The standard Dynamic Programming (DP) formulation can be used to solve Multi-Stage Optimization Problems (MSOP's) with additively separable objective functions. In this paper we consider a larger class of MSOP's monotonically backward functions; functions being special case We propose necessary and sufficient condition, utilizing generalization Bellman's equation, for solution MSOP, cost function, optimal. Moreover, show that proposed condition efficiently compute optimal solutions two...

In this paper, we propose an iterative method for using SOS programming to estimate the region of attraction a polynomial vector field, conjectured convergence which necessitates existence Lyapunov functions whose sublevel sets approximate true arbitrarily well. The main technical result paper is proof such function. Specifically, use Hausdorff distance metric analyze and in theorem demonstrate that n-times continuously differentiable maximal function implies any ε > 0, there exists...

In this paper we propose a convex Sum-of-Squares optimization problem for finding outer approximations of forward reachable sets nonlinear uncertain Ordinary Differential Equations (ODE’s) with either (or both) L2 or point-wise bounded input disturbances. To make our tight seek to minimize the volume approximation set. Our approach minimization is based on use determinant-like objective function. We provide several numerical examples including Lorenz system and Van der Pol oscillator.

In this paper, we consider the problem of dynamic programming when supremum terms appear in objective function. Such can represent overhead costs associated with underlying state variables. Specifically, form optimization be used to optimal scheduling batteries such as Tesla Powerwall for electrical consumers subject demand charges - a charge based on maximum rate electricity consumption. These reflect cost utility building and maintaining generating capacity. Unfortunately, show that...

In this paper we show that Sum-of-Squares optimization can be used to find optimal semialgebraic representations of sets. These sets may explicitly defined, as in the case discrete points or unions sets; implicitly attractors nonlinear systems. We define optimality sense minimum volume, while satisfying constraints include set containment, convexity, Lyapunov stability conditions. Our admittedly heuristic approach volume minimization is based on use a determinant-like objective function....

Triode radio-frequency amplifiers have come into extensive use for medium-high-frequency applications. The of triodes results from the reduced noise-equivalent resistance a triode amplifier as compared to multigrid-type tube. It is not possible with conventional circuits input grid circuit and output plate because this connection in excessive feedback which produces regeneration even oscillation. grounded-grid alleviates these difficulties by utilizing shield between or cathode circuit. Such...

The design, fabrication, and performance of a 0.4-W, 2 to 20 GHz distributed amplifier are described in this paper. Small-signal gain is 5 dB power-added efficiency 15%. fabricated on ion-implanted GaAs, achieves excellent through use series gate capacitors tapered drain line. Circuit layout optimization obtain process insensitivity first-pass design success discussed. A comparison made commercially available state-of-the-art 6 18 designed using conventional (lossy-mismatch) topology. shown...

This paper considers the problem of approximating "maximal" region attraction (the set that contains all asymptotically stable sets) any given locally exponentially nonlinear Ordinary Differential Equations (ODEs) with a sufficiently smooth vector field. Given exponential ODE differentiable field, we show there exists globally Lipschitz continuous converse Lyapunov function whose 1-sublevel is equal to maximal ODE. We then propose sequence d-degree Sum-of-Squares (SOS) programming problems...

We consider the problem of overbounding and underbounding both backward forward reachable set for a given polynomial vector field, nonlinear in state input, with semialgebriac initial conditions inputs constrained pointwise to lie semialgebraic set. Specifically, we represent using "value function" which gives optimal cost go an control problems if smooth satisfies Hamilton-JacobiBellman PDE. then show that there exist upper lower bounds this value function furthermore, these "sub-value"...

For any suitable Optimal Control Problem (OCP) there exists a value function, defined as the unique viscosity solution to Hamilton-Jacobi-Bellman (HJB) Partial-Differential-Equation (PDE), and which can be used design an optimal feedback controller for given OCP. In this paper, we approximately solve HJB-PDE by proposing sequence of Sum-Of-Squares (SOS) problems, each yields polynomial subsolution HJB-PDE. We show that resulting sub-solutions converges function OCP in L1 norm. Furthermore,...

We consider a general class of dynamic programming (DP) problems with nonseparable objective functions. show that for any problem in this class, there exists an augmented-state DP satisfies the principle optimality and solutions to which yield original problem. Furthermore, we identify subclass naturally forward separable functions state-augmentation scheme is tractable. extend framework stochastic problems, proposing suitable definition optimality. then apply resulting algorithms optimal...

A novel measurement is described which enables the full small-signal S-parameters of an RF/microwave transistor to be measured while it simultaneously driven and optimally tuned at a higher frequency as efficient, class B or C power amplifying stage. This capability allows classic, stability analysis amplifiers performed across frequencies below amplified carrier where parametrically pumped, subharmonic oscillations are often problem. Measured GaAs HBT (heterojunction bipolar transistor)...

In this paper, we propose a novel method for addressing Optimal Control Problems (OCPs) with input-affine dynamics and cost functions. This approach adopts Model Predictive (MPC) strategy, wherein controller is synthesized to handle an approximated OCP within finite time horizon. Upon reaching horizon, the re-calibrated tackle another approximation of OCP, updated based on final state information. To each instance, all non-polynomial terms are Taylor-expanded about current resulting...

This paper presents two novel algorithms for approximately projecting symmetric matrices onto the Positive Semidefinite (PSD) cone using Randomized Numerical Linear Algebra (RNLA). Classical PSD projection methods rely on full-rank deterministic eigen-decomposition, which can be computationally prohibitive large-scale problems. Our approach leverages RNLA to construct low-rank matrix approximations before projection, significantly reducing required numerical resources. The first algorithm...

Many dynamical systems described by nonlinear ODEs are unstable. Their associated solutions do not converge towards an equilibrium point, but rather some invariant subset of the state space called attractor set. For a given ODE, in general, existence, shape and structure sets ODE unknown. Fortunately, sublevel Lyapunov functions can provide bounds on ODEs. In this paper we propose new characterization that is well suited to problem finding minimal We show our non-conservative even when...

The increasing uptake of inverter based resources (IBRs) has resulted in many new challenges for power system operators around the world. high level complexity IBR generators makes accurate classical model-based stability analysis a difficult task. This letter proposes novel methodology solving problem estimating Region Attraction (ROA) nonlinear by combining model methods with modern data driven methods. Our method yields certifiable inner approximations ROA, typical to that methods, but...