Optimiality of Set-based Robust State Estimators

This repo includes all experimentes implemented in the paper "Theoretical Relationship of Set-Based Robust State Estimators: An Optimality Perspective" accepted by the IEEE Transactions on Automatic Control as the Technical Note. Please check the data and test some experiments to complement the paper.

Theoretical Relationship of Set-Based Robust State Estimators: An Optimality Perspective

Yidian Fan, Feng Xu

Contact: Yidian Fan ([email protected]), Feng Xu ([email protected])

Contents

• Pre-requisites
• Organization Structure
• Quick start
• News

Pre-requisites

• Example codes uses MATLAB R2019b or higher, MPT Toolbox, CVX
• Additional care should be taken to the MPT3 toolbox. We detected a bug in the file "mtimes.m", which is in the installation subdirectory "\tbxmanager\toolboxes\mpt\3.2.1\all\mpt3-3 2 1\mpt\modules\geometry\sets@Polyhedron\mtimes.m" of the MPT3 toolbox. This function deals with the operation that a scalar $\alpha$ times a polyhedron $P=\lbrace x|Ax\leq b\rbrace$. The 125th line makes $\alpha P=\lbrace x|Ax\leq \alpha b\rbrace$, which is only valid when $\alpha\geq 0$. The correct form is $\alpha P=\lbrace x|\frac{1}{\alpha}Ax\leq b\rbrace$, $\forall \alpha\neq 0$.

Organization Structure

This repo has one main directory with eight subdirectories, whose roles are listed as follows.

• Main directory: The main files that conduct large-scale numerical tests on the systems.
  • Twotank_intense_test.m: Run large-scale tests on the two-tank system.
  • Fourtank_intense_test.m: Run large-scale tests on the four-tank system.
  • main.m: Run one sampling case to debug or visualize the result.
  • Verification.m: Compute the approximation accuracy of the proposed algorithms by optimization programs.
• Data: The stored numerical results of the experiments described in the manuscript and Answer sheet.
• Error_data_twotank, Error_data_fourtank:The intermediate results of cases in which the MPT3 reports errors. These data are reserved for debugging and illustrating the limits of the MPT3.
• Warning_data_fourtank, Warning_data_twotank: The intermediate results of cases in which the numerical errors of the proposed algorithms are relatively large. The leading cause is the numerical inaccuracy of the MPT3 toolbox.
• Systems: The parameters of the numerical systems, including the two-tank, four-tank.
• Toolbox: Matlab functions utilized in the program.
  • Box.m: Generates the minimal interval that encloses a given polytope $P$, i.e., $\square(P)$.
  • Compute_SESs.m: Computes the state estimation sets (SES) $X_{k+1}^P$ and $X_{k+1}^I$ given the observer bundles $\mathbb{L}_k$ and $\mathbb{LS}_k$.
  • IO_opt.m: Solve $\mathbb{LS}_k$ of the linear-time invariant (LTI) system.
  • L_opt.m: Solve $\mathcal{L}(v;X_k)$ given the direction $v$ and the SES $X_k$. See Theorem 4.1 for details.
  • mRep.m: Solve the mH-Rep and mV-rep of a given polytope. This procedure is intended to reduce the representational complexity of each intermediate polytope in the SME process.
  • Normalize: Normalize the outward normals of the polytope ($|h_i|=1$) to make sure the numerical errors of all facets are of the same scale. See Sec. V in the manuscript.
  • Plot_sets.m: Plot the resulting SESs $X_{k+1}^M$, $X_{k+1}^P$, and $X_{k+1}^I$ of the LTI system.
  • Select_vertices.m: Solve the vertice $P(v;X_k)$ given the direction $v$ and SES $X_k$. See Lemma 2.4 in the paper.
  • Smatrix.m: Solve the coordinate transformation matrix $T_k=\mathcal{T}(v,L_k)$ given the direction $v$ and observer gain $L_k$. See Lemma 4.7 in the main content.
  • Support_func.m: Compute the support function $S(v;X)$.
  • SVO_opt.m: Solve $\mathbb{L}_k$ of the LTI system.
  • zonotope.m: Generates a zonotope with G-rep $Z(c,G)=\lbrace c+G\xi||\xi|_{\infty}\leq 1\rbrace$. This function can be utilized to generate symmetric and complicated uncertainty sets.

Quick start

• Run "main.m" to see the experimental result of one sampling case on the two-tank system. The numerical accuracy of the proposed algorithms is reported, and the resulting state estimation sets are plotted.
• Run "data_load.m" to visualize the outcomes of the large-scale numerical tests.

News

• [24-01-20] ✌️ Our work is accepted by the IEEE Transaction on Automatic Control.