Observational evidence that a feedback control system with proportional-integral-derivative characteristics is operating on atmospheric surface temperature at global scale

Figures & data

Table 1. Terms from climate science matched to control system terms.

Climate science term (IPCC, 2013)	Control systems term (Astrom and Murray, 2008)		As operationalised	Variable observed	Variable transformed	Comment on transformation
Temperature	Process variable			Observed (global surface) temperature (Temp)	Z_Temp
	Setpoint			Initial temperature of series	Z_Temp_setpoint	Setpoint is set up to equal zero
Driver	Disturbance			Level of atmospheric CO2 ( CO2)	Z_ CO2
	Error	Error = setpoint - disturbance			Z_Temp_setpoint minus Z_ CO2	As setpoint is defined as zero, Z_Temp_ setpoint minus Z_ CO2 reduces simply to minus Z_ CO2
	Controller output	Reverse error			Z_ CO2
	Proportional Controller output	Controller output			Z_ CO2
	Integral Controller output	Integral Controller output	Cumulative sum of Controller output		I_Z_ CO2
	Derivative Controller output	Derivative Controller output	First difference of Controller output		d_Z_ CO2

Fig. 1. Monthly data, Z-score base period November1958-December 1976 (period to left of vertical black line). Putative control system model for global surface temperature, selected elements: Disturbance (level of atmospheric CO₂) (black curve); control system setpoint for temperature (purple line); observed temperature (red curve).

Fig. 2. Monthly data, Z-score base period November 1958 - December 1976 (period to left of vertical black line). Putative control system model for global surface temperature, full set of elements: Disturbance (level of atmospheric CO₂) (black curve); control system setpoint for temperature (purple line); observed temperature (red curve); level of control system error (P_error) (green curve); integral (cumulative sum) of control system error (I_error) (orange curve); derivative (first difference) of control system curve (D_error) (brown curve).

Table 2. Results of ADF tests for stationarity, allowing for both drift and trend.

	Disturbance series	P_error series	I_error series	D_error series	Temp series
Order of polynomial trend determined for use in constructing the test	Same as P-error series	4^th-order	3^rd-order	Linear	Linear
Test indicated for use by order of polynomial trend of series		Schmidt-Phillips test (τ test)	Schmidt-Phillips test (τ test)	Augmented Dickey-Fuller test	Augmented Dickey-Fuller test
Value of test-statistic		−7.6313	−9.1785	−5.716	−0.17473
Critical value of test statistic for a significance level of 0.01		−4.85	−4.5	−3.9713	−7.03966
Conclusion: series is –		Stationary	Stationary	Stationary	Stationary

Table 3. Putative PID control system: ridge regression results.

	coefficient	std. error	z	p-value
Disturbance_Z5818	0.2630	0.0202	13.03	7.87E-39
P_Error_reZ5818	−0.2630	0.0202	−13.03	7.87E-39
I_Error_reZ5818	−0.2174	0.0369	−5.897	3.69E-09
L1m_13mma_D_Error_reZ5818	−0.2377	0.0218	−10.9	1.15E-27
const	0	NA	NA	NA

Penalty = 0.57.

Resid SD = 0.44516.

Table 4. Eviews ARDL estimation output for period November 1958 to February 2018 for temperature as a function of putative control system P_error, I_error and D_error: short-run model dynamic relationship.

Model dependent variable	Number of models evaluated	Selected model	No autocorrelation out to 36 lags	Adjusted R-squared	F-statistic	p-value
H46_Z5876	2	ARDL (2,0,0.0)	Nil	0.896911	1234.712	< 10-100

Table 5. Eviews ARDL estimation output for period November 1958 to February 2018 for temperature as a function of putative control system P_error, I_error and D_error: short-run model independent variables.

Model independent variables	Coefficient	Std. Error	t-Statistic	Prob.*
H46_Z5818(-1)	0.523	0.0368	14.222	1.57E-40
H46_Z5818(-2)	0.226	0.0367	6.162	1.21E-09
P_ERROR_REZ5818	−0.139	0.0334	−4.158	3.60E-05
I_ERROR_REZ5818	−0.046	0.0277	−1.675	0.0944
L1M_13MMA_D_ERROR_REZ581	−0.062	0.0173	−3.596	0.0003
C	0.002	0.0121	0.174	0.8621

Fig. 3. Fit with observed temperature of global surface temperature predicted from dynamic regression models involving various combinations of control system error terms. The PID model shows the best fit (highest Coefficient of Determination (R-squared) and lowest Akaike Information Criterion).

Table 6. Fit with observed temperature of global surface temperature predicted from dynamic regression models involving various combinations of control system error terms.

Error terms in model	Selected model	No autocorrelation out to:	Adjusted R-squared	Akaike information criterion
I	(2,0)	36 lags	0.8921	0.6188
D	(4,0)	37 lags	0.8930	0.6152
P	(2,0)	36 lags	0.8951	0.5900
ID	(2,0,0)	24 lags	0.8945	0.5972
PI	(2,0,0)	36 lags	0.8952	0.5912
PD	(2,0,0)	36 lags	0.8966	0.5769
PID	(2,0,0,0)	36 lags	0.8969	0.5758

Fig. 4. Monthly data, Z-score base period November 1958 - December 1976 (period to left of vertical black line). Predicted temperature from dynamic regression model of a control system for global surface temperature using P, I and D error terms (green curve); Disturbance to temperature (level of atmospheric CO₂) (black curve); observed temperature (red curve).

IPCC. 2013. Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press, Cambridge.

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