Multivariable analytic interpolation with complexity constraints: A modified Riccati approach

Cui, Yufang; Lindquist, Anders

doi:10.1109/cdc40024.2019.9030074

Cited by 5 publications

(3 citation statements)

References 36 publications

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“…This section illustrates the approach to address the analytic interpolation problem (18) employing the Covariance Extension Equation [19], [20], [21]. Without loss of generality, we set z 1 = 0 and W 1 = 1 2 I. Consequently, F (z) can be expressed as:…”

Section: Problemmentioning

confidence: 99%

“…In this paper, we build upon the foundational work of BK Ghosh, who effectively transformed the scalar case simultaneous stabilization problem into a Nevanlinna-Pick interpolation problem. We employ a more comprehensive interpolation approach based on our prior research on a Riccati-type method for analytic interpolation [20], [21], which is built upon algorithms for the partial stochastic realization problem [24], [25], [27], [28] and on [29]. This approach enables us to solve situations with derivative constraints and parameterize all potential solutions using a matrix polynomial.…”

Section: Introductionmentioning

confidence: 99%

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The Covariance Extension Equation: A Riccati-Type Approach to Analytic Interpolation

Cui

Lindquist

2022

IEEE Trans. Automat. Contr.

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In this paper, we tackle the significant challenge of simultaneous stabilization in control systems engineering, where the aim is to employ a single controller to ensure stability across multiple systems. We delve into both scalar and multivariable scenarios. For the scalar case, we present the necessary and sufficient conditions for a single controller to stabilize multiple plants and reformulate these conditions to interpolation constraints, which expand Ghosh's results by allowing derivative constraints. Furthermore, we implement a methodology based on a Riccatitype matrix equation, called the Covariance Extension Equation. This approach enables us to parameterize all potential solutions using a monic Schur polynomial. Consequently, we extend our result to the multivariable scenario and derive the necessary and sufficient conditions for a group of m × m plants to be simultaneously stabilizable, which can also be solved by our analytic interpolation method. Finally, we construct four numerical examples, showcasing the application of our method across various scenarios encountered in control systems engineering and highlighting its ability to stabilize diverse systems efficiently and reliably.

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Section: Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Covariance Extension Equation: A Riccati-Type Approach to Analytic Interpolation

Cui

Lindquist

2022

IEEE Trans. Automat. Contr.

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“…This idea was then used in [22] to attach the general problem presented above. A first attempt to generalize this method to multivariable analytic interpolation problems was then made in [23], and we shall pursue this inquiry in this paper.…”

Section: Introductionmentioning

confidence: 99%

The Covariance Extension Equation: A Riccati-type Approach to Analytic Interpolation

Cui¹,

Lindquist²

2020

Preprint

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Analytic interpolation problems with rationality and derivative constraints are ubiquitous in systems and control. This paper provides a new method for such problems, both in the scalar and matrix case, based on a non-standard Riccatitype equation. The rank of the solution matrix is the same as the degree of the interpolant, thus providing a natural approach to model reduction. A homotopy continuation method is presented and applied to some problems in modeling and robust control. We also address a question on the positive degree of a covariance sequence originally posed by Kalman.

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