A special form of the state transition equation

Most of the concepts in this chapter are developed under the assumption that the state transition equation, ${\dot x}= f(x,u)$ has the following form:  

$${\dot x}= \alpha^1(x) u_1 + \alpha^2(x) u_2 + \cdots + \alpha^m(x) u_m ,$$ (1)

in which each $\alpha^i(x)$ is a vector-valued function of x, and m is the dimension of U (or the number of inputs). The $\alpha^i$functions can also be arranged in an $n \times m$ matrix,

$$A(x) = [ \alpha^1(x) \;\; \alpha^2(x) \;\; \cdots \;\; \alpha^m(x) ] .$$

In this case, the state transition equation can be expressed as

$${\dot x}= A(x) u .$$

For the rest of the chapter, it will be assumed that the matrix A(x) is nonsingular. In other words, the rows of A(x) are linearly independent for all x. To determine if A(x) is nonsingular, one must find at least one $n \times n$ cofactor (or submatrix) of A(x) which has a nonzero determinant.