Observability

Definition

Consider a physical system modeled in state-space representation. A system is said to be observable if, for every possible evolution of state and control vectors, the current state can be estimated using only the information from outputs (physically, this generally corresponds to information obtained by sensors). In other words, one can determine the behavior of the entire system from the system's outputs. On the other hand, if the system is not observable, there are state trajectories that are not distinguishable by only measuring the outputs.

Linear time-invariant systems

For time-invariant linear systems in the state space representation, there are convenient tests to check whether a system is observable. Consider a SISO system with n state variables (see state space for details about MIMO systems) given by

: \dot{\mathbf{x}}(t) = \mathbf{A} \mathbf{x}(t) + \mathbf{B} \mathbf{u}(t) : \mathbf{y}(t) = \mathbf{C} \mathbf{x}(t) + \mathbf{D} \mathbf{u}(t)

Observability matrix

If and only if the column rank of the observability matrix, defined as

:\mathcal{O}=\begin{bmatrix} C \ CA \ CA^2 \ \vdots \ CA^{n-1} \end{bmatrix}

is equal to n, then the system is observable. The rationale for this test is that if n columns are linearly independent, then each of the n state variables is viewable through linear combinations of the output variables y. Observability is a sufficient and necessary condition for the design of continuous-time state observers.

Related concepts

Observability index

The observability index v of a linear time-invariant discrete system is the smallest natural number for which the following is satisfied: \text{rank}{(\mathcal{O}v)} = \text{rank}{(\mathcal{O}{v+1})}, where

: \mathcal{O}_v=\begin{bmatrix} C \ CA \ CA^2 \ \vdots \ CA^{v-1} \end{bmatrix}.

Unobservable subspace

The unobservable subspace N of the linear system is the kernel of the linear map G given by \begin{align} G \colon \mathbb{R}^{n} &\rightarrow \mathcal{C}(\mathbb{R};\mathbb{R}^n) \ x(0) &\mapsto C e^{A t} x(0) \end{align} where \mathcal{C}(\mathbb{R};\mathbb{R}^n) is the set of continuous functions from \mathbb{R} to \mathbb{R}^n . N can also be written as

: N = \bigcap_{k=0}^{n-1} \ker(CA^k)= \ker{\mathcal{O}}

Since the system is observable if and only if \operatorname{rank}(\mathcal{O}) = n, the system is observable if and only if N is the zero subspace.

The following properties for the unobservable subspace are valid:

• N \subset Ke(C)
• A(N) \subset N
• N= \bigcup { S \subset R^n \mid S \subset Ke(C), A(S) \subset N }

Detectability

A slightly weaker notion than observability is detectability. A system is detectable if all the unobservable states are stable.

Detectability conditions are important in the context of sensor networks.

Functional observability

Functional observability is a property that extends the classical notion of observability for cases in which full-state observability is not possible or required (due to lack of measurement signals and sensor placement). Rather than requiring full-state reconstruction, functional observability establishes the condition under which a linear functional \mathbf{z}(t) = \mathbf{F} \mathbf{x}(t) can still be estimated using solely information from the output signals.{{cite journal |last1=Fernando |first1=T. |last2=Trinh |first2=H. M.

: \operatorname{rank} \begin{bmatrix} \mathcal O \ \mathbf{F} \end{bmatrix} = \operatorname{rank} \mathcal O.

Functional observability is an important concept because it determines the sufficient and necessary condition under which a functional observer (also known as a Darouach observer {{cite journal |last=Darouach |first=M. |title=Existence and design of functional observers for linear systems |journal=IEEE Transactions on Automatic Control

Linear time-varying systems

Consider the continuous linear time-variant system

: \dot{\mathbf{x}}(t) = A(t) \mathbf{x}(t) + B(t) \mathbf{u}(t) , : \mathbf{y}(t) = C(t) \mathbf{x}(t). ,

Suppose that the matrices A, B and C are given as well as inputs and outputs u and y for all t \in [t_0,t_1]; then it is possible to determine x(t_0) to within an additive constant vector which lies in the null space of M(t_0,t_1) defined by : M(t_0,t_1) = \int_{t_0}^{t_1} \varphi(t,t_0)^{T}C(t)^{T}C(t)\varphi(t,t_0) , dt where \varphi is the state-transition matrix.

It is possible to determine a unique x(t_0) if M(t_0,t_1) is nonsingular. In fact, it is not possible to distinguish the initial state for x_1 from that of x_2 if x_1 - x_2 is in the null space of M(t_0,t_1).

Note that the matrix M defined as above has the following properties:

• M(t_0,t_1) is symmetric
• M(t_0,t_1) is positive semidefinite for t_1 \geq t_0
• M(t_0,t_1) satisfies the linear matrix differential equation :: \frac{d}{dt}M(t,t_1) = -A(t)^{T}M(t,t_1)-M(t,t_1)A(t)-C(t)^{T}C(t), ; M(t_1,t_1) = 0
• M(t_0,t_1) satisfies the equation :: M(t_0,t_1) = M(t_0,t) + \varphi(t,t_0)^T M(t,t_1)\varphi(t,t_0)

Observability matrix generalization

The system is observable in [t_0,t_1] if and only if there exists an interval [t_0,t_1] in \mathbb{R} such that the matrix M(t_0,t_1) is nonsingular.

If A(t), C(t) are analytic, then the system is observable in the interval [t_0,t_1] if there exists \bar{t} \in [t_0,t_1] and a positive integer k such that

: \operatorname{rank} \begin{bmatrix} & N_0(\bar{t}) & \ & N_1(\bar{t}) & \ & \vdots & \ & N_{k}(\bar{t}) & \end{bmatrix} = n,

where N_0(t):=C(t) and N_i(t) is defined recursively as

: N_{i+1}(t) := N_i(t)A(t) + \frac{\mathrm{d}}{\mathrm{d} t}N_i(t),\ i = 0, \ldots, k-1

Example

Consider a system varying analytically in (-\infty,\infty) and matricesA(t) = \begin{bmatrix} t & 1 & 0\ 0 & t^{3} & 0\ 0 & 0 & t^{2} \end{bmatrix},, C(t) = \begin{bmatrix} 1 & 0 & 1 \end{bmatrix}. Then \begin{bmatrix} N_0(0) \ N_1(0) \ N_2(0) \end{bmatrix} = \begin{bmatrix} 1 & 0 & 1 \ 0 & 1 & 0 \ 1& 0 & 0 \end{bmatrix} , and since this matrix has rank = 3, the system is observable on every nontrivial interval of \mathbb{R}.

Nonlinear systems

Given the system \dot{x} = f(x) + \sum_{j=1}^mg_j(x)u_j , y_i = h_i(x), i \in p. Where x \in \mathbb{R}^n the state vector, u \in \mathbb{R}^m the input vector and y \in \mathbb{R}^p the output vector. f,g,h are to be smooth vector fields.

Define the observation space \mathcal{O}_s to be the space containing all repeated Lie derivatives, then the system is observable in x_0 if and only if \dim(d\mathcal{O}_s(x_0)) = n, where

:d\mathcal{O}s(x_0) = \operatorname{span}(dh_1(x_0), \ldots , dh_p(x_0), dL{v_i}L_{v_{i-1}}, \ldots , L_{v_1}h_j(x_0)),\ j\in p, k=1,2,\ldots.

Early criteria for observability in nonlinear dynamic systems were discovered by Griffith and Kumar, Kou, Elliot and Tarn, and Singh.

There also exist an observability criteria for nonlinear time-varying systems.

Static systems and general topological spaces

Observability may also be characterized for steady state systems (systems typically defined in terms of algebraic equations and inequalities), or more generally, for sets in \mathbb{R}^n. Just as observability criteria are used to predict the behavior of Kalman filters or other observers in the dynamic system case, observability criteria for sets in \mathbb{R}^n are used to predict the behavior of data reconciliation and other static estimators. In the nonlinear case, observability can be characterized for individual variables, and also for local estimator behavior rather than just global behavior.

References

References

1. (1960). "On the general theory of control systems". IFAC Proceedings Volumes.
2. (1963). "Mathematical Description of Linear Dynamical Systems". Journal of the Society for Industrial and Applied Mathematics, Series A: Control.
3. Sontag, E.D., "Mathematical Control Theory", Texts in Applied Mathematics, 1998
4. "Controllability and Observability".
5. (November 2018). "A Weightedly Uniform Detectability for Sensor Networks". IEEE Transactions on Neural Networks and Learning Systems.
6. (2019). "On Boundedness of Error Covariances for Kalman Consensus Filtering Problems". IEEE Transactions on Automatic Control.
7. Brockett, Roger W.. (1970). "Finite Dimensional Linear Systems". John Wiley & Sons.
8. Eduardo D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems.
9. Lecture notes for Nonlinear Systems Theory by prof. dr. D.Jeltsema, [[Jacquelien Scherpen. prof dr. J.M.A.Scherpen]] and prof dr. A.J.van der Schaft.
10. (1971). "On the observability of nonlinear systems: I". Journal of Mathematical Analysis and Applications.
11. (1973). "Observability of nonlinear systems". Information and Control.
12. (1975). "Observability in non-linear systems with immeasurable inputs". International Journal of Systems Science.
13. Martinelli, Agostino. (2022). "Extension of the Observability Rank Condition to Time-Varying Nonlinear Systems". IEEE Transactions on Automatic Control.
14. (1981). "Observability and redundancy in process data estimation". Chemical Engineering Science.
15. (1981). "Observability and redundancy classification in process networks". Chemical Engineering Science.