Linear Time-invariant SISO Systems Research Articles

Model Evaluation for Robust Tracking Under Unknown Upper Bounds on Perturbations and Measurement Noise

This technical note addresses a problem of model evaluation in the framework of robust steady-state tracking in the l1 setup. The upper bounds on measurement noise, exogenous disturbance and coprime factor perturbations in a discrete-time linear time-invariant SISO system are assumed to be unknown. For a given nominal model and a given tracking controller, three different model validation tests are considered and the complexity of computing upper bounds on the worst-case steady-state tracking error, consistent with these tests, is analysed.

極配置―安定半径改善問題の一解法

This paper presents a robust controller design method for linear time-invariant SISO systems based on an optimization approach. The task is to enlarge real stability radius to counter plant parameter perturbations by using extra degree of freedom of the controller parameters in the pole placement scheme. The formula for the real stability radius is known as an infimum of a certain function and that makes its gradients discontinuous. It is shown that by some manipulations, the considered problem can be converted to a nonlinear optimization problem, to which a gradient-based optimization method becomes applicable. The case of complex stability radius is also considered, which is less sharp as a robust stability index than the real counterpart, but has a much simpler form. Numerical examples for both cases are presented to show that they actually work, and some comparisons are discussed, leading to a suggestion on the effective uses of these two stability radius computation methods.

Bond graph formulation of an optimal control problem for linear time invariant systems

A recent communication has proposed a conjectural procedure for representing a category of optimal control problems in bond graph language [W. Marquis-Favre, B. Chereji, D. Thomasset, S. Scavarda, Bond graph representation of an optimal control problem: the dc motor example, in: ICBGM' 05 International Conference of Bond Graph Modelling and Simulation, New Orleans, USA, January 23-27, 2005, pp. 239-244]. This paper aims at providing a fundamental theory for proving the effectiveness of this procedure. The class of problem that the procedure can deal with has been extended. Its application was formerly restricted to linear time invariant SISO system. The systems considered now are linear time invariant MIMO systems. The optimization objective is the minimization of dissipation and input. The developments concerning the optimal control problem are based on the Pontryagin maximum principle and the proof of the effectiveness of the procedure makes a broad use of the port-Hamiltonian concept. As a result, the bond graph representation of the given optimization problem enables the analytical system, which provides the optimal solution, to be derived. The work presented in this paper is the first step in research with perspectives towards formulating dynamic optimization problems in bond graph and, towards coupling this formulation with a sizing methodology using bond graph language and a state-space inverse model approach. This sizing methodology, however, is not the topic of this paper and thus is not presented here.

High-Gain Robust Control

We consider linear time-invariant SISO systems with parametric or non-parametric uncertainty in the transfer function. It is assumed that the system is robustly minimum-phase; we obtain necessary and sufficient conditions for the system to be robustly stabilisable by a high-gain controller and explicitly calculate the gain margin for interval systems. Thus, such systems need neither identification nor adaptation to be robustly stabilised and to attenuate disturbances. The case of unmodelled dynamics is more sophisticated; some models admit high-gain stabilisation while others do not.

Physical Interpretation of Zero Dynamics for Linear SISO Control Systems

A graphical method to study the zero dynamics of linear time-invariant SISO systems is presented based on bond graph representation and using causality and bicausality manipulations on the bond graph. It is shown that in the construction of the inverse bond graph which represents the inverse system model, bicausal bonds have some properties that allow to analyse independently the loops of subsystems in the inverse bond graph. The zeros of the direct system are studied as the poles of the inverse system using causal loops analysis.

Inversion of linear time-invariant SISO systems modelled by bond graph

Two different algorithms for deriving the inverse system state equations from a bond graph model are presented. The first method is based on the causal path analysis and it leads to the full-order inverse system. The second method which is procedural relies on the concept of bicausality and the state equations obtained from the resulting algorithm are those of a reduced inverse system. In both cases, some illustrative examples are given. The advantages of these methods are that they can easily be implemented in software using an algorithm of causality assignment and a procedure of causal paths analysis in a bond graph. Formal calculations of matrices are then avoided in the first case and also formal state transformations are not necessary to obtain the reduced inverse system in the second case.

Indirect Methods of Parameter Estimation of Discrete-Time Models

This paper considers the problem of estimating the discrete-time model (DTM) of a linear time-invariant SISO system from samples of its input-output data. It presents the results of numerical experiments with three different methods. One is the standard method in which the model is estimated directly in the shift operator (z or q) form. The other two are indirect in the sense that an intermediate model having close connections with the original continuous time (CT) version is first estimated and then transformed into the shift operator form. The delta operator form and the integral equation form realized via block pulse functions (BPF) are chosen here for estimation in the indirect methods. The results clearly indicate the superiority of the indirect methods especially in the presence of noise and with high sampling rates.