DESIGN OF ROBUST TWO-AXIS SYSTEM FOR STABILIZATION OF INFORMATION AND MEASURING DEVICES OPERATED ON GROUND VEHICLES



Design features of information and measuringrobust stabilization systems operated on the ground vehicles are considered.The mathematical description of the stabilization plant is derived. Designfeatures of the two-axis robust stabilization and tracking systems based on thestructural H∞-synthesis are researched.

Keywords: robust two-axis stabilization systems, informationand measuring devices, ground vehicles, ${{H}_{\infty }}$-synthesis

O.A. Sushchenko, professor, National Aviation University

Ukraine

Introduction. The basic motivation for improvement of performances of the stabilization systems operated on the vehicles is the real necessity in such measures. Information and measuring devices performances have been drastically increased last years [1]. These trends cannot be realized without appropriate progress of stabilization facilities. The modern approaches to design of the studied systems lie in using the robust control.

Many papers and textbooks, for example [2], deal with design of the robust systems. Design features of the systems for stabilization of information and measuring devices operated on the ground vehicles are represented in [3]. For the parametric optimization the controller structure is believed to be known based on experience of previous developments [4]. Approaches to the structural synthesis of the systems for stabilization and tracking of the information-measuring devices are represented in [5, 6].

Designof robust two-axis systems. In the general case, the system dynamic can be described by the Euler equations [7]. Based on the mathematical model of the one-axis system for stabilization of the information-measuring devices operated on the ground vehicles, which is represented in the [8], the model of the two-axis gyro stabilization system after some simplifications becomes

\[\dot{\alpha }={{\omega}_{x}}\cos \beta +{{\omega }_{z}}\sin \beta; \]

\[\dot{\beta}={{\omega }_{y}};\]

\[{{\dot{\alpha}}_{\text{e}}}={{\omega }_{\text{e}\alpha }};\]

\[{{\dot{\beta}}_{\text{e}}}={{\omega }_{\text{e}\beta }};\]

\[{{\dot{U}}_{\omega\alpha }}={{U}_{\omega \text{d}\alpha }};\]

\[{{\dot{U}}_{\omega\beta }}={{U}_{\omega \text{d}\beta }};\]

\[{{\dot{\omega}}_{x}}=[-({{J}_{z}}-{{J}_{y}}){{\omega }_{y}}{{\omega}_{z}}-{{M}_{frx}}\text{sign}{{\omega }_{x}}-{{M}_{unbx}}\cos \alpha+{{c}_{r}}({{\alpha }_{g}}-\alpha )/{{n}_{r}}]/{{J}_{x}};\]

\[{{\dot{\omega}}_{y}}=[-({{J}_{y}}-{{J}_{x}}){{\omega }_{x}}{{\omega}_{z}}-{{M}_{fry}}\text{sign}{{\omega }_{y}}-{{M}_{unby}}\cos \beta+{{k}_{spr}}(A-\beta )+\frac{{{c}_{r}}({{\beta }_{g}}-\beta)}{{{n}_{r}}}]/{{J}_{y}};\]

\[{{\dot{\omega}}_{e\alpha }}=\left[ -{{M}_{fre}}\text{sign}{{\omega }_{e\alpha}}+\frac{{{c}_{m}}}{{{R}_{w}}}{{U}_{\alpha }}+\frac{{{c}_{r}}({{\alpha }_{g}}-\alpha)}{{{n}_{r}}} \right]/{{J}_{e}};\]

\[{{\dot{\omega}}_{e\beta }}=\left[ -{{M}_{fre}}\text{sign}{{\omega }_{e\beta}}+\frac{{{c}_{m}}}{{{R}_{w}}}{{U}_{\beta }}+\frac{{{c}_{r}}({{\beta}_{g}}-\beta )}{{{n}_{r}}} \right]/{{J}_{e}};\]

\[{{\dot{U}}_{\alpha}}=[-{{U}_{\alpha }}+{{k}_{PWM}}{{U}_{PWM\alpha }}-{{c}_{ed}}{{\omega}_{e\alpha }}]/{{T}_{arm}};\]

\[{{\dot{U}}_{\beta}}=[-{{U}_{\beta }}+{{k}_{PWM}}{{U}_{PWM\beta }}-{{c}_{ed}}{{\omega }_{e\beta}}]/{{T}_{arm}};\]

\[{{\dot{U}}_{\omega d\alpha }}=[-2\nu {{T}_{0}}{{U}_{\omega e\alpha }}-{{U}_{\omega \alpha}}+{{k}_{ars}}{{\omega }_{x}}]/T_{0}^{2};\]

\[{{\dot{U}}_{\omega d\beta }}=[-2\nu {{T}_{0}}{{U}_{\omega e\beta }}-{{U}_{\omega \beta}}+{{k}_{ars}}{{\omega }_{y}}]/T_{0}^{2},\quad\quad\quad(1)\]

where $\alpha$, $\beta $ are turn angles of the platform with payload; ${{\omega }_{x}}$, ${{\omega }_{y}}$ are the platform angular rates; ${{\omega }_{\text{e}\alpha }}$, ${{\omega}_{\text{e}\beta }}$ are the rates of the engines mounted by axes $x$, $y$; ${{\alpha }_{e}}$, ${{\beta}_{e}}$ are the turn angles of the engines mounted by axes $x$, $y$; ${{U}_{\omega \alpha }}$, ${{U}_{\omega \beta }}$ are the output signals of the angular rate gyros by axes $x$, $y$; ${{U}_{\omega d\alpha }}$, ${{U}_{\omega d\beta }}$ are the derivatives of rate gyros signals; ${{J}_{x}}$, ${{J}_{y}}$, ${{J}_{z}}$ are the inertia moments of the platform with payload relative its own axes $x$, $y$, $z$; ${{M}_{frx}}$, ${{M}_{fry}}$ are nominal dry friction moments acting by the gimbals axes $x$, $y$; ${{M}_{unbx}}$, ${{M}_{unby}}$ are the unbalanced moments by the axes $x$, $y$; ${{k}_{spr}}$ is the rigidity coefficient of the spring compensator; $A$ is the initial angle of spring resetting; ${{c}_{r}}$ is the reducerrigidity; ${{\alpha}_{g}}$, ${{\beta }_{g}}$ are the turn angles of the platform taking intoaccount presence of the drive gap; ${{M}_{frex}}$, ${{M}_{frey}}$ are the nominal dry friction moments of engines by axes $x$, $y$; ${{c}_{m}}$ is the constant of the load moment; ${{R}_{\text{w}}}$ is the resistanceof the engine armature winding; ${{U}_{\alpha }}$, ${{U}_{\beta }}$ are thearmature voltages of engines mounted by the gimbals axes; ${{n}_{r}}$ is thereducer gear ratio; ${{T}_{arm}}$ is the time constant of the engine armature circuit; ${{k}_{PWM}}$ is the transfer constant of the linearized pulse widthmodulator; ${{U}_{PWM}}$ is the voltage at the pulse width modulator input; ${{c}_{ed}}$is the coefficient of proportionality between the engine angular rate and theelectromotive force; $\nu $ is therelative damping coefficient; ${{T}_{0}}$ is the time constant of the angularrate sensor, ${{k}_{ars}}$ is the transfer constant of the angularrate sensor.

In the represented non-linear equations (4) the angles ${{\alpha }_{g}}$, ${{\beta}_{g}}$ can be definedin accordance with the expressions

${{\alpha}_{g}}={{\alpha }_{e}}/{{n}_{p}}$, if $|{{\alpha }_{e}}/{{n}_{p}}-\alpha |\ge \ 0,5\Delta; $

${{\alpha }_{g}}=\alpha $, if $|{{\alpha }_{e}}/{{n}_{p}}-\alpha|\ \left\langle {} \right.\ 0,5\Delta; $

${{\beta }_{g}}={{\beta}_{e}}/{{n}_{p}}$, if $|{{\beta }_{e}}/{{n}_{p}}-\beta |\ \ge \ 0,5\Delta; $

${{\beta}_{g}}=\beta $, if $|{{\beta }_{e}}/{{n}_{p}}-\beta |\ \left\langle {}\right.\ 0,5\Delta, \quad\quad\quad(2)$

where $\Delta $ is the value of the experimentally determined system drive gap.

For further researches it is necessary to implement linearization of (1), (2) relative to the nominal values of the phase coordinates. Such linearization must include the following stages:

$1)$ linearization of the expressions for the friction and unbalanced moments of the engine and stabilization plant;

$2)$ neglect by the drive gap and the friction moments at the bearings of the gimbals and at the engine shaft;

$3)$ neglect by the zero drift of the angular rate sensor;

$4)$ assumption of smallness of the platform turn angles for linearization of the trigonometric functions. Then the setof equations (1) can be linearized and represented in the space of states by the quadruple f matrices:

The algorithm of the structural synthesis for the robust stabilization system. One of the modern approaches to the structural synthesis of the robust stabilizationand tracking system is the ${{H}_{\infty }}$-synthesis. Basic stages of such synthesis implementation are represented in [2]. The robust structural synthesis is based on solutions of two Riccati equations, check of some conditions and minimization of the mixed sensitivity function ${{H}_{\infty }}$-norm forthe system including the plant and controller and represented by the vector of outputs characterized the system quality, vector of inputs and vector of controls and observations [2]. The modern approach to the structural ${{H}_{\infty}}$-synthesis problem solution is based on forming of the desired frequency characteristics of the system. This is implemented by means of the augmented plant due to introduction of the weighting transfer functions. Such approach is called loop-shaping [2]. In this case, the optimization criterion represents the ${{H}_{\infty}}$-norm of the mixed sensitivity function of the augmented plant

where $\ {{W}_{1}},\ {{W}_{2}},\ {{W}_{3}}$ are the weighting transfer functions, $S,\ R,\ T$ are the sensitivity functions by the command signal, control and the complementary sensitivity function respectively.

Conclusions. The basic approaches to the robust structural synthesis of the two-axis of the system for stabilization of the information-measuring operated on the ground vehicles are represented. The mathematical description of the plant is obtained. The approach to the robust structural synthesis was researched.

References

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$2.$ Skogestad S. Multivariable Feedback Control/ Skogestad S., Postlethwaite I. –New York: Jonh Wiley, 1997. – 559 p.

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Mar 28, 2017