Yonsei University
Chae-Yeon Xon
Yonsei University, School of Economics
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Linear Dynamics and Modes · Chapter III-09

Poles, Cancellation, and Generalized Modes

Repeated characteristic roots can generate polynomial-times-exponential responses, and input-output maps can cancel internal modes. A determinant records algebraic roots; an observable transfer function records poles after cancellation. Generalized eigenvectors or Jordan chains are needed when a repeated observable pole remains.

Conceptual map

  1. III-09.01Transfer functions
  2. III-09.02Cancellations and multiplicity
  3. III-09.03Generalized modes
  4. III-09.04Jordan and realization views
  5. III-09.05Nearly repeated roots

1. Roots, poles, and measured modes

A root of a state-transition determinant becomes an observed mode only when the input excites it and the output loads on it. Repetition also matters: an observable Jordan chain generates a polynomial multiplying the usual exponential response.

For \(x_{t+1}=Ax_{t}+Bu_{t}\) and \(y_{t}=Cx_{t}\), the lag-generating transfer matrix is

\[H(q)=C(I-qA)^{-1}B.\](1)

The denominator \(\operatorname{det}(I-qA)\) lists internal characteristic roots. Common factors in the numerator and denominator cancel from a particular input-output channel.

Definition 1 · Observable pole order

A scalar channel has a pole of order \(r\) at \(q=1/\lambda\) when its reduced transfer function contains \((1-\lambda q)^{-r}\) with nonzero coefficient. Pole order is defined after cancellation.

2. Repeated poles produce polynomial factors

Proposition 1 · Coefficients of a repeated pole

For an integer \(r\ge1\),

\[(1-\lambda q)^{-r}=\sum _{t=0}^{\infty } \binom{t+r-1}{r-1}\lambda ^{t}q^{t}, \quad |\lambda q|<1.\](2)

The impulse coefficient is a degree-\(r-1\) polynomial in \(t\) times \(\lambda ^{t}\).

Proof. The geometric series gives \((1-x)^{-1}=\sum x^{t}\). Differentiating \(r-1\) times and dividing by (\(r-1\))! yields the binomial coefficient in (2). ∎

For the Jordan block \(J=\begin{bmatrix}\lambda & 1 \\ 0 & \lambda\end{bmatrix}\), one has \(J^{t}=\begin{bmatrix}\lambda ^{t} & t\lambda ^{t-1} \\ 0 & \lambda ^{t}\end{bmatrix}\). With \(B=e_{2}\) and \(C=e_{1}^{\top}\), the measured impulse response is \(t\lambda ^{t-1}\), the coefficient sequence of \(q/(1-\lambda q)^{2}\).

3. Two simple poles approaching one double pole

Let

\[H_{a,b}(q)=q/[(1-aq)(1-bq)].\](3)

Its coefficient at date \(t\ge1\) is \((a^{t}-b^{t})/(a-b)\). As \(b\) approaches \(a=\lambda\), this divided difference converges to \(t\lambda ^{t-1}\). For \(\lambda=0.8\) and \(t=5\), the repeated-pole response is \(5(0.8)^{4}=2.048\). Setting (\(a,b\))=(0.81,0.79) gives 2.04864.

The partial-fraction coefficients \(1/(a-b)\) and \(-1/(a-b)\) diverge as the poles coalesce even though their combined impulse response has a finite limit. Modal estimation can therefore be ill-conditioned near multiplicity.

4. Cancellation removes an internal mode

The rational expression

\[(1-\lambda q)/(1-\lambda q)^{2}=1/(1-\lambda q)\](4)

has an internal denominator with algebraic multiplicity two and a measured pole of order one. In state-space language, a two-dimensional realization can contain an unreachable or unobservable generalized direction. A minimal realization removes it.

Failure case · Determinant multiplicity reported as response multiplicity

\(\operatorname{det}(zI-A)\) alone does not identify the pole order of \(C(zI-A)^{-1}B\). The input and output maps must be included and exact or approximate cancellation must be assessed.

5. Implementation, exercises, and sources

Compute transfer numerators and denominators in exact arithmetic when coefficients permit. Reduce common factors before assigning modes. For estimated systems, inspect controllability and observability singular values and treat near-cancellation as a sensitivity range rather than a binary algebraic event.

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Exercises

  1. Derive \(J^{t}\) by induction and recover the transfer channel following (2).
  2. Verify the date-five values for \(\lambda=0.8\) and (\(a,b\))=(0.81,0.79). Repeat as the gap falls from \(10^{-2}\) to \(10^{-8}\).
  3. Construct a two-state realization of (4) and identify the uncontrollable or unobservable direction.
Partial solutions

1. Write \(J=\lambda I+N\) with \(N^{2}=0\). The binomial theorem gives \(J^{t}=\lambda ^{t}I+t\lambda ^{t-1}N\). 2. The close-pole response is the divided difference of \(x^{5}\); continuity of the derivative gives its limit \(5(0.8)^{4}\). Partial-fraction coefficients grow as the inverse pole gap, so calculate their combined response before interpreting numerical instability.

  1. Thomas Kailath, Linear Systems, Chapters 6–7.Minimal realizations and cancellation.
  2. Peter Lancaster and Miron Tismenetsky, The Theory of Matrices, Chapter 6.Jordan chains and matrix functions.
  3. Gene H. Golub and Charles F. Van Loan, Matrix Computations, Chapter 7.Eigenvalue sensitivity and invariant subspaces.

6. Audit checkpoint

State the realization, input, output, transfer convention, reduced numerator and denominator, pole multiplicities, and cancellation tolerance. Distinguish an internal characteristic root from a pole of the measured response.

7. Scope boundary

The chapter covers finite-dimensional discrete-time modal representations, repeated poles, and input-output cancellation. Statistical system identification and a full minimal-realization algorithm are outside scope.

Prerequisites