Control of a chain of integrators subject to actuator saturation and disturbances

In this paper, we study control of a chain of integrators under actuator saturation and non‐additive disturbances. We shall show that boundedness of the states can be ensured if the disturbances are matched and integral‐bounded, misaligned and magnitude‐bounded, or a combination of the two, using either a static or a dynamic low‐gain state feedback. This result is an extension of our earlier work on double integrator. Copyright © 2011 John Wiley & Sons, Ltd.     

Decentralized adaptive robust control for a class of large scale systems with uncertainties in the interconnections

The problem of decentralized control is considered for a class of time-varying large scale systems with uncertainties and external disturbances in the interconnections. In this paper, the upper bounds of the uncertainties and external disturbances are assumed to be unknown. The adaptation laws are proposed to estimate such unknown bounds, and by making use of the updated values of these unknown bounds, a class of decentralized linear and non-linear state feedback controllers are constructed. It is shown that by employing the proposed decentralized non-linear state feedback controllers, the solutions of the resulting adaptive closed-loop large scale system can be guaranteed to be uniformly bounded, and the states are uniformly asymptotically stable. By using the decentralized linear state feedback controllers, one can guarantee the uniform ultimate boundedness of the resulting adaptive closed-loop large scale system. Finally, a numerical example is given to demonstrate the validity of the results.     

Design of switched linear systems in the presence of actuator saturation and L-infinity disturbances

This paper considers the problem of disturbance tolerance/rejection of a switched system resulting from a family of linear systems subject to actuator saturation and L-infinity disturbances. For a given set of linear feedback gains, a given switching scheme and a given bound on the L-infinity norm of the disturbances, conditions are established, in terms of linear or bilinear matrix inequalities, under which a set of a certain form is invariant for a given switched linear system in the presence of actuator saturation and L-infinity disturbances, and the closed-loop system possesses a certain level of disturbance rejection capability. With these conditions, the design of feedback gains and switching scheme can be formulated and solved as constrained optimization problems. Disturbance tolerance is measured by the largest bound on the disturbances for which the trajectories starting from a given set remain bounded. Disturbance rejection is measured either by the L-infinity norm of the system output or by the system’s ability to steer its state into and/or keep it within a small neighborhood of the origin. In the event that all systems in the family are identical, the switched system reduces to a single system under a switching feedback law. Simulation results show that such a single system under a switching feedback law could have stronger disturbance tolerance/rejection capability than a single linear feedback law can.     

Design of high performance linear feedback laws for operation that extends into the nonlinear region of AMB systems

Existing active magnetic bearings (AMBs) operate in the linear region of the magnetic material flux density, which limits the utilization of the bearing capacity. In order to increase the utilization of the bearing capacity and enhance the performance of the AMB system, this paper develops a method for designing high performance linear feedback laws. The resulting feedback laws allow the AMB to operate in its nonlinear region and hence improve the closed-loop performance. We first establish an approximate nonlinear AMB current force response model, and place this nonlinear curve inside a sector formed by two piecewise linear lines. Based on the linear line segments in these two piecewise linear lines, we determine the maximum disturbance that can be tolerated by solving an optimization problem with linear matrix inequality (LMI) constraints. For a given level of disturbance under the maximum tolerable disturbance, we formulate and solve the problem of designing the linear feedback that achieves the highest level of disturbance rejection as another LMI problem. Both $L_2$ disturbances and $L_\infty$ disturbances are considered. Finally, we illustrate our design by both simulation and experimental results.     

Dynamic gain output feedback control of a class of uncertain feedforward nonlinear systems with uncertain time-delays and sensor noise

We consider a control problem of a class of uncertain feedforward nonlinear systems with uncertain time-delays under sensor noise. Time-delays and sensor noise can degrade the control accuracy and as a result, the boundedness of states and output is not guaranteed. To guarantee the boundedness of states and output, we develop an output feedback controller which involves a dynamic gain. Moreover, we show that the ultimate bounds of states and output can be arbitrarily small if DC component of sensor noise is zero.     

Input-to-state stability for a class of Lurie systems

We analyze input-to-state stability (ISS) for the feedback interconnection of a linear block and a nonlinear element. This study is of importance for establishing robustness against actuator nonlinearities and disturbances. In the absolute stability framework, we prove ISS from a positive real property of the linear block, by restricting the sector nonlinearity to grow unbounded as its argument tends to infinity. When this growth condition is violated, examples show that the ISS property is lost. The result is used to give a simple proof of boundedness for negative resistance oscillators, such as the van der Pol oscillator. In a separate application, we relax the minimum phase assumption of an earlier boundedness result for systems with nonlinearities that grow faster than linear.     

Modelling and control for a class of axially moving nonuniform system

In this paper, we deal with the active control problem for a class of axially moving system. The system is nonuniform due to the spatial-varying mass and spatiotemporally varying tension. The infinite dimensional model of the nonuniform system represented by the hybrid partial-ordinary differential equations is derived with consideration of the spatiotemporally varying distributed disturbance. To suppress the vibration of the nonuniform system, full state feedback boundary control is developed. For the case that the system states cannot be measured, output feedback boundary control is proposed by using the high-order observers for the estimation of the unmeasurable states. The disturbance observer is introduced to mitigate the effects of the boundary disturbance. With the proposed boundary control, the stability of the closed-loop nonuniform system is achieved through rigorous analysis and the uniform boundedness of the all closed-loop signals is guaranteed. Numerical simulations are performed to validate the performance of the control scheme proposed.     

Disturbance decoupling of switched linear systems

In this paper, we consider disturbance decoupling problems for switched linear systems. We will provide necessary and sufficient conditions for three different versions of disturbance decoupling, which differ based on which signals are considered to be the disturbance. In the first version, the exogenous input is considered as the disturbance, in the second, the switching signal and in the third both of them are considered as disturbances. All three versions of disturbance decoupling have direct counterparts for linear parameter-varying (LPV) systems, while the latter instance of the problem is relevant for disturbance decoupling of piecewise linear systems, as we will show. The solutions of the three disturbance decoupling problems will be based on geometric control theory for switched linear systems and will entail both mode-dependent and mode-independent static state feedback.     

Arbitrarily fast and robust tracking by feedback

The output of a singe-input-single-output linear feedback system with more than one pole in excess over the zeros in the loop transmission cannot track arbitrarily fast its input (by the root locus). In this work we extend the linear feedback so that some of the open loop poles may depend on the open loop gain; we call this new class quasi-linear feedback systems. We then derive time domain, pole-zero, and frequency domain conditions which ensure arbitrarily fast and robust tracking by quasi-linear feedback, for an arbitrary number of poles in excess over the zeros. We prove that in a particular case these conditions are equivalent, and that the boundedness in frequency of the closed loop transfer function is no longer necessary for achieving arbitrarily fast tracking. The robustness is to external disturbances and initial conditions, and the open loop has to be minimum phase. Some examples are presented which illustrate these results. They also show that this good performance can be obtained with a reduced control effort, and that quasi-linear feedback can alleviate the limitation on performance of non-minimum phase open loops.     

带有界扰动的一类大型互联非仿射非线性系统\=的鲁棒分散控制

对带有界扰动的一类大型互联非仿射非线性系统进行了分散状态反馈控制设计．通过子系统状态的线性变换,得到分散状态反馈控制律．当状态反馈控制律作用于该系统时,无扰动的闭环系统是渐近稳定的;当扰动较小时,系统的状态能够收敛到原点的一个小邻域内．     

Robust stabilization of uncertain systems with persistent disturbance and a class of non-linear actuators

Linear uncertain systems perturbed by persistent disturbances and driven by actuators subject to band-bounded nonlinearities are considered. A set of linear matrix inequality based sufficient conditions are derived for designing state feedback controllers which assure ultimate boundedness of closed-loop state trajectories. The effect of the non-linear actuators is identified to that of the considered persistent disturbances. The control purpose is to minimize the ultimate boundedness region to which the state trajectories are eventually confined, while tolerating disturbances of given or the largest magnitudes. Finally, an experimental system, an inverted pendulum on a Stewart platform making translational movement, is arranged to demonstrate the feasibility and effectiveness of the derived results.     

Deadtime compensation for nonlinear processes with disturbances

The control problem of time delay non-linear systems that are perturbed by disturbances is discussed. By coordinate transformations and feedback and prediction of the transformed states, we first linearize the perturbed non-linear systems into controllable quasi-linear systems with disturbances. Then, we can apply the well-developed linear control theory to stabilize the transformed systems. Thus, stable quasi-linear systems with time delay can be obtained. Furthermore, we may implement powerful deadtime compensation methods to study the performance of the proposed dynamic compensators, a Smith predictor and a new modified Smith predictor, for disturbance rejection. Finally, a typical non-linear chemical process, a continuous stirred tank reactor, is used as an example system to demonstrate the effectiveness of these deadtime compensators     

A high-gain scaling technique for adaptive output feedback control of feedforward systems

In this note, we propose an adaptive output feedback control design technique for feedforward systems based on our recent results on dynamic high-gain scaling techniques for controller design for strict-feedback systems. The system is allowed to contain uncertain functions of all the states and the input as long as the uncertainties satisfy certain bounds. Unknown parameters are allowed in the bounds assumed on the uncertain functions. If the uncertain functions involve the input, then the output-dependent functions in the bounds on the uncertain functions need to be polynomially bounded. It is also shown that if the uncertain functions can be bounded by a function independent of the input, then the polynomial boundedness requirement can be relaxed. The designed controllers have a very simple structure being essentially a linear feedback with state-dependent dynamic gains and do not involve any saturations or recursive computations. The observer utilized to estimate the unmeasured states is similar to a Luenberger observer with dynamic observer gains. The Lyapunov functions are quadratic in the state estimates, the observer errors, and the parameter estimation error. The stability analysis is based on our recent results on uniform solvability of coupled state-dependent Lyapunov equations. The controller design provides strong robustness properties both with respect to uncertain parameters in the system model and additive disturbances. This robustness is the key to the output feedback controller design. Global asymptotic results are obtained.     

Output tracking of constrained nonlinear processes with offset-free input-to-state stable fuzzy predictive control

This paper develops an efficient offset-free output feedback predictive control approach to nonlinear processes based on their approximate fuzzy models as well as an integrating disturbance model. The estimated disturbance signals account for all the plant-model mismatch and unmodeled plant disturbances. An augmented piecewise observer, constructed by solving some linear matrix inequalities, is used to estimate the system states and the lumped disturbances. Based on the reference from an online constrained target generator, the fuzzy model predictive control law can be easily obtained by solving a convex semi-definite programming optimization problem subject to several linear matrix inequalities. The resulting closed-loop system is guaranteed to be input-to-state stable even in the presence of observer estimation error. The zero offset output tracking property of the proposed control approach is proved, and subsequently demonstrated by the simulation results on a strongly nonlinear benchmark plant.     

Output feedback control with disturbance rejection of a class of nonlinear MIMO systems

Output feedback control with disturbance rejection is developed for a class of nonlinear multi-input-multi-output (MIMO) systems. The availability of state variables and the bound of disturbances are not required to be known in advance. In the design of an adaptive observer, a robust adaptive nonlinear state feedback controller using the estimated states is proposed. The control methodology is robust to bounded disturbances that are both constant and time-varying, with effective performance. The adaptive laws are derived based on the Lyapunov synthesis method; therefore closed-loop asymptotic stability is also guaranteed. Moreover, chattering can be reduced by the proposed design approach. Simulation results are included to illustrate the effectiveness of the proposed controller. The text was submitted by the authors in English.     

Robust control for a class of uncertain nonlinear systems with input quantization

In this paper, we propose a robust tracking control scheme for a class of uncertain strict‐feedback nonlinear systems. In these systems, the control signal is quantized by a class of sector‐bounded quantizers including the well‐known hysteresis quantizer and logarithmic quantizer. Compared with the existing results in input‐quantized control, the proposed scheme can control systems with non‐global Lipschitz nonlinearities and unmatched uncertainties caused by model uncertainties and external disturbances. It is shown that the designed robust controller ensures global boundedness of all the signals in the closed‐loop system and enables the tracking error to converge toward a residual, which can be made arbitrarily small. Copyright © 2015 John Wiley & Sons, Ltd.     

Robust Control for a Class of Time-delay Nonlinear Systems via Output Feedback Strategy

This paper studies the robust control problem for time-delay systems with complicated inherent nonlinearities and unknown disturbances. Based on the modified homogeneous domination method and by constructing the proper Lyapunov-Krasovskii (L-K) functional, output feedback controllers are successfully constructed to guarantee the boundedness of all the states of the closed-loop system. The convergence of the states is also realizable when the L₂ norm of the disturbance exists. The presented method is also extended to solve the control problem of nonholonomic time-delay system. Simulation examples are given to show the effectiveness of the proposed theory.     

Adaptive regulation of uncertain nonlinear systems by output feedback: A universal control approach

The problem of global state regulation via output feedback is investigated for uncertain nonlinear systems. The class of uncertain systems under consideration is assumed to be dominated by a bounding system which is linear growth in the unmeasurable states but can be a polynomial function of the system output, with unknown growth rates. To achieve global state regulation in the presence of parametric uncertainty, we propose a non-identifier based output feedback control scheme by employing the idea of universal control integrated with the design of a linear high-gain observer, whose gains are composed of two components, both of them are not constant and need to be dynamically updated. In particular, we explicitly design a universal output feedback controller which globally regulates all the states of the uncertain system while maintaining global boundedness of the closed-loop system.     

Output‐feedback tracking in causal nonminimum‐phase nonlinear systems using higher‐order sliding modes

Asymptotic output‐feedback tracking in a class of causal nonminimum phase uncertain nonlinear systems is addressed via sliding mode techniques. Sliding mode control is proposed for robust stabilization of the output tracking error in the presence of a bounded disturbance. The output reference profile and the unknown input/disturbance are supposed to be described by unknown linear exogenous systems of a given order. Local asymptotic stability of the output tracking error dynamics along with the boundedness of the internal states are proven. The unstable internal states are estimated asymptotically via the proposed multistage observer that is based on the method of extended system center. A higher‐order sliding mode observer/differentiator is used for the exact estimation of the input–output states in a finite time. The bounded disturbance is reconstructed asymptotically. A numerical example illustrates the efficiency of the proposed output‐feedback tracking approach developed for causal nonminimum phase nonlinear systems. Copyright © 2009 John Wiley & Sons, Ltd.     

Homogeneous control law design for output synchronization of networked Lur'e‐type systems with disturbances

In this paper, the homogeneous distributed control law is proposed such that the outputs of the networked Lur'e systems with disturbances are synchronized. A novel adaptive distributed state‐based disturbance compensator technique is utilized such that the disturbances can be offset asymptotically. The design of the distributed compensator is constructive in that some particular functions of the states of the Lur'e systems are included. The functions always exist for the considered Lur'e‐type system. Moreover, a theorem of networked Lur'e systems subject to disturbances is first presented to show the boundedness of the states of the closed‐loop system. Then, the synchronization is shown by the incremental passivity theory and graph theory. Finally, the effectiveness of the current method is illustrated by the networked Van der Pol oscillators with disturbances.