A new research of identification strategy based on particle swarm optimization and least square

Abstract

Within the heat and moisture system that is complex in the air-conditioning rooms of large space building, the existence of delay makes the stability cushion reduced, which thereby makes the estimated parameters more complex. In this paper, particle swarm optimization (PSO) is integrated with least square (LS) to improve least squares (short for PSOLS). LS, optimized by PSO, identifies the heat and moisture system parameters of the existence of delay in the air-conditioning rooms by sampling input and output data. In view of this delay system, the identification is an effective solution to nonlinear system which LS can not identify directly. The simulation results show that PSOLS is quite effective, and its global optimization has great potential.

Keywords

least square (LS) / particle swarm optimization (PSO) / system identification / air-conditioning room

Introduction

Heat and moisture transfer in air-conditioning rooms of the buildings plays an important role in air-conditioning load, building energy consumption, and thermal comfort of the living environment. The study on this matter is the foundation of the researches of air-conditioning load calculating method, building energy analysis technology, building energy-saving technologies, and thermal comfort of the living environment. The factors that impact the heat and moisture transfer in air-conditioning rooms are complex. Since only one theoretical modeling method cannot accurately analyze the heat and moisture transfer in different environments, which is quite necessary to control, the adoption of appropriate identification algorithm through experiments can improve it. Experiments and researches are based on the modern theory of heat and moisture transfer of air-conditioning rooms, trying to find the effective ways to improve the measures promote the development on its research and energy-saving technology, speed up China’s target of building energy conservation, improve the living environment, and benefit to the sustainable technological progress of building the field. This paper will combine particle swarm optimization (PSO) and least square (LS) into a PSOLS method, which samples input and output data of the online identification of delay in the system.

Selection for control model of heat and moisture system in air-conditioning rooms

As to the heat and moisture system of the air-conditioning room, the input and output of the transfer function can be used approximately with the transfer function of the first-order lag, which is not accurate enough obviously. Therefore, we consider the model with the system transfer function of the second-order lag to improve the accuracy of it. Simultaneously, applying the least squares identification algorithm, the transfer function of the second-order lag is different. The specific process is as follows:

Assume that the transfer function between input variables U and output variables Z is the system of the second-order lag:

(1){\displaystyle \frac{U(s)}{Z(s)}}={\displaystyle \frac{K_{\mathrm{p}}}{(T_{\mathrm{1}}s+\mathrm{1})(T_{\mathrm{2}}s+\mathrm{1})}}{\mathrm{e}}^{-\tau s}.

In Eq. (1), K_p is the proportional coefficient, T₁ and T₂ are system parameters, and \tau is the time-delay parameter.

Carry out identity transformation, the process is described as follows:

(2){\displaystyle \frac{A}{T_{\mathrm{1}}s+\mathrm{1}}}+{\displaystyle \frac{B}{T_{\mathrm{2}}s+\mathrm{1}}}\equiv {\displaystyle \frac{A(T_{\mathrm{2}}s+\mathrm{1})+B(T_{\mathrm{1}}s+\mathrm{1})}{(T_{\mathrm{1}}s+\mathrm{1})(T_{\mathrm{2}}s+\mathrm{1})}}\equiv {\displaystyle \frac{K_{\mathrm{p}}}{(T_{\mathrm{1}}s+\mathrm{1})(T_{\mathrm{2}}s+\mathrm{1})}}\Rightarrow \{\begin{array}{l}A={\displaystyle \frac{K_{\mathrm{p}}{T}_{\mathrm{1}}}{T_{\mathrm{1}}-T_{\mathrm{2}}}},\\ B=-{\displaystyle \frac{K_{\mathrm{p}}{T}_{\mathrm{2}}}{T_{\mathrm{1}}-T_{\mathrm{2}}}}.\end{array}

Then, the transfer function can be transformed into

(3){\displaystyle \frac{{\displaystyle \frac{K_{\mathrm{p}}{T}_{\mathrm{1}}}{T_{\mathrm{1}}-T_{\mathrm{2}}}}{\mathrm{e}}^{-\tau s}}{T_{\mathrm{1}}s+\mathrm{1}}}+{\displaystyle \frac{-{\displaystyle \frac{K_{\mathrm{p}}{T}_{\mathrm{2}}}{T_{\mathrm{1}}-T_{\mathrm{2}}}}{\mathrm{e}}^{-\tau s}}{T_{\mathrm{2}}s+\mathrm{1}}}={\displaystyle \frac{K_{\mathrm{p}}{T}_{\mathrm{1}}}{T_{\mathrm{1}}-T_{\mathrm{2}}}}{\displaystyle \frac{{\mathrm{e}}^{-\tau s}}{T_{\mathrm{1}}s+\mathrm{1}}}-{\displaystyle \frac{K_{\mathrm{p}}{T}_{\mathrm{2}}}{T_{\mathrm{1}}-T_{\mathrm{2}}}}{\displaystyle \frac{{\mathrm{e}}^{-\tau s}}{T_{\mathrm{2}}s+\mathrm{1}}}.

Assume that the period of data sampling is T, and nT = τ; after Z transform, the transfer function can be transformed into

(4){\displaystyle \frac{Y(z)}{U(z)}}={\displaystyle \frac{K_{\mathrm{p}}}{T_{\mathrm{1}}-T_{\mathrm{2}}}}{\displaystyle \frac{z^{-n}}{\mathrm{1}-{\mathrm{e}}^{-T/T_{\mathrm{1}}}{z}^{-\mathrm{1}}}}-{\displaystyle \frac{K_{\mathrm{p}}}{T_{\mathrm{1}}-T_{\mathrm{2}}}}{\displaystyle \frac{z^{-n}}{\mathrm{1}-{\mathrm{e}}^{-T/T_{\mathrm{2}}}{z}^{-\mathrm{1}}}}.

There is

(5)Y(z)(T_{\mathrm{1}}-T_{\mathrm{2}})-Y(z)(T_{\mathrm{1}}-T_{\mathrm{2}})\left({\mathrm{e}}^{-\frac{T}{T_{\mathrm{1}}}}+{\mathrm{e}}^{-\frac{T}{T_{\mathrm{2}}}}\right)z^{-\mathrm{1}}+Y(z)(T_{\mathrm{1}}-T_{\mathrm{2}}){\mathrm{e}}^{-\left(\frac{T}{T_{\mathrm{1}}}+\frac{T}{T_{\mathrm{2}}}\right)}{z}^{-\mathrm{2}}=U(z)K_{\mathrm{p}}{\mathrm{e}}^{-\frac{T}{T_{\mathrm{1}}}}{z}^{-(n+\mathrm{1})}-U(z)K_{\mathrm{p}}{\mathrm{e}}^{-\frac{T}{T_{\mathrm{2}}}}{z}^{-(n+\mathrm{1})}.

In the form of differential equations

(6)y(k)=\left({\mathrm{e}}^{-\frac{T}{T_{\mathrm{1}}}}+{\mathrm{e}}^{-\frac{T}{T_{\mathrm{2}}}}\right)y(k-\mathrm{1})-{\mathrm{e}}^{-\left(\frac{T}{T_{\mathrm{1}}}+\frac{T}{T_{\mathrm{2}}}\right)}y(k-\mathrm{2})+K_{\mathrm{p}}{\displaystyle \frac{K_{\mathrm{p}}}{T_{\mathrm{1}}-T_{\mathrm{2}}}}\left({\mathrm{e}}^{-\frac{T}{T_{\mathrm{1}}}}-{\mathrm{e}}^{-\frac{T}{T_{\mathrm{2}}}}\right)u(k-n-\mathrm{1}).

The precondition of applying least square is the identified object, which is linear, so the identification objects abide to the superposition principles of linear systems theory. The return air temperature is affected by the supply air temperature, outdoor temperature, and natural source temperature, and these three factors can be linear superposition. Therefore, a temperature system of air-conditioning room can be viewed as a multi-input single-output (MISO) system, which has three inputs and a single output. Each input affects transfer function of output identically, so the influence of three inputs on the output can also be written in their common forms, namely,

(7)y(k)={\theta }_{\mathrm{1}}y(k-\mathrm{1})+{\theta }_{\mathrm{2}}y(k-\mathrm{2})+{\theta }_{\mathrm{3}}{u}_{\mathrm{1}}(k-{\tau }_{\mathrm{1}})+{\theta }_{\mathrm{4}}{u}_{\mathrm{2}}(k-{\tau }_{\mathrm{2}})+{\theta }_{\mathrm{5}}{u}_{\mathrm{3}}(k-{\tau }_{\mathrm{3}}).

In Eq. (7), y is the return air temperature; u₁ is the outdoor temperature; u₂ is the supply air temperature; u₃ is natural source temperature; {\theta }_i, i = 1, 2, 3, 4, 5, are the system parameters that are to be identified; and {\tau }_i, i = 1, 2, 3, are the lag time that are to be identified (in the form of the multiplier of sampling period). They are all the parameters to be estimated [1-3].

PSOLS

Particle swarm optimization (PSO) [4-6]

PSO sources

PSO is a kind of evolution technology developed by J. Kennedy and R.C. Eberhart in 1995, which came from a simplified model of social simulation. The “swarm” of the PSO accord with five basic principles of colony intelligence proposed by M.M. Millonas in developing the model of artificial life. “Particle” is a concessive choice, because all group members need to be described as no quality, no size, and their speed and acceleration state need to be described.

Basic principle

Assuming that in a D-dimensional search space, there are n particles composing a particle swarm. In addition, the position of the ith particle is X_i=(x_{i\mathrm{1}},x_{i\mathrm{2}},\cdots ,x_{iD}), i=1, 2, …, n, which is a potential solution to the optimization. By substituting it into the objective function, we can calculate the corresponding fitness value, and the pros and cons of value can be measured. The current flight speed of the ith particle in the search space is V_i=(v_{i\mathrm{1}},v_{i\mathrm{2}},\cdots ,v_{iD}); the optimal position experienced by the ith particle is P_i=(p_{i\mathrm{1}},p_{i\mathrm{2}},\cdots ,p_{iD}), namely, the optimal position of individual history, recorded as P_Best; the optimal l position experienced by all particles is P_{\mathrm{g}}=(p_{\mathrm{g}\mathrm{1}},p_{\mathrm{g}\mathrm{2}},\cdots ,p_{\mathrm{g}D}), namely optimal position of overall history, recorded as G_Best. For the particles of each generation, the evolution equation of G_Best model is as follows:

(8)v_{i+\mathrm{1}}=v_i+c_{\mathrm{1}}\times r_{\mathrm{1}}\times (p_i-x_i)+c_{\mathrm{2}}\times r_{\mathrm{2}}\times (p_{\mathrm{g}}-x_i),

(9)x_{i+\mathrm{1}}=v_i+x_{i+\mathrm{1}}.

In Eq. (8), both c₁ and c₂ are the normal numbers, namely the acceleration factors, the appropriate acceleration factor is propitious for the rapid convergence algorithm and the detachment of the local extreme value; r₁ and r₂ are two random numbers changing within [0,1], and they ensure the diversity and random search of the particle swarm. On the right side of Eq. (8), the first part is the previous speed of the particles, and it enables the particles have expanding trend in the search space so that the algorithm has global search capability; the second part is cognitive portion, showing the progress that the particles draw their own experience knowledge; the third part is the social portion, showing the progress that the particles learn experience knowledge of other particles and incarnating information sharing and collaboration community in each particle.

Least square (LS)

LS is a method of the parameter, which is the square sum to be minimal of the different values (e(k)) between the actual observed value (y) and the model calculation value. If obtaining data from the experiments, such as \{u_{\mathrm{1}}(i),u_{\mathrm{2}}(i),\cdots ,u_n(i);y(i)\},, i=1, 2, …, n, the square sum of error is

(10)J(a_{\mathrm{0}},a_{\mathrm{1}},\cdots ,a_n)={\displaystyle \sum _{k=\mathrm{1}}^N{e}^{\mathrm{2}}(k)=}{\displaystyle \sum _{k=\mathrm{1}}^N{[y(k)-a_{\mathrm{0}}-a_{\mathrm{1}}{u}_{\mathrm{1}}(k)-\cdots -a_n{u}_n(k)]}^{\mathrm{2}}}.

The key of the estimation for system parameters is to calculate the minimal value of J({\widehat{a}}_{\mathrm{0}},{\widehat{a}}_{\mathrm{1}},\cdots ,{\widehat{a}}_n), and J({\widehat{a}}_{\mathrm{0}},{\widehat{a}}_{\mathrm{1}},\cdots ,{\widehat{a}}_n) is estimated value of the parameters. Wherever \left\{{\widehat{a}}_i\right\} is the least square estimation of \{a_i\}, J is a quadratic function of a₀, a₁, …, a_n. Therefore, the method of finding extreme value with multivariate functions is feasible.

Use of PSOLS

Combine PSO with LS and optimize LS by using PSO.

Based on Eq. (8), the G_Best model of PSO is improved by using general inertia weight, and the speed evolution equation is

(11)v_{i+\mathrm{1}}=w\times v_i+c_{\mathrm{1}}\times r_{\mathrm{1}}\times (p_i-x_i)+c_{\mathrm{2}}\times r_{\mathrm{2}}\times (p_{\mathrm{g}}-x_i),

(12)w=(x_{\max }-x_{\min })(t_{\max }-t_{\min })+x_{\min }.

In Eq. (11), w is inertia weight, which enables particles to maintain movement inertia. Moreover, w can be calculated by using Eq. (12) [5,6].

When the G_Best model of PSO is a local optimal point, the algorithm will be limited to local optimal point, and premature convergence phenomenon appears. The multi-function is limited to local optimal point more easily. Therefore, the best neighborhood position is introduced, which is L_Best, and the best neighborhood position of particle i is L_i=(l_{i\mathrm{1}},l_{i\mathrm{2}},\cdots ,l_{iD}) [7-9]. The speed evolution equation of PSO L_Best model is

(13)v_{i+\mathrm{1}}=w\times v_i+c_{\mathrm{1}}\times r_{\mathrm{1}}\times (p_i-x_i)+c_{\mathrm{2}}\times r_{\mathrm{2}}\times (l_i-x_i).

L_Best model has a strong global search capability. The existence of a number of neighborhoods will hold the optimal number of possible optimal seeds, which can greatly avoid the possibility of local optimum. However, the convergence speed is slow. A good search strategy should have a stronger global search capability in the initial search stage and find the best seeds as many as possible, while in the latter part of the search stage, it should have a stronger local search capability, improving the convergence rate and accuracy. Thus, combining the model of L_Best with G_Best and combining Eqs. (8), (9), (11), and (13), the speed update equation of PSO is

(14)v_{i+\mathrm{1}}=w\times v_i+c_{\mathrm{1}}\times r_{\mathrm{1}}\times (p_i-x_i)+r_{\mathrm{2}}\times (c_{\mathrm{2}}\times (l_i-x_i)+c_{\mathrm{3}}\times (p_{\mathrm{g}}-x_i)).

In Eq. (14), c₁=2, c₂+c₃=2. With progressing of the evolution, c₂ decreases, while c₃ increases. Therefore, there are many design means, which may be a linear change or a nonlinear change. The equation of the form of linear changes is as follows:

(15)c_{\mathrm{2}}=\mathrm{2}\times {\displaystyle \frac{\max -\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{r}}{\max }},

(16)c_{\mathrm{3}}=\mathrm{2}\times {\displaystyle \frac{\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{r}}{\max }}.

In Eqs. (15) and (16), max is the largest number of iteration, and iter is the current number of iteration.

For the three-input single-output system in Eq. (7), the algorithm flow is written as follows:

Step 1 Initialize the process, speed, position, P_Best, L_Best, and G_Best of each particle in the particle swarm (assume that the number of the swarm is n; the dimension of the space is D, which is also a parameter to be estimated). Moreover, the initialization equation of speed and location are

(17)V_{ij}=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}[\mathrm{0},\mathrm{1}]\times (V_j^{\mathrm{u}}-V_j^{\mathrm{l}})V_j^{\mathrm{l}},

(18)X_{ij}=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}[\mathrm{0},\mathrm{1}]\times (X_j^{\mathrm{u}}-X_j^{\mathrm{l}})X_j^{\mathrm{l}}.

In Eqs. (17) and (18), i = 1, 2, …, n, j = 1, 2, …, D, the speed upper limit of the jth dimension is V_j^{\mathrm{u}}, the lower limit is V_j^{\mathrm{l}}, the variable upper limit of the jth dimension is X_j^{\mathrm{u}}, and the lower limit is X_j^{\mathrm{l}}.

Step 2 Set the size of the neighborhood in the swarm and divide the neighborhood according to the random serial number of the particles.

Step 3 Calculate the fitness value of each particle. In order to observe and analyze conveniently, the value will be limited within the scope of [0,1]. The fitness function is

(19)f(k)={\displaystyle \frac{\mathrm{1}}{\sqrt{{\displaystyle \sum _{i=\mathrm{1}}^N{(y(k)-y_i(k))}^{\mathrm{2}}+\mathrm{1}}}}}.

Step 4 For each particle, compare its fitness value with optimal position of individual history (P_Best). If the current fitness value is optimum, use the current fitness value replace P_Best; otherwise, P_Best remains unchanged.

Step 5 For each particle, compare its fitness value with the best position (L_Best) in the neighborhood. If the current fitness value is optimum, use the current fitness value replace L_Best; otherwise, L_Best remains unchanged.

Step 6 For each particle, compare its fitness value with the optimal position of overall history (G_Best). If the current fitness value is optimum, use the current fitness value replace G_Best; otherwise, G_Best remains unchanged.

Step 7 Update the speed of each particle according to Eqs. (14)- (16). Determine whether the speed of particles exceeded the maximum speed, such as beyond the maximum speed.

Step 8 Update the position of each particle according to Eq. (9). Return to Step 3 if it fails to reach the end of the given conditions or the largest number of iteration.

Identification result and test

In this paper, condition data of the 4th Hall Art Gallery in the winter are adopted. The supply air temperature, outdoor temperature, and natural source temperature are viewed as the input, and the return air temperature of the air-conditioning room as the output. In January 20-27, 2006, the air-conditioning data of eight days, which are sampled every 15 minutes, are viewed as the input and output. Because the business time of Museum of Fine Arts during the day is from 9:00 to 17:00 and, at night, there always has not the air-conditioned circumstances, the recorded data from 9:00 to 17:00 are viewed as the air-conditioning data.

Equation (19) is the fitness function, in this example:

(20)y_i(k)={\theta }_{\mathrm{1}}{y}_i(k-\mathrm{1})+{\theta }_{\mathrm{2}}{y}_i(k-\mathrm{2})+{\theta }_{\mathrm{3}}{u}_{i\mathrm{1}}(k-{\tau }_{\mathrm{1}})+{\theta }_{\mathrm{4}}{u}_{i\mathrm{2}}(k-{\tau }_{\mathrm{2}})+{\theta }_{\mathrm{5}}{u}_{i\mathrm{3}}(k-{\tau }_{\mathrm{3}}).

In the optimization problems of this paper, (θ₁,θ₂,θ₃,θ₄,θ₅,τ₁,τ₂,τ₃) is the position of particles. As parameters are being estimated encode the real number form, the lag coefficient must be rounded before taken into calculation. The scale of PSO is 50; the scale of the neighborhood is 5; parameters c₁=2, c₂ and c₃ are calculated by Eqs. (15) and (16); and the weight w changes linearly in iteration number from 0.9 to 0.4, namely,

(21)w=\mathrm{0.9}-{\displaystyle \frac{(\mathrm{0.9}-\mathrm{0.4})\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{r}}{\max }}.

For the model, assume that the largest iteration number is 500. The results are shown in Table 1.

Through the analysis of SPSS v 13.0, the correlation of the output data of the model and actual output data, as well as their significance are shown in Table 2.

From the above results analysis, we can see that correlation coefficient with output of model data and the data of the actual sampling is up to 0.966, and the significant level is 0.001. It is very perfect.

Conclusions

According to characteristics of global optimal model and the local optimal model of PSO, combining their advantages, the LS has been improved and optimized. The system identification method PSOLS has a high global search capability, a quick convergence, and a high precision. Online parameter identification results obtained from the air-conditioning room of a Museum of Fine Arts shows that by optimizing the model parameters with PSOLS method, the result of identification is very good and efficient. The proposed method, which has a high global search capability, a quick convergence and a high precision, is very effective in nonlinear model for the online identification and parameter estimation.

References

[1] Wang J. The application of system identification about mathematical modeling in air-conditioning room. Dissertation for the Master’s Degree. Beijing: Beijing University of Civil Engineering and Architecture, 2007, 18-23 (in Chinese)

[2] Wang X F, Lu G Z. System Modeling and Identification. Beijing: Publishing House of Electronics Industry, 2004, 97-121 (in Chinese)

[3] Wang J M, Li X M. Modeling and estimating of the characteristic parameters for the air conditioning room of the VAV system. Computer Simulation, 2002, 19(4): 69-72 (in Chinese)

[4] Zeng J C, Jie Q, Cui Z H. Particle Swarm Optimization. Beijing: Science Press, 2004, 7-17 (in Chinese)

[5] Shi Y, Eberhart R. A modified particle swarm optimizer. In: Proceedings of the IEEE International Conference on Evolutionary Computation, Piscataway. 1998, 69-73

[6] Wang Z Q, Sun X, Zhang D X. A PSO-based multicast routing algorithm. In: Proceedings of the Third International Conference on Natural Computation, ICNC 2007. 2007, 4: 664-667

[7] Sun H, Zhang Z M, Ge H J. Application of PSO to improve multiple linear regression. Computer Engineering and Applications, 2007, 43(3): 43-45 (in Chinese)

[8] Wu L H, Wang Y N, Zeng Z F, Yuan X F. Parameter estimation of nonlinear systems model based on hybrid particles swarm optimization algorithm. Journal of System Simulation, 2006, 18(7): 1942-1945 (in Chinese)

[9] Wang X, Han C Z, Wang B W. Two new effective bidiagonalization least squares algorithms for nonlinear system identification. Acta Automatica Sinica, 1998, 24(1): 95-111 (in Chinese)

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg