Exponential Tracking Control of Nonlinear Systems with Actuator Nonlinearity

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1 Preprints of the 9th Worl Congress The International Feeration of Automatic Control Cape Town, South Africa. August -9, Exponential Tracking Control of Nonlinear Systems with Actuator Nonlinearity Zhengqiang Zhang Ruihua Wang Hanyong Shao School of Electrical Engineering an Automation, Qufu Normal University, Rizhao 7686, China School of Automation, Southeast University, Nanjing 96, China School of Electrical Engineering an Automation, Qufu Normal University, Rizhao 7686, China Abstract: In this paper, the problem of aaptive tracking control is aresse for a class of nonlinear systems with parametric uncertainty, unknown actuator nonlinearity an isturbance. Two type of actuator nonlinearities, that is, backlash-like hysteresis an symmetric eazone, are consiere simultaneously. Without constructing the inverse function of actuator nonlinearity, unifie control framework is establishe. An aaptive control scheme, capable of guaranteeing the exponential tracking with zero tracking error, is propose. Two simulation examples are provie to clarify an verify the propose approach. Keywors: Aaptive control, actuator nonlinearity, hysteresis, ea zone, tracking control.. INTRODUCTION Generally speaking, there are two esign methos to eliminate the effects of actuator nonlinearities. One is to construct the inverse function of actuator nonlinearity see, e.g. Tao an Kokotovic 99, 995); Zhou et al. 6, 7, )), an the other is to irectly esign robust aaptive controller without the inverse problem see, e.g. Chen et al. 8, ); Su et al. ); Zhou et al. ); Su et al. 3); Wen an Zhou 7); Su et al. 5); Wang an Su 6); Wang et al. ); Ibrir et al. 7); Hua et al. 8); Hua an Ding )). In the above two research irections, consierable efforts have been mae to mitigate the influences of actuator nonlinearity. For example, in orer to compensating for actuator nonlinearity, pioneering work was one in Tao an Kokotovic 99, 995, 996), where aaptive inverse methos were evelope. In Zhou et al. 6, 7, ), smooth inverse functions of ea-zone, backlash nonlinearity an Bouc-Wen hysteresis were respectively introuce an the aaptive backstepping output-feeback control schemes were propose. In orer to avoi the irect inversion of the hysteresis moel, in Chen et al. 8, ), the concept of implicit inversion was introuce. For iscrete an continuous linear systems with Prantl-Ishlinskii hysteresis, the approximate an perfect implicit inversions were respectively incorporate into aaptive control esigns. This work was supporte by the National Natural Science Founation of China uner Grants 67, 6739, 6379 an 6733, China Postoctoral Science Founation fune project uner Grant M565, Shanong Postoctoral Science Founation fune project uner Grant 33, the Founation of Key Laboratory of System Control an Information Processing, Ministry of Eucation, China, the Project supporte by the Young Core Instructor an Domestic Visitor Founation from the Eucation Department of Shanong Province, China an the Taishan Scholarship Project of Shanong Province, China. In Su et al. ), Zhou et al. ), Su et al. 3) an Wen an Zhou 7), the backlash nonlinearity was approximate by a continuous ifferential equation moel, which was calle as backlash-like hysteresis. Instea of constructing a hysteresis inverse, robust aaptive controllers were irectly esigne to minimize the effects of the hysteresis nonlinearity. Motivate by Su et al. ) an Zhou et al. ), robust aaptive control techniques were applie to eal with Prantl-Ishlinskii type hysteresis Su et al., 5; Wang an Su, 6). In Wang et al. ), Ibrir et al. 7), Hua et al. 8) an Hua an Ding ), symmetric an non-symmetric ea-zones were investigate, respectively. New escriptions on the ea-zone moels, that is, the combination of linear input function with constant or time-varying coefficients an a boune time-varying function, were introuce, an new control strategies were propose. Despite the great process in the control of ynamical systems with actuator nonlinearity, some challenging problems still remain. One of the main rawbacks in the current literature is that most of the propose robust aaptive tracking controllers o not prouce asymptotic tracking. Instea, the so-calle boune-error trajectory tracking is achieve see, e.g. Tao an Kokotovic 99, 995); Zhou et al. 6, 7); Chen et al. 8); Su et al. ); Wen an Zhou 7); Su et al. 5); Wang an Su 6); Wang et al. ); Ibrir et al. 7); Hua et al. 8); Hua an Ding )). It is note that asymptotic tracking was obtaine in Zhou et al. ) an Zhou et al. ) see Scheme in Zhou et al. )). However, it is worth pointing out that aitional cost ha to be pai. In Zhou et al. ), by constructing a new inverse function of the hysteresis, the tracking error was prove to converge to zero asymptotically. However, the parameters in the hysteresis moel were require to be known. In Copyright IFAC 8

2 9th IFAC Worl Congress Cape Town, South Africa. August -9, ut) vt) Fig.. Backlash-like hysteresis Zhou et al. ), two aaptive control schemes were propose by employing the backstepping approach. In the first scheme, asymptotic tracking was achieve. However, a iscontinuous sign function was introuce an the chattering phenomenon may occur. In the secon scheme, a smooth control scheme was presente by efining a novel ifferentiable function. However, perfect tracking coul not be ensure. It is well known that exacting output tracking has consierable theoretical an practical significance. Therefore, it is highly esirable to esign new aaptive compensation control scheme with zero tracking error an without the aforementione cost.. System moel. PROBLEM FORMULATION Consier a class of uncertain nonlinear systems with actuator nonlinearity x n) t) + a i Y i xt), ẋt),..., x n ) t)) = but) + t), ut) = Nvt)), ) where plant parameters a i are unknown constants, Y i are known smooth functions, control gain b is unknown constant, t) enotes boune external isturbance, N ) represents an actuator uncertainty, vt) is the applie control, an ut) is not available for measurement. In this paper, two type of actuator nonlinearity characteristics are consiere. Backlash-like hysteresis Su et al., ): u = α v cv u) + B v, ) where α, c an B are constants, an c > B. Fig. illustrates the backlash-like hysteresis ynamics escribe by ), where α =, c = 3.635, B =.35, vt) = 6.5 sin.3t), u) =. Symmetric ea-zone Tao an Kokotovic, 99): u = { mv br ), v b r,, b l < v < b r, mv b l ), v b l, 3) where m is the slope of the ea-zone characteristic, b r an b l represent the break points. Before proceeing further, we introuce two useful lemmas for the actuator nonlinearity moels. Lemma Su et al., ): The solution of ) is ut) = cvt) + v), ) αv v))sgn v) v) = u) cv)e αvsgn v)) +e an v) is boune. v v) B c)e αζsgn v) ζ, 5) Lemma Wang et al. ): The ea-zone moel 3) can be represente as an v) is boune. ut) = mvt) + v), 6) { mbr, v b r, v) = mv, b l < v < b r, 7) mb l, v b l, The control objective is to esign a control law vt) in ) such that all the close-loop signals are boune, while the system state vector X = x, ẋ,..., x n ) T exponentially tracks a specifie esire trajectory X = x, ẋ,..., x n ) T with zero tracking error, where x t) is a given reference signal. To this en, we make the following assumptions. Assumption : The reference signal x t) an its first n erivatives are known an boune. Assumption : The parameters of actuator nonlinearity such as α, c, B in the Backlash-like hysteresis an m, b r, b l in the ea-zone moel are unknown. Moreover, the uncertain parameters b, c, m are such that bc >, bm >. 3. ADAPTIVE CONTROL DESIGN Now, substituting ) or 6) into ) yiels where or x n) t) + a i Y i xt), ẋt),..., x n ) t)) = ρvt) + Dt), 8) ρ = bc, Dt) = b v) + t), 9) ρ = bm, Dt) = b v) + t). ) From Assumption, it follows that ρ is unknown but ρ >. Applying Lemmas - together with the bouneness of t), we know that Dt) is boune. Let the tracking error that is, e = e, e,..., e n T, e = X X, ) 9

3 9th IFAC Worl Congress Cape Town, South Africa. August -9, e = x x, e = ẋ ẋ,, e n = x n ) x n ). ) From 8) an ), the ynamics of the tracking error is governe by ė = Ae + B where a i Y i X) + ρvt) + Dt) x n),3) A =......, B =.. ) It is observe that A, B) is controllable. So is A+σ I, B), where σ > is a esign parameter. Thus, there exists a constant matrix k satisfying that A + σ I) + Bk is stable an hence P = P T > exists such that A + σ I) + Bk T P + P A + σ I) + Bk = Q, 5) where Q = Q T > is a given matrix, that is, A + Bk) T P + P A + Bk) = Q σ P < σ P.6) Before aaptive control scheme is presente, we efine some variables use in what follows: fx, t) = = sup t Yi X) + h i + et) + h +,7) Dt) + x n), 8) θ = max { a,..., a r, k, }, 9) θ = θ ρ, ) where h, h i, i =,,..., r, are positive esign constants, enotes the Eucliean norm of a vector. By Assumption an the bouneness of Dt), we know that oes exist. Then, the control law an parameter upate law are esigne as, respectively, vt) = e T P B ˆθ t)f X, t) e T P B tanh l e T P B expσ t) ˆθt)f + l expσ t), ) ˆθ = γ expσ t) e T P B fx, t), ) where l, σ are positive esign constants, γ > is aaptive gain, σ satisfies σ > σ, ˆθt) is the estimate of θ, ˆθ). Theorem : Consier the close-loop system consisting of the system ) with actuator nonlinearity ) or 3), control law ) an aaptive law ) base on Assumptions -. Then, all the close-loop signals remain boune, an the tracking error converges to zero exponentially with the rate of not less than σ. Proof: We first rewrite 3) as ė = A + Bk)e +B a i Y i X) + ρvt) + Dt) x n) Define a positive Lyapunov function with the estimation error ke 3). V = e T P e + ρ γ expσ t) θ t) ) θt) = θ ˆθt). 5) In view of 3)-5), the erivative of V is V = e T P A + Bk) + A + Bk) T P e σ ρ expσ t) θ ρ γ γ expσ t) θ ˆθ + e T P B a i Y i X) + ρvt) + Dt) x n) ke. 6) Substituting 6) an ) into 6) an noting ), we have V σ e T P e σ ρ expσ t) θ ρ θ e T P B fx, t) γ +e T P B a i Y i X) + Dt) x n) ke +ρe T P Bvt) = σ V ρ θ e T P B fx, t) + ρe T P Bvt) +e T P B a i Y i X) + Dt) x n) ke. 7) Noting the efinitions in 7)-), we have e T P B a i Y i X) + Dt) x n) ke ρ e T P B θ fx, t). 8) Combining 5), 7) an 8) implies that V σ V ρ θ e T P B fx, t) + ρe T P Bvt) +ρ e T P B θ fx, t) = σ V + ρe T P Bvt) + ρ e T P B ˆθfX, t). 9) Then, substituting ) into 9) results in V σ V + ρ e T P B ˆθfX, t) ρe T P B) ˆθ f X, t) e T P B tanh l e T P B expσ t) ˆθf + l expσ t) σ V + ρl expσ t) e T P B ˆθfX, t) e T P B ˆθfX, t) + l expσ t) σ V + ρl expσ t). 3) Thus, we obtain V t) V ) + ρl ) expσ t). 3) σ σ Owing to ), we conclue that

4 9th IFAC Worl Congress Cape Town, South Africa. August -9, e T P e V ) + ρl ) expσ t), 3) σ σ ρ γ expσ t) θ V ) + ρl ) expσ t),33) σ σ which further implies that e V ) + ρl σ σ λ min P ) γ V ) + θ ρ ) ρl σ σ exp σ t), 3). 35) Clearly, it can be seen from 3) that the tracking error converges to zero exponentially, an the convergence rate is not less than σ. Moreover, from 5) an 35), it follows that the parameter estimate ˆθt) is boune. By ), 3) an Assumption, it is shown that X is boune. Examining 7), we obtain the bouneness of fx, t). Next, we will prove vt) is boune. Using ), we get vt) ˆθfX, t) + κˆθ f X, t), 36) where κ =.785. Noting the bouneness of ˆθ an fx, t), we can obtain the bouneness of vt). Therefore, all the close-loop signals are boune. This completes the proof.. SIMULATION STUDIES Example : Consier the uncertain nonlinear system stuie in Zhou et al. 6), Su et al. ), Zhou et al. ), Su et al. 5) an Wang an Su 6) ẋ = a exp x) + but), 37) + exp x) where the parameters a =, b =, are assume to be unknown constants. The actuator nonlinearity is moele as backlash-like hysteresis ) with unknown parameters α =, c = 3.635, B =.35. The initial conition of the controlle plant 37) is set to be x) =.5. The objective is to esign the control v such that x can track the esire trajectory x =.5 sin.3t). Accoring to ), ) an 7), we have e = x x, A =, B =, exp x) fx, t) = + h + e + exp x) + h +,38) where h, h are chosen as h =., h =., respectively. In the simulation, we choose σ =., k =., Q =, γ = 3, l =., σ =.5, u) =, ˆθ) =. Thus, the solution of 5) is P = 5. The system responses are shown in Figs. -5. From Fig. 3, we can see that the tracking error converges to zero rapily. At the same time, the bouneness of control signal v is shown in Fig. 5, from which the large ifference between hysteresis input v an its output u can also be observe. The bouneness of other signals incluing plant state x an parameter estimate ˆθ is reveale in Figs. an, respectively. Plant state an reference signal Fig.. Example : plant state x an reference signal x Tracking error Fig. 3. Example : tracking error Parameter estimate Fig.. Example : parameter estimate x x

5 9th IFAC Worl Congress Cape Town, South Africa. August -9, Designe input an actual input Fig. 5. Example : esigne input v an actual input u Example : To further illustrate the effectiveness of our esign metho, we simulate the propose aaptive compensation controller on a nonlinear system escribe by Wang et al. ) ẍ = a exp x) + exp x) a ẋ + x) sinẋ).5a 3 x sin3t) + bu. 39) In this example, we consier the ea-zone moel 3). The plant parameters an ea-zone parameters are respectively given as a = a = a 3 = b =, m =, b r =.5, b l =.6, which are assume to be unknown. The initial conition of the controlle plant 39) is set to be x) =.5, ẋ) = 3.5. The reference signal is x =.5 sint). From ), ), ) an 7), it follows that e = e, e T, e = x x, e = x ẋ, A =, B =, fx, x, t) = exp x ) + h + exp x ) + x + x ) sinx ) + h +.5x sin3t) + h 3 + e + h +, ) where x = x, x = ẋ, h, h, h 3, h are selecte as h = h = h 3 = h =. Then, we apply the propose control scheme to this example. The esign parameters are chosen as follows: σ =., k =,., Q = 6 3; 3 6, γ =, l =, σ =., ˆθ) =. Solving 5), we have P = ) Accoring to Theorem, the control law an aaptive law can be erive. The simulation results are shown in Figs From these figures, similar conclusions on the signal bouneness an exponential tracking, as iscusse in Example, can be rawn. v u x an x \otx) an \otx)_ 5 5 \otx) \otx)_ 5 5 Fig. 6. Example : plant states an reference signals a) x, x ; b) ẋ, ẋ Tracking errors Fig. 7. Example : tracking errors Parameter estimate Fig. 8. Example : parameter estimate x x e e

6 9th IFAC Worl Congress Cape Town, South Africa. August -9, Designe input an ea zone region Fig. 9. Example : esigne input v an ea-zone region 5. CONCLUSION The problem of aaptive tracking control for a class of uncertain nonlinear systems with two possible actuator nonlinearities has been consiere. The plant parameters an the parameters of actuator nonlinearity are assume to be unknown. We have propose a class of aaptive controllers for tracking of ynamical signals. We have shown that by employing the presente aaptive tracking controller, the tracking error can be guarantee to ecrease to zero exponentially. REFERENCES Chen, X., Su, C. Y., & Fukua, T. 8). Aaptive control for the systems precee by hysteresis, IEEE Transactions on Automatic Control, 53), 9 5. Chen, X., Hisayama, T., & Su, C. Y. ). Aaptive control for uncertain continuous-time systems using implicit inversion of prantl-ishlinskii hysteresis representation, IEEE Transactions on Automatic Control, 55), Hua, C., Wang, Q., & Guan, X. 8). Aaptive tracking controller esign of nonlinear systems with time elays an unknown ea-zone input. IEEE Transactions on Automatic Control, 537), Hua, C., & Ding, S. X. ). Moel following controller esign for large-scale systems with time-elay interconnections an multiple ea-zone inputs. IEEE Transactions on Automatic Control, 56), Ibrir, S., Xie, W. F., & Su, C. Y. 7). Aaptive tracking of nonlinear systems with non-symmetric eazone input. Automatica, 33), Su, C. Y., Stepanenko, Y., Svoboa, J., & Leung, T. P. ). Robust aaptive control of a class of nonlinear systems with unknown backlash-like hysteresis, IEEE Transactions on Automatic Control, 5), 7 3. Su, C. Y., Oya, M., & Hong, H. 3). Stable aaptive fuzzy control of nonlinear systems precee by unknown backlash-like hysteresis, IEEE Transactions on Fuzzy Systems, ), 8. Su, C. Y., Wang, Q., Chen, X., & Rakheja, S. 5). Aaptive variable structure control of a class of nonlinear systems with unknown prantl-ishlinskii hysteresis, IEEE Transactions on Automatic Control, 5), Tao, G., & Kokotovic, P. V. 99). Aaptive control of plants with unknown ea-zones. IEEE Transactions on Automatic Control, 39), Tao, G., & Kokotovic, P. V. 995). Aaptive control of systems with unknown output backlash. IEEE Transactions on Automatic Control, ), Tao, G., & Kokotovic, P. V. 996). Aaptive Control of Systems with Actuator an Sensor Nonlinearities. New York: John Willey & Sons. Tao, G., & Lewis, F. L. ). Aaptive Control of Nonsmooth Dynamic Systems. Lonon: Springer-Verlag. Wen, C. & Zhou, J. 7). Decentralize aaptvie stabilizaiton in the presence of unknown backlash-like hystersis, Automatica, 33), 6. Wang, Q. & Su, C. Y. 6). Robust aaptive control of a class of nonlinear systems incluing actuator hysteresis with Prantl-Ishlinskii presentations, Automatica, 5), Wang, X. S., Su, C. Y. & Hong, H. ). Robust aaptive control of a class of linear systems with unknown eazone. Automatica, 3), 7 3. Zhou, J., Wen, C., & Zhang, Y. ). Aaptive backstepping control of a class of uncertain nonlinear systems with unknown backlash-like hysteresis, IEEE Transactions on Automatic Control, 9), Zhou, J., Wen, C., & Zhang, Y. 6). Aaptive output control of nonlinear systems with uncertain ea-zone nonlinearity. IEEE Transactions on Automatic Control, 53), 5 5. Zhou, J., Zhang, C., & Wen, C. 7). Robust aaptive output control of uncertain nonlinear plants with unknown backlash nonlinearity. IEEE Transactions on Automatic Control, 53), Zhou, J., Wen, C., & Li, T. ). Aaptive output feeback control of uncertain nonlinear systems with hysteresis nonlinearity, IEEE Transactions on Automatic Control, 57),