IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 18, NO. 3, MAY 2010
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Adaptive Robust Control for Servo Mechanisms With Partially Unknown States via Dynamic Surface Control Approach
Guozhu Zhang, Jie Chen, Member, IEEE, and Zhiping Lee
Abstract—In order to achieve high performance control for rvo mechanisms with electrical dynamics and unmeasurable states, an obrver-bad adaptive robust controller (ARC) is developed via dynamic surface control (DSC) technique. To reprent electrical dynamics, a third-order model is ud to describe the rvo mechanism. However, the third-order model brings some difficulties to obrver construction and recursive controller design. To solve this problem, we first transform the model into a particular form suitable for obrver design, and then construct a parameterized obrver to estimate the unmeasurable states. The state estimation is bad on the output and its derivatives, which can be acquired by an output differential obrver. Subquently, an obrver-bad ARC can be developed through DSC technique, with which the problem of “explosion of complexity” caud by backstepping method in the traditional ARC design can be overcome. A stability analysis is given, showing that our c
ontrol law can guarantee uniformly ultimate boundedness of the solution of the clod-loop system, and make the tracking error arbitrarily small. This scheme is implemented on a precision two-axis turntable. Experimental results are prented to illustrate the effectiveness and the achievable control performance of the propod scheme. Index Terms—Adaptive robust control (ARC), dynamic surface control (DSC), rvo mechanism, state obrver, two-axis turntable.
I. INTRODUCTION HE PERFORMANCE of rvo mechanisms is frequently deteriorated by external disturbances and nonlinearities (e.g., friction and cogging force). Moreover, there must be some parametric uncertainties in the plant model due to unavoidable modeling errors. When designing controllers for rvo mechanisms, all the factors mentioned above (i.e., disturbances, nonlinearities and parametric uncertainties) need considering. Adaptive robust control (ARC) propod by Yao and Tomizuka in [1] and [2] combines the advantages of adaptive control [3] and deterministic robust control (DRC) [4] and overcomes their practical performance limitations for a reasonably large class of nonlinear systems [5]. It has been proved that for the mi-strict feedback nonlinear systems, the ARC is not only able to attenuate the influence of disturbances and nonlinearities, but also to achieve asymptotic output tracking in
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在线读英语Manuscript received November 24, 2008; revid March 16, 2009. Manuscript received in final form June 05, 2009. First published August 18, 2009; current version published April 23, 2010. Recommended by Associate Editor Z. Wang. The authors are with the Department of Automatic Control, Beijing Institute of Technology, Beijing 100081, China (e-mail: zhangguozhu_bit@yahoo. com; chenjie@bit.edu; ). Color versions of one or more of the figures in this paper are available online at ieeexplore.ieee. Digital Object Identifier 10.1109/TCST.2009.2025265
the prence of parametric uncertainties only [1]. In [6], ARC is successfully applied to a third order rvo mechanism, but its control law is state dependent. In some cas, partial states of the plant such as motor current and angular velocity are unmeasurable, so that state feedback control laws will not be applicable. To solve this problem, the obrver-bad output feedback ARC is developed in [16] and [20] by combining state obrver with ARC design. Several applications of the output feedback ARC have been reported. In [7], an output feedback ARC is developed for a magnetic levitation system. In [8], the obrver-bad output feedback ARC is utilized to an epoxy core linear motor who current dynamics is negligible. However, the results are merely applicable to the plants with cond-order model, who structure is suitable for the obrver design. The ordinary thi
rd-order models of rvo mechanisms with current dynamics may lead to some difficulties in the obrver design. To this issue, a novel model transformation that facilitates the obrver construction is propod in this paper. Additionally, the commonly ud backstepping technique in traditional ARC design may result in the problem of “explosion of complexity”, especially for the plant with order larger than three [9], [10]. As shown in [6], the state feedback ARC developed by backstepping approach for a third-order plant is already quite complicated. If an obrver-bad ARC is designed, the computation will be more cumbersome. To eliminate “explosion of complexity” in backstepping design, the dynamic surface control (DSC) method was propod [10]. The DSC approach replaces the derivatives at each step of the traditional backstepping design by some first order filters. Becau of its convenience, the DSC technique has been ud in adaptive controller design [11], [12] and state feedback ARC design [13]. In this paper, we u DSC technique to simplify the design of an obrver-bad ARC, which is more complicated than the adaptive controller and state feedback ARC when synthesized by integrator backstepping approach. In this paper, the rvo mechanisms with current dynamics and partially unknown states are under investigation. The model of the rvo mechanism is first transformed into a particular form suitable for obrver design, and then a state obrver is designed so that the unknown states can be replaced by their estimates. Finally, the DSC technique is ud to design the adaptive robust controller. This paper is organized a
s follows. Dynamic model of the rvo mechanisms and problem formulation are prented in Section II. The propod ARC controller is shown in Section III. The clod-loop system stability is analyzed in Section IV. Experimental results are prented in Section V and conclusions are drawn in Section VI.
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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 18, NO. 3, MAY 2010
II. DYNAMIC MODELS AND PROBLEM FORMULATION A. Dynamic Models of Servo Mechanisms The dynamics of a dc motor rvo mechanism can be described as [6], [8] (1) where is the inertial sum of load and armature, is the motor’s output angle, is the electromagnetic torque, is the is the disturbance torque, is the viscous friction torque, is considered to have the friction coefficient. In general, following form [14]: (2) In the equation, is the level of the static friction torque, is the minimum level of Column friction torque, and and are empirical parameters ud to describe the Stribeck effect. A popular simplified model that relates the electromagnetic torque to the input voltage is given by [15] (3) where and are the resistance and induction of the armature, respectively,
is the motor current, is the input voltage, is the force constant, is the electromotive force coefficient. Defining the angle, angular velocity, and current as the state , from (1)–(3), the variables, i.e., entire system can be expresd in the state space form as (4)
B. Assumptions and Problem Statement For simplicity, the following notations will be ud: for the th component of the vector , for the minimum value of , for the maximum value of . The operation for two and vectors is performed in terms of the corresponding elements of the vectors. In general, the parameters of the model cannot be accurately determined. Thus, we assume, in this paper, that the uncertain parameters are in certain known intervals, as shown in assumption 1 and assumption 2. In addition, assumption 3 is made for . the desired motion trajectory , moreover and Assumption 1: are known. . Assumption 2: The disturbance is bounded, i.e., Assumption 3: The desired trajectory are continuous and with known compact t available, and , is a known positive constant. who size The control problem of this paper can be stated as follows: , the objective is to given the desired motion trajectory tracks synthesize a control input such that the output as cloly as possible in spite of various model uncertainties. Since the rvo mechanisms studied in this paper have partially unknown states, it is necessary to design controllers which are only dependent on the available states. III. ADAPTIVE ROBUST CONTROL WITH PARTIAL STATES FEEDBACK The method propo
d in this paper is motivated by the obrver-bad ARC developed in [16] and the ARC design via DSC technique prented in [13]. In order to reduce the dynamic uncertainties caud by the unmeasurable states, we first transform the model into a particular form, and then develop a parameterized obrver to estimate the unavailable states. Afterward, an adaptive robust controller relying on the available states and estimates of the unavailable states is synthesized by DSC technique. Owning to the DSC approach, the explosion of complexity in traditional backstepping design is avoided, and then the propod controller is less complicated than that developed by backstepping as in [1]. A. Model Normalization In order to design a parameterized obrver, we transform model (6) into a normalized form. From the cond equation of (6), we have (7) Substituting (7) into the third equation of (6), we obtain (6) (8)
is measureable, while In this paper, we assume that the state and (i.e., the angular velocity and current of the rvo mechanism) are unmeasureable. In order to linearly parameis approximated by terize model (4), the friction torque the quantity , where is chon as the following differentiable function [8], [21]: (5) in (5) should be chon to be a large posThe parameter can approxiitive number, so that the smooth function mate with adequately small residual error. The approxis . Substituting (5) imating error of into (4), we have
where is as follows:
. The definition of where (9)
ZHANG et al.: ADAPTIVE ROBUST CONTROL FOR SERVO MECHANISMS WITH PARTIALLY UNKNOWN STATES
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, . Following the design procedure of [3], one can define the following K-filters:
(10) and According to (10) and assumption 1, there exist satisfying , . Furtherand are functions of and . more, From assumption 2 and the boundedness of , it can be en and are also bounded. Therefore, model (8) has the that following properties. Property 1: , , where and are known. , , where and are Property 2: known. B. Discontinuous Projection as Define the unknown parameter t . Let denote the estimate of and reprent the estimation error (i.e., ). The discontinuous projection operator is defined as [17] if if otherwi. and and Then, the states of (14) can be rewritten as (16) where that is the estimation error. From (14) and (16) we know . The solution of this equation is (17) i
s the zero input respon of equation and for . Noting property 2 and that matrix is stable, one has (18) where (12) D. Output Differential Obrver where . P2 (13) and The variables and in the right-hand side of model (8) reprent the angular velocity and acceleration of the rvo and can mechanism, respectively. In the ideal ca, be computed by differentiating the output angel . However, becau the differential operator is nsitive to noi, the derivative of the output angle cannot be ud directly. Therefore, the output differential obrver [19] shown in (19) is ud to estimate the angular velocity and acceleration (19) (14) where , , , and (20) The pole placement method can be ud to adjust the bandwidth of (20), so that can track in any prescribed rate. Then, and will track and with desired fast respon, respectively. Let be the input and be the output, the transfer function of the output differential obrver is is a vector of unknown but bounded functions. where According to the above equations, since , and , we know that can be computed in the way described below: , ,
and
Therefore, the K-filters can be simplified as
(15)
(11)
If the adaptation law is given by , where is a diagonal positive definite matrix, then for any adaptation function , the projection mapping ud in (11) assures [18]. P1 If , then
C. Design of the State Obrver The last two equations of model (8) can be rewritten as
Then, by suitably choosing , one can synthesize the obrver matrix with arbitrarily fast convergence rate. Thus, such that there exists a symmetric positive definite matrix
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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 18, NO. 3, MAY 2010
E. Recursive Controller Design via DSC The design combines the DSC method with the ARC design is reprocedure. In the following, the unmeasurable state placed by its estimates and the estimation errors are handled by robust feedback to achieve a guaranteed robust performance. Step 1: First, a dynamic surface is defined as (21) where is the output tracking error, is any positive feedback gain. Since is a stable transfer function, if is small or converges to zero exponentially, the output tracking error will be small or converge to zero exponentially too. Differentiating (21) and noting (8), we have
The control input , which consists of two parts, is design as follows: (29) is the adaptive control term; is the robust control where term. Step 3: The adaptation law to update the parameter estimates is chon as
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(30) Remark 1: One smooth example of satisfying (24) can be any smooth function be found in the following way. Let or constant satisfying (31) Then, can be chon as [1], [2] (32) Similarly, an example of satisfying (25) is given by [20] (33)
(22) where . In the following, we u the ARC approach propod in [2] to cope with the parametric uncertainties and uncertain nonlinearity , in (22). If the filter state were the actual control input, as follows: one could synthesize for it a virtual control law
(23) Other smooth or continuous examples of worked out in the same way as in [6]–[8]. is the adaptive control term and is the robust where control term, is a positive design parameter. In (23), is lected to satisfy the following condition: (24) and is any continuous function satisfying (25) where and are positive parameters to be chon. Introduce a new variable and let pass through a first-order filter with time constant to obtain (26) Esntially, (24) shows that is synthesized to attenuate the effect of uncertain nonlinearities with known bound (i.e., ) to the level of control accuracy measur
ed by . Similarly, it can be en from (25) that is ud to counteract the effect . of state estimation error Step 2: The cond dynamic surface is defined as (27) From (15) and (23), the derivative of is (28) Thus (38) From (23) and (26), the derivative of is and can be
99宿舍六级成绩查询IV. STABILITY ANALYSIS OF THE CLOSED-LOOP SYSTEM Define a scalar as (34)
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(35) All terms in (35) can be dominated by some continuous functions, therefore, we have (36) where is a continuous function. Then, the following inequality can be obtained: (37)
ZHANG et al.: ADAPTIVE ROBUST CONTROL FOR SERVO MECHANISMS WITH PARTIALLY UNKNOWN STATES
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From (22), (23), (27), (30), and (34),
can be derived as
Noting (13) and (44), we have
(39) Then, we have
(47) (40) From (28) and (29), it follows that (41) Then, we obtain (42) Before the main result of stability is given, the ts and values which will be ud in the stability proof are defined below. For , define any where (45) and (48) into (47) yields Define a positive number satisfying (48) . Substituting
(49) Let (50)
where (43) Obviously, is a compact subt in , hence there must be a point corresponding to the supreme value of in . We denote , that is this supreme value as (44) Theorem 1: Considering system (6), if the control law is (29), adaptation law is (30) and assumption 1 3 are satisfied, then , for any initial states in , there exist positive parameters , , , and satisfying
then on . Thus, , then for all if bounded, so does , , , and . Define a positive definite function
is an invariant t, i.e., . Therefore, is satisfying (51)
In order to made a contradiction, we assume that there exist , so that when (52) where is an arbitrary positive number and is given by (53)neither是什么意思
(45) From (45), (47), (51), and (53), we know that such that all signals of the clod-loop system are uniformly ultimately bounded and the steady-state tracking error can be made arbitrarily small. Proof:
Considering the positive definite function in can (43), from (30), (38), (40), and (42), the derivative of be found as follows: (54) Multiplying 1 to both sides of (54), yields (55) Integrating (55) over , we have (56)swiss balance
(46)
Due to the boundedness of proven above, the function is upper bounded. From (52) and (56), it can be en that
is
