Adaptive robust control for DC motors with input saturation

Published in IET Control Theory and Applications Received on 0th July 00 Revised on 5th April 0 doi: 0.049/iet-cta.00.045 Adaptive robust control for DC motors with input saturation Z. Li, J. Chen G. Zhang M.G. Gan Beijing Institute of Control Engineering, 00080, People s Republic of China Beijing Key Laboratory of Automatic Control System, Beijing Institute of Technology, Beijing 0008, People s Republic of China E-mail: lizplst@gmail.com ISSN 75-8644 Abstract: The adaptive robust control ARC) for DC motors subjected to parametric uncertainties, disturbances and input saturation is considered in this study. To achieve high performance while keeping the control authority within saturation limit, a saturated ARC scheme is proposed. In this scheme, a variable-gain saturation function is introduced for the virtual control law, so that the amplitude of the virtual control and its derivative decrease when the control input approaches to the prescribed bound. Consequently, the virtual control and its derivative will not be excessively large, which is crucial for stabilising the system with a bounded input. We prove that the proposed controller cannot only assure global stability, but also provide desirable control performance, that is, the tracking error can be steered to the neighbourhood of the origin in finite time. Moreover, asymptotic tracking can be achieved in the presence of parametric uncertainties only. Finally, simulation results illustrate the effectiveness of the proposed controller. Introduction Input saturation is one of the most common non-linearities in physical systems, which can be found in almost all actuators. If the system operates with a control input beyond the saturation limit, deteriorated control performance appears, and even instability of the close loop occurs [, ]. Therefore the control problem of plants with saturation attracts much attention and research articles in the last decades contain a significant amount of new knowledge [3]. In [4], the controller design of a linear stable plant with input saturation is directly solved by a linear matrix inequality optimisation approach. In [5], composite quadratic Lyapunov function is utilised to construct the control law for linear plants subjected to input constraint. In [6], a novel saturated control structure was proposed to ensure the globally asymptotic stability using a set of linear coordinate transformations and multiple saturation-type functions. However, all the above results are based on the assumption that the plants are linear and exactly known, which is not satisfied in many applications. Taking servo systems for instance, the friction non-linearity, disturbance and parametric uncertainties influence the control performance [7]. It is necessary to consider these uncertainties when coping with the actuator saturation problem. In order to improve the control performance of the uncertain non-linear systems, an adaptive robust control ARC) method was proposed by Yao [8, 9]. It is performance-oriented and has strong performance in robustness [0 ]. However, in the traditional ARC the input saturation is rarely considered. Although the projection-type adaptation of ARC can alleviate integral windup problem caused by parameter adaptations [3], the previous ARC strategy [8, 9] does not guarantee global stability when the actuator saturation exists. To solve this problem, Gong and Yao [4] combined the nested saturated control design [6] and the ARC approach to achieve both global stability and high performance for the plants subjected to the matched parametric uncertainty and disturbance. Nevertheless, this method is based on a transformed state-space model, in which the model uncertainties have direct influence on every state. Hence, the resulting controller is usually conservatively designed to handle the enlarged model uncertainties. In [5, 6], Hong and Yao also investigated the ARC design for linear motors with input saturation. They introduced saturation functions to construct the virtual) control law in each recursive backstepping procedure, so that the output of the adaptive robust controller is bounded. However, in this method since the gain of virtual control is fixed, the derivative of virtual control may become excessively large, then a large control input is required to dominate the derivative of virtual control. In order to assure the stability of the close loop while keeping the control input within the limit, the gain of virtual control is constrained, and then the tracking error for virtual control, that is, z cannot be monotonically decreasing. That means the final control law may be conservative. In this paper, we propose a saturated adaptive robust control SARC) scheme to achieve high performance while considering the input saturation of the plant. To make the saturated controller less conservative, a variable-gain IET Control Theory Appl., 0, Vol. 5, Iss. 6, pp. 895 905 895 doi: 0.049/iet-cta.00.045 & The Institution of Engineering and Technology 0

saturation function in the virtual control law is introduced. Therefore the magnitude of virtual control and its derivative decrease when the control input gets close to the prescribed bound and the control saturation is avoided effectively. Moreover, comparing with the controller proposed in [5], the design procedure of our controller is more flexible, which leads to less conservative performance. This paper is organised as follows. The problem description is presented in Section. The design procedure of the proposed SARC is provided in Section 3. The stability proof and performance analysis are given in Section 4. Simulations are described and analysed in Section 5 and conclusions are outlined in Section 6. Dynamic models and problem formulations. Dynamic model of servo mechanisms The control of DC motors subjected to input saturation are investigated. The mechanical dynamics of a servo mechanism can be described by J q + T f + T l + T dis = T m ) where J is the inertial sum of load and armature; q is the motor output angle; T m is the electromagnetic torque; T f is the friction torque; T l is the unknown payload; T dis is the torque disturbance. In this paper, a simple friction model described by the Coulomb plus viscous model [7] is considered. the mechanical time constant J/B), therefore from 3) 5), we know that the electrical transients decay quite rapidly and Ldi/dt is very close to zero [8]. Thus, the dynamics of a servo mechanism can be simplified as [9] where q = K J satu, M u) K J q T c J sgn q) J T l + T dis ) 7) K = K T /R, K = K T K E + BR)/R 8) As seen, the DC motor model 7) is second order with nonlinearity and disturbance. Herein, we use a continuous function S f to approximate the discontinuous function sgnx ) [0]. The angle q is regarded as the system output y and the system state vector is defined by [x, x ] T = [q, q] T. Define T n as the mean value of T l + T dis and D as the lumped disturbance, that is, D ¼ T l + T dis T n + T c sgnx ) S f ))/J. Then, from 7), the following state-space form of the DC motor model can be obtained. ẋ = x ẋ = C u u x u S f + u 3 + D y = x u = sat u, M u ) where C and u j j ¼,, 3) are defined as follows 9) T f q) = T c sgn q) + B q ) In this equation, T c is the level of Coulomb friction torque, B is the viscous friction coefficient and the signum function sgn ) is defined by,. 0 sgn ) = 0, = 0,, 0 D = d J rad/s ), C = K J rad/s )/v) u = K J /s), u 3 = T n J rad/s ) u = T c J rad/s )/N m)),. Assumptions and problem formulations 0) From ) and ), one has J q + B q = T m T c sgn q) T l T dis s 3) The electromagnetic torque T m in ) is given by T m = K T i 4) where K T is the force constant. The current dynamics of a servo mechanism can be modelled as Ldi/dt + ir + K E q = satu, M u ) 5) In 5), R and L are the resistance and induction of the armature, respectively; i is the motor current amplitude; K E is the electromotive force coefficient; u is the input voltage; M u is the amplitude boundary of the input voltage; sat is the saturation function, which is defined as M u satu, M u ) = u M u u. M u M u u M u u, M u In general, the electrical constant L/R is small compared to 6) For simplicity, the following notations will be used: j for the jth component of the vector, ˆ for the estimate of, min for the minimum value of, and max for the maximum value of. The operation for two vectors is performed in terms of the corresponding elements of the vectors. Assumption : The parameter C in 9) is known and the extent of the unknown parameter u j, j ¼,, 3 is known, that is, 0, u jmin u j u jmax,whereu jmin and u jmax are known. Remark : This assumption is reasonable and of practical significance, because the two parameters K and J ) are more unlikely to change during a single operation in comparison with other parameters and can normally be estimated accurately online [5]. Thus, we assume that the parameter C in 9) is constant and known according to 0). In addition, to determine the bounds u jmin and u jmax,the rough value of u j can be obtained first from the product specifications or the off-line system identification. Then the lower and upper bounds can be set as less and more than 0 or 50% of this value, respectively. Assumption : The disturbance D is bounded, that is, D d, where d is some positive constant. 896 IET Control Theory Appl., 0, Vol. 5, Iss. 6, pp. 895 905 & The Institution of Engineering and Technology 0 doi: 0.049/iet-cta.00.045

With the dynamic models and assumptions above, the control problem of this paper can be stated as follows: Given the desired motion trajectory x d t) which is bounded with bounded derivatives up to the second order, the objective is to synthesise a control input u such that the output y ¼ x tracks x d t) as close as possible, while the input u remains in the prescribed bound [M u, M u ]. 3 ARC design for servo mechanisms with input saturation derivative below ds z ) dz = 0 z, L k z + L ) L L L z, L k L z, L k L z ) L L L z, L 0 L z 7) To solve the aforementioned control problem, a SARC method is proposed. The concrete design procedure is given as follows: Step : Define a variable z as z = x x d ) Here, z represents the output tracking error. From 9), we know ż = x ẋ d. In this step, we need to synthesise a virtual control law a satisfying these conditions: i) The tracking error z converges to zero globally when x ¼ a. ii) a should be bounded, which is necessary for the boundedness of the control input u. In view of the above two conditions, we design the virtual control law a as where z is given by a = ẋ d s z, z ) ) For clarity, according to 5) and 7), the smooth saturation function s z ) and its derivative are drawn in Figs. a and b. As we can see, both s z ) and ds z )/dz ) are continuous.. The definition of s is 0 z, L 0 )L + z ), L M z, L s z ) = L z, L 0 )L z ) L M z, L 0 L z 8) where L ¼ M /k, L ¼ L M / 0 )0, 0, ). Generally, 0 is chosen to be a very small positive number. Here, M, M and k are positive design parameters satisfying M. M k / 0 ) 9) and s z, z )is z = x a 3) which yields L. 0, so that s z ) is well defined. From 8), it is easy to check that the first-order derivative of s z, z ) = s z ) s z ) 4) where s and s are smooth functions with available firstorder derivatives defined as follows:. The definition of s is M z, L 0.5az + L ) M L z, L s z ) = k z L z, L 0.5aL z ) + M L z, L M L z 5) where L ¼ M /k ) + k /a) and L ¼ L k /a). In 5), M, a, k are positive design parameters satisfying M a. k 6) which yields L. 0, and then s z ) is well defined. From 5), it is easy to check that s z ) is a smooth saturated function. Furthermore, it has a continuous first-order Fig. s, s and their first-order derivatives a s bds /dz c s dds /dz IET Control Theory Appl., 0, Vol. 5, Iss. 6, pp. 895 905 897 doi: 0.049/iet-cta.00.045 & The Institution of Engineering and Technology 0

s z ) is as follows ds z ) dz = 0 z, L 0 )/M L z, L 0 L z, L 0 )/M L z, L 0 L z 0) where Proj ) represents the discontinuous projection operator. Its definition and properties can be seen in []. From the definition of w, s, s and s, we can conclude that u a is bounded. More specifically, according to the Cauchy Schwartz inequality, it follows that u a ẍ C d + k M. Graphically, s and its first-order derivative are drawn in Figs. c and d. As we can see, the value of s varies between zero and 0. Noting that s M and s ¼ s. s, one has s M ) which shows that s is bounded. In Assumption 3, we have stated the boundedness of ẋ d. Combining the boundedness of s and ẋ d with ), we know that a is also bounded. Remark : It should be noted that the virtual control law design in this paper is more flexible than that in [5]. In view of 4), the gain of virtual control k s z ) may vary between 0 and 0. Specifically, if s is a constant, the virtual control law a would be coherent to that in [5]. In other words, the controller proposed in this paper is more universal. From ) 3), the derivative of z can be derived as ż = z + a ẋ d = z s ) Noting ), from 9) and ), one has ż = ẋ ȧ = C u u x u S f + u 3 + D ẍ d + s z s ż + s z s ż = u z + wx) T u + C u ẍ d + s z s z s ) + s z s ż + D 3) where w ¼ [a, S f,] T, u ¼ [u, u, u 3 ] T. Here, we need to design a bounded control law u such that it is within the prescribed interval [M u, M u ]. Let the control law be u = u a + u s 4) where u a is the adaptive control term and u s is the robust control term. The concept of ARC is to compensate the known part of the model dynamics by using u a and to fight against various model uncertainties and disturbances by u s []. In order to render u within the saturation limits, u a and u s should be bounded and designed as follows u a = C ẍd wt û + s ) s z s 5) ) + ẋ d + M ) + u max + u max + u 3 max The robust control term u s is synthesised as 7) u s = s z )/C 8) where s is a saturated function M z, L s z ) = k z M L z, L L z 9) where M is designed as a positive constant. According to 9), u s is bounded by u s M /C 30) For clarity, the curve of s z ) is depicted in Fig.. From 4), 5) and 8), we have u = C ẍd w T û + s ) s z s s z ) Noting 7) and 30), then u u b M, M, k, u max ) = C ẍ d + k M + M + ẋ d + M ) + ) u max + u max + u 3 max 3) 3) If the controller parameters M, M, k and u max are designed such that u b M, M, k, u max ) M u, then the control input u = u will not cause saturation, which means that the in which the parameter estimate û is updated online whose adaptation law is given by û = Projû G wz ) 6) Fig. Curve of s z ) 898 IET Control Theory Appl., 0, Vol. 5, Iss. 6, pp. 895 905 & The Institution of Engineering and Technology 0 doi: 0.049/iet-cta.00.045

saturation non-linearity has no influence on the system as if it does not exist. From 3) and 3), one has ż = u z wx) T ũ + D) + s z s z s z ) + s z s ż 33) Because of the boundedness of w and Assumption, it can be assumed that w T ũ + D h 34) where h can be regarded as the bound of the total effect of model mismatch and unmodelled uncertainties. With this assumption, we will prove later that z t) can be made uniformly bounded if the parameters are designed to satisfy certain conditions as follows M h L. h/k k ) k L. h/k k ) 35a) 35b) 35c) Now we have finished the design procedures of the proposed SARC. In summary, the proposed SARC consists of: i) the control law given by 3), whose parameters satisfy 6), 9), 35a) 35c) and u b M u. ii) the adaptation law given by 6). From 3), we know that the purpose of introducing the saturated functions s and s is to make the control law u within saturation limit. But here comes a problem: what is the functionality of s? In view of 33), s can attenuate the influence of s / z )s z, since s becomes zero when z is larger than L. In the following section, we will demonstrate that the introduction of s, s and s into the SARC can make the close-loop system stable. 4 Stability proof and performance analysis [ h + )/k k k), h + )/k k k)] in a finite time. Part I: The evolution of z is discussed in three cases, that is, i) z. L ; ii) L, z L ; iii) h + )/ k k ), z L. To simplify this proof, the sets corresponding to the above three cases are defined as follows V 0 = {z, z : z. L } V = {z, z :L, z L } V = {z, z :h + )/k k ), z L } V 3 = {z, z : z h + )/k k )} Case : When z. L, from 0), ) and 3), we have s ¼ 0, s / z ) ¼ 0, s ¼ M. Noting that u. 0 Assumption ), according to 33) and 34), then z ż u z + h z M z M h) z 36) Since M h, 36) indicates that any trajectory with the initial state [z 0), z 0)] T in V 0 will reach the set V in a finite time t 0, which is bounded by t 0 z 0) L M h 37) Case : When L, z L, from 8), 0) and 9), it follows that s, s / z ¼ 0 /M ) and s ¼ k z. Noting that s M, one has s / z s 0. In accordance to 6) z ż = u z + w T ũ + D)z + s z s z k z + s z s z ż 38) which yields s ) s z z ż = u z + w T ũ + D)z Define a set V c as { V c = z, z : z h + k k k ), z h + } k k Noting 34), since + s z s z k z where, L k k k ) h,, L k k ) h,.. 0. Then we are going to prove that the error states can be steered into the set V c by SARC in finite time. Theorem : Suppose that the saturated adaptive robust controller proposed in Section 3 is applied to the plant 9). Then,. the controller guarantees that all signals in the close-loop system are bounded. Furthermore, the trajectory of the error states [z, z ] T reaches the prescribed set V c in a finite time and stay within thereafter.. the steady state of the tracking error z is bounded by z ) h/k k k )). Proof: The proof of ) is divided into two parts: Part I, z gets into the interval [h + )/k k ),h + )/k k )] in a finite time. Part II, with this property of z, z evolves into we have 0 s z s 0 and s z s, k z ż u z + h z +k z k z s / z )s ) k L k L h) z 0 39) Owing to k L k L + h, 39) indicates that any trajectory with the initial state [z 0), z 0)] T in V will reach the set V in a finite time t. Moreover, the upper bound of the reaching time t is t z 0) 0 ) k L k L h 40) IET Control Theory Appl., 0, Vol. 5, Iss. 6, pp. 895 905 899 doi: 0.049/iet-cta.00.045 & The Institution of Engineering and Technology 0

Case 3: When h + )/k k ), z L, from 8), 0) and 9), we have s ¼, s / z ) ¼ 0 and s ¼ k z. Noting z. h + )/k k ), from 33) and 34), one has z ż = u z + w T ũ + D)z + s z k z z w T ũ + D)z k k )z h + h) z = z 4) Owing to. 0, 4) indicates that any trajectory with the initial state [z 0), z 0)] T in V reaches the set V 3 in a finite time t 3. Furthermore, the upper bound of the reaching time t 3 is t 3 z 0) 4) From the analysis of the above three cases, we know that the trajectory of the [z, z ] T steps into V 3 in a finite time, no matter where the initial states are. That means z gets into the interval [h + )/k k ),h + )/k k )] within a finite time. Part II: After a finite time, z h + )/k k ). Under this condition, we discuss the evolution of z in two cases, that is, i) z. L, and ii) h + )/k k k )), z L. For simplicity, the set V 4 is defined as V 4 = {z, z : z L, z h + )/k k )} Note that V 4 is a subset of V 3. From ), one has z ż = z z s ) 43) Case : When z. L and z h + )/k k ), according to 4), 43) and z h + )/k k ), the following inequality holds. z ż k L h + ) z k k 44) Since h +, h +, L k k k ), 44) indicates that any trajectory with the initial states [z 0), z 0)] T in V 3 will reach the set V 4 in a finite time t 34. Furthermore, the upper bound of the reaching time t 34 is t 34 z 0)k k ) L k k k ) h 45) Case : When h + /k k k )), z L and z h + )/k k ), according to 4) and 43) z ż = k z + z z k z + h + z k k h + h + ) z k k k k = z k k 46) Since., 46) indicates that any trajectory with the initial state [z 0), z 0)] T in V 4 will reach V c in a finite time t 4c, which is bounded by t 4c z 0)k k ) 47) Combining the analysis of the above two cases, any trajectory with the initial states in V 3 will reach V c in finite time. From the deduction in Part I and Part II, we arrive at the conclusion that no matter what the initial states are, the trajectory of [z, z ] T can be steered to V c in a finite time. Therefore the states [z, z ] T are bounded. In addition, because of using the projection mapping, all the states of the adaptation law, that is, û, û and û 3 are bounded. Noting that a is bounded and x ¼ z + a, we know that x is also bounded. From the boundedness of z and x d, x is also bounded. Conclusively, all the states of the closeloop system are bounded. Hence, ) of Theorem has been proven. A Now, we prove ). According to the analysis of case 3) in Part I, after a finite time the tracking error z satisfies z ż = u z + w T ũ + D)z + s z k z z w T ũ + D)z k k )z k k k k z + h z k k h z + k k ) z 48) Define V as V = /)z and k s as k s ¼ k k. From 48), the derivative of V satisfies which yields V k s V + h k s 49) V t) exp k s t)v 0) + h ks [ exp k s t)] 50) It indicates that the steady state of z is bounded by z ) h/k s. Then, according to the first equation of 46), the steady state of the tracking error z is bounded by z ) h/k k s ) ¼ h/[k k k )]. Remark 3: From the proof of Theorem, we can see that if z 0). h + )/k k ), z decreases monotonically, and the trajectory of [z, z ] T will reach the set V 3 ¼ {z, z : z h + )/k k )} in finite time. Then the tracking error z will decay to the neighbourhood of zero. Theorem : With the saturated adaptive robust controller proposed in Section 3 applied to the plant 9), asymptotic output tracking can be achieved if the system is only subject to parametric uncertainty after a finite time, that is, D ¼ 0, t 0 for some t 0. Proof: Define a positive definite function V a as V a = /)z + /)ũ T G ũ According to Theorem, the trajectory of the states [z, z ] T 900 IET Control Theory Appl., 0, Vol. 5, Iss. 6, pp. 895 905 & The Institution of Engineering and Technology 0 doi: 0.049/iet-cta.00.045

will enter V c in finite time. If D ¼ 0, when [z, z ] T [ V c, the derivative of V a can be derived as V a = u z + w T ũz + s z z k z + ũ T G Projû Gwz ) u z k k )z 5) which indicates that z converges to zero asymptotically. Subsequently, the tracking error z will also converge to zero asymptotically according to ). The Theorem is proved. A 5 Simulations Consider the dynamic model given by 9) with the parameters selected as: J ¼ 0. kg. m, B ¼ 0.08 N. m /rad/s), T c ¼ 0.07 N. m, R a ¼ 5 V, K T ¼ 5N. m /A, K E ¼ 0. V/rad/ s), T l ¼ 0. N. m, M u ¼ V. The disturbance D is chosen as a uniformly distributed pseudo-random number with an amplitude less than 0.05 N/kg, that is, D ¼ 0. rand) 0.05) N/kg. Then the dynamic model of a servo system in the form of 9) can be derived as ẋ = x ẋ = 0 u.8x 0.7S f + + 0. rand ) 0.05 y = x u = satu, ) 5) The corresponding parameters of model 5) can be obtained as C = 0rad/s )/v), u =.8/s) u = 0.7rad/s )/N m)), u 3 = rad/s ) Assuming that these parameters are unknown, the parameter bounds are chosen as: u min ¼ [.5, 0.5, 0.5] T, u max ¼ [3,,.] T, d ¼ 0.. According to these prescribed bounds, by use of the design procedure in Section 3, the parameters of the SARC are synthesised as: a ¼ 500, k ¼ 00, M ¼.3, 0 ¼ 0.05, G ¼ diag[800, 60, 00]). The continuous function S f, which is used to approximate sgnx ), is selected as S f ¼ /p)atan900x ). In the simulations, two cases are considered. Case : The tracking performance will be demonstrated with a point-topoint desired trajectory. Case : The stabilisation with large initial error is considered. The purpose of this configuration is to illustrate the anti-windup effect of the proposed SARC. 5. Case : point-to-point trajectory The input signal is a point-to-point trajectory with a maximum angle of 0. rad, a maximum angular velocity of 0.4 rad/s, a maximum acceleration of rad/s, as depicted in Fig. 3. The initial states of the plant 5) are set to zeros. With the aforementioned SARC parameters and the property of input signal, the upper bound of the control input u b can be computed according to 3) as u u b = C ẍ d + k M + M + ẋ d + M ) + ) u max + u max + u 3 max = 0. + 5 0. +.3 + ) 0.4 + 0. ) + 3 + +. = 0.99739, 53) According to 53), the control input is within the saturation limit, which means that controller wind-up is avoided. As shown in Fig. 4, the output tracking performance of SARC is desirable. We can see that the tracking error is less than 0.5 0 4 rad when simulation time is larger than 5 s, and the amplitude of tracking error decreases gradually as time increases. That is because the controller parameters are tuned on line by the adaptation law. The parameter estimates are shown in Fig. 5. As seen, the parameter Fig. 3 Point-to-point trajectory IET Control Theory Appl., 0, Vol. 5, Iss. 6, pp. 895 905 90 doi: 0.049/iet-cta.00.045 & The Institution of Engineering and Technology 0

Fig. 4 Output tracking error of SARC with a point-to-point trajectory as input Fig. 5 Parameter estimates in Case Fig. 6 Control input of SARC in Case estimates evolve to the actual values gradually. Moreover, the estimates never exceed the prescribed bound owing to the use of projection operator. Fig. 6 shows the control input of SARC, whose amplitude is well within the preset bound M u ¼, hence control input wind-up never occurs. This is the most distinguished feature of the proposed SARC. 5. Case : stabilisation with large initial error In this case, the input signal is set to zero, while the initial condition of the plant is [x, x ] ¼ [0., 0.]. The same controller parameters as those given in Subsection 5. are utilised in this case. It is obvious that the 90 IET Control Theory Appl., 0, Vol. 5, Iss. 6, pp. 895 905 & The Institution of Engineering and Technology 0 doi: 0.049/iet-cta.00.045

upper bound of the control input u b accordingto3)as u u b = ẍ C d + k M + M can be computed = 0. 0 + 5 0. +.3 www.ietdl.org + ) 0 + 0. ) + 3 + +. ) + ẋ d + M ) + u max + u max + u max = 0.7607, 54) From 54), the control input satisfies u, M u, hence the control input wind-up is avoided. Fig. 7 Trajectories of x and x Fig. 8 Control inputs of Case a SARC b Ordinary ARC IET Control Theory Appl., 0, Vol. 5, Iss. 6, pp. 895 905 903 doi: 0.049/iet-cta.00.045 & The Institution of Engineering and Technology 0

Fig. 9 Trajectories of s, s and s The state evolution of the close-loop system is depicted in Fig. 7. Both states are steered to zero by SARC. The control input of SARC is shown in Fig. 8a. As we can see, the control input is within the bound M u, thus the saturation non-linearity has no influence on the close loop. This property is desired in our controller design. It should be noted that if we let s ¼ k z, s ¼, s ¼ k z, then the SARC will turn to be the ordinary ARC. Fig. 8b shows the control input of the ordinary ARC with the same parameters as those of the aforementioned SARC. We find that the control input of the ordinary ARC is saturated at the outset of the control process. Although the large control input dose not certainly cause the closeloop system unstable, it is harmful to the machine health. Hence the control input amplitude of the SARC is favourable. Fig. 9 shows the variations of s, s and s.asis depicted, at the startup s is saturated i.e. s ¼ M ) owing to the large magnitude of z. Besides, s is small and s is saturated at the startup, since z is larger than L in that phase. Noting s ¼ s s, from 3), we know that the control input amplitude is restrained at the startup. As the states [x, x ] get close to the origin, the amplitudes of z and z decrease, therefore s and s withdraw saturation, and s equals to M. Then, the SARC turns to be an ordinary ARC and recover the performance of an ordinary ARC without saturation. 6 Conclusions In this paper, we proposed a saturated adaptive robust controller for the DC servo system with input saturation. It embeds several saturation functions in the control law to keep the control input within the prescribed limit, and thus the control input wind-up is avoided. We proved that the proposed controller can not only assure the stability of the close-loop system, but also provide desirable control performance, that is, the tracking error can be steered to the neighbourhood of the origin. Moreover, asymptotic tracking can be achieved if the system is only subject to parametric uncertainty. In addition, the theoretical-analysis results were verified through simulation studies. 7 Acknowledgments Contract/grant sponsor: Beijing Education Committee Cooperation Building Foundation Project; National Science Fund for Distinguished Young Scholars; contract/grant number: XK0007053; 60950. 8 References Zaccarian, L., Teel, A.R.: A common framework for anti-windup, bumpless transfer and reliable designs, Automatica, 00, 38, 0), pp. 735 744 Galeani, S., Teel, A.R.: On performance and robustness issues in the anti-windup problem. Proc. IEEE Conf. on Decision and Control, Atlantis, 004, pp. 50 507 3 Bernstein, D.S., Michel, A.N.: A chronological bibliography on saturating actuators, Int. J. Robust Nonlinear Control, 995, 5, 5), pp. 375 380 4 Grimm, G., Postlethwaite, I., Teel, A.R., et al.: Linear matrix inequalities for full and reduced order anti-windup synthesis. Proc. American Control Conf., 00, pp. 434 439 5 Hu, T., Lin, Z.: Composite quadratic lyapunov functions for constrained control systems, IEEE Trans. Autom. Control, 003, 48, 3), pp. 440 450 6 Teel, A.R.: Global stabilization and restricted tracking for multiple integrators with bounded controls, Syst. Control Lett., 99, 8, ), pp. 65 7 7 Xu, L., Yao, B.: Adaptive robust motion control of linear motors for precision manufacturing, Mechatronics, 00,, 4), pp. 595 66 8 Yao, B., Tomizuka, M.: Adaptive robust control of SISO nonlinear systems in a semi-strict feedback form, Automatica, 997, 33, 5), pp. 893 900 9 Yao, B.: High performance adaptive robust control of nonlinear systems: a general framework and new schemes. Proc. IEEE Conf. on Decision and Control, 997, pp. 489 494 0 Zhu, X., Tao, G., Yao, B., et al.: Adaptive robust posture control of parallel manipulator driven by pneumatic muscles with redundancy, IEEE/ASME Trans. Mechatronics, 008, 3, 4), pp. 44 450 Zhu, X., Tao, G., Yao, B., Cao, J.: Adaptive robust posture control of a parallel manipulator driven by pneumatic muscle, Automatica, 008, 44, 9), pp. 48 57 Zhong, J., Yao, B.: Adaptive robust precision motion control of a piezoelectric positioning stage, IEEE Trans. Control Syst. Technol., 008, 6, 5), pp. 039 047 3 Yao, B., Majed, M.A., Tomizuka, M.: High performance robust motion control of machine tools: an adaptive robust control approach and comparative experiments, IEEE Trans. Mechatronics, 997,, ), pp. 63 76 4 Gong, J.Q., Yao, B.: Global stabilization of a class of uncertain systems with saturated adaptive robust control. Proc. IEEE Conf. on Decision and Control, Sydney, Australia, 000, pp. 88 887 904 IET Control Theory Appl., 0, Vol. 5, Iss. 6, pp. 895 905 & The Institution of Engineering and Technology 0 doi: 0.049/iet-cta.00.045

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