# Multi-Model Identification of HVAC System

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Proposed Simultaneous Clustering and Regression Method

^{N}

^{×M}.

_{1}from point a (center of all circles) is classified in cluster 1 (red dots in Figure 1), and the same for cluster 2 (black dots in Figure 1). To find the center (a) and each R

_{i}, one needs to solve an optimization problem:

_{i}, i = 1, …,h, defined as

_{i}contains the parameters of the linear model and x

_{k}is the corresponding repressors, and <•> denotes inner product operation. In this paper, the ARX model (where ${x}_{k}={\left[{y}_{k-1}.{y}_{k-2.}\cdots .{y}_{k-{n}_{y}}.{u}_{k}.{u}_{k-1}.\cdots .{u}_{k-{n}_{u}}\right]}^{T}$ and ${w}_{i}=\left[{a}_{1}^{i}.{a}_{2}^{i}.\cdots .{a}_{{n}_{y}}^{i}.{b}_{1}^{i}.{b}_{2}^{i}.\cdots .{b}_{{n}_{u}}^{i}\right]$) is used for HVAC system identification. In this model, ${y}_{k}$ and ${u}_{k}$ are system outputs and inputs, respectively; ${n}_{y}$ and ${n}_{u}$ denote orders of system outputs and inputs in the model, and ${a}_{j}^{i}$ and ${b}_{k}^{i}$ for $j=1.\cdots .{n}_{y}$ and $k=1.\cdots .{n}_{u}$ are model parameters associated with outputs and inputs terms, respectively.

_{k}, y

_{k}) is generated by model i, then its distance from that model will be the smallest compared to others. So, an intuitive way to determine a cluster for the point (x

_{k}, y

_{k}), is calculating the distances of the point from each of the models and assigning it to the cluster in which the corresponding model generates the lowest error. This way, the dynamic models are regarded as the centers, and the corresponding errors are used to determine the clusters.

_{k},y

_{k}) belongs to cluster i (model i). Therefore, each point has an association with all models but with different probabilities. In addition, R

_{i}determines the boundary of the cluster i. By supposing the point belongs to cluster i (x

_{k},y

_{k}) (i.e., ${z}_{k.i}=1$), in the residual space, R

_{i}can be interpreted as the cluster radius as $\left|\right|{r}_{i}\left|\right|{}^{2}\le {R}_{i}^{2}$. Given the system models (cluster centers), minimizing the cost function provides the optimal cluster boundary.

Algorithm 1 |

Step 1—Select proper values for m, h and construct regressor and observation vector $\left({x}_{k}.{y}_{k}\right)$.Step 2—Initialize the probability vector ${z}_{k.i}$ for each cluster i by a random number generator.Step 3—Iterate the algorithm until ${z}_{k.i}$ converges or termination criterion is satisfied. Step 4—Update residual values ${r}_{k.i}$ for each cluster i.Step 5—Update parameters ${\sigma}_{k.i}$, ${R}_{i}$ and ${w}_{i}$.Step 6—Update ${z}_{k.i}$ as ${z}_{k.i}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${{\displaystyle \sum}}_{s=1}^{h}{\left(\frac{{r}_{k.i}}{{r}_{k.s}}\right)}^{\frac{2m}{m-1}}$}\right.$Step 7—Go to step 3, and repeat until convergence. |

## 3. Gap Metric

_{1}and P

_{2}, can be defined by a gap metric. Let P

_{i}, i = 1, 2 be p × m rational transfer function matrices, and ${P}_{i}={N}_{i}{M}_{i}^{-1}={\tilde{M}}_{i}^{-1}{\tilde{N}}_{i},\hspace{1em}i=1,2$ denote the normalized right/left coprime factorizations of P

_{1}and P

_{2}, respectively. In the gap metric, the distance between the two plants P

_{1}and P

_{2}in the frequency domain is defined as:

- (1)
- $0\le \delta \left({P}_{1},{P}_{2}\right)\le 1$.
- (2)
- The gap metric is an extension of common distance measures between two linear systems such as the infinity norm. For instance, the distances between two systems ${P}_{1}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$ and ${P}_{2}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$s+0.1$}\right.$ in the sense of infinity norm and gap metric are infinity and 0.1, respectively.
- (3)
- One pro of the gap metric is that it measures the ‘distance’ in the closed-loop sense instead of the open-loop sense. In other words, a small distance between two systems in the gap metric sense means that there exists at least one feedback controller that stabilizes both systems and the distance between the closed loops is small in the infinity norm sense. For instance, for the distance between two systems ${P}_{1}=\raisebox{1ex}{$100$}\!\left/ \!\raisebox{-1ex}{$2s+1$}\right.$ and ${P}_{2}=\raisebox{1ex}{$100$}\!\left/ \!\raisebox{-1ex}{$2s-1$}\right.$ the gap metric is about 0.2 [13].

#### 3.1. Proper Number of Mode Selection

- (1)
- Initialize the operation regions and calculate the nonlinear measure index for each operating region. Select enough operation points in each operating region.
- (2)
- Collect enough data from the system around the operating point OPi, use the proposed Algorithm 1 and get the linear models Pi.
- (3)
- Compute the gap metric between all pairs of linear models P
_{i}and P_{j}(i.e., $\delta \left({P}_{i},{P}_{j}\right)\hspace{1em}i,j=1,\dots ,N$). - (4)
- Prescribe a threshold level τ. Cluster the local models that satisfy δj ≤ τ.

#### 3.2. Local Model Weights

_{i}, i = 1, …, N). The weights are updated at a discrete time interval (i). We assign the weights for each local model using the following formula

## 4. Simulation and Validation

#### 4.1. Simulation Results

#### 4.2. Test on Real-World Data

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Support vector-based clustering (SVC) schematic. Red and black dots belong to cluster 1 and 2, respectively.

**Figure 2.**The building ventilation schematic [15].

**Figure 3.**The fan control block [15].

Model Parameters | ||||||
---|---|---|---|---|---|---|

Cluster | ${\mathit{a}}_{1}^{\mathit{i}}$ | ${\mathit{a}}_{2}^{\mathit{i}}$ | ${\mathit{a}}_{3}^{\mathit{i}}$ | ${\mathit{b}}_{1}^{\mathit{i}}$ | ${\mathit{b}}_{2}^{\mathit{i}}$ | ${\mathit{b}}_{3}^{\mathit{i}}$ |

i = 1 | 2.967 | −2.935 | 0.9676 | 0.0274 | −0.05466 | 0.02726 |

i = 2 | 1.744 | −0.7323 | −0.0119 | −0.08569 | 0.2612 | −0.1751 |

i = 3 | 2.304 | −1.624 | 0.3201 | 0.07934 | −0.147 | 0.06769 |

i = 4 | 1.077 | 0.8135 | 0.89 | −0.0528 | 0.126 | −0.073 |

i = 5 | 2.927 | −2.855 | 0.9282 | 0.0168 | −0.0325 | 0.0157 |

Model Parameters | ||||||
---|---|---|---|---|---|---|

Cluster | ${\mathit{a}}_{1}^{\mathit{i}}$ | ${\mathit{a}}_{2}^{\mathit{i}}$ | ${\mathit{a}}_{3}^{\mathit{i}}$ | ${\mathit{b}}_{1}^{\mathit{i}}$ | ${\mathit{b}}_{2}^{\mathit{i}}$ | ${\mathit{b}}_{3}^{\mathit{i}}$ |

i = 1 | 2.967 | −2.935 | 0.9676 | 0.0274 | −0.05466 | 0.02726 |

i = 2 | 1.744 | −0.7323 | −0.0119 | −0.08569 | 0.2612 | −0.1751 |

i = 3 | 2.304 | −1.624 | 0.3201 | 0.07934 | −0.147 | 0.06769 |

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**MDPI and ACS Style**

Alipouri, Y.; Zhong, L.
Multi-Model Identification of HVAC System. *Appl. Sci.* **2021**, *11*, 668.
https://doi.org/10.3390/app11020668

**AMA Style**

Alipouri Y, Zhong L.
Multi-Model Identification of HVAC System. *Applied Sciences*. 2021; 11(2):668.
https://doi.org/10.3390/app11020668

**Chicago/Turabian Style**

Alipouri, Yousef, and Lexuan Zhong.
2021. "Multi-Model Identification of HVAC System" *Applied Sciences* 11, no. 2: 668.
https://doi.org/10.3390/app11020668