Adaptive boundary control for unstable parabolic PDEs—Part III: Output feedback examples with swapping identifiers

We develop output-feedback adaptive controllers for two benchmark parabolic PDEs motivated by a model of thermal instability in solid propellant rockets. Both benchmark plants are unstable, have infinite relative degree, and are controlled from the boundary. One plant has an unknown parameter in the PDE and the other in the boundary condition. In both cases the unknown parameter multiplies the measured output of the system, which is obtained with a boundary sensor located on the “opposite side” of the domain from the actuator. In comparison with the Lyapunov output-feedback adaptive controllers in Krstic and Smyshlyaev [(2005). Adaptive boundary control for unstable parabolic PDEs—Part I: Lyapunov design. IEEE Transactions on Automatic Control, submitted for publication], the controllers presented here employ much simpler update laws and do not require a priori knowledge about the unknown parameters. We show how our two benchmarks examples can be combined and illustrate the adaptive stabilization design by simulation.     

Adaptive control of PDEs

This paper presents several recently developed techniques for adaptive control of PDE systems. Three different design methods are employed—the Lyapunov design, the passivity-based design, and the swapping design. The basic ideas for each design are introduced through benchmark plants with constant unknown coefficients. It is then shown how to extend the designs to reaction-advection-diffusion PDEs in 2D. Finally, the PDEs with unknown spatially varying coefficients and with boundary sensing are considered, making the adaptive designs applicable to PDE systems with an infinite relative degree, infinitely many unknown parameters, and open loop unstable.     

Adaptive control with guaranteed transient and steady state tracking error bounds for strict feedback systems

Two robust adaptive control schemes for single-input single-output (SISO) strict feedback nonlinear systems possessing unknown nonlinearities, capable of guaranteeing prescribed performance bounds are presented in this paper. The first assumes knowledge of only the signs of the virtual control coefficients, while in the second we relax this assumption by incorporating Nussbaum-type gains, decoupled backstepping and non-integral-type Lyapunov functions. By prescribed performance bounds we mean that the tracking error should converge to an arbitrarily predefined small residual set, with convergence rate no less than a prespecified value, exhibiting a maximum overshoot less than a sufficiently small prespecified constant. A novel output error transformation is introduced to transform the original “constrained” (in the sense of the output error restrictions) system into an equivalent “unconstrained”one. It is proven that the stabilization of the “unconstrained” system is sufficient to solve the problem. Both controllers are smooth and successfully overcome the loss of controllability issue. The fact that we are only concerned with the stabilization of the “unconstrained” system, severely reduces the complexity of selecting both the control parameters and the regressors in the neural approximators. Simulation studies clarify and verify the approach.     

Adaptive neural control of nonlinear MIMO systems with unknown time delays

In this paper, a novel adaptive NN control scheme is proposed for a class of uncertain multi-input and multi-output (MIMO) nonlinear time-delay systems. RBF NNs are used to tackle unknown nonlinear functions, then the adaptive NN tracking controller is constructed by combining Lyapunov-Krasovskii functionals and the dynamic surface control (DSC) technique along with the minimal-learning-parameters (MLP) algorithm. The proposed controller guarantees uniform ultimate boundedness (UUB) of all the signals in the closed-loop system, while the tracking error converges to a small neighborhood of the origin. An advantage of the proposed control scheme lies in that the number of adaptive parameters for each subsystem is reduced to one, triple problems of “explosion of complexity”, “curse of dimension” and “controller singularity” are solved, respectively. Finally, a numerical simulation is presented to demonstrate the effectiveness and performance of the proposed scheme.     

Control of discrete time nonlinear systems with a time-varying structure

In this paper, we present a control methodology for a class of discrete time nonlinear systems that depend on a possibly exogenous scheduling variable. This class of systems consists of an interpolation of nonlinear dynamic equations in strict feedback form, and it may represent systems with a time-varying nonlinear structure. Moreover, this class of systems is able to represent some cases of gain scheduling control, Takagi-Sugeno fuzzy systems, as well as input-output realizations of nonlinear systems which are approximated via localized linearizations. We present two control theorems, one using what we call a “global” approach (akin to traditional backstepping), and a “local” approach, our main result, where backstepping is again used but the control law is an interpolation of local control terms. An aircraft wing rock regulation problem with varying angle of attack is used to illustrate and compare the two approaches.     

Adaptive output-feedback stabilization of non-local hyperbolic PDEs

We address the problem of adaptive output-feedback stabilization of general first-order hyperbolic partial integro-differential equations (PIDE). Such systems are also referred to as PDEs with non-local (in space) terms. We apply control at one boundary, take measurements on the other boundary, and allow the system’s functional coefficients to be unknown. To deal with the absence of both full-state measurement and parameter knowledge, we introduce a pre-transformation (which happens to be based on backstepping) of the system into an observer canonical form. In that form, the problem of adaptive observer design becomes tractable. Both the parameter estimator and the control law employ only the input and output signals (and their histories over one unit of time). Prior to presenting the adaptive design, we present the non-adaptive/baseline controller, which is novel in its own right and facilitates the understanding of the more complex, adaptive system. The parameter estimator is of the gradient type, based on a parametric model in the form of an integral equation relating delayed values of the input and output. For the closed-loop system we establish boundedness of all signals, pointwise in space and time, and convergence of the PDE state to zero pointwise in space. We illustrate our result with a simulation.     

Control of 1-D parabolic PDEs with Volterra nonlinearities, Part I: Design

Boundary control of nonlinear parabolic PDEs is an open problem with applications that include fluids, thermal, chemically-reacting, and plasma systems. In this paper we present stabilizing control designs for a broad class of nonlinear parabolic PDEs in 1-D. Our approach is a direct infinite dimensional extension of the finite-dimensional feedback linearization/backstepping approaches and employs spatial Volterra series nonlinear operators both in the transformation to a stable linear PDE and in the feedback law. The control law design consists of solving a recursive sequence of linear hyperbolic PDEs for the gain kernels of the spatial Volterra nonlinear control operator. These PDEs evolve on domains T_n of increasing dimensions n+1 and with a domain shape in the form of a “hyper-pyramid”, 0≤ξ_n≤ξ_n−1?≤ξ₁≤x≤1. We illustrate our design method with several examples. One of the examples is analytical, while in the remaining two examples the controller is numerically approximated. For all the examples we include simulations, showing blow up in open loop, and stabilization for large initial conditions in closed loop. In a companion paper we give a theoretical study of the properties of the transformation, showing global convergence of the transformation and of the control law nonlinear Volterra operators, and explicitly constructing the inverse of the feedback linearizing Volterra transformation; this, in turn, allows us to prove L² and H¹ local exponential stability (with an estimate of the region of attraction where possible) and explicitly construct the exponentially decaying closed loop solutions.     

A second-order maximum principle for discrete-time bilinear control systems with applications to discrete-time linear switched systems

A powerful approach for analyzing the stability of continuous-time switched systems is based on using optimal control theory to characterize the “most unstable” switching law. This reduces the problem of determining stability under arbitrary switching to analyzing stability for the specific “most unstable” switching law. For discrete-time switched systems, the variational approach received considerably less attention. This approach is based on using a first-order necessary optimality condition in the form of a maximum principle (MP), and typically this is not enough to completely characterize the “most unstable” switching law. In this paper, we provide a simple and self-contained derivation of a second-order necessary optimality condition for discrete-time bilinear control systems. This provides new information that cannot be derived using the first-order MP. We demonstrate several applications of this second-order MP to the stability analysis of discrete-time linear switched systems.     

Adaptive boundary control of a flexible manipulator with input saturation

In this study, we consider the anti-windup design as one of the approaches for the boundary control problem of a flexible manipulator in the presence of system parametric uncertainties, external disturbances and bounded inputs. The dynamics of the system are represented by partial differential equations (PDEs). Using the singular perturbation approach, the PDE model is divided into two simpler subsystems. With the Lyapunov's direct method, an adaptive boundary control scheme is developed to regulate the angular position and suppress the elastic vibration simultaneously and the adaptive laws are designed to compensate for the system parametric uncertainties and the disturbances. The proposed control scheme allows the application of smooth hyperbolic functions, which satisfy physical conditions and input restrictions, be easily realised. Numerical simulations demonstrate the effectiveness of the proposed scheme.     

The changing face of adaptive control: The use of multiple models

Adaptive systems that continuously monitor their own performance and adjust their control strategies to improve it, have been studied for over 50 years. The theory of such systems is now commonly referred to as classical adaptive control. Such control is now well established and is found to be satisfactory when the uncertainty in the system to be controlled (i.e. the plant) is small.During the past 15 years several attempts were made to extend this general methodology to systems with large uncertainties, by using multiple models to identify the plant. Among these, two general methods based on “switching” and “switching and tuning” have emerged as the leading contenders. Recently, a radically different approach was proposed by the authors (Han & Narendra, 2010b), in which the multiple models are used to play a significantly larger role in the decision making process, resulting in substantial improvement in performance.In this paper, which is tutorial in nature, the three methods based on multiple models are critically examined. At the same time, alternative methods using fixed and adaptive models are also proposed. In all cases, detailed simulation studies of adaptation in different environments are presented. Theoretical explanations are given, where available, for the wide spectrum of performances observed in the simulation studies.     

Adaptive neural dynamic surface control for servo systems with unknown dead-zone

An adaptive neural controller is proposed for nonlinear systems with a nonlinear dead-zone and multiple time-delays. The often used inverse model compensation approach is avoided by representing the dead-zone as a time-varying system. The “explosion of complexity” in the backstepping synthesis is eliminated in terms of the dynamic surface control (DSC) technique. A novel high-order neural network (HONN) with only a scalar weight parameter is developed to account for unknown nonlinearities. The control singularity and some restrictive requirements on the system are circumvented. Simulations and experiments for a turntable servo system with permanent-magnet synchronous motor (PMSM) are provided to verify the reliability and effectiveness.     

Output-feedback stabilization of an unstable wave equation

We consider the problem of stabilization of a one-dimensional wave equation that contains instability at its free end and control on the opposite end. In contrast to classical collocated “boundary damper” feedbacks for the neutrally stable wave equations with one end satisfying a homogeneous boundary condition, the controllers and the associated observers designed in the paper are more complex due to the open-loop instability of the plant. The controller and observer gains are designed using the method of “backstepping,” which results in explicit formulae for the gain functions. We prove exponential stability and the existence and uniqueness of classical solutions for the closed-loop system. We also derive the explicit compensators in frequency domain. The results are illustrated with simulations.     

An ISS-modular approach for adaptive neural control of pure-feedback systems

Controlling non-affine non-linear systems is a challenging problem in control theory. In this paper, we consider adaptive neural control of a completely non-affine pure-feedback system using radial basis function (RBF) neural networks (NN). An ISS-modular approach is presented by combining adaptive neural design with the backstepping method, input-to-state stability (ISS) analysis and the small-gain theorem. The difficulty in controlling the non-affine pure-feedback system is overcome by achieving the so-called “ISS-modularity” of the controller-estimator. Specifically, a neural controller is designed to achieve ISS for the state error subsystem with respect to the neural weight estimation errors, and a neural weight estimator is designed to achieve ISS for the weight estimation subsystem with respect to the system state errors. The stability of the entire closed-loop system is guaranteed by the small-gain theorem. The ISS-modular approach provides an effective way for controlling non-affine non-linear systems. Simulation studies are included to demonstrate the effectiveness of the proposed approach.     

Adaptive differential dynamic programming for multiobjective optimal control

An efficient numerical solution scheme entitled adaptive differential dynamic programming is developed in this paper for multiobjective optimal control problems with a general separable structure. For a multiobjective control problem with a general separable structure, the “optimal” weighting coefficients for various performance indices are time-varying as the system evolves along any noninferior trajectory. Recognizing this prominent feature in multiobjective control, the proposed adaptive differential dynamic programming methodology combines a search process to identify an optimal time-varying weighting sequence with the solution concept in the conventional differential dynamic programming. Convergence of the proposed adaptive differential dynamic programming methodology is addressed.     

Absolute stability of third-order systems: A numerical algorithm

The problem of absolute stability is one of the oldest open problems in the theory of control. Even for the particular case of second-order systems a complete solution was presented only very recently. For third-order systems, the most general theoretical results were obtained by Barabanov. He derived an implicit characterization of the “most destabilizing” nonlinearity using a variational approach. In this paper, we show that his approach yields a simple and efficient numerical scheme for solving the problem in the case of third-order systems. This allows the determination of the critical value where stability is lost in a tractable and accurate fashion. This value is important in many practical applications and we believe that it can also be used to develop a deeper theoretical understanding of this interesting problem.     

Sliding mode dynamics in continuous feedback control for distributed discrete-event scheduling

A continuous feedback control approach for real-time scheduling of discrete events is presented in this paper motivated by the need for control theoretic techniques to analyze and design such systems in distributed manufacturing applications. These continuous feedback control systems exhibit highly nonlinear and discontinuous dynamics. Specifically, when the production demand in the manufacturing system exceeds the available resource capacity then the control system “chatters” and exhibits sliding modes. This sliding mode behavior is advantageously used in the scheduling application by allowing the system to visit different schedules within an infinitesimal region near the sliding surface. In this paper, an analytical model is developed to characterize the sliding mode dynamics. This model is then used to design controllers in the sliding mode domain to improve the effectiveness of the control system to “search” for schedules with good performance. Computational results indicate that the continuous feedback control approach can provide near-optimal schedules and that it is computationally efficient compared to existing scheduling techniques.     

Adaptive extremum seeking control of nonlinear dynamic systems with parametric uncertainties

We pose and solve an extremum seeking control problem for a class of nonlinear systems with unknown parameters. Extremum seeking controllers are developed to drive system states to the desired set-points that extremize the value of an objective function. The proposed adaptive extremum seeking controller is “inverse optimal” in the sense that it minimizes a meaningful cost function that incorporates penalty on both the performance error and control action. Simulation studies are provided to verify the effectiveness of the proposed approach.     

Robust adaptive boundary control of a perturbed hybrid Euler-Bernoulli beam with coupled rigid and flexible motion

In this paper, a robust adaptive boundary controller is proposed to stabilize the coupled rigid-flexible motion of an Euler-Bernoulli beam in presence of boundary and distributed perturbations. Applying Hamilton’s principle, the dynamics of the hybrid beam model, including the actuators hub and the payload at its ends, is represented through four nonhomogeneous nonlinear partial differential equations (PDEs) subject to ordinary differential equations (ODEs) of boundary conditions. Using a Lyapunov-based control synthesis procedure, a robust nonlinear boundary controller is established that asymptotically stabilizes the perturbed beam vibration while regulating the rigid motion coordinates. A redesign of the proposed control laws produces a robust adaptive boundary controller that achieves control objectives in the presence of both parametric and modelling uncertainties. Control design is directly based on system PDEs without truncating the model so that instabilities from spillover effects are mitigated. The control inputs to the beam consist of three forces/torque applied to the actuators hub and a transverse force applied to the tip payload. Simulation results are used to investigate the efficiency of the proposed approach.

     

一类纯反馈力学系统的自适应模糊动态面控制

针对一类纯反馈形式的不稳定力学系统,提出自适应模糊动态面控制方法。在一般动态面控制的设计框架下,引入模糊系统逼近模型的未知函数,设计自适应律在线估计模糊系统权参数和模型未知参数,通过Lyapunov方法证明得出闭环系统半全局稳定。该策略避免了传统反演设计存在的“微分爆炸”现象,并且解决了纯反馈系统控制设计中通常存在的循环设计问题。仿真结果表明,控制系统能够克服不确定性,且能够简单有效地实现跟踪控制。     

An object class-uncertainty induced adaptive force and its application to a new hybrid snake

Object segmentation is of paramount interest in many imaging applications, especially, those involving numeric, symbolic, syntactic, or even high level cognitive knowledge perception. Among others, “snake”—an “active contour” model—is a popular boundary-based segmentation approach where a smooth curve is continuously deformed to lock onto an object boundary. The dynamics of a snake is governed by different internal and external forces. A major limitation of the present framework has been the difficulty of incorporating object-intensity driven features into snake dynamics so as to prevent uncontrolled expansion/contraction once the snake leaks through a weak boundary region. In this paper, a local-intensity-driven “adaptive force” is introduced into the model using object class-uncertainty theory. Given a priori knowledge of object/background intensity distributions, class-uncertainty theory yields object/background classification of every location and establishes its confidence level. It has been demonstrated earlier that confidence level is high inside homogeneous regions and low near boundaries. In the current paper, object class-uncertainty theory has been applied to control snake deformation leading to a new adaptive force acting outward (expanding) inside intensity-defined object regions and inward (squeezing) inside background regions. It has been demonstrated that the method possesses potential to resist uncontrolled expansion of a snake contour (for an expanding type) inside background after leaking through a weak boundary. Further, it has been shown that the adaptive force operates in a complementary fashion with the image intensity gradient by reducing its strength near boundaries using the confidence level of classification. Another major contribution of this paper is the formulation of a “hybrid snake” (HS)—a new model, where an initial contour is gradually deformed over a hybrid energy surface composed of some direct energies (e.g., internal energies) and other indirect energies contributed by local contour displacements over a force-field (e.g., image or user-constrained force-field). Applications of the proposed adaptive force-enabled HS on different phantom and real images have been presented and comparisons have been made with a conventional snake (CS). Finally, a quantitative comparison based on computer-generated phantoms at various levels of blur and noise has been provided.