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.

Appendices

Appendix 1 Application example

In this section the precision pointing H _∞ control design approach introduced in this article is applied to a simple case study. The aim is to design a SISO controller for one axis of a satellite with the plant, G = J ⁻¹ s ⁻², where the moment of inertia J = 4,500 kg m² and performance requirements are defined in Table 8. The requirements are given in mili-arcseconds (mas) and mili-Newton-meter (mNm).

Table 8 Performance requirements

The pointing error sources are modeled according to Eqs. (11) and (12). The measurement noise shaping filter W _n is shown in Fig. 13 and the disturbance noise shaping filter W _d in Fig. 14.

Fig. 13

Measurement noise shaping filter W _n

Fig. 14

Disturbance noise shaping filter W _d

Deriving the optimal closed-loop performance specification with the optimization in Sect. 6 yields the S- and T-specifications in Fig. 11. The specifications are put in context with the corresponding rms- and index-norm in Fig. 12. The limits of performance with respect to the system bandwidth are thus identified, which provides the basis for system trade-offs. At this point various specifications representing different hardware characteristics or design criteria can be traded.

As described in Sect. 7 the GS/T weighting scheme is used to set the specification bounds to ω _BW3 in Fig. 12, thus just complying with the RPE requirement, e _RPE,r , as stated in Table 7. This yields the closed-loop system in Fig. 15 with the specified and actual system performance listed in Table 9 and illustrated in Fig. 16.

Fig. 15

S- and T-specification with bounds and closed-loop transfer functions S and T

Fig. 16

Actual system $\sqrt {\tilde{F}_{{\rm RPE}} G_{{ee}} }$ versus specified $\sqrt {\tilde{F}_{\rm RPE} \hat{G}_{ee} }$

Table 9 System performance

The actual system performance is better than the specified due to the interpolation conditions applied during performance estimation in the transition region, $\omega_{cS} < \omega < \omega_{cT}$, as mentioned in Sect. 7.3. The resulting margin for the relative pointing performance error, e _RPE, is illustrated in $\tilde{F}_{\rm RPE} \hat{G}_{ee}$, is plotted with respect to the actual system PSD, $\tilde{F}_{\rm RPE} G_{ee}$. The more illustrative cumulated sum of the signal norm is shown in Fig. 17. The deviation of the actual PSD from the specified PSD for $\omega \to 0$ is due to residual spectral margins. This deviation could be reduced by selecting a more complex transfer function for W _e when performing the optimization in Sect. 6.

Fig. 17

Cumulated sum of actual $\left\| e \right\|_{\rm RPE}$versus specified $\left\| {\hat{e}} \right\|_{\rm RPE}$

Appendix 2 Performance measure derivation for sum of noise sources

The expected value of the sum of two stationary random noise processes $x_{k} (t)$ and $y_{k} (t)$ with zero mean value is:

$$ \begin{aligned} {{E}}\left[ {\left( {X_{k} (\omega ) + Y_{k} (\omega )} \right)\left( {X_{k} (\omega ) + Y_{k} (\omega )} \right)^{ * } } \right] \\ &= {E}\left[ {\left( {XX^{ * } + XY^{ * } + YX^{ * } + YY^{ * } } \right)} \right]\\ & = {E}\left[ {XX^{ * } } \right] + {E}\left[ {XY^{ * } } \right] + {E}\left[ {YX^{ * } } \right] + {E}\left[ {YY^{ * } } \right] \\&= G_{xx} (\omega ) + G_{xy} (\omega ) + G_{yx} (\omega ) + G_{yy} (\omega ) \hfill \\ \end{aligned} $$

(39)

with $X_{k} (\omega )$ and $Y_{k} (\omega )$ being the Fourier transforms of $x_{k} (t)$ and y _k (t), respectively, and $X_{k}^{ * } (\omega )$ and $Y_{k}^{ * } (\omega )$ being their complex conjugate. The expected value $E[ \cdot ]$ is taken over the ensemble index k. $G_{xx} (\omega )$ and $G_{yy} (\omega )$ are the PSD and G _xy (ω) and $G_{yx} (\omega )$ are the cross-spectral densities of $x_{k} (t)$ and $y_{k} (t)$.

If $G_{xy} (\omega )$ and $G_{yx} (\omega )$ are not known, lower and upper bound assumptions can be made. The upper bound assumption is that $x_{k} (t)$ and $y_{k} (t)$ are fully cross-correlated. In that case the following inequality applies:

$$ G_{xy} + G_{yx} \le G_{xx} + G_{yy} $$

(40)

The lower bound assumption is that x and y are not cross-correlated, meaning that:

$$ G_{xy} + G_{yx} = 0 $$

(41)

In the precision pointing control problem defined in Sects. 3 and 4 the PSD of the output signals, e and u, are the input noise PSD, $G_{nn}$ and $G_{dd}$, of w _n and w _d transformed by the closed loop system, ${\mathbf{H}}_{\rm CL}$, in Fig. 5 according to Eq. (2):

$$ \left[ {\begin{array}{*{20}c} {G_{ee} } & {G_{eu} } \\ {G_{ue} } & {G_{uu} } \\ \end{array} } \right] = \underbrace {{\left[ {\begin{array}{*{20}c} {H_{11} } & {H_{12} } \\ {H_{21} } & {H_{22} } \\ \end{array} } \right]}}_{{{\mathbf{H}}_{\text{CL}} }} \left[ {\begin{array}{*{20}c} {G_{nn} } & {G_{nd} } \\ {G_{dn} } & {G_{dd} } \\ \end{array} } \right]_{{}} \underbrace {{\left[ {\begin{array}{*{20}c} {H_{11}^{ * } } & {H_{21}^{ * } } \\ {H_{12}^{ * } } & {H_{22}^{ * } } \\ \end{array} } \right]}}_{{{\mathbf{H}}_{\text{CL}}^{ * } }} $$

(42)

The PSD at e and u written in terms of the transformed input PSD thus is:

$$ \begin{gathered} G_{ee} = H_{11} G_{nn} H_{11}^{ * } + H_{11} G_{nd} H_{12}^{ * } + H_{12} G_{dn} H_{11}^{ * } + H_{12} G_{dd} H_{12}^{ * } \hfill \\ G_{uu} = H_{21} G_{nn} H_{21}^{ * } + H_{21} G_{nd} H_{22}^{ * } + H_{22} G_{dn} H_{21}^{ * } + H_{22} G_{dd} H_{22}^{ * } \hfill \\ \end{gathered} $$

(43)

In Eqs. (28)–(31) the input noise processes, w _n and w _d , are assumed to be not cross-correlated because they originate from independent sources. Consequently, $G_{nd} = G_{dn} = 0$ and if ${\mathbf{H}}_{\rm CL}$ is augmented by $W_{n}$ and $W_{d}$ at its inputs, Eq. (43) can be simplified to:

$$ \left[ {\begin{array}{*{20}c} {G_{ee} } & {G_{eu} } \\ {G_{ue} } & {G_{uu} } \\ \end{array} } \right] = {\mathbf{H}}_{\rm CL} \left[ {\begin{array}{*{20}c} {W_{n} } & 0 \\ 0 & {W_{d} } \\ \end{array} } \right]_{{}} \left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ \end{array} } \right]_{{}} \left[ {\begin{array}{*{20}c} {W_{n}^{ * } } & 0 \\ 0 & {W_{d}^{ * } } \\ \end{array} } \right]_{{}} {\mathbf{H}}_{\rm CL}^{ * } $$

(44)

by remembering that in Eqs. (11) and (12) the PSD are written as $G_{nn} = W_{n} N_{ww} W_{n}^{*}$ and $G_{dd} = W_{d} N_{ww} W_{d}^{*} \,$ with $N_{ww} = 1$. The augmented ${\mathbf{H}}_{\rm CL}$ thus corresponds to ${\mathbf{H}}_{ 2 2}$ of Eq. (27). Substituting ${\mathbf{H}}_{\rm CL}$by ${\mathbf{H}}_{ 2 2}$ and simplifying Eq. (43) leads to

$$ \begin{array}{*{20}c} {G_{ee} }{ = H_{22\_11} H_{22\_11}^{ * } + H_{22\_12} H_{22\_12}^{ * } }{ = \left| {H_{22\_11} } \right|^{2} + \left| {H_{22\_12} } \right|^{2} } \\ {G_{uu} }{ = H_{22\_21} H_{22\_21}^{ * } + H_{22\_22} H_{22\_22}^{ * } }{ = \left| {H_{22\_21} } \right|^{2} + \left| {H_{22\_22} } \right|^{2} } \\ \end{array} $$

(45)

If the input noise processes are cross-correlated, but the cross-spectral densities $G_{nd}$ and $G_{dn}$ are unknown, a worst case $G_{ee}$ and $G_{uu}$ can be determined by setting the transformed $G_{nd}$ and $G_{dn}$ to its upper bound defined by Eq. (40):

$$ \begin{array}{*{20}c} {G_{ee} }{ = 2H_{22\_11} H_{22\_11}^{ * } + 2H_{22\_12} H_{22\_12}^{ * } }{ = 2\left| {H_{22\_11} } \right|^{2} + 2\left| {H_{22\_12} } \right|^{2} } \\ {G_{uu} }{ = 2H_{22\_21} H_{22\_21}^{ * } + 2H_{22\_22} H_{22\_22}^{ * } }{ = 2\left| {H_{22\_21} } \right|^{2} + 2\left| {H_{22\_22} } \right|^{2} } \\ \end{array} $$

(46)