JP2014153928A - Design method for gain scheduling controller - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a design method for a controller for a three-axial attitude control problem that globally secures stability and control performance.SOLUTION: A design method includes a characteristic formulation process of representing dynamic and kinematic characteristics of a system as an object of attitude control with a state equation based upon a linear parameter variable control theory; a simplification process of simplifying the state equation; and a design process of designing a gain scheduling controller based upon the simplified state equation. In the characteristic formulation process, the characteristics are represented with a state equation comprising a first partial coefficient matrix such that a coefficient matrix of a state variable changes with a scheduling variable related to a dynamic equation, a second partial coefficient matrix such that a coefficient matrix changes with a scheduling variable related to a kinematic equation, and a third partial coefficient matrix such that a coefficient matrix depends on neither of them. In the simplification process, parameter-dependent base conversion processing for performing base conversion on the state equation to convert the second partial coefficient matrix into a unit matrix is performed, and a state variable of a state equation after the processing is replaced with a virtual state quantity.

Description

  The present invention relates to a design method of a gain scheduling controller based on a linear parameter variable control theory.

≪LPV control theory≫
As an automatic control theory that theoretically supports feedback control, first, a classical control theory that designs a one-input one-output system in the frequency domain was established, and then a multi-input multi-output system designed in the time domain using state variables. Control theory was established. In modern control theory, the dynamic characteristics of the system to be controlled are expressed by a linear time-invariant state equation, but the actual control object is nonlinear and parameter-dependent, so the difference between theory and reality has become a problem. When a nonlinear and parameter-dependent system is to be controlled, it is often difficult to cover the entire operating range with a single feedback controller (hereinafter simply referred to as a controller). Cannot be obtained.

  Therefore, a linear parameter-variable control theory (LPV control theory) has been proposed as a control theory for handling control objects including nonlinear elements. In the LPV control theory, a controller (hereinafter referred to as an end point controller) having suitable characteristics at the end points of the variable range of each parameter, that is, at the upper limit and the lower limit, for some variable parameters of the system is designed in advance. The control target is represented by a linear system at each end point. Then, each end point controller is designed by applying linear control theory.

  In an operation stage in which feedback control is executed for a system in which the end point controller is designed, the variable parameter varies with time. Therefore, the value of the variable parameter is sequentially acquired, and a suitable controller corresponding to the acquired value of the variable parameter is calculated. This preferred controller calculates by the convex combination of end point controllers. Then, the gain of the controller changes corresponding to the change of the variable parameter with the passage of time, and the system is controlled by the controller. As described above, the method of controlling the system by changing the gain of the controller in accordance with the change of the variable parameter is called gain-scheduled control, and the variable parameter is also called a scheduling variable. LPV control is a technique for realizing gain scheduling control. In recent years, application of LPV control to various control plants has been studied.

In modern control theory, assuming that the control target is usually modeled by the following equation (1), a controller design method that feeds back state variables to the modeled system as in equation (2) is used. provide.
Here, x and u are vectors, x is a state variable, and u is an input of the system. A, B, and K are coefficient matrices. Equation (1) is a state equation representing the dynamic characteristics of the system, and Equation (2) represents a controller that feeds back the state variable x to the input u of the system. That is, K is the gain of the controller.

On the other hand, in LPV control theory, the control target is usually modeled by the following equation (3), and controller design that feeds back state variables to the modeled system as in equation (4) Apkarian, P., Gahinet, P., and Becker, G., “Self-scheduled H∞ Control of Linear Parameter-Varying Systems: a Design Example,” Automatica, Vol. 31, No. 9, 1995, pp. 1251-1261).

  Expression (3) differs from expression (1) in that the coefficient matrix A depends on the parameter κ. Similarly, the equation (4) differs from the equation (2) in that the controller gain K depends on the parameter κ. In general, κ is a vector composed of a plurality of parameters. The change of A according to κ represents the nonlinear and parameter-dependent characteristics of the system. Hereinafter, the dynamic characteristic of the system expressed in the form of Equation (3) is referred to as an LPV model or LPV system expression. Equation (4) represents gain scheduling control for changing K in accordance with κ. If A changes according to κ, it is reasonable to change the controller gain K also according to κ. Hereinafter, in order to clarify that the controller is based on gain scheduling control, the controller may be referred to as a GS controller.

≪Polytope expression≫
The LPV control theory provides a method for determining a suitable controller, that is, the controller gain K (κ) of equation (4) given the LPV model of equation (3) and the variation range of parameter κ. Let us touch on an important concept in designing the controller gain K (κ).

κ is generally a multidimensional variable. For ease of understanding, let κ be a three-dimensional variable such as equation (5).

And we know the minimum and maximum values that each element α, β, γ of κ can take,
Suppose that At this time, an arbitrary point κ exists in a hexahedron in which the coordinate values of each axis satisfy Equation (6) when the three-dimensional coordinates α, β, and γ are considered. Κ can be expressed in the form of a convex combination of the eight vertices (end points) of the hexahedron, as shown in Equation (7) below. This is called polytope (polyhedron) expression.

Here, the following relationship is established for the coupling coefficient λ _i .

As described above, when κ is expressed by the convex combination of the end points, each matrix A (κ) and B in Expression (3) is expressed as a polytope as follows.

In the system represented by Equation (3), the coefficient matrix A (κ) depends on κ and is nonlinear and parameter-dependent. However, if the control plant is expressed as a polytope as in Equation (9), the linear parameter is variable. Can be treated as a system.

Then, if the coefficient matrix A (κ) can be expressed as a polytope as shown in Equation (9), a suitable controller gain K _i is obtained in advance for each coefficient matrix A _{i at} each of the eight end points, and at the operation stage each time Λ _i is obtained according to the parameters α, β, and γ, and the controller gain K is calculated as a convex combination of K _i using the obtained λ _i, and the controller gain K (κ) at that time may be obtained.

≪Spacecraft attitude control≫
In recent years, attitude control of spacecraft has been studied over a wide range of fields. They deal with several types of actuators such as MED (Momentum Exchange Device or angular momentum exchange type devices, such as reaction wheels or control moment gyros) and external torque generators (eg gas jets or magnetic torquers). is there. External torque generators have the disadvantages of limited resources and low torque. Therefore, the present invention mainly focuses on MED that realizes attitude control.

When controlling the attitude of a spacecraft, most control laws use those based on the Lyapunov function (see, for example, Non-Patent Documents 1 to 3). A controller equipped with such a control side can easily guarantee global stability, but in most cases, attitude control considering control performance cannot be realized. For example, when Euler angles are used as kinematic parameters, there are few control laws that take control performance into account, and even if MRP (Modified Rodrigues Parameter) is used as a kinematic parameter, it is not possible to fully consider control performance. Have difficulty.
In order to solve such problems, some studies have applied LPV control theory to the spacecraft attitude control problem (see, for example, Non-Patent Documents 4 to 6).

Tsiotras, P., “Stabilization and Optimality Results for the Attitude Control Problem,” Journal of Guidance, Control, and Dynamics, Vol. 19, No. 4, July-August, 1996, pp. 772-779. Yoon, H., and Tsiotras, P., “Spacecraft Line-of-Sight Control Using a Single Variable-Speed Control Moment Gyro,” Journal of Guidance Control, and Dynamics, Vol. 29, No. 6, 2006, pp. 1295-1308. Chub, H., and Lappas, V., J., “Redundant Reaction Wheel Torque Distribution Yielding Instantaneous L2 Power-Optimal Attitude Control,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 4, July-August , 2009, pp. 1269-1276. Kwon, S., Shimomura, T., and Okubo, H., “Pointing control of spacecraft using two SGCMGs via LPV modeling theory,” Acta Astronautica, Vol. 68, 2011, pp. 1168-1175. Yamamoto, Y., and Shimomura, T., “Attitude Control of Spacecraft with VSCMGs using LPV Modeling Technique,” SICE Annual Conference, August, Tokyo, Japan, 2011, pp. 1676-1681. Gao, R., Ohtsubo, K., and Kajiwara, H., “LPV Design for a Space Vehicle Attitude Control Benchmark Problem,” SICE Annual Conference, August, Fukui, Japan, 2003, pp. 2764-2767.

However, most of these studies deal with the directional control problem (see, for example, Non-Patent Documents 4 and 5), and few studies have realized three-axis attitude control of spacecraft using LPV control theory. Even research that has achieved three-axis attitude control using LPV control theory (see, for example, Non-Patent Document 6) does not deal with spacecraft having MED.
The present invention has been made in consideration of the above circumstances, and by realizing the three-axis attitude control problem of a spacecraft having a plurality of MEDs using LPV control theory, global stability and control are achieved. The purpose is to design a controller that guarantees performance. However, the present invention is not limited to a spacecraft, and can be applied to various control objects represented by LPV models, such as aircraft, ships, automobiles, and robots.

  The present invention includes a characteristic formulating step that expresses the dynamic and kinematic characteristics of a system subject to attitude control by a state equation based on a linear parameter variable control theory, a simplification step that simplifies the state equation, A design process for designing a gain scheduling controller as a state feedback controller based on a simplified state equation, wherein the characteristic formulating step changes a coefficient matrix of a state variable according to a scheduling variable related to a dynamic equation Representing the characteristics of the system by a state equation comprising a first partial coefficient matrix to be processed, a second partial coefficient matrix that varies depending on a scheduling variable related to a kinematic equation, and a third partial coefficient matrix that does not depend on any scheduling variable, The process performs a basis transformation on the state equation to convert the second partial coefficient matrix into a unit matrix. A parameter-dependent basis conversion process, provides a method of designing a gain scheduling controller, characterized by replacing state variables associated kinematic equations virtual state quantity in the state equation of the parameter-dependent post-basis conversion process.

  According to the design method of the gain scheduling controller according to the present invention, the simplification process performs the parameter-dependent basis transformation for transforming the second partial coefficient matrix related to the kinematic equation into a unit matrix and performs the kinematic equation after the basis transformation. Therefore, the state equation is simplified, and the design of the gain scheduling controller based on the state equation is facilitated. That is, since the scheduling variable is reduced by the simplification process, it becomes possible to design a gain scheduling controller excellent in global stability and control performance in consideration of dynamic and kinematic characteristics of the system.

In the present invention, two new concepts of “parameter-dependent basis transformation” and “virtual state quantity” are used to facilitate controller design. “Parameter-dependent basis conversion” is a basis conversion for converting the second partial coefficient matrix into a unit matrix. For the "virtual state quantity", the state variables related to the kinematic equation after basis transformation are replaced with elements that have a differential integral relationship on both sides of the state equation, using the properties included in the null space of the coefficient matrix of the state equation Is. By introducing them, the original control object (kinematics + dynamics) can be handled as a simple LPV model, and the controller design becomes very simple. Controllers designed for simple LPV models automatically guarantee the same stability and control performance for the original control plant.
This invention will be described more specifically.
In the present invention, the system is an object of feedback control, and specific examples include, but are not limited to, spacecraft, aircraft, ships, automobiles, robots, etc., and are based on linear parameter variable control theory. The present invention can be applied to those whose characteristics are expressed by a state equation. In the embodiments described later, examples of spacecraft are described.
The linear parameter variable control theory is known as a control theory that deals with a controlled object including nonlinear elements. When a state equation based on a linear parameter variable control theory is given, a method for designing a state feedback controller for the system is known. However, it is not always possible to design, and the possibility of design depends on the complexity of the state equation. Depends on the number of dimensions of the scheduling variable.
The characteristic formulation process, simplification process, and design process of the present invention are executed by the designer as a whole. A designer usually uses a computer as a tool for efficiently performing each process. In particular, a computer is used in the design process. It should be noted that a series of steps can be automatically processed by a computer.

It is explanatory drawing which shows the structure of MED which concerns on embodiment of this invention. It is explanatory drawing which shows an example of the actuator arrangement | positioning of the spacecraft which concerns on embodiment of this invention. It is a graph which shows the result of the numerical simulation of the feedback system designed by this invention. It shows the transition of angular velocity. It is a graph which shows the result of the numerical simulation of the feedback system designed by this invention. The transition of improved Rodriguez parameters is shown. It is a graph which shows the result of the numerical simulation of the feedback system designed by this invention. It shows the transition of the spin speed of the wheel. It is a graph which shows the result of the numerical simulation of the feedback system designed by this invention. Shows the transition of control input (wheel acceleration). It is a graph which shows the result of the numerical simulation of the conventional feedback system. It shows the transition of angular velocity. It is a graph which shows the result of the numerical simulation of the conventional feedback system. The transition of improved Rodriguez parameters is shown. It is a graph which shows the result of the numerical simulation of the conventional feedback system. It shows the transition of the spin speed of the wheel. It is a graph which shows the result of the numerical simulation of the conventional feedback system. Shows the transition of control input (wheel acceleration).

Hereinafter, preferred embodiments of the present invention will be described.
The characteristic formulating step is a state equation where κ is a scheduling variable related to a dynamic equation and σ is a scheduling variable related to a kinematic equation.
(Ω and σ are state variables, u is the system input, and 0 is the zero matrix)
And the simplification step is a coefficient matrix of equation (10).
Conversion matrix for basis conversion of the second partial coefficient matrix G (σ) to unit matrix
Multiplied by the left side of both sides of equation (10)
And the simplification step replaces G ⁻¹ (σ) σ among the state variables of Equation (13) with a virtual state quantity ζ and does not include σ.
It may be what leads.
Further, each coefficient multiplied by the virtual state quantity ζ in the equation (14) may be zero.
Furthermore, the state variable ω may be an angular velocity, and the state variable σ may be an improved Rodrigues parameter.
σ may be another kinematic parameter such as a vector part of a quaternion.
Further, the gain scheduling controller performs three-axis attitude control by feeding back to three or more actuators, respectively, and the schedule variable is obtained by converting the drive amount of these actuators into the three-axis direction component of the airframe fixed coordinates. There may be. For example, when the actuator is a reaction wheel, the driving amount is a spin angular velocity. In that case, the schedule variable represents a vector of spin angular velocities by three-axis direction components of the aircraft fixed coordinates.
Furthermore, the method may further comprise the step of converting the gain scheduling controller designed for the system represented by the simplified equation of state into the original system to obtain the original system gain scheduling controller.
Preferred embodiments of the present invention include combinations of any of the plurality of embodiments shown here.

Hereinafter, the present invention will be described in more detail with reference to the drawings. In addition, the following description is an illustration in all the points, Comprising: It should not be interpreted as limiting this invention.
The structure of this specification is as follows. First, as a premise of the present invention, a dynamic equation of a control plant (specific example is a spacecraft having a plurality of MEDs as actuators) and a kinematic equation using MRP will be described. Next, consider the LPV model for the spacecraft three-axis attitude control problem. At that time, it is shown that a simple LPV model can be obtained by using parameter dependent basis transformation and virtual state quantity, which are major features of the present invention. Then, a numerical simulation using the LPV model is performed and compared with the conventional method.

≪Dynamic equation and kinematic equation≫
(Dynamic equation)
As described above, in the embodiment, a spacecraft is handled as a control target. It is assumed that the spacecraft is a rigid body and has n MEDs as shown in FIG. Body-fixed coordinate system F _B which is fixed to the body of the spacecraft orthogonal vectors
It is expressed by The i-th MED is arranged in three orthogonal unit vectors as shown in Fig. 1.

Given in. The first unit vector is a wheel spin axis, the second unit vector is a gimbal axis, and the third unit vector is a vector in an axis (torque axis) direction orthogonal to the spin axis and the gimbal axis. Assuming that each MED has the same moment of inertia, the total angular momentum H of the spacecraft can be written as follows.

Where ω is the angular velocity vector of the spacecraft, J is the inertia tensor of the entire spacecraft including MED, and I _ws is the spin axis
Wheel moment of inertia, I _cg is the gimbal axis
The sum of the gimbal structure and the moment of inertia of the wheel,

Is a vector whose components are wheel spin rates of each MED,
Is a vector whose component is the gimbal angle of each MED. In this specification, [] 'represents transposition. G _g and G _s are matrices given as follows.

That, G _g represents the unit vector of the gimbal axis of the MED in three axes of body-fixed coordinate system F _B, is a 3 × n matrix by arranging them. Similarly G _s is 3 × n matrix by arranging expressed in three axes of each unit vector of the spin axis of the MED F _B. These can be said to be a coordinate system conversion matrix for converting a coordinate system fixed to the actuator into a body fixed coordinate system F _B.
When the disturbance torque is not acting, the dynamic equation of the rigid spacecraft can be written as

The term ω × H in equation (17) represents the influence of the motion of the spacecraft axis on the inertial system. Where x × is any 3D vector
Is the matrix defined below.

In equation (18), the operator “x” is defined as an operator that converts an arbitrary three-dimensional vector x into a 3 × 3 matrix, but when an arbitrary three-dimensional vector y follows, the operation of x × y The result is equal to the outer product of the vectors x and y. The point that the outer product is an operator that defines an operation between two vectors, the operator “x” can be said to have defined this as an operator for one vector.

Substituting equation (15) into equation (17),

here,
Means differentiation on the airframe fixed coordinate system F _B. Actuator is a gimbal shaft
, The terms related to the actuator are expanded as follows.

However,

Means differentiation on a gimbal frame, that is, on a coordinate system fixed to the actuator. Second term in the second row of the above formula
Represents the effect of gimbal motion on the aircraft.

The second item of the above formula is shown in FIG.
From the relationship, it becomes like the third line. In the third line,
From the fourth line. Diag [Ω] in the fourth row represents a diagonal matrix having wheel spin rates Ω _{1 to} Ω _n of each MED as elements. Also,
Is used.

From the above, the dynamic equation of the spacecraft is
Is described. By simplifying this, the following dynamic equation is obtained.

However, in the above formula
This term is omitted because it is sufficiently small compared to other terms. You may be wondering if the term of the second derivative of the gimbal angle δ is really small. The following can be said about this. Most actuators originally have a limiter. For example, in CMG (Control Moment Gyro), there is an upper limit on the gimbal angular acceleration.

Equation (19) is a generalized description of the dynamic characteristics of a spacecraft equipped with multiple MED or SGVSCMG (Single-Gimbal Variable-Speed CMG). In equation (19),
(Corresponding to gimbal lock), dynamic characteristics when MED is RW (Reaction Wheel) can be obtained. Similarly,
(Corresponding to a constant wheel spin rate) provides dynamic characteristics when MED is CMG.

[Kinematic equation]
Next, the kinematic equations used in the LPV model are described. Stability spacecraft problem stopping the rotation by converging the body of the angular velocity omega for inertial zero, or result in problems to converge to the target angular velocity omega _r. The behavior of ω can be analyzed using dynamic equations. On the other hand, attitude control is a problem of stopping rotation while the aircraft is in a predetermined attitude, or a problem of causing the attitude to follow a changing target attitude angle. The attitude of the spacecraft is given by the direction of the body-fixed coordinate system F _B against inertial frame F _I. Aircraft deviation of the fixed coordinate system F _B and the inertial coordinate system F _I, i.e. to represent the attitude of the spacecraft is sufficient three-dimensional parameters. In the embodiment, MRP is used as a kinematic parameter (posture parameter). However, the essence of the present invention is not limited to MRP. The present invention can also be applied to other posture parameters such as Cayley-Rodrigues Parameter, 4-dimensional quaternion (also referred to as quaternion, Euler parameter), and generalized Rodrigues parameters.

Euler axis
And rotation angle θ, MRP can be expressed as follows (Non-patent Document 1 and Schaub, H., and Junkins, J., L., “Analytical Mechanics of Space Systems,” AIAA Education Series, AIAA, Reston, VA, Oct. 2003).
For MRP, the singular point is when θ = ± 2π, but such a situation is very unlikely except for tracking problems. The kinematic equation when using MRP can be written as follows.

However,

G (σ) is an inverse matrix except when θ = ± 2π:
It has.
Equation (21) represents the time change of the attitude parameter σ with respect to the angular velocity ω of the spacecraft.

≪LPV modeling and controller design≫
[LPV modeling]
As can be seen from Equation (19), the dynamic equation of the spacecraft is a non-linear equation and is difficult to handle as it is in LPV control theory. Therefore, Equation (19)
Linearize around and get the following LPV system.

First, transform equation (19),

Substituting equation (15) into this,
Under the condition of local linearization around the equilibrium point,
Therefore, if this is substituted into the above equation, the first and second items in parentheses in the first item on the left side will be zero,

It becomes. Or it can be written as follows according to the form of equation (3).
However,

It is. Here, the scheduling parameter κ is a function of the spin rate Ω and the gimbal angle δ. Note that δ is not explicitly expressed in A _d and B _d of the formula (25), but includes G _s and G _t . G _s and G _t is changed value when the gimbal angle δ is changed with respect to the machine body because it a representation in three-axis spin axis and the torque-axis direction of the unit vector a body-fixed coordinate system F _B. Therefore, κ is a function of δ.

When CMG or VSCMG is used as an actuator, since gimbal angle δ is changed, B _d (κ) depends on scheduling variable κ. When B _d (κ) depends on κ, it is difficult to implement gain scheduling control, so it is necessary to insert a pre-filter (Apkarian, P., Gahinet, P., and Becker, G., “Selfscheduled H∞ Control of Linear Parameter-varying Systems: A Design Example,” Automatica, Vol. 31, No. 9, 1995, pp. 1251-1261). In the case of RW, since the gimbal angle δ does not change, the δ derivative becomes zero, and the diag “Ω” of B _d (κ) in the equation (25) is deleted. Therefore, B _d (κ) is a constant matrix and does not depend on the scheduling variable. In the embodiment, the latter case is handled. In this case, the dynamic equation of the spacecraft can be written as

However, the operating region of κ and σ is represented by Κ and Σ. State variables
Then, the state space representation of the spacecraft three-axis attitude control problem is given below.

Or you can write:

However,
It is.

In Equation (28), ω and σ are both three-dimensional vectors, so x in Equation (29) is a six-dimensional vector, and Ω is an n-dimensional vector (where n is the number of MEDs as described above). U is an n-dimensional vector. A _d (κ) and G (σ) are each a 3 × 3 matrix, and B _d is a 3 × n matrix. Although G (σ) is non-linear, it can be covered with a convex hull, so equation (29) corresponding to equation (3) can be treated as an LPV model. In this case, since the dimension of the scheduling variable is the sum of the n dimensions of κ and the three dimensions of σ (n + 3), 2 ^{n + 3} LMIs (Linear Matrix Inequality) when designing the GS controller Must be solved simultaneously, but the κ dimension can be reduced to 3 dimensions in any case by appropriately selecting the elements (see “Reducing the number of dimensions by appropriate selection of scheduling variables” below) .
Therefore, 2 ⁶ = 64 LMIs are solved simultaneously for controller design. However, even if the number of LMIs to be solved simultaneously can be reduced in this way, it is still very difficult to solve 64 LMIs at the same time, and it is often impossible to design a controller.

[Introduction of parameter-dependent basis transformation and virtual state quantity]
Therefore, in the present invention, by introducing the virtual state quantity ζ and the parameter-dependent basis transformation matrix M (σ), the dimension of the scheduling variable is reduced to three dimensions, thereby realizing a simpler controller design.

Here, the virtual state quantity ζ and the parameter-dependent basis transformation matrix M (σ) are respectively defined as follows.
M (σ) is determined so as to convert G (σ) in the matrix of A (κ, σ) in Expression (30) into a unit matrix. The controller design in a similar system basis transformed by M (σ) has three dimensionality of scheduling variables. The controller in this similar system is polytopically expressed by a hexahedron with 2 ³ = 8 end points. Since the controller can be expressed by a convex combination of each end point controller (see “polytope expression” in the “Background Art” column), the controller design results in the design of each end point controller. However, a controller designed with a similar system must be operated after being converted back to the original system.
Performing basis transformation into equation (29) using M (σ) yields the following. Multiply both sides of equation (29) by M (σ) from the left,

Applying equations (31) and (32) to equation (33)
here,

It is. Substituting G ⁻¹ (σ) σ for ζ using the fact that the portion of G ⁻¹ (σ) σ is eliminated by the zero element of the coefficient matrix applied from the left,

Or you can write:

However,
And

Here are the points to keep in mind when performing the basis transformation of the above equation.

To obtain equation (35),
In spite of the
Was replaced by ζ. Such an operation is possible because the structure of the original control plant is used. As you can see from equation (34), it is the lower half of the state variable.

Is always erased by multiplication with the zero matrix.
That is, the lower half of the state variable
Can always be replaced with an arbitrary state variable. So the lower half of the state variable

It is possible to perform an operation for replacing the virtual state quantity ζ with. The original control plant expressed by Eq. (29) can be converted into a simple LPV model by making good use of the structure of the original control plant and introducing parameter-dependent basis conversion and virtual state quantities.

(Controller design)
From here, the GS (Gain-Scheduled) controller is designed for the simple LPV model expressed by Equation (36). That is, the controller is designed on a similar system. Equation (36) corresponds to the model of the control plant expressed by Equation (3). A controller corresponding to Equation (4) is designed for the modeled control plant. Then, by restoring the base transformation, it is shown that the GS controller obtained for Equation (36) is also effective for the original control plant. First, a generalized control plant (Generalized plant) for equations (29) and (36) is defined as follows. In order to consider whether the system can be controlled well when a disturbance enters the generalized control plant, the disturbance (see the following equation (38) and the equation (3)) representing the dynamic characteristics of the control plant. This is the addition of the item w) in (39).

First, a generalized control plant for equation (29)
A simple LPV model on a similar system obtained by introducing basis transformation and virtual state quantity into this, that is, a generalized control plant for Equation (36),

And
Where the coefficient matrix

Is the orthogonal condition
And positive definite condition
Often chosen to satisfy. Also,
Are the disturbance vector, the performance evaluation output for equation (29), and the performance evaluation output for equation (36), respectively.

At this time, the GS controller for the simple LPV model expressed by Equation (36)
To design. In an embodiment, global stability and
Design controllers that guarantee performance. Controller that satisfies these conditions
Is Lyapunov common solution
Is used as a solution to the following convex optimization problem (see “Asymptotic Stability” in the capture description section below).

However,

It is. Equations (40)-(42)
Under the constraints of equations (40)-(42)
Represents an optimization problem that finds the value of a parameter whose trace is lower than In addition, “*” at the upper right of the matrix in Equations (40) and (41) indicates that the transpose of the lower left element of the matrix is included in that portion.

If it is not guaranteed that the controller expressed by the convex combination of each end point controller in all the range that κ can take as shown in Equation (42), stabilizes the control plant at any κ in the polytope. Don't be. This stabilization condition is given by the Lyapunov inequality in equation (85).
If p is the number of endpoints, a simple LPV system is represented by the following polytope expression.
At this time, an endpoint controller that satisfies the following LMI problem:
think of.

Optimal solution
The end point controller
The GS controller is given as follows.

The Lyapunov common solution is used for this GS controller, which may result in a conservative design. Therefore, we design a GS controller using Lyapunov non-common solutions. If Lyapunov non-common solution is used, the following LMI, ie, equations (50) to (52) are satisfied
(See Shimomura, T., and Kubotani, T., “Gain-Scheduled Control under Common Lyapunov Functions: Conservatism Revisited,” Proc. Of 2005 American Control Conference, 2005, pp. 870-875).

Optimal solution
By using, a controller with less maintainability can be obtained. And the end point controller obtained by it
Is used to give a GS controller according to equation (49).

Each end point controller with i = 1,..., p is designed to stabilize the system at each end point. However, although the GS controller obtained in this way guarantees stability at each end point, it does not guarantee stability in the entire region in the polytope. It must be ensured that the controller represented by the convex combination of each endpoint controller as in equation (49) stabilizes the system at any k in the polytope. Global stability in all operational areas,
To guarantee performance,
After the decision, we need to solve the following LMI problem.

When this LMI problem is solvable, the GS controller designed for the simple LPV model expressed by Eq. (36) using the Lyapunov non-common solution satisfies the following LMI problem.

In this way, the GS controller represented by Equation (49) could be designed for a simple LPV model. And by reverting the parameter-dependent basis transformation
The GS controller K (κ, σ) for the original control plant can be obtained.

here,
It is.

Thus, the end point controller for the original control plant expressed by Equation (29) is
Given in. At that time, the GS controller Κ (κ, σ) satisfies the following LMI problem.

However,

Here, considering the structure of M (σ), B
It becomes.

Similarly,
(Usually disturbances only in the dynamic equation)
It is. Even if the equations (53) to (55) are not solvable, the global stability in the entire operation region can be guaranteed by confirming the solvability of the LMI shown below.

This completes the main results of the present invention. When applying to tracking problems, the same discussion should be made using deviation MRP. (For details, see Y. Yamamoto, and T. Shimomura, “Attitude Control of Spacecraft through A Simple LPV Model with A Virtual State Variable,” AIAA Guidance, Navigation, and Control Conference, AIAA 2012-5005)

≪Numerical simulation≫
In order to show the effectiveness of the present invention, numerical simulation results of the conventional method and the method according to the present invention are shown below. As a conventional method, a controller based on Lyapunov function using MRP was used. Table 1 shows the spacecraft parameters, initial conditions, and target attitude. An example of the actuator arrangement is shown in FIG. In the example of FIG. 2, four reaction wheels RW1 to RW4 are arranged as actuators.

Controller design variables
Was set as follows.

When the method according to the present invention (hereinafter referred to as the proposed method, see FIGS. 3 to 6), the attitude control of the spacecraft can be realized in about 100 [seconds]. On the other hand, when the conventional method based on the Lyapunov function (see FIGS. 7 to 10) is used, it takes 200 [seconds] or more to realize the attitude control. Considering that both control inputs are 10 [radians / second ² ], the control performance of the proposed method is excellent.

[Reduction of dimensionality by appropriate selection of scheduling variables]
In the following, it will be shown that controller design is facilitated by appropriately selecting scheduling variables. Considering a spacecraft with n RWs, the scheduling variable κ appears to have a dimension of n (when κ = Ω).
In this case, at least 2 ⁿ end points are required to cover a simple LPV model with a convex hull. Thus, if the scheduling variable is the spin rate Ω of each RW, controller design may be difficult. For example, when an actuator is redundantly arranged in preparation for an actuator failure, the number of end points of the convex hull increases, and it may be difficult to ensure global stability.

However, this problem can be solved by appropriately selecting the scheduling variable κ. By setting the scheduling variable to κ = G _s Ω instead of κ = Ω, the dimension of the scheduling variable can be reduced to 3. Incidentally, G _s is 3 × n matrix by arranging expressed in three axes F _B each unit vector of the spin axis of the RW. At this time, a simple LPV model can be covered with a convex hull with 2 ³ = 8 end points, and it is very easy to guarantee global stability.

(Convex coupling coefficient λ _i (κ))
The following shows how to obtain the convex coupling coefficient of the GS controller. _Let κ _imin and κ _{imax be} the minimum and maximum values of the scheduling variable κ _i . At this time, the scheduling variable κ _i can be expressed as follows.
The convex coupling coefficient λ _i (κ) can be calculated as shown in Table 2.

[Asymptotic stability]
1) Lyapunov stability theory System expressed by equations of state (72) and (73)
We discuss the asymptotic stability (x (t) → 0) from the standpoint of convergence of the Lyapunov function. The Lyapunov function V (t) is a function defined as the following positive definite function.

That is, V (t) takes a positive value for all x that are not zero. Here, a square target matrix P ′ = P corresponding to a positive definite function is called a positive definite matrix.
It is. On the state solution x (t)

If V holds, V (x (t)) decreases monotonically with increasing time.
From the condition of V (x) ≧ 0 shown in Equation (74),
Since P> 0 from equation (75), the state solution x (∞) = 0 must be satisfied in order to satisfy V (x (∞)) = 0. Therefore, the asymptotic stability of the system can be said.

2) Conditions for state asymptotic convergence If there is a positive definite solution P> 0 that satisfies the following matrix inequality, the asymptotic stability of the system is guaranteed.
From equations (72) and (73),
Substituting equation (79) into equation (78),

For x ≠ 0, the condition that the inequality of equation (80) holds is
It is.

3) Stability of variable parameter system For the linear system described above, the variable parameter system expressed by equations (82) and (83)
Discuss the stability. It can be considered that when the parameter κ changes, the Lyapunov function V (t), that is, P also changes (see equation (74)). Lyapunov function for a control plant that varies with κ
Think about ensuring stability.

If this is possible, it can be said that any set of plants corresponding to all possible values of κ is stable. The stability condition is the following inequality corresponding to equation (81) for all possible values of κ:
There is a positive definite matrix P (κ) that satisfies

4) Secondary stability In the parameter variable system, it can be considered that the positive definite matrix P related to the Lyapunov function V (t) changes when the parameter κ originally changes (see equation (84)). However, it is difficult to find P (κ) that changes according to κ. Therefore, we consider ensuring stability using P common to all the systems included in the plant set. If this is possible, the plant set is said to be secondary stable. Although secondary stability is a fairly severe requirement, it is useful in practice.

The secondary stability condition is that P (κ) in equation (85) is replaced with a common P for all values of κ,
It becomes.
5) Stability condition of polytope system Variable parameter system corresponding to equation (82) and expressed as polytope with p end points as follows
A suitable controller when executing feedback control for the above is the concept of LPV control that is calculated as a convex combination of end point controllers. The controller is represented as follows.
Consider the stability in this case.

Substituting equation (88) into equation (87),
Get. The plant stability condition is the following inequality corresponding to equation (85) for all λ _i (κ):
There is a common positive definite matrix P that satisfies

Organizing equation (91)
It becomes.
Here, when λ _i = 1, from equation (90), λ _j = 0 (j ≠ i), so the stability condition of equation (92) is
There is a common positive definite matrix P that satisfies Expression (93) holds for all i, that is, all end points.

Conversely, when equation (93) holds for all i, equation (92) holds because at least one λ _i is not zero. Therefore, the existence of a common P that satisfies the condition of Expression (93) at each end point is a secondary stability condition equivalent to Expression (92).
In Equation (93), _Ki and P are both matrix variables and include the product of both, so Equation (93) is not LMI but BMI (Bilinear Matrix Inequality). Therefore, it is difficult to solve the inequality of equation (93) as it is.

5) Conversion from BMI to LMI BMI is converted to LMI by applying the following variable conversion to equation (92).
First, multiply X: = P ^-1 from both sides of equation (93),
Furthermore, the variable conversion of W _i : = K _i X is performed,
In the formula (95), X and W _i are both but a parameter kappa, does not contain such a product is LMI, it can be solved. When W _i is obtained by solving equation (95), K _i is obtained by K _i = W _i X ⁻¹ .
The upper left element of the matrix inequality of Equation (47) represents the secondary stability condition corresponding to Equation (95), and is a matrix inequality extended to include performance guarantee.

  In addition to the embodiments described above, there can be various modifications of the present invention. These modifications should not be construed as not belonging to the scope of the present invention. The present invention should include the meaning equivalent to the scope of the claims and all modifications within the scope.

The present invention introduced a virtual state quantity ζ and a parameter-dependent basis transformation matrix M (σ) to show a new method for realizing spacecraft attitude control. By using the proposed method, it was shown that the original control plant can be converted into a simple LPV model that can be covered with fewer endpoints. It was shown that the GS controller for the original control plant can be obtained by restoring the basis transformation. It was also shown that a GS controller designed for a simple LPV model automatically guarantees global stability and control performance for the original control plant. By numerical simulation, we compared the proposed method with the conventional method and found that the proposed method shows better control performance. In the embodiment, RW is used as an actuator. However, the present invention can be applied to other actuators (CMG, VSCMG, etc.). In addition, H ₂ controller was designed at the time of controller design, but because this invention can design controller using LMI, other control objectives (H∞ performance, input saturation constraint, etc.) are satisfied at the same time. Controller design is possible.
However, the present invention is not limited to a spacecraft, and can be applied to various control plants expressed by LPV models such as aircraft, ships, automobiles, and robots.

11: Angular momentum exchange type device
RW1, RW2, RW3, RW4: Reaction wheel

Claims (6)

A characteristic formulation process that expresses the dynamic and kinematic characteristics of the system subject to attitude control by a state equation based on the linear parameter variable control theory;
A simplification process for simplifying the equation of state;
A design process for designing a gain scheduling controller as a state feedback controller based on a simplified state equation;
In the characteristic formulating step, the state variable coefficient matrix is changed to a first partial coefficient matrix that changes according to a scheduling variable related to a dynamic equation, a second partial coefficient matrix that changes depending on a scheduling variable related to a kinematic equation, and any scheduling variable. Represents the characteristics of the system with a state equation consisting of a third partial coefficient matrix that does not depend on
The simplification step includes a parameter-dependent basis transformation process for performing basis transformation on the state equation to convert the second partial coefficient matrix into a unit matrix, and a state variable relating to a kinematic equation in the state equation after the parameter-dependent basis transformation processing A design method for a gain scheduling controller, characterized in that is replaced with a virtual state quantity. The characteristic formulating step is a state equation where κ is a scheduling variable related to a dynamic equation and σ is a scheduling variable related to a kinematic equation.
(Ω and σ are state variables, u is the system input, and 0 is the zero matrix)
Represents the system,
The simplification process includes a coefficient matrix of equation (1)
Conversion matrix for basis conversion of the second partial coefficient matrix G (σ) to unit matrix
Equation of state after basis transformation by multiplying the left side of both sides of equation (1) by
And the simplification step replaces G ⁻¹ (σ) σ among the state variables in the equation (4) with a virtual state quantity ζ and does not include σ.
The design method according to claim 1, wherein:   The design method according to claim 2, wherein each coefficient multiplied by the virtual state quantity ζ in the equation (5) is zero.   The design method according to claim 2 or 3, wherein the state variable ω is an angular velocity, and the state variable σ is an improved Rodrigues parameter. The gain scheduling controller performs three-axis attitude control by providing feedback to three or more actuators,
5. The design method according to claim 1, wherein the schedule variable is obtained by converting the drive amount of the actuator into a three-axis component of airframe fixed coordinates.   6. The method according to claim 1, further comprising: converting a gain scheduling controller designed for a system represented by a simplified equation of state into an original system to obtain a gain scheduling controller of the original system. The design method according to one.