Urban Comprehensive Water Consumption: Nonlinear Control of Production Factor Input Based upon the C-D Function

Abstract

Utilizing the urban water demand function and the Cobb-Douglas (C-D) production function, an economic control model for the multi-input-multi-output (MIMO) nonlinear system was designed and implemented to describe urban comprehensive water consumption, where the urban water demand function was expressed as the product of the number of water users and per capita comprehensive water consumption, and the urban water supply function was expressed as a C-D production function. The control variables included capital investment and labor input for the urban water supply. In contrast to the Solow model, Shell model and aggregate model with renewable labor resources, the proposed model eliminated value constraints on investment and labor input in the state equations and hence avoided the difficulty in applying these models to urban water supply institutions. Furthermore, the feedback linearization control design (FLCD) method was employed to accomplish stability of the system. In contrast to the optimal control method, the FLCD method possesses an explicit solution of the control law and does not require the solution of a two-point boundary value problem of an ordinary differential equation, making the method more convenient for application. Moreover, two different scenarios of urban water consumption, one for the growth period and the other for the decline period, were simulated to demonstrate the effectiveness of the proposed control scheme.

Urban Industrial Water Supply and Demand: System Dynamic Model and Simulation Based on Cobb–Douglas Function

“…The impact of the resource attributes of water on the difficulty of obtaining water resources was mainly reflected in the economic control of the input and output efficiency of production factors. Therefore, from the perspective of water resources economics and economic management, the urban water supply can be expressed by the C-D production function [19,21]. where S(t) is the total industrial water supply at time t, K(t) is the capital stock, L(t) is the labor stock, A is the technical level coefficient, A > 0, α and β are the elasticity coefficients of the capital output and the elasticity coefficient of the labor output, 0 > α, β > 1.…”

“…First, many of the existing urban water supply and demand models [5,6,[18][19][20][21][22]25,26,32] are built around one city's overall water system, and there are few analyses on the supply and demand of water resources for individual industries.…”

“…Finally, in comparison with the models [18][19][20][21] that also incorporate the C-D production function in calculating the total water supply, this paper adds the method of system dynamics, which is capable of constructing an overall system modeling the urban industrial water. This enhanced model can be implemented to more intuitively and clearly analyze the water supply and water demand as well as their influencing factors and the adopted optimization process becomes more convenient to implement.…”

“…For the industrial water supply subsystem, we collect the data from Jiangsu Water Conservancy Yearbook and refer to the relevant literature [19,21]. For the system parameters, we set as follows: industry capital formation rate µ = 0.85, industry capital depreciation rate δ = 0.04, industrial effective labor formation rate γ = 0.98, industry labor force abolition rate σ = 0.02, capital stock of urban water For the values of the two elastic coefficients involved in the C-D production function, the initial parameter values for α and β were set as α = 0.7, β = 0.3, the same as that in Reference [19], and were brought into our system dynamic model.…”

“…Zhang et al [20] established a fuzzy credibility-constrained interval two-stage stochastic programming (FCITSP) model based on a C-D function, predicting three major industrial water distribution problems based on water demand. Li et al [21] designed the economic control model of multi-input and multi-output (MIMO) nonlinear systems to describe city's comprehensive water consumption, by using urban water demand function and C-D production function. Xu et al [22] used the C-D production function and agricultural output value, labor, agricultural inputs, capital investment, agricultural water, and irrigation water as efficiency measures to calculate the economic benefits of agricultural water resources in Hainan.…”