Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.

An adaptive PID like controller using mix locally recurrent neural network for robotic manipulator with variable payload

92 views

Published on

Being complex, non-linear and coupled system, the robotic manipulator cannot be effectively controlled using classical proportional integral derivative (PID) controller. To enhance the effectiveness of the conventional PID controller for the nonlinear and uncertain systems, gains of the PID controller should be conservatively tuned and should adapt to the process parameter variations. In this work, a mix locally recurrent neural network (MLRNN) architecture is investigated to mimic a conventional PID controller which consists of at most three hidden nodes which act as proportional, integral and derivative node. The gains of the mix locally recurrent neural network based PID (MLRNNPID) controller scheme are initi- alized with a newly developed cuckoo search algorithm (CSA) based optimization method rather than assuming randomly. A sequential learning based least square algorithm is then investigated for the on- line adaptation of the gains of MLRNNPID controller. The performance of the proposed controller scheme is tested against the plant parameters uncertainties and external disturbances for both links of the two link robotic manipulator with variable payload (TL-RMWVP). The stability of the proposed controller is analyzed using Lyapunov stability criteria. A performance comparison is carried out among MLRNNPID controller, CSA optimized NNPID (OPTNNPID) controller and CSA optimized conventional PID (OPTPID) controller in order to establish the effectiveness of the MLRNNPID controller.

Published in: Technology
  • Be the first to comment

  • Be the first to like this

An adaptive PID like controller using mix locally recurrent neural network for robotic manipulator with variable payload

  1. 1. Research Article An adaptive PID like controller using mix locally recurrent neural network for robotic manipulator with variable payload Richa Sharma a , Vikas Kumar a,n , Prerna Gaur b , A.P. Mittal b a Department of Electrical and Instrumentation Engineering, Thapar University Patiala, 147004, India b Instrumentation and Control Engineering Division, Netaji Subhas Institute of Technology, Dwarka, New Delhi 110078, India a r t i c l e i n f o Article history: Received 27 November 2015 Received in revised form 13 January 2016 Accepted 25 January 2016 Available online 23 February 2016 Keywords: Robotic manipulator Cuckoo search algorithm Artificial neural networks Recurrent neural networks On-line learning a b s t r a c t Being complex, non-linear and coupled system, the robotic manipulator cannot be effectively controlled using classical proportional-integral-derivative (PID) controller. To enhance the effectiveness of the conventional PID controller for the nonlinear and uncertain systems, gains of the PID controller should be conservatively tuned and should adapt to the process parameter variations. In this work, a mix locally recurrent neural network (MLRNN) architecture is investigated to mimic a conventional PID controller which consists of at most three hidden nodes which act as proportional, integral and derivative node. The gains of the mix locally recurrent neural network based PID (MLRNNPID) controller scheme are initi- alized with a newly developed cuckoo search algorithm (CSA) based optimization method rather than assuming randomly. A sequential learning based least square algorithm is then investigated for the on- line adaptation of the gains of MLRNNPID controller. The performance of the proposed controller scheme is tested against the plant parameters uncertainties and external disturbances for both links of the two link robotic manipulator with variable payload (TL-RMWVP). The stability of the proposed controller is analyzed using Lyapunov stability criteria. A performance comparison is carried out among MLRNNPID controller, CSA optimized NNPID (OPTNNPID) controller and CSA optimized conventional PID (OPTPID) controller in order to establish the effectiveness of the MLRNNPID controller. & 2016 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction In the past few years, the process industry and medical field are significantly captured by the robotic manipulators as these sys- tems have the ability of quick and accurate positioning. The precise control of end-effector is the prime requirement for the use of these systems. Over the time, several advanced control techniques have been developed to control these systems. The conventional PID controller is the widely deployed controller in the industry, due to its simple and easy design, almost generalized training rules and cost-effectiveness. The performance of conventional PID controller falls short in applications which are non linear and uncertain [1] while almost all industrial processes are nonlinear and uncertain. Such applications demand real time tuning/adap- tation of controller gains in an on-line manner and hence tuning of PID controller in such situations is a challenging task. The con- ventional tuning methods such as Zeigler–Nicholas and Coon– Cohen [2] cannot be applied to a highly non-linear, coupled and uncertain robotic manipulator. Moreover a 2-degree of freedom (2-DOF) robotic manipulator is a coupled system and requires tuning of two PID controllers simultaneously. With the advance- ments in the computation capabilities of the microprocessors and consequently computational techniques several authors have used nature inspired and evolutionary optimization methods such as Genetic Algorithm (GA) [3], Multi-Objective GA [4], Tabu Search [5], Particle Swarm Optimization (PSO) [6] and CSA [7] etc. for tuning of the conventional PID controller. These nature inspired and evolutionary optimization methods provide fixed optimal gains and using these fixed gains for uncertain and nonlinear processes it is very difficult to obtain the optimum performance. Therefore, an efficient and effective on-line tuning mechanism is required for the precise position control of an end effector. The introduction of fuzzy controllers has been significant in the field of control engineering due to their capability of nonlinear mapping; flexible and model-free design etc. These controllers are also able to cope up with uncertainties and nonlinearities [8–9]. The performance of classical PID controller can be enhanced to some extent due to its collaboration with fuzzy logic controllers (FLC) the conventional PID controller along with FLC can Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/isatrans ISA Transactions http://dx.doi.org/10.1016/j.isatra.2016.01.016 0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved. n Corresponding author. E-mail addresses: richasharma_7@yahoo.co.in (R. Sharma), sainivika@gmail.com (V. Kumar), prernagaur@yahoo.com (P. Gaur), mittalap@gmail.com (A.P. Mittal). ISA Transactions 62 (2016) 258–267
  2. 2. effectively deal with linear, nonlinear, higher order and uncertain systems [10–11]. In order to enhance the performance of conventional PID controller several authors have used fuzzy logic method to obtain the optimal gains of different structures of PID controller in an on- line manner for different plants such as induction machine, ship plant [12–14], AVR system [15], hybrid electric vehicle system [16] and microgrid [17]. Various authors investigated the use of fuzzy logic for tuning of different structures of PID controllers for robotic manipulators [18-22]. From the literature, it is clear that the performance of con- ventional PID or intelligent control techniques can be improved significantly by optimizing the critical parameters of these control methods using evolutionary optimization methods. GA and PSO are the mostly employed evolutionary optimization methods for the optimization of various intelligent controller parameters. Among the nature inspired optimization algorithms integrated with intelligent techniques, recently developed CSA based opti- mization method developed by Yang and Deb [23] has been established as a better optimization technique in literature in terms of convergence rate and optimization performance for the robotic manipulators [24–26]. CSA requires lesser number of parameters for the initial settings and the performance of this algorithm is also independent of parameters chosen [25–26]. Application of ANN in the areas of dynamical control systems has recently witnessed tremendous growth due to outstanding learning, adaptation and generalization capabilities. However implementation of fuzzy logic or ANN requires large number of parameters to be determined off line prior to the implementation [27]. Due to the unavailability of faster training algorithms for ANN, and convergence to local minima, the application of ANN in an on-line tuning of the controller remains less explored [28]. Cong and Liang proposed a mix locally recurrent neural network based PID like controller for motion control systems [29] using the conventional back propagation algorithm. The training time for the conventional gradient/back propagation based method was large and no systematic procedure for gain initialization was proposed. Moreover, the conventional gradient based tuning method suffers from certain limitations like convergence to a local minima, very large training time, and stability. The training time of the training algorithm depends on the learning rate as too small value of learning rate results in large training time and too large value of learning rate leads to stability problems. Based upon the faster training algorithm proposed by Huang et. al. [30], a mix locally recurrent neural network based PID like controller is implemented by Kumar et. al. for the precise position control of PMSM servo drives [31]. In their work the initial gains were not randomly assumed but were calculated using Moore–Penrose generalized inverse method which provides minimum norm least square (MNLS) solution. The training time for the proposed method is at least 100–1000 times faster than the conventional gradient based method. MNLS solution suffers from overtraining problems, as it tries to minimize the error for entire data set [30]. For an over trained network an on-line learning scheme cannot ensure the best tracking error. Also, the stability analysis of the proposed controller was not presented in both [30] and [31]. In order to overcome the limitations mentioned in [29] and [31], in the presented work all parameters of the MLRNNPID controller are optimized using CSA. Once the optimized parameters of the MLRNNPID controller are obtained, the weights of the hidden layer are re-adjusted using Moore Penrose generalized inverse. An on-line sequential learning algorithm based on the recursive least square solution is then derived for an on-line training of the MLRNNPID controller. The on-line sequential learning algorithm is quite fast and can never converge to a local minima as it is based on Moore–Pen- rose generalized inverse. The motivation of this work is to preserve the favorable char- acteristics of the conventional PID controller and integrate them with the learning and powerful approximation capabilities of the MLRNNPID controller for trajectory tracking control of a highly complex, non-linear and coupled 2- DOF robotic manipulator. Mix locally recurrent NN provide a partial memory to the NNPID con- troller, and hence the approximation and on-line learning capabilities of the NNPID controller will increase manifolds. The initial gains i.e. the weights of the hidden layer as well as output layer are calculated using CSA based optimization method which is established as a better optimization method as compared to GA and PSO [25]. An on- line sequential learning algorithm is then derived to tune the output weights of the NNPID controller in an on-line manner. The stability analysis of the presented control scheme is investigated out using Lyapunov's approach. Robustness testing of the proposed controller is presented under parametric uncertainty and external disturbances and finally a performance comparison is carried out among MLRNNNNPID controller, CSA tuned NNPID (OPTNNPID) controller and CSA tuned conventional PID (OPT PID) controller. 2. Mathematical model of TL-RMWVP The dynamic model of SCARA type TL-RMWVP, expressed in Eq. (1), has been given by Lin [32]. Fig. 1 represents the robotic manipulator with two rigid links and a payload at the tip. Also, Table 1 lists the parameters of TL-RMWVP plant for the simulation. Q11 Q12 Q21 Q22 " # €θ1 €θ2 " # þ P11 P21 " # þ f 1 f 2 " # þ g1q g2q " # ¼ τf ln1 τfln2 " # ð1Þ where Q11 ¼ I1 þI2 þm1l 2 c1 þm2l 2 1 þm2l 2 c2 þ2m2l1lc2 cos θ2 þm33l 2 1 þm33l 2 2 þ2m33l1l2 cos θ2 Q12 ¼ I2 þm2l 2 c2 þm2l1lc2 cos θ2 À Á þm33l 2 2 þm33l1l2 cos θ2 À Á Q21 ¼ Q12 Q22 ¼ I2 þm2l 2 c2 þm33l 2 2 P11 ¼ Àm2l1lc2 2 _θ1 þ _θ2 _θ2 sin θ2 Àm33l1l2 2 _θ1 þ _θ2 _θ2 sin θ2 P21 ¼ m2l1 _θ 2 1lc2 sin θ2 þm33l1 _θ 2 1lc2 sin θ2 f 1 ¼ b1q _θ1 f 2 ¼ b2q _θ2 g1q ¼ m1lc1gcos θ1 À Á þm2g lc2 cos θ1 þθ2 À Á þl1 cos θ1 À ÁÀ Á Fig. 1. Two-link planar robotic manipulator with payload attached. R. Sharma et al. / ISA Transactions 62 (2016) 258–267 259
  3. 3. þm33g l2 cos θ1 þθ2 À Á þl1 cos θ1 À ÁÀ Á g2q ¼ m2lc2gcos θ1 þθ2 À Á þm33lc2gcos θ1 þθ2 À Á where θ1 and θ2 are the positions of links; τf ln1 and τfln2 represent the control outputs; m1 and m2 are masses; l1 and l2 are the lengths; lc1 and lc2 presents the distances from the joints to their center of gravity; I1 and I2 represent lengthwise centroid inertia;b1q and b2q are the coefficients of friction at joints; f1 and f2 represent the coefficients of dynamic friction of Link1 and Link2 respectively. Also, m33 represents the mass of variable payload and its value varies between 0.5699 kg to 0.14172 kg as shown in Fig. 2. 3. MLRNN based PID controller For the past few years, the dynamic neural networks have been increasingly used in the area of robotic manipulator for designing of adaptive controllers because of their enhanced prediction and adaptation capabilities. Dynamic neural networks are classified as: Elman NN, Feed-forward networks with filters, globally recurrent NN, and locally recurrent NN. The MLRNN possess lesser number of parameters as compared to other dynamic NN. Hence MLRNN can be trained easily and training time is also small [33]. Apart from the lesser number of tuning parameters, the local feedback in mix locally recurrent NN provides partial memory to the network which further increases the approximation and learning cap- abilities of the network. Modeling of MLRNN-PID like controller created using mix locally recurrent NN is presented in the fol- lowing section. 3.1. MLRNN-PID controllers The structure of MLRNN based PID like controller is shown in Fig. 3. The structure consists of three nodes which act like pro- portional, integral and derivative nodes. The output of hidden layer neurons H1; H2 and H3 can be expressed in terms of elink1 and elink2 as follows: H1 kð Þ ¼ ϕ h11elink1 kð Þþh21elink2 kð Þð Þ H2 kð Þ ¼ ϕ h12elink1 kð Þþh22elink2 kð Þð ÞþH2 kÀ1ð Þ H3 kð Þ ¼ ϕ h13elink1ðkÞþh23elink2ðkÞþh13elink1ðkÀ1Þþh23elink2ðkÀ1Þð Þ 9 = ; ð2Þ where h ¼ h11h12h13 h21h22h23 # ; is the hidden layer weight matrix and Q ¼ Q11Q12Q13 Q21Q22Q23 # represents the output layer weights. If zÀ1 is a unit delay operator, the output of the node 2 as well as 3 can be represented using (2) as follows: H2 kð Þ ¼ ϕ h12elink1ðkÞÞþh22elink2ðkÞð Þ 1ÀzÀ 1 ð3Þ H3 kð Þ ¼ ϕ h13elink1ðkÞþh23elink2ðkÞð Þð1ÀzÀ1 Þ ð4Þ Considering the single input and single output system, the hidden layer output matrix can be expressed as: H ¼ H1 H2 H3 Â Ã The output weight matrix can be expressed as Q ¼ Q1 Q2 Q3 Â Ã The final output of the MLRNN-PID controller is τ ¼ Q1H1 kð ÞþQ2H2 kð ÞþQ3H3 kð Þ In matrix form, the above equation can be expressed in matrix form as τ ¼ HQ ð5Þ 3.2. On-line training algorithm The training algorithm for locally recurrent NN is developed in this section: E ¼ RÀHQ; E 2 ¼ ðRÀHQÞT ðRÀHQÞ E 2 ¼ RT RÀ2HRT Q þQ2 HT H The training of MLRNN-PID controller is required to adjust Q and H so that the error E 2 is minimized. ∂‖E‖2 ∂Q ¼ À2RT Hþ2QHT H ð6Þ ∂‖E‖2 ∂H ¼ À2Q ∂HT ∂w RT þ2Q2∂QT ∂x Q Table 1 Parameters for the SCARA type TL-RMWVP plant. Parameters Link1 Link2 Mass 0.392924 kg 0.094403 kg Acceleration due to gravity (g) 9.81 m/s2 9.81 m/s2 Length 0.2032 m 0.1524 m Distance from the joint to its center of gravity 0.104648 m 0.081788 m Lengthwise centroid inertia of link 0.0011411 kg m2 0.0020247 kg m2 Friction at joints 0.141231 N-m/radian/s 0.3530776 N-m/radian/s Fig. 2. Payload variations at tip. Fig. 3. MLRNN PID controller. R. Sharma et al. / ISA Transactions 62 (2016) 258–267260
  4. 4. The optimum value of H and Q is obtained when ∂‖E‖2 ∂h and ∂‖E‖2 ∂Q ¼ 0. From (6), the output weight matrix is calculated as Q ¼ RT H HT H or Q ¼ Hf T ð7Þ where Hf is the Moore–Penrose generalized inverse of matrix H, which gives minimum norm least square solution. If the weights of hidden layer are known; the weights of the output can be calcu- lated analytically in a single step using (7). In a traditional gradient based method the error has to back propagate recursively so the training time for conventional gradient based method is quite large. As the output weights are calculated in a single step the training speed can be several 100–1000 times greater than the conventional gradient based methods. In [30], and [31] the input weights are assumed randomly, in this work initial weights used are calculated using CSA. The CSA optimized output weights are used in MLRNNPID controller as output weights and the hidden layer weights are readjusted to obtain the minimum norm least square solution. The optimized weights of the output layer as calculated using CSA are used to reinitialize the hidden layer matrix using H0 ¼ Qf T ð8Þ The hidden layer weights hij are re-adjusted using (2), Once the hidden layer and output layer weights are obtained, for each incoming data chunk or single data a sequential learning algorithm which is based on the recursive least square solution is employed to update the weights of the MLRNNPID controller. The training algorithm is summarized in Fig. 4. Q0 ¼ ðHT 0H0ÞÀ 1 HT T ð9Þ For the next chunk of data, the following function is to be minimized H0 H1 # Q À T0 T1 # ð10Þ From (9), taking both data samples into account, the output weight matrix is represented as: Q1 ¼ K À1 1 H0 H1 #T T0 T1 # ð11Þ where K1 ¼ H0 H1 #T H0 H1 # ð12Þ For sequential learning we have to express Q1 as a function of Q0. Using (11) K1 can be expressed as K1 ¼ HT 0HT 1 h i H0 H1 # ð13Þ K1 ¼ K0 þHT 1H1 ð14Þ H0 H1 #T T0 T1 # ¼ HT 0T0 þHT 1T1 ¼ K0Kð À 1Þ 0 HT 0T0 þHT 1T1 ¼ K0Q0 þHT 1T1 ¼ ðK1 ÀQT 1Q1ÞG0 þQT 1Y1 ¼ K1Q0 ÀHT 1H1Q0 þHT 1T1 ð15Þ Q1 can be written using (15) and (11) Q1 ¼ K À1 1 K1Q0 ÀHT 1H1Q0 þHT 1T1 ¼ Q0 þK À 1 1 HT 1 T1 ÀH1Q0ð Þ ð16Þ From (14), the value of K for (kþ1)th data sample is given as Kk þ 1 ¼ Kk þHT k þ 1Hk þ 1 ð17Þ K À 1 k þ 1 ¼ ðKk þHT k þ1Hk þ 1ÞÀ 1 ð18Þ Each incoming data samples the Kk þ1 can be updated using (19) with the help of Woodbury formula [34]: K À 1 k þ 1 ¼ K À 1 k ÀK À1 k HT k þ 1ðIþHk þ1K À1 k HT k þ 1ÞHk þ1K À 1 k ð19Þ Using Pk þ 1 ¼ K À 1 k þ 1, (15) and (12) can be written in a simpler form as follows: Pkþ 1 ¼ Pk ÀPkHT k þ1ðIþHk þ 1PkHT k þ 1ÞÀ 1 Hk þ1Pk ð20Þ Qk þ 1 ¼ Qk þPk þ 1HT k þ 1ðTk þ1 ÀHkþ 1QkÞ ð21Þ 3.3. Optimization of MLRNNPID and conventional PID controller using Cuckoo search algorithm As robotic manipulator is a multi-input, multi-output plant, therefore the conventional PID controllers require six parameters Fig. 4. Flowchart for the implementation. R. Sharma et al. / ISA Transactions 62 (2016) 258–267 261
  5. 5. which are to be tuned before the implementation while the implementation of MLRNNPID controllers require twelve para- meters. Cuckoo search, a recently developed optimization method which gives faster convergence performance is used to tune the parameters of the MLRNNPID controller. Yang and Deb proposed an excellent technique namely CSA for optimization purpose and it is formed on the unique parasitic breeding behavior of cuckoos [23]. It is designed on the cuckoo's cunning planning of finding the other species nest. The cuckoos are in search of a nest in which other host bird has just laid its eggs [23,35]. Despite its nascent stage, CSA has emerged out as an efficient optimization technique because of its invincible features. The parameters used for the initialization of this algorithm are lesser as compared to famous GA and PSO techniques. The con- vergence rate is independent of all of its parameters [35–38]. The large steps taken during the searching procedure can make it more effective and powerful than other competitive techniques. It also possesses the well-renowned and significant elitism feature. The cuckoo birds have some unbelievable abilities such as copying the call of host bird; copying the color and pattern of eggs of hosts etc. [34–36]. CSA has been implemented to find different cutting parameters for the milling operation and it has been found effective than many other optimization methods such as hybrid PSO, hybrid immune algorithm, feasible direction method, hand- book recommendations, GA and ACO [37]. The algorithm is based on the Lévy flight for searching the space [23]. The detailed implementation of CSA can be obtained from [23,38]. The selection of appropriate objective function is necessary to use any optimization technique. For the present work, the objec- tive functions chosen are integral of absolute error (IAE) of both Link1 and Link2 for minimization and are expressed by Eqs. (22) and (23) respectively. The aggregate fitness function AOF is designed as the weighted sum of IAE of both the links. The reason behind the selection of these objective functions is to reduce the error between actual and desired trajectories. Of 1 ¼ Z j e1ðtÞjdt ð22Þ Of 2 ¼ Z je2ðtÞjdt ð23Þ AOF ¼ w1Of 1 þw2Of 2 ð24Þ w1 and w2 are the weights assigned to objective functions Of1 and Of2 respectively. 3.4. Stability analysis of the proposed controller The stability analysis of the proposed controller is presented in this section. Assume the non linear dynamical system is given by the fol- lowing equation yk þ1 ¼ f xkð Þþg xkð Þuk ð25Þ Assumption 1. : For the non-linear dynamical system described by (25), ( a constant Q such that if uk ¼ HT k Qk;[39] then Tkþ 1 ÀXk þ1 ¼ Δk, where Tk þ 1 is the desired trajectory, Xkþ 1 is the system output and Δk rεj is a small positive constant. The tracking position error can be expressed as: ek þ1 ¼ Xk þ 1 ÀTkþ 1 or ek þ1 ¼ HT Q þΔk ð26Þ Referring to the recursive least square algorithm as described above by (20)–(21) Pk þ 1 ¼ Pk À PkHT k þ 1Hkþ 1Pk IþHk þ 1PkHT k þ 1 ð27Þ Qk þ1 ¼ Qk þPkHT k ek þ 1 ð28Þ where P0 ¼ HT 0H0 T and Q0 ¼ HT 0H0 À 1 HT 0T If Δk oεj , then the RLS algorithm as derived in the earlier section can be modified using dead zone. We can rewrite (27) and (28) as follows [40]: Pk þ 1 ¼ Pk À γkβkPkÀ 1HT k HkPk IþHkPkHT k ð29Þ Qk þ1 ¼ Qk ÀγkβkPkþ 1HT kþ 1ekþ 1 ð30Þ where βk ¼ 1 1þγkHT k PkHk ð31Þ γk ¼ 1 ‖ek þ 1‖2 1þγkHkPk À 1HT kð Þ 2 oϵ2 0 otherwise 8 : ð32Þ Desired Trajectory for link1 PID1/ NNPID1/ OPTNNPID1 Desired Trajectory for link2 2-DOFRobotic Manipulator (SCARA) Torque1 Torque2 Theta 1 Theta 2 + - + - d1 d2 ++ ++ PID2/ NNPID2/ OPTNNPID2 Fig. 5. Block diagram of the proposed strategy. Table 2 IAE and optimized parameters for implemented control schemes. Parameters OPT PID (Kp, Ki, Kd) OPT PID MLRNNPID Hidden layer Output layer Hidden layer Output layer Link 1 400, 67.6405, 15.2305 0.984, 0.994, 1.0 400, 136.3997, 39.5656 1.469, 0.258, 2.31 395.99, 134.8173, 39.1254 Link 2 45.791, 20.28, 4.26 1.0, 0.82, 0.053 60.9486,12.3286, 4.1424 0.987,0.1037, 4.21 60.5018, 12.6301, 4.5960 IAE Link1 0.0097 0.005731 0.003049 Link2 0.01853 0.01467 0.006418 R. Sharma et al. / ISA Transactions 62 (2016) 258–267262
  6. 6. 3.4.1. Stability analysis by Lyapunov method According to Lyapunov stability criterion for non-linear and uncertain systems, if Lyapunov functionΔVk o0, the closed loop system is stable. Theorem 1. : Under the assumption 1, the system described by (25) is stable and the weights in the controller are updated as per (20)– (21). Then a stable closed loop system will satisfy the following: (i) The position tracking error between the reference input and the system output converges to the small neighborhood of zero. (ii) Qk is bounded and Qk þ 1 ÀQkk converges to zero. (iii) All the weight matrices in the RLS algorithm remains bounded. From (26) we can obtain HT k þ 1Qkþ 1 þΔk ¼ βk HT k þ1Qk þ1 þΔk ¼ βkek þ1 ð33Þ Using (33), (30) can be re-written as Qk þ 1 ¼ Qk ÀγkPkHkðHT k Qk þ1 þΔkÞ ð34Þ Using following matrix property P þMNð ÞÀ 1 ¼ P À1 ÀPÀ 1 MðIþNPÀ 1 MÀ 1 ÞNPÀ 1 PÀ 1 k þ 1 can be calculated as PÀ 1 kþ 1 ¼ P À1 k þγkHkHT k ð35Þ Considering the following Lyapunov function as inspired by [40] Vk þ1 ¼ QT k þ1PÀ 1 kþ 1Qk þ1 ð36Þ Substituting the value of PÀ 1 k þ1 from (35) in (36), the Lyapunov function can be written as: ¼ QT k þ1 P À1 k þγkHkHT k Qk þ 1 ¼ QT k þ 1PÀ 1 k Qk þ1 þγk HT k Qk þ1 2 ¼ Qk À γkβkPkHT k ek þ 1 T P À1 k Qk À γkβkPkHT k ek þ 1 þγk HT k Qk þ 1 2 ¼ Vk þγk HT k Qk þ1 2 À2 γkHT k Qk HT k Qk þΔk þγ2 k HkPk À1HT k 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0 5 10 Time (s) Position(radian) MLRNNPID Link 1 Ref. Link1 MLRNNPID Link2 Ref. Link2 Opt NNPID link1 Opt. NNPID Link2 Opt PID Link1 Opt. PID link2 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 -0.01 0 0.01 0.02 Time (s) ErrorLink1(radian) MLRNNPID OPTNNPID OPTPID 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 -0.01 0 0.01 0.02 Time(s) ErrorLink2(radian) MLRNNPID OPTNNPID PID Error 0 0.5 1 1.5 2 2.5 3 3.5 4 -2 0 2 4 6 Time (s) TorqueLink1(Nm) NNPID OPT NNPID OPT PID 0 0.5 1,0 1.5 2.0 2.5 3.0 3.5 4.0 -0.5 0 0.5 1 1.5 Time (s) TorqueLink2(Nm) MLRNNPID OPTNNPID OPT PID 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0 50 100 Time(s) MLRNNPIDController GainsVariationsLink2 Proportional Gain Derivative Gain Integral Gain 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0 200 400 600 Time(s) MLRNNPIDController GainVariation(link1) Proportional Gain Derivative Gain Integral Gain Fig. 6. Performance comparison of implemented control schemes under nominal parameters and variable payload (a) trajectory tracking, (b) tracking error for link 1, (c) tracking error for link 2, (d) controller output for link1, (e) controller output for link2, (f) variation in gains of MLRNNPID controller for link 2 and (g) variations in gains of MLRNNPID controller for link 1. R. Sharma et al. / ISA Transactions 62 (2016) 258–267 263
  7. 7.  HT k Qk þ1 þΔk 2 rVk Àγk HT k Qk þ 1 2 À2 γkHT k Qk þ 1Δk ¼ Vk þγkΔ2 k Àγk HT k Qk þ1 2 Àγk HT k Qk þ 1 þΔk 2 ¼ Vk þγkΔ2 k Àγk β2 k e2 k þ 1 ¼ Vk Àγk β2 k e2 k þ1 ÀΔ2 k Vk þ1 rVk Àγk β2 k e2 kþ 1 ÀΔ2 k ð37Þ Using (33), it can be concluded that Vk þ1 rVk Now from (32) and (36), we can obtain 1À 1 ε γ2 k β2 k e2 k þ 1 rVkþ 1 ÀVk Àγk 1 ε β2 k e2 k þ 1 ÀΔk 2 rVk þ1 ÀVk Summing up both sides from 0 to 1 1À 1 ε X1 k ¼ 0 γkβ2 k e2 k þ1 rV0 ¼ QT 0P À1 0 Q0 o1 ð38Þ From (38) it can be calculated γkβ2 k e2 k þ 1→0; As k→∞ ð39Þ From (32) and (39), γk ¼ 0; βk ¼ 18k4k0 From (38) we obtain ek þ1 o ffiffiffi ε p Δ0 8k4k0 ð40Þ (40) Implies that error is bounded. Since Vk rV0 it follows that γÀ1 Qk 2 ¼ QT k PÀ 1 0 Qk rQT k P À1 k Qk rQT 0P À1 0 Q0 ¼ γÀ 1 Q0 2 ð41Þ From (41), we can conclude that Qk r Q0 ;k Hence Qkk is bounded. 4. Simulation results The developed algorithm is implemented for two link rigid SACARA robot. The control scheme is shown in Fig. 5. The per- formance of the proposed controller scheme is compared with CSA optimized NNPID controller which is optimized in an off line mode and CSA optimized conventional PID controller. All simulations are carried out using Matlab software 2009b version and fixed step ODE 4 solver is used with a fixed time of 0.01 s. Payload of the manipulator is changed as shown in Fig. 2. The optimized con- troller parameters are summarized in Table 2. For the presented work, the trajectory used is a cubic poly- nomial type trajectory and is expressed as follows [21]: θRef flci tzð Þ ¼ zflc0 þzflc1 tzð Þþzflc2 tzð Þ2 þzflc3 tzð Þ3 ð42Þ Table 3 IAE values for link 1 and link 2 for 5% decrease in parameters. Parameter variation ( 5%) Decrease MLRNNPID OPTNNPID OPT PID Link 1 Link 2 Link 1 Link 2 Link 1 Link 2 Parameter 1: m1 0.002999 0.007621 0.005637 0.01853 0.008566 0.02472 Parameter 2: m2 0.003031 0.007624 0.005694 0.01853 0.008566 0.02465 Parameter 3:m1; m2 0.002981 0.007624 0.005601 0.01853 0.008513 0.02465 Parameter 4: b1q 0.003047 0.007621 0.005725 0.01853 0.008580 0.02474 Parameter 5: b2q 0.003047 0.007624 0.00573 0.01928 0.008598 0.02403 Parameter 6: b1q ; b2q 0.003045 0.007625 0.005725 0.01930 0.008579 0.02403 Parameter 7:b1q ; b2q ; m1; m2 0.002979 0.007625 0.005595 0.01932 0.008493 0.02395 Table 4 IAE values for link 1 and link 2 for 5% increase in parameters. Parameter variation (5%) Increase MLRNNPID OPTNNPID OPT PID Link 1 Link 2 Link 1 Link 2 Link 1 Link 2 Parameter 1: m1 0.003098 0.007621 0.005825 0.01853 0.008643 0.02472 Parameter 2: m2 0.003066 0.007619 0.005767 0.01853 0.008643 0.02472 Parameter 3:m1; m2 0.003115 0.007619 0.005861 0.01853 0.008686 0.02479 Parameter 4: b1q 0.003050 0.007621 0.005736 0.01853 0.008618 0.02472 Parameter 5:b2q 0.003049 0.007623 0.005731 0.01928 0.008600 0.02542 Parameter 6: b1q; b2q 0.002979 0.007626 0.005736 0.01928 0.008619 0.02542 Parameter 7:b1q ; b2q ; m1; m2 0.002979 0.007626 0.005866 0.01928 0.008706 0.02549 Fig. 7. Disturbance applied at link 1 and link 2. R. Sharma et al. / ISA Transactions 62 (2016) 258–267264
  8. 8. The constraints are: _θRef flci tzð Þ ¼ zflc1 þ2zflc2 tzð Þþ3zflc3 tzð Þ2 ð43Þ €θRef flci tzð Þ ¼ 2zflc2 þ6zflc3 tzð Þ ð44Þ where θRef flci represent the reference positions; i¼1, 2 represent Link1 and Link2 respectively; θRefflci ¼ 1radian and θRefflc2 ¼ 2radian for tz ¼ 2s; θRef flc1 ¼ 0:5radian and θRefflc2 ¼ 4radian for tz ¼ 4s; _θRflci ¼ 0radian=s for both tz ¼ 2s and tz ¼ 4s. 4.1. Performance evaluation under nominal parameters and variable payload The comparison of the simulation results for reference trajec- tory tracking, position error and controller output among the implemented controllers for both links are as shown in Fig. 6. A comparison of the IAE for all the three control schemes is sum- marized in Table 2. The IAE for the proposed MLRNNPID for link1 and link2 are 0.003049 rad and 0.006418 rad respectively, IAE for OPTNNPID controller is 0.005731 rad and 0.01467 rad respectively and for OPT PID controller IAE values for link 1 and link 2 are 0.0097 and 0.01853 rad respectively. Clearly IAE values for the proposed MLRNNPID controller are least and for OPT PID con- troller are highest among the three implemented control schemes. The proposed MLRNNPID control scheme has outperformed the OPTNNPID and OPTPID controller in terms of tracking error and IAE. The variations in the output weights of the MLRNNPID controller for link1 and link 2 are shown in Fig. 6(f) and (g) respectively. Some variations in the controller gains are seen during initial portions of the reference trajectory; the gains are maintained in the closed vicinity of the optimized gains as obtained for OPTPID and OPTNNPID controller. 4.2. Robustness testing In this section, the effect of various parameters variation on the trajectory tracking performance has been observed by evaluating the change in the IAE. Following variations in the parameter of the robotic manipulator are considered in both links: (i) 5% decrease in mass: link 1, link 2 and for both link 1 and link 2 simultaneously, coefficient of friction: link 1, link 2 and for both link 1 and link 2 simultaneously. (ii) 5% increase in the mass: link 1, link 2 and for both link 1 and link 2 simultaneously. Same variations have been introduced in coefficient of friction: for link 1, link 2 and also for both link 1 and link 2 simultaneously. The IAE values for both the cases are summarized in Tables 3 and 4 respectively. Clearly the change in IAE values is least for the proposed MLRNNPID con- troller for all the cases considered. MLRNNPID controller being adaptive in nature; is able to reject the effect of parameter varia- tions in an effective manner as there are small variations in the IAE values for the NNPID controller. The IAE values for OPT PID con- troller are the most affected due to the change in the parameters as considered above. Fig. 8. Performance evaluation of implemented control schemes under the effect of disturbances (a) trajectory tracking, (b) IAE for for link 1, (c) IAE for link 2, (d) controller output for link 1 and (e) controller output for link 2. R. Sharma et al. / ISA Transactions 62 (2016) 258–267 265
  9. 9. 4.3. Disturbance rejection The effect of external disturbances on trajectory tracking per- formance of the robotic manipulator has been observed for both links by adding a sinusoidal signal 1.0sin25tNm to the controller output for both links. The disturbance signal applied to both the links is shown in Fig. 7. The effect of external disturbance on the trajectory tracking, IAE for both links and controller output and for all implemented control schemes shown in Fig.8. The variations in the output layer weights of the MLRNNPID controller for link 1 and link 2 is shown in Fig. 9(a) and (b) respectively. The IAE values under the effect of disturbances are summarized in Table 5. For MLRNNPID controller the IAE values for disturbances in both links are 0.003817 and 0.01412 for link 1 and link 2 respectively. Clearly the MLRNNPID controller has outperformed conventional OPTPID controller and OPTNNPID controller in terms of trajectory tracking under nominal conditions as the IAE values of the MLRNNPID controller is smallest among all controllers. Moreover the MLRNNPID controller is more robust in comparison to OPTNNPID and OPTPID controller, also the performance of the MLRNNPID con- troller is least affected in the presence of disturbances. 5. Discussions In the presented work, a mix locally recurrent neural network's powerful approximation and learning capabilities are integrated with simplicity and effectiveness of the conventional PID con- troller to from an adaptive and robust MLRNNPID controller. A systematic approach is proposed for an on-line learning of the gains of the MLRNNPID controller. The initial gains are calculated using CSA based optimization scheme. The optimized initial weights are readjusted using Moore Penrose generalized inverse as given in (8). The optimization and re-initialization ensured that the mix locally recurrent neural network is free from convergence to a local minima and over training problems. The weights of the MLRNNPID controller are adjusted in an on-line manner using sequential learning methods which is based on the least square solution. The performance of the proposed controller is tested against parametric uncertainties in mass and coefficient of friction for both links. Moreover the external disturbances in sinusoidal form are added to both links and the performance of implemented controllers is compared taking IAE as the performance criteria. It is verified through simulations outcomes that MLRNNPID controller being adaptive in nature tends to minimize the effects of external disturbances and parametric uncertainties in an effective manner as compared to the conventional PID and optimized NNPID con- troller. With the proposed tuning procedure, the MLRNNPID con- troller is easy to implement and tune. 6. Conclusion In this work, a mix locally recurrent neural network based PID like controller scheme is implemented for the trajectory tracking control of a robotic manipulator. The tuning of NN based PID controller does not require the exact mathematical model of the process. The main advantage of the MLRNNPID controller is its simple structure, tre- mendous learning capabilities and more flexibility as it contains twelve tunable parameters as compared to the six parameters for conventional PID controller. The robotic manipulator is a nonlinear, time-varying and uncertain system. Also, the payload attached at the end is varying in nature. The on-line update of the parameters of MLRNNPID controller is necessary to deal with these complexities. The initial gains of MLRNNPID controller are calculated using cuckoo search algorithm, a sequential learning algorithm is then derived to tune weights of output layer in an on-line manner to deal with the highly complex, time varying and nonlinear dynamics of the robotic manipulator. It is verified through the results obtained that the per- formance of the proposed MLRNNPID controller is least affected due to parametric uncertainties and external disturbances. References [1] Reznik L, Ghanayem O, Bourmistrov A. PID plus fuzzy controller structures as a design base for industrial applications. Eng Appl Artif Intell 2003;16:227–36. [2] Astrom KJ, Wittenmark B. Adaptive control. New York: Addison-Wesley; 1995. [3] Jaung J-G, Haung M-T, Liu W-K. PID control using presearched genetic algo- rithms for a MIMO system. IEEE Trans Syst Man Cybern Part C: Appl Rev 2008;38(5):716–27. [4] Ayala HVH, Coelho LdS. Tuning of PID controller based on a multiobjective genetic algorithm applied to a robotic manipulator. Expert Syst Appl 2012;39:8968–74. [5] Bagis A. Tabu search algorithm based PID controller tuning for the desired system specifications. J Frankl Inst 2011;38(5):2798–812. [6] Gaing Z-L. A particle swarm optimization approach for optimum design of PID controller in AVR system. IEEE Trans Energy Convers 2004;19(2):384–91. [7] Gandomi AH, Yang X-S, Alavi AH. Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems. Eng Comput 2013;29:17–35. [8] Lee CC. Fuzzy logic in control systems: fuzzy logic controller- Part 1. IEEE Trans Syst Man Cybern 1990;20(2):404–18. [9] Ohtani Y, Yoshimura T. Fuzzy control of a manipulator using the concept of sliding mode. Int J Syst Sci 1996;27(2):179–86. [10] Wei L. Design of a hybrid fuzzy logic proportional plus conventional integral derivative controller. IEEE Trans Fuzzy Syst 1998;6(4):449–63. [11] Chen G. Conventional and fuzzy PID controllers: an overview. Int J Intell Control Syst 1996;1(2):235–46. [12] Zhao Z-Y, Tomizuka M, Isaka S. Fuzzy gain scheduling of PID controllers. IEEE Trans Syst Man Cybern 1993;23(5):1392–8. 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0 200 400 600 800 Time (s) MLRNNPIDOutput GainsForLink1 Proportional Derivative Integral Disturbance Applied to Link 1 at t=0.5s and removed at t=1.0s Disturbance Applied to Link 2 at t=1.5s and removed at t=2.0s Disturbance Applied to Link 1 Link 2 at t=3.0 and removed at t=3.5s 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0 50 100 Time (s) MLRNNPIDOutput GainsforLink2 Proportional Derivative Integral Disturbance Applied to Link 1 at t=0.5s and removed at t=1.0s Disturbance Applied to Link 2 at t=1.5s and removed at t=2.0s Disturbance Applied to Link 1 Link 2 at t=3.0 and removed at t=3.5s Fig. 9. MLRNNPID controller gain variations due to disturbance introduction for (a) link1 and (b) link 2. Table 5 IAE values for MLRNNPID, OPTNNPID and OPTPID scheme for disturbance. Disturbances (N-m) MLRNNPID OPTNNPID OPT PID Link 1 Link 2 Link 1 Link 2 Link 1 Link 2 Link1 0.003804 0.007621 0.007131 0.01852 0.009024 0.02472 Link2 0.003051 0.01407 0.005736 0.03710 0.008599 0.03105 Both Links 0.003817 0.01412 0.007212 0.03748 0.009023 0.03106 R. Sharma et al. / ISA Transactions 62 (2016) 258–267266
  10. 10. [13] Hazzab A, Bousserhane IK, Zerbo M, Sicard P. Real-time implementation of fuzzy gain scheduling of PI controller for induction motor machine control. Neural Process Lett 2006;24:203–15. [14] Yu K-W, Hsu J-H. Fuzzy gain scheduling PID control design based on particle swarm optimization method. In: Proceedings second international conference on innovative computing, information and control, ICICIC’07, 337. Kumamoto; 2007. [15] Devaraj D, Selvabala B. Real-coded genetic algorithm and fuzzy logic approach for real-time tuning of proportional-integral-derivative controller in auto- matic voltage regulator. IET Gener Transm Distrib 2009;3(7):641–9. [16] Syed FU, Kuang ML, Smith M, Okubo S, Ying H. Fuzzy gain scheduling proportional-integral control for improving engine power and speed behavior in a hybrid electric vehicle. IEEE Trans Veh Technol 2009;58(1):69–84. [17] Chaiyatham T, Ngamroo I. Alleviation of power fluctuation in a microgrid by electrolyzer based on optimal fuzzy gain scheduling PID control. IEEJ Trans Electr Electron Eng 2014;9:158–64. [18] Tang W, Chen G. A robust fuzzy PI controller for a flexible-joint robot arm with uncertainties. In: Proceedings of the Third IEEE Conference on Fuzzy Systems. Orlando, FL; 1994. p. 1554–9. [19] Sooraksa P, Chen G. Mathematical modeling and fuzzy control of a flexible- link robot arm. Math Comput Model 1998;27(6):73–93. [20] Er MJ, Sun YL. Hybrid fuzzy proportional-integral plus conventional derivative control of linear and nonlinear systems. IEEE Trans Ind Electron 2001;48 (6):1109–17. [21] Li W, Chang XG, Wahl FM, Farrell J. Tracking control of a manipulator under uncertainty by fuzzy PþID controller. Fuzzy Sets Syst 2001;122:125–37. [22] Meza JL, Santianez V, Soto R, Llama MA. Fuzzy self-tuning PID semiglobal regulator for robotic manipulators. IEEE Trans Ind Electron 2012;59(6): 2709–2717. [23] Yang XS, Deb S. Cuckoo search via Lévy flights. In: Proceedings of the world congress on nature biologically inspired computing. India; 2009. p. 210–4. [24] Sharma Richa, Gaur Prerna, Mittal AP. Performance evaluation of cuckoo search algorithm based FOPID controllers applied to a robotic manipulator with actuator. In: Proceedings of the International Conference on Advances in Computer Engineering and Applications (ICACEA). Ghaziabad, India: IMS Engineering College; 2015. [25] Sharma Richa, Rana KPS, Kumar Vineet. Comparative study of controller optimization techniques for a robotic manipulator. In: Proceedings of the third international conference on soft computing for problem solving, vol. 258; 2014. p 379–93. [26] Boussaïda I, Lepagnot J, Siarry P. A survey on optimization metaheuristics. Inf Sci 2013;237:82–117. [27] Uddin MN, Wen H. Development of a self-tuned neuro-fuzzy controller for induction motor drives. IEEE Trans Ind Appl 2007;43(4):1108–16. [28] Ho S-J, Shu LS, Ho SY. Optimizing fuzzy neural networks for tuning PID con- trollers using an orthogonal simulated annealing algorithm OSA. IEEE Trans Fuzzy Syst 2006;14(3):421–34. [29] Cong S, Liang Y. PID-Like neural network nonlinear adaptive control for uncertain multivariable motion control systems. IEEE Trans Ind Electron 2009;56(10):3872–9. [30] Rong HJ, Huang GB, Sundararajan N, Saratchandran P. Online sequential fuzzy extreme learning machine for function approximation and classification pro- blems. IEEE Trans Syst Man Cybern Part B: Cybern 2009;39(4):1067–72. [31] Kumar Vikas, Gaur Prerna, Mittal AP. ANN based self tuned PID like adaptive controller design for high performance PMSM position control. ESWA 2014;31 (6):7995–8002. [32] Lin F. Robust control design: an optimal control approach. England: John Wiley Sons Ltd; 2007. [33] Ruan D. Intelligent hybrid systems: fuzzy logic. New York: Neural Networks, and Genetic Algorithms Springer Science and Business Media; 1997. [34] Golub GH, Loan CFV. Matrix computations. 3rd ed.. Baltimore: MD: The Johns Hopkins Univ. Press; 1996. [35] Gandomi AH, Yang X-S, Alavi AH. Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems. Eng Comput 2013;29:17–35. [36] Tan WS, Hassan MY, Majid MS, Rahman HA. Allocation and sizing of DG using cuckoo search algorithm. In: Proceedings of the IEEE international conference on power and energy (PEcon). Kota Kinabalu Sabah, Malaysia; 2012. p. 133–8. [37] Yildiz AR. Cuckoo search algorithm for the selection of optimal machining parameters in milling operations. Int J Adv Manuf. Technol 2013;64(1–4):55–61. [38] Yang X-S, Deb S. Engineering optimization by cuckoo search. Int J Math Modell Numer Optim 2010;1(4):330–43. [39] Ge SS, Zhang J, Lee TH. Adaptive MNN control for a class of non-affine NAR- MAX system with disturbances. Syst Control Lett 2004;53:1–12. [40] Jiang H. Directly adaptive fuzzy control of discrete-time chaotic systems by least squares algorithm with dead-zone, vol. 62; 2010. p. 553–9. R. Sharma et al. / ISA Transactions 62 (2016) 258–267 267

×