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Multiobjective PID Control Design in Uncertain Robotic Systems Using Neural Network Elimination Scheme Chung-Shi Tseng, Member, IEEE, and Bor-Sen Chen, Fellow, IEEE
Abstract—In this paper, a proportional-plus-integral derivative (PID)-type controller incorporating neural network elimination scheme and sliding-mode control action for different objectives tracking performance, including 2 tracking performance, and regional pole constraints is developed in robotic systems under plant uncertainties and external disturbances. The adaptive neural networks are used to compensate the plant uncertainties. The sliding-mode control action is included to eliminate the effect of approximation error via neural network approximation. The sufficient conditions are developed for different objectives in terms of linear matrix inequality (LMI) formulations. The interesting combinations of different objectives are considered in this paper, which include PID tracking control design with regional pole constraints and mixed 2 PID tracking control design with regional pole constraints. These multiobjective PID control problems are characterized in terms of eigenvalue problem (EVP). The EVP can be efficiently solved by the LMI toolbox in Matlab. The proposed methods are simple and the PID control gain for different objectives can be obtained systematically. Simulation results indicate that the desired performance for the multiobjective control schemes of the uncertain robotic systems can be achieved using the proposed methods. Index Terms—Adaptive neural networks, eigenvalue problem (EVP), linear matrix inequality (LMI), multiobjective proportional-plus-integral derivative (PID) control, robotic control.
I. INTRODUCTION
I
N THE past two decades, the motion control of industrial manipulators has received a great deal of attention. Many approaches have been introduced to deal with this robotic control problem and various control algorithms have also been proposed in the literature [1], [2]. When the model is exactly known, the technique of feedback linearization in nonlinear systems, which is also called computed torque method, has been developed to deal with the control of robotics [3]. This method uses a nonlinear state feedback to exactly cancel the nonlinear terms, and then employs optimal control techniques or variable structure control (VSC) techniques to deal with the equivalent linear dynamic problem [1], [4], [5]. Manuscript received September 22, 2000; revised August 31, 2001. This work was supported by the National Science Council under Contract NSC 88-2213-E-007-069. This paper was recommended by Associate Editor R. A. Hess. C.-S. Tseng is with the Department of Electrical Engineering, Ming Hsin Institute of Technology, Hsinchu 30043, Taiwan R.O.C. (e-mail:
[email protected]). B.-S. Chen is with the Department of Electrical Engineering, National Tsing Hua University, Hsinchu 30401, Taiwan R.O.C. (e-mail:
[email protected]). Publisher Item Identifier S 1083-4427(01)09867-8.
Since uncertainties which may not be known a priori exactly are inevitable in practical robotic systems, (e.g., the load may vary while performing different tasks, the friction coefficients may change in different configurations, and some neglected nonlinearity, such as backlash, may appear as a disturbance at the control input), the robot arm receives unpredictable interference from the environment where it resides. Therefore, it is necessary to consider these effects due to plant uncertainties which contain structured (or parametric) uncertainties and unstructured uncertainties (or unmodeled dynamics), and external disturbances. In order to compensate these uncertainties in the robotic systems, many control strategies have been proposed. There are basically two underlying strategies to deal with the uncertain systems. One is the robust control strategy and the other is the adaptive control strategy. In the recent development of robust control algorithms [1], [2], [6], [7], if an a priori bound of uncertainty is known, then high-gain feedback laws or saturation-type controllers can be proposed to deal with the tracking of robot motion. The control law based on the upper bound of the uncertainties may lead to a conservative design. Moreover, precise bounds on the uncertainty are difficult to evaluate. On the other hand, in the recent development of adaptive control algorithms [8]–[11], the use of regressor matrix has become rather popular in adaptive control of robotic manipulators. In this situation, nonlinear dynamics of a rigid robot with unknown (or uncertain) system parameters are assumed to be expressed as a product of a regressor matrix and an unknown parameter vector, that is, the property of linearity in the system parameters is used in the derivation of the adaptive control results. Then, a parameter update law has been used to estimate the unknown parameters which are assumed to be constant or slowly varying. However, there are some potential difficulties associated with this approach, e.g., the unknown parameters may be quickly varying, the linear parameterization property may not hold, computation of the regressor matrix is a time-consuming task, and implementation also requires a precise knowledge of the structure of the entire robot dynamic model. Hence, the introduction of an alternative approach to deal with the adaptive control of robotic systems with uncertainties is interesting. Artificial neural networks offer the advantage of performance improvement through learning using parallel and distributed processing. These networks are implemented by massive connections among processing units and are attractive for wide applications in identification, signal processing, and control. Recently, adaptive neural network algorithms have been used
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to solve highly nonlinear control problems [12], [14]–[18]. In general, the conventional adaptive neural network algorithm has not attacked the problem of attenuation of the effects on the tracking error due to the approximation error via neural network approximation. However, the approximation error may deteriorate the tracking performance of the closed-loop neural network-based control system. Hence, it is still a challenge to introduce an additional performance criterion such that the closed-loop tracking performance is upgraded. In this paper, a proportional-plus-integral derivative (PID)-type controller incorporating neural network compensation and sliding-mode control action for different objectives tracking performance, tracking performance, including and regional pole constraints is developed in robotic systems under plant uncertainties and external disturbances. The popularity of PID controller can be attributed partly to their robust performance in a wide range of operating conditions and partly to their functional simplicity, which allows engineers to operate them in a simple, straightforward manner [19]. Therefore, the PID controller is widely used in industrial applications. However, the PID parameters are often tuned by experiences or simply by trial and error. In this paper, the PID parameters can be obtained systematically according to desired performances. In practical robotic systems, uncertainties which may effect the tracking performance are inevitable. In this paper, a neural network system is introduced to learn the uncertainties by an adaptive algorithm, that is, the adaptive neural networks are used to compensate the plant uncertainties. Actually, the adaptive neural networks can only approximate the plant uncertainties. The approximation error between adaptive neural networks and plant uncertainties do exist. Therefore, a sliding-mode control action is included to eliminate the effect of approximation error. The aim of this paper is to find a PID-type controller incorporating neural network compensation and sliding-mode control action such that the closed-loop robotic systems are stable. In addition to the stability, tracking performance is also an important issue in robotic control system design. In this paper, different objectives tracking performance, tracking performance, including and regional pole constraints are considered in robotic systems under plant uncertainties and external disturbances. The tracking performance is related to the linear quadratic (LQ) form of tracking error and control input [20], [21]. Since uncertainties are involved, it is not easily tractable by minimizing the performance index directly. Therefore, in this paper a suboptimal approach is taken by minimizing the upper bound tracking performance is of the performance index. The related to the attenuation property with respect to the external tracking control design has been disturbances. Recently, widely studied for its capability of disturbance attenuation tracking control design often results [22]–[25]. However, in a “high-gain” controller. In this situation, the closed-loop poles are needed to be constrained in a suitable stable region. Note that the regional pole constraints are also related to the closed-loop system performance including decay rate, max overshoot, rise time, settling time, etc. [26], [27]. The sufficient conditions are developed for different objectives in terms of linear matrix inequality (LMI) formulation. The
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interesting combinations for different objectives are introduced in this paper, which include PID tracking control design PID tracking with regional pole constraints and mixed control design with regional pole constraints. These multiobjective PID control problems are characterized in terms of eigenvalue problem (EVP). The EVP can be efficiently solved by the LMI toolbox in Matlab. The proposed methods are simple and the PID control gain for the multiobjective control schemes can be obtained systematically. The paper is organized as follows: The model description and problem formulation is presented in Section II. In Section III, adaptive algorithm and sliding-mode control for uncertain robotic systems are considered. In Section IV, LMI formulations for different design specifications are introduced, while multiobjective PID tracking control design of uncertain robotic systems is considered in Section V. In Section VI, simulation examples are provided to demonstrate the design procedures. Finally, concluding remarks are made in Section VII. II. MODEL DESCRIPTION AND PROBLEM FORMULATION The dynamic equations of a -joint robotic manipulator with revolute joints can be expressed as [1]–[3] (1) where the following notations apply: vectors of joint positions; matrix of moment inertia; vector of coriolis forces; vector of gravitational force; vector of applied torques; external disturbances. Remark 1: in the robot model (1) is symmetric posi1) The matrix . tive-definite for every 2) The external disturbances are assumed to be unknown but bounded. In practical robotic systems, the system parameter matrices , , and are not exactly known due to plant uncertainties or parameter variations. In this situation, the system parameter matrices in (1) can be expressed as follows:
(2) , , and are the nominal estimates of where , , and , respectively, and , , are the perturbed parameter matrices. and Therefore, the motion equations of the robotic manipulator in (1) can be expressed as follows: (3) Let us consider the following control law: (4)
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where PID-type control action to be designed; sliding-mode control to be designed; auxiliary control signal to be specified; desired trajectory vector which is assumed to be con). tinuously differentiable (at least By substituting (4) into (1), we obtain (5)
we note that PID controller is useful for reducing steady-state error and improving the transient response. Therefore, the tracking dynamics in (13) can be written as (15) The control signal is proposed to cancel the uncertain term . However, exact cancellation may not be possible since is unknown. A neural network is proposed to where is a vector conapproximate the uncertain term taining the tunable network parameters. Let
or (6) where
.. .
.. .
is defined as tracking error, i.e., (7)
and thus
.. .
(8)
..
.
..
.
..
.
.. . .. .
If we define
(16) where (9)
.. .
(10) and
..
.
..
.
..
.
.. .
and (11)
with error, i.e.,
and
is defined as integration of tracking
(12) then the tracking error dynamic equations can be expressed as follows:
The neural networks for are composed of nonlinear neurons in every hidden layer and linear neurons in the input and output layers. For the simplicity of design, for are put in the the adjustable weighting output layers of the following signal-output neural networks:
(13) where
(17) with
and Remark 2: It is easy to check that Let us specify the PID controller as
.. .
is controllable. and (14)
where
Remark 3: By the choice of (14), it is obvious that is corresponding to a standard PID-type control action. In general,
.. .
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for , where , according to the multilayer neural network approximation theorem [15], must be a nonconstant, bounded, and monotonically increasing continuous function. In this paper, the following hyperbolic tangent function is used:
By substituting (20) into (15), we obtain
(21) (18) is a function of the state . where Remark 4: in (17) are all of single1) The neural networks output network with one nonlinear layer containing hidden neurons. is 2) In general, the order of the neural network can be approximate as close given so that as possible. and are specified beforehand in this 3) The weighting paper. Let us define the optimal approximation parameter [14], [15] (19) Remark 5: obviously exists and explicit ex1) It should be noted that is not required since this pression for computation of value can be learned by using an adaptive algorithm in this paper. and denote the sets of suitable bounds on and . 2) 3) It is assumed that and never reach the boundary of and , otherwise, the projection algorithm which is described in the Remark 7 must be introduced to prevent the divergence of . 4) Unlike the conventional adaptive control, the adaptive neural networks do not require the linear parameterization property of robotic systems. By the optimal approximation in (19), we obtain
where (22) Our design procedure is divided into three steps. In the first step, an adaptive algorithm for updating is developed such is tuned to optimally cancel that the neural network . Under such a circumstance, the term the uncertain term will finally vanish. In the second step, a sliding-mode control is developed to eliminate the effect of . In the third step, sufficient conditions approximation error in terms of LMI formulations for the existence of the PID-type are developed for different objectives. The objeccontrol performance, performance, tives under consideration ( and regional constraints on the closed-loop poles) are described as follows. Performance: An LQ performance related to the tracking error and control action is considered as follows [20], [21]: (23) and . A straightforward obwhere jective is to minimize this performance. However, this objective is not easily tractable, since uncertainties involve in tracking error dynamics. Therefore, in this paper, a suboptimal approach is taken by minimizing the upper bound of the performance index. Performance: An performance is considered as follows [24], [25]:
(20) where tion error with
denotes optimal approximafor
where denotes a known (scale) bound on the optimal ap. proximation error is a known bound for Remark 6: It is assumed that . In practice, it is often hard to have a concrete idea , however, it is much easier to about the magnitude of begin with a rough, intuitive idea about this bound, and then iterate the design process and adjust it, until the bound is close to the right value [13]. Since neural networks are “universal can be made as small as possible by approximators,” a proper construction of the neural networks [14], [15]. It is represents the magnitude important to keep in mind that of error between uncertain dynamics and neural networks when the “best” parameters are used within the neural network system.
(24) , , and are prescribed where attenuation values which denote the worst case effect of the external disturbances on tracking error . The performance in (24) is that physical meaning of the effect of on must be attenuated below a desired level from the viewpoint of energy, no matter what is, i.e., the gain from to must be equal to or less than a prescribed value . In general, is chosen as a positive small value less than one for attenuation of . Regional Pole Constraints: The location of the in (21) effect the closed-loop poles of performance of the closed-loop system, i.e., the stability, the decay rate, the maximum overshoot, the rise time, and the settling time [26], [27]. Therefore, it is an interesting work for control engineer to design such that the closed-loop poles of the control gain
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lie in a suitable subregion of the left-half
where
plane. ..
III. THE ADAPTATION ALGORITHM AND SLIDING-MODE CONTROL FOR TRACKING OF UNCERTAIN ROBOTIC SYSTEMS As described in the above section, an adaptive algorithm for updating and a sliding-mode control are developed in this is tuned to opsection such that the neural networks and the effect of aptimally cancel the uncertain term is eliminated by , respectively. We first proximation error (thus consider the case when external disturbances ). Let us define the Lyapunov function for (21) as (25)
.. .
.
.. .
..
.
and
where and is easily verified that
denotes the th column of
, then it (31)
From (28), (30), and (31), we obtain (32)
By differentiating (25), we obtain
is controllable, there exists feedback matrix Since such that the poles of lie in the stable region. Therefore, by the Lyapunov stability theorem, there exists symmetric positive-definite such that (33) (26) Given
such
From (33), we conclude that there exists that
(34)
from (22) and the following update law (27)
we obtain (28) Remark 7: 1) If the constrained problem due to the opthat belongs to some timal approximation parameter preassigned compact set is considered, then additional tools concerning the projection algorithm can be used to analyze and this bounded problem. Suppose where and . Then the parameter law in (27) must be modified as follows [28]:
Remark 8: 1) It should be noted that even though is called a “sliding-mode” control, it does not guarantee that the state as traditionally trajectory will “slide” along the manifold guaranteed with nonadaptive sliding-mode control [18]. 2) If the are considered in external disturbances the sliding-mode control and their bounds are known, from (11) we obtain (35) is nonsingular]. If
which are also bounded [since for then the sliding-mode control
(36)
can be chosen as
(29) if ], otherwise
where
or [
with and . 2) By the projection algorithm in (29), the tunable parameters are bounded. It can be verified then for all [28]. that if If the sliding-mode control signal is chosen as .. .. .
.. .
. ..
.
.. .
and
.. . .. . (30)
..
.
.. .
.. . (37) .. . By the similar argument as above, it is easily verified that there such that . 3) The sliding-mode exists control can introduce a high-frequency signal, known as chattering phenomenon, to the plant which may excite unmodeled dynamics. To avoid this, the following smoothed control action
TSENG AND CHEN: MULTIOBJECTIVE PID CONTROL DESIGN
is considered [29]. The control is replaced by
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function in the sliding-mode which is defined as follows: if (38) otherwise.
where is a prescribed adaptive gain (constant) and and are initial values (constants) of and , respectively. in (43) is some conSince the latter term stant and according to the analysis above, we can formulate the Performance by minsuboptimal PID control problem with subject to (42) as follows: imizing the upper bound of
IV. LMI FORMULATION OF THE DESIGN SPECIFICATIONS In this section, sufficient conditions in terms of LMI formuare devellations for the existence of the PID-type control oped for different objectives, respectively. The objectives under tracking performance, tracking consideration include performance, and regional constraints on the closed-loop poles. A. PID Control With
Tracking Performance
An LQ performance related to the tracking error and control action is considered as
subject to
and (42)
(44)
, (42) is equivalent to
Note that, by defining
(45) and by the Schur complements [30], (45) is With equivalent to the following LMI:
(39) and . A straightforward objective is where to minimize this performance. However, this objective is not easily tractable, since uncertainties are involved in tracking error dynamics. Therefore, in this paper, a suboptimal approach is taken by minimizing the upper bound of the performance index. , we obtain In the case of
Therefore, the suboptimal mulated as
subject to
(46) problem in (44) can be refor-
and
(46)
(47)
Note that the minimization problem in (47) is not the standard form of the LMI problem. However, the minimization problem in (47) can be transformed into an LMI problem as the following such that procedure. By introducing a new variable (48) Note that (48) is equivalent to the following LMI: (49) and the minimizaTherefore, tion problem in (47) can be transformed into the following LMI problem:
subject to (46) (40) By the update law in (27) and the sliding-mode control in (30), we obtain
Note that if
(50)
Remark 9: The minimization problem in (50) is a standard LMI problem which is also called eigenvalue problem (EVP). The EVP can be efficiently solved using convex optimization technique such as interior point algorithm [30]. Software packages such as LMI toolbox in Matlab are developed for this purpose [31]. B. PID Control With
(41)
and (49)
Tracking Performance
In this subsection, we consider the effect of the external disperturbance on the tracking error with the following formance:
(42) then we obtain (43)
(51)
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where , , and is a prescribed attenuation value which denotes the worst case effect of on . From (51), we obtain
then we obtain the following
tracking performance:
(55) Note that, by defining
, (54) is equivalent to
(56) With equivalent to
and by the Schur complements [30], (56) is
(where
By the change of variable is equivalent the following LMI:
(57) ), (57)
(52) By the update law in (27) and the sliding-mode control in (30), we obtain
(58) tracking performance, we can To obtain better robust minimize subject to (58) as the following EVP:
subject to and
(58)
(59)
Remark 10: The effect of the external disturbances deteriorate not only the tracking performance but also the stability of the control systems. The boundedness of subject to the external disturbance is discussed in the Appendix. The boundedness of is guaranteed by the projection algorithm in Remark 7. C. PID Control With Regional Pole Constraints in (21) The location of the closed-loop poles of concern with the performance of the closed-loop system, i.e., the stability, the decay rate, the maximum overshoot, the rise time, and the settling time. Therefore, it is interesting work for such that the control engineers to design the control gain lie in a suitable subregion of closed-loop poles of the left-half plane. The interesting region for control purposes of complex number such that is the set and (53) Note that if
(60) as shown in Fig. 1. lie in the The LMI formulations for the poles of are characterized as the following LMIs [26], region such that [27]: if there exists symmetric (61)
(54)
(62)
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A.
PID Tracking Control With Regional Pole Constraints
PID control with regional The combination objectives of pole constraints can be characterized as the following EVP:
subject to (58)
and
(64)–(66)
(67)
PID Tracking Control With Regional Pole
B. Mixed Constraints
PID control The combination objectives of mixed with regional pole constraints for a prescribed attenuation level can be characterized as the following EVP:
subject to Fig. 1. Region of S ( ;
r; ).
(49), (57) and
and
(63) With
, the above LMIs are equivalent to
(46) (64)–(66)
(68)
Remark 12: From the analysis above, the most important ) by task in this paper is to find the and (and thus PID control with regional pole solving the EVP in (67) for PID control constraints problem or in (68) for mixed with regional pole constraints problem. If there exists a solution and in (67) or (68), then the PID control gain for , the adaptive update law and the sliding vector can be constructed at one stroke.
(64)
VI. SIMULATION EXAMPLES
(65)
Consider a two-link robotic manipulator, as shown in Fig. 2 [1]. The parameter matrices for the manipulator are
and
(66) Remark 11: 1) Other interesting regions for control purposes and the corresponding LMI formulations can be found in [26] and and [27]. 2) From the analysis above, if there exist for (64)–(66), then the poles of lie in the region . V. MULTIOBJECTIVE PID CONTROL DESIGN OF UNCERTAIN ROBOTIC SYSTEMS By the analysis in the previous sections, the interesting combination of different objectives is considered for the uncertain PID tracking robotic systems in this section including control with regional pole constraints and mixed PID tracking control with regional pole constraints which are described below.
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Fig. 2.
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 31, NO. 6, NOVEMBER 2001
Configuration of a two-link robotic manipulator.
where
and the shorthand notations , , , and are used. For the convenience of simulations, the nominal parameters , of the robotic manipulator are given as (kg), (m), (m), and m/s , and the and are perturbed as the following form: parameters and respectively. Note that the perturbations are up to 50% of their nominal estimates. The external disturbances are assumed to . The desired be square wave with magnitude 2 and period and , rereference trajectories are spectively. Obviously, the parameter uncertainties and external disturbances are extremely large. For the convenience of design, the parameters and are used for the following examples. PID tracking control design without reExample 1: ( gional pole constraints). To illustrate the importance of regional pole constraints, we PID control design without refirst consider the case of gional pole constraints. This can be done by solving the EVP in and the PID (59). In this case, we obtain control gain
which makes the poles of locate at , , , , , and . From a practical point of view, the control gain is too large to be implemented. PID tracking control design with regional Example 2: ( pole constraints). PID In the second example, we consider the case of control design with regional pole constraints. In the first step, we construct the neural networks. The neural network radial and basis functions are chosen to be with the following components:
Since the state variable choose
is not available in general, we . Moreover and
where and . Remark 13: 1) The number of basis functions in the neural networks heavily influence the complexity of a neural networks. In general, the larger is the number, the more complex is the neural networks and the higher is the cancellation performance of the neural networks. Hence, there is always a tradeoff between complexity and accuracy in the choice of the number of the basis functions. The choice is usually quite subjective and is based on some experiences. In the above design, seven basis functions for both neural networks and are chosen in which and for are selected as 10, the biases 7, 3, 0, 3, 7, and 10, respectively. On the other hand, the and for and weighting in the basis functions heavily influence the smoothness of the input–output surface determined by the neural networks. In general, the sharper is the basis function, the less smooth is the and input–output surface. The choice of the coefficients is also subjective and based on some experiences. Hence, for convenience, these weighting coefficients are all selected to be equal to one or zero. For the convenience of simulations, we choose , , , and . In the second step, we specify the parameters for and . sliding-mode control with In the third step, we can solve the EVP in (67) with pole . In this case, we constraints in the region of and the PID control gain obtain
which makes the poles of locate at , , , and . The simulation results are shown in Figs. 3–8. The trajectoare shown in Fig. 3, while the trajectories of ries of and and are shown in Fig. 4. The trajectories of and are are shown shown in Fig. 5, while the trajectories of and in Fig. 6. The control inputs are shown in Figs. 7 and 8. The PID control design with simulation results for the case of
TSENG AND CHEN: MULTIOBJECTIVE PID CONTROL DESIGN
Fig. 3. Trajectories of q (solid line); q for the H PID control with regional pole constraints (dash-dotted line); and q for the mixed H =H PID control with regional pole constraints (dotted line).
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Fig. 5. Trajectories of q (solid line); q for the H PID control with regional pole constraints (dash-dotted line); and q for the mixed H =H PID control with regional pole constraints (dotted line).
Fig. 6. Trajectories of q_ (solid line); q_ for the H PID control with regional pole constraints (dash-dotted line); and q_ for the mixed H =H PID control with regional pole constraints (dotted line). Fig. 4. Trajectories of q_ (solid line); q_ for the H PID control with regional pole constraints (dash-dotted line); and q_ for the mixed H =H PID control with regional pole constraints (dotted line).
regional pole constraints but without neural network compensation and sliding-mode control are shown in Figs. 9–12. The tracking performance for the case with neural network compensation is much better than the case without. Therefore, PID control design incorporating the neural network compensation and regional pole constraints clearly results in satisfactory tracking performance. PID control design with Example 3: (mixed regional pole constraints). In the third example, we consider the case of mixed PID control design with regional pole constraints. The first and second steps are the same as that in Example 2. In the third step, we can solve the EVP in (68) with pole constraints also in the re, initial condition , , , gion of
Fig. 7. Control input for the H PID control (dash-dotted line) and the mixed H =H PID control (dotted line).
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Fig. 8. Control input for the H PID control (dash-dotted line) and the mixed H =H PID control (dotted line).
Fig. 9. Trajectories of q (solid line) and q for the H PID control (dash-dotted line) and mixed H =H PID control (dotted line) with regional pole constraints without neural network compensation.
Fig. 11. Trajectories of q (solid line) and q for the H PID control (dash-dotted line) and mixed H =H PID control (dotted line) with regional pole constraints without neural network compensation.
Fig. 12. Trajectories of q_ (solid line) and q_ for the H PID control (dash-dotted line) and mixed H =H PID control (dotted line) with regional pole constraints without neural network compensation.
, , and a prescribed at. In this case, we obtain tenuation level and the PID control gain
Fig. 10. Trajectories of q_ (solid line) and q_ for the H PID control (dash-dotted line) and mixed H =H PID control (dotted line) with regional pole constraints without neural network compensation.
which makes the poles of locate at , , , and . For comparison, the simulation results of the mixed PID control design with regional pole constraints are also shown in Figs. 3–8. The simulation results for the case of mixed PID control design with regional pole constraints but without neural network compensation and sliding-mode control are also shown in Figs. 9–12. From the simulation PID control design with results, we observe that the regional pole constraints has fast decay response; however, it
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also has larger overshoot and control input. The characteristics tracking performance share the properties of of mixed tracking performance and tracking performance.
By the sliding-mode control in (30), we obtain
VII. CONCLUSIONS
(72)
In this paper, a PID-type controller incorporating neural network elimination scheme and sliding-mode control action tracking performance, for different objectives including tracking performance, and regional pole constraints is developed in robotic systems under plant uncertainties and external disturbances. The adaptive neural networks are used to cancel the plant uncertainties. The sliding-mode control action is included to eliminate the effect of approximation error. The tracking performance is related to the LQ form of tracking tracking performance is error and control input. The related to the attenuation property with respect to the external disturbances. The regional pole constraints are related to the closed-loop system performance including decay rate, max overshoot, rise time, settling time, etc. The sufficient conditions are developed for different objectives in terms of LMI formulation. The interesting combinations for different objectives are introduced in this paper, which PID control design with regional pole constraints include PID control design with regional pole conand mixed straints. These multiobjective PID control problems of the uncertain robotic systems are characterized in terms of EVP. The EVP can be efficiently solved by the LMI toolbox in Matlab. The proposed methods are simple and the PID control gain for different objectives can be obtained systematically. Simulation results indicate that the desired tracking performance for the multiobjective control schemes of the uncertain robotic systems can be achieved using the proposed methods.
Note that from (57) we obtain
(73) Therefore, we obtain (74) Since is nonsingular, i.e., bounded], we obtain
is bound [because is bounded, and is also
(75) where Whenever
denotes the minimal eigenvalue of
.
. According to a standard Lyapunov extension [32], [33], this demonstrates that the trajectories of the closed-loop system (21) are uniformly ultimately bounded (UUB) [34]. REFERENCES
APPENDIX In this Appendix, the boundedness of is discussed in the presence of external disturbances . Let us define the Lyapunov function for (21) as (69) By differentiating (69), we obtain
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H
(70)
H
By the update law in (27), we obtain
(71)
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H
[31] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox. Natick, MA: The MathWorks, 1995. [32] J. P. LaSalle, “Some extensions of Lyapunov’s second method,” IRE Trans. Circuit Theory, pp. 520–527, Dec. 1960. [33] K. Narendra and A. M. Annaswamy, “A new adaptation law for robust adaptation without persistent excitation,” IEEE Trans. Automat. Contr., vol. AC-32, pp. 134–145, 1987. [34] H. K. Khalil, Nonlinear Systems. Englewood Cliffs, NJ: Prentice-Hall, 1996.
Chung-Shi Tseng (M’01) received the B.S. degree from the Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan, R.O.C., in 1984, the M.S. degree from the Department of Electrical Engineering and Computer Engineering, University of New Mexico, Albuquerque, in 1987, and the Ph.D. degree in electrical engineering from National Tsing-Hua University, Hsinchu, Taiwan, in 2001. He is now an Associate Professor at Ming Hsin Institute of Technology and Commerce, Hsinchu. His research interests are in nonlinear robust control, adaptive control, fuzzy control, and robotics.
Bor-Sen Chen (M’82–F’01) received the B.S. degree from Tatung Institute of Technology, Taiwan, R.O.C., in 1970, the M.S. degree from National Central University, Taiwan, in 1973, and the Ph.D. degree from the University of Southern California, Los Angeles, in 1982. He was a Lecturer, Associate Professor, and Professor at Tatung Institute of Technology from 1973 to 1987. He is now a Professor at National Tsing Hua University, Hsinchu, Taiwan. His current research interests include control and signal processing. Dr. Chen has received the Distinguished Research Award from National Science Council of Taiwan four times. He is a Research Fellow of the National Science Council and the Chair of the Outstanding Scholarship Foundation.
