Controllability Gramian Estimation From Continuous-Time State Data
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Solution Overview
Problem
Existing methods for estimating controllability Gramians in systems lack effectiveness when a mathematical model is unknown, particularly for continuous-time systems, as they struggle with clarity and utilization of prior knowledge about continuous-time characteristics.
Innovation Solution
A data processing device and method that estimate the controllability Gramian using a data-driven approach by acquiring time-series state data, defining a system based on this data, and numerically solving equations to calculate the controllability Gramian without requiring a mathematical model, even in the presence of noise.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If model-based methods are used to calculate controllability Gramian, then calculation accuracy is improved, but data requirements increase and cannot be applied when system model is unknown
Solution Approach 1:
The patent replaces model-based calculation methods with a data-driven approach. Instead of using mathematical models of the system (mechanical representation), the invention directly processes state trajectory data to compute controllability Gramian, substituting the modeling mechanism with a direct data processing mechanism that works for unknown systems
Solution Approach 2:
The patent introduces an intermediary computational framework that bridges raw state trajectory data and controllability Gramian estimation. This intermediary process involves constructing data matrices from trajectories, solving linear equations, and deriving Gramian estimates without requiring direct system model knowledge
2Device complexity
If discrete-time models are used for data-driven estimation, then fewer decision variables are required, but physical information clarity deteriorates and prior knowledge utilization becomes difficult
Solution Approach 1:
The patent changes the time parameter representation from discrete to continuous. By formulating the data-driven method in continuous-time framework, the invention preserves physical information clarity and enables utilization of prior knowledge about continuous-time system characteristics while maintaining computational feasibility
Data Source
AI summary
A data processing device to estimate the limit:G(∞)[equation 140]of a controllability Gramian:G(t)[equation 138]defined by:G(t) :=∫ 0 tcAτBB ⊤eA⊤τdτ[equation 2]in:t=∞[equation 144]when:x˙(t)= Ax(t)+ Bu(t)[equation 1]holds, where:x(t)[equation 142]is an n-dimensional vector representing the state of a control object:u(t)[equation 139]is an m-dimensional vector representing the control input, A is an unknown n×n matrix and B is a known n×m matrix, comprises:a data acquisition unit that acquires a set of time-series state data:x([t11,t12],x11),x([t21,t22],x11),… ,x([tq1,tq2],xq1)[equation 5]for the following q time intervals:[ti1,ti2][equation 4](i =1,2,… ,q)when:u(t)≡0[equation 39]holds;a controllability Gramian calculation unit that defines:[equation 75]z(t)∈Rn expressed as:(13)z˙(t)=A⊤z(t)[equation 74]calculates:(16)z⊤(t2)Xz(t2)-z⊤(t1)Xz(t1)=-∫ t1 t2z⊤(t)BB Tz(t)dt[equation 81]z(t+ti1,ti1,xi1)=(E(t)E 0 -1)⊤xi1[equation 89]and estimates:G(∞)=X[equation 146]by numerically obtaining the solution X of the following linear equation:[Equation 6] xi1⊤(E(h)E0-1)X(E(h)E0-1)⊤xi1-xi1⊤Xxi1=-∫0 hxi1⊤(E(t)E0-1)BB⊤(E(t)E0-1)⊤xi1dt(i=1,2,… ,q)with respect to:[equation 157] E(t):=[x(?+?,?,x11x(?+?,?,x21)…x(t+tn1,tn1,xn1)][equation 158]E0:=[x11x21…xn1]; ?indicates text missing or illegible when filedandan output unit that outputs the input matrix when the controllability Gramian is maximized based on the estimated maximization condition.