Model predictive control step-response

The minimization of the quadratic performance index in Eq. (8-64), subject to the constraints in Eqs. (8-67) to (8-69) and the step response model in Eq. (8-61), can be formulated as a standard QP (quadratic programming) problem. Consequently, efficient QP solution techniques can be employed. When the inequality constraints in Eqs. (8-67) to (8-69) are omitted, the optimization problem has an analytical solution (Camacho and Bordons, Model Predictive Control, 2d ed., Springer-Verlag, New York, 2004 Maciejowski, Predictive Control with Constraints, Prentice-Hall, Upper Saddle River, N.J., 2002). If the quadratic terms in Eq. (8-64) are replaced by linear terms, an LP (linear programming) problem results that can also be solved by using standard methods. This MPC formulation for SISO control problems can easily be extended to MIMO problems. 

In this work, the influences of two different sets of manipulated inputs have been compared in the case of linear model predictive control of a simulated moving bed. The first one consisting in direct manipulation of flow rates of the SMB showed a very satisfactory behavior for set point tracking and feed disturbance rejection. The second one consists in manipulating the flow rates ratios over each SMB section. At the identification stage, this strategy proved to be more delicate as the step responses displayed important dynamic differences of the responses. However, when the disturbance concerns the feed flow rate, a better behavior is obtained whereas a feed concentration disturbance is more badly rejected. 

The implementation of model-predictive control requires four main steps -plant response testing, model analysis, commissioning, and training. 

A generalized framework for using step response models in model predictive control is presented in Chapter 20. Note that Fig. 7.14 illustrates the case where there is no time delay in the process model. When a time delay is present, the initial step (or impulse) coefficients are zero. For example, if there is a time delay of d sampling periods, then 5q, 5i,. .., 5 are zero in Eq. 7-48. Similarly, hi = 0 for 0 < i < d in Eq. 7-43. 

In principle, any type of process model can be used to predict future values of the controlled outputs. For example, one can use a physical model based on first principles (e.g., mass and energy balances), a linear model (e.g., transfer function, step response model, or state space-model), or a nonlinear model (e.g., neural nets). Because most industrial applications of MPC have relied on linear dynamic models, later on we derive the MPC equations for a single-input/single-output (SISO) model. The SISO model, however, can be easily generalized to the MIMO models that are used in industrial applications (Lee et al., 1994). One model that can be used in MPC is called the step response model, which relates a single controlled variable y with a single manipulated variable u (based on previous changes in u) as follows ... 

The empirical model of equation (11) predicted the response of the mechanistic model to a step change in initiator flow very closely (the average absolute deviation between the empirical model and mechanistic model was 0.8% of the response). Three algorithms have been considered for control of the downstream reactor modeled by equation (11). 

Early applications of MPC took place in the 1970s, mainly in industrial contexts, but only later MPC became a research topic. One of the first solid theoretic formulations of MPC is due to Richalet et al. [53], who proposed the so-called Model Predictive Heuristic Control (MPHC). MPHC uses a linear model, based on the impulse response and, in the presence of constraints, computes the process input via a heuristic iterative algorithm. In [23], the Dynamic Matrix Control (DMC) was introduced, which had a wide success in chemical process control both impulse and step models are used in DMC, while the process is described via a matrix of constant coefficients. In later formulations of DMC, constraints have been included in the optimization problem. Starting from the late 1980s, MPC algorithms using state-space models have been developed [38, 43], In parallel, Clarke et al. used transfer functions to formulate the so-called Generalized Predictive Control (GPC) [19-21] that turned out to be very popular in chemical process control. In the last two decades, a number of nonlinear MPC techniques has been developed [34,46, 57],... 

A mathematical model of the control system for erythropoiesis is presented. It is postulated that the rate of erythropoiesis is controlled by a hormone, erythropoietin, which is released from the kidney in response to reduced renal oxygen supply. Equations are developed relating erythropoietin release to arterial oxyhemoglobin concentration, and hemoglobin production to plasma erythropoietin concentration, with appropriate time delays. Effects of plasma volume changes during hypoxia are included. The model simulates the dynamic response of the erythropoietic system to a step decrease in the pOt of inspired air. Contributions of the parameters and relationships to the predicted response are analyzed. The model response compares favorably with experimental data obtained from mice subjected to different degrees of hypoxia. 

The previous analysis for SISO systems can be generalized to MIMO systems by using the Principle of Superposition. For simplicity, we first consider a process control problem with two outputs, y and yi, and two inputs, u and Ui The predictive model consists of two equations and four individual step-response models, one for each input-output pair ... 

The dimensions of the vectors and matrices in Eq. 20-36 are as follows. Both Y k + 1) and Y k + 1) are mP-dimensional vectors where m is the number of outputs and P is the prediction horizon. Also, AU(k) is an rM-dimensional vector where r is the number of manipulated inputs and M is the control horizon. Consequently, the dimensions of step-response matrix 5 are mP X rM. The MIMO model in (20-36) through (20-42) is the MIMO generalization of the SISO model in (20-24). It is also possible to write MIMO models in an alternative form, a generalization of Eqs. 20-34 and 20-35. An advantage of this alternative formulation is that the new dynamic matrix is partitioned into the individual SISO models, a convenient form for real-time predictions. 

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... 

Controlled strain is the preferred mode of operation for nonlinear studies. In step-strain experiments, an important source of experimental error is the deviation of the actual strain history from a perfect step. Laun [96] and Venerus and Kahvand [43] have discussed this problem and how it can be addressed. Gevgilili and Kalyon [ 100] found that the actual strain pattern generated by a popular coimnerdal rheometer in response to a command for a step was, in fact, a rather complex function of time. One approach that is of use in comparing data from any transient test with the predictions of a model is to record the actual, non-ideal, strain history and use this same history to calculate the model predictions. 

Closed-loop response to process disturbances and step changes in setpoint is simulated with the model of Kiparissides extended to predict the behavior of downstream reactors. Additionally, a self-optimizing control loop is simulated for conversion control of downstream reactors when the first reactor of the train is operating under closed-loop control with dead-time compensation. 

The success of MPC is based on a number of factors. First, the technique requires neither state space models (and Riccati equations) nor transfer matrix models (and spectral factorization techniques) but utilizes the step or impulse response as a simple and intuitive process description. This nonpara-metric process description allows time delays and complex dynamics to be represented with equal ease. No advanced knowledge of modeling and identification techniques is necessary. Instead of the observer or state estimator of classic optimal control theory, a model of the process is employed directly in the algorithm to predict the future process outputs. 

The experimental values of the three responses for this optimized resin are presented in Table 8, together with the values predicted by the empirical models (equations (3)-(5)). The particleboard properties (IB strength and FE) are sufficiently close to the predicted values, considering the inevitable variability induced by the use of industrial grade reagents, complex control of synthesis conditions (namely the pH history and monitoring of the viscosity in the condensation step) and the natural heterogeneity of the wood mix used for particleboards. 

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