State-Space Representation of
Dynamical Systems
1. State and State-Space Representation of Dynamical Systems
State: The state of a dynamical system refers to the set of variables that completely describe
the system at any given time. These variables include both internal conditions and external
inputs that affect the system.
State-Space Representation: This is a mathematical model that represents a system using a
set of first-order differential (or difference) equations. It is often written as:
x(t) = A x(t) + B u(t)
y(t) = C x(t) + D u(t)
Where:
- x(t) is the state vector
- u(t) is the input vector
- y(t) is the output vector
- A, B, C, D are system matrices that describe the dynamics.
2. Physical Variable Form, Phase Variable Form, and Jordan/Diagonal
Canonical Form
Physical Variable Form: This form uses the physical variables of the system (e.g., position,
velocity) to represent the state.
Phase Variable Form: In this form, the state variables are chosen to reflect the physical
quantities of the system in terms of generalized coordinates and their derivatives.
Jordan/Diagonal Canonical Form: These are specific forms of the system’s A matrix where
the matrix is diagonal or in a block diagonal form, making the system easier to analyze and
solve. The Jordan form is used when the system has repeated eigenvalues, and it helps in
finding the general solution of the system.
3. Conversion of Transfer Function to State-Space Model and Vice Versa
From Transfer Function to State-Space:
1. Express the transfer function in terms of the output over the input Y(s)/U(s).
2. Define the state variables as the derivatives of the output.
3. Use the standard procedure to form the matrices A, B, C, and D.
From State-Space to Transfer Function:
Given the state-space representation, the transfer function is obtained by:
G(s) = C (sI - A)^{-1} B + D
Where I is the identity matrix.
4. State Equation and Its Solution
The state equation describes how the system evolves over time. For a linear time-invariant
system:
x(t) = A x(t) + B u(t)
The solution to the state equation is:
x(t) = e^{At} x(0) + ∫(0 to t) e^{A(t-τ)} B u(τ) dτ
Where:
- e^{At} is the state transition matrix.
5. State Transition Matrix and Its Properties
The state transition matrix Φ(t) describes the evolution of the system state with respect to
time. It is defined as:
Φ(t) = e^{At}
Properties:
- Φ(0) = I (the identity matrix).
- Φ(t+s) = Φ(t)Φ(s).
- It is used to compute the system's response to initial conditions and inputs.
6. Computation of State Transition Matrix by Laplace Transform and
Cayley-Hamilton Method
Laplace Transform:
The state transition matrix can be computed using the Laplace transform of e^{At}:
L{e^{At}} = (sI - A)^{-1}
Cayley-Hamilton Method:
The Cayley-Hamilton theorem states that every square matrix satisfies its own
characteristic equation. This can be used to compute e^{At} by breaking it into a series
using the matrix's characteristic equation.
