By Alberto Isidori

This booklet makes a speciality of tools that relate, in a single shape or one other, to the “small-gain theorem”. it really is geared toward readers who're attracted to studying equipment for the layout of suggestions legislation for linear and nonlinear multivariable structures within the presence of version uncertainties. With labored examples all through, it comprises either introductory fabric and extra complex topics.

Divided into components, the 1st covers suitable features of linear-systems idea, the second one, nonlinear conception. with a purpose to deepen readers’ knowing, easier single-input–single-output platforms in general precede therapy of extra advanced multi-input–multi-output (MIMO) structures and linear platforms precede nonlinear platforms. This procedure is used all through, together with within the ultimate chapters, which clarify the newest complicated principles governing the stabilization, law, and monitoring of nonlinear MIMO platforms. significant layout difficulties are thought of, either within the presence of version uncertainties: asymptotic stabilization with a “guaranteed area 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 educating scholars, whereas the ultimate assurance of nonlinear MIMO structures bargains readers a primary coordinated therapy of thoroughly novel effects. The labored examples awarded give you the teacher with ready-to-use fabric to assist scholars to appreciate the mathematical theory.

Readers may be acquainted with the basics of linear-systems and keep an eye on concept. This e-book is a precious source for college students following postgraduate courses in structures and keep an eye on, in addition to engineers engaged on the keep an eye on of robot, mechatronic and tool systems.

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**Additional resources for Lectures in Feedback Design for Multivariable Systems**

**Example text**

21) can be expressed in the form H = L + MN for L= B A 0 , M= , −C T D −C T C −AT N = W −1 DT C W −1 BT . Suppose, by contradiction, that the matrix H has eigenvalues on the imaginary axis. By definition, there exist a 2n-dimensional vector x0 and a number ω0 ∈ R such that (jω0 I − L)x0 = MNx0 . Observe now that the matrix L has no eigenvalues on the imaginary axis, because its eigenvalues coincide with those of A and −AT , and A is by hypothesis stable. Thus (jω0 I − L) is nonsingular. Observe also that the vector u0 = Nx0 is nonzero because otherwise x0 would be an eigenvector of L associated with an eigenvalue at jω0 , which is a contradiction.

The space of all such functions, endowed with the so-called L2 norm, which is defined as 1 u(·) L2 ∞ := u(t) 2 dt 2 , 0 is denoted by UL2 . , T lim T →∞ 0 y(t) 2 dt < ∞. 1) This makes it possible to compare the L2 norms of input and output functions and define a concept of “gain” accordingly. 5, see the monographs [1–6], [11] and the survey [12]. © Springer International Publishing Switzerland 2017 A. 1007/978-3-319-42031-8_3 43 44 3 The Small-Gain Theorem for Linear Systems … with state x ∈ Rn , input u ∈ Rm , output y ∈ Rp .

This system can be “artificially” reduced to a system to which the stabilization procedure described in the previous section is applicable, by means of a simple strategy. 21) in which a0 , a1 , . . , ar−2 are design parameters. 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.