Piecewise Quadratic Lyapunov Functions Research Articles

Graph-based conditions for feedback stabilization of switched and LPV systems

This paper presents novel stabilizability conditions for switched linear systems with arbitrary and uncontrollable underlying switching signals. We distinguish and study two particular settings: (i) the robust case, in which the active mode is completely unknown and unobservable, and (ii) the mode-dependent case, in which the controller depends on the current active switching mode. The technical developments are based on graph-theory tools, relying in particular on the path-complete Lyapunov functions framework. The main idea is to use directed and labeled graphs to encode Lyapunov inequalities to design robust and mode-dependent piecewise linear state-feedback controllers. This results in novel and flexible conditions, with the particular feature of being in the form of linear matrix inequalities (LMIs). Our technique thus provides a first controller-design strategy allowing piecewise linear feedback maps and piecewise quadratic (control) Lyapunov functions by means of semidefinite programming. Numerical examples illustrate the application of the proposed techniques, the relations between the graph order, the robustness, and the performance of the closed loop.

New Results on Nonsynchronous-Observer-Based Output-Feedback Control of Fuzzy-Affine-Model-Based Discrete-Time Nonlinear Systems

This work investigates the non-synchronous observer-based output feedback control of nonlinear systems by T-S fuzzy affine models. In order to estimate the unmeasurable system states and handle the non-synchronization issue of premise variables between the observer/controller and the original system, a new non-synchronous piecewise fuzzy affine observer is firstly synthesized to estimate the unmeasurable system state variables. Via utilizing some elegant matrix inequalities with convexification techniques, new observer-based piecewise output feedback controller design results are proposed within the framework of piecewise quadratic Lyapunov functions. Simulation studies are given to illustrate the validity of the developed approach.

A New Design of Fuzzy Affine Model-Based Output Feedback Control for Discrete-Time Nonlinear Systems

The observer-based output feedback control problem for nonlinear systems is studied through Takagi-Sugeno fuzzy affine models. A novel piecewise fuzzy affine observer is designed to estimate the unmeasurable state variables. To remove the assumptions in the existing works that the system input matrices and/or output matrices should be common, full rank and uncertainties-free, a new approach is developed for the observer-based piecewise output feedback controller synthesis by making sufficient use of Young's inequality with its variance. Through fuzzy piecewise quadratic Lyapunov functions, improved and less conservative results on the output feedback controller design are obtained in terms of linear matrix inequalities, and the restrictions imposed in the existing results can be relaxed. Simulation studies are provided to confirm the effectiveness and conservatism reduction of the developed method.

Switching Control Strategy for DC–DC Converters Based on Polynomial Lyapunov Function and Sum-of-Squares Approach

The introduction of switched system theory opens new perspectives for the control design of dc/dc converters. Regarding the classical two-level converters, two switching control design methodologies have been developed based on the common quadratic Lyapunov function (CQLF) or piecewise quadratic Lyapunov function (PQLF). However, the CQLF-based design method is not applicable for the multilevel dc/dc converter (MC). As for the PQLF-based design method, it encounters the problem that the control design is very complex, involving a nonlinear optimization problem. To further advance the application of the switching system theory in power converters, a design framework based on the polynomial Lyapunov function and the sum-of-squares approach is proposed in this article. The proposed design method generalizes the CQLF-based one. Moreover, compared with the PQLF-based design method, the mathematical design procedure can be significantly simplified. Experimental validations on an MC, i.e., the three-level flying capacitor converter and a two-level Buck-Boost converter illustrate the versatility and the effectiveness of the proposed design framework. Moreover, comparative tests show the good control performance of the designed switching control strategy. The currently presented works are preliminary but could constitute an interesting direction for developing control solutions of power converters.

Fuzzy-Affine-Model-Based Sliding-Mode Control for Discrete-Time Nonlinear 2-D Systems via Output Feedback.

This work investigates the issue of output-feedback sliding-mode control (SMC) for nonlinear 2-D systems by Takagi-Sugeno fuzzy-affine models. Via combining with the sliding surface, the sliding-mode dynamical properties are depicted by a singular piecewise-affine system. Through piecewise quadratic Lyapunov functions, new stability and robust performance analysis of the sliding motion are carried out. An output-feedback dynamic SMC design approach is developed to guarantee that the system states can converge to a neighborhood of the sliding surface. Simulation studies are given to verify the validity of the proposed scheme.

ℋ2 reset controller design for linear systems using piecewise quadratic Lyapunov functions

SummaryThis study presents reset controller design conditions for multiple‐input and multiple‐output (MIMO) linear systems in terms of linear matrix inequalities (LMIs). The design conditions were developed using a piecewise quadratic Lyapunov function (PWQLF), leading to less conservative conditions than using a common Lyapunov function. Achieving LMI relaxation of the PWQLF‐based reset controller design conditions for MIMO linear systems is challenging because (1) dividing the flow set of the reset condition is highly complex, (2) equality constraints, which make the PWQLF continuous on the boundaries between adjacent flow sets, prevent the use of congruence transformation in matrix inequalities, and (3) indefiniteness of the matrices for regional constraints also makes it difficult to apply standard LMI relaxation techniques. To overcome these difficulties, this study introduces a two‐dimensional projection of the plant's output measurements and reset controller's states to construct a reset condition, allowing the PWQLF to be tractable in MIMO linear systems. Then, a new stability analysis condition is derived using the elimination lemma, overcoming the challenges in LMI relaxation of the controller design attributed to the equality constraints and indefinite matrices. An LMI‐based reset controller design condition is then proposed based on the proposed stability analysis. To account for transient behavior, tracking reset controller design problem is addressed. A numerical example is provided to demonstrate the effectiveness of the proposed method.

Output-based event-triggered control for discrete-time systems with three types of performance analysis

<abstract><p>This paper considers an output-based event-triggered control approach for discrete-time systems and proposes three new types of performance measures under unknown disturbances. These measures are motivated by the fact that signals in practical systems are often associated with bounded energy or bounded magnitude, and they should be described in the $ \ell_{2} $ and $ \ell_\infty $ spaces, respectively. More precisely, three performance measures from $ \ell_{q} $ to $ \ell_{p} $, denoted by the $ \ell_{p/q} $ performances with $ (p, q) = (2, 2), \ (\infty, 2) $ and $ (\infty, \infty) $, are considered for event-triggered systems (ETSs) in which the corresponding event-trigger mechanism is defined as a function from the measured output of the plant to the input of the dynamic output-feedback controller with the triggering parameter $ \sigma (>0) $. Such a selection of the pair $ (p, q) $ represents the $ \ell_{p/q} $ performances to be bounded and well-defined, and the three measures are natural extensions of those in the conventional feedback control, such as the $ H_\infty $, generalized $ H_2 $ and $ \ell_1 $ norms. We first derive the corresponding closed-form representation with respect to the relevant ETSs in terms of a piecewise linear difference equation. The asymptotic stability condition for the ETSs is then derived through the linear matrix inequality approach by developing an adequate piecewise quadratic Lyapunov function. This stability criterion is further extended to compute the $ \ell_{p/q} $ performances. Finally, a numerical example is given to verify the effectiveness of the overall arguments in both the theoretical and practical aspects, especially for the trade-off relation between the communication costs and $ \ell_{p/q} $ performances.</p></abstract>

Distributed Event-Triggered Synchronization of Interconnected Linear Two-Time-Scale Systems With Switching Topology.

This article investigates the synchronization problem of interconnected linear two-time-scale systems (TTSSs) with switching topology. By utilizing the Chang transformation, a distributed synchronization protocol is proposed with event-triggered communication. Static and dynamic event-triggered mechanisms are proposed successively, which both contain two separated event-triggering conditions corresponding to the slow and the fast subsystems. The existence of a strictly positive time period between any two successive transmissions is ensured regardless of the initial states. The main difficulty of this study lies in that the state jump and parametric uncertainty appear because of the system transformation. To overcome the difficulty, the system is first modeled as an uncertain hybrid system. Then, the control gain is properly designed by solving Riccati-like equations dependent on the rough bounds of the eigenvalues of communication graph Laplacians, and a piecewise quadratic Lyapunov function is proposed with which the jump caused by the switching topology is subtly evaluated. Sufficient conditions are thus established to achieve the event-triggered synchronization. Furthermore, the results are also extended to solve the synchronization problem of the interconnected impulsive linear TTSSs. Finally, three numerical examples are provided to demonstrate the effectiveness of the proposed theoretical results.

Event-triggered control for discrete-time piecewise affine systems

This work addresses the event-triggered control (ETC) of discrete-time piecewise affine systems. We propose a method to design a triggering strategy relying on an implicit representation of piecewise affine systems. Thanks to this implicit representation based on ramp functions, we propose a partition-dependent piecewise quadratic functions to define the trigger criterion and use a piecewise quadratic Lyapunov function candidate to derive conditions to certify the global exponential stability of the origin under the ETC strategy. Since the stability conditions can be expressed as linear matrix inequalities constraints, we propose a convex optimization solution to design the triggering function parameters and to compute the Lyapunov function to ensure the closed-loop stability and a reduction on the control updates. The approach is illustrated by numerical examples.

A piecewise ellipsoidal reachable set estimation method for continuous bimodal piecewise affine systems

In this work, the issue of estimation of reachable sets in continuous bimodal piecewise affine systems is studied. A new method is proposed, in the framework of ellipsoidal bounding, using piecewise quadratic Lyapunov functions. Although bimodal piecewise affine systems can be seen as a special class of affine hybrid systems, reachability methods developed for affine hybrid systems might be inappropriately complex for bimodal dynamics. This work goes in the direction of exploiting the dynamical structure of the system to propose a simpler approach. More specifically, because of the piecewise nature of the Lyapunov function, we first derive conditions to ensure that a given quadratic function is positive on half spaces. Then, we exploit the property of bimodal piecewise quadratic functions being continuous on a given hyperplane. Finally, linear matrix characterizations of the estimate of the reachable set are derived.

A variable gain impulsive observer for Lipschitz nonlinear systems with measurement noises

This paper investigates the variable gain impulsive observer design problem for Lipschitz nonlinear systems. It is assumed that the measurements are contaminated by noise and received by observer at aperiodic instants. To establish a tractable design condition for impulsive observers, the piecewise linear interpolation method is used to construct the variable gain function. To quantify the impact of the measurement noises and exogenous disturbance on the estimation error, a Lyapunov-based condition for establishing exponential input-to-state stability (EISS) property of the observation error dynamics is presented. Then it is shown that the EISS condition can be expressed as a set of linear matrix inequalities (LMIs) by introducing a piecewise quadratic Lyapunov function. A convex optimization problem is proposed in which the EISS gain is minimized. Comparisons with the existing methods show the effectiveness of the proposed design technique.

Asynchronous fault detection filtering design for continuous-time T-S fuzzy affine dynamic systems in finite-frequency domain

This paper investigates the problem of asynchronous fault detection filtering design for continuous-time Takagi-Sugeno (T-S) fuzzy affine dynamic systems in finite-frequency domain. The objective is to design an admissible piecewise affine filter such that the asymptotic stability of the filtering error system with the prescribed finite-frequency H−/H∞ performance can be guaranteed. It is assumed that the premise variables of the plant are unmeasurable so that the filter state transition and the plant state transition may be asynchronous. By applying the celebrated S-procedure, the generalized Kalman-Yakubovič-Popov lemma is extended such that the finite-frequency H−/H∞ performance of the fuzzy affine filtering error system is ensured. Furthermore, by utilizing piecewise quadratic Lyapunov functions, Projection lemma, and some matrix inequality linearization techniques, the finite-frequency fault detection filtering design approach is developed for the concerned T-S fuzzy affine dynamic system. It is shown that sufficient conditions for the existence of the fault detection filter are formulated as feasibility of a set of linear matrix inequalities. Finally, the effectiveness and advantages of the proposed design approach are illustrated by simulation studies.

Stability and performance analysis of hybrid integrator–gain systems: A linear matrix inequality approach

Hybrid integrator–gain systems (HIGS) are nonlinear elements designed for overcoming fundamental limitations of linear time-invariant integrators. This paper presents numerically robust conditions for time-domain stability and performance of HIGS-controlled systems. In particular, using piecewise quadratic (PWQ) Lyapunov functions defined over polyhedral subregions of the state-space, conditions for stability and computations of upper-bounds on the L2-gain and H2-norm are formulated as convex optimization problems in terms of numerically tractable linear matrix inequalities (LMIs). In order to improve accuracy and robustness, the LMIs are constructed in a manner to eliminate explicit equality constraints typically related to continuity of the PWQ functions. Novel conditions are presented that guide further refinement of the subregions over which the PWQ Lyapunov functions are defined in order to increase the accuracy of the approach. The effectiveness of the presented analysis tools is demonstrated through a numerical case-study on a motion system.

Observer-Based Output Feedback Control of Nonlinear 2-D Systems via Fuzzy-Affine Models

Through Takagi-Sugeno fuzzy affine models, the observer-based output feedback control of discrete-time nonlinear two-dimensional systems is investigated, where the two-dimensional system information evolves dynamically along two independent directions. A piecewise fuzzy affine observer is designed to estimate the unmeasurable system states, and two enhanced observer-based piecewise output feedback control approaches are then developed. By making sufficient use of Young’s matrix inequality with its variant, novel and less conservative observer-based piecewise output feedback controller design results are proposed via piecewise quadratic Lyapunov functions. Simulation studies on a doubly-indexed nonlinear process are provided to illustrate the effectiveness of the developed approaches.

Projection-based integrators for improved motion control: Formalization, well-posedness and stability of hybrid integrator-gain systems

In this paper we formally describe the hybrid integrator-gain system (HIGS), which is a nonlinear integrator designed to avoid the limitations typically associated with linear integrators. The HIGS keeps the sign of its input and output equal, thereby inducing less phase lag than a linear integrator, much like the famous Clegg integrator. The HIGS achieves the reduced phase lag by projection of the controller dynamics instead of using resets of the integrator state, which forms a potential benefit of this control element. To formally analyze HIGS-controlled systems, we present an appropriate mathematical framework for describing these novel systems. Based on this framework, HIGS-controlled systems are proven to be well-posed in the sense of existence and forward completeness of solutions. Moreover, we propose two approaches for analyzing (input-to-state) stability of the resulting nonlinear closed-loop systems: (i) circle-criterion-like conditions based on (measured) frequency response data, and (ii) LMI-based conditions exploiting a new construction of piecewise quadratic Lyapunov functions. A motion control example is used to illustrate the results.

Asymptotic Stability of Piecewise Affine Systems With Filippov Solutions via Discontinuous Piecewise Lyapunov Functions

Asymptotic stability of continuous-time piecewise affine systems defined over a polyhedral partition of the state space, with possible discontinuous vector field on the boundaries, is considered. In this article, the feasible Filippov solution concept is introduced by characterizing single-mode Caratheodory, sliding mode, and forward Zeno behaviors. Then, a global asymptotic stability result through a (possibly discontinuous) piecewise Lyapunov function is presented. The sufficient conditions are based on pointwise classifications of the trajectories which allow the identification of crossing, unreachable, and Caratheodory boundaries. It is shown that the sign and jump conditions of the stability theorem can be expressed in terms of linear matrix inequalities by particularizing to piecewise quadratic Lyapunov functions and using the cone-copositivity approach. Several examples illustrate the theoretical arguments and the effectiveness of the stability result.

Annular Stochastic Finite-Time Stability Using Piecewise Quadratic Lyapunov Functions

In this letter we present some new sufficient conditions for the annular stochastic finite-time stability of a class of stochastic linear time-varying systems. These new conditions are obtained adopting time-varying piecewise quadratic Lyapunov functions rather than the classical quadratic ones. The proposed approach allows us to extend the class of consider domains, which are typically limited to ellipsoidal domains. The proposed finite-time stability conditions can be converted into a feasibility problem based on a set of differential linear matrix inequalities. Two numerical examples are considered to perform a comparison with the previous results, and they show that the new proposed conditions are less conservative than the previous ones.

A Novel Fuzzy Output Feedback Dynamic Sliding Mode Controller Design for Two-Dimensional Nonlinear Systems

The system information of 2-D systems is usually in propagation along two independent directions. This article focuses on the issue of output feedback sliding mode control (SMC) for 2-D nonlinear systems through T-S fuzzy affine models. Associated with the sliding surface function, a singular system is established to describe the sliding mode dynamics. Based on piecewise quadratic Lyapunov functions, some new stability analysis results on the sliding mode dynamical system are attained, and the analysis conservatism can be further reduced. A robust output feedback dynamic SMC synthesis scheme is developed to guarantee the finite-time reachability of the sliding surface. It is also worthy of mentioning that the proposed dynamic SMC synthesis method can handle the mismatched disturbances, and the control input channels are permitted to have parameter uncertainties. Simulation studies are given to illustrate the validity of the developed scheme.

A new design of output feedback sliding mode controller for T‐S fuzzy‐affine systems

AbstractThis work focuses on output feedback sliding mode control (SMC) for nonlinear systems via T‐S fuzzy affine models. Associated with the sliding surface, a singular system is firstly proposed to describe the dynamical features of the sliding motion. Adopting some convexification techniques and based on piecewise quadratic Lyapunov functions (PQLFs) and a common quadratic Lyapunov function (CQLF), sufficient criteria for the asymptotic stability analysis of the sliding mode dynamical system are attained. Two output feedback SMC synthesis schemes are developed to drive the closed‐loop system states to the sliding surface in finite time. A simulation example is presented to show the validity of the proposed methods.

A Lyapunov Approach for Time-Bounded Reachability of CTMCs and CTMDPs

Time-bounded reachability is a fundamental problem in model checking continuous-time Markov chains (CTMCs) and Markov decision processes (CTMDPs) for specifications in continuous stochastic logics. It can be computed by numerically solving a characteristic linear dynamical system, but the procedure is computationally expensive. We take a control-theoretic approach and propose a reduction technique that finds another dynamical system of lower dimension (number of variables), such that numerically solving the reduced dynamical system provides an approximation to the solution of the original system with guaranteed error bounds. Our technique generalizes lumpability (or probabilistic bisimulation) to a quantitative setting. Our main result is a Lyapunov function characterization of the difference in the trajectories of the two dynamics that depends on the initial mismatch and exponentially decreases over time. In particular, the Lyapunov function enables us to compute an error bound between the two dynamics as well as a convergence rate. Finally, we show that the search for the reduced dynamics can be computed in polynomial time using a Schur decomposition of the transition matrix. This enables us to efficiently solve the reduced dynamical system by computing the exponential of an upper-triangular matrix characterizing the reduced dynamics. For CTMDPs, we generalize our approach using piecewise quadratic Lyapunov functions for switched affine dynamical systems. We synthesize a policy for the CTMDP via its reduced-order switched system that guarantees that the time-bounded reachability probability lies above a threshold. We provide error bounds that depend on the minimum dwell time of the policy. We demonstrate the technique on examples from queueing networks, for which lumpability does not produce any state space reduction, but our technique synthesizes policies using a reduced version of the model.<?vsp -2pt?>