Reduced Order Controller Design for Nonlinear Systems With Periodic Coefficients

2009 ◽  
Author(s):  
Amit P. Gabale ◽  
Subhash C. Sinha
Keyword(s):  
Controller Design ◽  
Linear Part ◽  
Eigenvalue Decomposition ◽  
Order System ◽  
Time Varying ◽  
Periodic Coefficients ◽  
Reduced Order ◽  
Periodic Linear ◽  
Time Invariant ◽  
Time Periodic

This study provides a methodology for reduced order controller design for nonlinear dynamic systems with time-periodic coefficients. System equations are represented by quasi-linear differential equations in state space, containing a time-periodic linear part and nonlinear monomials of states with periodic coefficients. The Lyapunov-Floquet (L-F) transformation is used to transform the time-varying linear part of the system into a time-invariant form. Eigenvalue decomposition of the time-invariant linear part can then be used to identify the dominant/ non-dominant dynamics of the system. The non-dominant states of the system are expressed as a nonlinear, time-periodic, manifold relationship in terms of the dominant states. As a result, the original large system can be expressed as a lower order system represented only by the dominant states. A reducibility condition is derived to provide conditions under which a nonlinear order reduction is possible. Then a proper coordinate transformation and state feedback can be found under which the reduced order system is transformed into a linear, time-periodic, closed-loop system. This permits the design of a time-varying feedback controller in linear space to guarantee the stability of the system. The proposed methodology is illustrated by designing a reduced order controller for a 4-dof, inverted pendulum subjected to a periodic follower force. Treatment for the time-invariant case is also included as a subset of the problem.

Order Reduction and Control of Large-Scale Linear Time Periodic Dynamical Systems

1999 ◽  
Author(s):  
Venkatesh Deshmukh ◽  
S. C. Sinha
Keyword(s):  
Large Scale ◽  
Linear Time ◽  
Parametric Instability ◽  
Order Reduction ◽  
System Stability ◽  
Order System ◽  
Control Laws ◽  
Reduced Order ◽  
Time Invariant ◽  
Time Periodic

Abstract This paper provides methodology for designing reduced order controllers for large-scale, linear systems represented by differential equations having time periodic coefficients. The linear time periodic system is first converted into a form in which the system stability matrix is time invariant. This is achieved by the application of Liapunov-Floquet transformation. Then a system called an auxiliary system is constructed which is a completely time invariant. Order reduction algorithms are applied to this system to obtain a reduced order system. The control laws are calculated for the reduced order system by minimizing the least square error between the auxiliary and the transformed system. These control laws when transformed back to time varying domain provide the desired control action. The schemes formulated are illustrated by designing full state feedback and output feedback controllers for a five mass inverted pendulum exhibiting parametric instability.

On the Analysis of Time-Periodic Nonlinear Hamiltonian Dynamical Systems

1995 ◽  
Author(s):  
Eric A. Butcher ◽  
S. C. Sinha
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Normal Form ◽  
Linear Part ◽  
Normal Form Theory ◽  
Mathieu’S Equation ◽  
Form Theory ◽  
Hamiltonian Dynamical Systems ◽  
Time Invariant ◽  
Time Periodic

Abstract In this paper, some analysis techniques for general time-periodic nonlinear Hamiltonian dynamical systems have been presented. Unlike the traditional perturbation or averaging methods, these techniques are applicable to systems whose Hamiltonians contain ‘strong’ parametric excitation terms. First, the well-known Liapunov-Floquet (L-F) transformation is utilized to convert the time-periodic dynamical system to a form in which the linear pan is time invariant. At this stage two viable alternatives are suggested. In the first approach, the resulting dynamical system is transformed to a Hamiltonian normal form through an application of permutation matrices. It is demonstrated that this approach is simple and straightforward as opposed to the traditional methods where a complicated set of algebraic manipulations are required. Since these operations yield Hamiltonians whose quadratic parts are integrable and time-invariant, further analysis can be carried out by the application of action-angle coordinate transformation and Hamiltonian perturbation theory. In the second approach, the resulting quasilinear time-periodic system (with a time-invariant linear part) is directly analyzed via time-dependent normal form theory. In many instances, the system can be analyzed via time-independent normal form theory or by the method of averaging. Examples of a nonlinear Mathieu’s equation and coupled nonlinear Mathieu’s equations are included and some preliminary results are presented.

Floquet Experimental Modal Analysis for System Identification of Linear Time-Periodic Systems

2007 ◽  
Author(s):  
Matthew S. Allen
Keyword(s):  
System Identification ◽  
Linear Time ◽  
Time Varying ◽  
Response Model ◽  
State Matrix ◽  
Linear Time Invariant ◽  
Response Data ◽  
Time Varying Systems ◽  
Time Invariant ◽  
Time Periodic

A variety of systems can be faithfully modeled as linear with coefficients that vary periodically with time or Linear Time-Periodic (LTP). Examples include anisotropic rotorbearing systems, wind turbines, satellite systems, etc… A number of powerful techniques have been presented in the past few decades, so that one might expect to model or control an LTP system with relative ease compared to time varying systems in general. However, few, if any, methods exist for experimentally characterizing LTP systems. This work seeks to produce a set of tools that can be used to characterize LTP systems completely through experiment. While such an approach is commonplace for LTI systems, all current methods for time varying systems require either that the system parameters vary slowly with time or else simply identify a few parameters of a pre-defined model to response data. A previous work presented two methods by which system identification techniques for linear time invariant (LTI) systems could be used to identify a response model for an LTP system from free response data. One of these allows the system’s model order to be determined exactly as if the system were linear time-invariant. This work presents a means whereby the response model identified in the previous work can be used to generate the full state transition matrix and the underlying time varying state matrix from an identified LTP response model and illustrates the entire system-identification process using simulated response data for a Jeffcott rotor in anisotropic bearings.

Observer and Controller Design Using Time-Varying Gains: Duality and Distinctions

2004 ◽  
Vol 127 (2) ◽  
pp. 267-274
Author(s):  
Vladimir Polotski
Keyword(s):  
State Feedback ◽  
Closed Loop ◽  
Dual Problem ◽  
Controller Design ◽  
Time Varying ◽  
Feedback Controllers ◽  
Observer Error ◽  
Time Varying Systems ◽  
Error Dynamics ◽  
Time Invariant

Stabilization of linear systems by state feedback is an important problem of the controller design. The design of observers with appropriate error dynamics is a dual problem. This duality leads, at first glance, to the equivalence of the responses in the synthesized systems. This is true for the time-invariant case, but may not hold for time-varying systems. We limit ourselves in this work by the situation when the system itself is time invariant, and only the gains are time varying. The possibility of assigning a rapidly decaying response without peaking is analyzed. The solution of this problem for observers using time-varying gains is presented. Then we show that this result cannot be obtained for state feedback controllers. We also analyze the conditions under which the observer error dynamics and the response of the closed loop time-varying controllers are equivalent. Finally we compare our results to recently proposed observer converging in finite time and Riccati-based continuous observer with limited overshoots.

Observer Design for Nonlinear Systems With Time-Periodic Coefficients via Normal Form Theory

2009 ◽  
Vol 4 (3) ◽  
Author(s):  
Yandong Zhang ◽  
S. C. Sinha
Keyword(s):  
Normal Form ◽  
Design Problem ◽  
Controller Design ◽  
Design Method ◽  
Observer Design ◽  
Periodic Systems ◽  
Normal Form Theory ◽  
Periodic Coefficients ◽  
System States ◽  
Time Periodic

For most complex dynamic systems, it is not always possible to measure all system states by a direct measurement technique. Thus for dynamic characterization and controller design purposes, it is often necessary to design an observer in order to get an estimate of those states, which cannot be measured directly. In this work, the problem of designing state observers for free systems (linear as well as nonlinear) with time-periodic coefficients is addressed. It is shown that, for linear periodic systems, the observer design problem is the duality of the controller design problem. The state observer is constructed using a symbolic controller design method developed earlier using a Chebyshev expansion technique where the Floquet multipliers can be placed in the desired locations within the unit circle. For nonlinear time-periodic systems, an observer design methodology is developed using the Lyapunov–Floquet transformation and the Poincaré normal form technique. First, a set of time-periodic near identity coordinate transformations are applied to convert the nonlinear problem to a linear observer design problem. The conditions for existence of such invertible maps and their computations are discussed. Then the local identity observers are designed and implemented using a symbolic computational algorithm. Several illustrative examples are included to show the effectiveness of the proposed methods.

Observer Design for Nonlinear Systems With Time Periodic Coefficients Via Normal Form Theory

2007 ◽  
Author(s):  
Yandong Zhang ◽  
S. C. Sinha
Keyword(s):  
Normal Form ◽  
Design Problem ◽  
Controller Design ◽  
Linear Time ◽  
Design Method ◽  
Observer Design ◽  
Periodic Systems ◽  
Normal Form Theory ◽  
Periodic Coefficients ◽  
Time Periodic

For most complex dynamic systems, it is not possible to measure all system states in a direct fashion. Thus for dynamic characterization and controller design purposes, it is often necessary to design an observer in order to get an estimate of those states which cannot be measured directly. In this work, the problem of designing state observers for free systems with time periodic coefficients is addressed. For linear time-periodic systems, it is shown that the observer design problem is the duality of the controller design problem. The state observer is constructed using a symbolic controller design method developed earlier using the Chebyshev expansion technique. For the nonlinear time periodic systems, the observer design is investigated using the Poincare´ normal form technique. The local identity observer is designed by using a set of near identity coordinate transformations which can be constructed in the ascending order of nonlinearity. These observer design methods are implemented using a symbolic computational algorithm and several illustrative examples are given to show the effectiveness of the methods.

Reduced-Order Model-Based Feedback Control of Flow Over an Obstacle Using Center Manifold Methods

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
Coşku Kasnakoğlu ◽  
R. Chris Camphouse ◽  
Andrea Serrani
Keyword(s):  
Boundary Control ◽  
Center Manifold ◽  
Controller Design ◽  
Tracking Error ◽  
Reduced Order Model ◽  
Order System ◽  
Linear Quadratic ◽  
Order Model ◽  
Control Effort ◽  
Reduced Order

In this paper, we consider a boundary control problem governed by the two-dimensional Burgers’ equation for a configuration describing convective flow over an obstacle. Flows over obstacles are important as they arise in many practical applications. Burgers’ equations are also significant as they represent a simpler form of the more general Navier–Stokes momentum equation describing fluid flow. The aim of the work is to develop a reduced-order boundary control-oriented model for the system with subsequent nonlinear control law design. The control objective is to drive the full order system to a desired 2D profile. Reduced-order modeling involves the application of an L2 optimization based actuation mode expansion technique for input separation, demonstrating how one can obtain a reduced-order Galerkin model in which the control inputs appear as explicit terms. Controller design is based on averaging and center manifold techniques and is validated with full order numerical simulation. Closed-loop results are compared to a standard linear quadratic regulator design based on a linearization of the reduced-order model. The averaging∕center manifold based controller design provides smoother response with less control effort and smaller tracking error.

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