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This paper introduces a model-based robust adaptive input-output control framework for a family of robotic systems that include under-actuation, nonholonomic, and constrained properties. The propos...

Zeros are of extreme importance in linear systems theory, especially unstable zeros, which degrade achievable performance. Furthermore, the presence of a zero in a state-space model implies the exi...

A key problem in process control is to decide which inputs should control which outputs. There are multiple ways to solve this problem, among them using gramian-based measures, which include the Ha...

The need for observers, or other means of state- or phase variables-on-line computations, is revisited from the perspective of structural integral reconstructors. The problem addressed is the regul...

Input–output linearisation (IOL) may encounter a singularity problem when applied to the tracking control of affine-form nonlinear system (AFNS), which may not have a well-defined relative degree. The singularity problem has occurred in the area of control for decades. In this paper, we incorporate the gradient dynamics (GD) into IOL, which leads to the GD-aided IOL method to solve the singularity problem, with the proposition of the loose condition on relative degree. Moreover, detailed theoretical analyses on tracking-error bound and convergence performance of the corresponding GD-aided IOL controller are presented. Simulations and comparisons substantiate that the proposed GD-aided IOL method is capable of completing the tracking-control task and conquering the singularity encountered in the AFNS.

In the linear time-invariant (LTI) framework, the transformation from an input–output equation into state space representation is well understood. Several canonical forms exist that realise the same dynamic behaviour. If the coefficients become time-varying however, the LTI transformation no longer holds. We prove by induction that there exists a closed-form expression for the observability canonical state space model, using binomial coefficients.

The combined problem of diagonal decoupling by output feedback with disturbance attenuation performance for any square system is formulated. A sufficient condition which is more general than the necessary and sufficient one of static state feedback for the existence of the controller is determined. The necessary and sufficient conditions for the controller and decoupled system that are both stable are presented. An analytical form and a numerical computable method for the controller are obtained for the first time when the condition is satisfied. The correctness of our method is tested by simulations for an example.

This article presents an input–output simulation approach to controlling multi-affine systems for linear temporal logic (LTL) specifications, which consists of the following steps. First, the state space is partitioned into rectangles, each of which satisfies atomic LTL propositions. Then, we study the control of multi-affine systems on rectangles, including the control based on the exit sub-region to drive all trajectories starting from a rectangle to exit through a facet and the control to stabilise the multi-affine system towards a desired point. With the proposed controllers, a finitely abstracted transition system is constructed which is shown to be input–output simulated by the rectangular transition system of the multi-affine system. Since the input–output simulation preserves LTL properties, the controller synthesis of the multi-affine system for LTL specifications is achieved by designing a nonblocking supervisor for the abstracted transition system and by implementing the resulting supervisor to...

This article studies the realisability property of discrete-time bilinear and quadratic input–output (i/o) equations in the classical state-space form. Constraints on the parameters of the i/o model are suggested that lead to realisable models. Using new formulae for computing basis vectors of certain vector spaces of differential one-forms, we present in this article the complete list of the third- and fourth-order realisable i/o bilinear models, and a new realisable subclass of an arbitrary order is suggested. Moreover, we provide the sufficient conditions of the second- and third-order realisable i/o quadratic models, respectively. All the developed theory and algorithms are illustrated by means of several examples.

The design of output feedback controllers that eliminate uncertainties caused by critical races in asynchronous sequential machines is considered. The objective is to build controllers that drive a race-afflicted machine so as to match a prescribed deterministic model. Necessary and sufficient conditions for the existence of such controllers are presented in terms of a numerical matrix derived from the given machine. When controllers exist, an algorithm for their construction is also provided. The discussion depends on the novel notion of ‘generalised state’, which helps represent the uncertainty created by critical races and facilitates the construction of controllers.

This article proposes a new approach to robustify an input–output linearisation controller. The robustification is achieved by estimating the uncertainties and external unmeasurable disturbances using a novel uncertainty and disturbance estimator. A significant feature of the proposed approach is that it does not need any information about the uncertainties. The stability of the system and the estimator is established. Effectiveness of the proposed approach is demonstrated through application to the wing rock motion control problem.

A new kind of network-based model called network-based feedback interconnection (NBFI) which is the natural generalisation of the networked control system (NCS) is proposed, and both the packet loss and the network-induced delay problems are studied in this article. The model of an NBFI is established under the framework of samplers and zero-order holds, which has been recognised as a usual modelling method for NCSs. By using the dissipativity theories, some sufficient conditions of the stability of NBFIs are obtained. The results on NBFIs are generalised to a wider class of systems called network-based cyclic feedback interconnections. A novel communication strategy called constant lag transmission (CLT) for NBFIs with arbitrary packet losses is also presented. Some numerical examples are given to illustrate the effectiveness of the proposed theories.

This article studies the realisability property of continuous-time quadratic input–output (i/o) equations in the classical state space form. Constraints on the parameters of the quadratic i/o model are suggested that lead to realisable models. The complete list of second- and third-order realisable i/o quadratic models is given and two subclasses of the n-th order realisable i/o quadratic systems are suggested. Our conditions rely basically upon the property that certain combinations of coefficients of the i/o equations are zero or not zero. We provide explicit state equations for realisable second-order quadratic i/o equations, and for one realisable subclass of quadratic i/o equations of arbitrary order.

This article considers pole assignment control of non-linear dynamic systems described by state-dependent parameter (SDP) models. The approach follows from earlier research into linear proportional-integral-plus methods but, in SDP system control, the control coefficients are updated at each sampling instant on the basis of the latest SDP relationships. Alternatively, algebraic solutions can be derived off-line to yield a practically useful control algorithm that is relatively straightforward to implement on a digital computer, requiring only the storage of delayed system variables, coupled with straightforward arithmetic expressions in the control software. Although the analysis is limited to the case when the open-loop system has no zeros, time delays are handled automatically. This article shows that the closed-loop system reduces to a linear transfer function with the specified (design) poles. Hence, assuming pole assignability at each sample, global stability of the non-linear system is guaranteed at...

In this paper we examine the model matching problem that concerns non-linear input–output discrete systems, containing products among delays of input and output signals, through a special factorisation. The algebraic framework of δe-operators and the star-product that we adopt describe these systems. Moreover, a certain procedure that resembles the Euclidean division allows us to discover the linear factors of those systems, with respect to the above mentioned operations. The entire approach is symbolically algorithmic and involves the use of suitable software.

This paper studies the realizability property of continuous-time bilinear input-output (i/o) equations in the classical state space form. Constraints on the parameters of the bilinear i/o model are suggested that lead to realizable models. The paper proves that the 2nd order bilinear i/o differential equation, unlike the discrete-time case, is always realizable in the classical state space form. The complete list of 3rd and 4th order realizable i/o bilinear models is given and two subclasses of realizable i/o bilinear systems are suggested. Our conditions rely basically upon the property that certain combinations of coefficients of the i/o equations are zero or not zero. We provide explicit state equations for all realizable 2nd and 3rd order bilinear i/o equations, and for one realizable subclass of bilinear i/o equations of arbitrary order.

This paper presents a new methodology for diagonal dominance of large-scale systems via eigenstructure assignment. For a given large-scale system in general form, an equivalent descriptor system in the input–output decentralized form is defined. Sufficient conditions for diagonal dominance of the closed-loop system are introduced. These conditions are in terms of the isolated subsystems. Based on them, interactions between subsystems can be considered as external disturbances for each isolated subsystem. Then a previously proposed approach is used for disturbance attenuation via dynamical output compensators based on complete parametric eigenstructure assignment. By attenuating disturbances, closed-loop poles of the overall system are placed in a desirable region, by assigning the eigenstructure of each isolated subsystem appropriately. The presented algorithm alleviates the necessity of choosing a suitable frequency in designing a pre-compensator, as required by previous methods. The designed controller ...

We give a new approach to the problem of stabilization of systems defined on the full time axis (the integers or the real numbers, as appropriate), which enables us to avoid the paradoxical situation observed by Georgiou and Smith. This approach enables us to treat directly the case of persistent signals, such as power-bounded signals, defined on the full time axis.

The problem of controlling time-invariant linear systems subject to parametric uncertainty and input saturation constraint, is dealt with using a pole placement adaptive controller. It is shown that the resulting adaptive control system can be described by a non-linear feedback scheme. Then, the l 2 -stability condition for the control system turns out to be the positive-realness of a transfer function involving the plant (estimated) model and the specified closed-loop poles. This makes it necessary to adapt on-line the desired closed-loop poles to the time-varying (estimated) model. The proposed adaptive controller keeps bounded all the closed loop sequences, whatever the initial conditions. Furthermore, it ensures a quite interesting output-reference tracking behaviour. More precisely, the quality of the tracking depends on the reference sequence rate: the slower the reference the better the tracking.

This paper develops an algorithm which, given an input-output differential equation, produces under appropriate circumstances a parametrization of the observation space of the underlying state-space system. The paper extends previous results by the authors to a more general setting, so that state-space systems, in which the controls enter nonlinearly, can be accommodated and, most importantly, the paper deals with implicitly defined systems at both the input-output and state-space level. Fliess and Hasler have demonstrated the importance of implicit input-output models, which yield only local state-space descriptions. Various realizations are obtained, depending on the properties of the derived candidate observation space. Examples of the algorithm are described, with the aid of an implementation using the computer algebra system AXIOM.

An identification method is presented for estimating the parameters of a discrete-time linear dynamic system excited by non-gaussian input signals using the fourth-order cumulants of the input and output signals, both of which are contaminated by additive gaussian noise. Two types of estimators of the fourth-order cumulants of the input and output signals are proposed for this method. The first is conventional. The second, which is new allows us only to have a recursive algorithm for computing the parameter estimators. The parameter estimators obtained by this algorithm are shown to be strongly consistent under certain weak conditions. Simulation examples are included to demonstrate the effectiveness of the proposed method.

Nonlinear systems affine in the input, and having a well-defined (vector) relative degree are considered. The invertibility of the input output dynamics can be used to achieve tracking of smooth desired trajectories if the associated ‘zero-dynamics’ are stable. We consider tracking in nonlinear systems with unacceptable zero-dynamics (i.e. non-minimum phase systems). The desired trajectories are assumed to be generated by some exosystem. We use sliding control to achieve tracking independent of disturbances entering in the channels of the input. The main idea is (i) to do an output-redefinition such that the zero-dynamics with respect to this new output are acceptable; (ii) to define a modified desired trajectory for the new output to track such that in the process, the original output tracks the original desired trajectory asymptotically.

A new identification method is proposed for estimating the parameters of a discrete-time linear dynamic system excited by non-gaussian inputs using kth order (k > 2) cumulants of input and output signals contaminated by additive (possibly coloured) gaussian noise. The parameter estimates obtained by this method are proved to be consistent under weak conditions. This method has an on-line algorithm for computing the parameter estimates, just like the least-squares method. A simulation example is included to demonstrate the effectiveness of this method.

A general and concise formulation is given of the loop transfer recovery (LTR) design problem based on recovery errors. Three types of recovery errors are treated, open loop recovery, sensitivity recovery and input-output recovery errors. The three corresponding versions of the asymptotic recovery problem turn out to be equivalent, since the minimization of the recovery errors all amount to the minimization of a certain matrix, the recovery matrix. Using the recovery error definitions, simple necessary and sufficient conditions for the controllers are derived both for the exact and asymptotic recovery cases. This general recovery formulation covers all known observer based compensator types as special cases. The conditions given in this setting are effectively the aim of all known LTR design methods. The recovery formulation is interpreted in terms of a model-matching problem as well, which is examined by means of the Q-parametrization. It is shown how the general controller obtained by the Q-parametrizat...

Upper bounds on the induced norm of the input/output operators in linear systems with L2 input and output signals, are provided. The main idea is to transform the original problem into a parametized set of indefinite LQ optimization problems. Using the classical maximum principle, the existence of solutions to the latter is equivalent to the solvability of a boundary value hamiltonian system or, equivalenty, of an ‘indefinite’ Riccati equation. Our time domain approach enables the treatment of a very general linear set-up, including time-varying, infinite-dimensional, finite- and infinite-horizon systems.

A rational expansion for non-linear input-output maps is introduced. This new method is based on the rational expansion of functions of several complex variables. If truncated, this series reduces to a ratio of truncated Volterra series. A 'feedback form' is presented.

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This paper is concerned with the stabilization or large-scale systems via an observer–controller compensator. The overall system is considered as a decentralized form containing many linear time-invariant multivariable subsystems. Using both the M-matrix property and analysis of the input-output stability, a new stability criterion for large-scale systems controlled by observer–controller compensators is derived. Finally, based on this stability criterion, a design algorithm is proposed for finding a stabilizing observer–controller compensator for large-scale systems.

Linear, time-invariant, completely controllable and observable systems can be described, for example, by state equations, transfer functions or time responses due to specific test signals. All these descriptions are uniquely related for a given system where different state representations are possible by use of equivalence transformations. However, a whole class of different systems, characterized by an infinite number of different pole-zero patterns, exhibits the same input-output behaviour within drawing accuracy, i.e. the plots coincide to the point of indistinguishability. This effect is demonstrated, and it may have implications on such fields as system modelling, model-order reduction and controller design.

The problem of transforming non-linear systems to linear systems is receiving much attention in the literature. In this paper, we present necessary and sufficient conditions such that a non-linear control system with output can be equivalent to a linear control system with linear output that is controllable and observable. The conditions depend on certain Lie derivatives of the output and can be verified in a finite number of steps. For simplicity, we consider only the single input/single output case.

One proposes a method for studying the input-output stability of some distributed parameter systems with space-dependent coefficients by investigating the stability of an ‘ associate system ’ with constant coefficients. This way provides an input-output stability condition, the application of which is very simple.

A rational transfer function model of the plant is generally desirable in feedback system design, when only plan input and output data are available over finite, sometimes incomplete, time intervals. This is especially so in a recent exact design technique for highly uncertain time-varying and non-linear plants. Here, the plants are replaced rigorously by an equivalent linear time-invariant plant set. Existing numerical techniques were found inadequate, especially in the high-frequency range, which is important in the design of feedback systems with large plant uncertainty. A technique was developed with excellent results, even for noisy data, unstable and non-minimum phase plants and severely truncated time intervals. The transfer function is calculated directly, without derivation of the input, output signal transforms. The operations involve repeated integrations of the data. Numerous examples are included.

By employing differential vector spaces, we obtain results for linear systems with constant or varying coefficients and with lumped or distributed parameters which were only known for constant linear systems described by rational transfer matrices. Questions such as the characterization of input and output, input-output equivalence, invertibility, feedback decoupling and realization are investigated.

In this paper a robust multivariable I-controller is designed for a class of distributed parameter systems, using only discrete-time input-output data. The I-controller has the same form as a robust multivariable I-controller given by Davison in the case of continuous-time input-output data. The construction of the I-controller is also based on step responses.

This paper presents two techniques for system order determination in continuous SISO systems. The first method is based on the behaviour of singular values of a specific non-square matrix, the elements of which are calculated from multiple integrals of input and output data. The second method uses shifted Legendre polynomials to produce a square matrix whose non-zero eigenvalues then identify the order of the system under consideration.

An input-output approach to adaptive motion control design of robot manipulators is presented. The main technical device in our approach is the passivity theory. This formulation provides a framework suitable for the design of new control and adaptation laws. A new control law which consists of a computed torque part and a feedforward compensation part is analysed using this approach.

The paper describes a general approach to state variable feedback pole assignment for linear discrete-time systems. This is based on the definition of a non-minimal state space (NMSS) whose state variables are defined only in terms of the present and past values of the system output and the past values of its input. In more conventional block-diagram terms, it can be interpreted as an extension to a proportional-integral (PI) control system in which linear feedback and forward path digital filters are introduced to ensure the required closed-loop pole assignment, For this reason, we refer to it as a proportional-integral-plus or PIP system. The paper discusses the conditions for pole assignability. Practical examples which illustrate the performance of the PIP system in fixed gain and self tuning/adaptive applications are also discussed.

In this paper, we investigate a model reference adaptive control problem for the class of distributed parameter systems generated by spectral operators. First we present an equivalent static input-output representation for the spectral system using filtered values generated from the inputs and the outputs. Next, we design and analyse a model reference adaptive control system for systems with unknown input and output distribution functions.

A procedure is presented which allows for the design of a sub-optimal feedback control law, for a linear, time-invariant system, which is described only from input-output data. According to this procedure, no mathematical model of the plant is required. in order to build the regulator, which is the optimal one in the class of time-invariant, output feedback regulators. It is shown that the gains of the sub-optimal feedback matrix can be directly found from the input—output data observed during the normal operating time of the system.

Some general properties of recently introduced inputs-output multistructural (overlapping) models are analysed and compared with the properties of the well-known input-output canonical forms. Algorithms for the transformation of a generic input-output model to a multi-structural one and for transformation between ‘ adjacent’ multistructural models are then proposed. The use of the models in on-line multivariate identification is also discussed and a numerical example given.

It is pointed out that the study of systems evolving at multiple time-scales is simplified by studying reduced-order models of these systems valid at specific time-scales. The present investigation is concerned with an extension of results on the time-scale decomposition of autonomous systems to that of input-output systems. The results are employed to study conditions under which positive realness of a transfer function is preserved under singular perturbation. Attention is given to the perturbation theory for linear operators, the multiple time-scale structure of autonomous linear systems, the input-output description of two time-scale linear systems, the positive realness of two time-scale systems, and multiple time-scale linear systems.

This paper, which is a survey and a compact reference on the commutativity of time-varying systems, gives the complete set of necessary and sufficient commutativity conditions for systems of any order. Original results are derived on Euler's systems, and explicit commutativity conditions are presented for fourth-order systems, which have not yet appeared in the literature.

Recursive input-output models for non-linear multivariate discrete-time systems are derived, and sufficient conditions for their existence are defined. The paper is divided into two parts. The first part introduces and defines concepts such as Nerode realization, multistructural forms and results from differential geometry which are then used to derive a recursive input-output model for multivariable deterministic non-linear systems. The second part introduces several examples, compares the derived model with other representations and extends the results to create prediction error or innovation input-output models for non-linear stochastic systems. These latter models are the generalization of the multivariable ARM AX models for linear systems and are referred to as NARMAX or Non-linear AutoRegressive Moving Average models with exogenous inputs.

Sliding mode control is a technique that has been successfully used for SISO nonlinear systems in controllable canonical form and for some particular multivariable system structures. It is the purpose of this paper to generalize the sliding mode approach to a larger class of multivariable systems and to illustrate the relationship between I/O linearization and the sliding mode approach when the error dynamics on the sliding surfaces are chosen to be linear and time-invariant. The developed technique is applied to a multivariable non-linear example.

The identification of a linear continuous-time model for a multivariable dynamic system from sampled input-output observations is considered. An augmented hybrid parametric method is proposed to overcome the interference of the coloured output noise in the sampled output. The parameters of the continuous-time process model are estimated from an augmented input-output realization which utilizes the dynamic information of the discrete-time noise model. Numerical examples are presented to illustrate how to obtain an adequate dynamic process model, considering the coloured output noise, by using a discrete-time noise model.

Control of internally delayed plants using only input-output signals is considered. Frequency domain techniques using two-variable polynomial algebra are used consistently. The restriction imposed on the plant is moderate. Pole placement as well as exact model matching is examined in detail. The control law obtained is simpler than that obtained via time domain techniques. The synthesis method differs in the case when A 0 ?0 and the case when A 0 =0. For the former case finite pole assignment and matching to a finite-dimensional reference model is realizable, but for the latter case it is not.

The open-loop input-output dynamics and the noise dynamics of a feedback controlled linear system perturbed by coloured noise admitting a Markov representation are identified in state variable form using a two-stage algorithm. This system is equivalent to an augmented system driven by white noise. First the input-output dynamics are identified through a stochastic approximation algorithm using superimposed white noise. Subtracting the model output from the system output yields correlated residuals which are then used to identify the noise dynamics using stochastic realization theory. An innovations representation is obtained that is equivalent to the above defined augmented system. The two stages are combined by a judicious coordinate transformation. The method can be applied on an operating feedback controlled process, regardless of the structure of the unknown suboptimal regulator.

A recursive second-order approximation approach for the optimization and control of steady-state systems is proposed. The method constructs the hessian matrix of the real performance by using a modification of the Broyden, Fletcher, Goldfart, and Shanno (BFGS) formula which only uses first-order information of the real process. The iterative direction can be obtained by solving a second-order approximate optimization problem which is determined by a modification of the BFGS formula. To obtain a new iterative point, Newton step and one-dimensional search updating strategies are employed. Global convergence and optimality of the derived algorithms are thoroughly investigated. In particular, the R-superlinear properties of the new approach are verified under local conditions. Numerical results show that the new approach is superior to the sequence model approximation (SMA) method.