## 6 Onboard Steering Commands via Minimal-Cost-Variance Control  
Each pLEO platform is presumably working to generate its own local IEM realization for pLEOT. To achieve this, each platform will adjust its local reference clock, such as an autonomous robust time scale defined by \(pLEOT(i)\), in relation to the IEM estimation. Alongside the controller update rate, denoted as \(h\), there is an additional tuning parameter called the controller gain, \(K_{i}\). This gain is used to adjust the corresponding OCXO signal response to match the IEM estimate. In general, a larger controller gain is expected to produce larger frequency corrections for any given offset of the OCXO from the IEM estimate, leading to a more accurate realization of the IEM. To optimize this responsiveness, the elements of the controller gain, \(K_{i}\), which dictate how effectively the autonomous remote time scale system operates on platform \(i\), are designed. Recall that the discrete-time state-space counterpart of a local IEM realization model (83) that is completely controllable is governed by  
\[s_{i}(k+1) =A_{i}(h)s_{i}(k)+B_{i}(h)u_{i}(k)+w_{i}(k)\,, \tag{91}\] \[z_{i}(k) =s_{i}(k)\,, \tag{92}\]  
where the states of the steering control system (91) include the phase and frequency of the steered OCXO clock onboard orbital platform \(i\). The steering commands, denoted as \(u_{i}(k)\), are sent to an actuator such as an MPS device. The output signals or measurements typically consist of the phase deviation \(z_{i}(k)\) between the local OCXO oscillator and the IEM estimation. The controller \(u_{i}(k)\) operates on the actuator to minimize the phase deviation \(z_{i}(k)\). Noise parameters associated with both the OCXO and IEM clocks are determined based on measured Allan deviation curves. The realization of these noises is represented by a zero-mean Gaussian process noise \(w_{i}(k)\), with its covariance matrix given by \(Q_{w_{i}}(h)=Q_{i,OCXO}(h)+Q_{IEM}(h)\). Moreover, one of several requirements can be established for clock steering on each platform in relation to the IEM estimation. For instance, the performance indicator suggests a convex cost function for selecting a steering policy. This insight can be represented by a quadratic cost function, based on the idea that the relative penalties assigned to phase and frequency deviations, as well as steering efforts concerning a local reference clock \(i\), can be optimized over a finite horizon, \([n_{0},N_{i}]\), for all time \(k\). To be credible, these penaltiesmust be responsive to the clock steering operations aimed at reducing phase and frequency deviations toward zero  
\[J_{i}(n_{0}) =\sum_{k=n_{0}+1}^{N_{i}}L_{i}(k,s_{i}(k),u_{i}(k-1)) \tag{93}\] \[=\sum_{k=n_{0}+1}^{N_{i}}\bigl{[}s_{i}^{T}(k)Q_{i}(k-1)s_{i}(k)+u_ {i}^{T}(k-1)R_{i}(k-1)u_{i}(k-1)\bigr{]} \tag{94}\]  
where the weighting coefficients associated with the steering efforts, \(R_{i}(k-1)\) and the phase and frequency deviations, \(Q_{i}(k-1)\) are positive and bounded. In general, if \(R_{i}\) is large compared to \(Q_{i}\), the penalty is large for the controller and/or actuator attempting to drive both phase and frequency deviations toward zero to rapidly. On the other hand, if \(Q_{i}\) is large compared to \(R_{i}\), the controller faces a small penalty for a large steering effort, and the system is driven toward zero more quickly. The risk management perspective of clock steering operations emphasizes the importance of effective risk handling. It specifically focuses on the feedback steering mechanism that shapes the probability distribution function of the chi-squared random variable referenced in Equation (94). This approach is crucial for ensuring the robustness and reliability of steering operations. The author has delved into various cost-cumulant control strategies, introducing a risk-value aware performance index defined as a linear manifold composed of a finite set of centralized moments linked to the performance measure affected by a chi-squared random variable.