State Inequality Constraints Research Articles

Robust parameter identification using parallel global optimization for a batch nonlinear enzyme-catalytic time-delayed process presenting metabolic discontinuities

In this paper, a nonlinear enzyme-catalytic time-delayed switched dynamical system is considered to describe batch culture of glycerol bioconversion to 1,3-propanediol induced by Klebsiella pneumoniae. This system can not only predict the exponential growth phase but also the lag and the stationary growth phases of batch culture since it contains two switching times for representing the starting moment of lag growth phase and the time when the cell specified growth rate reaches the maximum. The biological robustness is expressed in terms of the expectation and variance of the relative deviation. Our aim is to identify the switching times. To this end, a robust parameter identification problem is formulated, where the switching times are decision variables to be chosen such that the biological robustness measure is optimized. This problem, which is governed by the nonlinear system, is subject to a quality constraint and continuous state inequality constraints. Using a hybrid time-scaling transformation to parameterize the switching times into new parameters, an equivalently robust parameter identification problem is investigated. The continuous state inequality constraints are approximated by a conventional inequality constraint, yielding a sequence of approximate robust parameter identification subproblems. The convergence analysis of this approximation is also investigated. Owing to the highly complex nature of these subproblems, a parallel algorithm, based on simulated annealing, is proposed to solve these subproblems. From an extensive simulation study, it is observed that the obtained optimal switching times are satisfactory.

An Auxiliary Particle Filtering Algorithm With Inequality Constraints

For nonlinear non-Gaussian stochastic dynamic systems with inequality state constraints, this technical note presents an efficient particle filtering algorithm, constrained auxiliary particle filtering algorithm. To deal with the state constraints, the proposed algorithm probabilistically selects particles such that those particles far away from the feasible area are less likely to propagate into the next time step. To improve on the sampling efficiency in the presence of inequality constraints, it uses a highly effective method to perform a series of constrained optimization so that the importance distributions are constructed efficiently based on the state constraints. The caused approximation errors are corrected using the importance sampling method. This ensures that the obtained particles constitute a representative sample of the true posterior distribution. A simulation study on vehicle tracking is used to illustrate the proposed approach.

Adaptive nearly optimal control for a class of continuous-time nonaffine nonlinear systems with inequality constraints

The state inequality constraints have been hardly considered in the literature on solving the nonlinear optimal control problem based the adaptive dynamic programming (ADP) method. In this paper, an actor-critic (AC) algorithm is developed to solve the optimal control problem with a discounted cost function for a class of state-constrained nonaffine nonlinear systems. To overcome the difficulties resulting from the inequality constraints and the nonaffine nonlinearities of the controlled systems, a novel transformation technique with redesigned slack functions and a pre-compensator method are introduced to convert the constrained optimal control problem into an unconstrained one for affine nonlinear systems. Then, based on the policy iteration (PI) algorithm, an online AC scheme is proposed to learn the nearly optimal control policy for the obtained affine nonlinear dynamics. Using the information of the nonlinear model, novel adaptive update laws are designed to guarantee the convergence of the neural network (NN) weights and the stability of the affine nonlinear dynamics without the requirement for the probing signal. Finally, the effectiveness of the proposed method is validated by simulation studies.

Computational method for optimal machine scheduling problem with maintenance and production

This paper considers an optimal scheduling problem of maintenance and production for a machine. Firstly, the problem is formulated as a stochastic switched impulsive optimal control problem. However, there exists the stochastic disturbance in this model. Thus, it is difficult to solve the problem by conventional optimisation techniques. To overcome this difficulty, the stochastic switched impulsive optimal control problem is transformed into a deterministic switched impulsive optimal control problem with continuous state inequality constraints. Then, by combining a time-scaling transformation, a second-order smoothing technique and a penalty function method, an improved Newton algorithm is developed for solving this problem. Convergence results indicate that the algorithm is globally convergent with quadratic rate. Finally, two numerical examples are provided to illustrate the effectiveness of the developed algorithm.

Multi-objective optimization of nonlinear switched time-delay systems in fed-batch process

Maximization of productivity and minimization of consumption are two top priorities for biotechnological industry. In this paper, we model a fed-batch process as a nonlinear switched time-delay system. Taking the productivity of target product and the consumption rate of substrate as the objective functions, we present a multi-objective optimization problem involving the nonlinear switched time-delay system and subject to continuous state inequality constraints. To solve the multi-objective optimization problem, we first convert the problem into a sequence of single-objective optimization problems by using convex weighted sum and normal boundary intersection methods. A gradient-based single-objective solver incorporating constraint transcription technique is then developed to solve these single-objective optimization problems. Finally, a numerical example is provided to verify the effectiveness of the numerical solution approach. Numerical results show that the normal boundary intersection method in conjunction with the developed single-objective solver is more favourable than the convex weighted sum method.

Conversion of glycerol from biodiesel fuels to 1,3-propanediol by Citrobacter braakii TB-96

This paper proposes a new optimal control model for the production of 1,3-propanediol (1,3-PD) via microbial fed-batch fermentation. The proposed model is governed by a nonlinear multistage dynamic system with two modes: feeding mode, in which glycerol and alkali substrates are added continuously to the fermentor; and batch mode, in which no substrates are added to the fermentor. The non-standard objective function incorporates both the final 1,3-PD yield and the cost of changing the input feeding rate, which is the control variable for the fed-batch fermentation process. Continuous state inequality constraints are imposed to ensure that the concentrations of biomass, glycerol, and reaction products lie within specified limits. Using the constraint transcription method, we approximate the continuous state inequality constraints by a conventional inequality constraint to yield an approximate parameter optimization problem. We then develop a combined particle swarm and gradient-based optimization algorithm to solve this approximate problem. The paper concludes with simulation results.

Optimal control of a batch fermentation process with nonlinear time-delay and free terminal time and cost sensitivity constraint

In this paper, we consider a nonlinear time-delay dynamic system with uncertain system parameters to characterize the process of batch fermentation. Our goal is to design an optimal control scheme to maximize the productivity of 1,3-propanediol (1,3-PD). Accordingly, we introduce an optimal control problem governed by the nonlinear time-delay dynamic system, in which the control variables are the free terminal time of the batch fermentation process and the initial concentrations of biomass and glycerol. The optimal control problem is subject to a cost sensitivity constraint for ensuring that an acceptable level of the system performance is achieved and continuous state inequality constraints for ensuring that the concentrations of biomass, glycerol, 1,3-PD, acetate, ethanol lie within specified limits. Then, the optimal control problem with free terminal time is transformed, via a hybrid time-scaling strategy, into an equivalent problem with fixed terminal time, which is much preferred for numerical computation. Using the constraint transcription and local smoothing approximation techniques, we approximate these continuous state inequality constraints by conventional inequality constraints to yield an approximate optimal control problem. Because of the highly complex nature of this approximate problem, a parallel algorithm based on the filled function method is constructed to solve this approximate problem. Finally, it is observed that the optimal control obtained is satisfactory through numerical simulations.

Modelling and parameter identification of a nonlinear enzyme-catalytic time-delayed switched system and its parallel optimization

In this paper, we propose a nonlinear enzyme-catalytic time-delayed switched system with unknown state-delays, switching times and system parameters for describing the process of batch culture of glycerol bioconversion to 1,3-propanediol (1,3-PD) induced by Klebsiella pneumoniae (K. pneumoniae). Some important properties of the time-delayed switched system are discussed. In consideration of the difficulty in accurately measuring the concentration of intracellular substances and the absence of equilibrium points for the time-delayed switched system, we quantitatively define biological robustness of the intracellular substance concentrations for the entire process of batch culture. Our goal is to identify the unknown quantities. To this end, we formulate an optimization problem in which the cost function minimizes the defined biological robustness and the unknown state-delays, switching times and system parameters are regarded as decision variables. This optimization problem is subject to the time-delayed switched system, continuous state inequality constraints and parameter constraints. The hybrid time-scaling transformation, constraint transcription and local smoothing approximation techniques are used to convert this optimization problem to a sequence of approximate subproblems. In consideration of both the difficulty of finding analytical solutions and the highly complex nature of this optimization problem, we develop a parallel particle swarm optimization (PPSO) algorithm to solve these approximate subproblems. Finally, we explore the appropriateness of the optimal estimates for the state-delays, switching times and system parameters as well as the effectiveness of the parallel algorithm via numerical simulations.

Robust optimization for nonlinear time-delay dynamical system of dha regulon with cost sensitivity constraint in batch culture

Time-delay dynamical systems, which depend on both the current state of the system and the state at delayed times, have been an active area of research in many real-world applications. In this paper, we consider a nonlinear time-delay dynamical system of dha-regulonwith unknown time-delays in batch culture of glycerol bioconversion to 1,3-propanediol induced by Klebsiella pneumonia. Some important properties and strong positive invariance are discussed. Because of the difficulty in accurately measuring the concentrations of intracellular substances and the absence of equilibrium points for the time-delay system, a quantitative biological robustness for the concentrations of intracellular substances is defined by penalizing a weighted sum of the expectation and variance of the relative deviation between system outputs before and after the time-delays are perturbed. Our goal is to determine optimal values of the time-delays. To this end, we formulate an optimization problem in which the time delays are decision variables and the cost function is to minimize the biological robustness. This optimization problem is subject to the time-delay system, parameter constraints, continuous state inequality constraints for ensuring that the concentrations of extracellular and intracellular substances lie within specified limits, a quality constraint to reflect operational requirements and a cost sensitivity constraint for ensuring that an acceptable level of the system performance is achieved. It is approximated as a sequence of nonlinear programming sub-problems through the application of constraint transcription and local smoothing approximation techniques. Due to the highly complex nature of this optimization problem, the computational cost is high. Thus, a parallel algorithm is proposed to solve these nonlinear programming sub-problems based on the filled function method. Finally, it is observed that the obtained optimal estimates for the time-delays are highly satisfactory via numerical simulations.

Real-time constrained trajectory generation of mobile manipulators

This work offers the solution at the control feed-back level of the accurate positioning in a finite time of the end-effector whose mobile manipulator is subject to control and complex state constraints. We propose new forms of various terminal sliding modes (TSM’s) which result from the access to kinematic redundancy of the non-holonomic mechanical system. In order to incorporate control and state inequality constraints into trajectory generation law, both a suitably defined extended task error is introduced and exterior penalty function approach is utilized. In addition, to incorporate holonomic singularity avoidance condition, collision avoidance constraints for the whole mobile manipulator and its final velocity, a suitably defined projection term onto the null space of the extended Jacobian matrix has been introduced. Control limits are maintained by suitable choice of trajectory generator gains. The numerical simulation results carried out for a mobile manipulator consisting of a nonholonomic differentially steered wheeled mobile platform and a holonomic manipulator of two revolute kinematic pairs, operating both in a two-dimensional unconstrained work space and work space including the obstacles, illustrate performance of the proposed trajectory generators.

Automotive Engine Control with Rational Function Satisfying inequality Constraint

The major drive of current engine development is to increase fuel economy. It has become very essential to manage the restriction due to the boundaries which divide the engine operation conditions into the normal and the abnormal ones such as knocking and misfiring because an optimal operation condition is often close to such a boundary. It indicates that required controls must satisfy inequality state constraints derived from identified boundaries. This paper describes a model following control satisfying the inequality state constraint of knocking with an explicit rational function which is easy to be implemented on production ECUs. The proposed control design was successfully applied to a simple engine model.

Dynamic optimization for robust path planning of horizontal oil wells

This paper considers the three-dimensional path planning problem for horizontal oil wells. The decision variables in this problem are the curvature, tool-face angle and switching points for each turn segment in the path, and the optimization objective is to minimize the path length and target error. The optimal curvatures, tool-face angles and switching points can be readily determined using existing gradient-based dynamic optimization techniques. However, in a real drilling process, the actual curvatures and tool-face angles will inevitably deviate from the planned optimal values, thus causing an unexpected increase in the target error. This is a critical challenge that must be overcome for successful practical implementation. Accordingly, this paper introduces a sensitivity function that measures the rate of change in the target error with respect to the curvature and tool-face angle of each turn segment. Based on the sensitivity function, we propose a new optimization problem in which the switching points are adjusted to minimize target error sensitivity subject to continuous state inequality constraints arising from engineering specifications, and an additional constraint specifying the maximum allowable increase in the path length from the optimal value. Our main result shows that the sensitivity function can be evaluated by solving a set of auxiliary dynamic systems. By combining this result with the well-known time-scaling transformation, we obtain an equivalent transformed problem that can be solved using standard nonlinear programming algorithms. Finally, the paper concludes with a numerical example involving a practical path planning problem for a Ci-16-Cp146 well.

Computation of impulsive optimal control for 1,3-PD fed-batch culture

In this paper, in consideration of suddenly increasing of the glycerol and alkali, we propose a nonlinear impulsive system to describe the fed-batch culture of glycerol bioconversion to 1,3-propanediol (1,3-PD) induced by Klebsiella pneumonia (K. pneumonia). To maximize the concentration of 1,3-PD at the terminal time, we present an impulsive optimal control problem in which the final 1,3-PD yield is taken as the cost function, while the feeding volumes of glycerol and the feeding time points are taken as decision variables. Continuous state inequality constraints are imposed to ensure that the concentrations of biomass, glycerol, and reaction products lie within specified limits. We use the time scaling method to handle the variable feeding time points. The impulsive optimal control problem is approximated as a sequence of nonlinear optimization sub-problems through the application of the penalty function and barrier function method. A hybrid algorithm by integrating a particle swarm optimization with gradient-based optimization algorithm is proposed to seek the optimal feeding strategy. From extensive simulation study, it is observed that the obtained optimal strategy is highly satisfactory.

Two-Step Method for Dynamic Optimization of Inequality State Constrained Systems Using Iterative Dynamic Programming

A two-step method for the solution of dynamic optimization problems with state inequality constraints using iterative dynamic programming (IDP) is presented. In the first step, a preliminary control policy along with approximate values for the start and end points of the state-constrained arc is obtained. The final solution is obtained in the second step by using the preliminary control policy as an input along with an analytical expression for the control during the constrained portion of the state trajectory. The use of flexible stage lengths in IDP helps in accurately locating the start and end points of the state-constrained arc. For the examples considered, the method results in better values of the performance index than those previously reported in the literature. It also results in a significant reduction in computation time as compared to the existing method for the solution of problems with a single active state constraint and a single control using IDP.

Modeling and parameter identification for a nonlinear multi-stage system for dha regulon in batch culture

The bioconversion of glycerol to 1,3-propanediol (1,3-PD) is a complex bioprocess. In this paper, based on biological phenomena of different characteristics at different stages and the genetic regulation of dha regulon, we consider a fourteen-dimensional nonlinear multi-stage dynamic system with unknown time and system parameters for formulating the multi-stage cell growth in batch culture. Some important properties of the multi-stage system are discussed. Our goal is to identify the time and system parameters. To this end, we present a parameter identification problem in which the time and system parameters are decision variables and the cost function measures the discrepancy between experimental data and computational results, subject to the multi-stage system, parameter constraints and continuous state inequality constraints. The system sensitivity (the cost function’s gradient, namely, the derivative of the cost function with respect to the time and system parameters), which can be computed by solving an auxiliary initial value problem, can be regarded as the search direction of optimization algorithm. The identification problem is converted into a sequence of nonlinear programming subproblems through the application of the time-scaling transformation, the constraint transcription and local smoothing approximate techniques. Due to the highly complex nature of the identification problem, the computational cost is high. Thus, a parallel algorithm is proposed to solve these subproblems based on the novel combinations of system sensitivity and genetic algorithm. Finally, numerical results show that the multi-stage system can reasonably describe the process of batch culture.

Optimal 1,3-propanediol production: Exploring the trade-off between process yield and feeding rate variation

This paper proposes a new optimal control model for the production of 1,3-propanediol (1,3-PD) via microbial fed-batch fermentation. The proposed model is governed by a nonlinear multistage dynamic system with two modes: feeding mode, in which glycerol and alkali substrates are added continuously to the fermentor; and batch mode, in which no substrates are added to the fermentor. The non-standard objective function incorporates both the final 1,3-PD yield and the cost of changing the input feeding rate, which is the control variable for the fed-batch fermentation process. Continuous state inequality constraints are imposed to ensure that the concentrations of biomass, glycerol, and reaction products lie within specified limits. Using the constraint transcription method, we approximate the continuous state inequality constraints by a conventional inequality constraint to yield an approximate parameter optimization problem. We then develop a combined particle swarm and gradient-based optimization algorithm to solve this approximate problem. The paper concludes with simulation results.

Three-dimensional trajectory design for horizontal well based on optimal switching algorithms

This paper considers a three-dimensional trajectory design problem for horizontal well. The problem is formulated as an optimal control problem of switched systems with continuous state inequality constraints. Since the complexity of such constraints and the switching instants is unknown, it is difficult to solve the problem by standard optimization techniques. To overcome the difficulty, by a time-scaling transformation, a smoothing technique and a penalty function method, an efficient computational method is proposed for solving this problem. Convergence results show that, for a sufficiently large penalty parameter, any local optimal solution of the approximate problem is also a local optimal solution of the original problem. Two numerical examples are presented to illustrate the efficiency of the approach proposed.

Spatial statistical point prediction guidance for heating-rate-limited aeroassisted orbital transfer

Feedback control of constrained non-linear dynamical systems satisfying a certain optimality criterion and meeting a specified transfer objective in the state space is recognized as one of the most challenging problems in control theory. One approach to computing optimal feedback policies is the dynamic programming route of numerically solving the Hamilton–Jacobi–Bellman (HJB) partial differential equation directly. In this paper an alternate and more tractable dynamic programming approach, the optimal feedback synthesis method, is utilized. The effectiveness of this method is demonstrated through an explicit guidance scheme for the heating-rate-constrained maneuver of an Aeroassisted Transfer Vehicle (AOTV). In optimal feedback synthesis, a feedback chart is constructed from a family of open-loop extremals, thus ensuring optimality with respect to any initial condition in the family. This paper presents a solution to the AOTV optimal feedback synthesis problem using the Gaussian process spatial prediction method of universal kriging. A closed-form expression for a near-optimal guidance law is derived. Its performance is found to be very promising; initial atmospheric entry errors due to simulated thruster misfiring are seen to be accurately corrected while the algebraic state-inequality constraint is closely respected.

Non-degenerate necessary optimality conditions for the optimal control problem with equality-type state constraints

In this work, an optimal control problem with state constraints of equality type is considered. Novelty of the problem formulation is justified. Under various regularity assumptions imposed on the optimal trajectory, a non-degenerate Pontryagin Maximum Principle is proven. As a consequence of the maximum principle, the Euler---Lagrange and Legendre conditions for a variational problem with equality and inequality state constraints are obtained. As an application, the equation of the geodesic curve for a complex domain is derived. In control theory, the Maximum Principle suggests the global maximum condition, also known as the Weierstrass---Pontryagin maximum condition, due to which the optimal control function, at each instant of time, turns out to be a solution to a global finite-dimensional optimization problem.

Modelling and parameter identification for a hybrid dynamical system in microbial fed-batch culture

In this paper, we consider the modelling and identification in the fed-batch fermentation of glycerol by Klebsiella pneumoniae with open-loop glycerol input and pH logic control. Taking into account the hybrid characteristic of the fed-batch operation, we present a nonlinear hybrid dynamical system, which is developed by embedding the discrete process of adding glycerol and alkali into the dynamical system of batch culture, to formulate this fermentation process. Some important properties of the solution to the proposed system are then discussed, including the existence, uniqueness, boundedness and regularity. To estimate the unknown parameters in the system, a parameter identification problem subject to continuous state inequality constraints is proposed, and its identifiability is also proved. Subsequently, the parametric sensitivity functions of the system are given and utilized to obtain the requisite gradient information for further numerical computation. Finally, to solve the identification problem, a gradient-based algorithm is constructed in conjunction with constraint transcription and the smoothing approximation technique. Numerical simulations show that the validated hybrid system is fit for describing the fed-batch processes as observed from the experimental results.