Nonlinear L2 Gain Research Articles

Performance-based activation of anti-windup compensation for control of linear systems subject to actuator saturation

In this paper, we propose a performance-based activation mechanism for anti-windup compensation. The signals representing the transient performance of the closed-loop system are passed through a static performance compensator and then injected into the traditional anti-windup activation module and a performance-based anti-windup design framework is developed. The resulting equivalent controller contains the information of the anti-windup compensator and has the potential for improving the performance of the closed-loop systems. Under this new anti-windup design framework, sufficient conditions are established for estimating the domain of attraction and the nonlinear L2 gain of saturated systems. Based on these conditions, we formulate optimization problems to determine the optimal gains of anti-windup and performance compensators for enlarging the domain of attraction and reducing the nonlinear L2 gain. Simulation results show that, compared with the existing methods, our proposed approach has the ability to acquire a significantly larger estimate of the domain of attraction, a significantly tighter estimate of the nonlinear L2 gain, and significantly higher transient quality in reference tracking.

Stability and Performance Analysis of Saturated Systems Using an Enhanced Max Quadratic Lyapunov Function

This paper revisits the problem of estimating the domain of attraction and the nonlinear L2 gain of a saturated system with an algebraic loop. A max quadratic Lyapunov function, the maximum function of a group of quadratic Lyapunov functions, is proposed for use in the stability and performance analysis for a saturated system. The max quadratic Lyapunov function involves a partitioning of the state space. Exploiting the special properties of regions of the state space, we propose in this paper an enhanced max quadratic Lyapunov function, which results from adding a term that characterizes the regions of the state space to the max quadratic Lyapunov function. The matrix associated with the enhanced max quadratic Lyapunov function is not required to be positive definite, and thus a set of less conservative stability and performance conditions are established, from which a larger estimate of the domain of attraction and a tighter estimate of the nonlinear L2 gain can be obtained. Simulation results indicate the effectiveness and superiority of the proposed method.

Stability and performance analysis of saturated systems via partitioning of the virtual input space

This paper revisits the problem of estimating the domain of attraction and the nonlinear L2 gain for systems with saturation nonlinearities. We construct a virtual input space from the algebraic loop contained in systems, and divide this virtual input space into several regions. In one of these regions, none of the virtual inputs saturate. In each of the remaining regions, there is a unique virtual input that saturates everywhere with the time-derivative of its saturated signal being zero. These special properties of the virtual inputs in different regions of the virtual input space are combined with an existing piecewise quadratic Lyapunov function that contains the information of virtual input saturation to arrive at a set of less conservative stability and performance conditions, from which we can obtain a larger level set of the piecewise quadratic Lyapunov function as an estimate of the domain of attraction and a tighter scalar function of the bound on the L2 norm of the exogenous input as an estimate of the local nonlinear L2 gain. Simulation results indicate that the proposed approach has the ability to obtain a significantly larger estimate of the domain of attraction and a significantly tighter estimate of the nonlinear L2 gain than the existing methods.

Equivalences of Stability Properties for Discrete-Time Nonlinear Systems

Several qualitative equivalences are demonstrated between various robust stability properties for discrete-time nonlinear systems. In particular, two different summation-to-summation estimates are shown to be equivalent to the standard definitions of Input-to-State Stability (ISS) and integral ISS (iISS), respectively. Furthermore, ISS and iISS are shown to be qualitatively equivalent, via a nonlinear change of coordinates, to linear and nonlinear l2-gain properties, respectively. These equivalences, together with a previous result on the equivalence of 0-input global asymptotic stability and iISS provide interesting relationships between discrete-time robust stability properties that do not hold in continuous-time.

Performance bounds for nonlinear systems with a nonlinear ℒ2-gain property

Nonlinear ℒ2-gain is a finite gain concept that generalises the notion of conventional (linear) finite ℒ2-gain to admit the application of ℒ2-gain analysis tools of a broader class of nonlinear systems. The computation of tight comparison function bounds for this nonlinear ℒ2-gain property is important in applications such as small gain design. This article presents an approximation framework for these comparison function bounds through the formulation and solution of an optimal control problem. Key to the solution of this problem is the lifting of an ℒ2-norm input constraint, which is facilitated via the introduction of an energy saturation operator. This admits the solution of the optimal control problem of interest via dynamic programming and associated numerical methods, leading to the computation of the proposed bounds. Two examples are presented to demonstrate this approach.

State Feedback Controller Synthesis to Achieve a Nonlinear L2-gain Property

AbstractThis paper generalizes the framework of nonlinear H∞-control design methodology for state feedback synthesis. Instead of controlling a system to achieve a (linear) finite L2-gain from disturbance input to penalty output, it is aimed at achieving a more general class of nonlinear L2-gain bounds. This generalization is aimed at allowing the application of the L2-gain based control framework to a larger class of nonlinear systems. State feedback controllers achieving such a generalized performance objective are synthesized based on verification results developed for the nonlinear L2-gain property. An example is included to illustrate these results.

Stability and Performance for Saturated Systems via Quadratic and Nonquadratic Lyapunov Functions

In this paper, we develop a systematic Lyapunov approach to the regional stability and performance analysis of saturated systems in a general feedback configuration. The only assumptions we make about the system are well-posedness of the algebraic loop and local stability. Problems to be considered include the estimation of the domain of attraction, the reachable set under a class of bounded energy disturbances and the nonlinear L2 gain. The regional analysis is established through an effective treatment of the algebraic loop and the saturation/deadzone function. This treatment yields two forms of differential inclusions, a polytopic differential inclusion (PDI) and a norm-bounded differential inclusion (NDI) that contain the original system. Adjustable parameters are incorporated into the differential inclusions to reflect the regional property. The main idea behind the regional analysis is to ensure that the state remain inside the level set of a certain Lyapunov function where the PDI or the NDI is valid. With quadratic Lyapunov functions, conditions for stability and performances are derived as linear matrix inequalities (LMIs). To obtain less conservative conditions, we use a pair of conjugate non-quadratic Lyapunov functions, the convex hull quadratic function and the max quadratic function. These functions yield bilinear matrix inequalities (BMIs) as conditions for stability and guaranteed performance level. The BMI conditions cover the corresponding LMI conditions as special cases, hence the BMI results are guaranteed to be as good as the LMI results. In most examples, the BMI results are significantly better than the LMI results

Adaptive Back-Stepping Design of TCSC Robust Nonlinear Control for Power Systems

Abstract The back-stepping design methodology of the nonlinear control theory is known to provide an efficient constructive tool for designing non-linear controllers for a wide range of systems engineering applications where it is crucial to employ non-lineaz models of the plants. A novel adaptive and robust non-linear control design for electrical power systems employing thyristor controlled series compensation has been solved by using this methodology. For the case of single machine to infinite bus system with TCSC having internal and external disturbances, storage functions of the system aze constructed via adaptive back-stepping design and a novel non-linear L2 gain disturbance attenuation controller along with the parameter update law were derived. Simulation results have demonstrated that the proposed control design possesses asymptotic stability with superior performances. The construction of feedback control law via associated storage functions is consistently systematic, which makes the proposed ...

Nonlinear Control Systems: Analysis and Design

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.

Hamiltonian modelling and nonlinear disturbance attenuation control of TCSC for improving power system stability

To tackle the obstacle of applying passivity-based control (PBC) into power systems, an affine non-linear system widely existing in power systems is formulated as a standard Hamiltonian system using a pre-feedback method. The port controlled Hamiltonian with dissipation (PCHD) model of TCSC is then established corresponding with a revised Hamiltonian function. Furthermore, employing the modified Hamiltonian function directly as the storage function, a non-linear adaptive L2 gain control method is proposed to solve the problem of L2 gain disturbance attenuation for this Hamiltonian system with parametric perturbations. Finally, simulation results are presented to verify the validity of the proposed controller.

Design of a Robust H∞ PID Control for Industrial Manipulators

Industrial manipulators are under various limitations against high quality motion control; for example, both frictional and dynamic disturbances should be dealt with a simple PID control structure. A robust linear PID motion controller, called the reference error feedback (REF), is proposed, which solves the nonlinear L2-gain attenuation control problem for robotic manipulators. The stability, robustness, and performance tuning of the proposed controller are analyzed. Making use of the fact that the single parameter of the induced L2-gain γ controls the performance with stability attained, we propose a simple and stable method of performance tuning called “the square law.” The analytical results are verified through experiments of a six-degrees-of-freedom industrial manipulator. [S0022-0434(00)00104-0]

Nonlinear L2-Gain Suboptimal Control

A method of solving the nonlinear localL2-gain suboptimal control problem based on the chain-scattering approach is proposed. This problem requiresL2-gain of the closed loop to be less than or equal to one and the closed loop internal stability defined in the small signal input-output sense. We obtained sufficient conditions for the existence of a suboptimal controller in terms of state-space description of the plant as well as a local state-space parameterization of a class of controllers solving the localL2-gain suboptimal problem.