Vector dissipativity theory and stability of feedback interconnections for large-scale non-linear dynamical systems

Recent technological demands have required the analysis and control design of increasingly complex, large-scale non-linear dynamical systems. In analysing these large-scale systems, it is often desirable to treat the overall system as a collection of interconnected subsystems. Solution properties of the large-scale system are then deduced from the solution properties of the individual subsystems and the nature of the system interconnections. In this paper we develop an analysis framework for large-scale dynamical systems based on vector dissipativity notions. Specifically, using vector storage functions and vector supply rates, dissipativity properties of the composite large-scale system are shown to be determined from the dissipativity properties of the subsystems and their interconnections. Furthermore, extended Kalman–Yakubovich–Popov conditions, in terms of the subsystem dynamics and interconnection constraints, characterizing vector dissipativeness via vector system storage functions are derived. Finally, these results are used to develop feedback interconnection stability results for large-scale non-linear dynamical systems using vector Lyapunov functions.     

Dynamic modeling and control for aerial arm-operating of a multi-propeller multifunction aerial robot

We proposed a multi-propeller multifunction aerial robot that is constructed by a quadrotor with two multi-DOF arms to enable aerial robotic operations. This paper addresses the dynamics and control problems for aerial arm-operation. The dynamic modeling considering the coupling between the arms and main-body subsystems is investigated using the Lagrange approach. The dynamics of the system are partitioned into the main-body dynamics, the arm dynamics, and the interaction dynamics. A composite controller consisting of a main-body sub-controller and an arm sub-controller are presented. Each sub-controller is designed based on the partitioned dynamics. The main-body sub-controller is designed using trajectory linearization control technique. This composite controller is appropriate for real-time implementation due to its simplicity. An optimal planning strategy that minimizes the interaction between main-body subsystem and arm subsystem is proposed. Experimental results are presented, verifying the effectiveness of the composite controller.     

基于最大反馈线性化的两轮机器人平衡控制

为了对两轮直立式机器人进行平衡控制研究,首先应用一种拉格朗日方法对两轮直立式机器人系统进行动力学建模,以力矩作为模型控制输入并且考虑到了系统无滑动的非完整约束方程.求出了系统的最大相对阶并对非线性系统进行了部分反馈线性化,经过坐标变换和输出反馈得到了2个由两链积分器构成的子系统,外加3个代表系统内动态的非线性方程.根据...     

Variance results for identification of cascade systems

The objective of this contribution is to analyze statistical properties of estimated models of cascade systems. Models of such systems are important in for example cascade control applications. The aim is to present and analyze some fundamental limitations in the quality of an identified model of a cascade system under the condition that the true subsystems have certain common dynamics. The model quality is analyzed by studying the asymptotic (large data) covariance matrix of the Prediction Error Method parameter estimate. The analysis will focus on cascade systems with three subsystems. The main result is that if the true transfer functions of the first and second subsystem are identical, the output signal information from the second and third subsystems will not affect the asymptotic variance of the estimated model of the first subsystem. This result implies that for a cascade system with two subsystems, where the dynamics of the first subsystem is a factor of the dynamics of the second one, the output signal information from the second subsystem will not improve the asymptotic quality of the estimate of the first subsystem. The results are illustrated by some simple FIR examples.     

混合励磁电机系统输入输出解耦和线性化

讨论混合励磁电机系统的输入输出解耦和线性化问题.根据机电动力学原理,导出了混合励磁电机系统在与转子同步旋转的d-q坐标系中的动态方程.应用非线性系统几何理论,通过非线性状态反馈和坐标变换,实现了混合励磁电机系统的输入输出解耦控制和完全线性化.将原系统分解为3个线性子系统：d轴磁链子系统、q轴磁链子系统和转速子系统.仿真结果表明,基于输入输出线性化控制设计的混合励磁电机控制系统具有良好的动态性能.     

On-line co-ordination under uncertainty of weakly interacting dynamical systems

The problem of on-line co-ordination of a hierarchically controlled dynamical system under uncertainty by using the interaction balance principle is discussed. For the sake of simplicity, a two-level system consisting of two subsystems interconnected by means of buffer storages is considered. In order to avoid measure theoretic difficulties, only a finite set of typical disturbance realizations are taken to represent the uncertainty in the system. The uncertainty leads to a co-ordination method based on a ‘balance in the mean’ condition. It turns out that the weakness of the mutual interaction between the subsystems is an important concept. In the case considered the buffer storages contribute to the weakness of the interaction to a noticeable degree.     

平面倒立摆自适应滑模模糊控制

采用拉格朗日方程建立平面倒立摆的动力学模型，并将其在平衡位置进行线性化，得到了系统在X和Y两个正交控制方向解耦的近似模型．针对每一个控制方向上由互相耦合的基座小车定位子系统和摆杆镇定子系统组成的欠驱动系统，设计了自适应滑模模糊控制器，实现了基座小车沿圆周行走条件下摆杆的运动平衡控制．行走实验验证了所提出控制算法的有效性．     

Tracking trajectory in the workspace of rigid manipulators using distributed adaptive control strategy

This paper discusses the tracking trajectory in the workspace of rigid manipulators using distributed adaptive control strategy. This control strategy consists of two steps; first, the classical MIMO dynamical system is decomposed into a set of nonlinear interconnected subsystems. Each subsystem has one joint. Second, the distributed adaptive control strategy is introduced. This control strategy consists of controlling the last subsystem while assuming that the remaining subsystems are stable. Then, going backward to the second last subsystem, the same strategy is applied and so on until the first one. The system parameters are assumed to be unknown. An adaptive control is used to estimate these parameters. Indeed, the unknown parameters existing in the equation of motion of the last subsystem are first estimated and the control law is developed based on these estimated parameters. Then, going backward to the before last joint, the control law is developed using its own estimated parameters and the ones already estimated in the upper level subsystem. Asymptotical stability of the error dynamics is proved using Lyapunov approach. The developed algorithm is experimented on a 4 DOF hyper redundant articulated nimble adaptable trunk robot and compared with the classical computed torque approach. Good tracking in the workspace and joint space is obtained and effectiveness of the results is shown.     

Adaptive composite suboptimal control for linear singularly perturbed systems with unknown slow dynamics

In this article, using singular perturbation theory and adaptive dynamic programming (ADP) approach, an adaptive composite suboptimal control method is proposed for linear singularly perturbed systems (SPSs) with unknown slow dynamics. First, the system is decomposed into fast‐ and slow‐subsystems and the original optimal control problem is reduced to two subproblems in different time‐scales. Afterward, the fast subproblem is solved based on the known model of the fast‐subsystem and a fast optimal control law is designed by solving the algebraic Riccati equation corresponding to the fast‐subsystem. Then, the slow subproblem is reformulated by introducing a system transformation for the slow‐subsystem. An online learning algorithm is proposed to design a slow optimal control law by using the information of the original system state in the framework of ADP. As a result, the obtained fast and slow optimal control laws constitute the adaptive composite suboptimal control law for the original SPSs. Furthermore, convergence of the learning algorithm, suboptimality of the adaptive composite suboptimal control law and stability of the whole closed‐loop system are analyzed by singular perturbation theory. Finally, a numerical example is given to show the feasibility and effectiveness of the proposed methods.     

Decentralized overlapping control of a formation of unmanned aerial vehicles

Decentralized overlapping feedback laws are designed for a formation of unmanned aerial vehicles. The dynamic model of the formation with an information structure constraint in which each vehicle, except the leader, only detects the vehicle directly in front of it, is treated as an interconnected system with overlapping subsystems. Using the mathematical framework of the inclusion principle, the interconnected system is expanded into a higher dimensional space in which the subsystems appear to be disjoint. Then, at each subsystem, a static state feedback controller is designed to robustly stabilize the perturbed nominal dynamics of the subsystem. The design procedure is based on the application of convex optimization tools involving linear matrix inequalities. As a final step, the decentralized controllers are contracted back to the original interconnected system for implementation.     

Stability theory for nonnegative and compartmental dynamical systems with time delay

Nonnegative and compartmental dynamical system models are derived from mass and energy balance considerations and involve the exchange of nonnegative quantities between subsystems or compartments. These models are widespread in biological and physical sciences and play a key role in understanding these processes. A key physical limitation of such systems is that transfers between compartments is not instantaneous and realistic models for capturing the dynamics of such systems should account for material in transit between compartments. In this paper, we present necessary and sufficient conditions for stability of nonnegative and compartmental dynamical systems with time delay. Specifically, asymptotic stability conditions for linear and nonlinear nonnegative dynamical systems with time delay are established using linear Lyapunov–Krasovskii functionals.     

Modelling and control design for a magnetic levitation system

We study the problem of controlling the position of a platen levitated using linear motors in three-dimensional space. We develop a non-linear six-state model of the system and provide two non-linear controllers solving the set point stabilization problem. The first controller is derived by decomposing the model in two subsystems, applying feedback linearization to one of them, and using the invariance principle to prove attractiveness of the origin of the second subsystem. The second controller is found by dynamic feedback linearizing the entire system dynamics. In both cases we provide a rigorous procedure to determine the operating range of the device.     

A dynamics formulation of general constrained robots

The complexity of a standard compact-in-form Lagrangian dynamical expression is proportional to the fourth power of the number of degrees of freedom (DOF) of a robotic system. This fact challenges both simulation and control of robots with hyper degrees of freedom. In this paper, a systematic approach for deriving the dynamical expression of so-called general constrained robots is proposed. This proposed approach has two main features. First, it uses the subsystem dynamics such as the dynamics of joints and rigid links to construct the dynamical expression of the entire robotic system in a closed form. The complexity of the resulted dynamic expression is linearly proportional to the number of DOF of a robotic system. Second, it extends the standard dynamical form and properties of the conventional single-arm constrained robots to a class of more general robotic systems including the coordinated multiple-arm robotic systems. Three spaces, namely the general joint space, the general task space, and the extended subsystems space, are connected through corresponding velocity/force mapping matrices.An erratum to this article can be found at     

Dissipativity theory for nonnegative and compartmental dynamical systems with time delay

Nonnegative and compartmental dynamical system models are derived from mass and energy balance considerations that involve dynamic states whose values are nonnegative. These models are widespread in engineering and life sciences and typically involve the exchange of nonnegative quantities between subsystems or compartments wherein each compartment is assumed to be kinetically homogeneous. However, in many engineering and life science systems, transfers between compartments are not instantaneous and realistic models for capturing the dynamics of such systems should account for material in transit between compartments. Including some information of past system states in the system model leads to infinite-dimensional delay nonnegative dynamical systems. In this note, we develop dissipativity theory for nonnegative dynamical systems with time delay using linear storage functionals with linear supply rates. These results are then used to develop general stability criteria for feedback interconnections of nonnegative dynamical systems with time delay.     

An approach for decomposition and reduction of dynamical models of large-scale power systems

This paper describes an approach for decomposition and reduction of dynamical models of largo scale power systems. In this approach the system is decomposed into two subsystems, the first one is described by a linear model while the second is described by a non-linear model. This decomposition is based on a derived criteria for linearization by which we can know the two subsystems a priori. Further, the linear subsystem model is reduced by aggregation and hence the order of the system is reduced. The results of the validity of this approach as applied to two different largo power systems are indicated.     

Global stable tracking control of underactuated ships with input saturation

In this paper, tracking control of underactuated ship in the presence of input saturation is addressed. By dividing the tracking error dynamic system into a cascade of two subsystems, the torques in surge and yaw axes are designed separately using the backstepping technique. More specifically, we design the yaw axis torque in such a way that its corresponding subsystem is finite time stable, which makes it to be de-coupled from the second subsystem after a finite time. This enables us to design the torque in the surge axis independently. It is shown that the closed-loop system is stable and the mean-square tracking errors can be made arbitrarily small by choosing design parameters. Simulation results also verify the effectiveness of the proposed scheme.     

Robust Control for Two-Time-Scale Discrete Interval Systems

The problem of designing robust controller for discrete two-time-scale interval systems, conveniently represented using interval matrix notion, is considered. The original full order two-time-scale interval system is decomposed into slow and fast subsystems using interval arithmetic. The controllers designed independently to stabilize these two subsystems are combined to get a composite controller which also stabilizes the original full order two-time-scale interval system. It is shown that a state and output feedback control law designed to stabilize the slow interval subsystem stabilizes the original full order system provided the fast interval subsystem is asymptotically stable. The proposed design procedure is illustrated using numerical examples for establishing the efficacy of the proposed method.     

Quasi-ISS/ISDS observers for interconnected systems and applications

This paper considers interconnected nonlinear dynamical systems and studies observers for such systems. For single systems the notion of quasi-input-to-state dynamical stability (quasi-ISDS) for reduced-order observers is introduced and observers are investigated using error Lyapunov functions. It combines the main advantage of ISDS over input-to-state stability (ISS), namely the memory fading effect, with reduced-order observers to obtain quantitative information about the state estimate error. Considering interconnections quasi-ISS/ISDS reduced-order observers for each subsystem are derived, where suitable error Lyapunov functions for the subsystems are used. Furthermore, a quasi-ISS/ISDS reduced-order observer for the whole system is designed under a small-gain condition, where the observers for the subsystems are used. As an application, we prove that quantized output feedback stabilization for each subsystem and the overall system is achievable, when the systems possess a quasi-ISS/ISDS reduced-order observer and a state feedback law that yields ISS/ISDS for each subsystem and therefor the overall system with respect to measurement errors. Using dynamic quantizers it is shown that under the mentioned conditions asymptotic stability can be achieved for each subsystem and for the whole system.     

Decentralized adaptive iterative learning control for interconnected systems with uncertainties

In many applications,the system dynamics allows the decomposition into lower dimensional subsystems with interconnections among them.This decomposition is motivated by the ease and flexibility of the controller design for each subsystem.In this paper,a decentralized model reference adaptive iterative learning control scheme is developed for interconnected systems with model uncertainties.The interconnections in the dynamic equations of each subsystem are considered with unknown boundaries.The proposed controller of each subsystem depends only on local state variables without any information exchange with other subsystems.The adaptive parameters are updated along iteration axis to compensate the interconnections among subsystems.It is shown that by using the proposed decentralized controller,the states of the subsystems can track the desired reference model states iteratively.Simulation results demonstrate that,utilizing the proposed adaptive controller,the tracking error for each subsystem converges along the iteration axis.