A new set of invariants for linear systems--Application to reduced order compensator design (original) (raw)
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Model order reduction of linear time invariant systems
Advances in Radio Science, 2008
This paper addresses issues related to the order reduction of systems with multiple input/output ports. The order reduction is divided up into two steps. The first step is the standard order reduction method based on the multipoint approximation of system matrices by applying Krylov subspace. The second step is based on the rejection of the weak part of a system. To recognise the weak system part, Lyapunov equations are used. Thus, this paper introduces efficient solutions of the Lyapunov equations for port to port subsystems.
Structure Preserving Model Order Reduction of Linear Time-Invariant Control Systems
IFAC Proceedings Volumes, 2012
In this contribution a survey will be presented on the requirements of projection-based procedures of model order reduction of large-scale linear time-invariant systems to preserve certain structural properties like asymptotic stability, complete controllability, or complete observability. Also "algebraic" properties are discussed like steady-state accuracy, vanishing equation residuum, or separation of controller/observer design. The conditions on the projection matrices are shown explicitly. Some of these conditions coincide with respect to different structural properties. Additionally, it can be shown that truncation methods which are based on decoupling either of eigenmodes or of controllability and observability Gramian matrices preserve asymptotic stability, complete controllability, and complete observability always.
A New Technique For Reduced-Order Modelling of Linear Time-Invariant System
IETE Journal of Research, 2017
In this paper, a new technique for order reduction of linear time-invariant systems is presented. This technique is intended for both single-input single-output (SISO) and multi-input multi-output (MIMO) systems. Motivated by other reduction techniques, the new proposed reduction technique is based on modified pole clustering and factor division algorithm with the objective of obtaining a stable reduced-order system preserving all essential properties of the original system. The new technique is illustrated by three numerical examples which are considered from the literature. To evaluate the superiority and robustness of the new technique, the results of the proposed technique are compared with other well-known and recently developed order-reduction techniques like Routh approximation and Big Bang-Big Crunch algorithm. The comparison of performance indices shows the efficiency and powerfulness of the new technique.
The main objective of this paper, aims to apply model order reduction on a large scale system and design a Linear Quadratic Regulator (LQR) based controller, to analyse the performance indices, in the time and frequency domains. Control aspects of large scale systems (models with very high order) are a major concern, in the field of control systems. The order of the designed controller must be close to the order of the large scale system, or even more in most cases. As the order of the controller increases the control aspects of the system, it becomes even more complex. Evidently, there are many model order reduction techniques, that reduce the order of the higher order system, without losing the predominant characteristics. A linear quadratic regulator based design is an optimization tool, to derive an optimal controller by minimizing the cost function, based on the two weighting matrices Q and R, which weigh the state vector and the system input, respectively. The step, impulse and the frequency responses of the system with LQR controller are simulated in MatLab. In this paper, a single-input-single-output system (SISO) is considered, nevertheless due to the compatibility of LQR controller, with the state space equations, this study may be extended to multi-input-multi-output (MIMO) systems, provided the model order reduction techniques are chosen appropriately.
Circuits, Systems, and Signal Processing, 2019
A new model reduction method for the simplification and design of a controller for the linear time-invariant systems is proposed. An improved generalized pole clustering algorithm is employed in the proposed technique for obtaining the denominator of the reduced model. The numerator is computed with a simple mathematical technique available in the literature. The proposed method guarantees the stability in the reduced plant given that the full-order plant is stable and also retains the fundamental characteristics of the original model in the approximated one. This reduced model has been used for the design of compensator for the large-scale original plant by using a new algorithm. The compensator obtained by using the reduced model gives the approximately same time domain specification as compensator obtained by using large-scale original system, and the design of compensator by using the reduced model is comparatively easier. The results of the proposed algorithm are compared with existing methods of reduced-order modeling which show improvement in the performance error indices, time response characteristics and time domain specifications. The validity, effectiveness and superiority of the proposed technique have been demonstrated through some standard numerical examples.
On the use of reduced-order models in output feedback design of discrete systems
Automatica, 1985
The problem of designing output feedback controllers for discrete systems using reduced-order models is investigated in this paper. It is shown that neglecting nondominant modes does not generally guarantee the stability of the closed-loop system. Certain special structures have been identified for which the use of static and dynamic feedback controllers can provide a stabilizing effect. Examples are presented to illustrate these structures.
In this paper, to find a lower order model equivalent to a given higher order system, a simple method based on T.C Hsia's approximation method is proposed. To examine the accuracy of the proposed method, it is compared with the Hutton and Fried-land's approach which uses α and β parameters. To demonstrate the correctness and efficacy of the proposed method, a fourth order system is reduced to first and second order using both the methods. The various performance indices of the reduced systems, obtained from the graphs plotted in Matlab platform are compared and analysed.
Balanced model order reduction for systems depending on a parameter
2016
We provide an analytical framework for balanced realization model order reduction of linear control systems which depend on an unknown parameter. Besides recovering known results for the first order corrections, we obtain explicit novel expressions for the form of second order corrections for singular values and singular vectors. The final result of our procedure is an order reduced model which incorporates the uncertain parameter. We apply our algorithm to the model order reduction of a linear system of masses and springs with parameter dependent coefficients.
Integrated optimization of structures and LQR control systems for reduced order models
Proceedings of the 16th IFAC World Congress, 2005, 2005
In this study, integrated optimum design of structures and control systems is studied by using reduced order models. The structures and controllers are optimized simultaneously and successively. Since the degree-of-freedom (DOF) for structures is very large in practice, model order reduction techniques have to be employed at every controller design iteration during optimization in integrated optimum design approaches, that increase the CPU time and involves modeling errors. In this study, Subspace Based Identification (SBI) method is used as a model order reduction technique in frequency domain. It is shown that simultaneous optimization of structures and controllers by using LQR formulations can be achieved by an equivalent decoupled optimization problem where structures are optimized by shaping the structural singular values and following any control law of interest can be designed. Decoupled optimization of structures and controllers has certain advantages, especially for structures having large DOF.
Direct controller order reduction by identification in closed loop
Automatica, 2001
The paper 1 addresses the problem of directly estimating the parameters of a reduced order digital controller using a closed loop type identification algorithm. The algorithm minimizes the closed loop plant input error between the nominal closed loop system and the closed loop system using the reduced order controller. It is assumed that a plant model (if necessary validated in closed loop with the nominal controller) is available. One of the original features of this approach is that it can use either simulated or real data. The frequency bias distribution of the parameter estimates shows that the reduced order controller maintains the critical performance of the nominal closed loop system. A theoretical analysis is provided. Validation tests are proposed. Experimental results, obtained on an active suspension, illustrate the performance of the proposed algorithms.