Time-varying discrete linear systems: input-output operators; Riccati equations; disturbance attenuation.

*(English)*Zbl 0799.93035
Operator Theory: Advances and Applications. 68. Basel: Birkhäuser. 228 p. (1994).

The monograph is written as time-variant version of \(H^ \infty\) control theory which has been constructed for time-invariant case. The theory of time-variant discrete linear systems is systemized in this monograph in order to obtain the solution of disturbance attenuation problem. The monograph consists of five chapters and two appendices. Chapter 0. The motivation of writing this monograph is described. It is stressed that since the solution of robust stabilization reduces to solving disturbance attenuation problem, the latter problem occupies the central part of the former problem. It is also stated that Kalman-Szegö-Poppov-Yakubovich systems in \(J\) form must be investigated in order to solve the disturbance attenuation problem.

Chapter 1. Some basic notions on discrete-time linear systems with time- varying coefficients are introduced, where attention is focused on exponentially stable evolutions and exponentially dichotomic evolutions. Important results on Lyapunov equation are described in order to examine time-variant linear systems. Uniform controllability, stability, uniform observability and detectability are also described.

Chapter 2. An operator-based approach of input-output properties of discrete time-vriant linear systems is described. Node, a bounded operator between \(\ell^ 2(z)\) space of input sequence and \(\ell^ 2(z)\) space of output sequence, always plays a central role in solving different tasks of system compensation. Hankel operators, Toeplitz operators and structured stability radius are introduced. After defining Hankel singular values, all-pass nodes, \(J\)-unitary node as well as \(\gamma\)-contracting node are explained.

Chapter 3. The theory described in this chapter is the ground for the next chapter, which is devoted entirely to the disturbance attenuation problem. It is shown by defining Popov triplets that discrete-time Riccati equation is equivalent to the extended Kalman-Szegö-Popov- Yakubovich system. It is also shown that the discrete-time Riccati system has a stabilizing solution iff the Hamiltonian pair is dichotomic and disconjugate. By proving the positivity theorem, the contracting node problem is solved. It is shown by applying positivity theory that the closed loop system with good properties with respect to internal uncertainties can be constructed.

Chapter 4. In the first, problem formulation and basic assumptions are stated. It is shown that if stabilizing compensator exists the Kalman- Szegö-Popov-Yakubovich system has a stabilizing solution. Then, a solution to the disturbance attenuation problem is effectively constructed. The disturbance attenuation problem is successively reduced to simpler situations for which the solution can be easily constructed. The simplest case is that of the disturbance estimation problem, and a little more complicated situation is represented by the disturbance feedforward problem, or the output estimation problem.

Appendix A concerns with discrete-time stochastic control, while Appendix B deals with almost periodic discrete-time systems.

Chapter 1. Some basic notions on discrete-time linear systems with time- varying coefficients are introduced, where attention is focused on exponentially stable evolutions and exponentially dichotomic evolutions. Important results on Lyapunov equation are described in order to examine time-variant linear systems. Uniform controllability, stability, uniform observability and detectability are also described.

Chapter 2. An operator-based approach of input-output properties of discrete time-vriant linear systems is described. Node, a bounded operator between \(\ell^ 2(z)\) space of input sequence and \(\ell^ 2(z)\) space of output sequence, always plays a central role in solving different tasks of system compensation. Hankel operators, Toeplitz operators and structured stability radius are introduced. After defining Hankel singular values, all-pass nodes, \(J\)-unitary node as well as \(\gamma\)-contracting node are explained.

Chapter 3. The theory described in this chapter is the ground for the next chapter, which is devoted entirely to the disturbance attenuation problem. It is shown by defining Popov triplets that discrete-time Riccati equation is equivalent to the extended Kalman-Szegö-Popov- Yakubovich system. It is also shown that the discrete-time Riccati system has a stabilizing solution iff the Hamiltonian pair is dichotomic and disconjugate. By proving the positivity theorem, the contracting node problem is solved. It is shown by applying positivity theory that the closed loop system with good properties with respect to internal uncertainties can be constructed.

Chapter 4. In the first, problem formulation and basic assumptions are stated. It is shown that if stabilizing compensator exists the Kalman- Szegö-Popov-Yakubovich system has a stabilizing solution. Then, a solution to the disturbance attenuation problem is effectively constructed. The disturbance attenuation problem is successively reduced to simpler situations for which the solution can be easily constructed. The simplest case is that of the disturbance estimation problem, and a little more complicated situation is represented by the disturbance feedforward problem, or the output estimation problem.

Appendix A concerns with discrete-time stochastic control, while Appendix B deals with almost periodic discrete-time systems.

Reviewer: K.Ichikawa (Tokyo)

##### MSC:

93C99 | Model systems in control theory |

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

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

93C05 | Linear systems in control theory |

93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |