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Reference governors are add-on schemes that are used to modify trajectories to prevent controlled dynamical systems from violating constraints and so are playing an increasingly important role in aerospace, robotic, and other engineering applications. Here we present a novel reference governor design for systems whose polynomial constraints depend on unknown bounded parameters. This is a significant departure from earlier treatments of reference governors, where the constraints were linear or known, because here we transfer the uncertainties into the constraints instead of having them in the closed loop dynamics, which greatly simplifies the task of determining future evolution of the constraints. Unlike our earlier treatment of reference governors with polynomial constraints, which transformed the constraints into linear ones that depend on an augmented state of the system, here we transform the constraints into linear ones that depend on both the system's state and uncertain parameters. Convexity allows us to compute the maximal output admissible set for an uncertain pre-stabilized linear system. We show that it is sufficient to only consider the extreme values of the uncertain parameters when computing and propagating the polynomial constraints. We illustrate our method using an uncertain longitudinal dynamics for civilian aircraft, which is controlled using a disturbance compensation method and needs to satisfy input and state constraints, and where our reference governor method ensures that safety constraints are always satisfied.

We study a feedback stabilization problem for a first-order hyperbolic partial differential equation. The problem is inspired by the stabilization of equilibrium age profiles for an age-structured chemostat, using the dilution rate as the control. Two distinguishing features of the problem are that (a) the PDE has a multiplicative (instead of an additive) input and (b) the state is fed back to the inlet boundary. We provide a sampled-data feedback that ensures stabilization under arbitrarily sparse sampling and that satisfies input constraints. Our chemostat feedback does not require measurement of the age profile, nor does it require exact knowledge of the model., Comment: 34 pages, 3 figures

A new hybrid tracking controller for neuromuscular electrical stimulation is proposed. The control scheme uses sampled measurements and is designed by utilizing a numerical prediction of the state variables. The tracking error of the closed-loop system converges exponentially to zero and robustness to perturbations of the sampling schedule is exhibited. One of the novelties of our approach is the ability to satisfy a state constraint imposed by the physical system., Comment: 33 pages

We study the chemostat model for one species competing for one nutrient using a Lyapunov-type analysis. We design the dilution rate function so that all solutions of the chemostat converge to a prescribed periodic solution. In terms of chemostat biology, this means that no matter what positive initial levels for the species concentration and nutrient are selected, the long term species concentration and substrate levels closely approximate a prescribed oscillatory behavior. This is significant because it reproduces the realistic ecological situation where the species and substrate concentrations oscillate. We show that the stability is maintained when the model is augmented by additional species that are being driven to extinction. We also give an input-to-state stability result for the chemostat tracking equations for cases where there are small perturbations acting on the dilution rate and initial concentration. This means that the long term species concentration and substrate behavior enjoys a highly desirable robustness property, since it continues to approximate the prescribed oscillation up to a small error when there are small unexpected changes in the dilution rate function., Comment: 20 pages, 3 figures, accepted for publication in Mathematical Biosciences and Engineering on December 5, 2006

We provide general methods for explicitly constructing strict Lyapunov functions for fully nonlinear slowly time-varying systems. Our results apply to cases where the given dynamics and corresponding frozen dynamics are not necessarily exponentially stable. This complements our previous Lyapunov function constructions for rapidly time-varying dynamics. We also explicitly construct input-to-state stable Lyapunov functions for slowly time-varying control systems. We illustrate our findings by constructing explicit Lyapunov functions for a pendulum model, an example from identification theory, and a perturbed friction model., Comment: Accepted for publication in Mathematics of Control, Signals, and Systems (MCSS) on November 20, 2006

We provide explicit closed form expressions for strict Lyapunov functions for time-varying discrete time systems. Our Lyapunov functions are expressed in terms of known nonstrict Lyapunov functions for the dynamics and finite sums of persistency of excitation parameters. This provides a discrete time analog of our previous continuous time Lyapunov function constructions. We also construct explicit strict Lyapunov functions for systems satisfying nonstrict discrete time analogs of the conditions from Matrosov's Theorem. We use our methods to build strict Lyapunov functions for time-varying hybrid systems that contain mixtures of continuous and discrete time evolutions., Comment: 14 pages. Accepted for publication in Nonlinear Analysis: Hybrid Systems and Applications on September 6, 2006

We explicitly construct global strict Lyapunov functions for rapidly time-varying nonlinear control systems. The Lyapunov functions we construct are expressed in terms of oftentimes more readily available Lyapunov functions for the limiting dynamics which we assume are uniformly globally asymptotically stable. This leads to new sufficient conditions for uniform global exponential, uniform global asymptotic, and input-to-state stability of fast time-varying dynamics. We also construct strict Lyapunov functions for our systems using a strictification approach. We illustrate our results using a friction control example., Comment: 10 pages, 0 figues, revised and accepted for publication as a regular paper in Automatica in May 2006. To appear in October 2006 issue

We study the low-bias stabilization of active magnetic bearings (AMBs) subject to voltage saturation based on a recently proposed model for the AMB switching mode of operation. Using a forwarding-like approach, we construct a stabilizing controller of arbitrarily small amplitude and a control-Lyapunov function for the AMB dynamics. We illustrate our construction using a numerical example., Comment: 9 pages, 2 figures. IEEE Transactions on Control Systems Technology, accepted for publication in January 2006

For a broad class of nonlinear systems, we construct smooth control-Lyapunov functions whose derivatives along the trajectories of the systems can be made negative definite by smooth control laws that are arbitrarily small in norm. We assume our systems satisfy appropriate generalizations of the Jurdjevic-Quinn conditions. We also design state feedbacks of arbitrarily small norm that render our systems integral-input-to-state stable to actuator errors., Comment: 15 pages, 0 figures, accepted for publication in IEEE Transactions on Automatic Control in October 2005

We provide a new global small-gain theorem for feedback interconnections of monotone input-output systems with multi-valued input-state characteristics. This extends a recent small-gain theorem of Angeli and Sontag for monotone systems with singleton-valued characteristics. We prove our theorem using Thieme's convergence theory for asymptotically autonomous systems. An illustrative example is also provided., Comment: 17 pages, 2 figures, to appear in IEEE Transactions on Automatic Control

We study the stability properties of a class of time-varying nonlinear systems. We assume that non-strict input-to-state stable (ISS) Lyapunov functions for our systems are given and posit a mild persistency of excitation condition on our given Lyapunov functions which guarantee the existence of strict ISS Lyapunov functions for our systems. Next, we provide simple direct constructions of explicit strict ISS Lyapunov functions for our systems by applying an integral smoothing method. We illustrate our constructions using a tracking problem for a rotating rigid body., Comment: 6 pages, submitted for publication in June 2004

We show that any globally asymptotically controllable system on any smooth manifold can be globally stabilized by a state feedback. Since we allow discontinuous feedbacks, we interpret the solutions of our systems in the ``sample and hold'' sense introduced by Clarke-Ledyaev-Sontag-Subbotin (CLSS). Our work generalizes the CLSS Theorem which is the special case of our result for systems on Euclidean space. We apply our result to the input-to-state stabilization of systems on manifolds relative to actuator errors, under small observation noise., Comment: 25 pages, 0 figues, submitted for publication in October 2004

We provide new infinitesimal characterizations for strong invariance of multifunctions in terms of Hamiltonian inequalities and tangent cones. In lieu of the standard local Lipschitzness assumption on the multifunction, we assume a new feedback realizability condition that can in particular be satisfied by control systems that are discontinuous in the state variable. Our realization condition is based on H. Sussmann's unique limiting property, and allows a more general class of feedback realizations than is allowed by the recent strong invariance characterizations of Krastanov, Malisoff, and Wolenski. We also give new nonsmooth monotonicity characterizations for control systems that may be discontinuous in the state., Comment: 19 pages, 0 figures. For this revision, the author added a remark about an alternative nonconstructive proof of the main result

We provide a new sufficient condition for strong invariance for differential inclusions, under very general conditions on the dynamics, in terms of a Hamiltonian inequality. In lieu of the usual Lipschitzness assumption on the multifunction, we assume a feedback realization condition that can in particular be satisfied for measurable dynamics that are neither upper nor lower semicontinuous., Comment: 15 pages, 0 figures. For this revision, the authors added a remark about an alternative nonconstructive proof of the main result

We study the Hamilton-Jacobi equation for undiscounted exit time control problems with general nonnegative Lagrangians using the dynamic programming approach. We prove theorems characterizing the value function as the unique bounded-from-below viscosity solution of the Hamilton-Jacobi equation which is null on the target. The result applies to problems with the property that all trajectories satisfying a certain integral condition must stay in a bounded set. We allow problems for which the Lagrangian is not uniformly bounded below by positive constants, in which the hypotheses of the known uniqueness results for Hamilton-Jacobi equations are not satisfied. We apply our theorems to eikonal equations from geometric optics, shape-from-shading equations from image processing, and variants of the Fuller Problem., Comment: 29 pages, 0 figures, accepted for publication in NoDEA Nonlinear Differential Equations and Applications on July 29, 2002

We present new theorems characterizing robust Lyapunov functions and infinite horizon value functions in optimal control as unique viscosity solutions of partial differential equations. We use these results to further extend Zubov's method for representing domains of attraction in terms of partial differential equation solutions., Comment: 33 pages, 0 figures, accepted for publication in Dynamics of Continuous, Discrete, and Impulsive Systems Series A: Mathematical Analysis on August 16, 2002

We study nonlinear systems with observation errors. The main problem addressed in this paper is the design of feedbacks for globally asymptotically controllable (GAC) control affine systems that render the closed loop systems input to state stable with respect to actuator errors. Extensions for fully nonlinear GAC systems with actuator errors are also discussed., Comment: 15 pages, 0 figures, conference version submitted in February 2003, journal version submitted in January 2003. Revised and accepted for publication in SIAM Journal on Control and Optimization in August 2003