Flat Nonlinear Systems Research Articles

On the exact linearisation and control of flat discrete-time systems

The paper addresses the exact linearisation of flat nonlinear discrete-time systems by generalised static or dynamic feedbacks which may also depend on forward-shifts of the new input. We first investigate the question which forward-shifts of a given flat output can be chosen in principle as a new input, and subsequently how to actually introduce the new input by a suitable feedback. With respect to the choice of a feasible input, easily verifiable conditions are derived. Introducing such a new input requires a feedback which may in general depend not only on this new input itself but also on its forward-shifts. This is similar to the continuous-time case, where feedbacks which depend on time derivatives of the closed-loop input – and in particular quasi-static ones – have already been used successfully for the exact linearisation of flat systems since the nineties of the last century. For systems with a flat output that does not depend on forward-shifts of the input, it is shown how to systematically construct a new input such that the total number of the corresponding forward-shifts of the flat output is minimal. Furthermore, it is shown that in this case the calculation of a linearising feedback is particularly simple, and the subsequent design of a discrete-time flatness-based tracking control is discussed. The presented theory is illustrated by the discretised models of a wheeled mobile robot and a 3DOF helicopter.

Differential–geometric decomposition of flat nonlinear discrete-time systems

We prove that every flat nonlinear discrete-time system can be decomposed by coordinate transformations into a smaller-dimensional subsystem and an endogenous dynamic feedback. For flat continuous-time systems, no comparable result is available. The advantage of such a decomposition is that the complete system is flat if and only if the subsystem is flat. Thus, by repeating the decomposition at most n−1 times, where n is the dimension of the state space, the flatness of a discrete-time system can be checked in an algorithmic way. If the system is flat, then the algorithm yields a flat output which only depends on the state variables. Hence, every flat discrete-time system has a flat output which does not depend on the inputs and their forward-shifts. Again, no comparable result for flat continuous-time systems is available. The algorithm requires in each decomposition step the construction of state- and input transformations, which are obtained by straightening out certain vector fields or distributions with the flow-box theorem or the Frobenius theorem. Thus, from a computational point of view, only the calculation of flows and the solution of algebraic equations is needed. We illustrate our results by two examples.

A derivative-free nonlinear Kalman filtering approach using flat inputs

ABSTRACT Differential flatness theory has been widely used for both tracking control and state estimation of nonlinear systems. Recently, the concept of flat inputs has been proved to be a valuable tool for control design if the system is non-differentially flat. However, flat inputs have not been used yet to solve state estimation problems for these systems. By constructing flat inputs, we introduced a novel state estimation-based control strategy for both observable non-differentially flat nonlinear systems and differentially flat nonlinear systems whose flat output vector is not a measurable variable, as long as the internal dynamics of the system are stable. The efficiency of the proposed approach is illustrated through two numerical examples. Our findings indicated that we increased the class of systems to which the derivative-free nonlinear Kalman filtering based on differential flatness theory can be applied.

On the equivalence between ADRC and Flat Filter based controllers: A frequency domain approach

An equivalence is established between Extended State Observer (ESO) based Active Disturbance Rejection Control (ADRC) and Flat Filter (FF) controllers in the context of perturbed pure integration systems. These are known to constitute a most convenient paradigm for the control of SISO flat nonlinear systems. It is shown that a Reduced Order ESO (ROESO) based ADRC strategy has a unique equivalent Flat Filter counterpart. The reverse statement is not generally true. This equivalence is experimentally demonstrated to be an useful, practical, feature in FF controller design for the linear control of nonlinear flat systems.

On the Linearization of Flat Two-Input Systems by Prolongations and Applications to Control Design

In this paper we consider (x,u)-flat nonlinear control systems with two inputs, and show that every such system can be rendered static feedback linearizable by prolongations of a suitably chosen (new) control. This result is not only of theoretical interest, but has also important implications on the design of flatness based tracking controls. We show that a tracking control based on quasi-static state feedback can always be designed in such a way that only measurements of a (classical) state of the system, and not measurements of a generalized Brunovsky state, as reported in the literature, are required.

Unimodular Completion for Computation of Fractionally Flat Outputs for Linear Fractionally Flat Systems

The current paper presents a method for the computation of fractionally flat outputs for linear fractionally flat systems based on the notion of unimodular completion. This calculation method, which already exists for integer order non-linear flat systems, is extended for the class of fractionally linear flat systems by employing some fractional calculation properties. Two examples are used to validate the proposed extension.

New Method of Faults Diagnostic based on Neuro-Dynamic Sliding Mode for Flat Nonlinear Systems

This paper addresses the problem of simultaneous actuator, process and sensor Fault Detection and Isolation (FDI) for nonlinear system having flatness properties with the presence of disturbances and which are operating in closed-loop. In particular, the nonlinear system is corrupted with additive actuator, process or sensor faults with simultaneous occurrence. In this case, the residual signals might be sensitive to all of these faults that can appear in the system. The proposed FDI method is based on both input and parameter estimators that are designed in parallel. With the flatness property of such system, the design of these two estimators requires information on the measured outputs and their successive derivatives. To estimate these last one, a new scheme of the 2nd-order dynamic sliding mode differentiator is proposed. Residuals are next defined as the difference between the estimated and expected behavior. In order to isolate the faults, dynamic neural networks technique is employed. Besides, comparative study between this new differentiator and the well-known 2nd-order Levant’s differentiator is provided to show the pros and cons of the proposed FDI method. This latter is validated by the simulation results and is carried out on a three tank system.

On a linear input–output approach for the control of nonlinear flat systems

ABSTRACTThe 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 regulation, via simplified pure integration dynamics, of the large class of static, or dynamic, feedback linearisable systems (i.e. differentially flat systems), subject to additive exogenous disturbances and where nonlinearities are purposefully neglected. Integral reconstructors of the simplified input-to-flat output dynamics, directly lead to generalised proportional integral controllers, which are also shown to be equivalent to classical compensation networks. Their robust version, addressed here as flat filters, are found to be quite useful in the approximate (i.e. practical) control of perturbed differentially flat systems. The controller design is based on their simplified, perturbed, and pure integration dynamics. Flat filters evade for the need of nonlinear asymptotic observers and other, on-line, explicit output time derivatives computations. Simulation and challenging experimental examples are provided to validate this linear approach to the control of nonlinear flat systems.

Ultramodelos Globales y el Control por Rechazo Activo de Perturbaciones en Sistemas No lineales Diferencialmente Planos

Global ultra-models are introduced as a paradigm for vastly simplifying the design of observer-based linear output feedback control laws in output trajectory tracking problems for a large class of uncertain nonlinear flat systems. An ultra-model simply lumps, as an unknown time-varying disturbance, the combined effects of all uncertain state-dependent (i.e., endogenous) nonlinearities as well as those of external (i.e., exogenous) perturbation inputs present in the input-to-flat output model of the system. As a result, a global ultra-model based controller design only requires knowledge of the system order and of the nonlinear control input gain function. An extended high gain observer based control scheme is proposed for linear Active Disturbance Rejection (ADRC) for nonlinear differentially flat systems. An Illustrative case study example is presented.

Output feedback control of a mechanical system using magnetic levitation

This paper presents an application of a nonlinear magnetic levitation system to the problem of efficient active control of mass–spring–damper mechanical systems. An output feedback control scheme is proposed for reference position trajectory tracking tasks on the flexible mechanical system. The electromagnetically actuated system is shown to be a differentially flat nonlinear system. An extended state estimation approach is also proposed to obtain estimates of velocity, acceleration and disturbance signals. The differential flatness structural property of the system is then employed for the synthesis of the controller and the signal estimation approach presented in this work. Some experimental and simulation results are included to show the efficient performance of the control approach and the effective estimation of the unknown signals.

Flatness-based fault tolerant control

This paper presents a Fault Tolerant control approach for nonlinear flat systems. Flatness property affords analytical redundancy and permit to compute the states and control inputs of the system. Residual signals are computed by comparing real measures and the computed signals obtained using the differentially flat equations. Multiplicative and additive faults can be handled indistinctly. The redundant signals obtained with the differentially flat equations are used to reconfigure the faulty system. Feasibility of this approach is verified for additive faults in a three tank system.

Control Lineal Robusto de Sistemas No Lineales

In this arcicle, a linear observer-linear controller approach is proposed for the robust trajectory tracking task in a large class of nonlinear differentially flat systems, including multi-variable nonlinear flat systems. A non-phenomenological model of the input-to-flat-output dynamics is proposed which only retains the orders of the Kronecker integration subsystems and, the control input gain matrix, as key controller design elements. The additive influence of the rest of the nonlinear state dependent dynamics, including exogenous unknown perturbation inputs, is considered as unknown but uniformly absolutely bounded disturbance that is shown to be algebraically observable and it can, hence, be approximately determined, to any desired degree of accuracy, by means of a set of linear observers with suitable, self-updating, time-polynomial internal models of the unknown disturbances. The controller design task is reduced to that of canceling the additive disturbances while imposing a desired linear dynamics, via estimated state feedback, obtained from the proposed observer itself. A convincing simulation example dealing with rather complex nonlinear physical system is provided. Two experimental implementations on laboratory prototype systems are also reported.

Control lineal robusto de sistemas no lineales diferencialmente planos

In this arcicle, a linear observer-linear controller approach is proposed for the robust trajectory tracking task in a large class of nonlinear differentially flat systems, including multi-variable nonlinear flat systems. A non-phenomenological model of the input-to-flat-output dynamics is proposed which only retains the orders of the Kronecker integration subsystems and, the control input gain matrix, as key controller design elements. The additive influence of the rest of the nonlinear state dependent dynamics, including exogenous unknown perturbation inputs, is considered as unknown but uniformly absolutely bounded disturbance that is shown to be algebraically observable and it can, hence, be approximately determined, to any desired degree of accuracy, by means of a set of linear observers with suitable, self-updating, time-polynomial internal models of the unknown disturbances. The controller design task is reduced to that of canceling the additive disturbances while imposing a desired linear dynamics, via estimated state feedback, obtained from the proposed observer itself. A convincing simulation example dealing with rather complex nonlinear physical system is provided. Two experimental implementations on laboratory prototype systems are also reported.

On necessary and sufficient conditions for differential flatness

This paper is devoted to the characterization of differentially flat nonlinear systems in implicit representation, after elimination of the input variables, in the differential geometric framework of manifolds of jets of infinite order. We extend the notion of Lie-Bäcklund equivalence, introduced in Fliess et al. (IEEE Trans Automat Contr 44(5):922–937, 1999), to this implicit context and focus attention on Lie-Bäcklund isomorphisms associated to flat systems, called trivializations. They can be locally characterized in terms of polynomial matrices of the indeterminate \({\frac{d}{dt}}\) , whose range is equal to the kernel of the polynomial matrix associated to the implicit variational system. Such polynomial matrices are useful to compute the ideal of differential forms generated by the differentials of all possible trivializations. We introduce the notion of a strongly closed ideal of differential forms, and prove that flatness is equivalent to the strong closedness of the latter ideal, which, in turn, is equivalent to the existence of solutions of the so-called generalized moving frame structure equations. Two sequential procedures to effectively compute flat outputs are deduced and various examples and consequences are presented.

Trajectory tracking for the magnetic ball levitation system via exact feedforward linearisation and GPI control

The output feedback control of the popular magnetic ball levitation system is addressed from a suitable combination of several complementary viewpoints. We use: first, recent developments on exact feedforward linearisation controllers for nonlinear flat systems to substantially reduce the linear feedback controller efforts through pre-compensation. Second, an on-line ball velocity estimation strategy is proposed by using a model-based integral reconstructor, which is a linear combination of iterated integrals of the input and the output of the system, thus avoiding the use of traditional observers or noisy derivative estimations. Finally, we use a generalised proportional integral (GPI) controller which compensates the errors in the integral reconstructor and further bestows the enhanced robustness on the closed-loop system via output tracking error iterated integration feedback. This methodology only requires the measurements of the position of the levitated ball and of the control input voltage. The proposed feedback regulation scheme is shown to locally guarantee an asymptotically exponentially stable behaviour of the controlled ball position and, definitely, allows for the possibilities of safely carrying out the rest-to-rest trajectory tracking tasks on the ball position. The proposed output feedback controller is actually implemented on a laboratory prototype with excellent experimental results for, both, stabilisation and trajectory tracking tasks.

Flatness-Based Extremum-Seeking Control Over Periodic Orbits

In this note, an extremum-seeking controller is developed to steer a periodic system to orbits that maximize a functional of interest for a class of differentially flat nonlinear systems. The problem is posed as a real-time optimal trajectory generation problem in which the optimal orbit is computed using an extremum-seeking approach. Using the flatness property of the dynamics, the original dynamic optimization problem is transformed to a parameterized optimization problem which can be solved in real-time to approximate the optimal orbit. The control algorithm provides tracking of the optimal orbit. A drug delivery control problem is considered to demonstrate the application of the technique.

Real-time dynamic optimization of nonlinear systems: A flatness-based approach

In this paper, a real-time optimizing controller is developed to steer a differentially flat nonlinear control system to closed-loop trajectories that optimize a cost functional of interest. An interior-point optimization method with penalty function is used to formulate a real-time optimization scheme. The problem is posed as a real-time optimal trajectory generation problem in which the optimal trajectories are computed using an adaptive extremum-seeking approach. Simulation examples are used to demonstrate the application of the technique to various finite time dynamic optimization problems that arise in batch process control.

Sampled-data Control of a Class of Nonlinear Flat Systems With Application to Unicycle Trajectory Tracking

In this paper we propose a flatness-based nonlinear sampled-data control approach for the trajectory tracking of nonlinear differentially flat systems that can be expressed in cascade form. The nonlinear sampled-data control method relies on the flatness property for the generation of appropriate trajectories, with the design of one-step predictive control laws, and on controller discretization by means of an averaging-like method. In the paper we demonstrate that the causality problem that might arise in the implementation is avoided by using an estimator based on numerical integration techniques of sufficiently high order. Stability-like properties are proved. Numerical simulations show that the proposed sampled-data control law offers the best closed-loop performance when compared with nonlinear direct digital design for the trajectory tracking of a rotorcraft-like UAV modeled as the unicycle. The synthesis of the nonlinear sampled-data control law takes advantage of the feedback linearizability property of the unicycle model. Furthermore, the proposed nonlinear sampled-data control does not rely on approximated discretization techniques and is computed from exponentially convergent steering trajectories that result from the stabilization of the linearized unicycle model.

Tracking control for nonlinear flat systems by linear dynamic output feedback

In this contribution linear dynamic output feedback is used to achieve stable tracking for nonlinear flat systems. To this end a time varying differential operator representation resulting from a linearization of the plant about the reference trajectory is used. The control scheme is illustrated for a uniaxial vehicle model.

On flatness necessary and sufficient conditions

We give a characterization of differentially flat nonlinear systems in implicit representation, where the input variables are eliminated. Lie-Bäcklund isomorphisms associated to a flat system, called trivializations, can be locally characterized in terms of matrices polynomial with respect to d/dt. Such polynomial matrices are useful to compute the ideal of differential forms generated by the differentials of all possible trivializations. We introduce the notion of a strongly closed ideal of differential forms, and prove that flatness is equivalent to the strong closedness of the latter ideal.