Persistent Excitation Research Articles

Forward-looking persistent excitation in model predictive control

This work deals with the problem of integrating persistence of excitation into nonlinear constrained model predictive control to estimate uncertain parameters while guaranteeing a stable closed loop. We propose an adaptive tracking model predictive control and conditions which guarantee persistent excitation and uniformly bounded closed loop signals of nonlinear systems, despite bounded noise and parameter uncertainty. This is achieved by actively designing persistence of excitation through the computation of a reference trajectory around some nominal stationary point depending on the parameter estimate, opening up the opportunity of balancing excitation against other requirements. Appealing to the Total Stability Theorem, the results are local and solution evolves in a non-infinitesimal ball in state and estimated parameter.

A unified identification algorithm of FIR systems based on binary observations with time-varying thresholds

This paper is concerned with parameter estimation of finite impulse response (FIR) systems with binary observations. Combining a suitable design of the time-varying thresholds, a kind of sign-error type unified algorithm with projection is investigated for either deterministic systems or stochastic systems. The convergence properties of the studied algorithm are established under bounded persistent excitations. Specifically, for the case without noise, the square convergence rate is proved to be close to O1k2 with respect to the time step k. For the case with bounded noises, the upper bound of the estimation error is obtained, which depends on the bound of the noises and the lower bound of the input persistent excitation condition. For the case with independent and identically distributed (i.i.d.) stochastic noises, the estimate is shown to converge to the true parameter in the sense of mean square and almost surely. Besides, the mean square convergence rate of the estimation error is of the order O1k. Numerical examples are supplied to demonstrate the theoretical results.

Identification and U‐control of a state‐space system with time‐delay

SummaryThis article presents a state‐space model with time‐delay to map the relationship between known input‐output data for discrete systems. For the given input‐output data, a model identification algorithm combining parameter estimation and state estimation is proposed in line with the causality constraints. Consequently, this article proposes a least squares parameter estimation algorithm, and analyzes its convergence for the studied systems to prove that the parameter estimation errors converge to zero under the persistent excitation conditions. In control system design, the U‐model based control is introduced to provide a unilateral platform to improve the design efficiency and generality. A simulation portfolio from modeling to control is provided with computational experiments to validate the derived results.

Necessary and sufficient regressor conditions for the global asymptotic stability of recursive least squares

In recursive least squares (RLS), a persistently exciting (PE) regressor guarantees global asymptotic stability (GAS) of the estimation error relative to the zero equilibrium, and hence convergence of the parameter estimates to their true values. It is known, however, that PE is sufficient but not necessary for GAS, and thus GAS might be achieved even if the regressor is not PE. Since analyses based on PE are ubiquitous in the recursive estimation literature, the existence of non-PE regressors that ensure GAS raises the question of how PE can be directly generalized into a necessary and sufficient condition for the GAS of RLS, and whether or not such a generalization would provide a simple characterization of non-PE regressors for which RLS is GAS. In this paper, we introduce WPE, a direct generalization of PE that is necessary and sufficient for GAS, and explain how it can be understood as a “non-uniform” extension of PE to specific classes of summation windows and lower bound sequences. Next, we show that WPE is equivalent to a condition that emerges from extending certain proofs of non-negative series divergence, such as Oresme’s divergence proof for the harmonic series, to sequences of real symmetric positive-semidefinite matrices.

A new adaptive identification framework for nonlinear multi-input multi-output systems under colored noise

In this paper, an adaptive identification scheme is proposed for nonlinear multi-input multi-output systems with colored noise based on a novel parameter update law. With the help of the hierarchical principle, the identification model is decomposed into three sub-models in which the computational burden is reduced. For each sub-model, the identification algorithm is proposed to estimate the sub-model parameters. In the process of the identification algorithm design, considering the system information corrupted by the noise, an adaptive filter gain is exploited to extract helpful identification data, in which a filter is designed using the system data instead of the independent design. Based on several auxiliary filtered variables, the estimation error data are obtained, and a new parameter adaptive law with a variable learning gain is proposed according to the estimation error data. Compared with the classic parameter update law, the parameter estimation update is derived based on the estimation error information instead of other error information, such as prediction error information. Under the persistent excitation condition, all the estimated parameters converge to the true parameters. an example is used and two experiments are conducted to test the outstanding identification performance of the proposed algorithm in terms of convergence rate and identification accuracy.

Distributed Recursive Projection Identification with Binary-Valued Observations

This paper investigates a distributed recursive projection identification problem with binary-valued observations built on a sensor network, where each sensor in the sensor network measures partial information of the unknown parameter only, but each sensor is allowed to communicate with its neighbors. A distributed recursive projection algorithm is proposed based on a specific projection operator and a diffusion strategy. The authors establish the upper bound of the accumulated regrets of the adaptive predictor without any requirement of excitation conditions. Moreover, the convergence of the algorithm is given under the bounded cooperative excitation condition, which is more general than the previously imposed independence or persistent excitations on the system regressors and maybe the weakest one under binary observations. A numerical example is supplied to demonstrate the theoretical results and the cooperative effect of the sensors, which shows that the whole network can still fulfill the estimation task through exchanging information between sensors even if any individual sensor cannot.

Identifiability and Solvability in Inverse Linear Quadratic Optimal Control Problems

In this paper, the inverse linear quadratic (LQ) problem over finite time-horizon is studied. Given the output observations of a dynamic process, the goal is to recover the corresponding LQ cost function. Firstly, by considering the inverse problem as an identification problem, its model structure is shown to be strictly globally identifiable under the assumption of system invertibility. Next, in the noiseless case a necessary and sufficient condition is proposed for the solvability of a positive semidefinite weighting matrix and its unique solution is obtained with two proposed algorithms under the condition of persistent excitation. Furthermore, a residual optimization problem is also formulated to solve a best-fit approximate cost function from sub-optimal observations. Finally, numerical simulations are used to demonstrate the effectiveness of the proposed methods.

Regression Model-Based Flux Observer for IPMSM Sensorless Control with Wide Speed Range

A new linear regression form is derived for a flux observer and a position observer is designed. In general, the observability of the permanent-magnet synchronous motor is lost at zero speed. In this work, the proposed regressor vector contains current derivative terms in both directions (dq-axis), and it gives the chance for the model-based flux observer to operate at zero speed. When an excitation signal is injected into d and q axes with the proposed flux observer, it helps to satisfy the persistent excitation condition in the low-speed range. Therefore, the sensorless performance of the model-based is improved greatly, even at zero speed. However, it appears with a disturbance term, which depends on the derivative of the d-axis current. Thus, the disturbance does not vanish when an excitation signal is injected. In this work, the disturbance term is also taken care of in constructing an observer. It results in an observer which allows signal injection. Thus, high frequency signal can be injected in the low speed region and turned off when it is unnecessary as the speed increases. This model-based approach utilizes the signal injection directly without recurring to a separate high frequency model. In other words, it provides a seamless transition without switching to the other algorithm. The validity is demonstrated by simulation and experimental results under various load conditions near zero speed.

Event‐triggered optimal tracking control of multiplayer unknown nonlinear systems via adaptive critic designs

AbstractIn this article, the event‐triggered optimal tracking control problem for multiplayer unknown nonlinear systems is investigated by using adaptive critic designs. By constructing a neural network (NN)‐based observer with input–output data, the system dynamics of multiplayer unknown nonlinear systems is obtained. Subsequently, the optimal tracking control problem is converted to an optimal regulation problem by establishing a tracking error system. Then, the optimal tracking control policy for each player is derived by solving coupled event‐triggered Hamilton‐Jacobi (HJ) equation via a critic NN. Meanwhile, a novel weight updating rule is designed by adopting concurrent learning method to relax the persistence of excitation (PE) condition. Moreover, an event‐triggering condition is designed by using Lyapunov's direct method to guarantee the uniform ultimate boundedness (UUB) of the closed‐loop multiplayer systems. Finally, the effectiveness of the developed method is verified by two different multiplayer nonlinear systems.

Fault tolerant consensus of multiple nonholonomic chained‐form systems with actuator and communication faults

AbstractThis article investigates the fault tolerant consensus problem of multiple nonholonomic chained‐form systems with both actuator and communication faults. The systems are enabled to follow a reference trajectory and realize practical consensus, by applying distributed reference state observers and robust adaptive fault tolerant control (FTC) strategies. The proposed state observers are robust to communication faults and can estimate the actual reference trajectory for each system. Based on such an observer, a robust adaptive FTC scheme is further provided to address actuator faults. A sine function is injected into the FTC law to relax the persistent excitation condition to avoid uncontrollability of the two‐input chained‐form system when one of reference control input tends to zero. Simulation result of wheeled mobile robots shows the effectiveness of proposed FTC methods.

Event-Triggered Optimized Control for Nonlinear Delayed Stochastic Systems

This paper is concerned with the problem of event-triggered optimized control for uncertain nonlinear Itô-type stochastic systems with time-delay and unknown dynamic. By using fuzzy logic systems to approximate two unknown nonlinear functions with the delayed state and current state, respectively. The adaptive identifier is constructed to determine the stochastic system, and the optimized control is designed by using the identifier and adaptive dynamic programming (ADP) of actor-critic architecture. Almost all of the works are concentrated on ADP-based optimal control and it will inevitably cause the complexity of computation and requirements of persistence excitation (PE) assumption. In this paper, the ADP algorithm is obtained based on the negative gradient of a simple positive function (equivalent to the HJB equation), and so the proposed optimal control is simple and can release the PE assumption. Moreover, the event-triggered control approach is proposed to reduce computing burden and communication resources. Furthermore, we prove that the states of system and FLSs parameter errors are semi-globally uniformly ultimately bounded (SGUUB) in mean square via the adaptive identifier and the Lyapunov direct method as well as identifier-actor-critic architecture-based ADP algorithm. Finally, the effectiveness of the proposed method is illustrated through two numerical examples.

Limited‐information learning observer for simultaneous estimation of states and parameters

AbstractThe problem of simultaneous estimation of states and uncertainties by using limited information is investigated in this article. A sampled‐data learning observer (SLO) is presented for linear time‐invariant continuous systems, which can achieve successful estimation while only needs intermittent sampled‐data, saves computing resources, and does not require persistent excitation signals. The observer's demand for continuous measurement is reduced, that is, limited‐information is sufficient. Notably, the uncertainty estimation is performed by a learning equation with only simple addition operations, which is particularly suitable for actual digital system scenarios. Simulation results illustrate the effectiveness of the proposed SLO.

Finite excitation based robust adaptive observer for MIMO LTI systems

SummaryThe article addresses the problem of a robust adaptive observer design for multi‐input multi‐output linear time‐invariant plants subjected to bounded disturbances in the system dynamics and the output relation. An adaptive observer design is proposed based on the finite excitation leading to a switched estimation strategy thus relaxing persistence of excitation (PE) condition. The novel finite persistence excitation (f‐PE) condition introduced in this work is significantly milder than the PE condition in terms of excitation requirement and online verifiability. The switched estimator uses the finite excitation via input signal and ensures uniformly ultimately bounded stability of the estimation error dynamics. A multiple switching strategy, which continuously tracks the information richness (about the unknown parameters), is utilized to reduce the intermediate bounds and is proved to be instrumental in improving the result.

On key properties of the Lion's and Kreisselmeier's adaptation algorithms

SummaryThe article revises properties of two identification/adaptation algorithms proposed by Lion and Kreisselmeier more than 40 years ago to accelerate parametric convergence under regressor persistency of excitation (PE) condition. First, motivated by the paperwe demonstrate that these algorithms can provide asymptotic (not exponential) parametric convergence under a simple condition which is weaker than the requirement of PE. Second, it is shown that for some special choice of adaptation gain the algorithms can provide finite time parametric convergence if the regressor satisfies theinterval excitation(IE) condition that is even weaker than the condition of asymptotic convergence. Third, it is shown that for some certain structural requirements these algorithms can generate the high‐order time derivatives of the adjustable parameters. This property can be used for solution of a wide range of problems of adaptive control including, in particular, model reference adaptive control and modular backstepping design with high‐order tuners.

Neural network modelling reveals changes in directional connectivity between cortical and hypothalamic regions with increased BMI

Background/ObjectivesObesity has been ascribed to corticostriatal regions taking control over homeostatic areas. To test this assumption, we applied an effective connectivity approach to reveal the direction of information flow between brain regions and the valence of connections (excitatory versus inhibitory) as a function of increased BMI and homeostatic state.Subjects/MethodsForty-one participants (21 overweight/obese) underwent two resting-state fMRI scans: after overnight fasting (hunger) and following a standardised meal (satiety). We used spectral dynamic causal modelling to unravel hunger and increased BMI-related changes in directed connectivity between cortical, insular, striatal and hypothalamic regions.ResultsDuring hunger, as compared to satiety, we found increased excitation of the ventromedial prefrontal cortex over the ventral striatum and hypothalamus, suggesting enhanced top-down modulation compensating energy depletion. Increased BMI was associated with increased excitation of the anterior insula over the hypothalamus across the hunger and satiety conditions. The interaction of hunger and increased BMI yielded decreased intra-cortical excitation from the dorso-lateral to the ventromedial prefrontal cortex.ConclusionsOur findings suggest that excess weight and obesity is associated with persistent top-down excitation of the hypothalamus, regardless of homeostatic state, and hunger-related reductions of dorso-lateral to ventromedial prefrontal inputs. These findings are compatible with eating without hunger and reduced self-regulation views of obesity.

Online event-triggered adaptive critic design for multi-player zero-sum games of partially unknown nonlinear systems with input constraints

This paper focuses on the design of online event-triggered optimal control strategy for multi-player zero-sum games (MP-ZSGs) with control constraints when the system model is partially unknown. Non-quadratic functions are utilized to construct the cost functions under the condition of control constraints. The proposed algorithm is designed based on the framework of identifier-critic networks. The unknown drift dynamics model is reconstructed by an identifier neural network (INN) using the input and output data. The near-optimal event-based controls and time-based disturbances are designed by training a critic neural network (CNN). With the aid of the designed event-triggered mechanism (ETM), the needless computing and communication actions of the system signals have been reduced so as to save computing/communication resources. Meanwhile, to remove the persistence of excitation (PE) condition, the historical and current data are utilized to construct a modified tuning law of CNN. Theoretically, the uniform ultimate boundedness (UUB) properties of the system states and the critic weights errors are proved by Lyapunov approach. Moreover, the Zeno behavior is proved to be excluded under the designed triggering condition. Finally, the convergence and performance of the online method are verified by simulating a representative example.

Predefined-time parameter estimation via modified dynamic Regressor extension and mixing

This paper presents a novel switching predefined-time parameter identification algorithm with a relaxed excitation condition based on the dynamic regressor extension and mixing (DREM) method. DREM often requires the persistent excitation (PE) of the extended square regressor's determinant to ensure exponential parameter convergence. Unlike the classical DREM method, a new parameter identification algorithm configured with a two-layer filter technique is proposed under a relaxed initial excitation (IE) condition, rather than strict PE. A key point in choosing IE instead of PE is the introduction of a smooth switching function that dominates the pure integral action and filter behavior of the extended square regressor. The proposed algorithm relies on the predefined-time stability theorem and the settling-time of the identification algorithm is set a priori as a system parameter. The contributions of this paper are a novel switching predefined-time parameter estimation algorithm that 1) relaxes the stringent PE condition, 2) achieves predefined-time convergence, and 3) guarantees the monotonicity of each element of the parameter error inherited from the classical DREM method. Comparative simulation results are presented to illustrate the effectiveness of the proposed algorithm.

Reinforcement Learning-Based Approximate Optimal Control for Attitude Reorientation Under State Constraints

This article addresses the attitude reorientation problems of rigid bodies under multiple state constraints. A novel reinforcement learning (RL)-based approximate optimal control method is proposed to make the tradeoff between control cost and performance. The novelty lies in that it guarantees constraint handling abilities on attitude forbidden zones and angular velocity limits. To achieve this, barrier functions are employed to encode the constraint information into the cost function. Then, an RL-based learning strategy is developed to approximate the optimal cost function and control policy. A simplified critic-only neural network (NN) is employed to replace the conventional actor–critic structure once adequate data are collected online. This design guarantees the uniform boundedness of reorientation errors and NN weight estimation errors subject to the satisfaction of a finite excitation condition, which is a relaxation compared with the persistent excitation condition that is typically required for this class of problems. More importantly, all underlying state constraints are strictly obeyed during the online learning process. The effectiveness and advantages of the proposed controller are verified by both numerical simulations and experimental tests based on a comprehensive hardware-in-loop testbed.

Online event-based adaptive critic design with experience replay to solve partially unknown multi-player nonzero-sum games

This paper designs a novel online event-based near-optimal control scheme for multi-player nonzero-sum (NZS) games with partially unknown system dynamics. With the introduction of event-triggered mechanism (ETM), the repetitive computing burdens and communication loads of the signals are largely alleviated. Under the event-triggered mechanism, the framework of identifier-critic is structured to solve the coupled Hamilton–Jacobi equations (CHJEs) for NZS games. A feedforward neural network (FNN) is employed to construct an identifier to learn the unknown dynamics of the nonlinear system. The critic network for each player utilizes a modified tuning law, which is composed of three terms, to seek for the approximated optimal control schemes. Besides the conventional term derived from the gradient descent method, the second term derived from the experience replay (ER) technique utilizes the historical state data to update the critic network weight vector so as to remove the persistence of excitation (PE) condition. In addition, a novel term is added to stabilize the closed-loop systems. Owing to the stabilizing term, there is no need for the initial stabilizing control. In theory, the uniform ultimate boundedness (UUB) properties of the system states and the critic network weight errors are proved by utilizing Lyapunov theorem. Furthermore, the minimal intersample time is proved to be lower bounded with a positive constant which means that the Zeno behavior is excluded during the learning phases. Finally, we show the validity of the designed algorithm via simulating two examples.

Composite learning adaptive dynamic surface control for uncertain nonlinear strict‐feedback systems with fixed‐time parameter estimation under sufficient excitation

AbstractThis paper presents a novel practical fixed‐time parameter identification algorithm and a composite learning based practical fixed‐time adaptive dynamic surface control (DSC) scheme for nonlinear strict‐feedback systems subject to linear‐in‐parameters uncertainties. The convergence of conventional parameter estimation algorithms often requires a restrictive prerequisite termed persistent excitation (PE) condition. By contrast, a new fixed‐time parameter identification algorithm configured with two layer transformation technique is firstly proposed under relaxed sufficient excitation condition rather than strict PE condition. The key point of avoiding PE condition is by introducing a smooth switching function to adjust the forgetting factor in the filtered regressor dynamics. Instead of using the fractional power of the tracking errors to construct the control laws, a smooth hyperbolic tangent function based adaptive DSC scheme is designed such that the potential singular problem caused by time derivations of virtual control laws in back‐stepping algorithm is avoided. Integrating the parameter identification algorithm into the adaptive DSC scheme, a composite learning based control is formed to guarantee the practical fixed‐time convergence of parameter estimation errors and tracking errors. Comparative simulation results are given to illustrate the effectiveness of the proposed algorithm.