What are observability and controllability?

Observability and controllability are two major concepts of modern control system theory or state-space representation of systems. R. Kalman introduced these concepts in 1960. They can be roughly defined as follows.

What is observability?

The system must be observable to see what is going on inside it under observation. Thus, if a state is not observable, then the controller cannot determine its behavior from the system output and hence cannot use that state to stabilize the system.

Definition

Consider a dynamical system,

$$\begin{gathered} x\left(k+1\right)=Ax\left(k\right)+Cu\left(k\right) \\ y\left(k\right)=Bx\left(k\right)+Du\left(k\right) \end{gathered}$$

This is equation $\left(1\right)$, where $A\in R^{n\times n}, B\in R^{p\times n}, C\in R^{n\times m}, D\in R^{p\times m}$

The state model $\left(1\right)$ or the pair $\left(A,B\right)$ is said to be observable if any initial state can be uniquely determined from the knowledge of output $y\left(k\right)$ and input sequence $u\left(k\right), \mathrm{for} k\mathit{=}\mathit{0}\mathit{,}\mathit{1}\mathit{,}\mathit{2}\mathit{,}\mathit{.}\mathit{.}\mathit{.}\mathit{,}N\mathit{,}$ where $N$ is some finite time. Otherwise, the state model $\left(1\right)$ is unobservable.

Theorems on observability

• If the state model is in observable canonical form, then the system is observable.

• Kalman's test for observability: The state model $\left(1\right)$ or the pair $\left(A,B\right)$ is observable if the $np\times n$ observability matrix has rank $n$, i.e., full column rank. $U_0=\left[\begin{array}{c}B\\ BA\\ B{A}^2\\ ⋮\\ B{A}^{n-1}\end{array}\right]$

• The state model $\left(1\right)$ is observable, if the $n\times n$ observable Gramian matrix is non-singular for any non-zero finite N, i.e., $W_0=\sum _{i=0}^{n-1}{\left(A^i\right)}^T{B}^{T}B{A}^i=\sum _{i=0}^{n-1}{\left(A^{n-1-i}\right)}^T{B}^{T}B{A}^{n-1-i}$
• When $A$ has distinct eigenvalues and in Jordan/diagonal canonical form, the state model is observable if and only if none of the columns of $C$ contain zeros.
• Gilbert's test for observability: When $A$ has multiple order eigenvalues and in Jordan canonical form, then the state model is observable if and only if,
  a) each Jordan block corresponds to one distinct eigenvalue, and
  b) the elements of $C$ that correspond to the first column of each Jordan block are not all zero.

What is controllability?

The system must be controllable to do whatever we want with the given dynamic system under control input. Controllability deals with the possibility of forcing the system to a particular state by the application of a control input. If a state is uncontrollable, then no input will be able to control that state.

Let us discuss how the concepts of controllability and observability are related to linear systems of algebraic equations. It is well known that a solvable system of linear algebraic equations has a solution if and only if the rank of the system matrix is full. Observability and controllability tests will be connected to the rank tests of the controllability and observability matrices.

Definition

Again considering the equation $\left(1\right)$,

$$\begin{gathered} x\left(k+1\right)=Ax\left(k\right)+Cu\left(k\right) \\ y\left(k\right)=Bx\left(k\right)+Du\left(k\right) \end{gathered}$$

where $A\in R^{n\times n}, B\in R^{p\times n}, C\in R^{n\times m}, D\in R^{p\times m}$

• Complete state controllability: The state equation $\left(1\right)$ or the pair $\left(A,C\right)$ is said to be completely state controllable or simply state controllable if for any initial state $x\left(0\right)$ and any final state $x\left(N\right)$, there exists an input sequence $u\left(k\right), k=0,1,2,...,N,$ which transfers $x\left(0\right)$ to $x\left(N\right)$ for some finite $N$. Otherwise, the state equation $\left(1\right)$ is state uncontrollable.
• Complete output controllability: The system given in equation $\left(1\right)$ is said to be completely output controllable or simply output controllable if any final output $y\left(N\right)$ can be reached from any initial state $x\left(0\right)$ by applying an unconstrained input sequence $u\left(k\right), k=0,1,2,...,N,$ for some finite $N$. Otherwise, $\left(1\right)$ is not output controllable.

Theorems on controllability

• If the system has a single input and the state model is in controllable canonical form, then the system is controllable.
• Kalman's test for controllability: The state equation $\left(1\right)$ or the pair $\left(A,B\right)$ is state controllable if and only if the $n\times nm$ state controllability matrix has rank $n$, i.e., full row rank.       
• $U_c=\left[\begin{array}{ccccc}C& AC& A^{2}C& \cdots & A^{n-1}C\end{array}\right]$ The state equation (1) is controllable if the n × n controllability Gramian matrix is non-singular for any non-zero finite $N$.
• $W_C=\sum _{i=0}^{n-1}{A}^{i}C{C}^T{\left(A^i\right)}^T=\sum _{i=0}^{n-1}{A}^{n-1-i}B{B}^T\left(A^{n-1-i}\right)T$ When $A$ has distinct eigenvalues and in Jordan/diagonal canonical form, the state model is controllable if and only if all the rows of $B$ are non-zero.
• Gilbert's test to check controllability: When $A$ has multiple order eigenvalues and in Jordan canonical form, then the state model is controllable if and only if,
  a) each Jordan block corresponds to one distinct eigenvalue, and
  b) the elements of $B$ that correspond to last row of each Jordan block are not all zero.

Stabilizability and detectability

So far we have defined and studied the observability and controllability of the complete state vector. We have seen that the system is controllable (observable) if all components of the state vector are controllable (observable).

But a question arises here: Is there a need to observe and control all state variables?

In some applications, it is sufficient to take care of only the unstable components of the state vector. This leads to the definition of stabilizability and detectability, as follows:

Stabilizability: A linear system (continuous or discrete) is stabilizable if all unstable modes are controllable.

Detectability: A linear system (continuous or discrete) is detectable if all unstable modes are observable.

The concepts of stabilizability and detectability play very important roles in optimal control theory, and hence are studied in detail in advanced control theory courses.

Terms used above

• State of the system: The state of a dynamic system is the smallest set of variables that are also called state variables. The knowledge of these variables at $t=t_0$, together with the knowledge of the input variables $t\ge t_0$, determines the behavior of the system for any time $t\ge t_0$. ( $t$ is time and $t_0$ is the initial time.)
• State variables of the system: The variables that make up the smallest set of variables and are used to determine the state of the dynamic system are called state variables.
• State space of the system: State space is the $n-$dimensional space whose coordinates axes consist of the $x_1$ axis, $x_2$ axis,..., $x_n$ axis, where $x_{1,}{x}_{2, }{x}_{3, }..., x_{n, }$are state variables.
• State space equations of the system: In state-space analysis, three type of variables are used in the modelling of dynamic system: input variables, output variables, and state variables.
• State vector of the system: If we describe the behavior of a system by $n$ state variables, these $n$ state variables can be considered as the $n$components of a vector $x$. This vector $x$ is called a state vector.

Note: A dynamical system is a system of elements that changes over time. Some of its examples include the mathematical models that describe the swinging of a bob pendulum of clock, the flow of fluid through a pipe, and the motion of celestial bodies.

Common Mistakes

The Gram matrix or Gramian matrix is the Hermitian matrix of inner products of a set of vectors in an inner product space. It should not be confused with the Hermitian matrix.  Hermitian matrix (also called self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose.

Context and applications

This topic is significant in the professional exams for both graduate and postgraduate courses, especially for:

• Bachelor of Technology in Mechanical Engineering
• Associate Degree in Mathematics
• Bachelor's Degree in Mathematics
• Master's Degree in Mathematics
• Doctoral Degree in Mathematics

Related topics

• Digital Object Identifier system (DOI)
• Time-invariant systems
• Robotics
• MATLAB
• State-transition matrix

Practice problems

Q1. If it is possible to transfer the system state from any initial state to any desired state in finite interval of time, the system is called ___________.
a) controllable and observable
b) observable
c) controllable 
d) cannot be determined

Correct option: (c)

Explanation: The system must be controllable to be able to do whatever we want with the given dynamic system under control input. Controllability deals with the possibility of forcing the system to a particular state by application of a control input.

Q2. Kalman’s test is done for ________.
a) optimality
b) observability
c) observability and controllability
d) none of the above

Correct option: (c)

Explanation: The Kalman test is done for finding both observability and controllability.

Q3. If every state can be completely identified by measurements of the outputs at the finite time interval, the system is said to be ________.
a) observable
b) controllable
c) controllable and observable
d) cannot be determined

Correct option: (a)

Explanation: If every state can be completely identified by measurements of the outputs at the finite time interval, the system is said to be observable.

Q4. Gilbert’s test is done for ________.
a) optimality
b) observability
c) observability and controllability
d) none of the above

Correct option: (c)

Explanation: Gilbert's test is done for finding both observability and controllability.

Q5. Hermitian matrix is equal to its own __________.
a) inverse matrix
b) conjugate matrix
c) conjugate transpose matrix
d) transpose matrix

Correct option: (c)

Explanation: Hermitian matrix is a matrix that is equal to its own conjugate transpose matrix.