Linear Matrix Inequality Region Research Articles

Output feedback pole-placement in damping region for discrete-time uncertain polytopic systems

ABSTRACT Pole placement with specified damping and decay rate is one of the primary criteria for ensuring good transient response. This control design problem has known solutions for continuous-time systems in terms of the linear matrix inequality (LMI) region. However, a systematic solution for a discrete-time system is an open problem due to the non-convexity of the constant damping locus. This paper addresses the problem of placing the closed-loop poles of linear discrete-time uncertain polytopic systems in a constant damping region. The non-convex damping region is approximated to an elliptical segment that represents an LMI region. Criteria for designing state and output feedback (static as well as dynamic) controllers are then derived. The effectiveness of the proposed approach is demonstrated with examples including a boost converter to improve its transient behaviour in the presence of input voltage and load variations.

On the State-Feedback Controller Design for Polynomial Linear Parameter-Varying Systems with Pole Placement within Linear Matrix Inequality Regions

The present paper addresses linear parameter-varying systems with high-order time-varying parameter dependency known as polynomial LPV systems and their controller design. Throughout this work, a procedure ensuring a state-feedback controller from a parameterized linear matrix inequality (PLMI) solution is presented. As the main contribution of this paper, the controller is designed by considering the time-varying parameter rate as a tuning parameter with a continuous control gain in such a way that the closed-loop eigenvalues lie in a complex plane subset, with high-order time-varying parameters defining the system dynamics. Simulation results are presented, aiming to show the effectiveness of the proposed controller design.

Optimal control of satellite system model using Linear Matrix inequality approach

In this paper, a robust control strategy has been proposed for the satellite attitude control system, namely, ℋ∞ with regional pole constraints. This design method has been applied to eliminate the effects of disturbances and uncertainties, which are inevitable in aerospace applications. A convex optimization approach using Linear Matrix Inequalities (LMIs) has been utilized in this work. The concept of LMI region constraints have been used to guarantee the closed-loop poles lie within a defined region in the complex plane, which helps improve performance in addition to the robustness properties. Therefore, the designed controller can be viewed as a multi-objective controller that guarantees both robustness and performance. Simulation results demonstrate that the developed controller (ℋ∞ with LMI region constraint) performs better than only ℋ∞ and pole placement controllers, in terms of disturbance attenuation, transient response, and control effort.

Optimization of Robust LMI-Control Systems for Unstable Vertical Plasma Position in D-Shaped Tokamak

The paper is devoted to the synthesis, comparison, and optimization of robust LMI-control systems for the vertical plasma position in a D-shaped tokamak, specifically the T-15MD tokamak (Kurchatov Institute, Moscow, Russia). The novelty of this work is to find out the possibilities of LMI robust control systems, according to the criteria of the robust stability radius and control power peak at the rejection of a minor disruption type disturbance and a reference step signal using a unique unstable first-order plasma model. The plant under control consists of the connected in series plasma model with additive disturbance containing plant uncertainties, horizontal field coil (HFC), and actuator model as a multiphase rectifier. A set of robust controllers was designed by Linear Matrix Inequalities (LMI) method with pole placement in the LMI regions, state H2/H∞ performance, and output signal performance. The LMI theorems of the paper are directed to design the robust controllers and study the systems with the aim of eliminating the gap between theory and practice. The main achievement of this work consists in the optimization of robust control systems of the unstable plant with uncertain disturbance on the set of LMI synthesis approaches. The control systems have original quality criteria, such as control power and robust stability radius. The best control system on the basis of two criteria, namely, Dα,r,ϑ control system provides stabilization of the vertical plasma position on the real-time digital control testbed.

Stability and Stabilization of TS Fuzzy Systems via Line Integral Lyapunov Fuzzy Function

This paper is concerned with the stability and stabilization problem of a Takagi-Sugeno fuzzy (TSF) system. Using a non-quadratic function (well-known integral Lyapunov fuzzy candidate (ILF)) and some lemmas, new sufficient conditions are established as linear matrix inequalities (LMIs), which are solved with a stochastic fractal search (SFS). The main advantage of the technique used is its small conservatives. Motivated by the mean value theorem, a state feedback controller based on a non-quadratic Lyapunov function is designed. Unlike other approaches based on poly-quadratic Lyapunov candidates, stability conditions of the closed loop are obtained in LMI regions. It is important to highlight that the time derivatives of membership functions do not appear in the used line integral Lyapunov function, which is the well-known problem of poly-quadratic Lyapunov functions. A numerical example is given to show the advantages and the utility of the integral Lyapunov fuzzy candidate, which provides a wider feasibility region than other Lyapunov functions.

Tracking flight control of UAV based on multiple regions pole assignment method

This paper presents a tracking control problem for an unmanned aerial vehicle (UAV) using a multiple regions pole assignment method. To obtain the UAV control gains under the regional pole assignment method, the traditional robust control approach is to construct a closed-loop system and a specific linear matrix inequality (LMI) region in a complex plane, all poles in the system are configured in this specific region by solving LMI. Since the given LMI region is fixed with a large scale, further constraints on the LMI region can make poles configuring in a more appropriate position to obtain better control effects. Hence, based on the UAV tracking control problem, a multiple regions pole assignment method is proposed in this paper. By designing multiple LMI regions and a novel solving algorithm for pole assignment, the optimized tracking control scheme can be obtained. A practical example of quadrotor UAV tracking is given to illustrate the effectiveness of the proposed method.

High-gain interval observer for continuous–discrete-time systems using an LMI design approach

In this paper, a high-gain interval observer (HGIO) for a class of partially linear continuous-time systems with sampled measured outputs in the presence of bounded noise and additive disturbances is proposed. The design of the HGIO is formulated in the Linear Matrix Inequality (LMI) framework. The gain of the HGIO is designed to satisfy the cooperative property using a time-varying change of coordinates based on pole placement in separate LMI regions. Moreover, a procedure for designing the HGIO gain to minimise the effect of the noise and disturbance in the estimation is provided. The stability of the proposed HGIO is also guaranteed. The proposed approach is assessed in simulation using a numerical example.

Learning Controllers for Performance Through LMI Regions

In an open-loop experiment, an input sequence is applied to an unknown linear time-invariant system (in continuous or discrete time) affected also by an unknown-but-bounded disturbance sequence (with an energy or instantaneous bound); the corresponding state sequence is measured. The goal is to design directly from the input and state sequences a controller that enforces a certain performance specification on the transient behaviour of the unknown system. The performance specification is expressed through a subset of the complex plane where closed-loop eigenvalues need to belong, a so called LMI region. For this control design problem, we provide here convex programs to enforce the performance specification from data in the form of linear matrix inequalities (LMI). For generic LMI regions, these are sufficient conditions to assign the eigenvalues within the LMI region for all possible dynamics consistent with data, and become necessary and sufficient conditions for special LMI regions. In this way, we extend classical model-based conditions from a seminal work in the literature to the setting of data-driven control from noisy data. Through two numerical examples, we investigate how these data-based conditions compare with each other.

Robust Control for Helicopters Performance Improvement: an LMI Approach

This paper presents an LMI (Linear Matrix Inequalities) application for the design of robust controllers for multivariate systems that have multiple points of operation. Some systems change their parameters along time, then, it is necessary to switch the control for different operational points. The purpose of this controller is to ensure the stability and performance requirements of the system for different operating points with the same controller. The method uses the following concepts of predefined structures controller, LMI region, and polytopic systems. To validate the controller a linearized model of a helicopter was used. These helicopters belong to a system class of MIMO (Multiple-Input Multiple Outputs) type and present a complex dynamic in their flight modes, therefore, due to these features, this type of helicopter is a good model to implement and test the efficiency of the described method in this work. The results were satisfactory. Some limitations in its implementation were found and discussed. An LQG (Linear-Quadratic-Gaussian) controller was also designed for the same model of the helicopter just for comparison. Analyzing the settling time properties, the LMI controller presented a better response than the LQG controller.

Mapping prior information onto LMI eigenvalue‐regions for discrete‐time subspace identification

In subspace identification, prior information can be used to constrain the eigenvalues of the estimated state-space model by defining corresponding linear matrix inequality (LMI) regions. In this study, first the authors argue on what kind of practical information can be extracted from historical data or step-response experiments to possibly improve the dynamical properties of the corresponding model and, also, on how to mitigate the effect of the uncertainty on such information. For instance, prior knowledge regarding the overshoot, the period between damped oscillations and settling time may be useful to constrain the possible locations of the eigenvalues of the discrete-time model. Then, they show how to map the prior information onto LMI regions and, when the obtaining regions are non-convex, to obtain convex approximations.

Robust PI-D Controller Design for Uncertain Linear Polytopic Systems Using LMI Regions and $H_2$ Performance

The paper deals with the new design procedure for PID-D robust controller with state/output derivative feedback to stabilize the closed-loop system using linear matrix inequality (LMI) and LMI regions with common in the frame of $H_2$ performance quadratic cost functions known as $(Q,S,R)$ . The designed robust controller ensures robust properties of the closed loop system for linear time invariant and linear time variant gain scheduled plant descriptor systems. The robust controller design procedure is based on the extended original derivative of Lyapunov function and to obtain the convex conditions with respect to uncertainties, gain scheduled variables, and ensures that proposed new method guarantees less conservativeness to the design procedure we have introduced using the auxiliary matrices. The obtained results should be utilized to design power system stabilizer for synchronous generators and so on. The obtained robust PI-D controller with output and state/output derivative feedback for uncertain polytopic systems ensures that all closed-loop eigenvalues are lying in the defined LMI region. Two examples illustrate the effectiveness of the new design procedure.

LMI Pole Regions for a Robust Discrete-Time Pole Placement Controller Design

Herein, robust pole placement controller design for linear uncertain discrete time dynamic systems is addressed. The adopted approach uses the so called “D regions” where the closed loop system poles are determined to lie. The discrete time pole regions corresponding to the prescribed damping of the resulting closed loop system are studied. The key issue is to determine the appropriate convex approximation to the originally non-convex discrete-time system pole region, so that numerically efficient robust controller design algorithms based on Linear Matrix Inequalities (LMI) can be used. Several alternatives for relatively simple inner approximations and their corresponding LMI descriptions are presented. The developed LMI region for the prescribed damping can be arbitrarily combined with other LMI pole limitations (e.g., stability degree). Simple algorithms to calculate the matrices for LMI representation of the proposed convex pole regions are provided in a concise way. The results and their use in a robust controller design are illustrated on a case study of a laboratory magnetic levitation system.

Robust Control for the IPT System With Parametric Uncertainty Using LMI Pole Constraints

In the inductive power transfer (IPT) system, coil misalignment, load variation, and external disturbance will unavoidably cause power fluctuation and worsen system performances. Currently, robust control has been successfully introduced in the IPT system for promoting its stability and performances. However, due to the diversity of poles distribution, all possible IPT subsystems always show different overshoot, settling time, and resistance capacity to uncertainty factors. So, robust control design for the IPT system via pole placement constraints in linear matrix inequality (LMI) region is studied in this paper. An IPT system model with structured parametric uncertainty relevant to mutual inductance and load is first established. Based on the derived closed-loop system with a two-degrees-of-freedom control structure, a method of variable replacement is then proposed to transform the robust D-stability problem with pole region constraints into the LMI form, which can be solved to obtain a robust controller. After analyzing the influence of the LMI region change on poles distribution and system performances, an optimized robust controller is finally chosen. Experimental results show that a robust controller can reach the prescribed dynamic performances even if mutual inductance or load may change into another value, and also a good steady-state performance by resisting parametric uncertainty and external disturbance.

New results on restricted static output feedback controller design with regional pole placement

This study addresses the design of static output feedback (SOF) controller for continuous time linear systems. New sufficient conditions are derived for designing SOF controllers. The development involves suitable decomposition of Lyapunov matrices and deriving linear matrix inequality (LMI) criterion that ensures H ∞ performance. LMI criteria for pole placement in the LMI region are also derived. The proposed criteria can be used for restricted SOF controller design effectively. Several numerical examples are presented to demonstrate the effectiveness of the proposed results.

Observer based nonlinear control design for glucose regulation in type 1 diabetic patients: An LMI approach

Abstract This paper deals with the design of observer-based nonlinear control of blood glucose concentration (BGC) of Type 1 diabetes mellitus (T1DM) patients in a Linear Matrix Inequality (LMI) framework. The controller design relies on the information of the states obtained from a nonlinear observer. The control law is derived using feedback linearisation and regional pole placement technique. Further, a numerical optimisation method is proposed for the computation of the controller gains by capturing the relationship between transformed domain and original domain dynamics by iteratively tuning the circular LMI region parameters (‘q’ and ‘r’) such that the locations of closed-loop poles of the original nonlinear system are attained. The proposed controller can deliver robust closed-loop response of BGC within a specified range of parametric uncertainty and meal disturbances owing to the appropriately tuned bound of LMI region parameters. The performance of the proposed controller is tested for 100 virtual T1DM patients in the presence of parametric uncertainty and uncertain meal disturbance. Both severe hypoglycemia (

Uncertain T-S model-based robust controller design with [formula omitted]-stability constraints—A simulation study of quadrotor attitude stabilization

The problem of the attitude stabilization of a quadrotor unmanned aerial vehicle is considered in simulation. An uncertain-like Takagi–Sugeno model of the quadrotor, where the nonlinear entries which depend on the input motor voltages are reported in uncertainties, is proposed to avoid closed-loop algebraic loops. The design of a Parallel Distributed Compensation (PDC) control law is considered with D-stability constraints in order to improve the closed-loop transient response. In this context, new generic and relaxed Linear Matrix Inequality (LMI) conditions are proposed. These new PDC controller design conditions constitute an alternative to previously proposed D-stabilizing non-PDC controller ones. Indeed, non-PDC controllers require online inversions of time-varying matrices, which is a major drawback for embedded applications with low computational capabilities. The conservatism of the proposed LMI conditions with respect to previous results is illustrated through an academic example. Then, simulations of the quadrotor with usual prescribed LMI regions are proposed to show the effectiveness of the proposed conditions.

A new frequency-domain subspace algorithm with restricted poles location through LMI regions and its application to a wind tunnel test

ABSTRACTIn this paper, an innovative method is presented to identify models with a modified frequency-domain subspace method. This new approach allows to introduce in the subspace resolution constraints on the identified model poles location. Here, a very general formulation is proposed to take into account regions in continuous/discrete map. This formulation is based on an LMI (linear matrix inequalities) description where the stability domain represents a particular case. These LMI constraints are combined with the frequency-domain subspace resolution to obtain identified models whose poles are situated in the specified LMI regions. This approach is benchmarked with the Loewner one, which belongs to the class of frequency-domain input–output model identification and approximation methods. Besides the fact that they both belong to the data-driven model approximation class, they result to have slightly different objectives and show complementary performances. This discussion is illustrated in practice with experimentations that have been performed for the identification and control of the gust disturbance over a 2D wing span, from sub to transonic, in a wind tunnel facility.

Non-fragile reliable [formula omitted] for delta operator switched linear systems

This paper is concerned with the non-fragile reliable D-stabilization problem of a class of delta operator switched linear systems with actuator faults, in terms of linear matrix inequalities (LMIs). Firstly, to handle the determination problem of the decay rate of a delta operator system in the process of D-stabilizing, the theory of first-order LMI regions is proposed. Secondly, to deal with the uncertain matrices multiplication phenomenon appearing in non-fragile reliable control, a new approach is proposed. Based on the average dwell time technique and the two new methods mentioned above, the state feedback controller and the switching law are designed to guarantee that all the closed-loop poles of each mode lie in a specified disk and the closed-loop switched system is exponentially stable. Finally, the validity and feasibility of the proposed approach are illustrated by a flight control system example.

Robust fault detection observer design under fault sensitivity constraints

In this paper, a reduced-order robust fault detection observer (FDO) for continuous linear time-invariant systems with unknown inputs is designed by combining the algebraic unknown input decoupling principle with linear matrix inequalities (LMIs)-based optimization of fault sensitivity constraints. Algebraic unknown input decoupling is incorporated into the design of fault detection observers, thereby leading to a reduced-order robust FDO that is decoupled from unknown inputs. The gain matrix of the reduced-order robust FDO is determined from the optimal solution that maximizes the LMIs, which are based on the H_ index performance and the additional pole placement in a specified LMI region. This ensures that the designed robust FDO provides the optimal fault sensitivity on the residual. The efficiency of the proposed approach is evaluated through simulation studies and compared with the pole placement design and the mixed H_/H∞ observer given in the work of [9].

Partial Pole Placement in LMI Region

A new approach for pole placement of single-input system is proposed in this paper. Noncritical closed loop poles can be placed arbitrarily in a specified convex region when dominant poles are fixed in anticipant locations. The convex region is expressed in the form of linear matrix inequality (LMI), with which the partial pole placement problem can be solved via convex optimization tools. The validity and applicability of this approach are illustrated by two examples.