Robust H ∞ controller design for linear uncertain systems with delayed state and control

In this paper we consider linear time-varying systems subject to norm bounded, one block, structured uncertainties. First, given an unforced system, we find necessary and sufficient conditions for quadratic stability; then, considered a controlled system, we provide necessary and sufficient conditions for quadratic stabilizability both for the state and the output feedback case.

This article concerns the finite-time robust guaranteed cost control problem for a class of linear continuous-time singular systems with norm-bounded uncertainties. In this study, the problem is to design a state feedback controller such that the closed-loop system is finite-time stable, and the closed-loop cost function value is not more than a specified upper bound for all admissible uncertainties. By constructing an appropriate Lyapunov function, a sufficient condition for the finite-time robust stability of the system based on linear matrix inequality (LMI) is established. Furthermore, the sufficient condition for the existence of the guaranteed cost controller is formulated in terms of LMIs, which can make the closed-loop uncertain singular system finite-time robust stable. Finally, two numerical examples are given for illustration of the proposed theoretical results.

Quadratic stability for a class of switched singular linear systems is investigated. The convex combination technique of standard switched linear systems is extended to switched singular systems and a sufficient condition for quadratic stability derived. A stabilizing switching law is also designed. For a switched singular system with two subsystems this appears a necessary and sufficient condition for quadratic stability. Also, the switching law realization is discussed relative to avoiding zeno behavior. An illustrative example is given to demonstrate the validity of these new results.

This paper considers a subclass of hybrid systems, which is the discrete-time linear switching systems whose changes in the dynamics are determined by exogenous, uncontrollable events, and resets are present. In the presence of uncertain parameter and non-symmetrical constrained control, based on the results of nominal linear switching systems, a necessary and sufficient condition to test the robustness of a state feedback controller is given. The candidate state feedback controller for this test is that stabilizing nominal discrete-time linear switching systems. Also, the method in this paper can be extended to the case which has state constraints. At last, a numerical example is presented to illustrate the technique.

Discrete-time systems in state space subject to uncertainties in the state and input matrices as well as to finite-energy additive disturbances are considered. No statistical assumptions nor matching conditions are required. Necessary and sufficient conditions for quadratic stability by linear state feedback and output-disturbance attenuation below a prescribed value are derived independently of the uncertainty structure. These conditions are then applied to the case of convex polyhedral uncertainly and the robust synthesis problem is cast into a mathematical programming problem. Additional constraints like robust pole assignment in subregions of the complex plane can easily be fit into the design procedure. Numerical examples are provided.

Robust Gain-Scheduled PSD Controller Design from Educational Perspective

The paper deals with the implementation of optimal controller design for linear dynamic systems. The implementation is carried out with the use of quadratic cost function for both cases of continuous and discrete systems. The mathematical optimal control theory is applied to dynamical systems with considering the form of either vector differential equations in continuous time case, or difference equations in discrete time- or sampled data-case. The bulk of the work provides design techniques for synthesizing the optimal control structure by applying the Pontryagin's Maxi mum principle to linear systems with using Matrix Riccati Equation. The work is focused upon regulator problems; that is problems where the goal is to maintain the system states at zero level. The implemented techniques are tested by the application on fourth order practical example represents the linearized control model of two-reach river pollution System. Also comparative study between the continuous and discrete optimal regulator problems will be presented and discussed from a numerical point-of-view.

In this paper, a combination of integral sliding mode and composite nonlinear feedback method is proposed for fast and accurate robust tracking and model following of uncertain multiple-input multiple-output linear systems with multiple time-delays and external disturbances. The composite nonlinear feedback technique allows following the reference trajectory within a given accuracy and the invariance property of the integral sliding mode method rejects the disturbances and preserves the stability of the closed-loop system. By using the Lyapunov–Krasovskii functional, the conditions for asymptotic stabilization are obtained in the form of linear matrix inequalities. To improve the tracking performance for different reference amplitudes, a new nonlinear function is included in the control law and is optimally tuned using a modified random search algorithm. The selection of sliding surface and the existence of the sliding mode are two significant issues, which have been addressed. This scheme not only guarantees robustness against time-delays and uncertainties, but also avoids chattering phenomenon and reaching phase. Finally, the simulations are provided to verify the theoretical results. The results show that the proposed method leads to great improvement on the tracking error and the control effort compared to the available methods.

This paper is concerned with the method of impulsive controller design for linear systems with multiple-packet transmission. First, the mathematic model of respond closed-loop is built in terms of linear impulsive systems description. Second, on the basis of guaranteeing closed loop systems stability, the design method of the impulsive feedback controller and the design procedure are given. Finally, numerical simulation is provided to demonstrate that asymptotically stability of close-loop linear uncertainly impulsive system with multiple-packet transmission is guaranteed by using the impulsive controller.

Iterative learning control design for linear discrete-time systems with multiple high-order internal models

Controller design for scalar or multivariable systems whose models are unknown or highly complex are frequently based in practice on the use of a highly simplified approximate plant model. In such circumstances it is vital to be able to quantify the degree of uncertainty to be expected from the use of such a model for prediction of closed-loop characteristics. It is shown how frequency-domain design techniques can be simply extended to incorporate information deduced from the observed differences between open-loop plant and the approximate model step response to quantify this uncertainty and, in particular, to guarantee closed-loop stability and tracking of step demands. A modification of this analysis also yields the possibility of bounding the error in prediction of closed-loop transient performance. The approaches are all graphical in nature and are easily implemented in an interactive computer-aided-design mode.

In this article, a “Backward in Time Observer” has been designed for linear systems having time delays in output measurements and differences in the sampling rate of input and output measurements. With the proposed observer we are able to recover the performance of the system with no time delay and an equal sampling rate of input and output measurements. Different observers have been designed for time delayed systems by the researcher, but multi-rate time delayed systems have not been discussed so far, hence our proposed observer is able to cater for both of these problems in the system. The proposed observer is simulated for the seeker system of the missile for verification and illustration. Seeker system of missile is a linear time-invariant system having time delay in output measurement. Furthermore, the output measurement through IR camera of the seeker system has a lower sampling rate than its input signal from the gyro.

The robust H∞ control problem for time-varying uncertain linear systems with time-varying state, and control delays is considered. Based on the Riccati equation approach, we design a robust H∞ linear time-invariant state feedback control law, which stabilizes the system and reduces the effect of the disturbance input on the controlled output to a prescribed level for all admissible uncertainties. A sufficient condition for the existence of the. robust H∞ controller is derived and two synthesis methods are presented for a robust H∞ control law to uncertain time-delay systems. One is to construct a robust H∞ control law by an equivalent linear time-invariant structural description for the sufficient condition and the control law can be obtained by solving an algebraic Riccati equation (ARE). The other method is to transform the sufficient condition into a linear matrix inequality (LMF) problem and the control law can be obtained by solving the LMI. A numerical example is presented to illustrate the obtained results.

By complementarily fusing the robust stabilizability condition, the orthogonal-functions approach (OFA) and the hybrid Taguchi-genetic algorithm (HTGA), an integrative method is proposed in this paper to design the robust-stable and quadratic-optimal static output feedback controller such that i) the linear singular control system with structured parameter uncertainties is regular, impulse-free and asymptotically stable and ii) a quadratic finite-horizon integral performance index for the linear nominal singular control system can be minimized. Based on some essential properties of matrix measures, a new sufficient condition is presented for ensuring that the linear singular system with structured and quadratically-coupled structured parameter uncertainties is regular, impulse free and asymptotically stable. By using the OFA and the robust stabilizability condition, the dynamic-optimization problem for the robust-stable and quadratic-optimal static output feedback control design of the linear uncertain singular system is transformed into a static-constrained-optimization problem represented by the algebraic equations with constraint of robust stabilizability condition; thus greatly simplifying the robust-stable and quadratic-optimal static output feedback control design problem of the linear uncertain singular system. Then, for the static-constrained-optimization problem, the HTGA is employed to find the robust-stable and quadratic-optimal static output feedback controller of the linear uncertain singular control system. One design example of the robust-stable and quadratic-optimal static output feedback controller for a mass-spring-damper mechanical system with structured parameter uncertainties is given to demonstrate the applicability of the proposed integrative approach.

SummaryIn this paper, a set‐membership parity space approach for linear uncertain dynamic systems is proposed. First, a set of parity relations derived from the parity space approach is obtained by means of a transformation derived from the system characteristic polynomial. As a result of this transformation, parity relations can be expressed in regressor form. On the one hand, this facilitates the parameter estimation of those relations using a zonotopic set‐membership algorithm. On the other hand, fault detection is then based on checking, at every sample time, the non‐existence of a parameter value in the parameter uncertainty set such that the model is consistent with all the system measurements. The proposed approach is applied to two examples: a first illustrative case study based on a two‐tank system and a more realistic case study based on the wind turbine fault detection and isolation benchmark in order to evaluate its effectiveness. Copyright © 2014 John Wiley & Sons, Ltd.