Finite-time robust H _∞ control for high-speed underwater vehicles subject to parametric uncertainties and disturbances

Abstract

Traditional underwater vehicles are limited in speed due to dramatic friction drag on the hull. Supercavitating vehicles exploit supercavitation as a means to reduce drag and increase their underwater speed. Compared with fully wetted vehicles, the nonlinearities in modeling of cavitator, fin, and in particular, nonlinear planing force make the control design of supercavitating vehicles more challenging. By combining cascaded design and backstepping approach, this paper reformulates the supercavitating vehicle model into a novel cascade model with two error tracking subsystems. Based on linear matrix inequalities (LMIs) and by introducing the sector conditions of the nonlinear characteristics of planing force, a new finite-time robust $H_{\infty }$ control scheme is proposed for the error tracking subsystems which ensures that the closed-loop system is finite-time bounded and the effect of the disturbance input on the controlled output is reduced to a prescribed level. A sufficient condition is presented for the solvability of the design problem, which is further reduced to a feasibility problem of a set of LMIs. Simulations have been conducted for both initial and tracking responses to evaluate the performance and robustness of the proposed $H_{\infty }$ controller for all admissible uncertainties and the disturbance inputs.

Appendices

Appendix A

Consider the following time-invariant linear systems:

$${\dot{\mathbf{x}}}_{{\mathbf{1}}} (t) = {\mathbf{A}}_{{{\mathbf{11}}}} {\mathbf{x}}_{{\mathbf{1}}} (t) + {\mathbf{A}}_{{{\mathbf{12}}}} {\mathbf{x}}_{{\mathbf{2}}} (t)$$

(83)

$${\dot{\mathbf{x}}}_{{\mathbf{2}}} (t) = {\mathbf{A}}_{{{\mathbf{21}}}} {\mathbf{x}}_{{\mathbf{1}}} (t) + {\mathbf{A}}_{{{\mathbf{22}}}} {\mathbf{x}}_{{\mathbf{2}}} (t) + {\mathbf{Gw}}(t)$$

(84)

where ${\mathbf{x}}_{{\mathbf{1}}} (t) \in {\mathbb{R}}^{{n_{1} }}$ and ${\mathbf{x}}_{{\mathbf{2}}} (t) \in {\mathbb{R}}^{{n_{2} }}$ constitute the state variables, ${\mathbf{w}}(t) \in {\mathbb{R}}^{l}$ is the exogenous disturbance, and ${\mathbf{A}}_{{{\mathbf{11}}}} \in {\mathbb{R}}^{{n_{1} \times n_{1} }}$, ${\mathbf{A}}_{{{\mathbf{12}}}} \in {\mathbb{R}}^{{n_{1} \times n_{2} }}$, ${\mathbf{A}}_{{{\mathbf{21}}}} \in {\mathbb{R}}^{{n_{2} \times n_{1} }}$, ${\mathbf{A}}_{{{\mathbf{22}}}} \in {\mathbb{R}}^{{n_{2} \times n_{2} }}$, ${\mathbf{G}} \in {\mathbb{R}}^{{n_{2} \times l}}$.

Definition 1

(finite-time stability): The linear system

$${\dot{\mathbf{x}}}(t) = {\mathbf{Ax}}(t),\,{\mathbf{x}}(0) = {\mathbf{x}}_{{\mathbf{0}}}$$

(85)

is said to be finite-time stable with respect to $(c_{1} ,c_{2} ,T,R,d)$ with $c_{2} > c_{1} > 0$, $R > 0$, if ${\mathbf{x}}_{{\mathbf{0}}}^{{\mathbf{T}}} {\mathbf{Rx}}_{{\mathbf{0}}} \le c_{1}^{2}$ implies

$${\mathbf{x}}^{T} (t){\mathbf{Rx}}(t) < c_{2}^{2} ,\quad\forall t \in [0,T]$$

(86)

Definition 2

[finite-time boundedness (FTB)]: The linear systems (1) and (2) subject to an exogenous disturbance, $w(t)$, are said to be FTB with respect to $(c_{1} ,c_{2} ,T,{\mathbf{R}}_{{\mathbf{1}}} ,{\mathbf{R}}_{{\mathbf{2}}} ,d)$ with $c_{2} > c_{1} > 0$, $R_{1} > 0$, $R_{2} > 0$, if $\forall t \in [0,T]$,

$${\mathbf{x}}_{{\mathbf{1}}}^{{\mathbf{T}}} (0){\mathbf{R}}_{{\mathbf{1}}} {\mathbf{x}}_{{\mathbf{1}}} (0) + {\mathbf{x}}_{{\mathbf{2}}}^{{\mathbf{T}}} (0){\mathbf{R}}_{{\mathbf{2}}} {\mathbf{x}}_{{\mathbf{2}}} (0) < c_{1}^{2} \Rightarrow {\mathbf{x}}_{{\mathbf{1}}}^{{\mathbf{T}}} {\mathbf{R}}_{{\mathbf{1}}} {\mathbf{x}}_{1} + {\mathbf{x}}_{{\mathbf{2}}}^{{\mathbf{T}}} {\mathbf{R}}_{{\mathbf{2}}} {\mathbf{x}}_{{\mathbf{2}}} < c_{2}^{2}$$

(87)

Appendix B

Our approach relies on the assumption that the delay in the equations of motion can be eliminated by applying a suitable feedback. Then, the controllability analysis and the controller design can be performed for linear time invariant (LTI) systems. In addition, it is assumed that all states can be measured.

The SV equations can be written in compact form as:

$${\dot{\mathbf{x}}} = {\mathbf{f}}({\mathbf{x}}) + {\mathbf{g}}({\mathbf{x}}){\mathbf{u}}$$

(88)

$${\mathbf{y}} = {\mathbf{h}}({\mathbf{x}})$$

(89)

where ${\mathbf{x}} = [x_{1} \;\;x_{2} \;\;x_{3} \;\;x_{ 4} ]^{\rm T} = [z\;\;w\;\;\theta \;\;q]^{\rm T} \in {\mathbb{R}}^{4}$ is the state variable, ${\mathbf{u}} = [\delta_{\text{c}} \;\;\delta_{\text{f}} ]^{\rm T} \in {\mathbb{R}}^{2}$ is the control input, and ${\mathbf{y}} = [x_{1} \;\;x_{3} ]^{\rm T}$ is the output to be controlled by the control input ${\mathbf{u}}$.

From (18) and (19) in previous section, considering the nonlinear planing force, we have:

$${\mathbf{f}}({\mathbf{x}}) = \left[ {\begin{array}{*{20}c} {f_{1} } \\ {f_{2} } \\ {f_{3} } \\ {f_{4} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {x_{2} - Vx_{3} } \\ {\frac{{C_{\text{cav}} - nC_{\text{cav}} }}{mV}x_{2} + \frac{{C_{\text{cav}} - nC_{\text{cav}} }}{m}x_{ 3} + \frac{{ - C_{\text{cav}} L_{\text{c}} + nC_{\text{cav}} L_{\text{f}} }}{mV}x_{4} + g + \frac{{F_{\text{p}} }}{m}} \\ {x_{ 4} } \\ {\frac{{ - C_{\text{cav}} L_{\text{c}} + nC_{\text{cav}} L_{\text{f}} }}{{I_{\text{yy}} V}}x_{2} + \frac{{ - C_{\text{cav}} L_{\text{c}} + nC_{\text{cav}} L_{\text{f}} }}{{I_{\text{yy}} }}x_{ 3} + \frac{{C_{\text{cav}} L_{\text{c}}^{2} - nC_{\text{cav}} L_{\text{f}}^{2} }}{{I_{\text{yy}} V}}x_{4} - \frac{{F_{\text{p}} L_{\text{f}} }}{{I_{\text{yy}} }}} \\ \end{array} } \right]$$

(90)

$${\mathbf{g}}({\mathbf{x}}) = \left[ {\begin{array}{*{20}c} 0 & 0 \\ {\frac{{C_{\text{cav}} }}{m}} & { - \;\frac{{nC_{\text{cav}} }}{m}} \\ 0 & 0 \\ { - \frac{{C_{\text{cav}} L_{\text{c}} }}{{I_{\text{yy}} }}} & {\frac{{nC_{\text{cav}} L_{\text{f}} }}{{I_{\text{yy}} }}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{\mathbf{g}}_{1} } & {{\mathbf{g}}_{2} } \\ \end{array} } \right].$$

(91)

By choosing ${\mathbf{y}} = [x_{1} \;\;x_{3} ]^{\rm T}$, the Lie derivative of $h_{i}$ with respect to ${\mathbf{g}}_{i}$ is given by:

$$L_{{{\mathbf{g}}_{i} }} h_{i} ({\mathbf{x}}) = 0,\,i = 1,2$$

(92)

Using $L_{{\mathbf{f}}} h_{1} ({\mathbf{x}}) = x_{2} - Vx_{3}$ and $L_{\text{f}} h_{2} ({\mathbf{x}}) = x_{4}$, we obtain:

$$\begin{aligned} L_{{{\text{g}}_{1} }} L_{\text{f}} h_{1} ({\mathbf{x}}) &= C_{\text{cav}} /m \ne 0 \hfill \\ L_{{{\text{g}}_{ 1} }} L_{\text{f}} h_{2} ({\mathbf{x}}) &= - C_{\text{cav}} L_{\text{c}} /I_{\text{yy}} \ne 0 \hfill \\ L_{{{\text{g}}_{ 2} }} L_{\text{f}} h_{1} ({\mathbf{x}}) &= - nC_{\text{cav}} /m \ne 0 \hfill \\ L_{{{\text{g}}_{2} }} L_{\text{f}} h_{2} ({\mathbf{x}}) &= nC_{\text{cav}} L_{\text{f}} /I_{\text{yy}} \ne 0. \hfill \\ \end{aligned}$$

(93)

Obviously, systems (88) and (89) have a relative degree $\rho = 2$. Then, we should find the choosed nonlinear mapping whose Jacobian matrix ${\mathbf{E}}(x)$ is locally nonsingular.

Hence

$$rank\left( {{\mathbf{E}}({\mathbf{x}})} \right) = rank\left( {\left[ {\begin{array}{*{20}c} {L_{{{\mathbf{g}}_{1} }} L_{{\mathbf{f}}} h_{1} ({\mathbf{x}})} & {L_{{{\mathbf{g}}_{2} }} L_{{\mathbf{f}}} h_{1} ({\mathbf{x}})} \\ {L_{{{\mathbf{g}}_{1} }} L_{{\mathbf{f}}} h_{2} ({\mathbf{x}})} & {L_{{{\mathbf{g}}_{2} }} L_{{\mathbf{f}}} h_{2} ({\mathbf{x}})} \\ \end{array} } \right]} \right) = 2$$

(94)

New state variables for Eq. 88 are selected for analysis and control design:

$$z_{1} = h_{1} ({\mathbf{x}}) = x_{1}$$

(95)

$$z_{2} = L_{\text{f}} h_{1} ({\mathbf{x}}) = x_{2} - Vx_{3}$$

(96)

$$z_{3} = h_{2} ({\mathbf{x}}) = x_{3}$$

(97)

$$z_{4} = L_{\text{f}} h_{2} ({\mathbf{x}}) = x_{ 4}$$

(98)

Differentiating Eqs. 96 and 98 with respect to time along Eqs. 88 and 89 gives:

$$L_{\text{f}}^{2} h_{1} ({\mathbf{x}}) = f_{2} - Vf_{3} ,\,L_{\text{f}}^{2} h_{2} ({\mathbf{x}}) = f_{4} .$$

(99)

The foregoing equation shows clearly that the system is input–output linearizable, since the state feedback control

$${\mathbf{u}} = \left[ {\begin{array}{*{20}c} {u_{1} } \\ {u_{2} } \\ \end{array} } \right] = - {\mathbf{E}}^{ - 1} ({\mathbf{x}})\left[ {\begin{array}{*{20}c} {L_{\text{f}}^{2} h_{1} ({\mathbf{x}})} \\ {L_{\text{f}}^{2} h_{2} ({\mathbf{x}})} \\ \end{array} } \right] + {\mathbf{E}}^{ - 1} ({\mathbf{x}}){\mathbf{v}}$$

(100)

reduces the input–output map to

$$\varvec {\ddot{\textit{y}}} = \left[ \begin{aligned} {\ddot{\textit{z}}}_{1} \hfill \\ {\ddot{\textit{z}}}_{3} \hfill \\ \end{aligned} \right] = \left[ {\begin{array}{*{20}c} {L_{\text{f}}^{2} h_{1} ({\mathbf{x}})} \\ {L_{\text{f}}^{2} h_{2} ({\mathbf{x}})} \\ \end{array} } \right] + {\mathbf{Eu}} = {\mathbf{v}}$$

(101)

where ${\mathbf{v}} = \left[ {v_{1} \;\;v_{2} } \right]^{\rm T}$ is a fictitious input vector.

Based on the linear model of the SV, define commands of $z$ and $\theta$ as $z_{\text{d}}$ and $\theta_{\text{d}}$, respectively, the command tracking errors can be expressed as follows:

$$e_{\text{z}} = z - z_{\text{d}} ,\quad {\mkern 1mu} e_{\theta } = \theta - \theta_{\text{d}}$$

(102)

Considering the derivative of command tracking errors, then rewrite the errors into a vector form:

$${\mathbf{X}} = [e_{\text{z}} {\mkern 1mu} {\mkern 1mu} \dot{e}_{\text{z}} ]^{T} ,{\mkern 1mu} {\mathbf{Y}} = [e_{\theta } {\mkern 1mu} {\mkern 1mu} \dot{e}_{\theta } ]^{T}$$

(103)

The error equations of the SV can be written as:

$$\dot{\mathbf{{X}}} = {\mathbf{A}}_{{\mathbf{1}}} {\mathbf{X}} + {\mathbf{B}}_{{\mathbf{1}}} w_{1}$$

(104)

$$\dot{\mathbf{{Y}}} = {\mathbf{A}}_{{\mathbf{2}}} {\mathbf{Y}} + {\mathbf{B}}_{{\mathbf{2}}} w_{2}$$

(105)

where

${\mathbf{A}}_{{\mathbf{1}}} = {\mathbf{A}}_{{\mathbf{2}}} = \left[ {\begin{array}{*{20}c} 0 & 1 \\ 0 & 0 \\ \end{array} } \right]$, ${\mathbf{B}}_{{\mathbf{1}}} = {\mathbf{B}}_{{\mathbf{2}}} = \left[ \begin{aligned} 0 \hfill \\ 1 \hfill \\ \end{aligned} \right]$, $w_{1} = {\ddot{\textit{z}}} - {\ddot{\textit{z}}}_{\text{d}}$, and $w_{2} = \ddot{\theta } - \ddot{\theta }_{\text{d}}$.

As the feedback linearized system is completely controllable, it is apt to use the linear quadratic regulator technique to design a state feedback controller. The LQR solution formulae use solutions of two algebraic Riccati equations (AREs):

$${\mathbf{A}}_{{\mathbf{1}}}^{{\mathbf{T}}} {\mathbf{P}}_{{\mathbf{1}}} + {\mathbf{P}}_{{\mathbf{1}}} {\mathbf{A}}_{{\mathbf{1}}} - {\mathbf{P}}_{{\mathbf{1}}} {\mathbf{B}}_{{\mathbf{1}}} {\mathbf{R}}_{{\mathbf{1}}}^{{ - {\mathbf{1}}}} {\mathbf{B}}_{{\mathbf{1}}}^{{\mathbf{T}}} {\mathbf{P}}_{{\mathbf{1}}} + {\mathbf{Q}}_{{\mathbf{1}}} = 0$$

(106)

$${\mathbf{A}}_{{\mathbf{2}}}^{{\mathbf{T}}} {\mathbf{P}}_{{\mathbf{2}}} {\mathbf{ + P}}_{{\mathbf{2}}} {\mathbf{A}}_{{\mathbf{2}}} - {\mathbf{P}}_{{\mathbf{2}}} {\mathbf{B}}_{{\mathbf{2}}} {\mathbf{R}}_{{\mathbf{2}}}^{{ - {\mathbf{1}}}} {\mathbf{B}}_{{\mathbf{2}}}^{{\mathbf{T}}} {\mathbf{P}}_{{\mathbf{2}}} {\mathbf{ + Q}}_{{\mathbf{2}}} = 0$$

(107)

where the weighting functions ${\mathbf{Q}}_{{\mathbf{1}}} {\mathbf{ = Q}}_{{\mathbf{1}}}^{{\mathbf{T}}}$, ${\mathbf{Q}}_{{\mathbf{2}}} {\mathbf{ = Q}}_{{\mathbf{2}}}^{{\mathbf{T}}}$, ${\mathbf{R}}_{{\mathbf{1}}} {\mathbf{ = R}}_{{\mathbf{1}}}^{{\mathbf{T}}}$, and ${\mathbf{R}}_{{\mathbf{2}}} {\mathbf{ = R}}_{{\mathbf{2}}}^{{\mathbf{T}}}$ are positive matrix.

Then, two optimal state regulators are:

$$w_{1} = - {\mathbf{R}}_{{\mathbf{1}}}^{{ - {\mathbf{1}}}} {\mathbf{B}}_{{\mathbf{1}}}^{{\mathbf{T}}} {\mathbf{P}}_{{\mathbf{1}}} {\mathbf{X}}$$

(108)

$$w_{2} = - {\mathbf{R}}_{{\mathbf{2}}}^{{ - {\mathbf{1}}}} {\mathbf{B}}_{{\mathbf{2}}}^{{\mathbf{T}}} {\mathbf{P}}_{{\mathbf{2}}} {\mathbf{Y}}$$

(109)

The fictitious input vector is denoted as:

$${\mathbf{v}} = \left[ {\begin{array}{*{20}l} {\nu_{1} } \hfill \\ {\nu_{2} } \hfill \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {\ddot{{z}}_{\text{d}} } \hfill \\ {\ddot{\theta }_{\text{d}} } \hfill \\ \end{array} } \right] + \left[ {\begin{array}{*{20}l} { - {\mathbf{R}}_{{\mathbf{1}}}^{{ - {\mathbf{1}}}} {\mathbf{B}}_{{\mathbf{1}}}^{\text{T}} {\mathbf{P}}_{{\mathbf{1}}} {\mathbf{X}}} \hfill \\ { - {\mathbf{R}}_{{\mathbf{2}}}^{{ - {\mathbf{1}}}} {\mathbf{B}}_{{\mathbf{2}}}^{\text{T}} {\mathbf{P}}_{{\mathbf{2}}} {\mathbf{Y}}} \hfill \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {\ddot{\it z}} \hfill \\ {\ddot{\theta }} \hfill \\ \end{array} } \right].$$

(110)

The LQR optimal controller of the supercavitating vehicle is given by:

$$\left[ \begin{aligned} \delta_{\text{c}} \hfill \\ \delta_{\text{f}} \hfill \\ \end{aligned} \right] = - {\mathbf{E}}^{ - 1} ({\mathbf{x}})\left[ {\begin{array}{*{20}c} {L_{\text{f}}^{2} h_{1} ({\mathbf{x}})} \\ {L_{\text{f}}^{2} h_{2} ({\mathbf{x}})} \\ \end{array} } \right] + {\mathbf{E}}^{ - 1} ({\mathbf{x}})\left( {\left[ \begin{aligned} - {\mathbf{R}}_{{\mathbf{1}}}^{{ - {\mathbf{1}}}} {\mathbf{B}}_{{\mathbf{1}}}^{{\mathbf{T}}} {\mathbf{P}}_{{\mathbf{1}}} {\mathbf{X}} \hfill \\ - {\mathbf{R}}_{{\mathbf{2}}}^{{ - {\mathbf{1}}}} {\mathbf{B}}_{{\mathbf{2}}}^{{\mathbf{T}}} {\mathbf{P}}_{{\mathbf{2}}} {\mathbf{Y}} \hfill \\ \end{aligned} \right] + \left[ \begin{aligned} {\ddot{\it z}}_{\text{d}} \hfill \\ \ddot{\theta }_{\text{d}} \hfill \\ \end{aligned} \right]} \right).$$

(111)