# Design of Experiments for Control-Relevant Multivariable Model Identification: An Overview of Some Basic Recent Developments

^{*}

## Abstract

**:**

## 1. Introduction

(underlining added in quotation) where “detuning factor” refers to the optimal ratio of rotated PRBS input variances when input constraints are present in DOE. While the case of control-relevant DOE subject to input constrains is common and even though the result for the optimal rotated PRBS input variances shown above is both explicit and simple, it has remained largely unnoticed. In another instance, Featherstone and Braatz [13] state that, unfortunately, incorporating the integral controllability conditions (Equation (11) below below) directly in DOE is problematic, as the eigenvalue inequalities’.“Using quite advanced theoretical considerations, Darby and Nikolaou (2009) derived the detuning factor ${c}_{i}={\left(\frac{{\sigma}_{i}}{{\sigma}_{1}}\right)}^{\frac{2}{3}}$ for the $i$-th gain direction.”

“main weakness is that they consist of a coupling between the process model and the true process, which is highly cumbersome.”

- presents the mathematical problem formulation in a concise and clear manner;
- provides simple recipes to solve the problem either analytically or numerically; and
- illustrates the entire problem formulation/solution process through simulations on both textbook-level and realistic cases, to further help the user grasp the subject.

## 2. Control-Relevant DOE

## 3. DOE for Estimation of System Order in Subspace Identification

#### 3.1. Relevant Background on Subspace Identification

- (a)
- constructing certain matrices directly from input and output data;
- (b)
- performing a projection operation [33] on these matrices and, finally;
- (c)
- performing a singular value decomposition (SVD) of the resulting matrix.

#### 3.2. Pitfalls in Model-Order Estimation from Ordinary Experimental Data

#### 3.3. Why Ordinary Data May Easily Lead to the Wrong Order of an MIMO Model

#### 3.4. DOE for Accurate Estimation of Model Order: The Case for Rotated Inputs

#### 3.5. What Could Go Wrong with Rotated Inputs

#### 3.6. Adaptive DOE Employing Rotated Inputs

**Box 1.**Adaptive DOE for estimation of multivariable model order.

- Use standard PRBS inputs and generate data to obtain a preliminary model estimate with steady-state gain matrix $\hat{G}$.
- Obtain an SVD $\hat{G}={\hat{U}\hat{\Sigma}\hat{V}}^{T}$.
- Design an input $m$ based on a rotated PRBS input $\mathsf{\xi}$, Equations (6) and (7), and perform an identification experiment.
- Using data of the latest experiment, perform an SVD of the resulting matrix ${\rm Z}$ (Equation (2)) and choose the system order (select the matrices ${\Sigma}_{S}$ and ${\Sigma}_{N}$, Equation (3)) by checking where the singular values of ${\rm Z}$ demonstrate an abrupt transition from “large” to “small”.
- Using all experimental data collected, obtain matrix estimates $\hat{A},\hat{B},\hat{C},\hat{D}$ of a state-space model using standard SI formulas (e.g., taking Steps 4–6 in combined Algorithm 1 or 2 outlined in [30], p. 121 and p. 124, respectively), and estimate the new steady-state gain matrix $\hat{G}$.
- If $\hat{G}$ has satisfactorily converged, stop; else go to Step 2.

#### 3.7. Illustration of Adaptive DOE with Rotated Inputs for Accurate Model-Order Estimation

#### 3.8. Adaptive DOE in Comparison to Other DOE Approaches

#### 3.9. Summary: DOE for MIMO Model Order Estimation

## 4. DOE for Identification of Models Satisfying Integral Controllability

#### 4.1. Why Model Proximity to the Real Process Is Not Enough for Controller Design

#### 4.2. Integral Controllability: A Measure of Multivariable Controller Robustness

#### 4.3. Standard Approach to DOE for Multivariable Model Identification

- (1)
- D-optimal: minimize $\mathrm{det}({{C}_{m}}^{-1})$, equivalent to minimizing the volume of the uncertainty ellipsoids.
- (2)
- E-optimal: minimize ${\lambda}_{\mathrm{max}}({{C}_{m}}^{-1})$, equivalent to minimizing the maximum variance amongst all parameters.
- (3)
- A-optimal: minimize $\mathrm{trace}({{C}_{m}}^{-1})$, equivalent to minimizing average variance of the parameters.

#### 4.4. Making IC-Compliant DOE Feasible

#### 4.4.1. IC-Compliant DOE Subject to the Upper Bound on the Total Input/Output Variance: Analytical Solution

#### 4.4.2. IC-Compliant DOE for Minimizing Total Input/Output Variance Subject to IC: Analytical Solution

#### 4.4.3. IC-Compliant DOE Subject to Bounds on Individual Input and/or Output Variances: Numerical Solution

#### 4.5. DOE for the Identification of IC-Compliant Dynamic Models

- (a)
- The components ${z}_{i}(t)$ of $z(t)\in {\Re}^{{n}_{u}}$ are not correlated with each other, i.e., $E[{z}_{i}(t){z}_{j}(t-\tau )]=0$, $i\ne j$, for all time lags $\tau \ge 0$.
- (b)
- The auto-correlation function (equivalently, frequency spectrum) is the same for all ${z}_{i}(t)$, i.e., $E[{z}_{i}(t){z}_{i}(t-\tau )]={r}_{z}(\tau )$.

#### 4.6. DOE for Identification of Partially-Known IC-Compliant Models

#### 4.6.1. DOE for the Identification of an IC-Compliant Model Subject to Linear Equality Constraints on Each Row of the Steady-State Gain Matrix

#### 4.6.2. DOE for the Identification of an IC-Compliant Model Subject to Linear Equality Constraints Involving Multiple Rows of the Steady-State Gain Matrix

#### 4.6.3. Numerical Solution to the DOE Problem for the Identification of an IC-Compliant Model of a Partially-Known System

#### 4.7. Overview of DOE for the Identification of IC-Compliant Models

**Box 2.**Adaptive DOE for identification of IC-compliant models.

- Obtain a preliminary model estimate with steady-state gain matrix $\hat{G}$, from input-output data using standard PRBS inputs for limited time.
- Based on $\hat{G}$, solve the minimization problem associated with the relevant DOE subject to input and/or output constraints, as follows.
- Equations (31) and (32) for identification of a steady-state multivariable model.
- Equations (32) and (39) for identification of a dynamic multivariable model.
- Equations (42) and (40) (or (45) and (43)) for identification of a multivariable model of a partially-known system.

- Implement the inputs determined in Step 2 for limited time, and collect input-output data, to update $\hat{G}$.
- If the updated model does not satisfy IC, i.e., Equation (30) for Case a, Equation (35) for Case b and Equation (41) (or (44)) for Case c, respectively, go to Step 2. Else, stop.

#### 4.8. Illustrations of IC-Relevant DOE via Computer Simulations

#### 4.8.1. DOE for Minimization of Total Input/Output Variance Subject to IC

#### 4.8.2. DOE for IC Subject to Constraints on Individual Input and/or Output Variances

- IC-optimal rotated inputs, ${\xi}_{1},\text{}{\xi}_{2}$, are not necessarily uncorrelated as indicated by ${C}_{\xi}$ values for ICmin design.
- IC-optimal inputs, ${m}_{1},\text{}{m}_{2}$, are not necessarily highly correlated as indicated by Case #9 in Table 3.
- The optimal ratio, ${r}_{21}\stackrel{\wedge}{=}\sqrt{\mathrm{var}({\xi}_{2})}/\sqrt{\mathrm{var}({\xi}_{1})}$, may be quite smaller or larger than ${\sigma}_{1}/{\sigma}_{2}$ for ICmin design.
- Large values of $\beta $ and small values of $\mathrm{det}{C}_{m}$ for PRBSmax design indicate that PRBS inputs are neither IC-optimal nor D-optimal.
- The respective values of $\beta $ and $\mathrm{det}{C}_{m}$ for ICmin and Dmax designs are quite close to each other. These values indicate a trade-off between ICmin and Dmax designs for $\beta $ and $\mathrm{det}{C}_{m}$, supporting the claim that ICmin design sacrifices some accuracy in parameter estimates (lower value of $\mathrm{det}{C}_{m}$ for ICmin compared to that for Dmax) to achieve IC.

#### 4.8.3. DOE for the Identification of IC-Compliant Dynamic Models Subject to Constraints on Individual Input and/or Output Variances

- (a)
- a $2\text{}\times \text{}2$ ill-conditioned, high-purity distillation column;
- (b)
- a $2\text{}\times \text{}2$ well-conditioned distillation column;
- (c)
- a $5\text{}\times \text{}5$ fluidized catalytic cracking (FCC) reactor-regenerator system.

- (D1)
- Constrained D-optimal DOE, which seeks to maximize $\mathrm{det}{C}_{m}$ using a convex objective function, $\underset{{C}_{m}}{\mathrm{min}}\{-\mathrm{log}\mathrm{det}({C}_{m})\}$.
- (D2)
- Constrained IC-optimal DOE, which seeks to minimize $\beta $ (Equation (39)).
- (D3)
- DOE based on generalized binary noise (GBN) input [43], $m$, with only $\mathrm{var}({m}_{i})$ adjusted to satisfy input and/or output constraints, Equation (32). (this design for model identification is widely followed in industry).
- (D4)
- DOE based on GBN rotated input $\mathsf{\xi}$, with only $\mathrm{var}({\xi}_{i})$ adjusted to satisfy input and/or output constraints, Equation (32).

#### 4.8.4. DOE for the Identification of IC-Compliant Models of Partially-Known Systems

- (a)
- $5\text{}\times \text{}5$ fluidized catalytic cracking (FCC) reactor-regenerator system;
- (b)
- $2\text{}\times \text{}2$ two-stage absorber, respectively,

- (D1)
- IC-optimal DOE based on Equation (42) (or (45)), taking partial knowledge into account, Equation (40) (or (43)), and subject to input-output variance constraints, Equation (32).
- (D2)
- IC-optimal DOE based on Equation (22), without taking partial knowledge into account, and subject to input-output variance constraints, Equation (32).
- (D3)
- D-optimal DOE based on:$$\underset{{C}_{m}=Q{Q}^{T}}{\mathrm{min}}\left({\displaystyle \sum _{i=1}^{n}\mathrm{log}\left(\mathrm{det}({D}_{1,i})\right)}\right)$$
- (D4)
- D-optimal DOE based on:$$\underset{{C}_{m}=Q{Q}^{T}}{\mathrm{min}}\left(-\mathrm{log}\left(\mathrm{det}\left({C}_{m}\right)\right)\right)$$

- Since ${A}_{i}$ in Equation (42) is the information matrix, and larger values of $\mathrm{det}({A}_{i})$, $i=1,\dots ,5$ indicate more accurate parameter estimates. The value of $\mathrm{det}({A}_{i})$ for D-optimal design D4, which specifically targets maximization of $\mathrm{det}({A}_{i})$ via maximization of $\mathrm{det}({C}_{m})$, is larger than $\mathrm{det}({A}_{i})$ for design D2, which is IC-optimal. This shows that earlier IC satisfaction by IC-optimal designs is achieved at the cost of loss in accuracy of parameter estimates.
- The input and output pairs in experiments based on all of the above designs range from highly correlated to fairly uncorrelated. Therefore, the commonly used rule of thumb for control-relevant DOE to produce uncorrelated outputs is not always applicable when constraints are present.

#### 4.9. Summary: DOE for IC-Compliant Model Identification

## 5. Conclusions

## 6. Future Work

- Design of rotated PRBS inputs using a multivariable dynamic model rather than steady-state gain matrix for adaptive rotated DOE (cf. Section 3.8).
- Extension of the adaptive DOE framework to incorporate various chance constraints on outputs (cf. Section 3.8).
- Extension of IC-optimal design framework to incorporate constraints on inputs and/or outputs in the time domain (cf. Section 4.4, Section 4.5 and Section 4.6).
- Extend the MPCI framework of simultaneous model predictive control and identification [46] by incorporating the IC condition directly in the closed-loop objective function.
- Investigate the underlying theoretical reasons for agreement or disagreement between IC-optimal and D-optimal designs (cf. Section 4.4, Section 4.5 and Section 4.6).
- Extension of the IC-optimal DOE framework to other dynamic model forms such as state-space identified by the prediction-error or and SI method (cf. Section 4.5).
- Extension of IC-optimal design framework to incorporate partial knowledge in the form of inequality constraints (cf. Section 4.6).
- Extension of IC-optimal DOE to other kinds of IC, such as decentralized integral controllability (DIC) [39].

## Conflicts of Interest

## Appendix A

## Appendix B

## Appendix C

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**Figure 1.**Singular values of the matrix $Z$, Equation (3), resulting from input/output data collected for a $2\text{}\times \text{}2$ distillation column excited by pseudo random binary sequence (PRBS) inputs.

**Figure 2.**Singular values of the matrix $Z$, Equation (3), resulting from input/output data collected for a $5\times 5$ fluidized catalytic cracking (FCC) system excited by PRBS inputs.

**Figure 3.**Model order estimation for $2\text{}\times \text{}2$ distillation column using experimental data from a process excited by rotated PRBS inputs at the wrong rotation angle of $5\xb0$ in place of the correct rotation angle of $40\xb0$.

**Figure 4.**Order estimation for $2\text{}\times \text{}2$ distillation column using adaptive DOE with rotated PRBS inputs. A cut-off point is evident at two.

**Figure 5.**Order estimation for a $5\text{}\times \text{}5$ FCC system using adaptive DOE with rotated PRBS inputs. A cut-off point is evident at 15.

**Figure 6.**Closed-loop output response for step change in the set point of Output-1 (${y}_{1}^{sp}=1$) and no change in the set point of Output-2 (${y}_{2}^{sp}=0$) using controllers based on model ${\widehat{P}}_{1}$ (top) and ${\widehat{P}}_{2}$ (bottom), respectively.

**Figure 7.**Satisfaction (+1) or violation (−1) of IC for the identified model when $\chi =1$ (top row) and $\chi =0$ (bottom row). Integral controllability (IC) checked via Equation (11) for both cases.

**Figure 8.**Outputs, ${y}_{i}$, and inputs, ${m}_{i}$ ($i=1,2$), for IC optimal design when $\chi =1$ (top two rows) and $\chi =0$ (bottom two rows).

**Figure 9.**Identification time required for satisfaction of IC for an FCC reactor-regenerator system when inputs are produced from designs D1–D4. $J\stackrel{\wedge}{=}{\displaystyle \sum _{i=1}^{n}{r}_{A}\frac{{\Vert {\widehat{u}}_{i}\Vert}_{1}}{{\widehat{\sigma}}_{i}}}\sqrt{{\widehat{v}}_{i}^{T}{A}_{i}^{-1}{\widehat{v}}_{i}}<1$ (Equation (41)) for D1 and D3; $J\stackrel{\wedge}{=}c{\displaystyle \sum _{i=1}^{n}\frac{{\Vert {\widehat{u}}_{i}\Vert}_{1}}{{\widehat{\sigma}}_{i}}}\sqrt{{\widehat{v}}_{i}^{T}{({M}^{T}M)}^{-1}{\widehat{v}}_{i}}$ (Equation (21)) for D2 and D4.

**Figure 10.**Identification time required for satisfaction of IC for a two-stage absorber when inputs are produced from designs D1–D4. $J\stackrel{\wedge}{=}n{r}_{B}^{2}{\mu}_{\mathrm{max}}({B}^{-1}\Phi )$ (Equation (44)) for D1 and D3; $J\stackrel{\wedge}{=}c{\displaystyle \sum _{i=1}^{n}\frac{{\Vert {\widehat{u}}_{i}\Vert}_{1}}{{\widehat{\sigma}}_{i}}}\sqrt{{\widehat{v}}_{i}^{T}{({M}^{T}M)}^{-1}{\widehat{v}}_{i}}$ (Equation (21)) for D2 and D4.

**Table 1.**100-experiment averages for the design via Equation (26) on ${G}_{1}$; ${r}_{21}\stackrel{\wedge}{=}\sqrt{\mathrm{var}({\xi}_{2})/\mathrm{var}({\xi}_{1})}$.

Case | ${\mathit{r}}_{21}$ | $\mathbf{corr}({\mathit{m}}_{1},{\mathit{m}}_{2})$ | $\mathbf{corr}({\mathit{y}}_{1},{\mathit{y}}_{2})$ | $\mathbf{det}({C}_{\mathit{m}})$ |
---|---|---|---|---|

$x=1$ | 142 | 0.9999 | −0.0023 | 2.63 × 10^{−12} |

$x=0$ | 5.2 | 0.9292 | 0.9826 | 1.40 × 10^{−10} |

Design | Objective | Constraints |
---|---|---|

ICmin | $\underset{Q}{\mathrm{min}}\beta $ | Equation (32), $Q$ triangular |

Dmax | $\underset{{C}_{m}}{\mathrm{min}}[-\mathrm{log}\left(\mathrm{det}{C}_{m}\right)]$ | Equation (32), ${C}_{m}\succ \text{\hspace{0.17em}}0$ |

PRBSmax | $\underset{{C}_{m}}{\mathrm{min}}[-\mathrm{log}\left(\mathrm{det}{C}_{m}\right)]$ | Equation (32) ${C}_{m}\widehat{=}\mathrm{diag}({v}_{i})\succ \text{\hspace{0.17em}}0$ |

ξmax | $\underset{{C}_{m}}{\mathrm{min}}[-\mathrm{log}\left(\mathrm{det}{C}_{m}\right)]$ | Equation (32), ${v}_{i}/{v}_{j}={\widehat{\sigma}}_{j}^{2}/{\widehat{\sigma}}_{i}^{2}$, ${C}_{m}\widehat{=}\widehat{V}\mathrm{diag}({v}_{i}){\widehat{V}}^{T}\succ 0$ |

**Table 3.**Summary of results for experiment designs on ${G}_{1}$ and ${G}_{2}$. Corresponding variance value at its bound (active constraint in Equation (32)) is shown in bold and italics. Notation: ${C}_{m}$, ${C}_{y}$, ${C}_{\xi}$: covariance matrices of $m$, $y$, $\mathsf{\xi}$, respectively; ${r}_{21}\widehat{=}\sqrt{\mathrm{var}({\xi}_{2})}/\sqrt{\mathrm{var}({\xi}_{1})}$; $\beta $: Equation (31).

# | Bounds | Design | $\mathit{\beta}$ | $\mathbf{det}{\mathbf{C}}_{\mathit{m}}$ | ${\mathit{r}}_{21}$ | ${[{\mathbf{C}}_{\mathit{m}}]}_{11}$ | ${[{\mathbf{C}}_{\mathit{m}}]}_{12}$ | ${[{\mathbf{C}}_{\mathit{m}}]}_{22}$ | ${[{\mathbf{C}}_{\mathit{y}}]}_{11}$ | ${[{\mathbf{C}}_{\mathit{y}}]}_{12}$ | ${[{\mathbf{C}}_{\mathit{y}}]}_{22}$ | ${[{\mathbf{C}}_{\mathit{\xi}}]}_{11}$ | ${[{\mathbf{C}}_{\mathit{\xi}}]}_{12}$ | ${[{\mathbf{C}}_{\mathit{\xi}}]}_{22}$ | $\mathit{\rho}({\mathit{m}}_{1},{\mathit{m}}_{2})$ | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Ill-conditioned column, G_{1} | 1 | $\begin{array}{c}{[{C}_{y}]}_{11}\le 5\\ {[{C}_{y}]}_{22}\le 1\end{array}$ | ICmin | 1.206 | 6.63 × 10^{−5} | 222 | 0.92 | 0.91 | 0.90 | 5.0 | −0.07 | 1.0 | 6.4 × 10^{−5} | 7.1 × 10^{−3} | 1.8 | 0.99996 |

2 | Dmax | 1.207 | 6.64 × 10^{−5} | 218 | 0.90 | 0.89 | 0.88 | 5.0 | 0.0 | 1.0 | 6.6 × 10^{−5} | 7.1 × 10^{−3} | 1.8 × 10^{−5} | 0.99996 | ||

3 | PRBSmax | 7.014 | 1.78 × 10^{−9} | 1.01 | 4.3 × 10^{−5} | 0.0 | 4.2 × 10^{−5} | 0.64 | 0.80 | 1.0 | 4.2 × 10^{−5} | 5.4 × 10^{−7} | 4.2 × 10^{−5} | 0.00000 | ||

4 | ξmax | 1.562 | 1.33 × 10^{−5} | 142 | 0.26 | 0.26 | 0.26 | 1.0 | 0.0 | 1.0 | 2.6 × 10^{−5} | 0.0 | 5.2 × 10^{−1} | 0.99990 | ||

5 | $\begin{array}{c}{[{C}_{m}]}_{11}\le 0.15\\ {[{C}_{m}]}_{22}\le 1.5\\ {[{C}_{y}]}_{11}\le 5\\ {[{C}_{y}]}_{22}\le 5\end{array}$ | ICmin | 1.305 | 6.08 × 10^{−5} | 38.3 | 0.15 | 0.15 | 0.15 | 3.5 | 3.6 | 5.0 | 2.0 × 10^{−4} | 2.0 × 10^{−4} | 0.30 | 0.99864 | |

6 | Dmax | 1.321 | 6.24 × 10^{−5} | 37.5 | 0.15 | 0.15 | 0.15 | 4.1 | 3.9 | 5.0 | 2.2 × 10^{−4} | 1.5 × 10^{−3} | 0.30 | 0.99858 | ||

7 | PRBSmax | 4.691 | 4.44 × 10^{−8} | 1.01 | 2.1 × 10^{−4} | 0.0 | 2.1 × 10^{−4} | 3.2 | 4.0 | 5.0 | 2.1 × 10^{−4} | 2.7 × 10^{−6} | 2.1 × 10^{−4} | 0.00000 | ||

8 | ξmax | 2.051 | 4.47 × 10^{−6} | 142 | 0.15 | 0.15 | 0.15 | 0.58 | 0.0 | 0.58 | 1.5 × 10^{−5} | 0.0 | 0.30 | 0.99990 | ||

Well-conditioned column, G_{2} | 9 | $\begin{array}{c}{[{C}_{m}]}_{11}\le 4\mathrm{E}-5\\ {[{C}_{m}]}_{22}\le 8\mathrm{E}-5\end{array}$ | ICmin | 3.774 | 2.80 × 10^{−11} | 1.67 | 4.0 × 10^{−6} | 2.0 × 10^{−6} | 8.0 × 10^{−6} | 1.1 | 1.0 | 1.1 | 4.8 × 10^{−6} | −2.6 × 10^{−6} | 7.2 × 10^{−6} | 0.35527 |

10 | Dmax | 3.984 | 3.20 × 10^{−11} | 1.41 | 4.0 × 10^{−6} | 0.0 | 8.0 × 10^{−6} | 1.6 | 1.4 | 1.3 | 6.7 × 10^{−6} | −1.9 × 10^{−6} | 5.3 × 10^{−6} | 0.00000 | ||

11 | PRBSmax | 3.984 | 3.20 × 10^{−11} | 1.41 | 4.0 × 10^{−6} | 0.0 | 8.0 × 10^{−6} | 1.6 | 1.4 | 1.3 | 6.7 × 10^{−6} | −1.9 × 10^{−6} | 5.3 × 10^{−6} | 0.00000 | ||

12 | ξmax | 6.327 | 8.07 × 10^{−13} | 6.54 | 4.0 × 10^{−6} | 2.7 × 10^{−6} | 2.0 × 10^{−6} | 5.9 × 10^{−2} | 0.0 | 5.9 × 10^{−2} | 1.4 × 10^{−7} | 0.0 | 5.9 × 10^{−6} | 0.94848 | ||

13 | $\begin{array}{c}{[{C}_{m}]}_{11}\le 5\mathrm{E}-5\\ {[{C}_{m}]}_{22}\le 1\mathrm{E}-4\\ {[{C}_{y}]}_{11}\le 1\\ {[{C}_{y}]}_{22}\le 2\end{array}$ | ICmin | 1.514 | 2.99 × 10^{−10} | 4.70 | 5.0 × 10^{−5} | 3.8 × 10^{−5} | 3.5 × 10^{−5} | 1.0 | 0.56 | 1.6 | 4.2 × 10^{−6} | −6.1 × 10^{−6} | 8.1 × 10^{−5} | 0.91026 | |

14 | Dmax | 1.532 | 3.32 × 10^{−10} | 4.73 | 5.0 × 10^{−5} | 4.1 × 10^{−5} | 4.0 × 10^{−5} | 1.0 | 0.75 | 2.0 | 5.0 × 10^{−6} | −9.6 × 10^{−6} | 8.5 × 10^{−5} | 0.91350 | ||

15 | PRBSmax | 4.443 | 1.50 × 10^{−11} | 1.11 | 4.3 × 10^{−6} | 0.0 | 3.5 × 10^{−6} | 1.0 | 0.77 | 0.66 | 3.8 × 10^{−6} | 3.7 × 10^{−7} | 4.0 × 10^{−6} | 0.00000 | ||

16 | ξmax | 1.789 | 1.26 × 10^{−10} | 6.54 | 5.0 × 10^{−5} | 3.4 × 10^{−5} | 2.5 × 10^{−5} | 0.74 | 0.0 | 0.74 | 1.7 × 10^{−6} | 0.0 | 7.3 × 10^{−5} | 0.94848 |

**Table 4.**Skogestad and Morari column, ${G}_{1}(s)$. Results for different experiment designs. Generalized binary noise (GBN) signals in D3 and D4 are based on switching probability, ${p}_{sw}=0.05$, and mean switching time, $E\left[{T}_{sw}\right]=80\text{}\mathrm{min}$. Active constraints are shown in bold italics.

Design | $\mathit{\beta}$ | $\mathbf{det}{\mathbf{C}}_{\mathbf{u}}$ | $\mathbf{var}({\mathit{u}}_{1})$ | $\mathbf{var}({\mathit{u}}_{2})$ | $\mathbf{var}({\mathit{y}}_{1})$ | $\mathbf{var}({\mathit{y}}_{2})$ | ${\mathit{\rho}}_{12}$ |
---|---|---|---|---|---|---|---|

D1 s.t. Equation (50) | 3.968 | 1.081 × 10^{−4} | 0.738 | 0.736 | 1.00 | 1.00 | 0.99990 |

D2 s.t. Equation (50) | 3.968 | 1.081 × 10^{−4} | 0.738 | 0.736 | 1.00 | 1.00 | 0.99990 |

D3 s.t. Equation (50) | 221.2 | 1.447 × 10^{−8} | 1.218 × 10^{−4} | 1.187 × 10^{−4} | 0.64 | 1.00 | 0.00000 |

D4 s.t. Equation (50) | 3.968 | 1.081 × 10^{−4} | 0.738 | 0.736 | 1.00 | 1.00 | 0.99990 |

**Table 5.**Wood and Berry column, ${G}_{2}(s)$. Results for different experiment designs. GBN signals in D3 and D4 are based on switching probability, ${p}_{sw}=0.05$, and mean switching time, $E\left[{T}_{sw}\right]=20\text{}\mathrm{min}$. Active constraints are shown in bold italics.

Design | $\mathit{\beta}$ | $\mathbf{det}{\mathbf{C}}_{\mathbf{u}}$ | $\mathbf{var}({\mathit{u}}_{1})$ | $\mathbf{var}({\mathit{u}}_{2})$ | $\mathbf{var}({\mathit{y}}_{1})$ | $\mathbf{var}({\mathit{y}}_{2})$ | ${\mathit{\rho}}_{12}$ |
---|---|---|---|---|---|---|---|

D1 s.t. Equation (52) | 4.087 | 5.458 × 10^{−4} | 1.260 × 10^{−1} | 2.851 × 10^{−2} | 2.00 | 1.00 | 0.92087 |

D2 s.t. Equation (52) | 4.078 | 5.426 × 10^{−4} | 1.342 × 10^{−1} | 3.064 × 10^{−2} | 2.00 | 1.00 | 0.93171 |

D3 s.t. Equation (52) | 10.104 | 7.322 × 10^{−5} | 2.332 × 10^{−2} | 3.140 × 10^{−3} | 1.872 | 1.00 | 0.00000 |

D4 s.t. Equation (52) | 4.531 | 3.192 × 10^{−4} | 1.060 × 10^{−1} | 2.998 × 10^{−2} | 1.202 | 1.00 | 0.94848 |

**Table 6.**FCC reactor-regenerator system. Results for different experiment designs. GBN signals in D3 are based on switching probability, ${p}_{sw}=0.025$, and mean switching time, $E\left[{T}_{sw}\right]=40\text{}\mathrm{min}$. Active constraints are shown in bold italics.

Design | $\mathit{\beta}$ | $\mathbf{det}({\mathbf{C}}_{\mathbf{u}})$ | $\mathbf{var}({\mathit{u}}_{\mathit{i}})$, $\mathit{i}=1,\dots ,5$ | $\mathbf{var}({\mathit{y}}_{\mathit{i}})$, $\mathit{i}=1,\dots ,5$ | $\underset{1\le \mathit{i}\ne \mathit{j}\le 5}{\mathbf{max}}\left|{\mathit{\rho}}_{\mathit{i}\mathit{j}}\right|$ |
---|---|---|---|---|---|

D1 s.t. Equation (54) | 94.42 | 0.985 | 1.50, 0.45, 3.00, 1.50, 1.25 | 0.16, 0.65, 0.35, 0.10, 0.21 | 0.70 |

D2 s.t. Equation (54) | 90.47 | 0.791 | 1.50, 0.41, 3.00, 1.50, 1.21 | 0.16, 0.59, 0.35, 0.08, 0.20 | 0.75 |

D3 s.t. Equation (54) | 144.2 | 0.026 | 1.07, 0.07, 0.62, 1.50, 0.35 | 0.12, 0.14, 0.35, 0.03, 0.06 | 0.00 |

**Table 7.**Characterization of inputs and outputs for designs D1–D4 for a $5\text{}\times \text{}5$ FCC unit; active constraints are in bold.

Design | $\mathbf{det}({\mathbf{C}}_{\mathit{m}})$ | $\mathbf{det}({\mathbf{A}}_{\mathit{i}})$, $\mathit{i}=1,\dots ,5$ | $\mathbf{var}({\mathit{m}}_{\mathit{i}})$, $\mathit{i}=1,\dots ,5$ | $\mathbf{var}({\mathit{y}}_{\mathit{i}})$, $\mathit{i}=1,\dots ,5$ |
---|---|---|---|---|

D1 | $0.02$ | $1.49,0.60,0.49,0.01,0.88$ | $1.50,0.65,3.00,1.16,1.34$ | $0.22,0.35,0.65,0.17,0.19$ |

D2 | $1.01$ | $1.5,0.37,0.75,0.69,0.55$ | $1.50,0.42,3.00,1.50,1.04$ | $0.22,0.35,0.65,0.09,0.18$ |

D3 | $0.99$ | $1.5,0.53,0.66,0.75,0.77$ | $1.50,0.55,3.00,1.50,1.36$ | $0.23,0.35,0.65,0.14,0.20$ |

D4 | $1.16$ | $1.5,0.39,0.77,0.80,0.58$ | $1.50,0.41,3.00,1.50,1.17$ | $0.22,0.35,0.65,0.13,0.19$ |

**Table 8.**Input and output correlations matrices for designs D1–D4 for a $5\text{}\times \text{}5$ FCC unit.

Design | ${\mathit{R}}_{\mathit{m}}$ | ${\mathit{R}}_{\mathit{y}}$ |
---|---|---|

D1 | $\left[\begin{array}{ccccc}1& -0.08& -0.07& 0.24& 0.15\\ & 1& -0.87& 0.49& -0.55\\ & & 1& -0.84& 0.59\\ & & & 1& -0.34\\ & & & & 1\end{array}\right]$ | $\left[\begin{array}{ccccc}1& -0.09& -0.11& 0.04& 0.03\\ & 1& -0.01& 0.33& 0.86\\ & & 1& 0.49& 0.41\\ & & & 1& 0.70\\ & & & & 1\end{array}\right]$ |

D2 | $\left[\begin{array}{ccccc}1& -0.11& 0.002& 0.009& 0.082\\ & 1& -0.71& -0.23& -0.38\\ & & 1& 0.33& 0.41\\ & & & 1& 0.16\\ & & & & 1\end{array}\right]$ | $\left[\begin{array}{ccccc}1& 0.001& 0.01& -0.02& 0.14\\ & 1& 0.002& 0.50& 0.91\\ & & 1& 0.12& 0.26\\ & & & 1& 0.71\\ & & & & 1\end{array}\right]$ |

D3 | $\left[\begin{array}{ccccc}1& -0.19& 0.003& -3\times {10}^{-4}& 0.14\\ & 1& -0.80& 5\times {10}^{-8}& -0.53\\ & & 1& 6\times {10}^{-9}& 0.60\\ & & & 1& -3\times {10}^{-9}\\ & & & & 1\end{array}\right]$ | $\left[\begin{array}{ccccc}1& 4\times {10}^{-9}& -0.16& -0.04& 0.08\\ & 1& -1\times {10}^{-8}& 0.50& 0.89\\ & & 1& 0.37& 0.33\\ & & & 1& 0.77\\ & & & & 1\end{array}\right]$ |

D4 | $\left[\begin{array}{ccccc}1& -0.11& -1\times {10}^{-8}& 1\times {10}^{-9}& 0.08\\ & 1& -0.71& -9\times {10}^{-10}& -0.43\\ & & 1& -5\times {10}^{-10}& 0.50\\ & & & 1& 1\times {10}^{-9}\\ & & & & 1\end{array}\right]$ | $\left[\begin{array}{ccccc}1& -7\times {10}^{-9}& 2\times {10}^{-8}& -0.02& 0.13\\ & 1& 7\times {10}^{-9}& 0.51& 0.91\\ & & 1& 0.13& 0.26\\ & & & 1& 0.73\\ & & & & 1\end{array}\right]$ |

**Table 9.**Characterization of inputs and outputs designs D1–D4 for a $2\text{}\times \text{}2$ two-stage absorber; active constraints are in bold.

Design | $\mathbf{det}({\mathbf{C}}_{\mathit{m}})$ | $\mathbf{det}(\mathbf{B})$ | $\mathbf{var}({\mathit{m}}_{\mathit{i}})$ | $\mathbf{var}({\mathit{y}}_{\mathit{i}})$ | ${\mathit{\rho}}_{{\mathit{m}}_{1},}{}_{{\mathit{m}}_{2}}$ | ${\mathit{\rho}}_{{\mathit{y}}_{1},}{}_{{\mathit{y}}_{2}}$ |
---|---|---|---|---|---|---|

D1 | $9\text{}\times \text{}{10}^{-5}$ | $0.45$ | $0.5,0.5$ | $0.07,0.09$ | $0.99$ | $1.00$ |

D2 | $0.22$ | $0.18$ | $0.5,0.5$ | $0.07,0.09$ | $-0.35$ | $1.00$ |

D3 | $1\text{}\times \text{}{10}^{-8}$ | $0.45$ | $0.5,0.5$ | $0.07,0.09$ | $1.00$ | $1.00$ |

D4 | $0.25$ | $0.25$ | $0.5,0.5$ | $0.04,0.05$ | $-4\text{}\times \text{}{10}^{-9}$ | $0.79$ |

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**MDPI and ACS Style**

Misra, S.; Darby, M.; Panjwani, S.; Nikolaou, M.
Design of Experiments for Control-Relevant Multivariable Model Identification: An Overview of Some Basic Recent Developments. *Processes* **2017**, *5*, 42.
https://doi.org/10.3390/pr5030042

**AMA Style**

Misra S, Darby M, Panjwani S, Nikolaou M.
Design of Experiments for Control-Relevant Multivariable Model Identification: An Overview of Some Basic Recent Developments. *Processes*. 2017; 5(3):42.
https://doi.org/10.3390/pr5030042

**Chicago/Turabian Style**

Misra, Shobhit, Mark Darby, Shyam Panjwani, and Michael Nikolaou.
2017. "Design of Experiments for Control-Relevant Multivariable Model Identification: An Overview of Some Basic Recent Developments" *Processes* 5, no. 3: 42.
https://doi.org/10.3390/pr5030042