Risk-sensitive Cost Function Research Articles

Bayesian optimization assisted meal bolus decision based on Gaussian processes learning and risk-sensitive control

Effective postprandial glucose control is important to glucose management for subjects with diabetes mellitus. In this work, a data-driven meal bolus decision method is proposed without the need of subject-specific glucose management parameters. The postprandial glucose dynamics is learnt using Gaussian process regression. Considering the asymmetric risks of hyper- and hypoglycemia and the uncertainties in the predicted glucose trajectories, an asymmetric risk-sensitive cost function is designed. Bayesian optimization is utilized to solve the optimization problem, since the gradient of the cost function is unavailable. The proposed approach is evaluated using the 10-adult cohort of the FDA-accepted UVA/Padova T1DM simulator and compared with the standard insulin bolus calculator. For the case of announced meals, the proposed method achieves satisfactory and similar performance in terms of mean glucose and percentage time in [70, 180] mg/dL without increasing the risk of hypoglycemia. Similar results are observed for the case without the meal information (assuming that the patient follows a consistent diet) and the case of basal rate mismatches. In addition, a comparison with a run-to-run based method for the scenario of potentially incorrect meal carbohydrate counts is also performed, and the results show that the proposed method is more robust to the carbohydrate counting disturbances. At last, advisory-mode analysis is performed based on clinical data, which indicates that the method can determine safe and reasonable meal boluses in real clinical settings. The results verify the effectiveness and robustness of the proposed method and indicate the feasibility of achieving improved postprandial glucose regulation through a data-driven optimal control method.

Event-triggered robust state estimation for systems with unknown exogenous inputs

An event-triggered robust state estimation problem for linear time-varying systems subject to Gaussian noises and non-stochastic unknown exogenous inputs is investigated in this work. To design the estimator, the state estimation problem is formulated as an optimization problem with a risk-sensitive cost function. This problem is solved by constructing a reference probability measure, under which the cost function has a simpler form and an information state can be developed. The obtained robust state estimator is shown to have a recursive form parameterized by a Riccati-type time-varying matrix equation. The effectiveness of the proposed event-based robust state estimator is illustrated with numerical examples.

Lower Bounds on Exponential Moments of the Quadratic Error in Parameter Estimation

Considering the problem of risk-sensitive parameter estimation, we propose a fairly wide family of lower bounds on the exponential moments of the quadratic error, both in the Bayesian and the non-Bayesian regime. This family of bounds, which is based on a change of measures, offers considerable freedom in the choice of the reference measure, and our efforts are devoted to explore this freedom to a certain extent. Our focus is mostly on signal models that are relevant to communication problems, namely, models of a parameter-dependent signal (modulated signal) corrupted by additive white Gaussian noise, but the methodology proposed is also applicable to other types of parametric families, such as models of linear systems driven by random input signals (white noise, in most cases), and others. In addition to the well-known motivations of the risk-sensitive cost function (i.e., the exponential quadratic cost function), which is most notably, the robustness to model uncertainty, we also view this cost function as a tool for studying fundamental limits concerning the tail behavior of the estimation error. Another interesting aspect that we demonstrate in a certain parametric model is that the risk-sensitive cost function may be subjected to phase transitions, owing to some analogies with statistical mechanics.

Output‐feedback control for a class of Markovian jump non‐linear systems under a risk‐sensitive cost criterion with unknown transition probabilities

This study addresses the output-feedback control problem for a class of Markovian jump non-linear systems under a quadratic risk-sensitive cost function criterion. The transition probabilities of the Markov process are assumed to be completely unknown. By employing the high-gain scaling technique, common Lyapunov function method and backstepping technique, a control law is constructed that guarantees any arbitrary small risk-sensitive cost for a given risk-sensitive parameter. Moreover, the resulted closed-loop system solutions are bounded in probability. Compared with some previous results, this study does not require the uniform boundedness of the gain functions of the system noise, and the control law further achieves a zero risk-sensitive cost and asymptotically stable in the large for the closed-loop system solutions when the vector field of the disturbance vanishes at the origin. A numerical example is given to illustrate the proposed protocol.

Backstepping Controller Design for Nonlinear Stochastic Systems Under a Risk-Sensitive Cost Criterion

This paper develops a methodology for recursive construction of optimal and near-optimal controllers for strict-feedback stochastic nonlinear systems under a risk-sensitive cost function criterion. The design procedure follows the integrator backstepping methodology, and the controllers obtained guarantee any desired achievable level of long-term average cost for a given risk-sensitivity parameter $\theta$. Furthermore, they lead to closed-loop system trajectories that are bounded in probability, and in some cases asymptotically stable in the large. These results also generalize to nonlinear systems with strongly stabilizable zero dynamics. A numerical example included in the paper illustrates the analytical results.

The role of information state and adjoint in relating nonlinear output feedback risk-sensitive control and dynamic games

This paper employs logarithmic transformations to establish relations between continuous-time nonlinear partially observable risk-sensitive control problems and analogous output feedback dynamic games. The first logarithmic transformation is introduced to relate the stochastic information state to a deterministic information state. The second logarithmic transformation is applied to the risk-sensitive cost function using the Laplace-Varadhan lemma. In the small noise limit, this cost function is shown to be logarithmically equivalent to the cost function of an analogous dynamic game.