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Necessary and sufficient conditions for zero assignment by constant squaring down. (English) Zbl 0679.93012
Summary: The problem of zero assignment by constant squaring down (CZAP) is studied for minimal systems described by a transfer function G(s)$$\in {\mathbb{R}}^{m\times l}(s)$$, $$m>\ell$$, using tools from exterior algebra and algebraic geometry. Conditions for its solvability are given in terms of the numbers m, $$\ell$$, the Forney dynamical order $$\delta$$, and the rank $$\rho_{\delta}$$ of the Plücker matrix $$P_{\delta}.$$
For the cases $$\ell =l$$ or $$\ell =m-1$$, it is shown that G(s) is completely zero-assignable (CZA) if and only if $$\rho_{\delta}=\delta +1$$. For the cases $$\ell \neq 1$$, m-1, it is proved that G(s) is CZA, or generically zero-assignable (GZA), under a complex squaring down if and only if $$\ell (m-\ell)\geq \delta +1$$ and $$\rho_{\delta}=\delta +1.$$
The latter conditions also provide necessary conditions for the existence of a real solution. For a generic G(s), sufficient conditions for solvabilty of CZAP by a real squaring down are that $$\ell (m-\ell)\geq \delta +1$$ and that the number $g(a_ 0,...,a_{\ell -1})=(\delta +1)!/a_ 0!...a_{\ell -1}!\prod_{i>j}(a_ i-a_ j)$ is odd for sme set $$\{a_ 0,...,a_{\ell -1}\}$$ satisfying $\delta +1=\sum^{\ell -1}_{i=0}a_ i-\ell (\ell -1)/2\quad and\quad 0\leq a_ 0<a_ 1<...<a_{\ell -1}\leq m-1.$ Apart from the existence results, the present approach also allows the computation of the solutions (whenever they exist). Finally, it is proved that the only fixed zeros of CZAP are the zeros of G(s), and a procedure for computing the solutions of CZAP, whenever such solutions exist, based on an optimization problem is given.

##### MSC:
 93B55 Pole and zero placement problems 93B25 Algebraic methods 93B27 Geometric methods 93C05 Linear systems in control theory
##### References:
  Karcanias, N.; Giannakopoulos, C., Grassmann invariants: almost zeros and the determinantal zero, pole assignment problems of linear systems, Internat. J. control, 40, 673-698, (1984) · Zbl 0568.93010  Forney, G.D., Minimal bases of rational vector spaces with applications to multivariable linear systems, SIAM J. control optim., 13, 493-520, (1975) · Zbl 0269.93011  Karcanias, N.; Kouvaritakis, B., The output zeroing problem and its relationship to the invariant zero structure: A matrix pencil approach, Internat. J. control, 30, 395-415, (1979) · Zbl 0434.93018  Kouvaritakis, B.; MacFarlane, A.G.J., Geometric approach to analysis and synthesis of system zeros: part II: nonsquare systems, Internat. J. control, 23, 167-181, (1976) · Zbl 0337.93004  Marcus, M.; Minc, H., A survey of matrix theory and matrix inequalities, (1964), Allyn and Bacon Bostone · Zbl 0126.02404  Marcus, M., ()  Karcanias, N.; Giannakopoulos, C.; Hubbard, M., Almost zeros of a set of polynomials of $$R$$, Internat. J. control, 38, 1213-1238, (1983)  Hodge, W.V.D.; Pedoe, P.D., ()  Griffiths, P.; Harris, J., Principles of algebraic geometry, (1978), Wiley New York · Zbl 0408.14001  Giannakopoulos, C.; Kalogeropoulos, G.; Karcanias, N., The Grassmann variety of nondynamic compensators and the determinantal assignment problem of linear systems, Bull. Greek math. soc., 24, 35-57, (1983)  Bernstein, I., On the lusternick-Schnirelmann category of real Grassmannians, Proc. Cambridge philos. soc., 79, 129-139, (1976)  MacFarlane, A.G.J.; Kouvaritakis, B., A design technique for linear multivariable feedback systems, Internat. J. control, 25, 837-874, (1977) · Zbl 0361.93026  Rosenbrock, H.H., Computer-aided control system design, (1974), Academic London · Zbl 0328.93001  Kouvaritakis, B.; Edmunds, J.M., The characteristic frequency and characteristic gain design method for multivariable feedback systems, (), 229-246  Rosenbrock, H.H.; Rowe, B.A., Allocation of poles and zeros, Proc. IEE, 117, 1079, (1970)  Vardulakis, A.I.G., Zero placement and the “squaring down problem”: A polynomial matrix approach, Internat. J. control, 31, 821-832, (1980) · Zbl 0438.93031  Giannakopoulos, C., Frequency assignment problems of linear multivariable systems: an exterior algebra and algebraic geometry based approach, ()  Martin, C.; Hermann, R., Applications of algebraic geometry to systems theory: the macmillan degree and Kronecker indices, SIAM J. control optim., 16, 743-755, (1978) · Zbl 0401.93020  Brockett, R.N.; Byrnes, C.I., Multivariable Nyquist, root loci and pole placement: A geometric viewpoint, IEEE trans. automat. control, AC-26, 271-283, (1981) · Zbl 0462.93026  Byrnes, C.I.; Anderson, B.D.O., Output feedback and generic stabilizability, SIAM J. control optim., 22, 362-380, (1984) · Zbl 0557.93047  Giannakopoulos, C.; Karcanias, N., Pole assignment of strictly proper and proper linear systems by constant output feedback, Internat. J. control, 42, 543-565, (1985) · Zbl 0591.93023  Sain, M.K., The growing algebraic presence in systems engineering, IEEE proc., 64, 1, 99-111, (1976)  Sain, M.K.; Seshadri, V., Pole assignment and a theorem from exterior algebra, Proceedings of the 20th IEEE conference on decision and control, (1981), San Diego  Kailath, T., Linear systems, (1980), Prentice-Hall Englewood Cliffs, N.J · Zbl 0458.93025  Willems, J.C.; Hesselink, W.H., Generic properties of the pole placement problem, Proceedings of the 7th IFAC world congress, (1978), Helsinki  Rosenbrock, H.H., State space and multivariable theory, (1970), Nelson London · Zbl 0246.93010  Saberi, A.; Sannuti, P., Squaring down by static and dynamic compensators, IEEE trans. automat. control, AC-33, 358, (1988) · Zbl 0642.93027  Conte, G.; Perdon, A.M., Zeros of cascade compositions, () · Zbl 0634.93008  Wonham, W.M., Linear multivariable control: A geometric approach, (1979), Springer-Verlag New York · Zbl 0393.93024  Pierre, D.A., Optimization theory with applications, (1969), Wiley New York · Zbl 0205.15503  Marden, M., The geometry of zeros of a polynomial in a complex variable, Amer. math. soc., (1949) · Zbl 0038.15303  Karcanias, N.; Giannakopoulos, C., Grassmann matrices, decomposability of multivectors and the determinantal assignment problem, () · Zbl 0689.15014  Vardulakis, A.I.G.; Karcanias, N., On the stable exact model matching problem, Systems control lett., 5, No. 4, (1985) · Zbl 0559.93018  Vidyasagar, M., Control system synthesis: A factorization approach, (1985), MIT Press Cambridge, Mass · Zbl 0655.93001  Vardulakis, A.I.G.; Karcanias, N., Structure, Smith-mcmillan form and coprime MFD’s of a rational matrix inside a region $$P$$ = ω∪∞, Internat. J. control, Vol. 38, 927-957, (1983) · Zbl 0542.93010
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