Persistent Excitation Condition Research Articles

Adaptive Control With Guaranteed Transient Behavior and Zero Steady-State Error for Systems With Time-Varying Parameters

It is nontrivial to achieve global zero-error regulation for uncertain nonlinear systems. The underlying problem becomes even more challenging if mismatched uncertainties and unknown time-varying control gain are involved, yet certain performance specifications are also pursued. In this work, we present an adaptive control method, which, without the persistent excitation (PE) condition, is able to ensure global zero-error regulation with guaranteed output performance for parametric strict-feedback systems involving fast time-varying parameters in the feedback path and input path. The development of our control scheme benefits from generalized <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$t$</tex> -dependent and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$x$</tex> -dependent functions, a novel coordinate transformation and “congelation of variables” method. Both theoretical analysis and numerical simulation verify the effectiveness and benefits of the proposed method.

Performance analysis of the compressed distributed least squares algorithm

In this paper, we consider the distributed estimation problem of unknown high-dimensional sparse signals for a random dynamic system. We propose a compressed distributed algorithm by using the compressive sensing theory and the distributed least squares (LS) algorithm. Under a compressed cooperative persistent excitation condition, the upper bound of the estimation error is established which is positively related to the restricted isometry constant. Our results are obtained without relying on some stringent conditions such as independency or stationarity of the regression vectors. Finally, we provide a simulation example to show that the compressed distributed least squares algorithm has better performance than the regularized distributed LS algorithm with l 1 penalty for the estimation of high-dimensional sparse signals. • High-dimensional sparse signal. • Compressed distributed least squares. • System identification. • Random dynamic system

Off-policy reinforcement learning-based novel model-free minmax fault-tolerant tracking control for industrial processes

For industrial processes with external disturbance and actuator failure, off-policy reinforcement learning-based novel model-free minmax fault-tolerant control is proposed in this paper to solve H∞ fault-tolerant tracking control problem. An augmented model equivalent to the original system is constructed, and the state of the new augmented model is composed of state increment and tracking error of the original system. The original H∞ fault-tolerant tracking problem was transformed into the linear quadratic zero-sum game problem by establishing performance index function, and the Game Algebraic Riccati Equation (GARE) was established. Then Q function was introduced and the Off-policy reinforcement learning algorithm was designed. Different from the traditional model-based fault-tolerant control method, the proposed algorithm does not need the knowledge of system dynamics, and it can learn from the measured data of the system trajectory to solve the GARE. In addition, it is proved that the probing noise added to satisfy the persistent excitation condition does not cause bias. A simulation example of injection molding process is used to verify the effectiveness of the proposed algorithm.

Composite learning sliding mode control of uncertain nonlinear systems with prescribed performance

This paper explores the prescribed performance tracking control problem of nonlinear systems with triangular structure. To obtain the desired transient performance and precise estimations of uncertain terms, the techniques of neural network control, sliding mode control and composite learning control are incorporated into the proposed control method. The presented control strategy can ensure the tracking error converges to a prescribed small residual set. Compared with the persistent excitation condition required in the conventional adaptive control, the interval excitation condition needed in the proposed control approach is weak, which guarantees that the radial basis function neural networks approximate the unknown nonlinear terms more accurately. Finally, two simulation examples are exploited to manifest the effectiveness of the proposed approach.

Finite-time adaptive optimal consensus control for multi-agent systems subject to time-varying output constraints

In this paper, a finite-time optimal consensus control strategy is presented for unknown multi-agent systems (MASs) with the time-varying asymmetric output constraint. Different from existing results, the output constraint problem investigated here eliminates the requirements that constraint boundary functions must be strictly non-zero and have different signs, which is successfully handled by introducing special barrier functions. Moreover, to deal with disturbances well, a reinforcement learning (RL) with the critic-actor-disturbance structure is introduced. Meanwhile, the weights of neural networks are adjusted online by applying the gradient descent method to positive functions newly constructed, which not only significantly simplifies the algorithm but also eliminates the persistent excitation condition. For obtaining a fast convergence rate, the finite-time control technique is embedded into the RL algorithm, and an effective finite-time optimal control scheme is proposed to achieve the consistency of multi-agent system in a finite time. Finally, the effectiveness of the proposed protocol is demonstrated by two simulation examples.

Relaxed bearing rigidity and bearing formation control under persistence of excitation

This paper addresses the problem of time-varying bearing formation control in d(d≥2)-dimensional Euclidean space by exploring Persistence of Excitation (PE) of the desired bearing reference. A general concept of Bearing Persistently Exciting (BPE) formation defined in d-dimensional space is here fully developed. By providing a desired formation that is BPE, distributed control laws for multi-agent systems under both single- and double-integrator dynamics are proposed using bearing measurements (along with velocity measurements when the agents are described by double-integrator dynamics), which guarantee uniform exponential stabilization of the desired formation up to a translation. A key contribution of this work is to show that the classical bearing rigidity condition on the graph topology, required for achieving the stabilization of a formation up to a translation and a scale factor, is relaxed and extended in a natural manner by exploring PE conditions imposed either on a specific set of desired bearing vectors or on the whole desired formation. Simulation results are provided to illustrate the performance of the proposed control method.

A combined invariant-subspace and subspace identification method for continuous-time state–space models using slowly sampled multi-sine-wave data

The problem of identification of linear continuous-time (CT) state–space (SS) models directly from time-domain data is considered. Given multi-sine waves as excitation signals, this paper develops a new identification method called cISSIM (combined Invariant-Subspace and Subspace Identification Method), which is able to consistently identify CT SS models from slowly-sampled time-domain data in an error-in-variables framework. No prefiltering of the input/output signals is required which distinguishes cISSIM from other existing direct CT identification methods. Moreover, asymptotic unbiased estimates of the system state are obtained. A new persistence of excitation condition is also derived for cISSIM, which is shown to be related with the rank of a certain Khatri–Rao product.

Quantized‐output‐based least squares of ARX systems

SummaryThis paper considers the estimates of parameters of stochastic autoregressive exogenous input (ARX) systems with quantized data. The quantized error caused by quantized data brings difficulties to the parameter estimation of ARX systems. Therefore, I start with the boundedness of quantized error and design the size of quantized error thanks to good properties of the least squares. Then the influences of the noise of the system itself and the bounded noise caused by quantized error on the parameter estimation error are analyzed, respectively. The inputs of the system are designed to make the system satisfy the persistent excitation condition, so as to ensure the boundedness of the parameter estimation error. A numerical example is given to demonstrate the theorem.

Optimal detumbling strategy for a non-cooperative target with unknown inertial parameters using a space manipulator

Capturing tumbling target is considered as one of the greatest challenges in on-orbit service missions due to tumbling motion and parameter uncertainty of the space non-cooperative target. The tumbling motion of the target should be attenuated as quickly as possible before grasping to avoid the unexpected collisions and possible damages to the space manipulator or the target, which means the shortest-time detumbling trajectory is necessary. In many studies about the optimal detumbling trajectory planning the inertial parameters of the target have been assumed to be accurately known, which is usually difficult for a non-cooperative target so that parameter identification is needed before planning a optimal detumbling trajectory. In this paper, a novel strategy for parameter identification and detumbling of a tumbling target by a space manipulator is proposed. The strategy includes two phases. First, unknown inertial parameters of the target are identified in finite-time by a new estimation method with the assumption of interval excitation (IE), which relaxes the strict persistent excitation (PE) condition required in most prior studies. Second, the shortest-time detumbling trajectory of the tumbling target is generated by formulating a time-optimal control problem (OCP) in which the target attitude bounds and the limitations of the contact force as well as the relative velocity between the space manipulator and the target are represented as extended dynamical sub-systems and saturation functions. Then, the Calculus of Variations method is used to solve the OCP to obtain a high accuracy solution. During two phases, the contact force between the space manipulator and the target is utilized to generate external moment for identifying the inertia parameters of the target and detumbling the target. Meanwhile, a unified hybrid impedance control framework is applied to the space robot to track the desired contact force and motion trajectory. The whole strategy is verified by a space manipulator capturing a tumbling target, and numerical simulations demonstrate the effectiveness of the proposed methods.

On consistency of least squares algorithm for ARMAX time-delay models

Jointly identification of the sample delay and the parameters of time-delay discrete-time ARMAX models are addressed in this article. Like any identified problem, the proposed approach is based on the solution of an optimization problem. It consists then of two main steps: the first is the problem formulation and the second is the problem resolution. The problem formulation defines the criterion to be minimized according to an unknown parameter vector consisting of the sample delay and the parameters with their corresponding observation vector. Since the sample delay is of integer type, the obtained optimization problem cannot be efficiently solved using a real optimization algorithm. To overcome this difficulty, the rounding properties are used to transform the obtained problem into a real optimization problem. The solution is then done using the least square approach. Consistency of the estimates with their convergence rates is established under the persistent excitation condition. Finally, some numerical simulation and experimental results are presented to illustrate the performance of the proposed method.

Data-Driven Immersion and Invariance Adaptive Attitude Control for Rigid Bodies With Double-Level State Constraints

This article investigates the attitude control problem of a rigid body subject to attitude and angular rate constraints (double-level state constraints), and inertia uncertainties. A data-driven immersion and invariance (I&I) adaptive control scheme is proposed to tackle this technically challenging problem. As a stepping stone, a novel dynamically scaled I&I adaptive controller is designed to bypass the realizability condition that may not hold in the Lyapunov sense when considering angular rate constraints. Lyapunov stability analysis shows that this controller can enable the attitude errors and angular rates to converge asymptotically to zero for most initial conditions in the accessible space, while strictly obeying double-level state constraints with the help of two judiciously constructed potential functions. After that, to further relax the dependence of parameter convergence on the persistent excitation (PE) condition, the I&I adaptive law is extended to a data-driven counterpart through adding a learning term that is acquired by adopting the regressor filtering in conjunction with the dynamic regressor extension and mixing (DREM) procedure. The extended adaptive controller can not only preserve all the results obtained by the earlier proposed I&I adaptive controller, but also ensure asymptotic parameter convergence under a finite excitation condition much weaker than PE. In addition, benefiting from the DREM method and some special designs, the parameter convergence rates across all the parameter vector components can be flexibly tuned in an explicit way, and moreover, they are independent of the excitation level. Finally, simulation results are given to show the effectiveness of the proposed method.

Deterministic learning from neural control for a class of sampled-data nonlinear systems

This study investigates the deterministic learning and control issues for uncertain sampled-data nonlinear systems (SDNSs). The problem of how to acquire/learn knowledge from adaptive control for SDNSs with uncertain affine terms is studied. To be specific, an appropriate neural network-based (NNB) control strategy is first presented to ensure tracking performance. To further realize learning, the exponential stability (ES) of the integrated closed-loop system coupled with the estimation error of the NN weights is considered. As the uncertain affine term prevents learning from occurring, the integrated system is converted into a discrete linear time-varying (DLTV) perturbed system by employing the state conversion technique. Tracking convergence allows the persistent excitation condition (PEC) of the NNs to be established, which guarantees the ES of the integrated DLTV system. Thus, accurate modeling of closed-loop sampled dynamics is obtained. By reutilizing the experiential knowledge obtained, a knowledge-based controller is constructed for high-performance control. Finally, simulations are performed to verify the presented strategy.

Regularized adaptive Kalman filter for non-persistently excited systems

Most available methods for joint state-parameter estimation in discrete-time stochastic time-varying systems suffer from severe performance degradation if the system is not persistently excited. We revisit and extend a recently proposed adaptive Kalman filter in order to enhance its robustness against periods of poor excitation. A Levenberg–Marquardt-like regularization algorithm is integrated in the filter to preclude the estimator from becoming unreliable due to wind-up effects. It is proven that, under uniform complete observability–controllability conditions and a persistent excitation condition, the expectations of the state and parameter estimation errors tend to zero exponentially fast. Furthermore, their stability (in the sense of Lyapunov) is guaranteed even if the persistent excitation condition is not satisfied. The effectiveness of the algorithm is demonstrated using real sensor data from a vehicle motion estimation application. Unlike the original algorithm, the proposed regularized adaptive Kalman filter provides accurate and reliable estimates of the states and parameters despite periods of poor excitation.

Identification and adaptation with binary-valued observations under non-persistent excitation condition

Dynamical systems with binary-valued observations are widely used in information industry, technology of biological pharmacy and other fields. Though there have been much efforts devoted to the identification of such systems, most of the previous investigations are based on first-order gradient algorithm which usually has much slower convergence rate than the Quasi-Newton algorithm. Moreover, persistence of excitation (PE) conditions are usually required to guarantee consistent parameter estimates in the existing literature, which are hard to be verified or guaranteed for feedback control systems. In this paper, we propose an online projected Quasi-Newton type algorithm for parameter estimation of stochastic regression models with binary-valued observations and varying thresholds. By using both the stochastic Lyapunov function and martingale estimation methods, we establish the strong consistency of the estimation algorithm and provide the convergence rate, under a signal condition which is considerably weaker than the traditional PE condition and coincides with the weakest possible excitation known for the classical least square algorithm of stochastic regression models. Convergence of adaptive predictors and their applications in adaptive control are also discussed.

Dynamic learning‐based event‐triggered control of strict‐feedback systems with finite‐time prescribed performance

AbstractThis paper is concerned with the event‐triggered neural learning control for strict‐feedback nonlinear systems (SFNSs) subject to predefined tracking performance. A system transformation approach is utilized to transform the original SFNSs into the normal form, which makes it easily to verify the convergence of neural weights since only one neural network (NN) approximator is employed. Then, a novel finite‐time performance function is first proposed to characterize the performance constraints of tracking error, thus guaranteeing the prescribed tracking performance. On this basis, a new lemma is given, which is crucial to ensure the stability of the closed‐loop system. By using the error transformation technique, the constrained tracking control problem is converted into the stabilization problem of unconstrained error system. Subsequently, by introducing the first‐order sliding mode differentiator into the backstepping procedure and with the help of the high‐gain observer, a novel adaptive neural tracking control scheme is presented to ensure that the prescribed tracking performance is satisfied and all closed‐loop signals are ultimately bounded. After ensuring the satisfaction of the partial persistent excitation condition for radial basis function NN, the neural weight estimates are verified to converge to the ideal values, and then the convergent weights are stored as the constant values. By utilizing the stored knowledge, an event‐triggered neural learning controller is constructed to accomplish the same or similar control tasks, which can effectively improve the transient control performance, save the communication resource, and relieve the online computation at the same time. Finally, the effectiveness of the presented scheme is testified by numerical and practical examples.

Parameter Convergence in Adaptive Control in Presence of Unmatched Uncertainty

The problem of convergence of uncertainties to their true values is of great interest in adaptive control. It is desirable as it facilitates accurate online modeling of structured uncertainties and robust adaptation to disturbances. The adaptation law based on the classical gradient method requires the restrictive persistent excitation (PE) condition to be satisfied for parameter convergence. This paper deals with the relaxation of this PE condition for a class of nonlinear systems in the presence of unmatched uncertainty. The design of the proposed adaptive backstepping controller involves multiple filtered models that translate the restrictive PE condition to linear independence of the filtered regressor models. The proposed controller guarantees asymptotic convergence of the tracking and parameter estimation errors globally in the presence of unmatched uncertainty. Numerical simulations show the efficacy of the proposed controller.

Adaptive Observer with Enhanced Gain to Address Deficient Excitation

For joint estimation of state variables and unknown parameters, adaptive observers usually assume some persistent excitation (PE) condition. In practice, the PE condition may not be satisfied, because the underlying recursive estimation problem is ill-posed. To remedy the lack of PE condition, inspired by the ridge regression, this paper proposes a regularized adaptive observer with enhanced parameter adaptation gain. Like in typical ill-posed inverse problems, regularization implies an estimation bias, which can be reduced by using prior knowledge about the unknown parameters.

A high gain observer design for the three-phase shunt active power filter interfacing a photovoltaic source

In this article, a nonlinear observer is designed for a system consisting of a PVS photovoltaic source associated with a shunt active power filter (SAPF) and connected to a three-phase electrical network. In addition to improving the quality of energy in the distribution network (unity power factor), the use of SAPF also allows the injection of photovoltaic energy into the network. Since the system model is nonlinear and some variables are unknown (inaccessible to measurement), a high-gain observer is constructed. The advantage of this solution lies in the estimation of the unmeasured state variables (variables of the photovoltaic source). It is shown using theoretical analysis, under certain conditions of persistent excitation, that all estimation errors disappear exponentially. The effectiveness of the solution is verified by numerical simulation under Matlab/Simulink.

Linear Motor Command Tracking: A Novel Immersion and Invariance Adaptive Control Method with Arctangent-Function-Based Parameter Projection

This paper innovates an adaptive control law that unifies the Immersion and Invariance (I&I) principle and a novel arctangent-function-based smooth projection operator. Such an adaptive controller is exploited to accomplish the command tracking task of a parametric uncertain linear motor system. The uniqueness and advantage of the proposed method lie twofold. First and foremost, the novel nonaffine I&I scheme ensures the asymptotical convergence of the parameter estimation error despite the reference command's persistency of excitation conditions. Second, the arctangent-function-based smooth projection operator is a one-piece infinitely differentiable (C∞) function that requires no boundary layer construction nor piecewise-defined functions compared to existing smooth projection techniques. Simulation results are presented to examine the proposed adaptive controller. Furthermore, its performance (especially the transient one) is compared against a certainty-equivalence-based method.

Optimized Formation Control for Multi-Agent Systems Based on Adaptive Dynamic Programming Without Persistence of Excitation

In this letter, an adaptive dynamic programming (ADP) method is proposed for optimized formation control of second-order linear systems. The method exploits an actor-critic architecture, where an actor component is used to learn the optimal formation controller, and a critic component is used to learn the optimal value function. Generally, ADP requires a priori knowledge of persistence of excitation (PE) to guarantee the stability of the control system. However, the PE condition is hard to verify during the learning process and in practical applications. To this end, this letter redesigns the updating laws of the actor and critic components to ensure that the Bellman residual error can eventually approach to zero, and the stability of the control system can be guaranteed without introducing the PE and additional constraints. By using Lyapunov stability analysis, we prove that the proposed optimized formation scheme can achieve the desired optimizing performance. Finally, a simulation example is given to demonstrate the effectiveness of the proposed method.