Comparison between RST and PID controllers performance of a reduced order model and the original model of a hydraulic actuator dedicated to a semi-active suspension

2016 ◽  
Vol 13 (1) ◽  
pp. 66-71 ◽  
Author(s):  
Saad Babesse ◽  
Djameleddine Ameddah ◽  
Fouad Inel
Keyword(s):  
Transfer Functions ◽  
Linear Time ◽  
Active Suspension ◽  
High Order ◽  
Reduced Order Model ◽  
Hydraulic Actuator ◽  
Order Model ◽  
Content Type ◽  
Reduced Order ◽  
Real Process

Purpose In this paper, an effective method to calculate the reduced-order model (ROM) of high-order linear time-invariant system is elaborated; this is done by evaluating time moments of the original high-order model (HOM). Design/methodology/approach The developed method has been applied to a hydraulic actuator of antiroll bar mechanism dedicated to heavy vehicle semi-active suspension. And as the actuator is a large-scale system; and that in this case, the only control applied is a classical control and with trial and error procedure (like PID), the use of an order reduction method is necessary. Hence, the actuator that has an eighth-order transfer function with uncontrollable states has been approximated by fully controllable second-order model, which is suitable for feedback controllers (RST, LQR […]). The RST control is applied to control the roll angle of the actuator and simulations are carried out to show the effectiveness of the procedure. Findings It is clear that RST shows good tracking as compared to PID. For further work, the given RST controller has a discrete character and can be easily implemented on the real process and then as a further simulation, one can use another controller such as fractional adaptive controller. Originality/value In the recent years, the technological need of modeling order, thus the complexity of the systems, directed the researchers toward the reduction of order of these systems, not only to facilitate the analysis but also to find a suitable approximation of the high-order systems while keeping the same important characteristics as closely as possible. Several methods are available but they fail to give stable transfer functions or important characteristics of the original system.

scholarly journals Sampling-free parametric model reduction of systems with structured parameter variation

2018 ◽  
Author(s):  
Christopher Beattie ◽  
Serkan Gugercin ◽  
Zoran Tomljanović
Keyword(s):  
Model Reduction ◽  
Transfer Functions ◽  
Linear Time ◽  
Mass Matrix ◽  
Reduction Process ◽  
Parametric Model ◽  
Reduced Order Model ◽  
Computationally Efficient ◽  
Order Model ◽  
Reduced Order

We consider a parametric linear time invariant dynamical systems represented in state-space form as $$E \dot x(t) = A(p) x(t) + Bu(t), \\ y(t) = Cx(t),$$ where $E, A(p) \in \mathbb{R}^{n\times n}$, $B\in \mathbb{R}^{n\times m} $ and $C\in \mathbb{R}^{l\times n}$. Here $x(t)\in \mathbb{R}^{n} $ denotes the state variable, while $u(t)\in \mathbb{R}^{m}$ and $y(t)\in \mathbb{R}^{l}$ represent, respectively, the inputs and outputs of the system. We assume that $A(p)$ depends on $k\ll n$ parameters $p=(p_1, p_2, \ldots, p_k)$ such that we may write $$A(p)=A_0+U\,\diag (p_1, p_2, \ldots, p_k)V^T,$$ where $U, V \in \mathbb{R}^{n\times k}$ are given fixed matrices.We propose an approach for approximating the full-order transfer function $H(s;p)=C(s E -A(p))^{-1}B$ with a reduced-order model that retains the structure of parametric dependence and (typically) offers uniformly high fidelity across the full parameter range. Remarkably, the proposed reduction process removes the need for parameter sampling and thus does not depend on identifying particular parameter values of interest. Our approach is based on the classic Sherman-Morrison-Woodbury formula and allows us to construct a parameterized reduced order model from transfer functions of four subsystems that do not depend on parameters, allowing one to apply well-established model reduction techniques for non-parametric systems. The overall process is well suited for computationally efficient parameter optimization and the study of important system properties. One of the main applications of our approach is for damping optimization: we consider a vibrational system described by $$ \begin{equation}\label{MDK} \begin{array}{rl} M\ddot q(t)+(C_{int} + C_{ext})\dot q(t)+Kq(t)&=E w(t),\\ z(t)&=Hq(t), \end{array} \end{equation} $$ where the mass matrix, $M$, and stiffness matrix, $K$, are real, symmetric positive-definite matrices of order $n$. Here, $q(t)$ is a vector of displacements and rotations, while $ w(t) $ and $z(t) $ represent, respectively, the inputs (typically viewed as potentially disruptive) and outputs of the system. Damping in the structure is modeled as viscous damping determined by $C_{int} + C_{ext}$ where $C_{int}$ and $C_{ext}$ represent contributions from internal and external damping, respectively. Information regarding damper geometry and positioning as well as the corresponding damping viscosities are encoded in $C_{ext}= U\diag{(p_1, p_2, \ldots, p_k)} U^T$ where $U \in \mathbb{R}^{n\times k}$ determines the placement and geometry of the external dampers. The main problem is to determine the best damping matrix that is able to minimize influence of the disturbances, $w$, on the output of the system $z$. We use a minimization criteria based on the $\mathcal{H}_2$ system norm. In realistic settings, damping optimization is a very demanding problem. We find that the parametric model reduction approach described here offers a new tool with significant advantages for the efficient optimization of damping in such problems.

scholarly journals Event-triggered Discrete-time Sliding Mode Control for High-order Systems via Reduced-order Model Approach

2020 ◽  
Vol 53 (2) ◽  
pp. 6207-6212
Author(s):  
Kiran Kumari ◽  
Bijnan Bandyopadhyay ◽  
Johann Reger ◽  
Abhisek K. Behera
Keyword(s):  
Sliding Mode Control ◽  
Discrete Time ◽  
Sliding Mode ◽  
High Order ◽  
Reduced Order Model ◽  
Order Model ◽  
Reduced Order ◽  
Mode Control ◽  
Event Triggered ◽  
Model Approach

scholarly journals Order Reduction based on Coulomb’s and Franklin’s laws Algorithm

2021 ◽  
Author(s):  
Ram Kumar ◽  
Afzal Sikander
Keyword(s):  
Large Scale ◽  
Linear Time ◽  
Order Reduction ◽  
Absolute Error ◽  
Search Space ◽  
Engineering Problem ◽  
Reduced Order Model ◽  
Order Model ◽  
Reduced Order ◽  
Linear Time Invariant

Abstract The Coulomb and Franklin laws (CFL) algorithm is used to construct a lower order model of higher-order continuous time linear time-invariant (LTI) systems in this study. CFL is quite easy to implement in obtaining reduced order model of large scale system in control engineering problem as it employs the combined effect of Coulomb’s and Franklin’s laws to find the best values in search space. The unknown coefficients are obtained using the CFLA methodology, which minimises the integral square error (ISE) between the original and proposed ROMs. To achieve the reduced order model, five practical systems of different orders are considered. Finally, multiple performance indicators such as the ISE, integral of absolute error (IAE), and integral of time multiplied by absolute error were calculated to determine the efficacy of the proposed methodology. The simulation results were compared to previously published well-known research.

Model order reduction for quasi-magnetostatic problems

2017 ◽  
Vol 36 (6) ◽  
pp. 1783-1791
Author(s):  
Yuqing Xie ◽  
Lin Li ◽  
Shuaibing Wang
Keyword(s):  
Model Order Reduction ◽  
Order Reduction ◽  
Reduced Order Model ◽  
Order Method ◽  
Computational Accuracy ◽  
Order Model ◽  
Model Order ◽  
Content Type ◽  
Reduced Order ◽  
Magnetostatic Problem

Purpose To reduce the computational scale for quasi-magnetostatic problems, model order reduction is a good option. Reduced-order modelling techniques based on proper orthogonal decomposition (POD) and centroidal Voronoi tessellation (CVT) have been used to solve many engineering problems. The purpose of this paper is to investigate the computational principle, accuracy and efficiency of the POD-based and the CVT-based reduced-order method when dealing with quasi-magnetostatic problems. Design/methodology/approach The paper investigates computational features of the reduced-order method based on POD and CVT methods for quasi-magnetostatic problems. Firstly the construction method for the POD and the CVT reduced-order basis is introduced. Then, a reduced model is constructed using high-fidelity finite element solutions and a Galerkin projection. Finally, the transient quasi-magnetostatic problem of the TEAM 21a model is studied with the proposed reduced-order method. Findings For the TEAM 21a model, the numerical results show that both POD-based and CVT-based reduced-order approaches can greatly reduce the computational time compared with the full-order finite element method. And the results obtained from both reduced-order models are in good agreement with the results obtained from the full-order model, while the computational accuracy of the POD-based reduced-order model is a little higher than the CVT-based reduced-order model. Originality/value The CVT method is introduced to construct the reduced-order model for a quasi-magnetostatic problem. The computational accuracy and efficiency of the presented approaches are compared.

Control of Asymmetric Systems Using Complex Modes

1991 ◽  
Author(s):  
G. W. Fan ◽  
H. D. Nelson
Keyword(s):  
Normal Modes ◽  
Original System ◽  
High Order ◽  
Reduced Order Model ◽  
Order System ◽  
Matrix Transformations ◽  
Linear Quadratic ◽  
Order Model ◽  
Reduced Order ◽  
Complex Modes

Abstract The complex modal approach is introduced for the optimal vibration control (Linear Quadratic Regulator) of high-order nonsymmetric discrete systems. An LQ regulator is designed based on a reduced-order model obtained by neglecting high-frequency complex modes of the original system. The matrix transformations between physical coordinates and complex coordinates are derived. A 52 degree-of-freedom finite element based rotordynamic system is used for illustration. Simulation results show that an LQ regulator based on a reduced-order system obtained by using normal modes of a high-order system with asymmetric models can possibly destabilize the original system i.e., the spill-over problem (Ulsoy, 1984), however, this problem might be avoided by applying complex modes which provides a more accurate reduced-order model than obtained by normal modes. Comparison of the reduced-order models using normal modes and complex modes is presented. Frequency, time transient and steady state responses of the controlled and uncontrolled systems are also shown.

Active Suspension Control Synthesis Using Reduced Order Models That Account for the Time Delay Between Road Inputs

2006 ◽  
Author(s):  
R. Michael Van Auken
Keyword(s):  
Time Delay ◽  
Model Reduction ◽  
Active Suspension ◽  
High Order ◽  
Control Algorithms ◽  
Control Law ◽  
Order Model ◽  
Ground Vehicle ◽  
Reduced Order ◽  
Suspension Control

The control of wheeled ground vehicle suspension systems is well suited for analysis and refinement using multi-input multi-output (MIMO) control law synthesis methods for linear systems. Usually it is necessary and desirable to develop the control algorithms using a reduced order model of the system. Since such vehicles are also characterized by correlated road inputs with time delay between the front and rear wheels, it is also desirable to consider this delay during the model reduction process. If this delay is taken into consideration, then it may be possible to develop low order control algorithms which compensate for the vehicle modes that are disturbed by the road inputs, resulting in improved overall performance. This paper describes the application of model reduction to a model of a ground vehicle for active suspension control law synthesis. The vehicle is described by a high order MIMO model of a “half-car” with four rigid-body degrees of freedom and flexible body modes to account for structural vibration, plus additional states to represent colored noise road disturbance inputs. Fourth order MIMO models suitable for control law synthesis are then determined using internal balancing, taking into consideration the time delay between the front and rear wheels, followed by subsystem elimination. The performance of the vehicle (high order model) with the resulting low order active suspension control laws is then assessed.

scholarly journals Numerical investigation of the POD reduced-order model for fast predictions of two-phase flows in porous media

2019 ◽  
Vol 29 (11) ◽  
pp. 4167-4204 ◽  
Author(s):  
Jingfa Li ◽  
Tao Zhang ◽  
Shuyu Sun ◽  
Bo Yu
Keyword(s):  
Projection Methods ◽  
Reduced Order Model ◽  
Governing Equations ◽  
Order Model ◽  
Numerical Accuracy ◽  
Two Phase ◽  
Content Type ◽  
Reduced Order ◽  
Computational Speed ◽  
Speed Up

Purpose This paper aims to present an efficient IMPES algorithm based on a global model order reduction method, proper orthogonal decomposition (POD), to achieve the fast solution and prediction of two-phase flows in porous media. Design/methodology/approach The key point of the proposed algorithm is to establish an accurate POD reduced-order model (ROM) for two-phase porous flows. To this end, two projection methods including projecting the original governing equations (Method I) and projecting the discrete form of original governing equations (Method II) are respectively applied to construct the POD-ROM, and their distinctions are compared and analyzed in detail. It is found the POD-ROM established by Method I is inapplicable to multiphase porous flows due to its failed introduction of fluid saturation and permeability that locate on the edge of grid cell, which would lead to unphysical results. Findings By using Method II, an efficient IMPES algorithm that can substantially speed up the simulation of two-phase porous flows is developed based on the POD-ROM. The computational efficiency and numerical accuracy of the proposed algorithm are validated through three numerical examples, and simulation results illustrate that the proposed algorithm displays satisfactory computational speed-up (one to two orders of magnitude) without sacrificing numerical accuracy obviously when comparing to the standard IMPES algorithm that without any acceleration technique. In addition, the determination of POD modes number, the relative errors of wetting phase pressure and saturation, and the influence of POD modes number on the overall performances of the proposed algorithm, are investigated. Originality/value 1. Two projection methods are applied to establish the POD-ROM for two-phase porous flows and their distinctions are analyzed. The reason why POD-ROM is difficult to be applied to multiphase porous flows is clarified firstly in this study. 2. A highly efficient IMPES algorithm based on the POD-ROM is proposed to accelerate the simulation of two-phase porous flows. 3. Satisfactory computational speed-up (one to two orders of magnitude) and prediction accuracy of the proposed algorithm are observed under different conditions.

Optimal Output Feedback Control of Asymmetric Systems Using Complex Modes

1993 ◽  
Vol 115 (2) ◽  
pp. 307-313 ◽  
Author(s):  
G. W. Fan ◽  
H. D. Nelson ◽  
M. P. Mignolet
Keyword(s):  
Feedback Control ◽  
Output Feedback ◽  
Output Feedback Control ◽  
High Order ◽  
Reduced Order Model ◽  
Control Procedure ◽  
Order Model ◽  
Complex Mode ◽  
Reduced Order ◽  
Complex Modes

A Linear Quadratic Regulator (LQR)-based least-squares output feedback control procedure using a complex mode procedure is developed for the optimal vibration control of high-order asymmetric discrete system. An LQ Regulator is designed for a reduced-order model obtained by neglecting high-frequency complex modes of the original system. The matrix transformations between physical coordinates and complex mode coordinates are derived. The complex mode approach appears to provide more accurate reduced-order models than the normal mode approach for asymmetric discrete systems. The proposed least-squares output feedback control procedure takes advantage of the fact that a full-state feedback control is possible without using an observer. In addition, the lateral vibration of a high-order rotor system can be effectively controlled by monitoring one single location along the rotor shaft, i.e., the number of measured states can be much less than the number of eigenvectors retained in producing the reduced-order model while acceptable performance of the controller is maintained. The procedure is illustrated by means of a 52 degree-of-freedom finite element based rotordynamic system. Simulation results show that LQ regulators based on a reduced-order model with 12 retained eigenvalues can be accurately approximated by using feedback of four measured states from one location along the rotor shaft. The controlled and uncontrolled transient responses, using various numbers of measured states, of the original high-order system are shown. Comparisons of reduced-order model results using normal modes and complex modes are presented. The spillover problem is discussed for both collocated and noncollocated cases based on this same example.

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