Optimal control problems constraints

Bock I., Lovisek J. (1987) Optimal control problems for variational inequalities with controls in coefficients and in unilateral constraints. Apl. Mat. 32 (4), 301-314. 

These methods are efficient for problems with initial-value ODE models without state variable and final time constraints. Here solutions have been reported that require from several dozen to several hundred model (and adjoint equation) evaluations (Jones and Finch, 1984). Moreover, any additional constraints in this problem require a search for their appropriate multiplier values (Bryson and Ho, 1975). Usually, this imposes an additional outer loop in the solution algorithm, which can easily require a prohibitive number of model evaluations, even for small systems. Consequently, control vector iteration methods are effective only when limited to the simplest optimal control problems. 

While the reduced SQP algorithm is often suitable for parameter optimization problems, it can become inefficient for optimal control problems with many degrees of freedom (the control variables). Logsdon et al. (1990) noted this property in determining optimal reflux policies for batch distillation columns. Here, the reduced SQP method was quite successful in dealing with DAOP problems with state and control profile constraints. However, the degrees of freedom (for control variables) increase linearly with the number of elements. Consequently, if many elements are required, the effectiveness of the reduced SQP algorithm is reduced. This is due to three effects ... 

Therefore, for large optimal control problems, the efficient exploitation of the structure (to obtain 0(NE) algorithms) still remains an unsolved problem. As seen above, the structure of the problem can be complicated greatly by general inequality constraints. Moreover, the number of these constraints will also grow linearly with the number of elements. One can, in fact, formulate an infinite number of constraints for these problems to keep the profiles bounded. Of course, only a small number will be active at the optimal solution thus, adaptive constraint addition algorithms can be constructed for selecting active constraints. 

In contrast to the sequential solution method, the simultaneous strategy solves the dynamic process model and the optimization problem at one step. This avoids solving the model equations at each iteration in the optimization algorithm as in the sequential approach. In this approach, the dynamic process model constraints in the optimal control problem are transformed to a set of algebraic equations which is treated as equality constraints in NLP problem [20], To apply the simultaneous strategy, both state and control variable profiles are discretized by approximating functions and treated as the decision variables in optimization algorithms. 

The optimal control problem is to choose an admissible set of controls u(t), and final time, tF, to minimize the objective function, /, subject to the bounds on the controls and constraints. 

To pose the optimal control problem as a nonlinear programming (NLP) problem the controls u t) are approximated by a finite dimensional representation. The time interval [t0, r r] is divided into a finite number of subintervals (Ns), each with a set of basis functions involving a finite number of parameters u(t) = t, zj), te[( tj.i, tj), j = 1,2,. . J ], where tj = tF. The functions switching time tpj = 1,2,..., J. The control constraints now become ... 

Optimal Control. Optimal control is extension of the principles of parameter optimization to dynamic systems. In this case one wishes to optimize a scalar objective function, which may be a definite integral of some function of the state and control variables, subject to a constraint, namely a dynamic equation, such as Equation (1). The solution to this problem requires the use of time-varying Lagrange multipliers for a general objective function and state equation, an analytical solution is rarely forthcoming. However, a specific case of the optimal control problem does lend itself to analytical solution, namely a state equation described by Equation (1) and a quadratic objective function given by... 

Here, the temperature, pressure, and chemical potential are estimated at ambient conditions. For an optimal control problem, one must specify (i) control variables, volume, rate, voltage, and limits on the variables, (ii) equations that show the time evolution of the system which are usually differential equations describing heat transfer and chemical reactions, (iii) constraints imposed on the system such as conservation equations, and (iv) objective function, which is usually in integral form for the required quantity to be optimized. The value of process time may be fixed or may be part of the optimization. 

Clearly, formulation (PI2) is an optimal control problem with differential equation constraints, where the y s, and the temperature are the control profiles. The solution to this model will give us the optimal separation profile along the reactor. It is clear that 7(a) models the effect of separation within the reactor network. If all the elements of the vector 7(0 ) are the same (which implies that there is no relative separation between the species in the reactor), the second term for the governing differential equation vanishes, since... 

We have used sensitivity equation methods (Leis and Kramer, 1985) for gradient evaluation as these are simple and efficient for problems with few parameters and constraints. In general, the balance in efficiency between sensitivity and adjoint methods depends on the type of problem being addressed. Adjoint methods are particularly advantageous for optimal control problems in which the inputs are represented as a large number of piecewise constant input values and few interior point constraints exist. Sensitivity methods are preferable for problems with few parameters and many constraints. 

There are indications that the permeability of the precipitates is diminished due to the bimodal nature of the CSD of a seeded crystallizer. Thus, the minimization of the final mass of nucleation-formed crystals, m (tf), would be expected to favorably affect the rate of subsequent filtration. The corresponding optimal control problem involves the solution of the nonlinear programming problem with the following objective function and constraints ... 

The terms Pmax and Pmax denote the limiting capacities of the inspiratory muscles, and n is an efficiency index. The optimal Pmus(t) output is found by minimization of / subjects to the constraints set by the chemical and mechanical plants. Equation 11.1 and Equation 11.9. Because Pmusif) is generally a continuous time function with sharp phase transitions, this amounts to solving a difficult dynamic nonlinear optimal control problem with piecewise smooth trajectories. An alternative approach adopted by Poon and coworkers [1992] is to model Pmus t) as a biphasic function... 

In this chapter, we introduce the concept of Lagrange multipliers. We show how the Lagrange Multiplier Rule and the John Multiplier Theorem help us handle the equality and inequality constraints in optimal control problems. 

In Section 3.2.1 (p. 59), we had asserted the Lagrange Multiplier Rule that the optimum of the augmented J is equivalent to the constrained optimum of I. This rule is based on the Lagrange Multiplier Theorem, which provides the necessary conditions for the constrained optimum. We will first prove this theorem and then apply it to optimal control problems subject to different types of constraints. 

In most problems, a Lagrange multiplier can be shown to be related to the rate of change of the optimal objective functional with respect to the constraint value. This is an important result, which will be utilized in developing the necessary conditions for optimal control problems having inequality constraints. 

The above result can be readily generalized for the optimal control problem in which J is dependent on vectors y and u of state and control functions and is subject to m constraints, Ki = ki, i = 1,2,..., m. In this case, the Lagrange multipliers are given by... 

When solving an inequality-constrained optimal control problem numerically, it is impossible to determine which constraints are active. The reason is one cannot obtain a p, exactly equal to zero. This difficulty is surmounted by considering a constraint to be active if the corresponding p < a where a is a small positive number such as 10 or less, depending on the problem. Slack variables may be used to convert inequalities into equalities and utilize the Lagrange Multiplier Rule. 

Alternatively, increasing penalties may be applied on constraint violations during repeated applications of any computational algorithm used for unconstrained problems. We will use the latter approach in Chapter 7 to solve optimal control problems constrained by (in)equalities. 

The reason for the above provision is that if / = m, then the constraints would uniquely determine the control, and there would not be any optimal control problem remaining. 

Integral constraints could be equality or inequality constraints. We first consider integral equality constraints in an optimal control problem with free state and free final time. 

Consider the optimal control problem in the last section, but with integral equality constraints replaced with the inequality constraints... 

This is a simple method for solving an optimal control problem with inequality constraints. As the name suggests, the method penalizes the objective functional in proportion to the violation of the constraints, which are not enforced directly. A constrained problem is solved using successive applications of an optimization method with increasing penalties for constraint violations. This strategy gradually leads to the solution, which satisfies the inequahty constraints. 

The integral equality constraints of the optimal control problem in Section 7.2.4 (p. 214) may be transformed into differential equations. Using this approach,... 

Based on a multiple shooting method for parameter identification in differential-algebraic equations due to Heim [4], a new implementation of a direct multiple shooting method for optimal control problems has been developed, which enables the solution of problems that can be separated into different phases. In each of these phases, which might be of unknown length, the control behavior due to inequality constraints, the differential equations, even the dimensions of the state and/or the control space can differ. For the optimal control problems under investigation, the different phases are concerned with the different steps of the recipes. 

A constrained optimization problem subject to a DAE system, with or without inequality constraints, is referred to as a dynamic optimization problem or optimal control problem. This problem can be posed as follows with the DAEs (14.2 and 14.3) in semi-explicit form ... 

Two different optimal control problems have been studied with or without a production constraint on the by-products amount by-products amount inferior to 3.5% of the total production. In these problems, the temperature profile of the heat transfer fluid is discretised in five identical time intervals. Piecewise constant parameterisation of the temperature has then been adopted. Reactant addition flow rate has also been discretised in five intervals, but only the four last ones have the same size. Then, the time of the first interval and the value of the piecewise constant constitute the optimisation variables of the feed flow rate. The results associated to an optimal reaction carried out with a by-products constraint are given on figure 1. 

Other techniques employ a discretization approach whereby the optimal control problem is converted to an NLP through the discretization of all variables. This can be done using the finite difference and orthogonal collocations methods [22, 23]. The characteristic of the discretization method is that the optimization is carried out in the full space of the descretized variables and the discretized constraints are satisfied at the solution of the optimization problem only. This is therefore called the infeasible path approach. Another... 

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