##
**Inversion method in the discrete-time nonlinear control systems synthesis problems.**
*(English)*
Zbl 0822.93001

Lecture Notes in Control and Information Sciences. 205. Berlin: Springer- Verlag. xi, 155 p. (1995).

Discrete-time nonlinear control systems described by the equations
\[
\begin{cases} x(t+ 1)= f(x(t), u(t), w(t)),\;x(t_ 0)= x_ 0,\;t= 0,1,2,\dots\\ y(t)= h(x(t), u(t))\end{cases}\tag{1}
\]
are considered, where \(x\in X\subset \mathbb{R}^ n\), \(u\in U\subset \mathbb{R}^ m\), \(w\in W\subset \mathbb{R}^ r\), \(y\in Y\subset \mathbb{R}^ p\) denote the state, the control, the disturbance and the output, respectively, \(f: X\times U\times W\to X\) and \(h: X\times U\to Y\) are smooth mappings.

It is assumed that the whole state vector \(x\) can be measured. The structure and dynamic properties of (1) are changed by state or dynamic state feedbacks. A systematic approach based on system inversion is developed. The book consists of two parts. In the first part, special subsets of right invertible systems, the so-called \((d_ 1,\dots, d_ p)\)-forward time-shift right invertible systems are considered.

In the second part, the generalized notion of the forward time-shift right invertibility is introduced. The inversion method is applied to the solution of the model matching problem, the disturbance decoupling problem and input-output decoupling problem. The input-output linearization problem via static and dynamic state feedback is considered. At the end of each chapter bibliographical notes are given.

At the end of the book some open problems are discussed. The basic ideas are also applicable to continuous-time nonlinear systems, implicit systems, 2-D systems and distributed parameter systems.

It is assumed that the whole state vector \(x\) can be measured. The structure and dynamic properties of (1) are changed by state or dynamic state feedbacks. A systematic approach based on system inversion is developed. The book consists of two parts. In the first part, special subsets of right invertible systems, the so-called \((d_ 1,\dots, d_ p)\)-forward time-shift right invertible systems are considered.

In the second part, the generalized notion of the forward time-shift right invertibility is introduced. The inversion method is applied to the solution of the model matching problem, the disturbance decoupling problem and input-output decoupling problem. The input-output linearization problem via static and dynamic state feedback is considered. At the end of each chapter bibliographical notes are given.

At the end of the book some open problems are discussed. The basic ideas are also applicable to continuous-time nonlinear systems, implicit systems, 2-D systems and distributed parameter systems.

Reviewer: T.Kaczorek (Warszawa)

### MSC:

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

93C55 | Discrete-time control/observation systems |

93C10 | Nonlinear systems in control theory |

93B52 | Feedback control |

93A99 | General systems theory |

93A30 | Mathematical modelling of systems (MSC2010) |

93B18 | Linearizations |

93C73 | Perturbations in control/observation systems |