Realization of discrete-time nonlinear input-output equations: polynomial approach

Kotta, Ülle; Tõnso, Maris

doi:10.1109/wcica.2008.4592979

Cited by 14 publications

(6 citation statements)

References 28 publications

(26 reference statements)

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“…The greatest common left divisor (gcld) of two polynomial matrices V and W is a common left divisor which is a right multiple of every common left divisor of V and W . The proof is analogous with the discrete time case (see [16], Theorem 10) and therefore omitted. The gcld is, in general, not unique, since any two gclds, G 1 L and G 2 L , are related by definition as…”

Section: Polynomial Matricesmentioning

confidence: 99%

Polynomial accessibility condition for the multi-input multi-output nonlinear control system

Kotta¹,

Tõnso²,

Kawano³

2014

Proc. Estonian Acad. Sci.

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The paper presents a computation-oriented necessary and sufficient accessibility condition for the set of nonlinear higherorder input-output differential equations. The condition is presented in terms of the greatest common left divisor of two polynomial matrices, associated with the system of input-output equations. The basic difference from the linear case is that the elements of the polynomial matrices belong to a non-commutative polynomial ring. The condition found provides a basis for finding the accessible representation of the set of input-output equations, which is a suitable starting point for the construction of an observable and accessible state space realization. Moreover, the condition allows us to check the transfer equivalence of two nonlinear systems.The results of this paper are based on the conference paper [13]. Compared with [13], in the present paper irreducible system representation (accessible subsystem) is related to the recently introduced concept of transfer matrix of the nonlinear system [8,9], whereas that in [13] was based on the notion of an irreducible differential form, associated with the control system [6]. Moreover, the notion of the transfer equivalence of systems is now defined via the equality of transfer matrices, exactly like in the linear case, and the reduction problem is straightforwardly addressed. Furthermore, the proof of the main theorem is improved (while only the sketch of the proof was given in [13]) and the role of non-uniqueness of the greatest common left divisor to the solution is explained. Finally, comparison with alternative algebraic accessibility criteria is added. Note that the single-input single-output (SISO) case was studied in [27]. For discrete-time nonlinear systems the accessibility property was examined in [16], though the paper itself was focused on finding the minimal realization of the set of i/o equations and accessibility was studied only from the viewpoint of irreducibility of the i/o description of the system. The irreducibility problem for the SISO case has also been treated in [12], being a generalization of [27] to the systems defined on homogeneous time-scales.The paper is organized as follows. Section 2 describes the differential field and a polynomial matrix representation, associated to nonlinear systems. Using the polynomial matrix description, in Section 3 necessary and sufficient accessibility condition is given. Section 4 is devoted to the system reduction and concept of transfer equivalence. In Section 5 the obtained results are compared with the results of [4] and illustrated by two examples, one realizable in the state-space form and the other not. Section 6 draws the conclusions and drafts some future goals of study.

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Section: Polynomial Matricesmentioning

confidence: 99%

Polynomial accessibility condition for the multi-input multi-output nonlinear control system

Kotta¹,

Tõnso²,

Kawano³

2014

Proc. Estonian Acad. Sci.

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“…Note that even if the linear combination (5) is infinite, it nevertheless defines an element of E because, for all ω ∈ E , X, ω may be written as a sum with only finitely many nonzero terms; see (6). The delta-derivative X ∆ f and forward-shift X σ f of X ∈ E may be defined uniquely by the equations…”

Section: The Dual Space Of Vector Fieldsmentioning

confidence: 99%

“…It covers both the continuous-and discretetime cases in such a manner that those are the special cases of the formalism. However, it has to be stressed that in [5] the discrete-time system is described in terms of the difference operator unlike in the majority of papers where the system is described via the shift operator (see for example [2][3][4]6,7]). …”

Section: Introductionmentioning

confidence: 99%

Algebraic formalism of differential p-forms and vector fields for nonlinear control systems on homogeneous time scales

Bartosiewicz

Kotta

Pawłuszewicz³

et al. 2013

Proc. Estonian Acad. Sci.

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“…In a recent publication [29] for discrete-time nonlinear system whose description is based on the shift operator, explicit formulae are given to compute the basis vectors of the subspaces H1; . .…”

Section: Remark 22mentioning

confidence: 99%

Transfer Equivalence and Realization of Nonlinear Input-Output Delta-Differential Equations on Homogeneous Time Scales

Casagrande

Kotta

Tõnso

et al. 2010

IEEE Trans. Automat. Contr.

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Fig. 4. Constraint variables z (dotted), v (dashed), and v (solid) for the upper constraint of the uncertain active magnetic bearing system for the initial condition x(0) = [0:5; 0; 0] and the safety control law.while this would clearly happen without the invariance control. In Fig. 4 the time trajectories of the three constraints z 1;1 ; v 1;1 ; v 1;2 are shown. All three constraints are kept below zero at all times, hence the overall constraint is satisfied. The constraint v1;2 (solid line) would be the first to become positive, which is avoided by the invariance controller. VI. CONCLUSIONIn this paper, an algorithm is proposed to construct control laws for uncertain nonlinear systems under hard state constraints. Using a recursive design procedure, a set of safe initial conditions is constructed that can be rendered invariant by a suitable control action. A switching control is designed to guarantee positive invariance of the set and to satisfy a nominal control objective whenever possible. With some additional assumptions, stability of the closed loop is established.The current work focuses primarily on single input systems. However simulations show that a generalization to multiinput systems is possible. An important problem that remains open is with regards to the simultaneous satisfaction of state and input constraints. The backstepping approach seems to offer the flexibility to integrate both challenges into the controller design. Future research will aim to provide a sound theoretical analysis of this problem. ACKNOWLEDGMENTThe authors wish to thank the anonymous reviewers, who helped to improve the paper significantly. REFERENCES[1] M. Bürger and M. Guay, "Robust constraint satisfaction for continuous-time nonlinear systems," in Proc. 47th Conf.[8] P. Tsiotras and M. Arcak, "Low-bias control of amb subject to voltage saturation: State-feedback and observer designs," IEEE Trans.Abstract-Nonlinear control systems on homogeneous time scales are studied. First the concepts of reduction and irreducibility are extended to higher order delta-differential input-output equations. Subsequently, a definition of system equivalence is introduced which generalizes the notion of transfer equivalence in the linear case. Finally, the necessary and sufficient conditions are given for the existence of a state-space realization of a nonlinear input-output delta-differential equation.

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Realization of discrete-time nonlinear input-output equations: polynomial approach

Cited by 14 publications

References 28 publications

Polynomial accessibility condition for the multi-input multi-output nonlinear control system

Polynomial accessibility condition for the multi-input multi-output nonlinear control system

Algebraic formalism of differential p-forms and vector fields for nonlinear control systems on homogeneous time scales

Transfer Equivalence and Realization of Nonlinear Input-Output Delta-Differential Equations on Homogeneous Time Scales

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