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Transformation of LQR Weights for Discretization Invariant Performance of PI/PID Dominant Pole Placement Controllers

  • Kaushik Halder (a1), Saptarshi Das (a2) and Amitava Gupta (a1)

Summary

Linear quadratic regulator (LQR), a popular technique for designing optimal state feedback controller, is used to derive a mapping between continuous and discrete time inverse optimal equivalence of proportional integral derivative (PID) control problem via dominant pole placement. The aim is to derive transformation of the LQR weighting matrix for fixed weighting factor, using the discrete algebraic Riccati equation (DARE) to design a discrete time optimal PID controller producing similar time response to its continuous time counterpart. Continuous time LQR-based PID controller can be transformed to discrete time by establishing a relation between the respective LQR weighting matrices that will produce similar closed loop response, independent of the chosen sampling time. Simulation examples of first/second order and first-order integrating processes exhibiting stable/unstable and marginally stable open loop dynamics are provided, using the transformation of LQR weights. Time responses for set-point and disturbance inputs are compared for different sampling times as fraction of the desired closed loop time constant.

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Corresponding author

*Corresponding author. E-mails: saptarshi.das@ieee.org, s.das3@exeter.ac.uk

References

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1.Åström, K. J. and Hägglund, T., PID Controllers: Theory, Design, and Tuning, vol. 10 (Instrument Society of America, North Carolina, USA, 1995).
2.Cominos, P. and Munro, N., “PID controllers: recent tuning methods and design to specification,” IEEE Proc. Control Theory Appl. 149(1), 4653 (2002).
3.Ogata, K., Modern Control Engineering (Prentice Hall PTR, NJ, USA, 2001).
4.Anderson, B. D. and Moore, J. B., Optimal Control: Linear Quadratic Methods (Courier Corporation, Englewood Cliffs, NJ, USA, 2007).
5.Dorato, P. and Levis, A. H., “Optimal linear regulators: The discrete-time case,” IEEE Trans. Autom. Control 16(6), 613620 (1971).
6.Wang, Q.-G., Zhang, Z., Astrom, K. J. and Chek, L. S., “Guaranteed dominant pole placement with PID controllers,” J. Process Control 19(1), 349352 (2009).
7.Tang, W., Wang, Q.-G., Ye, Z. and Zhang, Z., “PID tuning for dominant poles and phase margin,” Asian J. Control 9(4), 466469 (2007).
8.Kang, H. I., “Design of dominant pole region assignment with PID controllers,” 2010 International Conference on Intelligent Computation Technology and Automation (ICICTA), Changsha, China, vol. 2 (2010) pp. 1922
9.Madady, A. and Reza-Alikhani, H.-R., “First-order controllers design employing dominant pole placement,” 2011 19th Mediterranean Conference on Control & Automation (MED), Corfu, Greece (2011) pp. 14981503.
10.Yinya, L., Andong, S. and Quoqing, Q., “Further results on guaranteed dominant pole placement with PID controllers,” 2011 30th Chinese Control Conference (CCC), Yantai, China (2011) pp. 37563760.
11.Velásquez, I. G., Yuz, J. I. and Salgado, M. E., “Optimal control synthesis with prescribed closed loop poles,” 2011 19th Mediterranean Conference on Control & Automation (MED), Corfu, Greece (2011) pp. 108113.
12.Das, S., Halder, K., Pan, I., Ghosh, S. and Gupta, A., “Inverse optimal control formulation for guaranteed dominant pole placement with PI/PID controllers,” 2012 International Conference on Computer Communication and Informatics (ICCCI), Coimbatore, India (2012) pp. 16.
13.Fujii, T., “A new approach to the LQ design from the viewpoint of the inverse regulator problem,” IEEE Trans. Autom. Control 32(11), 9951004 (1987).
14.Fujii, T. and Narazaki, M., “A complete solution to the inverse problem of optimal control,” 1982 21st IEEE Conference on Decision and Control, Orlando, FL, USA, vol. 21 (1982) pp. 289294.
15.Moylan, P. J. and Anderson, B., “Nonlinear regulator theory and an inverse optimal control problem,” IEEE Trans. Autom. Control 18(5), 460465 (1973).
16.Sugimoto, K., “Partial pole placement by LQ regulators: An inverse problem approach,” IEEE Trans. Autom. Control 43(5), 706708 (1998).
17.Choi, Y. and Chung, W. K., “Performance limitation and autotuning of inverse optimal PID controller for Lagrangian systems,” J. Dyn. Syst. Meas. Control 127(2), 240249 (2005).
18.Fujinaka, T. and Katayama, T., “Discrete-time optimal regulator with closed-loop poles in a prescribed region,” Inter. J. Control 47(5), 13071321 (1988).
19.He, J.-B., Wang, Q.-G. and Lee, T.-H., “PI/PID controller tuning via LQR approach,” Chem. Eng. Science, 55(13), 24292439 (2000).
20.Saha, S., Das, S., Das, S. and Gupta, A., “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), 36283642 (2012).
21.Saif, M., “Optimal linear regulator pole-placement by weight selection,” Inter J. Control 50(1), 399414 (1989).
22.Das, S., Pan, I., Halder, K., Das, S. and Gupta, A., “LQR based improved discrete PID controller design via optimum selection of weighting matrices using fractional order integral performance index,” Appl. Math Modell. 37(6), 42534268 (2013).
23.Das, S., Pan, I. and Das, S., “Multi-objective LQR with optimum weight selection to design FOPID controllers for delayed fractional order processes,” ISA Trans. 58, 3549 (2015).
24.Das, S. and Halder, K., “Missile attitude control via a hybrid LQG-LTR-LQI control scheme with optimum weight selection,” 2014 First International Conference on Automation, Control, Energy and Systems (ACES), Hooghly, India (2014) pp. 16.
25.Abdelaziz, T. H., “Pole placement for single-input linear system by proportional-derivative state feedback,” J. Dyn. Syst. Meas. Control 137(4), 041015 (2015).
26.Abdelaziz, T. H., “Stabilization of single-input LTI systems by proportional-derivative feedback,” Asian J. Control 17(6), 21652174 (2015).
27.Abdelaziz, T. H., “Stabilization of linear time-varying systems using proportional-derivative state feedback,” Trans. Inst. Meas Control 40(7), 21002115 (2017).
28.Ogata, K., Discrete-Time Control Systems, vol. 2 (Prentice Hall, Englewood Cliffs, NJ, 1995).
29.Franklin, G. F., Powell, J. D. and Workman, M. L., Digital Control of Dynamic Systems, vol. 3 (Addison-Wesley, Menlo Park, 1998).
30.Rao, V. G. and Bernstein, D. S., “Naive control of the double integrator,” IEEE Control Syst. 21(5), 8697 (2001).
31.Åstrröm, K. J. and Hägglund, T., “Benchmark systems for PID control,” IFAC Proc. Vol. 33(4), 165166 (2000).
32.Kristiansson, B. and Lennartson, B., “Robust tuning of PI and PID controllers: using derivative action despite sensor noise,” IEEE Control Syst. 26(1), 5569 (2006).
33.Åstrröm, K. J., Panagopoulos, H. and Hägglund, T., “Design of PI controllers based on non-convex optimization,” Automatica 34(5), 585601 (1998).
34.Isaksson, A. and Graebe, S., “Derivative filter is an integral part of PID design,” IEE Proc. Control Theory Appl. 149(1), 4145 (2002).
35.Yaniv, O. and Nagurka, M., “Robust PI controller design satisfying sensitivity and uncertainty specifications,” IEEE Trans. Autom. Control 48(11), 20692072 (2003).
36.Halder, K., Das, S., Dasgupta, S., Banerjee, S. and Gupta, A., “Controller design for networked Control systems—an approach based on L2 induced norm,” Nonlinear Anal. Hybrid Syst. 19, 134145 (2016).
37.Åstrröm, K. J. and Hägglund, T., Advanced PID Control (ISA-The Instrumentation, Systems and Automation Society, 2006).
38.Doyle, J. C., Francis, B. A. and Tannenbaum, A. R., Feedback Control Theory (Courier Corporation, 2013).
39.Srivastava, S., Misra, A., Thakur, S. and Pandit, V., “An optimal PID controller via LQR for standard second order plus time delay systems,” ISA Trans. 60, 244253 (2016).
40.Pan, I. and Das, S., “Design of hybrid regrouping PSO-GA based sub-optimal networked control system with random packet losses,” Memetic Comput. 5(2), 141153 (2013).
41.Pan, I., Mukherjee, A., Das, S. and Gupta, A., “Simulation studies on multiple control loops over a bandwidth limited shared communication network with packet dropouts,” 2011 IEEE Students’ Technology Symposium (TechSym) (2011) pp. 113118.

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