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Signal decoupling with preview in the geometric context: exact solution for nonminimum-phase systems. (English) Zbl 1136.93013
Summary: The problem of making the output of a discrete-time linear system totally insensitive to an exogenous input signal known with preview is tackled in the geometric approach context. A necessary and sufficient condition for exact decoupling with stability in the presence of finite preview is introduced, where the structural aspects and the stabilizability aspects are considered separately. On the assumption that structural decoupling is feasible, internal stabilizability of the minimal self-bounded controlled invariant satisfying the structural constraint $$\mathcal{V}_m$$ guarantees the stability of the dynamic feedforward compensator. However, if the structural decoupling is feasible, but $$\mathcal{V}_m$$ is not internally stabilizable, exact decoupling is nonetheless achievable with a stable feedforward compensator on the sole assumption that $$\mathcal{V}_m$$ has no unassignable internal eigenvalues on the unit circle, provided that the signal to be rejected is known with infinite preview. An algorithmic framework based on steering-along-zeros techniques, completely devised in the time domain, shows how to compute the convolution profile of the feedforward compensator in each case.

##### MSC:
 93B27 Geometric methods 93C55 Discrete-time control/observation systems 93C05 Linear systems in control theory 93B17 Transformations
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##### References:
 [1] DAVISON, E. J., and SCHERZINGER, B. M., Perfect Control of the Robust Servomechanism Problem, IEEE Transactions on Automatic Control, Vol. 32, No. 8, pp. 689–701, 1987. · Zbl 0625.93051 · doi:10.1109/TAC.1987.1104691 [2] QIU, L., and DAVISON, E. J., Performance Limitations of Nonminimum Phase Systems in the Servomechanism Problem, Automatica, Vol. 29, No. 2, pp. 337–349, 1993. · Zbl 0778.93053 · doi:10.1016/0005-1098(93)90127-F [3] HUNT, L. R., MEYER, G., and SU, R., Noncausal Inverses for Linear Systems, IEEE Transactions on Automatic Control, Vol. 41, No. 4, pp. 608–611, 1996. · Zbl 0864.93055 · doi:10.1109/9.489285 [4] DEVASIA, S., CHEN, D., and PADEN, B., Nonliner Inversion-Based Output Tracking, IEEE Transactions on Automatic Control, Vol. 41, No. 7, pp. 930–942, 1996. · Zbl 0859.93006 · doi:10.1109/9.508898 [5] GROSS, E., TOMIZUKA, M., and MESSNER, W., Cancellation of Discrete-Time Unstable Zeros by Feedforward Control, ASME Journal of Dynamic Systems, Measurement and Control, Vol. 116, No. 1, pp. 33–38, 1994. · Zbl 0800.93395 · doi:10.1115/1.2900678 [6] TSAO T. C., Optimal Feed forward Digital Tracking Controller Design, ASME Journal of Dynamic Systems, Measurement and Control, Vol. 116, No. 4, pp. 583–592, 1994. · Zbl 0850.93285 · doi:10.1115/1.2899256 [7] MARRO, G., and FANTONI, M., Using Preaction with Infinite or Finite Preview for Perfect or Almost Perfect Digital Tracking, Proceedings of the 8th Mediterranean Electrotechnical Conference, Vol. 1, pp. 246–249, 1996. [8] ZOU, Q., and DEVASIA, S., Preview-Based Stable-Inversion for Output Tracking of Linear Systems, ASME Journal of Dynamic Systems, Measurement and Control, Vol. 121, No 4, pp. 625–630, 1999. · doi:10.1115/1.2802526 [9] MARRO, G., PRATTICHIZZO, D., and ZZTTONI, E., Convolution Profiles for Right-Inversion of Multivariable Nonminimum Phase Discrete-Time Systems, Automatica, Vol. 38, No. 10, pp. 1695–1703, 2002. · Zbl 1011.93022 · doi:10.1016/S0005-1098(02)00088-2 [10] ZENG, G., and HUNT, L. R., Stable Inversion for Nonlinear Discrete-Time Systems, IEEE Transactions on Automatic Control, Vol. 45, No. 6, pp. 1216–1220, 2000. · Zbl 0972.93036 · doi:10.1109/9.863610 [11] WONHAM, W. M., Linear Multivariable Control: A Geometric Approach, 3rd Edition, Springer Verlag, New York, NY, 1985. · Zbl 0609.93001 [12] BHATTACHARYYA, S. P., Disturbance Rejection in Linear Systems, International Journal of Systems Science, Vol. 5, No. 7, pp. 931–943, 1974. [13] WILLIEMS, J. C., Feedforward Control, PID Control Laws, and Almost-Invariant Subspaces, Systems and Control Letters, Vol. 1, No. 4, pp. 277–282, 1982. · Zbl 0473.93032 · doi:10.1016/S0167-6911(82)80012-1 [14] IMAI, H., SHINOZUKA, M., YAMAKI, T., LI, D., and KUWANA, M., Disturbance Decoupling by Feedforward and Preview Control, ASME Journal of Dynamic Systems, Measurement and Control, Vol. 105, No. 3, pp. 11–17, 1983. · Zbl 0512.93029 · doi:10.1115/1.3139721 [15] BASILE, G., and MARRO, G., Self-Bounded Controlled Invariant Subspaces: A Straightforward Approach to Constrained Controllability, Journal of Optimization Theory and Applications Vol. 38, No. 1, pp. 71–81, 1982. · Zbl 0471.93008 · doi:10.1007/BF00934323 [16] SCHUMACHER, J. M., On a Conjecture of Basile and Marro, Journal of Optimization Theory and Applications, Vol. 41, No. 2, pp. 371–376, 1983. · Zbl 0517.93009 · doi:10.1007/BF00935232 [17] BASILE, G., MARRO, G., and PIAZZI, A., A New Solution to the Disturbance Localization Problem with Stability and Its Dual, Proceedings of the 1984 International AMSE Conference on Modelling and Simulation, Vol. 1.2, pp. 19–27, 1984. [18] BASILE, G., and MARRO, G., Controlled and Conditioned Invariants in Linear System Theory, Prentice Hall, Englewood Cliffs, New Jersey, 1992. · Zbl 0758.93002 [19] MARRO, G., and ZATTONI, E., Self-Bounded Controlled Invariant Subspaces in Model Following by Output Feedback: Minimal-Order Solution for Nonminimum-Phase Systems, Journal of Optimization Theory and Applications, Vol. 125, No. 2, pp. 409–429, 2005. · Zbl 1093.93004 · doi:10.1007/s10957-004-1857-5 [20] MARRO, G., and ZATTONI, E., Self-Bounded Controlled Invariant Subspaces in Model Following by Output Feedback, Technical Report DEIS-GCG-01-2004, University of Bologna, 2004. · Zbl 1093.93004
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