Optimal Control for Mechanical Vibration Systems Based on Second-Order Matrix Equations

Zhang, Jia Fan

doi:10.1006/mssp.2001.1441

Cited by 26 publications

(14 citation statements)

References 4 publications

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“…However, most of the above mentioned results mainly focus on the stability analysis of second‐order systems, very few studies have been done about the optimal control design of second‐order systems under a given quadratic performance index in the current works. The relevant articles published about the optimal control problem of second‐order linear systems are those of Ram and Inman and Zhang . In the work of Ram and Inman, the control input matrix is required to be an invertible square matrix, which should be impractical in applications.…”

Section: Introductionmentioning

confidence: 99%

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

Zhang

Liu

2019

Optim Control Appl Methods

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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.

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Section: Introductionmentioning

confidence: 99%

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

Zhang

Liu

2019

Optim Control Appl Methods

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“…Second-order systems capture the dynamic behavior of many natural phenomena, and have found applications in many fields, such as vibration and structural analysis ([1]- [3]), aerospace control ([4]- [7]), flexible structures ( [8], [9]), robotic systems ([10]- [12]), etc. However, when the actuator systems or the sensor systems involved in these second-order systems are modelled with a dynamical order not less than 1, then the dynamical models of these systems can be expressed by high-order systems.…”

Section: Introductionmentioning

confidence: 99%

Direct parametric control of fully-actuated high-order nonlinear systems—Normal case

Duan

2014

Proceeding of the 11th World Congress on Intelligent Control and Automation

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Full-actuation of a dynamical system really provides great potential for system control, and yet this potential is seldom utilized or even recognized. In this paper, the direct parametric approach for fully-actuated second-order nonlinear systems recently proposed is generalized to the case of fullyactuated high-order nonlinear systems. It is revealed that, with this proposed direct parametric approach, a fully-actuated system, no matter linear or nonlinear, can actually be turned into a constant linear system with desire eigenstructure by a state proportional plus derivative feedback controllers, and a general complete parametric expression for all such controllers are established based on the solution to a type high-order Sylvester matrix equations. What is more, in such a realization the approach also provides all the degrees of freedom which may be further utilized to improve the system performance.Index Terms-Fully-actuated high-order systems, direct parametric approach, eigenstructure, degree of freedom, nonlinear systems.

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“…Second-order systems capture the dynamic behavior of many natural phenomena, and have found applications in many fields, such as vibration and structural analysis ([1]- [3]), aerospace control ([4]- [7]), flexible structures ( [8], [9]), robotic systems ([10]- [12]), etc. Among most reported results, controls of such practical systems are carried out by using the first-order system framework.…”

Section: Introductionmentioning

confidence: 99%

Direct parametric control of fully-actuated second-order nonlinear systems—The normal case

Duan

2014

Proceedings of the 33rd Chinese Control Conference

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Fully-actuated second-order systems occur in many applications, such as robotics systems, aircraft systems, mechanical systems etc. Yet what does a fully-actuated system offer? This paper proposes a direct parametric approach which solves the control of a fully-actuated second-order system completely. It is revealed that, with this proposed approach, fully-actuated systems, no matter linear or nonlinear, can actually be turned into a constant linear system with desire eigenstructure by state proportional plus derivative feedback. What is more, in such a realization the approach also provides all the degrees of freedom which may be further utilized to improve the system performance. Direct general complete parametrization of such a controller is proposed based on the solution to a type second-order Sylvester matrix equations, application to a robotic system shows the great convenience of the proposed direct parametric approach.

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Optimal Control for Mechanical Vibration Systems Based on Second-Order Matrix Equations

Cited by 26 publications

References 4 publications

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

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

Direct parametric control of fully-actuated high-order nonlinear systems—Normal case

Direct parametric control of fully-actuated second-order nonlinear systems—The normal case

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Optimal Control for Mechanical Vibration Systems Based on Second-Order Matrix Equations

Cited by 26 publications

References 4 publications

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

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

Direct parametric control of fully-actuated high-order nonlinear systems&#x2014;Normal case

Direct parametric control of fully-actuated second-order nonlinear systems&#x2014;The normal case

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Direct parametric control of fully-actuated high-order nonlinear systems—Normal case

Direct parametric control of fully-actuated second-order nonlinear systems—The normal case