Internal model based tracking and disturbance rejection for stable well-posed systems.

*(English)*Zbl 1028.93012In this very important and useful paper the tracking and disturbance rejection problem for infinite-dimensional linear systems with reference and disturbance signals that are finite superpositions of sinusoids, is solved. The authors explore two approaches, both based on the internal model principle. In the first approach a low gain controller is used. These results are a partial extension of results given by Hämäläinen and Pohjolainen (here the plant is required to have an exponentially stable transfer function on the Callier-Desoer algebra). Main results: The authors only require the plant to be well-posed and exponentially stable. These conditions are sufficiently unrestrictive to be verifiable for many partial differential equations in more than one space variable. Moreover the second approach concerns the case where the second component of the plant transfer function (from control input to tracking error) is positive. In this case, a very simple stabilizing controller which comes from an internal model, but which does not require a low gain, is identified.

Finally the authors apply the proposed results to two problems involving systems modelled by partial differential equations: the problem of rejecting external noise in a model for structural acoustics, and a similar problem for two coupled beams.

Finally the authors apply the proposed results to two problems involving systems modelled by partial differential equations: the problem of rejecting external noise in a model for structural acoustics, and a similar problem for two coupled beams.

Reviewer: J.Lovíšek (Bratislava)

##### MSC:

93B51 | Design techniques (robust design, computer-aided design, etc.) |

93C73 | Perturbations in control/observation systems |

93C25 | Control/observation systems in abstract spaces |

##### Keywords:

well-posed linear system; tracking; internal model principle; input-output stability; exponential stability; dynamic stabilization; positive transfer function; optimizability; regulation; disturbance rejection; finite superpositions of sinusoids; low gain; structural acoustics; coupled beams
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\textit{R. Rebarber} and \textit{G. Weiss}, Automatica 39, No. 9, 1555--1569 (2003; Zbl 1028.93012)

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##### References:

[1] | Ammari, K.; Liu, Z.; Tucsnak, M., Decay rates for a beam with pointwise force and moment feedback, Mathematics of control, signals, and systems, 15, 229-255, (2002) · Zbl 1042.93034 |

[2] | Banks, H.T.; Demetriou, M.A.; Smith, R.C., Robustness studies for H∞ feedback control in a structural acoustic model with periodic excitation, International journal of robust and nonlinear control, 6, 453-478, (1996) · Zbl 0850.93220 |

[3] | Banks, H.T.; Smith, R.C., Feedback control of noise in a 2-D nonlinear structural acoustics model, Discrete and continuous dynamical systems, 1, 119-149, (1995) · Zbl 0872.93035 |

[4] | Chen, G.; Krantz, S.G.; Russell, D.L.; Wayne, C.E.; West, H.H.; Coleman, M.P., Analysis, designs, and behavior of dissipative joints for coupled beams, SIAM journal on control and optimization, 49, 1665-1693, (1989) · Zbl 0685.73046 |

[5] | Curtain, R., & Weiss, G. Stabilization of essentially skew-adjoint systems by collocated feedback. Submitted, available at www.ee.ic.ac.uk/CAP/ |

[6] | Davison, E.J., The robust control of a servomechanism problem for linear time invariant multivariable systems, IEEE transactions on automatic control, 21, 25-34, (1976) · Zbl 0326.93007 |

[7] | Davison, E.J., Multivariable tuning regulators: the feedforward and robust control of a general servomechanism problem, IEEE transactions on automatic control, 21, 35-47, (1976) · Zbl 0326.93008 |

[8] | Francis, B.A.; Wonham, W.M., The internal model principle for linear multivariable regulators, Applied mathematics and optimization, 2, 170-194, (1975) · Zbl 0351.93015 |

[9] | Hämäläinen, T.; Pohjolainen, S., Robust control and tuning problem for distributed parameter systems, International journal of robust and nonlinear control, 6, 479-500, (1996) · Zbl 0862.93036 |

[10] | Hämäläinen, T.; Pohjolainen, S., A finite dimensional robust controller for systems in the CD-algebra, IEEE transactions on automatic control, 45, 421-431, (2000) · Zbl 0972.93014 |

[11] | Ho, L.F.; Russell, D.L., Admissible input elements for systems in Hilbert space and a Carleson measure criterion, SIAM journal on control and optimization, 21, 614-640, (1983) · Zbl 0512.93044 |

[12] | Krein, S. G. (1971). In J. M. Danskin (Trans.), Linear differential equations in Banach space. Translations of Mathematical Monographs, Providence, RI: American Mathematical Society (in Russian). |

[13] | Lasiecka, I. (2002). Mathematical control theory of coupled PDEs. CBMS-NSF Regional Conference Series in Applied Mathematics. Philadelphia: SIAM. · Zbl 1032.93002 |

[14] | Logemann, H.; Bontsema, J.; Owens, D.H., Low-gain control of distributed parameter systems with unbounded control and observation, Control theory and advanced technology, 4, 429-446, (1988) |

[15] | Logemann, H., & Curtain, R. F. (2000). Absolute stability results for infinite-dimensional well-posed systems with applications to low-gain control. ESAIM: Control, Optimisation and Calculus of Variations, 5, 395-424, http://www.edpsciences.com/articles/cocv/abs/2000/01/cocvVol5-16. · Zbl 0964.93048 |

[16] | Logemann, H., & Mawby, A. D. (2001). Low-gain integral control of infinite dimensional regular linear systems subject to input hysteresis. In F. Colonius et al. (Eds.), Advances in math. systems theory (pp. 255-293). Boston: Birkhäuser. |

[17] | Logemann, H.; Owens, D.H., Low-gain control of unknown infinite-dimensional systemsa frequency-domain approach, Dynamics and stability of systems, 4, 13-29, (1989) · Zbl 0673.93068 |

[18] | Logemann, H.; Ryan, E.P.; Townley, S., Integral control of infinite-dimensional linear systems subject to input saturation, SIAM journal on control and optimization, 36, 1940-1961, (1998) · Zbl 0913.93031 |

[19] | Logemann, H.; Townley, S., Low-gain control of uncertain regular linear systems, SIAM journal on control and optimization, 35, 78-116, (1997) · Zbl 0873.93044 |

[20] | Logemann, H.; Townley, S., Discrete-time low-gain control of uncertain infinite-dimensional system, IEEE transactions on automatic control, 42, 22-37, (1997) · Zbl 0874.93056 |

[21] | Morgul, O., Stabilization and disturbance rejection for the beam equation, IEEE transactions on automatic control, 46, 1913-1918, (2001) · Zbl 1009.93040 |

[22] | Pohjolainen, S., Robust multivariable PI-controllers for infinite dimensional systems, IEEE transactions on automatic control, 27, 17-30, (1982) · Zbl 0493.93029 |

[23] | Pohjolainen, S., Robust controllers for systems with exponentially stable strongly continuous semigroups, Journal of mathematical analysis and applications, 111, 622-636, (1985) · Zbl 0577.93037 |

[24] | Pohjolainen, S.; Lätti, I., Robust controller for boundary control systems, International journal of control, 38, 1189-1197, (1983) · Zbl 0534.93043 |

[25] | Rebarber, R., Exponential stability of coupled beams with dissipative jointsa frequency domain approach, SIAM journal on control and optimization, 33, 1-28, (1995) · Zbl 0819.93042 |

[26] | Rebarber, R.; Weiss, G., Necessary conditions for exact controllability with a finite-dimensional input space, Systems & control letters, 40, 217-227, (2000) · Zbl 0985.93028 |

[27] | Salamon, D., Infinite dimensional systems with unbounded control and observation: a functional analytic approach, Transactions of the American mathematical society, 300, 383-431, (1987) · Zbl 0623.93040 |

[28] | Salamon, D., Realization theory in Hilbert space, Mathematical systems theory, 21, 147-164, (1989) · Zbl 0668.93018 |

[29] | Staffans, O.J., Quadratic optimal control of stable well-posed linear systems, Transactions of the American mathematical society, 349, 3679-3715, (1997) · Zbl 0889.49023 |

[30] | Staffans, O.J., Coprime factorizations and well-posed linear systems, SIAM journal on control and optimization, 38, 1268-1292, (1998) · Zbl 0919.93040 |

[31] | Staffans, O.J., Admissible factorizations of Hankel operators induce well-posed linear systems, Systems and control letters, 37, 301-307, (1999) · Zbl 0948.93014 |

[32] | Staffans, O.J.; Weiss, G., Transfer functions of regular linear systems, part iithe system operator and the Lax-Phillips semigroup, Transactions of the American mathematical society, 354, 3229-3262, (2002) · Zbl 0996.93012 |

[33] | Weiss, G., Admissibility of unbounded control operators, SIAM journal on control and optimization, 27, 527-545, (1989) · Zbl 0685.93043 |

[34] | Weiss, G., Admissible observation operators for linear semigroups, Israel J. math., 65, 17-43, (1989) · Zbl 0696.47040 |

[35] | Weiss, G., Transfer functions of regular linear systems, part icharacterizations of regularity, Transactions of the American mathematical society, 342, 827-854, (1994) · Zbl 0798.93036 |

[36] | Weiss, G., Regular linear systems with feedback, Mathematics of control, signals, and systems, 7, 23-57, (1994) · Zbl 0819.93034 |

[37] | Weiss, G.; Curtain, R.F., Dynamic stabilization of regular linear systems, IEEE transactions on automatic control, 42, 4-21, (1997) · Zbl 0876.93074 |

[38] | Weiss, G.; Rebarber, R., Optimizability and estimatability for infinite-dimensional linear systems, SIAM journal on control and optimization, 39, 1204-1232, (2001) · Zbl 0981.93032 |

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