Over the last thirty years an abundance of papers have been writ ten on adaptive dynamic control systems. Nevertheless, now it may be predicted with confidence that the adaptive mechanics, a new division, new line of inquiry in one of the violently developing fields of cybernetic mechanics, is emerging. The birth process falls far short of being com pleted. There appear new problems and methods of their solution in the framework of adaptive nonlinear dynamics. Therefore, the present work cannot be treated as a certain polished, brought-to-perfection school textbook. More likely, this is an…mehr
Over the last thirty years an abundance of papers have been writ ten on adaptive dynamic control systems. Nevertheless, now it may be predicted with confidence that the adaptive mechanics, a new division, new line of inquiry in one of the violently developing fields of cybernetic mechanics, is emerging. The birth process falls far short of being com pleted. There appear new problems and methods of their solution in the framework of adaptive nonlinear dynamics. Therefore, the present work cannot be treated as a certain polished, brought-to-perfection school textbook. More likely, this is an attempt to show a number of well known scientific results in the parametric synthesis of nonlinear systems (this, strictly speaking, accounts for the availability of many reviews), as well as to bring to notice author's developments on this question undoubtedly modern and topical. The nonlinear, and practically La grangian, systems cover a wide class of classical objects in theoretical mechanics, and primarily solid-body (robotic, gyroscopic, rocket-cosmic, and other) systems. And what is rather important, they have a direct trend to practical application. To indicate this discussion, I should like to notice that it does not touch upon the questions concerned with the linear and stochastic con trolobjects. Investigated are only nonlinear deterministic systems being in the conditions when some system parameters are either unknown or beyond the reach of measurement, or they execute an unknown limited and fairly smooth drift in time.
I Problems and methods of adaptive mechanical-system control.- 1. Adaptive Stabilization of Mechanical Systems by the Method of Recurrent Objective Inequalities.- 2. Searchless Self-Adjustable Adaptation and Control Systems.- 3. Rate Gradient Algorithms in the Problems of Adaptive Control of Mechanical Systems.- 4. Overview of some Methods and Results of Nonlinear Parametric Synthesis.- II Integral transformation method in the theory of adaptive systems.- 5. Synthesis of Dissipative and Stabilizing Systems of Adaptive Control.- 6. Adaptive Stabilization of Controlled Mechanical Systems in the Conditions of Unknown Parametric Drift.- 7. Optimum Stabilization of Holonomic and Nonholonomic Mechanical Systems.- 8. Parametric Universal Integral Tests in the Problem of Optimal Stabilization of Mechanical Systems.- II Integral transformation method in the theory of adaptive systems.- 9. Adaptive Optimization Synthesis: Equivalence, Suboptimality, and Robustness.- 10. Optimal Synthesis of Adaptive Mechanical Systems Imposed by General Constraints.- 11. Synthesis of Adaptive Controllable Information Systems Based on the Canonic Hamilton-Jacobi Transformation Method.- 12. Optimization of Adaptive Controllable Distributed Parameter Systems.- Appendices.- A-Lyapunov function method in the theory of controllable dynamic systems.- A.1 Basic definitions and notions. Lyapunov functions.- A.2 Basic theorems on stability.- A.2.1 Lyapunov theorems.- A.2.2 Homogeneous stability.- A.2.3 Stability in large.- A.2.4 Exponential stability.- A.2.5 Stability with constant perturbations.- A.2.6 Dissipative systems.- A.3 Link between the Lyapunov function method and optimal control.- A.4 Special questions of stability theory.- A.4.1 Trajectory stability.- A.4.2 Stability of periodic motions and orbital stability.- A.4.3 Vector Lyapunov functions.- B-Introduction to theory of singularly perturbed differential equations.- B.1 Tikhonov theorem.- B.2 Asymptotic expansions and representation accuracy estimation.- B.2.1 Preliminary remarks.- B.2.2 Asymptotic expansion of a regularly perturbed initial problem.- B.2.3 Asymptotic expansion of the solution to a singularly perturbed input problem.- B.2.4 Estimation of remaining term.- B.3 On stability of singularly perturbed systems.- B.3.1 Linear systems.- B.3.2 Nonlinear systems.- B.4 Decomposition of singularly perturbed systems on integral manifolds.- C-Pseudo-inversion and rectangular matrices.- C.1 Finite-dimensional spaces and linear manifolds.- C.2 Moore-Penrose pseudo-inversion.- C.3 Pseudo-inversion operation and skeleton matrix arrangement.- C.4 Methods of pseudo-inverse matrice calculation.- C.4.1 Computational procedure by Gram-Schmidt orthogonalization method.- C.4.2 Computational procedure for the Jordan-Gauss elimination method.- D-Approximate methods of solving Volterra integral and integro-differential equations.- D.1 Approximated Volterra integral equations.- D.2 Approximate solution to the Cauchy problem for Volterra integro-differential equations.- D.2.1 Preliminary integral transformations.- D.2.2 Solution of IDE by successive iterations.- D.2.3 Solution of IDE by parametric method.- D.2.4 Solution of IDE by quadrature method.- D.2.5 Solution of IDE by Chaplygin method.- D.3 Approximate solution of boundary problems for the Volterra integro-differential equations.- D.3.1 Solution of polylocal boundary problem.- D.3.2 Solution of the integral boundary problem.- D.3.3 Solution of IDE by the method of averaging functional correction.- 467.- 503.
I Problems and methods of adaptive mechanical-system control.- 1. Adaptive Stabilization of Mechanical Systems by the Method of Recurrent Objective Inequalities.- 2. Searchless Self-Adjustable Adaptation and Control Systems.- 3. Rate Gradient Algorithms in the Problems of Adaptive Control of Mechanical Systems.- 4. Overview of some Methods and Results of Nonlinear Parametric Synthesis.- II Integral transformation method in the theory of adaptive systems.- 5. Synthesis of Dissipative and Stabilizing Systems of Adaptive Control.- 6. Adaptive Stabilization of Controlled Mechanical Systems in the Conditions of Unknown Parametric Drift.- 7. Optimum Stabilization of Holonomic and Nonholonomic Mechanical Systems.- 8. Parametric Universal Integral Tests in the Problem of Optimal Stabilization of Mechanical Systems.- II Integral transformation method in the theory of adaptive systems.- 9. Adaptive Optimization Synthesis: Equivalence, Suboptimality, and Robustness.- 10. Optimal Synthesis of Adaptive Mechanical Systems Imposed by General Constraints.- 11. Synthesis of Adaptive Controllable Information Systems Based on the Canonic Hamilton-Jacobi Transformation Method.- 12. Optimization of Adaptive Controllable Distributed Parameter Systems.- Appendices.- A-Lyapunov function method in the theory of controllable dynamic systems.- A.1 Basic definitions and notions. Lyapunov functions.- A.2 Basic theorems on stability.- A.2.1 Lyapunov theorems.- A.2.2 Homogeneous stability.- A.2.3 Stability in large.- A.2.4 Exponential stability.- A.2.5 Stability with constant perturbations.- A.2.6 Dissipative systems.- A.3 Link between the Lyapunov function method and optimal control.- A.4 Special questions of stability theory.- A.4.1 Trajectory stability.- A.4.2 Stability of periodic motions and orbital stability.- A.4.3 Vector Lyapunov functions.- B-Introduction to theory of singularly perturbed differential equations.- B.1 Tikhonov theorem.- B.2 Asymptotic expansions and representation accuracy estimation.- B.2.1 Preliminary remarks.- B.2.2 Asymptotic expansion of a regularly perturbed initial problem.- B.2.3 Asymptotic expansion of the solution to a singularly perturbed input problem.- B.2.4 Estimation of remaining term.- B.3 On stability of singularly perturbed systems.- B.3.1 Linear systems.- B.3.2 Nonlinear systems.- B.4 Decomposition of singularly perturbed systems on integral manifolds.- C-Pseudo-inversion and rectangular matrices.- C.1 Finite-dimensional spaces and linear manifolds.- C.2 Moore-Penrose pseudo-inversion.- C.3 Pseudo-inversion operation and skeleton matrix arrangement.- C.4 Methods of pseudo-inverse matrice calculation.- C.4.1 Computational procedure by Gram-Schmidt orthogonalization method.- C.4.2 Computational procedure for the Jordan-Gauss elimination method.- D-Approximate methods of solving Volterra integral and integro-differential equations.- D.1 Approximated Volterra integral equations.- D.2 Approximate solution to the Cauchy problem for Volterra integro-differential equations.- D.2.1 Preliminary integral transformations.- D.2.2 Solution of IDE by successive iterations.- D.2.3 Solution of IDE by parametric method.- D.2.4 Solution of IDE by quadrature method.- D.2.5 Solution of IDE by Chaplygin method.- D.3 Approximate solution of boundary problems for the Volterra integro-differential equations.- D.3.1 Solution of polylocal boundary problem.- D.3.2 Solution of the integral boundary problem.- D.3.3 Solution of IDE by the method of averaging functional correction.- 467.- 503.
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