3.1 The State-Space Description

Any set of linear differential equations, relating the system variables x_i(t) and their r^th-order derivatives d^rx_i(t)/dt^r to the system outputs y_j(t) and inputs u_k(t), can be reduced to an equivalent set of first-order differential equations by the introduction of a set of additional variables (Rosenbrock (1970), Patel and Munro (1982)). This latter set of equations along with any algebraic equations can then be written as

where x(t) is an n × 1 vector of (state) variables, u(t) is a p × 1 vector of input functions, and y(t) is a q × 1 vector of outputs. The matrices A, B, C, and D have constant coefficients and are of dimension n × n, n × p, q × n, and q × p, respectively.

Example 3.1

For a simple fourth-order mass-spring system shown in Figure 3.1, the linearized equations of motion about an equilibrium point are

where the springs are characterized by spring rates r₁ and r₂ and the masses are represented by mass elements m₁ and m₂. The system is subjected to an input force u applied to mass m₁. The output of interest is the displacement y = x₂ of the mass m₂.

Figure 3.1. Simple mass-spring system.

By introducing additional variables x₃ and x₄ according to

these equations can be written in state-space form as

The state-space description of this system StateSpace[a, b, c, d] consists of the StateSpace object wrapper and the corresponding matrices.

Make sure the application is loaded.

In[1]:=

This is a state-space representation of the system.

In[2]:=

Out[3]=

The output format shows one matrix that contains the state-space system model, where the upper-left partition contains the matrix A, the upper-right partition contains the matrix B, the lower-left partition contains the matrix C, and the lower-right partition contains the matrix D. The superscripted letter identifies the StateSpace object, while the small subscripted bullet character denotes the continuous-time domain.