Lyapunov Functions, Stability and Input-to-State Stability Subtleties for Discrete-Time Discontinuous Systems

In this note we consider stability analysis of discrete-time discontinuous systems using Lyapunov functions. We demonstrate via simple examples that the classical second method of Lyapunov is precarious for discrete-time discontinuous dynamics. Also, we indicate that a particular type of Lyapunov condition, slightly stronger than the classical one, is required to establish stability of discrete-time discontinuous systems. Furthermore, we examine the robustness of the stability property when it was attained via a discontinuous Lyapunov function, which is often the case for discrete-time hybrid systems. In contrast to existing results based on smooth Lyapunov functions, we develop several input-to-state stability tests that explicitly employ an available discontinuous Lyapunov function.     

On integral input-to-state stability for a feedback interconnection of parameterised discrete-time systems

This paper addresses integral input-to-state stability (iISS) for a feedback interconnection of parameterised discrete-time systems involving two subsystems. Particularly, we give a construction for a smooth iISS Lyapunov function for the whole system from the sum of nonlinearly weighted Lyapunov functions of individual subsystems. Motivations for such a construction are given. We consider two main cases. The first one investigates iISS for the whole system when both subsystems are iISS. The second one gives iISS for the interconnected system when one of subsystems is allowed to be input-to-state stable. The approach is also valid for both discrete-time cascades and a feedback interconnection of iISS and static systems. Examples are given to illustrate the effectiveness of the results.     

Changing supply rates for input-output to state stable discrete-time nonlinear systems with applications

We present results on changing supply rates for input-output to state stable discrete-time nonlinear systems. Our results can be used to combine two Lyapunov functions, none of which can be used to verify that the system has a certain property, into a new composite Lyapunov function from which the property of interest can be concluded. The results are stated for parameterized families of discrete-time systems that naturally arise when an approximate discrete-time model is used to design a controller for a sampled-data system. We present several applications of our results: (i) a LaSalle criterion for input to state stability (ISS) of discrete-time systems; (ii) constructing ISS Lyapunov functions for time-varying discrete-time cascaded systems; (iii) testing ISS of discrete-time systems using positive semidefinite Lyapunov functions; (iv) observer-based input to state stabilization of discrete-time systems. Our results are exploited in a case study of a two-link manipulator and some simulation results that illustrate advantages of our approach are presented.     

Further Results on “Infinity Norms as Lyapunov Functions for Linear Systems”

This note continues on the results proposed in two previous papers for computing vector 1- or infin-norm based Lyapunov functions for linear discrete-time systems by presenting a finitely terminating algorithm for the construction of Lyapunov functions of that class. The algorithm utilizes the solution of a finite sequence of feasibility linear programs (LPs) with very few constraints or, equivalently, very simple algebraic tests in contrast to solving the original problem where a bilinear matrix equation with rank and norm constraint needs to be solved     

Robust performance analysis for linear continuous/discrete time systems with linear dependent and time-varying perturbations

In this paper, both continuous and discrete-time perturbed systems subject to linear dependent and time-varying perturbations are considered. The time domain performance which related to the convergence rate of the initial states is defined based on the continuous/discrete-time Lyapunov functions. The matrix measure is used to derive the robust performance of a perturbed system.     

A multivariable extension of the Tsypkin criterion using aLyapunov-function approach

For analyzing the stability of discrete-time systems containing a feedback nonlinearity, the Tsypkin criterion is the closest analog to the Popov criterion which is used for analyzing such systems in continuous time. Traditionally, the proof of this criterion is based upon input-output properties and function analytic methods. In this paper we extend the Tsypkin criterion to multivariable systems containing an arbitrary number of monotonic sector-bounded memoryless time-invariant nonlinearities, along with providing a Lyapunov function proof for this classical result     

Discrete-time Lyapunov-based small-gain theorem for parameterized interconnected ISS systems

Input-to-state stability (ISS) of a feedback interconnection of two discrete-time ISS systems satisfying an appropriate small gain condition is investigated via the Lyapunov method. In particular, an ISS Lyapunov function for the overall system is constructed from the ISS Lyapunov functions of the two subsystems. We consider parameterized families of discrete-time systems that naturally arise when an approximate discrete-time model is used in controller design for a sampled-data system.     

Explicit construction of quadratic lyapunov functions for the small gain,positivity, circle,and popov theorems and their application to robust stability. part II: Discrete-time theory

In a companion paper (‘Explicit construction of quadratic Lyapunov functions for the small gain, positivity, circle, and Popov theorems and their application to robust stability. Part I: Continuous-time theory’), Lyapunov functions were constructed in a unified framework to prove sufficiency in the small gain, positivity, circle, and Popov theorems. In this Part II, analogous results are developed for the discrete-time case. As in the continuous-time case, each result is based upon a suitable Riccati-like matrix equation that is used to explicitly construct a Lyapunov function that guarantees asymptotic stability of the feedback interconnection of a linear time-invariant system and a memoryless nonlinearity. Multivariable versions of the discrete-time circle and Popov criteria are obtained as extensions of known results. Each result is specialized to the linear uncertainty case and connections with robust stability for state-space systems is explored.     

Stabilisation of discrete-time polynomial fuzzy systems via a polynomial lyapunov approach

This paper deals with the problem of designing a controller for a class of discrete-time nonlinear systems which is represented by discrete-time polynomial fuzzy model. Most of the existing control design methods for discrete-time fuzzy polynomial systems cannot guarantee their Lyapunov function to be a radially unbounded polynomial function, hence the global stability cannot be assured. The proposed control design in this paper guarantees a radially unbounded polynomial Lyapunov functions which ensures global stability. In the proposed design, state feedback structure is considered and non-convexity problem is solved by incorporating an integrator into the controller. Sufficient conditions of stability are derived in terms of polynomial matrix inequalities which are solved via SOSTOOLS in MATLAB. A numerical example is presented to illustrate the effectiveness of the proposed controller.     

On infinity norms as Lyapunov functions for linear systems

The note presents a construction of the Lyapunov function defined by l_∞ⁿ vector norm for both continuous- and discrete-time linear systems. In contrast to the previous results, no restrictions on the positions or multiplicity of system matrix eigenvalues are required     

Robustness of stability through necessary and sufficient Lyapunov-like conditions for systems with a continuum of equilibria

The equivalence between robustness to perturbations and the existence of a continuous Lyapunov-like mapping is established in a setting of multivalued discrete-time dynamics for a property sometimes called semistability. This property involves a set consisting of Lyapunov stable equilibria and surrounded by points from which every solution converges to one of these equilibria. As a consequence of the main results, this property turns out to always be robust for continuous nonlinear dynamics and a compact set of equilibria. Preliminary results on reachable sets, limits of solutions, and set-valued Lyapunov mappings are included.     

Piecewise-affine Lyapunov functions for discrete-time linear systems with saturating controls

This paper is concerned with piecewise-affine (PWA) functions as Lyapunov function candidates for stability analysis of time-invariant discrete-time linear systems with saturating closed-loop control inputs. Using a PWA model of saturating closed-loop system, new necessary and sufficient conditions for a PWA function be a Lyapunov function are presented. Based on linear programming formulation of these conditions, an effective algorithm is proposed for construction of such Lyapunov functions for estimation of the region of local asymptotic stability. Compared to piecewise-linear functions, like Minkowski functions, PWA functions are more adequate to capture the dynamical effects of saturation nonlinearities, giving strictly less conservative results. The complexity of the proposed approach is polynomial in state dimension and exponential in saturating control dimension, being hence appropriate for problems with large state dimension but with few saturating inputs.     

基于区域极点约束的鲁棒H2控制

考虑了在区域极点约束下状态反馈的鲁棒H2控制问题.分别对含多面体不确定性的连续和离散系统进行了讨论.基于LMI,给出了存在参数相关的Lyapunov矩阵的充分条件.利用LMI凸优化方法的解,所得静态反馈控制器,不仅保证闭环系统的极点在-给定区域内,而且还使性能指标H2的一上界达到最小.     

Constructive Lyapunov stabilization of nonlinear cascade systems

We present a global stabilization procedure for nonlinear cascade and feedforward systems which extends the existing stabilization results. Our main tool is the construction of a Lyapunov function for a class of (globally stable) uncontrolled cascade systems. This construction serves as a basis for a recursive controller design for cascade and feedforward systems. We give conditions for continuous differentiability of the Lyapunov function and the resulting control law and propose methods for their exact and approximate computation     

A design methodology for switched discrete time linear systems with applications to automotive roll dynamics control

In this paper we consider the asymptotic stability of a class of discrete-time switching linear systems, where each of the constituent subsystems is Schur stable. We first present an example to motivate our study, which illustrates that the bilinear transform does not preserve the stability of a class of switched linear systems. Consequently, continuous time stability results cannot be transformed to discrete time analogs using this transformation. We then present a subclass of discrete-time switching systems that arise frequently in practical applications. We prove that global attractivity for this subclass can be obtained without requiring the existence of a common quadratic Lyapunov function (CQLF). Using this result, we present a synthesis procedure to construct switching stabilizing controllers for an automotive control problem, which is related to the stabilization of a vehicle’s roll dynamics subject to switches in the center of gravitys (CG) height.     

Transition probability bounds for the stochastic stability robustness of continuous- and discrete-time Markovian jump linear systems

This paper considers the robustness of stochastic stability of Markovian jump linear systems in continuous- and discrete-time with respect to their transition rates and probabilities, respectively. The continuous-time (discrete-time) system is described via a continuous-valued state vector and a discrete-valued mode which varies according to a Markov process (chain). By using stochastic Lyapunov function approach and Kronecker product transformation techniques, sufficient conditions are obtained for the robust stochastic stability of the underlying systems, which are in terms of upper bounds on the perturbed transition rates and probabilities. Analytical expressions are derived for scalar systems, which are straightforward to use. Numerical examples are presented to show the potential of the proposed techniques.     

Robust performance analysis and synthesis of linear polytopic discrete-time periodic systems via LMIs

A particular class of uncertain linear discrete-time periodic systems is considered. The problem of robust stabilization of real polytopic linear discrete-time periodic systems via a periodic state-feedback control law is tackled here, along with performance optimization. Using additional slack variables and the periodic Lyapunov lemma, an extended sufficient condition of robust stabilization is proposed. Based on periodic parameter-dependent Lyapunov functions, this last condition is shown to be always less conservative than the more classic one based on the quadratic stability framework. This is illustrated on a numerical example from the literature.     

Adaptive feedback synchronisation of complex dynamical network with discrete-time communications and delayed nodes

This paper considered the synchronisation of continuous complex dynamical networks with discrete-time communications and delayed nodes. The nodes in the dynamical networks act in the continuous manner, while the communications between nodes are discrete-time; that is, they communicate with others only at discrete time instants. The communication intervals in communication period can be uncertain and variable. By using a piecewise Lyapunov–Krasovskii function to govern the characteristics of the discrete communication instants, we investigate the adaptive feedback synchronisation and a criterion is derived to guarantee the existence of the desired controllers. The globally exponential synchronisation can be achieved by the controllers under the updating laws. Finally, two numerical examples including globally coupled network and nearest-neighbour coupled networks are presented to demonstrate the validity and effectiveness of the proposed control scheme.     

跳变双线性随机系统饱和执行器的镇定

对于一类具有Markov跳变参数的双线性离散随机系统,研究其饱和执行器问题.分别采用一般二次型Lyapunov函数、饱和关联Lyapunov函数进行系统随机稳定性分析,以椭圆不变集构造随机稳定域,提出两种依赖于模态跳变率的饱和状态控制器设计方法,两种方法均以线性矩阵不等式的形式给出.     

Universal controllers for robust control problems

In this paper we discuss the construction of “universal” controllers for a class of robust stabilization problems. We give a general theorem on the construction of these controllers, which requires that a certain nonlinear inequality is solvablepointwisely or, equivalently, that arobust control Lyapunov function does exist. The constructive procedure producesalmost smooth controllers. The robust control Lyapunov functions extend to uncertain systems the concept of control Lyapunov functions. If such a robust control Lyapunov function also satisfies a small control property, the resulting stabilizing controller is also continuous in the origin of the state space. Applications of our results range from optimal to robust control.