Consider the discrete-time generalized LTI plant with minimal state-space realization $$x_{k+1}=A_d x_k + B_{d1}w_k\\z_k=C_{d1}x_k+D_{d11}w_k\\y_k=C_{d2}x_k+D_{d21}w_k$$ For the Schur-stable $A_d$ problem of finding optimal H2-filter of the form $$\widehat{x}_{k+1}=A_f \widehat{x}_k+B_f y_k\\ \widehat{z}_k=C_f \widehat{x}_k+D_f y_k$$ is well known [1-3]. In [1-3] you can find a general methodology on how to reformulate this filter synthesis problem as convex optimization problem with LMI constraints. So my question is whether it is possible for Schur-unstable case to reformulate this filter synthesis problem as convex optimization problem with LMI constraints. Are there any known approaches for this important case?
Example:
Consider a simple system of two stochastic differential equations (Ornstein–Uhlenbeck process and it's integral): $$\begin{cases}dx_1=dx_2dt \\ dx_2=-\beta x_2dt + \sigma dw\end{cases}$$
By discretizing this system in time, we get: $A_d=\begin{bmatrix}1 & T \\0 & 1-\beta T \end{bmatrix}$, $B_{d1}=\alpha \begin{bmatrix}\sqrt{\frac{T^3}{3}} & 0 & 0 \\ \frac{1}{2}\sqrt{3T} & \frac{1}{2}\sqrt{T} & 0 \end{bmatrix}$, $\alpha=\sigma \sqrt{2\beta}$. Let also $C_{d1}=C_{d2}=\begin{bmatrix}1 & 0 \end{bmatrix}$, $D_{d11}=0$, $D_{d21}=\begin{bmatrix}0 & 0 &\sigma_n\end{bmatrix}$
Define filtering error $\epsilon_k=\widehat{x}_k-x_k$ and write down augmented system for variables $x_k, \epsilon_k$: $$\begin{bmatrix}x_{k+1} \\ \epsilon_{k+1} \end{bmatrix}=\begin{bmatrix}A_d & 0 \\A_f-A_d+B_fC_{d2} & A_f \end{bmatrix} \begin{bmatrix}x_k \\ \epsilon_k \end{bmatrix} + \begin{bmatrix}B_{d1} \\ B_f D_{d21}-B_{d1} \end{bmatrix}w_k$$ And now if we take for example $B_f=\begin{bmatrix}k_1 \\ k_2 \end{bmatrix}$ equal to steady-state Kalman gain and $A_f=\begin{bmatrix}1-k_1 & T \\-k_2 & 1 \end{bmatrix}$, we will see that augmented system has three stable modes (last 3 equations) and one unstable mode (first equation), decoupled(!) from stable modes. Thus, in such decoupled case we can freely remove from above augmented system equation(s) corresponding to unstable modes and get new stabilizable augmented system.
Thus, as we can see from this example, for filtering error plus some additional states we can get stabilizable system. But unfortunately we cannot apply the approach from [1-3] directly to this system (because now we have unwanted coupling of variables).
Literature
- Ryan James Caverly, James Richard Forbes. LMI Properties and Applications in Systems, Stability, and Control Theory. https://arxiv.org/abs/1903.08599 ,see p.136 and bellow
- C. Scherer et al. Multiobjective output-feedback control via LMI optimization. DOI: 10.1109/9.599969
- Izumi Masubuchi et al. LMI-based controller synthesis: A unified formulation and solution. DOI: 10.1002/(SICI)1099-1239(19980715)8:8<669::AID-RNC337>3.0.CO;2-W
The problem can certainly be reformulated as an LMI problem as you rightly mention but the problem will never be feasible because the system is not stable. In fact, you cannot even define the H2-norm for an unstable system which would be infinite.
If you look at [1], you can see that (5.7) is stable if and only if both the filter and the system are stable. There is no way around that.
A workaround is, instead of considering a filter, to consider an observer that will be able to stabilize the error dynamics even if the system itself is unstable. This is considered in Section 5.1.2 of [1]. You can see there that what matters is that $A_d-L_dC_{d2}$ is stable where $L_d$ is the gain of the observer. Such a stabilizing gain is known to exist if and only if the pair $(A_d,D_{d2})$ is detectable.