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.

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.

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.

Existence of Solutions of Riccati Differential Equations

2012 ◽  
Vol 134 (3) ◽  
Author(s):  
Sinan Kilicaslan ◽  
Stephen P. Banks
Keyword(s):  
Differential Equation ◽  
Nonlinear Systems ◽  
Existence Of Solutions ◽  
Linear Time ◽  
Sufficient Conditions ◽  
Necessary Condition ◽  
Time Varying ◽  
Riccati Differential Equation ◽  
Riccati Differential Equations ◽  
Time Varying Systems

A necessary condition for the existence of the solution of the Riccati differential equation for both linear, time varying systems and nonlinear systems is introduced. First, a necessary condition for the existence of the solution of the Riccati differential equation for linear, time varying systems is proposed. Then, the sufficient conditions to satisfy the necessary condition are given. After that, the existence of the solution of the Riccati differential equation is generalized for nonlinear systems.

A New Approach for Computing the Normal Forms of High Dimensional Nonlinear Systems

2010 ◽  
Author(s):  
Shuping Chen ◽  
Wei Zhang ◽  
Minghui Yao
Keyword(s):  
Nonlinear Systems ◽  
Normal Form ◽  
Cantilever Beam ◽  
Normal Forms ◽  
Nonlinear Differential Equations ◽  
Real Life ◽  
High Dimensional ◽  
Computation Method ◽  
Normal Form Theory ◽  
Averaged Equations

Normal form theory is very useful for direct bifurcation and stability analysis of nonlinear differential equations modeled in real life. This paper develops a new computation method for obtaining a significant refinement of the normal forms for high dimensional nonlinear systems. The method developed here uses the lower order nonlinear terms in the normal form for the simplifications of higher order terms. In the theoretical model for the nonplanar nonlinear oscillation of a cantilever beam, the computation method is applied to compute the coefficients of the normal forms for the case of two non-semisimple double zero eigenvalues. The normal forms of the averaged equations and their coefficients for non-planar non-linear oscillations of the cantilever beam under combined parametric and forcing excitations are calculated.

Control of chaos in nonlinear systems with time-periodic coefficients

2006 ◽  
Vol 364 (1846) ◽  
pp. 2417-2432 ◽  
Author(s):  
S.C Sinha ◽  
Alexandra Dávid
Keyword(s):  
Nonlinear Systems ◽  
Periodic Orbit ◽  
Nonlinear Control ◽  
State Feedback ◽  
State Feedback Control ◽  
Control Technique ◽  
Periodic Coefficients ◽  
Full State ◽  
Full State Feedback ◽  
Time Periodic

In this study, some techniques for the control of chaotic nonlinear systems with periodic coefficients are presented. First, chaos is eliminated from a given range of the system parameters by driving the system to a desired periodic orbit or to a fixed point using a full-state feedback. One has to deal with the same mathematical problem in the event when an autonomous system exhibiting chaos is desired to be driven to a periodic orbit. This is achieved by employing either a linear or a nonlinear control technique. In the linear method, a linear full-state feedback controller is designed by symbolic computation. The nonlinear technique is based on the idea of feedback linearization. A set of coordinate transformation is introduced, which leads to an equivalent linear system that can be controlled by known methods. Our second idea is to delay the onset of chaos beyond a given parameter range by a purely nonlinear control strategy that employs local bifurcation analysis of time-periodic systems. In this method, nonlinear properties of post-bifurcation dynamics, such as stability or rate of growth of a limit set, are modified by a nonlinear state feedback control. The control strategies are illustrated through examples. All methods are general in the sense that they can be applied to systems with no restrictions on the size of the periodic terms.

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.

New Analysis/Design of Generalized Discrete PI Controller via Discrete Time Delay Control for Nonlinear Systems

2020 ◽  
Author(s):  
Suresh B. Reddy
Keyword(s):  
Time Delay ◽  
Nonlinear Systems ◽  
Discrete Time ◽  
Linear Time ◽  
Sufficient Conditions ◽  
Pi Controller ◽  
Pid Controllers ◽  
Analysis And Design ◽  
Proportional Integral ◽  
Discrete Time Delay

Abstract Proportional-Integral (PI) and Proportional-Integral-Derivative (PID) controllers are among the most common schemes for control since their formulation nearly a century ago. They have been very successful in many applications, even as we have migrated from analog implementations to digital control systems. While there is rich literature for design and analysis of PI/PID controllers for linear time-invariant systems with modeled dynamics, the tools for analysis and design for nonlinear systems with unknown dynamics are limited, despite their known effectiveness. This paper extends previous observations about a form of discrete Time Delay Control’s equivalence to a generalized PI controller for more general canonical systems, with additional complimentary feedback linearization of known dynamics, as desired. In addition, sufficient conditions for Bounded Input-Bounded Output (BIBO) as well as exponential stability are developed in this paper for the form of discrete TDC that is closest to generalized discrete PI equivalent controller, for multi-input multi-output nonlinear systems, including nonaffine cases. Accordingly, design procedures are suggested for such discrete TDC, and generalized discrete PI controller for nonlinear systems.

On The Response of Linear Time-Periodic Systems Subjected to Deterministic and Stochastic Excitations

1996 ◽  
Vol 2 (2) ◽  
pp. 219-249 ◽  
Author(s):  
J.M. Spires ◽  
S.C. Sinha
Keyword(s):  
Chebyshev Polynomial ◽  
Linear Time ◽  
Original System ◽  
Invariant Form ◽  
Periodic Systems ◽  
Solution Technique ◽  
Stiffness Matrices ◽  
Original Time ◽  
Time Invariant ◽  
Time Periodic

In many situations, engineering systems modeled by a set of linear, second-order differential equations, with periodic damping and stiffness matrices, are subjected to external excitations. It has been shown that the fundamental solution matrix for such systems can be efficiently computed using a Chebyshev polynomial series solution technique. Further, it is shown that the Liapunov-Floquet transformation matrix associated with the system can be computed, and the original time-periodic system can be put into a time- invariant form. In this paper, these techniques are applied in finding the transient response of periodic systems subjected to deterministic and stochastic forces. Two formulations are presented. In the first formulation, the response of the original system is computed directly. In the second formulation, first the original system is transformed to a time-invariant form, and then the response is found by determining the response of the time-invariant system. Both formulations use the convolution integral to form an expression for the response. This expression can be evaluated numerically, symbolically, or through a Chebyshev polynomial expansion technique. Results for some time-invariant and periodic systems are included, as illustrative examples.

Further Reduction of Normal Forms for High Dimensional Nonlinear Systems and Application to a Composite Laminated Piezoelectric Plate

2013 ◽  
Vol 291-294 ◽  
pp. 2662-2665
Author(s):  
Shu Ping Chen ◽  
Wei Zhang
Keyword(s):  
Nonlinear Systems ◽  
Normal Forms ◽  
Nonlinear Oscillations ◽  
Piezoelectric Plate ◽  
Nonlinear Differential Equations ◽  
High Dimensional ◽  
Computation Method ◽  
Normal Form Theory ◽  
Averaged Equations ◽  
Form Theory

Normal form theory is robust and useful for direct bifurcation and stability analysis of nonlinear differential equations in real engineering problems. This paper develops a new computation method for obtaining a significant refinement of the normal forms for high dimensional nonlinear systems. In the theoretical model for the nonlinear oscillation of a composite laminated piezoelectric plate, the computation method is applied to compute the coefficients of the normal forms for the case of one double zero and a pair of pure imaginary eigenvalues. The algorithm is implemented in Maple V and the normal forms of the averaged equations and their coefficients for nonlinear oscillations of the composite laminated piezoelectric plate under combined parametric and transverse excitations are calculated.

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