Copyright 00 IFAC 5th Triennial World Congress, Barcelona, Spain QFT VERSUS CLASSICAL GAIN SCHEDULING: STUDY FOR A FAST FERRY J. Aranda, J. M. de la Cruz, J.M. Díaz, S. Dormido Canto Dpt. De Informática y Automática. Fac. Ciencias. UNED. Madrid. Spain Dpt. de Arquitectura de Ordenadores y Automática. Fac. Ciencias Físicas. U. Complutense. Madrid. Spain Fax: 34 9 398 66 97. Phone : 34 9 398 7 48. E -mail : jaranda#dia.uned.es Abstract: In this paper a comparative study of two different control strategies is done: Gain scheduling with classic controllers (PD and second order filter) versus QFT. Both of them are used to decrease motion sickness in a high speed ferry produced by the vertical acceleration associated with heave and pitch motion. Copyright 00 IFAC Keywords: Ship Control, PD, QFT.. INTRODUCTION The main problem for the development of high speed ferries is concerned with passenger comfort and vehicle safety. The vertical acceleration associated with roll, pitch and heave motion is the main cause of motion sickness. A decrease in the vertical acceleration associated with the roll motion can be easily obtained, while the vertical acceleration produced by the heave and pitch motion is more difficult to reduce. Shipbuilders are interested in obtaining mechanical actuators and control systems (O Hanlon and MacCawley, 974) to decrease the heave and pitch motion. The aim of this work is the development and application of different control design methodologies for a fast ferry, comparing a classical gain scheduling design and a Quantitative Feedback Theory (QFT) design. Several experiments (De la Cruz et al, 00) to obtain vertical dynamics models and to test controllers were done using a scaled-down (/5) replica of a fast ferry in the towing tank of CEHIPAR (Canal de Experiencias Hidrodinámicas de El Pardo, Madrid, Spain), a prestigious towing tank institution. Once the modelling stage of the vertical dynamics of a high speed ferry and the mechanical actuators had been completed then the next stage was to design a control system to position the actuators. The final goal was to decrease the vertical accelerations that produce motion sickness. The first design controllers were classical ones: P, PD, first order filters and second order filters, although in this paper only the results obtained with PD controllers and second order filters are shown. These controllers were tuned using a genetic algorithm at different ship speed and sea state number (SSN). A gain scheduling design was thus obtained. The next design controller was a robust one using QFT (Aranda et al., 00b). This controller considers the modelling uncertainties due to ship speed and sea state variations. In this paper, the gain scheduling design is compared with the QFT design.
. PROCESS MODEL A set of continuous linear models of the vertical dynamic of a high speed ferry were identified (De la Cruz et al., 998; Aranda et al., 000). In addition two different kinds of mechanical actuators were designed: a pair of flaps located in the stern and a T- Foil located in the bow. Linear and non-linear models for the actuators (Esteban et al., 000) were obtained. The process model (see Figure ) is the model of the vertical dynamic of a high speed ferry connected to the model of the actuators. This is a multivariable model with two manipulated variables: the set-point of the flaps position u F and the set-point of the T-Foil position u TF. The process has two controlled variables: the heave motion h and the pitch motion y. There is also one disturbance: wave height w. u F u TF w Process Model Fig.. Block diagram of the process. A continuous non-linear process model was implemented (Esteban et al., 000) in Simulink. This is a first good approximation to the real process, however the noise in the measured signal was not considered. Afterwards, a discrete non-linear process model (Esteban et al., 00) was implemented considering the noise. This is a very good approximation to the real process at just 40 knots, the nominal ship speed. This simulation environment makes possible to check the designed controllers before making real experiments in the CEHIPAR. For example, Figure shows the pitch motion obtained with the simulation environment and the pitch motion measured in the CEHIPAR at 40 knots and SSN5. PITCH (Degrees).5 0.5 0-0.5 - -.5 h y Using the linear model of the actuators, the process model can be expressed in the following equations: H P F + P TF + P3 W () Y s) P + P + P W ( ) () ( F TF 3 s where P ij (s) i, and j,,3 are transfer functions. The equation () shows that the pitch motion consists of three components that are functions of the wave frequency: wave height, flaps motion and T-foil motion. It can be demonstrated (Aranda et al,00b) that the main component to the pitch motion is the wave height and that the T-foil motion component is greater than the flaps motion component. For example, at V40 knots and SSN4 the T-foil motion component compensates 4% of the wave height component, while the flaps motion component compensates % of the wave height component. Therefore, the flaps motion component can be considered negligible. Thus, equation () can be simplified considering the position of the flaps as fixed. Y P( s) + D( s) (3) where P(s)P (s), U(s)U TF, D(s)P 3 (s)w(s). In order to study robustness properties, an interval plant P was obtained (Aranda et al, 00b). The interval plant is defined by the following family of linear invariant-time transfer functions P. P K ( s + a)( s + b) P ( s + 03.)( s +.8)( s + c)( s + ds + e) [ ] [ ] (4) K 0.87, 0.65 ; a 7.85, 6.67 ; b [ 0.06, 0.04 ]; c [ 0.44,0.49 ]; [ ] [ ] d 0.86,0.97 ; e.59,.80 The nominal plant P 0 identified at V40 knots and SSN4 is: ( s 7.85)( s + 0.04) 0.87 P0 ( s + 03.)( s +.8)( s + 0.49)( s + 0.86s +.8) (5) Figure 3 shows the block diagram of the system formed by the plant P and the controller G. - u G P + d y - -.5 0 0 30 40 50 60 70 TIME(seconds) Fig.. Pitch motion (solid line) obtained with the simulation environment and the pitch motion (broken line) measured in the CEHIPAR at 40 knots and SSN5. Fig. 3. Block diagram of the system. 3. SPECIFICATIONS The system (plant + controller) must fulfil the following specifications: robust stability, control
effort limitation and plant output disturbance rejection (sensitivity). 3. Robust Stability The system must be stable in all the possible work conditions. In order to fulfil the other specifications, an acceptable robust stability specification is a phase margin of 50º and a gain margin of.8. Thus, this specification can be expressed in the following equation (Borghesani et al., 994): P (., P P, ω [0, ) + P( (6) 3. Control effort limitation Motion sickness incidence (MSI) (Lloyd, 989; De la Cruz, 000) is defined by: where ± log MSI 00 0.5 ± erf ( J / g) # µ (0) 0 MSI 0.4 ( ) µ MSI 0.89 +.3 log 0 ω () and g is the gravity acceleration. If J value decreases it also produces a decrease in motion sickness incidence (MSI). From the robust performance point of view, a decrease in MSI can be associated with the sensitivity function of the system. The plant output disturbance rejection specification can thereby be expressed in the equation: The T-Foil angular position is limited because of cavitation problems inside the range [-5º,5º]. This specification can be expressed in the equation (Borghesani et al., 994): Y ( jω ) Se(, P P, ω Ω D( + P( ( s +.8)( s + 0.44)( s + 0.97s +.59)( s + 6.+ 3.6) Se ( s +.3)( s + 0.4)( s +.0s +.4)( s + 5.596s + 8.33) () U jω ) 5, P P, D( s) + P( ω Ω [,.5]( rad /sec) (7) where Ω is the range of frequencies where it finds the greater part of the energy spectrum of the wave height at V30,40 knots and SNN4,5. This specification does not guarantee that the T-Foil angular position in the non-linear process exceeds a 5º value, because if the disturbance modulus is greater than, the T-Foil angular position will exceed this value. However, this specification guaranteed that the T-Foil did not saturate too much at SSN4 and SSN5. 3.3 Plant output disturbance rejection The main goal was to decrease the vertical acceleration associated with the heave and pitch motion that produces motion sickness. The measured vertical acceleration at 40 metres ahead of the mass centre is acv40(t i ) i,..., N, where N is the total samples number. The vertical acceleration can be expressed as a sum of two accelerations: acv40(t i ) a VH (t i ) + a VP (t i ) (8) where a VH and a VP are the vertical accelerations produced by heave and pitch motion, respectively. The mean value of acv40 is: N J acv40 acv40( t i ) (9) N i S e is the result of studying (Aranda et al, 00b) the possible sensitivity functions that produced the greater decrease in MSI, considering the control effort limitation. 4. CONTROLLER DESIGN 4. Gain scheduling design Several classic controllers (P, PD, first order filter and second order filter) were tuned in four different work conditions (V,SSN). A gain scheduling design was obtained at V30,40 knots and SSN4,5. The PD controller transfer function is: G K P. T 0. T D D s + s + (3) The second order filter transfer function is : s K s G + as + b + cs + d (4) The tuning of these controllers was done (Aranda et al., 00a) by solving the following non-linear optimization problem: J ( θ ) min J (5) opt θ S where J is the mean vertical acceleration defined in (9) and S is the range of values allowed for the elements parameter vector θ.
4. QFT design Considering the family of transfer functions defined in (4), a robust controller was designed using QFT (Horowitz, 963; Yaniv, 999). The design must fulfil all the specifications defined in section 3. Inside the range Ω, the set Ω {,.5,.5,,.5} (rad/sec) was chosen to compute the output plant disturbance rejection bounds and the effort control limitation bounds. Furthermore, the system had to be stable in any frequency between zero and infinity. Hence, to generate the robust stability bounds was necessary to add some greater frequency to Ω. A possible election was to add the frequency 0 rad/sec. A new set Ω {,.5,.5,,.5, 0} (rad/sec) is defined. The QFT MATLAB toolbox was used to compute the different kinds of bounds. In the loop-shaping stage, we used the application IDESQLS (Dormido et al., 00) built in SYSQUAKE (Piguet, 999), an interactive design CAD tool for automatic control and signal processing. A complete explanation of this QFT design is given in Aranda et al., (00b). 5. RESULTS Table and Table show the tuned parameters for a PD controller and second order filter, respectively, in four different work conditions (V,SSN). Table : Tuned parameters for a PD controller in four different work conditions (V,SSN) (V,SSN) K P T D (30,4)..7 (30,5) 4.7. (40,4) 7.4.3 (40,5) 4.0.3 Table : Tuned parameters for a second order filter in four different work conditions (V,SSN) (V,SSN) K a b c d (30,4) 74.. 0.9. 5.9 (30,5) 7. 3..5.4 4. (40,4) 70..7.9.5 8.6 (40,5) 60.6.7 3.8 3. 55.8 Let L nd G P 0 the nominal open-loop transfer function where G is the second order filter at (40,4) whose coefficients are shown in Table. Figure 4 shows a Nichols chart with the intersection of all the bounds generated at the frequencies Ω, a different colour is used to represent each frequency (rad/sec): (red),.5 (blue),.5 (green), (yellow),.5 (cyan) and 0 (magenta). L nd is also drawn, the triangles represent L nd (jω i ) ω i Ω. Let L QFT G QFT P 0 the nominal open-loop transfer function where G QFT is the following designed QFT controller G QFT s s 0.3 s 0.65 8.7 ( + ) + + + +.7.8.8.74.74 s s s 0.66 s 0.5 + + + + + +.9 000 57 57 4.30 4.30 Figure 5 shows the Nichols chart with the intersection of all the bounds generated at the frequencies Ω, the same colour code as Figure 4 is used. L QFT is drawn as well. Table 3 shows a decrease in MSI obtained in simulation using: gain scheduling with a PD, gain scheduling with a second order filter, G QFT. Table 4 shows a decrease in MSI obtained in simulation using: nominal PD, nominal second order filter, G QFT. Table 3: A decrease in MSI obtained in simulation using: gain scheduling with a PD, gain scheduling with a nd order filter, G QFT. (V,SSN) PD nd order G QFT filter (30,4) 33. % 48.5 % 5.4 % (30,5).3 % 4.0 %.4 % (40,4) 38.3 % 49. % 3.8 % (40,5).8 % 5. % 4.0 % Table 4: A decrease in MSI obtained in simulation using: nominal PD, nominal nd order filter, G QFT. (V,SSN) Nominal PD Nominal nd order filter G QFT (30,4) 30.7 % 43.9 % 5.4 % (30,5).0 %.7 %.4 % (40,4) 38.3 % 49. % 3.8 % (40,5) 8.8 %.8 % 4.0 % Table 5 shows a decrease in MSI obtained with real experiments in CEHIPAR using: gain scheduling with a PD, gain scheduling with a second order filter, G QFT. Table 5: A decrease in MSI obtained with real experiments in CEHIPAR using: gain scheduling with a PD, gain scheduling with a nd order filter, G QFT. (V,SSN) PD nd order G QFT filter (40,4) 8.5 % -5. % 7.6 % (40,5) 6.6 % 3.4 % 4.5 %
Nichols Frequency Response 0 0 L nd (j) L nd (j) L nd (j.5) L nd (j.5) L nd (j.5) -0 L nd (j0) -40-300 -00-00 0 Fig 4. Nichols chart shows the intersection of all the bounds generated at the set of frequencies (rad/sec) Ω {(red),.5 (green),.5 (blue), (yellow),.5(cyan), 0(magenta)}. L nd ( is also shown, the triangles represent L nd (jω i ) with ω i Ω. Nichols Frequency Response 0 0-0 L QFT (j.5) L QFT (j.5) L QFT (j) L QFT (j0) L QFT (j) L QFT (j.5) -40-300 -00-00 0 Fig 5. Nichols chart shows the intersection of all the bounds generated at the set of frequencies (rad/sec) Ω {(red),.5 (green),.5 (blue), (yellow),.5(cyan), 0(magenta)}. L QFT ( is also shown, the triangles represent L QFT (jω i ) with ω i Ω.
6. CONCLUSION A comparative study of two different control strategies: Gain scheduling with classic controllers versus QFT was done. Both of them were used to decrease motion sickness incidence (MSI) in a high speed ferry. Gain scheduling with classic controllers does not fulfil the three specifications defined in Section 3 (see * Figure 4) for all P P, whereas QFT design fulfil all of them (see Figure 5). From Table 3 and Table 4, the following conclusions can be obtained: ) Gain scheduling with second order filter produces a greater decrease in MSI than gain scheduling with a PD, because second order filter has a greater number of tuning parameters. ) Using nominal controllers for all the work conditions does not produce a worse result than using gain scheduling. 3) QFT controller produces a smaller decrease in MSI, although it was designed considering all * P P and not only these four work conditions. QFT is a more conservative design because it always includes the worst possible case. However, the advantages of this design are shown in Table 5, where it produces a greater decrease in MSI in the real system than gain scheduling strategies. To improve the results with a gain scheduling strategy in the real system, it is necessary to tune the controllers again, which means that this strategy no longer has the advantage of being easier than QFT. Acknowledgements : This development was supported by CICYT of Spain under contracts DPI000-0386 - C03-0 and TAP97-0607 - C03-0 REFERENCES Aranda, J., J.M. de la Cruz, J.M. Díaz, B. de Ándres, P. Ruipérez, S. Esteban, J.M. Girón. (000). Modelling of a High Speed Craft by a nonlinear Least Squares Method with Constraints. Proceedings of 5th IFAC Conference on Manoeuvring and Control of Marine Crafts MCMC000. Aalborg.. pp. 7-3. Aranda, J., J.M. Diaz, P. Ruipérez, T.M. Rueda, E. López. (00). Decrease in of the motion sickness incidence by a multivariable classic control for a high speed ferry. Proceedings of CAMS 00 Control Applications in Marine Systems. Glasgow (United Kingdom). Aranda, J., J.M. De la Cruz, J.M. Diaz, P. Ruipérez. (00). Decrease in of the motion sickness by a QFT control on pitch in TF-0 ship. Technical Report (in Spanish) CRIBAV-0-05 of project TAP97-0607-C03-0 (ctb.dia.uned.es/cribav/). Borghesani, C., Y. Chait and O. Yaniv. (994). Quantitative Feedback Theory Toolbox - For use with MATLAB. st Edition. The Math Works. Inc. De la Cruz, J.M., J. Aranda, J.M. Díaz, P. Ruipérez, A. Marón. (998). Identification of the vertical plane motion model of a high speed craft by model testing in irregular waves. Proceedings of IFAC Conference CAMS'98 Control Applications in Marine Systems. Fukuoka. pp. 77-8. De la Cruz, J. M. (000). Evaluation. Technical Report (in Spanish) CRIBAV-0-04 of project TAP97-0607-C03-0 (ctb.dia.uned.es/cribav/). De la Cruz, J.M., De Lucas, P., Aranda, J.,Giron- Sierra, J.M.,Velasco, F., Marón, A. (00) A research on motion smoothing of fast ferries. Proceedings of CAMS 00 Control Applications in Marine Systems. Glasgow (United Kingdom). Dormido, S., J. Aranda, J.M. Díaz, S. Dormido Canto. (00). Interactive educational environment for design by QFT methodology. Proceedings of International Symposium on Quantitative Feedback Theory and Robust Frequency Domain Methods. Pamplona Esteban, S., J.M. Girón, J.M. de la Cruz, B. de Andres, J.M. Díaz, J. Aranda. (000). Fast Ferry Vertical Accelerations Reduction with Active Flaps and T-Foil.. Proceedings of 5th IFAC Conference on Manoeuvring and Control of Marine Crafts MCMC000. Aalborg. pp. 33-38. Esteban, S., B. de Andres, Girón-Sierra, Polo, O.R., Moyano, E. (00). A Simulation Tool for a Fast Ferry Design.. Proceedings of CAMS 00 Control Applications in Marine Systems. Glasgow (United Kingdom). Horowitz, I. M. (963). Synthesis of Feedback Systems. Academy Press, New York. Lloyd, A.R.J.M. (989). Seakeeping. Ship Behaviour in Rough Weather. Ellis Horwood. O Hanlon, J.F., MacCawley, M.E. (974). Motion Sickness incidence as a function of frequency and acceleration of vertical sinusoidal motion. AM. Piguet, Y. (999). Sysquake User Manual, version.0. Calerga. Lausanne (Switzerland) Yaniv, O. (999). Quantitative feedback design of linear and non-linear control systems. Kluwer Academic Publishers Boston/ Dordrecht/ London.