PID Passivity-Based Control of Nonlinear Systems with Applications. Romeo Ortega

. 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 ,

1 Introduction

Motivated