Small-gain Condition Research Articles

Incremental Stability of Discrete-Time First-Order Projection Elements

In this paper, we present and analyze a class of discrete-time and sampled-data controllers named first-order projection elements (FOPEs). FOPEs form a class of hybrid controllers that have an underlying linear base dynamics, which are combined with a projection operator that ensures that the input-output (I/O) pair is contained in a well-designed sector. The I/O sector-constraint enables performance benefits in various application domains including motion control of high-precision machines. After the introduction of FOPEs, we focus on incremental stability analysis of these hybrid controllers. We show that the properties of the projection operator ensure incremental stability of the open-loop FOPE dynamics in case of an underlying linear dynamics that is exponentially stable. In this case, we compute tight bounds on the incremental gain. Under an appropriate small-gain condition, we show incremental input-to-state stability of the sampled-data interconnection of a FOPE with a continuous-time linear time-invariant plant. For the case of projected integrator dynamics, we provide a counterexample showing that incremental stability can be compromised. We provide additional, necessary and sufficient conditions under which FOPEs with integrator dynamics are incrementally asymptotically stable. Numerical simulations provide a practical example of the established incremental properties for the trajectories of a dynamical system coming from an industrial motion control application.

A small-gain approach to inverse optimal adaptive control of nonlinear systems with unmodeled dynamics

This work formulates and solves the problem of inverse optimal adaptive control for nonlinear systems with parametric and dynamic uncertainties. A concept and sufficient conditions for solving this problem are first introduced and derived based on adaptive control Lyapunov function method. One difficulty obstructing current works to solve this problem is that inverse optimality and stability of the closed-loop system are not necessarily achieved simultaneously in the presence of both parametric uncertainties and unmodeled dynamics. To this end, a new auxiliary system and a cost functional that puts integral penalty on the state, control, parameter estimate and unmodeled dynamics are proposed to develop a new inverse optimal adaptive feedback approach. Then a small-gain approach is proposed to establish the links between inverse optimality and stability of the closed-loop system. It is proved that if inverse optimality is achieved and the small-gain condition holds, the closed-loop system is uniformly bounded and the system state converges to an arbitrarily small neighborhood of the origin. Finally, an example is given to illustrate the obtained results.

Contraction and [formula omitted]-contraction in Lurie systems with applications to networked systems

A Lurie system is the interconnection of a linear time-invariant system and a nonlinear feedback function. We derive a new sufficient condition for k-contraction of a Lurie system. For k=1, our sufficient condition reduces to the standard stability condition based on the bounded real lemma and a small gain condition. However, Lurie systems often have more than a single equilibrium and are thus not contractive with respect to any norm. For k=2, our condition guarantees a well-ordered asymptotic behavior of the closed-loop system: every bounded solution converges to an equilibrium, which is not necessarily unique. We demonstrate our results by deriving a sufficient condition for k-contraction of a general networked system, and then applying it to guarantee k-contraction in a Hopfield neural network, a nonlinear opinion dynamics model, and a 2-bus power system.

Finite-time stochastic integral input-to-state stability and its applications

In this paper, finite-time stabilization is investigated in depth for stochastic nonlinear systems with stochastic inverse dynamics. The contributions of this work are characterized by the following novel features: (i) Motivated by the concept of stochastic integral input-to-state stability (SiISS) and the existing stochastic finite-time stability theorem, a new concept of finite-time stochastic integral input-to-state stability (FT-SiISS) using Lyapunov function is first introduced, and an inclusion relationship among three types of stochastic stability, i.e., FT-SiISS, FT-SISS and SISS, is rigorously established. On the basis of FT-SiISS, we develop two new FT-SiISS small-gain conditions and discuss their relationship. Then, to analyze finite-time stability of stochastic nonlinear systems with FT-SiISS inverse dynamics, a stochastic finite-time stability theorem is improved. (ii) As the application of (i), we propose a unified finite-time control method, which can simultaneously handle stochastic nonlinear systems with FT-SiISS or FT-SISS inverse dynamics, and guarantee that the closed-loop system has an almost surely continuous solution, all the closed-loop signals are bounded almost surely, and its trivial solution is stochastically finite-time stable.

Robust I&I adaptive tracking control of systems with nonlinear parameterization: An ISS perspective

This paper studies the immersion and invariance (I&I) adaptive tracking problem for a class of nonlinear systems with nonlinear parameterization in the ISS framework. Under some mild assumptions, a novel I&I adaptive control algorithm is proposed, leading to an interconnection of an ISS estimation error subsystem and an ISS tracking error subsystem. Using an ISS small-gain condition, the desired uniform global asymptotic stability of the resulting interconnected “error” system can be achieved and a sum-type strict Lyapunov function can be explicitly constructed. Taking advantage of this ISS-based design framework, it is shown that the corresponding robustness with respect to the input perturbation can be rendered ISS. To remove the need to solve the immersion manifold shaping PDE, a new filter-based approach is proposed, which preserves the ISS-based design framework. Finally, we demonstrate the validness of the proposed framework on a tracking problem for series elastic actuators.

Output Synchronization of Linear Heterogeneous Multi-Agent Systems With Periodic Event-Triggered Output Communication

This paper addresses an event-triggered output synchronization problem of linear heterogeneous multi-agent systems (MASs) under directed graphs, where only the outputs not the internal information of the agents are allowed to be transmitted in the network. The event-triggered output synchronization problem is divided into two subproblems, the event-triggered consensus problem of a group of reference systems and the sampled-data output regulation problem of the local agent viewing its reference system as the exosystem. Under the intermittent output communication, these two subproblems are coupled with each other and need to be solved concurrently. By an event-triggered control law with a periodic ETM, cyclic small gain conditions are derived to guarantee the event-triggered output synchronization of the linear heterogeneous MASs. Two simulation examples presented finally show the effectiveness of the theoretical results.

Synchronization of Frequency-Modulated Multiagent Systems

Oscillation synchronization phenomenon is widely observed in natural systems through frequency modulated signals, especially in biological neural networks. Frequency modulation is also one of most widely used technologies in engineering. However, due to the technical difficulty, oscillations are always simplified as unmodulated sinusoidal-like waves in studying the synchronization mechanism in the bulky literature over the decades. It lacks mathematical principles, especially systems and control theories, for frequency modulated multi-agent systems. This paper aims to bring a new formulation of synchronization of frequency modulated multi-agent systems. It develops new tools to solve the synchronization problem by addressing three issues including frequency observation in nonlinear frequency modulated oscillators subject to network influence, frequency consensus via network interaction subject to observation error, and a well placed small gain condition among them. The architecture of the paper consists of a novel problem formulation, rigorous theoretical development, and numerical verification.

Decentralized Circular Formation Control of Nonholonomic Mobile Robots Under a Directed Sensor Graph

This paper investigates the circular formation control problem of multiple unicycle-type mobile robots under a directed sensor graph. The topology of the sensor graph among all robots is described by a directed graph containing a spanning tree, and the node denoting the center is only required to be globally reachable in the sensor graph. A dynamic control law is developed such that the multi-robot systems with speed constraint are able to globally converge to a circular formation prescribed by a stationary center, a radius, and a spacing configuration. Remarkably, the proposed control law does not rely on any communication, and only requires each robot to use local measurements with respect to its neighbors based on its local coordinate frame, which makes it feasible to implement this control law in a decentralized manner. The global asymptotic stability of the closed-loop multi-robot systems is established by small gain theorem of which the small-gain condition is guaranteed by a low gain parameter in the designed observer.

Adaptive cooperative output regulation of polytopic systems via spectrum estimation: An [formula omitted] perspective

This paper focuses on the fully distributed robust cooperative output regulation of polytopic multi-agent systems. Based on the internal model principle, the problem being studied is first converted into a stabilization problem of interconnected systems without external disturbance. Next, small-gain conditions are characterized to obtain an equivalent H∞ problem, where the upper bound of the H∞ performance index is closely associated with the spectral radius of the communication topology correlation matrix. To design adaptive fully distributed control protocols in both state and output feedback settings, a finite-time observer and a topology information estimation algorithm are proposed to avoid global information. Compared with existing related works, our methods do not require global information a priori. Finally, the proposed methodology is applied to numerical and practical examples of multi-pendulum systems to demonstrate the feasibility and applicability of the theoretical results.

A Lyapunov-Based ISS Small-Gain Theorem for Infinite Networks of Nonlinear Systems

In this article, we show that an infinite network of input-to-state stable (ISS) subsystems, admitting ISS Lyapunov functions, itself admits an ISS Lyapunov function, provided that the couplings between the subsystems are sufficiently weak. The strength of the couplings is described in terms of the properties of an infinite-dimensional nonlinear positive operator, built from the interconnection gains. If this operator induces a uniformly globally asymptotically stable (UGAS) system, a Lyapunov function for the infinite network can be constructed. We analyze necessary and sufficient conditions for UGAS and relate them to small-gain conditions used in the stability analysis of finite networks.

Event‐triggered adaptive stabilization control of stochastic nonlinear systems with unmodeled dynamics

AbstractIn this paper, we present a novel controller design method for stochastic nonlinear systems with unmodeled dynamics, uncertain parameters, and unknown covariance noise. In order to deal with these uncertainties, a new event‐based small gain controller is designed. The event‐triggered scheme can reduce the computational burden caused by disturbance and noise. By combining the technique of changing supply rate with small gain condition, a stochastic input‐to‐state practically stability Lyapunov function is obtained for the subsystem. Simultaneously, the changing supply function is used to treat with unmodeled dynamics. Based on backstepping process and adaptive control approach, the influence of unknown covariance noise is overcome. It is proved that the designed state‐feedback controller can ensure the overall system is stochastic input‐to‐state practically stability.

The exponential input-to-state stability property: characterisations and feedback connections

The exponential input-to-state stability (ISS) property is considered for systems of controlled nonlinear ordinary differential equations. A characterisation of this property is provided, including in terms of a so-called exponential ISS Lyapunov function and a natural concept of linear state/input-to-state L^2-gain. Further, the feedback connection of two exponentially ISS systems is shown to be exponentially ISS provided a suitable small-gain condition is satisfied.

A sufficient condition for k-contraction in Lurie systems

We consider a Lurie system obtained via a connection of a linear time-invariant system and a nonlinear feedback function. Such systems often have more than a single equilibrium and are thus not contractive with respect to any norm. We derive a new sufficient condition for k-contraction of a Lurie system. For k = 1, our sufficient condition reduces to the standard stability condition based on the bounded real lemma and a small gain condition. For k = 2, our condition guarantees well-ordered asymptotic behaviour of the closed-loop system: every bounded solution converges to an equilibrium, which is not necessarily unique. We apply our results to derive a sufficient condition for k-contractivity of a networked system.

On a Small Phase Theorem with Necessity over Phase Bounded Nonlinear Uncertainties⋆

In this paper, we consider the feedback interconnection of a linear time-invariant (LTI) system and phase bounded uncertainties that are possibly nonlinear and/or time-varying. We devise a small phase condition and show that it is necessary and sufficient for robust stability over such phase bounded uncertainties, in a similar spirit to the well-known small gain condition being necessary and sufficient for robust stability over norm bounded uncertainties. Moreover, we show that the small phase condition can be verified using a state space characterization via linear matrix inequalities (LMIs), which extends the sectored real lemma in previous work.

A Small-Gain Approach to Incremental Input-to-State Stability Analysis of Hybrid Integrator-Gain Systems

Incremental input-to-state stability plays an important role in the analysis of nonlinear systems, as it opens up the possibility for accurate performance characterizations beyond classical approaches. In this paper, we are interested in deriving conditions for incremental stability of a specific class of discontinuous dynamical systems containing a so-called hybrid integrator. Recently, it was shown that hybrid integrators have the potential for overcoming fundamental performance limitations of linear time-invariant control, thereby making them interesting for use in, e.g., high-precision motion control applications. The main contribution of this paper is to show that these hybrid integrators have incremental input-to-state stability properties, and that, under an incremental small-gain condition, the feedback interconnection of a hybrid integrator and a linear time-invariant plant is incrementally input-to-state stable.

From Small-Gain Theory to Compositional Construction of Barrier Certificates for Large-Scale Stochastic Systems

This article is concerned with a compositional approach for the construction of control barrier certificates for large-scale interconnected stochastic systems while synthesizing hybrid controllers against high-level logic properties. Our proposed methodology involves decomposition of interconnected systems into smaller subsystems and leverages the notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">control</i> <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sub-barrier certificates</i> of subsystems, enabling one to construct control barrier certificates of interconnected systems by employing some <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\max$</tex-math></inline-formula> -type small-gain conditions. The main goal is to synthesize hybrid controllers enforcing complex logic properties, including the ones represented by the accepting language of deterministic finite automata, while providing probabilistic guarantees on the satisfaction of given specifications in bounded-time horizons. To do so, we propose a systematic approach to first decompose high-level specifications into simple reachability tasks by utilizing automata corresponding to the complement of specifications. We then construct control sub-barrier certificates and synthesize local controllers for those simpler tasks and combine them to obtain a hybrid controller that ensures satisfaction of the complex specification with some lower bound on the probability of satisfaction. We finally apply our proposed techniques to a fully-interconnected Kuramoto network composed of 100 nonlinear oscillators.

A relaxed small-gain theorem for discrete-time infinite networks

We study input-to-state stability (ISS) of discrete-time networks of infinitely many finite-dimensional subsystems. We develop a so-called relaxed small-gain theorem for ISS with respect to a closed set and show the necessity of the proposed small-gain condition in case of exponentially ISS infinite networks. Moreover, we study the well-posedness of the interconnection based on the behavior of the individual subsystems. Finally, over-approximations of large-but-finite networks by infinite networks are studied and it is shown that all the stability and performance indices obtained for the infinite system can be transferred to the original finite one if each subsystem of the infinite network is individually ISS.

A small-gain theorem for set stability of infinite networks: Distributed observation and ISS for time-varying networks

We generalize a small-gain theorem for a network of infinitely many systems, recently developed in [Kawan et. al, IEEE TAC (2021)]. The generalized small-gain theorem addresses exponential input-to-state stability with respect to closed sets, which enables us to study diverse control-theoretic problems in a unified manner, and it also allows for agents to have infinitely many neighbors. The small-gain condition, expressed in terms of the spectral radius of a gain operator collecting all the information about the internal Lyapunov gains, has several useful characterizations which simplify the verification of the condition. To demonstrate applicability of our small-gain theorem, we apply it to the stability analysis of infinite time-varying networks as well as the design of distributed observers for infinite networks.

Compositional construction of abstractions for infinite networks of discrete-time switched systems

In this paper, we develop a compositional scheme for the construction of continuous abstractions for networks of infinitely many discrete-time switched systems. In particular, the constructed abstractions are themselves also continuous-space systems with potentially lower dimensions, which can be used as replacements of the original (also known as concrete) systems in the controller design process. Having designed a controller for the abstract system, it is refined to a more detailed one for the concrete system. We use the notion of so-called simulation functions to quantify the mismatch between the original system and its approximation. Each subsystem in the concrete network and its corresponding one in the abstract network are related through a notion of local simulation functions. We show that if the local simulation functions satisfy a spectral small-gain condition, then the aggregation of the individual simulation functions provides an overall simulation function quantifying the error between the overall abstract network and the concrete one. In addition, we show that our methodology results in a scale-free compositional approach for any finite-but-arbitrarily large networks obtained from truncation of an infinite network. We provide a systematic approach to construct local abstractions and simulation functions for networks of linear switched systems. In this case, the conditions are expressed in terms of linear matrix inequalities that can be efficiently computed. We illustrate the effectiveness of our approach through an application to AC islanded microgrids.

ISS small-gain criteria for infinite networks with linear gain functions

This paper provides a Lyapunov-based small-gain theorem for input-to-state stability (ISS) of networks composed of infinitely many finite-dimensional systems. We model these networks on infinite-dimensional ℓ ∞ -type spaces. A crucial assumption in our results is that the internal Lyapunov gains, modeling the influence of the subsystems on each other, are linear functions. Moreover, the gain operator built from the internal gains is assumed to be subadditive and homogeneous, which covers both max-type and sum-type formulations for the ISS Lyapunov functions of the subsystems. As a consequence, the small-gain condition can be formulated in terms of a generalized spectral radius of the gain operator. By an example, we show that the small-gain condition can easily be checked if the interconnection topology of the network has some sort of symmetry. While our main result provides an ISS Lyapunov function in implication form for the overall network, an ISS Lyapunov function in a dissipative form is constructed under mild extra assumptions.