Title: On the Popov-Belevitch-Hautus tests for functional observability and output controllability
Domain: Math

Question: How can the Popov–Belevitch–Hautus (PBH) tests be generalized to non-diagonalizable linear time-invariant (LTI) systems in order to provide necessary and sufficient conditions for functional observability and output controllability? | | In particular, how can one: | 1. Extend the PBH tests using Jordan decomposition and characterize both properties in the general case? | 2. Establish and formalize the duality between functional observability and output controllability, including minimal algebraic conditions? | 3. Derive theoretical implications for state estimation, control signal design, and eigenspace analysis? | 4. Translate the results into practical design guidelines for observer construction, controller implementation, pole assignment, and optimal sensor/driver placement?

Background: 1. LTI model with target output: \dot{x}=Ax+Bu,\; y=Cx,\; z=Fx. The challenge is estimating/controlling z when r \ll n. | 2. Functional observability ensures z(0) can be uniquely reconstructed from (y,u) over finite horizon. | 3. Output controllability ensures z(t) can be driven to any final value by some input u(t). | 4. Classical PBH tests cover only full-state observability/controllability and fail for non-diagonalizable A. | 5. Recent progress: generalized PBH for functional observability (using rank and Jordan decomposition), new PBH-type test for output controllability, and explicit duality linking the two ￼.

Definitions: 1. Observability matrix \mathcal{O}(C,A) = [C^T, (CA)^T,\dots,(CA^{n-1})^T]^T. | 2. Controllability matrix \mathcal{C}(A,B) = [B, AB, A^2B, \dots, A^{n-1}B]. | 3. Jordan decomposition: J = PAP^{-1} = \mathrm{diag}(J_1,\dots,J_m). | 4. Each Jordan submatrix J_k = \mathrm{diag}(J_{k1},\dots,J_{km_k}) encodes eigenvalue \lambda_k with algebraic multiplicity n_k and geometric multiplicity m_k. | 5. Lead columns: for each C_k or F_k aligned with J_k, the columns corresponding to the first column of each Jordan block J_{ki}; their linear independence is crucial for PBH equivalence. | 6. Compact notation: \mathcal{O}\equiv \mathcal{O}(C,A),\;\mathcal{C}\equiv \mathcal{C}(A,B)

Methodology: 1. Functional observability ： equivalence of | • \mathrm{rank}\big([\mathcal{O};F]\big)=\mathrm{rank}(\mathcal{O}), | • \mathrm{rank}\big([\mathcal{O};O(F,A)]\big)=\mathrm{rank}(\mathcal{O}), | • PBH-type rank test \mathrm{rank}\begin{bmatrix}\lambda I-A\\ C\\ F\end{bmatrix}=\mathrm{rank}\begin{bmatrix}\lambda I-A\\ C\end{bmatrix} for all \lambda, under LI lead columns assumption. | 2. Output controllability : equivalent to \mathrm{rank}(F\mathcal{C})=\mathrm{rank}(F); PBH form \mathrm{rank}[F(\lambda I-A)\; FB]=\mathrm{rank}(F) holds if (A,B) full-state controllable or with added eigenspace intersection condition. | 3. Eigenspace condition: require \ker(\mathcal{C}^T)\cap\mathrm{row}(F)\cap(\cup_i E_i)\neq \varnothing, ensuring that F intersects eigenvectors in uncontrollable directions. | 4. Duality: weak duality states FO implies dual OC; strong duality requires orthogonality F\,\mathrm{Im}(W)\perp F\,\ker(W) where W is the observability Gramian. | 5. Practical implications: Gramian-based expressions yield explicit observer/controller formulas; Jordan–PBH equivalence guides sensor and actuator placement

Answer: 1. Generalized PBH via Jordan decomposition (non-diagonalizable A) — characterization of both properties | Let \mathcal{O}:=O(C,A) and \mathcal{C}:=C(A,B). Write the Jordan form J=PAP^{-1}=\mathrm{diag}(J_1,\dots,J_m), and partition \(\bar C:=CP^{-1}=[C_1~\cdotsC_m]\), \bar F:=FP^{-1}=[F_1\cdots~F_m]. For each eigenvalue \lambda_k, the lead columns of F_k are the columns aligned with the first column of each Jordan block J_{k i}. Then: | • Functional observability (FO) of (C,A;F) holds iff any of the following equivalent conditions holds | \mathrm{rank}\!\begin{bmatrix}\mathcal{O}\\ F\end{bmatrix}=\mathrm{rank}(\mathcal{O})\quad\Longleftrightarrow\quad | \mathrm{rank}\!\begin{bmatrix}\mathcal{O}\\ O(F,A)\end{bmatrix}=\mathrm{rank}(\mathcal{O}). | Moreover, if all nonzero F_k have LI lead columns, FO is also equivalent to the PBH-type rank test (valid for non-diagonalizable A) | \mathrm{rank}\!\begin{bmatrix}\lambda I-A\\ C\\ F\end{bmatrix}=\mathrm{rank}\!\begin{bmatrix}\lambda I-A\\ C\end{bmatrix}\quad \forall\,\lambda\in\mathbb{C}. | When A is diagonalizable, the PBH form above is automatically equivalent to FO (no extra assumption needed). | • Output controllability (OC) of (A,B;F) admits the algebraic test | \mathrm{rank}(F\mathcal{C})=\mathrm{rank}(F)\quad(\text{necessary and sufficient}). | A PBH-type form is | \mathrm{rank}\, \big[\,F(\lambda I-A)\;\; FB\,\big] = \mathrm{rank}(F)\quad \forall\,\lambda\in\mathbb{C}, | which is equivalent to OC if (i) (A,B) is full-state controllable, or (ii) A is diagonalizable; in the general nondiagonalizable case it must be paired with the eigenspace condition stated in item 2) below. ￼ | 2. Duality and minimal algebraic conditions | There is weak duality: FO of (C,A;F) \Rightarrow OC of the dual (A^{\!\top},C^{\!\top};F). Strong duality holds (FO \Leftrightarrow dual OC) iff | F\,\mathrm{Im}(W)\ \perp\ F\,\ker(W), | where W=\int_0^{t_1} e^{A^{\!\top}t}C^{\!\top}Ce^{At}\,dt is the observability Gramian of (C,A). | For OC in the general case, the PBH rank condition must be accompanied by the eigenspace intersection requirement | \ker(\mathcal{C}^{\!\top})\ \cap\ \mathrm{row}(F)\ \cap\ \Big(\bigcup_{i=1}^m E_i\Big)\ \neq\ \varnothing, | where E_i=\{v: A^{\!\top}v=\lambda_i v\}. A convenient sufficient setting is \mathrm{row}(F)\supseteq \ker(\mathcal{C}^{\!\top}), which makes the PBH form necessary and sufficient for OC. ￼ | 3. Implications for estimation, control design, and eigenspace analysis | • Estimation (FO): FO \Leftrightarrow\ \mathrm{row}(F)\subseteq \mathrm{row}(\mathcal{O}). Equivalently, one can reconstruct z(0)=Fx(0) from (y,u) over a finite window; the Jordan-lead-column criterion pinpoints the eigen-Jordan chains that must be “seen” by C and F. In diagonalizable systems, this reduces to simple row-space tests per eigenvalue. | • Control signal design (OC): If W_F(t_1):=F W_c(t_1) F^{\!\top} (with W_c the controllability Gramian) is invertible, then the explicit control | u(t)= -\,B^{\!\top} e^{A^{\!\top}(t_1-t)} F^{\!\top}\, W_F^{-1}(t_1)\,\big(F e^{A t_1}x(0)-z(t_1)\big) | steers z from z(0) to z(t_1). Non-diagonalizable A requires verifying the eigenspace intersection above to ensure the PBH form suffices. | • Eigenspace diagnostics: Failure cases of the PBH-only test (without the intersection condition) occur exactly when F does not “hit” any eigenvector in \ker(\mathcal{C}^{\!\top}); conversely, LI lead columns of \bar F guarantee the FO-PBH equivalence even with Jordan chains. ￼ | 4. Practical design guidelines (observer, controller, pole assignment, sensor/driver placement) | • Observer construction (FO): First check \mathrm{rank}\big([\mathcal{O};F]\big)=\mathrm{rank}(\mathcal{O}). If A has Jordan blocks, verify LI lead columns of \bar F=FP^{-1}; if violated, redesign F (or augment outputs C) to cover the first vectors of each active Jordan chain. | • Controller implementation (OC): Prefer the algebraic test \mathrm{rank}(F\mathcal{C})=\mathrm{rank}(F). If using PBH, ensure either (a) (A,B) is full-state controllable, (b) A is diagonalizable, or (c) enforce \mathrm{row}(F)\supseteq \ker(\mathcal{C}^{\!\top}) (e.g., by shaping F or adding drivers so the intersection condition is satisfied). | • Pole assignment: Apply the FO/OC PBH tests per eigenvalue (or Jordan chain): an unobservable/uncontrollable target component at \lambda cannot be assigned independently—fix C,F (for FO) or B,F (for OC) to include the corresponding lead directions. | • Optimal sensor/driver placement: Use the PBH–Jordan view to pick rows of C (sensors) and rows of F (target functionals) that span lead columns of the relevant blocks (FO), and choose B and F so that \mathrm{row}(F) intersects \ker(\mathcal{C}^{\!\top}) along some eigenvector (OC); in diagonalizable cases, eigenvector coverage per \lambda_k suffices. ￼

Checklist: 1. Defines functional observability and output controllability in terms of (C,A;F) and (A,B;F). | 2. States the rank conditions for functional observability: | \mathrm{rank}\big([\mathcal{O};F]\big)=\mathrm{rank}(\mathcal{O}) and | \mathrm{rank}\big([\mathcal{O};O(F,A)]\big)=\mathrm{rank}(\mathcal{O}). | 3. Clarifies the PBH-type equivalence for FO under the linear independence of lead columns of F_k. | 4. Provides the algebraic test for OC: \mathrm{rank}(F\mathcal{C})=\mathrm{rank}(F). | 5. Establishes the PBH-type test for OC: \mathrm{rank}[F(\lambda I-A)\; FB]=\mathrm{rank}(F), with clear conditions on full-state controllability, diagonalizability, or eigenspace intersection. | 6. Identifies the eigenspace intersection condition: \ker(\mathcal{C}^T)\cap \mathrm{row}(F)\cap(\cup_i E_i)\neq\varnothing. | 7. Explains the weak duality (FO ⇒ dual OC) and the strong duality (FO ⇔ dual OC with Gramian orthogonality). | 8. Links FO and OC to Gramians: W_o for observability, W_c and W_F for controllability. | 9. Demonstrates the role of Jordan decomposition and lead columns in extending PBH to non-diagonalizable systems. | 10. Provides practical implications: observer construction, controller implementation, pole assignment, and sensor/driver placement strategies. | 11. Ensures symbol consistency: \mathcal{O},\mathcal{C}, J, J_k, F_k, E_i all used uniformly. | 12. Clarifies assumptions explicitly (e.g., rank(F)=r, LI of lead columns, row-space inclusion conditions).