Properties of State Transition Matrix

A rectangular arrangement of numbers in rows and columns is called a matrix. Control theory refers to the control of continuously operating dynamical systems in engineered processes and machines. The state-transition matrix is a matrix whose product with the state vector x at the time t0 gives x at a time t, where t0 denotes the initial time. This matrix is used to obtain the general solution of linear dynamical systems. It is represented by Φ. It is an important part of both the zero input and the zero state solutions of systems represented in state space. In this article, we will learn the important properties of the state transition matrix.

10 Important Properties of State Transition Matrix

1. It has continuous derivatives.

2. It is continuous.

3. It cannot be singular. Φ-1 (t, 𝜏) = Φ(𝜏, t)

Φ-1 (t, 𝜏) Φ(t, 𝜏) = I. I is the identity matrix.

4. Φ(t, t) = I ∀ t.

5. Φ (t2,t1) Φ(t1, t0) = Φ(t2, t0) ∀ t0≤ t1≤ t2

6. Φ(t, 𝜏) = U(t) U-1(𝜏).

U(t) is an n×n matrix. It is the fundamental solution matrix that satisfies the following equation with initial condition U(t0) = I.

U˙(t)=A(t)U(t)\dot{U}(t) = A(t) U(t)

7. It also satisfies the following differential equation with initial conditions Φ(t0, t0) = I

ϕ(t,t0)t=A(t)ϕ(t,t0)\frac{\partial \phi(t, t_0)}{\partial t} = A(t) \phi(t, t_0)

8. x(t) = Φ(t, 𝜏) x(𝜏)

9. The inverse will be the same as that of the state transition matrix just by replacing ‘t’ by ‘-t’. Φ-1(t) = Φ(-t).

10. Φ(t1 + t2) = Φ(t1)Φ(t2) .If t = t1 + t2, the resulting state transition matrix is equal to the multiplication of the two state transition matrices at t = t1 and t = t2.