Systems Of Relative Degree Research Articles

A behavioral approach to inversion-based control

Abstract A new simplified behavior theory is proposed to address inversion-based control for linear, nonminimum-phase SISO systems. The chosen space of signals is the set of piecewise C ∞ -functions and input–output pairs (as weak solutions) satisfy a differential–integral equation with additional smoothness requirements. A related key result is the output–input (or inverse) representation of the behavior set that leads to the solution of a general stable inversion problem where polynomially unbounded, noncausal desired outputs are allowed. It is shown that this problem has a solution if and only if the smoothness degree of the desired output is greater than or equal to the system relative degree minus one. When this straightforward condition is satisfied, a closed-form expression provides the inverse input. Then, an analysis on preaction and postaction control follows. Two examples are included showing the relevance of output signal design in control applications.

Improved sampled-data implementation of derivative-dependent control

Abstract We consider an LTI system of relative degree two that can be stabilized using the output and its derivative. The derivative is approximated using a finite difference, what leads to a time-delayed feedback. This feedback is analyzed using a Lyapunov-Krasovskii functional that compensates the derivative approximation error presented in an integral form. We show that if the derivative-dependent control exponentially stabilizes the system, then one can use consecutively sampled measurements to approximate the derivative and this approximation will preserve the stability if the sampling period is small enough. We provide linear matrix inequalities that allow to find admissible sampling period and can be used for robustness analysis with respect to system uncertainties. The results are demonstrated by two examples: 2D uncertain system and the Furuta pendulum.

Adaptive Control of Multivariable Systems with Reduced Knowledge of High Frequency Gain: Application of Dynamic Regressor Extension and Mixing Estimators

In this paper we provide a new solution to the problem of adaptive stabilization of linear multivariable systems of relative degree one with reduced prior knowledge of the high frequency gain—imposing only a full-rank condition to this matrix. Instrumental for the establishment of our result is the use of the parameter estimation technique of dynamic regressor extension and mixing recently proposed in the literature. This estimator is used to update the parameters of the classical output-feedback controller, hence the controller is smooth and does not require any projections nor overparameterizations.

Output-feedback variable gain super-twisting algorithm for arbitrary relative degree systems

ABSTRACTThis paper presents an output-feedback control strategy based on the variable gain super-twisting algorithm. The proposal achieves robust global/semi-global exact tracking results for plants with arbitrary relative degree. This is ensured in spite of parametric uncertainties and disturbances that may be state-dependent and time-varying. The construction of such controller is based on a (non)homogeneous higher order sliding-mode differentiator with dynamic gains. The gain adaptation schemes for the controller and differentiator are based on norm observers to overcome the lack of state measurement. The continuous nature of the obtained control signal alleviates the chattering phenomenon. The stability properties of the proposed controller are demonstrated by means of a Lyapunov function-based analysis. The theoretical results are verified through a simulation example, and experimentally tested on a seesaw module.

Global adaptive HOSM differentiators via monitoring functions and hybrid state-norm observers for output feedback

ABSTRACTAn algorithm for adaptation of the gains of higher-order sliding mode-based exact differentiators is developed for the case where the upper bound for the ρth derivative of the tracking error signal exists but it is unknown. Unlikely other publications in the literature, the developed adaptive algorithm based on monitoring functions guarantees global and exact tracking when used in closed-loop output feedback. In the closed-loop scenario, a global-exact and finite-time estimate for the variables is applied to construct the sliding surface of the proposed sliding mode controller. The class of uncertain systems of arbitrary relative degree (ρ ≥ 1) takes into account time-varying perturbations with unknown bounds and state-dependent nonlinearities satisfying a linear growth condition with any unknown rate. The norm of the unmeasured state is majorised by using a hybrid state-norm estimator. Numerical examples and an engineering application to wing rock control are presented in order to illustrate the properties and advantages of the novel adaptation approach for sliding mode control design.

P.4.f.001 - Brain structural correlates of subclinical obsessive-compulsive symptoms in healthy children

In this paper we provide a new solution to the problem of adaptive stabilization of linear multivariable systems of relative degree one with reduced prior knowledge of the high frequency gain—imposing only a full-rank condition to this matrix. Instrumental for the establishment of our result is the use of the parameter estimation technique of dynamic regressor extension and mixing recently proposed in the literature. This estimator is used to update the parameters of the classical output-feedback controller, hence the controller is smooth and does not require any projections nor overparameterizations.

Second order sliding mode-based MPPT control for photovoltaic applications

Abstract Taking into account the nonlinear feature of a PV array, a Maximum Power Point Tracking (MPPT) algorithm is required in a PV system leading the operation at the Maximum Power Point (MPP) and as a result maximizing the generated power. In this paper, a robust MPPT technique using second order sliding mode control strategy is presented. A super twisting algorithm is used to design second order sliding mode. This algorithm has been developed, in the case where the relative degree of system is equal to one, to eliminate chattering phenomenon. Second order sliding mode presents an adequate mean for control action of nonlinear systems since it overcomes chattering phenomenon of the classical first order sliding mode controller and ensures higher accuracy even with system imperfections. The stability analysis of the proposed controller is performed using a Lyapunov function. We compare second order sliding mode with classical first order sliding mode controller which are both designed for the same PV system. This comparison proves that the second order gives fast response and less chattering. The robustness and stability of the proposed controller are investigated against weather changes. Simulation results with real data, under MATLAB-SIMULINK, are given to demonstrate the effectiveness of the proposed approach.

Global robust stabilization of nonlinear cascaded systems with integral ISS dynamic uncertainties

The authors consider a global robust asymptotic stabilization problem (GRS) for cascaded systems having dynamic uncertainties that are not necessarily input-to-state stable (ISS). Specifically, a recursive Lyapunov design approach is developed by induction on the system relative degree, providing a smoothly globally stabilizing controller. It is applicable to nonlinear cascaded systems having multiple distinct iISS dynamic uncertainties. The proposed design is constructive and leads to an iISS-Lyapunov characterization in a superposition form for the closed-loop system.

Adaptive Generic Model Control of Multivariable Processes: Application to Semi-Batch Reactors with Different Relative Degrees

In this study, a multivariable Generic Model Control (GMC) approach is proposed based on input-output linear-in-parameters time series data-driven models. Adaptation of the model parameters is carried out at every sampling instant. For higher relative degree systems, two different definitions are used for output derivatives, yielding two versions of adaptive GMC for multivariable processes. The performance of the proposed control algorithms is illustrated by application to multivariable semi-batch reactors without and with coolant dynamics for control of temperature and one of the reactant concentrations. The study indicated that the adaptive GMC (AGMC) algorithms for higher relative degree multiple-input and multiple-output (MIMO) systems with a different relative degree have exhibited performance comparable to or better than the phenomenological model-based GMC with respect to both set point tracking and smooth input profiles, and also that the predictive version of AGMC (AGMC-II) has exhibited slightly lower integral square error (ISE) values compared to AGMC-I in case of multivariable semi-batch reactor with coolant dynamics.

ℒ2-gain tuning of variable structure SISO systems of relative degree n

ABSTRACTThe present paper extends the -gain analysis towards variable structure systems (VSS) with sliding-mode (SM) dynamics. The developed analysis is applied to feedback linearisable VSS, driven by standard SM algorithms of the first order. Constructive parameter-tuning of the resulting closed-loop system to possess the -gain less than a desired attenuation level is additionally deduced. The proposed tuning is shown to be capable not only of rejecting matched uniformly bounded disturbances but also of attenuating unbounded ones, including mismatched disturbances. Numerical simulations of an inverted pendulum, driven by a first-order SM controller, are involved to support the theory in the presence of the position measurement inaccuracy, being a source of a mismatched disturbance.

A framework for stability analysis of high-order nonlinear systems based on the CMAC method

A framework for analyzing the stability of a class of high-order minimum-phase nonlinear systems of relative degree two based on the characteristic model-based adaptive control (CMAC) method is presented. In particular, concerning the tracking problem for such high-order nonlinear systems, by introducing a consistency condition for quantitatively describing modeling errors corresponding to a group of characteristic models together with a certain kind of CMAC laws, we prove closed-loop stability and show that such controllers can make output tracking error arbitrarily small. Furthermore, following this framework, with a specific characteristic model and a golden-section adaptive controller, detailed sufficient conditions to stabilize such groups of high-order nonlinear systems are presented and, at the same time, tracking performance is analyzed. Our results provide a new perspective for exploring the stability of some high-order nonlinear plants under CMAC, and lay certain theoretical foundations for practical applications of the CMAC method.

Predictor-Based Adaptive Output Feedback Control: Application to Functional Electrical Stimulation of a Human Arm Model

A simulation study to control the motion of a human arm using muscle excitations as inputs is presented to validate a recently developed adaptive output feedback controller for a class of unknown multi-input multi-output (MIMO) systems. The main contribution of this paper is to extend the results of Nguyen and Leonessa (2014, “Adaptive Predictor-Based Output Feedback Control for a Class of Unknown MIMO Linear Systems,” ASME Paper No. DSCC2014-6214; 2014, “Adaptive Predictor-Based Output Feedback Control for a Class of Unknown MIMO Linear Systems: Experimental Results,” ASME Paper No. DSCC2014-6217; and 2015, “Adaptive Predictor-Based Output Feedback Control for a Class of Unknown MIMO Systems: Experimental Results,” American Control Conference, pp. 3515–3521) by combining a recently developed fast adaptation technique and a new controller structure to derive a simple approach for a class of high relative degree uncertain systems. Specifically, the presented control approach relies on three components: a predictor, a reference model, and a controller. The predictor is designed to predict the systems output for any admissible control input. A full state feedback control law is then derived to control the predictor output to approach the reference system. The control law avoids the recursive step-by-step design of backstepping and remains simple regardless of the system relative degree. Ultimately, the control objective of driving the actual system output to track the desired trajectory is achieved by showing that the system output, the predictor output, and the reference system trajectories all converge to each other. Thelen and Millard musculotendon models (Thelen, D. G., 2003, “Adjustment of Muscle Mechanics Model Parameters to Simulate Dynamic Contractions in Older Adults,” ASME J. Biomech. Eng., 125(1), pp. 70–77; Millard, M, Uchida, T, Seth, A, and Delp, Scott L., 2013, “Flexing Computational Muscle: Modeling and Simulation of Musculotendon Dynamics,” ASME J. Biomech. Eng., 135(2), p. 021005) are used to validate the proposed controller fast tracking performance and robustness.

Low-Complexity Prescribed Performance Control of Uncertain MIMO Feedback Linearizable Systems

A continuous, state-feedback controller guaranteeing zero overshoot, preselected bounds on the minimum admissible convergence rate and maximum allowable steady-state values of the output tracking error is proposed for a class of uncertain, multi-input, multi-output nonlinear systems. Contrary to the current state-of-the-art, a decisive step towards minimizing controller implementation complexity is taken by designing a static, approximation-free control scheme, defined by simple analytical expressions that scale well even for high relative degree systems. Simulations on a 2-dof robotic manipulator clarify and verify the approach.

Attitude control using higher order sliding mode

This paper proposes a chattering-free global finite-time convergent robust control method for the attitude control of a rigid spacecraft. The control method is formulated using higher order sliding mode control in integral sliding mode. The sliding mode order is higher than the system relative degree and therefore, chattering elimination is ensured. In the control design process, unmeasured sliding vector derivative is estimated using a robust exact differentiator. To verify the effectiveness of the proposed control method, simulations are performed in the presence of external disturbances and inertia uncertainty. The proposed controller successfully negates the effect of external disturbances and inertia uncertainty and attitude states attain the desired accuracy with good steady state precision and high convergence speed.

Adaptive disturbance rejection for unknown stable linear systems

The design of an adaptive output feedback compensator is addressed to reject biased multi-sinusoidal disturbances of unknown amplitudes, phases and frequencies, acting on unknown stable single-input single-output linear systems of unknown order and relative degree. A single biased sinusoidal disturbance is first considered. It is shown that the knowledge of the sign of the real part (or the imaginary part) of its transfer function at zero and at the disturbance frequency is sufficient for the explicit design of an adaptive compensator which guarantees local exponential convergence to zero of the system output and of the frequency estimation error. This result is generalized to the case of biased multi-sinusoidal disturbances. As a special case, an adaptive notch filter which provides the unknown frequencies contained in the disturbance is obtained.

Design of a nonlinear controller using feedback linearisation based on backstepping technique for magnetic levitation system

This study proposes a feedback linearisation based on the back-stepping method to design a nonlinear controller with a goal of improving both steady state and transient stability of a magnetic levitation system. The feedback linearisation based on back-stepping technique is the combination of the two techniques mentioned i.e. feedback linearisation and backstepping. It uses backstepping design process, designs a sequence of 'virtual' systems of relative degree one, reduces the relative degree by one by choosing a 'virtual' input, achieves passivity with respect to a 'virtual output', and the last 'virtual output' is used to close feedback loop. Unlike the direct feedback linearisation method, the proposed method does not require a linear controller. Moreover, there is no need to know the exact nonlinear model of the system. Back stepping process of the controller guarantees its robustness against disturbances and uncertainties.

Relative Degrees and Adaptive Feedback Linearization Control of T–S Fuzzy Systems

This paper presents a new study on the relative degrees of single-input and single-output T–S fuzzy systems in general noncanonical forms, and proposes a feedback linearization-based control design method for such systems. The study extends the system relative degree concepts, commonly used for the control of nonlinear systems, to general T–S fuzzy systems, derives various relative degree conditions for general T–S fuzzy systems, and establishes the relative degree dependent normal forms. A feedback linearization-based control design framework is developed for general T–S fuzzy systems using its normal form, to achieve closed-loop stability and asymptotic output tracking under relaxed design conditions. A new adaptive feedback linearization-based control scheme for T–S fuzzy systems in general noncanonical forms with parameter uncertainties is designed and analyzed. Some extensions of relative degrees and their possible application to robust adaptive control for noncanonical form T–S fuzzy systems are also demonstrated. An illustrative example is presented with simulation results to demonstrate the control system design procedure and to show the effectiveness of the proposed control scheme.

Adaptive parameter identification of linear SISO systems with unknown time-delay

An adaptive online parameter identification is proposed for linear single-input-single-output (SISO) time-delay systems to simultaneously estimate the unknown time-delay and other parameters. After representing the system as a parameterized form, a novel adaptive law is developed, which is driven by appropriate parameter estimation error information. Consequently, the identification error convergence can be proved under the conventional persistent excitation (PE) condition, which can be online tested in this paper. A finite-time (FT) identification scheme is further studied by incorporating the sliding mode scheme into the adaptation to achieve FT error convergence. The previously imposed constraint on the system relative degree is removed and the derivatives of the input and output are not required. Comparative simulation examples are provided to demonstrate the validity and efficacy of the proposed algorithms.

Real-time Model Predictive Control for Quadrotors

This paper presents a solution to on-board trajectory tracking control of quadrotors. The proposed approach combines the standard hierarchical control paradigm that separates the control into low-level motor control, mid-level attitude dynamics control, and a high-level trajectory tracking with a model predictive control strategy. We use dynamic reduction of the attitude dynamics and dynamic extension of the thrust control along with feedback linearisation to obtain a linear system of relative degree three that models force controlled position and trajectory tracking for the quadrotor. Model predictive control is then used on the feedback equivalent system and its control outputs are transformed back into the inputs for the original system. The proposed structure leads to a low complexity model predictive control algorithm that is implemented in real-time on an embedded hardware. Experimental results on different position and trajectory tracking control are presented to illustrate the application of the derived linear system and controllers.

Fractional order adaptive controller for stabilised systems via high‐gain feedback

Controllers based on fractional order calculus are gaining more and more interests from the control community. This type of controllers may involve fractional integration, fractional differentiation and/or fractional systems in their structure or implementation. They have been introduced in the control applications in a continuous effort to enhance the system control quality performances and robustness. In this study, a new scheme of fractional order adaptive controller via high-gain output feedback for a class of linear, time-invariant, minimum phase and single input-single output systems of relative degree one is proposed. The basic idea of the new design is a further modification in the adaptive proportional control law by the introduction of a fractional integration besides of the regular one of the squared output of the system in the adaptation gain of the control strategy. An analytical stability proof of the feedback control system is presented. The control quality enhancement of the proposed control scheme compared with the classical one has been presented through the simulation results of an illustrative example.