28. Observability and Detectability

28.1. Time-continuous linear systems

The solution of the linear system

\[\begin{split}\begin{aligned} \dot{\mathbf{x}} & = \mathbf{A} \mathbf{x} + \mathbf{B} \mathbf{u} \\ \mathbf{y} & = \mathbf{C} \mathbf{x} + \mathbf{D} \mathbf{u} \\ \end{aligned}\end{split}\]

with initial conditions \(\mathbf{x}(0) = \mathbf{x}_0\) reads

\[\mathbf{x}(t) = \boldsymbol\Phi(t,t_0) \mathbf{x}_0 + \int_{\tau=0}^{t} \boldsymbol\Phi(t,\tau) \mathbf{B}(\tau) \mathbf{u}(\tau) d \tau \ ,\]

and the output of the system reads

(28.1)\[\begin{split}\begin{aligned} \mathbf{y}(t) & = \mathbf{C}(t) \mathbf{x}(t) + \mathbf{D}(t) \mathbf{u}(t) = \\ & = \mathbf{C}(t) \boldsymbol\Phi(t,t_0) \mathbf{x}_0 + \mathbf{C}(t) \int_{\tau=0}^{t} \boldsymbol\Phi(t,\tau) \mathbf{B}(\tau) \mathbf{u}(\tau) d \tau + \mathbf{D}(t) \mathbf{u}(t) \ . \end{aligned}\end{split}\]

28.1.1. Observability

A system is observable if the initial state \(\mathbf{x}_0\) can be uniquely determined, knowing the output \(\mathbf{y}(t)\) and the input \(\mathbf{u}(t)\) over a finite time interval.

A relation between initial conditions as a function of the output and the input of the system can be retrieved from the expression (28.1),

\[\mathbf{C}(t) \boldsymbol\Phi(t,t_0) \mathbf{x}_0 = \mathbf{y}(t) - \left[ \mathbf{C}(t) \int_{\tau=0}^{t} \boldsymbol\Phi(t,\tau) \mathbf{B}(\tau) \mathbf{u}(\tau) d \tau + \mathbf{D}(t) \mathbf{u}(t) \right] =: \widetilde{\mathbf{y}}(t) \ ,\]

with the definition of the known combination of input and output on the RHS as \(\widetilde{\mathbf{y}}(t)\). This system of equations is usually not square, and only involves two time instants \(t_0\), \(t\). If the latter equation is multiplied by \(\boldsymbol\Phi^T(\tau,t_0) \mathbf{C}^T(\tau)\) (least square solution; \(L_2\) norm;…) and integrated from \(t_0\) to \(t\) in order to use all the information from \(t_0\) to \(t\)

\[\int_{\tau=t_0}^{t} \boldsymbol\Phi^T(\tau,t_0) \mathbf{C}^T(\tau) \mathbf{C}(\tau) \boldsymbol\Phi(\tau,t_0) d \tau \, \mathbf{x}_0 = \int_{\tau=t_0}^{t} \boldsymbol\Phi^T(\tau,t_0) \mathbf{C}^T(\tau) \widetilde{\mathbf{y}}(\tau) d \tau \ .\]

\[\mathbf{W}_o(t) \, \mathbf{x}_0 = \int_{\tau=t_0}^{t} \boldsymbol\Phi^T(\tau,t_0) \mathbf{C}^T(\tau) \widetilde{\mathbf{y}}(\tau) d \tau \ .\]

todo

• Discuss the pullback of the output to the state space (before \(\sim\) least-square inversion )

28.1.2. Detectability

A system is observable if all the unobservable modes are naturally stable, i.e. they decay to zero.

todo Check definition, and requirement for finite or infinite time range

Optimal input and controllability Gramian

Let’s solve the optimal output defined as the one with the maximum objective function \(\frac{1}{2}\int_{\tau=0}^{t} \left| \mathbf{y}(\tau) \right|^2\), subject to the equations of motion as constraints. todo Here the final time \(t\) is assumed to be known, but in general it’s not: include \(t\) in the optimization, or run several optimizations to find optimal \(t^*\). The solution of the free system - or the forced system, with the \(\widetilde{\mathbf{y}}(t)\) defined as the combination of output and input above - reads

\[\mathbf{x}(t) = \boldsymbol\Phi(t,t_0) \mathbf{x}_0 \ ,\]

so that the output is \(\mathbf{y}(t) = \mathbf{C} \boldsymbol\Phi(t,t_0) \mathbf{x}_0\). The objective function read

\[\begin{split}\begin{aligned} J(\mathbf{x}_0) & = \frac{1}{2} \int_{\tau=0}^{t} \mathbf{y}^T(\tau) \mathbf{y}(\tau) d \tau = \\ & = \frac{1}{2} \mathbf{x}_0^T \int_{\tau=0}^{t} \boldsymbol\Phi^T(t,t_0) \mathbf{C}^T \mathbf{C} \boldsymbol\Phi(t,t_0) \mathbf{x}_0 = \\ & = \frac{1}{2} \mathbf{x}_0^T \mathbf{W}_o(t) \mathbf{x}_0 \ . \end{aligned}\end{split}\]

• If \(\mathbf{x}_0 \in K(\mathbf{W}_o(t))\), i.e. belongs to the kernel of the observatility Gramian, thus \(J(\mathbf{x}_0) = 0\), i.e. the initial conditions has no influence on the output.

• If \(\mathbf{x}_0 \notin K(\mathbf{W}_o(t))\), i.e. does not belong to the kernel of the observatility Gramian, thus \(J(\mathbf{x}_0) > 0\).

• using singular value decomposition, it’s possible to find the initial condition producing the largest output. Let \(\mathbf{W}_o = \mathbf{U} \boldsymbol\Sigma \mathbf{U}^*\), the optimal initial condition with given \(L_2\) norm, e.g. \(\|\mathbf{x}_0\|_2 = 1\), is the singular vector \(\mathbf{u}_1\) associated with the largest singular value \(\sigma_1\), \(\mathbf{x}_0 = \mathbf{u}_1\), and the optimal output is

  \[\max_{\|\mathbf{x}_0\|_2 = 1} J(\mathbf{x}_0) = J(\mathbf{x}_0 = \mathbf{u}_1) = \frac{1}{2} \sigma_1 \ .\]

Observability Gramian satisfies the Lyapunov equation

\[ \dot{\mathbf{W}}_o(t) = \mathbf{C}^T(t) \mathbf{C}(t) + \mathbf{A}^T(t) \mathbf{W}_o(t) + \mathbf{W}_o(t) \mathbf{A}(t) \ , \]

with initial conditions \(\mathbf{W}_o(0) = \mathbf{0}\), as the integral vanishes with the same values of the lower and upper extremes of integration.

Lyapunov equation for the controllability Graminan

\[\begin{split}\begin{aligned} \frac{d}{dt} \mathbf{W}_o(t) & = \frac{d}{dt} \int_{\tau=0}^{t} \boldsymbol\Phi^T(t,\tau) \mathbf{C}^T(\tau) \mathbf{C}(\tau)\boldsymbol\Phi(t,\tau) d \tau = \\ & = \mathbf{C}(t) \mathbf{C}^T(t) + \int_{\tau=0}^{t} \partial_t \boldsymbol\Phi^T(t,\tau) \mathbf{C}^T(\tau) \mathbf{C}(\tau)\boldsymbol\Phi(t,\tau) d \tau + \int_{\tau=0}^{t} \boldsymbol\Phi^T(t,\tau) \mathbf{C}^T(\tau) \mathbf{C}(\tau) \partial_t \boldsymbol\Phi(t,\tau) d \tau = \\ & = \mathbf{C}(t) \mathbf{C}^T(t) + \mathbf{A}(t) \mathbf{W}_c(t) + \mathbf{W}_c(t) \mathbf{A}^T(t) \ , \end{aligned}\end{split}\]

as \(\boldsymbol\Phi(t,t) = \mathbf{I}\), and \(\partial_t \boldsymbol\Phi(t,\tau) = \mathbf{A}(t) \boldsymbol\Phi(t,\tau)\).

Observability for LTI systems

For LTI systems \(\boldsymbol\Phi(t,t_0) = e^{\mathbf{A}(t-t_0)}\). The controllability Gramian becomes

\[\mathbf{W}_o(t) = \int_{\tau=0}^{t} e^{\mathbf{A}^T (t-\tau)} \mathbf{C}^T \mathbf{C} e^{\mathbf{A} (t-\tau)} d \tau\]

A vector \(\mathbf{v}\) belongs to the null space of \(\left( e^{\mathbf{A}^T t} \mathbf{C}^T \right)^T\) if

\[0 = \mathbf{C} e^{\mathbf{A} t} \mathbf{v} \ .\]

Using the definition of the exponential matrix \(e^{\mathbf{A}t} := \sum_{n=0}^{+\infty} \frac{\mathbf{A}^n t^n}{n!}\), it follows

\[0 = \mathbf{C} \sum_{k=0}^{+\infty} \frac{\mathbf{A}^k t^k}{k!} \mathbf{v} \]

Thus…, a vector \(\mathbf{v}\) is in the null-space of \(\mathbf{W}_o\) (symmetric! Be precise and use orthogonality between range and kernel of the adjoint if it’s orthogonal to the rows of the matrix

\[\begin{split}\mathcal{O} = \begin{bmatrix} \mathbf{C} \\ \mathbf{C} \mathbf{A} \\ \dots \\ \mathbf{C} \mathbf{A}^{n-1} \end{bmatrix} \ ,\end{split}\]

i.e. \(\mathcal{O} \mathbf{v} = \mathbf{0}\). If the row space of matrix \(\mathcal{O}\) spans \(\mathbb{R}^n\), then the system is fully observable.

28.2. Time-discrete linear systems