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In the state-of-the-art literature on cryptography and control theory, there has been no systematic methodology of constructing cyber-physical systems that can achieve desired control performance while being protected against eavesdropping attacks. In this paper, we tackle this challenging problem. We first propose two novel notions referred to as sample identifying complexity and sample deciphering time in an encrypted-control framework. The former explicitly captures the relation between the dynamical characteristics of control systems and the level of identifiability of the systems while the latter shows the relation between the computation time for the identification and the key length of a cryptosystem. Based on these two tractable new notions, we propose a systematic method for designing the both of an optimal key length to prevent system identification with a given precision within a given life span of systems, and of an optimal controller to maximize both of the control performance and the difficulty of the identification. The efficiency of the proposed method in terms of security level and realtime-ness is investigated through numerical simulations. To the best of our knowledge, this paper first connect the relationship between the security of cryptography and dynamical systems from a control-theoretic perspective., 15 pages, 6 figures

Semidefinite and sum-of-squares (SOS) optimization are fundamental computational tools in many areas, including linear and nonlinear systems theory. However, the scale of problems that can be addressed reliably and efficiently is still limited. In this paper, we introduce a new notion of block factor-width-two matrices and build a new hierarchy of inner and outer approximations of the cone of positive semidefinite (PSD) matrices. This notion is a block extension of the standard factor-width-two matrices, and allows for an improved inner-approximation of the PSD cone. In the context of SOS optimization, this leads to a block extension of the scaled diagonally dominant sum-of-squares (SDSOS) polynomials. By varying a matrix partition, the notion of block factor-width-two matrices can balance a trade-off between the computation scalability and solution quality for solving semidefinite and SOS optimization problems. Numerical experiments on a range of large-scale instances confirm our theoretical findings., Accepted for publication as a regular paper at IEEE TAC. Code is available through https://github.com/zhengy09/SDPfw

The persistence of excitation (PE) condition is sufficient to ensure parameter convergence in adaptive estimation problems. Recent results on adaptive estimation in reproducing kernel Hilbert spaces (RKHS) introduce PE conditions for RKHS. This paper presents sufficient conditions for PE for the particular class of uniformly embedded reproducing kernel Hilbert spaces (RKHS) defined over smooth Riemannian manifolds. This paper also studies the implications of the sufficient condition in the case when the RKHS is finite or infinite-dimensional. When the RKHS is finite-dimensional, the sufficient condition implies parameter convergence as in the conventional analysis. On the other hand, when the RKHS is infinite-dimensional, the same condition implies that the function estimate error is ultimately bounded by a constant that depends on the approximation error in the infinite-dimensional RKHS. We illustrate the effectiveness of the sufficient condition in a practical example., Comment: 11 pages, 2 figures

This paper studies the control problem for safety-critical multi-agent systems based on quadratic programming (QP). Each controlled agent is modeled as a cascade connection of an integrator and an uncertain nonlinear actuation system. In particular, the integrator represents the position-velocity relation, and the actuation system describes the dynamic response of the actual velocity to the velocity reference signal. The notion of input-to-output stability (IOS) is employed to characterize the essential velocity-tracking capability of the actuation system. The uncertain actuation dynamics may cause infeasibility or discontinuous solutions of QP algorithms for collision avoidance. Also, the interaction between the controlled integrator and the uncertain actuation dynamics may lead to significant robustness issues. By using nonlinear control methods and numerical optimization methods, this paper first contributes a new feasible-set reshaping technique and a refined QP algorithm for feasibility, robustness, and local Lipschitz continuity. Then, we present a nonlinear small-gain analysis to handle the inherent interaction for guaranteed safety of the closed-loop multi-agent system. The proposed methods are illustrated by numerical simulations and a physical experiment.

This paper deals with the state estimation problem in discrete-event systems modeled with nondeterministic finite automata, partially observed via a sensor measuring unit whose measurements (reported observations) may be vitiated by a malicious attacker. The attacks considered in this paper include arbitrary deletions, insertions, or substitutions of observed symbols by taking into account a bounded number of attacks or, more generally, a total cost constraint (assuming that each deletion, insertion, or substitution bears a positive cost to the attacker). An efficient approach is proposed to describe possible sequences of observations that match the one received by the measuring unit, as well as their corresponding state estimates and associated total costs. We develop an algorithm to obtain the least-cost matching sequences by reconstructing only a finite number of possible sequences, which we subsequently use to efficiently perform state estimation. We also develop a technique for verifying tamper-tolerant diagnosability under attacks that involve a bounded number of deletions, insertions, and substitutions (or, more generally, under attacks of bounded total cost) by using a novel structure obtained by attaching attacks and costs to the original plant. The overall construction and verification procedure have complexity that is of O(|X|^2 C^2),where |X| is the number of states of the given finite automaton and C is the maximum total cost that is allowed for all the deletions, insertions, and substitutions. We determine the minimum value of C such that the attacker can coordinate its tampering action to keep the observer indefinitely confused while utilizing a finite number of attacks. Several examples are presented to demonstrate the proposed methods.

In this two-part paper we develop a unifying framework for the analysis of the feasibility of the power flow equations for DC power grids with constant-power loads. In Part I of this paper we present a detailed introduction to the problem of power flow feasibility of such power grids, and the associated problem of selecting a desirable operating point which satisfies the power flow equations. We introduce and identify all long-term voltage semi-stable operating points, and show that there exists a one-to-one correspondence between such operating points and the constant power demands for which the power flow equations are feasible. Such operating points can be found by solving an initial value problem, and a parametrization of these operating points is obtained. In addition, we give a full characterization of the set of all feasible power demands, and give a novel proof for the convexity of this set.

In this paper, we investigate the enforcement of opacity via supervisory control in the context of discrete-event systems. A system is said to be opaque if the intruder, which is modeled as a passive observer, can never infer confidentially that the system is at a secret state. The design objective is to synthesize a supervisor such that the closed-loop system is opaque even when the control policy is publicly known. In this paper, we propose a new approach for enforcing opacity using non-deterministic supervisors. A non-deterministic supervisor is a decision mechanism that provides a set of control decisions at each instant, and randomly picks a specific control decision from the decision set to actually control the plant. Compared with the standard deterministic control mechanism, such a non-deterministic control mechanism can enhance the plausible deniability of the controlled system as the online control decision is a random realization and cannot be implicitly inferred from the control policy. We provide a sound and complete algorithm for synthesizing a non-deterministic opacity-enforcing supervisor. Furthermore, we show that non-deterministic supervisors

International audience; For most networked systems found in the literature, the actuated boundary is usually located at one end. In this paper, we first consider the stabilization of a chain of two interconnected subsystems, actuated at the in-between boundary. Each subsystem corresponds to coupled hyperbolic partial differential equations. Such in-domain actuation leads to higher complexity, and represents a significant difference with existing results. Then, starting from a classical controllability condition, we design a state feedback control law for the considered class of systems. The proposed approach is based on the backstepping methodology. However, to deal with the complex structure of the system, we use Fredholm integral transforms instead of classical Volterra transforms. We prove the invertibility of such transforms using an original operator framework. The well-posedness of the backstepping kernel equations defining the transformations is also shown with the same arguments. By using a similar procedure, we are then able to design a Luenberger-type observer. Finally, we use the state estimation in the stabilizing controller to obtain an output-feedback law, and some test cases complete the paper.

With the wide application of cyber-physical systems, stealthy attacks on remote state estimation have attracted increasing research attention. Recently, various stealthy innovation-based linear attack models were proposed, in which the relaxed stealthiness was measured by the Kullback--Leibler divergence. This paper studies existing innovation-based linear attack strategies with relaxed stealthiness and concludes that all of them provided merely suboptimal solutions. The main reason is some oversight in solving the involved optimization problems: some covariance constraints were not perfectly handled. This paper provides the corresponding optimal solutions for those stealthy attacks. Both one-step and holistic optimizations of stealthy attacks are studied, and the worst-case attacks with and without zero-mean constraints are derived analytically, without the necessity to formulate semidefinite programming problems.

This paper aims to solve the problem of decentralized event-based finite-time control for a class of high-order interconnected nonlinear systems with an asymmetric output constraint. In order to achieve finite-time stability and reduce the update frequency of the controller, a novel controller and its event-trigger mechanism is proposed by adding a power integral technique and utilizing sign function. To fulfill the requirement of asymmetric output constraints, an asymmetric barrier Lyapunov function (BLF) is constructed, which is different from tantype and log-type BLFs. Based on the contradictory idea and finite-time stability theroy, it is proved that there is no Zeno phenomenon under this event-trigger mechanism and all state variables of the closed-loop system attenuate to the origin in the finite-time while the output signals never violate the constraint. Furthermore, the result in this paper is expanded to fixed-time control. Finally, a simulation example is given to demonstrate the effectiveness of the presented strategy

We propose decentralized primal-dual methods for cooperative multi-agent consensus optimization problems over both static and time-varying communication networks, where only local communications are allowed. The objective is to minimize the sum of agent-specific convex functions over conic constraint sets defined by agent-specific nonlinear functions; hence, the optimal consensus decision should lie in the intersection of these private sets. Under the strong convexity assumption, we provide convergence rates for sub-optimality, infeasibility, and consensus violation in terms of the number of communications required; examine the effect of underlying network topology on the convergence rates., A preliminary result of this paper was presented in arXiv:1706.07907 by Hamedani and Aybat. In this paper, we generalize our results to the setting where agent-specific constraints are defined by nonlinear functions rather than linear ones which greatly improves the modeling capability. This generalization requires a more complicated analysis which is studied in this separate arXiv submission

Opacity is a confidentiality property that holds if certain secret behavior of a system, typically represented by a predicate, cannot be revealed under any system evolution. Among other proposed methodologies, when opacity is violated, it can be enforced using insertion mechanisms, i.e., by inserting symbols before an actual system output (in real time as the system evolves) in order to replace observation sequences that lead to opacity violations with observation sequences that can be generated by system behavior that does not violate opacity. This paper focuses on opacity enforcement in discrete event systems modeled with finite state automata and proposes an extended insertion mechanism that can enforce opacity in a practical manner to a wide class of systems by inserting symbols before and after an actual system output. The paper also introduces event insertion constraints that require only certain specific symbols to be inserted before and after an actual system output. For each case, we obtain a necessary and sufficient condition (based on the construction of an appropriate verifier) for opacity-enforceability using the proposed extended insertion mechanism

This paper addresses the problem of learning the optimal control policy for a nonlinear stochastic dynamical. This problem is subject to the curse of dimensionality associated with the dynamic programming method. This paper proposes a novel decoupled data-based control (D2C) algorithm that addresses this problem using a decoupled, open-loop - closed-loop, approach. First, an open-loop deterministic trajectory optimization problem is solved using a black-box simulation model of the dynamical system. Then, closed-loop control is developed around this open-loop trajectory by linearization of the dynamics about this nominal trajectory. By virtue of linearization, a linear quadratic regulator based algorithm can be used for this closed-loop control. We show that the performance of D2C algorithm is approximately optimal. Moreover, simulation performance suggests a significant reduction in training time compared to other state of the art algorithms.

This paper proposes methods for identification of large-scale networked systems with guarantees that the resulting model will be contracting -- a strong form of nonlinear stability -- and/or monotone, i.e. order relations between states are preserved. The main challenges that we address are: simultaneously searching for model parameters and a certificate of stability, and scalability to networks with hundreds or thousands of nodes. We propose a model set that admits convex constraints for stability and monotonicity, and has a separable structure that allows distributed identification via the alternating directions method of multipliers (ADMM). The performance and scalability of the approach is illustrated on a variety of linear and non-linear case studies, including a nonlinear traffic network with a 200-dimensional state space., Comment: Preprint of full paper accepted for publication in IEEE Trans. Automatic Control

In this paper, we investigate the distributed state estimation problem for a nonlinear dynamical system. Relying upon the assumption of the existence of a block-triangular ``uniform observability" canonical form, local high-gain observers are used to stabilize the estimation error for the observable portion of the state and exchange of information between neighbors is used to achieve consensus in the estimation of the state of the unobservable part. The paper addresses first the case of an autonomous system, and then the case of a controlled dynamical system under the assumption that the exogenous input is accessible by each node in the network. Based on the designed state estimator, each distributed sensor node can achieve asymptotic estimation of the full state, with an arbitrarily fixed decay rate of the observation error.

In this paper, we consider cooperative multi-agent systems, where independent agents need to minimize a collective social cost. Given the coupling in the cost, the optimal strategy requires interaction and communications among all the agents in general. In this paper, we unveil important classes of social costs which are optimized by decentralized and selfish solutions, hence eliminating the need for an inter-agent communication network. In particular, we focus on a set of n independent agents coupled only through an overall cost that penalizes the divergence of each agent from the average collective behavior. Adopting input-output methods, we show that optimal decentralized and selfish solutions are possible in a variety of standard input-output cost criteria. These include the cases of l_1, l_2, l_infty induced, and H2 norms for any finite n. Moreover, if the cost includes nondeviation from average variables, the above results hold true as well for l_1, l_2, l_infty induced norms and any n, while they hold true for the normalized, per-agent square H2 norm, cost as n ! 1. The results of the paper demonstrate that selfish behavior can be socially optimal in nontr

This paper is concerned with the input-to-state stability (ISS) and stochastic input-to-state stability (SISS) of impulsive stochastic nonlinear systems. By constructing a general SISS-Lyapunov function, which takes the exponential SISS-Lyapunov function as a special case, the ISS criteria are obtained respectively, for destabilizing and stabilizing impulses. It should be stressed that we impose the average dwell-time (ADT) condition for destabilizing impulses and reverse average dwell-time (RADT) condition for stabilizing impulses to restrain the occurrence of impulses, which extends the fixed dwell-time condition in recent literature. Moreover, the ADT and RADT conditions in this paper correspond to the average impulsive interval method in most existing results under exponential ISS-Lyapunov condition. As a subsequent result, the ISS and SISS of impulsive stochastic interconnected systems on networks are also studied by the Lyapunov method and graph-theoretical technique. Finally, several illustrative examples together with their numerical simulations are provided to demonstrate theoretical

In this paper, the fault-tolerant consensus control problem is investigated for multi-agent systems with sensor faults. A first-order difference equation is utilized to describe the sensor fault, and an observer is designed to estimate the state and the fault simultaneously. For congestion mitigation and/or security enhancement purposes, the estimated state is first encrypted into a series of finite-level codewords by an encryption algorithm and then transmitted to other agents through a directed topology. After being received, the codewords are then decrypted by the corresponding decryption algorithm and subsequently utilized to design the consensus controller. By constructing a novel matrix norm along with its compatible vector norm, a necessary and sufficient condition is obtained for the existence of the controller. In the end, two simulation examples are given to demonstrate the validity of the results in this paper.

This paper develops a novel passive stochastic gradient algorithm. In passive stochastic approximation, the stochastic gradient algorithm does not have control over the location where noisy gradients of the cost function are evaluated. Classical passive stochastic gradient algorithms use a kernel that approximates a Dirac delta to weigh the gradients based on how far they are evaluated from the desired point. In this paper we construct a multi-kernel passive stochastic gradient algorithm. The algorithm performs substantially better in high dimensional problems and incorporates variance reduction. We analyze the weak convergence of the multi-kernel algorithm and its rate of convergence. In numerical examples, we study the multi-kernel version of the passive least mean squares (LMS) algorithm for transfer learning to compare the performance with the classical passive version.

This paper is concerned with the problem of exponential stability for semi-Markov jump stochastic nonlinear systems. It is well known that the semi-Markov chain is an extension of the Markov chain, whose sojourn time distribution depends on the current and next state, and is no longer limited to the exponential distribution. However, in existing works, the independence and the distribution function limitation are imposed on the sojourn time of semi-Markov jump systems. In this paper, by developing a new stochastic analysis method, the problem of exponential stability is investigated for semi- Markov jump stochastic nonlinear systems without additional constraints for the sojourn time. In addition, mode-dependent linear comparable relationships are assumed among Lyapunov like functions, which can effectively reduce the conservatism caused by mode-independent case. Two examples are presented to demonstrate the effectiveness of the proposed results.

This paper studies the multi-agent control problem for affine, rigid, and translational formation, with the aim of developing a unified distributed control strategy for global and finite-time convergence. Global stabilization of rigid formation in arbitrary dimensional spaces still remains open and challenging. This paper provides a general solution to it based on the sliding mode control idea. The control law design consists of two parts: the main control force regulates the trajectories of all agents to reach a sliding surface defined by an affine formation space in finite time and remain in it thereafter; the extra control force governs certain chosen leader agents towards the desired formation in the sliding surface. The paper then presents in detail two approaches for designing the extra control force, one based on distance constraints and the other based on relative position constraints. For the first time, the proposed sliding mode formation control laws solve the open problem of (almost) global and finite-time stabilization of affine, rigid and translational formations in any dimensional space.

The problems discussed in this paper are motivated by general ratio consensus algorithms, introduced by Kempe, Dobra, and Gehrke (2003) in a simple form as the push-sum algorithm, later extended by B\'en\'ezit et al. (2010) under the name weighted gossip algorithm. We consider a communication protocol described by a strictly stationary, ergodic, sequentially primitive sequence of non-negative matrices, applied iteratively to a pair of fixed initial vectors, the components of which are called values and weights defined at the nodes of a network. The subject of ratio consensus problems is to study the asymptotic properties of ratios of values and weights at each node, expecting convergence to the same limit for all nodes. The main results of the paper provide upper bounds for the rate of the almost sure exponential convergence in terms of the spectral gap associated with the given sequence of random matrices. It will be shown that these upper bounds are sharp. Our results complement previous results of Picci and Taylor (2013) and Iutzeler, Ciblat and Hachem (2013).

Comparing with traditional stochastic differential equations (SDEs) involving white noise, random differential equations (RDEs) with colored noise are claimed to have more practical meaning. This paper considers the event-triggered adaptive tracking control for RDE systems with coexisting parametric uncertainties and severe nonlinearities. Combining a tracking error-based dynamic gain with a relative threshold event triggered control mechanism (ETCM), the tracking control problem for the random systems is solved without Zeno behavior. The tracking error can be rendered small enough by tuning design parameters. First, a series of adaptive control laws are designed by using backstepping technique. Then, two special cases are considered and the main results are extended to MIMO systems. Finally, a simulation example confirms the validity of the results. To the best of the authors knowledge, this paper serves as the first attempt of event-based control for RDE systems.

This paper develops a decentralized actuator fault compensation scheme for hypersonic vehicles, using a model transform based backstepping control design. Different from the actuator dynamics represented by second-order ordinary differential equations (ODE) studied in the existing literature, a new actuator dynamics is studied in this paper, modeled by a fourth-order partial differential equation (PDE), i.e., Euler-Bernoulli beam (EBB) equation. An invertible model transformation is introduced to transform the EBB PDE-ODE cascade system into a heat-like PDE-ODE cascade system, and a new target system is constructed from a two-step backstepping procedure. A novel decentralized fault-tolerant control scheme is proposed for the heat-like PDE-ODE cascade system and the well-posedness of the system solutions is analyzed. The closed-loop system stability and the well-posedness property of the original EBB PDE-ODE cascade system are ensured. Simulation results illustrate the effectiveness of the proposed fault compensation scheme.

This paper addresses the problems of establishing structural stability and the L2 gain of 2D mixed continuous-discrete-time systems. The first contribution is to show that Lyapunov functions quadratically dependent on the frequency are exact for establishing structural stability. This is particularly important since the existing works that exploit Lyapunov functions provide a much larger upper bound on the dependence on the frequency or other parameters. The second contribution is to propose a novel linear matrix inequality (LMI) necessary and sufficient condition for establishing the existence of such Lyapunov functions. It is shown, analytically and through several examples, for both best and worst cases, that the numerical complexity of this novel condition is much smaller than that of the existing methods. The third contribution is to show that the proposed methodology can be used to establish upper bounds on the L2 gain, in particular, deriving a novel necessary and sufficient LMI condition based on Lyapunov functions quadratically dependent on the frequency. Lastly, the paper presents the generalization of the proposed methodology to non-mixed 2D systems.

In this paper, we provide a new Markovian-type model for stock price trend analysis at the transaction level, and illustrate its use for trading in conjunction with a controller which makes buy and sell decisions. Central to our formulation is a sequence of i.i.d. random variables T_k which corresponds to the number of transactions between reversals in price direction. For a trader, this is an important indicator of the duration of a trend. For processes with large T_k, there is an incentive to try and capitalize by buying stock when a temporary trend is up and selling when it is down. The extent to which this is possible is determined by a model parameter p_e, called the probability of efficiency, which indicates the likelihood that the bid, ask and current price are such that one can seamlessly enter or exit the market without slippage. The degree to which a trader can exploit trending behavior is quantified in our main result which provides the expected value of the trading gain resulting from a strategically constructed feedforward switching controller. The paper also includes an example illustrating application of the theory using historical data.

This paper is concerned with the design of a feedback control based on past states in order to make a given unstable hybrid stochastic differential equation (SDE) to be stable in distribution (stabilisation in distribution). This is the first paper in this direction. Under the global Lipschitz condition on the coefficients of the given unstable hybrid SDE, we will show that the stabilisation in distribution can be achieved by linear delay feedback controls. In particular, we discuss how to design the feedback controls in two structure cases: state feedback and output injection.

This paper studies the evolution of social power in influence networks with stubborn individuals. Based on the Friedkin-Johnsen opinion dynamics and the reflected appraisal mechanism, two models are proposed over issue sequences and over a single issue, respectively. These models generalize the original DeGroot-Friedkin (DF) model by including stubbornness. To the best of our knowledge, this paper is the first attempt to investigate the social power evolution of stubborn individuals basing on the reflected appraisal mechanism. Properties of equilibria and convergence are provided. We show that the models have same equilibrium social power and convergence property, where the equilibrium social power depends only upon interpersonal influence and individuals' stubbornness. Roughly speaking, more stubborn individual has more equilibrium social power. Moreover, unlike the DF model without stubbornness, we prove that for the models with stubbornness, autocracy can never be achieved, while democracy can be achieved under any network topology.

This paper studies the distributed optimal output agreement problem for multi-agent systems described by uncertain nonlinear models. By using the partial information of an objective function, the design aims to steer the outputs of the agents to an agreement on the optimal solution to the objective function. To solve this problem, this paper introduces distributed coordinators to calculate the desired outputs, and designs reference-tracking controllers for the agents to follow the desired outputs. To deal with the nonlinear uncertain dynamics, the closed-loop multi-agent system is considered as a dynamical network, and Sontag's input-to-state stability (ISS) is employed to characterize the interconnections. It is shown that output agreement in multi-agent nonlinear systems is achievable by means of distributed optimal controllers via a small-gain approach. The proposed design features a three-layer architecture, and the reference-tracking controllers can be implemented as successive nonlinear proportional-integral (PI) loops. A numerical example is employed to show the effectiveness of the design.

This paper studies cooperative tracking problem of heterogeneous Euler-Lagrange systems with an uncertain leader. Different from most existing works, system dynamic knowledge of the leader node is unaccessible to any follower node in our paper. Distributed adaptive observers are designed for all follower nodes, simultaneously estimate the state and parameters of the leader node. The observer design does not rely on the frequency knowledge of the leader node, and the estimation errors are shown to converge to zero exponentially. Moreover, the results are applied to general directed graphs, where the symmetry of Laplacian matrix does not hold. This is due to two newly developed Lyapunov equations, which solely depend on communication network topologies. Interestingly, using these Lyapunov equations, many results of multi-agent systems over undirected graphs can be extended to general directed graphs. Finally, this paper also advances the knowledge base of adaptive control systems by providing a main tool in the analysis of parameter convergence for adaptive observers.

Distributed optimization enables networked agents to cooperatively solve a global optimization problem even with each participating agent only having access to a local partial view of the objective function. Despite making significant inroads, most existing results on distributed optimization rely on noise-free information sharing among the agents, which is problematic when communication channels are noisy, messages are coarsely quantized, or shared information are obscured by additive noise for the purpose of achieving differential privacy. The problem of information-sharing noise is particularly pronounced in the state-of-the-art gradient-tracking based distributed optimization algorithms, in that information-sharing noise will accumulate with iterations on the gradient-tracking estimate of these algorithms, and the ensuing variance will even grow unbounded when the noise is persistent. This paper proposes a new gradient-tracking based distributed optimization approach that can avoid information-sharing noise from accumulating in the gradient estimation. The approach is applicable even when the {inter-agent interaction is} time-varying, which is key to enable the incorporation of a decaying factor in inter-agent interaction to gradually eliminate the influence of information-sharing noise. In fact, we rigorously prove that the proposed approach can ensure the almost sure convergence of all agents to the same optimal solution even in the presence of persistent information-sharing noise. The approach is applicable to general directed graphs. It is also capable of ensuring the almost sure convergence of all agents to an optimal solution when the gradients are noisy, which is common in machine learning applications. Numerical simulations confirm the effectiveness of the proposed approach., Comment: Accepted to IEEE Transactions on Automatic Control as a full paper. arXiv admin note: text overlap with arXiv:2202.01113

Optimization problems are essential for a broad spectrum of applications, and have been extensively studied in many aspects. Most existing literature on optimization algorithms does not consider systems that involve parameter uncertainties. This paper studies a class of stochastic adaptive optimization problems in which identification of unknown parameters and search of the optimal solutions must be performed simultaneously. Due to a fundamental conflict between parameter identifiability and optimality in such problems, we introduce a method of adding stochastic dither signals into the system, which provides sufficient excitation for estimating the unknown parameters, leading to adaptive optimization algorithms. Joint identification and optimization algorithms are developed and their simultaneous convergence properties of parameter estimation and optimization variable updates are proved. Under both noise-free and noisy observations, the corresponding convergence rates are established. The main results of this paper reveal certain fundamental relationships and trade-off among updating step sizes, dither magnitudes, parameter estimation errors, optimization accuracy, and convergence rates. Simulation case studies are used to illustrate the adaptive optimization algorithms and their main properties.

This paper studies the problem of signal tracking and disturbance rejection for sampled-data control systems, where the pertinent signals can reside beyond the so-called Nyquist frequency. In light of the sampling theorem, it is generally understood that manipulating signals beyond the Nyquist frequency is either impossible or at least very difficult. On the other hand, such control objectives often arise in practice, and control of such signals is much desired. This paper examines the basic underlying assumptions in the sampling theorem and pertinent sampled-data control schemes, and shows that the limitation above can be removed by assuming a suitable analog signal generator model. Detailed analysis of multirate closed-loop systems, zeros and poles are given, which gives rise to tracking or rejection conditions. Robustness of the new scheme is fully characterized; it is shown that there is a close relationship between tracking/rejection frequencies and the delay length introduced for allowing better performance. Examples are discussed to illustrate the effectiveness of the proposed method here.

In this paper we propose an observer for a class of Lipschitz nonlinear systems affected by time-varying and known measurement delays which is an improvement of the one presented in [9]. Under the assumption that the delay function is piece-wise continuous and differentiable we prove that exponential convergence to zero of the observation error can be achieved with any desired decay rate, by suitably tuning a gain vector. The delay bound achieved with the observer proposed here is less conservative than the one obtained in [9], as confirmed by numerical tests. For the sake of brevity, in this paper only one-step observers are considered. However, a cascade observer can be arranged to cope with arbitrarily long delays.

In our previous paper, we have considered the supervisory control of the probabilistic discrete event systems (PDESs) under the centralized framework. In this paper, we continue to investigate the decentralized probabilistic supervisory control issue of PDESs. The intersection-based control architecture is adopted, and the notion of the probabilistic intersection-based co-observability and its verification algorithm are presented. It is shown that this observability and the probabilistic controllability act as the necessary and sufficient conditions for the existence of the intersection-based decentralized probabilistic supervisors. Moreover, a working example is provided to illustrate the proposed concepts, the obtained results, and the potential application of the proposed control approaches.

In this paper, we consider a flow network that is described by a digraph, each edge of which can admit a flow within a certain interval, with nonnegative end points that correspond to lower and upper flow limits. We propose and analyze a distributed iterative algorithm for solving the so-called feasible circulation problem, which consists of computing flows that are within the given intervals at each edge and balance the total in-flow and the total out-flow at each node. Unlike previously proposed distributed algorithms that required bidirectional communication between pairs of nodes that share an edge in the flow network, the algorithm we propose can operate over any communication network, assuming the corresponding digraph that describes it is strongly connected. The proposed algorithm allows the nodes to asymptotically compute (with a geometric rate that depends on the specifics of the given flow network and communication topology) a solution to the feasible circulation problem, as long as such a solution exists. An important special case of the setting studied in this paper is the case where the digraph of the flow network matches the digraph of the communic

One of the technical assumptions in the above paper is flawed and requires a global rather than a local Lipschitz condition. All of the results in the paper hold under this strengthened assumption and all relevant examples in the paper satisfy this condition.

Model Predictive Control (MPC) algorithms have long been applied to nonlinear processes. In a quasi-Linear Parameter Varying (qLPV) setting, nonlinearities are included into bounded scheduling parameters, which are given as a function of endogenous variables; these scheduling parameters are a-priori unknown along a future prediction horizon, which complicates MPC design. To address this problem, literature points out two options: robust MPC approaches, considering the scheduling to be uncertain; or sub-optimal ones, that set values for these parameters along the horizon. With respect to the latter group, this paper proposes an extrapolation algorithm that estimates the future values of the qLPV scheduling parameters for a fixed prediction horizon of N steps; the method is based on a recursive procedure using simple Taylor expansions. Sufficient conditions for convergent extrapolation are presented with regard to the form and class of the scheduling function and the robustness of the MPC. Different benchmark examples from the literature are presented to illustrate the algorithm, which is also compared to state-of-the-art techniques.

In this paper, a novel control strategy namely disturbance observer-based control is first applied to stabilization and disturbance rejection for an anti-stable stochastic heat equation with Neumann boundary actuation and unknown boundary external disturbance generated by an exogenous system. A disturbance observer-based boundary control is designed based on the backstepping approach and estimation/cancellation strategy, where the unknown disturbance is estimated in real time by a disturbance observer and rejected in the closed-loop, while the in-domain multiplicative noise whose intensity is within a known finite interval is attenuated. It is shown that the resulting closed-loop system is exponentially stable in the sense of both mean square and almost surely. A numerical example is demonstrated to validate the effectiveness of the proposed control approach.

Signal Temporal Logic (STL) has been found an expressive language for describing complex, time-constrained tasks in several robotic applications. Existing methods encode such specifications by either using integer constraints or by employing set invariance techniques. While in the first case this results in MILP control problems, in the latter case designer-specific choices may induce conservatism in the robot's performance and the satisfaction of the task. In this paper a continuous-time receding horizon control scheme (RHS) is proposed that exploits the trade-off between task satisfaction and performance costs such as actuation and state costs, traditionally considered in RHS schemes. The satisfaction of the STL tasks is encoded using time-varying control barrier functions (CBFs) that are designed online, thus avoiding the integer expressions that are often used in literature. The recursive feasibility of the proposed scheme is guaranteed by the satisfaction of a time-varying terminal constraint that ensures the satisfaction of the task with pre-determined robustness. The effectiveness of the method is illustrated in a multi-robot simulation scenario. QC 20230706

Linear models with additive unknown-but-bounded input disturbances are extensively used to model uncertainty in robust control systems design. Typically, the disturbance set is either assumed to be known a priori or estimated from data through set-membership identification. However, the problem of computing a suitable input disturbance set in case the set of possible output values is assigned a priori has received relatively little attention. This problem arises in many contexts, such as in supervisory control, actuator design, decentralized control, and others. In this paper, we propose a method to compute input disturbance sets (and the corresponding set of states) such that the resulting set of outputs matches as closely as possible a given set of outputs, while additionally satisfying strict (inner or outer) inclusion constraints. We formulate the problem as an optimization problem by relying on the concept of robust invariance. The effectiveness of the approach is demonstrated in numerical examples that illustrate how to solve safe reference set and input-constraint set computation problems.

Given a stochastic dynamical system modelled via stochastic differential equations (SDEs), we evaluate the safety of the system through characterizations of its exit time moments. We lift the (possibly nonlinear) dynamics into the space of the occupation and exit measures to obtain a set of linear evolution equations which depend on the infinitesimal generator of the SDE. Coupled with appropriate semidefinite positive matrix constraints, this yields a moment-based approach for the computation of exit time moments of SDEs with polynomial drift and diffusion dynamics. However, the existing moment approach suffers from drawbacks which impede its applicability to the analysis of higher dimensional physical systems. To apply the existing approach, the dynamics of the systems are limited to polynomials of the state - excluding a large majority of real world examples. Computational scalability is also poor as the dimensionality of the state increases, largely due to the combinatorial growth of the optimization program. In this paper, we propose changes to the existing moment method to make feasible the safety analysis of higher dimensional physical systems. The restriction to polynomial dynamics is lifted by using a state augmentation method which allows one to generate the evolution equations for a broader class of nonlinear stochastic systems. We then reformulate the constraints of the optimization to mitigate the computational limitations associated with an increase in state dimensionality. We employ our methodology on two example processes to characterize their safety via expected exit times and demonstrate the ability to handle multi-dimensional physical systems that were previously unsupported by the existing SDP method of moments.

In this paper, we address the efficient implementation of moving horizon state estimation of constrained discrete-time linear systems. We propose a novel iteration scheme which employs a proximity-based formulation of the underlying optimization algorithm and reduces computational effort by performing only a limited number of optimization iterations each time a new measurement is received. We outline conditions under which global exponential stability of the underlying estimation errors is ensured. Performance guarantees of the iteration scheme in terms of regret upper bounds are also established. A combined result shows that both exponential stability and a sublinear regret which can be rendered smaller by increasing the number of optimization iterations can be guaranteed. The stability and regret results of the proposed estimator are showcased through numerical simulations.

Linear systems over the max-plus algebra provide a suitable formalism to model discrete event systems where synchronization, without competition, is involved. In this paper, we consider a formulation of the model matching problem for systems of such class, in which the output of a given system, called the plant, is forced, by a suitable input, to track exactly that of a given model. A necessary and sufficient condition for its solvability is obtained by making a suitable use of geometric methods in the framework of systems over the max-plus algebra.

The kinematics of many systems encountered in robotics, mechatronics, and avionics are naturally posed on homogeneous spaces, that is, their state lies in a smooth manifold equipped with a transitive Lie group symmetry. This paper proposes a novel filter, the Equivariant Filter (EqF), by posing the observer state on the symmetry group, linearising global error dynamics derived from the equivariance of the system, and applying extended Kalman filter design principles. We show that equivariance of the system output can be exploited to reduce linearisation error and improve filter performance. Simulation experiments of an example application show that the EqF significantly outperforms the extended Kalman filter and that the reduced linearisation error leads to a clear improvement in performance., 19 pages, 3 figures, submitted to IEEE TAC

We consider the problem of computing reachable sets directly from noisy data without a given system model. Several reachability algorithms are presented for different types of systems generating the data. First, an algorithm for computing over-approximated reachable sets based on matrix zonotopes is proposed for linear systems. Constrained matrix zonotopes are introduced to provide less conservative reachable sets at the cost of increased computational expenses and utilized to incorporate prior knowledge about the unknown system model. Then we extend the approach to polynomial systems and, under the assumption of Lipschitz continuity, to nonlinear systems. Theoretical guarantees are given for these algorithms in that they give a proper over-approximate reachable set containing the true reachable set. Multiple numerical examples and real experiments show the applicability of the introduced algorithms, and comparisons are made between algorithms., This paper is accepted at the IEEE Transactions on Automatic Control

A dual adaptive model predictive control (MPC) algorithm is presented for linear, time-invariant systems subject to bounded disturbances and parametric uncertainty in the state-space matrices. Online set-membership identification is performed to reduce the uncertainty and thus control affects both the informativity of identification and the system’s performance. The main contribution of the paper is to include this dual effect in the MPC optimization problem using a predicted worst-case cost in the objective function. This allows the controller to perform active exploration, that is, the control input reduces the uncertainty in the regions of the parameter space that have most influence on the performance. Additionally, the MPC algorithm ensures robust constraint satisfaction of state and input constraints. Advantages of the proposed algorithm are shown by comparing it to a passive adaptive MPC algorithm from the literature., IEEE Transactions on Automatic Control, 68 (5), ISSN:0018-9286, ISSN:1558-2523

This paper puts forward the concept that learning to take safe actions in unknown environments, even with probability one guarantees, can be achieved without the need for an unbounded number of exploratory trials. This is indeed possible, provided that one is willing to navigate trade-offs between optimality, level of exposure to unsafe events, and the maximum detection time of unsafe actions. We illustrate this concept in two complementary settings. We first focus on the canonical multi-armed bandit problem and study the intrinsic trade-offs of learning safety in the presence of uncertainty. Under mild assumptions on sufficient exploration, we provide an algorithm that provably detects all unsafe machines in an (expected) finite number of rounds. The analysis also unveils a trade-off between the number of rounds needed to secure the environment and the probability of discarding safe machines. We then consider the problem of finding optimal policies for a Markov Decision Process (MDP) with almost sure constraints. We show that the action-value function satisfies a barrier-based decomposition which allows for the identification of feasible policies independently of the reward process. Using this decomposition, we develop a Barrier-learning algorithm, that identifies such unsafe state-action pairs in a finite expected number of steps. Our analysis further highlights a trade-off between the time lag for the underlying MDP necessary to detect unsafe actions, and the level of exposure to unsafe events. Simulations corroborate our theoretical findings, further illustrating the aforementioned trade-offs, and suggesting that safety constraints can speed up the learning process., 16 pages, 7 figures. Submitted to TAC

In the past few years, Online Convex Optimization (OCO) has received notable attention in the control literature thanks to its flexible real-time nature and powerful performance guarantees. In this paper, we propose new step-size rules and OCO algorithms that simultaneously exploit gradient predictions, function predictions and dynamics, features particularly pertinent to control applications. The proposed algorithms enjoy static and dynamic regret bounds in terms of the dynamics of the reference action sequence, gradient prediction error, and function prediction error, which are generalizations of known regularity measures from the literature. We present results for both convex and strongly convex costs. We validate the performance of the proposed algorithms in a trajectory tracking case study, as well as portfolio optimization using real-world datasets.

This paper considers the distributed online convex optimization problem with time-varying constraints over a network of agents. This is a sequential decision making problem with two sequences of arbitrarily varying convex loss and constraint functions. At each round, each agent selects a decision from the decision set, and then only a portion of the loss function and a coordinate block of the constraint function at this round are privately revealed to this agent. The goal of the network is to minimize the network-wide loss accumulated over time. Two distributed online algorithms with full-information and bandit feedback are proposed. Both dynamic and static network regret bounds are analyzed for the proposed algorithms, and network cumulative constraint violation is used to measure constraint violation, which excludes the situation that strictly feasible constraints can compensate the effects of violated constraints. In particular, we show that the proposed algorithms achieve $\mathcal{O}(T^{\max\{\kappa,1-\kappa\}})$ static network regret and $\mathcal{O}(T^{1-\kappa/2})$ network cumulative constraint violation, where $T$ is the time horizon and $\kappa\in(0,1)$ is a user-defined trade-off parameter. Moreover, if the loss functions are strongly convex, then the static network regret bound can be reduced to $\mathcal{O}(T^{\kappa})$. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.