Convex Programs for Temporal Verification of Nonlinear Dynamical Systems

A methodology for safety verification of continuous and hybrid systems using barrier certificates has been proposed recently. Conditions that must be satisfied by a barrier certificate can be formulated as a convex program, and the feasibility of the program implies system safety in the sense that there is no trajectory starting from a given set of initial states that reaches a given unsafe region. The dual of this problem, i.e., the reachability problem, concerns proving the existence of a trajectory starting from the initial set that reaches another given set. Using insights from the linear programming duality appearing in the discrete shortest path problem, we show in this paper that reachability of continuous systems can also be verified through convex programming. Several convex programs for verifying safety and reachability, as well as other temporal properties such as eventuality, avoidance, and their combinations, are formulated. Some examples are provided to illustrate the application of the proposed methods. Finally, we exploit the convexity of our methods to derive a converse theorem for safety verification using barrier certificates

A new solution approach to polynomial LPV system analysis and synthesis

Based on sum-of-squares (SOS) decomposition, we propose a new solution approach for polynomial LPV system analysis and control synthesis problems. Instead of solving matrix variables over a positive definite cone, the SOS approach tries to find a suitable decomposition to verify the positiveness of given polynomials. The complexity of the SOS-based numerical method is polynomial of the problem size. This approach also leads to more accurate solutions to LPV systems than most existing relaxation methods. Several examples have been used to demonstrate benefits of the SOS-based solution approach

Safety Control Synthesis with Input Limits: a Hybrid Approach

We introduce a hybrid (discrete--continuous) safety controller which enforces strict state and input constraints on a system---but only acts when necessary, preserving transparent operation of the original system within some safe region of the state space. We define this space using a Min-Quadratic Barrier function, which we construct along the equilibrium manifold using the Lyapunov functions which result from linear matrix inequality controller synthesis for locally valid uncertain linearizations. We also introduce the concept of a barrier pair, which makes it easy to extend the approach to include trajectory-based augmentations to the safe region, in the style of LQR-Trees. We demonstrate our controller and barrier pair synthesis method in simulation-based examples.Comment: 6 pages, 7 figures. Accepted for publication at the 2018 American Controls Conference. Copyright IEEE 201

Nonlinear control synthesis by convex optimization

A stability criterion for nonlinear systems, recently derived by the third author, can be viewed as a dual to Lyapunov's second theorem. The criterion is stated in terms of a function which can be interpreted as the stationary density of a substance that is generated all over the state-space and flows along the system trajectories toward the equilibrium. The new criterion has a remarkable convexity property, which in this note is used for controller synthesis via convex optimization. Recent numerical methods for verification of positivity of multivariate polynomials based on sum of squares decompositions are used

Analysis of switched and hybrid systems - beyond piecewise quadratic methods

This paper presents a method for stability analysis of switched and hybrid systems using polynomial and piecewise polynomial Lyapunov functions. Computation of such functions can be performed using convex optimization, based on the sum of squares decomposition of multivariate polynomials. The analysis yields several improvements over previous methods and opens up new possibilities, including the possibility of treating nonlinear vector fields and/or switching surfaces and parametric robustness analysis in a unified way

Production of lignocellulosic ethanol from Lantana camara by bacterial cellulase of termite symbionts

Due to the rapid growth in population and industrialization, worldwide ethanol demand is increasing continuously. Lignocellulosic biomasses are most abundant and renewable sources of the world and they can act as a promising source for bioethanol production. The major objective of the work was to evaluate the effect of acid and steam pretreatment on Lantana camara for improved yield of bioethanol production by using cellulase production by bacteria isolated from termite gut and optimization of conditions required for maximum activity of cellulase enzyme. Cellulase producing bacteria isolated from termite’s gut screened by congored test. Cellulase activity was measured by DNS method. From the present study it is concluded that bacteria isolated from termite gut was producing maximum amount of cellulase enzyme after 40 hours (TCDB1, 2) and after 60 hours (TCDB 3). Cellulase enzyme produced by bacteria isolated from termite gut was found to have pH around 5, temperature 50°C for TCDB 1 and 70°C for TCDB 2 and 3 as optimum conditions .Activity of Cellulase enzyme produced by bacteria isolated from termite gut was found to be increased by the addition of 5 mM MnSO4 (all three TCDB) and MgSO4 (only TCDB3). It Possible to produce lignocellulosic bioethanol (11.66%) from Lantana camara after steam and acid pretreatment by using Cellulase for Saccharification (72 hours) and Saccharomyces cerevisiae for fermentation (72 hours). Bioethanol from lignocellulosic biomass is a globally accepted alternative fuel. The production of ethanol from Lantana camara would have the dual advantage of producing energy and serving as an effective method of weed management

Calcium Alginate Bead Based Scaffold For Drug Delivery And Tissue Engineering

Implants are medical devices that are fabricated in order to substitute, support or improve the biological structures. Generally, such implants are composed of biocompatible materials (synthetic polymers and natural biopolymers, metals, metal alloys) depending upon their applications. In this regard, we fabricated a novel three dimensional calcium alginate bead based scaffold implant for drug delivery and tissue engineering applications. The scaffold was designed by placing the alginate beads in layer by layer arrangement allowing hexagonal closed packing. The designed scaffold was analyzed for its physico-chemical characteristics using SEM, FTIR, swelling behavior and drug release kinetics. Further, biological characterizations such as biocompatibility and antibacterial activity of the scaffolds were also carried out. The fabricated scaffold had dense and compact packing of the calcium alginate beads resulting in better mechanical strength. Analysis of release kinetics of model drug (Rhodamine) showed that the release rate of the drug was dependent on the number of layers stacked over each other. The scaffold implant was found to be biocompatible. Thus, we decipher that the scaffold implant could be used for drug delivery and tissue engineering applications

Characterizations of safety in hybrid inclusions via barrier functions

This paper investigates characterizations of safety in terms of barrier functions for hybrid systems modeled by hybrid inclusions. After introducing an adequate definition of safety for hybrid inclusions, sufficient conditions using continuously differentiable as well as lower semicontinuous barrier functions are proposed. Furthermore, the lack of existence of autonomous and continuous barrier functions certifying safety, guides us to propose, inspired by converse Lyapunov theorems for only stability, nonautonomous barrier functions and conditions that are shown to be both necessary as well as sufficient, provided that mild regularity conditions on the system's dynamics holds

Analysis of the lactose metabolism in E. coli using sum-of-squares decomposition

We provide a system-theoretic analysis of the mathematical model of lactose induction in E.coli which predicts the level of lactose induction into the cell for specified values of external lactose. Depending on the levels of external lactose and other parameters, the Lac operon is known to have a low steady state in which it is said to be turned off and high steady state where it is said to be turned on. Furthermore, the model has been shown experimentally to exhibit a bi-stable behavior. Using ideas from Lyapunov stability theory and sum-of-squares decomposition, we characterize the reachable state space for different sets of initial conditions, calculating estimates of the regions of attraction of the biologically relevant equilibria of this system. The changes in the basins of attraction with changes in model parameters can be used to provide biological insight. Specifically, we explain the crucial role played by a small basal transcription rate in the Lac operon. We show that if the basal rate is below a threshold, the region of attraction of the low steady state grows significantly, indicating that system is trapped in the (off) mode, showing the importance of the basal rate of transcription