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doi: 10.3934/mcrf.2021001
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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 Early access 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 $
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Figure 4.2.  Square domain $ \Omega \subseteq \mathbb R^2 $
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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
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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
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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
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