State Delays Extraction in the Fractional-Order State-Space Model
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Fractional-order system
State-space representation
Fractional-order system
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State-space representation
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This article deals with the modeling of dynamical system in state-space representation. The state-space representation is a mathematical model of a physical system with the input, output and state variables composed by first-order differential equations. The state-space representation gives a suitable and compact way to model and analyze systems with multiple inputs and outputs. The article described the methodology for production of state of simple models of mechanical systems.
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State-space model is an efficient tool for describing multiple input multiple output system, but the parameters identification of state-space model is a complicated problem, because in many cases, the parameters and state variables are all unknown in the model. Aiming at the shortcomings of the traditional identification method, in this paper, combined the unknown parameters and state variables of the state space model into a new state variable, then the linear state space model equation can be transformed into a nonlinear equation, and extended Kalman filtering(EKF) algorithm is used to estimate the new state variables. In this way, we can implement double estimates of the unknown parameters and state variables. Doing simulation and analysis with Matlab, the results show that the method can realize parameter identification and state estimation of state-space model effectively, which has higher precision and accuracy.
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These notes are devoted to some methods used in the fractional calculus (theory of integration and differentiation of an arbitrary order) and to application of the fractional calculus to modelling and control of dynamical systems.
Fractional-order system
Time-scale calculus
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Fractional-order system
Fractional programming
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This paper uses several examples to show how the econometrics program RATS can be used to analyze state space models. It demonstrates Kalman filtering and smoothing, estimation of hyperparameters, unconditional and conditional simulation. It also provides a more complicated example where a dynamic simultaneous equations model is transformed into a proper state space representation and its unknown parameters are estimated.
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The state space representation of dynamic systems is a representation via a first order matrix differential equation. We shall introduce this by means of some examples. Inspec keywords: differential equations; state-space methods Other keywords: first order matrix differential equation; dynamic system; state space analysis; state space representation Subjects: Control system analysis and synthesis methods
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Many real dynamic systems are better characterized using a non-integer order dynamic model based on fractional calculus or, differentiation or integration of non-integer order. Traditional calculus is based on integer order differentiation and integration. The concept of fractional calculus has tremendous potential to change the way we see, model, and control the nature around us. Denying fractional derivatives is like saying that zero, fractional, or irrational numbers do not exist. In this paper, we offer a tutorial on fractional calculus in controls. Basic definitions of fractional calculus, fractional order dynamic systems and controls are presented first. Then, fractional order PID controllers are introduced which may make fractional order controllers ubiquitous in industry. Additionally, several typical known fractional order controllers are introduced and commented. Numerical methods for simulating fractional order systems are given in detail so that a beginner can get started quickly. Discretization techniques for fractional order operators are introduced in some details too. Both digital and analog realization methods of fractional order operators are introduced. Finally, remarks on future research efforts in fractional order control are given.
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