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

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

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

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Romeo Ortega

       José Guadalupe Romero

       Pablo Borja

       Alejandro Donaire

      Given a vector x colon equals left-bracket x 1 comma ellipsis comma x Subscript n Baseline right-bracket Superscript down-tack Baseline element-of double-struck upper R Superscript n, the symbol StartAbsoluteValue x EndAbsoluteValue denotes its Euclidean norm, i.e. StartAbsoluteValue x EndAbsoluteValue equals StartRoot x Superscript down-tack Baseline x EndRoot. We denote the ith element of x as x Subscript i. The ith element of the canonical basis of double-struck upper R Superscript n is represented by e Subscript i. To ease the readability, column vectors are also expressed as col left-parenthesis x 1 comma ellipsis comma x Subscript n Baseline right-parenthesis.

      Consider the matrix upper B element-of double-struck upper R Superscript n times m, then upper B Subscript i denotes the ith column of upper B, upper B Superscript k the kth row of upper B, and upper B Subscript i k the i kth element of upper B. Moreover, upper B Superscript down-tack denotes the transpose of upper B. Given a square matrix upper A element-of double-struck upper R Superscript n times n, sym left-parenthesis upper A right-parenthesis colon equals one half left-parenthesis upper A plus upper A Superscript down-tack Baseline right-parenthesis comma skew left-parenthesis upper A right-parenthesis colon equals one half left-parenthesis upper A minus upper A Superscript down-tack Baseline right-parenthesis. To simplify the notation, we express diagonal matrices as diag left-parenthesis a 1 comma ellipsis comma a Subscript n Baseline right-parenthesis, where a Subscript i are the diagonal elements of the matrix.

      The symbol upper I Subscript n denotes the n times n identity matrix. The symbol lamda Subscript i Baseline left-parenthesis upper A right-parenthesis refers to the ith eigenvalue of upper A. In particular, lamda Subscript max Baseline left-parenthesis upper A right-parenthesis, lamda Subscript min Baseline left-parenthesis upper A right-parenthesis denote the largest and the smallest eigenvalue of upper A, respectively. A matrix is said to be positive semidefinite if upper A equals upper A Superscript down-tack and x Superscript down-tack Baseline upper A x greater-than-or-equal-to 0 for all x element-of double-struck upper R Superscript n, and is said to be positive definite if the inequality is strict, i.e. x Superscript down-tack Baseline upper A x greater-than 0 for all x element-of double-struck upper R Superscript n Baseline reverse-solidus StartSet 0 EndSet. upper A is negative (semi)definite if negative upper A is positive (semi)definite. For a positive definite matrix upper A element-of double-struck upper R Superscript n times n and a vector x element-of double-struck upper R Superscript n, we denote the weighted Euclidean norm as double-vertical-bar x double-vertical-bar Subscript upper A Baseline colon equals StartRoot x Superscript down-tack Baseline upper A x EndRoot. 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 t is reserved to express time, where we assume t element-of double-struck upper R Subscript <hr><noindex><a href=Скачать книгу

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