Sabatier Agrawal Machado Advances in Fractional Calculus Episode 14 pptx

FLATNESS CONTROL OF A FRACTIONAL THERMAL SYSTEM 505 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 -200 -100 0 100 200 Bode Diagram of Open Loops (CRONE & PID) 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 -350 -300 -250 -200 -150 -100 Fig. 8. Comparison of both PID and CRONE open-loop Bode diagrams with different gain variations: 0 G /50 (grey), 0 G (solid ), 0 G Now, the system is studied in closed-loop so as to measure its immunity to different disturbances applied to its input ( U ) and its output (Y ). The ref U , the control obtained by the flatness principle using the chosen reference trajectory ref Y . The PID controller and the CRONE controller are both used in simulation. For this, we study the disturbances and gain variation influences on path tracking. For this, a 1° control input disturbance is applied at 500 s and a 3° U ref U Y Y ref Y U THERMAL SYSTEM CRONE or PID CONTROLLER  50 (dotted ).  50 (dash dotted), and G 0 6 Simulation Results control scheme is presented by Fig. 9, with, Fig. 9. Closed-loop control scheme. 506 0 500 1000 1500 2000 250 0 -5 0 5 10 15 20 25 30 Effective Output (°C) (PID & CRONE) Time (s ) 0 500 1000 1500 2000 250 0 0 1 2 3 4 5 6 7 8 9 10 System Input Control (V) (PID & CRONE) Time (s ) 0 500 1000 1500 2000 2500 -5 0 5 10 15 20 25 30 35 Effective Output (°C) (PID & CRONE) Tim e (s ) 0 500 1000 1500 2000 2500 0 1 2 3 4 5 6 7 8 9 System Input Control (V) (PID & CRONE) Time (s ) Fig. 11. Simulation with disturbances and no gain variation; path (dotted ), CRONE 0 500 1000 1500 2000 2500 -5 0 5 10 15 20 25 30 35 Effective Output (°C) (PID & CRONE) Tim e (s ) 0 500 1000 1500 2000 2500 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 System Input Control (V) (PID & CRONE) Time (s) Melchior, Cugnet, Sabatier, Poty, and Oustaloup output disturbance is applied at 1,500 s. Time responses are given for different gain variations (1, 50, and 80 times as much gain). CRONE (black), and PID (grey). Fig. 12. Simulation with disturbances and G0  50 gain variation; path (dotted ), (black), and PID (grey). (black) and PID (grey). Fig. 10. Simulation with no disturbance and no gain variation; path (dotted ), CRONE FLATNESS CONTROL OF A FRACTIONAL THERMAL SYSTEM 507 0 500 1000 1500 2000 2500 -5 0 5 10 15 20 25 30 35 Effective Output (°C) (PID & CRONE) Tim e (s ) 0 500 1000 1500 2000 2500 -2 -1.5 -1 -0.5 0 0.5 System Input Control (V) (PID & CRONE) Tim e (s ) Fig. 13. Simulation with disturbances and G0  80 gain variation; path (dotted ), Figure 10 shows the same path tracking for PID and CRONE controllers. In fact, the loop has no role in the nominal case. Figure 11 shows a good path tracking in presence of disturbances due to the loop. PID and CRONE have the same dynamic behaviour (same  cg). a better path tracking performance with the CRONE controller as well as beside the disturbances than gain variations, due to a quasi-constant phase around  cg in the Crone controller case. 7 Conclusion In this paper, a new robust path tracking design based on flatness and CRONE systems dynamic inversion was studied. Simulations with two different controllers (PID and CRONE) illustrated the robustness of the proposed path CRONE control can also be integrated in future designs. Flatness principle conceivable. This paper is a modified version of a paper published in proceedings of The authors would like to thank the American Society of Mechanical definitions used in control’s theory were reminded. Then, the fractional tracking strategy. The study of robust path tracking via a third-generation application through non-linear fractional systems dynamic inversion can be IDETC/CIE 2005, September 24–28, 2005, Long Beach, California, USA. CRONE (black) and PID (grey). The robustness study is presented by Figs. 12 and 13. We can see clearly a fractional system: a thermal testing bench. Firstly, flatness principle control approaches was presented. Therefore, this method was applied to Acknowledgment 508 Engineers (ASME) for allowing them to publish this revised contribution of Melchior, Cugnet, Sabatier, Poty, and Oustaloup an ASME article in this book. References 1. Fliess M, Lévine J, Martin Ph, Rouchon P (1992) Sur les systèmes non linéaires différentiellement plats, C. R. Acad. Sci. Paris, I-315:619–624. 2. Fliess M, Lévine J, Martin Ph, Rouchon P (1995) Flatness and defect of nonlinear systems: introductory theory and examples, Int. J. Control, 61(6): 1327–1361. 3. Ayadi M (2002) Contributions à la commande des systèmes linéaires plats de dimension finie, PhD thesis, Institut National Polytechnique de Toulouse. 4. Cazaurang F (1997) Commande robuste des systèmes plats, application à la commande d’une machine synchrone, PhD thesis, Université Bordeaux 1, Paris. 5. Lavigne L (2003) Outils d’analyse et de synthèse des lois de commande robuste des systèmes dynamiques plats, PhD thesis, Université Bordeaux 1, Paris. 6. Khalil W, Dombre E (1999) Modélisation, identification et commande des robots, 2ème édition, Editions Hermès, Paris. 7. Oustaloup A (1995) La dérivation non entière: théorie, synthèse et appli- cations, Traité des Nouvelles Technologies, série automatique, Editions Hermès, Paris. 8. Orsoni B (2002) Dérivée généralisée en planification de trajectoire et génération de mouvement, PhD thesis, Université Bordeaux, Paris. 9. Sabatier J, Melchior P, Oustaloup A (2005) A testing bench for fractional system education, Fifth EUROMECH Nonlinear Dynamics Conference (ENOC’05), Eindhoven, The Netherlands, August 7–12. 10. Cois O (2002) Systèmes linéaires non entiers et identification par modèle non entier: application en thermique, PhD thesis, Université Bordeaux 1, Paris. 11. Cugnet M, Melchior P, Sabatier J, Poty A, Oustaloup A. (2005) Flatness principle applied to the dynamic inversion of fractional systems, Third IEEE SSD’05, Sousse, Tunisia, March 21–24. 12. Oustaloup A, Sabatier J, Lanusse P (1999) From fractal robustness to the CRONE control, Fract. Calcul. Appl. Anal. (FCAA): Int. J. Theory Appl., 2(1):1–30, January. 13. Miller KS, Ross B (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York. 14. Podlubny I (1999) Fractional Differential Equations. Academic Press, San Diego. 15. Lévine J, Nguyen DV (2003) Flat output characterization for linear systems using polynomial matrices, Syst. Controls Lett., 48:69–75. 16. Bitauld L, Fliess M, Lévine J (1997) A flatness based control synthesis of linear systems and applications to windshield wipers, In Proceedings ECC’97, Brussels, July. FLATNESS CONTROL OF A FRACTIONAL THERMAL SYSTEM 509 17. Lévine J (2004) On necessary and sufficient conditions for differential flatness, In Proceedings of the IFAC NOLCOS 2004 Conference. 18. Samko SG, Kilbas AA, Maritchev OI (1987) Integrals and Derivatives of the Fractional Order and Some of Their Applications, Nauka I Tekhnika, Minsk, Russia. 19. Oldham KB, Spanier J (1974) The Fractional Calculus. Academic Press, New York, London. 20. Melchior P, Cugnet M, Sabatier J, Oustaloup A (2005) Flatness control: application to a fractional thermal system, ASME, IDETC/CIE 2005, September 24–28, Long Beach, California. ROBUSTNESS COMPARISON OF SMITH FRACTIONAL-ORDER CONTROL Patrick Lanusse and Alain Oustaloup cours de la Abstract Many modifications have been proposed to improve the Smith predictor structure used to control plant with time-delay. Some of them have been They are often based on the use of deliberately mismatched model of the plant predictor, IMC method. 1 Introduction In the context of the closed-loop control of time-delay systems, Smith [1] proposed a control scheme that leads to amazing performance which is impossible to obtain using common controller. It is now well known that such performance can be obtained for perfectly modeled systems only. When a system is uncertain or perturbed, trying to obtain high-performance, for instance settling times close to or lower than the time-delay value of the system, leads to lightly damped closed-loop system and sometimes to unstable E-mail: {lanusse, oustaloup}@laps.u-bordeaux1.fr proposed to enhance the robustness of Smith predictor-based controllers. and then the internal model control (IMC) method can be used to tune the con- troller. This paper compares the performance of two Smith predictor-based controllers including a mismatched model to the performance provided by a robustness and performance tradeoff. It is shown that even if it can simplify the design of (robust) controller, the use of an improved Smith predictor is not necessary to obtain good performance. Keywords Time-delay system, fractional-order controller, robust control, Smith © 2007 Springer. 511 PREDICTOR-BASED CONTROL AND LAPS, UMR 5131 CNRS, Bordeaux I University, ENSEIRB, 351 Libération, 33405 Talence Cedex, Tel: +33 (0)5 4000 2417, Fax: +33 (0)5 4000 66 44, fractional-order CRONE controller which is well known for managing well the in Physics and Engineering, 511–526. J. Sabatier et al. (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications 512 system. Then, many authors proposed to improve the design method of Smith predictors and provided what it is often presented as robust Smith predictors of the time-delay system to be controlled. Using such a model constrains to design the controller very carefully but does not really provide a meaningful degree of freedom to manage the robustness problem. Thus, Zhang and Xu freedom to tune the performance and the robustness of the controller. Even if one degree of freedom leads to a low order, and interesting controller, it can be thought that the performance obtained could be improved by using more degree of freedom. CRONE (acronym for Commande Robuste use of few high-level degrees of freedom. CRONE is a frequency-domain design approach for the robust control of uncertain (or perturbed) plants. The Section 2 presents the classical Smith predictor design and the approaches proposed by Wang and then by Zhang. Section 3 presents the CRONE approach and particularly its third generation. Section 4 proposes to make uncertain a time-delay system proposed by Wang, and then to compare the robustness and performance of Wang, Zhang and CRONE controllers. The structure of the classical Smith predictor (Fig. 1) includes the nominal model G 0 of the time-delay system G and the time-delay free model P 0 . G 0 (s) - P 0 (s) K(s) G(s) + + - - y u e Lanusse and Oustaloup [4] proposed to use the internal model control (IMC) [5] and one degree of Fractional-order control-system design provides such further degree of freedom [6–10]. For instance, d’Ordre Non Entier which means non-integer order robust control) control- system design [11–18] uses the integration fractional order which permits the 2 Smith Predictor-Based Control-Systems Fig. 1. Smith predictor structure. [2]. Wang et al. [3] proposed a design method based on a mismatched model plant uncertainties (or perturbations) are taken into account without dis- tinction of their nature, whether they are structured or unstructured. Using frequency uncertainty domains, as in the quantitative feedback theory (QFT) approach [19] where they are called template, the uncertainties are taken into account in a fully structured form without overestimation, thus leading to effi- cient controller because as little conservative as possible [20]. The closed-loop transfer function y/e is          sGsGsPsK sGsK sE sY 00 1   (1) If G 0 models the plant G perfectly, the closed-loop stability depends on the controller K and on the delay-free model P 0 only, and any closed-loop dynamic can be obtained. As it is impossible that G 0 can model G perfectly, it has been shown that the roll-off of transfer function (1) needs to be sufficient to avoid instability. Then, it is not really important to choose a high-order an accurate model G 0 for the control of an uncertain plant G. 0 G m , and P 0 by the first order part G m1 m system with a delay for the mismatched model   2 m 1 e s k sG ts     , (2) and uses  s k sG    1 m1 , (3) to design a low-order PID controller K. Using the relation between the IMC method and the Smith predictor structure, Zhang and Xu propose an analytical way to design controller K. G m 1 (s) K(s) + + - - y u e Q(s) G(s) G m (s) + - - d u The IMC controller Q equals:     sKsG sK sQ m1 1  (4) If G m approximates well the nominal plant G 0 , the nominal closed-loop transfer function y/e is close to the open-loop transfer function defined by:    sGsQsJ m  (5) Wang et al. propose to replace G by a deliberately mismatched model of G . Wang proposes a second-order Figure 2 presents the Smith predictor including the IMC controller Q. Fig. 2. Smith predictor with IMC controller. SMITH PREDICTOR AND FRACTIONAL-ORDER CONTROL 513 514 Zhang proposes to choose the user-defined transfer function J as   2 1 e s sJ ts     , (6) with  the time constant that can be tuned to achieved performance and robustness. Then, controller K is a PID controller given by:    ss s sK      11 1 2 2 (7) 3.1 Introduction The CRONE control-system design (CSD) is based on the common unity- feedback configuration (Fig. 3). The controller or the open-loop transfer function is defined using integro-differentiation with non-integer (or fractional) order. The required robustness is that of both stability margins and performance, and particularly the robustness of the peak value M r (called resonant peak) of the common complementary sensitivity function T(s). Three CRONE control design methods have been developed, successively extending the application field. If CRONE design is only devoted to the closed- from the parametric variations of the plant and from the controller phase variations around the frequency  cg , which can also vary. The first generation CRONE control proposes to use a controller without phase variation (fractional differentiation) around open loop gain crossover frequency  cg . Thus, the phase margin variation only results from the plant variation. This strategy has to be used when frequency  cg is within a frequency range where the plant phase is constant. In this range the plant variations are only gain like. Such a y(t) N (t) m u(t) - +  (s) + d (t ) u + d (t) y + (t) G(s) C(s) e F (t) y ref Lanusse and Oustaloup 3 CRONE CSD Principles Fig. 3. Common CRONE control diagram. second tracking problems. loop using the controller as one degree of freedom (DOF), it is obvious that a Second DOF (F, linear or not) could be added outside the loop for managing The variations of the phase margin (of a closed-loop system) come both [...]... +1 and +1.84 The uncertainty domains are computed for 120 pseudo-frequency v within the range 10 4, 102 and for nine log-spaced values of within the interval 12, 96 Figure 3 presents the nominal Nichols locus of the plant and its uncertainty domains [4] Even if the uncertainty domains of the plant to be controlled are almost vertical (gain like perturbation), it is interesting to use the third-generation... around this cutoff frequency Thus, the CRONE controller is defined 527 J Sabatier et al (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, 527–542 © 2007 Springer 528 Lanusse, Oustaloup, and Sabatier by a fractional- order n transfer function that can be considered as that of a fractional PIDn controller The second generation is also adapted... Manabe S (2003) Early development of fractional order control, DETC’2003, 2003 ASME Design Engineering Technical Conferences, Chicago, Illinois, Septembre 2–6 7 Podlubny I (1999) Fractional- order systems and PID-controllers, IEEE Trans Auto Control, 44(1):208– 214 8 Vinagre B, Chen YQ (2002) Lecture notes on fractional calculus applications in control and robotics, in: Vinagre Blas, YangQuan Chen, (ed.)... zeros of the nominal plant) with a predictive part e+ s Taking into account, internal stability for the nominal plant, stability for the perturbed plants and achievability of the controller, it is obvious that such a controller cannot be used Thus, the definition of (s) needs to be modified by including the nominal right half-plane zeros and the nominal time-delay: SMITH PREDICTOR AND FRACTIONAL- ORDER... arg (j ) B Fig 4 Generalized template in the Nichols plane The generalized template can be defined by an integrator of complex fractional order n whose real part determines its phase location at frequency cg, that is –Re/i(n) /2, and whose imaginary part then determines its angle to the vertical (Fig 5) The transfer function including complex fractional- order integration is: (s) cosh b ib a sign b... arg (jv) vB Fig 4 Generalized template in the Nichols plane The generalized template can be defined by an integrator of complex fractional order n whose real part determines its phase location at frequency vcg, that is –Re/i(n) /2, and whose imaginary part then determines its angle to the vertical (Fig 5) The transfer function including complex fractional- order integration is: sign b ( w) cosh b 2 vcg... is obvious that the settling time needs to be minimized Two roots of DC are lower than the gain crossover frequency vcg: 0 (the integrator) and – 0.0115 Its three others roots are included in : –0.129, –0.202 and –1.63 To add one more degree of freedom, D is chosen as a first degree polynomial Condition (30) imposes to include 3 further roots in They are determined by taking into account the frequency... performance Before designing the robust and performing control-system, it is sometimes difficult to translate the initial (time-domain) requirements to frequency-domain design specification and to set some of the open-loop parameters As the robust controller is designed only taking into account small-level exogenous signals, an anti-windup system often needs to be included [6] Using a laboratory plant... the multi singleinput single-output (SISO) approach If CRONE design is only devoted to the closed-loop using the controller as one degree of freedom (DOF), it is obvious that a second DOF (F, linear or not) could be added outside the loop for managing pure tracking problems [11, 12] Another solution is to implement the linear controller in a nonlinear way 530 Lanusse, Oustaloup, and Sabatier that provides... low-pass filter of integer order nh: Ch h s (13) nh s 1 N The optimal open-loop transfer function is obtained by the minimization of the robustness cost function J sup T j ,G - M r0 , (14) where Mr0 is the resonant peak set for the nominal parametric state of the plant, while respecting the following set of inequality constraints for all plants + (or parametric states of the plant) and for : inf T j G sup . (0)5 4000 66 44, fractional- order CRONE controller which is well known for managing well the in Physics and Engineering, 511–526. J. Sabatier et al. (eds.), Advances in Fractional Calculus: Theoretical. Mechanical definitions used in control’s theory were reminded. Then, the fractional tracking strategy. The study of robust path tracking via a third-generation application through non-linear fractional. template in the Nichols plane. The transfer function including complex fractional- order integration is: effort specifications at these frequencies using band-limited complex fractional- order integration: