# Performance Assessment of Predictive Control--A Survey
## 4 MPC Performance Assessment
### Model-Based Approaches  
Two approaches of the model-based assessment can be distinguished. The first one uses the so-called external benchmarking approach (Figure 6). Loop signals, such as controlled or manipulated variables, are used to evaluate benchmarking model. The second approach uses internal MPC signals and models (Figure 5). Minimum variance index allows measuring the current performance distance from the best available (optimal in minimum variance sense) control. The method uses normal process operation data to model a process and to calculate minimum variance benchmark. Thereby, a simple process model structure and the delay must be a priori known or estimated. The method calculates coefficients of the impulse response from noise-to-output transfer function with regressive models, as for instance of ARMA-type  
\[\eta_{0}=\frac{\sigma_{y}^{2}}{\sigma_{MV}^{2}}, \tag{4}\]  
where \(\sigma_{y}\) denotes CV variance, \(\sigma_{MV}\) is the minimum achievable variance, and \(\eta_{0}\geq 1\). DMC performance assessment using Harris-type index is addressed in [114]. Further works follow a similar path with other benchmarking approaches [115; 116]. [PERSON] et al. [117; 118] proposed the LQG benchmarking to estimate achievable variability reduction through control system improvement. [PERSON] and [PERSON] proposed using dynamic DMC performance bounds [119; 120] of a constrained Model Predictive Control system. They developed an index based on the constrained minimum variance controller. Such a performance bound has been calculated by using the proposed moving horizon approach. It converges to the unconstrained minimum variance performance bound, and the constraints on process variables become inactive. This method requires the process model, which is utilized to evaluate the constrained minimum variance controller. Stable inverse of the process model is the additional methodology assumption. Other consecutive methods have been proposed for the MPC benchmarking:  
* design-case approach [121], which uses the MPC controller criterion as the measure performance index \(J^{MPC}\);
* constraint benchmarking taking into account an economic performance assessment [122];
* Harris-based benchmarking [123] applied to the multivariate cases;  
Figure 5: MPC internal approach to the CPA. Figure 6: MPC external approach to the CPA. * multi-parametric quadratic programming analysis has been used to develop maps of minimum variance performance for constrained control over the state-space partition [124];
* predictive DMC structures used to compare and assess implemented as a single controller or as a supervisory level over PID regulatory control [125];
* orthogonal projection of the current output onto the space spanned by past outputs, inputs or setpoint using normal routine close loop data [126];
* the infinite-horizon MPC [65];
* Filtering and Correlation Analysis algorithm (FCOR) approach used to evaluate the minimum variance control problem and the performance assessment index [127]; and
* many others [122; 128; 129]. On-line Model Predictive Control performance benchmarking and monitoring is proposed in [130; 131]. Actually, various MPC structures may be used, but the research mostly focuses on DMC and GPC algorithms, as they are the most popular in practical implementations. The obtained performance index has similar formulation to the other minimum variance benchmarks  
\[\eta^{MPC}=\frac{J_{a}}{J_{o}^{MPC}}. \tag{5}\]  
The second group of the model-based approaches tries to utilize the already existing and evaluated MPC internal signals or knowledge on the model and cost-function. [PERSON] and [PERSON] [132] proposed comparing the expected value of the controller's internal objective function against its actual value over some assumed horizon. [PERSON] and [PERSON] [133] presented hybrid monitoring and diagnosis algorithm, in which historical, actual, and design model-based performances are compared.