Solving minimum norm problems using penalty functions and the gradient method

Hui, Stefen; Lillo, Walter E.; Żak, Stanisław H.

doi:10.1016/0005-1098(94)00072-q

Cited by 6 publications

(3 citation statements)

References 7 publications

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“…Nevertheless, in nonsquare right-invertible systems, i.e., full-rank plants with more input than output variables, we can obtain the desirable object-originated properties by involving nonunique generalized nonsquare right inverses into the PC design (PCD) processes [4], [25], [26], [27]. Nonetheless, in such a scenario, the broadly known and unique minimum-norm right T-inverse, which satisfies the four Moore-Penrose (MP) equations, has commonly been used in the well-established worldwide scientific literature since it has been treated as the optimal one, in general [28], [29], [30], [31], [32], [33], [34], [35].…”

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confidence: 99%

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

Feliks

Hunek

Stanimirović

2023

IEEE Trans. Syst. Man Cybern, Syst.

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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.

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confidence: 99%

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

Feliks

Hunek

Stanimirović

2023

IEEE Trans. Syst. Man Cybern, Syst.

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“…This can sometimes be inferred from the nature of the polynomial h(x). For example, when h(x) is 46th IEEE CDC, New Orleans, USA, Dec. [12][13][14]2007 FrA06.4…”

Section: The Infimum Of a Rational Functionmentioning

confidence: 99%

“…• The minimum norm problem, which is investigated in the literature intensively, turns out to be equivalent to finding the global optimum of a polynomially constrained optimization problem [13].…”

Section: Introductionmentioning

confidence: 99%

Rational optimization using sum-of-squares techniques

Lavaei

Sojoudi

Aghdam

2007

2007 46th IEEE Conference on Decision and Control

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Abstract-Motivated by many control applications, this paper deals with the global solutions of unconstrained optimization problems. First, a simple SOS method is presented to find the infimum of a polynomial, which can be handled efficiently using the relevant software tools. The main idea of this method is to introduce a perturbation variable whose approaching to zero results in a solution with any arbitrary precision. The proposed technique is then extended to the case of rational functions. The primary advantages of this approach over the existing ones are its simplicity and capability of treating problems for which the existing methods are not efficient, as demonstrated in three numerical examples.

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