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.

Control of Nonlinear Time-Periodic Systems Via Feedback Linearization

2005 ◽  
Author(s):  
Yandong Zhang ◽  
S. C. Sinha
Keyword(s):  
Control System ◽  
Periodic System ◽  
Feedback Linearization ◽  
Linear Method ◽  
Sufficient Conditions ◽  
Periodic Systems ◽  
Computation Method ◽  
Normal Form Theory ◽  
Periodic Coefficients ◽  
Time Periodic

The problem of designing controllers for nonlinear time periodic systems is addressed. The idea is to find proper coordinate transformations and state feedback under which the original system can be (approximately) transformed into a linear control system. Then a controller can be designed using the well-known linear method to guarantee the stability of the system. We propose two approaches for the feedback linearization of the nonlinear time periodic system. The first approach is designed to achieve local control of nonlinear systems with periodic coefficients desired to be driven either to a periodic orbit or to a fixed point. In this case the system equations can be represented by a quasi-linear system containing nonlinear monomials with periodic coefficients. Using near identity transformations and normal form theory, the original close loop problem is approximately transformed into a linear time periodic system with unknown gains. Then by using a symbolic computation method, the Floquet multipliers are placed in the desired locations in order to determine the control gains. We also give the sufficient conditions under which the system is feedback linearizable up to the rth order. The second approach is a generalization of the classical exact feedback linearization method for autonomous systems but applicable to general time-periodic affine systems. By defining a time-dependent Lie operator, the input-output nonlinear time periodic problem is transformed into a linear autonomous problem for which control system can be designed easily. A sufficient condition under which the system is feedback linearizable is also given.

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.

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.

scholarly journals Design Framework for Achieving Guarantees with Learning-Based Observers

Energies ◽  
2021 ◽  
Vol 14 (8) ◽  
pp. 2039
Author(s):  
Balázs Németh ◽  
Tamás Hegedűs ◽  
Péter Gáspár
Keyword(s):  
Controller Design ◽  
Design Method ◽  
Path Following ◽  
Front Wheel ◽  
Observer Design ◽  
Observation Error ◽  
Lateral Velocity ◽  
Controlled System ◽  
Velocity Signal ◽  
Dynamic Simulator

The paper proposes a novel framework for state observer design, in which learning-based observers are incorporated. The aim of the method is to provide a framework, which is able to guarantee the limitation of the observation error, even if the error of the learning-based observer under all scenarios cannot be verified. The framework is based on the robust H∞ design method, which is able to provide guarantees on the resulted observer. Moreover, the observer design process is extended with a controller design, which leads to a joint robust H∞ controller-observer design. In this paper the proposed method is applied on a vehicle control problem, such as lateral path following. In this problem the goal of the observer is to provide an accurate lateral velocity signal for the vehicle, which is used in the controlled system for the generation of front wheel steering angle. The effectiveness of the method is illustrated through simulation examples on high-fidelity vehicle dynamic simulator CarMaker.

Development of a Feedback Linearization Technique for Parametrically Excited Nonlinear Systems via Normal Forms

2006 ◽  
Vol 2 (2) ◽  
pp. 124-131 ◽  
Author(s):  
Yandong Zhang ◽  
S. C. Sinha
Keyword(s):  
Nonlinear Systems ◽  
Feedback Linearization ◽  
Linear Time ◽  
Sufficient Conditions ◽  
Original System ◽  
Computation Method ◽  
Normal Form Theory ◽  
Periodic Coefficients ◽  
Quasi Linear System ◽  
Time Periodic

The problem of designing controllers for nonlinear time periodic systems via feedback linearization is addressed. The idea is to find proper coordinate transformations and state feedback under which the original system can be (exactly or approximately) transformed into a linear time periodic control system. Then a controller can be designed to guarantee the stability of the system. Our approach is designed to achieve local control of nonlinear systems with periodic coefficients desired to be driven either to a periodic orbit or to a fixed point. The system equations are represented by a quasi-linear system containing nonlinear monomials with periodic coefficients. Using near identity transformations and normal form theory, the original close loop problem is approximately transformed into a linear time periodic system with unknown gains. Then by using a symbolic computation method, the Floquet multipliers are placed in the desired locations in order to determine the control gains. We also give the sufficient conditions under which the system is feedback linearizable up to the rth order.

scholarly journals Design of an optimal preview controller for linear discrete-time periodic systems

2021 ◽  
pp. 014233122110025
Author(s):  
Mengyuan Sun ◽  
Fucheng Liao ◽  
Jiamei Deng
Keyword(s):  
Control Problem ◽  
Discrete Time ◽  
Tracking Control ◽  
Reference Signal ◽  
Periodic Systems ◽  
Augmented System ◽  
The Difference ◽  
System States ◽  
Ordinary Time ◽  
Time Periodic

In this paper, the preview tracking control problem for linear discrete-time periodic systems is considered. First, to overcome the difficulty arising from periodicity of the system, the linear discrete-time periodic system is transformed into an ordinary time-invariant system by lifting method. Secondly, the difference between a system state and its steady-state value is used to derive an augmented system instead of the usual difference between system states. Then, the preview controller for the augmented system is proposed by the preview control theory, which solves the preview tracking control problem for the periodic systems. Moreover, an integrator is introduced to ensure that the output can track the reference signal without static error. Finally, the obtained results are illustrated by the simulation examples.

Some Ideas on the Local Control of Nonlinear Systems With Time-Periodic Coefficients

1999 ◽  
Author(s):  
Alexandra Dávid ◽  
S. C. Sinha
Keyword(s):  
Normal Form ◽  
Local Control ◽  
State Feedback ◽  
Feedback Controller ◽  
Periodic Systems ◽  
State Feedback Controller ◽  
Linearized System ◽  
Linear State ◽  
Time Invariant ◽  
Time Periodic

Abstract In this paper ideas on local control of linear and nonlinear time-periodic systems are presented. Our first goal is to stabilize the system far away from bifurcation points. In this case, the classical linear state feedback stabilization based on pole placement is generalized such that it is applicable to time-periodic systems. The linear state-feedback controller design involves computation of the fundamental solution matrix of the system in a symbolic form as function of the control parameters. Next, we focus on the bifurcation control of time-periodic systems. When the linearized system is in a critical case of stability (i.e. when it has Floquet multipliers on the unit circle of the complex plane) and the critical modes are uncontrollable in the linear sense, then a purely nonlinear state-feedback controller is designed to stabilize the equilibrium at the bifurcation point and ensure the stability of the bifurcated nontrivial solution. When the linearized system is linearly controllable, then it is shown that an appropriately chosen linear state-feedback control can also modify the nonlinear features of the bifurcations, such as stability or size of the limit cycles or quasi-periodic limit sets. The control techniques are based on a series of transformations that convert the system into a time-invariant form. First, the Lyapunov-Floquet transformation is used to make the linear part of the periodic system time-invariant. Then, time-periodic center manifold reduction and time-dependent normal form theory are applied to obtain the simplest nonlinear form of a system undergoing bifurcation. For most codimension one bifurcations the normal form is completely time-invariant and therefore, it is a rather simple task to choose the appropriate control gains. These ideas are illustrated by an example of a parametrically excited simple pendulum undergoing symmetry breaking bifurcation.

scholarly journals Harmonic transfer functions based controllers for linear time-periodic systems

2018 ◽  
Vol 41 (8) ◽  
pp. 2171-2184
Author(s):  
Elvan Kuzucu Hidir ◽  
Ismail Uyanik ◽  
Ömer Morgül
Keyword(s):  
State Space ◽  
Design Methodology ◽  
Performance Metrics ◽  
Transfer Functions ◽  
Controller Design ◽  
Linear Time ◽  
Data Driven ◽  
Periodic Systems ◽  
And Control ◽  
Time Periodic

The analysis, identification and control of periodic systems has gained increasing interest during the last few decades due to the increased use of dynamical systems that exhibit periodic motion. The vast majority of these studies focus on the analysis and control problem for a known state-space formulation of the linear time-periodic (LTP) system. On the other hand, there are also some studies that focus on data-driven identification of LTP systems with unknown state-space formulations. However, most of these methods provide numerical estimates for the harmonic transfer functions (HTFs) of an LTP system that are extremely difficult to work with during controller design. The goal of this paper is to provide a simple controller design methodology for unknown LTP systems by utilizing so-called HTFs estimates. To this end, we first build a mathematical basis of LTP controller design for known LTP systems using the Nyquist diagrams and analytically derived HTFs. We propose a new methodology to design P-, PD- and PID-type controllers for LTP systems using Nyquist diagrams and the eigenlocus of the HTFs. Having established the HTF-based controller design procedure, we extend our methodology to unknown LTP systems by presenting a new sum-of-cosine signal-based data-driven system identification method. We show that the proposed data-driven controller design method allows estimation of the HTFs and it provides simple tools for optimizing certain time-domain performance metrics. We provide numerical examples for both known and unknown LTP system cases to illustrate the performance of the proposed controller design methodology.

Stability Analysis and Controller Design for Linear Time Periodic Systems Using Normal Forms

2020 ◽  
Author(s):  
Susheelkumar C. Subramanian ◽  
Sangram Redkar
Keyword(s):  
Stability Analysis ◽  
Normal Forms ◽  
Controller Design ◽  
Linear Time ◽  
Identity Transformation ◽  
Feedback Controller ◽  
Linear Feedback ◽  
Periodic Systems ◽  
Linear Feedback Controller ◽  
Time Periodic

Abstract The investigation of stability bounds for linear time periodic systems have been performed using various methods in the past. The Normal Forms technique has been predominantly used for analysis of nonlinear equations. In this work, the authors draw comparisons between the Floquet theory and Normal Forms technique for a linear system with time periodic coefficients. Moreover, the authors utilize the Normal Forms technique to transform a linear time periodic system to a time-invariant system by using near identity transformation, similar to the Lyapunov Floquet (L-F) transformation. The authors employ an intuitive state augmentation technique, modal transformation and near identity transformations to enable the application of time independent Normal Forms directly without the use of detuning or book-keeping parameter. This method provides a closed form analytical expression for the state transition matrix with the elements as a function of time. Additionally, stability analysis is performed on the transformed system and the resulting transitions curves are compared with that of numerical simulation results. Furthermore, a linear feedback controller design is discussed based on the stability bounds and the implementation of an effective feedback controller for an unstable case is discussed. The theory is validated and verified using numerical simulations of temporal variation of a simple linear Mathieu equation.

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