Numerical algorithm for model reduction of polytopic uncertain linear systems

This paper is concerned with the robust fault detection (FD) problem for a class of polytopic uncertain linear systems driven by a Wiener process. It is assumed that the state matrix is affinely depending on unknown but bounded time-varying parameters. A switching mechanism is introduced to construct the robust FD filter with variable gains and to improve the FD performances which are inherent in the traditional FD filter with fixed gains. Finally, the filter design problem is formulated as a feasibility problem in terms of solving linear matrix inequalities (LMIs), and an example is presented to illustrate the proposed methodology.

In this paper, we consider nonlinear discrete time systems modeled with polytopic uncertain linear systems. We develop the CPWL (Canonical Piecewise Linear) approach proposed by Labit [1] to the case of discrete time systems associated with the new discrete time stability condition proposed by de Oliviera [2] then improved by Daafouz [3]. The obtained linear matrix inequalities condition leads to necessary and sufficient, and less conservative control design condition. This method (Discrete time -CPWL approach and poly-quadratic stability) takes also advantages of the systematic design of robust controllers in one shot and the insuring scheduling for the nonlinear discrete time system control.

This paper studies stability analysis of DC microgrids with uncertain constant power loads (CPLs). It is well known that CPLs have negative impedance effects, which may cause instability in a DC microgrid. Existing works often study the stability around a given equilibrium based on some nominal values of CPLs. However, in real applications, the equilibrium of a DC microgrid depends on the loading condition that often changes over time. Different from many previous results, this paper develops a framework that can analyze the DC microgrid stability for a given range of CPLs. The problem is formulated as a robust stability problem of a polytopic uncertain linear system. By exploiting the structure of the problem, we derive a set of sufficient conditions that can guarantee robust stability. The conditions can be efficiently checked by solving a convex optimization problem in polynomial time. The effectiveness and non-conservativeness of the proposed framework are demonstrated using a simulation example.

In this article, the robust model predictive control (RMPC) problem is investigated for a class of polytopic uncertain systems over high-rate networks whose signal exchanges are scheduled by the FlexRay protocol (FRP). During signal measurement, a high-rate network is applied to broadcast the data from the sensors to the controller efficiently. The FRP including the characteristics of event-triggered mechanism and the time-triggered mechanism is embedded into the high-rate network to regulate the data transmission in a circular period which can improve the flexibility of data transmission. With the aid of the Round-Robin and Try-Once-Discard protocols, a new expression of the measurement model is formulated by the use of certain data holding strategies. Subsequently, taking both high-rate networks and FRP into account, sufficient conditions are obtained by solving a time-varying terminal constraint set of an auxiliary optimization problem. In addition, an algorithm including both off-line and on-line parts is provided to find a sub-optimal solution. Lastly, two numerical simulations are carried out to substantiate the validity of the proposed RMPC strategy which is based on FRP and a high-rate network.

This study aims at developing new linear matrix inequality (LMI) conditions for robust κ-stability and κ-stabilisation of linear time-invariant uncertain systems in polytopic domains. An existing LMI condition for robust κ-stability is lifted into those of larger dimensions that are given as functions of a certain integer κ. Based on the consequent conditions, new sufficient conditions for the existence of a state-feedback control law ensuring the robust κ-stability of the closed-loop system is proposed in terms of LMIs. Finally, examples are given to illustrate the benefit of the proposed methods.

In this paper, Input-to-State Stability conditions are proposed for disturbed linear polytopic discrete-time systems. The conditions allow to optimize, independently in a certain extent, both the bounds on the decay rate and on the ISS gain, two central quantites. Indeed, the decay factor characterizes the transient behaviour of the state while the ISS gain characterizes the sensitivity with respect to the disturbances. The conditions are expressed in terms of tractable Matrix Inequalities. It is shown that the conditions are more general than existing ones proposed so far in the literature and thus less conservative. The conditions hold both for analysis or synthesis purposes. Two illustrative examples addressing the problem of polytopic observer design are given.

<div class="htmlview paragraph">As an important class of nonlinear system, polytopic bilinear system is investigated. Combined with the properties of convex polytope, the nonlinear control for polytopic bilinear system is formulated by synthesizing nonlinear <i>H</i><sub>∞</sub> controller which is designed for polytopic bilinear system at vertices. For a semi-active suspension system with controllable damping and variant stiffness elements, it is easily modeled as a polytopic bilinear system model. In this case, the desired nonlinear control properties are pursued in making effective use of the changeable damping property while the variant stiffness is taken as the affine parameter of polytopic model. Therefore, polytopic bilinear system model could be reduced to a feasible problem by polytopic convex decomposition. Then the control problem of bilinear system model is to find a solution of nonlinear <i>H</i><sub>∞</sub> control. Here, the merits in using nonlinear <i>H</i><sub>∞</sub> state feedback control are that it can confine the damping force output to 1<sup>st</sup> and 3<sup>rd</sup> phase planes where changeable damping force is produced and its nonlinear control output makes a fast state origin convergence in keeping small vibration and motion. In this paper, a practical control design demonstration for a polytopic bilinear suspension system is described after the theory description. The simulation study and experimental results demonstrate the effective control performance in improving ride comfort.</div>

The fault detection filter design problem of the high speed trains (HSTs) is systematically studied in this paper. First, an uncertain polytopic linear system is introduced to model the nonlinear uncertainty of the HSTs. And then a general structure of multi-objective H_/H ∞ fault detection filters is proposed. Necessary and sufficient conditions for both robustness condition referred to as H ∞ performance index and sensitivity condition expressed as H − performance index are given in the form of matrix inequality. In order to handle conflicting H − performance index and H ∞ performance index, a linear matrix inequality (LMI) based iterative algorithm is provided to obtain the desired filter. Finally, a numerical simulation is conducted to demonstrate the effectiveness of the proposed iterative algorithm.

In this study, an approach to design robust distributed model predictive control (MPC) is proposed for polytopic uncertain networked control systems with time delays. To reduce the computational complexity and improve the flexibility, the entire system is decomposed into multiple smaller dimensional subsystems. For each subsystem, the proposed robust distributed MPC algorithm requires solving multiple linear matrix inequality optimisation problems to minimise an upper bound on a robust performance objective. An augmented polytopic uncertainty description is invoked to handle the input delays. The conservativeness of distributed MPC algorithm is reduced by utilising a sequence of feedback control laws. An iterative on‐line algorithm for robust distributed MPC is developed to coordinate the distributed MPC controllers. Convergence and robust stability of the proposed distributed MPC are investigated. A numerical example is carried out to demonstrate the effectiveness of the proposed algorithm.

This paper investigates the problem of robust output feedback stabilization of uncertain Hamiltonian systems with saturating actuator and input delay. The uncertainties are assumed to be time invariant and belong to convex bounded polytopic domain. By employing Lyapunov-Krasovskii functional method and Wirtinger's inequality, some sufficient conditions are derived for stabilization of nonlinear Hamiltonian systems under consideration. A delay-independent condition in terms of linear matrix inequalities for control of the Hamiltonian systems is established first. And further, an output feedback controller synthesis scheme under the multiple Lyapunov functional framework is proposed which guarantees the robust stability of the closed-loop system when control saturation and delay effectively occur. Finally, numerical results are given to demonstrate the effectiveness of the proposed methodology.

In this paper, a class of discrete-time uncertain linear hybrid systems, which is affected by both parameter variations and exterior disturbances, is considered. The goal here is to synthesize a hybrid control law, which includes the continuous variable control law and the discrete event control signal, so as to guarantee the controlled invariance for a specific region in the hybrid state space. It turns out that the continuous variable control law is a piecewise linear state feedback control based on the partition of the given region's continuous variable state space projection. A numerical example is given for illustration.

This study is concerned with the problem of robust H 2 and H ∞ filtering for uncertain discrete-time linear systems. Two types of time-invariant parametric uncertainty, namely polytopic and ellipsoidal, are considered and represented by a linear fractional transformation structure. Obtained auxiliary variables by a convex optimisation problem, play the role of decoupling the Lyapunov variables and the robust filter parameters, in order to cast the problem into a linear matrix inequality-based optimisation problem. The design conditions are derived based on the quadratic separation concept and employing appropriate parameterisations for the corresponding set of multipliers. The merit of the methods presented in this study lies in their less conservatism than the existing methods for the polytopic uncertain systems, as well as presenting a new convex optimisation procedure for the robust filtering for the ellipsoidal uncertain systems.

The aim of this article is the state estimation of uncertain polytopic dynamic systems. The parametric uncertainties affecting the system are time varying, unknown and bounded with known bounds. The objective is to determine the state estimates consisting in the smallest interval containing the real-state value caused by the parametric uncertainties. This set will be characterised by the lower and upper bounds of the state trajectory. Given the uncertainty bounds, the set can be computed by a direct simulation of the system but a more accurate estimation is obtained with a Luenberger-type observer, fed with the system measurements. The proposed observer is designed to minimise the interval width of the estimates. The observer gains are obtained by solving an optimisation problem under LMI constraints. The efficiency of the proposed approach is illustrated by numerical examples.

This paper proposes \(H_{2}\)/\(H_{\infty }\) full state-feedback synthesis for robot manipulators, using uncertain polytopic linear parameter-varying (LPV) system modeling, with pole placement constraints to assign the poles of closed-loop system in a desired linear matrix inequality (LMI) region. The desired state trajectory of the system is used for generating an uncertain polytopic model of the system applying usual Lagrangian equations. The control gain matrix is derived by solving a set of LMIs to design a robust pole placement controller such that a prescribed mixed \(H_{2}\)/\(H_{\infty }\) performance is fulfilled and the response of the manipulator has a proper damping ratio. A sufficient condition is proposed to guarantee the asymptotic stability of the closed-loop uncertain polytopic LPV system against the uncertainties on the vertices. The proposed scheme is applied to controller synthesis of a two-degree-of-freedom manipulator trajectory-tracking problem. The simulation results show the effectiveness of the proposed controller.

Switched, piecewise and polytopic linear systems