Robust Attitude Orbit Control for Large Flimsy Appendages - IEEE Xplore
TuA08.5
43rd IEEE Conference on Decision and Control December 14-17, 2004 Atlantis, Paradise Island, Bahamas
Robust Attitude Orbit Control for Large Flimsy Appendages Michele Basso and Max Rotunno Abstract— The purpose of the paper is to present main results about design and simulation of an attitude orbit controller for a solar sail spacecraft with large flimsy appendages. The presence of a large number of uncertain flexible modes makes this problem suited for a robust control design based on Quantitative Feedback Theory.
I. INTRODUCTION Future space missions will most certainly involve spacecraft equipped with Large Flimsy Appendages (LFA). From a commercial point of view the rapid growth in space-based communications has fostered a new generation of spacecraft designed with higher power capabilities, larger solar arrays, and larger deployable payloads. Spacecraft that provide land-mobile communications services may have payload reflectors up to 15 meters in diameter, and solar arrays up to 20 meters long. For scientific missions there is renewed interest in solar sailing due to its potential for propellantless space propulsion [1]. Near-term applications of solar sailing technology include high-performance science missions to Mercury, Venus, and the inner solar system. Non-Keplerian orbits, high-velocity missions to the outer planets, and high-velocity interstellar precursor missions (all based on solar sailing technology) are also envisioned by space agencies around the world. In order to avoid that the above flexible appendages have a prohibitive mass, new structural concepts with very low stiffness are envisaged. These LFAs are thus characterized by closely spaced and lightly damped vibration modes, and may account for more than 99% of the total spacecraft rotational inertia. With dynamics dominated by these flexible structures, maintaining the required attitude pointing accuracy can be difficult, especially when periodic disturbances are present [1-3]. For example, during orbit adjust maneuvers, if the harmonics of the thruster pulsing frequency are close to those of the flexible modes, the resulting vibrations can be large. In these situations it is necessary for the control system to provide adequate disturbance rejection, so that the presence of the disturbances does not interfere with payload operation. In this context, Quantitative Feedback Theory (QFT) is a Michele Basso is with Dipartimento di sistemi e Informatica, Università di Firenze, via S. Marta 3, I-50139 Firenze, ITALY (e-mail: [email protected]). Max Rotunno is with Unmannned Technologies Research Institute, viale del Lido 37, I-00122 Roma, ITALY (e-mail: [email protected]).
0-7803-8682-5/04/$20.00 ©2004 IEEE
278
quite powerful tool to deal with uncertain systems [4]. It allows for a classical approach to the problem of design, without loosing the robustness objectives and leaving many degrees of freedom to the designer. Indeed, one of the advantages with respect to different control techniques (as, e.g., H-infinity) relies on its ability to cope with high-order uncertain systems without the need to design high-order controllers. In this paper, we consider QFT design of an Attitude Orbit Control (AOC) of a solar sail spacecraft. For its particular structure, the attitude dynamics largely depends on the presence of flexible modes which involve more than 40 uncertain parameters along each rotational axis. Since QFT is based on a frequency domain description of the plant uncertainty (the so-called templates), it is evident that gridding techniques of the above parameter set makes such an objective intractable. In this work we have pursued a different approach by suitably exploiting the plant additive structure together with an analytical description of each flexible mode, with the aim of generating the plant templates in a computationally efficient manner. The paper is organized as follows. In Section II we introduce the model of the spacecraft. The control system specifications are presented in Section III. Section IV develops an analytical approach for the computation of the plant templates, whereas Section V discusses the QFT design and the corresponding control system performance. Finally, in Section VI some conclusions are drawn. Deployment Module
Boom
Micro-Spacecraft Fig. 1. Solar Sail Spacecraft.
II. FLEXIBLE SPACECRAFT MODEL
The above model possesses an additive structure defined by
In this paper we consider the solar sail spacecraft shown in Fig. 1 in its deployed configuration. In its stowed configuration the spacecraft consists of a deployment module having dimensions 0.6m x 0. 6m x 0.8m prescribed by the ARIANE 5 Microsat Piggy-Back ASAP-5 constraints. Once in orbit, the spacecraft is deployed and in its final configuration it consists of a 40m x 40m square sail with four 28m diagonal booms to support four triangular sail segments, and a micro-spacecraft which is separated from the sail plane by a 10m mast. The sail structure is composed of three major elements: the booms, the sail film segments, and the central deployment module. The four supporting Carbon Fibre Reinforced Plastics (CFRP) booms are unrolled from the central deployment module and the four folded, triangular sail segments are released from the sail containers. The 10m collapsible mast is housed inside the micro-spacecraft in its stowed configuration. Attitude control and sensing is accomplished by Reaction Wheels and a Star Tracker, respectively. Both the actuators and sensors are assumed to be located in the deployment module. The above sensor provides an accuracy of 15/3600 deg. In order to define the flexibility characteristics of the solar sail spacecraft a Finite Element (FE) analysis has been performed using MSC/NASTRAN. The FE mesh of the solar sail, shown in Fig. 1, is made with a combination of rigid, beam, membrane, and lumped mass elements. The model features 1021 nodes and 942 elements. The four sail segments (made of Kapton) are rigidly connected to the deployment module and are pulled into tension by means of constant force devices that provide a nominal 3N tension force at the outer corner of each sail segment. The FE modeling enables us to calculate the transfer function P (s ) from torque input to angular position output. For the sake of simplicity, in the sequel we will only consider the pitch axis dynamics, characterized by closely spaced and lightly damped vibration modes with frequency, damping, and modal gains time-invariant and uncertain.
Fig. 2. Bode magnitude plot of system (1).
279
pk ,0
N
P (s ) = k =0
2
s + pk ,1s + pk ,2
,
(1)
where the mode k = 0 characterizes the rigid body dynamics, whereas the total number of elastic modes with contribution along the pitch axis is N = 14 . Table I shows the values for the nominal and the uncertain parameters of system (1), whose Bode magnitude plot for the nominal case is given in Fig. 2. TABLE I SYSTEM PARAMETERS Nominal Mode # Range k Values e-5 e-5 [8.5 ,9.1e-5] p0,0 = 8.8 0 [0,0] p0,1 = 0 p0,2 = 0 [0,0] [1.5e-5,5.3e-5] p1,0 = 3.1e-5 1 p1,1 = 6.1e-4 [7.9e-5,1.1e-3] p1,2 = 2.3e-1 [1.5e-1,3.2e-1] [0.6e-6,2.1e-6] p2,0 = 1.2e-6 2 p2,1 = 6.4e-4 [8.3e-5,1.2e-3] p2,2 = 2.6e-1 [1.6e-1,3.6e-1] [0.6e-5,2.2e-6] p3,0 = 1.3e-5 3 p3,1 = 6.7e-4 [8.7e-5,1.3e-3] p3,2 = 2.9e-1 [1.8e-1,4.0e-1] e-6 [0.8e-6,2.8e-6] p4,0 = 1.6 4 p4,1 = 7.0e-4 [9.0e-5,1.3e-3] p4,2 = 3.1e-1 [1.9e-1,4.2e-1] [4.4e-6,1.5e-5] p5,0 = 9.0e-6 5 [1.1e-4,1.6e-3] p5,1 = 8.3e-4 p5,2 = 4.4e-1 [2.7e-1,6.0e-1] [0.9e-7,3.0e-7] p6,0 = 1.8e-7 6 p6,1 = 9.0e-4 [1.2e-4,1.7e-3] p6,2 = 5.2e-1 [3.2e-1,7.2e-1] [4.9e-6,1.7e-5] p7,0 = 9.9e-6 7 p7,1 = 1.1e-3 [1.4e-4,2.0e-3] p7,2 = 7.3e-1 [4.5e-1, 1.0] e-7 [0.9e-7,3.0e-7] p8,0 = 1.8 8 p8,1 = 1.1e-3 [1.5e-4,2.1e-3] p8,2 = 8.4e-1 [5.2e-1,1.2] [0.9e-6,2.9e-6] p9,0 = 1.7e-6 9 p9,1 = 1.2e-3 [1.5e-4,2.2e-3] p9,2 = 9.0e-1 [5.6e-1,1.2] p10,0 = 9.1e-8 [4.5e-8,1.5e-7] 10 p10,1 = 1.3e-3 [1.7e-4, 2.4e-3] p10,2 = 1.0 [6.4e-1,1.4] p11,0 = 2.5e-9 [1.2e-9,4.2e-9] 11 p11,1 = 1.3e-3 [1.7e-4, 2.4e-3] p11,2 = 1.0 [6. 4e-1,1.4] p12,0 = 8.7e-8 [4.2e-8,1.5e-7] 12 p12,1 = 1.3e-3 [1.7e-4, 2.5e-3] p12,2 = 1.1 [6.9e-1,1.5] p13,0 = 3.9e-9 [1.9e-9,6.7e-9] 13 p13,1 = 1.4e-3 [1.9e-4, 2.7e-3] p13,2 = 1.3 [8.2e-1,1.8] p14,0 = 3.0e-7 [1.5e-7,5.1e-7] p14,1 = 1.5e-3 [1.9e-4, 2.7e-3] 14 p14,2 = 1.4 [8.4e-1,1.9]
III. QFT DESIGN SPECIFICATIONS The basic idea of QFT is to transform the design problem for the set of uncertain transfer functions P(s ) into a suitable problem for the unique nominal plant transfer function P0 (s ) [4]. Typically, the classical two degree of freedom (2 DOF) control scheme of Fig. 3 is employed, where the feedback compensator K (s ) is exploited to reduce the sensitivity to disturbances and to variations in the plant P(s ) , while the prefilter F (s ) gives the shape of closed-loop transmission from reference to output. More specifically, it is required to find a stable minimum-phase compensator K (s ) such that (2) L0 ( j X) = K ( j X)P0 ( j X) satisfies some suitable frequency bounds. To solve such a problem, one has to consider the following points: x for each frequency X , define the plant templates as the image (on the Nichols chart) of the set P( j X) ; x define the boundary B(X) (Horowitz bound) as the locus (open or closed curve) of the open loop nominal gain of Eq. (2) such that a given specification is satisfied for all plants of the uncertainty set and is exactly met (i.e., is tight) for at least one of the plants. The above bounds define a set of forbidden regions for the frequency response L0 ( j X) .
R(s)
Prefilter
Controller
S#
Specification
Value
1
Pointing Accuracy
0.1 deg
Overshoot
< 20 %
2 3 4
Maximum Complementary Sensitivity Function Peak Maximum Sensitivity Function Peak
0.005 Im (s)
0.01
2
> 0.035 Hz
6
Gain Margin
> 6dB
7
Phase Margin
> 30 deg
0
-0.01 -0.015
-6 -5
0 Re (s)
5
-0.02
-0.01
-4
x 10
0 Re (s)
0.01
0.02
-3 x 10 Template Boundary Z=0.78rad/s
Template Boundary Z=0.7rad/s 3 2
0.1 Im (s)
Im (s)
1 0
0 -1
-0.1
-0.2
-2 -0.2
-0.1
0 Re (s)
0.1
-3 -4
0.2
-2
0 Re (s)
Fig. 4. Elementary template for the elastic mode # 5.
280
< 3dB
Closed Loop Bandwidth
-0.005
-4
< 27dB
5
0.015
4
0
-(s)
Plant
TABLE II AOCS SPECIFICATIONS
Template Boundary Z=0.52rad/s
-2
P(s)
The specification for the AOCS problem that we need to take into account are summarized in Table II.
6
Im (s)
K(s)
Fig. 3. Controller structure.
-4 x 10 Template Boundary Z=0.5rad/s
0.2
U(s
F(s)
2 x 10
-3
0
10
-2
Radius
10
hand, we proceed by analytically computing the exact boundary of the elementary templates for each elastic mode (including the rigid body) using results derived from the Kharitonov Generalized Theorem (KGT) [5,6]. Such boundaries, shown in Fig. 4 at a given set of frequencies for Mode # 5, are always generated by arcs and segments. A rough measure of the mode uncertainty can be represented by the radius of the smallest circle (red in the figure) centered in the frequency response of the nominal plant (red cross) and containing the mode template. The additive structure of the plant allows us to compute the boundary of the plant templates by summing the boundaries of the elementary templates. This greatly simplifies the template problem. However, 14 elastic modes generate too many points in the final template for each single frequency at a usable specified resolution. Therefore, the only practical solution to pursue is to analytically evaluate the sum of the first elementary templates with the largest uncertainty radius for any given frequency, whereas taking the nominal values for the others. Figure 5 shows the above uncertainty radii for the system modes as a function of the frequency. More specifically, we have exploited a geometrical approach to efficiently compute the set of complex points containing the boundary of the sum of the 1st and the 2nd largest mode. This technique can be improved assuming the uncertainty of the remaining modes to be non-parametric (for instance, a circle with radius given by the sum of the uncertainty radii of the excluded modes). The plant templates for a set of frequency are reported on the Nichols chart of Fig. 6 and can be directly compared to the frequency response of the nominal plant (red
Measure of uncertainty
2
10
mode #0 mode #1 mode #2 mode #3 mode #4 mode #5 mode #6 mode #7 mode #8 mode #9 mode #10 mode #11 mode #12 mode #13 mode #14
-4
10
-6
10
-8
10
-10
10
10
-0.6
-0 .5
-0.4
10
-0.3
10
10
-0.2
10 Z [rad/s]
10
-0.1
10
0
0.1
10
10
0.2
0.3
10
Fig. 5. Uncertainty radii for mode templates.
IV. FREQUENCY TEMPLATES
OF FLEXIBLE MODES
One of the most important issues in control design is the use of an accurate description for the plant dynamics. Because QFT involves frequency-domain techniques, its design procedure requires one to define the plant dynamics only in terms of its frequency response. The term template is used to denote the collection of an uncertain plant's frequency responses at a given frequency. Gridding techniques are commonly used in many practical problems where only a few parameters are uncertain. Here, the plant model possesses 43 uncertain parameters varying over independent intervals. This makes the above gridding technique unpractical, as, for instance, the whole number of vertices of the parameter box is around 8.8e12. As an alternative, for the specific problem at Template Z =0.15
-50
-100
-150 -200
-150
-100 Phase (deg)
-50
Gain (dB)
-100
-150
-100 Phase (deg)
-100
-150
-50
0
-50
0
-100
Fig. 6. Plant templates.
281
-50
-50
-150 -200
0
-100 Phase (deg) Template Z =2
0
-50
-150 -200
Template Z =0.35
-50
-150 -200
0
Template Z =0.8
0
Gain (dB)
0
Gain (dB)
Gain (dB)
0
-150
-100 Phase (deg)
Bound Z=0.15[rad/s]
60
60
40
40
20
20
0 -20
-40
-60
-60 -200
-150
-100 -50 Phase (deg)
0
-80
50
Bound Z=0.8[rad/s]
80 60
60
40
40
20
20
0 -20
-40 -60 -150
-100 -50 Phase (deg)
0
-80
50
-100 -50 Phase (deg)
0
50
0
50
Bound Z=2[rad/s]
0
-60 -200
-150
-20
-40
-80
-200
80
Gain (dB)
Gain (dB)
0 -20
-40
-80
Bound Z=0.35[rad/s]
80
Gain (dB)
Gain (dB)
80
-200
-150
-100 -50 Phase (deg)
Fig. 7. Horowitz bounds and the nominal loop-gain L0 ( jX) .
crosses). V. CONTROL DESIGN AND PERFORMANCE The control system requirements (see Table II) together with an efficient description of the plant templates are the starting points for computing the Horowitz bounds of the QFT design problem [4]. From the specification on the pointing accuracy (S# 1), we need to constraint the DCgain of the feedback controller to satisfy (3) K (0) > 18 . This requirement also enforces S# 3, given the sensor accuracy of 15/3600 deg. Specification S# 2 is directly controlled by the prefilter F (s ) , whereas the remaining specifications can be translated into constraints on the amplitude of the sensitivity function, such that S ( j X) =
1 b ES (X) , 1 + K ( j X)P( j X)
(4)
where ¦£3dB ES (X) = ¦¤ ¦¦ 6dB ¥
X b 0.22 rad s X > 0.22 rad s
.
The Horowitz bounds of the nominal loop gain have been computed both for the sensitivity constraints in Eq. (4) and for the complementary sensitivity S# 3. However, it can be easily shown that for the given sensor accuracy the latter requirement is always achieved. Therefore, the
282
critical bounds which define the forbidden regions for the frequency response of the open-loop gain are only those concerned with the sensitivity and are depicted in Fig. 7 as shaded regions for a subset of significant frequencies. A. Feedback Controller Design Although specific automatic design procedures for QFT exist [7], the feedback compensator for the QFT AOCS problem has been designed using manual loop-shaping of the nominal loop-gain L0 ( jX) , taking into account at any chosen frequency the corresponding position of the forbidden regions (open or close) on the Nichols chart. In particular, choosing about 30 log-spaced frequencies in the range [0.1,2], we have designed the controller 1200(1 + 5s )(1 + 2s ) (5) (1 + s )(1 + 0.2s )(1 + 0.07s )2 which exploits 2 lead compensators (zero-pole networks) providing the required phase increase, and an additional low-pass double filter with cut-off frequency of 15 rad/s. The above controller provides the nominal loop-gain L0 ( j X) reported in Fig. 7. K (s ) =
B. Prefilter Design The prefilter has been designed in order to guarantee performance in the time domain. Therefore, from the analysis of the step response of the controlled system without prefilter, the following filter has been chosen
x 1000 uniformly randomly distributed plants over the vertices of the parameter box.
Step Response
1
TABLE III QFT CONTROLLER PERFORMANCE
Amplitude
0.8
0.6
0.4
0
20
40
60
80
100
120
140
160
180
Time (s)
(a)
Worst Plant
1000 Random Plants
Rise Time (s)
18.9
19.9
43rd IEEE Conference on Decision and Control December 14-17, 2004 Atlantis, Paradise Island, Bahamas
Robust Attitude Orbit Control for Large Flimsy Appendages Michele Basso and Max Rotunno Abstract— The purpose of the paper is to present main results about design and simulation of an attitude orbit controller for a solar sail spacecraft with large flimsy appendages. The presence of a large number of uncertain flexible modes makes this problem suited for a robust control design based on Quantitative Feedback Theory.
I. INTRODUCTION Future space missions will most certainly involve spacecraft equipped with Large Flimsy Appendages (LFA). From a commercial point of view the rapid growth in space-based communications has fostered a new generation of spacecraft designed with higher power capabilities, larger solar arrays, and larger deployable payloads. Spacecraft that provide land-mobile communications services may have payload reflectors up to 15 meters in diameter, and solar arrays up to 20 meters long. For scientific missions there is renewed interest in solar sailing due to its potential for propellantless space propulsion [1]. Near-term applications of solar sailing technology include high-performance science missions to Mercury, Venus, and the inner solar system. Non-Keplerian orbits, high-velocity missions to the outer planets, and high-velocity interstellar precursor missions (all based on solar sailing technology) are also envisioned by space agencies around the world. In order to avoid that the above flexible appendages have a prohibitive mass, new structural concepts with very low stiffness are envisaged. These LFAs are thus characterized by closely spaced and lightly damped vibration modes, and may account for more than 99% of the total spacecraft rotational inertia. With dynamics dominated by these flexible structures, maintaining the required attitude pointing accuracy can be difficult, especially when periodic disturbances are present [1-3]. For example, during orbit adjust maneuvers, if the harmonics of the thruster pulsing frequency are close to those of the flexible modes, the resulting vibrations can be large. In these situations it is necessary for the control system to provide adequate disturbance rejection, so that the presence of the disturbances does not interfere with payload operation. In this context, Quantitative Feedback Theory (QFT) is a Michele Basso is with Dipartimento di sistemi e Informatica, Università di Firenze, via S. Marta 3, I-50139 Firenze, ITALY (e-mail: [email protected]). Max Rotunno is with Unmannned Technologies Research Institute, viale del Lido 37, I-00122 Roma, ITALY (e-mail: [email protected]).
0-7803-8682-5/04/$20.00 ©2004 IEEE
278
quite powerful tool to deal with uncertain systems [4]. It allows for a classical approach to the problem of design, without loosing the robustness objectives and leaving many degrees of freedom to the designer. Indeed, one of the advantages with respect to different control techniques (as, e.g., H-infinity) relies on its ability to cope with high-order uncertain systems without the need to design high-order controllers. In this paper, we consider QFT design of an Attitude Orbit Control (AOC) of a solar sail spacecraft. For its particular structure, the attitude dynamics largely depends on the presence of flexible modes which involve more than 40 uncertain parameters along each rotational axis. Since QFT is based on a frequency domain description of the plant uncertainty (the so-called templates), it is evident that gridding techniques of the above parameter set makes such an objective intractable. In this work we have pursued a different approach by suitably exploiting the plant additive structure together with an analytical description of each flexible mode, with the aim of generating the plant templates in a computationally efficient manner. The paper is organized as follows. In Section II we introduce the model of the spacecraft. The control system specifications are presented in Section III. Section IV develops an analytical approach for the computation of the plant templates, whereas Section V discusses the QFT design and the corresponding control system performance. Finally, in Section VI some conclusions are drawn. Deployment Module
Boom
Micro-Spacecraft Fig. 1. Solar Sail Spacecraft.
II. FLEXIBLE SPACECRAFT MODEL
The above model possesses an additive structure defined by
In this paper we consider the solar sail spacecraft shown in Fig. 1 in its deployed configuration. In its stowed configuration the spacecraft consists of a deployment module having dimensions 0.6m x 0. 6m x 0.8m prescribed by the ARIANE 5 Microsat Piggy-Back ASAP-5 constraints. Once in orbit, the spacecraft is deployed and in its final configuration it consists of a 40m x 40m square sail with four 28m diagonal booms to support four triangular sail segments, and a micro-spacecraft which is separated from the sail plane by a 10m mast. The sail structure is composed of three major elements: the booms, the sail film segments, and the central deployment module. The four supporting Carbon Fibre Reinforced Plastics (CFRP) booms are unrolled from the central deployment module and the four folded, triangular sail segments are released from the sail containers. The 10m collapsible mast is housed inside the micro-spacecraft in its stowed configuration. Attitude control and sensing is accomplished by Reaction Wheels and a Star Tracker, respectively. Both the actuators and sensors are assumed to be located in the deployment module. The above sensor provides an accuracy of 15/3600 deg. In order to define the flexibility characteristics of the solar sail spacecraft a Finite Element (FE) analysis has been performed using MSC/NASTRAN. The FE mesh of the solar sail, shown in Fig. 1, is made with a combination of rigid, beam, membrane, and lumped mass elements. The model features 1021 nodes and 942 elements. The four sail segments (made of Kapton) are rigidly connected to the deployment module and are pulled into tension by means of constant force devices that provide a nominal 3N tension force at the outer corner of each sail segment. The FE modeling enables us to calculate the transfer function P (s ) from torque input to angular position output. For the sake of simplicity, in the sequel we will only consider the pitch axis dynamics, characterized by closely spaced and lightly damped vibration modes with frequency, damping, and modal gains time-invariant and uncertain.
Fig. 2. Bode magnitude plot of system (1).
279
pk ,0
N
P (s ) = k =0
2
s + pk ,1s + pk ,2
,
(1)
where the mode k = 0 characterizes the rigid body dynamics, whereas the total number of elastic modes with contribution along the pitch axis is N = 14 . Table I shows the values for the nominal and the uncertain parameters of system (1), whose Bode magnitude plot for the nominal case is given in Fig. 2. TABLE I SYSTEM PARAMETERS Nominal Mode # Range k Values e-5 e-5 [8.5 ,9.1e-5] p0,0 = 8.8 0 [0,0] p0,1 = 0 p0,2 = 0 [0,0] [1.5e-5,5.3e-5] p1,0 = 3.1e-5 1 p1,1 = 6.1e-4 [7.9e-5,1.1e-3] p1,2 = 2.3e-1 [1.5e-1,3.2e-1] [0.6e-6,2.1e-6] p2,0 = 1.2e-6 2 p2,1 = 6.4e-4 [8.3e-5,1.2e-3] p2,2 = 2.6e-1 [1.6e-1,3.6e-1] [0.6e-5,2.2e-6] p3,0 = 1.3e-5 3 p3,1 = 6.7e-4 [8.7e-5,1.3e-3] p3,2 = 2.9e-1 [1.8e-1,4.0e-1] e-6 [0.8e-6,2.8e-6] p4,0 = 1.6 4 p4,1 = 7.0e-4 [9.0e-5,1.3e-3] p4,2 = 3.1e-1 [1.9e-1,4.2e-1] [4.4e-6,1.5e-5] p5,0 = 9.0e-6 5 [1.1e-4,1.6e-3] p5,1 = 8.3e-4 p5,2 = 4.4e-1 [2.7e-1,6.0e-1] [0.9e-7,3.0e-7] p6,0 = 1.8e-7 6 p6,1 = 9.0e-4 [1.2e-4,1.7e-3] p6,2 = 5.2e-1 [3.2e-1,7.2e-1] [4.9e-6,1.7e-5] p7,0 = 9.9e-6 7 p7,1 = 1.1e-3 [1.4e-4,2.0e-3] p7,2 = 7.3e-1 [4.5e-1, 1.0] e-7 [0.9e-7,3.0e-7] p8,0 = 1.8 8 p8,1 = 1.1e-3 [1.5e-4,2.1e-3] p8,2 = 8.4e-1 [5.2e-1,1.2] [0.9e-6,2.9e-6] p9,0 = 1.7e-6 9 p9,1 = 1.2e-3 [1.5e-4,2.2e-3] p9,2 = 9.0e-1 [5.6e-1,1.2] p10,0 = 9.1e-8 [4.5e-8,1.5e-7] 10 p10,1 = 1.3e-3 [1.7e-4, 2.4e-3] p10,2 = 1.0 [6.4e-1,1.4] p11,0 = 2.5e-9 [1.2e-9,4.2e-9] 11 p11,1 = 1.3e-3 [1.7e-4, 2.4e-3] p11,2 = 1.0 [6. 4e-1,1.4] p12,0 = 8.7e-8 [4.2e-8,1.5e-7] 12 p12,1 = 1.3e-3 [1.7e-4, 2.5e-3] p12,2 = 1.1 [6.9e-1,1.5] p13,0 = 3.9e-9 [1.9e-9,6.7e-9] 13 p13,1 = 1.4e-3 [1.9e-4, 2.7e-3] p13,2 = 1.3 [8.2e-1,1.8] p14,0 = 3.0e-7 [1.5e-7,5.1e-7] p14,1 = 1.5e-3 [1.9e-4, 2.7e-3] 14 p14,2 = 1.4 [8.4e-1,1.9]
III. QFT DESIGN SPECIFICATIONS The basic idea of QFT is to transform the design problem for the set of uncertain transfer functions P(s ) into a suitable problem for the unique nominal plant transfer function P0 (s ) [4]. Typically, the classical two degree of freedom (2 DOF) control scheme of Fig. 3 is employed, where the feedback compensator K (s ) is exploited to reduce the sensitivity to disturbances and to variations in the plant P(s ) , while the prefilter F (s ) gives the shape of closed-loop transmission from reference to output. More specifically, it is required to find a stable minimum-phase compensator K (s ) such that (2) L0 ( j X) = K ( j X)P0 ( j X) satisfies some suitable frequency bounds. To solve such a problem, one has to consider the following points: x for each frequency X , define the plant templates as the image (on the Nichols chart) of the set P( j X) ; x define the boundary B(X) (Horowitz bound) as the locus (open or closed curve) of the open loop nominal gain of Eq. (2) such that a given specification is satisfied for all plants of the uncertainty set and is exactly met (i.e., is tight) for at least one of the plants. The above bounds define a set of forbidden regions for the frequency response L0 ( j X) .
R(s)
Prefilter
Controller
S#
Specification
Value
1
Pointing Accuracy
0.1 deg
Overshoot
< 20 %
2 3 4
Maximum Complementary Sensitivity Function Peak Maximum Sensitivity Function Peak
0.005 Im (s)
0.01
2
> 0.035 Hz
6
Gain Margin
> 6dB
7
Phase Margin
> 30 deg
0
-0.01 -0.015
-6 -5
0 Re (s)
5
-0.02
-0.01
-4
x 10
0 Re (s)
0.01
0.02
-3 x 10 Template Boundary Z=0.78rad/s
Template Boundary Z=0.7rad/s 3 2
0.1 Im (s)
Im (s)
1 0
0 -1
-0.1
-0.2
-2 -0.2
-0.1
0 Re (s)
0.1
-3 -4
0.2
-2
0 Re (s)
Fig. 4. Elementary template for the elastic mode # 5.
280
< 3dB
Closed Loop Bandwidth
-0.005
-4
< 27dB
5
0.015
4
0
-(s)
Plant
TABLE II AOCS SPECIFICATIONS
Template Boundary Z=0.52rad/s
-2
P(s)
The specification for the AOCS problem that we need to take into account are summarized in Table II.
6
Im (s)
K(s)
Fig. 3. Controller structure.
-4 x 10 Template Boundary Z=0.5rad/s
0.2
U(s
F(s)
2 x 10
-3
0
10
-2
Radius
10
hand, we proceed by analytically computing the exact boundary of the elementary templates for each elastic mode (including the rigid body) using results derived from the Kharitonov Generalized Theorem (KGT) [5,6]. Such boundaries, shown in Fig. 4 at a given set of frequencies for Mode # 5, are always generated by arcs and segments. A rough measure of the mode uncertainty can be represented by the radius of the smallest circle (red in the figure) centered in the frequency response of the nominal plant (red cross) and containing the mode template. The additive structure of the plant allows us to compute the boundary of the plant templates by summing the boundaries of the elementary templates. This greatly simplifies the template problem. However, 14 elastic modes generate too many points in the final template for each single frequency at a usable specified resolution. Therefore, the only practical solution to pursue is to analytically evaluate the sum of the first elementary templates with the largest uncertainty radius for any given frequency, whereas taking the nominal values for the others. Figure 5 shows the above uncertainty radii for the system modes as a function of the frequency. More specifically, we have exploited a geometrical approach to efficiently compute the set of complex points containing the boundary of the sum of the 1st and the 2nd largest mode. This technique can be improved assuming the uncertainty of the remaining modes to be non-parametric (for instance, a circle with radius given by the sum of the uncertainty radii of the excluded modes). The plant templates for a set of frequency are reported on the Nichols chart of Fig. 6 and can be directly compared to the frequency response of the nominal plant (red
Measure of uncertainty
2
10
mode #0 mode #1 mode #2 mode #3 mode #4 mode #5 mode #6 mode #7 mode #8 mode #9 mode #10 mode #11 mode #12 mode #13 mode #14
-4
10
-6
10
-8
10
-10
10
10
-0.6
-0 .5
-0.4
10
-0.3
10
10
-0.2
10 Z [rad/s]
10
-0.1
10
0
0.1
10
10
0.2
0.3
10
Fig. 5. Uncertainty radii for mode templates.
IV. FREQUENCY TEMPLATES
OF FLEXIBLE MODES
One of the most important issues in control design is the use of an accurate description for the plant dynamics. Because QFT involves frequency-domain techniques, its design procedure requires one to define the plant dynamics only in terms of its frequency response. The term template is used to denote the collection of an uncertain plant's frequency responses at a given frequency. Gridding techniques are commonly used in many practical problems where only a few parameters are uncertain. Here, the plant model possesses 43 uncertain parameters varying over independent intervals. This makes the above gridding technique unpractical, as, for instance, the whole number of vertices of the parameter box is around 8.8e12. As an alternative, for the specific problem at Template Z =0.15
-50
-100
-150 -200
-150
-100 Phase (deg)
-50
Gain (dB)
-100
-150
-100 Phase (deg)
-100
-150
-50
0
-50
0
-100
Fig. 6. Plant templates.
281
-50
-50
-150 -200
0
-100 Phase (deg) Template Z =2
0
-50
-150 -200
Template Z =0.35
-50
-150 -200
0
Template Z =0.8
0
Gain (dB)
0
Gain (dB)
Gain (dB)
0
-150
-100 Phase (deg)
Bound Z=0.15[rad/s]
60
60
40
40
20
20
0 -20
-40
-60
-60 -200
-150
-100 -50 Phase (deg)
0
-80
50
Bound Z=0.8[rad/s]
80 60
60
40
40
20
20
0 -20
-40 -60 -150
-100 -50 Phase (deg)
0
-80
50
-100 -50 Phase (deg)
0
50
0
50
Bound Z=2[rad/s]
0
-60 -200
-150
-20
-40
-80
-200
80
Gain (dB)
Gain (dB)
0 -20
-40
-80
Bound Z=0.35[rad/s]
80
Gain (dB)
Gain (dB)
80
-200
-150
-100 -50 Phase (deg)
Fig. 7. Horowitz bounds and the nominal loop-gain L0 ( jX) .
crosses). V. CONTROL DESIGN AND PERFORMANCE The control system requirements (see Table II) together with an efficient description of the plant templates are the starting points for computing the Horowitz bounds of the QFT design problem [4]. From the specification on the pointing accuracy (S# 1), we need to constraint the DCgain of the feedback controller to satisfy (3) K (0) > 18 . This requirement also enforces S# 3, given the sensor accuracy of 15/3600 deg. Specification S# 2 is directly controlled by the prefilter F (s ) , whereas the remaining specifications can be translated into constraints on the amplitude of the sensitivity function, such that S ( j X) =
1 b ES (X) , 1 + K ( j X)P( j X)
(4)
where ¦£3dB ES (X) = ¦¤ ¦¦ 6dB ¥
X b 0.22 rad s X > 0.22 rad s
.
The Horowitz bounds of the nominal loop gain have been computed both for the sensitivity constraints in Eq. (4) and for the complementary sensitivity S# 3. However, it can be easily shown that for the given sensor accuracy the latter requirement is always achieved. Therefore, the
282
critical bounds which define the forbidden regions for the frequency response of the open-loop gain are only those concerned with the sensitivity and are depicted in Fig. 7 as shaded regions for a subset of significant frequencies. A. Feedback Controller Design Although specific automatic design procedures for QFT exist [7], the feedback compensator for the QFT AOCS problem has been designed using manual loop-shaping of the nominal loop-gain L0 ( jX) , taking into account at any chosen frequency the corresponding position of the forbidden regions (open or close) on the Nichols chart. In particular, choosing about 30 log-spaced frequencies in the range [0.1,2], we have designed the controller 1200(1 + 5s )(1 + 2s ) (5) (1 + s )(1 + 0.2s )(1 + 0.07s )2 which exploits 2 lead compensators (zero-pole networks) providing the required phase increase, and an additional low-pass double filter with cut-off frequency of 15 rad/s. The above controller provides the nominal loop-gain L0 ( j X) reported in Fig. 7. K (s ) =
B. Prefilter Design The prefilter has been designed in order to guarantee performance in the time domain. Therefore, from the analysis of the step response of the controlled system without prefilter, the following filter has been chosen
x 1000 uniformly randomly distributed plants over the vertices of the parameter box.
Step Response
1
TABLE III QFT CONTROLLER PERFORMANCE
Amplitude
0.8
0.6
0.4
0
20
40
60
80
100
120
140
160
180
Time (s)
(a)
Worst Plant
1000 Random Plants
Rise Time (s)
18.9
19.9
