Feedback Interconnected Systems Research Articles

Partial stability of nonlinear dissipative feedback systems

Partially stable systems involve dynamical systems whose motion lie in a subspace of the state space resulting in system stability with respect to part of the system's states. In this paper, we develop partial stability theorems for nonlinear continuous-time and discrete-time dissipative feedback systems. Specifically, by invoking additional structural constraints on the forward loop and feedback loop system storage functions, we develop feedback interconnection partial stability results for dissipative nonlinear dynamical systems. Our results provide extensions of the positivity and small gain theorems for guaranteeing partial stability of feedback interconnected systems.

Phases of discrete-time LTI multivariable systems

In contrast to the well-developed gain analysis for multi-input-multi-output (MIMO) linear time-invariant (LTI) systems, the research on the phase analysis does not share the same status. In this paper, we introduce the phase response of a class of discrete-time (DT) LTI multivariable systems by exploiting a definition of matrix phases based on the numerical range. Positive frequency joint sectorial systems are also defined, which can generalize the positive real and negative imaginary systems. The interpretation of system phases is given in terms of the input and output signals of the system. A sectored real lemma is obtained to characterize the phase information from a state–space realization. Motivated by finding a phasic counterpart to the small gain theorem, we develop a small phase theorem for the internal stability of a closed-loop system. The phase properties are also investigated for parallel and feedback interconnected systems.

Dissipativity-based framework for stability analysis of aperiodically sampled nonlinear systems with time-varying delay

In this paper, we provide novel conditions for stability analysis of aperiodically sampled nonlinear control systems subjected to time-varying delay. The proposed approach can also deal with cases in which delay is larger than the sampling interval. It is applicable to a general class of nonlinear systems and provides sufficient criteria for stability that aid in making trade-offs between control performance and the bounds on sampling interval and delay. As a stepping stone, a preliminary and generic result based on dissipativity, is introduced to analyse the exponential stability of a class of feedback-interconnected systems. The nonlinear sampled-data system is remodelled to consider the effects of sampling and delay in the dissipativity framework, as perturbations to the nominal closed-loop system. This leads to constructive stability conditions for a continuous time closed-loop system given by the feedback interconnection of the nominal closed-loop system and an operator(s) that captures the effects of sampling and delay. For Linear Time-Invariant (LTI) systems, we recover simple Linear Matrix Inequality (LMI) and frequency domain conditions previously proposed in the robust control framework.

Event-Triggered Adaptive Control Design With Prescribed Performance for Macro–Micro Composite Positioning Stage

This article investigates the rapid vibration reduction problem for macro–micro composite positioning stage (MMCPS), which is of great engineering significance for microelectronics manufacturing. The MMCPS is modeled as a nonstrict feedback interconnected nonlinear system. Based on backstepping technique and radial basis function neural networks, an event-triggered adaptive control scheme with prescribed performance is developed. By designing performance function and coordinate transformation, the expected state convergence rate and precision are obtained. The relative threshold event-triggering mechanism is adopted to decrease the update frequency of control input. It is proved that all signals of the closed-loop system are semiglobally uniformly ultimately bounded. Finally, the simulation results are conducted to illustrate the effectiveness of the control algorithm.

Properties and stability analysis of discrete-time negative imaginary systems

This paper is concerned with discrete-time negative imaginary (DT-NI) functions. First, a new definition of DT-NI functions is introduced. Then, by means of the relations between discrete-time positive real and DT-NI functions, two different versions of DT-NI lemmas are established to characterize the DT-NI properties based on state-space realizations. Also, a necessary and sufficient condition is presented to guarantee the internal stability of positive feedback interconnected DT-NI systems. Meanwhile, some other properties of DT-NI functions are studied. Several numerical examples are presented to illustrate the main results of this paper. Compared to the previous results, our results remove the symmetric assumption in rational case.

Input–Output Stabilizing Controller Synthesis for SISO T–S Fuzzy Systems by Applying Large Gain Theorem

In this paper, using the large gain theorem, an input–output stabilization controller is presented for single input-single output Takagi–Sugeno (T–S) fuzzy systems. By large gain theorem, the feedback interconnected nonlinear system is stable if the product of minimum gain of subsystems is more than one. So first, a feedforward parallel distributed compensator (PDC) is designed to increase (T–S) fuzzy systems minimum gain by employing zero placement idea. The PDCs parameters are obtained by solving a set of LMIs. Then, the control loop is closed by using unit feedback loop. The controller structure is quite new and stability conditions are simple with fewer limitations so that for a wide class of nonlinear systems, linear controllers are acquired. Effectiveness of the proposed method is illustrated by simulation results.

Two-Dimensional Dissipative Control and Filtering for Roesser Model

This paper investigates the problems of two-dimensional (2-D) dissipative control and filtering for a linear discrete-time Roesser model. First, a novel sufficient condition is proposed such that the discrete-time Roesser system is asymptotically stable and 2-D $(Q,S,R)\hbox{-} \alpha$ -dissipative. Special cases, such as 2-D passivity performance and 2-D $H_{\infty} $ performance, and feedback interconnected systems are also discussed. Based on this condition, new 2-D $(Q,S,R)\hbox{-} \alpha$ -dissipative state-feedback and output-feedback control problems are defined and solved for a discrete-time Roesser model. The design problems of 2-D $(Q,S,R)\hbox{-} \alpha$ -dissipative filters of observer form and general form are also considered using a linear matrix inequality (LMI) approach. Two examples are given to illustrate the effectiveness and potential of the proposed design techniques.

Faà di Bruno Hopf algebra of the output feedback group for multivariable Fliess operators

Given two nonlinear input–output systems written in terms of Chen–Fliess functional expansions, i.e., Fliess operators, it is known that the feedback interconnected system is always well defined and in the same class. An explicit formula for the generating series of a single-input, single-output closed-loop system was provided by the first two authors in earlier work via Hopf algebra methods. This paper is a sequel. It has four main innovations. First, the full multivariable extension of the theory is presented. Next, a major simplification of the basic setup is introduced using a new type of grading that has recently appeared in the literature. This grading also facilitates a fully recursive algorithm to compute the antipode of the Hopf algebra of the output feedback group, and thus, the corresponding feedback product can be computed much more efficiently. The final innovation is an improved convergence analysis of the antipode operation, namely, the radius of convergence of the antipode is computed.

New results on stability analysis of delayed recurrent neural networks based on the integral quadratic constraints approach

This paper is concerned with the analysis of the stability of delayed recurrent neural networks. In contrast to the widely used Lyapunov–Krasovskii functional approach, a new method is developed within the integral quadratic constraints framework. To achieve this, several lemmas are first given to propose integral quadratic separators to characterize the original delayed neural network. With these, the network is then reformulated as a special form of feedback-interconnected system by choosing proper integral quadratic constraints. Finally, new stability criteria are established based on the proposed approach. Numerical examples are given to illustrate the effectiveness of the new approach.

Sum of Squares Based Input-to-State Stability Analysis of Polynomial Nonlinear Systems

Input-to-state stability (ISS) is an advanced concept of stability for nonlinear systems in the sense that the input to the system is taken into account. This paper proposes a numerically tractable condition to analyze ISS for polynomial nonlinear systems. The condition is formulated as a sum of squares (SOS) problem that is a feasibility problem including SOS polynomials. A solution of the problem is obtained by solving a semidefinite programming problem. The key technique is to formulate class K∞ functions, which have the inverse map on real nonnegative region, as polynomials. This paper also gives numerically tractable ISS conditions for the cascade connection of subsystems and for the feedback interconnected system. A few examples are shown to illustrate our analysis conditions.

Observers for interconnected nonlinear and linear systems

In this paper we begin by assuming that an observer with a corresponding quadratic-type Lyapunov function has been designed for a given nonlinear system. We then consider the problem that arises when the output of that nonlinear system is not directly available; instead, it acts as an input to a second, linear system from which a partial-state measurement is in turn available. We develop an observer design methodology for the resulting cascade interconnection, based on estimating the unavailable output together with the states of the linear system. We also extend this methodology to a more general class of feedback-interconnected systems. Under a set of technical assumptions, the overall error dynamics is proven to be globally exponentially stable if the gains are chosen to satisfy an H∞ condition. We illustrate application of the methodology by considering a navigation example based on integration of inertial and satellite measurements.

A population dynamics approach for the water distribution problem

The water distribution system (WDS) is composed of several elements, where flow control is one of the most important components needed in order to provide a satisfactory level of service. In order to achieve an adequate level of water in the distribution tanks, we need to dynamically control the flow. Here, we propose a population dynamics approach in order to control tanks, by allocating in them the maximum uniform volume. The feedback interconnected systems reach a stable equilibrium point for both approaches presented (i.e. replicator and supply dynamics), and more specific an asymptotically stable equilibrium point for the replicator dynamics case. The stability analysis uses some passivity concepts and classic Lyapunov theory for a closed-loop system that combines the population dynamics (controller), and the WDS (process). We show via simulations the operation under different scenarios.

Fault Tolerance Analysis for Switched Systems Via Global Passivity

In this brief, we introduce the passivity theory into the fault tolerance analysis for switched systems. We propose a ldquoglobal passivityrdquo concept which means that the total energy stored by the switched system is less than the total energy supplied from the outside. The individual passivity of each mode is not required, and the stability of the system can be achieved via the global energy dissipativity in the presence of faults. We further provide a ldquoperiodic fault tolerant passivityrdquo to check the fault tolerance easily. The obtained results are extended to feedback interconnected systems. A switched RLC circuit example is taken to illustrate the efficiency of the proposed results.

Nonlinear Modeling and Control of Human Heart Rate Response During Exercise With Various Work Load Intensities

The first objective of this paper is to introduce a nonlinear system to model the heart rate (HR) response during and after treadmill walking exercise. The model is a feedback interconnected system that has components to describe the central and peripheral local responses to exercise and their interactions. The parameters of the model were experimentally identified from subjects walking on a treadmill at different speeds. The stability of the obtained nonlinear model was mathematically proven. The modeling results demonstrate that the proposed model can be useful in examining the cardiovascular response to exercise. Based on the nonlinear model, the second objective is to present a computer-controlled treadmill system for the regulation of HR during treadmill exercise. The proposed nonlinear controller consists of feedforward and feedback components. The designed control system was experimentally verified and the results demonstrated that the proposed computer-controlled treadmill system regulated the HR of the experimental subjects according to two different exercising HR profiles, indicating that it can play an important role in the design of exercise protocols for individuals.

Stability robustness of feedback linearisable systems with input unmodelled dynamics

The stability robustness of a feedback linearisable system subject to unstructured uncertainty is studied, with respect to the concepts of local input-to-state stability (ISS), local input-to-output stability (IOS) and small-signal L2-stability. A nonlinear small-gain theorem for feedback interconnected systems that are characterised by both local ISS and IOS is established. This theorem is used to derive sufficient conditions for robust local ISS and IOS of the closed-loop system that is produced by the application of feedback linearising control to a nonlinear feedback linearisable system with unmodelled dynamics. Moreover, sufficient conditions for robust small-signal L2-stability are derived by employing both the aforementioned nonlinear small-gain theorem and the well known small-gain theorem for L2-stability. The conditions proposed are easily verifiable. A nontrivial illustrative example is included.

Cascaded control of feedback interconnected nonlinear systems: Application to robots with AC drives

In this paper we are interested in the design of global tracking controllers for feedback interconnected nonlinear systems with partial state measurements. We assume the forward subsystem is an underactuated Euler-Lagrange system and that, in the absence of the latter, a stabilizing controller is known for the feedback subsystem. This scenario, which appears in many practical applications with the forward subsystem representing the actuator dynamics, leads naturally to a classical cascaded (nested-loop) control scheme. Our main contribution is the establishment of conditions under which we can design an inner-loop controller for the Euler-Lagrange system such that global tracking is achieved. These are expressed in terms of actuator-sensor couplings, the “strength” of the subsystems interconnection, and the requirement of linear dependence on the unmeasurable variables. Interestingly enough, this analysis does not invoke the standard time-scale separation assumptions prevalent in cascaded schemes, but uses instead some “growth” conditions on the interconnections. The design is carried out using passivity arguments, and relies on energy shaping plus damping injection ideas. To insure asymptotic tracking of the inner loop the derivative of its reference, which is the output of the outer loop controller, is typically needed. To overcome this problem we add a nonlinear observer. The procedure is used to design an output feedback global tracking position controller for robot manipulators actuated by AC drives. To illustrate the generality of the method we consider a fairly large class of AC drives, which includes as particular cases induction, synchronous and stepper motors. Instrumental for the observer design is the utilisation of a new robot controller which is linear in the link velocities.

Reliable decentralized stabilization of feedforward and feedback interconnected systems

The synthesis of decentralized controllers that are reliable in the presence of breakdowns in the interconnections is considered for plants obtained via feedforward and feedback type interconnection schemes. The main objective is to determine the precise conditions on the plant which assure such reliable synthesis. It is shown that the central property for reliable decentralized stabilization is always an appropriate parity interlacing property among the poles of the interconnection subplants and certain blocking zeros of the overall plant. The conditions under which the stabilization of a two-channel plant is affected by the stabilization of its diagonal subplants are also obtained. The results also find application in multiple controller reliable synthesis problems. The discussion is restricted to two-channel plants.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>