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\[CapitalUDoubleDot]lle Kotta\ \>", "Author", CellChangeTimes->{{3.3877150737102766`*^9, 3.3877150764198074`*^9}, { 3.3877234144527607`*^9, 3.3877234158447623`*^9}, 3.387724094234042*^9, { 3.396188392847262*^9, 3.3961883946898193`*^9}, 3.4250199545936146`*^9, { 3.42528096340625*^9, 3.42528096675*^9}}], Cell["\<\ Institute of Cybernetics, Tallinn University of Technology, Akadeemia tee 21, \ Tallinn, 10918, Estonia maris@cc.ioc.ee, jbelikov@cc.ioc.ee, vkaparin@cc.ioc.ee, heli@cc.ioc.ee, \ kotta@cc.ioc.ee\ \>", "Affiliation", CellChangeTimes->{ 3.3877151029631763`*^9, 3.387723443194089*^9, 3.3877241009042325`*^9, { 3.3961883966124887`*^9, 3.3961883987154074`*^9}, {3.425280754640625*^9, 3.42528076359375*^9}, {3.425280810078125*^9, 3.425280831828125*^9}, { 3.425280954875*^9, 3.425281010734375*^9}, {3.42528115965625*^9, 3.425281167859375*^9}, {3.425281471359375*^9, 3.42528147290625*^9}}], Cell[CellGroupData[{ Cell["Abstract", "Section", CellChangeTimes->{{3.387715149597215*^9, 3.3877151508215218`*^9}}], Cell["\<\ In this talk we will introduce a package, NLControl, for modelling, analysis \ and synthesis of nonlinear control systems. Our package is oriented purely to \ nonlinear systems and deals with tasks not covered by Control Systems \ Professional. Only symbolic computation problems are handled in the package. \ Most functions in NLControl are based on the algebraic approach of \ differential one-forms and on the theory on non-commutative skew polynomial \ rings that provide universal and simple theoretical framework for addressing \ different nonlinear control problems. Linear algebraic approach is based on certain sequences of subspaces of \ differential one-forms, associated to the control system and on the \ classification of one-forms, for example, according to their relative \ degrees. The characteristics which make the algebraic approach useful, are \ its wide applicability (with proper modifications it applies both in \ discrete- and continuous-time cases), inherent simplicity (the procedures are \ transparent and most of them can be straightforwardly implemented in \ Mathematica), analogy with linear case (the analogy makes the package easier \ to use for engineers and students) and perhaps, most important, its \ universality (a single tool provides a key for solving several different \ control problems). The functions for solving control problems require to \ implement assistant functions that find forward- and backward-shifts, the \ derivative of one-forms, the inversive closure, associated to control system \ and integrate a set of one-forms. The second group of mathematical tools we used comes from the noncommutative \ polynomial theory. Noncommutative skew polynomials (also known as Ore \ polynomials) play in nonlinear control theory the similar role as the \ ordinary polynomials in the linear theory. Unfortunately, the existing \ general-purpose Ore polynomial package cannot be directly used by the \ following reason. In our case the variables appearing in coefficients of \ polynomials are not necessarily independent but relations are defined between \ them by the equations of control systems. Ignoring this fact may cause \ serious mistakes. So we wrote our own Ore package, including functions like \ addition and multiplication of Ore polynomials, finding their left and right \ quotient, reminder, GCD and LCM. It also enables to work with fractions of \ Ore polynomials and find the inverse of the Ore polynomials matrix. The main tasks the package can complete are listed below. 1. Check reducibility of the set of differential/difference input-output \ (i/o) equations and reduction them into the equivalent irreducible form. 2. Check realizability of the set of i/o equations in the state space form \ and find the minimal realization. It is also possible to find the i/o \ equations from the state equations. 3. Check realizability and find the minimal realization for the SISO \ composite systems, including series, parallel and feedback compositions. 4. Transform the generalized state equations into the classical state space \ form or, if not possible, lower the order of the input shifts/derivatives in \ the generalized equations. 5. Compute the transfer function both from the state equations and from the \ i/o equations. 6. Check (forward) accessibility of the system and decompose the state \ equations into the accessible and non-accessible subsystems. 7. Check observability of the system and decompose the state equations into \ the observable and unobservable subsystems. 8. Check identifiability of the system unknown parameters. 9. Transform the system equations into the normal form which is a good \ starting point for applying the inversion-based control algorithms and for \ computing the zero dynamics of the system. 10. Transform the state equations into the controller canonical form and \ solve the static state feedback linearization problem (full and partial). 11. Transform the state equations into the prime form, using the state \ feedback, state and output diffeomorphisms. 12. Discretization of the continuous-time system by using either Taylor \ series expansion or direct integration methods. Most implemented functions can handle both, discrete- and continuous-time \ systems, and additionally, some functions work also with systems, defined on \ homogeneous time scales. \ \>", "Text", CellChangeTimes->{ 3.3877151675704374`*^9, {3.387715200155057*^9, 3.387715259102409*^9}, { 3.387715316373866*^9, 3.3877153173272195`*^9}, 3.3877234012738104`*^9, { 3.3877234827609835`*^9, 3.387723513395033*^9}, {3.387723571959244*^9, 3.387723576816228*^9}, 3.387723687635578*^9, {3.3877241269540277`*^9, 3.387724131260607*^9}, {3.3877242593948393`*^9, 3.387724307774889*^9}, { 3.3880799362614255`*^9, 3.388079987475067*^9}, 3.39618841425698*^9, { 3.4250199669998646`*^9, 3.4250199744217396`*^9}, {3.425281321859375*^9, 3.42528134425*^9}}] }, Open ]] }, Open ]] }, WindowSize->{1272, 889}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, PageHeaders->{{ Cell[ TextData[{ CounterBox["Page"]}], "PageNumber"], None, Cell[ TextData[{ 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