Precision pointing H_{∞} control design for absolute, window-, and stability-time errors
Abstract
Achieving precision pointing performance plays a decisive role in future space missions. Pointing performance is specified by a set of requirements consisting of absolute, window- and stability-time errors. The European Cooperation for Space Standardization categorizes this set in the following pointing performance error indices: absolute performance error (APE), mean performance error (MPE), relative performance error (RPE), performance drift error (PDE) and performance reproducibility error (PRE). The analysis of pointing error indices in time-simulations is straightforward as error data time-series can be described by standard statistics. However, for design purposes there exists no method that can directly and systematically handle control loop performance in terms of window- and stability-time pointing error indices. In this article we extend the standard multi-objective H_{2}/H_{∞} control problem to explicitly take into account requirements on pointing error index performance. The main topic, however, is the derivation of a control design approach that subjects the closed-loop control system to only one matrix criterion, the H_{∞}-norm. Therefore an optimization is set up to map pointing error index requirements into closed-loop specifications. The advantage of this approach is that the derived closed-loop specifications serve as indicators for the direct identification of design drivers, limits of performance and eventually systematic design trade-offs. Unlike in multi-objective H_{2}/H_{∞} control design approaches, this can be done even before controller synthesis, and thus independently of the specified control problem feasibility. Moreover, the derived approach enables control design in the H_{∞} closed-loop shaping framework. Thus, various design objectives including pointing performance and robustness can be treated with one matrix criterion.
Keywords
Pointing performance Window-time error Stability-time error ECSS Precision pointing H_{inf}-control Closed-loop shaping Multi-objective H_{2}/H_{inf} control Mixed H_{2}/H_{inf} control1 Introduction
Future satellite missions, e.g. the ESA missions Euclid [1] and Geo-Oculus [2], require high precision pointing performance in the order of 10^{−3} arcsec. Depending on the payload observation objectives, pointing performance is specified for a set of requirements consisting of absolute, window-time and stability-time errors. This set of pointing error requirements is categorized by the European Cooperation for Space Standardization (ECSS) in the pointing performance error indices [3]: absolute performance error (APE), mean performance error (MPE), relative performance error (RPE), performance drift error (PDE) and performance reproducibility error (PRE). The error indices have their direct origin in payload observation requirements as shown in [3, 4, 5] and thus are crucial for mission success. That motivates the need for developing of a coherent pointing control design approach that enables systematic and multi-objective closed-loop specification and optimization, explicitly taking into account pointing error indices. The control problem in this article is referred to as multi-objective because it imposes more than one type of norm constraint on the closed-loop system, cf. [6, 7].
In [8, 9, 10] pointing error indices are optimized with classical control design methods in an iterative design process based on engineering judgment, but closed-loop performance is not directly specified in a coherent manner. Recent control design approaches in [11, 12] are based on the H_{∞}- or H_{2}/H_{∞}-techniques in [6, 7, 13, 14]. In these approaches the APE index is optimized directly, but not window- and stability-time error indices (MPE, RPE, PDE, PRE). In [5] a H_{2}/H_{∞}-control design approach with direct optimization of pointing error indices is suggested, but not further elaborated. These and other existing precision pointing control design approaches either optimize the pointing error indices iteratively or directly, but in case the required performance cannot be achieved there are no indicators for identifying the limits of performance and design drivers before controller synthesis.
This article introduces a coherent precision pointing control design approach, in which such indicators are derived systematically by imposing only H_{∞}-norm bounds on the closed-loop system. This leads to closed-loop specifications in form of sensitivity and complementary sensitivity function, similar to [15]. The bases for this approach are signal norms as measures for root mean square (rms) power in general and pointing error index performance in particular. The signal norms measuring APE, MPE, RPE, PDE, and PRE are determined by the pointing metrics^{1} derived in [4, 5, 16]. In order to apply the signal norm measures the output performance and input noise signals are described by their power spectral density (PSD). Thus, control design methods in the frequency domain based on signal and system norms can be applied, when modeling the control system as continuous-time linear time-invariant (LTI) system.
The precision pointing control problem structure suggests multi-objective H_{2}/H_{∞} design methods as those in [6, 7]. In this article we extend these methods to directly handle pointing error indices. However, to obtain a systematic and coherent design approach, we choose the H_{∞} closed-loop shaping design framework in [13, 14, 17]. In this regard, the idea is to translate the multi-objective H_{2}/H_{∞} control problem in a H_{∞} only control problem. The typical practice for doing so is to use the induced system rms-norm, which corresponds to the H_{∞}-norm [18]. This results in a worst case control problem formulation, in particular, if sufficient information is available to characterize the input signals of a control system, i.e. pointing error sources [19, 20], by their PSD. In this article we characterize input signals by their PSD and formulate the control problem in terms of signal and system norms. This allows us to develop a less conservative method for expressing various H_{2}-norm and H_{∞}-norm design objectives in the same framework subject to the same matrix criterion, the H_{∞}-norm. The multi-objective control problem can thus directly be solved with well-established H_{∞}-techniques in [21].
The main advantage of this design approach is that limits of performance, including pointing performance, and other design objectives can be pointed out by means of comparable closed-loop specifications. In this way, design drivers can be identified that provide the basis for design trade-offs in the specification process before controller synthesis, similar to the approach in [15]. This reduces time-consuming design iterations.
This article introduces general and pointing error index-specific signal and system norms that are relevant for satellite precision pointing control in Sect. 2. In Sect. 3 a typical multi-objective satellite precision pointing control problem is formulated and translated into the H_{∞} closed-loop shaping design framework in Sect. 4. In order to specify the control problem in that framework a pointing performance measure and estimate is derived in Sect. 5. The measure and estimate is the basis for the derivation of multi-objective and optimal closed-loop performance specifications in a H_{∞}-norm sense in Sect. 6. A control synthesis method is introduced in Sect. 7 for the thus-specified control problem. It enables the systematic exploitation of performance margins and finally the synthesis of the controller. Sections 5 and 6 together with the index-norm definition in Sect. 2 are the main contributions of this article.
2 Norms for precision pointing control design
Pointing performance error indices
Error index | Instantaneous \( e_{\rm{index}} \) |
---|---|
Absolute performance error (APE) | \( e(t) \) |
Mean performance error (MPE) | \( \bar{e}(t,\Updelta t) \) |
Relative performance error (RPE) | \( e(t) - \bar{e}(t,\Updelta t) \) |
Performance drift error (PDE) | \( \bar{e}(t,\Updelta t) - \bar{e}(t + \Updelta t_{\rm s} ,\Updelta t) \) |
Performance reproducibility error (PRE) | \( \bar{e}(t,\Updelta t) - \bar{e}(t + \Updelta t_{\rm s} ,\Updelta t) \) |
\( \Updelta t \) window-time \( \Updelta t_{\rm s} \) stability-time | \( \bar{e} (t,\Updelta t) = \left\langle {e(t)} \right\rangle_{\Updelta t} = \frac{1}{\Updelta t}\int\nolimits_{t - \Updelta t/2}^{t + \Updelta t/2} {e(t)_{{}} }\,{\rm d}t \) |
In the mathematical sense, PDE and PRE are identical as seen in Table 1. The difference is in the specification of requirements for these error indices, as explained in [3]. A PDE requirement specifies the MPE difference among two points in time, t and \( t + \Updelta t_{\rm s} \), within one observation period. The PRE specifies the MPE difference among two points in time, with the points being in different observation periods.
PSD is a powerful formalism to describe random stationary noise processes. The double-sided PSD in [unit^{2}/(rad s^{−1})] of \( x_{k} (t) \) is defined as \( S_{xx} (\omega ) \), based on which the single-sided PSD is given as \( G_{xx} (\omega ) = 2S_{xx} (\omega ) \) [23], with \( \omega \) being the frequency in [rad s^{−1}].
Moreover, the stationary random noise processes in this article are assumed to be not only zero mean, but also Gaussian distributed, as it is the case for many physical random processes due to the central limit theorem [23]. In that case the statistical distribution of a control system’s output signal, in general or pointing error index in specific, is fully described by the signal’s standard deviation, \( \sigma_{x} \) [23]. The level of confidence with respect to a requirement on the output signal can thus be written in terms of \( \sigma \). Common values for the level of confidence are \( 1\sigma = 68.3\,\% \), \( 2\sigma = 95.5\,\% \) and \( 3\sigma = 99.7\,\% \).
2.1 Performance signal norms
It shall be noted that the signal rms-norm in Eq. (3) has seminorm properties as defined in [13]; however, for stationary processes it can be treated as a full norm.
Pointing weighting functions
Error index | Metric | Rational weighting function \( \tilde{F}_{\rm metric} \) |
---|---|---|
APE | Absolute | \( \tilde{F}_{A} (s,\Updelta t) = 1 \) |
RPE | Windowed variance | \( \tilde{F}_{WV} (s,\Updelta t) = \frac{{s\Updelta t\left( {s\Updelta t + \sqrt {12} } \right)}}{{\left( {s\Updelta t} \right)^{2} + 6\left( {s\Updelta t} \right) + 12}} \) |
MPE | Windowed mean | \( \tilde{F}_{WM} (s,\Updelta t) = \frac{{2\left( {s\Updelta t + 6} \right)}}{{\left( {s\Updelta t} \right)^{2} + 6\left( {s\Updelta t} \right) + 12}} \) |
PDE, PRE | Windowed mean stability | \( \tilde{F}_{WMS} (s,\Updelta t,\Updelta t_{s} ) = \tilde{F}_{WM} (s,\Updelta t)\frac{{2s\Updelta t_{s} \left( {s\Updelta t_{s} + 6} \right)}}{{\left( {s\Updelta t_{s} } \right)^{2} + 6\left( {s\Updelta t_{s} } \right) + 12}} \) |
\( \Updelta t \) window-time \( \Updelta t_{\rm s} \) stability-time |
In [5] a stability-time weighting function, \( \tilde{F}_{\rm s} (s,\Updelta t_{\rm s} ) \), is defined for PDE and PRE. It is not included in Table 2 because it is equivalent to \( \tilde{F}_{WMS} \) for ∆t = 0.
Note that the evaluation of the frequency domain integral in Eq. (4) corresponds to the standard deviation because μ_{e} = 0.
2.2 System and induced system norms
It is based on Eq. (4) by formulating the relation of w and z in terms of their signal norms and thus PSD. According to Eq. (2), the output signal PSD is \( G_{zz} = \tilde{H}_{\rm CL} G_{ww} \tilde{H}_{\rm CL}^{ * } \) for a given input signal PSD, \( G_{ww} \), and the weighting function augmented closed-loop system, \( \tilde{H}_{\rm CL} = \tilde{F}_{\rm metric} H_{\rm CL} \).
Describing a system by its H_{∞}-norm is equivalent to approximating \( \left| {H_{\rm CL} (s)} \right| \) by a constant maximum gain bound, which according to Eqs. (5) and (6) represents the maximum rms- or index-gain of a system. However, if the PSD of the input signal is known, the H_{2}-norm, defined in [18], is the appropriate system norm for specifying the output signal performance. This is because unlike the H_{∞}-norm, the H_{2}-norm is not the worst case rms- or index-gain of a system, but the precise rms- or index-measure of the system response as will be explained hereafter.
3 Satellite precision pointing control problem
In precision pointing control design [8, 9, 10] performance requirements are generally imposed on output signal rms- and index-norms, and thus naturally specified in a H_{2}-norm sense, if the input signals of the control system are stationary random noise processes, cf. Sect. 2. In addition, there are requirements in a H_{∞}-norm sense e.g. representing uncertainties, unmodeled dynamics, and closed-loop bandwidth requirements, cf. [18]. Consequently, there are requirements and design objectives subjecting the channels of a control system to the H_{2}-norm and in addition to the H_{∞}-norm.
Design requirements with applicable norms
Requirement | Signal norm | System norm |
---|---|---|
Pointing error index | index-norm | H_{2}-norm |
Control effort | rms-norm | H_{2}-norm |
Robustness | – | H_{∞}-norm |
3.1 Input noise signal representation
3.2 Multi-objective H_{2}/H_{∞} control problem
The satellite precision pointing control problem in Fig. 3 on the right is mapped into the standard problem in Fig. 3 on the left by defining \( {\mathbf{w}}_{2} = [w_{n} ,w_{d} ]^{\rm T} \) and \( {\mathbf{z}}_{2} = [e_{\rm index} ,u]^{\rm T} \). The 2 × 1 unit-PSD signal \( {\mathbf{w}}_{2} \) is defined in Sect. 3.1. The 2 × 1 performance output signal, \( {\mathbf{z}}_{2} \), represents the pointing error index, e_{index}, and the control effort, u. Imposing requirements on the signal norm of \( {\mathbf{z}}_{2} \) is equivalent with subjecting the channel \( {\mathbf{w}}_{2} \to {\mathbf{z}}_{2} \) to the H_{2}-norm as shown in Eq. (8)–(10) because \( {\mathbf{w}}_{2} \) has unit-PSD. In the control problem formulated in this article, the channels \( {\mathbf{w}}_{\infty } \to {\mathbf{z}}_{\infty } \), subject to the H_{∞}-norm, are used to model satellite plant uncertainties. Without loss of generality, only one exemplary uncertainty is considered in the formulation of the control problem because the main focus is on pointing error index performance. The uncertainty related 1 × 1 input signal,\( w_{\infty } = w_{\Updelta } \), and the 1 × 1 output signal, \( z_{\infty } = z_{\Updelta } \), thus connect the perturbation, \( {\varvec{\Updelta}} \), satisfying the norm bound \( \left\| {\varvec{\Updelta}} \right\|_{\infty } < 1 \), back into the system.
Solving the optimization problem in Eq. (20) leads to a result if the robust stability constraint \( \left\| {{\text{H}}_{ 1 1} } \right\|_{\infty } < \gamma_{\infty } \) is feasible and if the performance requirement, expressed by the constraint \( \left\| {{\mathbf{F}}_{e} {\mathbf{H}}_{ 2 2} } \right\|_{ 2} < \gamma_{2} \), is fulfilled. The method in [6] solves the optimization problem in Eq. (20) with mixed H_{2}/H_{∞} design techniques that provide a suboptimal upper bound solution. This method is well defined and implemented in MATLAB^{®} by calling the command h2hinfsyn. Thus, the precision pointing control problem can be formulated as standard multi-objective control problem with the signal and system norms in Sect. 2 and solved with standard algorithms in MATLAB^{®}.
However, in case one of the constraints in Eq. (17) is not feasible, in practice there are no comparable indicators for determining the limits of performance and design drivers in a systematic manner. The aim of this article was to obtain such indicators. In order to achieve that, our approach is to impose design requirements by subjecting the closed-loop system to one common matrix criterion, the H_{∞}-norm. On that condition, closed-loop specifications, which serve as comparable indicators can be derived systematically. This enables the direct identification of design drivers, limits of performance, and eventually systematic design trade-offs even before controller synthesis and thus independently of the control problem feasibility. Moreover, the control design can thus be embedded in the H_{∞} closed-loop shaping framework, which in addition provides well-established tools for robust analysis and synthesis [21].
4 H_{∞} control problem formulation
In the preceding section the aim of deriving closed-loop specifications subjecting the control system to one matrix criterion, the H_{∞}-norm, is stated. In order to enable H_{∞} control design, it is therefore necessary to derive performance requirements in the H_{∞}-norm sense that reflect the rms- and index-norm requirements. This can be accomplished by bounding the H_{2}-norm and thus the rms- and index-norm in Eqs. (8) and (10), with the H_{∞}-norm. The problem of obtaining such a bound is called Norm Bounding Problem (NBP) in this article. In this section the NBP is formulated as a derivation problem of closed-loop specifications and thus weighting filters in the H_{∞} closed-loop shaping framework. A method that yields closed-loop specifications by deriving the respective weighting filters is introduced in Sect. 6 of this article.
4.1 H_{∞} closed-loop shaping framework
The input signal w_{∆} and output signal z_{∆} connect an exemplary unit-norm perturbation ∆ back into the system. In Fig. 5, a multiplicative perturbation, \( w_{\Updelta } = {\varvec{\Updelta}}W_{\Updelta } u_{d} \), that represents, e.g. unmodeled dynamics of the 1 × 1 nominal satellite plant transfer function, G, is chosen as an example. A comprehensive summary of uncertainties and their modeling is given in [13, 14, 21].
Note again that \( {\mathbf{M}}({\mathbf{P}},K) \) in Eq. (22) is \( {\mathbf{H}}({\mathbf{P}},K) \) of Eq. (16) augmented by the output weighting filters, \( W_{e} \) and \( W_{u} \), at e and u.
H_{∞} control design criteria
Nominal performance is satisfied if and only if | \( \left\| {{\mathbf{M}}_{22} } \right\|_{\infty } \le 1 \) |
Robust stability is satisfied if and only if | \( \left\| {M_{11} } \right\|_{\infty } \le 1 \) |
Robust performance is satisfied if | \( \left\| {\mathbf{M}} \right\|_{\infty } \le 1 \) |
4.2 Closed-loop specifications
Closed-loop specifications
Output channels | Input channels | ||
---|---|---|---|
\( w_{\Updelta } \) | \( w_{n} \) | w_{d} | |
\( z_{\Updelta } \) | \( T_{{\rm S}5} = \left[ {W_{\Updelta } } \right]^{ - 1} \) | \( T_{{\rm S}4} = G\left[ {W_{\Updelta } W_{n} } \right]^{ - 1} \) | \( S_{{\rm S}2} = \left[ {W_{\Updelta } W_{d} } \right]^{ - 1} \) |
\( z_{\rm index} \) | \( S_{{\rm S}3} = \left[ {W_{e} G} \right]^{ - 1} \) | \( T_{{\rm S}1} = \left[ {W_{e} W_{n} } \right]^{ - 1} \) | \( S_{{\rm S}1} = \left[ {W_{e} GW_{d} } \right]^{ - 1} \) |
\( z_{\rm rms} \) | \( T_{{\rm S}6} = \left[ {W_{u} } \right]^{ - 1} \) | \( T_{{\rm S}3} = G\left[ {W_{u} W_{n} } \right]^{ - 1} \) | \( T_{{\rm S}2} = \left[ {W_{u} W_{d} } \right]^{ - 1} \) |
All specifications S_{s} and T_{s} are explicit requirements on the closed-loop shape of the control system. They are defined by the weighting filters W and the plant G. More precisely, the input weighting filters are the spectral noise shaping filters W_{n} and W_{d} of Eqs. (11) and (12). The performance output weighting filters W_{u} and W_{e} serve as bounds on \( \left\| u \right\|_{\rm rms} \) and \( \left\| e \right\|_{\rm index} \). This is the case because \( W_{e} \) and \( W_{u} \) normalize e and u to have unit-PSD with the objective that \( \left\| u \right\|_{\rm rms} = u_{r} \) and \( \left\| e \right\|_{\rm index} = e_{{\rm index},r} \). Consequently, \( \left\| u \right\|_{\rm rms} \le u_{r} \) and \( \left\| e \right\|_{\rm index} \le e_{{\rm index},r} \) is satisfied nominally or robustly if the nominal performance criterion \( \left\| {{\mathbf{M}}_{22} } \right\|_{\infty } \le 1 \) or robust performance criterion \( \left\| {\mathbf{M}} \right\|_{\infty } \le 1 \) in Table 4 is satisfied, respectively, provided that \( {\mathbf{M}}_{22} \) is stable. The derivation of \( W_{e} \) and \( W_{u} \) is thus equivalent to solving the NBP. It is one of the main topics of this article and will be solved in Sect. 6. For the derivation of the weighting filter, W_{∆}, norming the perturbation ∆ to unity, one is referred to [13, 14, 21].
As mentioned in the introduction, robustness against uncertainties is part of the overall control problem, but not explicitly treated in the following sections to concentrate on the main topic, the specification of pointing error index performance.
5 Performance measure and estimation
5.1 Frequency domain performance measure
5.2 Frequency domain performance estimation
Closed-loop transfer matrix approximations
\( \omega \le \omega_{cS} \) | \( \omega_{cS} < \omega < \omega_{cT} \) | \( \omega \ge \omega_{cT} \) |
---|---|---|
\( {\hat{\mathbf{H}}} = \left[ {\begin{array}{*{20}c} {W_{n} } & {S_{\rm s} GW_{d} } \\ {G^{ - 1} W_{n} } & {W_{d} } \\ \end{array} } \right] \) | \( {\hat{\mathbf{H}}} = \left[ {\begin{array}{*{20}c} {2W_{n} } & {2GW_{d} } \\ {2G^{ - 1} W_{n} } & {2W_{d} } \\ \end{array} } \right] \) | \( {\hat{\mathbf{H}}} = \left[ {\begin{array}{*{20}c} {T_{\rm s} W_{n} } & {GW_{d} } \\ {T_{\rm s} G^{ - 1} W_{n} } & {T_{\rm s} W_{d} } \\ \end{array} } \right] \) |
The interpolation conditions are applied in frequency intervals to accurately approximate \( S_{\rm s} \) and \( T_{\rm s} \), and thus \( {\hat{\mathbf{H}}} \). This results in a piecewise definition of \( {\hat{\mathbf{H}}} \) in the frequency intervals: \( \omega \le \omega_{cS} \), \( \omega_{cS} < \omega < \omega_{cT} \) and \( \omega \ge \omega_{cT} \). The intervals are determined by the 0 dB crossing frequency, ω_{cT}, of \( T_{\rm s} \) and, ω_{cS}, of \( S_{\rm s} \). The interpolation of \( S_{\rm s} \) and \( T_{\rm s} \) is necessary to accurately estimate the rms- and index-norm based on \( {\hat{\mathbf{H}}} \). The property that typically S_{s} ≫ 1 for ω ≫ ω_{cS} and T_{s} ≫ 1 for ω ≪ ω_{cT} would otherwise result in a conservative and unrealistic estimation because actually \( S + T = 1 \).
The performance estimate in Eq. (30) is the index-norm estimate with respect to the pointing error indices in Table 1, and Eq. (31) is the rms-norm estimate of the control effort. Both estimates are the basis for the derivation of \( W_{e} \) and \( W_{u} \) in Sect. 6.
6 Derivation of optimal performance specification
In Sect. 3 a typical satellite precision pointing control problem is defined, which corresponds to the standard multi-objective H_{2}/H_{∞} design problem that can be solved with methods in [6, 7]. However, as discussed in Sect. 3, in case the performance requirements cannot be achieved with these methods, there are no indicators for identifying the limits of performance and design drivers. This drawback is accounted for in Sect. 4 by formulating an H_{∞} only control problem, which has the same design objectives but embedded in the H_{∞} closed-loop shaping framework. The embedding of the multi-objective H_{2}/H_{∞} design problem in this framework requires the derivation of closed-loop specifications in terms of input and output weighting filters. The input weighting filters W_{n} and W_{d} are specified in Sect. 3.1 and the derivation of performance weighting filters W_{e} and W_{u} at the system output is topic of this section. The performance measures and estimations in Sect. 5 are the basis for this derivation.
Note that hereafter the derivation of performance weighting filters in general will be shown by the example of the pointing performance weighting filter W_{e}. This is done without loss of generality because the performance signal measures in Eqs. (28) and (29) as well as their estimates in Eqs. (30) and (31) just differ by an additional LTI transfer function, the pointing metric weighting function, \( \tilde{F}_{\rm metric} \).
6.1 Optimization-based derivation
Hence, the aim is to derive performance requirements in the H_{∞}-norm sense that reflect the index-norm requirement, e_{index,r}, to enable H_{∞} control design. This can be accomplished by solving the NBP, which is equivalent to the derivation of the weighting filter W_{e} as shown in Sect. 4.
6.1.1 Formulation of optimization problem
A conservative solution to the NBP is the induced system norm in Eq.(6). It is a worst case bounding of the H_{2}-norm because it assumes that the PSD of the input signals are concentrated in an infinitesimal small frequency band. The method derived in this section solves the NBP by optimizing a parameterized \( \hat{G}_{ee} \) in Eq. (30) until \( \left\| {\hat{e}} \right\|_{\rm index} = e_{{\rm index},r} \). In simple words, the NBP is solved by giving the estimated standard deviation \( \hat{\sigma }_{\rm index} = \left\| {\hat{e}} \right\|_{\rm index} \) a parameterized shape in the frequency domain.
According to Eq. (30) \( \hat{G}_{ee} = \left| {\hat{H}_{22\_11} } \right|^{2} + \left| {\hat{H}_{22\_12} } \right|^{2} \), which means that an optimization of \( \hat{G}_{ee} \)corresponds to an optimization of the piecewise defined \( \hat{H}_{22\_11} \) and \( \hat{H}_{22\_12} \) in Table 6 and finally to an optimization of \( S_{\rm s} \) and \( T_{\rm s} \). The closed-loop specifications S_{s} and T_{s} in turn shape the closed-loop transfer matrix M such that a required pointing error signal index-norm is achieved if the selected H_{∞}-norm criterion in Table 4 is satisfied.
The constraints in Eq. (34) are necessary in addition to the objective function to get closed-loop specifications that are suitable for the performance estimation approach in Sect. 5.2, cf. [26].
6.1.2 Solution of quasilinear optimization problems
The quasilinear optimization problems are solved with the bi-section method. The steps of this method are illustrated in Fig. 9 and a detailed explanation on it is given in [25]. The application of standard solvers in [27] is yet to be investigated.
The solution of the optimization problem results in the closed-loop specifications S_{s} and T_{s} defined by the derived weighting filter \( W_{e} \). The specifications are optimal in the sense that the computation of \( \left\| {\hat{e}} \right\|_{\rm index} \) with S_{s} and T_{s} corresponds to e_{index,r} with an error smaller than \( \gamma_{e} \) for the predefined structure of \( W_{e} \) in Eq. (30). Consequently the magnitudes of S_{s} and T_{s} represent performance bounds indicating that at this magnitude shapes the pointing performance error estimate \( \left\| {\hat{e}} \right\|_{\rm index} \) corresponds to the requirement e_{index,r}. Meaning that any sensitivity and complementary sensitivity function below the magnitudes of S_{s} and T_{s} in Fig. 11 results in a closed-loop system with a performance signal rms-norm at u and index-norm at e below the respective requirements. The performance requirements are thus imposed on the closed-loop system not directly in terms of index-norm specifications, but as H_{∞}-norm specifications implicitly bounding the index-norm.
Note that the weighting filter derivation problem in terms of W_{u} can be equivalently casted into an optimization problem by substituting the respective equations. Further, note that the formulation of the optimization problem in Eqs. (33) and (34) can be extended by defining weighting filter transfer functions depending on more decision variables. This enables more complex spectral shapes and thus eventually less conservative closed-loop performance specifications. However, convergence of the optimization has to be proved for other spectral shapes of W_{e} or W_{u}.
7 Controller synthesis
In order to completely specify the H_{∞} optimization problem for controller synthesis the closed-loop specifications S_{s} and T_{s} in Table 5 are sufficient. However, the closed-loop specifications are formulated based on the input weighting filters W_{n} and W_{d} in Eqs. (11) and (12), which are generally high-order transfer functions to reflect complex PSD shapes of noise sources. This means that H_{∞} controller synthesis would result in high-order controllers because the order of the augmented transfer matrix M is directly reflected in the order of the controller, cf. [22]. In this respect S_{s} and T_{s} are bounded from below by low-order closed-loop specifications. This allows, in addition, the exploitation of potential performance margins and thus provides the basis for trade-offs as will be shown in this section.
Note that there exist also alternative approaches to achieve low-order controllers like fixed-order H_{∞} controller synthesis [21], but the approach with bounded specifications is preferred because it allows the exploitation of performance margins in a systematic manner.
7.1 Bounding of closed-loop specifications
The closed-loop specifications S_{bound} and T_{bound}, which bound all closed-loop specifications in Table 5 from below, are derived in the same way as in Sect. 4.2, but for \( {\mathbf{M}}_{zw} \le 1 \). This means that if the control loop complies with the bounded closed-loop specifications S_{bound} and T_{bound} it also complies with all individual closed-loop specifications in Table 5.
This applies only if \( W_{{\bar{u}}} \) is set to a small constant value such that the transfer matrix entries containing \( W_{{\bar{u}}} \)do not influence the magnitude shape of S and T. Note that \( W_{{\bar{u}}} \) is necessary to make the H_{∞}-synthesis problem feasible, cf. [22].
When bounding the closed specifications the criteria in [24] should be taken into account. The criteria state that \( \left| {S_{\rm s} } \right| \) and \( \left| {T_{\rm s} } \right| \) should be bounded by a magnitude ≤6 dB, but of course ≥0 dB to make the H_{∞}-synthesis problem feasible. The 6 dB magnitude bound is necessary to limit the peaks of \( \left| S \right| \) and \( \left| T \right| \), which results in stability margins of ≥6 dB (gain margin) and ≥30° (phase margin).
7.2 Exploiting performance margins
The closed-loop specifications S_{s} and T_{s} set the performance requirements e_{index,r} and u_{r} at the system outputs e and u in Fig. 5. In consequence, the specification bounds S_{bound} and T_{bound} have to be below all S- and T-specifications. If the 0 dB crossing frequency of S_{s} is smaller than the one of T_{s} and if both are sufficiently far apart, then there are performance margins that can be exploited by placing S_{bound} and T_{bound}. This allows further tuning of the closed-loop system with respect to other design objectives.
Note that the performance margin in Fig. 11 could even be increased by choosing a S_{bound} that follows the spectral shape of S_{S1} with the penalty of getting a controller with a higher order.
Characteristic closed-loop bandwidths
Outputs | Bandwidth | |||
---|---|---|---|---|
ω_{BW1} | ω_{BW2} | ω_{BW3} | ω_{BW4} | |
e | \( \left\| {\hat{e}} \right\|_{\rm index} = e_{{\rm index},r} \) | \( \left\| {\hat{e}} \right\|_{\rm index, min } \) | \( \left\| {\hat{e}} \right\|_{\rm index} = e_{{\rm index},r} \) | \( \left\| {\hat{e}} \right\|_{\rm index} > e_{{\rm index},r} \) |
u | \( \left\| {\hat{u}} \right\|_{\rm rms} < u_{{\rm rms},r} \) | \( \left\| {\hat{u}} \right\|_{\rm rms} < u_{{\rm rms},r} \) | \( \left\| {\hat{u}} \right\|_{\rm rms} < u_{{\rm rms},r} \) | \( \left\| {\hat{u}} \right\|_{\rm rms} = u_{{\rm rms},r} \) |
7.3 H_{∞} controller synthesis
The augmented and bounded transfer matrix in Eq. (36) completely specifies a H_{∞}-optimization problem. Standard MATLAB^{®} algorithms like hinfric or hinflmi exist in [21] to solve such problems.
A solution of the H_{∞}-optimization problem specified by S_{bound} and T_{bound} always guarantees a closed-loop system with better performance than a solution of an H_{∞}-optimization problem specified by all S_{s} and T_{s} in Table 5, provided that both solutions comply with the selected loop shaping objective in Table 4.
An additional performance margin originates from the interpolation of the transfer function \( {\hat{\mathbf{H}}} \) in the transition region, \( \omega_{cS} < \omega < \omega_{cT} \), by a constant 6 dB gain. The actual S and T might only obtain this gain at their magnitude peaks and not throughout the whole transition region. The estimated rms-norm and index-norm are thus upper bounds of the actual norms in that region. In case such a margin is unwanted [28] provides a possible approach to reduce it.
8 Conclusions
The design approach developed in this article enables a precision pointing control design cycle that is coherent with the pointing performance error indices defined in the ECSS standard [3], I.e., these requirements are explicitly taken into account in the design process by defining pointing error index-norm measures and estimates for absolute, window-time and stability-time errors.
In the beginning of this article a control design approach is presented, which extends standard multi-objective H_{2}/H_{∞} design [6, 7] to explicitly treat pointing error index requirements and other performance requirements based on the defined index- and rms-norm. However, we point out that with this approach in practice there are no comparable indicators for determining the limits of performance and design drivers in a systematic manner.
This drawback is accounted for by subjecting the closed-loop system to one common matrix criterion, the H_{∞}-norm, to impose design objectives and requirements. In terms of performance requirements, this is possible by bounding the H_{2}-norm and thus the index- and rms-norm, by the H_{∞}-norm. The main contribution of this article is the development of a method that achieves a less conservative bounding as this is the case when applying the induced system norm [18]. The basis of this method is the formulation of a quasilinear optimization problem to derive unique closed-loop specifications as hard requirements on the control system.
The derived closed-loop specifications serve as comparable indicators for the direct identification of design drivers, limits of performance and eventually systematic design trade-offs even before controller synthesis, and thus independently of the control problem feasibility. In addition, based on the indicators, a practical approach is introduced to exploit potential performance margins and design for a certain closed-loop bandwidth. Moreover, the control design is embedded in the H_{∞} closed-loop shaping framework, which provides well-established tools for robust analysis and synthesis [21].
In the control design of the future ESA mission Euclid, trade-offs with partially contradictory objectives are performed based on the approach described in this article, cf. [1]. The experience gained so far shows that the design cycle time is shortened considerably.
Footnotes
Notes
Acknowledgments
The results obtained and presented in this article have been developed under the ESA Network/Partnering Initiative with the title “Precision Pointing Control Design Under Agility Constraints“. Partners are the Institute of Flight Mechanics and Control of Universität Stuttgart, AOCS/GNC and Flight Dynamics Department of Astrium Satellites, Germany, and the Control Systems Division of ESA/ESTEC.
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