State Space Equation

Transfer function to state space.

State space equation. Recall that state space models of systems are not unique. The first equation is called the state equation and it has a first order derivative of the state variable s on the left and the state variable s and input s multiplied by matrices on the right. Together we then get a state space realization with matrices a b and c determined by the strictly proper part and matrix d determined by the constant.

This realization is equivalent to minreal ss sys where matrix a has the smallest possible dimension. To convert a transfer function into state equations in phase variable form we first convert the transfer function to a differential equation by cross multiplying and taking the inverse laplace transform assuming zero initial conditions. The state space realization of the constant is trivially.

In the linear state space system you provided the definition of u is missing. Let me give you a simpler example here. X 0 1 0 a 1 a 2 1 a 0 0 0 x t 0 0 1 u t displaystyle x begin bmatrix 0 1 0 a 1 a 2 1 a 0 0 0 end bmatrix x t begin bmatrix 0 0 1 end bmatrix u t.

State space modeling is also a kind of way to convert a differential equation into a set of matrix equation. A system has many state space representations therefore we will develop a few methods for creating state space models of systems. Before we look at procedures for converting from a transfer function to a state space model of a system let s first examine going from a differential equation to state space.

But the way we did in previous section differential equation meeting matrix was mainly for mathematical manipulation. Then we represent the differential equation in state space in phase variable form. Conversion to state space form is not uniquely defined in the siso case.

In order to solve an ode using ode45 you need to first define the function to describe the complete dynamics. The state space equations for the system will then be given by. In general u can be designed as a linear feedback control law such as u kx where k is a 50 by 50 matrix.