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Systems Theory

Observability, duality and the Kalman decomposition

The observability test with proof, the tanks-in-series example, the duality principle, the Kalman decomposition into four parts, transfer function and minimal realisations, detectability.

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The state of a system is almost never measured in full: some outputs are measured. The question symmetric to reachability is: by watching the output (and knowing the input) for some time, can the state be reconstructed? This is observability. In this lesson we look at Kalman’s observability test with proof, the duality principle, the Kalman decomposition into four parts and the very important fact that the transfer function sees only the reachable and observable part.

Definition and test

The system is observable if, given the input and the output over a finite interval, the initial state x_0 is uniquely determined. Since the contribution of the input to the output can be computed (it is the forced response), it is enough to study the free response: the question is whether two different initial states can give the same free output. Define the observability matrix

    \[ \mathcal{O} = \begin{pmatrix} C \\ CA \\ \vdots \\ CA^{n-1} \end{pmatrix} \]

Theorem. The system x(k+1) = Ax(k), y(k) = Cx(k) is observable if and only if \mathrm{rank}\,\mathcal{O} = n. In that case n output samples are enough.

Proof. The free output is y(k) = CA^kx_0. Collecting the first n samples:

    \[ \begin{pmatrix} y(0) \\ y(1) \\ \vdots \\ y(n-1) \end{pmatrix} = \mathcal{O}\,x_0 \]

The linear system in x_0 has a unique solution if and only if \mathcal{O} has rank n. Further samples do not help: by Cayley-Hamilton CA^k with k \ge n is a combination of rows already present. If the rank is less than n, every x_0 in the kernel of \mathcal{O} gives an identically zero output, and cannot be told apart from zero. ∎

In continuous time the test is the same: the derivatives y, \dot{y}, \ddot{y}, \dots at t = 0 equal Cx_0, CAx_0, CA^2x_0, \dots, and the reasoning is identical (or uses the observability Gramian). The kernel of \mathcal{O} is the unobservable subspace: the states that “cannot be seen” from the output.

Example. The two tanks in series of the lesson on reachability, A = \begin{pmatrix} -1 & 0 \\ 1 & -1 \end{pmatrix}. If the first is measured, C = (1\ \ 0), then CA = (-1\ \ 0): \mathcal{O} has rank 1, and the level of the second is invisible, because it does not affect the first. If the second is measured, C = (0\ \ 1), CA = (1\ \ -1): rank 2. The second tank “carries” information about the first, because the first feeds it. In general, what lies upstream of the sensor is well observed.

Duality

Comparing the two matrices: \mathcal{O}^T = (C^T\ \ A^TC^T\ \cdots\ (A^T)^{n-1}C^T), which is the reachability matrix of the system with matrices A^T and C^T in place of A and B.

Theorem (duality). The pair (A, C) is observable if and only if the pair (A^T, C^T) is reachable.

Proof. A matrix and its transpose have the same rank, and \mathcal{O}(A, C)^T = \mathcal{R}(A^T, C^T). ∎

Every result on reachability therefore translates into one on observability. In particular, just as reachability allows the eigenvalues to be assigned with state feedback, observability allows those of an observer, which estimates the state from the measurements, to be assigned (Automatic Control lesson on state space); designing one is designing the other on the dual system.

The Kalman decomposition

Combining the two properties, with a suitable change of coordinates the state splits into four parts:

  • RO: reachable and observable;
  • RŌ: reachable but not observable;
  • R̄O: not reachable but observable;
  • R̄Ō: neither reachable nor observable.
Kalman decomposition of a linear system
The input u acts only on the reachable parts, RO and R-O-bar; the output y collects only the observable parts, RO and R-bar-O. Only the RO part links input to output, and it alone determines the transfer function.

Theorem. The transfer function depends only on the reachable and observable part:

    \[ G(s) = C(sI - A)^{-1}B + D = C_{RO}(sI - A_{RO})^{-1}B_{RO} + D \]

Idea of the proof. In Kalman coordinates the matrix A is block triangular, B has zero components on the unreachable parts and C has zero components on the unobservable parts. The inverse of a block-triangular matrix is block triangular with the diagonal blocks inverted, and in the product C(sI - A)^{-1}B the zeros of B and C eliminate every term except that of the RO block. ∎

The consequences are far-reaching:

  • the eigenvalues of the other three parts do not appear as poles of G(s): they have cancelled with zeros. This is why the system in the example of the lesson on stability was BIBO stable but internally unstable;
  • a transfer function of order r describes a system with at least r states; a representation with exactly r states is called minimal, and it is minimal if and only if it is reachable and observable;
  • a hidden unstable mode (unreachable or unobservable) cannot be corrected by any controller using that input and that output: it must be fixed by changing actuators or sensors.

A system is called detectable if all its unobservable modes are asymptotically stable: it is the condition, dual to stabilisability, needed to build an observer whose error tends to zero.

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