Optimal control problems inequality constrained

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. 

This section deals with optimal control problems constrained by algebraic equalities and inequalities. 

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. 

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 ... 

The situation is quite different when inequality constraints are included in the MPC on-line optimization problem. In the sequel, we will refer to inequality constrained MPC simply as constrained MPC. For constrained MPC, no closed-form (explicit) solution can be written. Because different inequahty constraints may be active at each time, a constrained MPC controller is not linear, making the entire closed loop nonlinear. To analyze and design constrained MPC systems requires an approach that is not based on linear control theory. We will present the basic ideas in Section III. We will then present some examples that show the interesting behavior that MPC may demonstrate, and we will subsequently explain how MPC theory can conceptually simplify and practically improve MPC. 

As mentioned earlier, the developed algorithm employs dynopt to solve the intermediate problems associated with the local interaction of the agents. Specifically, dynopt is a set of MATLAB functions that use the orthogonal collocation on finite elements method for the determination of optimal control trajectories. As inputs, this toolbox requires the dynamic process model, the objective function to be minimized, and the set of equality and inequality constraints. The dynamic model here is described by the set of ordinary differential equations and differential algebraic equations that represent the fermentation process model. For the purpose of optimization, the MATLAB Optimization Toolbox, particularly the constrained nonlinear rninimization routine fmincon [29], is employed. 

A quadratic programming problem minimizes a quadratic function of n variables subject to m linear inequality or equality constraints. A convex QP is the simplest form of a nonlinear programming problem with inequality constraints. A number of practical optimization problems are naturally posed as a QP problem, such as constrained least squares and some model predictive control problems. 

The introduction of inequality constraints results in a constrained optimization problem that can be solved numerically using linear or quadratic programming techniques (Edgar et al., 2001). As an example, consider the addition of inequality constraints to the MFC design problem in the previous section. Suppose that it is desired to calculate the M-step control policy AU(k) that minimizes the quadratic objective function J in Eq. 20-54, while satisfying the constraints in Eqs. 20-59, 20-60, and 20-61. The output predictions are made using the step-response model in Eq. 20-36. This MFC... 

Infeasible calculations can occur if the calculations of Steps 5 and 6 are based on constrained optimization, because feasible solutions do not always exist (Edgar et al., 2001). Infeasible problems can result when the control degrees of freedom are reduced (e.g., control valve maintenance), large disturbances occur, or the inequality constraints are inappropriate for current conditions. For example, the allowable operating window in Fig. 19.6 could disappear for inappropriate choices of the y and yi limits. Other modifications can be made to ensure that the optimization problem always has a feasible solution (Kassmann et al., 2000). 

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