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Highlights • An approximate sampled-data model for nonlinear systems with backward triangle sample and hold function is first derived. • Obtaining a novel expression and stable conditions of zero dynamics in the obtained sampled-data model. • Zero dynamics of the resulting model can be arbitrarily placed by selecting proper parameters from the stable conditions. • Nonlinear systems using those methods in this paper have better control performance. Abstract It is well known that a good approximate sampled-data model is very important in sampled-data controller design of nonlinear continuous-time system because the exact sampled-data model of nonlinear continuous-time system is often unavailable. It is also a truth that unstable zero dynamics of sampled-data model greatly limit the control performance of the system. This paper investigates the properties of approximate sampled-data model and its zero dynamics, which generated during the process of discretization with backward triangle sample and hold (BTSH). Further, a more accurate sampled-data model is obtained for nonlinear systems in BTSH case. The zero dynamics form and stable conditions are also derived. Finally, numerical examples are provided to better show the results developed in this paper. [ABSTRACT FROM AUTHOR]

Highlights • Introduce a new nonlinear coordination method based on zero dynamics (ZD) approach. • Design a super twisting sliding mode control to improve performance of ZD method. • Coordinate controllers of DFIG and STATCOM in multi-machine power systems. • Enhance transient and voltage stability of inter-connected power systems. Abstract This paper proposes a new nonlinear control approach to coordinate doubly fed induction generator (DFIG)-based wind turbines and static synchronous compensator (STATCOM) controllers in multi-machine power systems. Its main objectives are to improve the transient and voltage stability of the inter-connected multi-machine power system via simultaneous design of the voltage of the DFIGs' rotors and reference current of STATCOM. The nonlinear approach is based on a multi input-multi output (MIMO) zero dynamics (ZD) technique augmented by a super twisting sliding mode control (STSMC) to reduce the effects of the linearization error. The proposed approach was implemented to the Kundur's four-machine two-area power system and the IEEE 39-bus power system. Its performance was assessed in the presence of various faulty scenarios and disturbances. The effectiveness of the proposed approach (ZD-STSMC) was evaluated via eigenvalue analysis, transient simulations and transient stability margin analysis (by critical clearing time (CCT)). The merits of the proposed design are further compared to those of the exact feedback linearization (EFL) approach. Enhancing the power and voltage stability of large scale power systems, damping power oscillations and providing faster post-fault recovery time are among the positive features of the proposed approach. [ABSTRACT FROM AUTHOR]

Abstract The zero dynamics of infinite-dimensional systems can be difficult to characterize. The zero dynamics of boundary control systems are particularly problematic. In this paper the zero dynamics of port-Hamiltonian systems are studied. A complete characterization of the zero dynamics for port-Hamiltonian systems with invertible feedthrough as another port-Hamiltonian system on the same state space is given. It is shown that the zero dynamics for any port-Hamiltonian system with commensurate wave speeds are a well-posed system, and are also a port-Hamiltonian system. Examples include wave equations with uniform wave speed on a network. A constructive procedure for calculation of the zero dynamics that can be used for very large system order is provided. [ABSTRACT FROM AUTHOR]

We study disturbance decoupling for linear differential–algebraic systems which are not necessarily regular. Compared to previous approaches, where state feedback is used, we use the concept of behavioral feedback which allows to study a larger class of systems. We derive geometric characterizations for solvability of the disturbance decoupling problem following the classical approach. Exploiting the freedom in the choice of the behavioral feedback we show that whenever disturbance decoupling can be achieved by behavioral feedback we may additionally achieve autonomous zero dynamics. Finally we solve Lebret’s twenty year old open problem concerning disturbance decoupling with output uniqueness using behavioral feedback. [ABSTRACT FROM AUTHOR]

Continuous-time systems are usually modelled by the form of ordinary differential equations arising from physical laws. However, the use of these models in practice and utilizing, analyzing or transmitting these data from such systems must first invariably be discretized. More importantly, for digital control of a continuous-time nonlinear system, a good sampled-data model is required. This paper investigates the new consistency condition which is weaker than the previous similar results presented. Moreover, given the stability of the high-order approximate model with stable zero dynamics, the novel condition presented stabilizes the exact sampled-data model of the nonlinear system for sufficiently small sampling periods. An insightful interpretation of the obtained results can be made in terms of the stable sampling zero dynamics, and the new consistency condition is surprisingly associated with the relative degree of the nonlinear continuous-time system. Our controller design, based on the higher-order approximate discretized model, extends the existing methods which mainly deal with the Euler approximation. [ABSTRACT FROM AUTHOR]

We present a formula that computes the output of an R-controllable, regular, single-input linear time-invariant implicit system in such a way that it has prescribed relative degree and zeros. The formula is inspired on different generalizations of Ackermann’s formula. A possible application is in the context of sliding-mode control of implicit systems where, as the first step, one can use the proposed formula to design a sliding surface with desired dynamic characteristics and, as the second step, apply a higher-order sliding-mode controller to enforce a sliding motion along the resulting sliding surface. [ABSTRACT FROM AUTHOR]

This paper seeks to provide a method for optimizing 3D bipedal walking with satisfactory energy efficiency, provable stability and one degree of underactuation. Following the studies of the planar biped RABBIT, we propose a 3D robot with massless line feet which serve as sagittal rotation axes when lying flat on the ground. Using this configuration and the control method based on virtual constraints and feedback linearization, the walking stability can be verified by the restricted scalar Poincaré map of the zero dynamics of the system, and periodic gaits can be obtained analytically as in the case of planar bipeds with point feet. A line-foot contact model is adopted to ensure one degree of underactuation. Unlike most published contact models of bipedal walking, this model also prevents possible yaw movements of the feet on the ground. In addition, we adopt an output function named “symmetry outputs” with the desired outputs parameterized by Bézier coefficients and postural parameters, and the gait optimization is performed using a SQP algorithm. According to the results obtained from a bipedal model with a height of 1.5 m and a weight of 39 kg, the optimization program is capable of calculating stable periodic gaits with a speed between 0.2 m/s and 1.5 m/s. [ABSTRACT FROM AUTHOR]

This article identifies previously unknown, fundamental constraints on the effectiveness of economic policy instruments. The constraints arise from unstable zero dynamics , which occur commonly in macroeconomic models, and require (in the linear, time-invariant case) that infinite sums of discounted future values of certain endogenous variables necessarily equal zero, irrespective of policy. Unstable zero dynamics potentially explain the “price puzzle,” and are exemplified by classic explanations of the effect of changes in the rate of money creation on interest rates. They furnish a mathematical formulation of elements of Minsky’s financial instability hypothesis. In the multivariable case, they give rise to constraints involving multiple endogenous variables, which relate to the Phillips curve and Okun’s Law, and provide a system-theoretic paradigm for phenomena such as stagflation and stagnation. [ABSTRACT FROM AUTHOR]

We study the concept of locally controlled invariant submanifolds for nonlinear differential–algebraic/descriptor systems. In contrast to classical approaches, we define controlled invariance as the property of solution trajectories to evolve in a given submanifold whenever they start in it. It is then proved that this concept is equivalent to the existence of a feedback which renders the closed-loop vector field invariant in the descriptor sense. This result is exploited to show that the outcome of the differential–algebraic version of the zero dynamics algorithm yields a maximal output zeroing submanifold. The latter is then used to characterize the zero dynamics of the system. In order to guarantee that the zero dynamics are locally autonomous (i.e., locally resemble the behavior of an autonomous dynamical system), sufficient conditions involving the locally maximal output zeroing submanifold are derived. [ABSTRACT FROM AUTHOR]

The paper is concerned with the properties of approximate sampled-data models and their zero dynamics, as the sampling period tends to zero, composed of a fractional order hold (FROH), a continuous-time multivariable plant and a sampler in cascade. The emphasis of this paper is the stability of discrete zero dynamics with the generalized gain β of the FROH, where we also present a condition to assure the stability of the sampling zero dynamics, which they have no counterpart in the underlying continuous-time system, of the resulting model. Similar to the linear case, the parameter β is the only factor in affecting the stability of discrete zero dynamics, and the appropriate β is determined to obtain the FROH that provides zero dynamics as stable as possible, or with improved stability properties even when unstable, for a given continuous-time multivariable plant. The study is also shown that the stability of the sampling zero dynamics is improved compared with a zero-order hold (ZOH). [ABSTRACT FROM AUTHOR]

Highlights: [•] The shunt hybrid active power filter is linearized and decoupled. [•] The external dynamics is exponentially stabilized at its reference state. [•] The zero dynamics can be stabilized by a novel PI controller. [•] The proposed control strategy is effective to eliminate harmonics. [Copyright &y& Elsevier]

Abstract: This paper presents the formulation of a novel block-backstepping based control algorithm to address the stabilization problem for a generalized nonlinear underactuated mechanical system. For the convenience of compact design, first, the state model of the underactuated system has been converted into the block-strict feedback form. Next, we have incorporated backstepping control action to derive the expression of the control input for the generic nonlinear underactuated system. The proposed block backstepping technique has further been enriched by incorporating an integral action additionally for enhancing the steady state performance of the overall system. Asymptotic stability of the overall system has been analyzed using Lyapunov stability criteria. Subsequently, the stability of the zero dynamics has also been analyzed to ensure the global asymptotic stability of the entire nonlinear system at its desired equilibrium point. The proposed control algorithm has been applied for the stabilization of a benchmarked underactuated mechanical system to verify the effectiveness of the proposed control law in real-time environment. [Copyright &y& Elsevier]

The concept of zero dynamics of a nonlinear system was introduced about thirty years ago as nonlinear analogue of the concept of transmission zero of the transfer function of a system. This paper reviews a few historical landmarks in the development of this concept and then addresses some current challenging issues, such as the development of global and coordinate free version of a standard output-feedback design paradigm, the analysis of problems of stabilization and tracking in the presence of unstable zero dynamics, and extensions to multivariable systems. [ABSTRACT FROM AUTHOR]

Abstract: FACTs devices are being used in transmission networks for increasing the power transfer limit and stability improvement. They also help damp out both local and inter-area low frequency oscillations. However, uncoordinated design of these devices with excitation systems may deteriorate the power system performance. Moreover, power system is a large, complex and nonlinear system, and the controllers that are designed based on linear control theories may have a detrimental effect on the system performance, especially when there are large disturbances occurring in the system. The design method of a nonlinear control technique, named zero dynamics is given in this paper to design the controllers of STATCOM and excitation systems coordinately for multi-machine power systems. This technique is able to provide the stability of both external and internal dynamic performances of the system. Simulations results clearly verify that the proposed method improves the power system stability. [Copyright &y& Elsevier]

Abstract: Chemical processes constitute strongly nonlinear systems and, for such systems, multiple steady-state solutions typically exist. In addition, the various steady state solutions are likely to differ in terms of stability and phase behaviors, which is an important consideration for practical applications. A chemical process is used in this paper to demonstrate how to analyze process stability and controllability. Finally, the conclusion is drawn that overall system stability and phase behaviors should be considered because the individual unit operations or subsystems differ from the total system in terms of these features. Therefore, the analysis of stability and controllability of process systems is important in terms of the design of inherently safer processes. [Copyright &y& Elsevier]

Abstract: Previous works for wheeled inverted pendulums usually eliminate nonholonomic constraint force in order to make the control design easier, under the assumption that the friction force from the ground is as large as needed. Nevertheless, such an assumption is unfeasible in practical applications. In this paper, adaptive robust motion/force control for wheeled inverted pendulums is investigated with parametric and functional uncertainties. The proposed robust adaptive controls based on physical properties of wheeled inverted pendulums make use of online adaptation mechanism to cancel the unmodelled dynamics. Based on Lyapunov synthesis, the proposed controls ensure that the system outputs track the given bounded reference signals within a small neighborhood of zero, and guarantee the semi-global uniform boundedness of all closed loop signals. The effectiveness of the proposed controls is verified through extensive simulations. [ABSTRACT FROM AUTHOR]

Abstract: The dynamic balance and motion control based on LS-SVM (least squares support vector machine) are considered for mobile wheeled inverted pendulums (WIP), in the presence of parametric and functional dynamics uncertainties. Based on Lyapunov synthesis, the proposed control mechanisms use the advantage of LS-SVM combined with on-line parameters estimation strategy in order to have an efficient approximation. Under the controller designed, we can ensure that the outputs of the system track the given bounded reference signals within a small neighborhood of zero, and guarantee semi-global uniform boundedness of all the closed loop signals. Simulation results are presented to verify the effectiveness of the proposed control. [ABSTRACT FROM AUTHOR]

Abstract: The hybrid joints can be switched to either active (actuated) or passive (under-actuated) mode as needed (), in this paper, dynamic coupling switching control incorporating Support Vector Machines (SVMs) is developed for wheeled mobile manipulators with hybrid joints. The hybrid actuated robot dynamics is a mixed under-actuated and actuated model. In order to approximate the high dimension unmodelled dynamics, the SVMs matrix and its operator are proposed. Considering the joint switching as an event, the event driven switching control strategy is used to ensure that the system outputs track the given bounded reference signals within a small neighborhood of zero, and guarantee semi-global uniform boundedness of all closed loop signals, and the switch stability. The effectiveness of the proposed controls is verified through extensive simulations. [Copyright &y& Elsevier]

Abstract: The nonlinear system model of an endoatmospheric missile whose attitude is controlled by tail fins and reaction jets is presented. By choosing the divert accelerations and rotational rates as output variables, the internal dynamics of the nonlinear system are derived and are proved to be bounded under a bounded input. The blending principle of the reaction jets and tail fins is addressed to ensure that the normal acceleration is principally maintained by angle-of-attack or sideslip angle. An autopilot is designed by using the feedback linearization technique. Results of a numerical simulation of an autopilot design example show the effectiveness of the proposed blending principle and the autopilot design. [Copyright &y& Elsevier]

Abstract: In this paper, adaptive fuzzy logic control of dynamic balance and motion is investigated for wheeled inverted pendulums with parametric and functional uncertainties. The proposed adaptive fuzzy logic control based on physical properties of wheeled inverted pendulums makes use of a fuzzy logic engine and a systematic online adaptation mechanism to approximate the unknown dynamics. Based on Lyapunov synthesis, the fuzzy control ensures that the system outputs track the given bounded reference signals to within a small neighborhood of zero, and guarantees semi-global uniform boundedness of all closed-loop signals. The effectiveness of the proposed control is verified through extensive simulations. [Copyright &y& Elsevier]

Abstract: Input–output feedback linearization provides a convenient means of extending linear control strategies such as output zeroing or pole placement to the case of nonlinear affine in the input systems, but such extensions cannot be applied in the presence of nonminimum phase characteristics. This paper overcomes this difficulty through the periodic use of a finite number of synthetic outputs which are so constructed as to define embedded dynamics with stable zero dynamics. The efficacy of the method is demonstrated by means of a numerical example. [Copyright &y& Elsevier]

Abstract: An image-based strategy for visual servo control of a class of dynamic systems is proposed. The class of systems considered includes dynamic models of unmanned aerial vehicles capable of quasi-stationary flight (hover and near hover flight). The control strategy exploits passivity-like properties of the dynamic model to derive a Lyapunov control algorithm using backstepping techniques. The paper extends earlier work (Hamel, T., & Mahony, R. (2002). Visual servoing of an under-actuated dynamic rigid-body system: An image based approach. IEEE Transactions on Robotics and Automation, 18(2), 187–198) where partial pose information was used in the construction of the visual error. In this paper the visual error is defined purely in terms of the image features derived from the camera input. Local exponential stability of the system is proved. An estimate of the basin of attraction for the closed-loop system is provided. [Copyright &y& Elsevier]

Abstract: This paper investigates maneuver regulation for single-input control-affine systems from a geometric perspective. The maneuver regulation problem is converted to output stabilization and necessary and sufficient conditions are provided to solve the latter problem by feedback linearizing the dynamics transverse to a suitable embedded submanifold of the state space. When specialized to the linear time invariant setting, this work recovers well-known results on output stabilization. [Copyright &y& Elsevier]

In this article, the internal dynamics of flat nonlinear SISO systems with respect to a real non-flat output is investigated in the coordinates of the flat output and its derivatives. The main result constitutes that the parameterization of the real output by the flat output and its derivatives up to a certain order forms a differential equation which is diffeomorphic to the aforementioned internal dynamics. [Copyright &y& Elsevier]

The present research work aims at the development of a systematic method to arbitrarily assign the zero dynamics of a nonlinear discrete-time real analytic system by constructing the requisite synthetic output maps. The problem under consideration is motivated by the need to adequately address the control problem of nonminimum-phase nonlinear discrete-time systems, since the latter represent a rather broad class of systems due to the well-known effect of sampling on the stability of zero-dynamics. In the proposed approach, the above control objective can be attained through: (i) a systematic computation of synthetic output maps that induce minimum-phase behavior while being statically equivalent to the original output maps (both vanish on the equilibrium manifold) and (ii) the subsequent integration into the methodological framework of currently available nonminimum-phase compensation schemes for nonlinear discrete-time systems that rely on output redefinition. The mathematical formulation of the zero-dynamics assignment problem is realized via a system of nonlinear functional equations, and a rather general set of necessary and sufficient conditions for solvability is derived. The solution to the above system of functional equations can be proven to be locally analytic, and this enables the development of a solution method that is easily programmable with the aid of a symbolic software package. The synthetic output maps that induce the prescribed zero dynamics for the original nonlinear discrete-time system can be explicitly computed on the basis of the solution to the aforementioned system of functional equations. [Copyright &y& Elsevier]

A method for the generation of globally attractive limit cycles for nonlinear systems is presented. It consists in designing an output that, when regulated through a suitable feedback, forces a limit cycle in the zero dynamics. Conditions are then given to ensure that a globally attractive limit cycle in the zero dynamics results in a globally attractive limit cycle in the whole system. The method is illustrated on the torque control of an induction motor. [Copyright &y& Elsevier]

The present research work aims at the development of a systematic method to arbitrarily assign the zero dynamics of a nonlinear system by constructing the requisite synthetic output maps. The minimum-phase synthetic output maps constructed can be made statically equivalent to the original output maps, and therefore, they could be directly used for nonminimum-phase compensation purposes. Specifically, the mathematical formulation of the problem is realized via a system of first-order nonlinear singular PDEs and a rather general set of necessary and sufficient conditions for solvability is derived. The solution to the above system of singular PDEs can be proven to be locally analytic and this enables the development of a series solution method that is easily programmable with the aid of a symbolic software package. The minimum-phase synthetic output maps that induce the prescribed zero dynamics for the original nonlinear system can be computed on the basis of the solution of the aforementioned system of singular PDEs. Moreover, static equivalence to the original output map can be readily established by a simple algebraic construction. [Copyright &y& Elsevier]

In two-input (or multi-input) nonlinear systems it may be possible to achieve feedback passivation of a chosen input using the second input (other inputs) to improve the stability properties of the first input''s zero dynamics. This ‘coordinated passivation’ approach is illustrated on a simplified model of a turbocharged diesel engine. [Copyright &y& Elsevier]

This paper discusses the applicability of a switching control scheme for a nonlinear system with ill-defined relative degree. The control scheme switches between exact and approximate input–output linearisation control laws. Unlike a linear system under a switching control scheme, the equilibria of a nonlinear system may change with the switching. It is pointed out that this is not sufficient to cause instability. When the region of the approximate linearisation control law is attractive to the exact zero dynamics, it is possible that the closed-loop system under the switching control scheme is still stable. The results in this paper shows that the switching control scheme proposed in Tomlin and Sastry (Systems Control Lett. 35(3) (1998) 145) is applicable for a wider class of nonlinear systems. [Copyright &y& Elsevier]

A new method of controlling nonlinear processes with a non-minimum-phase delay-free part is presented. Two control laws are derived for stable, multiple-input multiple-output processes. They are obtained by requesting an approximately linear, input–output response and exploiting the connections between model-predictive control and input–output linearization. Conditions under which the closed-loop system is asymptotically stable are given. The application and performance of the control laws are illustrated using numerical simulation of two chemical reactor examples that exhibit non-minimum-phase behavior. [Copyright &y& Elsevier]

The zero dynamics of linear systems with commensurate delays is studied. The zero dynamics is meant to be the dynamical equation that remains after compelling, by the input of the system, the system output to be identically equal to zero. Conditions are found out allowing to transform the system into a normal form that reveals its zero dynamics. By a straightforward use of the obtained decomposition, the disturbance decoupling problem is solved readily. An explanation is given on how to extend the obtained results to systems with distributed delays. Two examples of physical systems illustrate the effectiveness of the presented study. [ABSTRACT FROM AUTHOR]