##### Model Description

Feedback control systems in general exhibit inherent robustness-fragility tradeoffs. That is, by becoming very robust to a given set of disturbances for maintaining stability, feedback systems necessarily introduce hidden fragilities to disturbances outside this set. Even a small unanticipated disturbance can initiate cascading system-wide failures as a result. The model presented here demonstrates this phenomenon.

In Anderies et al. (2012), an agricultural production system is illustrated as an example of such feedback control systems. Here, farmers have an objective of continuously producing a certain level of yield (*r*) every year. They subtract their current yield (*y _{out})* from their target level (

*r*), which shows a gap (

*u*). Farmers refer to this gap (

*u*) to adjust the size of their cultivated area for the next crop season. When the output of cultivation in the next cycle is available, it is again compared with their target level (

*r*) to further adjust the size of their cultivated area. This cycle of "compare and adjust" is repeated indefinitely until the farmers reach their objective (zero gap).

From the perspective of institutions, it is clear that such feedback control mechanisms (i.e., sensing the yield and adjusting the cultivated area) are driven by the rules and norms adopted by the farmers. Institutions are thus critical for the robustness of such systems. However, if systems develop institutions that are too optimized to certain kinds of disturbances, they can become fragile to unfamiliar disturbances.

**Reference**

Anderies, John M, Elinor Ostrom, C. Folke, and B. Walker. 2012. “Aligning Key Concepts for Global Change Policy?: Robustness , Resilience , and Sustainability.” CSID Working Paper Series.

##### Scenarios

First, we illustrate the case of optimizing to a particular set of disturbance. View the auxilary variable 'yout' in the y-axis and time on the x-axis. Set the periodic cycle of all disturbances to 60 (*p _{1}*=60,

*p*=60) with amplitude 2 (

_{2}*d*). The default system is optimized to attenuate disturbances of this frequency. Set the gain of the controller (

_{max}*g*) to 10 (note: larger gain values more strongly attenuate the magnitude of disturbances). Gain values represent the degree of optimization. Target/reference harvest level is set to 5 by default. You will see that the yield (

*y*) is fairly stable/robust despite the presence of disturbance, maintaining the yield in the range of [4.5, 5.5].

_{out}Now, we insert a novel disturbance. Change the periodic cycles of the disturbances as the following: *p _{1}*=60,

*p*=20 with the same amplitude (

_{2}*d*=2). You will now see that the yields vary wildly between t=260~290. This is the time range when the disturbance of new frequency (

_{max}*p*=20) is being applied. This demonstrates that the default system is robust to the disturbances of frequency 1/60 but fragile to those with different frequencies.

_{2}$\Large y_{out}=d+a$ |

Net agricultural yield is the sum of disturbance and output from a cultivated land. |

$\Large u_{out}=r-x$ |

Controller signal (societal response to the yield) represents adjustments to the size of a cultivated area. |

$\Large \frac{dx}{dt}=k_{1}y-k_{2}x$ |

Rate of change in controller signal in response to the net yield. |

$\Large \frac{da}{dt}=gu$ |

Rate of change in output from a cultivated area. |

#Simple feadback model modified from Doyle. par k1=0.01,k2=0.1,g=0.1,dmax=2,rmax=0.5,omega=1 par switch=1,p1=60,p2=20,hfson=260,hfsoff=290 #functions f(x,a,b)=if(x<a)then(0)else(if(x<b)then(1)else(0)) #Equations and hidden variables------------------------ r=rmax d=dmax*v*(1-f(t,hfson,hfsoff)) + dmax*v1*f(t,hfson,hfsoff) y=d + a u=r - x #differential equations==------------------ # oscilators - shocks duo/dt=uo*(1 - uo^2 - v^2) - (2*Pi/p1)*v dv/dt=v*(1 - uo^2 - v^2) + (2*Pi/p1)*uo duo1/dt=uo1*(1 - uo1^2 - v1^2) - (2*Pi/p2)*v1 dv1/dt=v1*(1 - uo1^2 - v1^2) + (2*Pi/p2)*uo1 init uo=-1,v=0,uo1=-1,v1=0 #feedback system dx/dt=k1*y -k2*x da/dt=g*u aux yout=y aux uout=u aux rout=k2*r/k1 aux dist=d init x=0.5, a=5 @ yp=x, total=400, xhi=400, yhi=10, maxstor=100000 #@ yp=x, total=400, xhi=400, yhi=10, maxstor=100000 #done

Yu JHD, Arizona State University.

Bozicevic M, Arizona State University.