Nonlinear optimal and flatness-based control for debt–labor dynamics and an economy’s growth cycles

The paper proposes a nonlinear optimal control method for treating the control and stabilization problem of Keen’s macroeconomic model which describes the debt–labor interaction and the associated economic growth cycles. The primary state-space model of the debt–labor dynamics is considered to have as state variables the wages share, the employment share and the debt-share, while its control inputs are the rate of growth of the labor productivity, the rate of growth of the labor force and the debt’s interest rate. It is proven that the dynamic model of the debt–labor interaction is differentially flat and a nonlinear optimal control method is developed for it. To apply this control method, approximate linearization is performed with the use of Taylor-series expansion while an algebraic Riccati equation has to be solved at each sampling instance. The proposed control method avoids complicated changes of state variables and state-space model transformations while the control inputs it computes are used directly on the initial nonlinear model of the controlled system. It achieves optimality because of enabling convergence of the state variables of the debt–labor dynamics to the targeted setpoints under minimal variations of the control inputs. The paper’s nonlinear optimal control method is compared to flatness-based control implemented in successive loops.