m (t)) and FOS (P(t)) at different operating points are shown in figures from Figure 7.3 to Figure 7.9.
28.8.7 P(t) Cm(t).6.5.4.3.2. 2 3 4 5 6 7 8 9 Figure 7.3 Step responses of FOS and IOS at the operating point of 5.75cm The least square error between the reaction curves is.579. The corresponding FOTF at the operating point of 5.75 cm is G(s) =.7.s.2s.2.759 (7.7) 2 P(t) Cm(t).5.5 Figure 7.4 -.5 2 4 6 8 2 4 6 8 2 Step responses of FOS and IOS at the operating point of 3.5 cm
29 The least square error between the reaction curves is.2774. The corresponding FOTF at the operating point of 3.5 cm is G(s) = 4.7s.9 3.5s.8 2.67 (7.8) 3 2.5 P(t) Cm(t) 2.5.5 5 5 2 25 3 35 4 45 5 Figure 7.5 Step responses of FOS and IOS at the operating point of 2.75 cm The least square error between the reaction curves is.54. The corresponding FOTF at the operating of 2.75 cm is G(s) =.8 35s 6s.8 3.6372 (7.9)
3 4 3.5 3 2.5 P(t) Cm(t) 2.5.5 5 5 2 25 3 35 4 45 5 Figure 7.6 Step responses of FOS and IOS at the operating point of 27.25cm The least square error between the reaction curves is.975. The corresponding FOTF at the operating point of 27.25 cm is G(s) =.8 87s 6s.8 4.6885 (7.2) 5 4.5 4 3.5 3 P(t) cm(t) 2.5 2.5.5 2 3 4 5 6 7 8 9 Figure 7.7 Step responses of FOS and IOS at the operating point of 32cm
3 The least square error between the reaction curves is.9568. The corresponding FOTF at the operating point of 32 cm is G(s) =.8 5s 6s.8 5.43 (7.2) 5.5 5 4.5 4 P(t) Cm(t) 3.5 3 2.5 2.5.5 2 4 6 8 2 4 6 8 2 Figure 7.8 Step responses of FOS and IOS at the operating point of 38.25 cm The least square error between the reaction curves is.7235. The corresponding FOTF at the operating point of 38.25 cm is G(s) =.8 255s 6s.8 6.4576 (7.22)
32 7 6 5 P(t) Cm(t) 4 3 2 5 5 2 25 Figure 7.9 Step responses of FOS and IOS at the operating point of 46.875 cm The least square error between the reaction curves is.8575. The corresponding FOTF at the operating point of 46.875 cm is G(s) =.8 46s 6s.8 8.794 (7.23) 7.3 TUNING OF FOPI CONTROLLER FOR FRACTIONAL ORDER SYSTEM If the process is modeled as a FOS then the controller must be FOC. In fractional order controllers we have 3 choices that are FOPI, FOPD and FOPID. Based on the number of tuning parameters either the first three or all five constraints of the following can be considered for calculating the tuning parameters of the fractional order controller. The conditions should be specified by the user.
33. Phase margin constraint, Arg G j c Arg Gc j c Arg P j c m (7.24) 2. Gain crossover frequency constraint G j G j P j (7.25) c c c c 3. Robustness to loop gain variation constraint, which demands that the phase derivative with respect to the frequency is zero, namely, the phase Bode plot is flat, around the gain crossover frequency c. It means that the system is robust to loop gain changes and the overshoots of the responses are almost the same, d Arg G j d c (7.26) 4. To reject high-frequency noise, the closed loop transfer function must have a small magnitude at high frequencies hence, at some specified frequency h, its magnitude is to be less than some specified gain H. C( h) G( h) C( ) G( ) h h H (7.27)
34 5. To reject output disturbances and closely follow the references, the sensitivity function must have a small magnitude at low frequencies. Hence, at some specified frequency l, its magnitude is to be less than some specified gain N: C( ) G( ) l l N (7.28) A fractional order transfer function was taken from the literature (Ying Luo et al 2). They had designed FOPI controller for the FOTF. The same FOTF was considered with the same specifications. The FOPID controller was designed for a phase margin of 5degrees, gain crossover frequency of rad/sec, H of db and N of 2dB. The tuning parameters proportional gain K p, integral gain K i, derivative gain K d, integral power and derivative power are found using fmincon optimization function. The range of and is selected as to 2. considered The fractional order system with the following FOTF was P(s).4s.5 (7.29) The fractional order PID controller was tuned based on optimization of all the five constraints. controller is The transfer function of corresponding fractional order PID
35.375.6865 G (s).8.s (7.3) c.2 s The closed loop responses of FOPI and FOPID controllers were compared and the responses are shown in Figure 7. Comparisons of performance criteria are listed in Table 7.3. The robustness of the controller to the parameter (either process or controller) changes was checked by changing the proportional gain of the FOPID controller and the responses for various values of gains are shown in Figure 7.. From the simulation results it was observed that FOPID controller did not make a great change compared to FOPI controller. Since FOPI controller only needs tuning of three parameters, FOPI controller was selected for FOS in this research work..4.2 FOPID FOPI r(t).8.6.4.2 2 3 4 5 6 7 8 9 Figure 7. Closed loop step response of FOPID and FOPI controllers with FOS
36.4.2 gain=k*.9 gain=(.) gain=..8.6.4.2 2 4 6 8 2 4 6 8 2 Figure 7. Robustness checking of FOPID controller with FOS Table 7.3 Comparison of performance indices of FOPID and FOPI controllers Controller ISE IAE ITAE FOPID.5733.9.73 FOPI.5949.4.33 The three parameters (K p, K I and ) of fractional order PI controller were tuned based on the afore mentioned first three constraints. The FOPI controller was tuned for the phase cross over frequency of.43 rad / sec and phase margin of 2deg. The tuning parameters of resultant FOPI controller are listed in table 7.4.
37 Table 7.4 Tuning parameters of FOPI controller K p K i 6.5288.4233.9 The transfer function of tuned fractional order PI controller for the fractional order model is, C ( s) 6.5288.4233.9 s (7.3) With these tuning parameters the FOPI controller is connected in closed loop with FOS. The closed loop step responses of FOS with FOPI controller and IOS with IOPID controller are shown in Figure 7.2. The performance criteria are listed in table 7.5. COMPARISON OF CLOSED LOOP RESPONSE.6.4.2.8.6.4.2 -.2 IOPID FOR FOPDT FOPI FOR FOS SETPOINT -.4 2 4 6 8 2 4 6 8 2 TIME(sec) Figure 7.2 Closed loop step responses of FOS with FOPI controller and IOS with IOPID controller at the operating point of 2cm
38 Table 7.5 Comparison of performance criteria of FOS with FOPI controller and IOS with IOPID controller Controller Maximum Peak Overshoot (%) Settling Time (sec) ISE IOPID 62 68 7.3 FOPI 52 8.532 The above simulation results show that the fractional order system with fractional order proportional and integral controller results no overshoot, less ISE and settling time when compared to integer order modeling and control. The tuning parameters of IOPID controllers for FOPDT models at different operating points are listed in Table 7.6. The tuning parameters of FOPI controllers for the FOS models obtained using second method are listed in Table 7.7. The closed loop step responses of FOS with FOPI controller and IOS with IOPID controller are shown in figures from Figure 7.3 to Figure 7.9.
39 Table 7.6 Tuning parameters of IOPID controller for Integer Order Systems Level in cm 5.75 3.5 2.75 27.25 32. 38.25 46.875 Transfer function.699e.99s.554s.84s.97e 4.8667 s.67s 3.3e 2.823s.49s 3.843e 2.883s.5s 4.3573e 28.49s.49s 5.45e 39.57s 6.393e.33s 62.77s K c K i K d 4.76.88.4756 6.744 44.38.4853 3.377 9.8389 2.5384 43.568 45.883 3.263 52.87 72.8376 3.9235 63.986 2.537 4.76 88.699 333.8733 5.89
4 Table 7.7 Tuning parameters of FOPI controller for Fractional Order Systems Level in cm Transfer function K c K i 5.75.7.s.2s.2.759. 6.5 3.5 4.7s.9 3.5s.8 2.67. 32.5 2.75.8 35s 6s.8 3.6372.5 97.3 27.25.8 87s 6s.8 4.6885.8 79.8 32. 5s.8 6s.8 5.43 3. 258.6 38.25 255s.8 6s.8 6. 4576 5. 343.6 46.875.8 46s 6s.8 8.794 468.2
4 2.5 r(t) Ci(t) Cf(t).5 2 3 4 5 6 7 8 9 Figure 7.3 Step responses of FOS with FOPI controller and IOS with IOPID controller at 5.75cm.5 r(t) Ci(t) Cf(t).5 2 4 6 8 Figure 7.4 Step responses of FOS with FOPI controller and IOS with IOPID controller 3.5 cm
42 2.5 r(t) Ci(t) Cf(t).5 2 3 4 5 6 7 8 9 Figure 7.5 Step responses of FOS with FOPI controller and IOS with IOPID controller at 2.75 cm 2.5 r(t) Ci(t) Cf(t).5 2 3 4 5 6 7 8 9 Figure 7.6 Step responses of FOS with FOPI controller and IOS with IOPID controller at 27.25 cm
43.6.4.2 r(t) Ci(t) Cf(t).8.6.4.2 2 3 4 5 6 7 8 9 Figure 7.7 Step responses of FOS with FOPI controller and IOS with IOPID controller at 32 cm.6.4.2 r(t) Ci(t) Cf(t).8.6.4.2 2 3 4 5 6 7 8 9 Figure 7.8 Step responses of FOS with FOPI controller and IOS with IOPID controller at 38.25 cm
44 2.5 r(t) Ci(t) Cf(t).5 2 3 4 5 6 7 8 9 Figure 7.9 Step responses of FOS with FOPI controller and IOS with IOPID controller at 46.875 cm The performance criteria like M p, t s, ISE, IAE and ITAE of FOS with FOPI and IOS with IOPID controllers are listed in Table 7.8.
45 Table 7.8 Comparison of performance criteria of FOS with FOPI and IOS with IOPID controllers Level (Cm) Controller M p (%) t s (sec) ISE IAE ITAE 5.75 3.5 2.75 27.25 32. 38.25 46.875 IOPID 22..5.256.384.45 FOPI.97.35.37.339.6 IOPID 48.37.9.362.4776.289 FOPI.58.2.24.282.29 IOPID 5.46.7.2863.4467.823 FOPI 2.33.7.46.43.226 IOPID 5.82.5.2572.42.479 FOPI 3.25.8.39.399.22 IOPID 5.3.6.268.475.5 FOPI 3.7.8.35.387.2 IOPID 5.53.5.2586.433.483 FOPI 3.35.8.38.49.228 IOPID 52.69.4.233.36.8 FOPI 2.74.7.28.332.79 It is observed from the above results that the closed loop response of FOS with corresponding FOPI controller is performing well in all aspects compared to IOS with IOPID controller. 7.4 EXPERIMENTAL RESULTS A set point of 2 cm was given to the experimental setup when it was connected in closed loop with the FOPI controller. The tuning parameters of FOPI controller corresponding to the FOS modeled using first method were used. After it reached the steady state a disturbance was introduced through a sudden change in pump speed which is assumed due to voltage
46 fluctuation and once again the system was allowed to reach the steady state. The servo and regulatory responses are shown in Figure 7.2. The error graph corresponding to this response is shown in Figure 7.2. 2 8 6 4 r(t) c(t) d(t) 2 8 6 4 2 2 4 6 8 2 4 6 8 2 Figure 7.2 Servo and regulatory response of FOPI controller when the set point is 2cm 2 8 6 4 2-2 -4-6 2 4 6 8 2 4 6 8 2 Figure 7.2 Error graph when the set point is 2 cm
47 The closed loop response of FOPI controller was compared with the closed loop response of IOPI controller. Some of the time domain specifications are listed in Table 7.9 to show the efficiency of the modeling of the experimental set up as a FOS with properly tuned FOPI controller over integer order modeling with integer order controller. Table 7.9 Comparison of performance criteria of FOPI and IOPI controllers on experimental setup when the set point is 2 cm Controller Rise time (sec) Peak time (sec) Peak overshoot (%) Settling time(sec) ISE ZN-PI 42 278 33.5 948 5249 FOPI 65.5 8 4.833 378 2438 From the comparisons made it is observed that even though the fractional order modeling and control shows only small improvements in rise time, peak time and peak over shoot there is a great improvement in settling time which is more important. A set point of 32 cm was given to the experimental setup when it was connected in closed loop with the FOPI controller. The tuning parameters of FOPI controller corresponding to the FOS modeled using second method were used. After it reached the steady state a disturbance was introduced through a sudden change in pump speed and once again the system was allowed to reach the steady state. The servo and regulatory responses with IOPI controller tuned based on ZN and FOPI controller are shown in Figure 7.22 and Figure 7.23 respectively.
48 5 4 r(t) c(t) d(t) 3 2 2 4 6 8 2 4 Figure 7.22 Servo and regulatory response of IOPI control system when the set point is 32cm 4 3 c(t) d(t) r(t) 2 2 3 4 5 6 7 8 9 Figure 7.23 Servo and regulatory response of FOPI control system when the set point is 32cm
49 The closed loop servo and regulatory responses of FOPI controller and IOPI controller were compared. Some of the time domain specifications are listed in Table 7.9 to show the efficiency of the modeling of the experimental set up as a FOS with properly tuned FOPI controller over integer order modeling with integer order controller. Table 7. Comparison of performance criteria of FOPI and IOPI controllers on experimental setup when the set point is 32 cm Controller Rise time (sec) Peak time (sec) Peak overshoot (%) Settling time(sec) ISE ZN-PI 58 3 32.3 58 4362 FOPI 47 72 4.3 97 9 When the response of FOPI controller was compared with IOPI controller on the experimental setup, it was observed that the fractional order modeling and control shows very good improvements in rise time, peak time, over shoot and settling time. 7.5 CONCLUSIONS The spherical tank was modeled as non commensurate fractional order systems with and without dead time. From both simulation and experimental results it was observed that fractional order modeling and control improves the performance of the control system over integer order modeling and control. This was proved qualitatively as well as quantitatively.
5 It is concluded that among the two proposed methods of fractional order modeling, the FOS model developed using second method, with properly tuned FOPI controller produced good results in terms of performance criteria like rise time, maximum peak over shoot, settling time and ISE etc over the FOS, modeled by first method with properly tuned fractional order controllers.