American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Stochastic optimal control — A concise introduction
  • MCRF Home
  • This Issue
  • Next Article
    Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt damping under different types of geometric conditions
doi: 10.3934/mcrf.2021001

Sampled-data integral control of multivariable linear infinite-dimensional systems with input nonlinearities

Max E. Gilmore 1, , Chris Guiver 2,, and  Hartmut Logemann 1,

1. 

Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK
2. 

School of Engineering & the Built Environment, Edinburgh Napier University, Merchiston Campus, 10 Colinton Road, Edinburgh EH10 5DT, UK

* Corresponding author: Chris Guiver

Received  June 2020 Revised  November 2020 Published  January 2021

A low-gain integral controller with anti-windup component is presented for exponentially stable, linear, discrete-time, infinite-dimensional control systems subject to input nonlinearities and external disturbances. We derive a disturbance-to-state stability result which, in particular, guarantees that the tracking error converges to zero in the absence of disturbances. The discrete-time result is then used in the context of sampled-data low-gain integral control of stable well-posed linear infinite-dimensional systems with input nonlinearities. The sampled-date control scheme is applied to two examples (including sampled-data control of a heat equation on a square) which are discussed in some detail.

Keywords: Anti-windup methods, discrete-time, input-to-state stability, integral control, quantization, sampled-data control, saturation, well-posed infinite-dimensional systems.
Mathematics Subject Classification: Primary: 93C05, 93C10, 93C25, 93C35, 93C55, 93C57, 93D09, 93D15; Secondary: 93C20.
Citation: Max E. Gilmore, Chris Guiver, Hartmut Logemann. Sampled-data integral control of multivariable linear infinite-dimensional systems with input nonlinearities. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021001
References:
[1]

K. J. Åström and L. Rundquist, Integrator windup and how to avoid it, Amer. Control Conf., (1989), 1693–1698. Google Scholar

[2]

D.S. Bernstein and A.N. Michel, A chronological bibliography on saturating actuators, Internat. J. Robust Nonlinear Control, 5 (1995), 375-380.  doi: 10.1002/rnc.4590050502.  Google Scholar

[3]

C. Byrnes, D. Gilliam, V. Shubov and G. Weiss, Regular linear systems governed by a boundary controlled heat equation, J. Dynamical Control Systems, 8 (2002), 341-370.  doi: 10.1023/A:1016330420910.  Google Scholar

[4]

S.N. Chandler-Wilde, D.P. Hewett and A. Moiola, Interpolation of Hilbert and Sobolev spaces: Quantitative estimates and counterexamples, Mathematika, 61 (2015), 414-443.  doi: 10.1112/S0025579314000278.  Google Scholar

[5]

J. J. Coughlan, Absolute Stability Results for Infinite-Dimensional Discrete-Time Systems with Applications to Sampled-Data Integral Control, Ph.D thesis, University of Bath, 2007. Google Scholar

[6]

J.J. Coughlan and H. Logemann, Absolute stability and integral control for infinite-dimensional discrete-time systems, Nonlinear Anal., 71 (2009), 4769-4789.  doi: 10.1016/j.na.2009.03.072.  Google Scholar

[7]

E.J. Davison, Multivariable tuning regulators: The feedforward and robust control of a general servomechanism problem, IEEE Trans. Automatic Control, AC-21 (1976), 35-47.  doi: 10.1109/tac.1976.1101126.  Google Scholar

[8]

K-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Springer-Verlag, New York, 2000.  Google Scholar

[9]

T. Fliegner, H. Logemann and E.P. Ryan, Low-gain integral control of continuous-time linear systems subject to input and output nonlinearities, Automatica, 39 (2003), 455-462.  doi: 10.1016/S0005-1098(02)00238-8.  Google Scholar

[10]

T. Fliegner, H. Logemann and E.P. Ryan, Discrete-time low-gain control of linear systems with input/output nonlinearities, Internat. J. Robust Nonlinear Control, 11 (2001), 1127-1143.  doi: 10.1002/rnc.588.  Google Scholar

[11]

T. Fliegner, H. Logemann and E.P. Ryan, Low-gain integral control of well-posed linear infinite-dimensional systems with input and output nonlinearities, J. Math. Anal. Appl., 261 (2001), 307-336.  doi: 10.1006/jmaa.2000.7526.  Google Scholar

[12]

S. Galeani, S. Tarbouriech, M. Turner and L. Zaccarian, A tutorial on modern anti-windup design, Eur. J. Control, 15 (2009), 418-440.  doi: 10.3166/ejc.15.418-440.  Google Scholar

[13]

M.E. Gilmore, C. Guiver and H. Logemann, Stability and convergence properties of forced infinite-dimensional discrete-time Lur'e systems, Int. J. Control, 93 (2020), 3026-3049.  doi: 10.1080/00207179.2019.1575528.  Google Scholar

[14]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman (Advanced Publishing Program), Boston, 1985.  Google Scholar

[15]

C. Guiver, H. Logemann and S. Townley, Low-gain integral control for multi-input multi-output linear systems with input nonlinearities, IEEE Trans. Automat. Control, 62 (2017), 4776-4783.  doi: 10.1109/TAC.2017.2691301.  Google Scholar

[16]

C. Guiver, M. Mueller, D. Hodgson and S. Townley, Robust set-point regulation for ecological models with multiple management goals, J. Math. Biol., 72 (2016), 1467-1529.  doi: 10.1007/s00285-015-0919-7.  Google Scholar

[17]

Z. Ke, Sampled-Data Control: Stabilization, Tracking and Disturbance Rejection, Ph.D thesis, University of Bath, 2008. Google Scholar

[18]

Z. Ke, H. Logemann and S. Townley, Adaptive sampled-data integral control of stable infinite-dimensional linear systems, Systems Control Lett., 58 (2009), 233-240.  doi: 10.1016/j.sysconle.2008.10.015.  Google Scholar

[19]

H. Logemann and E.P. Ryan, Time-varying and adaptive discrete-time low-gain control of infinite-dimensional linear systems with input nonlinearities, Math. Control Signals Systems, 13 (2000), 293-317.  doi: 10.1007/PL00009871.  Google Scholar

[20]

H. Logemann and S. Townley, Discrete-time low-gain control of uncertain infinite-dimensional systems, IEEE Trans. Automat. Control, 42 (1997), 22-37.  doi: 10.1109/9.553685.  Google Scholar

[21]

H. Logemann and S. Townley, Low-gain control of uncertain regular linear systems, SIAM J. Control Optim., 35 (1997), 78-116.  doi: 10.1137/S0363012994275920.  Google Scholar

[22]

M. Morari, Robust stability of systems with integral control, IEEE Trans. Automatic Control, 30 (1985), 574-577.  doi: 10.1109/TAC.1985.1104012.  Google Scholar

[23]

S. Pohjolainen, Robust controller for systems with exponentially stable strongly continuous semigroups, J. Math. Anal. Appl., 111 (1985), 622-636.  doi: 10.1016/0022-247X(85)90239-2.  Google Scholar

[24]

S.A. Pohjolainen, Robust multivariable PI-controller for infinite-dimensional systems, IEEE Trans. Automatic Control, 27 (1982), 17-30.  doi: 10.1109/TAC.1982.1102887.  Google Scholar

[25]

W. Rudin, Functional Analysis, McGraw-Hill, Inc., New York, 1991.  Google Scholar

[26]

D. Salamon, Infinite-dimensional linear systems with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), 383-431.  doi: 10.2307/2000351.  Google Scholar

[27] O. Staffans, Well-Posed Linear Systems, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9780511543197.  Google Scholar
[28]

S. Tarbouriech and M. Turner, Anti-windup design: An overview of some recent advances and open problems, IET Control Theory Appl., 3 (2009), 1751-8644.  doi: 10.1049/iet-cta:20070435.  Google Scholar

[29]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Verlag AG, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

[30]

G. Weiss, Regular linear systems with feedback, Math. Control Signals Systems, 7 (1994), 23-57.  doi: 10.1007/BF01211484.  Google Scholar

show all references

References:
[1]

K. J. Åström and L. Rundquist, Integrator windup and how to avoid it, Amer. Control Conf., (1989), 1693–1698. Google Scholar

[2]

D.S. Bernstein and A.N. Michel, A chronological bibliography on saturating actuators, Internat. J. Robust Nonlinear Control, 5 (1995), 375-380.  doi: 10.1002/rnc.4590050502.  Google Scholar

[3]

C. Byrnes, D. Gilliam, V. Shubov and G. Weiss, Regular linear systems governed by a boundary controlled heat equation, J. Dynamical Control Systems, 8 (2002), 341-370.  doi: 10.1023/A:1016330420910.  Google Scholar

[4]

S.N. Chandler-Wilde, D.P. Hewett and A. Moiola, Interpolation of Hilbert and Sobolev spaces: Quantitative estimates and counterexamples, Mathematika, 61 (2015), 414-443.  doi: 10.1112/S0025579314000278.  Google Scholar

[5]

J. J. Coughlan, Absolute Stability Results for Infinite-Dimensional Discrete-Time Systems with Applications to Sampled-Data Integral Control, Ph.D thesis, University of Bath, 2007. Google Scholar

[6]

J.J. Coughlan and H. Logemann, Absolute stability and integral control for infinite-dimensional discrete-time systems, Nonlinear Anal., 71 (2009), 4769-4789.  doi: 10.1016/j.na.2009.03.072.  Google Scholar

[7]

E.J. Davison, Multivariable tuning regulators: The feedforward and robust control of a general servomechanism problem, IEEE Trans. Automatic Control, AC-21 (1976), 35-47.  doi: 10.1109/tac.1976.1101126.  Google Scholar

[8]

K-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Springer-Verlag, New York, 2000.  Google Scholar

[9]

T. Fliegner, H. Logemann and E.P. Ryan, Low-gain integral control of continuous-time linear systems subject to input and output nonlinearities, Automatica, 39 (2003), 455-462.  doi: 10.1016/S0005-1098(02)00238-8.  Google Scholar

[10]

T. Fliegner, H. Logemann and E.P. Ryan, Discrete-time low-gain control of linear systems with input/output nonlinearities, Internat. J. Robust Nonlinear Control, 11 (2001), 1127-1143.  doi: 10.1002/rnc.588.  Google Scholar

[11]

T. Fliegner, H. Logemann and E.P. Ryan, Low-gain integral control of well-posed linear infinite-dimensional systems with input and output nonlinearities, J. Math. Anal. Appl., 261 (2001), 307-336.  doi: 10.1006/jmaa.2000.7526.  Google Scholar

[12]

S. Galeani, S. Tarbouriech, M. Turner and L. Zaccarian, A tutorial on modern anti-windup design, Eur. J. Control, 15 (2009), 418-440.  doi: 10.3166/ejc.15.418-440.  Google Scholar

[13]

M.E. Gilmore, C. Guiver and H. Logemann, Stability and convergence properties of forced infinite-dimensional discrete-time Lur'e systems, Int. J. Control, 93 (2020), 3026-3049.  doi: 10.1080/00207179.2019.1575528.  Google Scholar

[14]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman (Advanced Publishing Program), Boston, 1985.  Google Scholar

[15]

C. Guiver, H. Logemann and S. Townley, Low-gain integral control for multi-input multi-output linear systems with input nonlinearities, IEEE Trans. Automat. Control, 62 (2017), 4776-4783.  doi: 10.1109/TAC.2017.2691301.  Google Scholar

[16]

C. Guiver, M. Mueller, D. Hodgson and S. Townley, Robust set-point regulation for ecological models with multiple management goals, J. Math. Biol., 72 (2016), 1467-1529.  doi: 10.1007/s00285-015-0919-7.  Google Scholar

[17]

Z. Ke, Sampled-Data Control: Stabilization, Tracking and Disturbance Rejection, Ph.D thesis, University of Bath, 2008. Google Scholar

[18]

Z. Ke, H. Logemann and S. Townley, Adaptive sampled-data integral control of stable infinite-dimensional linear systems, Systems Control Lett., 58 (2009), 233-240.  doi: 10.1016/j.sysconle.2008.10.015.  Google Scholar

[19]

H. Logemann and E.P. Ryan, Time-varying and adaptive discrete-time low-gain control of infinite-dimensional linear systems with input nonlinearities, Math. Control Signals Systems, 13 (2000), 293-317.  doi: 10.1007/PL00009871.  Google Scholar

[20]

H. Logemann and S. Townley, Discrete-time low-gain control of uncertain infinite-dimensional systems, IEEE Trans. Automat. Control, 42 (1997), 22-37.  doi: 10.1109/9.553685.  Google Scholar

[21]

H. Logemann and S. Townley, Low-gain control of uncertain regular linear systems, SIAM J. Control Optim., 35 (1997), 78-116.  doi: 10.1137/S0363012994275920.  Google Scholar

[22]

M. Morari, Robust stability of systems with integral control, IEEE Trans. Automatic Control, 30 (1985), 574-577.  doi: 10.1109/TAC.1985.1104012.  Google Scholar

[23]

S. Pohjolainen, Robust controller for systems with exponentially stable strongly continuous semigroups, J. Math. Anal. Appl., 111 (1985), 622-636.  doi: 10.1016/0022-247X(85)90239-2.  Google Scholar

[24]

S.A. Pohjolainen, Robust multivariable PI-controller for infinite-dimensional systems, IEEE Trans. Automatic Control, 27 (1982), 17-30.  doi: 10.1109/TAC.1982.1102887.  Google Scholar

[25]

W. Rudin, Functional Analysis, McGraw-Hill, Inc., New York, 1991.  Google Scholar

[26]

D. Salamon, Infinite-dimensional linear systems with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), 383-431.  doi: 10.2307/2000351.  Google Scholar

[27] O. Staffans, Well-Posed Linear Systems, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9780511543197.  Google Scholar
[28]

S. Tarbouriech and M. Turner, Anti-windup design: An overview of some recent advances and open problems, IET Control Theory Appl., 3 (2009), 1751-8644.  doi: 10.1049/iet-cta:20070435.  Google Scholar

[29]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Verlag AG, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

[30]

G. Weiss, Regular linear systems with feedback, Math. Control Signals Systems, 7 (1994), 23-57.  doi: 10.1007/BF01211484.  Google Scholar

Figure 4.1.  Quantization function $ q_\delta $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.2.  Square domain $ \Omega \subseteq \mathbb R^2 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.3.  Feasible sets of reference vectors $r = {\left( {\begin{array}{*{20}{l}} {{r_1}}&{{r_2}} \end{array}} \right)^T}$ for $ \phi_1 $ (darker grey) and $ \phi_2 $ (lighter gray) regions. The reference $ r $ in (4.8) is marked with a cross
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.4.  Model data as in (4.5), (4.6), (4.8) and (4.9). (a) Initial temperature profile $ z^0 $. (b) Temperature profile of solution $ z $ at time $ t = 20 $. (c) Outputs. (d) Held inputs. In panels (c) and (d), the solid and dashed lines correspond to $ \tau = 0.25 $ and $ \tau = 0.5 $, respectively. The dotted lines in panel (c) are the components of the reference
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.5.  Model data as in (4.5), (4.6), (4.8), (4.10) and (4.11). (a) Outputs. (b) Held inputs. In panels (a) and (b), the solid and dashed lines correspond to the external forcing $ v $ and $ 3 v $, respectively. The dotted lines in panel (a) are the components of the reference
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183
[2]

Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301
[3]

Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329
[4]

Peter Benner, Jens Saak, M. Monir Uddin. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 1-20. doi: 10.3934/naco.2016.6.1
[5]

Yves Dumont, Frederic Chiroleu. Vector control for the Chikungunya disease. Mathematical Biosciences & Engineering, 2010, 7 (2) : 313-345. doi: 10.3934/mbe.2010.7.313
[6]

Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 307-320. doi: 10.3934/naco.2020027
[7]

Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881
[8]

J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008
[9]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[10]

F.J. Herranz, J. de Lucas, C. Sardón. Jacobi--Lie systems: Fundamentals and low-dimensional classification. Conference Publications, 2015, 2015 (special) : 605-614. doi: 10.3934/proc.2015.0605
[11]

A. K. Misra, Anupama Sharma, Jia Li. A mathematical model for control of vector borne diseases through media campaigns. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1909-1927. doi: 10.3934/dcdsb.2013.18.1909
[12]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399
[13]

Xiaohong Li, Mingxin Sun, Zhaohua Gong, Enmin Feng. Multistage optimal control for microbial fed-batch fermentation process. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021040
[14]

John T. Betts, Stephen Campbell, Claire Digirolamo. Examination of solving optimal control problems with delays using GPOPS-Ⅱ. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 283-305. doi: 10.3934/naco.2020026
[15]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
[16]

Zengyun Wang, Jinde Cao, Zuowei Cai, Lihong Huang. Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2677-2692. doi: 10.3934/dcdsb.2020200
[17]

Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195
[18]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521
[19]

Sandrine Anthoine, Jean-François Aujol, Yannick Boursier, Clothilde Mélot. Some proximal methods for Poisson intensity CBCT and PET. Inverse Problems & Imaging, 2012, 6 (4) : 565-598. doi: 10.3934/ipi.2012.6.565
[20]

Rafael G. L. D'Oliveira, Marcelo Firer. Minimum dimensional Hamming embeddings. Advances in Mathematics of Communications, 2017, 11 (2) : 359-366. doi: 10.3934/amc.2017029

2019 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (29)
  • HTML views (59)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences