ABSTRACT
A number of stability concepts are used in the design of control systems. These definitions rely on a number of basic mathematical definitions. The first and most important of these are the definitions of the norms used in the subsequent sections. For further details, the reader is referred to Khalil [1] and Lewis et al. [2]. The norm of a vector x ∈ Rn or the 2-norm is defined as
x x x= = =
x2
. (6.1)
The p-norm is defined as
. (6.2)
The induced nom of a matrix is defined as
A A A= ( )lmax T , (6.3)
where λmax(⋅) is the maximum eigenvalue of the argument. The Frobenius norm of a matrix is defined as the root of the sum of the squares of all the elements of the matrix and its square is equivalent to
A A A
2 = ( ) . (6.4)
The function tr(⋅) refers to the trace of the matrix and is the sum of all the diagonal terms of the matrix argument. It satisfies the condition tr(A) = tr(AT ) or any A ∈ Rn × n. For any two matrices, B ∈ Rm × n and C ∈ Rn × m, tr(BC) = tr(CB). For any matrix B ∈ Rm × n and a positive definite matrix A ∈ Rn × n, tr(BABT) ≥ 0, which is equal to zero if and only if (iff ) B ∈ Rm × n is a zero matrix. The time derivative of a matrix function of time is as follows:
d dt
tr t tr d t
dt A
A( )( ) = ( )æ è çç
ö
ø ÷÷ . (6.5)
Two forms of input/output stability are applied extensively in nonlinear control synthesis:
1. Bounded-input, bounded-output (BIBO) stability A system is bounded-input, bounded-output stable if for every bounded input u t Mi ( ) < 1 for all t and for all i and for every bounded output y t Mj ( ) < 2 for all t and for all j, provided that the initial conditions are zero.
