Periodic Scenario Trees: A Novel Framework for Robust Periodic Invariance and Stabilization of Constrained Uncertain Linear Systems

TL;DR

The paper tackles robust stabilization of discrete-time constrained uncertain linear systems using periodic controllers. It introduces Finite Step Lyapunov Functions (FSLFs) and a scenario-tree viewpoint to synthesize both static gains and Linear Interpolating Tree Periodic Controllers (LITPC) with memory, ensuring robust exponential stability. For constrained systems, it derives convex LMIs to guarantee robust periodic invariants in the form of ellipsoids, and demonstrates that the resulting invariant sets can be larger than those from prior methods, improving terminal set design for MPC. The methods are non-anticipative and rely on offline computation with online interpolation, offering reduced conservatism and practical impact for robust MPC applications.

Abstract

This work proposes a new a framework for determining robust periodic invariant sets and their associated control laws for constrained uncertain linear systems. Necessary and sufficient conditions for stabilizability by periodic controllers are stated and proven using finite step Lyapunov functions for the unconstrained case. We then introduce a scenario tree interpretation of finite step Lyapunov functions for uncertain systems and show that this interpretation results in useful criteria for the design of robust stabilizing controllers. In particular, novel convex feasibility criteria for the synthesis of simple static controllers and what we call linear interpolating tree periodic controllers with memory are derived. It is proven that for a sufficiently large length of the period, a stabilizing linear interpolating tree periodic controller can always be found using the proposed criterion provided that the uncertain system is stabilizable by such controllers. In this sense, the presented synthesis method is non-conservative. The results are then extended to constrained uncertain linear systems and conditions for controllers that realize robust periodic invariant sets which are less conservative than those that result from the known methods in the literature are derived.

Figures (7)

• Figure 1: Scenario tree representation of $\mathcal{S}_{\mathcal{D}}$ for three time steps ($N=3$). The set $\mathbf{D}$ has two vertices, i.e., $\Gamma=\{1,2\}$. The quadratic functions at each stage (time step) are determined such that they are smaller than the quadratic function at their parent node. For the last stage ($t=N$), the symmetric positive definite matrix $P_0$ which was used for the initial state $x_0$ is used again.
• Figure 2: The figure shows a sketch of the ellipsoidal robust periodic invariance that results from the approach in canon_inv_1canon_inv_2, with $N=3$. The constraints are shown as dashed-dotted red boxes. If the state is inside the ellipsoid $\mathcal{P}_0$, applying $u_t=K_0x_t$ guarantees that the next state is inside $\mathcal{P}_1$, $\forall (A_t,B_t)\in \mathbf{D}$. At the next time step, $u_t=K_1x_t$ is applied to the plant which guarantees that the state at the next time step will be inside $\mathcal{P}_2$. At the next time step $u_t=K_2x_t$ moves the state back again to $\mathcal{P}_0$.
• Figure 3: The figure shows a sketch of the ellipsoidal robust periodic invariance resulting that results from the application of the LITPC with $N=3$ for an uncertain system with $n_d=2$. The constraints are shown by dashed-dotted red boxes. If the state is inside the ellipsoid $\mathcal{P}_0$, applying $u_t=K_0x_t$ guarantees that the next state is inside the convex hull of $\mathcal{P}^1_1$ and $\mathcal{P}^2_1$, $\forall (A_t,B_t)\in \mathbf{D}$. At the next time step, determining vectors $x^j_1 \in \mathcal{P}^j_1$ and scalars $\beta^j_1\geq0$ such that $\sum^2_{j=1}\beta^j_1=1$ and $x_t=\sum^2_{j=1}\beta^j_1x^j_1$, and applying $u_t=\sum^2_{j=1}\beta^j_1K^j_1x^j_1$ to the plant guarantees that the state at the next time step will be inside the convex hull of $\mathcal{P}^j_2$, for $j=\{1,2,3,4\}$. At the next time step, determining vectors $x^j_2 \in \mathcal{P}^j_2$ and scalars $\beta^j_2\geq0$ such that $\sum^4_{j=1}\beta^j_2=1$ and $x_t=\sum^4_{j=1}\beta^j_2x^j_2$, and applying $u_t=\sum^4_{j=1}\beta^j_2K^j_2x^j_2$ to the plant guarantees that the state at the next time step will be inside $\mathcal{P}_0$.
• Figure 4: Example \ref{['example_sets']}: The sets $\mathcal{P}_0$. The set obtained by the method from canon_inv_2 with $N=20$ (solid red). Sets obtained by the proposed static controller with $N=2$ (dashed blue) and $N=4$ (cross Green). Sets obtained by the proposed LITPC with $N=2$ (diamond grey) and $N=4$ (circle black).
• Figure 5: Example \ref{['example_sets']}: The sets $\mathcal{P}^j_t$ resulting from solving \ref{['optimization_differnt_K']} with $N=2$. The set $\mathcal{P}_0$ is shown in dashed red. The sets $\mathcal{P}^j_1$, $j \in \{ 1,2,3,4\}$ are shown in solid blue.

Theorems & Definitions (53)

• Definition 1
• Definition 2
• Remark 1
• Definition 3
• Definition 4
• Lemma 1
• proof
• Remark 2
• Remark 3
• Lemma 2