Analysis of unstable behavior of planing craft speed using the qualitative theory of dynamical systems

Abstract

This paper deals with the inherent instability observed in the speed of a planing type craft. In the case of displacement craft, the systems governing the speed are stable hence closed-loop control is trivial. In the case of planing craft, however, there may exist instability in their speed. By using the Qualitative Theory of Dynamical Systems (QTDS), this paper shows that there may exist a set of speeds in which planing craft are not able to achieve adequate stability. This instability problem cannot be acceptable in many applications (such as that examined in this paper, an Unmanned Surface Vehicle, USV, of planing craft type). The observed instability is explained by means of the appearance of bifurcations which bring new attractors to the state space, such as equilibrium points or limit cycles. This paper proposes a novel solution to manage the vessel instability behavior. This is done by way of increasing the droop characteristic in the propulsion thrust with respect to speed. By increasing the droop, the system becomes more robust. The key advantage of this approach is that it is achieved by way of modifying the propulsion controller rather than by changing the hydrodynamic profile of the vessel, the mass distribution or by adding extra control surface (i.e., flaps). Resulting in a more cost-effective control system. Furthermore, due to this method acting on propulsion and its control, it is compatible with the other methods aforementioned. Stability analysis is undertaken. This analysis is very general, because it considers a wide range of controller and propulsion systems. Open-loop control and analysis into different types of propulsion is also presented. The effect of each propulsion type on stability is explained. In addition, the effect in the control loop of the electro-mechanical actuators inaccuracy (dead-zone) has also been analyzed. The paper explains that this inaccuracy, though small, can make the speed oscillate in planing craft. A practical implementation of this analysis is validated by way of sea trials with a real planing craft.

Appendix

Proposition 1

Consider the system $\dot{x} = g\left( {x,\lambda } \right)$ where $x \in {\mathbb{R}}$ represents the state variables $,\lambda \in {\mathbb{R}}$ the external parameters and $g$ is a C ¹⁻ smooth function. Given any $\lambda_{i}$ , the system is stable, in the set $x \in \varOmega$ ,if the following conditions are satisfied:

• (a) There is only one equilibrium point for $\lambda_{i}$ . That means that there is only on value $x_{e} \in \varOmega$ which meets the next equation: $0 = g\left( {x_{e} ,\lambda_{i} } \right)$

• (b)
  $$ \left. {\frac{{{\text{d}}g}}{{{\text{d}}x}}} \right|_{{x_{e} }} < 0 $$

Proof

Since there is only one equilibrium point and regarding $g\left( {x,\lambda } \right)$ is a C¹⁻smooth function, the sign of $g$ only changes in $x_{e}$ in the set $x \in {{\varOmega }}$. Taking into account the condition b, it easy to verify that:

$$ \begin{aligned} & g\left( {x_{S} ,\lambda_{i} } \right)\left\langle {0\;\forall x_{S} } \right\rangle x_{e} \\ & g\left( {x_{I} ,\lambda_{i} } \right) > 0\;\forall x_{I} < x_{e} \\ \end{aligned} $$

Therefore, the system is stable. If the conditions are fulfilled in the set $x \in {\mathbb{R}}$, the system is globally stable.

Proposition 2

Consider the system $\dot{x} = g\left( {x,\lambda } \right)$ where $x \in {\mathbb{R}}$ represents the state variable $,\lambda \in {\mathbb{R}}$ the external parameters and $g$ is a C¹⁻smooth function. Given any $\lambda_{i}$, the system is stable in the set $x \in \varOmega$ if the following conditions are satisfied:

• (a) There are two points $x_{I} \in {{\varOmega }}$ and $x_{S} \in {{\varOmega }}$ such that $g\left( {x_{I} ,\lambda_{i} } \right) > 0$ and $g\left( {x_{S} ,\lambda_{i} } \right) < 0$

• (b)
  $$ \frac{{{\text{d}}g}}{{{\text{d}}x}} < 0 \forall x \in \varOmega $$

Proof

Firstly, it will be proven that there is only one equilibrium point. Applying the Mean Value Theorem, it should exist at point $x_{e} \subset \left( {x_{I} ,x_{S} } \right)$, which makes the function $g$ zero, hence $x_{e}$ is an equilibrium point. The next step is to demonstrate that $x_{e}$ is unique. Regarding the Rolle’s Theorem, the required condition to guarantee the existence of another equilibrium point $x_{j}$ is that there exist a point $x_{c} \in \left( {x_{e} , x_{j} } \right) \subset {{\varOmega }}$ such that:

$$ \left. {\frac{{{\text{d}}g\left( {x,\lambda_{i} } \right)}}{{{\text{d}}x}}} \right|_{{x_{c} }} = 0 $$

But this contradicts the condition b. Once it has been demonstrated that the system has only one equilibrium point and taking into account condition b, the system satisfied the requirements of Proposition 1. Therefore, the system is stable. If the conditions are fulfilled in the set $x \in {\mathbb{R}}$, the system is globally stable.