Unstable Zero Dynamics Research Articles

The limiting zero dynamics of discrete-time system based on forward triangle sample-and-hold.

Zero dynamics have crucial effect on system analysis and controller design. In the control analysis process, system performance is influenced by the unstable zero dynamics, greatly. This study concerns with the properties of limiting zero dynamics when the signal of controlled continuous-time systems was reconstructed by forward triangle sample-and-hold (FTSH). FTSH is a newly sample and hold method in the signal reconstruction area. However, more theoretical details about the limiting zero dynamics of the resulting discrete-time systems need to be revealed. Firstly, the framework for the limiting zero dynamics in the situation of sufficiently small or large sample period is introduced. Furthermore, this study provides the stable conditions of limiting zeros in the two different sampling situations. The results indicate that one can select a suitable variable parameter value of FTSH to replace the sampling zeros of discrete-time system locate inside the stable region. This paper also reveals the truth that FTSH has the outstanding advantage compared with the BTSH using theoretical analysis method. Finally, example simulations verify the effectiveness of the results in this study.

Some issues about FL and LAB servo control

In this paper, the servo control design methodologies based on Feedback Linearization (FL) and Linear Algebra Based (LAB) are applied to nonlinear systems with and without zero dynamics, analysing the case of unstable zero dynamics. Their advantages and drawbacks are discussed. Some examples are developed to illustrate the results.

A Novel Feedback Linearisation Control of Flyback Converter

Abstract This paper presents a novel approach for feedback linearisation in a continuous conduction mode (CCM) of the flyback converter. Due to the unstable zero dynamics, a flyback converter has highly non-linear behaviour. Flyback converters mostly use the indirect (current) control mechanism. In contrast, this paper shows a direct control of the output voltage of a flyback converter with feedback linearisation (a non-linear control method). In the designed controller, an error integrator is applied to improve the dynamic and steady-state behaviour of the controller. To design the feedback linearisation method, the state-space averaged model is determined. The converter and the proposed control are tested in a MatLab/Simulink environment, and the results are compared with other optimal controller methods. The results provide feedback about the efficiency and practical implementation of the proposed method.

Leader-Following Almost Output Consensus for Linear Heterogeneous Multiagent Systems With Disturbance-Affected Unstable Zero Dynamics by Output Feedback

A novel design of observer-based output feedback consensus protocols in a low-and-high gain feedback framework is proposed for a linear heterogeneous multiagent system, where the unstable zero dynamics of the follower agents is affected by disturbances. Conditions on the graph that represents the connectivity among the agents and on how the disturbances enter the zero dynamics of the follower agents are identified, under which the leader-following almost output consensus can be attained with the proposed protocols. More specifically, the proposed protocols achieve leader-following output consensus to an arbitrarily high level of accuracy, and the effect of the disturbances on the consensus errors is attenuated to an arbitrarily low level.

Output tracking control of non‐minimum phase systems using on‐line output redefinition approach based on barrier function method

This paper considers the output tracking problem of non-minimum phase non-linear systems using the inverse dynamics control which is impossible in conventional methods due to unstable zero dynamics. The proposed method is based on the output redefinition approach defined as a parametric function. The tracking error is decomposed into two terms. The first one named redefined output tracking error is the difference between the redefined output and the reference signal, and the second one is the distance between the redefined output and the system output. For the first term, due to minimum phase property of the system with the redefined output, an inverse dynamics control is used to guarantee the convergence of the redefined output tracking error to zero. To minimize the second term, the parameters of the redefined output are set by solving an optimization problem whose constraints include minimum phase conditions. To solve the optimization problem which satisfies all constraints, the barrier function method as an interior point method is adopted. The optimization parameters are set using steepest descent algorithm. Due to on-line updating the redefined output parameters, the related zero dynamics is time varying. Therefore, sufficient conditions for exponential stability of the resulting time-varying zero dynamics are obtained.

Parallel compensator for zero dynamics stabilization in fluidized bed spray granulation

This contribution is concerned with the stabilization of continuous fluidized bed spray granulation processes with external sieve mill cycle. To keep the overall mass of particles inside the granulation chamber in well-defined bounds these processes are in general operated with a mass controller. This configuration is well-known to become unstable for certain ranges of the milling diameter. Here, the instability results in a nonlinear limit cycle of the particle size distribution. As has been shown in a previous contribution the stability problem is not inherent to the process, which is open-loop stable over a wide range of operating conditions. In fact, the root cause for the instability is the mass controller, which interacts with the unstable zero dynamics of the granulation process. To overcome this problem a parallel compensator will be designed, which allows to stabilize the process zero dynamics and thus solves the problem of control induced instabilities.

A Differential-Flatness-Based Approach for Autonomous Helicopter Shipboard Landing

This paper presents a differential flatness-based approach to trajectory planning and tracking for helicopters and demonstrates this strategy for time-optimal autonomous helicopter shipboard landing. For planning and outer-loop tracking control, helicopter dynamics are captured by a simplified nonlinear model that is shown to be differentially flat, which enables the transformation of the original nonlinear system dynamics into an equivalent linear form with endogenous nonlinear constraints. For time-optimal trajectory planning, this equivalent model is used to construct a computationally-efficient optimization problem for free end-time end-state problems, where the decision-space is that of the trajectories of the flat outputs and their derivatives. While the attitude dynamics are stabilized by an inner loop controller, for tracking the planned trajectory a flatness-based outer loop control law is designed to stabilize the unstable zero dynamics. Further, analytical guarantees that relate inner loop tracking error to outer loop robust stability and tracking performance are derived. Simulations are performed on a high-fidelity numerical UH-60 helicopter model under ground effect and ship airwake disturbances, together with ship motion profiles collected from a naval environment to validate the proposed approach for realistic landing scenarios.

Suboptimal output consensus of a group of discrete-time heterogeneous linear non-minimum phase systems

This paper formulates the suboptimal output consensus problem for a discrete-time heterogeneous linear multi-agent system with unstable zero dynamics, where each agent possesses its own objective function and the sum of all these private objective functions, called the overall objective function, is to be minimized. Mild assumptions on the communication topology and the agent dynamics are made under which a parameterized distributed consensus protocol based on low gain feedback is proposed for each agent. The multi-agent system is shown to achieve suboptimal output consensus under the proposed protocols in the sense that the states of all agents remain bounded while their outputs converge to a pre-specified arbitrarily small neighborhood of the optimal point as long as the design parameter is chosen small enough. Simulation results are presented that demonstrate the proposed design.

Periodic Event-Triggered Control for a Class of Nonminimum-Phase Nonlinear Systems Using Dynamic Triggering Mechanism

In this paper, the output feedback based periodic event-triggered control problem is considered for a class of nonminimum-phase nonlinear systems via dynamic event-triggering mechanism. When only the sampled-data output is known, a new periodic event-triggered control method is proposed via output feedback and the control input updates based on a discrete-time dynamic event-triggering condition. Despite the unstable zero dynamics, the proposed control method can asymptotically stabilize the hybrid control systems by closing the loop only when it is necessary. In contrast to the continuous-time static one, the proposed dynamic triggering condition has several advantages including the easier digital implementation and the larger average inter-event time interval. The delicate analysis gives the explicit expression of the maximum allowable sampling period, and the global asymptotic stability can be achieved for the hybrid control systems. Finally, the effectiveness of the proposed periodic event-triggered control method is verified by a numerical simulation.

A Generalized Hold Based Countermeasure Against Zero-Dynamics Attack With Application to DC-DC Converter

Zero-dynamics attack (ZDA) is a model based cyber attack. It is stealthy in the sense that the existence of attack signal cannot be determined by monitoring the system output, and it is effective to systems that have unstable zero-dynamics. Several countermeasures against ZDA have been introduced, and the one employing the generalized hold (GH) is considered in this paper. The GH is a generalized version of zero-order hold that is commonly used in the digital control systems. In this paper, the lethality of ZDA and the effectiveness of GH as a countermeasure are demonstrated on a control system that involves a DC-DC converter. Through extensive simulations and experiments, the design of the proposed scheme is presented in detail and relevant practical issues are discussed.

Robust output tracking for the non-minimum phase over-actuated systems

Typically, if the system is right-invertible only, then this implies that the number of inputs is more than that of outputs as well as states of the system. Over-actuated systems belong to such class. In this paper, a systematic design of a reduced-order super-twisting control is proposed to track arbitrary reference signals for uncertain non-minimum phase over-actuated systems. The proposed method involves the transformation that resolves the redundancy in the control inputs so that system can be represented as a normally-actuated system with virtual control inputs. Further, unstable zero dynamics are virtually stabilized through one more transformation so that stable and virtually stable zero dynamic states can be excluded from the super-twisting control design. Finally, the computed virtual control magnitude is spread across all the actuators via a redistributed pseud-inverse type of control allocation so that the system can be operated within the actuator saturation limits.

Output tracking for a class of non-minimum phase nonlinear systems: A two-point boundary value problem formulation with a hybrid regulator

The asymptotic output tracking problem is studied for a class of non-minimum phase nonlinear systems without requiring the a priori knowledge, or even the existence, of a finite-dimensional exosystem that generates the prescribed periodic reference signal. The design strategy is illustrated in two steps. First, it is shown that the knowledge of a solution to a certain two-point boundary value problem involving the underlying zero-dynamics of the plant is instrumental and sufficient to construct a state feedback regulator that achieves boundedness of the trajectories and (exact) asymptotic tracking, hence completely circumventing the need for solving partial differential equations. Then, since the computation of the latter solution may be affected by numerical errors that are particularly detrimental in the presence of unstable zero-dynamics, the above architecture is robustified by means of an additional hybrid feedback loop whose trajectories converge to a solution of the two-point boundary value problem. Once the latter scheme has been established and discussed, the extension to the case of output feedback is presented, firstly in the specially structured case in which the input vector field depends only on the measured output and then extended to the generic case. The theory is then corroborated by means of a physically motivated numerical example involving an inverted pendulum on a cart. Interestingly, it is also shown that the solution provided by the hybrid scheme above coincides with the limiting solution of a suitably defined cheap optimal control problem.

Leader-following almost output consensus for linear multi-agent systems with disturbance-affected unstable zero dynamics

Abstract In this paper, we formulate and investigate the leader-following almost output consensus problem for linear multi-agent systems with disturbance affected unstable zero dynamics. Conditions on the communication topology, agent dynamics and the way the disturbances affect the zero dynamics are established under which low-and-high gain based consensus protocols are designed. These protocols are shown to achieve leader-following almost output consensus, that is, output consensus of the system can be achieved to an arbitrary level of accuracy with the states remain bounded in the absence of the disturbances, and when the system is operating in output consensus within the desired accuracy, the L 2 -gain from the disturbances to the difference between each follower agent’s output with and without the disturbances from the same initial condition can be made arbitrarily small. A numerical example is shown to verify the theoretical results.

Robust indirect adaptive sliding model control for Wiener nonlinear systems

ABSTRACTWiener model structures have attracted considerable attentions due to their powerful modelling abilities to solve industrial problems. This paper considers a new class of Wiener model structure, which consists of a linear transfer function and a radial basis function (RBF) network. Based on this structure, the unstable zero-dynamics, the added disturbances and the approximation errors problems can be dealt with. Since the model parameters are assumed unknown, we develop a new indirect adaptive control scheme. It is a consensus that the certainty equivalence principle is the key that connects the adaptation algorithm with the control law. This paper explains that, for indirect adaptive control of Wiener model, this principle may not be applied directly. The main contribution of this paper is to address this issue by presenting a rigorous theoretical analysis. Representative examples including a simulated pH process control problem are studied to test the control performance. Comparison results indicate that the proposed controller has much wider applicability than some alternative methods, especially for its improved performance and smoother adaptation.

Robust and Adaptive Path-Following Control of an Underactuated Ship

The design of a vessel path-following control system based on a full, realistic, nonlinear model is considered. The control objective is to force a surface, course-unstable vessel to track a predefined geometric path. We study an underactuated ship characterized only by a surge control force and yaw control moment, typical of many supply vessels. The assumption is made that the ship’s model parameters are unknown, while significant external disturbances and unmodeled dynamics exist. Therefore, the design procedures make use of robust and adaptive control techniques. The controller synthesis uses adaptive output feedback linearization and $H_{\infty }$ optimal control techniques. In this way, the proposed control scheme assures position tracking despite various uncertainties. Because the considered design method leads to a nonminimum phase system, the problem of how to stabilize unstable zero dynamics arises. The presented simulations are based on a realistic ship model in terms of the structure and experimentally identified parameters. The simulations illustrate the effectiveness of the proposed algorithms.

Byrnes-Isidori-based dynamic sliding-mode control for nonminimum phase hypersonic vehicles

This paper deals with the dynamic sliding-mode (DSM) control problem for nonminimum phase hypersonic vehicles (HSVs). When the elevator is the only control surface available for the altitude dynamics, the HSV model exhibits unstable zero dynamics, preventing the application of standard inversion-based control techniques. To solve this problem, a DSM control method based on Byrnes-Isidori (B-I) normalized form is proposed, achieving asymptotic tracking of velocity and Flight-Path-Angel (FPA) while stabilizing internal dynamics. First, for the pitch dynamics with nonminimum phase behavior, external dynamics and internal dynamics are determined by coordinate transformation to convert the longitudinal model to B-I normalized form, based on which a criterion of nonminimum phase property is given by the stability analysis of internal dynamics. Then, a DSM control method is proposed for the FPA subsystem of nonminimum phase, which transforms the output tracking problem into stabilization problem of an augmented system consisting of internal dynamics and dynamic compensator, making closed-loop pole adjustable, and thus improves the tracking performance. The principle of parameters determination is proposed, which is proved to achieve the stability of the system on the sliding surface. Besides, nonlinear disturbance observer is utilized to compensate the error caused by dynamic inversion control. The proposed method is compared with approximate backstepping control and is shown to have superior tracking accuracy as well as robustness from the simulation results. This paper may also provide a beneficial guidance for control design of other complex systems of nonminimum phase.

A Dynamic Feedback Framework for Control of Time-Delay Nonlinear Systems With Unstable Zero Dynamics

This paper presents a dynamic feedback framework for the control of time-delay cascade systems with unstable zero dynamics. Both dynamic state and output feedback control strategies are studied. Recognizing the difficulty of controlling time-delay systems via static feedback, we develop a systematic method, by taking advantage of dynamic feedback, for the design of delay-free, dynamic state and output feedback compensators that achieve global state regulation with stability. The controlled plants under consideration not only cover time-delay nonlinear systems in the normal form, but also include a class of time-delay cascade systems beyond the normal form. In the case of state feedback, the zero dynamics are not required to be “minimum phase” but satisfy certain regularity conditions. In the output feedback case, appropriate conditions are characterized for the zero dynamics so that the existence of a dynamic output controller is ensured. A key feature of the proposed control strategies is the utilization of dynamic gains to counteract the effect of time-delay nonlinearities and unstable zero dynamics.

Periodic adaptive stabilization of rapidly time-varying linear systems

Adaptive control deals with systems that have unknown and/or time-varying parameters. Most techniques are proven for the case in which any time variation is slow, with results for systems with fast time variations limited to those for which the time variation is of a known form or for which the plant has stable zero dynamics. In this paper, a new adaptive controller design methodology is proposed in which the time variation can be rapid and the plant may have unstable zero dynamics. Under the structural assumptions that the plant is relative degree one and that the plant uncertainty is a single scalar variable, as well as some mild regularity assumptions, it is proven that the closed-loop system is exponentially stable under fast parameter variations with persistent jumps. The proposed controller is nonlinear and periodic, and in each period the parameter is estimated and an appropriate stabilizing control signal is applied.

A novel approach to stable zero dynamics of sampled-data models for nonlinear systems in backward triangle sample and hold case

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.

Discrete-Time Neural Network Approach for Tracking Control of Spherical Inverted Pendulum

In recent years, the tracking problem of spherical inverted pendulum (SIP) system with multi-input, multi-output nature and unstable zero dynamics has been well addressed based on continuous-time nonlinear output regulation (NOR) theory. For the convenience of digital implementation, this paper further investigates the approximate NOR problem of the SIP system in discrete-time framework. The key for solving the discrete-time NOR problem lies in how to solve a set of algebraic functional equations known as discrete regulator equations (DRE). Since the equations are very complicated, the accurate solution of the DRE can not be obtained. In this paper, we first show that the DRE associated with the SIP system are solvable by center manifold theorem and then use neural network approach to tackle with the tracking problem. Finally, we compare our method with polynomial approximation method.