CPM for MPC

Model predictive control is based on real-time optimization of a cost function. Consequently, CPM methods that focus on the values of this cost function can be developed. The MPC cost function T(A ) is 

Here E[.] is the expectation operator and e k) and Au(A ) are computed from the data set under examination. The LQG benchmark [115], the historical performance benchmark [222], and the model-based performance benchmark [222, 347] are some of the methods that have been proposed in the literature for CPM of MPC. 

LQG-Benchmark The achievable performance of a linear system characterized by quadratic costs and Gaussian noise can be estimated by solving the linear quadratic Gaussian (LQG) problem. The solution can be plotted as a trade-off curve that displays the minimal achievable variance of the controlled variable versus the variance of the manipulated variable [115] which is used as a CPM benchmark. Operation close to optimal performance is indicated by an operating point near this trade-off curve. For multivariable control systems, H2 norms are plotted. The LQG objective function and the corresponding H2 norms are [115] 

The trade-off curve is obtained by calculating the H2 norms for different values of A and plotting ]Gw q versus ]]G j. Once the trade-off curve is calculated, the H2 norms under the existing control system are computed and compared to the optimal control represented by the trade-off curve. 

The LQG benchmark is limited to a special group of MPCs characterized by the equality of control (M) and prediction (P) horizons and lack of feedforward components and constraints. It may be considered as a limit of achievable performance in terms of input and output variance to evaluate 

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An overview of single-loop CPM is presented in Section 9.1. Section 9.2 surveys CPM tools for multivariable controllers. Monitoring of MPC performance and a case study based on MPC of an evaporator model and a supervisory knowledge-based system (KBS) is presented in Section 9.3 to illustrate the methodology. The extension of CPM to web and sheet processes is discussed in Section 10.3. 

Tools for controller performance assessment (CPA), CPM, and diagnosis are available for four types of MPCs by obtaining benchmarks for constrained cases and controllers including feedforward components, and establishing statistical analysis to the historical and model-based performance measures hi.st k) and ydes(k) (Table 9.1). 

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