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Feedback realization of open loop diagonalizers. (English) Zbl 0811.93027
For a given full row rank transfer matrix (rational matrix function) $$Z(s)$$ and a given $$p\times p$$ strictly proper diagonal matrix $$\Lambda(s)$$, the set of open loop $$\Lambda$$-diagonalizers, $$\text{OLD} (Z,\Lambda)$$ is defined as $\text{OLD} (Z,\Lambda):= \{\text{proper } M(s)\mid Z(s) M(s)= \Lambda\}.$ Assuming the $$Z(s)= C(sI- A)^{-1} B$$ is realized as the transfer function of a linear system \begin{aligned} \dot x(t) &= Ax(t)+ Bu(t), \qquad x(0)=0,\\ y(t) &= Cx(t),\\ y(s) &= Z(z) u(s)\end{aligned} (where $$u(s)$$ is the Laplace transform of $$u(t)$$), the author considers the problem of realizing elements of $$\text{OLD} (Z,\Lambda)$$ via implementation of a feedback control law of one of the following types:
i) Dynamic state feedback: $$u(s)=- F(s) x(s)+ Gv(s)$$, ii) Constant state feedback: $$u(s)=- Fx(s)+ Gv(s)$$, iii) Dynamic output feedback: $$u(s)=- Z_ c(s) y(s)+ Gv(s)$$, iv) Constant output feedback: $$u(s)=- Z_ c y)s_ + Gv(s)$$. In the above, $$F(s)$$ and $$Z_ c(s)$$ are proper and $$F$$ and $$Z_ c$$ are constant compensators in the feedback path, while $$G$$ is a full column rank constant precompensator. The first part of the paper discusses properties of open loop diagonalizers (old’s) which admit a feedback realization of one of the above types. Then dynamic (constant) output feedback decoupling problems are formulated as determining an open loop diagonalizer which admits the desired feedback realization. Finally, solutions to these problems are obtained by determining the conditions of existence for such open loop diagonalizers.

##### MSC:
 93B52 Feedback control
##### Keywords:
feedback; open loop diagonalizers; feedback realization
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##### References:
 [1] M. M. Bayoumi, T. L. Duffield: Output feedback decoupling and pole placement in linear time invariant systems. IEEE Trans. Automat. Control AC-22 (1977), 142-143. [2] J. Descusse J. F. Lafay, V. Kučera: Decoupling by restricted static state feedback: The general case. IEEE Trans. Automat. Control AC-29 (1983), 79-81. · Zbl 0543.93020 · doi:10.1109/TAC.1984.1103368 [3] J. Descusse J. F. Lafay, M. Malabre: Solution to Morgan’s problem. IEEE Trans. Automat. Control AC-33 (1988), 8, 732-739. · Zbl 0656.93018 · doi:10.1109/9.1289 [4] J. Descusse J. F. Lafay, M. Malabre: On the structure at infinity of linear block decouplable systems. IEEE Trans. Automat. Control AC-28 (1983), 1115-1118. · Zbl 0525.93016 · doi:10.1109/TAC.1983.1103187 [5] J. M. Dion, C. Commault: Minimal delay decoupling problem: Feedback implementation with stability. SIAM J. Control Optim. 20(1988), 1, 66-82. · Zbl 0646.93049 · doi:10.1137/0326005 [6] C. Commault J. Descusse J. M. Dion J. F. Lafay, M. Malabre: About new decoupling invariants: Essential orders. Internat. J. Control 13 (1986), 689-700. · Zbl 0611.93012 · doi:10.1080/00207178608933627 [7] J. M. Dion J. A. Torres, C. Commault: New feedback invariants and the block decoupling problem. Internat. J. Control 51 (1990), 1, 219-236. · Zbl 0695.93019 · doi:10.1080/00207179008934058 [8] V. Eldem, A. B. Ozguler: A solution to the diagonalization problem by constant precompensator and dynamic output feedback. IEEE Trans. Automat. Control AC-34 (1989), 10, 1061-1067. · Zbl 0695.93018 · doi:10.1109/9.35276 [9] G. D. Forney: Minimal basis of rational vector spaces with applications to multivariate linear systems. SIAM J. Control Optim. 13 (1975), 493-520. · Zbl 0269.93011 [10] J. Hammer, P. P. Khargonekar: Decoupling of linear systems by dynamical output feedback. Math. Systems Theory 17 (1984), 2, 135-157. · Zbl 0593.93013 · doi:10.1007/BF01744437 [11] M. L. J. Hautus, M. Heynmann: Linear feedback decoupling, transfer function analysis. IEEE Trans. Automat. Control AC-28 (1983), 823-832. · Zbl 0523.93035 · doi:10.1109/TAC.1983.1103320 [12] J. H. Howze: Necessary and sufficient conditions for decoupling using output feedback. IEEE Trans. Automat. Control AC-18 (1973), 44-46. · Zbl 0263.93032 · doi:10.1109/TAC.1973.1100205 [13] V. Kučera: Block decoupling by dynamic compensation with internal properness and stability. Problems Control Inform. Theory 12 (1983), 6, 379-389. · Zbl 0532.93032 [14] B. S. Morgan: The synthesis of linear multivariable systems by state feedback. Joint American Control Conference 64 (1964), 468-472. [15] A. B. Ozguler, V. Eldem: The set of open loop block diagonalizers of transfer matrices. Internat. J. Control 49 (1989), 1, 161-168. · Zbl 0689.93015 · doi:10.1080/00207178908961237 [16] W. A. Wolovich: Output feedback decoupling. IEEE Trans. Automat. Control AC-20 (1975), 148-149. · Zbl 0299.93017 · doi:10.1109/TAC.1975.1100840 [17] W. A. Wolovich, P. L. Falb: Invariants and canonical forms under dynamic compensation. SIAM J. Control Optim. 14 (1976), 996-1008. · Zbl 0344.93019 · doi:10.1137/0314063
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