Motivation

Problem

  • UAV is underactuated mechanical system, which complicate the control design stage even more.
  • UAV has a high nonliear and time-varying behavior.
  • It is constantly affected by aerodynamic disturbances.
  • Usually models subject to unmodelled dynamics and parametric uncertainties.

Solution

  • An advanced control strategy is required to achieve good performance in autonomous flight.
    • With high maneuverability and robustness w.r.t. external disturbances.
  • Nonlinear modelling techniques and modern nonlinear control theory are usually employed to achieve autonomous flight with high performance.
  • A good choice to reject these disturbances is the nonlinear H\(_\infty\) control theory.
    • Need to solve two Hamilton-Jacobi-Bellman-Isaacs partial differential equations (HJBI PDEs).
      • Replace the RIccati equations in the linear H\(_\infty\) control formulation.
      • But, the main problem is the absence of a general method to solve these equations.
    • Strongly model-dependent.

Contributions

  • Present an integral predictive and nonlinear robust control strategy to solve the path following problem for a quadrotor.
  • Hierarchical control structure:
    • A Model Predictive Controller (MPC) to track the reference trajectory.
      • Consider the integral of the position error.
    • A nonlinear H\(_\infty\) controller to stabilize the rotational movements.
      • Consider the integral of the error.
      • Cope with unknown disturbances.
  • Achieve a null steady-state error when sustained disturbances on the system.
  • Parametric and structural uncertainties are presented to corroborate the effectivemess and robustness of the control strategy.

Method

System Modelling

  • Body frame: \(B=\{B_1,B_2,B_3\}\).
  • World frame: \(E = \{E_x,E_y,E_z\}\).
    • Position \(p = [x,y,z]\).
  • Rotation matrix: \(R\).
  • Eulers: \(\eta\).
    • Roll: \(\phi\in(-\pi/2 < \phi < \pi/2)\).
    • Pitch: \(\theta\in(-\pi/2 < \theta < \pi/2)\).
    • Yaw: \(\psi\in(-\pi,\pi)\).

Controller

Goal: Control the quadrotor in presence of sustained external disturbances, parametric uncertainties and unmodelled dynamics.

Control strategy:

  • The reference trajectory is provided off-line by the Trajectory Generator block.
    • Translational movements: \([x_r,y_r,z_r]\).
    • Yaw angle is defined separately.
    • Reference control inputs: \([U_{1_r}, u_{x_r}, u_{y_r}]\).
    • There are no external disturbances.
  • Outer Loop: translational movements.
    • State-space predictive controller based on the error model (E-SSPC).
    • Include the integral of the position error.
      • Achieve null steady-state error.
    • Height \(z\) control: total thrust \(U_1\).
    • Pitch and roll control.
  • Inner Loop: rotational subsystem.
    • Nonlinear H\(_\infty\) controller.
    • Control the angle and angle velocities.
    • Using torques \(\tau\).
    • Include the integral of the angle error.
      • Achieve null steady-state error.

Nonlinear H\(_\infty\) Controller for attitude

Define the dynamic equation of a nonlinear system with unknown disturbance as:

\[\dot{x} = f(x,t) + g(x,t)u + k(x,t)d,\]

where

  • \(u\) is the vector of control inputs.
  • \(d\) is the vector of external diaturbances.
  • \(x\) is the vector of states.

Define the performance using the cost variable \(\zeta\):

\[\zeta = W\begin{bmatrix} h(x) \\ u \end{bmatrix},\]

where

  • \(h(x)\) represents a function of the vector of states to be controlled.
  • \(W\) is a weight matrix.

The optimal H\(_\infty\) problem can be posed as follows:

  • Find the smallest value \(\gamma^*\ge 0\), such that for any \(\gamma\ge \gamma^*\), there exists a state feedback \(u=u(x,t)\), such that the \(L_2\) gain from \(d\) to \(\zeta\) is less than or equal to \(\gamma\):
\[\int_0^T \lVert\zeta\rVert_2^2 dt \le \gamma^2 \int_0^T \lVert d \rVert_2^2 dt,\]

where \(\lVert\zeta\rVert_2^2 = \zeta'\zeta = \begin{bmatrix} h'(x) & u' \end{bmatrix}W'W\begin{bmatrix} h(x) \\ u \end{bmatrix}\).

\[W'W = \begin{bmatrix} Q & S \\ s' & R \end{bmatrix},\]

where

  • \(Q,R\) are symmetric positive definite, and \(W'W > O\).
  • \(Q - SR^{-1}S' > O\).

The optimal control signal \(u*(x,t)\) may be computed if there is a smooth solution \(V(x,t)\) to HJBI equation:

\[\frac{\partial V}{\partial t} + \frac{\partial'V}{\partial x}f(x,t) + \frac{1}{2} \frac{\partial'V}{\partial x} \left[ \frac{1}{\gamma^2}k(x,t)k'(x,t) - g(x,t)R^{-1}g'(x,t) \right]\frac{\partial V}{\partial x} - \frac{\partial'V}{\partial x}g(x,t)R^{-1}S'h(x) + \frac{1}{2}h'(x)(Q-SR^{-1}S')h(x) = 0.\]

Then, the optimal state feedback control law is derived as:

\[u^* = - R^{-1}\left( S'h(x) + g'(x,t) \frac{\partial V(x,t)}{\partial x} \right).\]

For quadrotor system, define the tracking error vector as:

\[x_\eta = \begin{bmatrix} \dot{\eta} - \dot{\eta}_r \\ \eta - \eta_r \\ \int(\eta - \eta_r)dt \end{bmatrix}.\]

E-SSPC for position

  • Two predictive controllers:
    • Control the height through the input \(U_1\).
    • Control \(x\) and \(y\) motions.

Define the state space as:

\[\dot{p}(t) = f(p(t),u_p(t)).\]

References

  • G. V. Raffo, M. G. Ortega, and F. R. Rubio, “An integral predictive/nonlinear H\(_\infty\) control structure for a quadrotor helicopter,” Automatica, vol. 46, no. 1, pp. 29–39, Jan. 2010.