0
Research Papers

Pole Placement for Time-Delayed Systems Using Galerkin Approximations

[+] Author and Article Information
Shanti S. Kandala

Department of Mechanical and
Aerospace Engineering,
Indian Institute of Technology Hyderabad,
Kandi,
Sangareddy, Telangana 502285, India

Thomas K. Uchida

Department of Bioengineering,
Stanford University,
James H. Clark Center,
318 Campus Drive,
Stanford, CA 94305

C. P. Vyasarayani

Department of Mechanical and
Aerospace Engineering,
Indian Institute of Technology Hyderabad,
Kandi,
Sangareddy, Telangana 502285, India
e-mail: vcprakash@iith.ac.in

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received February 27, 2018; final manuscript received December 24, 2018; published online January 29, 2019. Assoc. Editor: Dumitru I. Caruntu.

J. Dyn. Sys., Meas., Control 141(5), 051012 (Jan 29, 2019) (10 pages) Paper No: DS-18-1098; doi: 10.1115/1.4042465 History: Received February 27, 2018; Revised December 24, 2018

Many dynamic systems of practical interest have inherent time delays and thus are governed by delay differential equations (DDEs). Because DDEs are infinite dimensional, time-delayed systems may be difficult to stabilize using traditional controller design strategies. We apply the Galerkin approximation method using a new pseudo-inverse-based technique for embedding the boundary conditions, which results in a simpler mathematical derivation than has been presented previously. We then use the pole placement technique to design closed-loop feedback gains that stabilize time-delayed systems and verify our results through comparison to those reported in the literature. Finally, we perform experimental validation by applying our method to stabilize a rotary inverted pendulum system with inherent sensing delays as well as additional time delays that are introduced deliberately. The proposed approach is easily implemented and performs at least as well as existing methods.

Copyright © 2019 by ASME
Your Session has timed out. Please sign back in to continue.

References

Richard, J.-P. , 2003, “Time-Delay Systems: An Overview of Some Recent Advances and Open Problems,” Automatica, 39(10), pp. 1667–1694. [CrossRef]
Balachandran, B. , 2001, “Nonlinear Dynamics of Milling Processes,” Philos. Trans. R. Soc. A, 359(1781), pp. 793–819. [CrossRef]
Van Wiggeren, G. D. , and Roy, R. , 1998, “Communication With Chaotic Lasers,” Science, 279(5354), pp. 1198–1200. [CrossRef] [PubMed]
Nise, N. S. , 2000, Control Systems Engineering, 3rd ed., Wiley, New York.
Insperger, T. , and Stépán, G. , 2007, “Act-and-Wait Control Concept for Discrete-Time Systems With Feedback Delay,” IET Control Theory Appl., 1(3), pp. 553–557. [CrossRef]
Yi, S. , Nelson, P. W. , and Ulsoy, A. G. , 2010, Time-Delay Systems: Analysis and Control Using the Lambert W Function, World Scientific, Hackensack, NJ.
Yi, S. , Nelson, P. W. , and Ulsoy, A. G. , 2010, “Eigenvalue Assignment Via the Lambert W Function for Control of Time-Delay Systems,” J. Vib. Control, 16(7–8), pp. 961–982. [CrossRef]
Lavaei, J. , Sojoudi, S. , and Murray, R. M. , 2010, “Simple Delay-Based Implementation of Continuous-Time Controllers,” American Control Conference (ACC), Baltimore, MD, June 30–July 2, pp. 5781–5788.
Orosz, G. , Moehlis, J. , and Murray, R. M. , 2010, “Controlling Biological Networks by Time-Delayed Signals,” Philos. Trans. R. Soc. A, 368(1911), pp. 439–454. [CrossRef]
Bekiaris-Liberis, N. , Jankovic, M. , and Krstic, M. , 2013, “Adaptive Stabilization of LTI Systems With Distributed Input Delay,” Int. J. Adapt. Control Signal Process., 27(1–2), pp. 46–65. [CrossRef]
Michiels, W. , Engelborghs, K. , Vansevenant, P. , and Roose, D. , 2002, “Continuous Pole Placement for Delay Equations,” Automatica, 38(5), pp. 747–761. [CrossRef]
Michiels, W. , Vyhlídal, T. , and Zítek, P. , 2010, “Control Design for Time-Delay Systems Based on Quasi-Direct Pole Placement,” J. Process Control, 20(3), pp. 337–343. [CrossRef]
Vyhlídal, T. , Michiels, W. , and McGahan, P. , 2010, “Synthesis of Strongly Stable State-Derivative Controllers for a Time-Delay System Using Constrained Non-Smooth Optimization,” IMA J. Math. Control Inf., 27(4), pp. 437–455. [CrossRef]
Niu, J. , Ding, Y. , Zhu, L. , and Ding, H. , 2015, “Eigenvalue Assignment for Control of Time-Delay Systems Via the Generalized Runge–Kutta Method,” ASME J. Dyn. Syst. Meas. Control, 137(9), p. 091003. [CrossRef]
Burke, J. V. , Lewis, A. S. , and Overton, M. L. , 2005, “A Robust Gradient Sampling Algorithm for Nonsmooth, Nonconvex Optimization,” SIAM J. Optim., 15(3), pp. 751–779. [CrossRef]
Vanbiervliet, J. , Verheyden, K. , Michiels, W. , and Vandewalle, S. , 2008, “A Nonsmooth Optimisation Approach for the Stabilisation of Time-Delay Systems,” ESAIM: Control, Optim. Calculus Var., 14(3), pp. 478–493. [CrossRef]
Yi, S. , Nelson, P. W. , and Ulsoy, A. G. , 2013, “Proportional-Integral Control of First-Order Time-Delay Systems Via Eigenvalue Assignment,” IEEE Trans. Control Syst. Technol., 21(5), pp. 1586–1594. [CrossRef]
Wei, F. , Bachrathy, D. , Orosz, G. , and Ulsoy, A. G. , 2014, “Spectrum Design Using Distributed Delay,” Int. J. Dyn. Control, 2(2), pp. 234–246. [CrossRef]
Asl, F. M. , and Ulsoy, A. G. , 2003, “Analysis of a System of Linear Delay Differential Equations,” ASME J. Dyn. Syst. Meas. Control, 125(2), pp. 215–223. [CrossRef]
Jarlebring, E. , and Damm, T. , 2007, “The Lambert W Function and the Spectrum of Some Multidimensional Time-Delay Systems,” Automatica, 43(12), pp. 2124–2128. [CrossRef]
Yi, S. , Nelson, P. W. , and Ulsoy, A. G. , 2007, “Survey on Analysis of Time Delayed Systems Via the Lambert W Function,” Dyn. Contin., Discrete Impulsive Syst., Ser. A: Math. Anal., 14(S2), pp. 296–301.
Wahi, P. , and Chatterjee, A. , 2005, “Asymptotics for the Characteristic Roots of Delayed Dynamic Systems,” ASME J. Appl. Mech., 72(4), pp. 475–483. [CrossRef]
Vyasarayani, C. P. , 2012, “Galerkin Approximations for Higher Order Delay Differential Equations,” ASME J. Comput. Nonlinear Dyn., 7(3), p. 031004. [CrossRef]
Sadath, A. , and Vyasarayani, C. P. , 2015, “Galerkin Approximations for Stability of Delay Differential Equations With Distributed Delays,” ASME J. Comput. Nonlinear Dyn., 10(6), p. 061024. [CrossRef]
Kalmár-Nagy, T. , 2009, “Stability Analysis of Delay-Differential Equations by the Method of Steps and Inverse Laplace Transform,” Differ. Equations Dyn. Syst., 17(1–2), pp. 185–200. [CrossRef]
Insperger, T. , and Stépán, G. , 2011, Semi-Discretization for Time-Delay Systems: Stability and Engineering Applications, Springer, New York.
Wahi, P. , and Chatterjee, A. , 2005, “Galerkin Projections for Delay Differential Equations,” ASME J. Dyn. Syst. Meas. Control, 127(1), pp. 80–87.
Butcher, E. A. , Ma, H. , Bueler, E. , Averina, V. , and Szabo, Z. , 2004, “Stability of Linear Time-Periodic Delay-Differential Equations Via Chebyshev Polynomials,” Int. J. Numer. Methods Eng., 59(7), pp. 895–922. [CrossRef]
Breda, D. , Maset, S. , and Vermiglio, R. , 2005, “Pseudospectral Differencing Methods for Characteristic Roots of Delay Differential Equations,” SIAM J. Sci. Comput., 27(2), pp. 482–495. [CrossRef]
Wu, Z. , and Michiels, W. , 2012, “Reliably Computing All Characteristic Roots of Delay Differential Equations in a Given Right Half Plane Using a Spectral Method,” J. Comput. Appl. Math., 236(9), pp. 2499–2514. [CrossRef]
Mann, B. P. , and Patel, B. R. , 2010, “Stability of Delay Equations Written as State Space Models,” J. Vib. Control, 16(7–8), pp. 1067–1085. [CrossRef]
Khasawneh, F. A. , and Mann, B. P. , 2011, “A Spectral Element Approach for the Stability of Delay Systems,” Int. J. Numer. Methods Eng., 87(6), pp. 566–592. [CrossRef]
Sun, J.-Q. , 2009, “A Method of Continuous Time Approximation of Delayed Dynamical Systems,” Commun. Nonlinear Sci. Numer. Simul., 14(4), pp. 998–1007. [CrossRef]
Song, B. , and Sun, J.-Q. , 2011, “Lowpass Filter-Based Continuous-Time Approximation of Delayed Dynamical Systems,” J. Vib. Control, 17(8), pp. 1173–1183. [CrossRef]
Koto, T. , 2004, “Method of Lines Approximations of Delay Differential Equations,” Comput. Math. Appl., 48(1–2), pp. 45–59. [CrossRef]
Engelborghs, K. , and Roose, D. , 2002, “On Stability of LMS Methods and Characteristic Roots of Delay Differential Equations,” SIAM J. Numer. Anal., 40(2), pp. 629–650. [CrossRef]
Vyasarayani, C. P. , Subhash, S. , and Kalmár-Nagy, T. , 2014, “Spectral Approximations for Characteristic Roots of Delay Differential Equations,” Int. J. Dyn. Control, 2(2), pp. 126–132. [CrossRef]
Sadath, A. , and Vyasarayani, C. P. , 2015, “Galerkin Approximations for Stability of Delay Differential Equations With Time Periodic Coefficients,” ASME J. Comput. Nonlinear Dyn., 10(2), p. 021011. [CrossRef]
Sadath, A. , and Vyasarayani, C. P. , 2015, “Galerkin Approximations for Stability of Delay Differential Equations With Time Periodic Delays,” ASME J. Comput. Nonlinear Dyn., 10(6), p. 061008. [CrossRef]
Vyhlídal, T. , and Zítek, P. , 2014, “QPmR—Quasi-Polynomial Root-Finder: Algorithm Update and Examples,” Delay Systems: From Theory to Numerics and Applications. Advances in Delays and Dynamics, T. Vyhlídal , J.-F. Lafay , and R. Sipahi , eds. Vol. 1, Springer, Cham, Switzerland.
Breda, D. , Maset, S. , and Vermiglio, R. , 2015, Stability of Linear Delay Differential Equations: A Numerical Approach With MATLAB, Springer, New York.
Apkarian, J. , Lévis, M. , and Martin, P. , 2016, “Instructor Workbook: QUBE-Servo 2 Experiment for MATLAB/Simulink Users,” Quanser, Markham, ON, Canada, Report No. v 1.0.
Apkarian, J. , Karam, P. , and Lévis, M. , 2012, “Student Workbook: Inverted Pendulum Experiment for LabVIEW Users,” Quanser, Markham, ON, Canada, Report No. v 1.1.

Figures

Grahic Jump Location
Fig. 1

Characteristic roots of Eq. (27) using the QPmR algorithm, the PSD method, and the proposed pseudo-inverse-based Galerkin approximation method: (a) a = b =10−1, (b) a = b =10−3, and (c) a = b =10−6

Grahic Jump Location
Fig. 2

Errors obtained upon substituting into the characteristic equation (Eq. (28)) the rightmost eigenvalue computed using the QPmR algorithm, the PSD method, and the proposed pseudo-inverse-based Galerkin approximation method

Grahic Jump Location
Fig. 3

Stability diagram for the second-order DDE given by Eq. (29) obtained using the spectral tau method (red region) and the proposed pseudo-inverse-based Galerkin approximation method (blue lines)

Grahic Jump Location
Fig. 4

Rightmost characteristic roots of Eq. (30) using initial feedback gain k =0.8

Grahic Jump Location
Fig. 5

Rightmost characteristic roots of Eq. (30) with delay τ = 1 (a) and the variation of the five rightmost roots with respect to delay τ and (b) using optimal feedback gain k* = −3.5978

Grahic Jump Location
Fig. 6

Rightmost characteristic roots of Eq. (31) with delay τ = 5 (a) and the variation of the rightmost roots with respect to delay τ and (b) using initial feedback gains K=[0.719,1.04,1.29]T

Grahic Jump Location
Fig. 7

Rightmost characteristic roots of Eq. (31) with delay τ = 5 (a) and the variation of the rightmost roots with respect to delay τ and (b) using optimal feedback gains K*=[0.5473,0.8681,0.5998]T

Grahic Jump Location
Fig. 8

Rotary inverted pendulum apparatus, shown here with θ≈0 deg and γ≈180 deg

Grahic Jump Location
Fig. 9

Stable response of the inverted pendulum (γ) and rotary arm (θ) with inherent delay of 2 ms and feedback gains K=[−2,30,−2,2.5]T. An external disturbance is applied between 13 and 23 s.

Grahic Jump Location
Fig. 10

Variation of the rightmost roots of Eq. (37) with respect to delay τ using feedback gains K=[−2,30,−2,2.5]T. The critical delay is τ = 9.76 ms.

Grahic Jump Location
Fig. 11

Rightmost characteristic roots of Eq. (37) with delay (a) τ = 5 ms and (b) τ = 10 ms, using feedback gains K=[−2,30,−2,2.5]T

Grahic Jump Location
Fig. 12

Stable response of the inverted pendulum (γ) and rotary arm (θ) with total delay of τ = 2 + 7.5 = 9.5 ms and feedback gains K=[−2,30,−2,2.5]T

Grahic Jump Location
Fig. 13

Unstable response of the inverted pendulum (γ) and rotary arm (θ) with total delay of τ = 2 + 8 = 10 ms and feedback gains K=[−2,30,−2,2.5]T

Grahic Jump Location
Fig. 14

Stable response of the inverted pendulum (γ) and rotary arm (θ) with total delay of τ = 2 + 10 = 12 ms and optimal feedback gains K*=[−2.3443,31.3406,−1.1797,2.7717]T. An external disturbance is applied between 12 and 27 s.

Grahic Jump Location
Fig. 15

Variation of the rightmost roots of Eq. (37) with respect to delay τ using optimal feedback gains K*=[−2.3443,31.3406,−1.1797,2.7717]T. The critical delay is τ = 17.7 ms.

Grahic Jump Location
Fig. 16

Stable response of the inverted pendulum (γ) and rotary arm (θ) with total delay of τ = 2 + 15 = 17 ms and optimal feedback gains K*=[−2.3443,31.3406,−1.1797,2.7717]T

Grahic Jump Location
Fig. 17

Unstable response of the inverted pendulum (γ) and rotary arm (θ) with total delay of τ = 2 + 15.5 = 17.5 ms and optimal feedback gains K*=[−2.3443,31.3406,−1.1797,2.7717]T

Tables

Errata

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In