Generalization of Bass --- Gura Formula for Linear Dynamic Systems with Vector Control

The compact analytic formula of calculating the feedback law (controller matrix) coefficients is developed for solving the synthesis problem of modal controller providing desired pole placement by means of the fully measured state vector in linear dynamic systems with vector control. This formula represents the generalization of the known Bass --- Gura formula, used for synthesizing modal controllers in systems with scalar control, to systems with vector control. The obtained solution is applicable to systems with state-space dimension divisible by the number of control inputs and the matrix composed of the linearly independent first block columns of the Kalman controllability matrix by a number corresponding to the quantity of the mentioned multiplicity is reversible. To use the mentioned formula, it's not required to additionally transfer the described systems of the indicated class to special canonical forms. This formula may be applied to solve both numeric and analytic problems of modal control in mentioned class, independently on a specific ratio of state-vector and control-vector dimensions as well as on existence and multiplicity of real-value poles and complex-conjugate pairs of poles in original and desirable spectrums of state matrix. The examples are considered that prove the possibility of applying the generalized block-matrix Bass --- Gura formula to calculate modal controllers for the described class of systems with vector control

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Discrete Time Adaptive Control of Linear Dynamic Systems with a Two-Segment Piecewise-Linear Asymmetric Nonlinearity

10.23919/acc.1983.4788177 ◽

1983 ◽

Author(s):

Min-Chio Kung ◽

Baxter F. Womack

Keyword(s):

Adaptive Control ◽

Discrete Time ◽

Dynamic Systems ◽

Piecewise Linear ◽

Linear Dynamic Systems ◽

Linear Dynamic ◽

Asymmetric Nonlinearity

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On the Uncertainty Identification for Linear Dynamic Systems Using Stochastic Embedding Approach with Gaussian Mixture Models

Sensors ◽

10.3390/s21113837 ◽

2021 ◽

Vol 21 (11) ◽

pp. 3837

Author(s):

Rafael Orellana ◽

Rodrigo Carvajal ◽

Pedro Escárate ◽

Juan C. Agüero

Keyword(s):

Probability Density Function ◽

Probability Density ◽

Density Function ◽

Dynamic Systems ◽

Gaussian Mixture ◽

Error Model ◽

Fault Detection And Diagnosis ◽

Deterministic Models ◽

Linear Dynamic Systems ◽

Linear Dynamic

In control and monitoring of manufacturing processes, it is key to understand model uncertainty in order to achieve the required levels of consistency, quality, and economy, among others. In aerospace applications, models need to be very precise and able to describe the entire dynamics of an aircraft. In addition, the complexity of modern real systems has turned deterministic models impractical, since they cannot adequately represent the behavior of disturbances in sensors and actuators, and tool and machine wear, to name a few. Thus, it is necessary to deal with model uncertainties in the dynamics of the plant by incorporating a stochastic behavior. These uncertainties could also affect the effectiveness of fault diagnosis methodologies used to increment the safety and reliability in real-world systems. Determining suitable dynamic system models of real processes is essential to obtain effective process control strategies and accurate fault detection and diagnosis methodologies that deliver good performance. In this paper, a maximum likelihood estimation algorithm for the uncertainty modeling in linear dynamic systems is developed utilizing a stochastic embedding approach. In this approach, system uncertainties are accounted for as a stochastic error term in a transfer function. In this paper, we model the error-model probability density function as a finite Gaussian mixture model. For the estimation of the nominal model and the probability density function of the parameters of the error-model, we develop an iterative algorithm based on the Expectation-Maximization algorithm using the data from independent experiments. The benefits of our proposal are illustrated via numerical simulations.

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