Intelligent Control an Automation, 5, 6, -9 Publishe Online February 5 in SciRes. http://www.scirp.org/journal/ica http://x.oi.org/.436/ica.5.6 Robust Aaptive Control for a Class of Systems with Deazone Nonlinearity Nizar J. Ahma *, Mahmu J. Alnaser, Ebraheem Sultan, Khulou A. Alheni Faculty of Electronic Engineering echnology, College of echnological Stuies, he Public Authority for Applie Eucation an raining (PAAE), Kuwait City, Kuwait Email: * nj.ahma@paaet.eu.kw Receive 5 November 4; accepte 9 December 4; publishe 3 January 5 Copyright 5 by authors an Scientific Research Publishing Inc. his work is license uner the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4./ Abstract his paper presents a robust aaptive control scheme for a class of continuous-time linear systems with unknown non-smooth asymmetrical eazone nonlinearity at the input of the plant. he methoology is applie to hanle input eazone as well as unmeasurable isturbances simultaneously in strictly matche systems. he propose controller robustly cancels any resiual istortion cause by the inaccurate eazone cancellation scheme. he scheme is shown to successfully cancel the eazone s eleterious effect as well as eliminate other unmeasurable isturbances within the span of the input. he new controller ensures the global stability of all states an aaptations, an achieves asymptotic tracking. he asymptotic stability of the close-loop system is proven by Lyapunov arguments, an simulation results confirm the efficacy of the control methoology. Keywors Aaptive Control, Non-Symmetric Deazone, Har Nonlinearity. Introuction he significance of the eazone problem lies in the fact that it affects many physical an practical systems. Examples of such systems are the ones containing hyraulic or pneumatic values, electronic circuits an evices, temperature regulation circuits, an in actuators such as servo valves an DC motors. In most cases, the parameters of the eazone nonlinearity are unknown an continuously varying with time an temperature. Deazones exist in a number of inustrial applications specially the ones requiring high precision such as meical robots, semiconuctor manufacturing, an precision machine tools. It has been shown that eazone in actuators, * Corresponing author. How to cite this paper: Ahma, N.J., et al. (5) Robust Aaptive Control for a Class of Systems with Deazone Nonlinearity. Intelligent Control an Automation, 6, -9. http://x.oi.org/.436/ica.5.6
N. J. Ahma et al. such as hyraulic servo-valves, gives rise to limit cycling an instability. he avances reache in the area of aaptive compensation an control theory gave rise to increase interest in hanling the eazone problem. here have been many techniques that aresse the problem an have been shown to reuce if not eliminate the egraation of system performance resulting in an improve tracking accuracy an ensure stabilization of such systems. One sensible approach to counter the effect of the eazone, shown in Figure, was presente by [] which involve esigning an inverse eazone function to cancel its effect. he approach of esigning an inverse aaptive eazone compensator was thoroughly investigate in [] an [3] which was shown to improve performance. Lewis et al. in [4] propose a fuzzy logic type inverse eazone compensator, meanwhile, a neural network inverse compensator was esigne in [5]. Both approaches show clear improvement in reucing the tracking error. In [6], a new aaptive controller of linear or nonlinear systems with eazone is introuce without constructing a eazone inverse. Global an asymptotic tracking was achieve an simulation results were presente. An aaptive sliing moe control scheme use to offset a non-symmetrical eazone nonlinearity in continuous time was presente in [7]. he problem of chattering inherent with sliing moe control is hanle by allowing a small controlle tracking error. In recent years, many researchers aresse the eazone problem with encouraging results. In [8], a novel function was introuce to escribe eazone nonlinearity. o show the effectiveness of the propose equivalent function the authors combine it with vibration of a cantilever beam. In this paper we are motivate by the success of our earlier results eazone compensation of DC motor presente in [9] an []. he extension involves combining the eazone for a class of linear systems with uncertainties in the span of the input. he uncertainties are assume to be boune by a p th orer polynomial in the state of the system. A robust aaptive controller will compensate for the unmeasurable isturbances as well as any mismatch error in estimating the eazone parameters. he propose metho oes not require any knowlege of the eazone parameters or the specialize esign of an inverse eazone controller an only an upper boun of the eazone spacing which is easily etermine a priori.. he Problem Setup: Dynamics of a Non-Symmetrical Deazone Nonlinearity A common representation of a non-symmetrical eazone nonlinearity, shown in Figure, escribe in [] as follows mu r, if u> r DZ ( u ( t) ) =, if l < u < r () mu ( + l), if u< l where DZ ( u ( t )) enotes the output of eazone function preceing a plant input, m is the slope of the lines, ( r l) is the with of the eazone istance, an u( t ) is the input of the eazone block as shown in Figure. A more convenient representation of a non-symmetrical eazone was presente in [6] as where sat DZ u = u sat u () u represents a non-symmetrical saturation function given by Figure. Non-symmetric eazone nonlinearity at the input of a linear plant.
N. J. Ahma et al. r if u > r sat ( u) = u, if l < u < r l, if u < l A novel representation for () was presente in [] for an exact symmetrical eazone by efining = as the eazone spacing function as r l ( ) Consequently, the saturation function can be written as DZ u = m u + u u + (4) sat u = m u u + (5) In most applications the eazone parameters are unknown or time an temperature varying. Instea, for eveloping an inverse eazone function as in [9], we avocate a robust aaptive compensator that hanles the variability of the eazone parameters as part of an input isturbance. However, we outline some necessary an reasonable assumptions to be use in the proof of the efficacy of the propose control methoology: (A) he eazone parameters r > an l >. (A) he eazone parameters, an are boune as follows: r l [ min, max] an [ min, max] l l l. r r r (A3) Without any loss of generality, the slope of the eazone m is positive an is set to. (A4) he output of the eazone block DZ ( u ) is not available for measurement. Remark. Assumptions (A) an (A) are the actual physical attributes of a real inustrial eazone an is aopte in the literature [6]. herefore, the saturation function given by (4) can be shown to have an upper boun by closely analysing Equation (5). Using the general inequality rule a± b < a + b we have From Equations (6) from (7) we can state Multiplying by the slope m gives u u + (6) u+ u + (7) ( u u+ ) (8) m ( u u+ ) (9) he left han sie of (9) is the saturation function given by (5). In case of a non-symmetrical eazone function the upper-boun may be chosen by employing assumption (A) as follows where sat u () M m M = ( rmax lmin ) () he upper bouns will play a pivotal role to ensure the overall global stability of the close loop ynamics as will be emonstrate in the following section. (3) 3. Robust Aaptive Controller Design Consiering the following nonlinear system with input eazone nonlinearity escribe as y = Cx, { } x = Ax + B DZ u +ψ x ()
N. J. Ahma et al. where the matrices A an B are given by, A= B = An ψ ( x) represents the unmeasurable isturbance. he collective bouns can be expresse as ψ ( x) Let the reference moel to be tracke given by where lows: K p k k = k γ x (3) { } x = Ax + B Kx + r (4) m m m xn R an r is a reference signal. he tracking error ynamics x = x xm Inserting the eazone equation () into (5) yiels { ψ m } may be written as fol- x = Ax + B DZ u + x Kx r (5) { ψ m } x = Ax + B u sat u + x Kx r (6) herefore, for the class of systems escribe in (3) an eazone given in (7), we use the result state as Lemma RANDM in [] an moify it to ensure asymptotic convergence. he moifie controller is = + + ˆ + tanh ( + ) α β ρ (7) u t Kxm r B Px B Px a bt B Px where α, a, b >, ρ M, an P is the positive efinite symmetric solution of the Algebraic Riccati equation (ARE). he aaptation ˆβ is use to ensure robustness of the controller. he combine output of the compensator an the eazone nonlinearity may be written as = = + + ˆ + tanh ( + ) α β ρ (8) DZ u u sat u sat u Kxm r B Px B Px a bt B Px Inserting the propose control laws (7) an the output of the eazone block given by Equation (8) into the error ynamics (6) results in the close loop ynamics he aaptation law for ˆβ given by { ˆ α β ρtanh ψ } x = Ax + B sat u B Px B Px + a + bt B Px + x (9) ˆ β = ΓB Px () where Γ> is a constant scalar. heorem. For the plant escribe by () with input eazone (), an the robust aaptive control law (7) along with the aaptive upate law () will ensure the close-loop stability an bouneness of tracking error, hence reucing the effects of eazone. Proof. Using the following positive efinite control Lyapunov function Γ V = x Px + β () where ˆ β = β + B ifferentiating along the trajectories of the system yiels V = x Px + x Px + Γ ββˆ () ( Ax B{ DZ ( u) ( x) }) Px x P Ax B{ DZ ( u) ( x) } = + + ψ + + + ψ + Γ ββˆ (3) Substituting for the close loop ynamics given by (9) in (3) gives 3
N. J. Ahma et al. ( { ˆ α β ρtanh ψ }) x P Ax B{ sat ( u ) ˆ B Px B Px ( a bt) B Px ( x) } V = Ax + B sat u B Px B Px + a + bt B Px + x Px ( ) + + α β + ρtanh + + ψ + Γ ββˆ. Collecting terms an simplifying Rearranging terms we get { α ˆ β ψ } + x PB{ sat ( u ) + ρ ( a + bt) B Px } + V = x A P + PA x + x PB B Px B Px + x ˆ tanh Γ ββ. ( α ) ˆ β ψ + x PB{ sat ( u ) + ρ ( a + bt) B Px } + = + + V x A P PA PBB P x B Pxx PB x PB x tanh Γ ββˆ. he first term can be simplifie by solving the Algebraic Reccati Equation given by resulting in A P PA PBB P Q (4) (5) (6) + α = (7) ( ) V ˆ ˆ = x Qx βx PB + x PB ψ x + Γ ββ x PB sat ( u ) ρtanh ( a + bt) B Px (8) Replacing the aaptation law () an replacing ˆβ = β + β in (8) yiels ( β β ) β ψ ( ( ) ρtanh ) V = x Qx β x PB + x PB ( ψ ( x) ) x PB ( sat ( u ) tanh ρ a + bt B Px ) (9) V = x Qx + x PB + x PB + x PB x x PB sat u a + bt B Px (3) So far the first two terms are negative. As for the thir term we utilize the general inequality to establish proper bouns as follows Using the inequality (3) to moify (3) to become ab a + b x PB ψ x ς x PB + ς ψ x (3) ς x PB + ς ψ x ς x PB + ς γ x (3) herefore, the inequality of (3) can be incorporate in (3) as ( λmin ς γ) β ς ( ) ρtanh + + (33) V Q x x PB x PB x PB sat u a bt B Px λ By choosing the egree of freeom ς satisfying the conition min Q ς < an choosing β to be greatγ er than ς ensures that the first three terms of V negative. Meanwhile, the last term in (33) can be upper boune as follow ( ( ) ρ { }) ( ) ρ ( ( { })) x PB sat u tanh a + bt B Px x PB sat u tanh a + bt x PB (34) Utilizing the upper bouns on sat ( u ) < given by () an rewriting the right han sie of (34) M ( ) x PB ρ δ tanh a + bt x PB (35) where δ = ρ <.herefore the last term in (35) insures that V as long as M δ tanh a + bt x PB > (36) 4
N. J. Ahma et al. or Π ln + δ x PB > t (37) + δ ( a bt) o conclue, by choosing vanishing region as time increases. herefore, since ρ > M then V is renere negative an x PB as t an x PB converges to a close an x PB = cx + cx + + cnxn = implies that by choosing the c vector as the coefficient of a strictly Hurwitz polynomial will make the close loop system error asymptotically stable. For a more through conclusion of the proof one may refer to [3]. 4. Illustrative Example In this section, we illustrate the propose controller to compensate for a system with a eazone nonlinearity presente in [4] as x ( x ( + ( ) sin.5 sin ( 3 ) x = a a x + x x a x t + DZ u (38) 3 he parameter use for the simulation is shown in able. he plant (4) may be written in state space form by efining x = x an x = x then x = x (39) x ( x ( + ( ) sin.5 sin ( 3 ) x = a a x + x x a x t + DZ u 3 Resulting in A=, B = he solution of the ARE equation was chosen to be 3.7.36 P =, Q =.36 4.55 4 able. Parameters utilize in the example. Item Systems Physical Attributes Parameter Value Unit k p 4. PD Gain k v 3. PD gain 3 r. Deazone Right 4 l 5. Deazone Left 6 Γ. Gains 7 b, a, a, a 3. Scalars 9 ρ 4. Gain α. Gain a, b. Scalars 5
N. J. Ahma et al. he reference moel to be tracke is for a sinusoial reference trajectory given by x m xm = r t kp k + v xm r t t t = + sin ( ω ) + sin ( 5ω ) Meanwhile, the unmeasurable isturbance ψ ( x) can be collectively boune as x ( x ( + ( ) sin.5 sin ( 3 ) ψ x = a a x + x x ax t 3 p γ k k = x k Simulations of the system in (39) uner the aaptive control law (7) an () have been performe. he upper bouns on actuator actual spacing M = r l = 35 is assume unknown. In orer to emonstrate the performance improvement accomplishe by our propose metho, the system uner test given in (39) was use. he efficacy of the propose metho is proven by comparing its performance against the performance of a classic PD controller having equivalent gains. he complete parameters of the system uner test an controller gains are liste in able. he simulation results presente in Figure an Figure 3, clearly show the tracking performance for x an x states along with their respective reference trajectory. Figure 4 an Figure 5 emonstrate the tracking error for the states x an x in blue in aition to the same tracking errors for the system uner a PD controller in re. In both figures, the PD controller resulte in limit cycles where as the aaptive controller prove to be stable with no limit cycles an improve performance with a zero approaching tracking error. Figure 6 shows the control effort u ( t ) an DZ ( u ). In Figure 7 the evolution of the ˆβ which clearly emonstrates the bouneness of the aaptation. In Figure 8, the reference trajectory is change to emonstrates a superior a step response performance when compare with PD controller. While the step error is approaching zero for the system uner the propose aaptive controller, a steay state error is persistent with the PD controller. racking Performance (4) 5 4 Reference Signal r(t) r(t) Refernce X()-State Raians 3 X_ - X state - -3.5.5.5 3 ime (Secons) Figure. he tracking performance of x state of the system uner the propose robust controller. 6
N. J. Ahma et al. 5 racking Performance r(t) Refernce X()-State Ra/S 5 X() 5-5 - -5.5.5 ime (Secons) Figure 3. he tracking performance of x state of the system. 4 racking Performance PD Error X()-X()m error Raians racking Error 3 - - 3 4 5 ime (Secons) Figure 4. he aaptively compensate tracking error x (blue) vs. the same tracking error of the system uner a PD controller (re). 8 6 4 racking Performance PD Error X()-X()m error racking Error - -4-6 -8 - - -4 3 4 5 ime (Secons) Figure 5. he aaptively compensate tracking error x (blue) vs. the same tracking error of the system uner a PD controller (re). 7
N. J. Ahma et al. Control Signals 5 Control v(t) DZ(v) Output 5 N-m -5 - -5 -.5.5 Figure 6. he control effort ime (Secons) DZ u in re vs. v = u for the eazone compensate system. 5 Evolution of Aaptation Aaptation Beta 5 5 3 4 5 Figure 7. Evolution of the aaptation ˆβ. ime (Secons) 5 4 Step racking Error PD Aaptive Gain Unit Step Error 3 -.5.5.5 3 ime (Secons) Figure 8. Step tracking performance error for x for input r( t ) =. 8
N. J. Ahma et al. 5. Conclusion A robust aaptive controller is use to control a class of nonlinear systems with input eazone nonlinearity at the input. he robust controller was shown to be superior in performance when compare to a more conventional control metho such as a PD controller. he system uner the propose scheme has been shown to not only effectively stabilize a secon orer complex nonlinear system with isturbance, but also achieve asymptotic tracking. he avantage of the propose metho lies in the fact that no knowlege of the eazone parameters neee an only an upper boun for the eazone spacing is require. he aaptive eazone inverse controller is smoothly ifferentiable an can easily be combine with any of the avance control methoologies. he asymptotic stability of the close-loop system has been proven by using Lyapunov arguments an simulation results confirme the efficacy of the control methoology. Funing his work is supporte an fune by the Public Authority of Applie Eucation an raining, Research Project No. (S-4-3) t, Research itle (Aaptive Control of Systems with Output Deazone). References [] Recker, D.A. an Kokotovic, P.V. (993) Inirect Aaptive Nonlinear Control of Discrete-ime Systems Containing a Deazone. Proceeings of the 3n Conference on Decision an Control, 5-7 December 993, San Antonio, 647, 653. [] oa, G. an Kokotovic, P. (994) Aaptive Control of Plants with Unknown Dea-Zones. IEEE ransactions on Automatic Control, 39, 59-68. http://x.oi.org/.9/9.73339 [3] ao, G. an Kokotovic, P. (995) Discrete-ime Aaptive Control of Systems with Unknown Dea-Zones. International Journal of Control, 6, -7. http://x.oi.org/.8/779589889 [4] Lewis, F.L., im, W.K., Wang, L.-Z. an Li, Z.X. (999) Deazone Compensation in Motion Control System Using Aaptive Fuzzy Logic Control. IEEE ransactions on Control Systems echnology, 7, 73-74. http://x.oi.org/.9/87.799674 [5] Selmic, R.R. an Lewis, F.L. () Deazone Compensation in Motion Control Systems Using Neural Networks. IEEE ransaction of Automatic Control, 45, 6-63. [6] Wang, X.-S., Su, C.-Y. an Hong, H. (4) Robust Aaptive Control of a Class of Nonlinear Systems with Unknown Dea-Zone. Automatica, 4, 47-43. [7] Zhonghua, W., Bo, Y., Lin, C. an Shusheng, Z. (6) Robust Aaptive Deazone Compensation of DC Servo System. IEE Proc.-Control heory Application, 53, 79-73. [8] Seighi, H.M., Shirazi, K.H. an Zare, J. () Novel Equivelent Function for Deazone Nonlinearity: Applie to Analytical Solution of Beam Vibration Using He s Parameter Expaning Metho. Latin American Journal of Solis an Structures, 9, -. [9] Ahma, N.J., Alnaser, M.J. an Alsharhan, W.E. (3) Asymptotic racking of Systems with Non-Symmetrical Input Deazone Nonlinearity. International Journal of Automation an Power Engineering,, 87-9. [] Ahma, N.J., Ebraheem, H., Alnaser, M. an Alostath, J. () Aaptive Control of a DC Motor with Uncertain Deazone Nonlinearity at the Input. Control an Decision Conference (CCDC), 495-499. [] Seighi, H.M., Shirazi, K.H. an Zare, J. () Novel Equivelent Function for Deazone Nonlinearit: Apple to Analytical Solution of Beam Vibration Using He s Parameter Expaning Metho. Latin American Journal of Solis an Structures, 9, 443-45. http://x.oi.org/.59/s679-7854 [] Jain, S. an Khorrami, F. (995) Robust Aaptive Control of a Class of Nonlinear Systems: State an Output Feeback. Proceeing of the 995 American Control Conference, Seatle, 58-584. [3] Sankaranarayanan, S., Melkote, H. an Khorrami, F. (999) Aaptive Variable Structure Control of a Class of Nonlinear Systems with Nonvanishing Perturbations via Backstepping. Proceeings of the American Control Conference, San Diego, 449-4495. [4] Zhang,.P. an Feng, C.B. (997) Aaptive Fuzzy Sliing Moe Control for a Class of Nonlinear Systems. ACA Automatica Sinica, 3, 36-369. 9