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Novel techniques for the optimization and control of finite-time
processes in real-time are pursued. These are developed in the framework
of the Hamiltonian optimal control. Two methods are designed. The first
one constructs the reference control trajectory as an approximation of the
optimal control via the Riccati equations in an adaptive fashion based
on the solutions of a set of partial differential equations called the

Batch processes have received much attention during the past two decades due to developing chemical and pharmaceutical products, new polymers, and recent biotechnological applications. Usually, the engineers state that a batch process has three operative stages: start-up, batch run, and shutdown. While these three stages are widely studied by the engineers for each particular batch process, it is important to note that in a wide number of cases, the batch run stage is far from an optimal operation, supported only with the experience of operators and engineers.

Control techniques have been made inroad in the industry in order to improve the performance of the process. In batch context, interesting ideas such as controllability, observability, and stability have been introduced despite the duration of batch processes. Numerous control techniques have been adapted to the peculiarities of these kinds of systems. One example is the model predictive control (MPC). This method is essentially numerical, usually implemented on-line. Most successful industrial applications of MPC reported so far are in refining and petrochemical plants (see for instance [

Here, the optimal control paradigm is tackled, where the Hamiltonian formalism plays a major role. If the optimal control problem is regular, that is, a unique control

This problem cannot generally be solved in real-time due to the missing initial condition for the costate. Several numerical approaches have been designed to cope with this problem, and many of them were based on the shooting methods (see [

When input constraints are imposed to the problem, even in the linear control problem, the control law becomes nonlinear and phenomena like saturation appear. Many methods deal with this problem, for instance, control parametrization introduced by [

MPC strategies and other techniques based on nonlinear programming schemes are proposed in the literature to solve both optimal problems in real-time (see for instance [

The contributions of this paper are as follows. (i) A method to generate control strategies that can be used to obtain reference trajectories (or batch recipes) by means of a linear approximation of the nonlinear optimal solution using auxiliary matrices

The rest of the paper has the following structure. In Section

The dynamics of a fed-batch reactor usually is a nonlinear control system, which can be written in the affine form as follows:

In [

Only when the control action is calculated, the system considered will be a linear approximation of (

The costates

The main challenge here is either to solve the HCEs (

In order to find the missing boundary values

the Hamiltonian matrix

the augmented Hamiltonian system (a linear

For any

Now, a suboptimal control strategy for nonlinear systems can be developed based on the linearization (

Normally, this linearization is performed around an equilibrium point (or an steady-state point). And, this will work well near that point. However, the initial condition is far from the equilibrium and in fed-batch rectors equilibrium (or steady state) does not even exist as in continuous reactors. Here, the linearization will be calculated around the initial condition, so updating it in the running time is recommended.

In the following procedure, a basic adaptive scheme to update the controller gain as well as the matrices

Notice that

The goal in this section is to develop a suboptimal feedback control for the nonlinear systems (

Here, an on-line procedure to generate suboptimal feedback control laws is developed. To incorporate restrictions, the manner to solve the optimization problem was changed, basically modifying its horizon to infinite (see [

Note that the optimization horizon time was changed to infinity only to evaluate the cost function and to generate the control law, but the problem remains as a finite-time process. The finite-time horizon with input constraints case and on-line applications is still an open problem. Some first results in this context are illustrated in [

The solution to the optimal problem control described above is not easy to find, even when the system is linear. For these kinds of systems, the traditional Ricatti equations are no longer valid, and then a number of numerical approaches attempt to solve this problem like the MPC formulation [

Linear matrix inequalities (LMIs) are an alternative method to treat the optimization problem posed above with a low computational effort, at least in the infinite horizon set-up and for linear time-invariant systems.

A linear matrix inequality or LMI is a matrix inequality of the form:

In particular, the following control problems are handled as a single LMI-based optimization problem in this paper.

(i) If

A common procedure to convert convex nonlinear (matrix norm, quadratic, etc.) inequalities to a LMI form is through the well-known Schur complements; that is, a LMI of the form

Since

Note that to minimize the optimal control problem without constraints, it is equal to solve the ARE equation. And in [

(ii) Physical limitations inherent in process equipment invariably impose hard constraints on the manipulated variable

Considering that

Assuming that a

Consider that

Note that to find

One important advantage of the LMI is that the problem with multiple constrains can be expressed with multiple LMIs; that is,

To summarize, the optimal control problem with input constraints expressed by (

The control action that minimizes this problem can be expressed as

The procedure to implement these results in nonlinear systems is as follows.

The time instants

Consider a batch reactor with a nonlinear dynamic where an exothermic and irreversible second order chemical reaction

The reactor dynamics is modeled by the following equations:

Initially, the batch reactor is on the nominal values, and the goal is to control the reaction temperature without restrictions in

State trajectories for Method 1 and a comparison against the optimal solution with

Control trajectories for Method 1 and a comparison against the optimal solution with

Difference between optimal cost and suboptimal cost trajectories generated by Method 1.

Dynamically, the optimal solution has a worse behavior than that of the suboptimal trajectories because its state and control have more variability than the state and control of the suboptimal trajectories. However, the final error between the final desired state and the reached state is lower for the optimal solution as well as its total cost. Note that the solution with

The relative error between the optimal cost (

When the input constraints are active, two control strategies are proposed. A traditional predictive controller (MPC) according to [

Notice that neither the MPC nor the suboptimal controller with

State trajectories for Method 2 with

Note also that the saturation time with the suboptimal controller is less than that with the MPC, and this could be modified by setting the parameter

Figure

Control trajectories for Method 2 with

In this section, a fed-batch reactor for penicillin production is considered (see [

In this model

The goals for the control of the penicillin reactor were extracted from [

To reach in

The batch process includes an input constraint, which must be satisfied during the operation process

The strategy starts by computing the linearization of the system given by (

The two equations in (

In Figure

States and control trajectories for finite horizon set-up.

States and control trajectories for finite horizon set-up.

In this subsection, the results of Method 2 are described. The optimization problem via LMI is solved by traditional numerical methods (see [

The controller gain was updated four times according to the time instants

The parameter

States and control trajectories for the second method via LMIs in an infinite horizon set-up.

In this paper, complete control schemes have been developed and illustrated. The innovation in this paper is the novel way to obtain the control reference trajectory (or recipe), which is necessary in the context of batch reactors. These kinds of trajectories are constructed by using a set of new algebraic equations which are described above for the linear case. The other contribution is to handle constraints in the manipulated variable when the process is controlled and optimized. All tools developed to design come from modern optimal control theory based on the Hamiltonian formalism.

It is important to remark that the entire proposed schemes are designed to be implemented in real-time, with the controller parameters being updated on-line. This is because the resulting controller is based on the simplifications made of costate on the system, and, in consequence, a simple controller can be online-tuned. Thus, the two resulting strategies for tuning simple controllers are able to generate suboptimal trajectories such that, in the second case, different constraints can be satisfied.

Also, it is significant to comment that the first strategy was developed using finite horizon optimal control theory with the restrictions that support the definition of the objective function. Since the control action written in costate terms and assuming linearity in the costates, the resulting control law has time-varying gains. On the other hand, the second strategy is presented under infinite horizon optimal control framework and allows the inclusion of new constraints by means of LMI as for example maximum and minimum constraints in control variable. Maintaining the linearity in the co-states, it is possible to find a suboptimal control law, but with constant feedback gains. At this point, notice that the simplifications may lead to a suboptimal solution to the optimization problem. The counterpart of this is that the reduction in the complexity of the mathematical problem treatment leads to less computational effort.

Finally, the numerical results show a good performance in the controlled variables, with the suboptimal strategies, and, especially, in the second strategy, when the control is bounded. In general, all variables have a good performance inside the operation time fixed for the batch reactor achieving the requirements imposed upon the system.