Constraint Transcription Technique Research Articles

Optimal switching control of 1,3-propanediol fed-batch production with a cost on smooth feeding rate variation

In this paper, we propose a nonlinear switched control system for microbial 1,3-propanediol (1,3-PD) fed-batch production, where the smooth feeding rate of glycerol as well as the switching instants between batch process and feeding process are taken as control variables. To maximize 1,3-PD yield and minimize the fluctuation of feeding rate, we then present an optimal switching control problem with a cost on the smooth feeding rate variation and subject to path constraints reflecting the operational requirements. For solving this problem, we convert it to a series of parameter optimization problems by using a novel control parameterization scheme, a time-scaling transformation and a constraint transcription technique. An analytical expression of the total variation for the smooth feeding rate is also derived. On this basis, we develop a parallel Particle Swarm Optimization (PSO) algorithm together with the gradient-based optimization to solve the resulting problem. Finally, numerical simulations indicate that sacrificing a small amount of 1,3-PD can reduce the feeding rate variation significantly.

Robust parameter identification of a nonlinear impulsive time-delay system in microbial fed-batch process

In this paper, taking the feeding process as an impulsive form and considering the time-delay in fermentation process, we propose a nonlinear impulsive time-delay system to formulate the fed-batch culture in microbial 1,3-propanediol production. For this system, we discuss some important properties of the solution. To determine the optimal time-delay and kinetic parameters in the system, we then present a robust parameter identification model subject to continuous state inequality constraints. By the constraint transcription technique and a penalty method, we transform the robust parameter identification problem to a sequence of optimization problems with only box constraints. On this basis, we propose a parallel Particle Swarm Optimization algorithm to solve the resulting problem. Finally, the numerical results show that the impulsive time-delay system can well describe the fed-batch process and the developed algorithm is highly effective.

Time-delay optimal control of a fed-batch production involving multiple feeds

In this paper, we consider time-delay optimal control of 1, 3-propan-ediol (1, 3-PD) fed-batch production involving multiple feeds. First, we propose a nonlinear time-delay system involving feeds of glycerol and alkali to formulate the production process. Then, taking the feeding rates of glycerol and alkali as well as the terminal time of process as the controls, we present a time-delay optimal control model subject to control and state constraints to maximize 1, 3-PD productivity. By a time-scaling transformation, we convert the optimal control problem into an equivalent problem with fixed terminal time. Furthermore, by applying control parameterization and constraint transcription techniques, we approximate the equivalent problem by a sequence of finite-dimensional optimization problems. An improved particle swarm optimization algorithm is developed to solve the resulting optimization problems. Finally, numerical results show that 1, 3-PD productivity increases considerably using the obtained optimal control strategy.

Robust optimization for a nonlinear switched time-delay system with noisy output measurements using hybrid optimization algorithm

This paper considers the problem of using noisy output data to estimate unknown switching times and unknown system parameters arising in 1,3-PD batch production process of glycerol induced by Klebsiella pneumonia. We formulate the problem as a robust optimization problem in which the unknown quantities are decision variables to be chosen optimally, with the cost function penalizing the mean and variance of the relative error between the output of the system and the measured actual noisy system output. This problem is governed by a switched time-delay system subject to continuous state inequality constraints arising from engineering specifications. A new time-scaling transformation and constraint transcription technique are used then to convert this resulting problem into a sequence of approximate subproblems. A hybrid optimization algorithm combined genetic algorithm with gradient-based method is developed to solve these resulting subproblems. Finally, we explore the correctness of the optimal switching times and system parameters obtained, as well as the effectiveness of the proposed algorithm.

Robust bi-objective optimal control of 1,3-propanediol microbial batch production process

This paper considers optimal control of glycerol producing 1,3-propanediol (1,3-PD) in batch process with uncertain time-delay and kinetic parameters. Based on a dynamic system with uncertain time-delay and kinetic parameters describing the batch process, we propose a robust bi-objective optimal control model, in which 1,3-PD productivity and glycerol consumption rate, and their semi-relative sensitivities with respect to the uncertain parameters are all involved in the vector objective. The control variables in this problem are the initial concentrations of biomass and glycerol and the terminal time of the process. Then, by introducing auxiliary time-delay systems and performing a time-scaling transformation, we transform the robust bi-objective optimal control problem into an equivalent bi-objective optimal control problem in standard form. Furthermore, we convert the equivalent optimal control problem into a sequence of solvable single-objective optimal control problems by using the normalized normal constraint method together with the constraint transcription technique. To solve each of the resulting optimal control problems, we develop a novel gradient-based solution method. Finally, numerical results are provided to verify the effectiveness of the proposed solution approach.

Robust multi-objective optimal switching control arising in 1,3-propanediol microbial fed-batch process

This paper considers optimal control of glycerol producing 1,3-propanediol (1,3-PD) via microbial fed-batch fermentation. The fed-batch process is formulated as a nonlinear switched time-delay system. In general, the time-delay in the fed-batch process cannot be exactly estimated. Our goal is to design an optimal switching control scheme to simultaneously maximize 1,3-PD productivity and 1,3-PD yield under time-delay uncertainty. Accordingly, we propose a robust multi-objective optimal switching control model, in which two objectives, i.e., 1,3-PD productivity and 1,3-PD yield, and their sensitivities with respect to uncertain time-delay are considered in the objective vector. The control variables in this problem are the feeding rate of glycerol, the switching instants and the terminal time of the process. By introducing an auxiliary dynamic system to calculate the objective sensitivities and performing a time-scaling transformation, we obtain an equivalent multi-objective optimal switching control problem in standard form. We then convert the equivalent multi-objective optimal control problem into a sequence of single-objective optimal switching control problems by using a modified normal boundary intersection method. A novel gradient-based single-objective solver combining control parameterization with constraint transcription technique is developed to solve these resulting single-objective optimal control problems. Finally, numerical results are provided to verify the effectiveness of the proposed solution approach.

Bi-objective dynamic optimization of a nonlinear time-delay system in microbial batch process

In this paper, we propose a bi-objective dynamic optimization model involving a nonlinear time-delay system to optimize the 1,3-propanediol (1,3-PD) production in a microbial batch process, where the productivity of 1,3-PD and the consumption rate of glycerol are taken as the two objectives. The initial concentrations of biomass and glycerol, and the terminal time of the process are the decision variables. By a time-scaling transformation, we first transform the problem to the one with fixed terminal time but involving a new system with variable time-delay. The normalized normal constraint method is then used to convert the resulting problem into a sequence of single-objective dynamic optimization problems. A gradient-based optimization method incorporating the constraint transcription technique is developed to solve each of these single-objective dynamic optimization problems. Finally, numerical results are provided to demonstrate the effectiveness of the proposed solution method.

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.

Minimizing control variation in discrete-time optimal control problems

For a real practical system, a large fluctuation in the control signal is highly undesirable. To address this undesirable situation, we investigate a discrete-time optimal control problem subject to terminal state and all-time-step constraints on the state and control, where the cost function is the sum of terminal cost and the variation of the control signal. The variation of the control signal is expressed in terms of absolute value functions and hence is non-smooth. By a novel smooth transformation and the constraint transcription technique, this problem is approximated by a constrained discrete-time optimal control with the new cost function involves only smooth functions. A gradient-based computational method is then derived, which is supported by rigorous convergence analysis. Two examples are provided to demonstrate the effectiveness and advantages of the proposed method.

VISUAL MISER: An efficient user-friendly visual program for solving optimal control problems

The FORTRAN MISER software package has been used with great success over the past two decades to solve many practically important real world optimal control problems. However, MISER is written in FORTRAN and hence not user-friendly, requiring FORTRAN programming knowledge. To facilitate the practical application of powerful optimal control theory and techniques, this paper describes a Visual version of the MISER software, called Visual MISER. Visual MISER provides an easy-to-use interface, while retaining the computational efficiency of the original FORTRAN MISER software. The basic concepts underlying the MISER software, which include the control parameterization technique, a time scaling transform, a constraint transcription technique, and the co-state approach for gradient calculation, are described in this paper. The software structure is explained and instructions for its use are given. Finally, an example is solved using the new Visual MISER software to demonstrate its applicability.

Optimal control of a switched autonomous system with time delay arising in fed-batch processes

In this paper, we propose a switched autonomous system with time delay to model the 1,3-propanediol (1,3-PD) production in a fed-batch process. Taking the switching instants and the terminal time as the control variables, we formulate a constrained time-delayed optimal control (CTOC) problem to optimize the 1,3-PD production process. Using a time-scaling transformation and parametrizing the switching instants into new parameters, an equivalent CTOC problem is investigated. A numerical solution method is then developed to seek the optimal control strategy. This method is based on the constraint transcription technique and the gradients of the cost functional together with those of the constraints. Numerical simulation results show that the mass of 1,3-PD per unit time at the terminal time is increased considerably compared with the experiment results.

Modelling and parameter identification for a nonlinear time-delay system in microbial batch fermentation

Mathematical modelling and parameter identification of a microbial batch fermentation process is considered in this paper. In view of the existence of time delays, a nonlinear time-delay system is firstly proposed to formulate the fermentation process. Some important properties are also discussed. Taking the errors between the computational values and the experimental data as the cost function, a parameter identification model subject to continuous state constraints and parameter constraints is then presented. To seek the optimal time delay and the optimal kinetic parameters, an improved differential evolution algorithm in conjunction with the constraint transcription technique is developed. Finally, numerical results show that the model can describe the batch fermentation process reasonably.

Modelling and optimal control for a fed-batch fermentation process

The main control goal in fed-batch fermentation is to get a high concentration of production. In this paper, a new nonlinear dynamical system, in which the feed rate of glycerol is the control function and the switching instants between the batch and feed processes are the variables, is proposed to formulate the fed-batch fermentation process of glycerol bioconversion to 1,3-propanediol (1,3-PD). To maximize the concentration of 1,3-PD at the terminal time, an optimal control model involving the nonlinear system and subject to the continuous state constraints and the control constraint is presented. To seek the optimal solution of the constrained optimal control problem, the control parametrization enhancing transform together with the constraint transcription technique is first used to convert the constrained optimal control problem into a sequence of mathematical programming problems. An improved Particle Swarm Optimization is subsequently constructed to solve the resultant mathematical programming problem. Numerical results show that, by employing the obtained optimal strategy, 1,3-PD concentration at the terminal time can be increased considerably.

Approach for optimal power flow with transient stability constraints

Optimal power flow with transient stability (OTS) is a nonlinear optimisation problem in functional space that can be transformed to a static optimisation problem via constraint transcription techniques. However, the solution of the transformed OTS by ordinary numerical methods involves a massive calculation due to the stability constraints. Hence, the authors propose a penalty-based approach for OTS, in which the adjoint equation method is applied to evaluate the gradient of the penalty term associated with the stability constraints. It is demonstrated that the implementation of the adjoint equation method significantly reduces the computational burden in solving the OTS. A numerical test on a power system using prototype code is presented.

Recursive procedures for constrained optimisation problems and its application in signal processing

Many filter design problems in signal processing can be formulated as a quadratic programming problem with linear inequality constraints. The authors present new recursive procedures for solving this kind of problem. Using a constraint transcription technique, this inequality constrained quadratic programming problem can be approximated as an unconstrained minimisation problem. Two types of optimisation methods are developed to solve this unconstrained problem in a recursive adjusting manner. Analysis and simulation results on the proposed recursive procedures applied to the design of envelope-constrained filters are presented.