Abstract
In this paper the dead-beat control problem, by partial interconnection, of two-dimensional (2D) discrete behaviors, defined on the grid \(\mathbb{Z }_+\times \mathbb{Z }\) and having the time as (first) independent variable, is investigated. The possibility of driving to zero (in a finite number of “steps”) either all or part of the system variables, by means of a partial interconnection controller, proves to be equivalent to the reconstructibility of the variables that are not accessible for control. On the other hand, if we search for “admissible” dead-beat controllers, the only ones providing meaningful results in practice, we have to introduce the zero-time-controllability assumption. These two properties are just the necessary and sufficient conditions for the existence of an observer-based (admissible) dead-beat controller, which consists of a dead-beat observer, to estimate the relevant variables from the measured ones, and of a full interconnection dead-beat controller, acting on both the measured and the estimated variables.
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Notes
It is worth noticing that, so long as we restrict our attention to the estimation problem alone, it is reasonable to assume that \(\mathbf{w}_m\) and \(\mathbf{w}_r\) correspond to disjoint sets, namely they have no common variables. However, the choice of the relevant variables may also be imposed by control purposes, as it happens when dealing with observer-based controllers, as we will see in Sect. 7. When so, the two sets of variables are not necessarily disjoint. If this is the case, one can simply replace \(\mathbf{w}_r\) with \(\tilde{\mathbf{w}}_r= (\mathbf{w}_r, \mathbf{w}_{mr})\) and assume \(\mathbf{w}_m =(\mathbf{w}_{m,nr}, \mathbf{w}_{mr})\), where \(\mathbf{w}_r\) corresponds to the relevant (not measured) variables, \(\mathbf{w}_{m,nr}\) to the measured variables that are not relevant, and \(\mathbf{w}_{mr}\) are measured variables that are also relevant. In this case, we replace (4.2) with
$$\begin{aligned} \left[ \begin{array}{l@{\quad }l}R_r(\sigma _1,\sigma _2)&{}0\\ 0 &{}I\\ \end{array}\right] \tilde{\mathbf{w}}_r(h,k) = \left[ \begin{array}{l@{\quad }l}R_m(\sigma _1,\sigma _2)&{}0 \\ 0 &{} I\\ \end{array}\right] \mathbf{w}_m(h,k). \end{aligned}$$It is well-known Bisiacco and Valcher (2013), Oberst (1990), that if a behavior \(\mathfrak B \) is described by \(R_{I}(\sigma _1,\sigma _2)\mathbf{w}_{I}(h,k)=R_{\bar{I}}(\sigma _1,\sigma _2)\mathbf{w}_{\bar{I}}(h,k), (h,k) \in \mathbb{Z }_+ \times \mathbb{Z }\), then \(\mathcal{P }_I\mathfrak B = \mathrm{ker}(M_{\bar{I}}(\sigma _1,\sigma _2)R_{I}(\sigma _1,\sigma _2))\), where \(M_{\bar{I}}(z_1,z_2)\) is an MLA of \(R_{\bar{I}}(z_1,z_2)\). In the special case when \(R_{\bar{I}}(z_1,z_2)\) is of full row rank, its MLA is a void matrix and \(\mathcal{P }_I\mathfrak B =\ker 0\), which means that \(\mathcal{P }_I\mathfrak B =(\mathbb{R }^{\mathtt{w}_I})^{\mathbb{Z }_+\times \mathbb{Z }}\).
In the following, when we will need to distinguish between the two of them, we will denote a controller that acts by full interconnection by \(\mathcal{C }_{FI}\) and a controller that acts by partial interconnection by \(\mathcal{C }_{PI}\).
References
Bisiacco, M., & Valcher, M. E. (2008). Dead-beat estimation problems for 2D behaviors. Multidimensional Systems and Signal Processing, 19, 287–306.
Bisiacco, M., & Valcher, M. E. (2012). Dead-beat control in the behavioral approach. IEEE Transactions on Automatic Control, 57(9), 2163–2175.
Bisiacco, M., & Valcher, M.E. (2012). Dead-beat control of two-dimensional behaviors. In Proceedings of the 51st IEEE conference on decision and control, pp. 3195–3202, Maui, Hawaii, 2012.
Bisiacco, M., & Valcher, M. E. (2013). Zero-time-controllability and dead-beat control of two-dimensional behaviors. SIAM J. Control Optim., 51(1), 195–220.
Bisiacco, M., Valcher, M. E., & Willems, J. C. (2006). A behavioral approach to estimation and dead-beat observer design with applications to state-space models. IEEE Transactions on Automatic Control, 51(11), 1787–1797.
Fornasini, E., Rocha, P., & Zampieri, S. (1993). State space realization of 2-D finite-dimensional behaviours. SIAM Journal on Control and Optimization, 31, 1502–1517.
Fornasini, E., & Valcher, M. E. (1997). nD polynomial matrices with applications to multidimensional signal analysis. Multidimensional Systems and Signal Processing, 8, 387–407.
Napp, D., Rapisarda, P., & Rocha, P. (2011). Time-relevant stability of 2D systems. Automatica, 47(11), 2373–2382.
Ntogramatzidis, L., & Cantoni, M. (2012). Detectability subspaces and observer synthesis for two-dimensional systems. Multidimensional Systems and Signal Processing, 23, 79–96.
Oberst, U. (1990). Multidimensional constant linear systems. Acta Applicandae Mathematicae, 20, 1–175.
Oberst, U., & Scheicher, M. (2012). Time-autonomy and time-controllability of discrete multidimensional behaviors. International Journal of Control, 85(8), 990–1009.
Oberst, U., & Scheicher, M. (2013). The asymptotic stability of stable and time-autonomous discrete multidimensional behaviors. Mathematics of Control, Signals, and Systems, 25. doi:10.1007/s00498-013-0114-6.
Praagman, C., Trentelman, H. L., & Zavala Yoe, R. (2007). On the parametrization of all regularly implementing and stabilizing controllers. SIAM Journal on Control and Optimization, 45(6), 2035–2053.
Rapisarda, P., & Rocha, P. (2012). Lyapunov functions for time-relevant 2D systems, with application to first-orthant stable systems. Automatica, 48(9), 1998–2006.
Rocha, P. (1990). Structure and representation of 2-D systems. PhD thesis, University of Groningen, The Netherlands.
Rocha, P., & Wood, J. (2001). Trajectory control and interconnections of 1D and nD systems. SIAM Journal on Control and Optimization, 40(1), 107–134.
Sasane, A. J. (2004). Time-autonomy and time-controllability of 2-D behaviours that are tempered in the spatial direction. Multidimensional Systems and Signal Processing, 15, 97–116.
Sasane, A. J., & Cotroneo, T. (2002). Conditions for time-controllability of behaviours. International Journal of Control, 75, 61–67.
Sasane, A. J., Thomas, E. G. F., & Willems, J. C. (2002). Time-autonomy versus time-controllability. Systems and Control Letters, 45, 145–153.
Youla, D. C., & Gnavi, G. (1979). Notes on n-dimensional system theory. IEEE Transactionas on Circuits and Systems CAS, 26, 105–111.
Zerz, E., & Wagner, L. (2012). Finite multidimensional behaviors. Multidimensional Systems and Signal Processing, 24, 5–15.
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Bisiacco, M., Valcher, M.E. Partial interconnection and observer-based dead-beat control of two-dimensional behaviors. Multidim Syst Sign Process 26, 459–479 (2015). https://doi.org/10.1007/s11045-013-0252-5
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DOI: https://doi.org/10.1007/s11045-013-0252-5