Unknown input observer design and analysis for networked control systems

The insertion of communication networks in the feedback loops of control systems is a defining feature of modern control systems. These systems are often subject to unknown inputs in a form of disturbances, perturbations, or attacks. The objective of this paper is to design and analyse an observer for networked dynamical systems with unknown inputs. The network effect can be viewed as either a perturbation or time-delay to the exchanged signals. In this paper, we (1) review an unknown input observer (UIO) design for a non-networked system, (2) derive the networked unknown input observer (N_etUIO) dynamics, (3) design a N_etUIO such that the effect of higher delay order terms are nullified and (4) establish stability-guaranteeing bounds on the networked-induced time-delay and perturbation. The formulation and results derived in this paper can be generalised to scenarios and applications where the signals are perturbed due to a different source of perturbation or delay.     

Adaptive Control for State Synchronization of Nonlinear Haptic Telerobotic Systems with Asymmetric Varying Time Delays

In this paper, we introduce a new adaptive controller design scheme for nonlinear telerobotic systems with varying time delays where the delays and their variation rates are unknown. The designed controller has the ability to synchronize the state behaviors of the local and the remote robots. In this paper, asymptotic stability in the presence of varying time delays is of interest. Using the proposed controller, asymptotic stability of the bilateral telerobotic system subject to any bounded yet unknown varying delay with a bounded yet unknown rate of change can be guaranteed. Besides the varying time delay, the proposed adaptive controller has the ability to adapt to the parameter variations in the local and the remote robots’ dynamics. It is shown that position and velocity errors between the local and the remote manipulators converge to the zero asymptotically, thus ensuring teleoperation transparency. Experimental and simulation results with a pair of PHANToM haptic devices and a pair of planar manipulators under varying time delays in the communication channel demonstrate the effectiveness of the proposed scheme.     

Static output feedback control via PDE boundary and ODE measurements in linear cascaded ODE–beam systems

This paper addresses the problem of feedback control design for a class of linear cascaded ordinary differential equation (ODE)–partial differential equation (PDE) systems via a boundary interconnection, where the ODE system is linear time-invariant and the PDE system is described by an Euler–Bernoulli beam (EBB) equation with variable coefficients. The objective of this paper is to design a static output feedback (SOF) controller via EBB boundary and ODE measurements such that the resulting closed-loop cascaded system is exponentially stable. The Lyapunov’s direct method is employed to derive the stabilization condition for the cascaded ODE–beam system, which is provided in terms of a set of bilinear matrix inequalities (BMIs). Furthermore, in order to compute the gain matrices of SOF controllers, a two-step procedure is presented to solve the BMI feasibility problem via the existing linear matrix inequality (LMI) optimization techniques. Finally, the numerical simulation is given to illustrate the effectiveness of the proposed design method.     

Output-feedback adaptive control of a wave PDE with boundary anti-damping

We develop an adaptive output-feedback controller for a wave PDE in one dimension with actuation on one boundary and with an unknown anti-damping term on the opposite boundary. This model is representative of a torsional stick–slip instability in drillstrings in deep oil drilling, as well as of various acoustic instabilities. The key feature of the proposed controller is that it requires only the measurements of boundary values and not of the entire distributed state of the system. Our approach is based on employing Riemann variables to convert the wave PDE into a cascade of two delay elements, with the first of the two delay elements being fed by control and the same element in turn feeding into a scalar ODE. This enables us to employ a prediction-based design for systems with input delays, suitably converted to the adaptive output-feedback setting. The result’s relevance is illustrated with simulation example.     

Periodic signal analysis by maximum likelihood modeling of orbits of nonlinear ODEs

This paper treats a new approach to the problem of periodic signal estimation. The idea is to model the periodic signal as a function of the state of a second-order nonlinear ordinary differential equation (ODE). This is motivated by Poincare theory, which is useful for proving the existence of periodic orbits for second-order ODEs. The functions of the right-hand side of the nonlinear ODE are then parameterized by a multivariate polynomial in the states, where each term is multiplied by an unknown parameter. A maximum likelihood algorithm is developed for estimation of the unknown parameters, from the measured periodic signal. The approach is analyzed by derivation and solution of a system of ODEs that describes the evolution of the Cramer-Rao bound over time. This allows the theoretically achievable accuracy of the proposed method to be assessed in the ideal case where the signals can be exactly described by the imposed model. The proposed methodology reduces the number of estimated unknowns, at least in cases where the actual signal generation resembles that of the imposed model. This in turn is expected to result in an improved accuracy of the estimated parameters.     

Boundary stabilization of a cascade of ODE‐wave systems subject to boundary control matched disturbance

In this paper, we are concerned with a cascade of ODE‐wave systems with the control actuator‐matched disturbance at the boundary of the wave equation. We use the sliding mode control (SMC) technique and the active disturbance rejection control method to overcome the disturbance, respectively. By the SMC approach, the disturbance is supposed to be bounded only. The existence and uniqueness of solution for the closed‐loop via SMC are proved, and the monotonicity of the ‘reaching condition’ is presented without the differentiation of the sliding mode function, for which it may not always exist for the weak solution of the closed‐loop system. Considering that the SMC usually requires the large control gain and may exhibit chattering behavior, we then develop an active disturbance rejection control to attenuate the disturbance. The disturbance is canceled in the feedback loop. The closed‐loop systems with constant high gain and time‐varying high gain are shown respectively to be practically stable and asymptotically stable. Then we continue to consider output feedback stabilization for this coupled ODE‐wave system, and we design a variable structure unknown input‐type state observer that is shown to be exponentially convergent. The disturbance is estimated through the extended state observer and then canceled in the feedback loop by its approximated value. These enable us to design an observer‐based output feedback stabilizing control to this uncertain coupled system. Copyright © 2016 John Wiley & Sons, Ltd.     

Control of an unstable reaction–diffusion PDE with long input delay

A Smith Predictor-like design for compensation of arbitrarily long input delays is available for general, controllable, possibly unstable LTI finite-dimensional systems. Such a design has not been proposed previously for problems where the plant is a PDE. We present a design and stability analysis for a prototype problem, where the plant is a reaction–diffusion (parabolic) PDE, with boundary control. The plant has an arbitrary number of unstable eigenvalues and arbitrarily long delay, with an unbounded input operator. The predictor-based feedback design extends fairly routinely, within the framework of infinite-dimensional backstepping. However, the stability analysis contains interesting features that do not arise in predictor problems when the plant is an ODE. The unbounded character of the input operator requires that the stability be characterized in terms of the H₁ (rather than the usual L₂) norm of the actuator state. The analysis involves an interesting structure of interconnected PDEs, of parabolic and first-order hyperbolic types, where the feedback gain kernel for the undelayed problem becomes an initial condition in a PDE arising in the compensator design for the problem with input delay. Space and time variables swap their roles in an interesting manner throughout the analysis.     

带有大输出时滞的ODE–热方程级联系统的观测器设计

为了估计系统的状态并补偿输出时滞,本文提出一种新颖的观测器设计方法应用在常微分方程(ODE)-热方程级联系统中.不同于传统观测器的设计方法,该方法的核心思想是将大时滞划分为若干段小时滞,再设计一串级联的子观测器逐步地估计系统的状态.这种方法的优势在于对于大时滞仍有效.本文通过backstepping-like方法实现了原系统与目标系统的等价变换,结合Lyapunov-Krasovskii方法和线性矩阵不等式理论,得到了误差系统指数稳定的结果.最后,通过仿真实例验证了级联观测器的有效性.     

Stabilization of a general linear heat‐ODE system coupling at an intermediate point

This paper considers the stabilization of a general linear heat‐ODE system coupling at an intermediate point which is an extension from our previous work on stabilizing a coupled system of a heat PDE and a second‐order ODE. A novel backstepping transformation is proposed which can overcome the difficulty of using the one proposed in our previous work to a more general system considered in the current paper. Existence of smooth kernels in both the forward and inverse transformations is proved. Also, we show that forward and inverse transformations are a mutually inverse pair governed by the kernels equations. Finally, the effectiveness of the controller design is demonstrated with some numerical examples for heat‐ODE systems. Copyright © 2017 John Wiley & Sons, Ltd.     

Static output feedback control design for linear MIMO systems with actuator dynamics governed by diffusion PDEs

This paper deals with the problem of static output feedback (SOF) control design for a class of diffusion partial differential equation (PDE) and ordinary differential equation (ODE) cascades, where the ODE model is used to describe the dynamics of the multi-input and multi-output (MIMO) plant and the diffusion PDE model is employed to represent the dynamics of actuators. The objective of this paper is to develop a simple as well as effective SOF controller via the Lyapunov's direct method such that the resulting closed-loop system is globally exponentially stable. By constructing a quadratic Lyapunov function, the sufficient condition on the globally exponential stability of the closed-loop cascaded system is presented in terms of linear matrix inequality (LMI). Then, an LMI-based design method of the SOF controller is developed on the basis of the obtained stability analysis result. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed design method.     

On one numerical method for solving some self-similarityproblems of gas-dynamics on a multiprocessor

In this paper we consider mathematical models of some problems of natural science, for example, self-similarity problems of gas-dynamics giving rise to boundary problems of first order ordinary differential equations (ODE) with one parameter. The boundary problems of first order ODE with one parameter are considered in [1, 2], where iterative methods based on the implementation of Newton's Method, are presented. Next, an iterative method for solving the boundary value problem of the first order system of ODE with one parameter on a multiprocessor type SIMD [3] is shown. The convergence of this process is proved and the speed of convergence is estimated. The feasibility of this method is illustrated for the one dimensional instability movement of gas arising from the movement of the piston in presence of a volume source (volume channel) of mass, impulse and energy in gas. Finally the results are given.     

Output feedback control of general linear heterodirectional hyperbolic PDE-ODE systems with spatially-varying coefficients

This paper presents a backstepping solution for the output feedback control of general linear heterodirectional hyperbolic PDE-ODE systems with spatially varying coefficients. Thereby, the ODE is coupled to the PDE in-domain and at the uncontrolled boundary, whereas the ODE is coupled with the latter boundary. For the state feedback design, a two-step backstepping approach is developed, which yields the conventional kernel equations and additional decoupling equations of simple form. In order to implement the state feedback controller, the design of observers for the PDE-ODE systems in question is considered, whereby anti-collocated measurements are assumed. Exponential stability with a prescribed convergence rate is verified for the closed-system pointwise in space. The resulting compensator design is illustrated for a 4 × 4 heterodirectional hyperbolic system coupled with a third-order ODE modelling a dynamic boundary condition.     

Adaptive fuzzy control for unknown nonlinear time-delay systems with virtual control functions

In this paper, adaptive fuzzy control is presented for a class of unknown nonlinear timedelay systems with virtual control functions. By employing fuzzy logic systems and the technique of delay replacement, dynamic surface control (DSC) design approach can be carried out with both unknown delay signals and nonlinearities. This is different from the existing results, which are used to make limitations on the time-delays. It is proved that the proposed design method is able to guarantee semiglobal uniform ultimate boundedness (SGUUB) of all signals in the closed-loop system, with arbitrary small tracking error by appropriately choosing design constants.     

Optimal determination of rate coefficients in multiple-reaction systems

An efficient methodology is developed for parameter estimation and is applied by fitting 6 unknown rate coefficients. The estimation procedure is generally applicable to any system, although development has currently been limited to first-order systems of ordinary differential equations (ODE), such as those describing multiple chemical reactions. The objective is to find parameter values so as to minimize the sum of squared error (SSE), where each error term is the difference between the calculated system solution at a point and a selected data value. Since the calculated solution is generally quite nonlinear, an iterative solution is required. At each iteration, parameter values are supplied, the system is solved, and the SSE is determined. In addition, efficient algorithms require the SSE gradient (with respect to the vector of unknown parameters) in order to provide updated parameter estimates. Using conventional techniques, determination of this gradient involves solution of an ODE system for each parameter to be estimated. ff more than a few parameters are involved, the cost could be prohibitive. However, a procedure using adjoint operators is developed in which the SSE gradient can be calculated by solving only one additional ODE system, regardless of the number of parameters being optimized. Combined with a quasi-Newton updating system, an efficient methodology results. This methodology has been applied to a set of six chemical reactions describing the aqueous speciation (hydrolysis) of iodine.     

Linear parameter-varying discrete time-delay systems: Stability and l 2-gain controllers

In this paper, we investigate a class of linear parameter-varying discrete time-delay (LPVDTD) systems where the state-space matrices depend on time-varying parameters and the delay is unknown but bounded. We treat both notions of quadratic stability based on a single quadratic Lyapunov function and affine quadratic stability using parameter-dependent Lyapunov functions. In both cases, we develop LMI-based results of stability testing for time-delay as well as delayless discrete-time systems. Then, we design state-feedback controllers which guarantee quadratic stability and an induced l ₂-norm bound. For the case of dynamic output feedback control, we use a parameter-independent quadratic Lyapunov-Krasovskii function to develop LMI-based solvability conditions which are evaluated at the extreme points of the admissible parameter set. Throughout the paper, complementary results for linear parameter-varying discrete (LPVD) systems without delay are presented.     

Global tracking of nonlinear systems with simultaneous unknown time‐varying state and input delays

This paper presents the design of a robust control law for a class of nonlinear dynamical systems subjected to parametric uncertainty and simultaneous unknown, variable state and input delays. A novel controller is developed, which consists of a filtered tracking error and the integral of previous values of control input where the limits of integration are dependent on the known bound of the input delay. Lyapunov‐Krasovskii functionals–based stability analysis guarantees a global uniformly ultimately bounded tracking result where sufficient conditions on controller gains and maximum allowable delay are derived. The performance and robustness of the controller are evaluated by simulation on a two‐link robot manipulator for different combinations of time‐varying state and input delays.     

A galerkin/neural-network-based design of guaranteed cost control for nonlinear distributed parameter systems.

This paper presents a Galerkin/neural-network- based guaranteed cost control (GCC) design for a class of parabolic partial differential equation (PDE) systems with unknown nonlinearities. A parabolic PDE system typically involves a spatial differential operator with eigenspectrum that can be partitioned into a finite-dimensional slow one and an infinite-dimensional stable fast complement. Motivated by this, in the proposed control scheme, Galerkin method is initially applied to the PDE system to derive an ordinary differential equation (ODE) system with unknown nonlinearities, which accurately describes the dynamics of the dominant (slow) modes of the PDE system. The resulting nonlinear ODE system is subsequently parameterized by a multilayer neural network (MNN) with one-hidden layer and zero bias terms. Then, based on the neural model and a Lure-type Lyapunov function, a linear modal feedback controller is developed to stabilize the closed-loop PDE system and provide an upper bound for the quadratic cost function associated with the finite-dimensional slow system for all admissible approximation errors of the network. The outcome of the GCC problem is formulated as a linear matrix inequality (LMI) problem. Moreover, by using the existing LMI optimization technique, a suboptimal guaranteed cost controller in the sense of minimizing the cost bound is obtained. Finally, the proposed design method is applied to the control of the temperature profile of a catalytic rod.     

On-Line Symbolic Constraint Embedding for Simulation of Hybrid Dynamical Systems

In this paper we present a simulator designed to handle multibody systems with changing constraints, wherein the equations of motion for each of its constraint configurations are formulated in minimal ODE form with constraints embedded before they are passed to an ODE solver. The constraint-embedded equations are formulated symbolically according to a re-combination of terms of the unconstrained equations, and this symbolic process is undertaken on-line by the simulator. Constraint-embedding undertaken on-the-fly enables the simulation of systems with an ODE solver for which constraints are not known prior to simulation start or for which the enumeration of all constraint conditions would be unwieldy because of their complexity or number. Issues of drift associated with DAE solvers that usually require stabilization are sidestepped with the constraint-embedding approach. We apply nomenclature developed for hybrid dynamical systems to describe the system with changing constraints and to distinguish the roles of the forward dynamics solver, a collision detector, and an impact resolver. We have prototyped the simulator in MATLAB and demonstrate the design using three representative examples.