Stochastic Contraction Research Articles

Arbitrary 6-DOF large-angle tracking control for autonomous underwater vehicles via stochastic contraction stability and rotation matrix-based attitude representation

Arbitrary 6-DOF large-angle tracking control for autonomous underwater vehicles via stochastic contraction stability and rotation matrix-based attitude representation

An Enhanced IHHO-LSTM Model for Predicting Online Public Opinion Trends in Public Health Emergencies

Social networks accelerate information communication in public health emergencies. Some negative information may cause an outbreak of public opinion crisis. Accurately predicting online public opinion trends can help the relevant departments take timely and effective measures to cope with risks. Therefore, this research proposes a prediction model incorporating the swarm intelligence optimization algorithm and the deep learning method. In this model, we improve the Harris Hawks Optimization (HHO) algorithm by introducing the Cauchy distribution function, the stochastic contraction exponential function, and the adaptive inertia weight. Then we utilize the improved HHO (IHHO) algorithm to optimize the hyperparameters of the deep learning method LSTM, including the learning rate and the number of neurons in the hidden layer. Finally, we construct the IHHO-LSTM model to make predictions in three public health emergencies. The experiments verify that the proposed model outperforms other single and hybrid models. The MAPE values reduce by 78.34%, 54.46%, and 46.42% relative to the average values of the three single models. Compared with the mean values of the two hybrid models, the MAPE values decrease by 47.69%, 18.45%, and 5.78%. The IHHO-LSTM model can be applied to public opinion early warning and reversal identification, providing a reference in public opinion management.

A stochastic contraction mapping theorem

A stochastic contraction mapping theorem

Stochastic contracts and subjective evaluations

AbstractSubjective evaluations are widely used, but call for different contracts from classical moral‐hazard settings. Previous literature shows that contracts require payments to third parties. I show that the (implicit) assumption of deterministic contracts makes payments to third parties necessary. This article studies incentive contracts with stochastic compensation, like payments in stock options or uncertain arbitration procedures. These contracts incentivize employees without the need for payments to third parties. In addition, stochastic contracts can be more efficient and can make the principal better off compared to deterministic contracts. My results also address the puzzle about the prevalence of labor contracts with stochastic compensation.

Stochastic contraction of myosin minifilaments drives evolution of microridge protrusion patterns in epithelial cells.

Actin-based protrusions vary in morphology, stability, and arrangement on cell surfaces. Microridges are laterally elongated protrusions on mucosal epithelial cells, where they form evenly spaced, mazelike patterns that dynamically remodel by fission and fusion. To characterize how microridges form their highly ordered, subcellular patterns and investigate the mechanisms driving fission and fusion, we imaged microridges in the maturing skin of zebrafish larvae. After their initial development, microridge spacing and alignment became increasingly well ordered. Imaging F-actin and non-muscle myosin II (NMII) revealed that microridge fission and fusion were associated with local NMII activity in the apical cortex. Inhibiting NMII blocked fission and fusion rearrangements, reduced microridge density, and altered microridge spacing. High-resolution imaging allowed us to image individual NMII minifilaments in the apical cortex of cells in live animals, revealing that minifilaments are tethered to protrusions and often connect adjacent microridges. NMII minifilaments connecting the ends of two microridges fused them together, whereas minifilaments oriented perpendicular to microridges severed them or pulled them closer together. These findings demonstrate that as cells mature, cortical NMII activity orchestrates a remodeling process that creates an increasingly orderly microridge arrangement.

Neural Stochastic Contraction Metrics for Learning-Based Control and Estimation

We present Neural Stochastic Contraction Metrics (NSCM), a new design framework for provably-stable robust control and estimation for a class of stochastic nonlinear systems. It uses a spectrally-normalized deep neural network to construct a contraction metric, sampled via simplified convex optimization in the stochastic setting. Spectral normalization constrains the state-derivatives of the metric to be Lipschitz continuous, thereby ensuring exponential boundedness of the mean squared distance of system trajectories under stochastic disturbances. The NSCM framework allows autonomous agents to approximate optimal stable control and estimation policies in real-time, and outperforms existing nonlinear control and estimation techniques including the state-dependent Riccati equation, iterative LQR, EKF, and the deterministic neural contraction metric, as illustrated in simulation results.

Stochastic Contracts and Subjective Evaluations

Stochastic Contracts and Subjective Evaluations

Robust Controller Design for Stochastic Nonlinear Systems via Convex Optimization

This article presents ConVex optimization-based Stochastic steady-state Tracking Error Minimization (CV-STEM), a new state feedback control framework for a class of Itô stochastic nonlinear systems and Lagrangian systems. Its innovation lies in computing the control input by an optimal contraction metric, which greedily minimizes an upper bound of the steady-state mean squared tracking error of the system trajectories. Although the problem of minimizing the bound is nonconvex, its equivalent convex formulation is proposed utilizing SDC parameterizations of the nonlinear system equation. It is shown using stochastic incremental contraction analysis that the CV-STEM provides a sufficient guarantee for exponential boundedness of the error for all time with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\bf \mathcal {L}_2}$</tex-math></inline-formula> -robustness properties. For the sake of its sampling-based implementation, we present discrete-time stochastic contraction analysis with respect to a state- and time-dependent metric along with its explicit connection to continuous-time cases. We validate the superiority of the CV-STEM to PID, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_\infty$</tex-math></inline-formula> , and baseline nonlinear controllers for spacecraft attitude control and synchronization problems.

QTorch: The quantum tensor contraction handler.

Classical simulation of quantum computation is necessary for studying the numerical behavior of quantum algorithms, as there does not yet exist a large viable quantum computer on which to perform numerical tests. Tensor network (TN) contraction is an algorithmic method that can efficiently simulate some quantum circuits, often greatly reducing the computational cost over methods that simulate the full Hilbert space. In this study we implement a tensor network contraction program for simulating quantum circuits using multi-core compute nodes. We show simulation results for the Max-Cut problem on 3- through 7-regular graphs using the quantum approximate optimization algorithm (QAOA), successfully simulating up to 100 qubits. We test two different methods for generating the ordering of tensor index contractions: one is based on the tree decomposition of the line graph, while the other generates ordering using a straight-forward stochastic scheme. Through studying instances of QAOA circuits, we show the expected result that as the treewidth of the quantum circuit’s line graph decreases, TN contraction becomes significantly more efficient than simulating the whole Hilbert space. The results in this work suggest that tensor contraction methods are superior only when simulating Max-Cut/QAOA with graphs of regularities approximately five and below. Insight into this point of equal computational cost helps one determine which simulation method will be more efficient for a given quantum circuit. The stochastic contraction method outperforms the line graph based method only when the time to calculate a reasonable tree decomposition is prohibitively expensive. Finally, we release our software package, qTorch (Quantum TensOR Contraction Handler), intended for general quantum circuit simulation. For a nontrivial subset of these quantum circuits, 50 to 100 qubits can easily be simulated on a single compute node.

Deterministic versus stochastic contracts in a dynamic principal-agent model

I show that deterministic dynamic contracts between a principal and an agent are always at least as profitable to the principal as stochastic ones, if the so-called first-order approach in dynamic mechanism design is satisfied. The principal commits, while the agent's type evolution follows a Markov process. My results demonstrate, even when allowing for potential correlation of stochastic contracts across periods that the usual restriction in the literature to deterministic contracts is admissible, as long as the first-order approach is valid.

Stochastic contraction based online estimation of second order wiener system

Wiener system is a block oriented model, having a linear time-invariant dynamic system followed by a memory-less nonlinearity. To design a stochastic estimator for online estimation of Wiener system of second order, this paper utilizes differential mean value theorem and the results of stochastic contraction theory. The asymptotic convergence of proposed estimator is derived by using contraction theory related to semi-contracting systems. The boundedness and convergence of the parameter and state estimates have been shown analytically. The introduced method has potentials to estimate accurately states and parameters of Wiener model simultaneously from the noisy output of the system and unknown structure of nonlinearity. Numerical simulation of the stochastic estimator is presented to justify the claim by considering the two examples of the real world system with an additive measurement noise.

Stochastic contraction-based observer and controller design algorithm with application to a flight vehicle

This paper focuses on a stochastic version of contraction theory to construct observer-controller structure for a flight dynamic system with noisy velocity measurement. A nonlinear stochastic observer is designed to estimate the pitch rate, the pitch angle, and the velocity of an aircraft example model using stochastic contraction theory. Estimated states are used to compute feedback control for solving a tracking problem. The structure and gain selection of the observer is carried out using Itô's stochastic differential equations and the contraction theory. The contraction property of integrated observer-controller structure is derived to ensure the exponential convergence of the trajectories of closed-loop nonlinear system. The upper bound of the state estimation error is explicitly derived and the efficacy of the proposed observer-controller structure has been shown through the numerical simulations.

Observer Design for Stochastic Nonlinear Systems via Contraction-Based Incremental Stability

This paper presents a new design approach to nonlinear observers for Ito stochastic nonlinear systems with guaranteed stability. A stochastic contraction lemma is presented which is used to analyze incremental stability of the observer. A bound on the mean-squared distance between the trajectories of original dynamics and the observer dynamics is obtained as a function of the contraction rate and maximum noise intensity. The observer design is based on a non-unique state-dependent coefficient (SDC) form, which parametrizes the nonlinearity in an extended linear form. The observer gain synthesis algorithm, called linear matrix inequality state-dependent algebraic Riccati equation (LMI-SDARE), is presented. The LMI-SDARE uses a convex combination of multiple SDC parametrizations. An optimization problem with state-dependent linear matrix inequality (SDLMI) constraints is formulated to select the coefficients of the convex combination for maximizing the convergence rate and robustness against disturbances. Two variations of LMI-SDARE algorithm are also proposed. One of them named convex state-dependent Riccati equation (CSDRE) uses a chosen convex combination of multiple SDC matrices; and the other named Fixed-SDARE uses constant SDC matrices that are pre-computed by using conservative bounds of the system states while using constant coefficients of the convex combination pre-computed by a convex LMI optimization problem. A connection between contraction analysis and L_2 gain of the nonlinear system is established in the presence of noise and disturbances. Results of simulation show superiority of the LMI-SDARE algorithm to the extended Kalman filter (EKF) and state-dependent differential Riccati equation (SDDRE) filter.

On convergence of the unscented Kalman–Bucy filter using contraction theory

Contraction theory entails a theoretical framework in which convergence of a nonlinear system can be analysed differentially in an appropriate contraction metric. This paper is concerned with utilising stochastic contraction theory to conclude on exponential convergence of the unscented Kalman–Bucy filter. The underlying process and measurement models of interest are Itô-type stochastic differential equations. In particular, statistical linearisation techniques are employed in a virtual–actual systems framework to establish deterministic contraction of the estimated expected mean of process values. Under mild conditions of bounded process noise, we extend the results on deterministic contraction to stochastic contraction of the estimated expected mean of the process state. It follows that for the regions of contraction, a result on convergence, and thereby incremental stability, is concluded for the unscented Kalman–Bucy filter. The theoretical concepts are illustrated in two case studies.

Design and convergence analysis of stochastic frequency estimator using contraction theory

This study investigates the design and analysis of an estimator for unknown frequencies of a sinusoid in the presence of additive noise. A dynamic stochastic estimator is proposed to ensure simultaneous globally convergent estimation of the state and the frequencies of a sinusoid comprising multiple frequencies. Approach given in this study exploits the results of stochastic contraction theory and the observers. The concept of contraction theory related to semi-contracting systems is used to show the asymptotic convergence of the proposed non-linear estimator. The boundedness and convergence of the state and frequencies estimates for all initial conditions and frequency values has been shown analytically. The proposed estimator is generalised to estimate n-unknown frequencies of a given noisy sinusoid. Numerical simulations of estimator are presented for different combinations of frequencies to justify the claim.

Endogenous information and stochastic contracts

A growing literature analyzes revenue-maximizing contracts for situations in which agents can acquire private information before they decide whether to join the contract. It is conjectured that the results also apply to the more natural scenario where information can be acquired either before or after signing. This paper shows that, in fact, the latter scenario is more favorable for the principal. Using stochastic contracts, she can induce information acquisition with some probability and yet appropriate the generated surplus.

Contracting with Subjective Evaluations and Communication

Should principals explain and justify their evaluations? In this paper the principal's evaluation is private information, but she can provide justification by sending a costly cheap-talk message. I show that the principal explains her evaluation to the agent if the evaluation turns out to be bad. The justification assures the agent that the principal has not distorted the evaluation downwards. In equilibrium, the wage increases in the agent's performance, when the principal justifies her evaluation. For good performance, however, the principal pays a constant high wage without justification. Furthermore, no payments to third parties are necessary if stochastic contracts are feasible.

Contracting with Subjective Evaluations and Communication

Should principals explain and justify their evaluations? In this paper the principal’s evaluation is private information, but she can provide some justifications by sending a costly cheap-talk message. Indeed, it is optimal for the principal to explain her evaluation to the agent if and only if the evaluation turns out to be bad. The justification assures the agent that the principal has not distorted the evaluation downwards. In equilibrium, the wage increases in the performance of the agent, as long as the principal provides a justification. For good performances, however, the principal pays a given high wage without providing justifications. This wage pattern fits empirical observations that subjective evaluations are lenient and discriminate poorly between good performances. I show that this pattern is part of the optimal contract instead of biased behavior. Furthermore, it is possible to implement the optimal contract in an ex-post budget-balanced way if stochastic contracts are feasible. JEL classifications: D82, D86, J41, M52

A stochastic contraction principle

A stochastic contraction principle

Optimal Contracts when Enforcement is a Decision Variable

This paper analyzes choice-theoretic costly enforcement in an intertemporal contracting model with a differentially informed investor and entrepreneur. An intertemporal contract is modeled as a mechanism in which there is limited commitment to payment and enforcement decisions. The goal of the analysis is to characterize the effect of choice-theoretic costly enforcement on the structure of optimal contracts. The paper shows that simple debt is the optimal contract when commitment is limited and costly enforcement is a decision variable (Theorem 1). In contrast, stochastic contracts are optimal when agents can commit to the ex-ante optimal decisions (Theorem 2). The paper also shows that the costly state verification model can be viewed as a reduced form of an enforcement model in which agents choose payments and strategies as part of a perfect Bayesian Nash equilibrium.