March  2022, 12(1): 17-47. doi: 10.3934/mcrf.2021001

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

* Corresponding author: Chris Guiver

Received  June 2020 Revised  November 2020 Published  March 2022 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.

Citation: Max E. Gilmore, Chris Guiver, Hartmut Logemann. Sampled-data integral control of multivariable linear infinite-dimensional systems with input nonlinearities. Mathematical Control and Related Fields, 2022, 12 (1) : 17-47. 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.

[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. ByrnesD. GilliamV. 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-WildeD.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. FliegnerH. 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. FliegnerH. 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. FliegnerH. 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. GaleaniS. TarbouriechM. 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. GilmoreC. 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. GuiverH. 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. GuiverM. MuellerD. 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. KeH. 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. ByrnesD. GilliamV. 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-WildeD.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. FliegnerH. 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. FliegnerH. 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. FliegnerH. 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. GaleaniS. TarbouriechM. 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. GilmoreC. 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. GuiverH. 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. GuiverM. MuellerD. 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. KeH. 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.

Figure 4.1.  Quantization function $ q_\delta $
Figure 4.2.  Square domain $ \Omega \subseteq \mathbb R^2 $
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 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 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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