Linear State Feedback Control Research Articles

Prescribed-Time Stabilization of Singularly Perturbed Systems

Dear Editor, This letter investigates the prescribed-time stabilization of linear singularly perturbed systems. Due to the numerical issues caused by the small perturbation parameter, the off-the-shelf control design techniques for the prescribed-time stabilization of regular linear systems are typically not suitable here. To solve the problem, the decoupling transformation techniques for time-varying singularly perturbed systems are combined with linear time-varying high gain feedback design techniques. A composite linear time-varying state feedback controller is designed, and the existence of the time-varying Chang transformation matrix for decoupling the slow and fast dynamics is guaranteed. As a result, the prescribed-time stability is ensured. Finally, a numerical example is provided to illustrate the effectiveness of the results.

Autonomous cooperative formation control of underactuated USVs based on improved MPC in complex ocean environment

A dual model predictive control (DMPC) method based on virtual trajectory is proposed in this paper, in order to achieve autonomous cooperative formation control of underactuated unmanned surface vehicles (USVs) in complex ocean environment. Firstly, the formation tracking error model of the USVs is designed, and considering the large initial state tracking error of USVs, a virtual transition trajectory is put forward to guide the design of tracking controller. To solve the mutation problem caused by misalignment of the follower USV's desired trajectory in the early stage, an improved differential tracker is introduced into virtual leader USV to smooth the desired trajectory of the follower USV. In addition, a dual mode switching strategy is designed to decide when and how to quit the transition of the virtual USV. A nonlinear disturbance observer is introduced to compensate the complex marine environment interference. Finally, by introducing terminal penalty function and linear state feedback controller, the Lyapunov function is constructed to prove the control stability of the proposed model predictive control method in finite time domain, and the effectiveness and reliability of the proposed method are verified by simulation experiments.

Finite time settling controller for a hierarchically linearized quadcopter model

In this paper, we apply finite-time settling control to a hierarchical linearized quadcopter model and verify its robustness against model errors by simulation. Hierarchical linearization is one of the exact linearization method and is linearizable on all points. Finite-time settling control has the property of finite-time stability, which perfectly matches the equilibrium point in a finite-time range. This property increases robustness near the equilibrium point. Thus, it is expected to improve the stability of drones that are vulnerable to wind disturbances and model errors. As a comparison, linear state feedback control with exponential stability is used. The feedback gain is calculated using linear quadratic regulator(LQR), which is known to have high robustness. The simulation results show that the finite-time settling controller has higher the trajectory tracking performance than the linear quadratic regulator.

CONTROL OF DYNAMIC TRIM FOR PLANING VESSELS WITH INTERCEPTORS IN TERMS OF COMFORT AND MINIMUM DRAG

Nowadays, interceptors are often used to decrease total resistance and enhance comfort by reducing dynamic trim for high-speed planing vessels. They can be controlled manually as well as automatically by using a suitable closed-loop control system. Thus, in the present study, an automatically controllable system is presented to minimize the total resistance by reducing the dynamic trim in calm water. To reach this aim, a mathematical model which can represent the 2 degree of freedom vertical motion of a prismatic planing vessel is presented. The coefficients used in the model are calculated by using the Savitsky method. The standard dynamic trim angle and the optimum ones in terms of resistance are calculated by using the same method. For control action, a linear full state feedback control strategy (linear quadratic regulator) is applied, and instantaneous blade heights are found considering the change in forward speed. Therefore, the control-oriented model is able to change the blade height to reach the optimum trim angle in terms of the total resistance of the vessel for different forward speeds and speed profiles. The results show that the designed linear quadratic regulator control strategy is successful for reference trim tracking problems.

Analysis and Design of Secure Sampled-Data Control Subject to Denial-of-Service Attacks

This study addresses the issue of secure control design for cyber-physical systems (CPS) against denial of service (DoS) attacks. We take into account a continuous-time linear system with a convex quadratic performance measure and a sampled linear state feedback control. DoS attacks impose constraints on the CPS, where packets may be jammed between the sensor and controller by a malicious entity, potentially resulting in system instability and performance degradation. We assume that the attacker can perform DoS attacks with a limited time and frequency due to energy restrictions. We devise an efficient procedure using the linear matrix inequality approach to compute an upper bound on the performance degradation brought on by the DoS attack. We also propose a redesign of the controller to minimize this performance degradation. Finally, a simulation example illustrates the computation of the performance degradation under a bounded DoS attack and the design of a secure controller. Simulation results show that the designed controller effectively keeps the feedback loop’s performance and stability under attack.

Autonomous Landing of an UAV Using H∞ Based Model Predictive Control

Possibly the most critical phase of an Unmanned Air Vehicle (UAV) flight is landing. To reduce the risk due to pilot error, autonomous landing systems can be used. Environmental disturbances such as wind shear can jeopardize safe landing, therefore a well-adjusted and robust control system is required to maintain the performance requirements during landing. The paper proposes a loop-shaping-based Model Predictive Control (MPC) approach for autonomous UAV landings. Instead of conventional MPC plant model augmentation, the input and output weights are designed in the frequency domain to meet the transient and steady-state performance requirements. Then, the H∞ loop shaping design procedure is used to synthesize the state-feedback controller for the shaped plant. This linear state-feedback control law is then used to solve an inverse optimization problem to design the cost function matrices for MPC. The designed MPC inherits the small-signal characteristics of the H∞ controller when constraints are inactive (i.e., perturbation around equilibrium points that keep the system within saturation limits). The H∞ loop shaping synthesis results in an observer plus state feedback structure. This state estimator initializes the MPC problem at each time step. The control law is successfully evaluated in a non-linear simulation environment under moderate and severe wind downburst. It rejects unmeasured disturbances, has good transient performance, provides an excellent stability margin, and enforces input constraints.

Asynchronous Guaranteed Cost Control of 2-D Markov Jump Roesser Systems.

This article is concerned with the problem of guaranteed cost control for 2-D Markov jump Roesser systems with mismatched modes. The hidden Markov model is introduced to describe the asynchronous phenomenon caused by mismatched modes, and an asynchronous linear state-feedback control law is designed based on this model. With the help of the 2-D Lyapunov function and linear matrix inequality (LMI) techniques, sufficient conditions are established to ensure the asymptotic stability of the concerned system with a bound of the predefined guaranteed cost under three different boundary conditions, respectively. Finally, an algorithm that concludes the design processes of the optimal asynchronous guaranteed cost control law is proposed, and a numerical example is provided to verify its effectiveness.

Robust Precision Position Tracking of Planar Motors Using Min-Max Model Predictive Control

In this article, a min-max model predictive control (MPC) method of planar motors is proposed for the first time to achieve robust precision position tracking, which has a low computational burden and strong capability to deal with the problems of stability, robustness, optimization, and input constraints. A state-space model with a homogeneous state equation is built to describe the dynamics of the time-varying reference trajectory. Combining the state-space model of the reference trajectory and that of the planar motor, an augmented state-space model is established to obtain an error state formulation. Then, using the error state formulation, a min-max optimal control problem subject to the constraints on bounded uncertainty, stability, and control input is developed. Moreover, applying the theory of asymptotically stable invariant ellipsoids and employing the nested invariant ellipsoids, the explicitly linear state-feedback control laws are obtained using a linear-matrix-inequalities based offline control algorithm. Finally, the min-max MPC is applied to a planar motor system developed in the laboratory for an experimentally comparative study. The results demonstrate the effectiveness of the proposed min-max MPC of planar motor for robust precision position tracking applications.

Domain stabilization in probability in a fixed time for nonlinear stochastic systems via feedback control

AbstractThis paper is concerned with the domain stabilization in probability in a fixed time for nonlinear stochastic systems. More specifically, given a target domain , a probability and a fixed time , we aim to find suitable state feedback control laws such that the trajectory of the resulting closed‐loop system, starting from outside , will reach the target domain in the fixed time with the minimum probability . To this end, we prove a local existence and uniqueness result of solutions to nonlinear stochastic systems under mild conditions, and derive upper bounds of the mean of the first hitting time with respect to under various rigorous conditions. By these results, a Lyapunov‐based controller design is given to guarantee the domain stability for nonlinear stochastic systems with affine control inputs. Besides, inequalities of quadratic forms focusing on semi‐linear stochastic systems are used to design a linear state feedback controller for achieving the domain stability in probability. Several examples are given for demonstration.

Sliding-Mode Control of Full-State Constraint Nonlinear Systems: A Barrier Lyapunov Function Approach

This study presents the design of a robust control based on the sliding-mode theory to solve both; the stabilization and the trajectory tracking problems of nonlinear systems subjected to a class of full-state restrictions. The selected nonlinear system satisfies a standard Lagrangian structure affected by nonparametric uncertainties. A barrier Lyapunov function is used to ensure the state constraints by designing a time-varying gain, which guarantees the fulfillment of the predefined state constraints even under external perturbations. The proposed design methodology for the barrier sliding-mode control (BSMC) ensures the convergence of the sliding surface in finite time to the origin. Consequently, the asymptotic convergence of the states to the corresponding equilibrium point is achieved. The finite-time stability of the origin in the closed-loop system with the proposed controller has been demonstrated using the second Lyapunov stability method. The suggested controller was evaluated on a two-link robotic manipulator. Then, the obtained results showed better stabilization and tracking performances (while the restrictions are satisfied) than the traditional first-order sliding-mode or linear state feedback controllers.

Non-overshoot tracking control for positive continuous-time linear systems

This article focuses on state and output tracking control for positive continuous-time linear systems without overshoot. Linear state feedback controllers are designed respectively to ensure both the system state, and output can unilateral track the reference state and output. In addition, using Lyapunov direct method, the exponential convergence rate of tracking system is calculated. A numerical example is given to demonstrate the effectiveness of the proposed method.

On interval observer design for active Fault Tolerant Control of Linear Parameter-Varying systems

This paper proposes an active Fault Tolerant Control (FTC) scheme for polytopic uncertain Linear Parameter-Varying (LPV) systems subject to uncertainties and actuator faults. First, a fault estimation interval observer is designed to estimate the system state and the actuator fault. A novel approach is developed using the L ∞ norm to attenuate the effects of the uncertainties and to improve the accuracy of the proposed observer. Then, based on the fault estimation information, the FTC strategy is designed using a linear state feedback control law and H ∞ technique to compensate actuator faults and maintain system performance and stability, even under faulty conditions. Finally, the effectiveness of the proposed method is demonstrated by its application to a vehicle lateral dynamic nonlinear model. • Fault Tolerant Control using interval state estimation and robust stabilization. • Set-membership framework for the design of Linear Parameter Varying systems. • Fault estimation observer design for descriptor system. • L ∞ norm-based design method to attenuate the effect of fault and uncertainties and to improve the estimation performance.

Distributed Differential Games for Control of Multi-Agent Systems

Motivated by the challenges arising in the field of multi-agent system (MAS) control, we consider linear heterogeneous MAS subject to local communication and investigate the problem of designing distributed controllers for such systems. We provide a game theoretic framework for systematically designing distributed controllers, taking into account individual objectives of the agents and their possibly incomplete knowledge of the MAS. Linear state-feedback control laws are obtained via the introduction of a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">distributed differential game</i> , namely, the combination of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">local non-cooperative differential games</i> , which are solved in a decentralized fashion. Conditions for stability of the MAS are provided for the special cases of acyclic and strongly connected communication graph topologies. These results are then exploited to provide stability conditions for general graph topologies. The proposed framework is demonstrated on a tracking synchronization problem associated with the design of a distributed secondary voltage controller for microgrids and on a numerical example.

Hidden and Coexisting Attractors in a Novel 4D Hyperchaotic System with No Equilibrium Point

The investigation of chaotic systems containing hidden and coexisting attractors has attracted extensive attention. This paper presents a four-dimensional (4D) novel hyperchaotic system, evolved by adding a linear state feedback controller to a 3D chaotic system with two stable node-focus points. The proposed system has no equilibrium point or two lines of equilibria, depending on the value of the constant term. Complex dynamical behaviors such as hidden chaotic and hyperchaotic attractors and five types of coexisting attractors of the simple 4D autonomous system are investigated and discussed, and are numerically verified by analyzing phase diagrams, Poincaré maps, the Lyapunov exponent spectrum, and its bifurcation diagram. The short unstable cycles in the hyperchaotic system are systematically explored via the variational method, and symbol codings of the cycles with four letters are realized based on the topological properties of the trajectory projection on the 2D phase space. The bifurcations of the cycles are explored through a homotopy evolution approach. Finally, the novel 4D system is implemented by an analog electronic circuit and is found to be consistent with the numerical simulation results.

Stabilization of the cart-inverted-pendulum system using trivial state-feedback to output-feedback control conversion

A new linear observer-free output-feedback controller with five adjustable parameters is proposed to stabilize the cart-inverted-pendulum system (CIP) at the unstable equilibrium point. The controller architecture is deduced from a trivial conversion of the linear state-feedback controller that is obtained using a two-step method. First, based on a set of cart change variables, a slightly modified state-feedback controller is developed. Then, the output-feedback controller is obtained through the judicious combination of the cart step reference input internal model and a convenient open-loop state estimator with the above modified state-feedback controller. The local stability of the output-based control system is conducted using the signature formulas method to get simplified conditions. A partial single parameter tuning method and optimal global single parameter tuning method are proposed for adjusting the controller gains to maximize a new efficiency-based objective function. Numerical simulations are first conducted to reveal the simplicity of output-feedback controller design using the partial tuning method, where the state-feedback gains are assumed to be known. Then, an optimal output-feedback controller is designed using the global tuning method. The proposed output-feedback controller is equivalent in terms of performance efficiency to the best five-parameter output-feedback two PID controller.

First Order Approximation of Model Predictive Control Solutions for High Frequency Feedback

The lack of computational power on mobile robots is a well-known challenge when it comes to implementing a real-time MPC scheme to perform complex motions. Currently the best solvers are barely able to reach 100 Hz for computing the control of a whole-body legged model, while modern robots are expecting new torque references in less than 1 ms. This problem is usually tackled by using a handcrafted low-level tracking control whose inputs are the low-frequency trajectory computed by the MPC. We show that a linear state feedback controller naturally arises from the optimal control formulation and can be used directly in the low-level control loop along with other sensitivities of relevant time-varying parameters of the problem. When the optimal control problem is solved by DDP, this linear controller can be computed for cheap as a by-product of the backward pass, and corresponds in part to the classical Riccati gains. A side effect of our proposition is to show that Riccati gains are valuable assets that must be used to achieve an efficient control and that they are not stiffer than the optimal control scheme itself. We propose a complete implementation of this idea on a full-scale humanoid robot and demonstrate its importance with real experiments on the robot Talos.

Spectral stabilization of linear transport equations with boundary and in-domain couplings

In this work, the problem of stabilization of general systems of linear transport equations with indomain and boundary couplings is investigated. It is proved that the unstable part of the spectrum is of finite cardinal. Then, using the pole placement theorem, a linear full state feedback controller is synthesized to stabilize the unstable finite dimensional part of the system. Finally, by a careful study of semigroups, we prove the exponential stability of the closed-loop system. As a by product, the linear control constructed before is saturated and a fine estimate of the basin of attraction is given.

Robust Feedback Control for Discrete-Time Systems Based on Iterative LMIs with Polytopic Uncertainty Representations Subject to Stochastic Noise

This paper deals with the design of linear observer-based state feedback controllers with constant gains for a class of nonlinear discrete-time systems in the form of a quasi-linear representation in presence of stochastic noise. For taking into account nonlinearities in the design of linear observer-based state feedback controllers, a polytopic modeling approach is investigated. An optimization problem is formulated to reduce the sensitivity of the controlled system towards stochastic input, state, and output noise with a predefined covariance. Due to the nonlinearities, the separation principle does not hold, thus, the controller and the observer have to be designed simultaneously. For this purpose, a Lyapunov-based method is used, which provides, in addition to the controller and observer gains, a stability proof for the nonlinear closed loop in a predefined polytopic domain. In general, this leads to nonlinear matrix inequalities. To solve these nonlinear matrix inequalities efficiently, we propose an approach based on linear matrix inequalities (LMIs) with a superposed iteration rule. When using this iterative LMI approach, a minimization task can be solved additionally, which desensitizes the closed loop to stochastic noise. The proposed method additionally enables the consideration of different linear closed loop structures by a unified Lyapunov-based framework. The efficiency of the proposed approach is demonstrated and compared with a classical LQG approach for a nonlinear overhead traveling crane.

A novel multi-objective optimization algorithm for Pareto design of a fuzzy full state feedback linearization controller applied on a ball and wheel system

The control and stabilization of a ball and wheel system around the equilibrium point are challenging tasks because it is an underactuated, nonlinear, and open-loop unstable plant. In this paper, Pareto design of a Fuzzy Full State Feedback Linearization Controller (FFSFLC) for the ball and wheel system based upon a novel multi-objective optimization algorithm is introduced. To this end, at first, a full state feedback linearization approach is employed to stabilize the dynamics of the system. Next, appropriate fuzzy systems are determined to tune the control gains. Then, a new multi-objective optimization algorithm is utilized to promote the proposed control scheme. This optimization algorithm is a combination of Simulated Annealing (SA) and Artificial Bee Colony (ABC) approaches benefiting advantages of the non-dominated Pareto solutions. To evaluate the capabilities of the suggested algorithm, its optimal solutions of several standard test functions are compared with those of five renowned multi-objective optimization algorithms. The results confirm that the proposed hybrid algorithm yields closer non-dominated Pareto solutions to the true optimal Pareto front with shorter runtimes than other algorithms. After selecting proper objective functions, multi-objective optimization of FFSFLC for the ball and wheel system is performed, and the results are compared with previous works. Simulations illustrate that the proposed strategies can accurately converge the system states to the desired conditions and yield superior robustness against disturbance signals in comparison with former studies.

Responsive surging, heading and diving controls of autonomous underwater vehicle based on brute forcing and smoothing of controllers

There are many types of controllers had been used to control Autonomous Underwater Vehicle (AUV) such as Proportional Integral Derivative (PID), Linear Quadratic Regulator (LQR), state feedback linearization, integrator back-stepping, and Sliding-Mode Control (SMC). However, for PID and SMC in particular, it is difficult to determine the optimal control design parameters. The objective of this study is to design and develop a responsive motion control system with optimal parameters for an AUV. The contribution of this paper is in term of introducing a filter to smooth reference signal and proposing a brute forcing technique to find optimal controller parameters. The methodology starts with modeling the AUV, estimating the unknown parameters from a real AUV model, designing a control system based on PI and SMC methods, and finally optimizing the controller parameters. The controller design was onto controlling surge speed using PI, heading using SMC, and diving using SMC. Simulation-wise, the developed control system has an average value of 93.89 % of responsiveness to track desired trajectory while 82.33 % of responsiveness without using the smoothing filter. The tested input signals were unit step, ramp, parabolic, and sinusoidal.