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This paper introduces a novel state estimator designed for discrete-time nonlinear dynamical systems that encompass unknown-but-bounded uncertainties, along with state linear inequality and nonlinear equality constraints. Our algorithm is based on constrained zonotopes (CZs) and employs a DC programming approach (where DC stands for difference of convex functions). Recently, techniques such as mean value extension and first-order Taylor extension have been adapted from zonotopes to facilitate the propagation of CZs across nonlinear mappings. While the resulting algorithms, known as CZMV and CZFO, achieve higher precision compared to the original zonotopic versions, they still exhibit sensitivity to the wrapping and dependency effects inherent in interval arithmetic. These interval-related challenges can be mitigated through the use of DC programming, as it allows for the determination of approximation error bounds by solving optimization problems. One direct advantage of this technique is the elimination of the dependency effect. Our set-membership filter, referred to as CZDC, provides an alternative solution to CZMV and CZFO. To showcase the effectiveness of our proposed approach, we conducted experiments using CZDC on two numerical examples. [ABSTRACT FROM AUTHOR]

This paper establishes exponential convergence rates for a class of primal–dual gradient algorithms in distributed optimization without strong convexity. The convergence analysis is based on a carefully constructed Lyapunov function. By evaluating metric subregularity of the primal–dual gradient map, we present a general criterion under which the algorithm achieves exponential convergence. To facilitate practical applications of this criterion, several simplified sufficient conditions are derived. We also prove that although these results are developed for the continuous-time algorithms, they carry over in a parallel manner to the discrete-time algorithms constructed by using Euler's approximation method. [ABSTRACT FROM AUTHOR]

Various efficient matrix inequalities have recently been proposed to deal with the stability analysis of linear systems with time-varying delays. This paper provides more insights on the relationship between some of them. We present an equivalent formulation of Moon et al.’s inequality, allowing us to discover strong links not only with the most recent and efficient matrix inequalities such as the reciprocally convex combination lemma and also its relaxed version but also with some previous inequalities such as the approximation inequality introduced in Shao (2009) or free-matrix-based inequality. More especially, it is proved that these existing inequalities can be captured as particular cases of Moon et al.’s inequality. Examples show the best tradeoff between the reduction of conservatism and the numerical complexity. [ABSTRACT FROM AUTHOR]

This technical paper presents a distributed continuous-time algorithm to solve multi-agent optimization problem with the team objective being the sum of all local convex objective functions while subject to an equality constraint. The optimal solutions are achieved within fixed time which is independent of the initial conditions of agents. This advantage makes it possible to off-line preassign the settling time according to task requirements. The fixed-time convergence for the proposed algorithm is rigorously proved with the aid of convex optimization and fixed-time Lyapunov theory. Finally, the algorithm is valuated via an example. [ABSTRACT FROM AUTHOR]

This paper investigates an iterative approach to solve the general rank-constrained optimization problems (RCOPs) defined to optimize a convex objective function subject to a set of convex constraints and rank constraints on unknown rectangular matrices. In addition, rank minimization problems (RMPs) are introduced and equivalently transformed into RCOPs by introducing a quadratic matrix equality constraint. The rank function is discontinuous and nonconvex, thus the general RCOPs are classified as NP-hard in most of the cases. An iterative rank minimization (IRM) method, with convex formulation at each iteration, is proposed to gradually approach the constrained rank. The proposed IRM method aims at solving RCOPs with rank inequalities constrained by upper or lower bounds, as well as rank equality constraints. Proof of the convergence to a local minimizer with at least a sublinear convergence rate is provided. Four representative applications of RCOPs and RMPs, including system identification, output feedback stabilization, and structured H 2 controller design problems, are presented with comparative simulation results to verify the feasibility and improved performance of the proposed IRM method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and discuss the Data-Driven Inversion-Based Control ( D 2 -IBC) method for nonlinear control system design. The method relies on a two degree-of-freedom architecture, with a nonlinear controller and a linear controller running in parallel, and does not require any detailed physical knowledge of the plant to control. Specifically, we use input/output data to synthesize the controller by employing convex optimization tools. We show the effectiveness of the proposed approach on a benchmark simulation example, regarding control of the Duffing system. [ABSTRACT FROM AUTHOR]

This paper studies distributed optimization problem over a fixed network. We develop and analyze an accelerated distributed gradient descent method, named Acc-DGDlm, which utilizes gradient tracking technique and local memory. Specifically, we add two memory slots per agent to store two past estimates, namely, an estimate of the optimal solution and an estimate of the average gradient. For strongly convex and smooth functions, Acc-DGDlm achieves a linear convergence rate of O (C k) for some constant 0 < C < 1 when the fixed stepsize is sufficiently small and the coefficient θ of past variables satisfies 0 ≤ θ < 1. Compared to the related works where both the stepsize and the momentum coefficient should belong to intervals determined by global parameters, we eliminate the dependence of θ on global parameters, which makes θ easy to be chosen in practice. We also provide a theoretical analysis showing that including local memory can decrease the convergence factor C and thus speed up the convergence. Besides, numerical experiments with distributed estimation problems show that Acc-DGDlm converges faster in comparison with state-of-the-art methods, especially for sparse networks. [ABSTRACT FROM AUTHOR]

This paper presents a theoretical framework for the design and analysis of gradient descent-based algorithms for coverage control tasks involving robot swarms. We adopt a multiscale approach to analysis and design to ensure consistency of the algorithms in the large-scale limit. First, we represent the macroscopic configuration of the swarm as a probability measure and formulate the macroscopic coverage task as the minimization of a convex objective function over probability measures. We then construct a macroscopic dynamics for swarm coverage, which takes the form of a proximal descent scheme in the L 2 -Wasserstein space. Our analysis exploits the generalized geodesic convexity of the coverage objective function, proving convergence in the L 2 -Wasserstein sense to the target probability measure. We then obtain a consistent gradient descent algorithm in the Euclidean space that is implementable by a finite collection of agents, via a "variational" discretization of the macroscopic coverage objective function. We establish the convergence properties of the gradient descent and its behavior in the continuous-time and large-scale limits. Furthermore, we establish a connection with well-known Lloyd-based algorithms, seen as a particular class of algorithms within our framework, and demonstrate our results via numerical experiments. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the high-performance robust control of discrete-time linear time-invariant (LTI) systems with semi-algebraic uncertainty regions. It is assumed that a robustly stabilizing static controller is given whose gain depends polynomially on the uncertain variables. The problem of tuning this parameter-dependent gain with respect to a prescribed quadratic cost function is formulated as a sum-of-squares (SOS) optimization. This method leads to a near-optimal controller whose performance is better than that of the initial controller. It is shown that the results derived in the present work encompass the ones obtained in a recent paper. The efficacy of the results is elucidated by an example. [Copyright &y& Elsevier]

A strongly convex function naturally induces a gradient flow that is contractive. This paper is a short investigation on when the converse to the previous statement holds. That is, given a differential equation that is contractive, does there exist a strongly convex function that induces the differential equation? We show that, if sufficient smoothness of the vector field is assumed, then the contractivity of such a differential equation with a symmetric Jacobian is equivalent to the existence of a strongly convex function which induces the differential equation as its gradient flow. [ABSTRACT FROM AUTHOR]

This paper considers a class of generalized convex games where each player is associated with a convex objective function, a convex inequality constraint and a convex constraint set. The players aim to compute a Nash equilibrium through communicating with neighboring players. The particular challenge we consider is that the component functions are unknown a priori to associated players. We study two distributed computation algorithms and analyze their convergence properties in the presence of data transmission delays and dynamic changes of network topologies. The algorithm performance is verified through demand response on the IEEE 30-bus Test System. Our technical tools integrate convex analysis, variational inequalities and simultaneous perturbation stochastic approximation. [ABSTRACT FROM AUTHOR]

Spectral estimation can be performed using the so called THREE-like approach. Such method leads to a convex optimization problem whose solution is characterized through its dual problem. In this paper, we show that the dual problem can be seen as a new parametric spectral estimation problem. This interpretation implies that the THREE-like solution is optimal in terms of closeness to the correlogram over a certain parametric class of spectral densities describing ARMA models, enriching in this way its meaningfulness. [ABSTRACT FROM AUTHOR]

The aim of this paper is to extend the LQG design for linear system to the case of switched linear systems in continuous time. The main result provides a control Lyapunov function and a dynamic output feedback law leading to sub-optimal solutions. Practically, the dynamic output feedback is easy to apply and the design procedure is effective if there exists at least one controllable and observable convex combination of the subsystems. Practical applications concern the large class of power converters. [ABSTRACT FROM AUTHOR]

Abstract: The paper studies the problem of making Getz’s bicycle model traverse a strictly convex Jordan curve with bounded roll angle and bounded speed. The approach to solving this problem is based on the virtual holonomic constraint (VHC) method. Specifically, a VHC is enforced making the roll angle of the bicycle become a function of the bicycle’s position along the curve. It is shown that the VHC can be automatically generated as a periodic solution of a scalar periodic differential equation, which we call virtual constraint generator. Finally, it is shown that if the curve is sufficiently long as compared to the height of the bicycle’s center of mass and its wheel base, then the enforcement of a suitable VHC makes the bicycle traverse the curve with a steady-state speed profile which is periodic and independent of initial conditions. An outcome of this work is a proof that the constrained dynamics of a Lagrangian control system subject to a VHC are generally not Lagrangian. [Copyright &y& Elsevier]

In this paper, a class of optimization problems is investigated, where the objective function is the sum of N convex functions viewed as local functions and the constraints are N nonidentical closed convex sets. Additionally, it is aimed to solve the considered optimization problem in a distributed manner and thus a sequence of time-varying unbalanced directed graphs is introduced first to depict the information connection topologies. Then, the novel push-sum based constrained optimization algorithm (PSCOA) is developed, where the new gradient descent-like method is applied to settle the involved closed convex set constraints. Furthermore, the rigorous convergence analysis is shown under some standard and common assumptions and it is proved that the developed distributed discrete-time algorithm owns a convergence rate of O ( ln t t ) in general case. Specially, the convergence rate of O ( 1 t ) can be further obtained under the assumption that at least one objective function is strongly convex. Finally, simulation results are given to demonstrate the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: The presence of abrupt changes, such as impulsive and load disturbances, commonly occur in applications, but make the state estimation problem considerably more difficult than in the standard setting with Gaussian process disturbance. Abrupt changes often introduce a jump in the state, and the problem is therefore readily and often treated by change detection techniques. In this paper, we take a different approach. The state smoothing problem for linear state space models is here formulated as a constrained least-squares problem with sum-of-norms regularization, a generalization of -regularization. This novel formulation can be seen as a convex relaxation of the well known generalized likelihood ratio method by Willsky and Jones. Another nice property of the suggested formulation is that it only has one tuning parameter, the regularization constant which is used to trade off fit and the number of jumps. Good practical choices of this parameter along with an extension to nonlinear state space models are given. [Copyright &y& Elsevier]

Abstract: This paper proposes a state-space solution to weight optimization problem in loop-shaping control. A pointwise-in-frequency weight optimization framework is transformed into convex optimization searches that are independent of frequency. The introduced optimization problem therefore avoids gridding of the frequency space and consequently, the inaccuracies attributed to fitting transfer functions to magnitude data. In this optimization problem, the order of the weight is specified ‘a priori’, thus facilitating the synthesis of low-order controllers, which is desirable from an implementation perspective. The proposed solution algorithm simultaneously synthesizes a robust stabilizing controller and a loop-shaping weight (pre-compensator) that maximize the robust stability margin subject to constraints on the performance and the singular values of the weight. Three numerical examples are given to illustrate the effectiveness of the technique. [Copyright &y& Elsevier]

Abstract: This paper studies the fundamental problems: whether an affine system affected by additive disturbances is robustly transferable from a source set (simplex) to a target set (polytope) and whether it is robustly stabilizable with its state constrained in a simplex. First, a necessary and sufficient condition is derived for the existence of affine feedback control that solves the robust reachability problem. Further investigation is provided for two situations relying on whether the union of the source set and the target set is convex or non-convex. For the former one, a necessary and sufficient condition is obtained in the form of linear inequalities, while for the latter, several computationally feasible sufficient conditions are found. Second, we show that robust stabilization subject to a state constraint is equivalent to find a feasible solution to a linear equation. Once it is known that either of the problems has a solution by checking the derived conditions, design of control laws is then straightforward. [Copyright &y& Elsevier]

Abstract: This paper proposes a model predictive control (MPC) approach to the periodic implementation of the optimal solutions of a class of resource allocation problems in which the allocation requirements and conditions repeat periodically over time. This special class of resource allocation problems includes many practical energy optimization problems such as load scheduling and generation dispatch. The convergence and robustness of the MPC algorithm is proved by invoking results from convex optimization. To illustrate the practical applications of the MPC algorithm, the energy optimization of a water pumping system is studied. [ABSTRACT FROM AUTHOR]

Abstract: The concept of asymptotic stability is studied for a class of semi-linear distributed parameter dissipative systems with nonlinearity defined on a convex subset of the state space. This is achieved by using infinite-dimensional Banach state space description. Stability criteria are established, which are based on a weaker technical condition of the -dissipativity. These theoretical results are applied to a class of transport–reaction processes. Different types of nonlinearities are studied by adapting the criteria given in the early portions of the paper. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the problems of robust and  filtering for 2-dimensional (2-D) discrete-time linear systems described by a Fornasini–Marchesini second model with matrices that depend affinely on convex-bounded uncertain parameters. By a suitable transformation, the system is represented by an equivalent difference-algebraic representation. A parameter-dependent Lyapunov function approach is then proposed for the design of 2-D stationary discrete-time linear filters that ensure either a prescribed performance or  performance for all admissible uncertain parameters. The filter designs are given in terms of linear matrix inequalities. Numerical examples illustrate the effectiveness of the proposed filter design methods. [Copyright &y& Elsevier]

Many large-scale and distributed optimization problems can be brought into a composite form in which the objective function is given by the sum of a smooth term and a nonsmooth regularizer. Such problems can be solved via a proximal gradient method and its variants, thereby generalizing gradient descent to a nonsmooth setup. In this paper, we view proximal algorithms as dynamical systems and leverage techniques from control theory to study their global properties. In particular, for problems with strongly convex objective functions, we utilize the theory of integral quadratic constraints to prove the global exponential stability of the equilibrium points of the differential equations that govern the evolution of proximal gradient and Douglas–Rachford splitting flows. In our analysis, we use the fact that these algorithms can be interpreted as variable-metric gradient methods on the suitable envelopes and exploit structural properties of the nonlinear terms that arise from the gradient of the smooth part of the objective function and the proximal operator associated with the nonsmooth regularizer. We also demonstrate that these envelopes can be obtained from the augmented Lagrangian associated with the original nonsmooth problem and establish conditions for global exponential convergence even in the absence of strong convexity. [ABSTRACT FROM AUTHOR]

This paper presents two distributed event-triggered algorithms under directed communication networks to solve a class of convex optimization problems such as the economic dispatch problem (EDP) with equality constraint, while the objective of optimization is the sum of all locally convex functions. One is distributed continuous-time event-triggered optimization algorithm, and the other is discrete-time algorithm based on iteration scheme. Continuous communication is not required by adopting event-triggered communication strategy. It means that the communication cost can be reduced and unnecessary waste of network resources can be avoided. Moreover, the convergence for the proposed algorithms are rigorously proved with the aid of Lyapunov stability theory under the strongly connected and weight-balanced network topology. Finally, four numerical simulations show the effectiveness and advantages of the two novel distributed event-triggered optimization algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we exploit the possibility of combining nonsmooth control jointly with projected gradient approaches to solve convex optimization problems with linear equality constraints. Our development gives rise to a sign projected gradient method employing only "coarse" information, which has finite-time convergence capability and preserves the simplicity and elegance of gradient descent. Our primary objective is to understand the design principle of the projection matrix in the nonsmooth setting and to investigate some key features of the method, such as convergence time. Specifically, we offer a set of convergence results via nonsmooth analysis, revealing that the algorithm guarantees finite-time convergence for a wide range of cost functions, including strongly convex, strictly convex, and convex functions, provided that the proposed design criteria and conditions are satisfied. The results suggest that the choice of the projection matrix might not be unique, which motivates the optimal design issue of the projection matrix to minimize convergence time. We show that the problem can be formulated as a convex optimization problem, which can be solved readily by existing optimization tools. We discuss the possibility of employing the proposed method to solve unconstrained optimization problems, which in turn gives an initial feasible solution to the sign projected gradient flow. We apply the algorithm to solve a formation control problem, and the numerical results show the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

This paper treats the control design of a state-dependent switching function for discrete-time switched affine systems, assuring global asymptotic stability of a limit cycle. This limit cycle is determined according to criteria of interest related to the steady-state behaviour of the system trajectories, as for instance, its mean value, oscillation amplitude and frequency. More specifically, based on a time-varying convex Lyapunov function, a min-type switching rule is determined in order to orchestrate the state trajectories, evolving from any initial condition, towards the desired limit cycle. Most of the available methodologies are able to guide the state trajectories to a set of attraction containing the equilibrium point, assuring its practical stability. However, although important, the behaviour of the trajectories inside this set is not considered. Differently, the proposed technique assures an appropriate steady-state behaviour by adequately designing the limit cycle. Moreover, the proposed conditions take into account H 2 and H ∞ guaranteed costs, assuring also a suitable transient response. All the conditions are expressed in terms of linear matrix inequalities being simple to solve by readily available tools. Illustrative examples, one of them concerning the voltage regulation of a DC–DC multicellular converter, are used for validation and comparison. [ABSTRACT FROM AUTHOR]

In this paper, we revisit primal–dual dynamics for convex optimization and present a generalization of the dynamics based on the concept of passivity. We hypothesize that supplying a stable zero to one of the integrators in the dynamics allows one to eliminate the assumption of strict convexity on the cost function. Then, we prove this hypothesis based on the passivity paradigm together with the invariance principle for Carathéodory systems. Additionally, we show that the presented algorithm is also a generalization of existing augmented Lagrangian-based primal–dual dynamics and discuss the benefit of the presented generalization in terms of noise reduction and convergence speed. Finally, the presented algorithm is demonstrated through simulation of lighting control in a building. [ABSTRACT FROM AUTHOR]