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Inspired by the work of Tsiamis et al. \cite{tsiamis2022learning}, in this paper we study the statistical hardness of learning to stabilize linear time-invariant systems. Hardness is measured by the number of samples required to achieve a learning task with a given probability. The work in \cite{tsiamis2022learning} shows that there exist system classes that are hard to learn to stabilize with the core reason being the hardness of identification. Here we present a class of systems that can be easy to identify, thanks to a non-degenerate noise process that excites all modes, but the sample complexity of stabilization still increases exponentially with the system dimension. We tie this result to the hardness of co-stabilizability for this class of systems using ideas from robust control., Comment: 7 pages, 2 figures, accepted by CDC 2023

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., Comment: Minor differences between the implementation and the originally provided descriptions are corrected, ensuring better clarity and accuracy of the content

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.

Aiming at an obstacle avoidance problem with dynamic constraints for Unmanned Surface Vehicle (USV), a method based on Circle Grid Trajectory Cell (CGTC) is proposed. Firstly, the ship model and standardization rules are constructed to develop and constrain the trajectory, respectively. Secondly, by analyzing the properties of the circle grid, the circle grid tree is produced to guide the motion of the USV. Then, the kinematics and dynamics of the USV are considered through the on-line trajectory generator by designing a relational function that links the rudder angle, heading angle, and the central angle of the circle grid. Finally, obstacle avoidance is achieved by leveraging the on-line trajectory generator to choose a safe, smooth, and efficient path for the USV. The experimental results indicate that the proposed method can avoid both static and dynamic obstacles, have better performance in terms of distance cost and steering cost comparing with the related methods, and our method only takes 50% steering cost of the grid-based method; the collision avoidance path not only conforms to the USV dynamic characteristic but also provides a reference of steering command.

Learning how to effectively control unknown dynamical systems is crucial for intelligent autonomous systems. This task becomes a significant challenge when the underlying dynamics are changing with time. Motivated by this challenge, this paper considers the problem of controlling an unknown Markov jump linear system (MJS) to optimize a quadratic objective. By taking a model-based perspective, we consider identification-based adaptive control for MJSs. We first provide a system identification algorithm for MJS to learn the dynamics in each mode as well as the Markov transition matrix, underlying the evolution of the mode switches, from a single trajectory of the system states, inputs, and modes. Through mixing-time arguments, sample complexity of this algorithm is shown to be $\mathcal{O}(1/\sqrt{T})$. We then propose an adaptive control scheme that performs system identification together with certainty equivalent control to adapt the controllers in an episodic fashion. Combining our sample complexity results with recent perturbation results for certainty equivalent control, we prove that when the episode lengths are appropriately chosen, the proposed adaptive control scheme achieves $\mathcal{O}(\sqrt{T})$ regret, which can be improved to $\mathcal{O}(polylog(T))$ with partial knowledge of the system. Our proof strategy introduces innovations to handle Markovian jumps and a weaker notion of stability common in MJSs. Our analysis provides insights into system theoretic quantities that affect learning accuracy and control performance. Numerical simulations are presented to further reinforce these insights.

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 $\epsilon$ and $\eta$ 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 $\mathcal{O}(\epsilon + \eta)$ and $\mathcal{O}((\epsilon + \eta)^2)$ respectively., Comment: 17 pages, 8 figures

The essence of the path planning problems is multi-modality constraint. However, most of the current literature has not mentioned this issue. This paper introduces the research progress of path planning based on the multi-modality constraint. The path planning of multi-modality constraint research can be classified into three stages in terms of its basic ingredients (such as shape, kinematics and dynamics et al.): Route Planning, Trajectory Planning and Motion Planning. It then reviews the research methods and classical algorithms, especially those applied to the Unmanned Surface Vehicle (USV) in every stage. Finally, the paper points out some existing problems in every stage and suggestions for future research.

In this work, we consider the problem of mode clustering in Markov jump models. This model class consists of multiple dynamical modes with a switching sequence that determines how the system switches between them over time. Under different active modes, the observations can have different characteristics. Given the observations only and without knowing the mode sequence, the goal is to cluster the modes based on their transition distributions in the Markov chain to find a reduced-rank Markov matrix that is embedded in the original Markov chain. Our approach involves mode sequence estimation, mode clustering and reduced-rank model estimation, where mode clustering is achieved by applying the singular value decomposition and k-means. We show that, under certain conditions, the clustering error can be bounded, and the reduced-rank Markov chain is a good approximation to the original Markov chain. Through simulations, we show the efficacy of our approach and the application of our approach to real world scenarios.

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.