State-Space Models

State-Space Model Representations

State-space models rely on linear differential equations or difference equations to describe system dynamics. Control System Toolbox™ software supports SISO or MIMO state-space models in continuous or discrete time. State-space models can include time delays. You can represent state-space models in either explicit or descriptor (implicit) form.

State-space models can result from:

Use ss model objects to represent state-space models.

Explicit State-Space Models

Explicit continuous-time state-space models have the following form:

dx/dt= A x + B u= C x + D u

where x is the state vector. u is the input vector, and y is the output vector. A, B, C, and D are the state-space matrices that express the system dynamics.

A discrete-time explicit state-space model takes the following form:

x[n+1]=Ax[n]+ B u[n]

y[n]=Cx[n]+ D u[n]

where the vectors x[n], u[n], and y[n] are the state, input, and output vectors for the nth sample.

Descriptor (Implicit) State-Space Models

A descriptor state-space model is a generalized form of state-space model. In continuous time, a descriptor state-space model takes the following form:

E(dx/dt)= A x + B u= C x + D u

where x is the state vector. u is the input vector, and y is the output vector. A, B, C, D, and E are the state-space matrices.

Commands for Creating State-Space Models

Use the commands described in the following table to create state-space models.

Create State-Space Model From Matrices

This example shows how to create a continuous-time single-input, single-output (SISO) state-space model from state-space matrices using ss.

Create a model of an electric motor where the state-space equations are:

dx/dt= A x + B u

y= C x + D u

where the state variables are the angular position θ and angular velocity dθ/dt:

x=[θ dθ/dt]

u is the electric current, the output y is the angular velocity, and the state-space matrices are:

A=[0  1

    -5 −2], B=[0 3], C=[0 1], D=[0].

To create this model, enter:

A = [0 1;-5 -2];
B = [0;3];
C = [0 1];
D = 0;
sys = ss(A,B,C,D);

sys is an ss model object, which is a data container for representing state-space models.