7.2 State Transition Matrix and System Response

The state transition matrix is a powerful tool in dynamic systems analysis. It maps a system's state from one time to another, helping predict future behavior. This concept is crucial for understanding how systems evolve over time and respond to inputs.

By using the state transition matrix, we can solve for system responses, analyze stability, and examine transient and steady-state behaviors. This knowledge is essential for designing and controlling dynamic systems in various engineering applications.

State Transition Matrix

Concept and Properties

• The state transition matrix, denoted as $\Phi(t, t₀)$, maps the state vector from an initial time $t₀$ to a future time $t$ in a linear time-invariant (LTI) system
• Unique for a given LTI system and initial conditions
• Satisfies the following properties:
  • $\Phi(t₀, t₀) = I$, where $I$ is the identity matrix
  • Composition property or semigroup property: $\Phi(t₂, t₀) = \Phi(t₂, t₁) \Phi(t₁, t₀)$
  • Inverse property: $\Phi(t, t₀)⁻¹ = \Phi(t₀, t)$
• Solution to the matrix differential equation: $d\Phi(t, t₀)/dt = A(t)\Phi(t, t₀)$, where $A(t)$ is the system matrix

Matrix Differential Equation

• The state transition matrix is the solution to the matrix differential equation: $d\Phi(t, t₀)/dt = A(t)\Phi(t, t₀)$
  • $A(t)$ represents the system matrix, which characterizes the dynamics of the LTI system
  • The matrix differential equation describes the evolution of the state transition matrix over time
• Initial condition for the matrix differential equation is $\Phi(t₀, t₀) = I$
  • Ensures that the state transition matrix maps the initial state vector to itself at the initial time $t₀$

Solving for the State Transition Matrix

Matrix Exponential Method

• For LTI systems with a constant system matrix $A$, the state transition matrix can be solved using the matrix exponential: $\Phi(t, t₀) = e^{A(t - t₀)}$
  • The matrix exponential is defined as a power series: $e^A = I + A + \frac{A^2}{2!} + \frac{A^3}{3!} + ...$
  • Simplifies the computation of the state transition matrix for LTI systems with constant coefficients
• Example: Given a system matrix $A = \begin{bmatrix} -2 & 1 \\ 0 & -3 \end{bmatrix}$, the state transition matrix can be calculated as $\Phi(t, t₀) = e^{A(t - t₀)}$

Laplace Transform Method

• The Laplace transform method can be used to find the state transition matrix: $\Phi(t, t₀) = L⁻¹[(sI - A)⁻¹]$
  • $L⁻¹$ denotes the inverse Laplace transform
  • $(sI - A)⁻¹$ is the resolvent matrix, which can be found using Cramer's rule or other matrix inversion techniques
• Useful for systems with time-varying coefficients or when the matrix exponential is difficult to compute
• Example: For a system with matrix $A = \begin{bmatrix} 0 & 1 \\ -2 & -3 \end{bmatrix}$, the state transition matrix can be found by taking the inverse Laplace transform of $(sI - A)⁻¹$

Other Methods

• Time-varying systems: The state transition matrix can be obtained by solving the matrix differential equation using numerical methods (Runge-Kutta) or power series expansion
• Cayley-Hamilton theorem: Express the state transition matrix as a linear combination of powers of the system matrix $A$ and the identity matrix $I$
• Diagonalizable systems: $\Phi(t, t₀) = Ve^{\Lambda(t - t₀)}V⁻¹$, where $V$ is the modal matrix and $\Lambda$ is the diagonal matrix of eigenvalues
  • Modal matrix $V$ consists of eigenvectors of the system matrix $A$
  • Diagonal matrix $\Lambda$ contains the eigenvalues of $A$ on its diagonal

System Response with the State Transition Matrix

State Vector Calculation

• The system response, or state vector $x(t)$, can be determined using the state transition matrix and the initial state vector $x(t₀)$: $x(t) = \Phi(t, t₀)x(t₀)$
  • The state transition matrix maps the initial state vector to the state vector at any future time $t$
  • Allows for the prediction of the system's state given an initial condition
• For systems with inputs, the state vector can be calculated using the convolution integral: $x(t) = \Phi(t, t₀)x(t₀) + \int_{t₀}^t \Phi(t, \tau)B(\tau)u(\tau) d\tau$
  • $B(\tau)$ is the input matrix, and $u(\tau)$ is the input vector
  • The convolution integral accounts for the effect of inputs on the system's state over time

Output Vector Determination

• The output vector $y(t)$ can be determined using the state vector and the output matrix $C$: $y(t) = Cx(t)$
  • The output matrix $C$ relates the system's state to its observable outputs
  • Allows for the calculation of the system's output given its state vector
• Example: For a system with state vector $x(t) = \begin{bmatrix} 2e^{-t} \\ 3e^{-2t} \end{bmatrix}$ and output matrix $C = \begin{bmatrix} 1 & 0 \end{bmatrix}$, the output vector is $y(t) = Cx(t) = 2e^{-t}$

System Stability and Behavior

Stability Analysis

• The stability of an LTI system can be determined by analyzing the eigenvalues of the system matrix $A$, which are also the eigenvalues of the state transition matrix $\Phi(t, t₀)$
  • Eigenvalues provide insight into the long-term behavior of the system
  • The eigenvalues of $A$ and $\Phi(t, t₀)$ are the same because $\Phi(t, t₀) = e^{A(t - t₀)}$
• System stability categories:
  • Asymptotically stable: All eigenvalues have negative real parts
  • Marginally stable: All eigenvalues have non-positive real parts, and those with zero real parts are distinct
  • Unstable: Any eigenvalue has a positive real part
• Example: A system with eigenvalues $\lambda_1 = -2$ and $\lambda_2 = -1$ is asymptotically stable because both eigenvalues have negative real parts

Transient and Steady-State Behavior

• The state transition matrix can be used to analyze the transient behavior of the system
  • Transient behavior refers to the system's response immediately after an input or disturbance
  • Characteristics such as settling time, rise time, and overshoot can be examined by analyzing the time-dependent terms in the matrix exponential
• The steady-state behavior of the system can be determined by evaluating the state transition matrix as $t \to \infty$
  • Steady-state behavior describes the system's response long after an input or disturbance
  • The steady-state behavior depends on the eigenvalues of the system matrix $A$
  • Systems with eigenvalues that have negative real parts will have a steady-state response that decays to zero as $t \to \infty$
• Example: For a system with state transition matrix $\Phi(t, t₀) = \begin{bmatrix} e^{-2t} & 0 \\ 0 & e^{-3t} \end{bmatrix}$, the transient behavior is characterized by exponential decay, and the steady-state response is zero