Lecture 20: system-level synthesis
Goal: reformulate optimal control problems for linear systems in novel way.
References
- Anderson, J., Doyle, J. C., Low, S. H. & Matni, N. System level synthesis. Annu. Rev. Control 47, 364–393 (2019).
Problem formulation
Consider the discrete-time linear time-varying (DT-LTV) system
\[ x_{t+1}=A_{t} x_{t}+B_{t} u_{t}+w_{t}, \quad x_{t} \in \mathbb{R}^{n}, u_{t} \in \mathbb{R}^{p} \]
and suppose \(u_{t}=K_{t}\left(x_{0}, x_{1}, \cdots, x_{t}\right)\), i.e. feedback is causal, and let \(K_{t}\) be linear.
Define \(x=\left[\begin{array}{c}x_{0} \\ x_{1} \\ \vdots \\ x_{T}\end{array}\right], u=\left[\begin{array}{c}u_{0} \\ u_{1} \\ \vdots \\ u_{T}\end{array}\right], w=\left[\begin{array}{c}w_{0} \\ w_{1} \\ \vdots \\ w_{T}\end{array}\right]\) and
\[ K=\left[\begin{array}{cccc} K^{0,0} & 0 & \cdots & 0 \\ K^{1,1} & K^{1,0} & \cdots & 0 \\ \vdots & & \ddots & 0 \\ K^{T,T} & \cdots & K^{T,1} & K^{T,0} \end{array}\right] \]
so that \(u_t = K_t x\).
Furthermore, define
\[ A=\operatorname{diag}\left( A_0, A_1, \dots, A_{T-1}, 0\right), \]
\[ B=\operatorname{diag}\left( B_0, B_1, \dots, B_{T-1}, 0\right), \]
\[ Z=\left[\begin{array}{ccc} 0 & \cdots & \cdots & 0 \\ I & 0 & 0 & \vdots \\ 0 & I & 0 & \vdots \\ \vdots & \ddots & \ddots & \vdots \\ 0 & \cdots & I & 0 \end{array}\right] \]
so that the dynamics over the entire time horizon are specified by the linear equation
\[ x = Z A x + Z B u + w = Z(A + B K) x + w; \]
Note that \(A, B, K, Z\) are all block lower-triangular.
Rearranging yields \[ x = (I - Z(A + BK))^{-1} w \] \[ u = K (I - Z(A + BK))^{-1} w \]
Q: verify that \(K (I - Z(A + BK))^{-1}\) is block lower-triangular (BLT).
The product of BLT is BLT, and the inverse of BLT is BLT.
System responses
We can write \(\left[\begin{array}{c}x \\ u\end{array}\right]=\left[\begin{array}{c}\Phi_{x} \\ \Phi_{u}\end{array}\right] w\) with \(\Phi_{x}, \Phi_{u}\) BLT.
Theorem 2.1: 1. \([I-Z A-Z B]\left[\begin{array}{c}\Phi_{x} \\ \Phi_{u}\end{array}\right]=I\) for all systen responses; 2. if \(\Phi_{x}, \Phi_{u}\) are BLT then \(K=\Phi_{u} \Phi_{x}^{-1}\) is the controller.
Proof: \(\Phi_{x}=(I-Z(A+B K))^{-1}\) and \(\Phi_{u}=K(I-Z(A+B K))^{-1}\) so it is easy to verify that \((I-Z A) \Phi_{x}-Z B \Phi_{u}=I\). (Note that \(\Phi_{x}^{t, 0}=I\) so the inverse exists.)
\[ \begin{aligned} & \left(I-Z\left(A+B \Phi_{u} \Phi_{x}^{-1}\right)\right)^{-1}=\left[\left((I-Z A) \Phi_{x}-Z B \Phi_{u}\right) \Phi_{x}^{-1}\right]^{-1} \\ & =\Phi_{x}\left[(I-Z A) \Phi_{x}-Z B \Phi_{u}\right]^{-1}=\Phi_{x} \\ \end{aligned} \] so \(x=\Phi_{x} w\). We have \(u=\Phi_{u} \Phi_{x}^{-1} x=\Phi_{u} \Phi_{x}^{-1} \Phi_{x} w=\Phi_{u} w\ \square\).
Optimal control
Now the system response is parameterized by \(\Phi_{x}\) & \(\Phi_{u}\) so the closed-loop dynamics become linear constraints in an optimal control problem!
Example: (LQR) let \(w \sim \mathcal{N}\left(0, \Sigma_{w}\right)\) be iid and consider \[ \begin{aligned} \min _{x_{t}, u_{t}} & \sum_{t=0}^{T} E\left[x_{t}^{\top} Q x_{t}+u_{t}^{\top} R_{t} u_{t}\right] \\ \text { s.t. } & x_{t+1}=A x_{t}+B u_{t}+w_{t}. \end{aligned} \] This problem can be rewritten \[\begin{aligned} \min_{\Phi_x,\Phi_u} & \left\| \left[\begin{array}{cc} Q^{1/2} & 0 \\ 0 & R^{1/2} \end{array}\right]\left[\begin{array}{c}\Phi_x \\ \Phi_u \end{array} \right] \Sigma_w^{1/2} \right\|_F^2 \\ & \text{s.t.}\ \ \left[\begin{array}{cc}I - Z A \\ - Z B \end{array} \right] \left[\begin{array}{c}\Phi_x \\ \Phi_u \end{array} \right] = I \end{aligned}\]which is a convex quadratic program in \(\Phi_x, \Phi_u\)! Therefore there are tractable algorithms to find the global optimum for large-scale problems.
Aside: if gave worried about inverting \(\Phi_{x}\), fear not! With \(\hat{w}_{t}=x_{t+1}-\hat{x}_{t+1}\) the equations
\[ u_{t}=\sum_{\tau=1}^{T} \Phi_{u}(t, \tau) \hat{w}_{t-\tau},\ \hat{x}_{t+1}=\sum_{\tau=1}^{T-1} \Phi_{x}(t+1, \tau+1) \hat{w}_{t-\tau} \]
implement \(u=K x=\Phi_{u} \Phi_{x}^{-1} x\).
Takeaway: SLS allows reformulation of optimal (robust) control problems to be more amenable to constrained/distributed settings.