Observability

In control theory, observability is a measure of how well internal states of a system can be inferred from knowledge of its external outputs. The observability and controllability of a linear system are mathematical duals. The concept of observability was introduced by Hungarian-American engineer Rudolf E. Kálmán for linear dynamic systems.[1][2] A dynamical system designed to estimate the state of a system from measurements of the outputs is called a state observer or simply an observer for that system.

Definition

Consider a physical system modeled in state-space representation. A system is said to be observable if, for any 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 ${\displaystyle n}$ state variables (see state space for details about MIMO systems) given by

${\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} \mathbf {x} (t)+\mathbf {B} \mathbf {u} (t)}$

${\displaystyle \mathbf {y} (t)=\mathbf {C} \mathbf {x} (t)+\mathbf {D} \mathbf {u} (t)}$

Observability matrix

If the row rank of the observability matrix, defined as

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

is equal to ${\displaystyle n}$, then the system is observable. The rationale for this test is that if ${\displaystyle n}$ rows are linearly independent, then each of the ${\displaystyle n}$ state variables is viewable through linear combinations of the output variables ${\displaystyle y}$.

Observability index

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

${\displaystyle {\mathcal {O}}_{v}={\begin{bmatrix}C\\CA\\CA^{2}\\\vdots \\CA^{v-1}\end{bmatrix}}.}$

Unobservable subspace

The unobservable subspace ${\displaystyle N}$ of the linear system is the kernel of the linear map ${\displaystyle G}$ given by[3]

${\displaystyle {\begin{aligned}G\colon \mathbb {R} ^{n}&\rightarrow {\mathcal {C}}(\mathbb {R}$ ;\mathbb {R} ^{n})\\x(0)&\mapsto Ce^{At}x(0)\\\end{aligned}}}

where ${\displaystyle {\mathcal {C}}(\mathbb {R}$ ;\mathbb {R} ^{n})} is the set of continuous functions from ${\displaystyle \mathbb {R} }$ to ${\displaystyle \mathbb {R} ^{n}}$. ${\displaystyle N}$ can also be written as [3]

${\displaystyle N=\bigcap _{k=0}^{n-1}\ker(CA^{k})=\ker {\mathcal {O}}}$

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

The following properties for the unobservable subspace are valid:[3]

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

Detectability

A slightly weaker notion than observability is detectability. A system is detectable if all the unobservable states are stable.[4]

Detectability conditions are important in the context of sensor networks.[5][6]

Nonlinear observers

sliding mode and cubic observers[7] can be applied for state estimation of time invariant linear systems, if the system is observable and fulfills some additional conditions.

Linear time-varying systems

Consider the continuous linear time-variant system

${\displaystyle {\dot {\mathbf {x} }}(t)=A(t)\mathbf {x} (t)+B(t)\mathbf {u} (t)\,}$

${\displaystyle \mathbf {y} (t)=C(t)\mathbf {x} (t).\,}$

Suppose that the matrices ${\displaystyle A}$, ${\displaystyle B}$ and ${\displaystyle C}$ are given as well as inputs and outputs ${\displaystyle u}$ and ${\displaystyle y}$ for all ${\displaystyle t\in [t_{0},t_{1}];}$ then it is possible to determine ${\displaystyle x(t_{0})}$ to within an additive constant vector which lies in the null space of ${\displaystyle M(t_{0},t_{1})}$ defined by

${\displaystyle M(t_{0},t_{1})=\int _{t_{0}}^{t_{1}}\phi (t,t_{0})^{T}C(t)^{T}C(t)\phi (t,t_{0})dt}$

where ${\displaystyle \phi }$ is the state-transition matrix.

It is possible to determine a unique ${\displaystyle x(t_{0})}$ if ${\displaystyle M(t_{0},t_{1})}$ is nonsingular. In fact, it is not possible to distinguish the initial state for ${\displaystyle x_{1}}$ from that of ${\displaystyle x_{2}}$ if ${\displaystyle x_{1}-x_{2}}$ is in the null space of ${\displaystyle M(t_{0},t_{1})}$.

Note that the matrix ${\displaystyle M}$ defined as above has the following properties:

• ${\displaystyle M(t_{0},t_{1})}$ is symmetric
• ${\displaystyle M(t_{0},t_{1})}$ is positive semidefinite for ${\displaystyle t_{1}\geq t_{0}}$
• ${\displaystyle M(t_{0},t_{1})}$ satisfies the linear matrix differential equation

${\displaystyle {\frac {d}{dt}}M(t,t_{1})=-A(t)^{T}M(t,t_{1})-M(t,t_{1})A(t)-C(t)^{T}C(t),\;M(t_{1},t_{1})=0}$

• ${\displaystyle M(t_{0},t_{1})}$ satisfies the equation

${\displaystyle M(t_{0},t_{1})=M(t_{0},t)+\phi (t,t_{0})^{T}M(t,t_{1})\phi (t,t_{0})}$[8]

Observability matrix generalization

The system is observable in [${\displaystyle t_{0}}$,${\displaystyle t_{1}}$] if and only if there exists an interval [${\displaystyle t_{0}}$,${\displaystyle t_{1}}$] in ${\displaystyle \mathbb {R} }$ such that the matrix ${\displaystyle M(t_{0},t_{1})}$ is nonsingular.

If ${\displaystyle A(t),C(t)}$ are analytic, then the system is observable in the interval [${\displaystyle t_{0}}$,${\displaystyle t_{1}}$] if there exists ${\displaystyle {\bar {t}}\in [t_{0},t_{1}]}$ and a positive integer k such that[9]

${\displaystyle rank{\begin{bmatrix}&N_{0}({\bar {t}})&\\&N_{1}({\bar {t}})&\\&:&\\&N_{k}({\bar {t}})&\end{bmatrix}}=n,}$

where ${\displaystyle N_{0}(t):=C(t)}$ and ${\displaystyle N_{i}(t)}$ is defined recursively as

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

Example

Consider a system varying analytically in ${\displaystyle (-\infty ,\infty )}$ and matrices

${\displaystyle A(t)={\begin{bmatrix}t&1&0\\0&t^{3}&0\\0&0&t^{2}\end{bmatrix}}}$, ${\displaystyle C(t)={\begin{bmatrix}1&0&1\end{bmatrix}}.}$

Then ${\displaystyle {\begin{bmatrix}N_{0}(0)\\N_{1}(0)\\N_{2}(0)\end{bmatrix}}={\begin{bmatrix}1&0&1\\0&1&0\\1&0&0\end{bmatrix}}}$ , and since this matrix has rank = 3, the system is observable on every nontrivial interval of ${\displaystyle \mathbb {R} }$.

Nonlinear systems

Given the system ${\displaystyle {\dot {x}}=f(x)+\sum _{j=1}^{m}g_{j}(x)u_{j}}$, ${\displaystyle y_{i}=h_{i}(x),i\in p}$. Where ${\displaystyle x\in \mathbb {R} ^{n}}$ the state vector, ${\displaystyle u\in \mathbb {R} ^{m}}$ the input vector and ${\displaystyle y\in \mathbb {R} ^{p}}$ the output vector. ${\displaystyle f,g,h}$ are to be smooth vector fields.

Define the observation space ${\displaystyle {\mathcal {O}}_{s}}$ to be the space containing all repeated Lie derivatives, then the system is observable in ${\displaystyle x_{0}}$ if and only if ${\displaystyle {\textrm {dim}}(d{\mathcal {O}}_{s}(x_{0}))=n}$.

Note: ${\displaystyle d{\mathcal {O}}_{s}(x_{0})=\mathrm {span} (dh_{1}(x_{0}),\ldots ,dh_{p}(x_{0}),dL_{v_{i}}L_{v_{i-1}},\ldots ,L_{v_{1}}h_{j}(x_{0})),\ j\in p,k=1,2,\ldots .}$[10]

Early criteria for observability in nonlinear dynamic systems were discovered by Griffith and Kumar,[11] Kou, Elliot and Tarn,[12] and Singh.[13]

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 ${\displaystyle \mathbb {R} ^{n}}$.[14][15] 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 ${\displaystyle \mathbb {R} ^{n}}$ 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.

See also

• Controllability
• Identifiability
• State observer
• State space (controls)

References

1. Kalman R. E., "On the General Theory of Control Systems", Proc. 1st Int. Cong. of IFAC, Moscow 1960 1481, Butterworth, London 1961.
2. Kalman R. E., "Mathematical Description of Linear Dynamical Systems", SIAM J. Contr. 1963 1 152
3. Sontag, E.D., "Mathematical Control Theory", Texts in Applied Mathematics, 1998
4. http://www.ece.rutgers.edu/~gajic/psfiles/chap5traCO.pdf
5. Li, W.; Wei, G.; Ho, D. W. C.; Ding, D. (November 2018). "A Weightedly Uniform Detectability for Sensor Networks". IEEE Transactions on Neural Networks and Learning Systems. 29 (11): 5790–5796. doi:10.1109/TNNLS.2018.2817244. PMID 29993845. S2CID 51615852.
6. Li, W.; Wang, Z.; Ho, D. W. C.; Wei, G. (2019). "On Boundedness of Error Covariances for Kalman Consensus Filtering Problems". IEEE Transactions on Automatic Control. 65 (6): 2654–2661. doi:10.1109/TAC.2019.2942826. S2CID 204196474.
7. Pasand, Mohammad Mahdi Share (2020). "Luenberger-type cubic observers for state estimation of linear systems". International Journal of Adaptive Control and Signal Processing. n/a (n/a): 1148–1161. arXiv:1909.11978. doi:10.1002/acs.3125. ISSN 1099-1115. S2CID 202888832.
8. Brockett, Roger W. (1970). Finite Dimensional Linear Systems. John Wiley & Sons. ISBN 978-0-471-10585-5.
9. Eduardo D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems.
10. Lecture notes for Nonlinear Systems Theory by prof. dr. D.Jeltsema, prof dr. J.M.A.Scherpen and prof dr. A.J.van der Schaft.
11. Griffith E. W. and Kumar K. S. P., "On the Observability of Nonlinear Systems I, J. Math. Anal. Appl. 1971 35 135
12. Kou S. R., Elliott D. L. and Tarn T. J., Inf. Contr. 1973 22 89
13. Singh S.N., "Observability in Non-linear Systems with immeasurable Inputs, Int. J. Syst. Sci., 6 723, 1975
14. Stanley G.M. and Mah, R.S.H., "Observability and Redundancy in Process Data Estimation, Chem. Engng. Sci. 36, 259 (1981)
15. Stanley G.M., and Mah R.S.H., "Observability and Redundancy Classification in Process Networks", Chem. Engng. Sci. 36, 1941 (1981)

External links

• "Observability". PlanetMath.
• MATLAB function for checking observability of a system
• Mathematica function for checking observability of a system