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Design of State Feedback LQR Based Dual Mode Fractional-Order PID Controller using Inertia Weighted PSO Algorithm: For Control of an Underactuated System

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Abstract

Several control strategies are proposed and developed to enhance the performance of the various underactuated systems, particularly inverted pendulum. This paper presents the dual-mode fractional-order control with a reference model for pitch and angle control of an inverted pendulum. An inertia weighted PSO is utilized for optimal tuning of the FOPID parameters to ensure an optimal balance between local and global search. A cost function of this algorithm is framed based on the error between the reference model output and actual system output along with time-domain performance criteria. The reference model-based tuning improves the performance of the controller and steady-state error tracking. In addition, an optimal state feedback LQR is implemented using pole placement design in the feedback to stabilize and improve the robustness of the inverted pendulum. Compared to the conventional PID, the proposed structure illustrates the FOPID structure has significant performance improvement in both with and without disturbance conditions.

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Abbreviations

\(F\) :

Applied force to cart (N)

\(X\) :

Position of the cart from the reference (m)

\(\theta\) :

Pendulum angle concerning the vertical (rad)

\(M\) :

Mass of the cart (kg)

\(m\) :

Mass of the pendulum (kg)

\(L\) :

Length of the pendulum (m)

\(J\) :

Moment of inertia of pendulum (kg.m2),

\(b\) :

Coefficient of cart friction (Ns/m)

\(g\) :

Acceleration owing to gravity (m/s2).

\(P\) :

Vertical force exerted by the cart on the base of the pendulum (N)

\(N\) :

Horizontal force exerted by the cart on the base of the pendulum (N)

\(\phi\) :

Minor diverge (\(\theta = \pi + \phi\)) from equilibrium (rad)

\(\chi\) :

Order of the fractional system

\(\Gamma \left( \cdot \right)\) :

Gamma function

\(\ell\) :

Lower limit of fractional-order integro-differential operator

\(\varepsilon\) :

Upper limit of fractional-order integro-differential operator

\(K_{{\text{p}}}\) :

Proportional gain

\(K_{{\text{i}}}\) :

Integral gain

\(K_{{\text{d}}}\) :

Derivative gain

\(\lambda\) :

Integer-order

\(\mu\) :

Derivative order

\(x\) :

State vector of the system

\(\dot{x}\) :

Differentiation of state variable

\(\, A\) :

System matrix \(\left( {R^{n \times n} } \right)\)

\(B\) :

Input matrix \(\left( {R^{n \times m} } \right)\)

\(C\) :

Output matrix \(\left( {R^{p \times n} } \right)\)

\(D\) :

Transition matrix \(\left( {R^{p \times m} } \right)\)

\(u\) :

Control input vector

\(Q\;{\text{and}}\;R\) :

Weighting matrices of LQR

\(y\) :

Output vector

\(K\) :

Optimal control gain

\(\beta_{i}\) :

Characteristic polynomial \(\left( {i = 1,2 \ldots n} \right)\)

\(\alpha_{i}\) :

Eigenvalues \(\left( {i = 1,2 \ldots n} \right)\)

\(\vartheta_{i}\) :

Desired poles

\(T^{ + }\) :

Transformation matrix

\(M\) :

Controllability matrix

\(P\) :

Riccati matrix

\(K_{P}\) :

State feedback gain matrix

\(\left( {B^{T} } \right)^{ + }\) :

Pseudoinverse matrix

\(D\) :

Dimensions

\(f\) :

Function

\(J\) :

Objective function

\(v_{i}\) :

Velocity vector of particle i

\(x_{i}\) :

Position vector of particle i

\(P_{best,i}\) :

Personal best position of particle i

\(G_{best}\) :

Global best particle position

\(c_{1} ,c_{2}\) :

Accelerated constants

\(rand_{1} ,rand_{2}\) :

Random numbers

\(w\) :

Inertia weight constant

\(w_{{{\text{start}}}}\) :

Starting value of the inertia weight

\(w_{{{\text{end}}}}\) :

Final value of the inertia weight

\(\delta\) :

Weightage function

\(t\) :

Current iteration of the algorithm

\(t_{\max }\) :

User-specified maximum iteration of the algorithm

\(\omega\) :

Frequency

\(G_{{\text{R}}}\) :

Transfer function of the reference model

\(G_{m}\) :

Process transfer function

\(G_{m1}\) :

Process transfer function or model at various working condition

\(C(s)\) :

Controller transfer function

\(e\) :

Error signal

\(T\) :

Simulation time

\(t_{{\text{r}}}\) :

Rise time

\(t_{{\text{s}}}\) :

Settling time

E ss :

Steady-state error

IP:

Inverted pendulum

P:

Proportional

PI:

Proportional integral

PID:

Proportional integral derivative

PID:

Proportional derivative

FO:

Fractional order

IO:

Integral order

NN:

Neural networks

FOPID:

Fractional-order proportional integral derivative

PSO:

Particle swarm optimization

IWPSO:

Inertia weight particle swarm optimization

LDIW:

Linearly decreasing inertia weight

LQR:

Linear quadratic regulator

FOMCON:

Fractional-order modeling and control

CRONE:

Commande robuste d’ordre non entier

LTI:

Linear time-invariant

ARE:

Algebraic Riccati equation

SD:

Standard deviation

ISE:

Integral square error

References

  1. L.B. Prasad, B. Tyagi, H.O. Gupta, Optimal control of nonlinear inverted pendulum system using PID controller and LQR: performance analysis without and with disturbance input. Int. J. Autom. Comput. 11(6), 661–670 (2014). https://doi.org/10.1007/s11633-014-0818-1

    Article  Google Scholar 

  2. F.H. Guaracy, R.L. Pereira, C.F. de Paula, Robust stabilization of inverted pendulum using ALQR augmented by second-order sliding mode control. J. Control Autom. Electr. Syst. 28(5), 577–584 (2017). https://doi.org/10.1007/s40313-017-0332-0

    Article  Google Scholar 

  3. B. Beigzadehnoe, Z. Rahmani, A. Khosravi, B. Rezaie, Control of interconnected systems with sensor delay based on decentralized adaptive neural dynamic surface method. Proc. Inst. Mech. Eng. Part I: J. Syst. Control Eng. 1, 1 (2020). https://doi.org/10.1177/0959651820966529

    Article  Google Scholar 

  4. J. Huang, M. Ri, D. Wu, S. Ri, Interval type-2 fuzzy logic modeling and control of a mobile two-wheeled inverted pendulum. IEEE Trans. Fuzzy Syst. 26(4), 2030–2038 (2017). https://doi.org/10.1109/tfuzz.2017.2760283

    Article  Google Scholar 

  5. M. Babaie, Z. Rahmani, B. Rezaie, Design of a switching controller for unstable systems under variable disturbance. Int. J. Dyn. Control 6(4), 1706–1718 (2018). https://doi.org/10.1007/s40435-018-0394-2

    Article  MathSciNet  Google Scholar 

  6. S. Irfan, A. Mehmood, M.T. Razzaq, J. Iqbal, Advanced sliding mode control techniques for inverted pendulum: Modelling and simulation. Eng. Sci. Technol. Int. J. 21(4), 753–759 (2018). https://doi.org/10.1016/j.jestch.2018.06.010

    Article  Google Scholar 

  7. S.H. Zabihifar, A.S. Yushchenko, H. Navvabi, Robust control based on adaptive neural network for Rotary inverted pendulum with oscillation compensation. Neural Comput. Appl. 1, 1–13 (2020). https://doi.org/10.1007/s00521-020-04821-x

    Article  Google Scholar 

  8. M.A. Johnson, M.H. Moradi, PID Control (Springer-Verlag, 2005)

    Book  Google Scholar 

  9. G.W. Leibniz, Letter from Hanover, Germany, 30 Sept. 1695 to GAV 2. L'Hospital Lebnizen Mathematische Schriften., Olms Verl., Hildesheim, 1962, 301–302.

  10. J.T. Machado, V. Kiryakova, F. Mainardi, Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1140–1153 (2011). https://doi.org/10.1016/j.cnsns.2010.05.027

    Article  MathSciNet  MATH  Google Scholar 

  11. I. Birs, C. Muresan, I. Nascu, C. Ionescu, A survey of recent advances in fractional order control for time delay systems. IEEE Access 7, 30951–30965 (2019). https://doi.org/10.1109/ACCESS.2019.2902567

    Article  Google Scholar 

  12. I. Podlubny, Fractional-order systems and PI/sup/spl lambda//D/sup/spl mu//-controllers. IEEE Trans. Autom. Control 44(1), 208–214 (1999). https://doi.org/10.1109/9.739144

    Article  MATH  Google Scholar 

  13. M.F. Silva, J.T. Machado, A.M. Lopes, Fractional order control of a hexapod robot. Nonlinear Dyn. 38(1), 417–433 (2004). https://doi.org/10.1007/s11071-004-3770-8

    Article  MATH  Google Scholar 

  14. R. Duma, P. Dobra, M. Trusca, Embedded application of fractional order control. Electron. Lett. 48(24), 1526–1528 (2012)

    Article  Google Scholar 

  15. R. Rajesh, Optimal tuning of FOPID controller based on PSO algorithm with reference model for a single conical tank system. SN Appl. Sci. 1(7), 1–14 (2019). https://doi.org/10.1007/s42452-019-0754-3

    Article  Google Scholar 

  16. V. Pommier, J. Sabatier, P. Lanusse, A. Oustaloup, CRONE control of a nonlinear hydraulic actuator. Control. Eng. Pract. 10(4), 391–402 (2002). https://doi.org/10.1016/S0967-0661(01)00154-X

    Article  Google Scholar 

  17. D. Valerio, J.S. Da Costa, Ninteger: a non-integer control toolbox for MatLab. Proceedings of fractional differentiation and its applications, Bordeaux. 2004.

  18. A. Tepljakov, FOMCON: fractional-order modeling and control toolbox. In Fractional-order Modeling and Control of Dynamic Systems (107–129). Springer. 2017. https://doi.org/10.1007/978-3-319-52950-9_6

  19. S. Das, I. Pan, S. Das, Multi-objective LQR with optimum weight selection to design FOPID controllers for delayed fractional order processes. ISA Trans. 58, 35–49 (2015). https://doi.org/10.1016/j.isatra.2015.06.002

    Article  Google Scholar 

  20. R. Ranganayakulu, G.U.B. Babu, A.S. Rao, D.S. Patle, A comparative study of fractional order PIλ/PIλDµ tuning rules for stable first order plus time delay processes. Resour.-Efficient Technol. 2, S136–S152 (2016). https://doi.org/10.1016/j.reffit.2016.11.009

    Article  Google Scholar 

  21. Z. Chen, X. Yuan, B. Ji, P. Wang, H. Tian, Design of a fractional order PID controller for hydraulic turbine regulating system using chaotic non-dominated sorting genetic algorithm II. Energy Convers. Manag. 84, 390–404 (2014). https://doi.org/10.1016/j.enconman.2014.04.052

    Article  Google Scholar 

  22. M. Pirasteh-Moghadam, M.G. Saryazdi, E. Loghman, A. Kamali, F. Bakhtiari-Nejad, Development of neural fractional order PID controller with emulator. ISA Trans. 106, 293–302 (2020). https://doi.org/10.1016/j.isatra.2020.06.014

    Article  Google Scholar 

  23. C. Li, N. Zhang, X. Lai, J. Zhou, Y. Xu, Design of a fractional-order PID controller for a pumped storage unit using a gravitational search algorithm based on the Cauchy and Gaussian mutation. Inf. Sci. 396, 162–181 (2017). https://doi.org/10.1016/j.ins.2017.02.026

    Article  Google Scholar 

  24. Z. Bingul, O. Karahan, Comparison of PID and FOPID controllers tuned by PSO and ABC algorithms for unstable and integrating systems with time delay. Optim. Control Appl. Methods 39(4), 1431–1450 (2018). https://doi.org/10.1002/oca.2419

    Article  MathSciNet  MATH  Google Scholar 

  25. D. Guha, P.K. Roy, S. Banerjee, S. Padmanaban, F. Blaabjerg, D. Chittathuru, Small-signal stability analysis of hybrid power system with quasi-oppositional sine cosine algorithm optimized fractional order PID controller. IEEE Access 8, 155971–155986 (2020). https://doi.org/10.1109/ACCESS.2020.3018620

    Article  Google Scholar 

  26. L. Chaib, A. Choucha, S. Arif, Optimal design and tuning of novel fractional order PID power system stabilizer using a new metaheuristic Bat algorithm. Ain Shams Eng. J. 8(2), 113–125 (2017). https://doi.org/10.1016/j.asej.2015.08.003

    Article  Google Scholar 

  27. R. Rajesh, S.N. Deepa, Design of direct MRAC augmented with 2 DoF PIDD controller: an application to speed control of a servo plant. J. King Saud Univ.-Eng. Sci. 32(5), 310–320 (2020). https://doi.org/10.1016/j.jksues.2019.02.005

    Article  Google Scholar 

  28. F. Zhao, G.Q. Zeng, K.D. Lu, EnLSTM-WPEO: Short-term traffic flow prediction by ensemble LSTM, NNCT weight integration, and population extremal optimization. IEEE Trans. Veh. Technol. 69(1), 101–113 (2019). https://doi.org/10.1109/TVT.2019.2952605

    Article  Google Scholar 

  29. M. Črepinšek, S.H. Liu, M. Mernik, Exploration and exploitation in evolutionary algorithms: a survey. ACM Comput. Surv. (CSUR) 45(3), 1–33 (2013). https://doi.org/10.1145/2480741.2480752

    Article  MATH  Google Scholar 

  30. G.Q. Zeng, X.Q. Xie, M.R. Chen, J. Weng, Adaptive population extremal optimization-based PID neural network for multivariable nonlinear control systems. Swarm Evol. Comput. 44, 320–334 (2019). https://doi.org/10.1016/j.swevo.2018.04.008

    Article  Google Scholar 

  31. S.N. Deepa, G. Sugumaran, Model order formulation of a multivariable discrete system using a modified particle swarm optimization approach. Swarm Evol. Comput. 1(4), 204–212 (2011). https://doi.org/10.1016/j.swevo.2011.06.005

    Article  Google Scholar 

  32. S. Khan, M. Kamran, O.U. Rehman, L. Liu, S. Yang, A modified PSO algorithm with dynamic parameters for solving complex engineering design problem. Int. J. Comput. Math. 95(11), 2308–2329 (2018). https://doi.org/10.1080/00207160.2017.1387252

    Article  Google Scholar 

  33. P.A. Digehsara, S.N. Chegini, A. Bagheri, M.P. Roknsaraei, An improved particle swarm optimization based on the reinforcement of the population initialization phase by scrambled Halton sequence. Cogent Eng. 7(1), 1737383 (2020). https://doi.org/10.1080/23311916.2020.1737383

    Article  Google Scholar 

  34. F. Rajabi, B. Rezaie, Z. Rahmani, A novel nonlinear model predictive control design based on a hybrid particle swarm optimization-sequential quadratic programming algorithm: Application to an evaporator system. Trans. Inst. Meas. Control. 38(1), 23–32 (2016). https://doi.org/10.1177/0142331214561917

    Article  Google Scholar 

  35. R.C. Eberhart, Y. Shi, Comparing inertia weights and constriction factors in particle swarm optimization, in Proceedings of the 2000 congress on evolutionary computation, 2000, 1, 84–88. https://doi.org/10.1109/CEC.2000.870279

  36. Y. Shi, R.A. Eberhart, A modified particle swarm optimizer. In 1998 IEEE international conference on evolutionary computation proceedings. IEEE world congress on computational intelligence, IEEE, 1998, 69–73https://doi.org/10.1109/ICEC.1998.699146

  37. R.C. Eberhart, Y. Shi, Tracking and optimizing dynamic systems with particle swarms, in Proceedings of the 2001 congress on evolutionary computation, IEEE, 2001, 1, 94–100. https://doi.org/10.1109/CEC.2001.934376

  38. J. Xin, G. Chen, Y. Hai, A particle swarm optimizer with multi-stage linearly-decreasing inertia weight, in 2009 International joint conference on computational sciences and optimization, IEEE, 1, 505-508https://doi.org/10.1109/CSO.2009.420

  39. M.A. Arasomwan, A.O. Adewumi, On the performance of linear decreasing inertia weight particle swarm optimization for global optimization. Sci. World J. 1, 1–12 (2013). https://doi.org/10.1155/2013/860289

    Article  Google Scholar 

  40. M.S. Arumugam, M.V.C. Rao, On the performance of the particle swarm optimization algorithm with various inertia weight variants for computing optimal control of a class of hybrid systems. Discret. Dyn. Nat. Soc. 1, 1–7 (2006). https://doi.org/10.1155/DDNS/2006/79295

    Article  MathSciNet  MATH  Google Scholar 

  41. R.S. Barbosa, J.T. Machado, I.M. Ferreira, Tuning of PID controllers based on Bode’s ideal transfer function. Nonlinear Dyn. 38(1), 305–321 (2004). https://doi.org/10.1007/s11071-004-3763-7

    Article  MATH  Google Scholar 

  42. R. Pradhan, S.K. Majhi, J.K. Pradhan, B.B. Pati, Antlion optimizer tuned PID controller based on Bode ideal transfer function for automobile cruise control system. J. Ind. Inf. Integr. 9, 45–52 (2018). https://doi.org/10.1016/j.jii.2018.01.002

    Article  Google Scholar 

  43. G.J. Wang, C.T. Fong, K.J. Chang, Neural-network-based self-tuning PI controller for precise motion control of PMAC motors. IEEE Trans. Industr. Electron. 48(2), 408–415 (2001). https://doi.org/10.1109/41.915420

    Article  Google Scholar 

  44. R. Rajesh, P.H. Krishnan, Effects of adaption gain in direct model reference adaptive control for a single conical tank system. Digit. Signal Process. 10(5), 77–82 (2018)

    Google Scholar 

  45. W. Wang, K. Nonami, Y. Ohira, Model reference sliding mode control of small helicopter XRB based on vision. Int. J. Adv. Rob. Syst. 5(3), 26 (2008). https://doi.org/10.5772/5604

    Article  Google Scholar 

  46. M.R. Mortazavi, A. Naghash, Pitch and flight path controller design for F-16 aircraft using combination of LQR and EA techniques. Proc. Inst. Mech. Eng. Part G: J. Aerosp. Eng. 232(10), 1831–1843 (2018). https://doi.org/10.1177/0954410017703144

    Article  Google Scholar 

  47. E.V. Kumar, G.S. Raaja, J. Jerome, Adaptive PSO for optimal LQR tracking control of 2 DoF laboratory helicopter. Appl. Soft Comput. 41, 77–90 (2016). https://doi.org/10.1016/j.asoc.2015.12.023

    Article  Google Scholar 

  48. S. Saha, S. Das, S. Das, A. Gupta, A conformal mapping based fractional order approach for sub-optimal tuning of PID controllers with guaranteed dominant pole placement. Commun. Nonlinear Sci. Numer. Simul. 17(9), 3628–3642 (2012). https://doi.org/10.1016/j.cnsns.2012.01.007

    Article  MathSciNet  MATH  Google Scholar 

  49. S. Das, I. Pan, K. Halder, S. Das, A. Gupta, LQR based improved discrete PID controller design via optimum selection of weighting matrices using fractional order integral performance index. Appl. Math. Model. 37(6), 4253–4268 (2013). https://doi.org/10.1016/j.apm.2012.09.022

    Article  MathSciNet  Google Scholar 

  50. Ö. Oral, L. Çetin, E. Uyar, A novel method on selection of Q and R matrices in the theory of optimal control. Int. J. Syst. Control 1(2), 1 (2010)

    Google Scholar 

  51. M.C. Priess, R. Conway, J. Choi, J.M. Popovich, C. Radcliffe, Solutions to the inverse LQR problem with application to biological systems analysis. IEEE Trans. Control Syst. Technol. 23(2), 770–777 (2014). https://doi.org/10.1109/TCST.2014.2343935

    Article  Google Scholar 

  52. C.W. Anderson, Learning to control an inverted pendulum using neural networks. IEEE Control Syst. Mag. 9(3), 31–37 (1989). https://doi.org/10.1109/37.24809

    Article  Google Scholar 

  53. S. Bhoi, N.C. Rout, B.C. Rout, (2017) MGWO Meta Heuristic Algorithm vs. Classical Tuning Method of FOPID Controller for Inverted Pendulum, in 2017 International Conference on Innovations in Control, Communication and Information Systems (ICICCI) (1–7). IEEE, 2017. https://doi.org/10.1109/ICICCIS.2017.8660898

  54. A. Ateş, C. Yeroglu, Optimal fractional order PID design via Tabu Search based algorithm. ISA Trans. 60, 109–118 (2016). https://doi.org/10.1016/j.isatra.2015.11.015

    Article  Google Scholar 

  55. C. Ma, Y. Hori, Fractional-order control: Theory and applications in motion control [past and present]. IEEE Ind. Electron. Mag. 1(4), 6–16 (2007). https://doi.org/10.1109/MIE.2007.909703

    Article  Google Scholar 

  56. A.H. Heidari, S. Etedali, M.R. Javaheri-Tafti, A hybrid LQR-PID control design for seismic control of buildings equipped with ATMD. Front. Struct. Civ. Eng. 12(1), 44–57 (2018). https://doi.org/10.1007/s11709-016-0382-6

    Article  Google Scholar 

  57. A. Jose, C. Augustine, S.M. Malola, K. Chacko, Performance study of PID controller and LQR technique for inverted pendulum. World J. Eng. Technol. 3(02), 76 (2015). https://doi.org/10.4236/wjet.2015.32008

    Article  Google Scholar 

  58. F. Marini, B. Walczak, Particle swarm optimization (PSO): A tutorial. Chemomet. Intell. Lab. Syst. 149, 153–165 (2015). https://doi.org/10.1016/j.chemolab.2015.08.020

    Article  Google Scholar 

  59. J.S. Chiou, S.H. Tsai, M.T. Liu, A PSO-based adaptive fuzzy PID-controllers. Simul. Model. Pract. Theory 26, 49–59 (2012). https://doi.org/10.1016/j.simpat.2012.04.001

    Article  Google Scholar 

  60. S.K. Mishra, D. Chandra, Stabilization of inverted cart-pendulum system using controller: a frequency-domain approach. Chin. J. Eng. (2013). https://doi.org/10.1155/2013/962401

    Article  Google Scholar 

  61. E. Garcia-Gonzalo, J.L. Fernandez-Martinez, A brief historical review of particle swarm optimization (PSO). J. Bioinf. Intell. Control 1(1), 3–16 (2012). https://doi.org/10.1166/jbic.2012.1002

    Article  Google Scholar 

  62. C. Guimin, J. Jianyuan, H. Qi, Study on the strategy of decreasing inertia weight in particle swarm optimization algorithm. J.-Xian Jiaotong Univ. 40(1), 53 (2006)

    Google Scholar 

  63. N. Adhikary, C. Mahanta, Integral backstepping sliding mode control for underactuated systems: swing-up and stabilization of the Cart-Pendulum System. ISA Trans. 52(6), 870–880 (2013). https://doi.org/10.1016/j.isatra.2013.07.012

    Article  Google Scholar 

  64. E.S. Varghese, A.K. Vincent, V. Bagyaveereswaran, Optimal control of inverted pendulum system using PID controller, LQR and MPC. IOP Conf. Ser.: Mater. Sci. Eng. 263(5), 1–16 (2017)

    Google Scholar 

  65. I. Siradjuddin, Z. Amalia, B. Setiawan, F. Ronilaya, E. Rohadi, A. Setiawan, C. Rahmad, S. Adhisuwignjo, Stabilising a cart inverted pendulum with an augmented PID control scheme. MATEC Web of Conf. EDP Sci. 197, 1–7 (2018). https://doi.org/10.1051/matecconf/201819711013

    Article  Google Scholar 

  66. Minouchehr, N. and Hosseini-Sani, S.K., 2015, October. Design of Model Predictive Control of two-wheeled inverted pendulum robot. In 2015 3rd RSI International Conference on Robotics and Mechatronics (ICROM), IEEE, 2015, 456–462. https://doi.org/10.1109/ICRoM.2015.73 67827

  67. Y.J. Wang, Determination of all feasible robust PID controllers for open-loop unstable plus time delay processes with gain margin and phase margin specifications. ISA Trans. 53(2), 628–646 (2014). https://doi.org/10.1016/j.isatra.2013.12.037

    Article  Google Scholar 

  68. F. Hao, G.Q. Zeng, K.D. Lu, EnLSTM-WPEO: Short-term traffic flow prediction by ensemble LSTM, NNCT weight integration, and population extremal optimization. IEEE Trans. Vehic. Technol. 69(1), 101–113 (2019). https://doi.org/10.1109/TVT.2019.2952605

    Article  Google Scholar 

  69. M. Morari, E. Zafiriou, Robust process control. Morari, 1989.

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Authors and Affiliations

  1. Department of Engineering Design, Indian Institute of Technology Madras, Chennai, 600036, India

    Rajesh Ramakrishnan

  2. Department of Electrical Engineering, National Institute of Technology Arunachal Pradesh, Yupia, 791112, India

    Deepa Subramaniam Nachimuthu

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Ramakrishnan, R., Subramaniam Nachimuthu, D. Design of State Feedback LQR Based Dual Mode Fractional-Order PID Controller using Inertia Weighted PSO Algorithm: For Control of an Underactuated System. J. Inst. Eng. India Ser. C 102, 1403–1417 (2021). https://doi.org/10.1007/s40032-021-00756-x

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