Input-Output Feedback Linearization: Case Study on 2—Contractive Zero Dynamics

This paper deals with output feedback stabilisation of a class of nonlinear systems with nonlinearity of the unmeasured state variables. The nonlinearity of the unmeasured state complicates the zero dynamics of the nonlinear system and makes the control design more difficult. In this paper, the zero dynamics analysed for the class of nonlinear systems with respect to the original output and then with respect to a virtual output. It is demonstrated that nonlinear systems with unstable zero dynamics may have stable zero dynamics with the virtual output. Based on the analysis, conditions are then identified for nonlinear systems which can be stabilised by control design with estimated state, including the estimated virtual output, and this removes a restriction of a recent result that requires the transformed system has no zero dynamics. Examples are included to demonstrate that the proposed control design can be used to stabilise nonlinear systems with unstable nonlinear zero dynamics.

One of the approaches to sampled-data controller design for nonlinear continuous-time systems consists of obtaining an appropriate model and then proceeding to design a controller for the model. Then, it is important to derive a good approximate sampled-data model because the exact sampled- data model for nonlinear systems is often unavailable to the controller designers. Recently, Yuz and Goodwin have proposed an accurate sampled-data model which includes extra zero dynamics, so-called the sampling zero dynamics, corresponding to the relative degree of the continuous-time nonlinear system. This paper shows that a more accurate sampled-data model is required for a controlled Van der Pol system with the relative degree two. The reason is that the closed-loop system becomes unstable when a controller design method based on cancellation of the zero dynamics is applied, and the phenomenon seems related to the instability of the sampling zero dynamics of the more accurate sampled-data model. Further, this paper derives a more accurate model than that of Yuz and Goodwin for continuous-time nonlinear systems with the relative degree two, and presents a condition which assures the stability of the sampling zero dynamics of the obtained model.

In this paper, we study the best achievable performance of output regulation for an uncertain nonlinear system under disturbances. The nonlinear system is a stabilizable nonminimum phase system and its zero dynamics consists of a robust input-to-state stable component and an unstable but stabilizable component. It is assumed that the uncertain parameter vector and disturbances only enter the robust input-to-state stable component of the zero dynamics and the linearizable part of the system. In the latter, the uncertain functions satisfy lower triangular structure. The performance of this problem is measured by the energy of output signal of above system. It is shown in this paper that the best achievable performance of output regulation under the worst uncertain parameter vector and disturbances for the nonlinear system is only determined by the minimum energy of stabilizing the unstable part of its zero dynamics.

Stability of Zero Dynamics of Sampled-Data Nonlinear Systems

Introduction. 1.1 Linear Time-Invariant Systems. 1.2 Nonlinear Systems. 1.3 Equilibrium Points. 1.4 First-Order Autonomous Nonlinear Systems. 1.5 Second-Order Systems: Phase-Plane Analysis. 1.6 Phase-Plane Analysis of Linear Time-Invariant Systems. 1.7 Phase-Plane Analysis of Nonlinear Systems. 1.8 Higher-Order Systems. 1.9 Examples of Nonlinear Systems. 1.10 Exercises. Mathematical Preliminaries. 2.1 Sets. 2.2 Metric Spaces. 2.3 Vector Spaces. 2.4 Matrices. 2.5 Basic Topology. 2.6 Sequences. 2.7 Functions. 2.8 Differentiability. 2.9 Lipschitz Continuity. 2.10 Contraction Mapping. 2.11 Solution of Differential Equations. 2.12 Exercises. Lyapunov Stability I: Autonomous Systems. 3.1 Definitions. 3.2 Positive Definite Functions. 3.3 Stability Theorems. 3.4 Examples. 3.5 Asymptotic Stability in the Large. 3.6 Positive Definite Functions Revisited. 3.7 Construction of Lyapunov Functions. 3.8 The Invariance Principle. 3.9 Region of Attraction. 3.10 Analysis of Linear Time-Invariant Systems. 3.11 Instability. 3.12 Exercises. Lyapunov Stability II: Nonautonomous Systems. 4.1 Definitions. 4.2 Positive Definite Functions. 4.3 Stability Theorems. 4.4 Proof of the Stability Theorems. 4.5 Analysis of Linear Time-Varying Systems. 4.6 Perturbation Analysis. 4.7 Converse Theorems. 4.8 Discrete-Time Systems. 4.9 Discretization. 4.10 Stability of Discrete-Time Systems. 4.11 Exercises. Feedback Systems. 5.1 Basic Feedback Stabilization. 5.2 Integrator Backstepping. 5.3 Backstepping: More General Cases. 5.4 Examples. 5.5 Exercises. Input-Output Stability. 6.1 Function Spaces. 6.2 Input-Output Stability. 6.3 Linear Time-Invariant Systems. 6.4 Lp Gains for LTI Systems. 6.5 Closed Loop Input-Output Stability. 6.6 The Small Gain Theorem. 6.7 Loop Transformations. 6.8 The Circle Criterion. 6.9 Exercises. Input-to-State Stability. 7.1 Motivation. 7.2 Definitions. 7.3 Input-to-State Stability (ISS) Theorems. 7.4 Input-to-State Stability Revisited. 7.5 Cascade Connected Systems. 7.6 Exercises. Passivity. 8.1 Power and Energy: Passive Systems. 8.2 Definitions. 8.3 Interconnections of Passivity Systems. 8.4 Stability of Feedback Interconnections. 8.5 Passivity of Linear Time-Invariant Systems. 8.6 Strictly Positive Real Rational Functions. Exercises. Dissipativity. 9.1 Dissipative Systems. 9.2 Differentiable Storage Functions. 9.3 QSR Dissipativity. 9.4 Examples. 9.5 Available Storage. 9.6 Algebraic Condition for Dissipativity. 9.7 Stability of Dissipative Systems. 9.8 Feedback Interconnections. 9.9 Nonlinear L2 Gain. 9.10 Some Remarks about Control Design. 9.11 Nonlinear L2-Gain Control. 9.12 Exercises. Feedback Linearization. 10.1 Mathematical Tools. 10.2 Input-State Linearization. 10.3 Examples. 10.4 Conditions for Input-State Linearization. 10.5 Input-Output Linearization. 10.6 The Zero Dynamics. 10.7 Conditions for Input-Output Linearization. 10.8 Exercises. Nonlinear Observers. 11.1 Observers for Linear Time-Invariant Systems. 11.2 Nonlinear Observability. 11.3 Observers with Linear Error Dynamics. 11.4 Lipschitz Systems. 11.5 Nonlinear Separation Principle. Proofs. Bibliography. List of Figures. Index.

In order to fully consider the nonlinear characteristics of the doubly-fed induction generator (DFIG) fault condition, we propose a ZD-STASMC method of zero dynamics (ZD) combined with super-twisting adaptive sliding mode control (STASMC) to improve the transient voltage stability of DFIG, which applies the adaptive STASMC to improve the ZD method for multiple input multiple output (MIMO) systems. The use of ZD method decouples the internal and external dynamics of the original system, which simplifies the control design tasks. To further enhance voltage regulation in the presence of parameter uncertainties and external disturbances, the ZD approach is combined with the STASMC through which the transient performance of DFIG is improved by constructing auxiliary input, by reducing the influence of modeling errors and various errors in the actual process. The use of the improved ZD technique can reduce the influence of linearization error and enhance the transient performance of DFIG. Through the simulation model under the three-phase faults, it is verified that the proposed ZD-STASMC control has better transient performance during faults, and can recover stability faster after fault removal, compared with the ZD control method and the conventional proportion integration (PI) control. The proposed strategy can effectively enhance the fault traversing capability of DFIG.

The analysis and design of linear dynamic feedback sequential machines implemented with matrix memories, such as read only memories, is presented. Such machines differ from direct feedback sequential machines with no external inputs in that they may enter or exit any trap state or loop at any specified time; each particular machine is defined by an external control input. This control input may come from the external world or the output of another memory or a processing computer. Consequently, these machines may be considered as adaptive machines. Additionally, they are programmable, the same machine may be designed to satisfy requirements of a variety of applications in discrete-time control systems.

This paper analyzes the passivity and feedback passivity of discrete-time-switched nonlinear systems with passive and nonpassive modes that are affine in the control input. When a nonpassive mode is active, the increase in storage function is not necessarily bounded by the energy supplied to the switched system at every time step. Therefore, a switched system with at least one nonpassive mode is defined to be nonpassive in the classical passivity theory. In this paper, we propose a framework to analyze the passivity of such switched systems in a more general sense. We consider switched nonlinear systems which are affine in the control input and may consist of passive, feedback passive modes, and modes which cannot be rendered passive using feedback. In the proposed framework, we prove that a switched nonlinear system is locally feedback passive if and only if its zero dynamics are locally passive. A lower bound on the ratio of total activation time between (feedback) passive and nonfeedback passive modes is obtained to guarantee passive zero dynamics. Finally, we prove that two important properties of classical passivity still hold for the proposed passivity definition, that is: 1) output feedback control can be used to stabilize the switched system, and 2) parallel and negative feedback interconnections of two such passive systems are also passive.

Considering the problem for inverse system method in EMU AC induction traction motor linear decoupling, the zero dynamics subsystem will be separated from the original dynamic system through coordinate transformation. Firstly, a getting method for zero dynamics of the multiple input multiple output nonlinear system is discussed when γ＜ n. Second, the zero dynamics analysis for five order nonlinear model of asynchronous traction motor which base on the stationary coordinate system is given by using inverse decoupling method. The analysis results show that if the stability of the zero dynamics can be ensured, then the entire linearization of original nonlinear system is not necessary, need only partial linearization which effect on the external dynamic portion. The inverse decoupling process of asynchronous traction motor can be simplified by this conclusion. DOI : http://dx.doi.org/10.11591/telkomnika.v12i1.3354

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In the path-following problem formulated in this note, it is required that the error between the system output and the desired geometric path eventually be less than any prespecified constant. If in a nonlinear multiple-input–multiple-output (MIMO) system the output derivatives do not enter into its zero dynamics, a condition relating path geometry and stabilizability of the zero dynamics is given under which a solution to this problem exists. The solution is obtained by combining input-to-state stability and hybrid system methodologies. </para>

This paper presents a novel stability analysis of the zero dynamics of nonlinear dynamic models of spacecraft with fuel slosh. Mathematical models based on a Lagrange formulation are used, with the pendulum analogy of a typical spacecraft with fuel slosh dynamics, to include the behavior of the liquid dynamics coupled to the spacecraft. With both analytical and numerical results, it is shown that in the presence of fuel sloshing, the zero dynamics of both single-input single-output and multi-input multi-output spacecraft models are not asymptotically stable or even unstable at their nominal equilibrium point. This study indicates that special design considerations are needed for nonlinear feedback motion control of spacecraft with fuel sloshing uncertainties, to handle the nonminimum phase characteristics of nonlinear spacecraft models, which are shown by the phase portraits of their zero dynamics. The development of suitable control designs for such nonlinear systems is currently under investigation.

The exact feedback linearization technique with partial state feedback and stable zero dynamics ensures local exponential stability. However, the method requires an accurate model, without uncertainties. The paper proposes an extension of the method for input-affine nonlinear systems with stable zero dynamics and matched uncertainties. Moreover, the weighted closed-loop sensitivity function is imposed using the ℒ<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf>-gain analysis. The advantage of the proposed extension is that it ensures both robust stability and performance (tailored to reference tracking) for the uncertain nonlinear system affected by the limitations of zero dynamics. The two methodologies are then illustrated on a given academic example.

Zero dynamics of nonlinear control systems with symmetries are studied in this paper. It is shown that the zero dynamics of a symmetric system is also symmetric and admits a special form of cascade decomposition. A local zero dynamics is always possible to be extended to the semiglobal case. The relation between the zero dynamics and the quotient system is also discussed.

Lateral Acceleration Control Design of a Non-Linear Homing Missile

We study the zero dynamics and funnel control for linear passive electrical circuits. We show that asymptotic stability of the zero dynamics can be characterized by criteria on the circuit topology. Thereafter we consider the output regulation problem for electrical circuits by funnel control. We show that for circuits with asymptotically stable zero dynamics, the funnel controller achieves tracking of a class of reference signals within a pre‐specified funnel. This result can be relaxed to the case of non‐autonomous zero dynamics by requiring the reference trajectory to evolve within a certain subspace. (© 2014 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)