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Important Properties of State Transition Matrix | Mathematical Expression

A state transition matrix is ​​a fundamental concept used to describe the Fundamental evolution of a linear time-invariant system in a state space representation. The state transition matrix is often represented by Ф(t). In this article, we will Go Through What is State Transition Matrix, What is Linear time-invariant System, the General Representation State Transition Matrix, and the Mathematical expression for the state transition matrix, and At last we will go through Solved examples of State Transition Matrix with its Application, Advantages, Disadvantages, and FAQs.

Table of Content

What is the State Transition Matrix?

A state transition matrix is ​​a fundamental concept used to describe the Fundamental evolution of a linear time-invariant system in a state space representation. The state transition matrix is often represented by Ф(t). The state transition matrix is also called the “system matrix” or “state matrix“. It is a matrix that relates the current state of the system to its initial state and it is a key component in the solution of linear time-invariant systems. The state transition matrix provides a mathematical tool to analyze and predict the behavior of a linear time-invariant system over time, given its initial conditions and inputs. It is a critical component in control system analysis and design, enabling engineers to understand system dynamics, stability, and controllability.

What is a Linear Time-Invariant System (LTI System)?

A linear time-invariant (LTI) system is a fundamental concept in the control system, control theory, and signal processing. It describes a class of systems that exhibits two key characteristics linearity and time – invariance. Now let us discuss both concepts in detail.

Properties of Linear Time-Invariant System

Given are the two Properties of Linear Time-Invariant System

What is Linearity?

Linearity means that the system follows the principle of superposition . In other words , if we apply a linear combination of inputs to the system , the output is the same linear combination of the individual responses to each input . Mathematically . for an linear time invariant system , if you have an input signal x1(t) and another input signal x2(t), and you apply them to a system , the response to x1(t)+x2(t) is the same as the response to x1(t) added to the response to x2(t).

If y1(t) is the response of x1(t) and y2(t) is the response of x2(t) , then for a LTI system:

y1(t) + y2(t) = LTI[x1(t)] + LTI[x2(t)]

What is Time-Invariance?

Time – invariance means that the system’s properties and behaviour don not change with time. In other words , the system’s response to an input signal at any given time is the same as it’s response to the same input signal at any other time . Mathematically , if you apply an input signal x(t) the response at time time t, denoted as y(t) , is the same as the response at a later time t+T.

If y(t) is the response of x(t) , then for an LTI system:
y(t) = LTI[x(t)]
→y(t+T) = LTI[x(t+T)]

These two properties , linearity and time – invariance makes LTI systems particularly mathematically tractable and amenable to analysis. They allow for the use of techniques like convolution , laplace transform and the state transition matrix to analyze and design control systems and signal processing systems .

LTI (Linear time invariant) systems can be described using state – space representation , transfer function and differential or difference equations , depending on the context and the specific system .They are widely used in various engineering and scientific fields for modeling and controlling dynamic systems , such as electrical circuits , mechanical systems, chemical processes and communication systems. Now let us look at the mathematical equation of the linear time invariant system (LTI system) in continuous time and discrete time systems.

Continuous -Time LTI System

Mathematical equation of continuous time LTI system,
a_0\frac{dy(t)}{dt}+a_1y(t)=b_0\frac{dx(t)}{dt}+b_1x(t)
where:
y(t) is the output of the system.
x(t) is the input to the system.
a0 , b0 , a1 , b1  are the constant co-efficients that depends on the system dynamics .

Discrete -Time LTI System

a0 y[n] + a1 y[n-1] + a2 y[n-2]+............+ am y[n-m] = b0 x[n] + b1 x[n-1] + b2 x[n-2] + .............+ bm x[n-m]
where :
y[n] is the output at discrete time step n .
x[n] is the input at discrete time step n.
a0,a1,a2........ am and b0,b1,b2...........bm are constant co-efficients that determine the system's behaviour.

General Representation of State Transition Matrix

Generally , state transition matrix can be represented as follows:

P_{ij}= [Tex]\begin{vmatrix} p_{11}& p_{12}& .& .& .& p_{1n}&\\ p_{21}& p_{22}& .& .& .& p_{2n}& \\ .& .& .& .& .& .&\\ .& .& .& .& .& .&\\ .& .& .& .& .& .&\\ p_{n1}& p_{n2}& .& .& .& p{nn}& \end{vmatrix} [/Tex]

where :

Pij is the probabiIlty of transitioning from state i to state j .

Sum of probabilities in each row is equal to 1. (i.e., \underset{j = 1}{\overset{n}{\sum }}Pij = 1 where n is the number of states).

Mathematical Expression for the State Transition Matrix

Mathematically , the state transition matrix can be represented as follows :

If you have a linear time – invariant system represented in state -space as :

ẋ(t) = A * x(t) + B * u(t)

where:

ẋ(t) is the derivative of the state vector x(t) with respect to time .

A is the system matrix .

B is the input matrix.

u(t) is the control input .

Then, the state transition matrix  Ф(t)  satisfies the following equation  ẋ(t) = A * x(t) + B * u(t) , where   Ф(t) is a matrix 
such that x(t)= Ф(t)*x(0).
In the above equation , x(t) represents the state of the system at time t, and x(0) is  the initial state at time t=0.

State Transition Matrix in Exponential Form

The below expression represents the state transition matrix in the exponential form .

Ф(t) = e^{( A * t)}where :Ф(t) is the state transition matrix .A is a system matrix .

Ф(t) can be obtained by the Inverse Laplace Transform form of the Ф(s) , where Ф(s) = [sI – A]-1 (i.e., inverse of [sI-A]).

 Ф(t) = L-1 { Ф(s) }

Steps to Evaluate the State Transition Matrix

  1. Find the matric [sI – A ] where I is the identity matrix and A is the system matrix
  2. Find the inverse of the matrix [sI – A ] . (inverse = adj(matrix) / det(matrix). [implies : [sI – A]-1 = Ф(s)
  3. Apply the laplcae transform to Ф(s) .
  4. Ф(t) = L-1 { Ф(s) } which is the state transition matrix .
Note: Either the system matrix A will be given or conditions/instructions to find the system matrix  will be given.

Now let us discuss an example to calculate the state transition matrix with given system matrix.

Example to Calculate State Transition Matrix

Question

Calculate the state transition matrix where system matrix is given as ,

A=\begin{vmatrix} 0& -2& \\ 1& -3& \end{vmatrix}

Solution

System matrix A=\begin{vmatrix} 0& -2& \\ 1& -3& \\ \end{vmatrix}

Given A=\begin{vmatrix} 0& -2& \\ 1& -3& \\ \end{vmatrix}

[sI-A]=\begin{vmatrix} s& 2& \\ -1& s+3& \\ & & \end{vmatrix}

Where I=\begin{vmatrix} i& 0& \\ 0& 1& \\ \end{vmatrix}

Ф(s)=[sI-A]^{-1}=\frac{1}{det(sI-A)}adj(sI-A)

adj(matrix)=\begin{vmatrix} s+3& -2& \\ 1& s& \\ \end{vmatrix}

where matrix=[sI-A]

det(matrix)=s(s+3)-(-2)

=s^2+3s+2

det(matrix)=(s+2)(s+1)

Ф(s)=\frac{1}{(s+2)(s+1)}\begin{vmatrix} s+3 & -2& \\ 1& s& \\ \end{vmatrix}

Ф(s)=\frac{1}{(s+2)(s+1)}\begin{vmatrix} \frac{s+3}{(s+2)(s+1)} & \frac{-2}{(s+2)(s+1)}& \\ \frac{1}{(s+2)(s+1)}& \frac{s}{(s+2)(s+1)}& \\ \end{vmatrix}

Ф(t)= L^{-1} { Ф(s) }=\begin{vmatrix} 2e^{-t}-e^{-2t}& -2e^{-t}+2e^{-2t}& \\ e^{-t}-e^{-2t}& -e^{-t}+2e^{-2t}& \\ \end{vmatrix}

Properties of State Transition Matrix

  • The state transition matrix is invertible . The inverse Ф-1(t) allows for backward integration.
  • The eigen values of the state transition matrix are related to the poles of the system.
  • The state transition matrix is independent of the initial conditions of the system.
  • State transition matrix is time invariant (i.e., Ф(t1 + Δt) = Ф(t1) *Ф(Δt) .
  • It satisfies the semi group property (i.e., Ф(t1 + t2) = Ф(t1) *Ф(t2) .
  • State transition matrix exhibits linearity property (i.e., if x1(t) and x2(t) are the solutions of a equation then c1*x1(t)+c2*x2(t) is also a solution ).
  • The state transition matrix describes the deterministic evolution of the system over the time .
  • The inverse of the state transition matrix at time t is equal to the state transition matrix at time -t (i.e., Ф-1(t) = Ф(-t) .
  • If the state transition matrix is evaluated at time t=0 , it is equal to identity matrix . (i.e., Ф(t) = I at t=0).
  • Ф^k(t) = [Ф(t)]k .

Advantages of State Transition Matrix

  • The state transition matrix provides a compact and elegant way to represent the dynamics of linear time – invariant systems .
  • State transition matrix simplifies the analysis and modelling of complex dynamic systems.
  • It allows for the accurate prediction of the system’s state at any future time , given its initial state and input .
  • It is essential in control and estimation problems.
  • It can be used to analyze the system’s stability , controllability and observability properties.
  • The state transition matrix satisfies the superposition principle making it easy to analyze the response of a system.
  • Well – suited for analyzing linear time – invariant systems.

Disadvantages of state transition matrix

  • The state transition matrix is specifically designed for linear time – invariant systems and may not be directly applicable to non – linear or time- variant systems , which are common in the real world.
  • The calculation of the state transition matrix for large systems can be computationally intensive and may involve solving matrix exponential equation , which can be challenging.
  • The state transition matrix relies on the assumption of linearity , which is not always valid in practical systems .
  • Derivatives from linearity may lead to inaccuracies in predictions.

Applications of State Transition Matrix

  • The state transition matrix is extensively used in control systems for designing and analyzing feedback control systems, including stability analysis , controller design and stable estimation.
  • It plays a crucial role in estimation and filtering (Kalman filtering and extended kalman filtering ).
  • State transition matrix is used to understand and model the behaviour of dynamic systems.
  • The state transition matrix is applied in aerospace engineering for modelling and analysing the behaviour of aircraft and space craft.
  • It is used in the analysis of the macroeconomics variables and forecasting.
  • It is used in the analysis of the dynamic signals and systems, particularly in applications like digital filtering and spectral analysis.

Conclusion

In this article , we have learned about what is state transition matrix , different mathematical forms of it such as general form , exponential form. We also learnt about the evaluation of state transition matrix from the system matrix (i.e., the process to calculate the state transition matrix from the system matrix A) . We also learnt about the properties of the state transition matrix . We also learnt about the applications , advantages and disadvantages of the state transition matrix .

FAQs on State Transition Matrix

What is state transition matrix ?

State transition matrix is a fundamental concept used ti describe the time evolution of a linear time – invariant system in state space representation .State transition matrix is often represented by Ф(t) . The state transition matrix is also referred as the “system matrix” or “state matrix”.

How a state transition matrix represented ?

State transition matrix is often represented by Ф(t)

What is the mathematical representation of state transition matrix ?

The state transition matrix Ф(t) satisfies the following equation ẋ(t) = A * x(t) + B * u(t) ,

where Ф(t) is a matrix such that x(t)= Ф(t)*x(0).

In the above equation , x(t) represents the state of the system at time t, and x(0) is the initial state at time t=0.

What is the representation of the state transition matrix in exponential form ?

The below expression represents the state transition matrix in the exponential form .Ф(t) = e^( A * t)where :Ф(t) is the state transition matrix .A is a system matrix .Ф(t) can be obtained by the inverse laplace transform form of the Ф(s) , where Ф(s) = [sI – A]-1 (i.e., inverse of [sI-A]). Ф(t) = L-1 { Ф(s) }

What are the steps to evaluate the state transition matrix ?

Find the matrix [sI – A ] where I is the identity matrix and A is the system matrix. Find the inverse of the matrix [sI – A ] . (inverse = adj(matrix) / det(matrix). [implies : [sI – A]-1 = Ф(s) Apply the laplace transform to Ф(s) .Ф(t) = L-1 { Ф(s) } which is the state transition matrix .Note : Either the system matrix A will be given or conditions/instructions to find the system matrix will be given.

Discuss few applications of the state transition matrix ?

  • The state transition matrix is extensively used in control systems for designing and analyzing feedback control systems, including stability analysis , controller design and stable estimation.
  • It plays a crucial role in estimation and filtering (Kalman filtering and extended kalman filtering ).
  • State transition matrix is used to understand and model the behaviour of dynamic systems.

 


Reffered: https://www.geeksforgeeks.org

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