Conceptual map
- III-07.01Linear state equations
- III-07.02Stability
- III-07.03Impulse-response maps
- III-07.04Reachability and observability
- III-07.05Compression criteria
1. A response system has three maps
Stability describes what the transition law does to an existing state. Policy analysis also needs a map from interventions into the state and a map from the state into measured targets.
Consider the discrete-time linear system
Here \(A\in \mathbb{R}^{n\times n}\), \(B\in \mathbb{R}^{n\times m}\), and \(C\in \mathbb{R}^{p\times n}\). The system is asymptotically stable when the spectral radius \(\rho(A)\) is below one. The finite-horizon response operator for a one-time input is
The horizon-\(H\) response space is \(\operatorname{range}(K_{H})\). Its dimension is \(\operatorname{rank}(K_{H})\). With Euclidean input and output metrics, the squared singular values of \(K_{H}\) are the eigenvalues of its input Gram matrix \(K_{H}^{\top}K_{H}\).
2. Reachability, observability, and compression
The reachable subspace through horizon \(H\) is the column span of \([B,AB,\ldots,A^{H-1}B]\). The unobservable subspace through the same horizon is the kernel of the stacked matrix \([C^{\top},(CA)^{\top},\ldots,(CA^{H-1})^{\top}]^{\top}\).
A state displacement \(d\) has zero measured consequence at all dates \(h=0,\ldots,H-1\) exactly when \(d\) lies in
The quotient \(\mathbb{R}^{n}/N_{H}\) is the smallest linear state that reconstructs every tested future output.
Proof. The stacked observability matrix sends \(d\) to the vector of future output displacements. Its kernel is (3). Any linear state representation that reconstructs those outputs must identify no pair separated by a direction outside this kernel, so its row space must contain the row space of the stacked matrix. The quotient therefore has minimal dimension equal to the stacked matrix rank. ∎
3. One stable transition, three response spaces
Set \(A=\operatorname{diag}(0.8,0.4,0.2)\) and \(H=3\). All three modes are stable. First take \(B=I\) and \(C=I\). The response operator has rank three and singular values 1.43164, 1.08885, and 1.02059.
Keep \(B=I\) and replace the target by \(C\)=[1,1,0]. The three response rows are [1,1,0], [0.8,0.4,0], and [0.64,0.16,0]. Their stack has rank two and singular values 1.76186 and 0.36200. With \(C\)=[1,0,0], the rank falls to one and the sole positive singular value is 1.43164.
The same matrix \(A\) produces response dimensions three, two, and one. Its spectral radius and unconditional decay rate are unchanged. A stability calculation therefore cannot determine how many intervention directions enter the measured target.
4. Horizon and economic weights
For positive-semidefinite date weights \(W_{h}\), replace the Euclidean Gram matrix by
A direction \(v\) has total weighted response \(v^{\top}G_{H}v\). Discounting, outcome units, and intervention units enter this matrix. Comparisons across models require the same horizon and metrics.
5. Implementation, exercises, and sources
Build response blocks by repeated multiplication, stack them, and compute a rank-revealing SVD. Report the scale used for the rank tolerance. For a long horizon, update the Gram matrix recursively and monitor its smallest retained eigenvalue rather than constructing a large dense stack.
Download the volume verification script →Exercises
- Reproduce the three ranks and singular-value lists in the example.
- For \(C\)=[1,1,0], find a basis for the response-null state directions at horizons one, two, and three.
- Choose a stable nondiagonal \(A\) with one slowly decaying hidden mode and construct \(B\), \(C\) that expose or suppress it.
Partial solutions
1. For \(B=C=I\), each column norm is \(\sqrt{1+\lambda ^{2}+\lambda ^{4}}\). With \(C\)=[1,1,0], diagonalize the \(2\times2\) Gram matrix \(\begin{bmatrix}2.0496 & 1.4224 \\ 1.4224 & 1.1856\end{bmatrix}\). 2. At horizon one the kernel is span{(1,\(-1\),0),(0,0,1)}. At horizons two and three it is span{(0,0,1)} because the first two response rows are independent.
- R. E. Kalman (1960), “On the General Theory of Control Systems,” Proceedings of the First IFAC Congress.Reachability and observability.
- Thomas Kailath, Linear Systems, Chapters 2, 6.Response operators and minimal realization.
- Chi-Tsong Chen, Linear System Theory and Design, Chapters 5–6.Discrete-time stability and Gramians.
6. Audit checkpoint
Report \(A\), \(B\), \(C\), the horizon, input and output metrics, response rank, singular values, and numerical tolerance. Separate transition stability from reachability, observability, and response compression.
7. Scope boundary
The chapter covers finite-dimensional linear time-invariant systems and finite-horizon response spaces. Nonlinear controllability, robust control, and stochastic realization require further structure.