Adaptive neural control design for nonlinear distributed parameter systems with persistent bounded disturbances

Event-Triggered Optimal Control for Temperature Field of Roller Kiln Based on Adaptive Dynamic Programming

Temperature field control is crucial for the comprehensive performance of Ni-Co-Mn layered cathode material that is the most important part of lithium-ion batteries. Starting from the aspect of a class of distributed parameter systems described by highly dissipative partial differential equations (PDEs), an event-triggered optimal control (ETOC) method based on adaptive dynamic programming (ADP) for the roller kiln temperature field is proposed. First, we formulate the event-triggered control problem of the temperature field under the general framework of PDE systems. Then, an event-triggered condition is designed based on the stability of the closed-loop PDE system, which also guarantees the upper bound of the performance index. Subsequently, ADP technology is adopted to realize the ETOC, where the critic network is employed to approximate the optimal value function. Since the studied system can be regarded as an impulsive dynamic system with flow dynamics and jump dynamics simultaneously, the stability of the impulsive dynamic system combined with the ADP-based closed-loop PDE system is proved. Finally, simulation results on the temperature field verify the effectiveness of the proposed method.

Distal-less homeobox genes Dlx5/6 regulate Müllerian duct regression

Dlx5 and Dlx6 encode distal-less homeodomain transcription factors that are present in the genome as a linked pair at a single locus. Dlx5 and Dlx6 have redundant roles in craniofacial, skeletal, and uterine development. Previously, we performed a transcriptome comparison for anti-Müllerian hormone (AMH)-induced genes expressed in the Müllerian duct mesenchyme of male and female mouse embryos. In that study, we found that Dlx5 transcripts were nearly seven-fold higher in males compared to females and Dlx6 transcripts were found only in males, suggesting they may be AMH-induced genes. Therefore, we investigated the role of Dlx5 and Dlx6 during AMH-induced Müllerian duct regression. We found that Dlx5 was detected in the male Müllerian duct mesenchyme from E14.5 to E16.5. In contrast, in female embryos Dlx5 was detected in the Müllerian duct epithelium. Dlx6 expression in Müllerian duct mesenchyme was restricted to males. Dlx6 expression was not detected in female Müllerian duct mesenchyme or epithelium. Genetic experiments showed that AMH signaling is necessary for Dlx5 and Dlx6 expression. Müllerian duct regression was variable in Dlx5 homozygous mutant males at E16.5, ranging from regression like controls to a block in Müllerian duct regression. In E16.5 Dlx6 homozygous mutants, Müllerian duct tissue persisted primarily in the region adjacent to the testes. In Dlx5-6 double homozygous mutant males Müllerian duct regression was also found to be incomplete but more severe than either single mutant. These studies suggest that Dlx5 and Dlx6 act redundantly to mediate AMH-induced Müllerian duct regression during male differentiation.

Boundary Control of Linear Uncertain 1-D Parabolic PDE Using Approximate Dynamic Programming

This paper develops a near optimal boundary control method for distributed parameter systems governed by uncertain linear 1-D parabolic partial differential equations (PDE) by using approximate dynamic programming. A quadratic surface integral is proposed to express the optimal cost functional for the infinite-dimensional state space. Accordingly, the Hamilton-Jacobi-Bellman (HJB) equation is formulated in the infinite-dimensional domain without using any model reduction. Subsequently, a neural network identifier is developed to estimate the unknown spatially varying coefficient in PDE dynamics. Novel tuning law is proposed to guarantee the boundedness of identifier approximation error in the PDE domain. A radial basis network (RBN) is subsequently proposed to generate an approximate solution for the optimal surface kernel function online. The tuning law for near optimal RBN weights is created, such that the HJB equation error is minimized while the dynamics are identified and closed-loop system remains stable. Ultimate boundedness (UB) of the closed-loop system is verified by using the Lyapunov theory. The performance of the proposed controller is successfully confirmed by simulation on an unstable diffusion-reaction process.

Data-Driven H∞ Control for Nonlinear Distributed Parameter Systems

The data-driven H∞ control problem of nonlinear distributed parameter systems is considered in this paper. An off-policy learning method is developed to learn the H∞ control policy from real system data rather than the mathematical model. First, Karhunen-Loève decomposition is used to compute the empirical eigenfunctions, which are then employed to derive a reduced-order model (ROM) of slow subsystem based on the singular perturbation theory. The H∞ control problem is reformulated based on the ROM, which can be transformed to solve the Hamilton-Jacobi-Isaacs (HJI) equation, theoretically. To learn the solution of the HJI equation from real system data, a data-driven off-policy learning approach is proposed based on the simultaneous policy update algorithm and its convergence is proved. For implementation purpose, a neural network (NN)- based action-critic structure is developed, where a critic NN and two action NNs are employed to approximate the value function, control, and disturbance policies, respectively. Subsequently, a least-square NN weight-tuning rule is derived with the method of weighted residuals. Finally, the developed data-driven off-policy learning approach is applied to a nonlinear diffusion-reaction process, and the obtained results demonstrate its effectiveness.

Adaptive optimal control of highly dissipative nonlinear spatially distributed processes with neuro-dynamic programming

Highly dissipative nonlinear partial differential equations (PDEs) are widely employed to describe the system dynamics of industrial spatially distributed processes (SDPs). In this paper, we consider the optimal control problem of the general highly dissipative SDPs, and propose an adaptive optimal control approach based on neuro-dynamic programming (NDP). Initially, Karhunen-Loève decomposition is employed to compute empirical eigenfunctions (EEFs) of the SDP based on the method of snapshots. These EEFs together with singular perturbation technique are then used to obtain a finite-dimensional slow subsystem of ordinary differential equations that accurately describes the dominant dynamics of the PDE system. Subsequently, the optimal control problem is reformulated on the basis of the slow subsystem, which is further converted to solve a Hamilton-Jacobi-Bellman (HJB) equation. HJB equation is a nonlinear PDE that has proven to be impossible to solve analytically. Thus, an adaptive optimal control method is developed via NDP that solves the HJB equation online using neural network (NN) for approximating the value function; and an online NN weight tuning law is proposed without requiring an initial stabilizing control policy. Moreover, by involving the NN estimation error, we prove that the original closed-loop PDE system with the adaptive optimal control policy is semiglobally uniformly ultimately bounded. Finally, the developed method is tested on a nonlinear diffusion-convection-reaction process and applied to a temperature cooling fin of high-speed aerospace vehicle, and the achieved results show its effectiveness.

Direct Adaptive Neural Control for a Class of Uncertain Nonaffine Nonlinear Systems Based on Disturbance Observer

In this paper, the direct adaptive neural control is proposed for a class of uncertain nonaffine nonlinear systems with unknown nonsymmetric input saturation. Based on the implicit function theorem and mean value theorem, both state feedback and output feedback direct adaptive controls are developed using neural networks (NNs) and a disturbance observer. A compounded disturbance is defined to take into account of the effect of the unknown external disturbance, the unknown nonsymmetric input saturation, and the approximation error of NN. Then, a disturbance observer is developed to estimate the unknown compounded disturbance, and it is established that the estimate error converges to a compact set if appropriate observer design parameters are chosen. Both state feedback and output feedback direct adaptive controls can guarantee semiglobal uniform boundedness of the closed-loop system signals as rigorously proved by Lyapunov analysis. Numerical simulation results are presented to illustrate the effectiveness of the proposed direct adaptive neural control techniques.

Feedback optimal control of distributed parameter systems by using finite-dimensional approximation schemes

Optimal control for systems described by partial differential equations is investigated by proposing a methodology to design feedback controllers in approximate form. The approximation stems from constraining the control law to take on a fixed structure, where a finite number of free parameters can be suitably chosen. The original infinite-dimensional optimization problem is then reduced to a mathematical programming one of finite dimension that consists in optimizing the parameters. The solution of such a problem is performed by using sequential quadratic programming. Linear combinations of fixed and parameterized basis functions are used as the structure for the control law, thus giving rise to two different finite-dimensional approximation schemes. The proposed paradigm is general since it allows one to treat problems with distributed and boundary controls within the same approximation framework. It can be applied to systems described by either linear or nonlinear elliptic, parabolic, and hyperbolic equations in arbitrary multidimensional domains. Simulation results obtained in two case studies show the potentials of the proposed approach as compared with dynamic programming.

Moving-horizon state estimation for nonlinear systems using neural networks

Moving-horizon (MH) state estimation is addressed for nonlinear discrete-time systems affected by bounded noises acting on system and measurement equations by minimizing a sliding-window least-squares cost function. Such a problem is solved by searching for suboptimal solutions for which a certain error is allowed in the minimization of the cost function. Nonlinear parameterized approximating functions such as feedforward neural networks are employed for the purpose of design. Thanks to the offline optimization of the parameters, the resulting MH estimation scheme requires a reduced online computational effort. Simulation results are presented to show the effectiveness of the proposed approach in comparison with other estimation techniques.

Design of a neural network adaptive controller via a constrained invariant ellipsoids technique

In safety critical applications, control architectures based on adaptive neural networks (NNs) must satisfy strict design specifications. This paper presents a practical approach for designing a mixed linear/adaptive model reference controller that recovers the performance of a reference model, and guarantees the boundedness of the tracking error within an a priori specified compact domain, in the presence of bounded uncertainties. The linear part of the controller results from the solution of an optimization problem where specifications are expressed as linear matrix inequality constraints. The linear controller is then augmented with a general adaptive NN that compensates for the uncertainties. The only requirement for the NN is that its output must be confined within pre-specified saturation limits. Toward this end a specific NN output confinement algorithm is proposed in this paper. The main advantages of the proposed approach are that requirements in terms of worst-case performance can be easily defined during the design phase, and that the design of the adaptation mechanism is largely independent from the synthesis of the linear controller. A numerical example is used to illustrate the design methodology.