Rational Matrix Equations in Stochastic Control

Banach spaces and nonlinear operators......Page 0
1.1 Stochastic integrals......Page 18
1.2 Stochastic differential equations......Page 22
1.3 Ito’s formula......Page 23
1.4 Linear stochastic differential equations......Page 24
1.5 Stability concepts......Page 27
1.6 Mean-square stability and robust stability......Page 36
1.7 Stabilization of linear stochastic control systems......Page 38
1.7.1 Stabilizability......Page 39
1.7.2 A Riccati type matrix equation......Page 43
1.8 Some remarks on stochastic detectability......Page 45
1.9.1 Population dynamics in a random environment......Page 48
1.9.2 The inverted pendulum......Page 49
1.9.3 A two-cart system......Page 50
1.9.4 An automobile suspension system......Page 51
1.9.5 A car-steering model......Page 53
1.9.6 Satellite dynamics......Page 56
1.9.7 Further examples......Page 58
2.1 Linear quadratic optimal stabilization......Page 60
2.2 Worst-case disturbance: A Bounded Real Lemma......Page 64
2.3 Disturbance attenuation......Page 67
2.3.1 Disturbance attenuation by static linear state feedback......Page 68
2.3.2 Systems with bounded parameter uncertainty......Page 73
2.3.3 Disturbance attenuation by dynamic output feedback......Page 75
3.1 Ordered Banach spaces......Page 78
3.2 Positive and resolvent positive operators......Page 79
3.2.1 Spectral properties......Page 80
3.2.2 Equivalent characterizations of resolvent positivity......Page 84
3.3 Linear mappings on the space of Hermitian matrices......Page 85
3.3.1 Representation of mappings between matrix spaces......Page 86
3.3.2 Completely positive operators......Page 90
3.4 Lyapunov operators and resolvent positivity......Page 92
3.5.1 Direct solution......Page 100
3.5.2 The case of simultaneous triangularizability......Page 101
3.5.3 Low-rank perturbations of Lyapunov equations......Page 102
3.5.4 Iterative solution with di>erent splittings......Page 106
3.5.5 Ljusternik acceleration......Page 108
3.5.6 Conclusions......Page 110
3.6 Recapitulation: Resolvent positivity, stability and detectability......Page 111
3.7 Minimal representations......Page 115
4 Newton’s method......Page 119
4.1 Concave maps......Page 120
4.2 Resolvent positive operators and Newton’s method......Page 121
4.2.1 A modified Newton iteration......Page 126
4.3 The use of double Newton steps......Page 129
4.4.1 L²-sensitivity optimization of realizations......Page 132
4.4.2 A non-symmetric Riccati equation......Page 134
4.4.3 The standard algebraic Riccati equation......Page 136
5 Solution of theRiccati equation......Page 138
5.1.1 Riccati operators and the definite constraint......Page 139
5.1.2 The indefinite constraint......Page 141
5.1.4 Some comments......Page 142
5.1.5 Algebraic Riccati equations from deterministic control......Page 145
5.1.6 A duality transformation......Page 147
5.1.7 A regularity transformation......Page 148
5.2.1 The Riccati operator......Page 149
5.2.2 The dual operator......Page 153
5.3.1 The Riccati equation with definite constraint......Page 160
5.3.2 The Riccati equation with indefinite constraint......Page 170
5.4.1 Newton’s method......Page 177
5.4.2 Computation of stabilizing matrices......Page 179
5.4.3 A nonlinear fixed point iteration......Page 180
5.5.1 The two-cart system......Page 182
5.5.2 The automobile suspension system......Page 185
5.5.3 The car-steering problem......Page 189
A Hermitian matrices and Schur complements......Page 195
References......Page 199
Index......Page 210