The nonlinear control problem of quadrotor UAVs which perform cooperative transportation of payloads is treated with the use of nonlinear optimal and multi-loop flatness-based control methods. The load is suspended with a link from a cart which is turn in connected through cables with two quadrotors. The aim is to compute the flight path and the control inputs of the quadrotors that will allow to lift the load and move it to any desirable final position. First, the dynamic model of the cable-suspended load is obtained through Euler-Lagrange analysis. Despite underactuation the associated nonlinear optimal control problem is solved, thus allowing to compute the lift forces of the cables that enable the load to move on the vertical plane until it reaches the targeted position. These forces are also applied with opposite sign to the quadrotors’ side through joints at the other end of the cables. Thus, the dynamic model of the quadrotors is updated by including in it additional drag forces which are due to the tension of the cables. The flight paths for the two quadrotors that enables to bring the suspended load to its final position are also computed. Next, for each quadrotor the nonlinear control and path following problem is solved, taking into account the cable-induced drag forces effects. To this end, a flatness-based control approach which is implemented in successive loops is applied to each quadrotor. The state-space model of each quadrotor UAV is separated into subsystems, which are connected between them in cascading loops. Each one of these subsystems can be viewed independently as a differentially flat system and control about it can be performed with inversion of its dynamics as in the case of input-output linearized flat systems. The state variables of the second subsystem become virtual control inputs for the first subsystem. In turn, exogenous control inputs are applied to the second subsystem. The whole control method is implemented in two successive loops and its global stability properties are also proven through Lyapunov stability analysis. The whole procedure is repeated at each sampling instance, that is (i) solution of the nonlinear optimal control problem for the transportation of the payload (ii) computation of the drag forces which are exerted on the UAVs due to lifting the load, (iii) solution of the multi-loop flatness-based control problem for the individual UAVs. This control method allows each quadrotor to follow precisely the defined flight path and finally achieves to bring the load to the targeted position.

The article proposes a novel solution to the control problem of centrifugal gas compressors which are driven by three-phase induction motors (IMs) and three-phase permanent magnet synchronous motors (PMSMs), through a novel flatness-based control scheme which is implemented in successive loops. By re-arranging state variables and by splitting suitably the state vector of the IM-driven gas-compressor and of the PMSM-driven gas compressor into subsystems one arrives at writing the associated state-space models in the triangular (strict feedback) form. For the latter state-space description it is possible to solve the control and stabilization problem using chained control loops. The state-space model of the IM-driven gas-compressor and of the PMSM-driven gas-compressor is decomposed into cascading subsystems which satisfy differential flatness properties. For these subsystems virtual control inputs are computed, capable of inverting their dynamics and of eliminating the associated tracking error. The control inputs which are actually applied to the complete nonlinear form of the IM-actuated gas-compressor and of the PMSM-actuated gas compressor are computed from the last subsystem of the chained state-space description. These control inputs incorporate in a recursive manner all virtual control inputs which were computed from the individual subsystems included in the initial state-space equation. The control inputs that should be applied to the nonlinear system so as to assure that all state vector elements will converge to the desirable setpoints are obtained at each iteration of the control algorithm by tracing backwards the subsystems of the chained state-space model. The method is of proven global stability and ensures fast and accurate tracking of the reference setpoints by the state variables of the IM-driven gas compressor and of the PMSM-driven gas compressor.