Robust d-stability analysis for linear uncertain systems by unified approach using the delta operator

In this paper, the problem of robust performance analysis of interconnected uncertain systems with hierarchical structure is investigated. The computational load associated with such problems does not allow a direct application of robustness analysis usual tools. To overcome this difficulty, we exploit the hierarchical structure of the problem and propose an algorithm to perform robustness analysis using integral quadratic constraint “propagation” along the hierarchical structure. This algorithm allows us to establish a tradeoff between computation time required to perform the analysis and the conservatism of the obtained results. Furthermore, it is possible to perform parallel computation using the proposed algorithm.

Aiming at high-speed signal sampling system with internal parameter perturbation and external disturbance, a novel delta operator saturated variable rate reaching law based on sigmoid function is presented, and soft sliding mode controller design is addressed for delta operator systems with uncertainties in the paper. The structural feature of uncertain delta operator system is concerned. Soft sliding mode controller design for continuous-time system and discrete-time system can be converted to the case of delta operator system. The stability of uncertain delta operator soft sliding mode control (SMC) system is guaranteed by Lyapunov stability theory. Soft sliding mode controller design for uncertain delta operator systems is achieved by the proposed reaching law under a certain matching condition. It is illustrated that delta operator soft SMC system by the proposed reaching law in a simulation example can not only reach the scope of the switching plane in limited time, but also reduce system chattering effectively. The system on stability and process steady is better than that of delta operator exponential reaching law and delta operator variable rate reaching law. At the same time, the system has a good completely robustness compared with delta operator statured reaching law.

We consider the problem of robust performance analysis when some of the exogenous inputs acting on the system are assumed to be fixed and known, while others are unknown but bounded. In the special case when all the exogenous inputs are fixed and known, we extend previous results for the unstructured case to the general MIMO case where the perturbation is allowed to be structured. In this case we present both upper and lower bounds for the measure of robust performance. In the more general case, where both fixed and worst case inputs act on the system, we propose a new sufficient condition for robust performance. All these conditions are readily computable, and yield much less conservative results than one would obtain from applying standard worst-case analysis methods.

The state feedback design for singularly perturbed systems described in Delta operator is considered. The composite state feedback controller for slow and fast subsystems is designed by using the direct method. The obtained results can bring previous conclusions of continuous and discrete time systems into the unified Delta framework. A simulation example is presented to demonstrate the validity and efficiency of the design.

This paper investigates the robust stabilization of uncertain time-delayed systems via delta operator approach. By introducing model transportation, the linear delta operator system with time-varying delays is reformulated into an interconnected system for which the uncertainties can become easy to deal with. Based on a two-term approximation and scaled small gain theorem, a new delay-dependent sufficient condition of state feed-back stabilization for an uncertain delta operator time-delayed system is established by using a novel Lyapunov-Krasovskii functional. The condition obtained can unify some previously suggested relevant methods seen in literature for achieving asymptotic stabilization of both continuous and discrete systems into the delta operator framework. Numerical examples presented explicitly demonstrate the advantages and effectiveness of the proposed method.

A delta operator based Discrete Sliding Mode Fuzzy Controller (DSMFC) for dc-dc buck Converter is described. The discrete systems by means of shift operator turns into mathematically disordered at an unexpected sampling rates. In order to face this difficulty, the delta operator is introduced. The delta operator can convert continuous-time system into discrete-time system and vice-versa. The main aim is to increase the rate of response, develop efficiency, get excellent Robustness and chattering phenomenon inhibition. This method is easy to implement and the overall system is efficient and cost-effective. During discretization, to circumvent any data damage, quick sampling has to be done which can be achieved with the help of delta operator because of its well defined mathematical properties even at rapid sampling.

This paper combines a Smith Predictor with an approach for graphically determining all Proportional-Integral-Derivative (PID) controllers in either the Continuous-Time (CT) or Discrete-Time (DT) domains that meet Gain Margin (GM) and Phase Margin (PM) specifications. Smith Predictors (SP) are often used when designing controllers for systems that exhibit "relatively" large delays that may cause the system's stability and/or performance to deteriorate. The PID controller gains, namely Proportional gain (K <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> ), Integral gain (K <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> ) and Derivative gain (K <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</inf> ), will be determined graphically using only the frequency response of the system's components, i.e., plant with delay and SP structure. The inclusion of a SP along with a PID controller can significantly improve stability margins and performance when compared to relying solely on a PID controller. The improvement can be observed even if there is a mismatch between the actual process and its corresponding SP model. By using the delta operator, the procedure outlined applies to both CT and DT systems, hence, the unified approach designation. Additionally, this procedure will ensure meeting predesign requirements in terms of gain and phase margins. The stability boundaries of the PID controller gains will be determined graphically where within the boundaries, nominal stability is guaranteed and stability margins requirements are met.

The problem of designing H∞ filters for delta operator formulated systems with low sensitivity to filter coefficient variations is investigated. The delta operator provides a theoretically unified formulation of continuous-time and discrete-time systems and also has the advantage of better numerical properties at high sampling rates. In addition to the standard H∞ criterion, the H∞ norm of the sensitivity function is introduced in order to improve the designed filter's insensitivity to the filter coefficient variations. Finsler's Lemma is used to derive novel sufficient conditions which are adapted to treat this multiplicative objective optimisation problem in a potentially less conservative framework. Linear matrix inequality conditions are obtained for the existence of admissible filters with respect to additive/multiplicative coefficient variations based on two different types of sensitivity measures. Finally, the effectiveness of the proposed method is illustrated by a numerical example. It is shown that the delta operator approach offers better coefficient sensitivity than the traditional shift operator approach when the sampling rate is high.

This paper focuses on uncertain pendulum-like systems subject to norm constraint structured uncertainties, and addresses the robust gradient-like behaviour analysis and synthesis problems for such systems. By applying the structured singular value of an operator to structured uncertainty, the results on robust analysis for the gradient-like behaviour of uncertain pendulum-like systems are obtained. Based on the analysis results, using KYP lemma, a dynamic output feedback controller for the closed loop system guaranteeing gradient-like behaviour is designed and the controller parameters are explicitly expressed by a set of feasible solutions of corresponding linear matrix inequalities. Finally two numerical examples are presented to demonstrate the applicability and validity of the proposed approach.

In this paper, we investigate the problem of robust exponential stability analysis for uncertain linear systems with discrete interval time-varying delay, distributed time-varying delay and nonlinear perturbations. Based on constructing an augmented Lyapunov–Krasovskii functional with some parameter, decomposition technique of coefficient matrix, mixed model transformation with Leibniz–Newton formula and utilization of zero equations, new delay-range-dependent robust exponential stability criteria are derived in terms of linear matrix inequalities (LMIs). Numerical examples are given to show the superiority of our results to those in the literature.

This paper will present controller designs for two-time scale systems with a unified approach using delta operators that guarantee stability with a known H<SUP>(infinity</SUP> ) norm bound and also allow the incorporation of reliability with respect to failures in some control channels. Here, sensor (or measurement) outages case will be studied. By such a unified approach, the procedure and result for the solution are shown simpler and more accurate than those of discrete system using shift operators.

Robust analysis and control for a class of uncertain nonlinear discrete-time systems

In our previous paper 4) -12), we have studied “Theory of Invariance” for nonlinear continuous-time systems with their inputs appearing linearly. The aim of this paper is to derive necessary and sufficient conditions that inputs never effect outputs (so called conditions of “Invariance”) for nonlinear discrete-time and continuous-time systems without the assumption of linearity in inputs. Applying these conditions to the decoupling problem, we obtained necessary and sufficient conditions which yield decoupling control laws for both the continuous-time systems and the discrete-time systems. The conditions of decoupling given by van der Schaft 13) and Grizzle 14) via geometric approach are different in forms from ours. But, it seems difficult to derive control laws from their conditions because the integration of partial deferential equations is needed there.

The authors consider the problem of robust stability analysis and stabilization of uncertain two-dimensional (2-D) discrete systems described by the Fornasini-Marchesini second local state-space (LSS) model. Sufficient conditions for the uncertain 2-D discrete systems to be robustly stable are given in terms of linear matrix inequalities (LMI's). The LMI approach is also introduced to solve the robust stabilization problem of the uncertain 2-D systems via static-state feedback. Furthermore, sufficient conditions for the 2-D uncertain systems to be free of overflow oscillations under a saturation arithmetic are established.

This paper presents an algebraic theory for analysis and design of linear multivariable feedback systems. The theory is developed in an algebraic setting sufficiently general to include, as special cases, continuous and discrete time systems, both lumped and distributed. Designs are implemented by construction of a controller with two vector inputs and one vector output. Use of controllers of this type is shown to generate convenient stability results, and convenient global parametrizations of all I/O maps and all disturbance-to-output maps achievable, for a given plant, by a stabilizing compensator. These parametrizations are then used to show that any such I/O map and any such disturbance-to-output map may be simultaneously realized by choice of an appropriate controller. In the special case of lumped systems, it is shown that the design theory. can be reduced to manipulations involving polynomial matrices only. The resulting design procedure is thus shown to be more efficient computationally. Finally, the problem of asymptotically tracking a class of input signals is considered in the general algebraic setting. It is shown that the classical results on asymptotic tracking can be generalized to this setting. Additionally, sufficient conditions for robustness of asymptotic tracking, and robustness of stability are developed.