A stochastic model of Escherichia coli AI-2 quorum signal circuit reveals alternative synthesis pathways

Quorum sensing (QS) is an important determinant of bacterial phenotype. Many cell functions are regulated by intricate and multimodal QS signal transduction processes. The LuxS/AI-2 QS system is highly conserved among Eubacteria and AI-2 is reported as a ‘universal' signal molecule. To understand the hierarchical organization of AI-2 circuitry, a comprehensive approach incorporating stochastic simulations was developed. We investigated the synthesis, uptake, and regulation of AI-2, developed testable hypotheses, and made several discoveries: (1) the mRNA transcript and protein levels of AI-2 synthases, Pfs and LuxS, do not contribute to the dramatically increased level of AI-2 found when cells are grown in the presence of glucose; (2) a concomitant increase in metabolic flux through this synthesis pathway in the presence of glucose only partially accounts for this difference. We predict that ‘high-flux' alternative pathways or additional biological steps are involved in AI-2 synthesis; and (3) experimental results validate this hypothesis. This work demonstrates the utility of linking cell physiology with systems-based stochastic models that can be assembled de novo with partial knowledge of biochemical pathways

A Methodology for the Synthesis of Robust Control Systems for Multivariable Sampled-Data Processes

The problem of the synthesis of multivariable controllers which are robust with respect to model-plant mismatch is addressed. A two-step design procedure based on the Internal Model Control (IMC) structure is used. In the first step the IMC controller is designed assuming no modeling error, and in the second step the IMC filter is designed to preserve the closed-loop characteristics in spite of model-plant mismatch. Two alternatives are provided for the first step. One of them allows the designer to satisfy structural performance specifications, in terms of the structure of the closed-loop interactions, their magnitude and duration. The closed-loop transfer function matrix is directly designed. The method requires only standard linear algebra operations and includes the construction of the IMC or the feedback controller in state-space. The second approach involves the minimization of the appropriately weighted H2-norm of the sensitivity transfer function matrix, that relates the errors to the external inputs (setpoints or disturbances). A method is given for the meaningful selection of a full matrix weight so that the H2-error is minimized for a set of external input directions and their linear combinations. The procedure is extended to open-loop unstable systems. In both approaches, special care is taken to avoid intersample rippling. The design of the filter in the second step is formulated as an optimization problem over the filter parameters. The objective function is constructed by using the Structured Singular Value theory so that the maximum singular value of the sensitivity transfer function remains bounded in spite of modeling error. The selection of the frequency bound is based on the properties of the design that was obtained in the first step. Analytic gradient expressions have been developed for the objective function. The optimization problem is an unconstrained one, solved with standard gradient search techniques. An iterative method for the selection of the appropriate sampling time is proposed, which explicitly takes into account model uncertainty information and performance specifications.</p

State Estimation Nonlinear QDMC with Input-Output Models

A State Estimation NLQDMC algorithm is presented for use with nonlinear input-output models. The proposed algorithm extends the state estimation NLQDMC [5] to nonlinear models identified based on input-output information. The algorithm preserves the computational advantages of [5] when compared to the other algorithms based on nonlinear programming techniques. The illustrating example demonstrates the usage of tuning parameters and points out the benefits and shortcomings of the algorithm

Nonlinear Quadratic Dynamic Matrix Control with State Estimation

Quadratic Dynamic Matrix Control (QDMC) with state estimation is presented for use with nonlinear process models. This formulation extends Garcia's nonlinear version of QDMC to open- loop unstable nonlinear processes and allows for better disturbance rejection. It also extends Ricker's linear state space formulation with state estimation to nonlinear systems. Stability and better performance is observed when compared to the algorithm without state estimation in rejecting disturbances for processes operating at unstable steady state setpoints, as illustrated with two simple examples. The algorithm requires that only a Quadratic Program be solved on-line. The modest computational requirements make it attractive for industrial implementation. the effectiveness of the approach is demonstrated by its successful application to the temperature control of a semibatch polymerization reactor. A model and related control requirements for this problem were presented at the 1990 AIChE Annual Meeting in a session on "Industrial Challenge Problems in Process Control.

State Estimation Model Based Algorithm for On-line Optimization and Control of Batch Processes

Batch/semi-batch processes are highly nonlinear and involve complex reaction mechanisms. Model-plant mismatch always exists. The lack of rapid direct or indirect measurements of the properties to be controlled makes the control task difficult. It is the usual practice to follow the prespecified setpoint profiles for process variables for which measurements are available, in order to obtain desired product properties. Modeling error can be the cause of bad performance when optimal profiles computed for the model, are implemented on the actual plant. In this paper, a state estimation model based algorithm is presented for on-line modification of the optimal profile and control with the goal of obtaining the desired properties at the minimum batch time. The effectiveness of the algorithm is demonstrated by its application to bulk polymerization of styrene

On the Quadratic Stability of Constrained Model Predictive Control

Analytic and numerical methods are developed in this paper for the analysis of the quadratic stability of Constrained Model Predictive Control (CMPC). According to the CMPC algorithm, each term of the closed-form of control law corresponding to an active constraint situation can be decomposed to have an uncertainty block, which is time varying over the control period. By analytic method, if a quadratic Lyapunov function can be found for the CMPC closed-loop system with uncertainty blocks in the feedback control law by solving a Riccati type equation, then the control system is quadratic stable. Since no rigorous solving method has been found, this Riccati type equation is solved by a trial-and- error method in this paper. A numerical method that does not solve the Riccati type equation, the Linear Matrix Inequality (LMI) technique, was found useful in solving this quadratic stability problem. Several examples are given to show the CMPC quadratic stability analysis results. It is also noticeable that the quadratic stability implies a similarity to a contraction

The Closed-Form Control Laws of the Constrained Model Predictive Control Algorithm

The Analysis of quadratic Stability and strongly Hperformance of Model Predictive Control (MPC) with hard constraints (or called Constrained Model Predictive Control (CMPC)) can be accomplished by reformulating the hard constraints of CMPC. From the CMPC algorithm, each term of the closed-form of CMPC control law corresponding to an active constraint situation can be decomposed to have an uncertainty block, which is time varying over the control period. The control law also contains a bias from the bounds of the constraints which cause difficulty in stability and performance analysis. An alternative way to avoid this difficulty is to reformulate the hard constraints to adjustable constraints with time varying adjustable weights on the adjustable variables added to the on-line objective function. The time varying weights in the adjustable constraint control law make the control action just the same as the hard constrained control. Theoretical derivatives and examples are given. The same reformulation is applied to the softened constraint cases.On the analysis of the quadratic stability and strongly H performance, the control system for hard constraint control law without bias satisfies the stability and performance criteria if and only if the control system for adjustable constraint control law with time varying adjustable weights satisfies the same criteria. The details will be shown in the technical reports on quadratic stability and strongly Hperformance analysis, which are in preparation

Observer Based Nonlinear Quadratic Dynamic Matrix Control for State Space and I/O Models

Observer based nonlinear QDMC algorithm is presented for use with nonlinear state space and input-output models. The proposed algorithm is an extension of Nonlinear Quadratic Dynamic Matrix Control (NLQDMC) by Garcia (1984) and its extension by Gattu and Zafiriou (1992a). Garcia proposed an extension of linear Quadratic Dynamic Matrix Control (QDMC) to nonlinear processes. Although a nonlinear model is used, only a single Quadratic Program (QP) is solved on-line. Gattu and Zafiriou extended this formulation to open-loop unstable systems, by incorporating a Kalman filter. The requirement of solving only one QP on-line at each sampling time makes this algorithm an attractive option for industrial implementation. This extension of NLQDMC to open-loop unstable systems was ad hoc and did not address the problem of offset free tracking and disturbance rejection in a general state space setting. Independent white noise was added to the model states to handle unstable processes. The approach can stabilize the system but leads to an offset in the presence of persistent disturbances. To obtain offset free tracking Gattu and Zafiriou added a constant disturbance to the predicted output as done in DMC-type algorithms. This addition is ad hoc and does not result from the filtering/prediction theory. The proposed algorithm eliminates the major drawbacks of the algorithm presented by Gattu and Zafiriou and extends that algorithm for nonlinear models identified based on input-output information. An algorithm schematic is presented for measurement delay cases. The algorithm preserves the computational advantages when compared to the other algorithms based on nonlinear programming techniques. The illustrating examples demonstrate the usage of tuning parameters for unstable and stable systems and points out the benefits and short comings of the algorithm