Change Of State Coordinates Research Articles

Realization of time-delay systems

The paper addresses the problem of transforming a single-input single-output nonlinear retarded time-delay system, described by an input–output equation, in the traditional observable state space form. The solution is generalized from the delay-free case and depends on integrability of certain submodule of differential 1-forms. The integrability conditions are improved to make them constructive. Finally, it is explained why one may obtain two realizations, which are not connected by bi-causal change of state coordinates.

Simultaneous state and unknown input reconstruction using cascaded high-gain observers

ABSTRACTThis paper presents a systematic routine for jointly reconstruct the state variables and the unknown inputs (UIs) for a large class of nonlinear MIMO systems. After an appropriate change of state coordinates, a set of cascade high-gain observers (CHGO) are designed in such a way that each of them provides an estimation of only one component of the UI vector, except the last one which gives a final adjustment of the whole state variables. Such design achieves a boundary of the state estimation error which can be arbitrarily small by properly specifying the sole synthesis parameter. An illustrative example verifies the effectiveness of the proposed scheme.

Cascade High Gain Observers Based State and Unknown Input Reconstruction

This paper presents a systematic routine for jointly reconstruct the state variables and the unknown inputs for a large class of nonlinear MIMO systems. After an appropriate change of state coordinates, a set of cascade high gain observers are designed in such a way that each of them provides an estimation of only one component of the unknown input vector, except the last one which gives a final adjustment of the whole state variables. Such design achieves a boundary of the state estimation error which can be arbitrarily small by properly specifying the sole synthesis parameter. An illustrative example verifies the effectiveness of the proposed scheme.

Nonlinear dynamic image-based visual servoing of a quadrotor

This paper proposes a dynamic image-based visual servoing (IBVS) control law for a quadrotor unmanned aerial vehicle (UAV) equipped with a single fixed on-board camera facing downward. The motion control problem is to regulate the relative lateral position of the vehicle to a stationary target located on the ground. The control law is termed dynamic as it is based on the dynamics and kinematics of the vehicle. The proposed design uses a nonlinear input-dependent change of state coordinates and its error dynamics are proven to be locally exponentially stable with an estimate provided for the region of attraction. Experimental and simulation results demonstrate the method's ease of on-board implementation, performance, and robustness. The simulation and experimental results include a comparison with an established dynamic IBVS method. This comparison shows the proposed method can provide similar performance with the benefit of reduced complexity.

Cascade Control for a Rigid-Flexible Two-link Robot using Sliding Modes

AbstractThe dynamical model of a two-link planar robot with one rigid-link, one flexible forearm and a payload attached to its tip, is represented in a cascade-like fashion using a convenient change of state coordinates which is found by observing the structure of the inertia matrix of the system model. The new system representation makes possible to devise a way to stabilize the overall system by designing a suitable trajectory for the angular positions of the joint shafts, in terms of the new state coordinates. These trajectories of the joints are used as virtual control signals that allow the controlling of the end effector position. Some simulation results are presented in which sliding mode control is employed to successfully perform regulation and trajectory tracking of the end effector.

High-Gain State Feedback Analysis Based on Singular System Theory

We consider linear, time-invariant state-space systems under high-gain state feedback. The analysis is couched in terms of singular system theory and Grassman manifolds. Our work is distinguished from that of other authors by the fact that we do not allow a gain-dependent state coordinate change. Simple necessary and sufficient conditions are proven under which a singular system is a high-gain limit of a given state-space system. It is shown that the feedback matrix achieves a limit on an appropriate Grassmanian, so infinite gains constitute well-defined mathematical objects. The special cases of minimum-order stable and zeroth-order limits are studied in depth, including an analysis of solution behavior. Finally, the classical cheap control problem is interpreted within the context of our results.

Control Bifurcations

A parametrized nonlinear differential equation can have multiple equilibria as the parameter is varied. A local bifurcation of a parametrized differential equation occurs at an equilibrium where there is a change in the topological character of the nearby solution curves. This typically happens because some eigenvalues of the parametrized linear approximating differential equation cross the imaginary axis and there is a change in stability of the equilibrium. The topological nature of the solutions is unchanged by smooth changes of state coordinates so these may be used to bring the differential equation into Poincare/spl acute/ normal form. From this normal form, the type of the bifurcation can be determined. For differential equations depending on a single parameter, the typical ways that the system can bifurcate are fully understood, e.g., the fold (or saddle node), the transcritical and the Hopf bifurcation. A nonlinear control system has multiple equilibria typically parametrized by the set value of the control. A control bifurcation of a nonlinear system typically occurs when its linear approximation loses stabilizability. The ways in which this can happen are understood through the appropriate normal forms. We present the quadratic and cubic normal forms of a scalar input nonlinear control system around an equilibrium point. These are the normal forms under quadratic and cubic change of state coordinates and invertible state feedback. The system need not be linearly controllable. We study some important control bifurcations, the analogues of the classical fold, transcritical and Hopf bifurcations.

137 擬似線形化法によるテザー衛星システムの非線形制御

This paper discusses the control problems of deployment and retrieval of a tethered satellite system. First, a linear tangent model of the tethered satellite system which is independent of its operating point is derived by using a pseudolinearization technique. Then, a nonlinear control law is designed based on the linear tangent model and a nonlinear change of state coordinates. The present feedback control law is simulated numerically, and the results show excellent controlled response of the tethered subsatellite system.

Normal Forms and Bifurcations of Discrete Time Nonlinear Control Systems

The quadratic and cubic normal forms of discrete time nonlinear control systems are presented. These are the normal forms with respect to the group of state coordinate changes and invertible state feedbacks. We introduce the concept of a control bifurcation for such systems. A control bifurcation takes place at an equilibrium where there is a loss of linear stabilizability in contrast to a classical bifurcation, which typically takes place at an equilibrium where there is a loss of linear stability. We present the analogous control bifurcations to the well-known classical bifurcations; the fold, the transcritical, the flip, and the Neimark--Sacker bifurcations. When the loop is closed, a control bifurcation can lead to a classical bifurcation.

Nonlinear Observer Design in the Siegel Domain

We extend the method of Kazantzis and Kravaris [Systems Control Lett., 34 (1998), pp. 241--247] for the design of an observer to a larger class of nonlinear systems. The extended method is applicable to any real analytic observable nonlinear system. It is based on the solution of a first-order, singular, nonlinear PDE. This solution yields a change of state coordinates which linearizes the error dynamics. Under very general conditions, the existence and uniqueness of the solution is proved. Lyapunov's auxiliary theorem and Siegel's theorem are obtained as corollaries. The technique is constructive and yields a method for constructing approximate solutions.

Nonlinear Observer Design in the Siegel Domain Through Coordinate Changes

We extend the method of Kazantzis and Kravaris (Kazantzis and Kravaris, 1998) for the design of an observer to a larger class of nonlinear systems. The extended method is applicable to any real analytic observable nonlinear system. It is based on the solution of a first-order nonlinear PDE. This solution yields a change of state coordinates which linearizes the error dynamics. Under very general conditions, the existence and uniqueness of the solution is proved. Siegel's theorem is obtained as a corollary. The technique is constructive and yields a method for constructing approximate solutions.

A Procedure Towards Linearizing Dynamic Feedback

Passing via a generalized nonlinear controller form (resp. “normal form”) involving the system inputs and their time derivatives in a block triangular way, a procedure is derived for the calculation of so-called (partially) linearizing outputs. The underlying conditions are thus sufficient for (partial) linearizability by dynamic state feedback and generalized change of state coordinates.

Input-output pseudolinearization for nonlinear systems

An input-output pseudolinearized system has the property that its family of linearizations about constant operating points has input-output behavior that is independent of the operating point. We develop a necessary and sufficient condition for existence of a state feedback and state coordinate change that transforms a given system into a pseudo-normal form that is input-output pseudolinearized. These ideas are applied to obtain new approaches to approximate tracking and approximate inversion of nonlinear systems. >

On Constructing Nonlinear Observers

Two algorithmic approaches to constructing nonlinear observers currently exist. Here one of these is improved by finding an explicit solution to the partial differential equations describing the change of state coordinates, thereby avoiding expensive bracket computations. This simplifies the algorithm and has implications for system identification.

The effect of sampling on linear equivalence and feedback linearization

We investigate the effect of sampling on linearization for continuous time systems. It is shown that the discretized system is linearizable by state coordinate change for an open set of sampling times if and only if the continuous time system is linearizable by state coordinate change. Also, it is shown that linearizability via digital feedback imposes highly nongeneric constraints on the structure of the plant, even if this is known to be linearizable with continuous-time feedback. For n = 2, we show, under the assumption of completeness of ad FG , that if the discretized system is lineariable by state coordinate change and feedback, then the continuous time affine complete analytic system is linearizable by state coordinate change only. Also, we suggest a method of proof when n ≥ 3.

Non-linear observer design by observer canonical forms

This paper studies the non-linear observer design problem by observer canonical forms. We present necessary and sufficient conditions in terms of the ranks of matrices for the existence of a non-linear state coordinate change under which a time-variable system is transformed into an observer canonical form. These conditions give the multiple versions of Li and Tao (1986) for the single-output case. Then a comparison is made with the Lie algebraic conditions obtained by Krener and Respondek (1985), and modified by the authors in a recent paper (Xia and Gao 1987).

Feedback linearization of discrete-time systems

We give necessary and sufficient conditions for a nonlinear discrete-time system to be locally linearizable by a change of state coordinates and a feedback. The conditions are expressed in terms of controllability distributions associated to the system.

Linearization of discrete-time systems

Abstract We characterize the equivalence of single-input single-output discrete-time nonlinear systems to linear ones, via a state-coordinate change and with or without feedback. Four cases are distinguished by allowing or disallowing feedback as well as by including the output map or not; the interdependence of these problems is analyzed. An important feature that distinguishes these discrete-time problems from the corresponding problem in continuous-time is that the state-coordinate transformation is here directly computable as a higher composition of the system and output maps. Finally, certain connections are made with the continuous-time case.

Approximate linearization by state feedback and coordinate change

A nonlinear system can always be approximated to first order by linear systems. It has been shown by Jakubczyk and Respondek [4] and Hunt and Su [3] that certain nonlinear systems are the exact transforms of linear systems under nonlinear state coordinate change and nonlinear state feedback. In this paper we give necessary and sufficient conditions for a nonlinear system to be approximated to higher order by the transform of a linear system. The use of this technique in the design of nonlinear compensators has been suggested recently by the author [6].

Linear-quadratic discrete-time control and constant directions

The general linear-quadratic discrete-time minimization problem is studied, in which no restrictions are placed on the singularity or otherwise of certain matrices, or the appearance of cross-product terms in the performance indices. Constant directions are characterized in a number of ways, and an algorithm is presented for replacing a prescribed problem with constant directions by one of lower state and/or control dimension, lacking constant directions. The replacement is achieved using a series of state and input co-ordinate basis changes, and allows simplification of the calculations obtaining the optimal control law and performance index of the original problem.