We consider the problem of learning a realization for a linear time-invariant (LTI) dynamical system from input/output data. Given a single input/output trajectory, we provide finite time analysis for learning the system's Markov parameters, from which a balanced realization is obtained using the classical Ho-Kalman algorithm. By proving a stability result for the Ho-Kalman algorithm and combining it with the sample complexity results for Markov parameters, we show how much data is needed to learn a balanced realization of the system up to a desired accuracy with high probability. ‡ Version 2 has two improvements: First, paper now uses spectral radius rather than largest singular value hence applies to a larger class of systems. Secondly, new sample complexity bounds are provided for approximating the system's Hankel operator via estimated Markov parameters. These bounds leverage stability and treat the system as if it has a logarithmic order.Hence, if we had access to infinitely many independent (y t , u t−k ) pairs, our task could be accomplished by a simple averaging. In this work, we will show that, one can robustly learn these matrices from a small amount of data generated from a single realization of the system trajectory. The challenge is efficiently using finite and dependent data points to perform reliable estimation. Observe that, our problem is identical to learning the concatenated matrix G defined asNext section describes our input and output data. Based on this, we formulate a least-squares procedure that estimates G. The estimateĜ will play a critical role in the identification of the system matrices.1 Balanced realizations give a representation of the system in a basis that orders the states in terms of their effect on the input/output behavior. This is relevant for determining the system order and for model reduction [23]. 2 While we assume diagonal covariance throughout the paper, we believe our proof strategy can be adapted to arbitrary covariance matrices.

In an aircraft electric power system (EPS), a supervisory control unit must actuate a set of switches to distribute power from generators to loads, while satisfying safety, reliability and real-time performance requirements. To reduce expensive re-design steps in current design methodologies, such a control problem is generally addressed based on minor incremental changes on top of consolidated solutions, since it is difficult to estimate the impact of earlier design decisions on the final implementation. In this paper, we introduce a methodology for the design space exploration and virtual prototyping of EPS supervisory control protocols, following the platform-based design (PBD) paradigm. Moreover, we describe the modeling infrastructure that supports the methodology. In PBD, design space exploration is carried out as a sequence of refinement steps from the initial specification towards a final implementation, by mapping higher-level behavioral models into a set of library components at a lower level of abstraction. In our flow, the system specification is captured using SysML requirement and structure diagrams. State-machine diagrams enable verification of the control protocol at a high level of abstraction, while lowerlevel hybrid models, implemented in Simulink, are used to verify properties related to physical quantities, such as time, voltage and current values. The effectiveness of our approach is illustrated on a prototype EPS control protocol design.

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Motivated by the challenge of developing control software provably meeting specifications for real world problems, this paper applies formal methods to adaptive cruise control (ACC). Starting from a Linear Temporal Logic specification for ACC, obtained by interpreting relevant ACC standards, we discuss in this paper two different control software synthesis methods. Each method produces a controller that is correct-byconstruction, meaning that trajectories of the closed-loop systems provably meet the specification. Both methods rely on fixed-point computations of certain set-valued mappings. However, one of the methods performs these computations on the continuous state space whereas the other method operates on a finitestate abstraction. While controller synthesis is based on a lowdimensional model, each controller is tested on CarSim, an industry-standard vehicle simulator. Our results demonstrate several advantages over classical control design techniques. First, a formal approach to control design removes potential ambiguity in textual specifications by translating them into precise mathematical requirements. Second, because the resulting closed-loop system is known a priori to satisfy the specification, testing can then focus on the validity of the models used in control design and whether the specification captures the intended requirements. Finally, the set from where the specification (e.g., safety) can be enforced is explicitly computed and thus conditions for passing control to an emergency controller are clearly defined.

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Abstraction-based, hierarchical approaches to control synthesis from temporal logic specifications for dynamical systems have gained increased popularity over the last decade. Yet various issues commonly encountered and extensively dealt with in control systems have not been adequately discussed in the context of temporal logic control of dynamical systems, such as inter-sample behaviors of a sampled-data system, effects of imperfect state measurements and unmodeled dynamics, and the use of time-discretized models to design controllers for continuous-time dynamical systems. We discuss these issues in this paper. The main motivation is to demonstrate the possibility of accounting for the mismatches between a continuous-time control system and its various types of abstract models used for control synthesis. We do this by incorporating additional robustness measures in the abstract models. Such robustness measures are gained at the price of either increased nondeterminism in the abstracted models or relaxed versions of the specification being realized. Under a unified notion of abstraction, we provide concrete means of incorporating these robustness measures and establish results that demonstrate their effectiveness in dealing with the above mentioned issues.

Abstract-This paper addresses the problem of robust identification of a class of discrete-time affine hybrid systems, switched affine models, in a set membership framework. Given a finite collection of noisy input/output data and some minimal a priori information about the set of admissible plants, the objective is to identify a suitable set of affine models along with a switching sequence that can explain the available experimental information, while optimizing a performance criteria (either minimum number of switches or minimum number of plants). Our main result shows that this problem can be reduced to a sparsification form, where the goal is to maximize sparsity of a given vector sequence. Although in principle this leads to an NP-hard problem, as we show in the paper, efficient convex relaxations can be obtained by exploiting recent results on sparse signal recovery. These results are illustrated using two non-trivial problems arising in computer vision applications: video-shot and dynamic texture segmentation.

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