Dual Control Problem Research Articles

Distributed aperiodic sampled‐data implement for dual objective control of energy storage systems

AbstractThis article solves the dual objective control problem for an energy storage system by distributed aperiodic sampled‐data controller under both connected static network and jointly‐connected switching network. The proposed sample‐and‐hold controller is composed of the leaderless consensus algorithm, which aims at state‐of‐energy balancing, and the leader‐following distributed observer, which aims at reference power tracking. By resorting to the generalized Krasovskii‐LaSalle theorem, it is proven that the closed‐loop system is uniformly exponentially stable using the so‐called limiting solution. There are two distinguished features of the proposed control scheme. First, the designs of the leaderless consensus algorithm and the leader‐following distributed observer can be accomplished independently. Second, the effectiveness of the control law can be guaranteed by properly tuning the feedback gains even for some extreme cases of drastic changes on sensing and sampling. Comprehensive simulation results are shown to examine the performance of the proposed control implement.

Credit risk contagion and optimal dual control—An SIS/R model

In order to help regulators to control credit risk contagion effectively when the financial crisis happens, this paper first proposes a new nonlinear SIS/R (Susceptible–Infected–Susceptible/Removed) model to describe the real dynamic of risk spreading, which is more reasonable compared with traditional epidemic models. We provide a comprehensive nonlinear analysis of monotonicity, threshold conditions, stability properties, and asymptotic convergence for the SIS/R model. Moreover, we formulate a time-varying optimal dual control problem considering the macro-prudential tools and the easy monetary policy. By applying Pontryagin’s minimum principle, we obtain the existence of a solution and its uniqueness for sufficient small regulatory deadlines. At last, the theoretical results are confirmed by numerical simulations. We show the effectiveness of the optimal dual regulatory strategy against the constant dual strategy and no control strategy. The sensitivity of the control cost to the model parameters is also shown.

Distributed dual objective control of an energy storage system under jointly connected switching networks

This paper studies a dual objective control problem for an energy storage system (ESS) consisting of multiple independently-controlled energy storage units (ESUs). The power output of the entire ESS is designed to meet its reference, and meanwhile the state-of-energy (SOE) of all the ESUs maintains to be balanced. In practice, the communication among ESUs might be unreliable and thus is modeled by jointly connected switching networks. By making use of the so-called Generalized Krasovskii–LaSalle Theorem together with a novel two-step recursive analysis method, it is rigorously proven that the dual objective control problem subject to jointly connected switching networks can be solved by a consensus-based distributed control algorithm.

A Nonparametric Dual Control Algorithm of Multidimensional Objects with Interval-Valued Observations

We focus on the dual interval control problem of multidimensional objects with delay. We propose a new nonparametric algorithm. In such a case, it is not necessary to determine a parametric structure of the investigated object. Another difficulty lies in the complex nature of the decision-making field as it might not be flexible or convenient for decision-makers to exactly quantify their opinions with crisp numbers. Due to this fact, we introduce the interval-valued observations into the algorithm by means of the single-level constraint interval arithmetic. The results of computational experiments illustrate the effectiveness of the algorithm in the case of using intervals instead of crisp values.

A Dual Control Approach to Solve Exploration Versus Exploitation Trade-Offs in the Design of Personalized Physical Exercise Sessions

Using biofeedback in medical therapies has proven to be effective for adapting patient behaviors while keeping the patients engaged and motivated in an exercise session. This paper considers general problems in personalized exercise sessions where the input is opportune biofeedback and the session goal is to maximize a particular exercise effect. Due to the individual differences between patients and their physiological signals, however, personalized patient models also need to be identified. With the two objectives: 1) maximize a training effect with minimal control effort, and 2) identify the individualized patient model, we have a typical exploration vs. exploration trade-off. Control problems of this form are called dual control problems. In this paper, we formulate a dual control problem for a personalized exercise session and test the approach against classical optimal control and optimal experimental design approaches in an illustrative example of performing Kegel exercises where the control and identification goals conflict with each other.

Cooperative Distributed GPU Power Capping for Deep Learning Clusters

The recent GPU-based clusters that handle deep learning (DL) tasks have the features of GPU device heterogeneity, a variety of deep neural network (DNN) models, and high computational complexity. Thus, the traditional power capping methods for CPU-based clusters or small-scale GPU devices cannot be applied to the GPU-based clusters handling DL tasks. This article develops a cooperative distributed GPU power capping (CD-GPC) system for GPU-based clusters, aiming to minimize the training completion time of invoked DL tasks without exceeding the limited power budget. Specifically, we first design the frequency scaling approach using the online model estimation based on the recursive least square method. This approach achieves the accurate tuning for DL task training time and power usage of GPU devices without needing offline profiling. Then, we formulate the proposed FS problem as a Lagrangian dual decomposition-based economic model predictive control problem for large-scale heterogeneous GPU clusters. We conduct both the NVIDIA GPU-based lab-scale real experiments and real job trace-based simulation experiments for performance evaluation. Experimental results validate that the proposed system improves the power capping accuracy to have a mean absolute error of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$&lt; \!1\%$</tex-math></inline-formula> , and reduces the deadline violation ratio of invoked DL tasks by 21.5% compared with other recent counterparts.

Understanding the dual formulation for the hedging of path-dependent options with price impact

We consider a general path-dependent version of the hedging problem with price impact of Bouchard et al. (SIAM J. Control Optim. 57 (2019) 4125–4149), in which a dual formulation for the super-hedging price is obtained by means of PDE arguments, in a Markovian setting and under strong regularity conditions. Using only probabilistic arguments, we prove, in a path-dependent setting and under weak regularity conditions, that any solution to this dual problem actually allows one to construct explicitly a perfect hedging portfolio. From a pure probabilistic point of view, our approach also allows one to exhibit solutions to a specific class of second order forward backward stochastic differential equations, in the sense of Cheridito et al. (Comm. Pure Appl. Math. 60 (2007) 1081–1110). Existence of a solution to the dual optimal control problem is also addressed in particular settings. As a by-product of our arguments, we prove a version of Itô’s lemma for path-dependent functionals that are only C0,1 in the sense of Dupire.

Effective Approximation Methods for Constrained Utility Maximization with Drift Uncertainty

In this paper, we propose a novel and effective approximation method for finding the value function for general utility maximization with closed convex control constraints and partial information. Using the separation principle and the weak duality relation, we transform the stochastic maximum principle of the fully observable dual control problem into an equivalent error minimization stochastic control problem and find the tight lower and upper bounds of the value function and its approximate value. Numerical examples show the goodness and usefulness of the proposed method.

Total probability and Bayes formulae for joint multidimensional-matrix Gaussian distributions

This paper is devoted to the development of a mathematical tool for obtaining the Bayesian estimations of the parameters of multidimensional regression objects in their finite-dimensional multidimensional-matrix description. Such a need arises, particularly, in the problem of dual control of regression objects when multidimensional-matrix mathematical formalism is used for the description of the controlled object. In this paper, the concept of a one-dimensional random cell is introduced as a set of multidimensional random matrices (in accordance with the “cell array” data type in the Matlab programming system), and the definition of the joint multidimensional-matrix Gaussian distribution is given (the definition of the Gaussian one-dimensional random cell). This required the introduction of the concepts of one-dimensional cell of the mathematical expectation and two-dimensional cell of the variance-covariance of the one-dimensional random cell. The integral connected with the joint Gaussian probability density function of the multidimensional matrices is calculated. The two formulae of the total probability and the Bayes formula for joint multidimensional-matrix Gaussian distributions are given. Using these results, the Bayesian estimations of the unknown coefficients of the multidimensional-matrix polynomial regression function are obtained. The algorithm of the calculation of the Bayesian estimations is realized in the form of the computer program. The results represented in the paper have theoretical and algorithmic generality.

Multicriteria Optimization of Induced Norms of Linear Operators: Primal and Dual Control and Filtering Problems

Multicriteria Optimization of Induced Norms of Linear Operators: Primal and Dual Control and Filtering Problems

Finite horizon portfolio selection with durable goods

We study the consumption and portfolio selection problem of a finitely lived agent who derives utility from the stock of durable goods. We show that the agent’s effective relative risk aversion implied by the optimal portfolio tends to decline and approaches zero, as the planning horizon approaches, whereas the agent exhibits constant effective relative risk aversion in the infinite horizon model of Hindy and Huang (1993). The existence of the stock of durable goods acts as buffer stock and induces the highly risk-tolerant attitude. We approach this problem using successive transformations. We transform our problem by applying an isomorphism proposed by Schroder and Skiadas (2002) to a singular control problem involving the choice of a monotone increasing consumption process. We next transform the problem into a dual singular control problem using the dual martingale method. We then transform the dual singular control problem into an optimal stopping problem. We analyze the variational inequality arising from the optimal stopping problem and provide an integral representation of optimal strategies.

Тotal probability formula for vector Gaussian distributions

The total probability formula for continuous random variables is the integral of product of two probability density functions that defines the unconditional probability density function from the conditional one. The need for calculation of such integrals arises in many applications, for instant, in statistical decision theory. The statistical decision theory attracts attention due to the ability to formulate the problems in a strict mathematical form. One of the technical problems solved by the statistical decision theory is the problem of dual control that requires calculation of integrals connected with the multivariate probability distributions. The necessary integrals are not available in the literature. One theorem on the total probability formula for vector Gaussian distributions was published by the authors earlier. In this paper we repeat this theorem and prove a new theorem that uses more familiar form of the initial data and has more familiar form of the result. The new form of the theorem allows us to obtain the unconditional mathematical expectation and the unconditional variance-covariance matrix very simply. We also confirm the new theorem by direct calculation for the case of the simple linear regression.

Optimal control and duality-based observer design for a hyperbolic PDEs system with application to fixed-bed reactor

This paper is devoted to the design of an optimal infinite-dimensional Luenberger observer combined with a linear-quadratic state feedback controller for a system of hyperbolic PDEs. The design is based on the duality fact between the control design and the observer design. Both the original linear-quadratic and dual control problems have been solved by using the associated Riccati equations. A general algorithm that combines the designed observer together with the (estimated) state-feedback controller has been developed. The theoretical development has been applied to a fixed-bed reactor to validate the performances of the designed observer-controller via numerical simulation. Estimation and control of the temperature and the reactant concentration in a fixed-bed reactor is investigated by using the developed algorithm, which lead to express the jacket temperature (manipulated variable) as a feedback of the estimated temperature and concentration in the reactor.

Finite Horizon Portfolio Selection with Durable Goods

We study the consumption and portfolio selection problem of a finitely lived agent who derives utility from the stock of durable goods. We show that the agent's effective relative risk aversion implied by the optimal portfolio tends to decline and approaches zero, as the planning horizon approaches, whereas the agent exhibits constant effective relative risk aversion in the infinite horizon model of Hindy and Huang (1993). The existence of the stock of durable goods acts as buffer stock and induces the highly risk-tolerant attitude. We approach this problem using successive transformations. We transform our problem by applying an isomorphism proposed by Schroder and Skiadas (2002) to a singular control problem involving the choice of a monotone increasing consumption process. We next transform the problem into a dual singular control problem using the dual martingale method. We then transform the dual singular control problem into an optimal stopping problem. We analyze the variational inequality arising from the optimal stopping problem and provide an integral representation of optimal strategies.

Cooperative Control of A Flywheel Energy Storage System with Identical Damping

Motivated by the work of Cai and Hu (2018), this paper considers the dual objective control problem of a flywheel energy storage system targeting simultaneous state-of-energy balancing and reference power tracking. It is first shown that, in the presence of flywheel damping, the steady state solution subject to the dual control objective exists and is defined by a virtual dynamic system. Second, under the identical damping condition, it is proven that the control law proposed in Cai and Hu (2018) can still solve the dual objective control problem.

Robust Adaptive Control with Active Learning for Fed-Batch Process based on Approximate Dynamic Programming

Batch process is often subject to a high degree of uncertainty in raw material quality and other initial feedstock conditions. One of the key objectives in operating a batch process is achieving consistent performance and constraint satisfaction in the presence of these uncertainties. This study presents a method for optimal control of a fed-batch process, which can actively and robustly cope with system uncertainty. As in dual control, the method aims to achieve an optimal balance between control actions (exploitation) and probing actions (exploration), leading to improved process performance by actively reducing system uncertainty. An optimal solution of the dual control problem can be found by stochastic dynamic programming but it is computationally intractable in most practical cases. In this study, an approximate dynamic programming (ADP) method for solving the dual control problem is tailored to a batch process which involves non-stationary and nonlinear dynamics. Rewards are formulated to maximize a given end objective while satisfying path constraints. Performance of the ADP-based dual controller is tested on a fed-batch bioreactor with two uncertain parameters.

Co-Design of Dual Security Control and Communication for Nonlinear CPS Under DoS Attack

The less conservative co-design problem of dual security control and communication for CPS was studied for a class of nonlinear CPS with an actuator fault and DoS attack based on the DETCS. First, combining with the edge computing thought, data filtering and partial computing tasks were assigned to the network edge devices. The DoS attack with limited energy is converted into a special time delay; thus, the discrete event trigger condition, the actuator fault estimation and regulation, and the robustness for DoS attack can be considered uniformly. A dual security control framework of non-uniformly sampled data system was proposed under the DETCS for CPS, and the T-S fuzzy model of closed-loop nonlinear CPS with the co-existence of fault and attack was established. Second, with the help of time-delay system theory, a robust observer for state and fault estimation and the co-design method of dual security control and communication with the actuator fault and DoS attack were obtained by constructing an appropriate Lyapunov-Krasovskii function and using affine Bessel-Legendre inequality. Finally, the validity and feasibility of the theoretical results were verified by a classical quadruple-tank system.

A Further Study on the Cooperative Control of Energy Storage Systems under Unreliable Communication Network

In Cai and Hu (2018), a dual objective control problem for an energy storage system was solved by a distributed control scheme which can achieve both state-of-energy balancing and power tracking. However, it relies on the assumption that the communication network is reliable and fixed. In this work, we further consider the same problem for the case that the communication network is unreliable and switched. It is proven that, under certain connectivity condition on the communication network, the same control law as in Cai and Hu (2018) can still achieve the dual control objective.

Structured exploration in the finite horizon linear quadratic dual control problem

This paper presents a novel approach to synthesize dual controllers for unknown linear time-invariant systems with the tasks of optimizing a quadratic cost while reducing the uncertainty. To this end, a synthesis problem is defined where the feedback law has to simultaneously gain knowledge of the system and robustly optimize the cost. By framing the problem in a finite horizon setting, the trade-offs arising when the tasks include both identification and control are formally captured in the optimization problem. Results show that efficient exploration strategies are achieved when the structure of the problem is exploited.

Dual Control for Stochastic Linear MIMO Systems With Parameter Uncertainty

This paper considers a dual control problem for stochastic linear MIMO (Multiple Input Multiple Output) systems with parameter uncertainty. A novel dual adaptive control law for MIMO systems is proposed. The design is based on the innovation of the dual control cost function, which was originally developed for conventional adaptive control of linear systems. However, the design process is modified and developed to cater to the stochastic MIMO case. This is a more challenging problem because the superposition of parameter uncertainty and the MIMO property makes the problem more intractable. As in all dual adaptive strategies, it leads to a control law that balances out the need for caution, due to parameter uncertainty, with the conflicting requirement of probing that acts to quickly reduce parameter uncertainty, which is the nature of dual control. The proposed control law has two parts. One reflects the goal of regulating the output, and the second reflects the ability to handle uncertain parameters. A learning factor is introduced to balance these two parts to obtain a control law that can be applied to the original system. In the simulation examples, the uncertain parameters can be estimated quickly and accurately for the unknown but constant case and the abrupt parameter case. Furthermore, it is shown that the novel dual control law is superior to other control strategies by comparing the performance of the cost function in a statistical sense.