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

IEEE Trans. Syst. Man Cybern, Syst.

Stanimirović

2023

Application of generalized inverses in solving the inverse model control (IMC)-oriented minimum-energy perfect control design (PCD) problem for linear time-invariant multi-input/multi-output systems governed by the discrete-time d-state-space structure is presented in this article. For this reason, an appropriate class of polynomial generalized inverses is investigated. Moreover, it can be stated that the nonunique right σ -inverse, based on properly selected so-called degrees of freedom (DOFs), outperforms the well-known unique Moore-Penrose (MP) minimum-norm right T-inverse in terms of the energy consumption of perfect control (PC) input signals. However, the analytical confirmation of such an intriguing statement has only been established for the special class of the single-delayed plants with a zero reference value. Moreover, because of the complexity of the IMC, the objects with a time delay d > 1 having a nonzero setpoint have never been analytically explored in regard to the PC energy context until now. Thus, the newly introduced analytical methods defined in this article allow us to designate the proper forms of σ -inverse-related DOFs that guarantee the minimum-energy PCD for the entire set of LTI multivariable nonsquare systems with the delay d ≥ 1. Moreover, the new original results, supported by numerical examples, strongly contest the well-established control and systems theory canons related to the optimal minimum-energy-originated peculiarity of the MP pseudoinverse.

“…After the solution of defined performance index (3), the general d-state-space-related PC formula is obtained [1] with any time delay d ≥ 1…”

Section: State-space Perfect Control Paradigmmentioning

confidence: 99%

Section: State-space Perfect Control Paradigmmentioning

confidence: 99%

See 1 more Smart Citation

Application of Generalized Inverses in the Minimum-Energy Perfect Control Theory

Feliks

IEEE Trans. Syst. Man Cybern, Syst.

Stanimirović

2023

A New Generalized Θ-Inverse vs. Moore-Penrose Structure: A Comparative Control-Oriented Practical Investigation

2021

IEEE Access

Self Cite

A new non-unique Θ-inverse of non-square polynomial matrices is presented in this paper. It is shown that the above inverse specializes to the unique Moore-Penrose one under several specific assumptions. Due to the existence of some degrees of freedom, the Θ-inverse outperforms the Moore-Penrose inverse in various inverse-related problems covering the multivariable control theory. In many scenarios the former inverse can stabilize a closed-loop plant being unstable by application of the Moore-Penrose formula. Practical and theoretical simulation examples confirm the correctness of the proposed method.

Robust Fractional-Order Perfect Control for Non-Full Rank Plants Described in the Grünwald-Letnikov IMC Framework

Feliks

2021

Fract Calc Appl Anal

The advanced analytical study in the field of fractional-order non-full rank inverse model control design is presented in the paper. Following the recent results in this matter it is certain, that the inverse model control-oriented perfect control law can be established for the non-full rank integer-order systems being under the discrete-time state-space reference with zero value. It is shown here, that the perfect control paradigm can be extended to cover the multivariable non-full rank plants governed by the more general Grünwald-Letnikov discrete-time state-space model. Indeed, the postulated approach significantly reduces both iterative and non-iterative computational effort, mainly derived from the approximation of the Moore-Penrose inverse of the non-full rank matrices to finally be inverted. A prevention provided by the new method excludes the detrimental matrix behavior in the form of singularity, often avoided due to the observed ill-conditioned sensitivity. Thus, the new defined robust fractional-order non-full rank instance of such control strategy, supported by the pole-free mechanism, gives rise to the introduction of the general unified non-full rank perfect control-originated theory. Numerical algorithms with simulation investigation clearly confirm the innovative peculiarities provided by the manuscript.