Milano (Italy) August - September, 11 Dea Zone Moel Base Aaptive Backstepping Control for a Class of Uncertain Saturate Systems Seyye Hossein Mousavi Alireza Khayatian School of Electrical an Computer Engineering, Shiraz University, Shiraz, Iran s.h.mousavi@gmail.com khayatia@shirazu.ac.ir Abstract: In this paper, a ea zone base moel of saturation phenomenon is propose, which enables to moel iverse kins of saturation, such as har limitte an soft limitte ones. Due to the use of a linear in parameter approach, the propose moel is consistent with the available aaptive control techniques in the literature. Besies, base on the propose moel, an aaptive controller is esigne for a class of nonlinear saturate systems, in which, the shape of saturation phenomenon is assume to be unknown. The effectiveness of the propose metho an its robustness against change in initial conition an refernce signal is evaluate via simulate examples. Keywors: Aaptive control; Saturate system; Deazone base moelling. 1. INTRODUCTION Saturation is a phenomenon that exists in a vast range of practicalcontrolsystems,suchaselectricalmachines(jang [9]), robot maipulators (Huang et al. []) an mems (Jagannathan an Hamee []). The presence of such input nonlinearity may result in egraation of control performance an even worse, it may lea to instability of thecloseloopsystem.hence,controlofsaturatesystems is of a great significance. However, ue to complexity of the problem, very limitte works have been reporte in the literature. Yan et al. [7] stuie the problem of robust stabilization for uncertain time elay systems subjecte to saturating actuator.uner certain conitions, a ynamic compensator, which uses only the accessible output variables, is synthesize to achieve the stabilization. In Zhong [], moel reference aaptive control problem for single-input single-output minimum phase systems with input saturation was consiere, where a sufficient conition is presente which becomes necessary in some cases. Moreover the close loop sytem can have global stability uner certain assumptions. In most of the works cite before, the control problem has been consiere for linear systems an uner certain limitting assumptions, the global stability is satisfie. To see more works on the linear systems, one can refer to Chaoui et al. [199], Chaoui et al. [1] an Fliegner et al. [3]. Gao an Selmic [] propose neural net (NN)-base actuator saturation compensation scheme for a class of nonlinear systems, presente in Brunovsky canonical form. The actuator saturation is assume to be unknown an the saturation compensator is inserte into a feeforwar path. Besies, the operation of the esigne controller was inicate by simulation results. In Zhou an Wen [], base on backstepping technique, a controller was esigne for uncertain nonlinear systems in the presence of input saturation. The evelope controller oes not require uncertain parameters within a known compact set. however, to achieve an aaceptable tracking performance, proper ajustment of system parameters is not a simple task. In this paper as inspire by Su et al. [], a new eazone operator base moel of saturation phenomenon is propose, that is able to moel iverse kins of saturation such as har limitte an soft limitte ones. Applying this moel, provies enough egree of freeom for the esigner to hanle a trae-off between moelling accuracy an computation complexity. Moreover using a linear in parameter approach in the moelling, makes it consistent with the available aaptive control techniques. Besies, an aaptive backstepping controller is esigne for a class of nonlinear systems which contains an unknown saturation as input nonlinearity. In aition, the simulation results are implemente for a Spring-Mass-Damper to consier the performance of the esigne controller. The rest of this paper is organize as follows: In section, the problem formulation is presente, In section 3, the propse saturation moel is introuce an the aaptive control esign is explaine in section. Moreover, section shows simulation results to illustrate the efficiency of our propose metho. Finally conclusions are given in section.. PROBLEM FORMULATION Consier a class of nonlinear saturate systems escribe by: Copyright by the International Feeration of Automatic Control (IFAC) 19
Milano (Italy) August - September, 11 where x (n) + a j Y j (x,ẋ,...,x (n1) ) = bw (1) a j (for j = 1,...,N) an b are some unknown constants an Y j (for j = 1,...,N) are some known functions. Furthermore, w = sat(u) is the saturate control input efine as: z r ( ) is a ea zone operator efine by: z r (u) = max(ur,min(,u+r)) () The operation of ea zone operator is illustrate in Fig.3 w = { wsat u w sat u w sat u w sat w sat u w sat () where w sat is an unknown constant parameter, w is illustrate in Fig.1. Fig. 1. Ieal saturation operator. Assumption 1. Sign of b is known an without loss of generality it is assume to be positive. The aim of this paper is to esign a control law for u, such thatthesignalxfollowsaesirereferencesignalx ;while the close loop system is stable. 3. DEAD ZONE OPERATOR BASED MODEL OF SATURATION Fig. 3. Deazone operator. Remark 1. To moel a saturation operator, ifferent kins of ensity functions such as Guassian function can be consiere as ρ(r), the parameters of which, shoul be approximate base on experimental ata. To clarify the ea zone operator base moel of saturation, an example is epicte in Fig., in which, ρ(r) is consiere as : 1 r R = ρ(r) = () r It is clear from Fig., at u = R, the ouput of saturation operator enters its saturate region. Remark. To calculate the saturate value(v sat ), we have : v sat = lim u v = rρ(r)r (7) Consier the following saturation block iagram: Fig.. Saturation block. φ( ), enotes the saturation operator. Similar to the work presente in Su et al. [], it is propose that φ( ) can be moele with the following relation : where v = φ(u) = ρ u ρ(r)z r (u)r (3) ρ(r)isaensityfunction,vanishingatafinitehorizon R an satisfying ρ(r) r >. ρ is a constant parameter, calculate from: ρ = ρ(r)r () Fig.. Propose saturation moel.. CONTROLLER DESIGN In this section, base on the propose saturation moel, an aaptive controller is esigne an the stability of the close loop system is guarantee via Lyapunov stability theorem. To achieve our goal, the system ynamic is expresse in a normal form: ẋ i = x i+1 for i = 1,...,n1 ẋ n = a j Y j ( x)+bw () y = x 1 19
Milano (Italy) August - September, 11 where x = [x 1,...,x n ] T R n is the state vector an w is the saturate control input, moele by the propose saturation moel: w = ρ u ρ(r)z r (u)r (9) in which ρ an ρ(r) are suppose to be unknown. Substituting (9) into () resluts in: ẋ i = x i+1 for i = 1,...,n1 ẋ n = a j Y j (x 1,...,x n1 )+ρ u (1) ρ (r)z r (u)r y = x 1 where ρ = bρ, ρ (r) = bρ(r) (11) To esign a controller base on aaptive backstepping technique, new state variables are efine as follows: { z1 = x 1 x z i = x i x (i1) (1) α i1 for i =,...,n α i is calle virtual control law an calculate from: { α1 = c 1 α i = c i z i z i1 +α i1 for i =,...,n1 (13) c i (for i = 1,...,n) are some positive arbitrary real numbers. Base on the above efinitions an consiering β = (ρ ) 1, the propose aaptive backstepping base controller is: u = ˆβu 1 (1) u 1 = z n1 +x (n) + α n1 c n z n ˆρ (r,t)z r (u)r â j Y j (1) where ˆρ(r,t) an â j (for i = 1,...,N) are aaptation function an parameters which are upate by: â j = k j Y j z n for j = 1,...,N (1) ˆβ = k β u 1 z n (17) t ˆρ (r,t) = k ρ z r (u)z n (1) In the above equation k i >,k β > an k ρ > are arbitrary aaptation gains. In the following theorem, the summary of the esigne controller is expresse an the stability proof is given. Theorem 1. Consier a class of saturate nonlinear systems, escribe by equation (1). If assumption 1 is hol, applying control laws (1) an (1) an the aaptation laws (1)-(1) guarantees the stability of the close loop system. Moreover lim z i (19) t Hence, the output signal x 1 tracks the esire reference signal x. Proof. Accoring to equations (1) an (1), one can achieve the following relations: z 1 = z +α 1 z = z 3 +α α 1. () ż n1 = z n +α n1 α n z n = ẋ n x (n) α n1 Defining the aaptation parametere errors as: ã i = a i â i fori = 1,...,N β = β ˆβ ρ (r,t) = ρ (r) ˆρ (r,t) an introucing the following Lyapunov function: V = 1 n z 1 i + ã j + 1 ρ k j k (r,t)r ρ + ρ k β β (1) () The time erivative of V along the trajectories of system () is V = (z +α 1 )+z (z 3 +α α 1 )+... +z n1 (z n +α n1 α n ) +z n ( a j Y j +ρ u ρ (r)z r (u)r x (n) α n1 ) ρ k β ˆβ β 1 âj ã j 1 ˆρ k j k (r,t) ρ (r,t)r ρ Substituting (13) into (3) results in: n1 V = c i zi +z n ( a j Y j +ρ u ρ (r)z r (u)r x (n) α n1 ) 1 âj ã j 1 ˆρ k j k (r,t) ρ (r,t)r ρ bρ ˆβ β k β applying the control law (1) into (): (3) () 191
Milano (Italy) August - September, 11 n V = c i zi 1 ã j ( â j k j Y j z n ) k j ρ k β β( ˆβ +kβ u 1 z n ) 1 k ρ ρ (r,t)( ˆρ (r,t)k ρ z r (u)z n )r () factor an m is the mass value. In these set of simulations, the system parameters are slecete as: m = 1. kg, c = Ns m, k = N m an w = sat(u) is characterize as : { 1 u 1 w = u 1 u 1 (31) 1 u 1 Finally, using the aaptation laws (1)-(1) leas to: n V = c i zi () Hence the close loop system is stable. Moreover Barbalat slemma(khalil[])statesthatthetrajectoriesof z i will asymptotically converge to the origin as time tens to infinity. Consequently the output signal x 1 tracks the esire reference signal x. Remark 3. Base on implicit function theorem (Khalil []), the control law can be calculate from equations (1) an (1) as: u = ˆβ( z n1 +x (n) + α n1 c n z n ˆρ (r,t)z r (u)r ) â j Y j (7) Remark. To implement the control law, the integral term is approximate in the following form: ˆρ (r,t)z r (u)r in which r is a step size an M ˆρ (i r,t) r () M = R r (9) The parameter r plays an important role in ajusting a balance between estimation accuracy an computation complexity. Choosing a larger value for r leas to a coarser estimation an a higher level of computation complexity. On the contrary, a smaller value, results in a more accurate estimation, while more computation is neee.. SIMULATION RESULTS In this section, the performance of the propose control metho is illustrate by simulation results. The propose controller is applie to a n orer Spring-Mass-Damper system which is escribe by(zhou an Wen []): ẋ 1 = x ẋ = k m x 1 c m x + 1 m w (3) y = x 1 where, x 1 an x are the position an velocity of the mass, respectively, k is the spring constant, c is the amping Our propose controller an conventional clampe aaptive controller are simulate an compare to represent the efficiency of the propose metho. In the conventional aaptive clampe controller, the saturation phenomenon is ignore in the controller esign process. As a result, it woul be a convetional aaptive controller, the output of which is clampe by a saturation phenomenon, before being exerte to the system. 1- The propose saturation moel base aaptive controller: To implement the controller, the control law is calculate from the following equation: u = ˆβ( +ẍ + α 1 c z â 1 x 1 â x u sc ) u sc = 1 ˆρ (r,t)z r (u)r 1 (3) ˆρ (i r,t) r (33) where R = 1 an r =.1. Moreover, the aaptation laws are: â 1 =.x 1 z (3) â =.x z (3) ˆβ = u 1 z (3) t ˆρ (r,t) =.1z r (u)z (37) The reference signal is selecte as x = sin. c 1 an c are set to 1 an to have a better convergence, these parameters are swiche to 1 at time instant t = s. To show the robustness of our propose metho against initial coniton changes, the initial conitions are set to x 1 () = x () =. Simulation results are epicte in Figs. an. It is obvious from Fig. that the output tracking error tens to zero in the course of time. 1 1 3 3 Fig.. System output tracking error, while x = sin an the propose controller is applie. In aition, to investigate the performance of propose controller in tracking a wie range of reference signals, the 19
1 1 3 3 Preprints of the 1th IFAC Worl Congress Milano (Italy) August - September, 11 1 1 w 1 1 1 1 3 3 Fig.. Saturate control signal, while x = sin an the propose controller is applie. simulation is carrie out once again,while x = sin + cos(t) an all the other parameters are ajuste similar to the previous simulation. Figs.7 an show that, like the preceing simulation, the system output tracking error converges to zero properly an the saturate input (i.e.w) is a smooth signal. Fig. 9. System output tracking error, while x = sin an the conventional controller is applie. w 1 1 1 1 1 1 3 3 Fig. 1. Saturate control signal, while x = sin an the conventional aaptive controller is applie. 1 1 3 3 Fig. 7. System output tracking error, while x = sin+ cos(t) an the propose controller is applie. 1 1 1 1 3 w 1 1 1 1 3 3 Fig. 11. System output tracking error, while x = sin+ cos(t) an the conventional aaptive controller is applie. 1 Fig..Saturatecontrolsignal,whilex = sin+cos(t) an the propose controller is applie. - The conventional clampe aaptive controller: In this part, an aaptive controller is esigne, while the input saturation phenomenon is ignore in the system equation (3), which leas to a conventional aaptive controller. The only iffernce between this part an the previous one is that, here, the correction term ue to propose esign, as in equation (3), has been set to zero: u sc = As the preceing part, simulations are carrie out for two mentione reference signals. All the parameters are selecte similar to part 1. SimulationresultsareshowninFigs.9-1.Itisclear,while the conventional aaptive controlle is applie, the output signal is not able to track the esire signal properly. Furthermore, the saturate input signal (i.e.w) switches with a high frequency, which enangers the actuator. w 1 1 1 1 1 3 Fig. 1. Saturate control signal, while x = sin + cos(t) an the conventional aaptive controller is applie.. CONCLUSION In this paper, a new ea zone base moel of saturation was propose to moel a wie range of saturation phenomenon. Due to use of a linear in parameter approach, themaincharacteristicofthismoelisitsconsistencywith the available aaptive control techniques. Moreover, using 193
Milano (Italy) August - September, 11 the propose saturation moel, an aaptive backstepping base controller was esigne for a class of nonlinear saturate systems. Finally, some simulations were carrie out on a Spring-Mass-Damper system to show the efficiency of our proposr metho against reference signal an initial conition change. REFERENCES F. Z. Chaoui, F. Giri, L. Dugar, J. M. Dion, an M. Msaa. Aaptive tracking with saturating input an controller integral action. IEEE Trans. Automat. Contr., 3:13 13, 199. F. Z. Chaoui, F. Giri, an M. MSaa. Asymptotic stabilization of linear plants in the presence of input an output saturations. Automatica, 37:37, 1. T. Fliegner, H. Logemann, an E. P. Ryan. Low-gain integralcontrolofcontinuoustimelinearsystemssubject to input an output nonlinearities. Automatica, 39:, 3. W.GaoanR.R.Selmic. Neuralnetworkcontrolofaclass of nonlinear systems with actuator saturation. IEEE Trans. Neural Netw., 17:17 1,. C.Q. Huang, X.F. Peng, C.Z. Jia, an J.D. Huang. Guarantee robustness/performance aaptive control with limite torque for robot manipulators. Mechatronics, 1:1,. S. Jagannathan an M. Hamee. Aaptive forcebanlancing control of mems gyroscope with actuator limits. Proceeings of American Control Conference, pages 1 17,. J.O.Jang. Neuro-fuzzynetworkssaturationcompensation of c motor systems. Mechatronics, 19:9 3, 9. H.K.Khalil. Nonlinear Systems. Prentice-Hall,Englewoo Cliffs, NJ,. C. Y. Su, Q. Wang, X. Chen, an S. Rakheja. Robust aaptivecontrolofaclassofnonlinearsystemsincluing actuator hysteresis with prantlishlinskii presentations. Automatica, :9 7,. J. J. Yan, J. S. Lin, an T. L. Liao. Robust ynamic compensator for a class oftime elaysystems containing saturating control input. Chaos, Solitons an Fractals, 31:13 131, 7. Y.S. Zhong. Globally stable aaptive system esign for minimum phase siso plants with input saturation. Automatica 1 () 139 17, 1:139 17,. J. Zhou an C. Wen. Robust aaptive control of uncertain nonlinear systems in the presence of input saturation. Proceeings of 1th IFAC Symposium on System Ientification,. J. Zhou an C. Wen. Aaptive Backstepping Control of Uncertain Systems. Nonsmooth Nonlinearities, Interactions or Time-Variations. Springer-Verlag, Berlin Heielberg,. 19