Automation and Robotics 368 (15) Average values of obtained a and β B and other directly measured coefficients are listed in the Table 2. Table 2. Parameters used in the considered system Due to the considered levitation system being naturally unstable and having a very fast response, it is difficult to validate the developed model directly. Therefore, a simple PID feedback controller is developed to keep the considered system operating properly. The mathematical model is validated by comparing the simulated closed-loop control system and the real controlled system afterwards (Yang et al. (2007)). 4. Design and implementation of PID controllers 4.1 Empirical PID controller By using the obtained nonlinear model, an analog PID controller is developed and manually tuned based on the Ziegler-Nichols PID tuning method. Then the developed PID controller is discretized with a sampling frequency of 480 Hz, which is determined by the NI DAQ card used for the digital implementation. The implemented controller has the form (16) where T, K p , T i and T d are sampling period, P, I, and D coe±cients, respectively. e(k) is the displacement tracking error. The simulation of the closed-loop control system using the empirical PID controller is shown in Fig.8. It can be observed that the controlled system has a reasonable response time and good tracking capacity. 4.2 Automatic tuning of PID controller using GA algorithms From our preliminary investigation (Pedersen & Yang (2006); Yang & Pedersen (2006)), it turned out that the PID controller can be automatically tuned using the multi-objective non- dominated sorting genetic algorithm (NSGA-II) based on the nonlinear system model. Model-Based Control of a Nonlinear One Dimensional Magnetic Levitation with a Permanent-Magnet Object 369 The performance induced by different PID-controller parameters are evaluated by the following criteria based on the step response: • Overshoot (M p ); • Rise time (t r ); • Settling time (t s ); and • Integrated absolute error (IAE). An illustration of the performance measures is given in Fig. 9. Each of these performance measures will be included as objectives to be minimized as their inter-dependence will depend highly on the nonlinear system expressed by (10) and (12). Fig. 9. Performance measures for step response The non-dominated sorting genetic algorithm (NSGA-II) developed in (Deb et al. (2000)) is a multi-objective algorithm, which can evolve a set of non-dominated solutions that are all equally well suited for solving the specific problem given the performance measures specified. Many of the NSGA-II run-time parameters used for here are the same as the NSGA-II default values (Pedersen & Yang (2006); Yang & Pedersen (2006)), such as Table. 3. Parameters used for running NSGA-II In the simulation, The range for K p is set to [-1000,0]. The ranges for T i and T d are both set to [0,15]. With respect to the computational complexity of the simulations, a population size of 50 individuals was chosen along with a maximum number of generations of 150. Besides from the use of the 4 objectives a constraint on the allowable amount of overshoot has also been formulated as only values below 100% was allowed. The distribution of K p , T i and T d for the case where the outliers have been removed is illustrated in Fig. 10. It is quite obvious that there is a large grouping of individuals for small values of T i and K p values below -800. A simulation of a typical controller from this cluster, with parameters as K p = -800.46, T i = 0.021 and T d = 0.06, is shown in Fig. 11. The corresponding performance measures for this individual are IAE=5 ⋅ 10 -4 , M p = 84.82%, t r = 21ms and t s = 0.425s. It can be observed that the system response consists of a fast Automation and Robotics 370 oscillation on top of a slower one. The fast rise time is mainly due to the size of K p which is obviously very aggressive towards positional errors. Fig. 10. Plot of parameters K p , T i and T d for last generation Fig. 11. System step response in simulation 4.3 LabView Implementation The developed controllers are implemented in NI LabView environment on a PC running Windows XP. Therefore some attention needs to be paid on the real-time issues. For instance, the connection between the external devices and the LabView environment is setup manually, even though the DAQ assistant in LabView could more easily create the Model-Based Control of a Nonlinear One Dimensional Magnetic Levitation with a Permanent-Magnet Object 371 communication line. However, our experiences showed that the DAQ Assistant is quite time consuming, no matter if it is used inside or outside the timed loop (Sønderskov & Østerö (2007); Yang et al. (2007)). Another real-time issue relevant to the Windows XP operating system. It is well known that Windows XP gives priority to different processes that are executed. For example, just moving the mouse is sometimes enough to slow down the execution of LabView code. In order to solve this real-time problem, the timed loop structure is used in the LabView program, which guarantees that the LabView code should be executed within the defined time period. Furthermore, In order to check the sampling rate issues, a sampling frequency calculator is constructed as shown in Fig. 12. A front panel of the developed controller is shown in Fig.13. Fig. 12. Sampling frequency calculator with front panel indicators Fig. 13. Front panel of the developed controller 5. Testing results and discussions The simulated performance of the closed-loop control system using the empirical PID controller is shown in Fig. 8. The same controller is implemented in the LabView program and tested with the physical setup. One test result based on the same set of set-points as for simulation is shown in Fig. 14. It can be observed that in principle the controlled physical Automation and Robotics 372 system has quite similar performance as the simulation model. However, it is also obvious that the controlled physical system has much shorter response time and much larger overshot and oscillation compared with the simulated system performance. The reasons for these deviations could be explained in the following perspectives: • Imprecise sensor measurement. The optical position sensor is very sensitive to light disturbances; • Frequent switchings of the MOSFET IRFZ44. The frequent on-off switchings of current due to this MOSFET can directly lead to oscillations in real tests (Yang et al. (2007)); • Imprecise sampling rates of DAQ card and PID computation due to the real-time problem of Windows XP operating system. This could cause synchronization problems in data acquisition and control computation; • the approximation of system coefficients. For example, in a strict sense, the system coefficient β B should be displacement dependent. However, we assume it is always constant due to simplicity. The consistency between simulation and real tests could be improved if above problems could be solved or moderated. By softly changing the set-points, e.g., filtering the rectangular set- points, the controlled physical system shows a better performance as shown in Fig. 15. It can be observed that the large overshot that appeared in Fig. 14 has disappeared. Fig. 14. Response of the controlled physical setup One test result using the same control coefficients directly from NSGA-II tuning is shown in Fig. 16. Compared with the simulation result shown in Fig. 11, this implemented controller has quite similar behavior as simulation study. However, it is also obvious that the fast dynamic has much larger amplitude than it does in simulation, which could be due to the following facts: • The designed closed-loop system is obviously under-damped; • The influence from the external disturbances, e.g., the air flow around the ball etc; • Model uncertainties and unprecise position measurements. More analysis of these issues will be one part of our future work. Model-Based Control of a Nonlinear One Dimensional Magnetic Levitation with a Permanent-Magnet Object 373 Fig. 15. Response of the controlled physical setup with soft changes Fig. 16. Step response of the controlled setup using the NSGA-II tuned controller 6. Conclusion The modeling and control of a 1-D magnetic levitation system with a permanent magnet object is investigated. The feature of the moving permanent magnet is explored using an experimental method and it is modeled through curve fitting technique. The entire system model is derived based on the electromagnetic theory and afterward system coefficients are identified through designed experiments. The developed model is validated through performance comparison of the closed-loop model and the controlled physical system. The PID control is chosen as the control structure at this stage regarding the fact: (1) it is simple and require few computation resources; (2) The developed PID controllers only need the position information, with no need for the current measurement and speed estimation, such that the potential degradation of the system performance due to quantization (Barie & Chiasson (1996)) can be minimized; The developed controllers are implemented in the LabView environment based on a PC running Windows XP. The real-time issues are managed by additional programs. Both simulation and real tests showed a clear consistency and a good system performance. Furthermore, The investigation of using genetic algorithms to automatically tune PID controller shows a potential to use this artificial intelligence method for supporting the control design for complicated nonlinear systems. Automation and Robotics 374 7. Acknowledgement The authors would like to thank René M. Sønderskov, Kim S. Østerö, Niels A. Pedersen, Stefan K. Greisen and Jette R. Hansen for their contributions in system development and laboratory tests. 8. References Special issue on magnetic bearing control. IEEE Control Systems Technology, Sept 1996. W. Barie and J. Chiasson. Linear and nonliear state-space controllers for magnetic levitation. Int. J. of Systems Science, 27(11):1153-1163, 1996. K. Deb, A. Pratap, and S. Moitra. A fast elitist non-dominated sorting genetic algorithm for multi-objective optimization: Nsga-ii. Parrallel Problem Solving from Nature - PPSN VI, pages 849-858, 2000. NSGA-II code available at KanGAL website: http://www.iitk.ac.in/kangal. L. Gentili and L. Marconi. Robust nonlinear disturbance suppression of a magnetic levitation system. Automatica, (39):735-742, 2003. A. Isidori. Nonlinear Control Systems. New York: Springer-Verlag, 1989. W. Kim. High-Precision Planar Magnetic Levitation. Phd thesis, Massachusetts Institute of Technology, June 1997. W. Kim, D.L. Trumper, and J.H. Lang. Modeling and vector control of planar magnetic levitator. IEEE Trans. on Industry Applications, 34(6):1254-1262, Nov/Dec 1998. V.A. Oliveira, E.F. Costa, and J.B. Vargas. Digital implementation of a megnetic suspension control system for laboratory experiments. IEEE Trans. on Education, 42(4):315-322, Nov. 1999. G.K.M. Pedersen and Z. Yang. Multi-objective pid-controller tuning for a magnetic levitation system using nsga-ii. In Maarten Keijzer, editor, Proceedings of Genetic and Evolutionary Computation Conference - GECC0 2006, pages 1737-1744, Seattle, Washington, USA, Jul 2006. ACM. T.L. Simpson. Effect of a conducting shield on the inductance of an air-core solenoid. IEEE Trans. on Magnetics, 35(1):508-515, Jan 1999. R.M. Sønderskov and K.S. Østerö. Malecos: Magnetic levitation control systems. 7th semester project report, Aalborg University Esbjerg, Denmark, Jan 2007. M.T. Thompson. Electrodynamic magnetic suspension - models, scaling laws, and experimental results. IEEE Trans. on Education, 43(3):336-342, Aug. 2000. M. Varella, E. Calloni, L.Di Fiore, L. Milano, and N. Arnaud. Feasibility of a magnetic suspension for second generation gravitational wave interferometers. Astroparticle Physics, (21):325-335, 2004. R. Wisniewski and J. Stoustrup. Periodic h-2 synthesis for spacecraft attitude control with magnetorquers. Journal of Guidance Control and Dynamics, 27(5):874-881, 2004. T.H. Wong. Design of a magnetic levitation control system - an undergraduate project. IEEE Trans. on Education, E-29(4):196-200, Nov 1986. H. Woodson and J. Melcher. Electromechanical Dynamics - part I: Discrete Systems. Wiley, New York, 1968. Z. Yang and G.K.M. Pedersen. Automatic tuning of pid controller for a 1-d levitation system using a genetic algorithm: a real case study. In Proceedings of the 2006 IEEE International Symposium on Intelligent Control, pages 3098-3103, Munich, Germany, Oct 2006. IEEE. Z. Yang, G.K.M. Pedersen, and J.H. Pedersen. Modeling and control of one-dimensional magnetic levitation system with a permanent-magnet object. In Proceedings of the 13 th IEEE/IFAC International Conference on Methods and Models in Automation and Robotics, pages 723-729, Szczecin, Poland, Aug 2007. IEEE. 22 Nonlinear Adaptive Tracking-Control Synthesis for General Linearly Parametrized Systems Zenon Zwierzewicz Department of Applied Mathematics Szczecin Maritime University Poland 1. Introduction A common problem of engineering practice is to cope with mathematical models of objects with only partly known structure. The model may e.g. involve some unknown (linear or nonlinear) functions that depend on the kind of object (of a given class to which the model refers) and/or of its operation conditions. As an example we take an affine model of SISO system u ⋅ + = )()( xβxαx  (1a) )(xhy = (1b) where y, x, u denote output, state and control variables respectively, α and β are smooth vector fields on n R and RRh n →: a smooth function. It is assumed here also that the functions α and β are unknown or may be estimated with a considerable inaccuracy. Considering the system (1) it is possible (under certain conditions (Fabri & Kadrikamanathan, 2001; Sastry & Isidori, 1989)) to obtain a direct input-output relation between u and y, by successive differentiation y with respect of time having ugfy r )()( )( xx += (2) where r denotes a system relative degree. The whole approach could be well systematized and explained using the concept of Lie derivatives (Isidori, 1989) . In this chapter the system (1) is uncertain in the sense it is linearly parametrized, or in other words, the unknown functions α i and β i are assumed to be linear combinations of some known model related functions which represents our elementary knowledge on the model. It is easy to prove (see appendix) that if the functions α i and β i of system (1a) are of the form of linear combinations of some known functions α i and β i i.e. )()( 1 1 xαxα i m i i a ∑ = = ; )()( 2 1 xβxβ i m i i b ∑ = = (3) Automation and Robotics 376 where a i , b i are real unknown parameters then the scalar functions f , g of system (2) may be represented in similar form: )()()( 0 1 1 1 xxx fff i n i i += ∑ = θ ; )()()( 0 1 2 2 xxx ggg i n i i += ∑ = θ (4) with 1 i θ , 2 i θ unknown parameters and i f , i g (called here model basis functions) again known trough the α i and β i (see appendix). There are a huge amount of nonlinear systems that might be modeled in general form (1),(3). Using described above model transformation one can obtain a parametric model of the form (2),(4) in relative easy way (see section 3.2). The model in this form, referred below as a transformed model, was considered in many papers. One of the known method of tracking control synthesis in the case when we have a rough estimate of the model (2) functions, is a sliding mode control law (Slotine & Li, 1991). The alternative is to use adaptation (for model in the form (2),(4)) which offers more subtle policy but requires more advanced theory. In our approach the unknown functions f and g of the transformed model are, as it turned out, linear combinations of some known model related basis functions i.e. some elementary knowledge of the model is assumed. The assumption above may, however, be substantially relaxed via applying, as basis functions, some sort of known approximators (Fabri & Kadrikamanathan, 2001; Tzirkel-Hancock & Fallside, 1992). As an example one may adopt a neuro-approximator with Gaussian radial basis functions (Sanner & Slotine, 1992). Systems of this sort are referred to as functional adaptive (Fabri & Kadrikamanathan, 2001) and represent a new branch of intelligent control systems. In the real-world applications, however, it seems purposeful to assume that we have at our disposal some (often very limited) knowledge, on the considered plant or process, that should be exploited in reasonable way. In this paper the accent is put-on the later issue. This chapter is concerned with the problem of adaptive tracking system control synthesis for the described above class (1),(3) of uncertain systems. It has been proven that proportional state feedback plus parameters adaptation via the model basis function concept are able to assure system asymptotic stability. This form of controller permits on-line compensation of unknown model nonlinearities which leads to satisfactory tracking performance. The presented theory is illustrated by the example of ship path-following control system (Zwierzewicz, 2007ab). It is worth to observe that affine model description (1) is taken here without loss of generality. The general nonlinear system ),( uxFx =  (5a) )( xhy = (5b) may be easy expressed in this form by augmenting it with input integrator vu =  which leads to new state [ ] T T a uxx = . Now considering v as a new input the above system is in the form (1). The chapter is organized as follows. In section 2., an appropriate portion of the theory is shortly presented, which utility (in the next section) is then verified via an example of ship path-following control system. The next sections contain results of the relevant system simulations, remarks and conclusion. Nonlinear Adaptive Tracking-Control Synthesis for General Linearly Parametrized Systems 377 2. Adaptive tracking control synthesis The control objective is to force the plant (1) output vector Tr yyy ],,,[ )1( − = "  y to follow a specified desired trajectory Tr dddd yyy ],,,[ )1( − = "  y with state vector x remaining bounded. It is moreover assumed that reference input d y and its r derivatives are bounded and known as well as that the system zero dynamics is globally exponentially stable (minimum phase condition) . As the model (1),(3) can be transformed to the form (2),(4) thus, in what follows, our considerations will be referred to the later form. 2.1 The case of exact model It is assumed in this section that the nonlinear functions f and g of model (2) are known and n Rg ∈∀≠ xx ,0)( . A substitution of control law )( )( x x g vf u + − = (6) in the system (2) results in exact cancellation of both nonlinearities ( f(x) and g(x) ) which yields vy r = )( (7) To find control v(t) stabilizing this linear system, a standard poles location technique can be used. If v is chosen as eeyv r r r d 1 )1()( μμ −−−= − " (8) where y d denotes the reference input which y is required to track, d yye − = : denotes the output tracking error and coefficients i μ are chosen such that 0:)( 1 1 =+++=Γ − ssss r r r μμ " is Hurwitz polynomial in the Laplace variable s, then the tracking error and its derivatives converge to zero asymptotically, because the closed-loop dynamics reduce to the equation 0 1 )1()( =+++ − eee r r r μμ " (9) which, by virtue of the choice of coefficients i μ is asymptotically stable (Fabri & Kadrikamanathan, 2001; Sastry & Isidori, 1989; Tzirkel-Hancock & Fallside 1992). 2.2 The case with functional uncertainty Let us consider now the case when functions f and g are unknown but have the form (4) with ,1 , 1 1 ni i "= θ , ,1 , 2 2 ni i "= θ unknown ‘true’ parameters and the )(x i f , )(x i g known model basis functions. At time t our estimates of the functions f and g are respectively [...]... kinematical model (compare the first two equations of model (39)) while (28b) and (28c) are in fact the Norrbin ship model (Fossen, 1994; Lisowski, 1981) whose standard form 382 Automation and Robotics Tψ + F (ψ ) = kδ (29) can be transformed into the relevant equations of (28) via definition ψr = r and substitution of Φ=− F (⋅) and c = k / T T The first equation of kinematics, in the model (28), is omitted...378 Automation and Robotics n1 n2 ˆ ˆ f ( x ) = ∑ θi1 (t ) fi ( x ) + f 0 ( x ) ; ˆ ˆ g ( x ) = ∑ θi2 (t )gi ( x ) + g 0 ( x ) i =1 (11) i =1 ˆ ˆ with θ i1 , θ i2 standing for the estimates of the parameters θ i1 , θ i2 respectively at time t Since substitution in the system (2) the control law u= ˆ − f ( x) + v ˆ g ( x) (12) no longer guarantees exact cancellation and whereby a resulting... parameters of the m.s Compass Island model are adopted The units of time, length and angle are respectively one minute, one nautical mile and one radian The parameters were determined as follows a = 1.084 /min, b=0.62min, c = 3.553 rad/min, r1 = -0.0375 nm/rad, r2=0, f = 0.86 /min, W= 0.067 nm/rad2, S=0.215 nm/min2 The maximum speed of rudder and rudder angle are 3.8 deg/s, and 35 deg, respectively The... h( x ) = ∇h( x )α ( x ) (40) 388 Automation and Robotics [ ] where ∇h denotes the gradient of h(x) i.e ∂h / ∂x1 ∂h / ∂x n Lie derivative is scalar so the process of taking Lie derivatives could be chained and is denoted as follows Liα h( x ) = ∇( Liα−1h( x ))α ( x ) (41) Lβ Liα h( x ) = ∇( Liα h( x )) β ( x ) (42) Differentiating y in eqation (1) with respect to time and using Lie derivatives we get... zero (Sastry & Bodson, 1989) One can now observe that adaptive reconstruction of functions f and g in the formula (11) may be interpreted as an extra control leading to much more exact cancellation of system (2) nonlinearities, which in turn make the resulting system closer to linear (see Fig 1) 380 Automation and Robotics Fig 1 Model basis functions adaptive control scheme 3 Adaptive ship path-following... ahead of time (Sastry & Isidori, 1989; Wang & Hill, 2006) 20 18 16 ship trajectories 14 X Axis 12 10 8 current 6 4 2 0 original ship position -1 0 1 2 Y Axis 3 4 5 Fig 3 Ship trajectories, constant current 70 60 Heading [deg] 50 40 30 20 10 0 -10 -20 0 10 20 Fig 4 Ship headings versus time 30 40 50 Time [min] 60 70 80 90 386 Automation and Robotics 40 30 Angle [deg] 20 10 0 -10 -20 -30 -40 0 10 20 30... n22 − θ n22 )]T (16) are model basis functions and ˆ θ 1 = [(θ11 − θ11 ) 1 ˆ1 ˆ (θ n1 − θ n1 )]T ; θ 2 = [(θ12 − θ12 ) are vectors of parameters [ ] T T T Moreover θ = θ1T θ 2T ; w = [ w1 w 2 ]T Theorem The closed-loop system (2), (12) and (8) after introduction of parameter update law, θ = −εw (17) yields bounded y(t) asymptotically converging to yd(t) Proof: Differentiating (13) and multiplying by... coordinate yr and the heading ψr as the path-following errors corresponding to the given segment For curvilinear reference path the local (relative) coordinate system should be tangent to the path at the point that is closest to the actual ship position This system has to be then shifted and rotated from time step to time step in such a way, that it remains tangent to the reference path and that the... (35) as well as parameters update law (17) (Fig 1) In Fig 3 the path to be followed (preset) is a broken line defined by the way points (0,0); (0,10); (4,12) and (4, 20) The original ship position, its heading and angular velocity are (0,0.5), 60° and 0 rad/min respectively The adopted distance scale is 1 nm while the nominal ship velocity is 0.25 nm/min In the simulation a transversal current has been,... bounded ε and θ However, to verify that ε → 0 as t → ∞ we use Barbalat’s lemma (Slotine & Li, 1991) To check the uniform continuity of V it is enough to prove that the second derivative of V i.e V = −2k d εε = −2k d ε (−k d ε + θ T w ) is bounded This in turn needs ε and yd w , a continuous function of x are bounded, it is implied that y (25) to be bounded Note that if is bounded These facts and assumed . Automation and Robotics 368 (15) Average values of obtained a and β B and other directly measured coefficients are listed in the. of the 13 th IEEE/IFAC International Conference on Methods and Models in Automation and Robotics, pages 723-729, Szczecin, Poland, Aug 2007. IEEE. 22 Nonlinear Adaptive Tracking-Control. equations of model (39)) while (28b) and (28c) are in fact the Norrbin ship model (Fossen, 1994; Lisowski, 1981) whose standard form Automation and Robotics 382 δ ψ ψ kFT = + )( 