Integral control of infinite-dimensional systems in the presence of hysteresis : an input-output approach

Logemann, Hartmut; Ryan, Eugene P.; Shvartsman, Ilya

ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 3, pp. 458-483.

This paper is concerned with integral control of systems with hysteresis. Using an input-output approach, it is shown that application of integral control to the series interconnection of either (a) a hysteretic input nonlinearity, an $L^2$-stable, time-invariant linear system and a non-decreasing globally Lipschitz static output nonlinearity, or (b) an $L^2$-stable, time-invariant linear system and a hysteretic output nonlinearity, guarantees, under certain assumptions, tracking of constant reference signals, provided the positive integrator gain is smaller than a certain constant determined by a positivity condition in the frequency domain. The input-output results are applied in a general state-space setting wherein the linear component of the interconnection is a well-posed infinite-dimensional system.

[1] H.T. Banks, R.C. Smith and Y. Wang, Smart Material Structures: Modeling, Estimation, Control. Masson, Paris (1996). | Zbl

[2] M. Brokate, Hysteresis operators, in Phase Transitions and Hysteresis, A. Visintin Ed., Springer, Berlin (1994) 1-38. | Zbl

[3] M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, New York (1996). | MR | Zbl

[4] C.I. Byrnes, D.S. Gilliam, V.I. Shubov and G. Weiss, Regular linear systems governed by a boundary controlled heat equation. J. Dynam. Control Syst. 8 (2002) 341-370. | Zbl

[5] R.F. Curtain, H. Logemann and O. Staffans, Stability results of Popov-type for infinite-dimensional systems with applications to integral control. Proc. London Math. Soc. 86 (2003) 779-816. | Zbl

[6] T. Fliegner, H. Logemann and E.P. Ryan, Low-gain integral control of well-posed infinite-dimensional linear systems with input and output nonlinearities. J. Math. Anal. Appl. 261 (2001) 307-336. | Zbl

[7] R.B. Gorbet, K.A. Morris and W.L. Wang, Passivity-based stability and control of hysteresis in smart actuators. IEEE Trans. Control Systems Technology 9 (2001) 5-16.

[8] B.Z. Guo and Z.C. Shao, Regularity of a Schrödinger equation with Dirichlet control and colocated observation, Syst. Control Lett. 54 (2005) 1135-1142. | Zbl

[9] B.Z. Guo and Z.C. Shao, Regularity of an Euler-Bernoulli plate with Neumann control and colocated observation, J. Dynam. Control Syst. 12 (2006) 405-418. | Zbl

[10] B.Z. Guo and X. Zhang, The regularity of the wave equation with partial Dirichlet control and colocated observation, SIAM J. Control Optim. 44 (2005) 1598-1613. | Zbl

[11] F. Ikhouane and J. Rodellar, A linear controller for hysteretic systems. IEEE Trans. Auto. Control 51 (2006) 340-344.

[12] F. Ikhouane, V. Mañosa and J. Rodellar, Adaptive control of a hysteretic structural system. Automatica 41 (2005) 225-231. | Zbl

[13] U. Jönson, Stability of uncertain systems with hysteresis nonlinearities. Int. J. Robust Nonlinear Control 8 (1998) 279-293. | Zbl

[14] M.A. Krasnosel'Skii and A.V. Pokrovskii, Systems with Hysteresis. Springer, Berlin (1989).

[15] I. Lasiecka and R. Triggiani, The operator $B^{*}L$ for the wave equation with Dirichlet control. Abstract Appl. Anal. 2004 (2004) 625-634. | Zbl

[16] H. Logemann and A.D. Mawby, Low-gain integral control of infinite-dimensional regular linear systems subject to input hysteresis, F. Colonius et al. Eds., Birkhäuser, Boston, Advances in Mathematical Systems Theory (2001) 255-293.

[17] H. Logemann and A. Mawby, Discrete-time and sampled-data low-gain integral control of infinite-dimensional linear systems in the presence of input hysteresis. SIAM J. Control Optim. 41 (2002) 113-140. | Zbl

[18] H. Logemann and E.P. Ryan, Time-varying and adaptive integral control of infinite-dimensional regular systems with input nonlinearities, SIAM J. Control Optim. 38 (2000) 1120-1144. | Zbl

[19] H. Logemann and E.P. Ryan, Systems with hysteresis in the feedback loop: existence, regularity and asymptotic behaviour of solutions. ESAIM: COCV 9 (2003) 169-196. | Numdam | Zbl

[20] H. Logemann, E.P. Ryan and S. Townley, Integral control of linear systems with actuator nonlinearities: lower bounds for the maximal regulating gain. IEEE Trans. Auto. Control 44 (1999) 1315-1319. | Zbl

[21] R. Rebarber and G. Weiss, Internal model based tracking and disturbance rejection for stable well-posed systems, Automatica 39 (2003) 1555-1569. | Zbl

[22] D. Salamon, Control and Observation of Neutral Systems. Pitman, London (1984). | MR | Zbl

[23] D. Salamon, Infinite-dimensional linear systems with unbounded control and observation: a functional analytic approach. Trans. Amer. Math. Soc. 300 (1987) 383-431. | Zbl

[24] O.J. Staffans, Well-Posed Linear Systems. Cambridge University Press, Cambridge (2005). | MR | Zbl

[25] O.J. Staffans and G. Weiss, Transfer functions of regular linear systems, part II: The system operator and the Lax-Phillips semigroup. Trans. Amer. Math. Soc. 354 (2002) 3229-3262. | Zbl

[26] X. Tan and J.S. Baras, Modeling and control of hysteresis in magnetostrictive actuators. Automatica 40 (2004) 1469-1480. | Zbl

[27] G. Tao and P.V. Kokotović, Adaptive Control of Systems with Actuator and Sensor Nonlinearities. John Wiley, (1996) | MR | Zbl

[28] M. Tucsnak and G. Weiss, How to get a conservative well-posed system out of thin air, part II: Controllability and stability. SIAM J. Control Optim. 42 (2003) 907-935. | Zbl

[29] G. Weiss, Transfer functions of regular linear systems, part I: Characterization of regularity. Trans. Amer. Math. Soc. 342 (1994) 827-854. | Zbl

[30] G. Weiss and R. Rebarber, Optimizability and estimatability for infinite-dimensional linear systems. SIAM J. Control Optim. 39 (2000) 1204-1232. | Zbl