Least squares-based iterative identification methods for linear-in-parameters systems using the decomposition technique

By extending the least squares-based iterative (LSI) method, this paper presents a decomposition-based LSI (D-LSI) algorithm for identifying linear-in-parameters systems and an interval-varying D-LSI algorithm for handling the identification problems of missing-data systems. The basic idea is to apply the hierarchical identification principle to decompose the original system into two fictitious sub-systems and then to derive new iterative algorithms to estimate the parameters of each sub-system. Compared with the LSI algorithm and the interval-varying LSI algorithm, the decomposition-based iterative algorithms have less computational load. The numerical simulation results demonstrate that the proposed algorithms work quite well

Networked Control System Design and Parameter Estimation

Networked control systems (NCSs) are a kind of distributed control systems in which the data between control components are exchanged via communication networks. Because of the attractive advantages of NCSs such as reduced system wiring, low weight, and ease of system diagnosis and maintenance, the research on NCSs has received much attention in recent years. The first part (Chapter 2 - Chapter 4) of the thesis is devoted to designing new controllers for NCSs by incorporating the network-induced delays. The thesis also conducts research on filtering of multirate systems and identification of Hammerstein systems in the second part (Chapter 5 - Chapter 6). Network-induced delays exist in both sensor-to-controller (S-C) and controller-to-actuator (C-A) links. A novel two-mode-dependent control scheme is proposed, in which the to-be-designed controller depends on both S-C and C-A delays. The resulting closed-loop system is a special jump linear system. Then, the conditions for stochastic stability are obtained in terms of a set of linear matrix inequalities (LMIs) with nonconvex constraints, which can be efficiently solved by a sequential LMI optimization algorithm. Further, the control synthesis problem for the NCSs is considered. The definitions of H₂ and H∞ norms for the special system are first proposed. Also, the plant uncertainties are considered in the design. Finally, the robust mixed H₂/H∞ control problem is solved under the framework of LMIs. To compensate for both S-C and C-A delays modeled by Markov chains, the generalized predictive control method is modified to choose certain predicted future control signal as the current control effort on the actuator node, whenever the control signal is delayed. Further, stability criteria in terms of LMIs are provided to check the system stability. The proposed method is also tested on an experimental hydraulic position control system. Multirate systems exist in many practical applications where different sampling rates co-exist in the same system. The l₂-l∞ filtering problem for multirate systems is considered in the thesis. By using the lifting technique, the system is first transformed to a linear time-invariant one, and then the filter design is formulated as an optimization problem which can be solved by using LMI techniques. Hammerstein model consists of a static nonlinear block followed in series by a linear dynamic system, which can find many applications in different areas. New switching sequences to handle the two-segment nonlinearities are proposed in this thesis. This leads to less parameters to be estimated and thus reduces the computational cost. Further, a stochastic gradient algorithm based on the idea of replacing the unmeasurable terms with their estimates is developed to identify the Hammerstein model with two-segment nonlinearities. Finally, several open problems are listed as the future research directions

A classification of techniques for the compensation of time delayed processes. Part 2: Structurally optimised controllers

Following on from Part 1, Part 2 of the paper considers the use of structurally optimised controllers to compensate time delayed processes

A non-uniform predictor-observer for a networked control system

The final publication is available at Springer via http://dx.doi.org/10.1007/s12555-011-0621-5This paper presents a Non-Uniform Predictor-Observer (NUPO) based control approach in order to deal with two of the main problems related to Networked Control Systems (NCS) or Sensor Networks (SN): time-varying delays and packet loss. In addition, if these delays are longer than the sampling period, the packet disordering phenomenon can appear. Due to these issues, a (scarce) nonuniform, delayed measurement signal could be received by the controller. But including the NUPO proposal in the control system, the delay will be compensated by the prediction stage, and the nonavailable data will be reconstructed by the observer stage. So, a delay-free, uniformly sampled controller design can be adopted. To ensure stability, the predictor must satisfy a feasibility problem based on a time-varying delay-dependent condition expressed in terms of Linear Matrix Inequalities (LMI). Some aspects like the relation between network delay and robustness/performance trade-off are empirically studied. A simulation example shows the benefits (robustness and control performance improvement) of the NUPO approach by comparison to another similar proposal. © ICROS, KIEE and Springer 2011.This work was supported by the Spanish Ministerio de Ciencia y Tecnologia Projects DPI2008-06737-C02-01 and DPI2009-14744-C03-03, by Generalitat Valenciana Project GV/2010/018, by Universidad Politecnica de Valencia Project PAID06-08.Cuenca Lacruz, ÁM.; García Gil, PJ.; Albertos Pérez, P.; Salt Llobregat, JJ. (2011). A non-uniform predictor-observer for a networked control system. International Journal of Control, Automation and Systems. 9(6):1194-1202. doi:10.1007/s12555-011-0621-5S1194120296K. Ogata, Discrete-time Control Systems, Prentice-Hall, Inc., Upper Saddle River, NJ, USA, 1987.Y. Tipsuwan and M. Chow, “Control methodologies in networked control systems,” Control Eng. 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Piza, “A delay-dependent dual-rate PID controller over an Ethernet network,” IEEE Trans. Ind. Informat., vol. 7, no. 1, pp. 18–29, Feb. 2011.Y. Tian and D. Levy, “Compensation for control packet dropout in networked control systems,” Inform. Sci., vol. 178, no. 5, pp. 1263–1278, 2008.Y. Zhao, G. Liu, and D. Rees, “Modeling and stabilization of continuous-time packet-based networked control systems.” IEEE Trans. Syst., Man, Cybern. B, vol. 39, no. 6, pp. 1646–1652, Dec. 2009.X. Zhao, S. Fei, and C. Sun, “Impulsive controller design for singular networked control systems with packet dropouts,” Int. J. Control, Autom., and Syst., vol. 7, no. 6, pp. 1020–1025, 2009.H. Gao and T. Chen, “H ∞ estimation for uncertain systems with limited communication capacity,” IEEE Trans. Autom. Control, vol. 52, no. 11, pp. 2070–2084, 2007.S. Oh, L. Schenato, P. Chen, and S. Sastry, “Tracking and coordination of multiple agents using sensor networks: System design, algorithms and experiments,” Proc. of the IEEE, vol. 95, no. 1, pp. 234–254, 2007.M. Moayedi, Y. Foo, and Y. Soh, “Optimal and suboptimal minimum-variance filtering in networked systems with mixed uncertainties of random sensor delays, packet dropouts and missing measurements,” Int. J. Control, Autom., and Syst., vol. 8, no. 6, pp. 1179–1188, 2010.W. Zhang, M. Branicky, and S. Phillips, “Stability of networked control systems,” IEEE Control Syst. Mag., vol. 21, no. 1, pp. 84–99, 2001.J. Hespanha, P. Naghshtabrizi, and Y. Xu, “A survey of recent results in networked control systems,” Proc. of the IEEE, vol. 95, no. 1, pp. 138–162, 2007.J. Baillieul and P. Antsaklis, “Control and communication challenges in networked real-time systems,” Proc. of the IEEE, vol. 95, no. 1, pp. 9–28, 2007.P. Garcia, P. Castillo, R. Lozano, and P. Albertos, “Robustness with respect to delay uncertainties of a predictor-observer based discrete-time controller,” Proc. of the 45th IEEE Conf. on Decision and Control, pp. 199–204, 2006.C. Lien, “Robust observer-based control of systems with state perturbations via LMI approach,” IEEE Trans. Autom. Control, vol. 49, no. 8, pp. 1365–1370, 2004.A. Sala, “Computer control under time-varying sampling period: an LMI gridding approach,” Automatica, vol. 41, no. 12, pp. 2077–2082, Dec. 2005.J. Li, Q. Zhang, Y. Wang, and M. Cai, “H ∞ control of networked control systems with packet disordering,” IET Control Theory Appl., vol. 3, no. 11, pp. 1463–1475, March 2009.Y. Zhao, G. Liu, and D. Rees, “Improved predictive control approach to networked control systems,” IET Control Theory Appl., vol. 2, no. 8, pp. 675–681, Aug. 2008.K. Astrom, “Event based control,” Analysis and Design of Nonlinear Control Systems, pp. 127–147, 2007.A. Cuenca, P. García, K. Arzén, and P. Albertos, “A predictor-observer for a networked control system with time-varying delays and non-uniform sampling,” Proc. Eur. Control Conf., pp. 946–951, 2009.J. Xiong and J. Lam, “Stabilization of linear systems over networks with bounded packet loss,” Automatica, vol. 43, no. 1, pp. 80–87, 2007.H. Song, L. Yu, and A. Liu, “H ∞ filtering for network-based systems with communication constraints and packet dropouts,” Proc. of the 7th Asian Control Conf., pp. 220–225, 2009.P. Garcia, A. Gonzalez, P. Castillo, R. Lozano, and P. Albertos, “Robustness of a discrete-time predictor-based controller for time-varying measurement delay,” Proc. of the 9th IFAC Workshop on Time Delay Systems, 2010.J. Sturm, “Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones,” Optimization methods and software, vol. 11, no. 1, pp. 625–653, 1999.T. Henningsson and K. Astrom, “Log-concave observers,” Proc. of the 17th Int. 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State-space self-tuner for on-line adaptive control

Dynamic systems, such as flight vehicles, satellites and space stations, operating in real environments, constantly face parameter and/or structural variations owing to nonlinear behavior of actuators, failure of sensors, changes in operating conditions, disturbances acting on the system, etc. In the past three decades, adaptive control has been shown to be effective in dealing with dynamic systems in the presence of parameter uncertainties, structural perturbations, random disturbances and environmental variations. Among the existing adaptive control methodologies, the state-space self-tuning control methods, initially proposed by us, are shown to be effective in designing advanced adaptive controllers for multivariable systems. In our approaches, we have embedded the standard Kalman state-estimation algorithm into an online parameter estimation algorithm. Thus, the advanced state-feedback controllers can be easily established for digital adaptive control of continuous-time stochastic multivariable systems. A state-space self-tuner for a general multivariable stochastic system has been developed and successfully applied to the space station for on-line adaptive control. Also, a technique for multistage design of an optimal momentum management controller for the space station has been developed and reported in. Moreover, we have successfully developed various digital redesign techniques which can convert a continuous-time controller to an equivalent digital controller. As a result, the expensive and unreliable continuous-time controller can be implemented using low-cost and high performance microprocessors. Recently, we have developed a new hybrid state-space self tuner using a new dual-rate sampling scheme for on-line adaptive control of continuous-time uncertain systems

지도 및 비지도 학습을 이용한 로봇 머니퓰레이터 충돌 감지

학위논문(박사) -- 서울대학교대학원 : 공과대학 기계항공공학부, 2022.2. 박종우.사람과 공유된 구조화되지 않은 동적 환경에서 작동하는 협업 로봇 머니퓰레이터는 날카로운 충돌(경성 충돌)에서 더 긴 지속 시간의 밀고 당기는 동작(연성 충돌)에 이르기까지의 다양한 충돌을 빠르고 정확하게 감지해야 한다. 모터 전류 측정값을 이용해 외부 조인트 토크를 추정하는 동역학 모델 기반 감지 방법을 사용할 경우, 정확한 마찰 파라미터 모델링 및 식별과 같은 모터 마찰에 대한 적절한 처리가 필요하다. 올바르게 적용하면 매우 효과적이지만, 동역학과 마찰 파라미터를 모델링 및 식별하고 여러 개의 감지 임계값을 수동으로 설정하는 데에는 상당한 노력이 필요하기 때문에 대량 생산되는 산업용 로봇에 이를 적용하기는 어렵다. 또한 적절한 식별 후에도 동역학에 백래시, 탄성 등 모델링되지 않은 효과나 불확실성이 여전히 존재할 수 있다. 본 논문에서는 순수 모델 기반 방법의 구현 어려움을 피하고 불확실한 동역학적 효과를 보상하는 수단으로 로봇 머니퓰레이터를 위한 총 네 가지의 학습 기반 충돌 감지 방법을 제안한다. 두 개의 방법은 학습을 위해 충돌 및 비충돌 동작 데이터가 모두 필요한 지도 학습 알고리즘(서포트 벡터 머신 회귀, 일차원 합성곱 신경망 기반)을 사용하며 나머지 두 개의 방법은 학습을 위해 비충돌 동작 데이터만을 필요로 하는 비지도 이상치 탐지 알고리즘(단일 클래스 서포트 벡터 머신, 오토인코더 기반)에 기반한다. 로봇 동역학 모델과 모터 전류 측정값만을 필요로 하며 추가적인 외부 센서나 마찰 모델링, 여러 개의 감지 임계값에 대한 수동 조정은 필요하지 않다. 먼저 지도 및 비지도 감지 방법을 학습시키고 검증하는 데 사용되는, 6자유도 협업 로봇 머니퓰레이터를 이용해 수집된 로봇 충돌 데이터를 설명한다. 우리가 고려하는 충돌 시나리오는 경성 충돌, 연성 충돌, 비충돌 동작으로, 경성 및 연성 충돌은 모두 동일하게 충돌로 간주한다. 감지 성능 검증을 위한 테스트 데이터는 총 787건의 충돌과 62.4분의 비충돌 동작으로 이루어져 있으며, 이는 로봇이 랜덤 점대점 6관절 동작을 수행하는 동안 수집된다. 데이터 수집 중 로봇의 끝단에는 미부착, 3.3 kg, 5.0 kg의 세 가지 유형의 페이로드를 부착한다. 다음으로, 수집된 테스트 데이터를 이용해 지도 감지 방법의 감지 성능을 실험적으로 검증한다. 실험 결과는 지도 감지 방법이 가벼운 네트워크를 이용해 광범위한 경성 및 연성 충돌을 실시간으로 정확하게 감지할 수 있음을 보여주며, 이를 통해 모델 파라미터의 불확실성과 측정 노이즈, 백래시, 변형 등 모델링되지 않은 효과까지 보상됨을 알 수 있다. 또한 서포트 벡터 머신 회귀 기반 방법은 하나의 감지 임계값에 대한 조정만 필요하며 일차원 합성곱 신경망 기반 방법은 하나의 아웃풋 필터 파라미터에 대한 조정만 필요한데, 두 방법 모두 직관적인 감도 조정이 가능하다. 나아가 일련의 시뮬레이션 실험을 통해 지도 감지 방법의 일반화 성능을 실험적으로 검증한다. 마지막으로, 동일한 테스트 데이터에 대해 비지도 감지 방법의 감지 성능과 일반화 성능 또한 검증한다. 실험 결과는 비지도 감지 방법 또한 가벼운 계산과 하나의 감지 임계값에 대한 조정만으로 다양한 경성 및 연성 충돌을 실시간으로 강인하게 감지할 수 있음을 보여주며, 이를 통해 모델링되지 않은 마찰을 포함한 불확실한 동역학적 효과를 비지도 학습으로도 보상할 수 있음을 알 수 있다. 지도 감지 방법이 더 나은 감지 성능을 보이지만, 비지도 감지 방법은 학습을 위해 비충돌 동작 데이터만을 필요로 하며 발생할 수 있는 모든 유형의 충돌에 대한 정보를 필요로 하지 않기 때문에 대량 생산되는 산업용 로봇에 더 적합하다.Collaborative robot manipulators operating in dynamic and unstructured environments shared with humans require fast and accurate detection of collisions, which can range from sharp impacts (hard collisions) to pulling and pushing motions of longer duration (soft collisions). When using dynamics model-based detection methods that estimate the external joint torque with motor current measurements, proper treatment for friction in the motors is required, such as accurate modeling and identification of friction parameters. Although highly effective when done correctly, modeling and identifying the dynamics and friction parameters, and manually setting multiple detection thresholds require considerable effort, making them difficult to be replicated for mass-produced industrial robots. There may also still exist unmodeled effects or uncertainties in the dynamics even after proper identification, e.g., backlash, elasticity. This dissertation presents a total of four learning-based collision detection methods for robot manipulators as a means of sidestepping some of the implementation difficulties of pure model-based methods and compensating for uncertain dynamic effects. Two methods use supervised learning algorithms – support vector machine regression and a one-dimensional convolutional neural network-based – that require both the collision and collision-free motion data for training. The other two methods are based on unsupervised anomaly detection algorithms – a one-class support vector machine and an autoencoder-based – that require only the collision-free motion data for training. Only the motor current measurements together with a robot dynamics model are required while no additional external sensors, friction modeling, or manual tuning of multiple detection thresholds are needed. We first describe the robot collision dataset collected with a six-dof collaborative robot manipulator, which is used for training and validating our supervised and unsupervised detection methods. The collision scenarios we consider are hard collisions, soft collisions, and collision-free, where both hard and soft collisions are treated in the same manner as just collisions. The test dataset for detection performance verification includes a total of 787 collisions and 62.4 minutes of collision-free motions, all collected while the robot is executing random point-to-point six-joint motions. During data collection, three types of payloads are attached to the end-effector: no payload, 3.3 kg payload, and 5.0 kg payload. Then the detection performance of our supervised detection methods is experimentally verified with the collected test dataset. Results demonstrate that our supervised detection methods can accurately detect a wide range of hard and soft collisions in real-time using a light network, compensating for uncertainties in the model parameters as well as unmodeled effects like friction, measurement noise, backlash, and deformations. Moreover, the SVMR-based method requires only one constant detection threshold to be tuned while the 1-D CNN-based method requires only one output filter parameter to be tuned, both of which allow intuitive sensitivity tuning. Furthermore, the generalization capability of our supervised detection methods is experimentally verified with a set of simulation experiments. Finally, our unsupervised detection methods are also validated for the same test dataset; the detection performance and the generalization capability are verified. The experimental results show that our unsupervised detection methods are also able to robustly detect a variety of hard and soft collisions in real-time with very light computation and with only one constant detection threshold required to be tuned, validating that uncertain dynamic effects including the unmodeled friction can be successfully compensated also with unsupervised learning. Although our supervised detection methods show better detection performance, our unsupervised detection methods are more practical for mass-produced industrial robots since they require only the data for collision-free motions for training, and the knowledge of every possible type of collision that can occur is not required.1 Introduction 1 1.1 Model-Free Methods 2 1.2 Model-Based Methods 2 1.3 Learning-Based Methods 4 1.3.1 Using Supervised Learning Algorithms 5 1.3.2 Using Unsupervised Learning Algorithms 6 1.4 Contributions of This Dissertation 7 1.4.1 Supervised Learning-Based Model-Compensating Detection 7 1.4.2 Unsupervised Learning-Based Model-Compensating Detection 8 1.4.3 Comparison with Existing Detection Methods 9 1.5 Organization of This Dissertation 14 2 Preliminaries 17 2.1 Introduction 17 2.2 Robot Dynamics 17 2.3 Momentum Observer-Based Collision Detection 19 2.4 Supervised Learning Algorithms 21 2.4.1 Support Vector Machine Regression 21 2.4.2 One-Dimensional Convolutional Neural Network 23 2.5 Unsupervised Anomaly Detection 25 2.6 One-Class Support Vector Machine 26 2.7 Autoencoder-Based Anomaly Detection 28 2.7.1 Autoencoder Network Architecture and Training 28 2.7.2 Anomaly Detection Using Autoencoders 29 3 Robot Collision Data 31 3.1 Introduction 31 3.2 True Collision Index Labeling 31 3.3 Collision Scenarios 35 3.4 Monitoring Signal 36 3.5 Signal Normalization and Sampling 37 3.6 Test Data for Detection Performance Verification 39 4 Supervised Learning-Based Model-Compensating Detection 43 4.1 Introduction 43 4.2 SVMR-Based Collision Detection 44 4.2.1 Input Feature Vector Design 44 4.2.2 SVMR Training 45 4.2.3 Collision Detection Sensitivity Adjustment 46 4.3 1-D CNN-Based Collision Detection 50 4.3.1 Network Input Design 50 4.3.2 Network Architecture and Training 50 4.3.3 An Output Filtering Method to Reduce False Alarms 53 4.4 Collision Detection Performance Criteria 54 4.4.1 Area Under the Precision-Recall Curve (PRAUC) 54 4.4.2 Detection Delay and Number of Detection Failures 54 4.5 Collision Detection Performance Analysis 56 4.5.1 Global Performance with Varying Thresholds 56 4.5.2 Detection Delay and Number of Detection Failures 57 4.5.3 Real-Time Inference 60 4.6 Generalization Capability Analysis 60 4.6.1 Generalization to Small Perturbations 60 4.6.2 Generalization to an Unseen Payload 62 5 Unsupervised Learning-Based Model-Compensating Detection 67 5.1 Introduction 67 5.2 OC-SVM-Based Collision Detection 68 5.2.1 Input Feature Vector 68 5.2.2 OC-SVM Training 70 5.2.3 Collision Detection with the Trained OC-SVM 70 5.3 Autoencoder-Based Collision Detection 70 5.3.1 Network Input and Output 71 5.3.2 Network Architecture and Training 71 5.3.3 Collision Detection with the Trained Autoencoder 72 5.4 Collision Detection Performance Analysis 74 5.4.1 Global Performance with Varying Thresholds 75 5.4.2 Detection Delay and Number of Detection Failures 75 5.4.3 Comparison with Supervised Learning-Based Methods 80 5.4.4 Real-Time Inference 83 5.5 Generalization Capability Analysis 83 5.5.1 Generalization to Small Perturbations 84 5.5.2 Generalization to an Unseen Payload 85 6 Conclusion 89 6.1 Summary and Discussion 89 6.2 Future Work 93 A Appendix 95 A.1 SVM-Based Classification of Detected Collisions 95 A.2 Direct Estimation-Based Detection Methods 97 A.3 Model-Independent Supervised Detection Methods 101 A.4 Generalization to Large Changes in the Dynamics Model 102 Bibliography 106 Abstract 112박

Modeling, identification, estimation and adaptation for the control of power-generating kites

Large-scale kites, flying high-force crosswind trajectories, have been proposed for wind power generation. A two phase operational cycle generates net positive power using a ground-based motor/generator. In the traction phase the kite flies a high-force trajectory while reeling out the generator-connected tethers. A low-force retraction phase reels in the tethers and returns the kite to the start of the cycle. Highly variable conditions and significant uncertainty in the dynamics pose challenges to autonomous, well-controlled flight. The control task is divided into trajectory generation and tracking components and the most uncertain parameters in the model are identified online. The control structure uses these parameters in a robust framework resulting in an experimentally verified adaptive control scheme

Combined state and parameter estimation for Hammerstein systems with time-delay using the Kalman filtering

This paper discusses the state and parameter estimation problem for a class of Hammerstein state space systems with time-delay. Both the process noise and the measurement noise are considered in the system. Based on the observable canonical state space form and the key term separation, a pseudo-linear regressive identification model is obtained. For the unknown states in the information vector, the Kalman filter is used to search for the optimal state estimates. A Kalman-filter based least squares iterative and a recursive least squares algorithms are proposed. Extending the information vector to include the latest information terms which are missed for the time-delay, the Kalman-filter based recursive extended least squares algorithm is derived to obtain the estimates of the unknown time-delay, parameters and states. The numerical simulation results are given to illustrate the effectiveness of the proposed algorithms

Optimized state feedback regulation of 3DOF helicopter system via extremum seeking

In this paper, an optimized state feedback regulation of a 3 degree of freedom (DOF) helicopter is designed via extremum seeking (ES) technique. Multi-parameter ES is applied to optimize the tracking performance via tuning State Vector Feedback with Integration of the Control Error (SVFBICE). Discrete multivariable version of ES is developed to minimize a cost function that measures the performance of the controller. The cost function is a function of the error between the actual and desired axis positions. The controller parameters are updated online as the optimization takes place. This method significantly decreases the time in obtaining optimal controller parameters. Simulations were conducted for the online optimization under both fixed and varying operating conditions. The results demonstrate the usefulness of using ES for preserving the maximum attainable performance

State estimation, system identification and adaptive control for networked systems

A networked control system (NCS) is a feedback control system that has its control loop physically connected via real-time communication networks. To meet the demands of `teleautomation', modularity, integrated diagnostics, quick maintenance and decentralization of control, NCSs have received remarkable attention worldwide during the past decade. Yet despite their distinct advantages, NCSs are suffering from network-induced constraints such as time delays and packet dropouts, which may degrade system performance. Therefore, the network-induced constraints should be incorporated into the control design and related studies. For the problem of state estimation in a network environment, we present the strategy of simultaneous input and state estimation to compensate for the effects of unknown input missing. A sub-optimal algorithm is proposed, and the stability properties are proven by analyzing the solution of a Riccati-like equation. Despite its importance, system identification in a network environment has been studied poorly before. To identify the parameters of a system in a network environment, we modify the classical Kalman filter to obtain an algorithm that is capable of handling missing output data caused by the network medium. Convergence properties of the algorithm are established under the stochastic framework. We further develop an adaptive control scheme for networked systems. By employing the proposed output estimator and parameter estimator, the designed adaptive control can track the expected signal. Rigorous convergence analysis of the scheme is performed under the stochastic framework as well