Bilinear Matrix Inequalities Research Articles

Application of Lyapunov matrix inequality based unsymmetrical saturated control to a multi-vectored propeller airship

The problem of the design of a controller for a multi-vectored propeller airship is addressed. The controller includes anti-windup that takes into account unsymmetrical actuator constraints. First, a linear transformation is applied to transform the unsymmetrical constraints into symmetric constraints with an amplitude-bounded exogenous disturbance. Then, a stability condition based on a quadratic Lyapunov function for the saturated closed-loop system is proposed. The condition considers both amplitude-bounded and energy-bounded exogenous disturbances. Thus, the controller design problem is transformed into a convex optimization problem expressed in a bilinear matrix inequality form. Two controller design methods were applied: one-step controller and traditional anti-windup controller. The one-step method obtains the controller and the anti-windup compensator in one step while the anti-windup controller method separates this process into the linear controller design and the compensator design. Simulation results showed that both controllers enlarge the stability zone of the saturation system and have good tracking performance. It is shown that the anti-windup controller design method not only has a larger region of stability, but the demanded actuator output exceeds the constraints less and has a smaller anti-windup coefficient matrix compared to the one-step method.

Optimization of Noise Shaping Filter for Quantizer With Error Feedback

Noise shaping filters for quantizers with error feedback are designed to mitigate the effects of quantization errors. In this paper, we prove that if the transfer function from the quantization error to the signal-of-interest has minimum phase and there is no constraint on the feedback signal, then the scaled inverse of the transfer function is the optimal noise shaping filter. Next, we design a noise shaping filter to minimize the variance or the $l_{2}$ norm of the error in the signal-of-interest under a constraint on the variance or the $l_{2}$ norm of the feedback signal, which can be expressed as bilinear matrix inequalities (BMIs). Although the BMIs are not convex, we prove that the minimization of the error variance under the constraint on the variance of the feedback signal can be cast into a convex optimization problem. This enables us to design optimal noise shaping filters using numerical methods. We formulate our design problem as a convex optimization problem using extended linear matrix inequalities to obtain noise shaping filters, except for the special case. Examples are provided to demonstrate the effectiveness of the designed noise shaping filters.

Robust feedback control of the underactuated Inertia Wheel Inverted Pendulum under parametric uncertainties and subject to external disturbances: LMI formulation

This paper proposes a robust feedback controller using Linear Matrix Inequalities (LMIs) formulation for the stabilization of an underactuated mechanical system, namely the Inertia Wheel Inverted Pendulum (IWIP), in its upright position. Such mechatronic system is subject to state constraints, external disturbances and norm-bounded parametric uncertainties. The main idea to solve the stabilization problem lies in the use of the S-procedure Lemma. Such problem is then transformed into a solving problem of Bilinear Matrix Inequalities (BMIs). Through the Schur complement Lemma and the Matrix Inversion Lemma, a linearization procedure is employed to transform the BMIs into LMIs. Some improvements and comparisons with other LMI-based design techniques without state constraints are developed and discussed. An extensive portfolio of numerical studies is presented. The effectiveness and robustness of the proposed feedback controller toward uncertainties in the friction parameters and external disturbances are illustrated through simulation results.

The distributionally robust complementarity problem

We investigate the linear complementarity problem with uncertain parameters (ULCP) which affect the linear mapping affinely or quadratically. Assuming that the distribution of the uncertain parameters belongs to some ambiguity set with prescribed partial information, we formulate the ULCP as a distributionally robust optimization reformulation named as the distributionally robust complementarity problem (DRCP), which minimizes the worst case of an expected complementarity measure with a joint chance constraint that the probability of the linear mapping being nonnegative is not less than a given level. Applying the cone dual theory and S-procedure, we conservatively approximate the DRCP as a nonlinear semidefinite programming (NSDP) with bilinear matrix inequalities, which can be solved by the NSDP solver PENLAB. The preliminary numerical test on a constrained stochastic linear quadratic control problem shows that the DRCP as well as the corresponding solution method is promising.

Network synthesis and parameter optimization for vehicle suspension with inerter

In order to design a comfortable-oriented vehicle suspension structure, the network synthesis method was utilized to transfer the problem into solving a timing robust control problem and determine the structure of “inerter–spring–damper” suspension. Bilinear Matrix Inequality was utilized to obtain the timing transfer function. Then, the transfer function of suspension system can be physically implemented by passive elements such as spring, damper, and inerter. By analyzing the sensitivity and quantum genetic algorithm, the optimized parameters of inerter–spring–damper suspension were determined. A quarter-car model was established. The performance of the inerter–spring–damper suspension was verified under random input. The simulation results manifested that the dynamic performance of the proposed suspension was enhanced in contrast with traditional suspension. The root mean square of vehicle body acceleration decreases by 18.9%. The inerter–spring–damper suspension can inhibit the vertical vibration within the frequency of 1–3 Hz effectively and enhance the performance of ride comfort significantly.

Control design of non‐homogeneous Markovian jump systems via relaxation of bilinear time‐varying matrix inequalities

This study addresses the issue of control synthesis for discrete-time non-homogeneous Markov jump systems. Here, the concept of multiple boundary sets (BSs) is made to represent time-varying transition probabilities of the system operation mode, and another kind of probabilities is exploited to describe the switching among the multiple BSs. Overall, the H ∞ stabilisation condition is first derived in terms of bilinear time-varying matrix inequalities with two kinds of probabilities that affect the evolvement of the modes. Then, to derive a finite number of solvable conditions from the stabilisation condition, this study introduces a relaxation technique capable of directly integrating some available constraints on the probabilities without resorting to any over-bounding technique.

New observer-based control design for mismatched uncertain systems with time-delay

Abstract In this paper, the state estimation problem for a class of mismatched uncertain time-delay systems is addressed. The estimation uses observer-based control techniques. The mismatched uncertain time-delay systems investigated in this study include mismatched parameter uncertainties in the state matrix and in the delayed state matrix. First, based on a new lemma with appropriately choosing Lyapunov functional, new results for stabilization of mismatched uncertain time-delay systems are provided on the basis of a linear matrix inequality (LMI) framework and the asymptotic convergence properties for the estimation error is ensured. Second, the control and observer gains are given from single LMI feasible solution which can overcome the drawback of the bilinear matrix inequalities approach often reported in the literature. Finally, a numerical example is used to demonstrate the efficacy of the proposed method.

Robust control of two parallel‐connected permanent magnet synchronous motors fed by a single inverter

Two parallel-connected permanent magnet synchronous motors (PMSMs) fed by a single inverter is investigated in this study. A robust controller is designed for this system via interval matrix, and in terms of bilinear matrix inequalities, sufficient conditions for the existence of the robust controller are derived. The proposed controller not only can deal with the uncertainty of load, but also has robust to the parameter perturbation. The effectiveness of the proposed controller is proved by theory analyses and computer simulation. Besides, the simulation results show that the control performance of the proposed control strategy is better than that of the conventional master–slave strategy.

Finite dimensional disturbance observer based control for nonlinear parabolic PDE systems via output feedback

This paper considers the problem of finite dimensional disturbance observer based control (DOBC) via output feedback for a class of nonlinear parabolic partial differential equation (PDE) systems. The external disturbance is generated by an exosystem modeled by ordinary differential equations (ODEs), which enters into the PDE system through the control channel. Motivated by the fact that the dominant dynamic behavior of parabolic PDE systems can be characterized by a finite number of degrees of freedom, the modal decomposition technique is initially applied to the PDE system to derive a slow subsystem of finite dimensional ODEs. Subsequently, based on the slow subsystem and the exosystem, a disturbance observer (DO) and a slow mode observer (SMO) are constructed to estimate the disturbance and the slow modes. Moreover, an observation spillover observer (OSO) is also constructed to cancel approximately the effect of the observation spillover. Then, a finite dimensional DOBC design via output feedback is developed to estimate and compensate the disturbance, such that the closed-loop PDE system is exponentially stable in the presence of the disturbance. The condition for the existence of the proposed controller is given in terms of bilinear matrix inequality. Two algorithms based on the linear matrix inequality (LMI) technique are provided for solving control and observer gain matrices of the proposed controller. Finally, the developed design method is applied to the control of a one-dimensional diffusion-reaction process to illustrate its effectiveness.

H∞ output-feedback fuzzy proportional-integral control of fully delayed input/output systems

This paper presents the output-feedback fuzzy proportional-integral (PI) controller design for uncertain nonlinear systems with both fully delayed input and output. Based on the Takagi–Sugeno (T–S) fuzzy model representation, the output-feedback PI control is realized via parallel distributed PI compensation and novel LMI gain design. Although the T–S fuzzy PI controller is simple, asymptotic output regulation is assured to overcome the effect of uncertainty, state delay, and full input/output delays. When considering disturbance and measurement noise, the control performance is achieved by robust gain design. Furthermore, state observers and bilinear matrix inequality conditions are removed in this paper. Finally, time-delay Chua׳s circuit system and a continuous-time stirred tank reactor are taken as applications to show the expected performance.

Global asymptotic stabilisation of rational dynamical systems based on solving BMI

ABSTRACTIn this paper, the global asymptotic stabiliser design of rational systems is studied in detail. To develop the idea, the state equations of the system are transformed to a new coordinate via polynomial transformation and the state feedback control law. This in turn is followed by the satisfaction of the linear growth condition (i.e. Lipschitz at zero). Based on a linear matrix inequality solution, the system in the new coordinate is globally asymptotically stabilised and then, leading to the global asymptotic stabilisation of the primary system. The polynomial transformation coefficients are derived by solving the bilinear matrix inequality problem. To confirm the capability of this method, three examples are highlighted.

Linear Matrix Inequality Method for a Quadratic Performance Index Minimization Problem with a class of Bilinear Matrix Inequality Conditions

There are a lot of design problems of control system which are expressed as a performance index minimization under BMI conditions. However, a minimization problem expressed as LMIs can be easily solved because of the convex property of LMIs. Therefore, many researchers have been studying transforming a variety of control design problems into convex minimization problems expressed as LMIs. This paper proposes an LMI method for a quadratic performance index minimization problem with a class of BMI conditions. The minimization problem treated in this paper includes design problems of state-feedback gain for switched system and so on. The effectiveness of the proposed method is verified through a state-feedback gain design for switched systems and a numerical simulation using the designed feedback gains.

Output feedback control for singular Markovian jump systems with uncertain transition rates

The static output feedback stabilisation problems for singular Markovian jump system are studied. Transition rates are assumed to be generally uncertain, namely, they could be unknown or only their estimate values are known. A criterion on the existence of an output feedback controller is established in terms of bilinear matrix inequalities (BMIs) to guarantee the closed-loop systems are stochastically admissible. To solve the BMIs in the criteria, the other criterion is presented by using a less conservative linearisation method, and an algorithm is proposed to solve the related conditions. Finally, three numerical examples are given to show the effectiveness of their methods.

Design of Observer-Based Robust Power System Stabilizers

<p>Power systems are subject to undesirable small oscillations that might grow to cause system shutdown and consequently great loss of national economy. A model to describe power system dynamics for different loads is derived in the norm-bounded form. The first controller design is based on the derived model to achieve robust stability against load variation. The design is based on a new Bilinear matrix inequality (BMI) condition. The BMI optimization is solved interatively in terms of Linear Matrix Inequality (LMI) framework. The condition contains a symmetric positive definite full matrix to be obtained, rather than the commonly used block diagonal form. The difficulty in finding a feasible solution is thus alleviated. The resulting LMI is of small size, easy to solve. The second PSS design shifts the closed loop poles in a desired region so as to achieve a favorite settling time and damping ratio via a non-iterative solution to a set of LMIs. Simulation results based on single-machine and multi-machine power system models verify the ability of the proposed PSS to satisfy control objectives for a wide range of load conditions.</p>

Improved synthesis conditions for mixed gain-scheduling control subject to uncertain scheduling parameters

ABSTRACTThe vast majority of the existing work in gain-scheduling (GS) control literature assumes perfect knowledge of scheduling parameters. Generally, this assumption is not realistic since for practical control applications measurement noises are unavoidable. In this paper, novel synthesis conditions are derived to synthesise robust GS controllers with mixed performance subject to uncertain scheduling parameters. The conditions are formulated in terms of parameterised bilinear matrix inequalities (PBMIs) that depend on varying parameters inside multi-simplex domain. The conditions provide practical GS controllers independent of the derivatives of scheduling parameters. That is, the designed controllers are feasible for implementation. Since bilinear matrix inequality problems are intractable, an iterative PBMI algorithm is developed to solve the developed synthesis conditions. By the virtue of this algorithm, conservativeness reduction is achieved with few iterations. Examples are presented to illustrate the effectiveness of the developed conditions. Compared to other design methods from literature, the developed conditions achieve better performance.

The control of drilling vibrations: A coupled PDE-ODE modeling approach

Abstract The main purpose of this contribution is the control of both torsional and axial vibrations occurring along a rotary oilwell drilling system. The model considered consists of a wave equation coupled to an ordinary differential equation (ODE) through a nonlinear function describing the rock-bit interaction. We propose a systematic method to design feedback controllers guaranteeing ultimate boundedness of the system trajectories and leading consequently to the suppression of harmful dynamics. The proposal of a Lyapunov-Krasovskii functional provides stability conditions stated in terms of the solution of a set of linear and bilinear matrix inequalities (LMIs, BMIs). Numerical simulations illustrate the efficiency of the obtained control laws.

Optimal worst case solutions for state estimation of power system with uncertain network parameters

This paper addresses the problem of power system state estimation under the condition that transmission line network parameters are unknown but bounded. A robust estimation in the sense of an optimal worst case solution is determined. Data collected via remote terminal units, i.e. voltage magnitude, power flow, and power injection, are used as measurement quantities. The state variables are bus voltage phasors expressed in rectangular coordinates. This makes it possible to express the relations between measured data and state variables as quadratic functions. The proposed formulation based on the structured robust least squares optimization yields a minimization problem with bilinear matrix inequality constraints. A solution method based on semidefinite programming is also presented. Some test results on the standard IEEE test systems are given. © 2016 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

Robust fault tolerant control of distributed networked control systems with variable structure

This paper studies the robust fault tolerant control problem of distributed network control systems (DNCSs). DNCSs are defined as the networked control systems (NCSs) consist of several sub-systems. Each sub-system contains its own sensor, controller and actuator. This kind of systems widely exist in the real world, such as the urban water supply network, smart grid and so on. Three main aspects of DNSCs are investigated in this paper. Firstly, we consider the variable structure problem of DNCSs. Meanwhile, the network problems including time delays, packet-dropouts and quantization errors are considered. Finally, the actuator fault tolerant control problem due to the possible poor working environment is also studied. A series of two-mode-dependent controllers are proposed to stabilize such DNCSs and meet the robust closed-loop performance. By using the Lyapunov–Krasovskii (L-K) function, sufficient conditions on the existence of such controllers are presented in terms of bilinear matrix inequalities (BMIs). A cone complementarity linearization (CCL) algorithm is given to solve proposed BMIs. Finally, a simulation example is exploited to show the effectiveness of the proposed method.

Consensus of heterogeneous linear multi-agent systems: Linear-transformation-based partial stability approach

We deal with the state consensus problem of a general heterogeneous linear multi-agent system under a time-invariant and directed communication topology. First we adopt a general linear consensus protocol consisting of two parts. One is a state feedback of the agent for independently regulating its dynamics, and the other is a cooperative term in a generalized feedback form of the relative states between the agents. Then we propose a state-linear-transformation to equivalently transform the state consensus problem into a partial stability problem. Therefore, the result from the partial stability theory is applied to derive a sufficient and necessary algebraic criterion of consensus convergence, which is expressed in terms of the Hurwitz stability of a real matrix constructed from the parameters of both the agents’ models and the protocol. Meanwhile, an analytical formula of the consensus function is presented. Based on the criterion, we propose a design procedure of the gain matrices in the protocol by solving a bilinear matrix inequality.

A Distributionally Robust Co-Ordinated Reserve Scheduling Model Considering CVaR-Based Wind Power Reserve Requirements

The reserve scheduling problem becomes more difficult to handle when wind power is increasing at a rapid rate in power systems and the complete information on the stochasticity of wind power is hard to be obtained. In this paper, considering the uncertainty on the probability distribution (PD) of the wind power forecast error (WPFE), a distributionally robust co-ordinated reserve scheduling (DRCRS) model is proposed, aiming to minimize the total procurement cost of conventional generation and reserve, while satisfying the security requirement over all possible PDs of WPFE. In this model, a distributionally robust formulation based on the concept of conditional value-at-risk (CVaR) is presented to obtain the reserve requirement of wind power. In addition, to achieve tractability of the scheduling model, the random variable that refers to WPFE in the scheduling model is eliminated, equivalently converting the stochastic model into a deterministic bilinear matrix inequality problem that can be effectively solved. Case studies based on the IEEE-39 bus system are used to verify the effectiveness of the proposed method. The results are compared with the normal distribution based co-ordinated reserve scheduling (NDCRS) method that assumes WPFE is of normal distribution.