Gerhard Jank

Solution and asymptotic behavior of coupled Riccati equations in jump linear systems

A new necessary and sufficient condition for the existence of a positive semidefinite solution of coupled Riccati equations occurring in jump linear systems is derived. By verifying a Riccati inequality it is shown that such a solution exists; in addition two numerical algorithms are given to compute it. An example is given to illustrate the proposed method. >

On global existence of solutions to coupled matrix Riccati equations in closed-loop Nash games

Presents comparison and global existence results for solutions of coupled matrix Riccati differential equations appearing in closed loop Nash games and in mixed H/sub 2//H/sub /spl infin//-type problems. Convergence of solutions is established for the diagonal case, solutions of the corresponding algebraic equations are discussed using numerical examples.

On the solution of discrete-time Markovian jump linear quadratic control problems

Abstract A necessary and sufficient condition for the existence of a positive-semidefinite solution of the coupled algebraic discrete-time Riccati-like equation occurring in Markovian jump control problems is derived. By verifying a simple matrix inequality, it is shown that such a solution exists and can be obtained as a limit of a monotonic sequence. This leads to a straightforward numerical algorithm for the computation of the solution. An example is given to illustrate the proposed method.

Meromorphe funktionen, die mit ihrer ersten und zweiten ableitung einen endlichen wert teilen

Uber diese Arbeit wurde auf einem “Tag der Funktionenthcorie” am 15. und 16.6.1984 an der RWTH Aachen vom zweiten Verfasser berichtet. Recently E. Mues and N. Steinmetz and independently G. G. Gundersen proved that if a nonconstant meromorphic function f shares two different non-zero values a and b (counting multiplicities) with its first derivative, then f′=f.ln this paper the following theorem is proved: If a nonconstant meromorphic function f shares a finite non-zero value (counting multiplicities) with f′ and f″ then f′ = f.In the case of an entire function the conditions can be weakened considerably.

Existence and comparison theorems for algebraic Riccati equations and Riccati differential and difference equations

We present comparison and global existence theorems for solutions of generalized matrix Riccati differential and difference equations. Moreover we obtain existence and comparison results for the maximal solutions of the corresponding generalized algebraic Riccati equations. For the symplectic matrix Riccati differential equation we derive sufficient conditions ensuring the global existence of the solutions of the corresponding initial value problems.

Necessary conditions for constant solutions of coupled Riccati equations in Nash games

Abstract The asymptotic behaviour of solutions as t → ∞ for coupled matrix Riccati equations occurring in open-loop linear-quadratic Nash games is studied in this paper. A general formula representing all possible solutions is given. Necessary conditions for constant real solutions are derived and an estimate for the rate of convergence is obtained. Two examples illustrate these results.

Generalized Riccati difference and differential equations

Abstract In this paper we investigate generalized Riccati differential and difference equations obtained from standard Riccati equations by adding a semidefinite perturbation term. For such equations we give results on the monotonic dependence of the solutions on the coefficients and initial values as well as results on convergence of solutions.

Existence and uniqueness of open-loop Stackelberg equilibria in linear-quadratic differential games

We present existence and uniqueness results for a hierarchical or Stackelberg equilibrium in a two-player differential game with open-loop information structure. There is a known convexity condition ensuring the existence of a Stackelberg equilibrium, which was derived by Simaan and Cruz (Ref. 1). This condition applies to games with a rather nonconflicting structure of their cost criteria. By another approach, we obtain here new sufficient existence conditions for an open-loop equilibrium in terms of the solvability of a terminal-value problem of two symmetric Riccati differential equations and a coupled system of Riccati matrix differential equations. The latter coupled system appears also in the necessary conditions, but contrary to the above as a boundary-value problem. In case that the convexity condition holds, both symmetric equations are of standard type and admit globally a positive-semidefinite solution. But the conditions apply also to more conflicting situations. Then, the corresponding Riccati differential equations may be of H∞-type. We obtain also different uniqueness conditions using a Lyapunov-type approach. The case of time-invariant parameters is discussed in more detail and we present a numerical example.

An LPV Modeling and Identification Approach to Leakage Detection in High Pressure Natural Gas Transportation Networks

In this paper a new approach to gas leakage detection in high pressure natural gas transportation networks is proposed. The pipeline is modelled as a Linear Parameter Varying (LPV) System driven by the source node massflow with the gas inventory variation in the pipe (linepack variation, proportional to the pressure variation) as the scheduling parameter. The massflow at the offtake node is taken as the system output. The system is identified by the Successive Approximations LPV System Subspace Identification Algorithm which is also described in this paper. The leakage is detected using a Kalman filter where the fault is treated as an augmented state. Given that the gas linepack can be estimated from the massflow balance equation, a differential method is proposed to improve the leakage detector effectiveness. A small section of a gas pipeline crossing Portugal in the direction South to North is used as a case study. LPV models are identified from normal operational data and their accuracy is analyzed. The proposed LPV Kalman filter based methods are compared with a standard mass balance method in a simulated 10% leakage detection scenario. The Differential Kalman Filter method proved to be highly efficient.

Disturbance attenuation of linear quadratic OL-Nash games on repetitive processes with smoothing on the gas dynamics

This article presents necessary and sufficient results for existence and uniqueness of an equilibrium of a N-player disturbed Nash game with quadratic performance criteria and an affine repetitive process with smoothing describing the two dimensional 2D-system dynamics, under open loop information pattern. The gas dynamics in a single pipeline is modelled in this theoretical framework, and an algorithm withdrawn from the exposed procedure to calculate the equilibrium point is applied to a simple network example.