Homogeneous Polynomial Lyapunov Functions Research Articles

Output-feedback control design for Takagi-Sugeno fuzzy systems through Lyapunov functions depending polynomially on the states

This paper proposes a new strategy based on linear matrix inequalities (LMIs) for the synthesis of state- and output-feedback controllers through parallel distributed compensation for continuous-time Takagi-Sugeno fuzzy systems. The main novelty of the proposed design technique is the use of homogeneous polynomial Lyapunov functions of degree larger than two on the states, generalizing the results based on quadratic functions. Parametrized in terms of a constant matrix, those homogeneous polynomial Lyapunov functions provide less conservative results in terms of stability and decay rate of trajectories when dealing with membership functions whose time-derivatives have unknown bounds. The synthesis procedure is formulated in terms of a locally convergent iterative algorithm, where a set of LMIs defined in an augmented parameter space is solved at each iteration. Numerical examples are used to compare the proposed method with quadratic stabilizability techniques available in the literature, illustrating the advantages of higher degree Lyapunov functions.

[formula omitted] output tracking control for polynomial parameter-varying systems via homogeneous Lyapunov methods

This paper investigates the H∞ output tracking control for polynomial parameter-varying systems—a class of nonlinear parameter-varying systems. Based on the homogeneous polynomial Lyapunov function (HPLF), a state feedback controller is presented to make the augmented system (composed of the original system and the tracking error system) robust exponentially stable, so that the output of the original system can track the reference signal and satisfy the H∞ output tracking performance. The conservativeness of the theoretical results is reduced due to the following facts: (1) the HPLF can depend on the full states and time-varying parameters; and (2) there are no constraints on the input coefficient matrix and the inverse of the Lyapunov matrix. In particular, when the system states and time-varying parameters are confined to a compact set, local results are obtained based on the generalized S-procedure. The solvable conditions are given in terms of state-and-parameter-dependent linear matrix inequalities, which can be solved by sum of squares (SOS) techniques. Finally, the feasibility and effectiveness of the proposed method are verified by a numerical example and the heading control of unmanned surface vehicle.

Hierarchy of Quadratic Lyapunov Functions for Linear Time-Varying and Related Systems

A method for constructing homogeneous polynomial Lyapunov functions is presented for linear time-varying or switched-linear systems and the class of nonlinear systems that can be represented as such. The method uses a simple recursion based on the Kronecker product to generate a hierarchy of related dynamical systems, whose first element is the system under study and the second element is the well-known Lyapunov differential equation. It is then proven that a quadratic Lyapunov function for the system at one level in the hierarchy, which can be found via semidefinite programming, is a homogeneous polynomial Lyapunov function for the system at the base level in the hierarchy. Searching for Lyapunov functions of the foregoing kind is equivalent to searching for homogeneous polynomial Lyapunov functions via the formulation of sum-of-squares programs. The quadratic perspective presented in this paper enables the easy development of procedures to compute bounds on pointwise-in-time system metrics, such as peak norms, system stability margins, and many other performance measures. The applications of the theory to analyzing an aircraft model, on the one hand, and an experimental aerospace vehicle, on the other hand, are presented. The theory can be comprehended with a first course on state-space control systems and an elementary knowledge of convex programming.

Heading control based on extended homogeneous polynomial Lyapunov function

AbstractFor the nonlinear parameter‐varying (NPV) model of unmanned surface vehicle with the consideration of the velocities on yaw and surge as well as wave disturbances, a robust control method is proposed based on extended homogeneous polynomial Lyapunov function (EHPLF) to regulate heading for the superior performance on the rapidity, accuracy, and robustness. First, a NPV model of heading error is established to design a general form of a state feedback controller with a robust performance. Second, a Lyapunov matrix with full states and varying parameter is constructed to derive the robust global exponential stability conditions by Euler's homogeneity relation for the NPV system, known as the EHPLF stability conditions. Third, since the EHPLF stability conditions consist of a set of nonlinear coupled inequalities that cannot be directly solved by sum of squares (SOS) toolboxes, they are decoupled with matrix transformations to obtain the EHPLF‐SOS stability conditions, which is solved for the parameters of the state feedback controller. Finally, the simulation results indicate that EHPLF method exhibits a superior performance on dynamic, steady‐state, and robustness.

Interval type-2 polynomial fuzzy fault detection scheme with a multi-order homogeneous polynomial Lyapunov functions considering unmeasurable premise variables

A polynomial fuzzy fault detection scheme for sampled-output-measurements-based interval type-2 (IT2) polynomial-fuzzy-model-based (PFMB) systems is investigated in this paper, where the uncertainties in the premise variables (PVs) and membership functions (MFs) are described by IT2 fuzzy sets. Fully or partially unmeasurable PVs cause the parameter matrices of the polynomial fuzzy fault detection observer (PFFDO) to rely on the estimated states and the corresponding mismatching problems are further considered. Lyapunov stability theory is carried out with a novel multi-order homogenous polynomial Lyapunov functions (MHPLF) to introduce more information of the states when eliminating the partial derivatives, and the time-delays introduced by sampled-output measurements are handled by L-K functions. Unlike the membership-function-independent (MFI) approaches, the membership-function-dependent (MFD) approaches carry the information of the MFs for the relaxation of the stability constraints. Corresponding stable constraints in sum-of-squares (SOS) form are given to hold the asymptotic stability of the fault detection system with H∞ performance γ. A numerical example with many cases illustrates the effectiveness of the proposed techniques in uncertainty handling and conservativeness reduction, while an inverted pendulum example verifies the feasibility of the method on physical systems.

An Output-Feedback Design Approach for Robust Stabilization of Linear Systems With Uncertain Time-Delayed Dynamics in Sensors and Actuators

In this paper, we propose a control approach for the robust stabilization of linear time-invariant (LTI) systems with non-negligible sensor and actuator dynamics subject to time-delayed signals. Our proposition is based on obtaining an augmented model that encompasses the plant, sensor, and actuator dynamics and also the time-delay dynamic effect. We make use of the Padé Approximation for modeling the time-delay impact on the feedback loop. Since the actual plant state variables are not available for feedback, the sensor outputs, which represent a subset of the augmented system state variables, are used for composing a static output-feedback control law. The robust controller gains are computed by means of a two-stage strategy based on linear matrix inequalities (LMI). For obtaining less conservative conditions we consider the use of homogeneous-polynomial Lyapunov functions (HPLF)– and other decision variables– of arbitrary degree. In our proposition, we also take into account the inclusion of a minimum decay rate criterion in order to improve closed-loop system transient response. Disturbance rejection is also addressed through extensions to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathscr {H}_{2}$ </tex-math></inline-formula> guaranteed cost minimization. The effectiveness of the proposed strategy is attested in the design of a controller for the lateral axis dynamics of an aircraft and other academic examples.

Robust Stability Analysis of Linear Parameter-Varying Systems With Markov Jumps

This article presents conditions to assure the mean-square stability of linear parameter-varying systems with Markov jumps. The model dynamics are driven not only by a Markov chain but also by time-varying parameters that take values in a polytopic set. No assumption is imposed on how the parameters vary within the polytopic set, i.e., the variation rate can be arbitrarily fast. The proposed conditions stem from a homogeneous polynomial Lyapunov function in the state space, adapted to account for Markov jumps. The stability certificate is sought through linear matrix inequalities. Numerical examples illustrate this article’s contribution.

Robust fuzzy observer‐based fault‐tolerant control: A homogeneous polynomial Lyapunov function approach

In this paper, a homogeneous polynomial Lyapunov function (HPLF) is employed in robust observer-based Integrated Fault-Tolerant Control (IFTC) of nonlinear systems modelled by the polynomial fuzzy model (PFM). This makes the design problem benefit from the conservative reduction property of polynomial Lyapunov functions (PLFs) and simultaneously avoid the emergence of non-convex terms due to differentiation of the polynomial Lyapunov matrix. As a result, the less conservative sum of square (SOS) conditions are obtained in the form of polynomial matrix inequalities (PMI), which are solved via SOSTOOLS. For the first time, a non-quadratic Lyapunov function is used to design a polynomial fuzzy unknown-input observer that estimates the system's states and actuator faults in the presence of model uncertainties and external disturbances. The effectiveness of the proposed approach in providing more relaxed and less conservative results along with the better fault tolerance is illustrated through three simulating examples and compared with the quadratic Lyapunov function (QLF) approach.

Exponential stability of discrete-time systems with slowly time-varying delays

Slowly time-varying delays are seldom, but do need to be, considered in the context of discrete-time systems. This paper addresses the exponential stability issue of discrete-time systems with slowly time-varying delays. The basic idea is to transform, by utilizing the switching transformation approach, the original system with slowly time-varying delays into an equivalent switched system with special switching signal. Different types of delays correspond to different types of switching signals, and the stability issue of the original system is converted into that of a switched system. It is the first time that the method of switched homogeneous polynomial Lyapunov function is applied to general delayed systems. Some sufficient exponential stability conditions for the original system are proposed in several situations. It is numerically shown that the conservativeness of the proposed conditions reduces as the degree of the switched homogeneous polynomial Lyapunov function increases.

Stability Analysis of Discrete-time System with Slowly Time-varying Delays

This paper addresses the stability issue of discrete-time systems with slowly time-varying delays. The basic idea is to convert the stability of the original system into an equivalent switched system, that is, each subsystem of the switched system corresponds to a constant delay of the original system in the entire delay interval via a switching transform approach. Moreover, the method of switched homogeneous polynomial Lyapunov function of higher order is proposed. By this approach, the results proposed are less conservative than the reported results. Finally, an example is provided to illustrate the advantages of the proposed stability results.

Stability Analysis of Switched Continuous-time Systems with Stable and Unstable Subsystems

This paper addresses the stability issue of switched continuous-time linear systems composed of stable and unstable subsystems. By using a class of homogeneous polynomial Lyapunov functions, sufficient stability criterion ensuring the globally asymptotic stability of the considered systems under dwell-time constraint. An example is provided to illustrate the theoretical results in this paper.

Homogeneous polynomial Lyapunov functions for the analysis of polytopic parameter-dependent descriptor systems

In this paper, we present an extension of the homogeneous polynomial Lyapunov functions (HPLFs) approach previously developed for uncertain standard systems to the case of uncertain continuous-time descriptor ones. By employing a new power transformation of the state vector with respect to its dynamic and algebraic parts, new necessary and sufficient admissibility analysis conditions are developed based on such a class of Lyapunov functions. The presented results deal with time-varying parameters with both unbounded variation rates, based on parameter-independent HPLFs, and bounded variation rates lying inside a polytope by exploiting polytopic parameter-dependent HPLFs. Numerically, well tractable sufficient LMI conditions are also derived for both cases. A comparison with results from the literature is performed, showing the reduction of conservatism throughout the novel conditions presented in this article.

Stability and stabilisation of heterogeneous switched systems with arbitrary switching constraints

Heterogeneous switched systems are a class of special hybrid systems containing different dimensional subsystems. To research the stability and stabilisation of heterogeneous switched systems, a novel approach is presented to design static output feedback controllers for asymptotically stabilising heterogeneous continuous-time and discrete-time switched systems under arbitrary switching rules. This approach, which has less computation complexity, includes a main diagonal square block common Lyapunov function for stability analysis of heterogeneous switched systems and a main diagonal square block common homogeneous polynomial Lyapunov function for stabilisation of heterogeneous switched systems. In the study of stabilisation of heterogeneous switched systems, a two-step iterative method including two convex optimisation functions is established by sums of squares to express existing properties condition and solvable properties condition for determining block controller. At last, a practical example and two numerical examples are presented to show the feasibility of the proposed results.

Stability and Stabilization of Heterogeneous Switched Systems with Mode-Dependent Average Dwell Time via Homogeneous Polynomial Lyapunov Functions Approach

This work researches the problem of searching for multiple homogeneous polynomial Lyapunov functions (HPLFs) for heterogeneous switched linear systems. First, a nonconvex optimization condition is constructed to study the stability property of heterogeneous switched systems, where each Lyapunov function candidate reduces dimension to their corresponding matrix eigenvalue. Based on the stability analysis condition, a controller-dependently multiple HPLFs condition is introduced to determine controllers and explores locally minimum mode-dependent average dwell time (LMMDADT). Additionally, the existing properties condition and solvable properties condition of controllers are given in the form of HPLFs. At last, a practical example and a contrast example are both presented to show feasibility of the proposed results.

Necessary and Sufficient Conditions for $2p$th Moment Stability of Several Classes of Linear Stochastic Systems

This technical note presents necessary and sufficient conditions for $2p$ th moment stability of several widely used linear stochastic systems. By the matrix derivative operator and vectorial Ito's formula, we first derive two extended systems: one is described by stochastic differential equation (SDE) and the other by ordinary differential equation (ODE). It is shown that the stochastic system is $2p$ th-moment exponentially stable if and only if the SDE is mean-square exponentially stable or the ODE is exponentially stable. Subsequently, combining the quasi-periodic homogeneous polynomial Lyapunov function methods with the proposed techniques, a series of nonconservative stability criteria are established for linear impulsive, sampled-data, and switched stochastic systems under dwell-time constraints. A numerical example illustrates the proposed theoretical results.

Polynomial static output feedback H∞ control via homogeneous Lyapunov functions

SummaryIn this paper, we study a polynomial static output feedback (SOF) stabilization problem with H∞ performance via a homogeneous polynomial Lyapunov function (HPLF). It is shown that the quadratic stability ascertaining the existence of a single constant Lyapunov function becomes a special case. With the HPLF, the proposal is based on a relaxed two‐step sum of square (SOS) construction where a stabilizing polynomial state feedback gain K(x) is returned at the first stage and then the obtained K(x) gain is fed back to the second stage, achieving the SOF closed‐loop stabilization of the underlying polynomial fuzzy control systems. The SOS equations obtained thus effectively serve as a sufficient condition for synthesizing the SOF controllers that guarantee polynomial fuzzy systems stabilization. To demonstrate the effectiveness of the proposed polynomial fuzzy SOF H∞ control, benchmark examples are provided for the new approach.

Iterative Homogeneous Polynomial Lyapunov Functions Approach for Stabilizing Minimum Mode-Dependent Average Dwell Time Switched Systems via Output Feedback

A novel approach is proposed in this paper to design static output feedback controllers for asymptotically stabilizing continuous-time switched linear systems with minimum mode-dependent average dwell time (MMDADT). This approach adopts an iterative algorithm containing the solution of a traversal algorithm function and three optimization functions, the decision variables of which are established by the sum of squares (SOS) of matrix polynomials. The traversal algorithm function takes advantage of a polynomially parameter-bounded condition, allowing us to get the lower-bound of the Homogeneous Polynomial Lyapunov Functions’ (HPLFs) derivative. The first optimization function is a non-convex optimization function, which expresses a sufficient condition for stability analysis and computing MMDADT. The second and third optimization functions are both convex optimization functions, which are used to calculate the polynomially mode-dependent output feedback controllers. Two numerical examples are presented in order to show the feasibility of the proposed results.

Convex Synthesis of Robust Controllers for Linear Systems With Polytopic Time-Varying Uncertainty

The design of robust controllers for linear systems with structured time-varying uncertainty is a fundamental problem in control systems. This problem is still open because the available methods for robust analysis lead to nonconvex optimization whenever design variables are present. This paper proposes a novel approach that overcomes this difficulty. Specifically, continuous-time linear systems with polytopic time-varying uncertainty are considered. The basic problem consists of designing a robust static output feedback controller ensuring robust asymptotical stability. A novel approach is proposed through the introduction of robust stabilizability functions (RSFs), i.e., functions able to establish whether a controller is robustly asymptotically stabilizing. In particular, RSFs based on homogeneous polynomial Lyapunov functions (HPLFs) are searched for. The proposed approach requires the solution of two convex optimization problems with constraints expressed as linear matrix inequalities (LMIs). For any size of the LMIs, the proposed approach provides a sufficient condition for the existence of a sought controller. Moreover, this condition is also necessary when the size of the LMIs is large enough. Several extensions of the proposed methodology are presented, which deal with the synthesis of controllers ensuring desired decay rate, fixed-order dynamic output controllers, and controllers with the minimum norm.

Estimation of asymptotic stability regions via composite homogeneous polynomial Lyapunov functions

In this article, we present a new method to estimate the asymptotic stability regions for a class of nonlinear systems via composite homogeneous polynomial Lyapunov functions, where these nonlinear systems are approximated as a convex hull of some linear systems. Since level set of the composite homogeneous polynomial Lyapunov functions is a union set of several homogeneous polynomial functions, the composite homogeneous polynomial Lyapunov functions are nonconservative compared with quadratic or homogeneous polynomial Lyapunov functions. Numerical examples are used to illustrate the effectiveness of our method.

Establishing robust stability of discrete-time systems with time-varying uncertainty: The Gram-SOS approach

This paper addresses the problem of establishing robust asymptotical stability of discrete-time linear systems polynomially affected by time-varying uncertainty confined into a polytope. A linear matrix inequality (LMI) condition for establishing robust asymptotical stability is proposed by introducing a novel approach for establishing the existence of a common homogeneous polynomial Lyapunov function (HPLF). This approach consists, firstly, of introducing a Gram matrix built with respect to the state and parametrized by an arbitrary vector function of the uncertainty, and secondly, of requiring that a transformation of the introduced Gram matrix is a sum of squares (SOS) of matrix polynomials. The approach, hence, is referred to as a Gram-SOS approach. It is shown that the proposed LMI condition is sufficient for any degree of the HPLF candidate, that includes quadratic robust stability as a special case, and that is also necessary for a sufficiently large degree of the HPLF candidate. Numerical examples also show that the proposed LMI condition can outperform alternative ones in terms of conservatism and computational burden.