Convergence Properties of Iterative Learning Control Processes in the Sense of the Lebesgue‐P Norm

This paper addresses the convergence issue of first‐order and second‐order PD‐type iterative learning control schemes for a type of partially known linear time‐invariant systems. By taking advantage of the generalized Young inequality of convolution integral, the convergence is analyzed in the sense of the Lebesgue‐p norm and the convergence speed is also discussed in terms of Q_p factors. Specifically, we find that: (1) the sufficient condition on convergence is dominated not only by the derivative learning gains, along with the system input and output matrices, but also by the proportional learning gains and the system state matrix; (2) the strictly monotone convergence is guaranteed for the first‐order rule while, in the case of the second‐order scheme, the monotonicity is maintained after some finite number of iterations; and (3) the iterative learning process performed by the second‐order learning scheme can be Q_p‐faster, Q_p‐equivalent, or Q_p‐slower than the iterative learning process manipulated by the first‐order rule if the learning gains are appropriately chosen. To manifest the validity and effectiveness of the results, several numerical simulations are conducted. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

PDα‐type iterative learning control for fractional‐order linear continuous‐time switched systems

For a class of fractional‐order linear continuous‐time switched systems specified by an arbitrary switching rule, this paper proposes a PD^α‐type fractional‐order iterative learning control algorithm. For systems disturbed by bounded measurement noise, the robustness of PD^α‐type algorithm is first discussed in the iteration domain and the tracking performance is analyzed. Next, a sufficient condition for monotone convergence of the algorithm is studied when external noise is absent. The results of analysis and simulation illustrate the feasibility and effectiveness of the proposed control algorithm.     

具有反馈信息的迭代学习控制律在Lebesgue-p范数意义下的收敛性

针对一类线性时不变系统, 提出了具有反馈信息的PD-型(Proportional-derivative-type)迭代学习控制律, 利用卷积的推广的Young不等式, 分析了控制律在Lebesgue-p范数意义下的单调收敛性. 分析表明, 收敛性不但决定于系统的输入输出矩阵和控制律的微分学习增益, 而且依赖于系统的状态矩阵和控制律的比例学习增益; 进一步, 当适当选取反馈增益时, 反馈信息可加快典型的PD-型迭代学习控制律的单调收敛性. 数值仿真验证了理论分析的正确性和控制律的有效性.     

An Extended State Observer Based Fractional Order Sliding‐Mode Control for a Novel Electro‐Hydraulic Servo System with Iso‐Actuation Balancing and Positioning

An extended state observer based fractional order sliding‐mode control (ESO‐FOSMC) is proposed in this study, with consideration of the strong nonlinear characteristics of a new electro‐hydraulic servo system with iso‐actuation balancing and positioning. By adopting the fractional order calculus theory, a fractional order proportional–integral–derivative (PID)‐based sliding mode surface was designed, which has the ability to obtain an equivalent positioning control with fractional order kinetic characteristics. By introducing the integral term into the sliding mode surface, it was found to be beneficial in reducing the steady‐state errors, as well as improving the precision of the control system. Also, by using the fractional order calculus to replace the integral calculus, the form of the convergence is improved; the system transfer of energy is slowed down; and the chattering of the system is greatly weakened. The extended state observer was designed to observe the real‐time disturbances, and also to generate the compensation control commands which are added to the FOSMC to achieve the dynamic compensation. By means of numerical simulations, the dynamic and static characteristics of the sliding mode control system were compared with those of the FOSMC and ESO‐FOSMC. The experimental results show that the ESO‐FOSMC system could effectively restrain the external disturbances and achieve higher control precision, as well as better control quantity without chattering. The semi‐physical simulations based experimental tests also demonstrated that the proposed ESO‐FOSMC outperformed the FOSMC in terms of system robustness and control precision, which could have a stable control of the gun system quickly and accurately.     

Fractional‐order–dependent global stability analysis and observer‐based synthesis for a class of nonlinear fractional‐order systems

This paper focuses on proposing novel conditions for stability analysis and stabilization of the class of nonlinear fractional‐order systems. First, by considering the class of nonlinear fractional‐order systems as a feedback interconnection system and applying small‐gain theorem, a condition is proposed for L₂‐norm boundedness of the solutions of these systems. Then, by using the Mittag‐Leffler function properties, we show that satisfaction of the proposed condition proves the global asymptotic stability of the class of nonlinear fractional‐order systems with fractional order lying in (0.5, 1) or (1.5, 2). Unlike the Lyapunov‐based methods for stability analysis of fractional‐order systems, the new condition depends on the fractional order of the system. Moreover, it is related to the H_∞‐norm of the linear part of the system and it can be transformed to linear matrix inequalities (LMIs) using fractional‐order bounded‐real lemma. Furthermore, the proposed stability analysis method is extended to the state‐feedback and observer‐based controller design for the class of nonlinear fractional‐order systems based on solving some LMIs. In the observer‐based stabilization problem, we prove that the separation principle holds using our method and one can find the observer gain and pseudostate‐feedback gain in two separate steps. Finally, three numerical examples are provided to demonstrate the advantage of the novel proposed conditions with the previous results.     

Finite‐time angular velocity observers for rigid‐body attitude tracking with bounded inputs

The attitude tracking of a rigid body without angular velocity measurements is addressed. A continuous angular velocity observer with fractional power functions is proposed to estimate the angular velocity via quaternion attitude information. The fractional power gains can be properly tuned according to a homogeneous method such that the estimation error system is uniformly almost globally finite‐time stable, irrespective of control inputs. To achieve output feedback attitude tracking control, a quaternion‐based nonlinear proportional‐derivative controller using full‐state feedback is designed first, yielding uniformly almost globally finite‐time stable of the attitude tracking system as well as bounded control torques a priori. It is then shown that the certainty equivalent combination of the observer and nonlinear proportional‐derivative controller ensures finite‐time convergence of the attitude tracking error for almost all initial conditions. The proposed methods not only avoid high‐gain injection, as opposed to the semi‐global results, but also overcome the unwinding problem associated with some quaternion‐based observers and/or controllers. Numerical simulations are presented to verify the effectiveness of the proposed methods. Copyright © 2016 John Wiley & Sons, Ltd.     

Hybrid‐impulsive second‐order sliding mode control: Lyapunov approach

Dynamic system of relative degree two controlled by discontinuous‐hybrid‐impulsive feedback in the presence of bounded perturbations is considered. The state feedback impulsive‐twisting control exhibits a uniform exact finite time convergence to the second‐order sliding mode with zero convergence time. The output feedback discontinuous control augmented by a simplified hybrid‐impulsive functions provides uniform exact convergence with zero convergence time of the system's states to a real second‐order sliding mode in the presence of bounded perturbations. Only ‘snap’ knowledge of the output derivative, that is, the knowledge of the output derivative in isolated time instants, is required. The output feedback hybrid‐impulsive control with practically implemented impulsive actions asymptotically drives the system's states to the origin. The Lyapunov analysis of the considered hybrid‐impulsive‐discontinuous system proves the system's stability. The efficacy of the proposed control technique is illustrated via computer simulations. Copyright © 2016 John Wiley & Sons, Ltd.     

State estimation for nonlinear conformable fractional‐order systems: A healthy operating case and a faulty operating case

The issue of estimating states for classical integer‐order nonlinear systems has been widely addressed in the literature. Yet, generalization of existing results to the fractional‐order framework represents a fertile area of research. Note that, recently, a new and advantageous type of fractional derivative, the conformable derivative, was defined. So far, the general query of designing observers for conformable fractional‐order systems has not been investigated. In addition, it has been proved in the literature that some important tools for stability analysis of fractional‐order systems are valid using the conformable derivative concept, but invalid using other fractional derivative concepts. Motivated by the cited facts, this paper presents a first‐state estimation scheme for fractional‐order systems under the conformable derivative concept. A healthy operating case and a faulty operating case are treated. In this paper, a version of Barbalat's lemma, which is invalid using the well‐known Caputo derivative, is exploited to prove the convergence of the estimation errors. In order to validate the theoretical results, a numerical example is studied in the simulation section.     

Lyapunov Functional Approach to Stability Analysis of Riemann‐Liouville Fractional Neural Networks with Time‐Varying Delays

This paper is concerned with the globally asymptotic stability of the Riemann‐Liouville fractional‐order neural networks with time‐varying delays. The Lyapunov functional approach to stability analysis for nonlinear fractional‐order functional differential equations is discussed. By constructing an appropriate Lyapunov functional associated with the Riemann‐Liouville fractional integral and derivative, the asymptotic stability criteria of fractional‐order neural networks with time‐varying delays and constant delays are derived. The advantage of our proposed method is that one may directly calculate the first‐order derivative of the Lyapunov functional. Two numerical examples are also presented to illustrate the validity and feasibility of the theoretical results. With the increasing of the order of fractional derivatives, the state trajectories of neural networks show that the speeds of converging toward zero solution are faster and faster.     

Trajectory Controllability of Fractional Integro‐Differential Systems in Hilbert Spaces

In this paper, sufficient conditions for trajectory controllability of nonlinear fractional integro‐differential systems involving Caputo fractional derivative of order α∈(1,2] in finite and as well as in infinite dimensional Hilbert spaces are obtained. Our tools of study include set‐valued functions, theory of monotone operators and α‐order cosine family of operators. The main results are well illustrated with the aid of examples.     

A Novel Sliding Mode Control for a Class of Second‐order Mechanical Systems

This paper considers the tracking control problem of a class of second‐order mechanical systems involving parametric uncertainty and external disturbance by a sliding mode control (SMC) without reaching phase. Specifically, an SMC strategy with modified variable‐gain proportional–integral–derivative (PID)‐type sliding function is proposed, by which the existence of a sliding mode throughout an entire response of the system starting from the initial time instance is ensured. Meanwhile, the introduction of a variable gain in the sliding function design effectively solves the dilemma between quicker response and smaller overshoot. The effectiveness of the proposed strategy is verified by both theoretical analysis and simulation results.     

On the sliding‐mode control of fractional‐order nonlinear uncertain dynamics

This paper deals with applications of sliding‐mode‐based fractional control techniques to address tracking and stabilization control tasks for some classes of nonlinear uncertain fractional‐order systems. Both single‐input and multi‐input systems are considered. A second‐order sliding‐mode approach is taken, in suitable combination with PI‐based design, in the single‐input case, while the unit‐vector approach is the main tool of reference in the multi‐input case. Sliding manifolds containing fractional derivatives of the state variables are used in the present work. Constructive tuning conditions for the control parameters are derived by Lyapunov analysis, and the convergence properties of the proposed schemes are supported by simulation results. Copyright © 2015 John Wiley & Sons, Ltd.     

A computational method for solving two‐dimensional nonlinear variable‐order fractional optimal control problems

In this paper, a new class of two‐dimensional nonlinear variable‐order fractional optimal control problems (V‐OFOCPs) is introduced where the variable‐order fractional derivative is defined in the Caputo type. The general procedure for solving theses systems is expanding the state variable and the control variable based on the Legendre cardinal functions in the matrix form. Hence, we derive their operational matrix of derivative (OMD) and operational matrix of variable‐order fractional derivative (OMV‐OFD). More significantly, some properties of these basis functions are proved to be exploited in our approach. Using these achieved results, we simply expand the matrix form of the nonlinear performance index in terms of the Legendre cardinal functions and subsequently convert it to an algebraic equation. We emphasize that it is a valuable advantage of applying cardinal functions in approximation theory. Then, we implement the OMD and the OMV‐OFD of the Legendre cardinal functions to transform the variable‐order fractional dynamical system to a system of algebraic equations. Next, the method of constrained extremum is applied to adjoin the constraint equations including the given dynamical system and the initial‐boundary conditions to the performance index by a set of undetermined Lagrange multipliers. Finally, the necessary conditions of the optimality are derived as a system of nonlinear algebraic equations including the unknown coefficients of the state variable, the control variable and the Lagrange multipliers. The applicability and efficiency of the proposed approach are investigated through the various types of test problems.     

Time‐varying feedback for stabilization in prescribed finite time

This paper provides a time‐varying feedback alternative to control of finite‐time systems, which is referred to as “prescribed‐time control,” exhibiting several superior features: (i) such time‐varying gain–based prescribed‐time control is built upon regular state feedback rather than fractional‐power state feedback, thus resulting in smooth (C^m) control action everywhere during the entire operation of the system; (ii) the prescribed‐time control is characterized with uniformly prespecifiable convergence time that can be preassigned as needed within the physically allowable range, making it literally different from not only the traditional finite‐time control (where the finite settling time is determined by a system initial condition and a number of design parameters) but also the fixed‐time control (where the settling time is subject to certain constraints and thus can only be specified within the corresponding range); and (iii) the prescribed‐time control relies only on regular Lyapunov differential inequality instead of fractional Lyapunov differential inequality for stability analysis and thus avoids the difficulty in controller design and stability analysis encountered in the traditional finite‐time control for high‐order systems.     

Optimal fractional order PIλDμ controller for stabilization of cart‐inverted pendulum system: Experimental results

The cart‐inverted pendulum is a non‐minimum phase system having right half s‐plane pole and zero in close vicinity to each other. Linear time invariant (LTI) classical controllers cannot achieve satisfactory loop robustness for such systems. Therefore, in the present work the fractional order PI^λD^μ (FOPID) controller is addressed for robust stabilization of the system, since fractional order controller design allows more degrees of freedom compared to its integer order counterparts by virtue of its two parameters λ and μ. The controller parameters are tuned by three evolutionary optimization techniques. In order to select the controller parameters optimally, a novel non‐linear fitness function using integral time square error (ITSE), settling‐time, and rise time is proposed here. The control algorithm is implemented successfully in real‐time. Moreover, stability analysis of the system compensated with a fractional order controller is presented using Riemann surface. Robustness of the physical cart‐inverted pendulum system towards multiplicative gain variations and plant parameter variations is verified. In this regard, it is shown that the fractional order controller provides satisfactory robust performance in both simulation and real‐time system.     

optimization‐based fractional‐order PID controllers design

In this paper we propose a fractional‐order proportional‐integral‐derivative controller design based on the solution of an model matching problem for fractional first‐order‐plus‐dead‐time processes. Starting from the analytical solution of the problem, we show that a fractional proportional‐integral‐derivative suboptimal controller can be obtained. Guidelines for the tuning of the controller parameters are given in order to address the robust stability issue and to obtain the required performance. The main differences with respect to the integer‐order case are highlighted. Simulation results show that the design methodology is effective and allows the user to consider process with different dynamics in a unified framework. Copyright © 2013 John Wiley & Sons, Ltd.     

Robust control under worst‐case uncertainty for unknown nonlinear systems using modified reinforcement learning

Reinforcement learning (RL) is an effective method for the design of robust controllers of unknown nonlinear systems. Normal RLs for robust control, such as actor‐critic (AC) algorithms, depend on the estimation accuracy. Uncertainty in the worst case requires a large state‐action space, this causes overestimation and computational problems. In this article, the RL method is modified with the k‐nearest neighbor and the double Q‐learning algorithm. The modified RL does not need the neural estimator as AC and can stabilize the unknown nonlinear system under the worst‐case uncertainty. The convergence property of the proposed RL method is analyzed. The simulations and the experimental results show that our modified RLs are much more robust compared with the classic controllers, such as the proportional‐integral‐derivative, the sliding mode, and the optimal linear quadratic regulator controllers.     

Gradient‐based back‐propagation dynamical iterative learning scheme for the neuro‐fuzzy inference system

In this paper, a gradient‐based back propagation dynamical iterative learning algorithm is proposed for structure optimization and parameter tuning of the neuro‐fuzzy system. Premise and consequent parameters of the neuro‐fuzzy model are initialized randomly and then tuned by the proposed iterative algorithm. The learning algorithm is based on the first order partial derivative of the output with respect to the structure parameters. The first order derivative of the model output with respect to the structure parameters determines the sensitivity of the model to structure parameters. The sensitivity values are then used to set the tuning factors and parameters updating step sizes. Therefore, an adaptive dynamical iterative scheme is achieved which adapts the learning procedure to the current state of the performance during the optimization process. Larger tuning step sizes make the convergence speed higher and vice versa. In this regard, this parameter is treated according to the calculated sensitivity of the model to the parameter. The proposed learning algorithm is compared with the least square back propagation method, genetic algorithm and chaotic genetic algorithm in the neuro‐fuzzy model structure optimization. Smaller mean square error and shorter learning time are sought in this paper, and the performance of the proposed learning algorithm is versified regarding these criteria.     

Robust PID‐PSD Controller Design: BMI Approach

A new robust proportional‐integral‐derivative (PID)–proportional‐sum‐derivative (PSD) controller design method based on linear (bilinear) matrix inequalities (LMI, BMI) is proposed for uncertain affine linear system. The design procedure guarantees the parameter dependent quadratic stability, and guaranteed cost control with a new quadratic cost function (LQRS) including the derivative term for the state vector as a tool to influence the overshoot and response rate. The second approach to the PSD controller design procedure is based on a Lyapunov function with a special term corresponding to the time‐delay part of the control algorithm. The results obtained are illustrated on three examples to show the robust PID, PSD control design procedure and the influence of the choice of matrix S in the extended cost function.     

Consensus Problem with a Reference State for Fractional‐Order Multi‐Agent Systems

In this paper, the consensus problem of fractional‐order multi‐agent systems with a reference state is studied under fixed directed communication graph. At the beginning, the convergence speeds of fractional‐order multi‐agent systems are investigated based on the Mittag‐Leffler function. Then, a common consensus control law and a consensus control law based on error predictor are proposed, and it is shown that the consensus tracking can be achieved using the above control laws when a communication graph has a directed spanning tree. Finally, the convergence speeds of fractional‐order systems are compared, and it is discovered that the convergence of systems is faster using the control law based on error predictor than using the common one.