CEAS Space Journal

, Volume 4, Issue 1–4, pp 13–30 | Cite as

Precision pointing H control design for absolute, window-, and stability-time errors

Original Paper

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 H2/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 H2/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 Hinf-control Closed-loop shaping Multi-objective H2/Hinf control Mixed H2/Hinf control 

1 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 H2/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 H2/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 metrics1 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 H2/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 H2/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 H2-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.

A step-by-step overview of the precision pointing H control design approach is illustrated in Fig. 1. It is mainly based on the design methodology in [15, 22] with the techniques in [13,14] and adapted to precision pointing needs by the optimization of performance specifications in Sect. 6. The design approach is applied to a case study in Appendix 1. The figures in this article are based on the design data of this case study. Robustness is not regarded in the case study to keep it simple and clearly state the main topic of this article, i.e. the pointing error index performance. Nevertheless, robustness issues are regarded in the formulation of the control problem in Sects. 3 and 4 to show their general interfacing in the design process. Moreover, the approach in this article is derived for typical satellite attitude control systems, which can be decoupled in SISO LTI systems per satellite axis as shown in [10].
Fig. 1

Precision pointing control design approach

2 Norms for precision pointing control design

This section summarizes the basics of signal performance in a condensed manner and defines norms for measuring it. Pointing performance is of particular interest because a pointing error has different absolute, window-time and stability-time properties as illustrated in Fig. 2. In [3], these properties are categorized in the performance error indices outlined in Table 1.
Fig. 2

Pointing performance error indices

Table 1

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.

Frequency domain metrics are used to measure pointing error index performance for closed-loop specification. The metrics are restricted to signals that are stationary random noise processes, \( x_{k} (t) \), as they are defined in [23], with zero mean value:
$$ \mu_{x} = E[(x_{k} (t)] = 0 $$
(1)
where \( E[ \cdot ] \) is the expected value over the ensemble index k.

PSD is a powerful formalism to describe random stationary noise processes. The double-sided PSD in [unit2/(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].

In this article input noise signals, \( w(t) \), and output noise signals, \( z(t) \), of a control system are considered to be stationary random noise processes. Describing \( w(t) \) and \( z(t) \) by their PSD, a control system can be analyzed in terms of its PSD transformation properties [23]:
$$ {\mathbf{G}}_{zz} (\omega ) = {\mathbf{H}}_{\rm{CL}} (s){\mathbf{G}}_{ww} (\omega ){\mathbf{H}}_{\rm{CL}}^{ * } (s) $$
(2)
where \( {\mathbf{H}}_{\rm CL} (s) \) is the time-continuous LTI transfer matrix of a closed-loop control system and \( {\mathbf{H}}_{\rm CL}^{*} (s) = {\mathbf{H}}_{\rm CL}^{\rm T} ( - s) \) is its conjugate transpose with \( s = j\omega \) being a complex variable. \( {\mathbf{G}}_{ww} \) and \( {\mathbf{G}}_{zz} \) are the PSD at the control system input and output, respectively. Note that for SISO systems Eq. (2) reduces to \( G_{zz} = \left| {H_{\rm CL} } \right|^{2} G_{ww} \).

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

A measure for the size of a signal, xk(t), being a stationary random noise processes, is the rms-norm as defined in [13]:
$$ \left\| x \right\|_{rms} := \left( {\sigma_{x}^{2} + \mu_{x}^{2} } \right)^{1/2} = \left( {E[x_{k} (t)^{2} ]} \right)^{1/2} = \left( {\frac{1}{2\pi }\int\limits_{0}^{\infty } {G_{xx} (\omega )\,{\rm d}\omega } } \right)^{1/2} $$
(3)

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.

Whenever performance shall be measured in line with the pointing error indices defined in [3], the analysis of the pointing error signal, e, with respect to instantaneous time, window-time and stability-time is required [19, 20]. In the following appropriate signal norms for measuring pointing error indices are introduced. The signal norms are determined by the pointing metrics defined in [4, 5]. The pointing metrics rely on index specific PSD weighting functions, \( F_{\rm metric} \). In order to perform LTI analysis, rational approximations, \( \tilde{F}_{\rm metric} \), of the weighting functions are given in [5] and summarized in Table 2, such that \( F_{\rm metric} (\omega ) \cong \left| {\tilde{F}_{\rm metric} (s)} \right|^{2} \).
Table 2

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.

Analogous to Eq. (3), and with the rational weighting functions defined in Table 2, pointing error index measures can be formulated as spectral weighted signal rms-norms. This leads to the definition of the pointing error index-norm:
$$ \left\| e \right\|_{\rm index} : = \sigma_{\rm index} = \left( {\frac{1}{2\pi }\int\limits_{0}^{\infty } {F_{\rm metric} (\omega )G_{ee} (\omega )\,{\rm d}\omega } } \right)^{1/2} $$
(4)

Note that the evaluation of the frequency domain integral in Eq. (4) corresponds to the standard deviation because μe = 0.

2.2 System and induced system norms

Signal and in particular pointing error signal performance is measured with norms. Therefore, induced system norms are important as analytical design tool for specifying the closed-loop behavior of the control system. According to [18] the induced rms-norm is defined as
$$ \begin{array}{*{20}c} {\left\| {H_{\rm CL} (s)} \right\|_{\rm rms,ind} = \left\| {H_{\rm CL} (s)} \right\|_{\infty } : = \mathop {\sup }\limits_{{\left\| w \right\|_{\rm rms} \ne 0}} \frac{{\left\| z \right\|_{\rm rms} }}{{\left\| w \right\|_{\rm rms} }} = \mathop {\sup }\limits_{{\left\| w \right\|_{\rm rms} \ne 0}} \frac{{\left\| {H_{\rm CL} w} \right\|_{\rm rms} }}{{\left\| w \right\|_{\rm rms} }}} \hfill \\ \end{array} $$
(5)
where w and z are the respective input and output signals of \( H_{\rm CL} \) as previously defined.
The induced index-norm is defined likewise:
$$ \begin{array}{*{20}c} {\left\| {\tilde{H}_{\rm CL} (s)} \right\|_{\rm index,ind} = \left\| {\tilde{H}_{\rm CL} (s)} \right\|_{\infty } : = \mathop {\sup }\limits_{{\left\| w \right\|_{\rm rms} \ne 0}} \frac{{\left\| e \right\|_{\rm index} }}{{\left\| w \right\|_{\rm rms} }} = \mathop {\sup }\limits_{{\left\| w \right\|_{\rm rms} \ne 0}} \frac{{\left\| {\tilde{H}_{\rm CL} w} \right\|_{index} }}{{\left\| w \right\|_{\rm rms} }}} \hfill \\ \end{array} $$
(6)

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, \( \tilde{H}_{\rm CL} = \tilde{F}_{\rm metric} H_{\rm CL} \).

The induced rms- and index-norm of \( H_{\rm CL} \) correspond to the H-norm, which is defined as the maximum magnitude over frequency of that system [18]:
$$ \left\| {H_{\rm CL} (s)} \right\|_{\infty } : = \mathop {\sup }\limits_{\omega } \left| {H_{\rm CL} (j\omega )} \right| $$
(7)

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 H2-norm, defined in [18], is the appropriate system norm for specifying the output signal performance. This is because unlike the H-norm, the H2-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.

The PSD, \( G_{ww} (\omega ) \), of w can be written as \( G_{ww} = W_{w} N_{ww} W_{w}^{ * } \) with the white noise PSD being \( N_{ww} (\omega ) = 1 \) and the spectral shaping filter, \( W_{w} (s) \). Hence, according to the PSD system transformation in Eq. (2), the rms-norm of a control system output signal is
$$ \left\| z \right\|_{\rm rms} = \left( {\frac{1}{2\pi }\int\limits_{0}^{\infty } {\left| {H_{\rm CL} (s)} \right|^{2} } G_{ww} (\omega )\,{\rm d}\omega } \right)^{1/2} = \left( {\frac{1}{2\pi }\int\limits_{0}^{\infty } {\left| {H_{\rm CL} (s)W_{w} (s)} \right|^{2} }\,{\rm d}\omega } \right)^{1/2} $$
(8)
with \( H_{\rm CL} H_{\rm CL}^{ * } = \left| {H_{\rm CL} } \right|^{2} \) and \( W_{w} W_{w}^{ * } = \left| {W_{w} } \right|^{2} \). Equation (8) is thus equivalent to the H2-norm of \( H_{\rm CL} \) augmented by \( W_{w} \):
$$ \left\| {H(s)} \right\|_{2} = \left\| {H_{\rm CL} (s)W_{w} (s)} \right\|_{2} $$
(9)
with \( H(s) = H_{\rm CL} (s)W_{w} (s) \). Accordingly, replacing \( H_{\rm CL} (s) \) by \( \tilde{H}_{\rm CL} (s) \) defines the pointing error index-norm or pointing error index related H2-norm:
$$ \left\| e \right\|_{\rm index} = \left( {\frac{1}{2\pi }\int\limits_{0}^{\infty } {\left| {\tilde{H}_{\rm CL} (s)} \right|^{2} } G_{ww} (\omega )\,{\rm d}\omega } \right)^{1/2} = \left( {\frac{1}{2\pi }\int\limits_{0}^{\infty } {\left| {\tilde{H}(s)} \right|^{2} }\, {\rm d}\omega } \right)^{1/2} $$
(10)
with \( \tilde{H} = \tilde{F}_{\rm metric} H_{\rm CL} W_{w} \). The norms in Eqs. (8) and (10) are the performance measures used in the control design approach developed in this article.

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 H2-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 H2-norm and in addition to the H-norm.

In this article we consider the precision pointing control problem of a typical satellite with uncertainties in the plant and high performance requirements on the control effort, u, and pointing error index, eindex, outputs. The input signals are measurement noise, n, and disturbance noise, d, both being stationary random noise processes. The design objective is to guarantee at least nominal performance on the respective outputs and robust stability with respect to plant uncertainties. The applicable signal and system norms for formulating the satellite precision pointing control problem are summarized in Table 3, on the basis that the satellite plant is a LTI system and the input signals are stationary random noise processes.
Table 3

Design requirements with applicable norms

Requirement

Signal norm

System norm

Pointing error index

index-norm

H2-norm

Control effort

rms-norm

H2-norm

Robustness

H-norm

3.1 Input noise signal representation

The first step to formulate the control problem in terms signal and system norms is to normalize the input noise sources such that their PSD attain the power of one. This is necessary to reduce the input signal to a real positive number and augment information about its frequency content on the control system plant. According to Eq. (2) the noise source PSD can be written as
$$ G_{nn} (\omega ) = W_{n} (s)N_{ww} (\omega )W_{n}^{*} (s) \, = \left| {W_{n} (s)} \right|^{2} $$
(11)
$$ G_{dd} (\omega ) = W_{d} (s)N_{ww} (\omega )W_{d}^{*} (s) \, = \left| {W_{d} (s)} \right|^{2} $$
(12)
where \( W_{n} (s) \) and \( W_{d} (s) \) are spectral shaping filters, and \( N_{ww} (\omega ) = 1 \) is an ideal white noise PSD. The input noise signals can thus be written as
$$ d = W_{d} w_{d} {\text{ and }}n = W_{n} w_{n} $$
(13)
with \( w_{n} \) and \( w_{d} \) being the normalized input noise signals with the same unit-PSD, \( N_{ww} \). In this way the system noise inputs are not only normalized, but also their frequency domain behavior and thus time-correlation is taken explicitly into account in the control problem formulation to enhance precision.

3.2 Multi-objective H2/H control problem

The system norms in Table 3 imply the standard multi-objective H2/H control problem in Fig. 3 on the left. In this context, the input signals \( {\mathbf{w}}_{\infty } \) and output signals \( {\mathbf{z}}_{\infty } \) are system channels subject to the H-norm. The input signals \( {\mathbf{w}}_{2} \) and output signals \( {\mathbf{z}}_{2} \) are system channels subject to the H2-norm.
Fig. 3

Multi-objective H2/H precision pointing 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, eindex, 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 H2-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.

In Fig. 3 K is the controller and \( {\tilde{\mathbf{P}}} \) is the 3 × 3 block-structured time-continuous LTI transfer matrix of the generalized and perturbed plant augmented by \( W_{n} \) and \( W_{d} \):
$$ \left[ {\begin{array}{*{20}c} {z_{\Updelta } } \\ e \\ u \\ v \\ \end{array} } \right] = \underbrace {{\left[ {\begin{array}{*{20}c} {\tilde{P}_{11\_11} } & {\tilde{P}_{11\_12} } & {\tilde{P}_{11\_13} } & {\tilde{P}_{12\_1} } \\ {\tilde{P}_{11\_21} } & {\tilde{P}_{11\_22} } & {\tilde{P}_{11\_23} } & {\tilde{P}_{12\_2} } \\ {\tilde{P}_{11\_31} } & {\tilde{P}_{11\_32} } & {\tilde{P}_{11\_33} } & {\tilde{P}_{12\_3} } \\ {\tilde{P}_{21\_1} } & {\tilde{P}_{21\_2} } & {\tilde{P}_{21\_3} } & {\tilde{P}_{22} } \\ \end{array} } \right]}}_{{{\tilde{\mathbf{P}}}}}\;\left[ {\begin{array}{*{20}c} {w_{\Updelta } } \\ {w_{n} } \\ {w_{d} } \\ u \\ \end{array} } \right] $$
(14)
with v being the controller input. Closing the lower linear fractional transformation (LFT) gives the 3 × 3 time-continuous LTI transfer matrix:
$$ {\mathbf{H}}({\tilde{\mathbf{P}}},K) = {\tilde{\mathbf{P}}}_{11} + {\tilde{\mathbf{P}}}_{12} K\left( {I - \tilde{P}_{22} K} \right){\tilde{\mathbf{P}}}_{21} $$
(15)
Augmenting the output e of H by \( \tilde{F}_{\rm metric} \) describes the time-continuous LTI transfer matrix of the system in Fig. 3 on the right:
$$ \left[ {\begin{array}{*{20}c} {z_{\Updelta } } \\ {e_{\rm index} } \\ u \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & {\tilde{F}_{\rm metric} } & 0 \\ 0 & 0 & 1 \\ \end{array} } \right]_{{}} \underbrace {{\left[ {\begin{array}{*{20}c} {H_{11} } & {H_{12\_1} } & {H_{12\_2} } \\ {H_{21\_1} } & {H_{22\_11} } & {H_{22\_12} } \\ {H_{21\_2} } & {H_{22\_21} } & {H_{22\_22} } \\ \end{array} } \right]}}_{{{\mathbf{H(P},K)}}} \left[ {\begin{array}{*{20}c} {w_{\Updelta } } \\ {w_{n} } \\ {w_{d} } \\ \end{array} } \right] $$
(16)
Finally, the multi-objective precision pointing control problem is formulated in terms of the H2/H system norm requirements imposed on sub-matrices of Eq. (16) by the following constraints:
$$ \left\| {{\text{H}}_{ 1 1} } \right\|_{\infty } < \gamma_{\infty } \quad \text{and} \quad \left\| {{\mathbf{F}}_{e} {\mathbf{H}}_{ 2 2} } \right\|_{ 2} < \gamma_{2} $$
(17)
with
$$ {\mathbf{F}}_{\rm e} = \text{diag}\left( {\tilde{F}_{\rm metric} ,1} \right) $$
(18)
The requirements in Eq. (17) are typically contradictory, meaning that a smaller \( \gamma_{\infty } \) results in a bigger \( \gamma_{2} \) and vice versa [18]. In order to determine Pareto-optimal controllers several approaches exist, e.g. in [6, 7]. These approaches solve the multi-objective control problem by transforming it into a convex optimization problem with the objective function:
$$ a\gamma_{\infty } + b\gamma_{2} $$
(19)
where a and b are real positive parameters.
A typical multi-objective precision pointing control problem is to achieve a certain pointing error index and control effort performance requirement as well as robust stability. This defines the following optimization problem, a combination of Eqs. (17) and (19):
$$ \begin{array}{*{20}c} {\text{minimize}} & {a\gamma_{\infty } + b\gamma_{2} } & {} & {} \\ {\text{subject to}} & {\left\| {{\text{H}}_{ 1 1} } \right\|_{\infty } < \gamma_{\infty } } & {\text{and}} & {\left\| {{\mathbf{F}}_{e} {\mathbf{H}}_{ 2 2} } \right\|_{ 2} < \gamma_{2} } \\ \end{array} $$
(20)

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 H2/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 H2-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 precision pointing control design problem can be formulated in the H closed-loop shaping framework, outlined in [13, 14, 17], if it is possible to explicitly treat in addition to the H-norm also the H2-norm requirements in Table 3. This is achieved by solving the NBP for the H2-norm imposed on the respective transfer functions of the closed-loop system transfer matrix \( {\mathbf{M}} \) in Fig. 4. The system in Fig. 4 corresponds in principle to the system in Fig. 3, but it is adapted to allow the bounding of the H2-norm by the H-norm. The differences of the system in Fig. 4 with respect to the system in Fig. 3 are described hereafter.
Fig. 4

H-norm bounded precision pointing control problem

In Fig. 4 the input signals, \( {\mathbf{w}}_{2\infty } = [w_{n} ,w_{d} ]^{\rm T} \), are the noise signals defined in Eq. (13) and are equivalent to \( {\mathbf{w}}_{2} \) in Sect. 3.2. The performance output signals, \( {\mathbf{z}}_{2\infty } = [z_{\rm index} ,z_{\rm rms} ]^{\rm T} \), are the weighted pointing error, \( z_{\rm index} = W_{e} e \), and the weighted control effort, \( z_{\rm rms} = W_{u} u \), as shown in Fig. 5.
Fig. 5

Precision pointing 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].

The weighting filters, \( W_{e} \) and \( W_{u} \), in Fig. 5 normalize the PSD of the outputs signals e and u to obtain unity. The normalization is performed with the optimization in Sect. 6 such that the H2-norm of the respective transfer functions to \( {\mathbf{z}}_{2\infty } \) and from \( {\mathbf{w}}_{2\infty } \) are bounded with the H-norm. The weighting filter, \( W_{e} \), in Fig. 6 thus directly represents the different pointing error index requirements, \( e_{{\rm index},r} \), on e. In the following it is assumed without loss of generality that there is only one pointing error index requirement, meaning that \( W_{e} \) is a 1 × 1 transfer function. The weighting filter, \( W_{u} \), is also a 1 × 1 transfer function and represents the control effort requirement, \( u_{r} \), on u. The requirements \( e_{{\rm index},r} \) and \( u_{r} \) are both real positive numbers.
Fig. 6

Weighting filter, We, representing pointing error index requirements

The set-up in Fig. 5 corresponds to the H closed-loop shaping framework. Our formulation of the control problem in this framework is similar to the so-called signal-based loop shaping in [13], but explicitly taking into account the pointing error indices. It is named signal-based because the generalized and perturbed plant \( {\tilde{\mathbf{P}}} \) in Eq. (14) is augmented by the input weighting filters \( W_{n} \) and \( W_{d} \) that represent the signals’ PSD shape. In Fig. 5, the outputs, e and u, of \( {\tilde{\mathbf{P}}} \) are in addition augmented by the weighting filters, \( W_{e} \) and \( W_{u} \), which represent signal index- and rms-norm requirements, as mentioned in the previous paragraph. This leads to the following adapted 4 × 4 augmented and perturbed plant transfer matrix P in Fig. 5:
$$ \left[ {\begin{array}{*{20}c} {z_{\Updelta } } \\ {z_{\rm index} } \\ {z_{\rm rms} } \\ v \\ \end{array} } \right] = \underbrace {{\left[ {\begin{array}{*{20}c} 0 & 0 & {W_{\Updelta } W_{d} } & {W_{\Updelta } } \\ {W_{e} G} & 0 & {W_{e} GW_{d} } & {W_{e} G} \\ 0 & 0 & 0 & {W_{u} } \\ { - G} & {W_{n} } & { - GW_{d} } & { - G} \\ \end{array} } \right]}}_{{\mathbf{P}}} \left[ {\begin{array}{*{20}c} {w_{\Updelta } } \\ {w_{n} } \\ {w_{d} } \\ u \\ \end{array} } \right] $$
(21)
The lower LFT \( {\mathbf{M}}({\mathbf{P}},K) \) in Fig. 5 is the following 3 × 3 augmented closed-loop transfer matrix, with unit-PSD input and output signals:
$$ \left[ {\begin{array}{*{20}c} {z_{\Updelta } } \\ {z_{\rm index} } \\ {z_{\rm rms} } \\ \end{array} } \right] = \underbrace {{\left[ {\begin{array}{*{20}c} { - W_{\Updelta } T} & { - W_{\Updelta } TG^{ - 1} W_{n} } & {W_{\Updelta } SW_{d} } \\ { - W_{e} GS} & {W_{e} TW_{n} } & { - W_{e} GSW_{d} } \\ { - W_{u} T} & { - W_{u} TG^{ - 1} W_{n} } & { - W_{u} TW_{d} } \\ \end{array} } \right]}}_{{{\mathbf{M(P}},K)}} \left[ {\begin{array}{*{20}c} {w_{\Updelta } } \\ {w_{n} } \\ {w_{d} } \\ \end{array} } \right] $$
(22)
where the closed-loop sensitivity function is:
$$ S(s): = \left[ {I + G(s)K(s)} \right]^{ - 1} $$
(23)
and the complementary sensitivity function is
$$ T(s): = G(s)K(s)\left[ {I + G(s)K(s)} \right]^{ - 1} $$
(24)
In order to keep notation simple, the transfer matrix in Eq. (22) is written as 2 × 2 block-structured matrix:
$$ \left[ {\begin{array}{*{20}c} {z_{{{\Updelta}}} } \\ {{\mathbf{z}}_{2\infty } } \\ \end{array} } \right] = \underbrace {{\left[ {\begin{array}{*{20}c} {M_{11} } & {{\mathbf{M}}_{12} } \\ {{\mathbf{M}}_{21} } & {{\mathbf{M}}_{22} } \\ \end{array} } \right]}}_{{{\mathbf{M(P}},K)}} \left[ {\begin{array}{*{20}c} {w_{{{\Updelta}}} } \\ {{\mathbf{w}}_{2\infty } } \\ \end{array} } \right] $$
(25)

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.

The general precision pointing H control problem is now to find a controller K that stabilizes P for a stable \( {\varvec{\Updelta}} \), satisfying \( \left\| {\varvec{\Updelta}} \right\|_{\infty } < 1 \), and that achieves the closed-loop performance requirements at e and u. These objectives are equivalent with the robust control design criteria of nominal performance and robust stability, which are guaranteed if the respective H-norm bounds on the sub-matrices of \( {\mathbf{M}} \) are fulfilled and if K stabilizes the nominal unperturbed subsystem, \( {\mathbf{M}}_{22} \). As stated in Table 4, not only robust stability and nominal performance, but also robust performance can be achieved by means of H-norm bound specifications on \( {\mathbf{M}} \). If the robust performance criterion is satisfied also robust stability and nominal performance are satisfied as can be seen in Table 4 and Eq. (25) [17].
Table 4

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

Performance requirements in general and pointing performance error requirements in particular are imposed on the control system by deriving closed-loop specifications in form of sensitivity function, S, and complementary sensitivity function, T. The closed-loop specifications are derived with the objective to satisfy the selected criterion in Table 4. Imposing the criterion on the transfer matrix M means that all entries in the matrix must attain a H-norm of one or less. Applying now the worst case H-norm criteria,\( \left\| \ldots \right\|_{\infty } = 1 \), and remembering that the input and output signal PSD are normalized to unity, the entries in M can be re-arranged. Analogous to the approach in [15], sensitivity function specifications, Ss, and complementary sensitivity function specifications, Ts, can thus be formulated. That means that the closed-loop specifications in Table 5 are derived by rearranging the channel-wise transfer functions in Eq. (22). Note that KS is substituted by TG−1 to get closed-loop specifications that do not explicitly depend on the controller.
Table 5

Closed-loop specifications

Output channels

Input channels

\( w_{\Updelta } \)

\( w_{n} \)

wd

\( z_{\Updelta } \)

\( T_{{\rm S}5} = \left[ {W_{\Updelta } } \right]^{ - 1} \)

\( T_{{\rm S}4} = G\left[ {W_{\Updelta } W_{n} } \right]^{ - 1} \)

\( S_{{\rm S}2} = \left[ {W_{\Updelta } W_{d} } \right]^{ - 1} \)

\( z_{\rm index} \)

\( S_{{\rm S}3} = \left[ {W_{e} G} \right]^{ - 1} \)

\( T_{{\rm S}1} = \left[ {W_{e} W_{n} } \right]^{ - 1} \)

\( S_{{\rm S}1} = \left[ {W_{e} GW_{d} } \right]^{ - 1} \)

\( z_{\rm rms} \)

\( T_{{\rm S}6} = \left[ {W_{u} } \right]^{ - 1} \)

\( T_{{\rm S}3} = G\left[ {W_{u} W_{n} } \right]^{ - 1} \)

\( T_{{\rm S}2} = \left[ {W_{u} W_{d} } \right]^{ - 1} \)

All specifications Ss and Ts 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 Wn and Wd of Eqs. (11) and (12). The performance output weighting filters Wu and We 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].

The control design objectives are fulfilled if \( S(s) \) and \( T(s) \) comply with all specifications, \( S_{\rm s} (s) \) and \( T_{\rm s} (s) \), in Table 5 after controller synthesis, which is the case if \( \left| S \right| \le \left| {S_{\rm s} } \right| \) and \( \left| T \right| \le \left| {T_{\rm s} } \right| \). However, the limiting property of every control system is that \( S + T = 1 \), which imposes on any combination of \( S_{\rm s} \) and \( T_{\rm s} \) the following criterion:
$$ \left| {S_{\rm s} } \right| + \left| {T_{\rm s} } \right| \ge 1 $$
(26)
for the specified control problem to be feasible. The criterion in Eq. (26) is derived based on the triangle inequality stating that \( \left| {S_{\rm s} } \right| + \left| {T_{\rm s} } \right| \ge \left| S \right| + \left| T \right| \ge \left| {S + T} \right| = S + T = 1 \) because\( \left| S \right| \le \left| {S_{\rm s} } \right| \) and \( \left| T \right| \le \left| {T_{\rm s} } \right| \). In contrast, the closed-loop specifications are conflicting if \( \left| {S_{\rm s} } \right| + \left| {T_{\rm s} } \right| < 1 \). Plotting the magnitudes of \( S_{\rm s} \) and \( T_{\rm s} \) thus is a practical approach to examine the feasibility of the specified control problem. In addition, and independent of the control problem feasibility, the magnitude plots are practical indicators for identifying performance limitations and design drivers.

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

The solution of the NBP and thus the derivation of \( W_{e} \) and \( W_{u} \) requires a measure for the estimation of the signal norms \( \left\| u \right\|_{\rm rms} \) and \( \left\| e \right\|_{\rm index} \) at the performance outputs e and u. The underlying closed-loop system for computing \( \left\| u \right\|_{\rm rms} \) and \( \left\| e \right\|_{\rm index} \) is shown in Fig. 7.
Fig. 7

Closed-loop system performance measure

The closed-loop system in Fig. 7 corresponds to \( {\mathbf{H}}_{22} \) in Eq. (16) with the transfer matrix:
$$ \left[ {\begin{array}{*{20}c} e \\ u \\ \end{array} } \right] = \underbrace {{\left[ {\begin{array}{*{20}c} {H_{22\_11} } & {H_{22\_12} } \\ {H_{22\_21} } & {H_{22\_22} } \\ \end{array} } \right]}}_{{{\mathbf{H}}_{22} }} \left[ {\begin{array}{*{20}c} {w_{n} } \\ {w_{d} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {TW_{n} } & { - GSW_{d} } \\ { - TG^{ - 1} W_{n} } & { - TW_{d} } \\ \end{array} } \right] \left[ {\begin{array}{*{20}c} {w_{n} } \\ {w_{d} } \\ \end{array} } \right] $$
(27)

5.1 Frequency domain performance measure

The performance measure for a pointing error index, eindex, or the control effort, u, relies on the system transformation of the input noise source PSD of wn and wd in Fig. 7. If the noise sources are not cross-correlated, the pointing error is the H2-norm of the sum of the respective system matrix entries in Eqs. (8) and (10) as shown in the derivations of Appendix 2. The index-norm at the output channel e in Eq. (27) is thus
$$ \begin{array}{*{20}l} {\left\| e \right\|_{\rm index} }{ = \left( {\frac{1}{2\pi }\int\limits_{0}^{\infty } {F_{\rm metric} G_{ee}\,{\rm d}\omega } } \right)^{1/2} } \\ {}{ = \left( {\frac{1}{2\pi }\int\limits_{0}^{\infty } {\left( {{}_{{}}\left| {\tilde{F}_{\rm metric} H_{22\_11} } \right|^{2} + \left| {\tilde{F}_{\rm metric} H_{22\_12} } \right|^{2} } \right)_{{}} }\, {\rm d}\omega } \right)^{1/2} } \\ \end{array} $$
(28)
and the control effort rms-norm is
$$ \begin{array}{*{20}l} {\left\| u \right\|_{\rm rms} }{ = \left( {\frac{1}{2\pi }\int\limits_{0}^{\infty } {G_{uu}\,{\rm d}\omega } } \right)^{1/2} } \\ {}{ = \left( {\frac{1}{2\pi }\int\limits_{0}^{\infty } {\left( {{}_{{}}\left| {H_{22\_21} } \right|^{2} + \left| {H_{22\_22} } \right|^{2} } \right)_{{}} }\, {\rm d}\omega } \right)^{1/2} } \\ \end{array} $$
(29)

5.2 Frequency domain performance estimation

The evaluation of the rms- and index-norm requires the knowledge of the closed-loop system H in Eq. (27), and thus of the controller K. Since K is not known in the specifications phase it can be approximated by the transfer matrices \( {\hat{\mathbf{H}}} \) in Table 6. The approximation \( {\hat{\mathbf{H}}} \) is obtained by substituting S and T in H by \( S_{\rm s} \) and \( T_{\rm s} \) or the constant values defined by the interpolation conditions in [22].
Table 6

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, ωcT, of \( T_{\rm s} \) and, ωcS, 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 Ss ≫ 1 for ω ≫ ωcS and Ts ≫ 1 for ω ≪ ωcT would otherwise result in a conservative and unrealistic estimation because actually \( S + T = 1 \).

The closed-loop system approximations in Fig. 8 are derived based on the interpolated closed-loop specifications. In [22] the interpolated sensitivity function \( S \cong S_{\rm s} \) for \( \omega \le \omega_{cS} \) and interpolated complementary sensitivity function \( T \cong T_{\rm s} \) for \( \omega \ge \omega_{cT} \). Likewise, \( S \cong 1 \) for \( \omega \ge \omega_{cT} \) and \( T \cong 1 \) for \( \omega \le \omega_{cS} \) by considering that \( S + T = 1 \) because \( T \to 0 \) for \( \omega \to \infty \) and \( S \to 0 \) for \( \omega \to 0 \). Unlike in [22] for the interval \( \omega_{cS} < \omega < \omega_{cT} \) a worst-case approximation of \( S \cong 2 \)(6 dB) and \( T \cong 2 \)(6 dB) is taken into account. The worst-case approximation at 6 dB is based on the triangle inequality stating that \( \left| S \right| + \left| T \right| \ge \left| {S + T} \right| = 1 \) and on the condition that \( \left| {S_{\rm s} } \right| \) and \( \left| {T_{\rm s} } \right| \) are typically bounded at 6 dB magnitude, cf. Sect. 7.1.
Fig. 8

Closed-loop system approximation

Fig. 9

Optimization based weighting filter derivation process

Fig. 10

GS/T weighting scheme

Fig. 11

S- and T-specifications with bounds

Thus, by substituting \( {\hat{\mathbf{H}}} \) into the Eqs. (28) and (29), the pointing index-norm is estimated by:
$$ \begin{array}{*{20}l} {\left\| {\hat{e}} \right\|_{\rm index} }{ = \left( {\frac{1}{2\pi }\int\limits_{0}^{\infty } {F_{\rm metric} \hat{G}_{ee} \, {\rm d}\omega } } \right)^{1/2} } \\ {}{ = \left( {\frac{1}{2\pi }\int\limits_{0}^{\infty } {\left( {{}_{{}}\left| {\tilde{F}_{\rm metric} \hat{H}_{22\_11} } \right|^{2} + \left| {\tilde{F}_{\rm metric} \hat{H}_{22\_12} } \right|^{2} } \right)_{{}} } \,{\rm d}\omega } \right)^{1/2} } \\ \end{array} $$
(30)
and the control effort rms-norm by:
$$ \begin{array}{*{20}l} {\left\| {\hat{u}} \right\|_{\rm rms} }{ = \left( {\frac{1}{2\pi }\int\limits_{0}^{\infty } {\hat{G}_{uu} \, {\rm d}\omega } } \right)^{1/2} } \\ {}{ = \left( {\frac{1}{2\pi }\int\limits_{0}^{\infty } {\left( {{}_{{}}\left| {\hat{H}_{22\_21} } \right|^{2} + \left| {\hat{H}_{22\_22} } \right|^{2} } \right)_{{}} }\, {\rm d}\omega } \right)^{1/2} } \\ \end{array} $$
(31)

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 H2/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 H2/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 Wn and Wd are specified in Sect. 3.1 and the derivation of performance weighting filters We and Wu 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 We. 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, eindex,r, to enable H control design. This can be accomplished by solving the NBP, which is equivalent to the derivation of the weighting filter We 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 H2-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 Ss and Ts 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 weighting filter We is the term that is not defined in the \( \left\| {\hat{e}} \right\|_{\rm index} \) related closed-loop specifications SS1 and TS1 in Table 5. The idea is thus to define a parameter dependent performance weighting filter:
$$ W_{e} (s,k_{e} ,\omega_{ce} ) = k_{e} (s + \omega_{ce} )^{m} $$
(32)
that is used to optimize \( \hat{G}_{ee} \). The parameter \( k_{e} \) is the filter gain, \( \omega_{ce} \) is the filter corner frequency and \( s = j\omega \) a complex variable. The parameters \( k_{e} \) and \( \omega_{ce} \) are used to shape SS1 and TS1 to tune \( \left\| {\hat{e}} \right\|_{\rm index} \). In the example of Appendix 1 the transfer function We in Eq. (32) is chosen to be a second-order high-pass by setting \( m = 2 \). This results in good noise cut-off properties of T with respect to measurement noise n and reasonable stability margins due to a roll-off of 40 dB/decade [24] because We shapes T based on the specification TS1.
The multivariable optimization problem with the variables \( k_{e} \) and \( \omega_{ce} \) is finally formulated as minimization of the objective function:
$$ \left\| {\delta \hat{e}} \right\|_{\rm index} = \left| {\left\| {\hat{e}} \right\|_{\rm index} - e_{{\rm index},r} } \right| $$
(33)
subject to the constraints:
$$ \begin{array}{*{20}c} {\left| {T_{\rm s} } \right|} \hfill & \ge \hfill & {6\,{\text{dB}}} \hfill & \forall \hfill & {\omega \ll \omega_{cT} } \hfill \\ {\left| {S{}_{S}} \right|} \hfill & \ge \hfill & {6\,{\text{dB}}} \hfill & \forall \hfill & {\omega \gg \omega_{cS} } \hfill \\ {\omega_{ce} } \hfill & \ge \hfill & {\omega_{cS} } \hfill & {} \hfill & {} \hfill \\ \end{array} $$
(34)
where \( \omega_{cT} (k_{e} ,\omega_{ce} ) \) and \( \omega_{cS} (k_{e} ) \). Note that \( \omega_{cS} \) only depends on \( k_{e} \) because of the constraint that \( \omega_{ce} \ge \omega_{cS} \).
In this way \( \left\| {\hat{e}} \right\|_{\rm index} \)is made compliant with the pointing error requirement \( e_{{\rm index},r} \) by minimizing \( \left\| {\delta \hat{e}} \right\|_{\rm index} \). The optimization problem in Eqs. (33) and (34) is quasilinear for \( e_{{\rm index},r} = 0 \). The definition of quasilinearity is given in [25]. The proof for quasilinearity is given in [26] for the parameterized transfer function \( W_{e} (s,k_{e} ,\omega_{ce} ) = k_{e} (s + \omega_{ce} )^{m} \). The number of decision variables is thereby limited to two in order to restrict the parameter space for solving the optimization problem. In the example of Fig. 12 the quasilinearity of the optimization problem is graphically illustrated. However, if \( e_{{\rm index},r} \ne 0 \), the objective function has two minima as can be seen in Fig. 12. This has to be considered when solving the optimization problem to get a unique and reproducible solution.
Fig. 12

rms- and index-norm performance estimate versus bandwidth

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 process for solving the optimization problem in Eqs. (33) and (34) is split in three steps to find both minima of the objective function, cf. [26]. In the first step the optimization process in Fig. 9 is performed to identify the decision variables that minimize \( \left\| {\hat{e}} \right\|_{\rm index} \). The closed-loop bandwidth, \( \omega_{\rm BW} \), that corresponds to that minimum is \( \omega_{\rm{BW, min}} \approx \omega_{{cT, {\rm min}}} (k_{e} ,\omega_{ce} ) \). In the next step the same optimization problem is formulated, but with the objective that \( \left\| {\hat{e}} \right\|_{\rm index} = e_{{\rm index},r} \). This objective is obviously only feasible if the minimum of the pointing error index estimate, \( \left\| {\hat{e}} \right\|_{\rm index, min} \), is smaller than \( e_{{\rm index},r} \). If this is the case, \( \omega_{cT} (k_{e} ,\omega_{ce} ) \) and thus implicitly the decision variables are additionally constrained by
$$ \omega_{cT} \ge \omega_{{{cT},{\rm min} }} {\text{ or }}\omega_{cT} \le \omega_{{{cT},{\rm min} }} $$
(35)
which results in two additional quasilinear optimization problems with either of the constraints in Eq.(35). The above-described solution process is clearly understood by looking at the example in Fig. 12. In this figure, and in general, one minimum for \( \left\| {\hat{e}} \right\|_{\rm index} = e_{{\rm index},r} \) is at frequencies \( \omega_{cT} \approx \omega_{\rm BW} \) higher than \( \omega_{\rm cT, min} \approx \omega_{{rm BW}2} \) and the other lower than that frequency.

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 Ss and Ts 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 Ss and Ts corresponds to eindex,r with an error smaller than \( \gamma_{e} \) for the predefined structure of \( W_{e} \) in Eq. (30). Consequently the magnitudes of Ss and Ts represent performance bounds indicating that at this magnitude shapes the pointing performance error estimate \( \left\| {\hat{e}} \right\|_{\rm index} \) corresponds to the requirement eindex,r. Meaning that any sensitivity and complementary sensitivity function below the magnitudes of Ss and Ts 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 Wu 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 We or Wu.

7 Controller synthesis

In order to completely specify the H optimization problem for controller synthesis the closed-loop specifications Ss and Ts in Table 5 are sufficient. However, the closed-loop specifications are formulated based on the input weighting filters Wn and Wd 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 Ss and Ts 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 GS/T weighting scheme, shown in Fig. 10 and specified in [22], is chosen to set lower bounds on the closed-loop specifications. The advantage of this weighting scheme is that it does not invert the plant when performing H-synthesis because G is included in the weighting scheme. The transfer matrix \( {\mathbf{M}}_{zw} \) of the GS/T weighting scheme with unit-PSD input and output signals, w and z, is:
$$ \left[ {\begin{array}{*{20}c} {z_{1} } \\ {z_{2} } \\ \end{array} } \right] = \underbrace {{\left[ {\begin{array}{*{20}c} {W_{{\bar{u}}} KSW_{T} } & { - W_{{\bar{u}}} TW_{S} } \\ {TW_{T} } & { - SGW_{S} } \\ \end{array} } \right]}}_{{{\mathbf{M}}_{zw} }} \left[ {\begin{array}{*{20}c} {w_{1} } \\ {w_{2} } \\ \end{array} } \right] $$
(36)

The closed-loop specifications Sbound and Tbound, 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 Sbound and Tbound it also complies with all individual closed-loop specifications in Table 5.

In Eq. (36) Sbound and Tbound are defined by two weighting filters. The weighting filter \( W_{T} \), which directly shapes the magnitude of T because:
$$ T_{\rm bound} = \left[ {W_{T} } \right]^{ - 1} $$
(37)
and the weighting filter \( W_{S} \), which shapes the magnitude of SG because:
$$ S_{\rm bound} = \left[ {W_{S} G} \right]^{ - 1} $$
(38)

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 Ss and Ts set the performance requirements eindex,r and ur at the system outputs e and u in Fig. 5. In consequence, the specification bounds Sbound and Tbound have to be below all S- and T-specifications. If the 0 dB crossing frequency of Ss is smaller than the one of Ts and if both are sufficiently far apart, then there are performance margins that can be exploited by placing Sbound and Tbound. 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 Sbound that follows the spectral shape of SS1 with the penalty of getting a controller with a higher order.

In Fig. 12 the estimates \( \left\| {\hat{e}} \right\|_{\rm index} \) and \( \left\| {\hat{u}} \right\|_{\rm rms} \) are plotted with respect to the control system bandwidth in the lower graph and put into context with the performance related closed loop specifications, in the upper graph. The estimates rely on the evaluation of Eqs.(30) and (31) by setting Ss and Ts of \( {\hat{\mathbf{H}}} \) in Table 6 to Sbound and Tbound. The resulting characteristic bandwidths in Table 7 are the basis for performance and bandwidth trade-offs.
Table 7

Characteristic closed-loop bandwidths

Outputs

Bandwidth

ωBW1

ωBW2

ωBW3

ωBW4

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 Sbound and Tbound always guarantees a closed-loop system with better performance than a solution of an H-optimization problem specified by all Ss and Ts 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 H2/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 H2-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

  1. 1.

    In this article the frequency domain representation of the pointing metrics derived in [4, 5, 16] is considered, which is the integral over the pointing error PSD weighted by a function Fmetric, cf. Sect. 2.

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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Copyright information

© CEAS 2013

Authors and Affiliations

  1. 1.Institute of Flight Mechanics and Control (iFR)Universität StuttgartStuttgartGermany
  2. 2.ESA/ESTECNoordwijkThe Netherlands
  3. 3.Astrium SatellitesFriedrichshafenGermany

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