Controllability of discrete bilinear systems with bounded control

Abstract

minors of R ; A [ [ R ; p t ] is the Mcl\Iillan degree of the pole pn of R and IVote Added in Proof: Recent work shows that condition (9) rea [ R ] = ~~= l A I R ; p k ] [24]. quires that the return difference I + F&(s) not have a zero of Corollary 1: Let (?be defined by (3) and let L Y~ be defined by Fact 3. transmission at, P, = 1 7 % ' ' . J 1271. Under these conditions, for any k E { 1, 2,. . .,l] for which Re P k > 0, REFEREXCES det X k ( p s ) # 0 if and only if where in the triangular Hankel matrix, t.he Rka's are the coefficient matrices of R k defined by (4). Proof: From (17) and (19)-(20),m ) l ; hence and for any integer Bk > -,t lim (spk)'*det [I + F&(s)] = 0. Hence, by Lemma 2, (9) is true if and only if Q[det [ I f Fk] : p k ] = 7s. Let [ A , B, C , 01 be any minimal realization of the strictly proper element R of CnXn(s) defined by (15); then, because of the coprime factorization (16), det D ( s ) = c det (SI --4) [lti], [20]. S o a d e t (SI -<J) = +[RI (s) = II~.= (spk)A[R;pkI 1231, 1231, 12.11. Comparing with ( l i ) , i t follon-s t.hat. 7r. = I [ R ; p k ] . The last equality of (29) is established by observing that in (1.5) F is nonsingular, and by using a result of [26]. Renmrks: 1) The conditions of [26, theorem 9-10] are a special case of those of Theorem 1 above: indeed, for the lumped rase, i.e., Gp(s) E &Xn n CnXn(s), we have m P k -l[&;pn] = A [ R ; p t l , for k = 1,2,. . ' , I and Chen's condit.ions are equivalent to (8) and Q[det [ I + F 6 ] ; p~; ] = A[r$;p1~1, for I; = 1,2,...,1 .2) The conditions of [ 11, theorem 31 can be show1 to be equivalent to (8) and (29) (for details see [15] ).3) The equivalence of Corollary 1 is st.ated only for interior poles (i.e., Re p k > 0). 1 : is still valid for poles on the boundary (i.e., Re pk = 0) whenever Gp(s) is meromorphic in a neighborhood of such p s . This will always be the case for differential delay systems.

Nonlinear controls for a class of discrete‐time bilinear systems

“…for any non-zero x in R n : Assumption A2 [7] guarantees that system (1) is controllable [8] at any non-zero x: Notice that under the Assumption A1, the zero-state detectability in Reference [6] is equivalent to the controllability Assumption A2 here.…”

Estimation for Bilinear Stochastic Systems

“…A recently developed stochastic model for air pollution [63], [64] involves partial differential equations which, when discretized, become discrete-time bilinear equations (see [65], [66] for discussions of discrete-time bilinear systems). The advection-diffusion model of [63], [64], which is a generalization of Qc and qc are the mean and the zero-mean stochastic component of the pollution source rate; and K c is the eddy diffusivity.…”

On controllability of discrete-time bilinear systems

“…The main interest of bilinear systems lies in the fact that many important processes, not only in engineering, but also in biology, socio-economics, and ecology, can be modeled by bilinear systems [2][3][4][5][6][7][8][9][10][11]. The fundamental property of bilinear systems, say, controllability, has been extensively studied for both continuous-time and discrete-time cases [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. Since a discrete-time approximation to a bilinear system is essential for computer simulation or digital control purpose [28][29][30], a natural question is that whether the discretization changes the system controllability.…”

“…Firstly, the controllability problem of system (2) is completely solved by introducing the necessary and sufficient conditions, which, to the best knowledge of the authors, are new. Note that in the existing literature [23][24][25][26][27], mainly, [24,27], either the sufficient condition of [24] or the necessary and sufficient condition of [27] is not able to discuss the controllability of (2). Secondly, we show that, in comparison with the controllability criterion of this paper, the counterexample presented by [1] is a special case of the proposed necessary and sufficient conditions.…”