Optimal control of second‐order and high‐order descriptor systems

Abstract

This study investigates the optimal control problem of second-order descriptor systems. The optimal control is characterized by using a new second-order generalized Riccati equation, which is directly derived in terms of the original coefficient matrices of the system. Under some assumption conditions, applying matrix's singular value decomposition and matrix transformation, a nonlinear generalized Riccati matrix equation is transformed into a linear matrix equation, and the optimal control gain can be determined by solving the linear matrix equation. Furthermore, relevant results are also extended to the optimal control problems for the high-order descriptor system. Finally, several simulation examples and the comparison with the existing linearization method are provided to illustrate the effectiveness of the developed approach. KEYWORDSgeneralized Riccati equation, optimal control, second-order singular system, singular value decomposition INTRODUCTIONDescriptor systems (or singular systems, differential-algebraic equations) are more natural mathematical descriptions than normal state space systems for the modeling of many real-world physical systems, including electrical circuit systems, economic system, mechanical systems, and other areas. 1,2 In the past decades, many significant basic theories of normal state space systems have been fluently generalized to descriptor systems, for instance, controllability and observability, 3,4 Lyapunov stability, 5,6 and H ∞ control. 7,8 Optimal control is an important branch of the modern control theory, in the last decades, optimal control problems of first-order descriptor systems have been studied extensively, many significant results have been achieved (see other works 9-14 ). Thus, it can be seen that the research on the optimal control problem of first-order descriptor systems has already developed a relatively mature theory.As the research moves along, many scholars find that in practical applications, a large number of physical systems are more suitable to model by second-order or high-order differential algebraic equations. [15][16][17] Second-order systems have received wide attention from different fields in mechanical vibration systems, 18-20 robotics control, and control of large flexible space structures 2,19 in the last few decades. Meanwhile, a large number of works have been published in the field of the analysis and synthesis of second-order descriptor systems, such as controllability and observability conditions, 15 stability conditions, 21,22 robust pole placement problem of second-order descriptor systems with velocity and acceleration feedback, 23-25 partial eigenstructure assignment of second-order undamped vibration systems, 26 output-feedback control, 27,28 and robust H ∞ control of uncertain mechanical systems. 29 Recently, Moysis et al 30 have determined the state response of the linear constrained mechanical system described by a high-order matrix differential equations.

H∞ model following control of linear parameter‐varying descriptor systems

“…The vectors u in (t) ∈ R m is the input of the filter (7), z o (t) ∈ R p is the output of the filter (8), and y o (t) ∈ R q is the output of the filter (9). The LPV descriptor system (4)-( 6) with the low pass filters ( 7)-( 9) can be written as ( 1)- (3), where…”

“…The problem of designing different types of controllers for different classes of descriptor systems has been considered by many researchers. Examples of these controllers are state‐feedback, 2,3 linear quadratic state‐feedback, 4 state‐feedback, 5 state‐feedback, 6 robust state‐feedback, 7 output‐feedback, 8,9 dynamic output‐feedback (DOF), 10 output‐feedback, 11 DOF, 12‐14 DOF, 15 mixed / DOF, 16 dissipative DOF, 17 observer based controller, 18 decentralized static output‐feedback, 19 adaptive controller, 20 Hamiltonian controller, 21 neural network controller, 22 static‐output preview controller, 23 fuzzy controller, 24 sliding‐mode controller, 25 and model predictive controller 26 …”

Numerical approach for partial eigenstructure assignment problems in singular vibrating structure using active control

“…It is known that the singular second-order vibrating structure is widely existed in engineering practice (Abdelaziz, 2015; Duan, 2010; Yu and Zhang, 2016; Zhang and Zhang, 2019). The vibrating structure with the mass matrix M being sparse is often considered as a singular second-order system in modeling.…”

Minimum norm partial eigenstructure assignment problems in high‐order system via feedback control

“…High‐order systems often have complex‐valued eigenvalues and eigenvectors, even if the coefficient matrices are assumed to be real. One may obtain a real‐valued spectral decomposition of characteristic polynomials and by using a suitable procedure given in References 4. As a result, the coefficient matrices of system () can be characterized by employing the real representation of eigenstructure information when all eigenvalues are assumed to be simple.…”

“…with A i ∈ R n×n for i = 0 ∶ k. High-order systems are regarded as more general description forms of linear systems, and are developed from traditional state space systems and descriptor systems. [1][2][3][4][5] The multiagent systems, power systems, and vibrating structure can be modeled as second-order systems, while the three-axis dynamic flight motion simulator systems and the cantilever beam models are often described, respectively, as third-order and fourth-order systems using finite element technique. [6][7][8][9][10][11][12][13] If k = 2, then the systems arise mainly in vibration structure.…”