Optimal Synthesis of a Satellite Attitude Control System under Constraints on Control Torques and Velocities of Reaction Wheels

Abstract

The problem of optimal synthesis of a nonlinear system’s control law parameters of satellite attitude control under constraints is considered. The maximum stability degree of a system is considered as an optimality criterion. The reaction wheels’ control moments and angular velocities achievable within the drives’ physical characteristics are considered as constraints. A linear model of the system converted from the original nonlinear model was used to solve the problem. The decomposition method, based on the transition from real time to relative time, is used to separate the tasks of optimality criterion maximization and constraints consideration. A method of synthesizing the optimal form of system transient process according to the maximum stability degree criterion is developed. An analytical method for determining the optimal values of control law parameters is proposed, based on the structural features of the linear differential equations system. A method that takes into account constraints on control moments and angular velocities of reaction wheels, based on transition scale from relative time to real time and its calculation algorithm, ensuring that conditions of all constraints are met, is developed. An example of solving the problem of optimal system synthesis is considered. The results demonstrate the effectiveness and simplicity of the proposed methods and algorithm for targeted selection of a system’s technical characteristics, taking into account constraints on reaction wheels’ maximum angular velocities and control moments values.

1. Introduction

Engineering practice mainly applies methods based on the use of linear models obtained by linearization of dynamic equations or by converting them into a linear form for the optimal synthesis of satellite attitude control systems (SACS), the dynamics of which are described by nonlinear differential equations.

Quite a large number of works have been devoted to methods of optimal synthesis of nonlinear SACS based on linear models obtained by linearization of dynamic equations, among which the most well-known and frequently cited papers can be highlighted [1,2,3,4,5,6,7,8,9,10,11,12,13]. These fundamental methods of optimal synthesis of nonlinear SACS include the State-Dependent Algebraic Riccati Equation (SDARE) method [1,2,3], the State-Dependent Ricatti Equation (SDRE) method [4,6,7,8], the optimal control method [5,9], and the method of pole placement using $H\infty$ and $H2$ norms [10]. At the same time, the nonlinear feedback linearization method is often used for dynamic equations linearization [1,2,3,11,12,13].

From the point of view of taking into account non-linearities in SACS, these optimal synthesis methods can be divided into two groups: the first takes into account only the non-linearities of the equations of dynamics [2,3,4,5,6,8]; the second, along with the non-linearities of the equations of dynamics, also accounts for the non-linearities associated with the saturation effect in actuators [1,7,9,10,11,12,13]. In these papers, integral criteria for the minimum mean square error, and criteria for the minimum energy consumption for control and the minimum time, are considered as criteria for optimal synthesis. In the second group of works, along with optimality criteria, the possibilities of taking into account constraints on the maximum control moments created by reaction wheels and on their maximum angular velocities are also considered.

In particular, in Wie, B. and Lu, J. [1], the problem of the fastest possible reorientation of a rigid spacecraft within the physical capabilities of the drives is investigated. The problem is solved by the methods of optimal control theory according to minimum time criterion, taking into account constraints on the control moment increase rate. This task requires the use of powerful computing tools with high speed and a large amount of memory to implement complex control algorithms and to solve the attitude control problem of large satellites. For the design of micro- and nanosatellites, the perspectives for the use of which are constantly expanding, simplicity and low complexity of control algorithms are more in demand as engineering requirements for the developed method of optimal synthesis of parameters of SACS.

In Romero, A.G. [7], a satellite angular position and orbit control system is considered, which can be successfully designed based on linear control theory if the satellite has slow angular movements and small position maneuvers. However, linearized models are not capable of reflecting all the disturbances due to the influence of nonlinear terms present in dynamics and in actuators with large and fast maneuvers (for example, saturation).

In Zhou, H. [9], a reorientation maneuver controller is presented, optimal in minimum time criterion, with saturation constraints for the control and kinetic moments of reaction wheels. The proposed control scheme consists of two parts. The first part presents the construction of an optimal trajectory without feedback, taking into account the dynamics of actuator and saturation of control and kinetic moments. The problem of optimal synthesis of SACS is solved by the methods of optimal control theory according to the minimum time criterion, taking into account constraints on control torque increase rate. The approach mentioned above requires the use of powerful computing tools and implementation of complex control algorithms. The second part is a feedback control law for tracking an optimized reference trajectory based on error and reaction wheel dynamics. However, in this part of the optimal synthesis of SACS parameters, constraints on control torque increase rate are not taken into account.

In Wu, B. [10], a mixed controller with feedback on the output $H2/H\infty$ is proposed to control the microsatellite’s orientation with inertia moment uncertainties and external disturbances. The controller is designed based on a linear orientation dynamics model. The characteristics of $H\infty$ provide stability to inertia moment uncertainties and disturbances suppression. The characteristics of $H2$ allows for avoiding undesirable effect of reaction wheels’ saturation. In addition, the closed-loop poles can be forced into some sector of a stable half-plane to obtain well-damped transient responses. The disadvantage of this approach is that constraints associated with reaction wheels’ saturation are not explicitly formulated through the physical characteristics of reaction wheel drives but indirectly through requirements to place the characteristic equation roots in a certain area of the complex half-plane. This creates difficulties in transition from requirements for reaction wheel drives’ physical characteristics to requirements to place characteristic equation roots, since the placement of roots is not expressed explicitly through reaction wheel drives’ physical characteristics.

In Shan, G. [11], the problem of satellite’s constraints on control torque and angular velocity is discussed, as well as the relationship between control parameter and velocity upper limit, and constraints on control parameters. However, the paper does not provide an algorithm for synthesizing control law parameters based on explicit analytical dependencies between constraints and used drives’ technical characteristics.

In Dong, Y. [12], an adaptive finite-time controller is proposed for rapid satellite attitude maneuver, and the general global finite-time stability of the system is proved by the Lyapunov method. In general, the standard sliding mode controller can be considered as a mature control method with high resistance to disturbances; however, the convergence rate should be improved when solving the problem of rapid orientation maneuver. The key to increasing the convergence rate of the system is the correct design of angular velocity trajectory, i.e., the transition from the problem of synthesizing control law parameters to the optimal control problem.

In Hu, Q. [13], a new nonlinear control design with constraints on drive’s given velocity and torque is investigated to stabilize the orientation of a rigid spacecraft. In particular, nonlinear feedback control is developed by explicitly taking into account constraints on individual angular velocity components as well as external disturbances. However, the abovementioned studies have been derived under implicit assumptions that actuators are capable of providing any requested joint torque.

Thus, the well-known and widely used methods of optimal synthesis of parameters of SACS, firstly, are based on their linearized mathematical models. The disadvantages of using linearized equations are their approximation of SACS dynamics, and the question regarding global asymptotic stability of the initial nonlinear equations system remains unsolved, as its global asymptotic stability cannot be derived from the asymptotic stability of linearized system. Secondly, the methods used for the optimal synthesis of parameters of SACS do not consider all possible optimization criteria, such as the criterion of the maximum stability degree of the system, which is very important in the presence of system parameter uncertainties and uncontrolled external disturbances.

Therefore, there is a great theoretical and practical interest in, firstly, the use of linear equivalents of nonlinear equations of dynamics of SACS and, secondly, the use of new criteria for the optimal synthesis of parameters of SACS such as the maximum stability degree.

In Moldabekov, M. et al. [14], nonlinear equations of dynamics of SACS with a PD controller are investigated and the possibility of converting them into a linear system of differential equations with time–variable parameters is shown. Possibilities of using the linear form of equations of dynamics of SACS for analysis of its stability and control law parameters synthesis according to required quality indicators of transient processes of satellite orientation are shown. In Moldabekov, M. et al. [15], the results of a study of dynamics of a PD–controlled SACS—the kinematics of which are described by equations in quaternions—are presented. Based on Cauchy’s theorem on existence and uniqueness of the solution of differential equations system (Wirkus, A.S. and Swift, J.R. [16]) and on the theorem of mechanical system kinetic moment conservation (Knudsen, J.M. and Hjorth, P.G. [17]), the conversion of dynamic nonlinear equations of SACS into a system of linear differential equations with time-variable parameters is obtained. A study of linear model stability of SACS is carried out, and it is shown that if the roots of characteristic equation of system are placed using the condition of maximum stability degree, then global asymptotic stability of the system is achieved.

The proposed article considers the problem of optimal synthesis of control law parameters of a nonlinear SACS with a PD controller under constraints associated with the saturation effect of reaction wheels. The solution of the optimal synthesis problem is based on the use of a system linear model proposed by Moldabekov, M. et al. [15]. Simplicity, low labor intensity, and the possibility of taking into account a drive’s physical capabilities are considered as engineering requirements for the developed method of solving the problem. A new criterion, the maximum stability degree of control system, is considered as an optimality criterion for control law parameters; further, as constraints, the maximum absolute values of reaction wheels’ control moments and angular velocities are considered, which are achievable within the drive’s physical capabilities.

2. Linear Form of the Mathematical Model of the Nonlinear Satellite Attitude Control System

The mathematical model of the nonlinear SACS in linear form was obtained by Moldabekov, M. et al. [15]. While constructing the mathematical model, satellite and reaction wheels were considered as a single closed mechanical system, whose total kinetic moment is defined as

$${\overline{h}}_B\left(\overline{\omega },{\overline{\omega }}_W\right)={\overline{h}}_{BS}\left(\overline{\omega }\right)+{\overline{h}}_{BW}\left({\overline{\omega }}_W\right),$$

(1)

where ${\overline{h}}_{BS}\left(\overline{\omega }\right)=I_{S\overline{\omega }}$ and ${\overline{h}}_{BW}\left({\overline{\omega }}_W\right)=I_W{\overline{\omega }}_W$ are the vectors of the kinetic moments of the satellite and reaction wheels, respectively; $\overline{\omega }={\left[{\omega }_1,{\omega }_2,{\omega }_3\right]}^T$ and ${\overline{\omega }}_W={\left[{\omega }_{W1},{\omega }_{W2},{\omega }_{W3}\right]}^T$ are the angular velocities of the satellite and reaction wheels, respectively; $I_S=diag\left\{I_{S1},I_{S2},I_{S3}\right\}$ and $I_W=diag\left\{I_{W1},I_{W2},I_{W3}\right\}$ are diagonal (3 × 3) matrices of the inertia tensors of the satellite and reaction wheels, respectively (all parameters are expressed in the body-fixed coordinate system). It was assumed that the moments of external forces are close to zero and the control torques of reaction wheels in a body-fixed coordinate system are $\overline{T}=I_W{\dot{\overline{\omega }}}_W$.

The system dynamics are described using the standard form equations given by Sidi M. [18]:

$$\dot{\overline{\omega }}=-{I_S}^{-1}\left[S\left(\overline{\omega }\right){\overline{h}}_B\left(\overline{\omega },{\overline{\omega }}_W\right)+{I_W\dot{\overline{\omega }}}_W\right]$$

(2)

where the vector product operator is

$$S\left(\overline{\omega }\right)=\left[\begin{array}{ccc}0& -{\omega }_3& {\omega }_2\\ {\omega }_3& 0& -{\omega }_1\\ -{\omega }_2& {\omega }_1& 0\end{array}\right]$$

(3)

The kinematics are described using quaternion equations by Wie, B. and Lu, J. [1]:

$${\displaystyle \frac{d}{dt}}{\overline{q}}_S={{\displaystyle \frac{1}{2}}\Omega \left(\overline{\omega }\right)\overline{q}}_S={\displaystyle \frac{1}{2}}\left[\begin{array}{cc}-S\left(\overline{\omega }\right)& \overline{\omega }\\ -{\overline{\omega }}^T& 0\end{array}\right]{\overline{q}}_S,$$

(4)

where ${\overline{q}}_S={\left[q_1,q_2,q_3,q_4\right]}^T\equiv {[{\overline{q}}^T,q_4]}^T$.

A linear control law corresponding to a PD regulator is adopted:

$$-{\dot{\overline{h}}}_{BW}=-I_W{\dot{\overline{\omega }}}_W=-D\overline{\omega }-K\overline{q}$$

(5)

where $D=diag\left\{d_1,d_2,d_3\right\}$ and $K=diag\left\{k_1,k_2,k_3\right\}$ are arbitrary ($3\times 3$) matrices with unknown parameters of the control law. The complete nonlinear system of SACS motion equations in normal Cauchy form is as follows:

$$\left[\begin{array}{c}\dot{\overline{\omega }}\\ \dot{\overline{q}}\\ q_4\end{array}\right]=\left[\begin{array}{ccc}-{\displaystyle \frac{D}{I_S}}+{\displaystyle \frac{S\left[{\overline{h}}_B\left(\overline{\omega },{\overline{\omega }}_W\right)\right]}{I_S}}& -{\displaystyle \frac{K}{I_S}}& 0\\ 0& -{\displaystyle \frac{1}{2}}S\left(\overline{\omega }\right)& {\displaystyle \frac{1}{2}}\overline{\omega }\\ 0& -{\displaystyle \frac{1}{2}}{\overline{\omega }}^T& 0\end{array}\right]\left[\begin{array}{c}\overline{\omega }\\ \overline{q}\\ q_4\end{array}\right].$$

(6)

In Moldabekov, M. et al. [15], it was shown that a nonlinear system of motion, described in Equation (6), can be transformed into a system of linear differential equations with time-varying parameters of the form

$$\dot{X}=\left[A+B\left(t\right)\right]X$$

(7)

where

$$A=\left[\begin{array}{ccc}-{\displaystyle \frac{D}{I_S}}+{\displaystyle \frac{S\left({\overline{h}}_I^0\right)}{I_S}}& -{\displaystyle \frac{K}{I_S}}& 0\\ {\displaystyle \frac{1}{2}}E& 0& 0\\ 0& 0& 0\end{array}\right],B\left(t\right)=\left[\begin{array}{ccc}{\displaystyle \frac{S\left[\Delta {\overline{h}}_B\left(t\right)\right]}{I_S}}& 0& 0\\ {\displaystyle \frac{1}{2}}S\left(\overline{q}\right)-{\displaystyle \frac{1}{2}}E\left(1-q_4\right)& 0& 0\\ -{\displaystyle \frac{1}{2}}{\overline{q}}^T& 0& 0\end{array}\right],X={\left[{\omega }_1,{\omega }_2,{\omega }_3,q_1,q_2,q_3,q_4\right]}^T,$$

(8)

${\overline{h}}_I\left(t_0\right)\equiv {\overline{h}}_I^0=const$ is the total kinetic moment of the closed mechanical system in the inertial coordinate system, which is constant according to the theorem of conservation of kinetic moment [13];

${\overline{h}}_B\left(t\right)={\overline{h}}_B\left(\infty \right)+\Delta {\overline{h}}_B\left(t\right)={\overline{h}}_I^0+\Delta {\overline{h}}_B\left(t\right),t\in \left[t_0,\infty \right)$ is the total kinetic moment of the closed mechanical system in the body-fixed coordinate system.

3. Stability and Quality of Control Processes

In Moldabekov, M. et al. [15], it was provided that for asymptotic stability of a nonlinear system (6), it is necessary and sufficient that a linear homogeneous system of differential equations with constant coefficients be asymptotically stable

$$\dot{X}=AX,$$

(9)

and the following condition must be met:

$$\underset{t\to \infty }{lim}B\left(t\right)=0.$$

(10)

According to Demidovich, B.P. [19], these statements lead to the important conclusion—which is important for the task being solved—that stability analysis and control processes’ quality of SACS, described by the nonlinear system of differential Equation (6), can be carried out based on an analysis of solutions of the differential equations’ linear system with constant parameters (9) and, consequently, based on an analysis of its characteristic equation’s roots location:

$$det\left(A-sE\right)=0.$$

(11)

The elements of matrix A depend on the elements of the matrices D and K of the control law (5) as well as on the initial values of the total angular momentum ${\overline{h}}_I\left(t_0\right)$:

$$-{\overline{C}}_m\le {\overline{h}}_I\left(t_0\right)\equiv \overline{C}\le {\overline{C}}_m,$$

(12)

where $\overline{C}={\left[C_1,C_2,C_3\right]}^T,{\overline{C}}_m={\left[C_m,C_m,C_m\right]}^T,C_m=max\left|{\overline{h}}_I\left(t_0\right)\right|$ are the maximum absolute values of the satellite’s angular momentum (1) at the initial moment in time.

In the special case when the initial conditions for the total angular momentum of the satellite and reaction wheels ${\overline{h}}_I\left(t_0\right)$ are zero, the linear system of Equation (9) takes a truncated form:

$$\dot{X}=A^{*}X,$$

(13)

where

$$A^{*}=\left[\begin{array}{ccc}-{\displaystyle \frac{D}{I_S}}& -{\displaystyle \frac{K}{I_S}}& 0\\ {\displaystyle \frac{1}{2}}E& 0& 0\\ 0& 0& 0\end{array}\right].$$

(14)

The elements of the $A^{*}$ matrix depend solely on the D and K parameters of the control law (5), and the characteristic polynomial of the truncated system of differential Equation (13) has the form

$$det\left(A^{*}-sE\right)=\sum _{i=0}^6{a}_i{s}^i,$$

(15)

where

$$\begin{array}{cc}\hfill a_6& =1,a_5={\displaystyle \frac{d_1}{I_{S1}}}+{\displaystyle \frac{d_2}{I_{S2}}}+{\displaystyle \frac{d_3}{I_{S3}}},\hfill \\ \hfill a_4& ={\displaystyle \frac{d_1{d}_2}{I_{S1}{I}_{S2}}}+{\displaystyle \frac{d_1{d}_3}{I_{S1}{I}_{S3}}}+{\displaystyle \frac{d_2{d}_3}{I_{S2}{I}_{S3}}}+{\displaystyle \frac{k_1}{2I_{S1}}}+{\displaystyle \frac{k_2}{2I_{S2}}}+{\displaystyle \frac{k_3}{2I_{S3}}},\hfill \\ \hfill a_3& ={\displaystyle \frac{d_1}{2I_{S1}}}\left({\displaystyle \frac{k_2}{I_{S2}}}+{\displaystyle \frac{k_3}{I_{S3}}}\right)+{\displaystyle \frac{d_2}{2I_{S2}}}\left({\displaystyle \frac{k_1}{I_{S1}}}+{\displaystyle \frac{k_3}{I_{S3}}}\right)+{\displaystyle \frac{d_3}{2I_{S3}}}\left({\displaystyle \frac{k_1}{I_{S1}}}+{\displaystyle \frac{k_2}{I_{S2}}}\right)+{\displaystyle \frac{d_1{d}_2{d}_3}{I_{S1}{I}_{S2}{I}_{S3}}},\hfill \\ \hfill a_2& ={\displaystyle \frac{k_1{k}_2}{4I_{S1}{I}_{S2}}}+{\displaystyle \frac{k_1{k}_3}{4I_{S1}{I}_{S3}}}+{\displaystyle \frac{k_2{k}_3}{4I_{S2}{I}_{S3}}}+{\displaystyle \frac{d_1{k}_2{d}_3+k_1{d}_2{d}_3+d_1{d}_2{k}_3}{2I_{S1}{I}_{S2}{I}_{S3}}},\hfill \\ \hfill a_1& ={\displaystyle \frac{k_1{k}_2{d}_3+d_1{k}_2{k}_3+k_1{d}_2{k}_3}{4I_{S1}{I}_{S2}{I}_{S3}}},a_0={\displaystyle \frac{k_1{k}_2{k}_3}{8I_{S1}{I}_{S2}{I}_{S3}}}.\hfill \end{array}$$

(16)

The characteristic polynomial of the complete system of differential Equation (9)—the elements of matrix A, which also depend on the initial values of the satellite’s total angular momentum ${\overline{h}}_I\left(t_0\right)$—has the form

$$det\left(A-sE\right)=\sum _{i=0}^6{b}_i{s}^i,$$

(17)

where

$$\begin{array}{cc}\hfill b_6& =a_6,b_5=a_5,b_4=a_4+{\displaystyle \frac{{C_1}^2}{I_{S2}·I_{S3}}}+{\displaystyle \frac{{C_2}^2}{I_{S1}·I_{S3}}}+{\displaystyle \frac{{C_3}^2}{I_{S1}·I_{S2}}},\hfill \\ \hfill b_3& =a_3+{\displaystyle \frac{1}{I_{S1}{I}_{S2}{I}_{S3}}}\left({C_1}^2{d}_1+{C_2}^2{d}_2+{C_3}^2{d}_3\right),\hfill \\ \hfill b_2& =a_2+{\displaystyle \frac{1}{2I_{S1}{I}_{S2}{I}_{S3}}}\left({C_1}^2{k}_1+{C_2}^2{k}_2+{C_3}^2{k}_3\right),\hfill \\ \hfill b_1& =a_1,b_0=a_0.\hfill \end{array}$$

(18)

Thus, stability and quality of control processes of SACS are generally determined by both control law parameters and initial values of satellite total kinetic moment. Below, we will use the result obtained in Moldabekov, M. et al. [15]: if control law parameters D,K (5) are such that the characteristic equation of truncated system (13) has real, multiple, and negative roots, then a linear system with constant parameters (9) is globally asymptotically stable for any non-zero initial values of the total kinetic moment of satellite and reaction wheels, given by inequalities in (12).

4. Decomposition of Tasks of Obtaining Required Shape and Rapidity of Transient Process

The coefficients $a_i,i=\left(\overline{0,6}\right)$ of characteristic polynomial (15) determine both the form and rapidity of transient process in SACS. The relationships between values of polynomial coefficients and these control quality indicators are described by transcendental expressions, which do not allow for obtaining explicit analytical estimates of both the required form and rapidity of transient process simultaneously. The problem can be solved by decomposing it into two sub-tasks and sequentially solving them: achieving the optimal form of transient process in the first step and then achieving required rapidity of its execution in the second step, if we use the normalized characteristic equation of the control system by Besekersky, V.A. and Popov, E.P. [20]. We use the concept of geometric mean root for this purpose:

$${\Omega }_0=\sqrt[6]{|s_1{s}_2{s}_3{s}_4{s}_5{s}_6|}=\sqrt[6]{{\displaystyle \frac{a_0}{a_6}}}=\sqrt[6]{a_0},$$

(19)

where $s_i,(i=\overline{1,6})$ are the roots of the characteristic equation of the system (13):

$$\sum _{i=0}^6{a}_i{s}^i=0.$$

(20)

Let us proceed in Equation (20) to a new complex value p by substituting $p={\Omega }_{0}s$. As a result, we obtain the normalized characteristic equation:

$$p^6+A_5{p}^5+A_4{p}^4+A_3{p}^3+A_2{p}^2+A_{1}p+1=0,$$

(21)

where $A_i={\displaystyle \frac{a_i{\Omega }_0^i}{a_0}},(i=\overline{1,5})$ are dimensionless coefficients that do not change within the time scale, i.e., they do not characterize the speed of the SACS and only determine the form of its transient process. When returning to the original complex variable, the original characteristic Equation (20) takes the form of

$$s^6+A_5{\Omega }_0{s}^5+A_4{\Omega }_0^2{s}^4+A_3{\Omega }_0^3{s}^3+A_2{\Omega }_0^4{s}^2+A_1{\Omega }_0^{5}s+{\Omega }_0^6=0.$$

(22)

A geometric mean root ${\Omega }_0$ can serve as a measure of the speed of transient processes. The use of the normalized characteristic Equation (21) allows us to build the transient process of the SACS in relative time $\tau ={\Omega }_{0}t$. According to the theorem of time-scale compression, the process’s form remains unchanged with a change in the time scale, discussed by Besekersky, V.A. and Popov, E.P. [20]. In the first step, this allows us to search for the control law parameters to abstract from the speed requirements of the SACS and solve the problem of obtaining the required transient process form independently of the speed requirements. If the quality of the transient process meets the requirements of its form, the required speed of the transient process can be ensured by appropriately choosing the value of ${\Omega }_0$, i.e., by changing the time scale of the process.

5. Optimal Synthesis of the Form of the Transient Process

As shown above in paragraph 4, for optimal synthesis of the transient process form of SACS, it is necessary to solve the first subtask of obtaining transient process optimal form in relative time $\tau ={\Omega }_{0}t$. As a criterion for the optimality of the control law parameters, consider the maximum stability degree of the SACS. For a stable system, where all roots of the normalized characteristic equation have negative real parts, the stability degree of the SACS is defined as the absolute value of the real part of the root closest to the imaginary axis:

$$\eta =\underset{1\le i\le 6}{min}\left|Re{p}_i\right|,$$

(23)

where $p_i=p_i\left(D,K\right),\left(i=\overline{1,6}\right)$ are functions of the control law parameters. The maximum stability degree for the optimal system is defined as

$${\eta }^0=\underset{D,K}{max}\eta \left(D_{\tau }^0,K_{\tau }^0\right),$$

(24)

where $D_{\tau }^0,K_{\tau }^0$ are the optimal values of the control law parameters in the relative time $\tau$.

For the constraints on the control moments and angular velocities of the reaction wheels, it is necessary to consider their maximum absolute values in real time t:

$${|T}_j^m|\le {|T}_{jp}^m|,{|\omega }_{Wj}^m|\le {|\omega }_{Wj}^{mp}|,(j=\overline{1,3}),$$

(25)

where ${|T}_{jp}^m|,{|\omega }_{Wj}^{mp}|\left(j=\overline{1,3}\right)$ are the highest values of the control torques and the angular velocities of the reaction wheels achievable within the physical characteristics of the actuators. It is known from Besekersky, V.A. and Popov, E.P. [20] that the maximum stability degree of the SACS is achieved if the roots of the normalized characteristic Equation (21) are real, negative, and multiple: $p_i=-1,(i=\overline{1,6})$, i.e., the dimensionless coefficients of the normalized characteristic Equation (21) must be binomial and equal to

$$A_5=6,A_4=15,A_3=20,A_2=15,A_1=6.$$

(26)

It is possible to determine the optimal values of the control law parameters $D_{\tau }^0=d_{\tau 1}^0,d_{\tau 2}^0,d_{\tau 3}^0,K_{\tau }^0=k_{\tau 1}^0,k_{\tau 2}^0,k_{\tau 3}^0$ using the values of (26) for the coefficients of the normalized characteristic Equation (21), which correspond to the optimal form of the transient processes in the SACS in the relative time $\tau$. To do this, it is necessary to solve a system of six nonlinear algebraic Equation (16) with respect to the six unknown parameters of the control law $d_1,d_2,d_3,k_1,k_2,k_3$, using the values of the coefficients of the normalized characteristic equation:

$$\begin{array}{cc}\hfill 6& ={\displaystyle \frac{d_1}{I_{S1}}}+{\displaystyle \frac{d_2}{I_{S2}}}+{\displaystyle \frac{d_3}{I_{S3}}},15={\displaystyle \frac{d_1{d}_2}{I_{S1}{I}_{S2}}}+{\displaystyle \frac{d_1{d}_3}{I_{S1}{I}_{S3}}}+{\displaystyle \frac{d_2{d}_3}{I_{S2}{I}_{S3}}}+{\displaystyle \frac{k_1}{2I_{S1}}}+{\displaystyle \frac{k_2}{2I_{S2}}}+{\displaystyle \frac{k_3}{2I_{S3}}},\hfill \\ \hfill 20& ={\displaystyle \frac{d_1}{2I_{S1}}}\left({\displaystyle \frac{k_2}{I_{S2}}}+{\displaystyle \frac{k_3}{I_{S3}}}\right)+{\displaystyle \frac{d_2}{2I_{S2}}}\left({\displaystyle \frac{k_1}{I_{S1}}}+{\displaystyle \frac{k_3}{I_{S3}}}\right)+{\displaystyle \frac{d_3}{2I_{S3}}}\left({\displaystyle \frac{k_1}{I_{S1}}}+{\displaystyle \frac{k_2}{I_{S2}}}\right)+{\displaystyle \frac{d_1{d}_2{d}_3}{I_{S1}{I}_{S2}{I}_{S3}}},\hfill \\ \hfill 15& ={\displaystyle \frac{k_1{k}_2}{4I_{S1}{I}_{S2}}}+{\displaystyle \frac{k_1{k}_3}{4I_{S1}{I}_{S3}}}+{\displaystyle \frac{k_2{k}_3}{4I_{S2}{I}_{S3}}}+{\displaystyle \frac{d_1{k}_2{d}_3+k_1{d}_2{d}_3+d_1{d}_2{k}_3}{2I_{S1}{I}_{S2}{I}_{S3}}},\hfill \\ \hfill 6& ={\displaystyle \frac{k_1{k}_2{d}_3+d_1{k}_2{k}_3+k_1{d}_2{k}_3}{4I_{S1}{I}_{S2}{I}_{S3}}},1={\displaystyle \frac{k_1{k}_2{k}_3}{8I_{S1}{I}_{S2}{I}_{S3}}}.\hfill \end{array}$$

(27)

The system of six nonlinear algebraic equations with six unknowns (27) generally does not have an analytical solution and, therefore, is solved using iterative methods. The drawback of these methods is the dependence of the obtained solution on the choice of the initial approximation of values of unknowns; thus, obtaining an accurate solution using iterative methods is not guaranteed. Due to this, an analytical method for determining the optimal values of control law parameters is proposed below.

6. Analytical Method for Determining the Optimal Values of the Parameters of the Control Law

Let us show that the analytical solution of the system of Equation (27) with respect to the optimal values of control law parameters can be obtained by taking into account properties of matrix $A^{*}$ (14) of truncated system (13). Indeed, from the form of the matrix $A^{*}$, it follows that by permuting the rows, it can be transformed into a quasi-diagonal matrix:

$$\begin{array}{cc}\hfill A^{**}& =\left[\begin{array}{ccccccc}0& {\displaystyle \frac{1}{2}}& 0& 0& 0& 0& 0\\ -{\displaystyle \frac{d_1}{I_{S1}}}& -{\displaystyle \frac{k_1}{I_{S1}}}& 0& 0& 0& 0& 0\\ 0& 0& 0& {\displaystyle \frac{1}{2}}& 0& 0& 0\\ 0& 0& -{\displaystyle \frac{d_2}{I_{S2}}}& -{\displaystyle \frac{k_2}{I_{S2}}}& 0& 0& 0\\ 0& 0& 0& 0& 0& {\displaystyle \frac{1}{2}}& 0\\ 0& 0& 0& 0& -{\displaystyle \frac{d_3}{I_{S3}}}& -{\displaystyle \frac{k_3}{I_{S3}}}& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right],X^{*}=\left[\begin{array}{c}{\omega }_1\\ q_1\\ {\omega }_2\\ q_2\\ {\omega }_3\\ q_3\\ q_4\end{array}\right]\hfill \end{array}$$

(28)

Therefore, the normalized characteristic polynomial of the matrix $A^{**}$ can be represented as

$$\det \left(A^{**}-pE\right)=\prod _{i=1}^3\left(p^2+{\displaystyle \frac{d_i}{I_{Si}}}p+{\displaystyle \frac{k_i}{{2I}_{Si}}}\right)$$

(29)

The factorization of the characteristic polynomial (15) of the truncated system (13) into three identical factors of the form (29) means that in the case of zero initial conditions for the total kinetic moment of the satellite and the reaction wheels, the rotational motions of the satellite around the three coordinate axes are independent of each other. This makes it possible to decompose the problem of synthesizing the parameters of the SACS into three subproblems of synthesizing second-order linear control systems with identical normalized characteristic equations of the form

$$p^2+A_{1}p+1=0,$$

(30)

where $A_1={\displaystyle \frac{d_j}{I_{Sj}}},1={\displaystyle \frac{k_j}{2I_{Sj}}},(j=\overline{1,3})$. With multiple real, negative roots of the normalized characteristic Equation (21) $p_i=-1,(i=\overline{1,6})$, the unknown coefficient of the characteristic Equation (30) is equal to $A_1=2$. From here, desired optimal values of control law parameters, which correspond to the maximum degree of stability ${\eta }^0$ of SACS, are easily determined and equal to

$$d_{\tau j}^0=k_{\tau j}^0=2I_{Sj},(j=\overline{1,3)}.$$

(31)

These values of the control law parameters ensure the optimal shape of the transient process according to the criterion of the maximum stability degree $\eta$ in relative time $\tau$. When returning to the previous complex variable, i.e., when returning to real time t, the normalized characteristic Equation (30) takes the form

$$s^2+2{\Omega }_{0}s+{\Omega }_0^2=0.$$

(32)

Comparing the coefficients of the characteristic Equations (30) and (32), it follows that the sought optimal values of the control law parameters, corresponding to the maximum stability degree ${\eta }^0$ (24) of the SACS in real time t, should be equal to

$$d_j^0=d_{\tau j}^0·{\Omega }_0,k_j^0=k_{\tau j}^0·{\Omega }_0^2,\left(j=\overline{1,3}\right).$$

(33)

7. Consideration of Constraints on Control Torques and Angular Velocities of Reaction Wheels

Based on the optimal values of the control law parameters (31), we can construct the transient process and determine the highest values of the control moments and angular velocities of the reaction wheels $T_j^{\tau }$ and ${\omega }_{Wj}^{\tau }\left(j=\overline{1,3}\right)$ in relative time $\tau$. At the same time, constraints on the maximum values of reaction wheels’ control moments and angular velocities (25) in the transient process must be carried out in real time t. Therefore, it is necessary to determine the relationships between the angular accelerations, control moments $T_j^{\tau }$ and $T_j\left(j=\overline{1,3}\right)$, and the angular velocities ${\omega }_{Wj}^{\tau }$ and ${\omega }_{Wj}\left(j=\overline{1,3}\right)$ of the reaction wheels in relative time $\tau$ and real time t. To determine these relationships, we use the known solution of a linear, homogeneous, second-order differential equation describing the rotational motion of a rigid body with the characteristic Equation (32) in real time t by Wirkus, A.S. and Swift, J.R. [16]:

$$\phi \left(t\right)=e^{{-\Omega }_{0}t}·{\phi }_0\left(C_1+C_{2}t\right),t\in \left[0,\infty \right),$$

where $\phi \left(t\right)$ is the angular position of the body around the axis of rotation, $\phi \left(0\right)={\phi }_0$. For initial conditions, where the initial kinetic moment of the body is zero, the solutions in real and relative time, respectively, are

$$\begin{array}{c}\hfill \phi \left(t\right)=e^{-{\Omega }_{0}t}·{\phi }_0\left(1+{\Omega }_{0}t\right),\\ \hfill \omega \left(t\right)=-{\Omega }_0{\phi }_0{e}^{{-\Omega }_{0}t}t,\\ \hfill \dot{\omega }\left(t\right)={\Omega }_0^2{\phi }_0{e}^{{-\Omega }_{0}t}\left({\Omega }_{0}t-1\right),t\in \left[0,\infty \right).\end{array}$$

(34)

and

$$\begin{array}{c}\hfill \phi \left(\tau \right)=e^{-\tau }·{\phi }_0\left(1+t\right),\omega \left(\tau \right)=-{\phi }_0{e}^{-\tau }·\tau ,\\ \hfill \dot{\omega }\left(\tau \right)={\phi }_0{e}^{-\tau }\left(\tau -1\right),\tau \in \left[0,\infty \right).\end{array}$$

(35)

The maximum absolute values of the angular velocities and accelerations of the body in real time t and relative time $\tau$ are determined by the following expressions:

$$\begin{array}{c}\hfill {\omega }^m=\underset{0\le t<\infty }{max}\left|\omega \left(t\right)\right|={\left|\omega \left(t\right)\right|}_{{\Omega }_{0}t=1}={\Omega }_0{\phi }_0{e}^{-1},\\ \hfill {\omega }_{\tau }^m=\underset{0\le \tau <\infty }{max}\left|\omega \left(\tau \right)\right|={\left|\omega \left(\tau \right)\right|}_{\tau =1}={\phi }_0{e}^{-1},\end{array}$$

(36)

and

$$\begin{array}{c}\hfill {\dot{\omega }}^m=\underset{0\le t<\infty }{max}\left|\dot{\omega }\left(t\right)\right|={\left|\dot{\omega }\left(t\right)\right|}_{t=0}={\Omega }_0^2{\phi }_0,\\ \hfill {\dot{\omega }}_{\tau }^m=\\ \hfill \underset{0\le \tau <\infty }{max}\left|\dot{\omega }\left(\tau \right)\right|={\left|\dot{\omega }\left(\tau \right)\right|}_{\tau =0}={\phi }_0.\end{array}$$

(37)

Comparing the expressions for the angular velocities and accelerations of the body in real time and relative time in Equations (36) and (37), we obtain

$${\omega }^m={\Omega }_0·{\omega }_{\tau }^m,{\dot{\omega }}^m={\Omega }_0^2{\dot{\omega }}_{\tau }^m.$$

(38)

Let us use the equalities in (38) to determine the relationship between the highest absolute values of angular velocities and control torques of reaction wheels in relative time and real time: $\left|{\omega }_{Wj}^{\tau m}\right|$ and $\left|{\omega }_{Wj}^m\right|,\left|T_{j\tau }^m\right|$ and $\left|T_j^m\right|,\left(j=\overline{1,3}\right)$. For the absolute values of the angular velocities and control moments of the reaction wheels, considering Equation (38), we have

$$\begin{array}{c}\hfill \left|{\omega }_{Wj}\right|={\Omega }_0\left|{\omega }_{Wj}^{\tau }\right|,\\ \hfill \left|T_j\right|=I_{Wj}\left|{\dot{\omega }}_{Wj}\right|=I_{Wj}\left|{\Omega }_0^2{\dot{\omega }}_{Wj}^{\tau }\right|={\Omega }_0^2{I}_{Wj}\left|{\dot{\omega }}_{Wj}^{\tau }\right|={\Omega }_0^2\left|T_{j\tau }\right|,\left(j=\overline{1,3}\right).\end{array}$$

(39)

Hence, for the highest absolute values of the angular velocities and control moments of the reaction wheels in relative time and real time, we have

$$\left|{\omega }_{Wj}^m\right|={\Omega }_0\left|{\omega }_{Wj}^{\tau m}\right|,\left|T_j^m\right|={\Omega }_0^2\left|T_{j\tau }^m\right|,\left(j=\overline{1,3}\right).$$

(40)

From the equations in (40), it follows that when transitioning from relative time to real time, the highest absolute values of the angular velocities of the reaction wheels increase by ${\Omega }_0$ times and the highest absolute values of the control moments of the reaction wheels increase by ${\Omega }_0^2$ times. The value of the transition scale ${\Omega }_0$ between relative time $\tau$ and real time t, considering the equations in (40), can be determined using the following algorithm:

(1)

Solve the system of nonlinear differential Equation (6) or the system of linear differential equations with variable parameters (7) under given nonzero initial conditions at the optimal values of the control law parameters (31) in relative time $\tau$.

(2)

Construct graphs of the obtained transient processes for the control moments $T_{j\tau },\left(j=\overline{1,3}\right)$ and the angular velocities ${\omega }_{Wj}^{\tau },\left(j=\overline{1,3}\right)$ of the reaction wheels in relative time $\tau$.

(3)

Determine the highest absolute values of the control moments $\left|T_{j\tau }^m\right|,\left(j=\overline{1,3}\right)$ and the angular velocities ${|\omega }_{Wj}^{\tau m}|,(j=\overline{1,3})$ of the reaction wheels on the obtained graphs of the transient processes in relative time $\tau$.

(4)

Determine the achievable maximum absolute values of the control moments ${|T}_{jp}^m|,(j=\overline{1,3})$ and the angular velocities $\left|{\omega }_{Wj}^{mp}\right|,\left(j=\overline{1,3}\right)$ of the reaction wheels in real time t from a technical characteristic of the drives.

(5)

Determine the values of the transition scale ${\Omega }_0$ from relative time $\tau$ to real time t using Equation (40):

$${\Omega }_{0j}^{\left(1\right)}={\displaystyle \frac{\left|{\omega }_{Wj}^{mp}\right|}{\left|{\omega }_{Wj}^{\tau m}\right|}},{\Omega }_{0j}^{\left(2\right)}=\sqrt{{\displaystyle \frac{\left|T_{jp}^m\right|}{\left|T_{j\tau }^m\right|}}},\left(j=\overline{1,3}\right).$$

(41)

(6)

We choose the smallest from the six obtained values of the transition scale ${\Omega }_0$, which ensures that all constraints are met (25):

$${\Omega }_0=\underset{1\le j\le 3}{min}\left\{{\Omega }_{0j}^{\left(1\right)},{\Omega }_{0j}^{\left(2\right)}\right\}.$$

(42)

(7)

We determine the optimal values of the parameters of the control law (33) for real time t and solve the system of differential Equation (6) or differential Equation (7) with given non-zero initial conditions in real time t based on the selected scale value ${\Omega }_0$.

(8)

We plot the obtained transients and determine the maximum absolute values of the control moments ${|T}_j^m|,(j=\overline{1,3})$ and angular velocities ${|\omega }_{Wj}^m|,(j=\overline{1,3})$ of the reaction wheels in real time t.

(9)

We check fulfillment of conditions (25) for constraints of the optimal synthesis problem and compare the minimum values of ${\Omega }_{0j}^{\left(1\right)}$ and ${\Omega }_{0j}^{\left(2\right)}$ for $j=\overline{1,3}$.

(10)

If the minimum values of ${\Omega }_{0j}^{\left(1\right)}$ and ${\Omega }_{0j}^{\left(2\right)}$ for $j=\overline{1,3}$ are equal to each other, this means that limits on the maximum angular velocity and the maximum control torques are reached simultaneously and we accept this value as transition scale.

(11)

If the minimum values of ${\Omega }_{0j}^{\left(1\right)}$ and ${\Omega }_{0j}^{\left(2\right)}$ for $j=\overline{1,3}$ are not equal, this means that restriction is achieved either only according to angular velocity or only according to the maximum control torque. In this case, it is possible to optimize reaction wheels’ inertia moments. Here, we accept the minimum transition scale among the minimum values of ${\Omega }_{0j}^{\left(1\right)}$ and ${\Omega }_{0j}^{\left(2\right)}$ for $j=\overline{1,3}$.

8. Numerical Example

In this example, a rigid microsatellite with three reaction wheels installed in three body-fixed axes is the researched object. Data for a rigid microsatellite are as follows (Zhou, H., et al. [9]): $I_S=\mathrm{diag}\left\{I_{S1}=6.63,I_{S2}=8.90,I_{S3}=9.63\right\}{\mathrm{kgm}}^2$, $I_W=\mathrm{diag}\left\{I_{W1}=I_{W2}=I_{W3}=0.000169\right\}{\mathrm{kgm}}^2$. Maximum absolute values of control torques and angular velocities of reaction wheels are ${|T}_{jp}^m|=5.05·{10}^{-3}\mathrm{Nm}(j=\overline{1,3})$ and $\left|{\omega }_{Wj}^{mp}\right|=710\mathrm{rad}/\mathrm{s},\left(j=\overline{1,3}\right)$. Initial conditions are $q_1\left(0\right)=q_2\left(0\right)=q_3\left(0\right)=q_4\left(0\right)=1/2;\overline{\omega }\left(0\right)=0;{\overline{\omega }}_W\left(0\right)=0$. The optimal values (31) of parameters $d_{\tau j}^0,k_{\tau j}^0(j=\overline{1,3)}$ of the control law are determined based on the requirement that the normalized characteristic equation $det\left(A^{*}-sE\right)=0$ has multiple roots $p_i=-1,(i=\overline{1,6})$ or the binomial coefficients in (26). Reaction wheels’ maximum absolute values of the control moments $\left|T_{j\tau }^m\right|\left(j=\overline{1,3}\right)$ and angular velocities ${|\omega }_{Wj}^{\tau m}|(j=\overline{1,3})$ are determined from the graphs of the solution of the system of differential Equation (7) with given non-zero initial conditions at optimal values of the parameters of the control law (31) in relative time $\tau$, shown in Figure 1a. As a result of steps 4–6 of the algorithm, the transition scale from relative time to real time was determined to be ${\Omega }_0=0.022899$ and the optimal values of the control law parameters (33) were determined based on this value. Graphs of transients for control moments and angular velocities of reaction wheels in real time t, obtained by solving the system of differential Equation (7) with given non-zero initial real-time t conditions, are shown in Figure 1b. Based on the results of steps 7–8 of the algorithm, the optimal values of control law parameters as well as the maximum absolute values of control moments and angular velocities are determined, which are presented in

Table 1. The optimal values of control law parameters and the maximum absolute values of control moments and angular velocities.

j	1	2	3
$\left|{\omega }_{Wj}^{\tau m}\right|$, rad/s	15,614.3	20,960.4	22,679.6
$\left|T_{j\tau }^m\right|$, Nm	6.63	8.9	9.63
${\Omega }_{0j}^{\left(1\right)}$	0.04547	0.03387	0.03131
${\Omega }_{0j}^{\left(2\right)}$	0.027599	0.02382	0.022899
$d_j^0$, kgm2	0.30365	0.40762	0.44105
$k_j^0$, kgm2	0.00695	0.00933	0.0101
$\left|{\omega }_{Wj}^m\right|$, rad/s	357.448	479.832	519.189
$\left|T_j^m\right|$, Nm	$3.47678·{10}^{-3}$	$4.66717·{10}^{-3}$	$5.05·{10}^{-3}$

.

An analysis of the data in Table 1 shows that all conditions (25) for the constraints of the optimal synthesis problem are satisfied. At the same time, as follows from the comparison between the values of ${\Omega }_{0j}^{\left(1\right)}$ and ${\Omega }_{0j}^{\left(2\right)}$ from the third and fourth rows of Table 1, the constraints associated with the maximum values of angular velocity ${|\omega }_{Wj}^{\tau m}|(j=\overline{1,3})$ are fulfilled with a margin of $0.03131/0.022899=1.36$ times. This allows reducing the moments of inertia of the reaction wheels by a factor of $1.36$ to $0.000169/1.36=0.000124{\mathrm{kgm}}^2$. Graphs of transient processes for reaction wheels’ control moments and angular velocities in real time t, obtained by solving the system of differential Equation (7) with lower values of reaction wheels’ inertia moments, $I_W=diag\left\{I_{W1}=I_{W2}=I_{W3}=0.000124\right\}{\mathrm{kgm}}^2$, are shown in Figure 1c. The results of steps 7–8 of the algorithm are presented in

Table 2. The optimal values of control law parameters and the maximum absolute values of control moments and angular velocities.

j	1	2	3
$\left|{\omega }_{Wj}^{\tau m}\right|$, rad/s	21,348.8	28,658.2	31,008.9
$\left|T_{j\tau }^m\right|$, Nm	6.63	8.9	9.63
${\Omega }_{0j}^{\left(1\right)}$	0.03326	0.02477	0.022899
${\Omega }_{0j}^{\left(2\right)}$	0.02759	0.02382	0.022899
$d_j^0$, kgm2	0.30365	0.40762	0.44105
$k_j^0$, kgm2	0.00695	0.00933	0.0101
$\left|{\omega }_{Wj}^m\right|$, rad/s	488.372	655.582	709.355
$\left|T_j^m\right|$, Nm	$3.47678·{10}^{-3}$	$4.66717·{10}^{-3}$	$5.05·{10}^{-3}$

.

An analysis of the data in Table 2 shows that all conditions (25) for constraints of the optimal synthesis problem are met. At the same time, as follows from the comparison of values of ${\Omega }_{0j}^{\left(1\right)}$ and ${\Omega }_{0j}^{\left(2\right)}$ from the third and fourth rows of Table 2, constraints associated with the maximum values of angular velocity are performed simultaneously with the constraints associated with the maximum control moments, and this is achieved by reducing reaction wheels’ required moment of inertia.

A comparison of data in Table 1 and Table 2, and graphs of transient processes in Figure 1b,c, show that with a 1.36-fold decrease in the reaction wheels’ moments of inertia $I_{Wj}\left(j=\overline{1,3}\right)$, the maximum values of reaction wheels’ angular velocities ${|\omega }_{Wj}^{\tau m}|(j=\overline{1,3})$ in relative time $\tau$ increase by 1.36 times. The maximum control moments $\left|T_{j\tau }^m\right|\left(j=\overline{1,3}\right)$ remain unchanged. The scales of ${\Omega }_{0j}^{\left(1\right)}\left(j=\overline{1,3}\right)$ decreased by 1.36 times, and scales of ${\Omega }_{0j}^{\left(2\right)}\left(j=\overline{1,3}\right)$ remained unchanged. The optimal values of control law parameters $d_j^0,k_j^0\left(j=\overline{1,3}\right)$ remained unchanged. The maximum values of control moments $\left|T_j^m\right|\left(j=\overline{1,3}\right)$ remained unchanged. The maximum values of reaction wheels’ angular velocities $\left|{\omega }_{Wj}^m\right|\left(j=\overline{1,3}\right)$ in real time increased by 1.36 times and reached boundary value simultaneously with $\left|T_j^m\right|\left(j=\overline{1,3}\right)$ for the maximum control torque. At the same time, as follows from Figure 1b,c, the angular velocities of the satellite ${\omega }_j\left(t\right)\left(j=\overline{1,3}\right)$ in Figure 1b,c as well as the control moments $T_j\left(t\right)\left(j=\overline{1,3}\right)$ in Figure 1b,c coincide in both cases. As for the shapes of transient processes in angular velocities of reaction wheels ${\omega }_{Wj}\left(t\right)\left(j=\overline{1,3}\right)$, they are also the same in both cases but their values for each real-time value t increase by 1.36 times. Thus, the algorithm allows minimizing the required values of reaction wheels’ inertia moments, without violating constraints of the problem of optimal synthesis of SACS control law parameters.

9. Conclusions

(1)

The problem of optimal synthesis of control law parameters of a nonlinear SACS under constraints associated with reaction wheels’ saturation effect is formulated. The maximum stability degree of SACS is considered as a criterion for the optimality of control law parameters, and the maximum absolute values of control moments and angular velocities of reaction wheels, which are achievable within drives’ physical capabilities, are considered as constraints. A linear model of SACS, obtained by authors of the article by equivalent conversation of the original nonlinear model, was used to solve the problem.

(2)

Decomposition of the problem of synthesizing control law parameters into two subtasks is carried out: obtaining required form of transient process and obtaining required rapidity of SACS. To solve the decomposition problem, the concept of a normalized characteristic equation of SACS was used, which made it possible to solve the subtask of obtaining the required form of transient process, regardless of the subtask of obtaining required rapidity.

(3)

The method for synthesis of the optimal form of the transient process of SACS according to criterion of the maximum stability degree has been developed. The synthesis method uses the optimal location of normalized characteristic equation roots of SACS according to criterion of the maximum stability degree. According to known roots of the normalized characteristic equation, its dimensionless coefficients are determined, which are included in a system of six nonlinear algebraic equations with respect to six optimal values of control law parameters of SACS. The synthesis method provides its global asymptotic stability and required control quality indicators, along with optimization of system parameters.

(4)

An analytical method is proposed for solving a system of nonlinear algebraic equations with six unknown parameters of control law relative to the desired optimal values of control law parameters. The method is based on the use of features of linear differential equations system matrix structure of dynamics of SACS. The optimal values of control law parameters according to criterion of the maximum stability degree are expressed analytically, directly through the known dynamic parameters of the system.

(5)

A method taking into account constraints on reaction wheels’ control moments and angular velocities in the problem of control law parameters optimal synthesis, based on use of the transition scale from relative time to real time has been developed. The relations between angular accelerations, control moments, and angular velocities of reaction wheels in relative time and real time are found, as well as the corresponding relations for the maximum absolute values of reaction wheels’ angular velocities and control moments in relative time and real time.

(6)

An algorithm calculating the transition scale from relative time to real time, which ensures the fulfillment of conditions of all constraints in the problem of control law parameters’ optimal synthesis of SACS, has been developed. Simultaneously, while calculating transition scale from relative time to real time, the algorithm allows minimizing required values of reaction wheels’ inertia moments, without violating constraints of the problem of optimal synthesis of SACS control law parameters.

(7)

An example of solving the optimal synthesis problem of SACS is considered, the results of which demonstrate the wide possibilities of proposed methods and algorithm for technical characteristics-targeted selection of engines used, taking into account constraints. It is shown that the constraint accounting method allows minimization of the reaction wheels’ required inertia moment from condition achievement of the maximum values of reaction wheels’ control moments and angular velocities simultaneously.