Linear State Feedback Control Research Articles

Stable active running of a planar biped robot using Poincare map control

This work formulates the active limit cycles of bipedal running gaits for a compliant leg structure as the fixed point of an active Poincare map. Two types of proposed controllers stabilize the Poincare map around its active fixed point. The first one is a discrete linear state feedback controller designed with appropriate pole placement. The discrete-time control first uses purely constant torques during stance and flight phase, then discretizes each phase into smaller constant-torque intervals. The other controller is an invariant manifold based chaos controller: a generalized Ott, Grebogi and Yorke controller having a linear form and a nonlinear form. Both controllers can stabilize active running gaits on either even or sloped terrains. The efficiency of these controllers for bipedal running applications are compared and discussed.

State feedback control and variable step size MPPT algorithm of three-level grid-connected photovoltaic inverter

In this paper, the state feedback linearization control technique is applied for controlling the power factor of a grid-connected multi-level photovoltaic inverter. By applying this technique, the nonlinear state model of three-level inverter – in the d–q reference frame – will be transformed into two equivalent linear subsystems. Hence, the pole placement linear control loops can be applied on these linear subsystems in order to separately control the grid power factor and the dc link voltage of the inverter. The control system includes also a Maximum Power Point Tracker (MPPT) based on variable step size incremental conductance algorithm. Compared with conventional fixed step size method, the variable step MPPT improves the speed and the tracking accuracy. It has been shown that the three-level inverter allows reduction of the THD of grid voltage and current as well as the reduction of the blocking voltage of the inverter switches.

Improving Stability Margin in Electric Power Systems by Linear Quadratic Regulator and Disturbance Model

This article deals with the design of a voltage controller using state feedback with integral action for an electric power system. The control system is tailored to achieve zero error of the voltage magnitude. First the transient response and the attraction region of the non-linear model is improved by using a state-feedback linear quadratic regulator controller. On the basis of this design, the integral action is designed as a second loop according to the internal model controller. Both feedback control systems are compared in terms of the robustness of the closed-loop dynamic response and the existing coupling between active and reactive powers when the load reactive power is changed. Simulation results show that the integral controller achieves zero voltage error even for the non-linear system.

Robust Control of Uncertain Singular Systems with Distributed Delays

Robust control for a class of singular systems with distributed delaysand uncertainties is researched. Firstly, a linear state-feedback controller isdesigned. Secondly, without model transformation and matrices decomposition, Asufficient condition for the studied systems, which makes sure the systems tobe regular, impulse free and asymptotically stable, is given by constructingthe appropriate Lyapunov functionals in terms of linear matrixinequalities(LMIs) method and integral inequality technique. The obtained results can be shown in LMIs andsovled easily by LMI toolbox in Matlab. At last, A numerical example is givento illustrate the validity of the method

Global exponential dissipativity and stabilization of memristor-based recurrent neural networks with time-varying delays

This paper addresses the global exponential dissipativity of memristor-based recurrent neural networks with time-varying delays. By constructing proper Lyapunov functionals and using M-matrix theory and LaSalle invariant principle, the sets of global exponentially dissipativity are characterized parametrically. It is proven herein that there are 22n2−n equilibria for an n-neuron memristor-based neural network and they are located in the derived globally attractive sets. It is also shown that memristor-based recurrent neural networks with time-varying delays are stabilizable at the origin of the state space by using a linear state feedback control law with appropriate gains. Finally, two numerical examples are discussed in detail to illustrate the characteristics of the results.

Compliant Leg Architectures and a Linear Control Strategy for the Stable Running of Planar Biped Robots

This paper investigates two fundamental structures for biped robots and a control strategy to achieve stable biped running. The first biped structure contains straight legs with telescopic springs, and the second one contains knees with compliant elements in parallel with the motors. With both configurations we can use a standard linear discrete-time state-feedback control strategy to achieve an active periodic stable biped running gait, using the Poincare map of one complete step to produce the discrete-time model. In this case, the Poincare map describes an open-loop system with an unstable equilibrium, requiring a closed-loop control for stabilization. The discretization contains a stance phase, a flight phase and a touch-down. In the first approach, the control signals remain constant during each phase, while in the second approach both phases are discretized into a number of constant-torque intervals, so that its formulation can be applied easily to stabilize any active biped running gait. Simulation results with both the straight-legged and the kneed biped model demonstrate stable gaits on both horizontal and inclined surfaces.

Maximal perturbation bounds for robust stabilizability of fractional-order systems with norm bounded perturbations

This paper investigates the maximal perturbation bound problem for robust stabilizability of the fractional-order system with two-norm bounded perturbations or infinity-norm bounded perturbations. Firstly, a necessary condition and several sufficient conditions for robust stabilization are derived. Secondly, linear matrix inequality approaches for computing the maximal robust stabilizability perturbation bound of such perturbed fractional-order system with a linear state feedback controller, simultaneously obtaining the corresponding linear state feedback stabilizing controller are presented. With the help of the linear matrix inequality solvers, we can easily obtain the maximal robust stabilizability perturbation bound and the corresponding linear state feedback stabilizing controller. Finally, simulation examples are given to demonstrate the effectiveness of the proposed approaches.

Robust control for fractional-order four-wing hyperchaotic system using LMI

This paper reports a new fractional-order four-dimensional (4D) smooth autonomous system which is special since it can generate a four-wing hyperchaotic attractor but has only one zero equilibrium. The issue of robust control for the fractional-order hyperchaotic system with uncertainties and interference is considered. On the basis of the fractional Lyapunov stability theorem, a linear state feedback controller is derived to guarantee robust Mittag–Leffler stability of the fractional-order hyperchaotic system. The control strength matrix is obtained by solving the linear matrix inequality (LMI). The effectiveness and robustness of the proposed scheme is verified via numerical simulation.

Robust and resilient finite-time bounded control of discrete-time uncertain nonlinear systems

Finite-time state-feedback stabilization is addressed for a class of discrete-time nonlinear systems with conic-type nonlinearities, bounded feedback control gain perturbations, and additive disturbances. First, conditions for the existence of a robust and resilient linear state-feedback controller for this class of systems are derived. Then, using linear matrix inequality techniques, a solution for the controller gain and the maximum allowable bound on the gain perturbation is obtained. The developed controller is robust for all unknown nonlinearities lying in a known hypersphere with an uncertain center and all admissible disturbances. Moreover, it is resilient against any bounded perturbations that may alter the controller’s gain by at most a prescribed amount. The paper is concluded with a numerical example showcasing the applicability of the main result.

Receptance method for active vibration control of a nonlinear system

This paper presents the application of the receptance method to nonlinear systems for active vibration control. The method, previously established for linear systems, is extended to a class of single-degree-of-freedom nonlinear systems that can be characterised using describing functions. A significant advantage of the receptance method is that there is no requirement to know the system parameters such as mass, damping and stiffness terms, typically obtained using finite element methods. The method is particularly advantageous for nonlinear systems, since there is no requirement for nonlinear identification. A linear state feedback controller is applied to an example of a single-degree-of-freedom Duffing oscillator, to assign the peak resonance to a prescribed value using the established Sherman–Morrison receptance method. It is then demonstrated that an iterative form of the Sherman–Morrison receptance method is required for the accurate assignment of this peak resonance, in order to account for changes in the open-loop receptance. Both harmonic balance and Volterra series representations are investigated to approximate the receptance in the complex domain, and their advantages and disadvantages are discussed in a numerical example.

A global sliding mode controller for missile electromechanical actuator servo system

A global sliding mode controller (GSMC) is proposed for the missile electromechanical actuator (EA) servo system, where exists high uncertainties, such as parameter variations and external disturbances. By the design of an optimal integral switching function based on optimal linear quadratic regulator (LQR) theory, the initial state of system is set on the switching surface, and the optimal sliding mode motion is produced. The proposed GSMC is composed of an optimal linear state feedback controller (OLSFC), and a fuzzy nonlinear robust controller (FNRC), which can be designed respectively. The OLSFC, generated by the designed switching function, intends to minimise a quadratic performance index, and then improves the dynamic performance of system. Meanwhile, the FNRC employs a fuzzy decision maker (FDM), which estimates the upper bound of uncertainties as FNRC’s gain adaptively, and then makes GSMC robust and control input smooth. With the computer simulations on an EA experiment plant, it presents that the proposed scheme possesses good tracking precision, effective suppression against chattering at control input, and strong robustness against system uncertainties.

Optimal control of Takagi–Sugeno fuzzy-model-based systems representing dynamic ship positioning systems

Orthogonal function approach (OFA) and the hybrid Taguchi-genetic algorithm (HTGA) are used to solve quadratic finite-horizon optimal controller design problems in both a fuzzy parallel distributed compensation (PDC) controller and a non-PDC controller (linear state feedback controller) for Takagi–Sugeno (TS) fuzzy-model-based control systems for dynamic ship positioning systems (TS-DSPS). Based on the OFA, an algorithm requiring only algebraic computation is used to solve dynamic equations for TS-fuzzy-model-based feedback and is then integrated with HTGA to design quadratic finite-horizon optimal controllers for TS-DSPS under the criterion of minimizing a quadratic finite-horizon integral performance index, which is also converted to algebraic form by the OFA. Integration of OFA and HTGA in the proposed approach enables use of simple algebraic computation and is well adapted to the computer implementation. Therefore, it facilitates design tasks of quadratic finite-horizon optimal controllers for the TS-DSPS. The applicability of the proposed approach is demonstrated in the example of a moored tanker designed using quadratic finite-horizon optimal controllers.

NetSimplex: Controller Fault Tolerance Architecture in Networked Control Systems

The assurance of reliability becomes increasingly challenging as the complexity of networked control systems (NCS) rapidly increases. Simplex architecture was designed to tolerate control software design and implementation. This architecture consists of a high assurance controller (HAC) and a high performance controller (HPC). The HAC uses the linear state feedback control to create a large maximum stability region (MSR). The HPC aims at achieving a better control performance and may use any design. However, the plant's states under HPC must stay within the MSR, or the control is switched to HAC.

An optimization algorithm for designing a stabilizing controller for linear time-varying uncertain systems with state delays

This paper investigates the problem of designing a linear memoryless state feedback control to stabilize a class of linear uncertain systems with state delays. Each uncertain parameter and each delay under consideration may take arbitrarily large values. In such a situation, the locations of uncertain entries in the system matrices play an important role. So far, obtained conditions show that if a system has a particular geometric configuration called a triangular antisymmetric configuration, then the system is stabilizable however large the given bounds of uncertain variations might be. However, the procedure for designing a stabilizing controller has still not been fully examined although the proof was successfully completed. The objective of this paper is to provide a novel optimization algorithm for designing a controller to stabilize a linear uncertain delay system independently of the given bounds of uncertain parameters and time delays. An illustrative example verifies the usefulness of the proposed method.

Study on Indefinite Stochastic Linear Quadratic Optimal Control with Inequality Constraint

This paper studies the indefinite stochastic linear quadratic (LQ) optimal control problem with an inequality constraint for the terminal state. Firstly, we prove a generalized Karush-Kuhn-Tucker (KKT) theorem under hybrid constraints. Secondly, a new type of generalized Riccati equations is obtained, based on which a necessary condition (it is also a sufficient condition under stronger assumptions) for the existence of an optimal linear state feedback control is given by means of KKT theorem. Finally, we design a dynamic programming algorithm to solve the constrained indefinite stochastic LQ issue.

Global Output Feedback Stabilization of a Class of Nonlinear Systems via Linear Sampled-Data Control

In the literature, it has been proved that under a lower-triangular linear growth condition, a class of uncertain nonlinear systems can be globally stabilized by a linear state feedback controller (Tsinias) and later by a linear output feedback controller (Qian and Lin), both in the continuous-time form. This technical note shows that the same continuous-time system under the same assumption can be globally stabilized by a sampled-data output feedback controller whose observer and control law are discrete-time and linear, and hence can be easily implemented by computers.

Controllability and Controller-Observer Design for a Class of Linear Time-Varying Systems

In this paper a class of linear time-varying control systems is considered. The time variation consists of a scalar time-varying coefficient multiplying the state matrix of an otherwise time-invariant system. Under very weak assumptions of this coefficient, we show that the controllability can be assessed by an algebraic rank condition, Kalman canonical decomposition is possible, and we give a method for designing a linear state-feedback controller and Luenberger observer.

Decentralized MRAC for large-scale interconnected systems with time-varying delays and applications to chemical reactor systems

The model reference adaptive control problem is investigated for a class of large-scale systems with time-varying delays. The considered systems have mismatched delay functions and matched interconnections. Firstly, a state coordinate transformation is employed to convert the original error system into a cascade system. Secondly, a delay-dependent virtual linear state feedback controller is developed to stabilize the first subsystem. Based on the virtual controller, a memoryless state feedback controller is constructed for the second subsystem. By choosing new Lyapunov Krasovskii functional, we show that the designed decentralized continuous adaptive controller renders that the solutions of the closed-loop system converge exponentially to a bounded region. Finally, the theoretic achievements are applied to the control design of a chemical reactor system with two subsystems. The control results show the effectiveness of the proposed method.

Truncated predictor feedback for linear systems with long time-varying input delays

In this paper we study the problem of stabilizing a linear system with a single long time-varying delay in the input. Under the assumption that the open-loop system is stabilizable and not exponentially unstable, a finite dimensional static time-varying linear state feedback controller is obtained by truncating the predictor based controller and by adopting the parametric Lyapunov equation based controller design approach. As long as the time-varying delay is exactly known and bounded, an explicit condition is provided to guarantee the stability of the closed-loop system. It is also shown that the proposed controller achieves semi-global stabilization of the system if its actuator is subject to either magnitude saturation or energy constraints. Numerical examples show the effectiveness of the proposed approach.

Minkowski terminal cost functions for MPC

This technical communique delivers a systematic procedure for obtaining a suitable terminal cost function for model predictive control based on Minkowski cost functions. It is shown that, for any given stabilizing linear state feedback control law and associated λ-contractive proper C-set, there always exists a non-trivial scaling of the λ-contractive proper C-set such that the associated Minkowski function satisfies the standard MPC terminal cost stability inequality.