The Structure of Multivariable Nonlinear Systems

Isidori, Alberto

doi:10.1007/978-3-319-42031-8_9

Alberto Isidori⁴

Part of the book series: Advanced Textbooks in Control and Signal Processing ((C&SP))

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Abstract

The purpose of this chapter is to analyze, in a multi-input multi-output nonlinear system (having the same number of input and output components), the notion of invertibility. A major consequence of such property is the existence of a change of variables that plays, for a multivariable system, a role equivalent to the change of variable leading to the normal form of a single-input single-output system. A class of special relevance consists of those invertible systems in which it is possible to force, by means of state feedback, a linear input–output behavior. A further subclass is that of those systems for which a (multivariable version of the concept of) relative degree can be defined.

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Notes

1.
The algorithm in question, originally introduced by Silverman in [4] to the purpose of analyzing the structure of the “zeros at infinity” of the transfer function matrix of a multivariable linear system, was extended in [5, 6] to the purpose of analyzing the property of invertibility of a nonlinear system. Additional features of such algorithm are discussed in [7] and [10]. Related results on nonlinear invertibiliy can be found in [8, 9].
2.
For consistency, set also $J_1(x)=M_1(x)$.
3.
Clearly, if at step $k^*$ one finds that $\rho _{k^*}=m$, then the partitions of $L_{k^*}(\cdot )$ and $M_{k^*}(\cdot )$ are trivial partitions, in which the upper blocks coincide with the matrices themselves, and this explains the notation used in the following formula.
4.
It suffices to choose a $C^k$ function with k large enough so that all derivatives of y(t) appearing in the developments which follow are defined and continuous.
5.
If $\rho _1=0$, the formula that follows must be replaced by
$$\begin{aligned} Q_1(y_{[1,\rho _1]}^{(1)},x)=H_1(x). \end{aligned}$$
6.
If $\rho _2-\rho _1=0$, the formula that follows must be replaced by
$$Q_2(y_{[1,\rho _1]}^{(1)},y_{[1,\rho _2]}^{(2)},x)=H_2(y_{[1,\rho _1]}^{(1)},y_{[1,\rho _1]}^{(2)},x)+ \tilde{F}_{21}(y_{[1,\rho _1]}^{(1)},x)[-y_{[1,\rho _1]}^{(1)} + H_1^\prime (x)],$$
while if $\rho _2=\rho _1=0$ we have
$$Q_2(y_{[1,\rho _1]}^{(1)},y_{[1,\rho _2]}^{(2)},x)=H_2(y_{[1,\rho _1]}^{(1)},y_{[1,\rho _1]}^{(2)},x). $$
7.
See [1, pp. 109–112].
8.
The general case only requires appropriate notational adaptations.
9.
For a proof, see [2] and also [1, pp. 114–115]. Related results can also be found in [11].
10.
The notation used in this formula is defined as follows. Recall that Z(x) is a $(n-d)\times 1$ vector, whose ith component is the function $z_i(x)$, and that g(x) is a $n\times m$ matrix, whose jth column is the vector field $g_j(x)$. The vector $L_fZ(x)$ is the $(n-d)\times 1$ vector whose ith entry is $L_fz_i(x)$ and the matrix $L_gZ(x)$ is the $(n-d)\times m$ matrix whose entry on the ith row and jth column is $L_{g_j}z_i(x)$.
11.
See, again, [2] and also [1, pp. 115–118].
12.
If $\tau _1(x)$ and $\tau _2(x)$ are vector fields defined on $\mathbb R^n$ and $\lambda (x)$ is a real-valued function defined on $\mathbb R^n$, the property in question consists in the identity
$$ L_{[\tau _1,\tau _2]}\lambda (x) = L_{\tau _1}L_{\tau _2}\lambda (x) - L_{\tau _2}L_{\tau _1}\lambda (x).$$
Therefore, if $L_{\tau _i}\lambda (x)=0$ for $i=1,2$, then $L_{[\tau _1,\tau _2]}\lambda (x) =0$ also.
13.
See [3].
14.
The matrix $W_k(x)$ is the $m\times m$ matrix whose entry on the ith row and jth column is $L_{g_j}L_f^{k-1}h_i(x)$. With reference to the functions $\tilde{f}(x)$ and $\tilde{\beta }(x)$ defined in (9.3), note that, for any $m\times 1$ vector $\lambda (x)$, we have $L_{\tilde{f}}\lambda (x)= L_f\lambda (x) + [L_g\lambda (x)]\alpha (x)$ and $L_{\tilde{g}}\lambda (x)= [L_g\lambda (x)]\beta (x)$.
15.
For consistency, we rewrite the matrix $F_1(x)$ determined at the first step of the algorithm as $F_{11}(x)$.
16.
Simply take the derivative of both sides along f(x), use the fact that $L_f L_{\tilde{f}}^kh(x)= L_{\tilde{f}}^{k+1}h(x) + L_{\tilde{g}}L_{\tilde{f}}^kh(x)\hat{\alpha }(x)$ and the fact that the $K_i$’s are constant.
17.
We consider in what follows the case in which $\rho _1>0$ and $\rho _2>0$. The other cases require simple adaptations of the arguments.
18.
There is no abuse of notation in using here symbols (specifically $r_1,\ldots , r_m$ in (i) and (ii), and B(x) in (9.47)) identical to symbols used earlier in Sect. 9.4. As a matter of fact, we will show in a moment that the notations are consistent.

References

A. Isidori, Nonlinear Control System II (Springer, New York, 1999)
Book MATH Google Scholar
B. Schwartz, A. Isidori, T.J. Tarn, Global normal forms for MIMO systems, with applications to stabilization and disturbance attenuation. Math. Control Signals Syst. 12, 121–142 (1999)
Article MathSciNet MATH Google Scholar
A. Isidori, A. Ruberti, On the synthesis of linear input-output responses for nonlinear systems. Syst. Control Lett. 4, 19–33 (1984)
Article MathSciNet MATH Google Scholar
L.M. Silverman, Inversion of multivariable nonlinear systems. IEEE Trans. Autom. Control 14, 270–276 (1969)
Article Google Scholar
R.M. Hirschorn, Invertibility for multivariable nonlinear control systems. IEEE Trans. Autom. Control 24, 855–865 (1979)
Article MathSciNet MATH Google Scholar
S.N. Singh, A modified algorithm for invertibility in nonlinear systems. IEEE Trans. Autom. Control 26, 595–598 (1981)
Article MathSciNet MATH Google Scholar
A. Isidori, C.H. Moog, in Modelling and Adaptive Control, ed. by C.I. Byrnes, A. Kurzhanski. On the Nonlinear Equivalent of the Notion of Transmission Zeros. Lecture Notes in Control and Communication, vol. 105 (Springer, 1988), pp. 146–158
Google Scholar
M.D. Di Benedetto, J.W. Grizzle, C.H. Moog, Rank invariants of nonlinear systems. SIAM J. Control Optim. 27, 658–672 (1989)
Article MathSciNet MATH Google Scholar
W. Respondek, in Nonlinear Controllability and Optimal Control, ed. by H.J. Sussman. Right and Left Invertibility of Nonlinear Control Systems (Marcel Dekker, New York, 1990), pp. 133–176
Google Scholar
D. Liberzon, Output-input stability implies feedback stabilization. Syst. Control Lett. 53, 237–248 (2004)
Article MathSciNet MATH Google Scholar
A. Liu, Z. Lin, On normal forms of nonlinear systems affine in control. IEEE Trans. Autom. Control 56(2), 239–253 (2011)
Article MathSciNet Google Scholar

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Dipartimento di Ingegneria Informatica, Automatica e Gestionale, Università degli Studi di Roma “La Sapienza”, Rome, Italy
Alberto Isidori

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Isidori, A. (2017). The Structure of Multivariable Nonlinear Systems. In: Lectures in Feedback Design for Multivariable Systems. Advanced Textbooks in Control and Signal Processing. Springer, Cham. https://doi.org/10.1007/978-3-319-42031-8_9

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DOI: https://doi.org/10.1007/978-3-319-42031-8_9
Published: 13 August 2016
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-42030-1
Online ISBN: 978-3-319-42031-8
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