CN105204507A - Hovercraft course made good robust control method containing input time lag - Google Patents

Abstract

The invention discloses a hovercraft course made good robust control method containing an input time lag. The method comprises the following steps: 1, obtaining state vectors of the motion of a hovercraft through a sensor system of the hovercraft, wherein the state vectors comprise the longitudinal velocity u, the side velocity v and the prow rotation angular velocity r; 2, according to data collected by the sensor system, obtaining the course made good phi of current motion of the hovercraft. 3, according to reaction time after actuator entry input by a controller, introducing an input time lag link, and establishing a state-space model containing the input time lag; 4, transmitting the course made good to a Robust controller, obtaining a control instruction, transmitting the control instruction to an actuator, and achieving control over the course made good. According to the hovercraft course made good robust control method, the input time lag link is introduced, so that the hovercraft model is more accurate, system analysis and synthesis are facilitated, the designed controller is greater in practicability and robustness, lower in conservative property and easy to achieve, and the genetic algorithm ensures the optimum of design of the Robust controller.

Description

A kind of track laying air cushion vehicle comprising input delay to robust control method

Technical field

The invention belongs to track laying air cushion vehicle to control field, particularly relate to a kind of track laying air cushion vehicle comprising input delay to robust control method.

Background technology

Air cushion vehicle is a kind of novel high performance ship, has important effect in the industry.The type boats and ships, due to special structure and mode of motion, thus have a great difference with conventional ship.For conventional ship, its side direction water resistance is comparatively large, and be difficult to produce the drift of obvious side, thus bow is little to the difference between angle and course made good, so just can be replaced the control of direction of motion by Heading control.But for air cushion vehicle, because its headway is fast, side drift easily occurs when being subject to the interference of inertia effects or wind, and its bow has bigger difference usually to angle and speed of a ship or plane direction, and the control therefore for course made good simply can not be summed up as flight tracking control.

2011 Master's thesis " research of air cushion vehicle course made good expert system for control " use expert system for track laying air cushion vehicle to controlling, but the real-time of the method is poor, do not consider the delay of control inputs, for hovermarine practicality and the robustness deficiency all to some extent of high-speed cruising.

Some parameters of hovermarine are larger by the impact of surrounding environment, and this just causes its modeling can there is certain uncertainty, and thus, the course made good for hovermarine carries out robust control and has important practical significance.

Summary of the invention

The object of this invention is to provide a kind of have high stability and robustness, the track laying air cushion vehicle comprising input delay to robust control method.

The track laying air cushion vehicle comprising input delay to a robust control method, comprise the following steps,

Step one: the state vector being obtained hovermarine motion by the sensing system of hovermarine, comprises longitudinal velocity u, side velocity v and heading rate r;

Step 2: the data gathered according to sensing system, obtain the course made good of hovermarine current kinetic

Step 3: enter the reaction time after actuator according to controller input, introduce input delay link, set up the state-space model comprising input delay:

$\left\{\begin{array}{c}\stackrel{\·}{x}\left(t\right)=Ax\left(t\right)+B_1\mathrm{\τ}(t-d(t\left)\right)+B_{2}b\left(t\right)\\ y\left(t\right)=Cx\left(t\right)\end{array}\right.$

Wherein m=M ^t>0 is inertial matrix, and D>0 is linear damping matrix, and τ is controlled quentity controlled variable, and b is environmental interference, and d (t) is Time-varying time-delays, meets:

$\left\{\begin{array}{c}0<d\left(t\right)<d\\ \stackrel{\&CenterDot;}{d}\left(t\right)<\mathrm{\&mu;}\end{array}\right.$

Wherein d is the upper bound of Time-varying time-delays, and μ is time lag change restriction;

Step 4: send course made good to robust controller, obtains steering order and sends actuator to, realizes course made good and controls.

The track laying air cushion vehicle that the present invention is a kind of comprises input delay to robust control method, can also comprise:

1, robust controller is:

τ(t)＝Kx(t)

According to Lyapunov-Krasovskii stability theory, H-infinity control theory and hovermarine to the actual needs controlled, structure Lyapunov-Krasovskii function, is met the adequate condition of the Delay-Dependent of state feedback controller:

Wherein, P>0, Q ₁>0, Q ₂>0, Z>0, N ^t=(N ₁ ^tn ₂ ^tn ₃ ^tn ₄ ^t0) L ^t=(L ₁ ^tl ₂ ^tl ₃ ^tl ₄ ^t0), N _i(i=1,2,3,4), L _i(i=1,2,3,4) are matrix, T>0, and the T in the matrix upper right corner represents this transpose of a matrix, σ=γ ²b _max+ V (0), b _maxfor the maximal value of environmental interference energy, y _maxfor maximum drift angle, V (0) is the value of Lyapunov-Krasovskii function V (t) when t=0;

Utilize genetic algorithm to obtain the optimum solution of controller gain K, optimization aim is:

$\begin{array}{cc}\underset{K_i}{min}& \mathrm{\&gamma;}\end{array}$

Optimal conditions is the adequate condition of the Delay-Dependent of controller.

Beneficial effect:

At track laying air cushion vehicle in control, when the control signal of controller enters in the process of actuator, the phenomenon that inevitable life period postpones, for hovermarine at a high speed, especially responsive for such Time Delay, the control performance that the course made good that ignoring this Time Delay can affect hovermarine controls, even causes the instability of control system, thus, when carrying out modeling analysis to hovermarine, introducing time lag item, in theory and reality, all there is very important function and significance.

Genetic algorithm is a kind of effective optimization intelligent algorithm, can ensure the globally optimal solution of the problem of being optimized, use it in the solution procedure of LMI, can guarantee the optimality of obtained controller gain.

The present invention introduces input delay link and makes the model of hovermarine more accurate, be conducive to the analysis and synthesis of system, design controller have more practicality and robustness, conservative property lower and be easy to realize, genetic algorithm then ensure that the optimum of robust Controller Design.

Accompanying drawing explanation

Accompanying drawing 1 course made good and bow to angle, the calculated relationship of drift angle;

Accompanying drawing 2 course made good control system block diagram;

Accompanying drawing 3 genetic algorithm process flow diagram.

Embodiment

Below in conjunction with accompanying drawing, the present invention is described in further details.

A kind of track laying air cushion vehicle comprising input delay is to robust control method, comprise the following steps: the motion state being obtained boats and ships by hovermarine sensing system, the course made good of current ship motion is obtained through course made good counter, according to the motion model of hovermarine, establish the state-space model comprising input delay, construct suitable Lyapunov-Krasovskii function, obtain the robust Controller Design condition of LMI form, use genetic algorithm for solving LMI problem, obtain controller gain.And then by actuator for track laying air cushion vehicle to regulating.

Consider controller input enter actuator after reaction time, introduce input delay link, according to the account form of course made good, set and control to export, the four-degree-of-freedom motion model of hovermarine is converted into the state-space model of the hovermarine comprising input delay, meanwhile, according to drift angle, course made good and the bow relation to angle, set specific output quantity, obtain model and the course made good counter of system:

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{x}\left(t\right)=Ax\left(t\right)+B_1\mathrm{\&tau;}(t-d(t\left)\right)+B_{2}b\left(t\right)\\ y\left(t\right)=Cx\left(t\right)\end{array}\right.$

Wherein integration item can be decided to be the constant of certain limit by the bow set by hovermarine to angle gauge.

Construct suitable Lyapunov-Krasovskii function, use free-form curve and surface method to obtain the robust Controller Design condition of LMI form, use genetic algorithm optimization computing controller design conditions, obtain optimum controller gain.

1. pair hovermarine establishes the state-space model comprising input delay

2. design course made good counter, obtain course made good angle by the online data of sensing system

3. construct Lyapunov-Krasovskii function, obtain the robust Controller Design condition of LMI form

4. use genetic algorithm for solving LMI problem, obtain controller gain

The invention discloses a kind of track laying air cushion vehicle comprising input delay to robust control method, solve track laying air cushion vehicle to kinds of robust control problems.Consider that the steering order of controller enters the account form of the reaction time after actuator and course made good, set up the state-space model comprising the hovermarine of input delay, and devise the course made good counter being easy to realize.In order to ensure that controller meets certain robust performance under certain interference, construct suitable Lyapunov-Krasovskii function, introduce free-form curve and surface, obtain the lower and Controller gain variations condition of the LMI form that parameter is less of conservative property, utilize genetic algorithm to obtain optimum controller gain, and then control the course made good of hovermarine by actuator.The present invention introduces input delay link and makes the model of hovermarine more accurate, be conducive to the analysis and synthesis of system, design controller have more practicality and robustness, conservative property lower and be easy to realize, genetic algorithm then ensure that the optimum of robust Controller Design.

1. obtained the state vector of hovermarine motion by the sensing system of hovermarine, mainly comprise the longitudinal velocity u of hovermarine, side velocity v and heading rate r etc.

2. course made good counter design:

By reference to the accompanying drawings 1, obtain the computing formula of course made good:

Wherein: course made good

ψ-bow is to angle

β-drift angle

Whole control system obtains the longitudinal velocity of hovermarine, side velocity and bow to angle by sensing system, can obtain according to physical significance: $\mathrm{\&beta;}=\arctan \frac{v}{u},\stackrel{\&CenterDot;}{\mathrm{\&psi;}}=r$

Therefore course made good counter can be obtained by the variable in hovermarine four-degree-of-freedom model:

Wherein integration item can be decided to be the constant of certain limit by the bow set by hovermarine to angle gauge, can reduce the time that course made good calculates like this.With reference to the accompanying drawings 2, after obtaining the current course made good angle of hovermarine, compare with the course made good preset, thus obtain the modified value of course made good.

3. structure comprises the track laying air cushion vehicle of Time Delay to the state-space model controlled

According to the motion model of hovermarine, selection mode variable is x=(uvr) ^t, set up the state-space model of hovermarine:

$\stackrel{\&CenterDot;}{x}\left(t\right)=Ax\left(t\right)+B_1\mathrm{\&tau;}\left(t\right)+B_{2}b\left(t\right)---\left(3\right)$

Wherein m=M ^t>0 is boats and ships inertial matrix, and D>0 is linear damping matrix, and τ is controlled quentity controlled variable, and b is environmental interference, and environmental interference bounded, concrete form is:

$\left|\right|b|{|}_2=\sqrt{{\&Integral;}_0^{\mathrm{\&infin;}}b{\left(t\right)}^{T}b\left(t\right)dt}<\mathrm{\&infin;}$

Because control objectives reduces drift angle, the computing formula according to drift angle can obtain, and needs the difference of longitudinal velocity and the side velocity exported as far as possible little, therefore, selects output quantity:

y(t)＝Cx(t)(4)

Wherein C=(-110)

In sum, the state-space model of hovermarine is obtained:

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{x}\left(t\right)=Ax\left(t\right)+B_1\mathrm{\&tau;}+B_{2}b\left(t\right)\\ y\left(t\right)=Cx\left(t\right)\end{array}\right.---\left(5\right)$

In order to press close to engineering reality, we consider the input delay of controller to actuator, the linear model of hovermarine can change into:

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{x}\left(t\right)=Ax\left(t\right)+B_1\mathrm{\&tau;}(t-d(t\left)\right)+B_{2}b\left(t\right)\\ y\left(t\right)=Cx\left(t\right)\end{array}\right.---\left(6\right)$

Wherein d (t) is Time-varying time-delays, meets:

$\left\{\begin{array}{c}0<d\left(t\right)<d\\ \stackrel{\&CenterDot;}{d}\left(t\right)<\mathrm{\&mu;}\end{array}\right.---\left(7\right)$

Wherein d is the upper bound of Time-varying time-delays, and μ is time lag change restriction.

4. robust Controller Design condition

This Controller gain variations problem is expressed as a standard H-infinity problem: design a state feedback controller

τ(t)＝Kx(t)(8)

This control makes: hovermarine state-space model closed-loop system (9) Asymptotic Stability, and under zero initial condition, meets || T _yb|| _∞< γ, wherein T _ybrepresent that environment interferes with the closed loop transfer function, controlling to export, and have 0< γ < ∞.

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{x}\left(t\right)=Ax\left(t\right)+B_{1}Kx(t-d(t\left)\right)+B_{2}b\left(t\right)\\ y\left(t\right)=Cx\left(t\right)\end{array}\right.---\left(9\right)$

The actual needs that we control according to Lyapunov-Krasovskii stability theory, H-infinity control theory and hovermarine course made good, construct suitable Lyapunov-Krasovskii function, introduce the conservative property that free-form curve and surface reduces result, be met the adequate condition of the Delay-Dependent of the state feedback controller design that above-mentioned H-infinity typical problem requires:

Wherein, P>0, Q ₁>0, Q ₂>0, Z>0, N ^t=(N ₁ ^tn ₂ ^tn ₃ ^tn ₄ ^t0)

L ^t=(L ₁ ^tl ₂ ^tl ₃ ^tl ₄ ^t0), matrix N _i(i=1 _,2 _,3 _,4), L _i(i=1 _,2 _,3 _,4) be the matrix of any appropriate dimension, T>0, the T in the matrix upper right corner represents this transpose of a matrix, σ=γ ²b _max+ V (0), b _maxfor the maximal value of environmental interference energy, y _maxfor maximum drift angle, V (0) is the value of designed Lyapunov-Krasovskii function V (t) when t=0, and

$V\left(t\right)=x^T\left(t\right)Px\left(t\right)+{\&Integral;}_{t-d}^t{x}^T\left(s\right)Q_{1}x\left(s\right)ds+{\&Integral;}_{t-d\left(t\right)}^t{x}^T\left(s\right)Q_{2}x\left(s\right)ds+{\&Integral;}_{-d}^0{\&Integral;}_{t+\mathrm{\&theta;}}^t{\stackrel{\&CenterDot;}{x}}^T\left(s\right)Z\stackrel{\&CenterDot;}{x}\left(s\right)dsd\mathrm{\&theta;}$

$\mathrm{\&Pi;}=\left(\begin{array}{ccccc}{\mathrm{\&Pi;}}_{11}& {\mathrm{\&Pi;}}_{12}& {\mathrm{\&Pi;}}_{13}& {\mathrm{\&Pi;}}_{14}& {\mathrm{\&Pi;}}_{15}\\ *& {\mathrm{\&Pi;}}_{22}& {\mathrm{\&Pi;}}_{23}& {\mathrm{\&Pi;}}_{24}& 0\\ *& *& {\mathrm{\&Pi;}}_{33}& {\mathrm{\&Pi;}}_{34}& 0\\ *& *& *& {\mathrm{\&Pi;}}_{44}& {\mathrm{\&Pi;}}_{45}\\ *& *& *& *& 0\end{array}\right)$

${\mathrm{\&Pi;}}_{11}=TA+A^{T}T+N_1+N_1^T+Q_1+Q_2,{\mathrm{\&Pi;}}_{12}=L_1-N_1+N_2^T+{\mathrm{TB}}_{1}K$

${\mathrm{\&Pi;}}_{13}=N_3^T-Q_1-L_1,{\mathrm{\&Pi;}}_{14}=P-T+N_4^T+A^{T}T,{\mathrm{\&Pi;}}_{15}={\mathrm{TB}}_2$

${\mathrm{\&Pi;}}_{22}=(\mathrm{\&mu;}-1)Q_2-N_1+L_2-N_1^T+L_2^T,{\mathrm{\&Pi;}}_{23}=L_3^T-L_2-N_3^T$

${\mathrm{\&Pi;}}_{24}=L_4^T-N_4^T+K^T{B}_1^{T}T,{\mathrm{\&Pi;}}_{33}=-Q_1-L_3-L_3^T,{\mathrm{\&Pi;}}_{34}=-L_4^T$

Π ₄₄=dZ-2T, Π ₄₅=TB ₂, " * " represents that this element is the transposition of symmetry element

$\widetilde{\mathrm{\&Pi;}}=\left(\begin{array}{cccc}{\mathrm{\&Pi;}}_{11}& {\mathrm{\&Pi;}}_{12}& {\mathrm{\&Pi;}}_{13}& {\mathrm{\&Pi;}}_{14}\\ *& {\mathrm{\&Pi;}}_{22}& {\mathrm{\&Pi;}}_{23}& {\mathrm{\&Pi;}}_{24}\\ *& *& {\mathrm{\&Pi;}}_{33}& {\mathrm{\&Pi;}}_{34}\\ *& *& *& {\mathrm{\&Pi;}}_{44}\end{array}\right)$

$\widehat{\mathrm{\&Pi;}}=\left(\begin{array}{ccccc}{\mathrm{\&Pi;}}_{11}+C^{T}C& {\mathrm{\&Pi;}}_{12}& {\mathrm{\&Pi;}}_{13}& {\mathrm{\&Pi;}}_{14}& {\mathrm{\&Pi;}}_{15}\\ *& {\mathrm{\&Pi;}}_{22}& {\mathrm{\&Pi;}}_{23}& {\mathrm{\&Pi;}}_{24}& 0\\ *& *& {\mathrm{\&Pi;}}_{33}& {\mathrm{\&Pi;}}_{34}& 0\\ *& *& *& {\mathrm{\&Pi;}}_{44}& {\mathrm{\&Pi;}}_{45}\\ *& *& *& *& -{\mathrm{\&gamma;}}^2\end{array}\right)$

5. realize Controller gain variations by genetic algorithm

From above-mentioned design conditions, in the condition of a given maximum drift angle, the Controller gain variations condition that one meets control objectives can be obtained, for given hovermarine and mission requirements, we can obtain the parameter of specific Controller gain variations condition, have good applicability and wide range of application.At this, we utilize genetic algorithm to be optimized calculating to these LMIs, can ensure the optimum solution obtaining controller gain, as shown in Figure 3.

Optimization aim:

Constraint condition: formula (10), (11), (12), (13)

Obtain controller gain: K

6. with reference to the accompanying drawings 2, realize the robust control for the course made good of hovermarine by actuator.

Introduce detailed process of the present invention in detail below:

1. set up four-degree-of-freedom hovermarine motion model:

$\left\{\begin{array}{c}m(\stackrel{\&CenterDot;}{u}-vr)=F_x\\ m(\stackrel{\&CenterDot;}{v}+ur)=F_y\\ I_x\stackrel{\&CenterDot;}{p}=M_x\\ I_z\stackrel{\&CenterDot;}{r}=M_z\end{array}\right.$

Wherein, I _z-hovermarine is around the moment of inertia of Z axis

I _x-hovermarine is around the moment of inertia of X-axis

The quality of m-hovermarine

The longitudinal velocity of u-hovermarine

The side velocity of v-hovermarine

The heading rate of r-hovermarine

The angular velocity in roll of p-hovermarine

F _xthe power of X-direction suffered by-hovermarine

F _ythe power of Y direction suffered by-hovermarine

M _xthe suffered moment around X-axis of-hovermarine

M _zthe suffered moment around Z axis of-hovermarine

2. course made good counter design:

Obtain the computing formula of course made good:

Wherein: course made good

ψ-bow is to angle

β-drift angle

Can obtain according to physical significance: $\mathrm{\&beta;}=arctan\frac{v}{u},\stackrel{\&CenterDot;}{\mathrm{\&psi;}}=r$

Therefore course made good counter can be obtained by the variable in hovermarine four-degree-of-freedom model:

Wherein integration item can be decided to be the constant of certain limit by the bow set by hovermarine to angle gauge.

3. be converted into the state-space model comprising input delay:

Set condition variable: x=(uvr) ^t, η=(ξ ζ ψ) ^t

Then have:

$\left\{\begin{array}{c}M\stackrel{\&CenterDot;}{x}+Dx=\mathrm{\&tau;}+b\\ \stackrel{\&CenterDot;}{\mathrm{\&eta;}}=S\left(\mathrm{\&psi;}\right)x\end{array}\right.$

Wherein ξ is the northern position of hovermarine at east northeast coordinate system, and ζ is the eastern position of hovermarine at east northeast coordinate system, and ψ is that the bow of hovermarine is to, M=M ^t>0 is boats and ships inertial matrix, and D>0 is linear damping matrix, and τ is controlled quentity controlled variable, and S (ψ) is transition matrix, and b is environmental interference, and environmental interference bounded, concrete form is:

$\left|\right|b|{|}_2=\sqrt{{\&Integral;}_0^{\mathrm{\&infin;}}b{\left(t\right)}^{T}b\left(t\right)dt}<\mathrm{\&infin;}$

$S\left(\mathrm{\&psi;}\right)=\left(\begin{array}{ccc}cos\mathrm{\&psi;}& -sin\mathrm{\&psi;}& 0\\ sin\mathrm{\&psi;}& \cos \mathrm{\&psi;}& 0\\ 0& 0& 1\end{array}\right)$

Because control objectives reduces drift angle, the computing formula according to drift angle can obtain, and needs the difference of longitudinal velocity and the side velocity exported as far as possible little, therefore, selects output quantity:

y(t)＝Cx(t)

Wherein C=(-110)

In sum, the state-space model of hovermarine is obtained:

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{x}\left(t\right)=Ax\left(t\right)+B_1\mathrm{\&tau;}\left(t\right)+B_{2}b\left(t\right)\\ y\left(t\right)=Cx\left(t\right)\end{array}\right.$

Wherein $A=-\frac{D}{M},B_1=B_2=-\frac{1}{M},$

When we consider controller to the input delay of actuator, the linear model of hovermarine can change into:

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{x}\left(t\right)=Ax\left(t\right)+B_1\mathrm{\&tau;}(t-d(t\left)\right)+B_{2}b\left(t\right)\\ y\left(t\right)=Cx\left(t\right)\end{array}\right.$

Wherein d (t) is Time-varying time-delays, meets:

$\left\{\begin{array}{c}0<d\left(t\right)<d\\ \stackrel{\&CenterDot;}{d}\left(t\right)<\mathrm{\&mu;}\end{array}\right.$

Wherein d is the upper bound of Time-varying time-delays, and μ is time lag change restriction.

This control problem is expressed as a standard H-infinity problem: design a state feedback controller

τ(t)＝Kx(t)

This control makes: hovermarine state-space model closed-loop system Asymptotic Stability, and under zero initial condition, meets || T _yb|| _∞< γ, wherein T _ybrepresent that environment interferes with the closed loop transfer function, controlling to export, and have 0< γ < ∞.

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{x}\left(t\right)=Ax\left(t\right)+B_{1}Kx(t-d(t\left)\right)+B_{2}b\left(t\right)\\ y\left(t\right)=Cx\left(t\right)\end{array}\right.---\left(14\right)$

4. construct Lyapunov-Krasovskii function:

In order to excavate the information of time lag as far as possible, construct Lyapunov-Krasovskii function:

V(t)＝V ₁(t)+V ₂(t)+V ₃(t)(15)

Wherein:

V ₁(t)＝x ^T(t)Px(t)

$V_2\left(t\right)={\&Integral;}_{t-d}^t{x}^T\left(s\right)Q_{1}x\left(s\right)ds+{\&Integral;}_{t-d\left(t\right)}^t{x}^T\left(s\right)Q_{2}x\left(s\right)ds$

$V_3\left(t\right)={\&Integral;}_{-d}^0{\&Integral;}_{t+\mathrm{\&theta;}}^t{\stackrel{\&CenterDot;}{x}}^T\left(s\right)Z\stackrel{\&CenterDot;}{x}\left(s\right)dsd\mathrm{\&theta;}$

P > 0, Q is met in formula ₁> 0, Q ₂> 0, Z > 0,

5. the stability criterion of Controller gain variations is obtained by Lyapunov stability theory:

By designed Lyapunov-Krasovskii function along (14) differential:

$\begin{array}{c}\stackrel{\&CenterDot;}{V}\left(t\right)={\stackrel{\&CenterDot;}{x}}^T\left(t\right)Px\left(t\right)+x^T\left(t\right)P\stackrel{\&CenterDot;}{x}\left(t\right)+x^T{Q}_{1}x\left(t\right)+x^T\left(t\right)Q_{2}x\left(t\right)\\ -x^T\left(t-d\right)Q_{1}x\left(t-d\right)-\left(1-\stackrel{\&CenterDot;}{d}\left(t\right)\right)x^T\left(t-d\left(t\right)\right)Q_{2}x\left(t-d\left(t\right)\right)\\ +d{\stackrel{\&CenterDot;}{x}}^T\left(t\right)Z\stackrel{\&CenterDot;}{x}\left(t\right)-{\&Integral;}_{t-d}^t{\stackrel{\&CenterDot;}{x}}^T\left(s\right)Z\stackrel{\&CenterDot;}{x}\left(s\right)ds\end{array}$

From formula:

$\left\{\begin{array}{c}-\left(1-\stackrel{\&CenterDot;}{d}\left(t\right)\right)\&le;-\left(1-\mathrm{\&mu;}\right)\\ {\&Integral;}_{t-d}^t{\stackrel{\&CenterDot;}{x}}^T\left(s\right)Z\stackrel{\&CenterDot;}{x}\left(s\right)ds={\&Integral;}_{t-d\left(t\right)}^t{\stackrel{\&CenterDot;}{x}}^T\left(s\right)Z\stackrel{\&CenterDot;}{x}\left(s\right)ds+{\&Integral;}_{t-d}^{t-d\left(t\right)}{\stackrel{\&CenterDot;}{x}}^T\left(s\right)Z\stackrel{\&CenterDot;}{x}\left(s\right)ds\end{array}\right.$

Can obtain thus:

$\begin{array}{c}\stackrel{\&CenterDot;}{V}\left(t\right)\&le;{\stackrel{\&CenterDot;}{x}}^T\left(t\right)Px\left(t\right)+x^T\left(t\right)P\stackrel{\&CenterDot;}{x}\left(t\right)+x^T{Q}_{1}x\left(t\right)+x^T\left(t\right)Q_{2}x\left(t\right)\\ -x^T\left(t-d\right)Q_{1}x\left(t-d\right)-\left(1-\mathrm{\&mu;}\right)x^T\left(t-d\left(t\right)\right)Q_{2}x\left(t-d\left(t\right)\right)\\ +d{\stackrel{\&CenterDot;}{x}}^T\left(t\right)Z\stackrel{\&CenterDot;}{x}\left(t\right)-{\&Integral;}_{t-d}^t{\stackrel{\&CenterDot;}{x}}^T\left(s\right)Z\stackrel{\&CenterDot;}{x}\left(s\right)ds+{\&Integral;}_{t-d}^{t-d\left(t\right)}{\stackrel{\&CenterDot;}{x}}^T\left(s\right)Z\stackrel{\&CenterDot;}{x}\left(s\right)ds\end{array}$

If $\mathrm{\&chi;}{\left(t\right)}^T=\left(\begin{array}{ccccc}{x}^T\left(t\right)& x^T(t-d(t))& x^T(t-d)& {\stackrel{\&CenterDot;}{x}}^T\left(t\right)& b^T\left(t\right)\end{array}\right),$ Introduce free-form curve and surface:

$2\mathrm{\&chi;}{\left(t\right)}^{T}N\&lsqb;x\left(t\right)-x(t-d(t\left)\right)-{\&Integral;}_{t-d\left(t\right)}^t\stackrel{\&CenterDot;}{x}\left(s\right)\&rsqb;=0$

$2\mathrm{\&chi;}{\left(t\right)}^{T}L\&lsqb;x(t-d(t\left)\right)-x(t-d)-{\&Integral;}_{t-d}^{t-d\left(t\right)}\stackrel{\&CenterDot;}{x}\left(s\right)\&rsqb;=0---\left(16\right)$

$2\left(x^T\right(t)T+{\stackrel{\&CenterDot;}{x}}^T(t\left)T\right)(-\stackrel{\&CenterDot;}{x}(t)+Ax(t)+B_{1}Kx(t-d\left(t\right))+B_{2}b(t\left)\right)=0---\left(17\right)$

Wherein N ^t=(N ₁ ^tn ₂ ^tn ₃ ^tn ₄ ^t0), L ^t=(L ₁ ^tl ₂ ^tl ₃ ^tl ₄ ^t0), matrix N _i(i=1,2,3,4), L _ithe matrix that (i=1,2,3,4) are any appropriate dimension, T>0.:

$\begin{array}{c}\stackrel{\&CenterDot;}{V}\left(t\right)\&le;\mathrm{\&chi;}{\left(t\right)}^T\&lsqb;\mathrm{\&Pi;}+{\mathrm{dNZ}}^{-1}{N}^T+{\mathrm{dLZ}}^{-1}{L}^T\&rsqb;\mathrm{\&chi;}\left(t\right)\\ -{\&Integral;}_{t-d\left(t\right)}^t\&lsqb;{\mathrm{\&chi;}}^T\left(t\right)N+\stackrel{\&CenterDot;}{x}\left(s\right)Z\&rsqb;Z^{-1}{\&lsqb;{\mathrm{\&chi;}}^T\left(t\right)N+\stackrel{\&CenterDot;}{x}\left(s\right)Z\&rsqb;}^{T}ds\\ -{\&Integral;}_{t-d\left(t\right)}^t\&lsqb;{\mathrm{\&chi;}}^T\left(t\right)L+\stackrel{\&CenterDot;}{x}\left(s\right)Z\&rsqb;Z^{-1}{\&lsqb;{\mathrm{\&chi;}}^T\left(t\right)L+\stackrel{\&CenterDot;}{x}\left(s\right)Z\&rsqb;}^{T}ds\end{array}$

≤x(t) ^T［Π+dNZ ^-1N ^T+dLZ ^-1LT］x(t)(18)

$\mathrm{\&Pi;}=\left(\begin{array}{ccccc}{\mathrm{\&Pi;}}_{11}& {\mathrm{\&Pi;}}_{12}& {\mathrm{\&Pi;}}_{13}& {\mathrm{\&Pi;}}_{14}& {\mathrm{\&Pi;}}_{15}\\ *& {\mathrm{\&Pi;}}_{22}& {\mathrm{\&Pi;}}_{23}& {\mathrm{\&Pi;}}_{24}& 0\\ *& *& {\mathrm{\&Pi;}}_{33}& {\mathrm{\&Pi;}}_{34}& 0\\ *& *& *& {\mathrm{\&Pi;}}_{44}& {\mathrm{\&Pi;}}_{45}\\ *& *& *& *& 0\end{array}\right)---\left(19\right)$

Wherein ${\mathrm{\&Pi;}}_{11}=TA+A^{T}T+N_1+N_1^T+Q_1+Q_2,$ Π ₁₂＝L ₁-N ₁+N ₂ ^T+TB ₁K

${\mathrm{\&Pi;}}_{13}=N_3^T-Q_1-L_1,{\mathrm{\&Pi;}}_{14}=P-T+N_4^T+A^{T}T,{\mathrm{\&Pi;}}_{15}={\mathrm{TB}}_2$

${\mathrm{\&Pi;}}_{22}=(\mathrm{\&mu;}-1)Q_2-N_1+L_2-N_1^T+L_2^T,{\mathrm{\&Pi;}}_{23}=L_3^T-L_2-N_3^T$

${\mathrm{\&Pi;}}_{24}=L_4^T-N_4^T+K^T{B}_1^{T}T,{\mathrm{\&Pi;}}_{33}=-Q_1-L_3-L_3^T,{\mathrm{\&Pi;}}_{34}=-L_4^T$

Π ₄₄=dZ-2T, Π ₄₅=TB ₂, " * " represents that this element is the transposition of symmetry element

Closed-loop system Asymptotic Stability is equivalent to: Π+dNZ ^-1n ^t+ dLZ ^-1l ^t<0

Mend lemma by Schur can be equivalent to by this stable condition:

$\left(\begin{array}{ccc}\mathrm{\&Pi;}& dN& dL\\ *& -dZ& 0\\ *& *& -dZ\end{array}\right)<0---\left(20\right)$

Therefore closed-loop system asymptotically stable condition when b (t)=0 is:

$\left(\begin{array}{ccc}\widetilde{\mathrm{\&Pi;}}& dN& dL\\ *& -dZ& 0\\ *& *& -dZ\end{array}\right)<0---\left(21\right)$

Wherein $\widetilde{\mathrm{\&Pi;}}=\left(\begin{array}{cccc}{\mathrm{\&Pi;}}_{11}& {\mathrm{\&Pi;}}_{12}& {\mathrm{\&Pi;}}_{13}& {\mathrm{\&Pi;}}_{14}\\ *& {\mathrm{\&Pi;}}_{22}& {\mathrm{\&Pi;}}_{23}& {\mathrm{\&Pi;}}_{24}\\ *& *& {\mathrm{\&Pi;}}_{33}& {\mathrm{\&Pi;}}_{34}\\ *& *& *& {\mathrm{\&Pi;}}_{44}\end{array}\right)$

6. set up H-infinity performance index:

In this step, under we will set up zero initial condition, the H-infinity performance index criterion of this controller.

$J={\&Integral;}_0^{\mathrm{\&infin;}}\&lsqb;y^T\left(t\right)y\left(t\right)-{\mathrm{\&gamma;}}^2{b}^T\left(t\right)b\left(t\right)\&rsqb;dt---\left(22\right)$

Then can obtain:

$J\&le;{\&Integral;}_0^{\mathrm{\&infin;}}\&lsqb;y^T\left(t\right)y\left(t\right)-{\mathrm{\&gamma;}}^2{b}^T\left(t\right)b\left(t\right)+\stackrel{\&CenterDot;}{V}\left(t\right)\&rsqb;dt---\left(23\right)$

Be equivalent to:

$y^T\left(t\right)y\left(t\right)-{\mathrm{\&gamma;}}^2{b}^T\left(t\right)b\left(t\right)+\stackrel{\&CenterDot;}{V}\left(t\right)\&le;{\mathrm{\&chi;}}^T\left(t\right){\mathrm{\&Pi;}}_J\mathrm{\&chi;}\left(t\right)---\left(24\right)$

${\mathrm{\&Pi;}}_J=\left(\begin{array}{ccc}\widehat{\mathrm{\&Pi;}}& dN& dL\\ *& -dZ& 0\\ *& *& -dZ\end{array}\right)<0---\left(25\right)$

Wherein $\widehat{\mathrm{\&Pi;}}=\left(\begin{array}{ccccc}{\mathrm{\&Pi;}}_{11}+C^{T}C& {\mathrm{\&Pi;}}_{12}& {\mathrm{\&Pi;}}_{13}& {\mathrm{\&Pi;}}_{14}& {\mathrm{\&Pi;}}_{15}\\ *& {\mathrm{\&Pi;}}_{22}& {\mathrm{\&Pi;}}_{23}& {\mathrm{\&Pi;}}_{24}& 0\\ *& *& {\mathrm{\&Pi;}}_{33}& {\mathrm{\&Pi;}}_{34}& 0\\ *& *& *& {\mathrm{\&Pi;}}_{44}& {\mathrm{\&Pi;}}_{45}\\ *& *& *& *& -{\mathrm{\&gamma;}}^2\end{array}\right)$

Work as Π _jh-infinity performance can be ensured during <0.

We set up maximum drift angle y now _maxcondition, can be obtained by (24) integration can obtain:

$V\left(t\right)-V\left(0\right)<{\mathrm{\&gamma;}}^2{\&Integral;}_0^t{b}^T\left(t\right)b\left(t\right)dt<{\mathrm{\&gamma;}}^2\left|\right|b|{|}_2^2$

Due to $\left|\right|b|{|}_2=\sqrt{{\&Integral;}_0^{\mathrm{\&infin;}}b{\left(t\right)}^{T}b\left(t\right)dt}=b_{max}<\mathrm{\&infin;},$ Order $x^T\left(t\right)Px\left(t\right)<{\mathrm{\&gamma;}}^2{b}_{max}+V\left(0\right)=\mathrm{\&sigma;},$ Then:

$\begin{array}{c}\underset{t>0}{\max }{\left|y\left(t\right)\right|}^2=\underset{t>0}{\max }\left|\right|x^T\left(t\right)C^{T}Cx\left(t\right)|{|}_2\\ =\underset{t>0}{\max }\left|\right|x^T\left(t\right)-P^{1/2}{P}^{-1/2}{C}^T{\mathrm{CP}}^{-1/2}{P}^{1/2}x\left(t\right)\left|\right|<\mathrm{\&sigma;}\&CenterDot;{\mathrm{\&lambda;}}_{\max }\left(P^{-1/2}{C}^T{\mathrm{CP}}^{-1/2}\right)\end{array}$

λ _maxthe eigenvalue of maximum of () representing matrix, this condition can by σ P ^-1/2c ^tcP ^1/2<y _maxi obtains,

Mend lemma by Schur can obtain:

$\left(\begin{array}{cc}-I& \sqrt{\mathrm{\&sigma;}}C\\ *& -y_{\max }^{2}P\end{array}\right)<0---\left(26\right)$

So far, the device design conditions that is under control (20), (21), (25), (26)

7. calculate LMI optimum solution by genetic algorithm

From above-mentioned design conditions, in the condition of a given maximum drift angle, can obtain the Controller gain variations condition that meets control objectives, we utilize genetic algorithm to be optimized calculating to these LMIs, can ensure the optimum solution obtaining ride gain.

Optimization aim:

Constraint condition: formula (20), (21), (25), (26)

8. obtain controller gain: K

Claims (2)

1. the track laying air cushion vehicle comprising input delay to a robust control method, it is characterized in that: comprise the following steps,

Step one: the state vector being obtained hovermarine motion by the sensing system of hovermarine, comprises longitudinal velocity u, side velocity v and heading rate r;

Step 2: the data gathered according to sensing system, obtain the course made good of hovermarine current kinetic

Step 3: enter the reaction time after actuator according to controller input, introduce input delay link, set up the state-space model comprising input delay:

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{x}\left(t\right)=Ax\left(t\right)+B_1\mathrm{\&tau;}(t-d(t\left)\right)+B_{2}b\left(t\right)\\ y\left(t\right)=Cx\left(t\right)\end{array}\right.$

Wherein m=M ^t>0 is inertial matrix, and D>0 is linear damping matrix, and τ is controlled quentity controlled variable, and b is environmental interference, and d (t) is Time-varying time-delays, meets:

$\left\{\begin{array}{c}0<d\left(t\right)<d\\ \stackrel{\&CenterDot;}{d}\left(t\right)<\mathrm{\&mu;}\end{array}\right.$

Wherein d is the upper bound of Time-varying time-delays, and μ is time lag change restriction;

Step 4: send course made good to robust controller, obtains steering order and sends actuator to, realizes course made good and controls.

2. a kind of track laying air cushion vehicle comprising input delay according to claim 1 to robust control method, it is characterized in that: described robust controller is:

τ(t)＝Kx(t)

According to Lyapunov-Krasovskii stability theory, H-infinity control theory and hovermarine to the actual needs controlled, structure Lyapunov-Krasovskii function, is met the adequate condition of the Delay-Dependent of state feedback controller:

Wherein, P>0, Q ₁>0, Q ₂>0, Z>0, $N^T=\left(\begin{array}{ccccc}{N_1}^T& {N_2}^T& {N_3}^T& {N_4}^T& 0\end{array}\right)$ $L^T=\left(\begin{array}{ccccc}{L_1}^T& {L_2}^T& {L_3}^T& {L_4}^T& 0\end{array}\right),$ N _i(i=1,2,3,4), L _i(i=1,2,3,4) are matrix, T>0, and the T in the matrix upper right corner represents this transpose of a matrix, σ=γ ²b _max+ V (0), b _maxfor the maximal value of environmental interference energy, y _maxfor maximum drift angle, V (0) is the value of Lyapunov-Krasovskii function V (t) when t=0;

Utilize genetic algorithm to obtain the optimum solution of controller gain K, optimization aim is:

$\begin{array}{cc}\underset{K_i}{min}& \mathrm{\&gamma;}\end{array}$

Optimal conditions is the adequate condition of the Delay-Dependent of controller.