Polytopic Parameter Uncertainties Research Articles

Dynamic output feedback model predictive control for asynchronous Markovian jump systems under try‐once‐discard protocol

AbstractThis paper is concerned with the dynamic output feedback robust model predictive control (RMPC) problem for a class of discrete‐time asynchronous Markovian jump systems (MJSs) subject to polytopic parameter uncertainties and hard constraints on states and inputs. A hidden Markov model is employed to describe the asynchronous phenomenon between the detected modes and the system modes. For the purpose of improving the network utilization as well as reducing the transmission burden and avoiding data collisions, the try‐once‐discard (TOD) protocol is deployed in the shared communication network between sensor nodes and the controller node. Considering the difficulty of obtaining the system state in the practice, the dynamic output‐feedback control in the framework of RMPC is adopted, and the worst‐case optimization problem over the infinite moving horizon is formulated for the performance analysis and control synthesis. By means of the Lyapunov‐like function approach, free weighting matrices and inequality techniques such as the ‐procedure are introduced to deal with the non‐convexity caused by couplings between decisive variables and the negative influence resulting from the data orchestration is mitigated. Based on the delicate establishments, a certain upper bound of the objective function is employed to construct an auxiliary optimization problem with solvability to find the desired controllers, and sufficient conditions are obtained to guarantee that the MJSs under the proposed RMPC‐based controllers are mean‐square stable. Finally, an illustrative example is used to demonstrate the validity of the proposed methods.

Experimental validation of an interval observer-based sensor fault detection strategy applied to a biohydrogen production dark fermenter

In this paper, the problem of designing an interval observer applied to a biohydrogen production dark fermenter is addressed, in order to: (i) estimate the glucose and biomass concentrations, and the hydrogen flow rate produced, (ii) reduce the influence of the unknown inlet glucose concentration and the model uncertainties, and (iii) detect the presence of faults in the hydrogen flow rate sensor. Based on literature reports, we propose an interval observer structure, constituted by a Luenberger observer and an interval predictor, for a class of linear systems. The interval observer offers three estimations: the upper and lower estimations provided by the interval predictor, and one estimation given by the Luenberger observer. The upper and lower estimations provide bounds at any instant of time for the state vector of the perturbed/uncertain biohydrogen production bioreactor model from an appropriate initial condition. The design conditions presented in this work are based on control H∞ in combination with pole placement and polytopic parametric uncertainties. The observer design is proposed as a semi-definite optimization problem subject to Linear Matrix Inequalities, which do not depend on the cooperativity property. Based on the proposed observer structure, we also present two fault detection schemes for the same class of uncertain systems to detect the occurrence of sensor faults: one makes use of the logic comparison between the set threshold and the residual produced by the Luenberger observer; the other considers adaptive thresholds for the output signal by using the upper and lower estimates of the interval predictor. Finally, the proposed approaches are validated with experimental data from a biohydrogen production dark fermenter.

Sampled-Data Model Predictive Control

This paper focus on model predictive control (MPC) design in the context of sampled-data control systems with full-state measurements. It is shown that recent results on this area can be successfully generalized to cope with sampled-data MPC. The open-loop plant is subjected to polytopic parameter uncertainty and at sampling times a controlled output variable satisfies a set of convex constraints. A guaranteed H2 performance index with infinity horizon is minimized such as the feedback control preserves asymptotic stability and feasibility. The design conditions are expressed though differential linear matrix inequalities (DLMIs). Continuous-time systems are treated with no kind of discrete-time modeling approximation. Comparisons with classical methods from the literature dealing with continuous-time systems are presented and discussed. Examples are included for illustration.

Iterative Solution of Linear Matrix Inequalities for the Combined Control and Observer Design of Systems with Polytopic Parameter Uncertainty and Stochastic Noise

Most research activities that utilize linear matrix inequality (LMI) techniques are based on the assumption that the separation principle of control and observer synthesis holds. This principle states that the combination of separately designed linear state feedback controllers and linear state observers, which are independently proven to be stable, results in overall stable system dynamics. However, even for linear systems, this property does not necessarily hold if polytopic parameter uncertainty and stochastic noise influence the system’s state and output equations. In this case, the control and observer design needs to be performed simultaneously to guarantee stabilization. However, the loss of the validity of the separation principle leads to nonlinear matrix inequalities instead of LMIs. For those nonlinear inequalities, the current paper proposes an iterative LMI solution procedure. If this algorithm produces a feasible solution, the resulting controller and observer gains ensure robust stability of the closed-loop control system for all possible parameter values. In addition, the proposed optimization criterion leads to a minimization of the sensitivity to stochastic noise so that the actual state trajectories converge as closely as possible to the desired operating point. The efficiency of the proposed solution approach is demonstrated by stabilizing the Zeeman catastrophe machine along the unstable branch of its bifurcation diagram. Additionally, an observer-based tracking control task is embedded into an iterative learning-type control framework.

Robust PD-type iterative learning control for discrete systems with multiple time-delays subjected to polytopic uncertainty and restricted frequency-domain

This paper proposes the PD-type iterative learning control (ILC) for multiple time-delays systems with polytopic parameter uncertainty. Based on repetitive process framework, the system under study is equivalently converted into a class of uncertain repetitive processes with multiple time-delays. This approach accounts for effective inclusion of both time and trial domain objectives and hence some requirements on transient dynamics and trial-to-trial error convergence are incorporated for robust design procedures. Additionally, this approach can easily avoid the need for computation with very large dimensioned matrices as it is required for the lifting approach. Also, the proposed controller is designed with the generalized Kalman-Yakubovich-Popov lemma to ensure the monotonic trial-to-trial error convergence in finite frequency domain. This allows us to reduce the conservatism inherent to entire frequency range approaches since the reference signal spectrum reside in a known frequency range. Moreover, the sufficient conditions for the convergence of the resulting scheme are expressed by linear matrix inequalities and hence they are amenable to effective algorithmic solution. Finally, numerical simulations of different scenarios are presented to illustrate the effectiveness of the proposed method. In particular, to highlight the potential interest in PD-type ILC the robust tracking performance is compared with the results for P and D types of ILC.

Blending based multiple-model adaptive control of multivariable systems with application to lateral vehicle motion control

This paper develops multiple fixed model blending based adaptive parameter identification schemes for multi-input multi-output (MIMO) systems with polytopic parameter uncertainty. The developed identification schemes are proven to be asymptotically stable for uncertain linear time-invariant (LTI) MIMO systems, and is shown to provide fast adaptation for even uncertain linear time-varying (LTV) systems. Furthermore, utilizing the proposed parameter identification schemes, a linear-quadratic (LQ) optimal multiple-model adaptive control (MMAC) scheme is developed for linear MIMO systems with polytopic uncertainties. The proposed MMAC scheme is proven to be asymptotically stable for LTI MIMO systems and applied to tracking control of uncertain lateral vehicle dynamics. A set of simulation test results are presented to verify the stability, effectiveness, and comparison of the proposed MMAC scheme.

Regional Stabilization of Input-Delayed Uncertain Nonlinear Polynomial Systems

This paper addresses the problem of local stabilization of nonlinear polynomial control systems subject to time-varying input delay and polytopic parameter uncertainty. A linear matrix inequality approach based on the Lyapunov–Krasovskii theory is proposed for designing a nonlinear polynomial state feedback controller ensuring the robust local uniform asymptotic stability of the system origin along with an estimate of its region of attraction. Two convex optimization procedures are presented to compute a stabilizing controller ensuring either a maximized set of admissible initial states for given upper bounds on the delay and its variation rate or a maximized lower bound on the maximum admissible input delay considering a given set of admissible initial states. Numerical examples demonstrate the potentials of the proposed stabilization approach.

Robust Receding Horizon Control with Free Control Moves for Polytopic Parametric Uncertainty Systems Subject to Multiple Input Delays

A robust receding horizon control (RHC) with free control moves is applied to polytopic parametric uncertainty systems with multiple input delays and unstable system matrices. A difficulty in the previous robust RHC work is that the free control moves are unsuitable for the system with input time delay, which is overcome in this paper by the design based on the augmented state. As a result, the synthesis of local control based on augmented feedback is given offline to alleviate the online computation burden. The free control moves before the augmented feedback are the online decision variables, which are solved by minimizing a sequence of nonnegative scalars online. The recursive feasibility is guaranteed by adopting the augmented state space equation. By adjusting the robust positively invariant set, the stability of the closed-loop system is guaranteed. Simulation results demonstrate that the proposed algorithm improves the control performance effectively.

Parameter-dependent robust H∞ filtering for uncertain two-dimensional discrete systems in the FM second model

Abstract This paper deals with the problem of robust $H_{\infty }$ filtering for uncertain two-dimensional discrete systems in the Fornasini–Marchesini second model with polytopic parameter uncertainties. Firstly, a new $H_{\infty }$ performance criterion is derived by exploiting a new structure of the parameter-dependent Lyapunov function. Secondly, based on the criterion obtained, a new condition for the existence of a robust $H_{\infty }$ filter that ensures asymptotic stability, and a prescribed $H_{\infty }$ performance level of the filtering error system, for all admissible uncertainties is established in terms of linear matrix inequalities. Finally, two examples are given to illustrate the effectiveness and advantage of the proposed method.

Low-complexity stabilizing PWA controllers for linear systems with parametric uncertainties

Explicit MPC often results in a large number of irregular partitions in the feasible region as the dimension of the system increases and the storage requirement for the control and region parameters often limit its applications. In this paper, we consider a class of discrete-time linear systems with polytopic parametric uncertainties and provide a robust control Lyapunov based synthesis method to obtain robust low-complexity PWA controllers on regular partitionings. By implementing a refinement procedure, we can fit the PWA feedback control law in each regular partitioning based on feasibility of linear programming problems, which preserves stability, constraint satisfaction, and certain performance requirement. Numerical examples will demonstrate the effectiveness of the approach.

A Synthesis Approach to Output Feedback MPC for LPV Model With Bounded Disturbance

The model with polytopic parametric uncertainty and bounded disturbance is controlled by the approach named dynamic OFRMPC (Output Feedback Robust Model Predictive Control). A key knob for control performance and region of attraction for this approach is the selection of Lyapunov matrix. A Lyapunov matrix, which does not have structural restriction, is proposed. In the ICCA (Iterative Cone Complementary Approach), which is invoked in optimizing the control law parameters, the starting up steps are designed as a variant CCA (Cone Complementary Approach). ICCA designs an outer loop, over CCA, for searching the minimum cost bound, while the variant CCA omits the outer loop by adding the cost bound in CCA objective function. This starting up can reduce the computational burden. The suboptimal dynamic OFRMPC (where CCA is avoided) is discussed, and a previous approach is re-formulated. A numerical example is given to show the advantages of the proposed approach.

Asynchronous Constrained Resilient Robust Model Predictive Control for Markovian Jump Systems

In this article, the asynchronous resilient robust model predictive control (RMPC) problem is investigated for Markovian jump systems (MJSs) subject to polytopic parameter uncertainties and hard constraints on states and inputs. A hidden Markov model with partially available mode detection probabilities is introduced to characterize the asynchronous phenomenon between the detected modes and the system modes. A novel resilient control law is formulated such that, in the framework of the RMPC approach, the MJSs are mean-square stable in spite of partly accessible mode detection probabilities. Subsequently, by resorting to the stochastic analysis technique, sufficient conditions are derived, where the requirement of the terminal invariant set is met and the upper bound of the worst-case infinite horizon cost function is obtained. Moreover, by means of solving certain auxiliary optimization problems, the explicit expression of the desired controller is parameterized. Finally, simulation examples are presented to verify the validity of the proposed methods.

Robust master-slave synchronization of chaos in a one-sided 1-DoF impact mechanical oscillator subject to parametric uncertainties and disturbances

This paper addresses the problem of robust master-slave synchronization of chaos in a 1-DoF impact mechanical oscillator with a single rigid constraint. The master is considered to be with nominal parameters, whereas the slave impacting system is considered to be subject to polytopic parametric uncertainties and disturbances. We adopt a state-feedback controller for the robust stabilization of the master-slave synchronization error and the Linear Matrix Inequality (LMI) approach for the design of stability conditions. We use the S-procedure Lemma in order to only consider the regions within which the two impacting systems evolve. Thus, the stability conditions are formulated in terms of Bilinear Matrix Inequalities (BMIs). Via some technical Lemmas and congruence transformations, we transform these BMIs into LMIs, which are numerically traceable. An improved LMI-based optimization problem is proposed to solve the problem of high gains of the controller. Finally, the robustness of the proposed state-feedback feedback controller towards parametric uncertainties and disturbances is presented through simulation results showing then the master-slave synchronization of chaos in the one-sided 1-DoF impact mechanical oscillator.

Reachable set estimation for linear systems with time-varying delay and polytopic uncertainties

This study is concerned with the problem of reachable set estimation for linear systems with time-varying delays and polytopic parameter uncertainties. Our target is to find an ellipsoid that contains the state trajectory of linear system as small as possible. Specifically, first, in order to utilize more information about the state variables, the RSE problem for time-delay systems is solved based on an augmented Lyapunov-Krasovskii functional. Second, by dividing the time-varying delay into two non-uniformly subintervals, more general delay-dependent stability criteria for the existence of a desired ellipsoid are derived. Third, the integral interval is decomposed in the same way to estimate the bounds of integral terms more exactly. Fourth, an optimized integral inequality is used to deal with the integral terms, which is based on distinguished Wirtinger integral inequality and Reciprocally convex combination inequality. This can be regard as a new method in the delay systems. Finally, three numerical examples are presented to demonstrate the effectiveness and merits of the theoretical results.

Output feedback robust MPC using general polyhedral and ellipsoidal true state bounds for LPV model with bounded disturbance

ABSTRACTThis paper considers the dynamic output feedback robust model predictive control (MPC) for a system with both polytopic model parametric uncertainty and bounded disturbance. For this topic, the techniques for handling the unknown true state are crucial, and the strict guarantee of the input/output/state constraints requires replacing the true state by its bounds in the optimisation problems. Previously, in the separate works, we (i) gave the general polyhedral bound; (ii) proposed the general ellipsoidal bound; (iii) applied some special polyhedral bounds to tighten the ellipsoidal bound since the latter is crucial for guaranteeing recursive feasibility. In this paper, (i)–(iii) are unified, and the up-to-date least conservative treatment of the true state bound is given, so the control performance can be greatly improved. The contribution mainly lies in overcoming the difficulties in developing technical details for the unification. A numerical example is given to illustrate the effectiveness of the new method.

Provision for Guaranteed Inertial Response in Diesel-Wind Systems via Model Reference Control

Frequency performance has been a crucial issue for islanded microgrids. On the one hand, most distributed energy resources (DER) are converter-interfaced and do not inherently respond to frequency variations. On the other hand, current inertia emulation approach cannot provide guaranteed response. In this paper, a model reference control based inertia emulation strategy is proposed for diesel-wind systems. Desired inertia can be precisely emulated through the proposed strategy. A typical frequency response model with parametric inertia is set to be the reference model. A measurement at a specific location delivers the information about the disturbance acting on the diesel-wind system to the reference model. The objective is for the speed of the diesel generator to track the reference so that the desired inertial response is realized. In addition, polytopic parameter uncertainty will be considered. The control strategy is configured in different ways according to different operating points. The parameters of the reference model are scheduled to ensure adequate frequency response under a pre-defined worst case. The controller is implemented in a nonlinear three-phase diesel-wind system fed microgrid using the Simulink software platform. The results show that exact synthetic inertia can be emulated and adequate frequency response is achieved.

Iterative Learning Control of an Electrostatic Microbridge Actuator With Polytopic Uncertainty Models

In this brief, a robust control design is presented for an electrostatic microbridge actuator. For this system, a spatially distributed electrostatic force serves as control input. Moreover, a spatially distributed measurement of the bridge displacement is assumed to be available. For an accurate tracking of a reference trajectory—repeated periodically during the operation of the microbridge—an iterative learning control (ILC) approach is proposed based on so-called wave repetitive processes. The design procedure represents an efficient combination of linear matrix inequalities and an appropriate parameter optimization. By explicitly considering polytopic parameter uncertainty, the ILC becomes robust against uncertain parameters such as the squeeze film damping coefficient, the mass density, and the time constant of the electrostatic actuator. Convincing simulation results provide a numerical validation of the proposed ILC scheme as a prestage for a future experimental implementation.

Stabilization of a class of nonlinear discrete time systems with time varying delay

The stability and stabilization of a class of nonlinear discrete time delayed systems (NDTDS) with time-varying delay and norm-bounded nonlinearity are investigated. Based on discrete time Lyapunov-Krasovskii functional method, a sufficient delay-dependent condition for asymptotic stability of nonlinear systems is offered. Then, this condition is used to design a new efficient delayed state feedback controller (DSFC) for stabilization of such systems. These conditions are in the linear matrix inequality (LMI) framework. Illustrative examples confirm the improvement of the proposed approach over the similar cases. Furthermore, the obtained stability and stabilization conditions will be extended to uncertain discrete time delayed systems (UDTDS) with polytopic parameter uncertainties and also with norm-bounded parameter uncertainties.

Periodically Time-Varying ${\cal H}_{\infty}$ Memory Filter Design for Discrete-Time LTI Systems With Polytopic Uncertainty

This technical note focuses on the development of a general periodically time-varying memory filter (PTVMF) to improve the robust <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">${\cal H}_{\infty}$</tex></formula> filtering performance for discrete-time linear time-invariant (LTI) systems with polytopic parameter uncertainties. A sufficient condition to design the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">${\cal H}_{\infty}$</tex></formula> -PTVMF is given in terms of convex optimizations expressed as linear matrix inequalities (LMIs). Finally, examples demonstrate its potential less conservatism.

Robust H ∞ directional control for a sampled-data autonomous airship

A robust H∞ directional controller for a sampled-data autonomous airship with polytopic parameter uncertainties was proposed. By input delay approach, the linearized airship model was transformed into a continuous-time system with time-varying delay. Sufficient conditions were then established based on the constructed Lyapunov-Krasovskii functional, which guarantee that the system is mean-square exponentially stable with H∞ performance. The desired controller can be obtained by solving the obtained conditions. Simulation results show that guaranteed minimum H∞ performance γ=1.4037 and fast response of attitude for sampled-data autonomous airship are achieved in spite of the existence of parameter uncertainties.