Читать книгу PID Passivity-Based Control of Nonlinear Systems with Applications - Romeo Ortega - Страница 15

Given a vector , the symbol denotes its Euclidean norm, i.e. . We denote the th element of as . The th element of the canonical basis of is represented by . To ease the readability, column vectors are also expressed as .

Consider the matrix , then denotes the th column of , the th row of , and the th element of . Moreover, denotes the transpose of . Given a square matrix , . To simplify the notation, we express diagonal matrices as , where are the diagonal elements of the matrix.

The symbol denotes the identity matrix. The symbol refers to the th eigenvalue of . In particular, , denote the largest and the smallest eigenvalue of , respectively. A matrix is said to be positive semidefinite if and for all , and is said to be positive definite if the inequality is strict, i.e. for all . is negative (semi)definite if is positive (semi)definite. For a positive definite matrix and a vector , we denote the weighted Euclidean norm as . The notation used for constant matrices is directly extended to the nonconstant case.

Unless something different is stated, all the functions treated in this book are assumed to be smooth. Moreover, the symbol is reserved to express time, where we assume . Then, given a function that depends on time, the symbol denotes the differentiation with respect to time of , i.e. where . The and norms of signals are denoted and , respectively.

Given a function and a vector , we define the differential operator and . For a function , we define the th element of its Jacobian matrix as . When it is clear from the context, we omit the subindex of . Given a distinguished element , we define the matrix .

Throughout this book, we consider nonlinear systems described by differential equations of the form

(1)

where is the state vector, , , is the control vector, is an output of the system defined via the mappings and , and is the input matrix, which is full rank. In the sequel, we will refer to this system as or system.

We also consider the case of port‐Hamiltonian systems when the vector field may be factorized as

(2)

where is the Hamiltonian, and , with and , are the interconnection and damping matrices, respectively. To simplify the notation in the sequel, we define the matrix ,