1. Kalman criterion I 005
1.1. Introduction 005
1.2. Linear systems with unconstrained control 006
2. Kalman criterion II 011
2.1. General theorem 011
2.2. Kalman controls 014
3. Kalman criterion III 017
3.1. Relations between attainable sets for constrained and unconstrained systems 017
3.2. Convex sets and support functions 020
3.3. Support function of attainable set 021
4. Kalman criterion IV 023
4.1. More about convex sets 023
4.2. More on support functions 023
4.3. Kalman duality 025
5. Brammer’s criterion 027
6. Stochastic approach 031
6.1. Least square method 031
6.1.1. Problem statement 031
6.1.2. Algorithm 032
6.1.3. Properties of LSM 033
6.1.4. Gaussian case 034
6.1.5. Student distribution 035
6.1.6. Maximum likelihood and least square 036
6.2. Stochastic Integrals 036
6.2.1. Definition of the integral 0 036
6.2.1.1. Integration of simple non-anticipating functions 037
6.2.1.2. Approximation by simple functions 037
6.3. Extension of Stochastic Integral 041
6.3.1. Doob inequality 041
6.3.2. Applications of Doob inequality 042
6.4. Ito formula 045
6.4.1. Statement of Ito lemma 045
6.4.2. Vector version of Ito formula 047
6.4.3. First applications of Ito formula 048
6.4.4. Proof of Ito formula 050
6.5. Stochastic Differential Equations 052
6.5.1. Cauchy Problem 052
6.5.2. Proof of Cauchy–Lipschitz theorem for SDE 054
6.5.3. Dependence on parameters 056
6.6. Diffusion Processes 058
7. Pole placement 063
7.1. Cauchy formula 063
4
7.2. Remarks on Cauchy formula 066
7.2.1. Remark 1 066
7.2.2. Remark 2 067
7.3. Exercises 069
8. Elements of Nonlinear Controllability 073
8.1. Lie brackets 073
8.2. Rashevsky – Chow theorem 075
8.3. H?rmander theorem 078
9. Black box 079
10. Examples 085
10.1. A cart under bounded force 085
10.2. Harmonic oscillator 086
11. White Noise I 089
11.1. Reminder on the Probability Theory 089
11.1.1. Measure and integral 091
11.1.2. Independence and conditional expectation 092
11.1.3. Characteristic functions and Gaussian vectors 093
12. White Noise II 097
12.1. Stochastic processes 097
12.2. Construction of the white noise 099
13. White Noise III 105
13.1. Why the white noise is white? 105
13.2. Quantum mechanics and functional integrals 107
13.3. Girsanov’s theorem 111
14. Kalman Filter I 113
14.1. Equations for the conditional density 113
14.2. Kalman filter 117
14.2.1. Heisenberg algebra 117
14.2.2. Lax type equations 119
14.2.3. Riccatti equation 120
15. Kalman Filter II 121
15.1. Filtering problem 121
15.2. Innovation process 121
15.3. Equations for the conditional mean 123
15.4. Riccati equation 126
15.5. Kalman-Bucy filter 127
References 131

Special topics in control theory

Издания