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Sampled-data integral control of multivariable linear infinite-dimensional systems with input nonlinearities
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 |
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.
References:
[1] |
K. J. Åström and L. Rundquist, Integrator windup and how to avoid it, Amer. Control Conf., (1989), 1693–1698. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
K-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Springer-Verlag, New York, 2000. |
[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. |
[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. |
[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. |
[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. |
[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. |
[14] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman (Advanced Publishing Program), Boston, 1985. |
[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. |
[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. |
[17] |
Z. Ke, Sampled-Data Control: Stabilization, Tracking and Disturbance Rejection, Ph.D thesis, University of Bath, 2008. |
[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. |
[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. |
[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. |
[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. |
[22] |
M. Morari,
Robust stability of systems with integral control, IEEE Trans. Automatic Control, 30 (1985), 574-577.
doi: 10.1109/TAC.1985.1104012. |
[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. |
[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. |
[25] |
W. Rudin, Functional Analysis, McGraw-Hill, Inc., New York, 1991. |
[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. |
[27] |
O. Staffans, Well-Posed Linear Systems, Cambridge University Press, Cambridge, 2005.
doi: 10.1017/CBO9780511543197.![]() ![]() ![]() |
[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. |
[29] |
M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Verlag AG, 2009.
doi: 10.1007/978-3-7643-8994-9. |
[30] |
G. Weiss,
Regular linear systems with feedback, Math. Control Signals Systems, 7 (1994), 23-57.
doi: 10.1007/BF01211484. |
show all references
References:
[1] |
K. J. Åström and L. Rundquist, Integrator windup and how to avoid it, Amer. Control Conf., (1989), 1693–1698. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
K-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Springer-Verlag, New York, 2000. |
[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. |
[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. |
[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. |
[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. |
[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. |
[14] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman (Advanced Publishing Program), Boston, 1985. |
[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. |
[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. |
[17] |
Z. Ke, Sampled-Data Control: Stabilization, Tracking and Disturbance Rejection, Ph.D thesis, University of Bath, 2008. |
[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. |
[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. |
[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. |
[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. |
[22] |
M. Morari,
Robust stability of systems with integral control, IEEE Trans. Automatic Control, 30 (1985), 574-577.
doi: 10.1109/TAC.1985.1104012. |
[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. |
[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. |
[25] |
W. Rudin, Functional Analysis, McGraw-Hill, Inc., New York, 1991. |
[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. |
[27] |
O. Staffans, Well-Posed Linear Systems, Cambridge University Press, Cambridge, 2005.
doi: 10.1017/CBO9780511543197.![]() ![]() ![]() |
[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. |
[29] |
M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Verlag AG, 2009.
doi: 10.1007/978-3-7643-8994-9. |
[30] |
G. Weiss,
Regular linear systems with feedback, Math. Control Signals Systems, 7 (1994), 23-57.
doi: 10.1007/BF01211484. |





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