Zhe Du

Haldun Balim, Zhe Du, Samet Oymak et al.

Transformer models have shown great success in natural language processing; however, their potential remains mostly unexplored for dynamical systems. In this work, we investigate the optimal output estimation problem using transformers, which generate output predictions using all the past ones. Particularly, we train the transformer using various distinct systems and then evaluate the performance on unseen systems with unknown dynamics. Empirically, the trained transformer adapts exceedingly well to different unseen systems and even matches the optimal performance given by the Kalman filter for linear systems. In more complex settings with non-i.i.d. noise, time-varying dynamics, and nonlinear dynamics like a quadrotor system with unknown parameters, transformers also demonstrate promising results. To support our experimental findings, we provide statistical guarantees that quantify the amount of training data required for the transformer to achieve a desired excess risk. Finally, we point out some limitations by identifying two classes of problems that lead to degraded performance, highlighting the need for caution when using transformers for control and estimation.

Zhe Du, Necmiye Ozay, Laura Balzano

In this paper, we consider the problem of online identification of Switched AutoRegressive eXogenous (SARX) systems, where the goal is to estimate the parameters of each subsystem and identify the switching sequence as data are obtained in a streaming fashion. Previous works in this area are sensitive to initialization and lack theoretical guarantees. We overcome these drawbacks with our two-step algorithm: (i) every time we receive new data, we first assign this data to one candidate subsystem based on a novel robust criterion that incorporates both the residual error and an upper bound of subsystem estimation error, and (ii) we use a randomized algorithm to update the parameter estimate of chosen candidate. We provide a theoretical guarantee on the local convergence of our algorithm. Though our theory only guarantees convergence with a good initialization, simulation results show that even with random initialization, our algorithm still has excellent performance. Finally, we show, through simulations, that our algorithm outperforms existing methods and exhibits robust performance.

Zhe Du, Laura Balzano, Necmiye Ozay

Switched systems are capable of modeling processes with underlying dynamics that may change abruptly over time. To achieve accurate modeling in practice, one may need a large number of modes, but this may in turn increase the model complexity drastically. Existing work on reducing system complexity mainly considers state space reduction, yet reducing the number of modes is less studied. In this work, we consider Markov jump linear systems (MJSs), a special class of switched systems where the active mode switches according to a Markov chain, and several issues associated with its mode complexity. Specifically, inspired by clustering techniques from unsupervised learning, we are able to construct a reduced MJS with fewer modes that approximates well the original MJS under various metrics. Furthermore, both theoretically and empirically, we show how one can use the reduced MJS to analyze stability and design controllers with significant reduction in computational cost while achieving guaranteed accuracy.

Zhe Du, Yahya Sattar, Davoud Ataee Tarzanagh et al.

Real-world control applications often involve complex dynamics subject to abrupt changes or variations. Markov jump linear systems (MJS) provide a rich framework for modeling such dynamics. Despite an extensive history, theoretical understanding of parameter sensitivities of MJS control is somewhat lacking. Motivated by this, we investigate robustness aspects of certainty equivalent model-based optimal control for MJS with quadratic cost function. Given the uncertainty in the system matrices and in the Markov transition matrix is bounded by $ε$ and $η$ respectively, robustness results are established for (i) the solution to coupled Riccati equations and (ii) the optimal cost, by providing explicit perturbation bounds which decay as $\mathcal{O}(ε+ η)$ and $\mathcal{O}((ε+ η)^2)$ respectively.