تجزیه و تحلیل حساسیت سیستم های کنترل آبشار LP-MPC

Model predictive control (MPC) has found wide application in the chemical process industry as well as other industrial sectors. Commercial MPC systems are typically implemented in conjunction with a steady-state linear or quadratic programming optimizer, whose key functions are to track the economic optimum and to provide feasible set-points to the model predictive controller. The two-level system is complementary to real-time optimization which typically utilizes more complex models and is executed less frequently. Despite the widespread adoption of LP-MPC systems, occurrences of poor performance have been reported, where large variations in the computed set-points were observed. In this paper, we analyze the sensitivity of the LP solution to variation in the LP model bias, through which feedback to the LP layer occurs. We consider both multi-input, single-output (MISO) and multi-input, multi-output (MIMO) systems. Principles are illustrated through graphical representation as well as case studies. The performance of the two-level LP-MPC closed-loop system is evaluated and explained using results of the LP sensitivity analysis.

Model predictive control (MPC) is arguably the advanced control algorithm of choice in the chemical process industry, and has made inroads into other industrial sectors as well [10] and [13]. A dynamic model is utilized within the control algorithm to predict the effect of future plant inputs on the controlled outputs. Future inputs are computed in accordance with a performance objective, typically as the solution of an optimization problem. The inputs corresponding to the first control interval are implemented, and the calculation process repeated at the end of the sampling period, with the model predictions adjusted using the difference between the measured and predicted outputs. Details of the algorithm may be found, inter alia, in [4], [9] and [13]. Industrial MPC systems are generally implemented in conjunction with a linear programming (LP) or quadratic programming (QP) steady-state optimizer [5], [13], [17], [19] and [20]. The LP (or QP) typically uses a static model consistent with the dynamic MPC model, and is executed at the same frequency as the model predictive controller. The plant economic optimum may shift due to disturbances; thus the steady-state LP (QP) provides a bridge between a higher-level and less frequently executed real-time optimization (RTO) layer and the model predictive controller by making set-point adjustments in response to changing conditions between RTO executions. The LP formulation may involve minimization of the deviation between the set-points and target values determined by the real-time optimizer, or optimization of an economic criterion directly. Fig. 1 illustrates the location of the steady-state LP (QP) within a plant automation hierarchy, as given in [19]. We note that the process would typically include local PID-type controllers and that several variants are possible, such as the presence of a plant unit optimization layer between the LP and plant-wide RTO layers [5] and [13]. Full-size image (10 K) Fig. 1. LP (QP) within process automation hierarchy [19]. Figure options Despite the apparent success of two-level LP-MPC systems, instances of poor performance have been reported [6] and [16]. Shah et al.[16], in the context of control performance monitoring, describe an industrial application in which the variation in the set-points exceeds that of the corresponding controlled variables. Kozub[6] also reports set-points being noisy relative to their controlled variables, and excessive variation in the set of inputs which are at their constraints. This motivates an investigation into the potential causes of such behavior. Ying and Joseph [19] provide stability theorems for LP-MPC and QP-MPC cascade control systems with no plant/model mismatch. Consideration of model uncertainty is included in a case study based on the Shell Standard Control Problem [12]. Kassmann et al.[5] present a formulation for robust steady-state target calculation. For elliptic uncertainty on the model parameters, the target calculation problem takes the form of a second order cone program (SOCP) which the authors solve using software based on a primal-dual interior point algorithm. Steady-state targets are also computed in [11] and [14]. However, this is driven primarily not by economics, but rather to provide steady-state values for the plant inputs and states for inclusion in an MPC formulation in which deviations of the states and inputs from corresponding steady-states are weighted by positive definite matrices. Feedback in the two-level LP-MPC configuration occurs through a bias term in the steady-state LP model. In this paper, we analyze the sensitivity of the LP solution to variations in the LP model bias. We consider first the LP level separately for multi-input, single-output (MISO) and multi-input, multi-output (MIMO) systems. This analysis is coupled with graphical representations to provide insights into the sensitivity effects. Thereafter, the performance of a two-level LP-MPC cascade system is evaluated, with observed performance related to earlier sensitivity results

Industrial implementations of MPC typically include a steady-state optimization layer to track the economic optimum and to provide feasible set-points to the model predictive controller. While two-level LP-MPC systems are relatively widely applied, reports of excessive set-point variation motivate the investigation of potential causes of such behavior. The present study focused on the sensitivity of the LP solution to variations in the LP model bias term, since the bias is updated prior to each LP execution based on the difference between measured and predicted plant outputs. A sensitivity analysis was conducted first for multi-input, single output systems. The steady-state LP model has only one bias term in this case, and its effect on the LP solution is conveniently summarized graphically. A key observation is that if the output is not at one of its limits, it varies linearly with the bias term with gain of unity, provided that the LP solution is unique. However, changes in the inputs may exceed those of the bias term if the gain relating the output to the corresponding input is less than unity. MIMO systems were explored through a detailed analysis of a two-input, two-output system. A graphical analysis was developed by expressing the problem in terms of the inputs only. Scenarios in which the solution is defined by two output constraints, and an output and an input constraint, were considered. In the former case, the set-points remain constant for small perturbations in the bias term. However, in the latter case, variations in the bias may result in amplified variations in the output that is not at its bound, depending on the magnitudes of the process gains. Simulation of the two-level LP-MPC system was conducted to evaluate the response of the two-level closed-loop system to a step disturbance combined with measurement noise. The results of the sensitivity analysis were used to explain the observed performance of the two-level system. The focus of this paper has been on analysis of the sensitivity of LP-generated set-points to variations in the bias term induced, for example, by measurement and/or process noise. A question that naturally arises is how adverse amplification effects can be avoided or mitigated. One possibility would be the use of an alternative control structure. Another option might be to filter the bias update; however, the impact on the overall performance of the two-level LP-MPC system would need to be evaluated. A systematic investigation of these and other factors within a design framework would be a potentially useful avenue for future research.