New Iterative Learning Control Algorithm Using Learning Gain Based on <i>σ</i> Inversion for Nonsquare Multi-Input Multi-Output Systems

2020

IEEE Access

In this paper, the powerful issue of a non-full rank inverse model control investigation is provided. It is broadly known, that the inverse-based methodology is associated with the full-rank control systems only. However, following the recently obtained authors' results in this matter, it should be concluded, that for single-delayed non-full rank state-space plants, the inverse model control-related perfect control expression can also be established. It is shown here, that for all right-and left-oriented multivariable LTI non-full rank systems, including the square ones, governed by the discrete-time statespace domain arranging the zero-reference value, the maximum-speed pole-free instance of such inverse model control strategy can be achieved. Thus, the new non-full rank algorithm does not coincide with the z-transfer-function scenario, which additionally sounds the intriguing peculiarity. The innovative content of the manuscript, that has never been seen before, is strongly supported by the numerical examples. Henceforth, the presented methodology covers the entire set of LTI MIMO state-space-oriented plants in the discrete-time domain, which is also a consequence of conducted research investigation in the past. INDEX TERMS Non-full rank, perfect control procedure, Moore-Penrose inverse, generalized inverses, skeleton factorization, pole-free delayed plants, discrete-time state-space MIMO structures

“…Proof. Immediately, after considering the entire investigations, in particular the expressions (26) and (27).…”

Section: Non-full Rank Imc Structurementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A New Non-Full Rank Algorithm for the IMC-Derived d-Step MIMO Structures in the Pole-Free State Space

Feliks

2020

IEEE Access

“…T-inverse considered, however, as the 'minimum-energy' inverse, cannot be applied regarding a wide class of cases where the control inputs remain unstable under perfect control law. The recently introduced σ-inverse and H-inverse with various types of so-called degrees of freedom offer powerful tools to be applied regarding the above class of problems [24,25]. It is interesting that the first inverse constitutes an extension of minimum-norm right T-inverse in which degrees of freedom are organized as separate matrices being both in parameter as well as polynomial form.…”

Section: Selection Of Nonunique Right Inversementioning

confidence: 99%

A New Stability Theory for Grünwald–Letnikov Inverse Model Control in the Multivariable LTI Fractional-Order Framework

Wach

2019

Symmetry

The new general theory dedicated to the stability for LTI MIMO, in particular nonsquare, fractional-order systems described by the Grünwald–Letnikov discrete-time state–space domain is presented in this paper. Such systems under inverse model control, principally MV/perfect control, represent a real research challenge due to an infinite number of solutions to the underlying inverse problem for nonsquare matrices. Therefore, the paper presents a new algorithm for fractional-order perfect control with corresponding stability formula involving recently given H- and σ -inverse of nonsquare matrices, up to now applied solely to the integer-order plants. On such foundation a new set of stability-related tools is introduced, among them the key role played by so-called control zeros. Control zeros constitute an extension of transmission zeros for nonsquare fractional-order LTI MIMO systems under inverse model control. Based on the sets of stable control zeros a minimum-phase behavior is specified because of the stability of newly defined perfect control law described in the non-integer-order framework. The whole theory is complemented by pole-free fractional-order perfect control paradigm, a special case of fractional-order perfect control strategy. A significant number of simulation examples confirm the correctness and research potential proposed in the paper methodology.

“…However, in the case of examination of nonsquare MIMO systems with different number of input and output signals, we can effectively impact the robustness of the IMC-related perfect control schemes. It can only be done for rightinvertible plants described by both input-output and statespace LTI multivariable frameworks [26]- [28]. The higher number of input than output runs of nonsquare MIMO systems guarantees possibility of influencing this intriguing control law.…”

Section: Introductionmentioning

confidence: 99%

A New Geometric-Oriented Minimum-Energy Perfect Control Design in the IMC-Based State-Space Domain

Feliks

2020

IEEE Access

A new geometric approach providing the minimum-energy issue for inverse model controlrelated perfect regulation of linear time-invariant multi-input/single-output plants described in the discretetime state-space framework is proposed in the paper. Recent results have shown that the minimum-norm T-inverse does not guarantee the minimum-energy perfect control design, which has been confirmed by heuristic studies only. The new proposal, postulated throughout the manuscript, certifies the potential of nonunique σ-inverse regarding the minimum-energy behavior of inverse model control-based structures. After application of the proposed geometric approach dedicated to some class of state-space systems, we can precisely calculate the total energy of the multivariable perfect control runs. Thus, the analytical new methodology allows to obtain the minimum-energy inverse model control schemes, what constitutes the main accomplishment of the paper. Additionally, the aim of future analytical exploration covering the entire class of right-invertible state-space systems is clearly focused. INDEX TERMS Geometric solution, perfect control, minimum-energy problem, inverses of nonsquare matrices, discrete-time state-space domain, LTI MIMO.