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Register a new userConvergence Certificate for Stochastic Derivative-Free Trust-Region Methods based on Gaussian Processes
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In many machine learning applications, one wants to learn the unknown objective and constraint functions of an optimization problem from available data and then apply some technique to attain a local optimizer of the learned model. This work considers Gaussian processes as global surrogate models and utilizes them in conjunction with derivative-free trust-region methods. It is well known that derivative-free trust-region methods converge globally---provided the surrogate model is probabilistically fully linear. We prove that glspl{gp} are indeed probabilistically fully linear, thus resulting in fast (compared to linear or quadratic local surrogate models) and global convergence. We draw upon the optimization of a chemical reactor to demonstrate the efficiency of gls{gp}-based trust-region methods.
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In this paper, a distributed learning leader-follower consensus protocol based on Gaussian process regression for a class of nonlinear multi-agent systems with unknown dynamics is designed. We propose a distributed learning approach to predict the residual dynamics for each agent. The stability of the consensus protocol using the data-driven model of the dynamics is shown via Lyapunov analysis. The followers ultimately synchronize to the leader with guaranteed error bounds by applying the proposed control law with a high probability. The effectiveness and the applicability of the developed protocol are demonstrated by simulation examples.
This work attempts to approximate a linear Gaussian system with a finite-state hidden Markov model (HMM), which is found useful in solving sophisticated event-based state estimation problems. An indirect modeling approach is developed, wherein a state space model (SSM) is firstly identified for a Gaussian system and the SSM is then used as an emulator for learning an HMM. In the proposed method, the training data for the HMM are obtained from the data generated by the SSM through building a quantization mapping. Parameter learning algorithms are designed to learn the parameters of the HMM, through exploiting the periodical structural characteristics of the HMM. The convergence and asymptotic properties of the proposed algorithms are analyzed. The HMM learned using the proposed algorithms is applied to event-triggered state estimation, and numerical results on model learning and state estimation demonstrate the validity of the proposed algorithms.
In the interaction between control and mathematics, mathematical tools are fundamental for all the control methods, but it is unclear how control impacts mathematics. This is the first part of our paper that attempts to give an answer with focus on solving linear algebraic equations (LAEs) from the perspective of systems and control, where it mainly introduces the controllability-based design results. By proposing an iterative method that integrates a learning control mechanism, a class of tracking problems for iterative learning control (ILC) is explored for the problem solving of LAEs. A trackability property of ILC is newly developed, by which analysis and synthesis results are established to disclose the equivalence between the solvability of LAEs and the controllability of discrete control systems. Hence, LAEs can be solved by equivalently achieving the perfect tracking tasks of resulting ILC systems via the classic state feedback-based design and analysis methods. It is shown that the solutions for any solvable LAE can all be calculated with different selections of the initial input. Moreover, the presented ILC method is applicable to determining all the least squares solutions of any unsolvable LAE. In particular, a deadbeat design is incorporated to ILC such that the solving of LAEs can be completed within finite iteration steps. The trackability property is also generalized to conventional two-dimensional ILC systems, which creates feedback-based methods, instead of the common used contraction mapping-based methods, for the design and convergence analysis of ILC.
This paper presents a scheme for dual robust control of batch processes under parametric uncertainty. The dual-control paradigm arises in the context of adaptive control. A trade-off should be decided between the control actions that (robustly) optimize the plant performance and between those that excite the plant such that unknown plant model parameters can be learned precisely enough to increase the robust performance of the plant. Some recently proposed approaches can be used to tackle this problem, however, this will be done at the price of conservativeness or significant computational burden. In order to increase computational efficiency, we propose a scheme that uses parameterized conditions of optimality in the adaptive predictive-control fashion. The dual features of the controller are incorporated through scenario-based (multi-stage) approach, which allows for modeling of the adaptive robust decision problem and for projecting this decision into predictions of the controller. The proposed approach is illustrated on a case study from batch membrane filtration.
We develop in this work a numerical method for stochastic differential equations (SDEs) with weak second order accuracy based on Gaussian mixture. Unlike the conventional higher order schemes for SDEs based on It^o-Taylor expansion and iterated It^o integrals, the proposed scheme approximates the probability measure $mu(X^{n+1}|X^n=x_n)$ by a mixture of Gaussians. The solution at next time step $X^{n+1}$ is then drawn from the Gaussian mixture with complexity linear in the dimension $d$. This provides a new general strategy to construct efficient high weak order numerical schemes for SDEs.
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