7. Nonlinear Tracking Controller Design for Unmanned Helicopters

The override of the f ^B components in the \(\vec {i}_{{B}}\) and \(\vec{j}_{{B}}\) directions of the body-fixed frame achieves the decoupling of the helicopter external force and moment model. The work reported in [47] indicates that if the complete description of the force vector given in (7.8) is used, then the state space dynamics of the nonlinear helicopter model cannot be input–output linearizable and the zero-dynamics of the system will be unstable. If the system dynamics are not input–output linearizable most of the standard control methodologies will be inapplicable. If the proposed approximation takes place, the helicopter nonlinear model becomes full state linearizable by considering the position and the yaw as outputs. To the authors knowledge, there is not any controller design in the literature that is based on the exact model and in all case studies this approximation is performed. The use of the approximated model also took place in Chap. 6 indicating that for the helicopter control problem only practical stability can be achieved based on the approximated model.

The function f(t,s) is Lipschitz in s∈ℝ ⁿ if it is piecewise continuous in t and satisfies the Lipschitz condition: for every s ₁,s ₂∈ℝ ⁿ and a positive constant ^f L (called Lipschitz constant). If the function f(t,s) is Lipschitz in s, then the system \(\dot{s}=f(t,s)\) with s(t ₀)=s _o has a unique solution for every t [43].

Note that ρ _3,3=ρ _1,1 ρ _2,2−ρ _1,2 ρ _2,1.

Based on [43] a continuous function \(\alpha_{\mathcal{K}}(s):[0\ \infty )\rightarrow[0\ \infty)\) belongs to the class \(\mathcal {K}\) if it is strictly increasing and \(\alpha_{\mathcal{K}}(0)=0\). A continuous function \(\beta_{\mathcal{KL}}(s_{1},s_{2}):[0\ \infty)\times [0\ \infty)\rightarrow[0\ \infty)\) belongs to the class \(\mathcal{KL}\) if, for each fixed s ₂, the mapping \(\beta _{\mathcal{KL}}(s_{1},s_{2})\) belongs to the class \(\mathcal{K}\) with respect to s ₁ and for each fixed s ₁, the mapping \(\beta_{\mathcal{KL}}(s_{1},s_{2})\) is decreasing with respect to s ₂ and \(\beta_{\mathcal{KL}}(s_{1},s_{2})\rightarrow0\) as s ₂→∞.