Bilinear Matrix Inequality Condition Research Articles

Nonlinear optimization filters for stochastic time‐varying convex optimization

AbstractWe look at a stochastic time‐varying optimization problem and we formulate online algorithms to find and track its optimizers in expectation. The algorithms are derived from the intuition that standard prediction and correction steps can be seen as a nonlinear dynamical system and a measurement equation, respectively, yielding the notion of nonlinear filter design. The optimization algorithms are then based on an extended Kalman filter in the unconstrained case, and on a bilinear matrix inequality condition in the constrained case. Some special cases and variations are discussed, notably the case of parametric filters, yielding certificates based on LPV analysis and, if one wishes, matrix sum‐of‐squares relaxations. Supporting numerical results are presented from real data sets in ride‐hailing scenarios. The results are encouraging, especially when predictions are accurate, a case which is often encountered in practice when historical data is abundant.

Reset control and ℒ2-gain analysis of piecewise-affine systems

This paper is concerned with reset controller design and L 2 -gain stability of piecewise-affine systems under the framework of hybrid systems. Firstly, stability conditions of piecewise-affine systems under dynamic state-feedback control are established through bilinear matrix inequality conditions. Secondly, a reset controller with reset rules under the hybrid systems framework is proposed in the sense of Lyapunov and sufficient conditions for exponential and L 2 -gain stability of the closed-loop systems are provided. Different from the piecewise-affine systems with the dynamic state-feedback controller, the reset controller is designed such that the L 2 -gain performance of piecewise-affine systems can be enhanced. Thirdly, an LMI approach is proposed to avoid the difficulty of solving the bilinear matrix inequalities. Furthermore, robustness to inflations of the flow and jump sets is established, and robustness to the norm-bounded uncertainties is proposed. Finally, numerical simulations including a robot arm system, an inverted pendulum system and the Chua's circuit system are provided to illustrate the results.

Leader-following consensus of heterogeneous linear multi-agent systems: New results based on linear transformation method

This paper studies the leader-following state consensus problem for heterogeneous linear multi-agent systems under fixed directed communication topologies. First, we propose a consensus protocol consisting of four parts for high-order multi-agent systems, in which different agents are allowed to have different gain matrices so as to increase the degree of design freedom. Then, we adopt a state linear transformation, which is constructed based on the incidence matrix of a directed spanning tree of the communication topology, to equivalently transform the state consensus problem into a partial variable stability problem. Meanwhile, the results of the partial variable stability theory are used to derive a sufficient and necessary consensus criterion, expressed as the Hurwitz stability of a real matrix. Then, this criterion is further expressed as a bilinear matrix inequality condition, and, based on this condition, an iterative algorithm is proposed to find the gain matrices of the protocol. Finally, numerical examples are provided to verify the effectiveness of the proposed protocol design method.

Observer-based trajectory tracking control with preview action for a class of discrete-time Lipschitz nonlinear systems and its applications

In this article, the observer-based preview tracking control problem is investigated for a class of discrete-time Lipschitz nonlinear systems. To convert the observer-based trajectory tracking problem into a regulation problem, the classical difference technique is used to construct an augmented error system containing tracking error signal and previewable reference knowledge. Then, a state feedback controller with specific structures is taken into consideration. Sufficient design condition is established, based on the Lyapunov function approach, to guarantee the asymptotic stability of the closed-loop system. By means of some special mathematical derivations, the bilinear matrix inequality condition is successfully transformed into a tractable linear matrix inequality. Meanwhile, the gains of both observer and tracking controller can be computed simultaneously only in one step. As for the original system, the developed tracking control law is composed of an integrator, an observer-based state feedback controller, and a preview action term related to the reference signal. Finally, two numerical examples are provided to demonstrate the effectiveness of the theoretical method.

Observer-Based Fuzzy Fault-Tolerant Control for Nonlinear Parabolic PDEs

This paper investigates fault-tolerant control design for a class of nonlinear parabolic distributed parameter systems (DPSs) with actuator faults. Under the averaged measurements, an observer-based fuzzy fault-tolerant control law is constructed. Sufficient conditions in terms of bilinear matrix inequalities conditions are derived to ensure internal exponential stability and disturbance attenuation performance of system simultaneously by using Lyapunov–Krasovskii method. An iterative optimization algorithm is proposed to find feasible solutions. An example is given to illustrate the efficiency of main results.

Static Output Feedback Stabilization of Discrete-Time Linear Systems With Stochastic Dynamics Determined by an Independent Identically Distributed Process

This letter discusses an extension of our earlier results about state feedback stabilization toward static output feedback stabilization for discrete-time linear systems with stochastic dynamics determined by an independent and identically distributed (i.i.d.) process. A completely linearized necessary and sufficient inequality condition for static output feedback synthesis is known to be difficult to obtain even in the case with deterministic systems. Hence, this letter derives with the so-called S-variable approach a bilinear matrix inequality condition that can be effectively solved through exploiting the result of state feedback synthesis. An illustrative numerical example is also given for such synthesis.

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.

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.

Observer-based robust control for a flexible launch vehicle

This article investigates the problem of robust stabilization for a flexible launch vehicle. Since the launch vehicle suffers from parametric uncertainties, bending modes, and external wind disturbances simultaneously, an observer-based methodology is provided to address these negative factors. The proposed method can guarantee the stability of the closed-loop system and minimize the H∞ performance index. Additional regional pole placement constraints are imposed on the feedback gain matrices to improve the transient performance of the system. A two-step strategy is proposed to solve the involving bilinear matrix inequality problem. Compared with existing methods, which mainly depend on introducing additional constraints to linearize the bilinear matrix inequality conditions, the proposed strategy can reduce the conservatism and is suitable for engineering practice. The simulation results for one operating point and nonlinear model illustrate the validity and effectiveness of the proposed control method.

Robust H∞ control of uncertain switched systems defined on polyhedral sets with Filippov solutions

This paper considers the control problem of a class of uncertain switched systems defined on polyhedral sets known as piecewise linear systems where, instead of the conventional Carathéodory solutions, Filippov solutions are studied. In other words, in contrast to the previous studies, solutions with infinite switching in finite time along the facets and on faces of arbitrary dimensions are also taken into account. Firstly, established upon previous studies, a set of linear matrix inequalities are brought forward which determines the asymptotic stability of piecewise linear systems with Filippov solutions. Subsequently, bilinear matrix inequality conditions for synthesizing a robust controller with a guaranteed H∞ performance are presented. Furthermore, these results has been generalized to the case of piecewise affine systems. Finally, a V–K iteration algorithm is proposed to deal with the aforementioned bilinear matrix inequalities. The validity of the proposed method is verified through the analysis of two simulation examples.