2016
48
2
0
0
Partial Eigenvalue Assignment in Discrete-time Descriptor Systems via Derivative State Feedback
Partial Eigenvalue Assignment in Discrete-time Descriptor Systems via Derivative State Feedback
2
2
A method for solving the descriptor discrete-time linear system is focused. For easily, it is converted to a standard discrete-time linear system by the definition of a derivative state feedback. Then partial eigenvalue assignment is used for obtaining state feedback and solving the standard system. In partial eigenvalue assignment, just a part of the open loop spectrum of the standard linear systems are reassigned, while leaving the rest of the spectrum invariant and for reassigning, similarity transformation is used. Using partial eigenvalue assignment is easier than using eigenvalue assignment. Because by partial eigenvalue assignment, size of matrices and state and input vectors are decreased and stability is kept, too. Also concluding remarks and an algorithm are proposed to the descriptions will be obvious. At the end, convergence of state and input vectors in the descriptor system to balance point (zero) are showed by figures in a numerical example.
1
A method for solving the descriptor discrete-time linear system is focused. For easily, it is converted to a standard discrete-time linear system by the definition of a derivative state feedback. Then partial eigenvalue assignment is used for obtaining state feedback and solving the standard system. In partial eigenvalue assignment, just a part of the open loop spectrum of the standard linear systems are reassigned, while leaving the rest of the spectrum invariant and for reassigning, similarity transformation is used. Using partial eigenvalue assignment is easier than using eigenvalue assignment. Because by partial eigenvalue assignment, size of matrices and state and input vectors are decreased and stability is kept, too. Also concluding remarks and an algorithm are proposed to the descriptions will be obvious. At the end, convergence of state and input vectors in the descriptor system to balance point (zero) are showed by figures in a numerical example.
65
74
Sakineh
Mirassadi
Sakineh
Mirassadi
Ph.D. Student, Department of Mathematics, Shahrood University of Technology
Ph.D. Student, Department of Mathematics,
Iran
s.mirassadi@shahroodut.ac.ir
Hojat
Ahsani Tehrani
Hojat
Ahsani Tehrani
Associate Professor, Department of Mathematics, Shahrood University of Technology
Associate Professor, Department of Mathematics,
Iran
hahsani@shahroodut.ac.ir
Descriptor Discrete-Time System
Derivative State Feedback
Partial Eigenvalue Assignment
Converge to Balance Point
[[1] Armentano, V. A.; “Eigenvalue Placement for Generalized Linear Systems,” Systems and Control Letter, Vol. 4, No. 4, pp. 199–202, 1984. ##[2] Bernhard, P.; “On Singular Implicit Linear Dynamical Systems,” SIAM Journal on Control and Optimization, Vol. 20, No. 5, pp. 612–623, 1982. ##[3] Bunse-Gerstner, A.; Nichols, N. and Mehrmann, V.; “Regularization of Descriptor Systems by Derivative and Proportional State Feedback,” SIAM J. Matrix Anal, Vol. 13, No. 1, pp. 46–67, 1992. ##[4] Campbell, S. L.; Meyer, C. D. and Rose, N. J.; “Applications of the Drazin Inverse to Linear Systems of Differential Equations with Singular Constant Coefficients,” SIAM Journal on Applied Mathematics, Vol. 31, No. 3, pp. 411–425, 1976. ##[5] Cobb, J. D.; “Feedback and Pole Placement in Descriptor Variable Systems,” Int. J. Control, Vol. 33, No. 6, pp. 1135–1146, 1981. ##[6] Dai, L.; “Singular Control System,” Lecture Notes in Control and Information Sciences, Springer, Berlin, Vol. 118, 1989. ##[7] Datta, B. N.; “Numerical Methods for Linear Control Systems Design and Analysis,” Academic Press, New York, 2002. ## [8] Datta, B. N. and Sarkissian, D. R.; “Partial Eigenvalue Assignment in Linear Systems: Existence, Uniqueness and Numerical Solution,” Mathematical Theory of Networks and Systems, Notre Dame, 2002. ##[9] Fletcher, L. R.; “Eigenstructure Assignment by Output Feedback in Descriptor Systems,” IEE Proceedings, Vol. 135, No. 4, pp. 302–308, 1988. ## [10] Gilnther, M. and Rentrop, P.; “Multirate Row- Methods and Latency of Electric Circuits,” Tech. Rep. TUM-M9208, Mathematisches Inst., Technische Univ. MiJnchen, 1992. ##[11] Hopeld, J. J.; “Neural Networks and Physical Systems with Emergent Collective Computational Abilities,” Proc. Natl. Acad. Sci., USA, Vol. 79, No. 8, pp. 2554–2558, 1982. ##[12] Kaczorek, T.; “Checking of the Positivity of Descriptor Linear Systems by the Use of the Shuffle Algorithm,” Archives of Control Sciences, Vol. 21, No. 3, pp. 287–298, 2011. ##[13] Kaczorek, T.; “Positivity of Descriptor Linear Systems with Regular Pencils,” Archives of Electrical Engineering, Vol. 61, No. 1, pp. 101–113, 2012. ##[14] Kaczorek, T.; “Applications of the Drazin Inverse to the Analysis of Descriptor Fractional Discrete-Time Linear Systems with Regular Pencils,” Int. J. Appl. Math. Comput. Sci., Vol. 23, No. 1, pp. 29–33, 2013. ##[15] Kaczorek, T.; “Positivity and Asymptotic Stability of Descriptor Linear Systems with Regular Pencils,” Archives of Control Sciences, Vol. 24, No. 2, pp. 193–205, 2014. ##[16] Kaczorek, T.; “Analysis of Descriptor Roesser Model with the use of Drazin Inverse,” Int. J. Appl. Math. Comput. Sci., Vol. 25, No. 3, pp. 539–546, 2015. ##[17] Karbassi, S. M.; “An Algorithm for Minimizing the Norm of State Feedback Controllers in Eigenvalue Assignment,” Computers and Mathematics with Applications, Vol. 41, No. 10, pp. 1317–1326, 2001. ##[18] Karbassi, S. M. and Bell, D. J.; “Parametric Time-Optimal Control of Linear Discrete-Time Systems by State Feedback-Part 1: Regular Kronecker Invariants,” International Journal of Control, Vol. 57, No. 4, pp. 817–830, 1993. ##[19] Karbassi, S. M.; and Bell, D. J.; “Parametric Time-Optimal Control of Linear Discrete-Time Systems by State Feedback-Part 2: Irregular Kronecker Invariants,” International Journal of Control, Vol. 57, No. 4, pp. 831–883, 1993. ## [20] Lewis, F. L.; “A Survey of Linear Singular Systems,” Circuits Systems Signal Process, Vol. 5, No. 1, pp. 3–36, 1986. ##[21] Ozcaldiran, K. and Lewis, F. L.; “A Geometric Approach to Eigenstructure Assignment for Singular Systems,” IEEE, Trans. Automatic Control, Vol. 32, No. 7, pp. 626–632, 1987. ##[22] Rahimi, M.; Naserpour, A. and Karbassi, S. M.; “Eigenvalue Assignment in State Feedback Control for Uncertain Systems,” Journal of Scientific Research and Development, Vol. 2, No. 1, pp. 73–75, 2015. ##[23] Ramadan, M. A. and El-Sayed, E. A.; “Partial Eigenvalue Assignment Problem of High Order Control Systems Using Orthogonality Relations,” Computers and Mathematics with Applications, Vol. 59, No. 6, pp. 1918–1928, 2010. ##[24] Saad, Y.; “Projection and Deflation Methods for Partial Pole Assignment in Linear State Feedback,” IEEE Trans. Automat. Control, Vol. 33, No. 3, pp. 290–297, 1988. ##[25] Schmidt, T. and Hou, M.; “Bollringgetriebel,” Internal Rep., Sicherheitstechnische Regelungsund MeBtechnik, Bergische Univ., GH Wuppertal, Wuppertal, FRG, 1992. ##[26] Varga, A.; “On stabilization Methods of Descriptor Systems,” Systems and Control Letters, Vol. 24, No. 2, pp. 133–138, 1995. ##[27] Zhai, G. and Xu, X.; “A Unified Approach to Stability Analysis of Switched Linear Descriptor Systems under Arbitrary Switching,” Int. J. Apple. Math. Comput. Sci., Vol. 20, No. 2, pp. 249–259, 2010. ##[28] Zhoua, L. and Lub, G.; “Detection and Stabilization for Discrete-Time Descriptor Systems via a Limited Capacity,” Atomtica, Vol. 45, No. 10, pp. 2272–2277, 2009.##]
Potentials of Evolving Linear Models in Tracking Control Design for Nonlinear Variable Structure Systems
Potentials of Evolving Linear Models in Tracking Control Design for Nonlinear Variable Structure Systems
2
2
Evolving models have found applications in many real world systems. In this paper, potentials of the Evolving Linear Models (ELMs) in tracking control design for nonlinear variable structure systems are introduced. At first, an ELM is introduced as a dynamic single input, single output (SISO) linear model whose parameters as well as dynamic orders of input and output signals can change through the time. Then, the potential of ELMs in modeling nonlinear time-varying SISO systems is explained. Next, the potential of the ELMs in tracking control of a minimum phase nonlinear time-varying SISO system is introduced. For this mean, two tracking control strategies are proposed respectively for (a) when the ELM is known perfectly and (b) when the ELM model has uncertainties but dynamic orders of the input and output signals are fixed. The methodology and superiority of the proposed tracking control systems are shown via some illustrative examples: speed control in a DC motor and link position control in a flexible joint robot.
1
Evolving models have found applications in many real world systems. In this paper, potentials of the Evolving Linear Models (ELMs) in tracking control design for nonlinear variable structure systems are introduced. At first, an ELM is introduced as a dynamic single input, single output (SISO) linear model whose parameters as well as dynamic orders of input and output signals can change through the time. Then, the potential of ELMs in modeling nonlinear time-varying SISO systems is explained. Next, the potential of the ELMs in tracking control of a minimum phase nonlinear time-varying SISO system is introduced. For this mean, two tracking control strategies are proposed respectively for (a) when the ELM is known perfectly and (b) when the ELM model has uncertainties but dynamic orders of the input and output signals are fixed. The methodology and superiority of the proposed tracking control systems are shown via some illustrative examples: speed control in a DC motor and link position control in a flexible joint robot.
75
92
Ahmad
Kalhor
Ahmad
Kalhor
Assistant Professor, School of Electrical and Computer Engineering, University of Tehran
Assistant Professor, School of Electrical
Iran
akalhor@ut.ac.ir
Nima
Hojjatzadeh
Nima
Hojjatzadeh
M.Sc. Student, Faculty of New Sciences and Technologies, University of Tehran
M.Sc. Student, Faculty of New Sciences and
Iran
nhojjatzadeh@ut.ac.ir,
Alireza
Golgouneh
Alireza
Golgouneh
M.Sc. Student, Faculty of New Sciences and Technologies, University of Tehran
M.Sc. Student, Faculty of New Sciences and
Iran
golgouneh@ut.ac.ir
Evolving linear model
nonlinear time-varying systems
tracking control system
[[1] Zames, G.; “Feedback and Optimal Sensitivity: Model Reference Transformations, Multiplicative Seminorms and Approximate Inverses,” IEEE Transaction on Automatic Control, Vol. 26, No. 2, pp. 301–320, 1981. ##[2] Helton, J. W.; “Orbit Structure of the Mobius Transformation Semi-Group Action on H-Infinity (Broadband Matching),” Adv. in Math. Suppl. Stud., Vol. 3, pp. 129–197, 1978. ##[3] Bakule, L.; Rehák, B. and Papík, M.; “Decentralized Image-Infinity Control of Complex Systems with Delayed Feedback,” Automatica, Vol. 67, No. 3, pp. 127–131, 2016. ##[4] Rojas, C. R.; Oomen, T.; Hjalmarsson, H. and Wahlberg, B.; “Analyzing Iterations In Identification With Application To Nonparametric H∞-Norm Estimation,” Automatica, Vol. 48, No. 11, pp. 2776– 2790, 2012. ##[5] Yang, C. D. and Taic, H. C.; “Synthesis of μ Controllers Using Statistical Iterations,” Asian Journal of Control, Vol. 4, No. 3, pp. 331–310, 2002. ##[6] Taher, S. A.; Akbari, S.; Abdolalipour, A. and Hematti, R.; “Robust Decentralized Controller Design for UPFC Using μ–Synthesis,” Communications in Nonlinear Science and Numerical Simulation, Vol. 15, No. 8, pp. 2149–2161, 2010. ##[7] Zinober, A. S. I.; “Deterministic Control of Uncertain Systems,” IEE Control Engineering Series, 1991. ## [8] Khalil, H. K.; “Nonlinear Systems,” Prentice Hall, NewJercy, 3rd Edition, 2002. ##[9] Zhou, Q.; Yao, D.; Wang, J. and Wu, C.; “Robust Control of Uncertain Semi-Markovian Jump Systems Using Sliding Mode Control Method Original,” Applied Mathematics and Computation, Vol. 286, pp. 72–87, 2016. ##[10] Barambones, O. and Alkorta, P.; “Vector Control for Induction Motor Drives Based on Adaptive Variable Structure Control Algorithm,” Asian Journal of Control, Vol. 12, No. 5, pp. 640–649, 2010. ##[11] Ling, R.; Wu, M.; Dong, Y. and Chai, Y.; “High Order Sliding-Mode Control for Uncertain Nonlinear Systems with Relative Degree Three,” Communications in Nonlinear Science and Numerical Simulation, Vol. 17, No. 8, pp. 3406–3416, 2012. ##[12] Nasiri, R. and Radan, A.; “Adaptive Ole- Placement Control of 4-Leg Voltage-Source Inverters for Standalone Photovoltaic Systems,” Renewable Energy, Vol. 36, No. 7, pp. 2032–2042, 2011. ##[13] Yang, Q.; Xue, Y.; Yang, S. X. and Yang, W.; “An Auto-Tuning Method for Dominant-Pole Placement Using Implicit Model Reference Adaptive Control Technique,” Journal of Process Control, Vol. 22, No. 3, pp. 519–526, 2012. ##[14] Madady, A.; “ A Self-Tuning Iterative Learning Controller for Time Variant Systems,” Asian Journal of Control, Vol. 10, No. 6, pp. 666–677, 2008. ##[15] Ahmed, M. S.; “Neural Net Based MRAC for a Class of Nonlinear Plants,” Neural Networks, Vol. 13, No. 1, pp. 111–124, 2000. ##[16] Guo, J.; Tao, G. and Liu, Y.; “A Multivariable MRAC Scheme with Application to a Nonlinear Aircraft Model,” Automatica, Vol. 47, No. 4, pp. 804–812, 2011. ##[17] Mohideen, K. A.; Saravanakumar, G.; Valarmathi, K.; Devaraj, D. and Radhakrishnan, T. K.; “Real-Coded Genetic Algorithm for System Identification and Tuning of a Modified Model Reference Adaptive Controller for a Hybrid Tank System,” Applied Mathematical Modeling, Vol. 37, No. 6, pp. 3829–384, 2013. ##[18] Rugh, W. J. and Shamma, J. S.; “Research on Gain Scheduling,” Automatica, Vol. 36, No. 10, pp. 1401–1425, 2000. ##[19] Wu, F.; Packard, A. and Balas, G.; “Systematic Gain-Scheduling Control Design: A Missile Autopilot Example,” Asian Journal of Control, Vol. 4, No. 3, pp. 341–34, 2002. ##[20] Horowitz, I.; Smay, J. and Shapiro, A.; “A Synthesis Theory for Self-Oscillating Adaptive Systems (SOAS) Original Research Article,” Automatica, Vol. 10, No. 4, pp. 381–392, 1974. ##[21] Olivier, J. C.; Loron, L.; Auger, F. and Le- Claire, J. C.; “Improved Linear Model of Self Oscillating Systems Such as Relay Feedback Current Controllers,” Control Engineering Practice, Vol. 18, No. 8, pp. 927–935, 2010. ##[22] Vargas, J. F. and Ledwich, G.; “Variable Structure Control for Power Systems Stabilization,” International Journal of Electrical Power and Energy Systems, Vol. 32, No. 2, pp. 101–107, 2010. ##[23] Sumar, R.; Coelho, A. and Goedtel, A.; “Multivariable System Stabilization via Discrete Variable Structure Control,” Control Engineering Practice, Vol. 40, No. 4, pp. 71–80, 2015. ##[24] Landau, I. D.; “Combining Model Reference Adaptive Controllers and Stochastic Self-Tuning Regulators,” Automatica, Vol. 18, No. 1, pp. 77–84, 1982. ##[25] Landau, I. D. and Karimi, A.; “A Unified Approach to Model Estimation and Controller Reduction (Duality and Coherence),” European Journal of Control, Vol. 8, No. 6, pp. 561–572, 2002. ##[26] Kasabov, N.; “DENFIS: Dynamic Evolving Neural Fuzzy Inference System and its Application for Time Series Prediction,” IEEE Transaction on Fuzzy Systems, Vol. 10, No. 2, pp. 144–154, 2002. ##[27] Angelov, P. and Filev, D.; “An Approach to Online Identification of Takagi Sugeno Fuzzy Models,” IEEE Transaction on Systems, Man and Cybernetics Part B, Vol. 34, No. 1, pp. 484–498, 2004. ##[28] Angelov, P. and Zhou, X.; “Evolving Fuzzy Systems from Data Streams in Real-Time,” International Symposium on Evolving Fuzzy Systems, pp. 29–35, 2006. ##[29] Angelov, P.; Filev, D. and Kasabov, N.; “Evolving Intelligent Systems: Methodology and Applications,” John Wiley and Sons, Chapter 2, pp. 21–50, 2010. ##[30] Lughofer, E. D.; “FLEXFIS: A Robust Incremental Learning Approach for Evolving Takagi–Sugeno Fuzzy Models,” IEEE Transaction Fuzzy Systems, Vol. 16, No. 6, pp. 1393–1410, 2008. ##[31] Kalhor, A.; Araabi, B. N. and Lucas, C.; “Online Extraction of Main Linear Trends for Nonlinear Time Varying Processes,” Information Sciences, Vol. 220, pp. 22–33, 2013. ##[32] Kalhor, A., Iranmanesh, H. and Abdollahzade, M.; “Online Modeling of Real-World Time Series through Evolving AR Models,” IEEE International Conference on Fuzzy systems, FUZIEEE, 2012. ##[33] Kalhor, A.; Araabi, B. N. and Lucas, C.; “A New Systematic Design for Habitually Linear Evolving TS Fuzzy Model,” Journal of Expert Systems with Applications, Vol. 39, No. 2, pp. 1725–1736, 2012. ##[34] Jang, R.; “ANFIS: Adaptive Network-Based Fuzzy Inference System,” IEEE Transaction on Systems, Man and Cybernetics, Vol. 23, No. 3, pp. 665–685, 1993. ##[35] Nelles, O.; “Nonlinear System Identification,” Springer, New York, pp. 365–366, 2001. ##[36] Kalhor, A.; Araabi, B. N. and Lucas, C.; “Reducing the Number of Local Linear Models in Neuro–Fuzzy Modeling: A Split and Merge Clustering Approach,” Applied Soft Computing, Vol. 11, No. 8, pp. 5582–5589, 2011. ##[37] Robinson, J. C.; “An Introduction to Ordinary Differential Equations,” Cambridge University Press, Cambridge, UK, 2004. ##[38] SIemon, G. R. and Straughen, A.; “Electric Machines, Addison,” Wesley, Reading, MA, 1980.##]
Analysis of critical paths in a project network with random fuzzy activity times
Analysis of critical paths in a project network with random fuzzy activity times
2
2
Project planning is part of project management, which is relates to the use of schedules such as Gantt charts to plan and subsequently report progress within the project environment. Initially, the project scope is defined and the appropriate methods for completing the project are determined. In this paper a new approach for the critical path analyzing a project network with random fuzzy activity times has been proposed. The activity times of a project are assumed to be random fuzzy. Linear programming formulation has been applied to determine the critical path. The critical path method (CPM) problem has been solved using the expected duration optimization model and the mean-variance model, along with Liu’s definition for random fuzzy variables. Furthermore, a numerical example problem is solved for illustrating the proposed method.
1
Project planning is part of project management, which is relates to the use of schedules such as Gantt charts to plan and subsequently report progress within the project environment. Initially, the project scope is defined and the appropriate methods for completing the project are determined. In this paper a new approach for the critical path analyzing a project network with random fuzzy activity times has been proposed. The activity times of a project are assumed to be random fuzzy. Linear programming formulation has been applied to determine the critical path. The critical path method (CPM) problem has been solved using the expected duration optimization model and the mean-variance model, along with Liu’s definition for random fuzzy variables. Furthermore, a numerical example problem is solved for illustrating the proposed method.
93
102
Abolfazl
Kazemi
Abolfazl
Kazemi
Assistant Professor, Faculty of Industrial and Mechanical Engineering, Islamic Azad University (Qazvin Branch)
Assistant Professor, Faculty of Industrial
Iran
abkaazemi@gmail.com
Ahmad
Talebi
Ahmad
Talebi
M.Sc., Faculty of Industrial and Mechanical Engineering, Islamic Azad University (Qazvin Branch)
M.Sc., Faculty of Industrial and Mechanical
Iran
a.talebi@yahoo.com
Mahsa
Oroojeni Mohammad Java
Mahsa
Oroojeni Mohammad Javad
Ph.D., Department of Mechanical and Industrial Engineering, Northeastern University, Boston, USA
Ph.D., Department of Mechanical and Industrial
Iran
oroojeni.m@husky.neu.edu
Critical Path Method (CPM)
Activity Times
Random Fuzzy Time
Triangular Fuzzy Numbers
Normal Distribution
[[1] Chanas, S.; Dubois, D. and Zielin´-Ski, P.; “On the Sure Criticality of Tasks in Activity Networks with Imprecise Durations,” IEEE Transactions on Systems, Man and Cybernetics–Part B: Cybernetics, Vol. 4, No. 32, pp. 393–407, 2002. ##[2] Chanas, S. and Zielin´-Ski, P.; “Critical Path Analysis in the Network with Fuzzy Activity Times,” Fuzzy Sets and Systems, Vol. 122, No. 2, pp. 195–204, 2001. ##[3] Chanas, S. and Zielin´-Ski, P.; “The Computational Complexity of the Criticality Problems in a Network with Interval Activity Times,” European Journal of Operational Research, Vol. 136, No. 2, pp. 541–550, 2002. ##[4] Chanas, S. and Zielin´-Ski, P.; “On the Hardness of Evaluating Criticality of Activities in a Planar Network with Duration Intervals,” Operations Research Letters, Vol. 31, No. 1, pp. 53–59, 2003. ##[5] Chanas, S. and Kamburowski, J.; “The Use of Fuzzy Variables in PERT,” Fuzzy Set Systems, Vol. 5, No. 1, pp. 11–9, 1981. ##[6] Chen, S. P.; “Analysis of Critical Paths in a Project Network with Fuzzy Activity Times,” European Journal of Operational Research, Vol. 183, No. 1, pp. 442–459, 2007. ##[7] Tseng, C. and KO, P.; “Measuring Schedule Uncertainty for a Stochastic Resource-Constrained Project Using Scenario-Based Approach with Utility- Entropy Decision Model,” Journal of Industrial and Production Engineering, pp. 1–10, 2016. ##[8] Ding, C. and Zhu, Y.; “Two Empirical Uncertain Models for Project Scheduling Problem,” Journal of the Operational Research Society, Vol. 66, No. 9 , pp. 1471–1480, 2015. ##[9] Elmaghraby, S.; “On Criticality and Sensitivity in Activity Networks,” International Journal of Production Research, Vol. 127 No. 2, pp. 220–38, 2000. ##[10] Hassanzadeh, R.; Mahdavi, I.; Mahdavi-Amiri, N. and Tajdin, A.; “A Genetic Algorithm for Solving Fuzzy Shortest Path Problems with Mixed Fuzzy Arc Lengths,” Mathematical and Computer Modelling, Vol. 57, No. 1, pp. 84–99, 2013. ##[11] Hasuike, T.; Katagiri, H. and Ishii, H.; “Portfolio Selection Problems with Random Fuzzy Variable Returns,” Fuzzy Sets and Systems, Vol. 160, pp. 2579–2596, 2009. ##[12] Hillier, F.S. and Lieberman, G. J.; “Introduction to Operations Research,” McGraw-Hill, Singapore, 7th ed., 2001. ##[13] Kaur, P. and Kumar, A.; “Linear Programming Approach for Solving Fuzzy Critical Path Problems with Fuzzy Parameters,” Applied Soft Computing, Vol. 21, pp. 309–319, 2014. ##[14] Ke, H. and Liu, B.; “Project Scheduling Problem with Mixed Uncertainty of Randomness and Fuzziness,” European Journal of Operational Research, Vol. 183, No. 9, pp. 135–147, 2007. ##[15] Kelley, J. E.; “Critical Path Planning and Scheduling–Mathematical Basis,” Operational Research, Vol. 9, No. 3, pp. 296–320, 1961. ##[16] Li, X. and Liu, B.; “New Independence Definition of Fuzzy Random Variable and Random Fuzzy Variable,” World Journal of Modelling and Simulation, Vol. 2, No. 5, pp. 338–342, 2006. ##[17] Li, X.; Qin, Z. and Kar, S.; “Mean-Variance- Skewness Model for Portfolio Selection with Fuzzy Returns,” European Journal of Operational Research, Vol. 202, No. 1, pp. 239–247, 2010. ##[18] Lin, L.; Lou, T. and Zhan, N. “Project Scheduling Problem with Uncertain Variables,” Applied Mathematics, Vol. 5, pp. 685–690, 2014. ##[19] Liu, B.; “Theory and Practice of Uncertain Programming,” Physica-Verlag, Heidelberg, 2002. ##[20] Liu, B.; “Uncertainty Theory: An Introduction to its Axiomatic Foundations,” Springer-Verlag, Berlin, 2004. ##[21] Madhuri, K. U.; Saradhi, B. P. and Shankar, N. R.; “Fuzzy Linear Programming Model for Critical Path Analysis,” Int. J. Contemp. Math. Sciences, Vol. 8, No. 2, pp. 93–116, 2013. ##[22] Malcolm, D. G.; Roseboom, J. H.; Clark, C. E. and Fazar, W.; “Application of a Technique for Research and Development Project Evaluation,” Operational Research, Vol. 7, pp. 646–69, 1959. ##[23] Guide, A.; “Project Management Body of Knowledge (PMBOK® GUIDE),” Project Management Institute, 2001. ##[24] Sadjadi, S. J.; Pourmoayed, R. and Aryanezhad, M. B.; “A Robust Critical Path in an Environment with Hybrid Uncertainty,” Applied Soft Computing, Vol. 12, No. 3, pp. 1087–1100, 2012. ##[25] Van-Slyke, R. M.; “Monte-Carlo Method and the PERT Problem,” Operational Research, Vol. 11, No. 5, pp. 839–60, 1963. ##[26] Yakhchali, S. H. and Ghodsypour, S. H.; “On the Latest Starting Times and Criticality of Activities in a Network with Imprecise Durations,” Appllied Mathemathical Modelling, Vol. 34, No. 8, pp. 2044– 2058, 2010. ##[27] Zadeh, L. A.; “The Concept of a Linguistic Variable and its Application to Approximate Reasoning,” Information Sciences, Vol. 8, No. 3, pp. 199–249, 1975. ##[28] Zammori, F. A.; Braglia, M. and Frosolini, M.; “A Fuzzy Multi-Criteria Approach for Critical Path Definition,” International Journal of Project Management, Vol. 27, No. 3, pp. 278–291, 2009. ##[29] Zareei, A.; Zaerpour, F.; Bagherpour, M.; Noora, A. and Vencheh, A.; “A New Approach for Solving Fuzzy Critical Path Problem Using Analysis of Events,” Expert Systems with Applications, Vol. 38, No. 3, pp. 87–93, 2011. ##[30] Zielin´-Ski, P.; “On Computing the Latest Starting Times and Floats of Activities in a Network with Imprecise Durations,” Fuzzy Sets and Systems, Vol. 150, No. 3, pp. 53–76, 2005.##]
A risk adjusted self-starting Bernoulli CUSUM control chart with dynamic probability control limits
A risk adjusted self-starting Bernoulli CUSUM control chart with dynamic probability control limits
2
2
Usually, in monitoring schemes the nominal value of the process parameter is assumed known. However, this assumption is violated owing to costly sampling and lack of data particularly in healthcare systems. On the other hand, applying a fixed control limit for the risk-adjusted Bernoulli chart causes to a variable in-control average run length performance for patient populations with dissimilar risk score distributions in monitoring clinical and surgical performance. To solve these problems, a self-starting scheme is proposed based on a parametric bootstrap method and dynamic probability control limits for the risk-adjusted Bernoulli cumulative sum control charts. The advantage of the proposed control charts lies in the use of probability control limits when any assumptions about the patients’ risk distributions and process parameter. Simulation studies show that both proposed schemes have good performance under various shifts.
1
Usually, in monitoring schemes the nominal value of the process parameter is assumed known. However, this assumption is violated owing to costly sampling and lack of data particularly in healthcare systems. On the other hand, applying a fixed control limit for the risk-adjusted Bernoulli chart causes to a variable in-control average run length performance for patient populations with dissimilar risk score distributions in monitoring clinical and surgical performance. To solve these problems, a self-starting scheme is proposed based on a parametric bootstrap method and dynamic probability control limits for the risk-adjusted Bernoulli cumulative sum control charts. The advantage of the proposed control charts lies in the use of probability control limits when any assumptions about the patients’ risk distributions and process parameter. Simulation studies show that both proposed schemes have good performance under various shifts.
103
110
Majid
Aminnayeri
Majid
Aminnayeri
Associate Professor, Department of Industrial Engineering, Amirkabir University of Technology
Associate Professor, Department of Industrial
Iran
mjnayeri@aut.ac.ir
Fatemeh
Sogandi
Fatemeh
Sogandi
Ph.D. Student, Department of Industrial Engineering, Amirkabir University of Technology
Ph.D. Student, Department of Industrial Engineerin
Iran
f.sogandi1990@gmail.com
Average Run Length
Self-Starting Monitoring
Bernoulli Process
Probability Control Limits
Surgical Performance
[[1] Chou, S. C.; “Statistical Process Control for Health Care,” International Journal for Quality in Health Care, Vol. 14, No. 5, pp. 427-428, 2002. ##[2] Tennant, R.; Mohammed, M. A.; Coleman, J. J. and Martin, U.; “Monitoring Patients Using Control Charts: A Systematic Review,” International Journal for Quality in Health Care, Vol. 19, No. 4, pp. 187- 194, 2007. ##[3] Woodall, W. H.; Adams, B. M.; Benneyan, J. C.; “The Use of Control Charts in Healthcare,” Statistical Methods in Healthcare, Wiley, in Faltin, F.; Kenett, R.; Ruggeri, F. Eds., pp. 251-267, 2011. ##[4] Lim, T. O.; “Statistical Process Control Tools for Monitoring Clinical Performance,” International Journal for Quality in Health Care, Vol. 15, No. 1, pp. 3-4, 2003. ##[5] Coory, M.; Duckett, S. and Sketcher-Baker, K.; “Using Control Charts to Monitor Quality of Hospital Care with Administrative Data,” International Journal for Quality in Health Care, Vol. 20, No. 1, pp. 31-39, 2008. ##[6] Joner, M. D.; Woodall, W. H.; Reynolds, M. R.; “Detecting a Rate Increase Using a Bernoulli Scan Statistic,” Statistics in Medicine, Vol. 27, No. 14, pp. 2555-2575, 2008. ##[7] De-Leval, M. R., François, K.; Bull, C.; Brawn, W. B. and Spiegelhalter, D.; “Analysis of a Cluster of Surgical Failures,” The Journal of Thoracic and Cardiovascular Surgery, Vol. 107, No. 3, pp. 914- 924, 1994. ##[8] Steiner, S. H.; Cook, R. and Farewell, V.; “Monitoring Paired Binary Surgical Outcomes Using Cumulative Sum Charts,” Statistics in Medicine, Vol. 18, No. 1, pp. 69-86, 1999. ##[9] Shu, L.; Jiang, W.; Tsui, K. L.; “A Comparison of Weighted CUSUM Procedures that Account for Monotone Changes in Population Size,” Statistics in Medicine, Vol. 30, No. 7, pp. 725-741, 2011. ##[10] Sun, R. J.; Kalbfleisch, J. D.; Schaubel, D. E.; “A Weighted Cumulative Sum (WCUSUM) to Monitor Medical Outcomes with Dependent Censoring,” Statistics in Medicine, Vol. 33, No. 18, pp. 3114- 3129, 2014. ##[11] Lim, T. O.; Soraya, A.; Ding, L. M. and Morad, Z.; “Assessing Doctors’ Competence: Application of CUSUM Technique in Monitoring Doctors’ Performance,” International Journal for Quality in Health Care, Vol. 14, No. 3, pp. 251-258, 2002. ##[12] Steiner, S. H.; Cook, R. J.; Farewell, V. T. and Treasure, T.; “Monitoring Surgical Performance Using Risk-Adjusted Cumulative Sum Charts,” Biostatistics, Vol. 1, No. 4, pp. 441-452, 2000. ##[13] Jones, M. A.; Steiner, S. H.; “Assessing the Effect of Estimation Error on Risk-Adjusted CUSUM Chart Performance,” International Journal for Quality in Health Care, Vol. 24, No. 2, pp. 176-181, 2012. ##[14] Rossi, G.; Del-Sarto, S. and Marchi, M.; “A New Risk-Adjusted Bernoulli Cumulative Sum Chart for Monitoring Binary Health Data,” Statistical Methods in Medical Research, 2014. ##[15] Taseli, J. C. and Benneyan, A.; “Cumulative Sum Charts for Heterogeneous Dichotomous Events,” Industrial Engineering Research Conference Proceedings, pp. 1754-1759, 2008. ##[16] Gombay, E.; Hussein, A. and Steiner, S. H.; “Monitoring Binary Outcomes Using Risk-Adjusted Charts: a Comparative Study,” Statistics in Medicine, Vol. 30, No. 23, pp. 2815-2826, 2008. ##[17] Zeng, L.; “Risk-Adjusted Performance Monitoring in Healthcare Quality Control,” Quality and Reliability Management and Its Applications, Springer, London, pp. 27-45, 2016. ##[18] Tang, X.; Gan, F. F. and Zhang, L.; “Risk- Adjusted Cumulative Sum Charting Procedure Based on Multiresponses,” Journal of the American Statistical Association, Vol. 110, No. 509, pp. 16-26, 2016. ##[19] Cook, D. A.; Coory, M. and Webster, R. A.; “Exponentially Weighted Moving Average Charts to Compare Observed and Expected Values for Monitoring Risk-Adjusted Hospital Indicators,” BMJ Quality and Safety, Vol. 20, No. 6, pp. 469–474, 2011. ##[20] Fang, Y. Y.; Khoo, M. B.; Teh, S. Y. and Xie, M.; “Monitoring of Time between Events with a Generalized Group Runs Control Chart,” Quality and Reliability Engineering International, 2015. ##[21] Pan, X. and Jarrett, J. E.; “The Multivariate EWMA Model and Health Care Monitoring,” International Journal of Economics and Management Sciences, 2014. ##[22] Zhang, X. and Woodall, W. H.; “Dynamic Probability Control Limits for Risk-Adjusted Bernoulli CUSUM Charts,” Statistics in Medicine, Vol. 34, No. 25, pp. 3336-3348, 2014. ##[23] Shen, X.; Tsung, F.; Zou, C. and Jiang, W.; “Monitoring Poisson Count Data with Probability Control Limits When Sample Sizes are Time- Varying,” Naval Research Logistics, Vol. 60, No. 8, pp. 625-636, 2011. ##[24] Paynabar, K.; Jin, J. and Yeh, A.; “Phase I Risk Adjusted Control Charts for Monitoring Surgical Performance with Considering Categorical Covariates,” Journal of Quality Technology, Vol. 44, No. 1, pp. 39-53, 2012. ##[25] Asadayyoobi, N. and Niaki, S. T. A.; “Monitoring Patient Survival Times in Surgical Systems Using a Risk-Adjusted AFT Regression,” Quality Technology and Quantitative Management Chart, 2015 [Accepted]. ##[26] Mohammadian, F.; Niaki, S. T. A. and Amiri, A.; “Phase I Risk-Adjusted Geometric Control Charts to Monitor Health Care Systems,” Quality and Reliability Engineering International, 2014. ##[27] Hawkins, D. M.; “Self-Starting CUSUM Charts for Location and Scale,” The Statistican, Vol. 36, No. 1, pp. 299-315, 1987. ##[28] Hawkins, D. M. and Maboudou-Tchao, E. M.; “Self-Starting Multivariate Exponentially Weighted Moving Average Control Charting,” Technometrics, Vol. 49, No. 1, pp. 199-209, 2007. ##[29] Shen, X.; Tsui, K. L.; Woodall, W. H. and Zou, C.; “Self-Starting Monitoring Scheme for Poisson Count Data with Varying Population Sizes,” Technometrics, 2015 [Accepted]. ##[30] Szarka, J. L. and Woodall, W. H.; “A Review and Perspective on Surveillance of Bernoulli Processes,” Quality and Reliability Engineering International, Vol. 27, No. 6, pp. 735-752, 2007. ##[31] Parsonnet, V.; Dean, D. and Berstein, A. D.; “A Method of Uniform Stratification of Risk for Evaluating the Results of Surgery in Acquired Adult Heart Disease,” Circulation, Vol. 79, No. 6, pp. 3-12, 1989. ##[32] Tian, W. M.; Sun, H. Y.; Zhang, X. and Woodall, W. H.; “The Impact of Varying Patient Populations on the in Control Performance of the Risk-Adjusted CUSUM Chart,” International Journal for Quality in Health Care, Vol. 27, No. 1, pp. 31-36, 2015.##]
Time-Invariant State Feedback Control Laws for a Special Form of Underactuated Nonlinear Systems Using Linear State Bisection
Time-Invariant State Feedback Control Laws for a Special Form of Underactuated Nonlinear Systems Using Linear State Bisection
2
2
Linear state bisection is introduced as a new method to find time-invariant state feedback control laws for a special form of underactuated nonlinear systems. The specialty of the systems considered is that every unactuated state should be coupled with at least two directly actuated states. The basic idea is based on bisecting actuated states and using linear combinations with adjustable parameters to stabilize the unactuated states. These linear combinations make the underactuated system virtually fullyactuated, making it suitable to be stabilized with well-known nonlinear control methods, like feedback linearization. In addition to its simplicity, one of the main contributions of this method is that it can be applied to systems with more than one unactuated state. Three underactuated systems are considered: an asymmetric rigid body, a planar rigid body with an unactuated internal degree of freedom and a system with two degrees of underactuation. It is shown through simulations that the proposed control laws can be effectively used to stabilize the special form of underactuated systems considered.
1
Linear state bisection is introduced as a new method to find time-invariant state feedback control laws for a special form of underactuated nonlinear systems. The specialty of the systems considered is that every unactuated state should be coupled with at least two directly actuated states. The basic idea is based on bisecting actuated states and using linear combinations with adjustable parameters to stabilize the unactuated states. These linear combinations make the underactuated system virtually fullyactuated, making it suitable to be stabilized with well-known nonlinear control methods, like feedback linearization. In addition to its simplicity, one of the main contributions of this method is that it can be applied to systems with more than one unactuated state. Three underactuated systems are considered: an asymmetric rigid body, a planar rigid body with an unactuated internal degree of freedom and a system with two degrees of underactuation. It is shown through simulations that the proposed control laws can be effectively used to stabilize the special form of underactuated systems considered.
111
122
Rouzbeh
Moradi
Rouzbeh
Moradi
Ph.D. Student, Aerospace Research Institute, Ministry of Science, Research and Technology
Ph.D. Student, Aerospace Research Institute,
Iran
roozbeh_moradi_aerospace@yahoo.com
Alireza
Alikhani
Alireza
Alikhani
Assistant Professor, Aerospace Research Institute, Ministry of Science, Research and Technology
Assistant Professor, Aerospace Research Institute,
Iran
aalikhani@ari.ac.ir
Mohsen
Fathi Jegarkandi
Mohsen
Fathi Jegarkandi
Assistant Professor, Aerospace Department, Sharif University of Technology
Assistant Professor, Aerospace Department,
Iran
fathi@sharif.edu
Underactuation
Feedback linearization
State bisection
[[1] Choukchu-Braham, A.; Cherki, B.; Djemai, M. and Busawon, K.; “Analysis and Control of Underactuated Mechanical Systems,” Springer Science and Business Media, 2014. ##[2] Olfati-Saber, R.; “Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles,” Ph.D. Thesis, MIT University, 2001. ##[3] Spong, M. W.; “Underactuated Mechanical Systems,” Control Problems in Robotics and Automation, Lecture Notes in Control and Information Sciences, Vol. 230, pp. 135-150, 1998. ##[4] Liu, Y. and Yu, H.; “A Survey of Underactuated Mechanical Systems,” IET Control Theory and Applications, Vol. 7, No. 7, pp. 921-935, 2013. ##[5] Voytsekhovsky, D. A. and Hirschorn, R. M.; “Stabilization of Single-Input Nonlinear Systems Using Higher Order Term Compensating Sliding Mode Control,” International Journal of Robust and Nonlinear Control, Vol. 18, No. 4-5, pp. 468-480, 2008. ##[6] Dixon, W. E.; Behal, A.; Dawson, D. M. and Nagarkatti, S. P.; “Nonlinear Control of Engineering Systems: A Lyapunov-Based Approach,” Birkhäuser Basel, 2003. ##[7] Rahman, E. A. A.; Nayfeh, A. H. and Masoud, Z. N.; “Dynamics and Control of Cranes: A Review,” Journal of Vibration and Control, Vol. 9, No 7, pp. 863-908, 2003. ##[8] Block, D. J.; Astrom, K. J. and Spong, M. W.; “The Reaction Wheel Pendulum,” Synthesis Lectures on Controls and Mechatronics, Vol. 1, No. 1, pp. 1-105, 2007. ##[9] Ghommam, J.; Mnif, F.; Benali, A. and Derbel, N.; “Asymptotic Backstepping Stabilization of an Underactuated Surface Vessel,” IEEE Transactions on Control Systems Technology, Vol. 14, No. 6, pp. 1150-1157, 2006. ##[10] Huang, J.; Chuan-Jiang, L. I.; Guang-Fu, M. A. and Gang, L.; “Generalized Inversion Based Attitude Control for Underactuated Spacecraft,” Acta Automatica Sinica, Vol. 39, No. 3, pp. 285-292, 2013. ##[11] Brockett, R. W.; “Asymptotic Stability and Feedback Stabilization,” Defense Technical Information Center, Harvard University, 1983. ##[12] Reyhanoglu, M.; Cho, S.; Harris, N. and McClamroch, N. H.; “Discontinuous Feedback Control of a Special Class of Underactuated Mechanical Systems,” International Journal of Robust and Nonlinear Control, Vol. 10, No. 4, pp. 265-281, 2000. ##[13] Morin, P. and Samson, C.; “Time-Varying Exponential Stabilization of the Attitude of a Rigid Spacecraft with Two Controls,” IEEE Conference on Decision and Control, Vol. 4, pp. 3938-3993, 1995. ##[14] Acosta, J.; Ortega, R.; Astolfi, A. and Mahindrakur, A. D.; “Interconnection and Damping Assignment Passivity-Based Control of Mechanical Systems with Underactuation Degree One,” IEEE Transactions on Automatic Control, Vol. 50, No. 12, pp. 1936-1955, 2005. ##[15] Sidi, M. J.; “Spacecraft Dynamics and Control, A Practical Engineering Approach,” Cambridge Aerospace Series, 2000. ##[16] Reyhanoglu, M.; “Discontinuous Feedback Stabilization of the Angular Velocity of a Rigid Body with Two Control Torques,” 35th IEEE Conference on Decision and Control, Vol. 3, pp. 2692-2694, 1996. ##[17] Wang, D.; Jia, Y.; Jin, L. and Xu, S.; “Control Analysis of an Underactuated Spacecraft under Disturbance,” Acta Astronautica, Vol. 83, pp. 44-53, 2013. ##[18] Reyhanoglu, M.; Cho, S.; Harris, N. and McClamroch, N. H. and Kolmanovsky, I.; “Discontinuous Feedback Control of a Planar Rigid Body with an Unactuated Degree of Freedom,” IEEE Conference on Decision and Control, Vol. 1, pp. 433- 438, 1998.##]
Adaptive Leader-Following and Leaderless Consensus of a Class of Nonlinear Systems Using Neural Networks
Adaptive Leader-Following and Leaderless Consensus of a Class of Nonlinear Systems Using Neural Networks
2
2
This paper deals with leader-following and leaderless consensus problems of high-order multi-input/multi-output (MIMO) multi-agent systems with unknown nonlinear dynamics in the presence of uncertain external disturbances. The agents may have different dynamics and communicate together under a directed graph. A distributed adaptive method is designed for both cases. The structures of the controllers simplify their implementation and reduce computational cost. Unknown nonlinearities are estimated by a radial basis function neural network (RBFNN). The ultimate boundness of the closed-loop system is guaranteed through Lyapunov stability analysis by introducing a suitably driven adaptive rule. Finally, the simulation results verify performance of the proposed control method.
1
This paper deals with leader-following and leaderless consensus problems of high-order multi-input/multi-output (MIMO) multi-agent systems with unknown nonlinear dynamics in the presence of uncertain external disturbances. The agents may have different dynamics and communicate together under a directed graph. A distributed adaptive method is designed for both cases. The structures of the controllers simplify their implementation and reduce computational cost. Unknown nonlinearities are estimated by a radial basis function neural network (RBFNN). The ultimate boundness of the closed-loop system is guaranteed through Lyapunov stability analysis by introducing a suitably driven adaptive rule. Finally, the simulation results verify performance of the proposed control method.
123
138
Bahram
Karimi
Bahram
Karimi
Associated Professor, Department of Electrical Engineering, Malek-e Ashtar University of Technology
Associated Professor, Department of Electrical
Iran
bkarimi@mut-es.ac.ir
Hassan
Ghiti Sarand
Hassan
Ghiti Sarand
Assistant Professor, Department of Electrical Engineering, Malek-e Ashtar University of Technology
Assistant Professor, Department of Electrical
Iran
hghsarand@mut-es.ac.ir
Adaptive control
Consensus
MIMO systems
Neural Networks
multi-agent systems
[[1] Ren, W.; Beard, R. W. and Atkins, E. M.; “Information Consensus in Multivehicle Cooperative Control,” IEEE Control Syst. Mag., Vol. 27, No. 2, pp. 71–82, 2007. ##[2] Olfati-Saber, R. and Murray, R. M.; “Consensus Problems in Networks of Agents with Switching Topology and Time-Delays,” IEEE Trans. Autom. Control, Vol. 49, No. 9, pp. 1520–1533, 2004. ##[3] Ren, W. and Beard, R. W.; “Distributed Consensus in Multi-Vehicle Cooperative Control: Theory and Applications,” Springer-Verlag, London, 2008. ##[4] Ren, W.; Beard, R. W. and Chen, Y.; “High-order and Model Reference Consensus Algorithms in Cooperative Control of Multivehicle Systems,” J. Dyn. Syst., Meas., Control, Vol. 129, No. 5, pp. 678– 688, 2007. ##[5] Ni, W.; Wang, X. and Xiong, C.; “Consensus Controllability, Observability and Robust Design for Leader-Following Linear Multi-Agent Systems,” Automatica, Vol. 49, No. 7, pp. 2199–2205, 2013. ##[6] Li, Z.; Duan, Z. and Chen, G.; “Dynamic Consensus of Linear Multi-Agent Systems,” IET Control Theory Appl., Vol.5, No.1, pp.19–28, 2011. ##[7] Wang, L. and Wang, Q. G.; “A General Approach for Synchronization of Nonlinear Networked Systems with Switching Topology,” In. J. Syst. Sci., Vol. 44, No. 12, pp. 2199–2210, 2013. ##[8] Yu, W.; Chen, G. and Cao, M.; “Consensus in Directed Networks of Agents with Nonlinear Dynamics,” IEEE Trans. Autom. Control, Vol. 56, No. 6, pp. 1436–1441, 2011. ##[9] Wen, G.; Rahmani, A. and Yu, Y.; “Consensus Tracking for Multi-Agent Systems with Nonlinear Dynamics under Fixed Communication Topologies,” Proc. World Congr. Eng. Comput. Sci. (WCECS), San Francisco, USA, Vol. 1, pp. 19–21, 2011. ##[10] Li, Z.; Liu, X. and Fu, M.; “Global Consensus Control of Lipschitz Nonlinear Multi-Agent Systems,” 18th IFAC World Congr., Milano, Italy, pp. 10056–10061, 2011. ##[11] Yu, W.; Chen, G.; Cao, M. and Kurths, J.; “Second-Order Consensus for Multiagent Systems with Directed Topologies and Nonlinear Dynamics,” IEEE Trans. Syst., Man, Cybern., Syst., Part B: Cybernetics, Vol. 40, No. 3, pp. 881–891, 2010. ##[12] Ren, W.; “Distributed Leaderless Consensus Algorithms for Networked Euler– Lagrange Systems,” Int. J. Control, Vol. 82, pp. 2137–2149, 2009. ##[13] Hui, Q.; Haddad, M. M. and Bhat, S.; “Finite- Time Semistability and Consensus for Nonlinear Dynamical Networks,” IEEE Trans. Autom. Control, Vol. 53, No. 8, pp. 1887–1900, 2008. ##[14] Suiyang, K.; Xie, L. and Man, Z.; “Robust Finite- Time Consensus Tracking Algorithm for Multirobot Systems,” IEEE/ASME Trans. Mechatronics, Vol. 14, No. 2, pp. 219–228, 2009. ##[15] Dong, Y. and Huang, J.; “Cooperative Global Output Regulation for a Class of Nonlinear Multi- Agent Systems,” IEEE Trans. Autom. Control, Vol. 59, No. 5, pp. 1348–1354, 2014. ##[16] Wen, G.; Duan, Z.; Chen, G. and Yu, W.; “Consensus Tracking of Multi-Agent Systems with Lipschitz-Type Node Dynamics and Switching Topologies,” IEEE Trans. Circuits Syst. I, Reg. Papers, Vol. 61, No. 2, pp. 499–511, 2014. ##[17] Wen, G.; Peng, Z.; Rahmani, A. and Yu, Y.; “Distributed Leader-Following Consensus for Second-Order Multi-Agent Systems with Nonlinear Inherent Dynamics,” Int. J. Systems Sci., Vol. 45, No. 9, pp. 1892–1901, 2014. ##[18] Zhang, W.; Wang, Z. and Guo, Y.; “Backstepping- Based Synchronisation of Uncertain Networked Lagrangian Systems,” Int. J. Systems Sci., Vol. 45, No. 2, pp. 145–158, 2014. ##[19] Hou, Z.; Cheng, L.; and Tan, M.; “Decentralized Robust Adaptive Control for the Multiagent System Consensus Problem Using Neural Networks,” IEEE Trans. Syst., Man, Cybern.,Syst., Part B: Cybernetics, Vol. 39, No. 3, pp. 636–647, 2009. ##[20] Cheng, L.; Hou, Z. G.; Tan, M.; Lin, Y. and Zhang, W.; “Neural-Network Based Adaptive Leader-Following Control for Multiagent Systems with Uncertainties,” IEEE Trans. Neural Netw., Vol. 21, No. 8, pp. 1351–1358, 2010. ##[21] Das, A. and Lewis, F. L.; “Distributed Adaptive Control for Synchronization of Unknown Nonlinear Networked Systems,” Automatica, Vol. 46, No. 2, pp. 2014–2021, 2010. ##[22] Liu, Y. and Jia, Y.; “Adaptive Consensus Protocol for Networks of Multiple Agents with Nonlinear Dynamics Using Neural Networks,” Asian J. Control, Vol. 14, No. 5, pp. 1328–1339, 2012. ##[23] Cui, R.; Ren, B. and Ge, S. S.; “Synchronised Tracking Control of Multi-Agent System with High- Order Dynamics,” IET Control Theory Appl., Vol. 6, No. 5, pp. 603–614, 2012. ##[24] Zhang, H. and Lewis, F. L.; “Adaptive Cooperative Tracking Control of Higher-Order Nonlinear Systems with Unknown Dynamics,” Automatica, Vol. 48, No. 7, pp. 1432–1439, 2012. ##[25] Das, A. and Lewis, F. L.; “Cooperative Adaptive Control for Synchronization of Second-Order Systems with Unknown Nonlinearities,” Int. J. Robust Nonlinear Control, Vol. 21, No. 13, pp. 1509– 1524, 2011. ##[26] Zouand, A. M. and Kumar, K. D.; “Neural Network-Based Adaptive Output Feedback Formation Control for Multi-Agent Systems,” J. Nonlinear Dynamics, Vol. 70, No. 2, pp. 1283–1296, 2012. ##[27] Zou, A. M.; Kumar, K. D. and Hou, Z. G.; “Distributed Consensus Control for Multi-Agent Systems Using Terminal Sliding Mode and Chebyshev Neural Networks,” Int. J. Robust Nonlinear Control, Vol. 23, pp. 334–357, 2013. ##[28] Peng, Z.; Wang, D.; Sun, G. and Wang, H.; “Distributed Cooperative Stabilisation of Continuous–Time Uncertain Nonlinear Multi-Agent Systems,” Int. J. Systems Sci., Vol. 45, No. ,10, pp. 2031–2041, 2013. ##[29] Xu, H. and Ioannou, P. A.; “Robust Adaptive Control for a Class of MIMO Nonlinear Systems with Guaranteed Error Bounds,” IEEE Trans. Autom. Control, Vol. 48, No. 5, pp. 728–742, 2003. ##[30] Bechlioulis, C. P. and Rovithakis, G. A.; “Prescribed Performance Adaptive Control for Multi- Input Multi-Output Affine in the Control Nonlinear Systems,” IEEE Trans. Autom. Control, Vol. 55, No.5, pp. 1220–1226, 2010. ##[31] Frikha, S.; Djemel, M. and Derbel, N.; “Observer Based Adaptive Neuro-Sliding Mode Control for MIMO Nonlinear Systems,” Int. J. Control, Automation, Syst., Vol. 8, No. 2, pp. 257–265, 2010. ##[32] Slotine, J. J. and Li, W.; “Applied Nonlinear Control,” Upper Saddle River, Englewood Cliffs, NJ: Prentice–Hall, Vol. 199, No. 1, 1991. ##[33] Qu, Z.; “Cooperative Control of Dynamical Systems: Applications to Autonomous Vehicles,” Springer Science and Business Media, London, 2009. ##[34] Lian, J.; Hu, J. and Zak, S. H.; “Variable Neural Adaptive Robust Control: A Switched System Approach,” IEEE Trans. Neural Networks Learning Systems, Vol. 26, No. 5, pp. 903–15, 2015. ##[35] Karimi, B.; Menhaj, M. B.; Karimi–Ghartemani, M. and Saboori, I.; “Decentralized Adaptive Control of Large–Scale Affine and Nonaffine Nonlinear Systems,” IEEE Trans. Instrum. Meas., Vol. 58, No. 8, pp. 2459–2467, 2009. ##[36] Kar, I. and Behera, J.; “Direct Adaptive Neural Control for Affine Nonlinear Systems,” Appl. Soft Computing, Vol. 9, No. 2, pp. 756–764, 2008. ##[37] Khalil, H. K.; “Nonlinear Systems,” Upper Saddle River, NJ: Prentice–Hall, 3rd ed., 2002. ##[38] Qin, J.; Yu, C.; Gao, H. and Wang, X.; “Leaderless Consensus Control of Dynamical Agents under Directed Interaction Topology,” 50th IEEE Conference on Decision and Control and European Control Conference (CDC–ECC), Orlando, FL, USA, pp. 12–15, 2011. ##[39] Almeida, J.; Silvestre, C.; Pascoal, A. M.; “Cooperative Control of Multiple Surface Vessels with Discrete–Time Periodic Communications,” Int. J. Robust and Nonlinear Control, Vol. 22, No. 4, pp. 398–419, 2012. ##[40] Chung, S. J.; “Nonlinear Control and Synchronization of Multiple Lagrangian Systems with Application to Tethered Formation Flight Spacecraft,” Ph.D. Dissertation, Department of Aeronautics and Astronautics, MIT, 2007.##]