Yoav Sharon

Publications

1. Sharon, Y. and Margaliot, M. "Third-order nilpotency, nice reachability and asymptotic stability", Journal of Differential Equations, 233:136-150, 2007

   Abstract: We consider an affine control system whose vector fields span a third-order nilpotent Lie algebra. We show that the reachable set at time T using measurable controls is equivalent to the reachable set at time T using piecewise-constant controls with no more than four switches. The bound on the number of switches is uniform over any final time T. As a corollary, we derive a new sufficient condition for stability of nonlinear switched systems under arbitrary switching. This provides a partial solution to an open problem posed in [1].

   Conference version:

   Sharon, Y. and Margaliot, M., "Third-Order Nilpotency, Finite Switchings and Asymptotic Stability", 44th IEEE Conference in Decision and Control, CDC-ECC '05, 2005.

1. Sharon, Y. and Liberzon, M. "Input-to-State Stabilization with Minimum Number of Quantization Regions", 46th IEEE Conference in Decision and Control, 2007

   Abstract: We study control systems where the state measurements are quantized and time-sampled, and an unknown disturbance is being applied. We present a dynamic quantization scheme that switches between three modes of operation. We show that by using this scheme with a continuous static feedback controller we achieve a closed-loop system which has the Input-to-State Stability property (ISS). Our design does not use any characterization of the disturbance; as long as the disturbance is bounded the system will remain stable. We show that three quantization regions per dimension is sufficient to achieve the ISS property, and furthermore we show that the ISS property is achievable using a data rate that is arbitrarily close to the minimum required data rate when no disturbance is applied.
   - Slides
   - Earlier version (with proofs)

2. Sharon, Y. and Liberzon, M. "Input-to-State Stabilization with Quantized Output Feedback", Hybrid Systems: Computation and Control (HSCC 2008) conference, 2008

   Abstract: We study control systems where the output subspace is covered by a finite set of quantization regions, and the only information available to a controller is which of the quantization regions currently
   contains the system's output. We assume the dimension of the output subspace is strictly less than the dimension of the state space. The number of quantization regions can be as small as 3 per dimension of the output subspace. We show how to design a controller that stabilizes such a system, and makes the system robust to an external unknown disturbance in the sense that the closed-loop system has the Input-to-State Stability property. No information about the disturbance is required to design the controller. Achieving the ISS property for continuous-time systems with quantized measurements requires a hybrid approach, and indeed our controller consists of a dynamic, discrete-time observer, a continuous-time state-feedback stabilizer, and a switching logic that switches between several modes of operation. Except for some properties that the observer and the stabilizer must possess, our approach is general and not restricted to a specific observer or stabilizer. Examples of specific observers that possess these properties are included.

   - Slides

1. Sharon, Y., Wright, J. and Ma, Y. "Computation and Relaxation of Conditions for Equivalence between l1 and l0 Minimization", submitted to the 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2007

   Abstract: In this paper, we investigate the exact conditions under which the `1 and `0 minimizations arising in the context of sparse error correction or sparse signal reconstruction are equivalent. We present a much simplified condition for verifying equivalence, which leads to a provably correct algorithm that computes the exact sparsity of the error or the signal needed to ensure equivalence. Our algorithm is combinatorial, and for large matrices it can only be used to estimate, for signals with a certain sparsity, the probability that the l1 and l0 minimizations are equivalent. For small matrices, however, it produces the exact result in a reasonably short time, up to 106 times faster than the only other algorithm known for this problem. We present an example application where such small matrices are needed, and for which our algorithm can greatly assist with system design. We also show how, in the case when the encoding matrix is imbalanced an optimal diagonal rescaling matrix can be computed via linear programming, so that the rescaled system enjoys the widest possible equivalence.

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