State Feedback Gain Research Articles

Distributed controller design and performance optimization for discrete‐time linear systems

SummaryThis article addresses the problem of distributed controller design for linear discrete‐time systems. The problem is posed using the classical framework of state feedback gain optimization over an infinite‐horizon quadratic cost, with an additional sparsity constraint on the gain matrix to model the distributed nature of the controller. An equivalent formulation is derived that consists in the optimization of the steady‐state solution of a matrix difference equation, and two algorithms for distributed gain computation are proposed based on it. The first method consists in a step‐by‐step optimization of said difference matrix equation, and allows for fast computation of stabilizing state feedback gains. The second algorithm optimizes the same matrix equation over a finite time window to approximate asymptotic behavior and thus minimize the infinite‐horizon quadratic cost. To assess the performance of the proposed solutions, simulation results are presented for the problem of distributed control of a quadruple‐tank process, as well as a version of that problem scaled up to 40 interconnected tanks.

A Parameter Eigenstructure Assignment Design with Any Desired State Vector in Linear Time-Invariant System

The parametric design problem for eigenstructure assignment with arbitrary expected state vector is studied via state feedback in linear time invariant systems. The main idea of the considered problem is to design a parametric form of the state feedback, so that the designed closed-loop system has the desired eigenvalues, the desired eigenvectors and arbitrary expected state vector. Based on the parametric solution for a class of Sylvester matrix equation, the parametric expression of all the state feedback gain matrices is given in this paper. The free parameters including in the state feedback gain matrix can be further used to meet other performance requirements for the control system design. Finally, a simulation experiment is given to illustrate the effectiveness and simplicity of the proposed parametric method for the state feedback eigenstructure assignment problem with arbitrary expected state vector in the linear time invariant systems.

Observer‐based synthesis of linear parameter‐varying mixed sensitivity controllers

SummaryThis article presents a computationally efficient way of synthesizing linear parameter‐varying (LPV) controllers. It reviews the possibility of a separate observer and state feedback synthesis with guaranteed performance and shows that a standard mixed sensitivity problem can be solved in this way. The resultant output feedback controller consists of an LPV observer, augmented with dynamic filters to incorporate integral control and roll‐off properties, and an LPV state feedback gain. It is thus highly structured, which is beneficial for implementation. Moreover, it does not depend on scheduling parameter rates regardless of whether parameter‐dependent Lyapunov matrices are used during synthesis. A representative control design for active flutter suppression on an aeroelastic unmanned aircraft demonstrates the benefits of the proposed method in comparison with state‐of‐the‐art LPV output feedback synthesis.

Superstabilization of Descriptor Continuous-Time Linear Systems via State-Feedback Using Drazin Inverse Matrix Method

In this paper the descriptor continuous-time linear systems with the regular matrix pencil ( E , A ) are investigated using Drazin inverse matrix method. Necessary and sufficient conditions for the stability and superstability of this class of dynamical systems are established. The procedure for computation of the state-feedback gain matrix such that the closed-loop system is superstable is given. The effectiveness of the presented approach is demonstrated on numerical examples.

Interval Analysis of the Eigenvalues of Closed-Loop Control Systems with Uncertain Parameters

Uncertainty caused by a parameter measurement error or a model error causes difficulties for the implementation of the control method. Experts can divide the uncertain system into a definite part and an uncertain part and solve each part using various methods. Two uncertainty problems of the control system arise: problem A for the definite part—how does one find out the optimal number and position of actuators when the actuating force of an actuator is smaller than the control force? Problem B for the uncertain part—how does one evaluate the effect of uncertainty on the eigenvalues of a closed-loop control system? This paper utilizes an interval to express the uncertain parameters and converts the control system into a definite part and an uncertain part using interval theory. The interval state matrix is constructed by physical parameters of the system for the definite part of the control system. For Problem A, the paper finds out the singular value element sensitivity of the modal control matrix and reorders the optimal location of the actuators. Then, the paper calculates the state feedback gain matrix for a single actuator using the receptance method of pole assignment and optimizes the number and position of the actuators using the recursive design method. For Problem B, which concerns the robustness of closed-loop systems, the paper obtains the effects of uncertain parameters on the real and imaginary parts of the eigenvalues of a closed-loop system using the matrix perturbation theory and interval expansion theory. Finally, a numerical example illustrates the recursive design method to optimize the number and location of actuators and it also shows that the change rate of eigenvalues increases with the increase in uncertainty.

New gain‐scheduled static output feedback controller design strategy for stability and transient performance of LPV systems

In this study, a new static output feedback (SOF) control for application to linear parameter-varying (LPV) systems is proposed. The asymptotic stability of LPV systems is ensured based on the proposal of a design strategy for gain-scheduled static output feedback (GS-SOF) controllers. The approach selected for the design of the GS-SOF gains is based on a two-stage method, which consists of obtaining a state feedback gain in the first stage, and then in the second stage, using that information to derive the desired GS-SOF controller. In the controller design, the proposed strategy considers performance improvement regarding the specification of the minimum decay rate. The solution for the investigated problem is presented in terms of linear matrix inequalities (LMI) obtained using Finsler's Lemma. Less conservative LMI conditions are also proposed by considering parameter-dependent slack variables. For illustration purposes, the proposed method is implemented in practice for the design of GS-SOF controllers for an active suspension system. In the experiments, it is assumed that only one of its four system state variables is available for the measurement. The dynamic performance achieved in the practical implementation of the designed controllers demonstrates the efficiency of the method.

Event-triggered control co-design for rational systems

This work deals with the problem of designing stabilizing event-triggered state-feedback controllers for rational systems. Using differential algebraic representations and Lyapunov theory techniques, LMI-based conditions are derived to ensure regional asymptotic stability of the origin. These conditions are then cast into a convex optimization problem to the co-design of the event generator parameters and the state-feedback gain in order to reduce the controller updates while ensuring the asymptotic stability of the origin with respect to a given set of admissible initial conditions. The proposed methodology is illustrated by means of a numerical example.

Stabilizability and Bipartite Containment Control of Multi-Agent Systems Over Signed Directed Graphs

This paper investigates the stabilizability and bipartite containment control problem of general linear multi-agent systems over signed directed graphs, where the negative edges indicate the antagonistic interactions between agents. By employing the proposed bipartite containment control protocols over signed directed graphs, and combining the linear system stabilization theory and the signed Laplacian matrix, the necessary condition for the stabilization of multi-agent systems with multiple leaders is discussed. According to this necessary condition, a leader-follower matching method is presented to establish leader-follower multi-agent system networks. Based on the leader-follower topology established above and the designed state feedback gain, the stability of multi-agent systems is proved by using the system error and the algebraic Riccati equation. It is shown that the followers can gradually converge into the convex hull spanned by the states and the sign-inverted states of leaders, and the bipartite containment control of multi-agent systems can be realized. The simulation results verify the feasibility of the theoretical analysis.

Model Reference Control for Linear Time-Varying Systems: A Direct Parametric Approach

This paper investigates model reference control for linear time-varying (LTV) systems. It is shown that the problem can be decomposed into two sub-problems: a state feedback stabilization problem of LTV systems and a feed-forward compensation problem. Firstly, the sufficient condition for the existence of the model reference control is deduced. The condition concerns two coefficient matrices such that two matrix equations are met simultaneously, and the state feedback gain matrix stabilizes LTV systems according to the unified spectral theory. Secondly, generally complete parametrization of the model reference tracking controller including a state feedback controller and a feed-forward compensator is proposed based on the solution to a type of generalized Sylvester matrix equations. Furthermore, a general procedure for solving the model reference control problem is also provided. Finally, a numerical example shows the feasibility and effectiveness of the parametric design approach to the model reference control in LTV systems.

Finite-Time Stabilization for a Class of Nonlinear Singular Systems

The finite-time stabilization problem for a class of nonlinear singular systems is studied. Under the assumption that the considered system is impulse controllable, a sufficient condition is provided for the design of a state feedback control law guaranteeing the finite-time stability of the closed-loop system, and an explicit expression of the state feedback gain is also given. The proposed criterion is expressed in terms of strict matrix inequalities which is easy to be verified numerically. A numerical example is given to illustrate the effectiveness of the proposed method.

Adaptive control designed by online solving for Riccati and Lyapunov equations with nonlinear flight body

The aim of this paper is to control the path for nonlinear missile model in the pitch channel using model reference adaptive control (MRAC) and L1 adaptive control with linear quadratic regulator time-varying (LQRTV) strategy. Linear time-varying (LTV) model is designed where their parameters are varying with time. The nonlinear flying body, LTV model, and two adaptive control structures are modelled mathematically in the MATLAB-Simulink environment. LQRTV optimal control is designed using LTV model by online solving of Riccati Equation to get time-varying state feedback gain K(t). Adaptive control structures are designed using closed loop LTV model by online solving of Lyapunov equation to get time-varying Lyapunov gain matrix P(t). LQRTV and Lyapunov weight matrixes are tuned by Simulink design optimisation method. The results of two adaptive controls with the nonlinear flying body are compared. The wind effect and the dynamic uncertainties are researched.

Phase-Relaxed-Passive Full State Feedback Gain Limits for Series Elastic Actuators

Full state feedback (FSF) controllers for series elastic actuators bridge the gap between impedance and admittance controllers. For humanoid robots-which must stably accomplish both stiff and soft behaviors-FSF controllers are ideal candidate joint controllers. However, previous work on FSF gain tuning has used a nonlinear passivity analysis which does not account for time delay and can result in unstable behavior. In this article, we introduce a phase-based frequency domain approach to limiting these gains. This strategy can guarantee phase-relaxed passivity according to a relaxation parameter-an a priori bound on the regenerative efficiency of springs appearing in the environment. A simple demonstration illustrates how relaxed phase passivity behaves differently than strict passivity in how it handles the load inertia, allowing stiffness beyond the passive limit.

Local state-feedback stabilization of continuous-time Takagi-Sugeno fuzzy systems

Abstract This paper deals with the problem of local state-feedback stabilization for continuous-time nonlinear systems represented by Takagi-Sugeno (T-S) fuzzy models. The approach is based on a polytopic representation for the gradient of the membership functions but, differently from most of the available methods, bounds for the time-derivatives of the membership functions are not required. A two step strategy is proposed for the control design. First, a sufficient condition provides a stabilizing state-feedback gain for the dual system. Although there is no guarantee of stability for the original system, the controller is used as an initial condition for the second step of the method. If a feasible solution is found, a stabilizing state-feedback controller and an estimate of the domain of attraction are certified by means of a fuzzy Lyapunov function with polynomial dependence on the membership functions. The proposed conditions, given in terms of parameter-dependent linear matrix inequalities (LMIs) with a scalar search, can be solved by LMI relaxations with optimization variables considered as homogeneous polynomials of fixed degree. Examples based on T-S models borrowed from the literature illustrate that the method performs better than other existing approaches in terms of providing stabilizing gains associated with larger estimates for the domain of attraction.

A Robust Hybrid of a Feedback Linearization Technique and an Interval Type-2 Fuzzy Control System for the Flapping Angle Dynamics of a Biomimetic Aircraft

We introduce a new configuration of a robust and adaptive autopilot system for a model-scale flapping-wing aircraft. The system is specifically designed to achieve high performance flapping angle tracking in the face of large uncertainties. To describe the dynamics of the system, we leverage the benefits of both first principle modeling and data-driven approach (system identification technique). We introduce a high-performance robust and adaptive nonlinear control system by means of a feedback linearization (FL) technique, supported with an interval Type-2 fuzzy system due to its ability to accommodate the footprint-of-uncertainties (FoUs). While the first stage of our nonlinear control system is to cancel some predictable nonlinearities using an FL technique, the second phase of control is to accommodate the existing uncertainties in the system by way of an interval Type-2 fuzzy control technique, e.g., due to imperfect cancelation and modeling errors. This way, the stability and the robustness of the closed-loop control system can be guaranteed. We quantify the relative merit of our hybrid control system with respect to an FL technique, supported with a fixed gain state feedback controller and a Type-1 fuzzy system. Lastly, we also conduct stability analysis of the overall closed-loop control system.

Finite-time stabilization of linear systems by bounded linear time-varying feedback

Time-varying features are generally considered to be detrimental to the analysis and design of control systems. This paper establishes methods to design bounded linear time-varying (LTV) controllers such that the control performance of a linear time-invariant (LTI) system can be improved, that is, the finite-time stability of the closed-loop system can be obtained. Specifically, for an LTI control system, by using the solution to a parametric Lyapunov equation (PLE), a bounded LTV controller containing a suitable time-varying parameter is designed. By fully exploiting properties of the solution to the PLE, it is shown that the closed-loop system is finite-time stable. Both state feedback and observer based output feedback, in which both the observer gain and the state feedback gain are time-varying, are considered. As a consequence, the finite-time semi-global stabilization and the fixed-time (prescribed finite-time) stabilization problems for linear systems by bounded controls are solved. The established method is utilized to the design of the spacecraft rendezvous control system and its effectiveness is verified by simulations.

Linear Quadratic Optimal Control Design: A Novel Approach Based on Krotov Conditions

This paper revisits the problem of synthesizing the optimal control law for linear systems with a quadratic cost. For this problem, traditionally, the state feedback gain matrix of the optimal controller is computed by solving the Riccati equation, which is primarily obtained using calculus of variations- (CoV-) based and Hamilton–Jacobi–Bellman (HJB) equation-based approaches. To obtain the Riccati equation, these approaches require some assumptions in the solution procedure; that is, the former approach requires the notion of costates and then their relationship with states is exploited to obtain the closed-form expression of the optimal control law, while the latter requires a priori knowledge regarding the optimal cost function. In this paper, we propose a novel method for computing linear quadratic optimal control laws by using the global optimal control framework introduced by V. F. Krotov. As shall be illustrated in this article, this framework does not require the notion of costates and any a priori information regarding the optimal cost function. Nevertheless, using this framework, the optimal control problem gets translated to a nonconvex optimization problem. The novelty of the proposed method lies in transforming this nonconvex optimization problem into a convex problem. The convexity imposition results in a linear matrix inequality (LMI), whose analysis is reported in this work. Furthermore, this LMI reduces to the Riccati equation upon imposing optimality requirements. The insights along with the future directions of the work are presented and gathered at appropriate locations in this article. Finally, numerical results are provided to demonstrate the proposed methodology.

Stochastic LQ control under asymptotic tracking for discrete systems over multiple lossy channels

This study addresses the asymptotic tracking problem subjected to linear quadratic (LQ) constraints for linear discrete-time systems, where packet dropout occurs in actuating channels. To solve this objective control problem, the controller-coding co-design approach is adopted, i.e. the controller, encoder and decoder are designed for taking full advantage of the network resource collaboratively, thereby achieving better transmission of control signals. A stabilisability condition in the mean square sense that reveals the fundamental limitation among the H 2 norm of the plant, data arrival rates and coding matrices is first derived. Then, a solvability condition is conducted to handle the additional stochastic LQ control objective by a modified discrete-time algebraic Riccati equation, and an iterative algorithm is also given for designing the corresponding state feedback gain and coding matrices. Relied on such design, the asymptotic tracking constraint is further fulfilled through solving a Sylvester equation, and the feedforward gain related to tracking is parameterised. Finally, a simulation with the implementation of the design method on two cooperative robots is included to show the effectiveness of the current results.

Stability radius formulation of L σ ‐gain in positive stabilisation of regular and time‐delay systems

This study initially considers the relationship between stability radius and -gain of linear time-invariant positive systems. The -, -, and -gains of an asymptotically stable positive system are characterised in terms of stability radii and useful bounds are derived. The authors show that the structured perturbation of a stable matrix can be regarded as a closed-loop system with uncertainty structure represented by the unknown static output feedback. This makes it possible to relate the -gains in terms of closed-form expression available for stability radii of Metzler matrices. The authors generalise the above connection for positive-delay systems as well. Performance characterisation and computation of -gains are also given based on linear programming for and linear matrix inequality (LMI) for . The importance of this characterisation becomes evident when state feedback controllers are designed for regular and time-delay systems with positivity constraints. In particular, they show that positive stabilisation with maximum stability radius for the case of can be considered as an -gain minimisation, which can be solved by LMI. This inherently achieves the performance criterion and establishes a link to the reported iterative convex optimisation approaches that have been developed for the cases of and . A significant result of this study is the derivation of bounds for -gains and the unique commonality among the optimal state feedback gain matrices in obtaining -gains of the stabilised system.

Robust Control for Renewable-Integrated Power Networks Considering Input Bound Constraints and Worst Case Uncertainty Measure

Uncertainty from renewable energy and loads is one of the major challenges for stable grid operation. Various approaches have been explored to remedy these uncertainties. In this paper, we design centralized or decentralized state-feedback controllers for generators while considering worst-case uncertainty. Specifically, this paper introduces the notion of $\mathcal{L}_{\infty}$ robust control and stability for uncertain power networks. Uncertain and nonlinear differential algebraic equation model of the network is presented. The model includes unknown disturbances from renewables and loads. Given an operating point, the linearized state-space presentation is given. Then, the notion of $\mathcal{L}_{\infty}$ robust control and stability is discussed, resulting in a nonconvex optimization routine that yields a state feedback gain mitigating the impact of disturbances. The developed routine includes explicit input-bound constraints on generators' inputs and a measure of the worst-case disturbance. The feedback control architecture can be centralized, distributed, or decentralized. Algorithms based on successive convex approximations are then given to address the nonconvexity. Case studies are presented showcasing the performance of the $\mathcal{L}_{\infty}$ controllers in comparison with automatic generation control and $\mathcal{H}_{\infty}$ control methods.

Output feedback robust MPC for linear systems with norm-bounded model uncertainty and disturbance

This paper presents an output feedback robust model predictive control strategy for linear systems with norm-bounded model parametric uncertainty and disturbance. In order to reduce the computational burden and enhance the control performance, in the optimization problem we optimize the estimator state matrix instead of the state estimator gain. The infinite horizon control moves and estimated states are parameterized by a dynamic output feedback control law, where its state feedback gain and estimator state matrix are optimized at each sampling instant. By appropriately refreshing the estimation error set, the optimization problem is shown to be recursively feasible and the convergence of the augmented state is guaranteed. A numerical example is given to show the effectiveness of the proposed approach.