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**Adaptive control in large-scale systems: searchless techniques.
(Adaptivnoe upravlenie v slozhnykh sistemakh: bespoiskovye metody.)**
*(Russian.
English summary)*
Zbl 0732.93046

Moskva: Nauka. 296 p. R. 4.60 (1990).

Contents: Ch. 1. Problems and methods of adaptive control; Ch. 2. Speed gradient algorithms; Ch. 3. Design of continuous time adaptive systems; Ch. 4. Design of discrete time adaptive systems; Ch. 5. Adaptive system decomposition based on movements separation; Ch. 6. CAD of adaptive systems; Ch. 7. Adaptive control of nuclear reactors; Ch. 8. Adaptation in system analysis.

After a brief over-view (Ch. 1) the author pays the main attention to an idea of speed gradient algorithm. The idea is very simple. If a goal of control is given by the inequality \[ (I)\quad Q(x,t)\leq \Delta,\quad \Delta >0 \] then the control law \[ (2)\quad \dot u=-\Gamma \nabla_ u\omega (x,u,t),\quad \omega =\frac{\partial Q}{\partial t}+\nabla^ T_ xQ\cdot F(x,u,t),\quad \Gamma >0 \] can give good results for a system \[ (3)\quad \dot x=F(x,u,t). \] The function \(\omega\) is named as the speed gradient. Under some conditions the algorithm guarantees the asymptotic (t\(\to \infty)\) fulfillment of the goal inequality. Some estimates of transient regime are given too. Various realizations of that general idea are used for analysis and synthesis of adaptive control systems and algorithms of identification. For example, if the matrices A, B are unknown but the state vector x(t) is observable as output of the system \[ \dot x=Ax+Bu \] then it is recommended to look for estimates \(A^*,B^*\) by the algorithm \[ \overset \circ A^*=-\gamma Hex^ T,\quad \overset \circ B=-\gamma Heu,\quad H=H^ T>0;\quad \gamma >0,\quad e=x^*-x; \]

\[ \overset \circ x^*=Gx^*+(A-G)x+Bu,\quad HG+G^ TH<0. \] The following part (Ch. 4) is devoted to discrete time algorithms with main attention to the algorithms of gradient projection type which were investigated in detail by V. Jacubovich and his team which the author belongs to. The singular perturbation method is used in Ch. 5 as a powerful tool for the investigation of continuous and discrete adaptive systems. Ch. 6 contains a description of CAD software “AVANS”, which enables to analyze large-scale systems and to organize some parametric optimization schemes. As an example of adaptive system design the problem of field control in a nuclear reactor is investigated in Ch. 7.

In conclusion (Ch. 8) some formal models of population control theory are presented to illustrate a possibility to use the speed gradient algorithm. All results are presented in a rigorous form with good qualitative comments. Proofs are taken out in appendix.

After a brief over-view (Ch. 1) the author pays the main attention to an idea of speed gradient algorithm. The idea is very simple. If a goal of control is given by the inequality \[ (I)\quad Q(x,t)\leq \Delta,\quad \Delta >0 \] then the control law \[ (2)\quad \dot u=-\Gamma \nabla_ u\omega (x,u,t),\quad \omega =\frac{\partial Q}{\partial t}+\nabla^ T_ xQ\cdot F(x,u,t),\quad \Gamma >0 \] can give good results for a system \[ (3)\quad \dot x=F(x,u,t). \] The function \(\omega\) is named as the speed gradient. Under some conditions the algorithm guarantees the asymptotic (t\(\to \infty)\) fulfillment of the goal inequality. Some estimates of transient regime are given too. Various realizations of that general idea are used for analysis and synthesis of adaptive control systems and algorithms of identification. For example, if the matrices A, B are unknown but the state vector x(t) is observable as output of the system \[ \dot x=Ax+Bu \] then it is recommended to look for estimates \(A^*,B^*\) by the algorithm \[ \overset \circ A^*=-\gamma Hex^ T,\quad \overset \circ B=-\gamma Heu,\quad H=H^ T>0;\quad \gamma >0,\quad e=x^*-x; \]

\[ \overset \circ x^*=Gx^*+(A-G)x+Bu,\quad HG+G^ TH<0. \] The following part (Ch. 4) is devoted to discrete time algorithms with main attention to the algorithms of gradient projection type which were investigated in detail by V. Jacubovich and his team which the author belongs to. The singular perturbation method is used in Ch. 5 as a powerful tool for the investigation of continuous and discrete adaptive systems. Ch. 6 contains a description of CAD software “AVANS”, which enables to analyze large-scale systems and to organize some parametric optimization schemes. As an example of adaptive system design the problem of field control in a nuclear reactor is investigated in Ch. 7.

In conclusion (Ch. 8) some formal models of population control theory are presented to illustrate a possibility to use the speed gradient algorithm. All results are presented in a rigorous form with good qualitative comments. Proofs are taken out in appendix.

Reviewer: A.A.Pervozvanskij (St.Petersburg)

### MSC:

93C40 | Adaptive control/observation systems |

93C10 | Nonlinear systems in control theory |

93A15 | Large-scale systems |

93C15 | Control/observation systems governed by ordinary differential equations |

93C55 | Discrete-time control/observation systems |