Monotone Nonlinear Systems Research Articles

Hybrid Hu-Storey type methods for large-scale nonlinear monotone systems and signal recovery

We propose two hybrid methods for solving large-scale monotone systems, which are based on derivative-free conjugate gradient approach and hyperplane projection technique. The conjugate gradient approach is efficient for large-scale systems due to low memory, while projection strategy is suitable for monotone equations because it enables simply globalization. The derivative-free function-value-based line search is combined with Hu-Storey type search directions and projection procedure, in order to construct globally convergent methods. Furthermore, the proposed methods are applied into solving a number of large-scale monotone nonlinear systems and reconstruction of sparse signals. Numerical experiments indicate the robustness of the proposed methods.

Deterministic Safety Guarantees for Learning-Based Control of Monotone Nonlinear Systems Under Uncertainty

Deterministic Safety Guarantees for Learning-Based Control of Monotone Nonlinear Systems Under Uncertainty

Novel Stability Conditions for Nonlinear Monotone Systems and Consensus in Multiagent Networks

We introduce a novel definition of monotonicity, termed “type-K” in honor of Kamke, and study nonlinear type-K monotone dynamical systems possessing the plus-subhomogeneity property, which we call “K-subtopical” systems after Gunawardena and Keane. We show that type-K monotonicity, which is weaker than strong monotonicity, is also equivalent to monotonicity for smooth systems evolving in continuous-time, but not in discrete-time. K-subtopical systems are proved to converge toward equilibrium points, if any exists, generalizing the result of Angeli and Sontag about convergence of topical systems' trajectories toward the unique equilibrium point when strong monotonicity is considered. The theory provides an new methodology to study the consensus problem in nonlinear multi-agent systems (MASs). Necessary and sufficient conditions on the local interaction rule of the agents ensuring the K-subtopicality of MASs are provided, and consensus is proven to be achieved asymptotically by the agents under given connectivity assumptions on directed graphs. Examples in continuous-time and discrete-time corroborate the relevance of our results in different applications

The vanishing discount problem for monotone systems of Hamilton-Jacobi equations: a counterexample to the full convergence

<abstract><p>In recent years there has been intense interest in the vanishing discount problem for Hamilton-Jacobi equations. In the case of the scalar equation, B. Ziliotto has recently given an example of the Hamilton-Jacobi equation having non-convex Hamiltonian in the gradient variable, for which the full convergence of the solutions does not hold as the discount factor tends to zero. We give here an explicit example of nonlinear monotone systems of Hamilton-Jacobi equations having convex Hamiltonians in the gradient variable, for which the full convergence of the solutions fails as the discount factor goes to zero.</p></abstract>

A Scaled Dai–Yuan Projection-Based Conjugate Gradient Method for Solving Monotone Equations with Applications

In this paper, we propose two scaled Dai–Yuan (DY) directions for solving constrained monotone nonlinear systems. The proposed directions satisfy the sufficient descent condition independent of the line search strategy. We also reasonably proposed two different relations for computing the scaling parameter at every iteration. The first relation is proposed by approaching the quasi-Newton direction, and the second one is by taking the advantage of the popular Barzilai–Borwein strategy. Moreover, we propose a robust projection-based algorithm for solving constrained monotone nonlinear equations with applications in signal restoration problems and reconstructing the blurred images. The global convergence of this algorithm is also provided, using some mild assumptions. Finally, a comprehensive numerical comparison with the relevant algorithms shows that the proposed algorithm is efficient.

Two new Hager–Zhang iterative schemes with improved parameter choices for monotone nonlinear systems and their applications in compressed sensing

Notwithstanding its efficiency and nice attributes, most research on the Hager–Zhang (HZ) iterative scheme are focused on unconstrained minimization problems. Inspired by this and recent extensions of the one-parameter HZ scheme to system of nonlinear monotone equations, two new HZ-type iterative methods are developed in this paper for solving system of monotone equations with convex constraint. This is achieved by developing two HZ-type search directions with new parameter choices combined with the popular projection method. The first parameter choice is obtained by minimizing the condition number of a modified HZ direction matrix, while the second choice is realized using singular value analysis and minimizing the spectral condition number of a nonsingular HZ search direction matrix. Interesting properties of the schemes include solving non-smooth problems and satisfying the inequality that is vital for global convergence. Using standard assumptions, global convergence of the schemes are proven and numerical experiments with recent methods in the literature, indicate that the methods proposed are promising. The effectiveness of the schemes are further demonstrated by their applications to sparse signal and image reconstruction problems, where they outperform some recent schemes in the literature.

Data-Driven Model Reduction of Monotone Systems by Nonlinear DC Gains

In this paper, we develop data-driven model reduction methods for monotone nonlinear control systems based on a nonlinear version of the dc gain. The nonlinear dc gain is a function of the amplitude of the input and can be used to evaluate the importance of each state variable. In fact, the nonlinear dc gain is directly related to the infinity-induced norm of the system as well as a notion of output reachability. Given the dc gain, model reduction is performed by either truncating not-so-important state variables or aggregating state variables having similar importance. Under such truncation and clustering, monotonicity and boundedness of the nonlinear dc gain are preserved; moreover, these two operations can be approximately performed based on simulation or experimental data alone. This empirical model reduction approach is illustrated by an example of a gene regulatory network.

Distributed Nonlinear Control Design Using Separable Control Contraction Metrics

This paper gives convex conditions for the synthesis of a distributed control system for large-scale networked nonlinear dynamic systems. It is shown that the technique of control contraction metrics can be extended to this problem by utilizing separable metric structures, resulting in controllers that only depend on information from local sensors and communications from immediate neighbors. The conditions given are pointwise linear matrix inequalities, and are necessary and sufficient for linear positive systems and certain monotone nonlinear systems. Distributed synthesis methods for systems on chordal graphs are also proposed based on semidefinite program decompositions. The results are illustrated on a problem of vehicle platooning with heterogeneous vehicles, and a network of nonlinear dynamic systems with over 1000 states that is not feedback linearizable and has an uncontrollable linearization.

Contraction Analysis of Monotone Systems via Separable Functions

In this paper, we study incremental stability of monotone nonlinear systems through contraction analysis. We provide sufficient conditions for incremental asymptotic stability in terms of the Lie derivatives of differential one-forms or Lie brackets of vector fields. These conditions can be viewed as sum- or max-separable conditions, respectively. For incremental exponential stability, we show that the existence of such separable functions is both necessary and sufficient under standard assumptions for the converse Lyapunov theorem of exponential stability. As a by-product, we also provide necessary and sufficient conditions for exponential stability of positive linear time-varying systems. The results are illustrated through examples.

Exponential Stability of Two-Dimensional Homogeneous Monotone Systems With Bounded Directional Delays

One-dimensional (1-D) monotone systems have received considerable attention recently due to their wide applicability and interesting mathematical properties. One of these special properties is that, for LTI monotone systems, exponential stability is insensitive to time-delays. Some extensions to 1-D nonlinear monotone systems based on conditions of homogeneity have also been reported. In this paper, we study the problem of exponential stability of discrete-time 2-D nonlinear monotone systems described by the Roesser model with time-varying delays. Specifically, based on the monotone property and homogeneity of the associated vector fields, necessary and sufficient delay-independent exponential stability conditions are derived. The magnitudes of delays are also taken into deriving an explicit estimation of the exponential decay rate which correlates the impact of delays on the system performance. Two examples are given to demonstrate the effectiveness of the obtained results.

Stability Analysis of Monotone Systems via Max-Separable Lyapunov Functions

We analyze stability properties of monotone nonlinear systems via max-separable Lyapunov functions, motivated by the following observations: first, recent results have shown that asymptotic stability of a monotone nonlinear system implies the existence of a max-separable Lyapunov function on a compact set; second, for monotone linear systems, asymptotic stability implies the stronger properties of D-stability and insensitivity to time delays. This paper establishes that for monotone nonlinear systems, equivalence holds between asymptotic stability, the existence of a max-separable Lyapunov function, D-stability, and insensitivity to bounded and unbounded time-varying delays. In particular, a new and general notion of D-stability for monotone nonlinear systems is discussed, and a set of necessary and sufficient conditions for delay-independent stability are derived. Examples show how the results extend the state of the art.

A new projection-based algorithm for solving a large-scale nonlinear system of monotone equations

The projection based methods are an efficient and applicable family of derivative free methods for solving nonlinear monotone systems. This paper proposes a new projection method for solving a system of large-scale nonlinear monotone equations. The new algorithm, in each iteration, by using a modified conjugate gradient direction, constructs an appropriate hyperplane which strictly separate the current approximation from the solution set of the problem. Then the new approximation is determined by projecting the current point onto the separating hyperplane. The global convergence and the linear convergence rate of the proposed algorithm are proved under standard assumptions. Preliminary numerical experiments indicate that the proposed algorithm is promising.

On Existence of Separable Contraction Metrics for Monotone Nonlinear Systems

Finding separable certificates of stability is important for tractability of analysis methods for large-scale networked systems. In this paper we consider the question of when a nonlinear system which is contracting, i.e. all solutions are exponentially stable, can have that property verified by a separable metric. Making use of recent results in the theory of positive linear systems and separable Lyapunov functions, we prove several new results showing when this is possible, and discuss the application of to nonlinear distributed control design via convex optimization.

Delay-Independent Asymptotic Stability in Monotone Systems

Monotone systems comprise an important class of dynamical systems that are of interest both for their wide applicability and because of their interesting mathematical properties. It is known that under the property of quasimonotonicity time-delayed systems become monotone, and some remarkable properties have been reported for such systems. These include, for example, the fact that for linear systems global asymptotic stability of the undelayed system implies global asymptotic stability for the delayed system under arbitrary bounded delays. Nevertheless, extensions to nonlinear systems have thus far relied primarily on the conditions of homogeneity and subhomogeneity, and it has been conjectured that these can be relaxed. Our aim in this paper is to show that this is feasible for a general class of nonlinear monotone systems by deriving convergence results in which simple properties of the undelayed system lead to delay-independent stability. In particular, one of our results shows that if the undelayed system has a convergent trajectory that is unbounded in all components as $t\rightarrow-\infty$ , then the system is globally asymptotically stable for arbitrary bounded time-varying delays. This follows from a more general result derived in the paper that allows to quantify delay-independent regions of attraction, which can be used to prove global asymptotic stability for various classes of systems. These also recover various known delay-independent stability results that are discussed within the paper.

FR type methods for systems of large-scale nonlinear monotone equations

A large class of iterative methods for solving nonlinear monotone systems is developed in recent years. In this paper we propose some new FR type directions in the frame of algorithm which is a combination of conjugate gradient approach and hyperplane projection technique. Derivative-free, function-value-based line search combined with projection procedure is used for globalization strategy. Numerical performances of methods with different search directions are compared.

Spectral DY-Type Projection Method for Nonlinear Monotone Systems of Equations

In this paper, we propose a spectral DY-type projection method for nonlinear monotone system of equations, which is a reasonable combination of DY conjugate gradient method, the spectral gradient method and the projection technique. Without the differentiability assumption on the system of equations, we establish the global convergence of the proposed method, which does not rely on any merit function. Furthermore, this method is derivative-free and so is very suitable to solve large-scale nonlinear monotone systems. The preliminary numerical results show the feasibility and effectiveness of the proposed method. Mathematics subject classification: 65F10, 65K05.

Positive invariance, monotonicity and comparison of nonlinear systems

In this paper, the interrelation between positive invariance, monotonicity and comparison of iterated nonlinear systems defined in partially ordered sets is studied. First, necessary and sufficient conditions guaranteeing the positive invariance of sets defined by relations of the form v ( x ) ≤ w with respect to nonlinear systems are established. Then, various characterizations of monotone nonlinear systems are developed and a connection between positive invariance and monotonicity is established. Finally, necessary and sufficient conditions for a, not necessarily monotone, nonlinear system to be a comparison system are established. No specific algebraic structure is necessary for developing the main results of this work. Numerical examples illustrating the applicability of all these results to the case of discrete-time dynamical systems are also given.

Computing reachable sets for uncertain nonlinear monotone systems

We address nonlinear reachability computation for uncertain monotone systems, those for which flows preserve a suitable partial orderings on initial conditions. In a previous work Ramdani (2008) [22], we introduced a nonlinear hybridization approach to nonlinear continuous reachability computation. By analysing the signs of off-diagonal elements of system’s Jacobian matrix, a hybrid automaton can be obtained, which yields component-wise bounds for the reachable sets. One shortcoming of the method is induced by the need to use whole sets for addressing mode switching. In this paper, we improve this method and show that for the broad class of monotone dynamical systems, component-wise bounds can be obtained for the reachable set in a separate manner. As a consequence, mode switching no longer needs to use whole solution sets. We give examples which show the potentials of the new approach.

Generalized Gagliardo–Nirenberg estimates and differentiability of the solutions to monotone nonlinear parabolic systems

In this paper, we are concerned with interior differentiability of weak solutions u to nonlinear parabolic systems with natural growth and coefficients uniformly monotone in Du. Making use of estimates of Gagliardo---Nirenberg's type in generalized Sobolev spaces, we show that u belongs to $$L^2(-a, 0, H^2(B(\sigma), {\mathbb{R}}^N)) \cap H^1(-a, 0, L^2(B(\sigma), {\mathbb{R}}^N))$$ (see Theorem 3).

Asymptotic Tracking for Constrained Monotone Systems

This note addresses tracking control of nonlinear discrete-time monotone systems subject to input and state constraints. Forcing saturation on a previously designed controller may, in general, lead to destabilization or, at least, result in constraint violation and performance losses. Hereby, it is shown that for a certain class of nonlinear monotone systems it is possible to design a static nonlinear output feedback which, saturated among suitable state-dependent bounds, is able to guarantee constraint satisfaction and asymptotic tracking of piecewise constant references, with a moderate online computational burden. Simulation experiments concerning the synthesis of a protein demonstrate the effectiveness of the proposed control strategy.