Optimal decentralized wavelength control in light sources for lithography

The goal of the multi-prism configuration is to minimize the error ($\lambda-\hat{\lambda}$) between the reference ($\lambda$) and actual ($\hat{\lambda}$) wavelength. In reality, this wavelength error is $\lambda-\hat{\lambda}=K_{OP}(y_{P}-y_{P_{\textup{ref}}})+K_{OS}(y_{S}-y_{S_{\textup{ref}}})$, a combination of errors incurred from both prism positions. Without loss of generality, the cost function $J$ is defined as:

where $Q\succeq 0$ is a weighting on the states of the overall system, $\rho_{P}\succ 0$ and $\rho_{S}\succ 0$ penalize the control signals for the PZT and the stepper respectively. The state weighting matrix $Q$ is a function of the optics gains $K_{OP}$ and $K_{OS}$, and allows for coupling between the states of the two sub-systems: PZT $+$ Prism $3$, and Stepper $+$ Prism $4$. Since the PZT is a finer actuator with a limited range, $\rho_{P}>\rho_{S}$ holds in practice. Note that we are considering the infinite-horizon case in this paper for simplicity, with the finite horizon cost briefly discussed in Section 5.1.

To the best of our knowledge, existing state-of-the-art control strategies lack communication between each other, with both prisms controlled separately in a discontinuous manner [27, 3]. However, since both prisms simultaneously control the wavelength, we establish two-way communication between the controllers as shown in Fig. 4.

In the next section, we present the solution to Eq. 8. It is worth pointing out that there is no restriction on the time-delays and all four delay terms in $\mathcal{K}$ could be different from each other. The solution would still hold provided the delays satisfy the triangle inequality [33, 34, 32, 35].

We begin with the setup in Eq. 8. Define the control cost matrix $R\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\left(\begin{smallmatrix}\rho_{P}&0\\ 0&\rho_{S}\end{smallmatrix}\right)$ and performing Cholesky decomposition on $Q$ and $R$, we form the regulated output $z$ for wavelength control. We define the global plant

Now re-defining the cost Eq. 6 into the operator theoretic framework we obtain:

Eq. 9 is exactly the problem setup for Lemma 1. Using the block-diagonal structure of $\mathcal{G}^{\lambda}$, and $\forall\;\mathcal{K}\in\mathcal{S}_{\tau}$, we have $\mathcal{K}\mathcal{G}^{\lambda}\mathcal{K}\in\mathcal{S}_{\tau}$, which is the defining property of quadratic invariance [33]. The rest of the solution process follows the same steps as in ​​[36, Thm. 43]. Briefly, we leverage the block-diagonal structure of $\mathcal{W}^{\lambda}$ to divide Eq. 9 into two smaller sub-problems. Using the loop-shifting technique [38, 39] iteratively, we compensate for $\tau_{1}$, followed by $\tau_{2}-\tau_{1}$ in both the sub-problems. This results into two non-delayed LQG synthesis problems that are solved by standard optimal control techniques [37].

The solution to Eq. 8 generates an optimal controller, which can be implemented at the sub-system level. Leveraging the existing agent-level controller results for decentralized LQG problems ​​[36, Chap. 3], we implement a continuous-time version of our controller in Fig. 5. We introduce some notation used specifically for this implementation. $0$ is a matrix of zeros with dimensions based on context of use. $I_{n_{P}}$ is an identity matrix of size $n_{P}\times n_{P}$. $C_{1_{:P}}$ corresponds to the first $n_{P}$ columns of $\left(\begin{smallmatrix}Q^{\frac{\mathsf{T}}{2}}&0\end{smallmatrix}\right)^{\mathsf{T}}$. Similarly $D_{{12}_{:P}}$ is the first $m_{P}$ columns of $\left(\begin{smallmatrix}R^{\frac{\mathsf{T}}{2}}&0\end{smallmatrix}\right)^{\mathsf{T}}$.

Fig. 5 represents a continuous-time implementation of the optimal controller for the PZT $+$ Prism $3$ sub-system. While the calculation of the Kalman gain remains the same as for a non-delayed, non-decentralized LQG problem, the delays alter LQR gain computations. Given a combination $\textup{ARE}\left(A,B,C,D\right)$ and if the Riccati assumptions hold [37], there is a unique stabilizing solution $X\succ 0$ for

where gain $F\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}-\left(D^{\mathsf{T}}D\right)^{-1}\left(B^{\mathsf{T}}X+D^{\mathsf{T}}X\right)$, and $A+BF$ is Hurwitz. So we evaluate the Kalman and LQR gains as $L_{P}^{\mathsf{T}}=\textup{ARE}\left(A_{P}^{\mathsf{T}},B_{1_{P}}^{\mathsf{T}},C_{2_{P}}^{\mathsf{T}},D_{{21}_{P}}^{\mathsf{T}}\right)$ and $\tilde{F}_{P}=\textup{ARE}\left(A_{P},\tilde{B}_{2_{PS}},\tilde{C}_{1_{:P}},D_{{12}_{:P}}\right)$. $\tilde{B}_{2_{PS}}$ and $\tilde{C}_{1_{:P}}$ are modifications of ${B}_{2_{P}}$, ${C}_{1_{:P}}$ and are delay-dependent [36, 26]. The top block is a state-space matrix representation of the sub-system, which transmits wavelength measurements via the LAM. These are multiplied by the inverse of the optics gain $K_{OP}$ (not shown in Fig. 5) to generate $y_{P}$. However, we have a measurement delay incurred during this processing along-with any LAM delay, which are represented by the lumped delay term $\tau_{1}$. Note that the number of states in the sub-controller is equal to the total number states of the overall system, i.e., $n_{P}+n_{S}$. The Kalman filter (red) predicts the states of both the sub-systems based on available information. For the PZT $+$ Prism $3$, the available information are $\{e^{-s\tau_{1}}x_{P},\;e^{-s\tau_{2}}x_{S}\}$. The signal $v_{SP}$ already contains delayed (by $\tau_{1}$ s) information regarding the states of the Stepper $+$ Prism $4$ sub-system and is further delayed by a $\left(\tau_{2}-\tau_{1}\right)$ s of transmission time. The FIR blocks $\Pi_{u_{P}}$ (feed-forward) and $\Pi_{b_{P}}$ (feedback) together compensate for the continuous time-delays, without any Padé approximations.

While the controller works for a infinite horizon problem, the light sources seldom produce bursts over an infinite horizon. A finite horizon cost defined as

makes sense in such a case. The results from the infinite-horizon case are transferable. However the gains are no longer time-invariant and we can use dynamic programming approaches to solve the Hamiltonian-Jacobi-Bellman (HJB) equation associated with the differential Riccati equations to obtain $L_{P}(t)$ and $\tilde{F}_{P}(t)$.

An alternate approach to the above wavelength control approach is to begin with a discretized version of the plant itself. For sampling period $h$ satisfying the Nyquist criterion, a straightforward transformation of the plant dynamics occurs: $A_{i}\mapsto e^{A_{i}h}$, $B_{2_{ii}}\mapsto A_{i}^{-1}(e^{A_{i}h}-I)B_{2_{ii}}$ as $A_{i}$ is non-singular, $B_{1_{ii}}B_{1_{ii}}^{\mathsf{T}}\mapsto\int_{\tau=0}^{h}e^{A\tau}B_{1_{ii}}B_{1_{ii}}^{\mathsf{T}}e^{A^{\mathsf{T}}\tau}\mathrm{d}\tau$, $D_{{21}_{ii}}D_{{21}_{ii}}^{\mathsf{T}}\mapsto\frac{D_{{21}_{ii}}D_{{21}_{ii}}^{\mathsf{T}}}{h}$ for $i\in\{P,S\}$, with rest of the terms remaining the same. The corresponding Kalman and LQR gains are obtained by solving a discrete ARE for the infinite horizon cost case. Akin to Section 5.1, we can solve for time-varying gains using the Bellman equation for the finite-horizon cost for discrete decentralized LQG problem [43]. This approach loses the elegance and exactness of handling continuous time-delays due to the underlying approximation involved. However the structure of the discrete implementation remains similar to Fig. 5 with a discrete Kalman filter, discrete LQR, finite-dimensional FIR blocks, and time delays.

Here we show that the controller architectures provided in Figs. 5 and 6 incur the lowest cost in comparison to existing approaches for wavelength control. Current control techniques utilize a discontinuous approach, where the coarse actuator serves to desaturate the finer actuator [3]. Further the amount of wavelength target change determines whether a single actuator or a multi-actuator combination is used for control [3]. The decentralized control method allows for synchronization of the control action of both the sub-systems, ensuring a more continuous level of control. We define the average cost incurred by any sub-optimal strategies as $J_{\textup{dec,del}}+\Delta$, where $J_{\textup{dec,del}}$ is the average optimal cost for a global controller $\mathcal{K}_{\textup{dec,del}}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\left(\begin{smallmatrix}e^{-s\tau_{1}}\mathcal{K}_{P}&0\\ 0&e^{-s\tau_{1}}\mathcal{K}_{S}\end{smallmatrix}\right)$ given the cost function defined in Eq. 6, and $\Delta$ is the additional cost due to sub-optimal implementations: for instance, open-loop control of the stepper. Note that the measurement delay $\tau_{1}$ is already considered in this framework. Defining the cost $J_{\textup{cen,del}}$ for $\left(\begin{smallmatrix}e^{-s\tau_{1}}\mathcal{K}_{P}&e^{-s\tau_{1}}\mathcal{K}_{SP}\\ e^{-s\tau_{1}}\mathcal{K}_{PS}&e^{-s\tau_{1}}\mathcal{K}_{S}\end{smallmatrix}\right)$, we have $J_{\textup{cen,del}}<J_{\textup{dec,del}}$ from results in ​​[26, Thm. 17] and ​​[44, Lem. 9]. The final step is to establish that the cost $J_{\textup{cen},\tau_{2}}$ defined for $\left(\begin{smallmatrix}e^{-s\tau_{1}}\mathcal{K}_{P}&e^{-s\tau_{2}}\mathcal{K}_{SP}\\ e^{-s\tau_{2}}\mathcal{K}_{PS}&e^{-s\tau_{1}}\mathcal{K}_{S}\end{smallmatrix}\right)$ satisfies $J_{\textup{cen,del}}<J_{\textup{cen},\tau_{2}}<J_{\textup{dec,del}}$. While $J_{\textup{cen,del}}<J_{\textup{cen},\tau_{2}}$ is implicit from ​​[26, Thm. 17]. Using a simple limit argument and $J_{\textup{cen},\tau_{2}}$ being a monotonic non-decreasing function with increasing delays ​​[45, Prop. 6], we establish $J_{\textup{cen},\tau_{2}}<J_{\textup{dec,del}}$. Indeed for $\lim_{\left(\tau_{2}-\tau_{1}\right)\to\infty}$, $\mathcal{K}\to\mathcal{K}_{\textup{dec,del}},$ and $J_{\textup{cen},\tau_{2}}\to J_{\textup{dec,del}}$. Thus $J_{\textup{cen},\tau_{2}}<J_{\textup{dec,del}}+\Delta$ establishes the ‘optimal’ performance of the decentralized controllers in comparison to existing approaches.

The light source open loop frequency data has several narrow-band, periodic components at known frequencies, some of which are aliased [27]. One of the primary sources of such wavelength disturbances are blowers in the gas chamber that cause acoustic waves inside the chamber, which couple into the optics [3]. For a given sampling rate, periodic disturbances can be modeled as sinusoidal signals, and this disturbance model can be integrated with the state-space formulation in Eqs. 5 and 4 as part of the PZT $+$ Prism $3$ sub-system. However, this could induce the loss of unobservability for the estimation and uncontrollability for the regulator problems in the augmented state-space system. This is not a hard constraint for solving AREs in the finite-horizon case; while workarounds exist for the infinite-horizon case. We can add a small, non-zero value to disturbance matrices for $B_{2}$ and $C_{2}$ to satisfy observability and controllability. With the prior changes, we can solve the optimal decentralized control problem for the augmented system, while simultaneously compensating for these periodic disturbances. Alternately, there exists a rich literature on implementable disturbance cancellation techniques, which are beyond the scope of this paper.

The discrete-time formulations of Figs. 5 and 6 consider a constant sampling rate. However practical considerations could require a variable sampling rate, due to laser feature specifications or the sampler characteristics. Further hardware or other restrictions could require a sampling rate different from the control rate [27]. However the elegant observer-regulator separation structure for the decentralized control problem (red-blue box separation) allows for implementation of modified Kalman filters and modified LQRs to handle these variations. It is worth noting that the LQR gain is solely delay-dependent, with only certain elements of this matrix being functions of $\tau_{1}$ and $\tau_{2}-\tau_{1}$. This allows for faster tuning in case of time-varying delays.