Transforms for continuous time system modeling

ABSTRACTRecursive identification of a physically parameterized model of a single link flexible robot arm is considered. The parameters are estimated using a recursive prediction error method applied to a linear continuous time model structure and discrete time data. The algorithm is applied to real data from an industrial robot, and three important parameters are identified using only measurements of the motor angle. The experiments show that the particular parameter that is estimated will have big influence on the algorithm behavior.

A general framework for modeling of a time-varying continuous-time SISO system from its sampled input and output while retaining the system parameters with their physical interpretation is presented. The theory can be specialized to the Poisson moment functional approach, the integrated sampling approach, the instantaneous sampling approach or the use of state variable filters. In all methods, the initial conditions can be removed by applying an appropriate discrete-time operator. Digital filtering is used to explicitly or implicitly reconstruct the time-derivatives of the sampled continuous-time signals involved. A thorough study of the approximations resulting from converting the continuous-time model to a discrete-time version is presented. It is shown how these errors can be controlled and that system parameter estimates can be obtained with an arbitrary accuracy. The digital filtering approach to integrated sampling and instantaneous sampling exhibit optimal properties among all methods that fit in the general framework. Moreover, using digital filter designs instead of numerical integration allows slower sampling. A maximum likelihood estimator is derived for time-invariant systems in an errors-in-variables stochastic framework. Finally, the theory is verified through simulations.

A higher-order integrator approach is proposed to obtain an approximate discrete-time transfer function for uncertain continuous systems having interval uncertainties. Thanks to simple algebraic operations of this approach, the resulting discrete model is a rational function of the uncertain parameters. The problem of non-linearly coupled coefficients of exponential nature in the exact discrete-time transfer function is therefore circumvented. Furthermore, interval structure of the uncertain continuous-time system is preserved in the resulting discrete model by using this approach. Formulas to obtain the lower and upper bounds for the discrete interval system are derived, so that existing robust results in the discrete-time domain can be easily applied to the discretized system. Digital simulation and design for the continuous-time interval plant can then be performed based on the obtained discrete-time interval model.

The paper concerns stability analysis of hybrid systems. The main contribution is the proposed approach of discrete-time modeling of continuous-time hybrid systems. Stability analysis and robustness properties are derived from linear Lyapunov theory in discrete time. The method is especially suited for time-switched systems but can also incorporate more complex switch structures. It is also shown how periodic switch sequences fit into the analysis. The resulting problem formulations are given as linear matrix inequalities (LMIs). As an alternative to the S-procedure it is shown how to reduce conservativeness with a reduction of the LMIs state dimension when the switch regions are hyper planes. An example shows that the number of LMIs and LMI variables are reduced compared to a continuous-time approach.

Choice of Basis Functions for Continuous and Discrete System Modelling

Simulation models of continuous systems are in fact discrete computational models. This is the result of applying numerical methods for integrating the differential equations generally used as mathematical models of continuous systems. Under such conditions the values of all variables in the simulation model are computed for discrete values of the independent variable time , separated by a fixed or variable step , according to the integration method used. Each value of the independent variable time for which the simulation model state changes can be considered as the time point when an event takes place. Such point of view leads to an obvious conclusion: continuous systems simulation can be performed using a discrete event simulator. This approach is very important in those cases when the considered system has a combined (discrete event and continuous) model and the available simulation tool does not provide for continuous systems modeling.

Due to the changes which may occur in their parameters, systems are usually demonstrated by some subsystems for different conditions. This paper employs Modified Petri Nets (MPN) to model theses subsystems and makes it simple to analyze them. In this method, first, the continuous transfer function is converted to a discrete transfer function and then, by MPN, system is modeled and analyzed. All subsystems can be modeled and used in state control or cascade control loops. Here, the focus is mainly on the conception as well as the definitions of the new unified representation model for continuous control systems. Simulation results show that the new method for modeling continuous systems works effectively.

The stability of an inverted pendulum is a textbook example of control. The easiest case is to put the pendulum on a cart and apply feedback force control on it to keep the upright position stable. This paper compares the no delay case (feed‐in‐time control: an anticipatory effect) and the delay differential equation approach. Then we study both continuous and discrete time systems. The main aim of the work is to investigate the behaviour of such systems at the stability boundaries by using numerical simulation. The principal points of interest are how continuous time systems differ from discrete time system at a bifurcation point and how time delay or an anticipatory feed‐in‐time control acts on its behaviour. Other exciting questions are how sampling delay can be taken into consideration and is bifurcation a robust phenomenon.

Neimark-Sacker bifurcation theory is used to analyze a differential equation system. A three dimensionel continuous time system is investigated here. Generally, forward Euler scheme is used for discretization from continuous time system to discrete time system. But, in this study Nonstandard Finite Difference (NSFD) scheme is applied to discrete the continuous time system to discrete time system. After that, we examine topological structure of equation. Then, we give the conditions of local stability of this system around feasible fixed point. After, we show analitically that discretized system undergoes Neimark-Sacker bifurcation when one of the system parameter varies near its critical value. We confirm the existence of Neimark-Sacker bifurcation via explicit Flip and Neimark-Sacker bifurcation criterion. And we determine the direction of bifurcation with the help of center monifold theory and bifurcation theory. We carry out numerical simulation to approve our analitical findings.

In this paper, a decentralized controller is designed for cross coupled MIMO Twin rotor aerodynamic system (TRAS). The decentralized controller design is done by decoupling the MIMO system into two decoupled, a pitch and a yaw system. Then 2-DOF controller is designed using control technique in continuous and discrete time. The performance of these controllers are tested through extensive simulations in MatLAB and results are compared between discrete and continuous time systems.

Sliding Mode Control in Discrete Time Linear Systems

The contributions of this paper are threefold. We first explore the differences in uncertainty representations and control designs between continuous time systems and discrete time systems, particularly the high frequency interpretation does not work for discrete time systems. It is shown that uncertainties appeared in sampled data systems may be highly structured, however they do not fall into any category which is currently available in the literature. Secondly, we define a measure on these uncertainties which confines a neighborhood to the nominal model in terms of frequency content, and then we propose a control design method which guarantees the stability of the closed loop system in the face of uncertainties. Last, we consider a generalization of a classical continuous time approach in representing the multiplicative uncertainty to sampled data systems by using a slightly modified sampler and present a method to calculate a frequency bound on the uncertainty in sampled data systems.

This paper demonstrates the development of a generic nonlinear model for describing the dynamics of an autonomous hybrid system and it is validated by conducting experiments on real-time experimental set up as one of the contribution. A cylindrical three-tank hybrid system is considered for this purpose. Recently, the study of dynamics of hybrid system and its control strategies for performance improvement have become very significant in a perspective where in most of such studies combine the behavior of both continuous and discrete time systems. As a second contribution, such a model is used for the design of an efficient inverse dynamics controller (IDC) for controlling the output variable of the system to the desired set point and also a robust, derivative-free state estimator for predicting all the states of the system in presence of noise and parameter uncertainties. The effectiveness of the controller and estimator is illustrated by conducting simulation studies and experiments with real-time experimental setup.

This chapter presents time-domain analysis of continuous-time systems. It develops representation of signals in terms of impulses. The notions of linearity, time-invariance, causality, stability, memorability, and invertibility are introduced. It has shown that the input-output relationship for linear time-invariant (LTI) continuous systems is described in terms of a convolution integral. The differential equation representation of LTI continuous systems and classical solutions of differential equations are also presented. Next, block-diagram representation of LTI continuous-time systems is introduced. Furthermore, a brief discussion on singularity functions is provided. Finally, the state-space representation of continuous-time LTI systems is described.

This paper presents an idea on how to express equilibrium processes that shall be regarded as infinitely fast within equation-based languages for continuous systems. The expression of such processes can help to solve the corresponding non-linear systems of equations at initialization and during simulation since it enables the use of continuation methods.