Uncertain LTI Systems Research Articles

Exponentially stable adaptive optimal control of uncertain LTI systems

SummaryA novel method of an adaptive linear quadratic (LQ) regulation of uncertain continuous linear time‐invariant systems is proposed. Such an approach is based on the direct self‐tuning regulators design framework and the exponentially stable adaptive control technique developed earlier by the authors. Unlike the known solutions, a procedure is proposed to obtain a non‐overparametrized regression equation (RE) with respect to the unknown controller parameters from an initial RE of the LQ‐based reference tracking control system. On the basis of such result, an adaptive law is proposed, which under mild regressor finite excitation condition provides monotonous convergence of the LQ‐controller parameters to an adjustable set of their true values, which bound is defined only by the machine precision. Using the Lyapunov‐based analysis, it is proved that the mentioned law guarantees the exponential stability of the closed‐loop adaptive optimal control system. The simulation examples are provided to validate the theoretical contributions.

Sliding Mode Controllers Design based on Control Lyapunov functions for uncertain LTI systems

In this paper, a design method of sliding controllers based on control Lyapunov functions for uncertain LTI systems is proposed. Systems with matched non-vanishing disturbances, unmatched uncertainties and uncertain control coefficient matrix are considered. It is shown that whenever a robust nonlinear controller exists that renders the system quadratically stabilizable, in the absence of matched non vanishing disturbances, there is also an appropriate First-Order Sliding-Mode controller stabilizing the origin, that also renders the equilibrium point robustly stable in presence of perturbations. This shows that Sliding-Mode controllers are not more restrictive than other controllers, at least for the class of uncertain systems considered. Moreover, the design based on control Lyapunov functions provides a natural sliding surface, having an asymptotically stable sliding dynamics, so that with the corresponding sliding mode controller ensures global asymptotic stability of the uncertain system. The efficiency of the proposed SMC control is validated experimentally on a Furuta pendulum benchmark.

Robust design for 3-DoF anti-windup framework based on QFT

On the basis of the fact that all signals in the practical system are always bounded, this paper proposes a practical QFT-based approach for the synthesis of anti-windup compensation for uncertain LTI systems. Meanwhile, a modified version of the circle criterion is developed for saturated systems with parametric uncertainty to reduce the conservativeness of the anti-windup compensator design within the non-interfering anti-windup framework. In the context of QFT, it means enlarging the allowed zone of the compensator remarkably in the Nichols chart compare to the traditional version of the circle criterion. As a result, the less conservative stability boundary provides more room for the compensator synthesis, making the performance improvement possible. Numerical examples demonstrate the effectiveness and the performance improvement as expected of the proposed technique based on QFT for the anti-windup compensator design.

A robust predictive observer-based integral control law for uncertain LTI systems under external disturbance

The integral control is an effective way to track non-zero constant reference. In this paper, to utilize a robust predictive integral control strategy in the output-feedback case, the control requirement and the goals are formulated in a suitable form. Then, by introducing a novel matrix transformation, the controller designing issue is translated into an optimization problem subjected to some constraints. Such constraints are represented in terms of linear matrix inequality (LMI). By solving a minimization problem, the gains of the integral controller are determined and updated at some time instants. Finally, several numerical simulations are given to evaluate the advantages of the suggested robust control method over similar ones. The outcomes confirm the superiority of the proposed idea compared to the existing approach.

Solution to negative-imaginary control problem for uncertain LTI systems with multi-objective performance

This paper addresses the problem of dynamic output feedback controller synthesis in LMI framework for LTI generalized plants with negative-imaginary uncertainties. The proposed technique transforms the controller synthesized nominal closed-loop system into a strongly strict negative-imaginary system and exploits the DC loop gain condition pertaining to negative-imaginary theory to establish internal stability in presence of negative-imaginary uncertainties. Apart from making the closed-loop system robustly stable, the designed output feedback controller also facilitates multi-objective performance of the closed-loop system such as minimizing the H2 and H∞ norms, reducing the peak overshoot, increasing the decay rate, optimizing the settling time, etc. Illustrative examples have been studied to demonstrate the usefulness of the proposed control scheme. A thorough μ-analysis subjected to ‘mixed’ two-block perturbations is included in the examples to show that the designed controller can also facilitate a certain level of robust H∞ performance in presence of stable NI/SNI uncertainties.

Direct Adaptive Optimal Control for Uncertain Continuous-Time LTI Systems Without Persistence of Excitation

This brief presents a novel direct adaptive optimal controller design for uncertain continuous-time linear time-invariant systems. The optimal gain parameter, obtained from the Riccati equation, is continuously estimated without using knowledge of the system dynamics, rather information rich past, and current data along the system trajectory is used for parameter estimation. This approach guarantees parameter convergence to the close neighborhood of the optimal controller by relaxing the restrictive persistence of excitation condition, typically required for achieving parameter convergence in approximate/adaptive optimal control methods. A Lyapunov-based analysis establishes the uniformly ultimately bounded stability of the designed controller. Further a simulation example demonstrates the efficacy of the proposed result.

Discrete‐time higher‐order sliding mode control of systems with unmatched uncertainty

SummaryThe research on discrete‐time higher‐order sliding mode has received a considerable attention recently. Systems with unmatched uncertainties are common in practice; however, the existing discrete‐time higher‐order sliding mode control algorithms are designed considering only matched uncertainty. This paper proposes a technique to design discrete‐time higher‐order sliding mode control for an uncertain LTI system in the presence of unmatched uncertainty. The proposed technique is numerically simulated and experimentally validated on an electromechanical rectilinear plant. Various experiments are conducted considering the several operational conditions of electromechanical systems in industries to verify the performance of the proposed controller.

Discrete higher order sliding mode: concept to validation

A formal definition and design methodology for discrete-time higher-order sliding mode (DHOSM) based on finite difference method is proposed. A generalised reaching law is proposed to achieve r-order sliding mode in discrete time, and its behaviour and stability are analysed. The proposed reaching law is used to design a higher-order sliding mode control for an uncertain LTI system. The concept is experimentally validated on a real-time rectilinear plant and is compared with the existing algorithms on the basis of sensitivity function and ( p , q r ) -continuity. The results and comparison show that the proposed sliding algorithm has superior disturbance rejection capability as well as better continuity. This work is the first attempt to formally define and design DHOSM as opposed to discretisation of continuous time higher-order sliding mode.

Optimal discrete higher‐order sliding mode control of uncertain LTI systems with partial state information

SummaryThe concept of discrete higher‐order sliding mode has received increased attention in the recent literature. This paper presents an optimal discrete higher‐order sliding mode control for an uncertain discrete LTI system using partial state information, which has been missing in literature. A new technique is proposed to design an optimal time‐varying higher‐order sliding surface and control input through the minimization of a quadratic performance index. Moreover, disturbance estimation technique is utilized to modify the control algorithm to reduce the width of the discrete higher‐order sliding mode band. The proposed algorithm is experimentally validated on a rectilinear plant. Copyright © 2017 John Wiley & Sons, Ltd.

Exponential Decay Rate Conditions for Uncertain Linear Systems Using Integral Quadratic Constraints

This technical note develops linear matrix inequality (LMI) conditions to test whether an uncertain linear system is exponentially stable with a given decay rate $\alpha$ . These new $\alpha$ - exponential stability tests are derived for an uncertain system described by an interconnection of a nominal linear time-invariant system and a “troublesome” perturbation. The perturbation can contain uncertain parameters, time delays, or nonlinearities. This technical note presents two key contributions. First, $\alpha$ - exponential stability of the uncertain LTI system is shown to be equivalent to (internal) linear stability of a related scaled system. This enables derivation of $\alpha$ - exponential stability tests from linear stability tests using integral quadratic constraints (IQCs). This connection requires IQCs to be constructed for a scaled perturbation operator. The second contribution is a list of IQCs derived for the scaled perturbation using the detailed structure of the original perturbation. Finally, connections between the proposed approach and related work are discussed.

Robust state-feedback control of uncertain LPV systems: An LMI-based approach

In this paper, the problem of designing an LPV state-feedback controller for uncertain LPV systems that can guarantee some desired bounds on the H∞ and the H2 performances and that satisfies some desired constraints on the closed-loop poles location is considered. In the proposed approach, the vector of varying parameters is used to schedule between uncertain LTI systems. The resulting idea consists in using a double-layer polytopic description so as to take into account both the variability due to the parameter vector and the uncertainty. The first polytopic layer manages the varying parameter and is used to obtain the vertex uncertain systems, where the vertex controllers are designed. The second polytopic layer is built at each vertex system so as to take into account the model uncertainties and add robustness into the design step. Under some assumptions, the problem reduces to finding a solution to a finite number of LMIs, a problem for which efficient solvers are available nowadays. The solution to the multiobjective design problem is found both in the case when a single fixed Lyapunov function is used and when multiple parameter-varying Lyapunov functions are used. The validity and performance of the theoretical results are demonstrated through a numerical example.

Control of Uncertain Nonlinear Systems Using an Uncertainty and Disturbance Estimator

In this paper, we extend a recently developed method (Zhong and Rees, 2004, “Control of Uncertain LTI Systems Based on an Uncertainty and Disturbance Estimator,” ASME J. Dyn. Syst., Meas., Control, 126(4), pp. 905–910), for controlling uncertain linear systems using an uncertainty and disturbance estimator, to uncertain nonlinear systems. Further the control proposed in this paper removes the drawback of large initial control underlying the method. The stability of the overall system of the plant and the estimator is proved. The efficacy of the method is illustrated by simulation.

Direct data-driven filter design for uncertain LTI systems with bounded noise

This paper investigates the filter design problem for linear time-invariant dynamic systems when no mathematical model is available, but a set of initial experiments can be performed where also the variable to be estimated is measured. Instead of using the initial experimental data to identify a model on the basis of which a filter is designed, these data are used to directly design a filter. Assuming norm-bounded disturbances and noises, a Set Membership formulation is followed. For classes of filters with exponentially decaying impulse response, approximating sets are determined that guarantee to contain all the solutions to the optimal filtering problem, where the aim is the minimization of the induced norm from disturbances to the estimation error. A method is proposed for designing almost-optimal linear filters with finite impulse response, whose worst-case filtering error is at most twice the lowest achievable one. In the H ∞ SISO case, an efficient technique is presented, that allows the evaluation of bounds on the guaranteed worst-case filtering error of the designed filter. Numerical examples illustrate the effectiveness of the proposed solution.

Robust analysis of Demeter benchmark via quadratic separation

AbstractThe purpose of this paper is to give numerical illustrations of recent results obtained in the area of robust analysis of uncertain parametric LTI systems. The idea is to highlight theoretical as well as numerical issues pertaining to real life applications of sophisticated robust analysis of spaceborne control systems. The framework of quadratic separation applied to interconnected implicit linear transformations and uncertain operators as defined in Peaucelle et al. (2007) is renewed. Improved robust stability tests involving more complex parameter-dependent Lyapunov functions are applied on a benchmark from the space industry. It is based on the problem of attitude control of a flexible satellite developed at CNES and it is used to analyze the relevance as well as some shortcomings of the proposed approach.

Explicit Model Predictive Control for Systems with Linear Parameter-Varying State Transition Matrix

In this paper we propose a closed-loop min-max MPC algorithm based on dynamic programming, to compute explicit control laws for systems with a linear parameter-varying state transition matrix. This enables the controller to exploit parameter information to improve performance compared to a standard robust approach where no uncertainty knowledge is used, while keeping the benefits of fast online computations. The off-line computational burden is similar to what is required for computing explicit control laws for uncertain or nominal LTI systems. The proposed control strategy is applied to the controlled Hénon map to draw a comparison, in terms of complexity and control performance, with a controller based on a piecewise affine approximation.

Control of Uncertain LTI Systems Based on an Uncertainty and Disturbance Estimator

This paper proposes a robust control strategy for uncertain LTI systems. The strategy is based on an uncertainty and disturbance estimator (UDE). It brings similar performance as the time-delay control (TDC). The advantages over TDC are: (i) no delay is introduced into the system; (ii) there are no oscillations in the control signal; and (iii) there is no need of measuring the derivatives of the state vector. The robust stability of LTI-SISO systems is analyzed, and simulations are given to show the effectiveness of the UDE-based control with a comparison made with TDC.

Necessary and sufficient conditions for robust identification of uncertain LTI systems

Robust identification of uncertain systems arises whenever a chosen family of models does not completely describe reality. In these situations the issue of unmodeled dynamics gains significance in addition to random measurement noise. To deal with such mixed stochastic/deterministic settings we introduce a novel notion for robust consistency, which requires that the expectation (with respect to noise) of the worst-case (with respect to unmodeled dynamics) identification error asymptotically approach zero. It turns out that this notion leads to transparent necessary and sufficient conditions. We show that robust consistency holds, if and only if there is an instrument-input-pair capable of annihilating the residual error as well as stochastic noise. An extension of this result to the well-known “bounded but unknown” noise model shows that if we were to remove a set of Lebesgue measure zero, the error bound asymptotically approaches zero.

Comments on “Robust Iterative Learning Control Design is Straightforward for Uncertain LTI Systems Satisfying the Robust Performance Condition”

For original paper see ibid., vol.48, p.101-6 (2003). The article comments on the result of the above paper. The authors of the original paper showed that Theorem 1 is only the sufficient condition of the convergence in the sense of the L/sub 2/-norm. In this note, Theorem 1 is proved to be the not only sufficient but also necessary condition using some mathematical manipulation.

Robust iterative learning control design is straightforward for uncertain LTI systems satisfying the robust performance condition

This note demonstrates that the design of a robust iterative learning control is straightforward for uncertain linear time-invariant systems satisfying the robust performance condition. It is shown that once a controller is designed to satisfy the well-known robust performance condition, a convergent updating rule involving the performance weighting function can be directly obtained. It is also shown that for a particular choice of this weighting function, one can achieve a perfect tracking. In the case where this choice is not allowable, a sufficient condition ensuring that the least upper bound of the /spl Lscr//sub 2/-norm of the final tracking error is less than the least upper bound of the /spl Lscr//sub 2/-norm of the initial tracking error is provided. This sufficient condition also guarantees that the infinity-norm of the final tracking error is less than the infinity-norm of the initial tracking error.

LMI-BASED INTEGRATION OF ROBUST H∞-CONTROL AND RFD FOR LTI SYSTEMS

In this contribution, problems related to the integrated design of observer-based robust H∞ controller and robust fault detection (RFD) system is studied for uncertain LTI systems with both modelling errors and unknown input. We first propose to use an ideal solution, which is in fact an optimization solution of fault detection filter achieved on the assumption of the system model is perfectly known (i.e. without modelling errors), as the reference model of robust fault detection filter design for uncertain systems. The integrated design can then be formulated as a two-objective H∞–optimization problem which is solved using LMI techniques. The basic idea behind our study is that the robust H∞ controller share a common observer with the fault detection filter; by introducing a reference residual model, the feedback controller and robust fault detection filter can be designed together via solving a two-objective H∞–optimization problem; finally an LMI approach to integrated design is developed. An example is also presented to illustrate the proposed approach.