A relaxed coupling method for algebraically constrained mechanical systems

Multibody System Dynamics - A coupling method is presented that aims at computing the dynamics of constrained mechanical systems connected by algebraic constraints. A relaxed coupling method is...     

Doubly-indexed dynamical systems: State-space models and structural properties

Doubly-indexed dynamical systems provide a state space realization of two-dimensional filters which includes previous state models. Algebraic criteria for testing structural properties (reachability, observability, internal stability) are introduced.This work was supported by CNR-GNAS     

Stability analysis of non-linear dynamical systems

This paper examines the possibility of using non-linear comparison systems in the stability analysis of non-linear dynamical systems. Stability criteria and estimates of the stability region of non-linear comparison systems are established. It is shown that the use of non-linear comparison systems may lead to less conservative results than those obtained by linear ones.     

Hierarchical optimisation for non-linear dynamical systems with non-separable cost functions

In this paper the previous hierarchical optimisation algorithm of Hassan and Singh for non-linear interconnected dynamical systems with separable cost functions is extended to the case of non-linear and non-separable cost functions. This ensures that any decomposition could be used and makes the new algorithm suitable for the optimisation of general non-linear problems.     

Passivity-based robust feedback control for non-linear systems with input dynamical uncertainty

This paper addresses the robust feedback control problem for a class of non-linear systems with uncertain input dynamics. The main objective is to develop a passivity-based systematic design approach for this kind of uncertain system. First, a passivity condition is presented for a non-linear system in feedback interconnection form, and then it is shown that with the help of this condition, a state feedback control law can be designed to render the uncertain system passive. Moreover, the extensions of the passivation controller are investigated in two cases where the uncertainty allows unknown control direction and there exists the external disturbance, respectively. It will be shown that an adaptive controller with a Nussbaum-type function can be incorporated into the passivation controller to deal with the unknown control direction and the L ₂-gain performance can be achieved by gain re-assignment of the passivation controller. Finally, a numerical example is given to demonstrate the proposed approach.     

An adaptive multiscale basis method for modelling discrete-time non-linear dynamical systems

Based on locally applicable autoregressive moving average with exogenous (ARMAX) models this paper proposes two new adaptive modelling schemes for discrete-time non-linear dynamic systems based on the concept of a multiscale which originates in approximation theory. A general result on the convergence of modelling based on the multiscale basis functions with a least square estimated is derived. With the advantage of the multiresolution nature of the multiscale basis, the new modelling scheme can be used to capture both the global and local characteristics of a non-linear system over distinct scales. The algorithms also provide a trade-off between the model structure/model size and the modelling error, one of which emphasizes the modelling error with variable model structure whilst the other stresses a desired model size with better modelling error. Data-based modelling examples are used to demonstrate the effectiveness of the multiscale modelling schemes with different multiscale basis.     

State-space models for infinite-dimensional systems

Distributed effects are present in almost all physical systems. In some cases these can be safely ignored but there are many interesting problems where these effects must be taken into account. Most infinite dimensional systems which are important in control theory are specifiable in terms of a finite number of parameters and hence are, in principle, amenable to identification. The state-space theory of infinite dimensional systems has advanced greatly in the last few years and is now at a point where real applications can be contemplated. The realizability criteria provided by this work can be employed effectively in the first step of the identification procedure, i.e., in the selection of an appropriate infinite dimensional model. We show that there exists a natural classification of nonrational transfer functions, which is based on the character of their singularities. This classification has important implications for the problem of finite dimensional approximations of infinite dimensional systems. In addition, it reveals the class of transfer functions for which there exist models with spectral properties closely reflecting the properties of the singularities of the transfer functions. The study of models with infinitesimal generators having a connected resolvent sheds light on some open problems in classical frequency response methods. Finally, the methods used here allow one to see the finite dimensional theory itself more clearly as the result of placing it in the context of a larger theory.     

State-space approach to non-linear model reference adaptive control

A new method is developed for tracking in non-linear delay-differential systems when the exact knowledge of the system parameters are not known. The method is based on securing the stability in the state spaces of plant and reference models. The results obtained are quite general in nature and can be applied to many physical systems.     

Diagnosis of unknown parametric faults in non-linear stochastic dynamical systems

A new detection and isolation scheme for unknown parametric faults in non-linear stochastic systems is presented that is particularly suited for small parametric changes. The proposed residual is the moving angle between two stochastic processes: the error derived from the reference model and the expected error in case a certain fault has occurred. Since the fault is unknown, fault classes are defined. Most of the faults that belong to such a fault class can be detected and isolated by just testing one representative of this fault class. As the moving angle itself is a stochastic process, an estimator is designed and characterised. Conditions for the detectability and isolability of fault classes are given for this estimator based on hypotheses tests. Finally, the theoretical results are confirmed by simulations and Monte Carlo tests.     

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.     

混合值逻辑控制系统的状态空间方法

对于混合值逻辑控制系统,提出确定一个正则子空间的补空间的充分必要条件.对于正则子空间已有的判别 准则,给出新的证明,并提出构造补空间的新方法.对于子空间,设计计算其友好子空间的新算法,引入没有正则性假 设的不变子空间的概念,获得判别不变子空间的一系列充分必要条件.     

Time-domain performance based non-linear state feedback control of constrained linear systems

This article describes a method to design a non-linear state feedback controller that meets a set of time-domain specifications not attainable by linear state feedback. Using a constrained polynomial interpolation technique, an input signal is computed that satisfies the desired time-domain constraints on the input and state-trajectories. The computed input is constructed by non-linear combinations of the states, such that a non-linear state feedback law is obtained. Stability of the resulting closed-loop polynomial system is analysed using sum-of-squares techniques. An illustrative example is presented, showing that the proposed non-linear controller outperforms the best linear static state feedback. To validate the proposed method, experiments on a fourth-order motion system have been carried out.     

Eliminating constraint drift in the numerical simulation of constrained dynamical systems

By means of the Udwadia–Kalaba approach we propose an explicit equation of constrained motion developed to simulate constrained dynamical systems without error accumulation due to constraint drift. The basic idea is to embed a small virtual force and a small virtual impulse to the equation of motion, in order to avoid the drift typically experienced in constrained multibody simulations. The embedded correction terms are selected to minimally alter the dynamics in an acceleration and kinetic energy norm sense. The formulation allows one to use a standard ODE solver, avoiding the need for iterative constraint stabilization. The equation is based on the pseudoinverse of a constraint matrix such that it can be used under redundant constraints and kinematic singularities. The proposed method takes into account the finite word-length of the computational environment, and also accommodates possibly inconsistent initial conditions.     

State-space constrained optimal control problems with control variables appearing linearly

This paper deals with the state-space constrained optimal control problems with control variables appearing linearly by the concept of decomposition. To solve this continuous optimal control problem, we first discretize the time and replace the system of differential equations by difference equations. For this resulting discrete optimal control problem, fixing the value of state variables reduces the given problem to a finite number of independent linear programming problems which are parameterized by the value of state variables. From this point of view, after para. meterizing by the value of state variables, we outer-linearize the resulting itifimal valuo functions in the minimond and apply the relaxation strategy to the new constraints arising as a consequence of outer-linearization. An algorithm is proposed which requires baek-and-forth iteration between a master problem and a finite number of linear programming subproblems. Finite convergence of this algorithm follows directly from the finite number of constraints of the master problem.     

Third-order approximate Kushner filter for a non-linear dynamical system

The analytical and numerical solutions of the non-linear exact Kushner filter are not possible, since the mean and variance evolutions are infinite dimensional and require the knowledge of the higher-order moment evolutions. The approximate filters seem to preserve some of the qualitative characteristics of the exact filter. In this paper, evolutions of conditional mean and conditional covariance of the third-order approximate filter for estimating the states of a non-linear dynamical system, especially accounting state-dependent and state-independent noise perturbations, are derived. In this analysis, we make a comparison of this filter with second-order Gaussian filter discussed in standard textbooks on non-linear filtering. This paper discusses Duffing filter, by taking up two different non-linear observation equations to demonstrate the effectiveness of the higher-order filters, i.e. third order, and second-order filters. Most notably, this paper is about examining the ability of the higher-order filters for estimating the states of the stochastically perturbed non-linear dynamical systems. In addition, the analytical findings are supported with numerical work generated by a simple, but effective, finite difference scheme.     

State-space analysis and identification for a class of hysteretic systems

In this paper we present results on the twin subjects of system analysis and system identification for a class of state-space realizable dynamic systems under the influence of hysteresis. The class of systems in question consists of models in the form of a linear time-invariant dynamic system in series with a differential model of hysteresis. It will be demonstrated that under fairly light constraints on the differential model of hysteresis, it is possible to design a series of experiments leading towards the identification of the full state-space realization. The approach is tested successfully on a high-precision mechanical translation system affected by hysteresis.     

Observers for Lipschitz non-linear systems

In an interesting paper R. Rajamani and Y. M. Cho have proposed a systematic methodology to design observers. They have introduced a new problem: relation between distance to unobservability and convergence of 'Luenberger-like' observers. A result for the convergence of the observer has also been given. They have presented a quantity denoted by i, claimed as the distance to unobservability. We show that this number is not actually the distance from the system to the set of unobservable systems. Moreover, the result for the observer is incorrect. We provide a counterexample for the result of convergence. In this paper results for convergence are obtained, with additional strengthened hypothesis, to correct Rajamani and Cho's result. The results are used to design an observer for a, now classical, single-link flexible robot joint. The behaviour of the observer to noisy output is quite satisfactory.     

High-gain observers for non-linear systems

State estimation for non-linear dynamic systems is discussed. A high-gain injection from the output variables is used to attenuate to any desired degree, the effect of the non-linear terms on the estimation errors, which can be made arbitrarily small, from a particular observable form. Some Liapunov arguments are used to study the stability properties of the resulting error dynamics. It is shown that any completely observable multi-input multi-output non-linear system, not necessarily linear in the input variables, can be transformed, by means of a change of coordinates depending on the input variables, in such an observable form. In particular, the solvability of partial differential equations is not needed for the design of this transformation.     

Set-point changing for non-linear systems

In many control systems, we sometimes face a situation such that we have to change the set-point of the system in the course of operation. Theoretically ‘set-point changing’ means letting the system be at another equilibrium point. If the system is linear, this operation presents no problem because the properties of linear systems are always global. However, in non-linear systems, this type of global operation has many unsolved problems. We only know experimentally that, if the system is locally stable at every equilibrium point, then we can slowly change the set-point without exciting unstable motion. This paper proposes a theoretically guaranteed method of changing the set-point of non-linear systems and showing a sufficient condition needed for this operation.     

A note on the control of uncertain linear dynamical systems with constrained control input

A method is presented to design a stabilizing saturated state feedback controller for linear continuous-time systems with bounded uncertainty acting via the input channel.