By Alberto Isidori
This e-book specializes in tools that relate, in a single shape or one other, to the “small-gain theorem”. it really is geared toward readers who're drawn to studying equipment for the layout of suggestions legislation for linear and nonlinear multivariable platforms within the presence of version uncertainties. With labored examples all through, it contains either introductory fabric and extra complex topics.
Divided into elements, the 1st covers correct facets of linear-systems concept, the second one, nonlinear conception. so as to deepen readers’ realizing, less complicated single-input–single-output platforms as a rule precede therapy of extra complicated multi-input–multi-output (MIMO) structures and linear structures precede nonlinear platforms. This procedure is used all through, together with within the ultimate chapters, which clarify the most recent complicated principles governing the stabilization, law, and monitoring of nonlinear MIMO structures. significant layout difficulties are thought of, either within the presence of version uncertainties: asymptotic stabilization with a “guaranteed zone of appeal” of a given equilibrium element and asymptotic rejection of the influence of exogenous (disturbance) inputs on chosen regulated outputs.
Much of the introductory tutorial fabric during this ebook has been constructed for instructing scholars, whereas the ultimate assurance of nonlinear MIMO platforms deals readers a primary coordinated therapy of thoroughly novel effects. The labored examples offered give you the teacher with ready-to-use fabric to assist scholars to appreciate the mathematical theory.
Readers could be acquainted with the basics of linear-systems and keep an eye on idea. This ebook is a invaluable source for college kids following postgraduate courses in platforms and regulate, in addition to engineers engaged on the keep watch over of robot, mechatronic and gear systems.
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Additional info for Lectures in Feedback Design for Multivariable Systems
S. = π u0 ω0 2 π T (jω0 )u0 2 . ω0 1 2 1 2 u0 2 T (jω0 )u0 2 . s. s. , T (jω0 )u0 ≤ γ¯ u0 , or, bearing in mind the definition of norm of a matrix,5 T (jω0 ) ≤ γ¯ . Define now the quantity T H∞ := sup T (jω) , ω∈R which is called the H∞ norm of the matrix T (jω). Observing that ω0 in the above inequality is arbitrary, it is concluded T H∞ ≤ γ¯ . 5) is stable and the H∞ norm of its frequency response matrix is bounded from above by the number γ¯ . 5 Recall that the norm of a matrix T ∈ Rp×m is defined as T = sup u =0 Tu = max Tu .
10 This system can be formally viewed as a system having relative degree 1, with input u and output θ . To this end, in fact, it suffices to set ⎞ z ⎜ ξ1 ⎟ ⎟ ζ =⎜ ⎝ ··· ⎠ ξr−1 ⎛ and rewrite the system as ⎛ ⎛ ⎞ ⎞ A00 (μ) a˜ 01 (μ) a˜ 02 (μ) · · · a˜ 0,r−1 (μ) a˜ 0r (μ) ⎜ 0 ⎜ 0 ⎟ ⎟ 0 1 ··· 0 ⎜ ⎜ ⎟ ⎟ ⎟ζ + ⎜ ··· ⎟θ · · · · · · · ζ˙ = ⎜ ⎜ ⎜ ⎟ ⎟ ⎝ 0 ⎝ 0 ⎠ ⎠ 0 0 ··· 1 0 −a0 −a1 · · · −ar−2 1 θ˙ = A10 (μ) a˜ 11 (μ) a˜ 12 (μ) · · · a˜ 1,r−1 (μ) ζ + a˜ 1r (μ)θ + b(μ)u. The latter has the structure of a system in normal form 10 These coefficients can be easily derived as follows.
5) is nonsingular and T0 B = 0. We use now the matrix T introduced above to define a change of variables. 4)) it is natural to choose different notations for the first n − r new state variables and for the last r new state variables, setting ⎞ ⎛ ⎞ C ξ1 ⎜ ξ2 ⎟ ⎜ CA ⎟ ⎟ ⎜ ⎟ ξ =⎜ ⎝· · ·⎠ = ⎝ · · · ⎠ x. ξr CAr−1 ⎛ z = T0 x, To determine the equations describing the system in the new coordinates, we take the derivatives of z and ξ with respect to time. For the former, no special structure is found and we simply obtain z˙ = T0 x˙ = T0 (Ax + Bu) = T0 Ax + T0 Bu.