Zero Dynamics Research Articles

Asymptotic Properties of Sampling Zero Dynamics for Nonlinear Systems in the Case of Time Delay and Relative Degree Two

It is well-known that stability of zero dynamics is often inevitable to the controller design. And most real world plants often involve a time delay. This paper investigates the zero dynamics, as the sampling period tends to zero, of a sampled-data model composed of a zero-order hold (ZOH), a continuous-time plant with a time delay and a sampler in cascade. We first present how an approximate sampled-data model can be obtained for the nonlinear system with relative degree two, and the local truncation error between the output of obtained model and the true system output is of order , where T is the sampling period and r is the relative degree. Furthermore, we also propose the additional zero dynamics in the sampling process, which are called the sampling zero dynamics, and the condition for assuring the stability of sampling zero dynamics for the desired model is derived. The results presented here generalize a well-known notion of sampling zero dynamics from the linear case to nonlinear systems.

Funnel control for electrical circuits

AbstractWe study the zero dynamics and funnel control for linear passive electrical circuits. We show that asymptotic stability of the zero dynamics can be characterized by criteria on the circuit topology. Thereafter we consider the output regulation problem for electrical circuits by funnel control. We show that for circuits with asymptotically stable zero dynamics, the funnel controller achieves tracking of a class of reference signals within a pre‐specified funnel. This result can be relaxed to the case of non‐autonomous zero dynamics by requiring the reference trajectory to evolve within a certain subspace. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Asymptotic properties of zero dynamics for nonlinear discretized systems with time delay via Taylor method

Most real world plants often operate in continuous-time case and involve time delay. These models are typically described by ordinary differential equations. However, to utilize and analyze these data, the control signals must first be discretized. In this paper, a new discretization method for obtaining an approximate sampled-data model of a nonlinear continuous-time system with time delay is proposed. The presented approach is based on the Taylor method and zero-order hold assumption, which can be used to approximate the system output and its derivatives in such a way as to obtain a local truncation error between the output of the resulting sampled-data model and true continuous-time system output. More importantly, on the basis of this discretized representation, we explicitly derive the mathematical structure of nonlinear discrete-time zero dynamics in the case of time delay. The main contribution is to analyze the stability of sampling zero dynamics for the proposed model with time delay. The ideas presented here generalize the well-known results from the linear system to nonlinear case.

Tracking Control for Nonlinear Minimum Phase Systems

In this paper, we proposed the robust output tracking control scheme for a class of MIMO nonlinear dynamical systems using output feedback linearization method. Because the states of zero dynamics are not accessible, we consider these states as bounded uncertainties(or disturbances) under the assumption that zero dynamics is stable(or nonlinear minimum phase) and also we have no priori knowledge concerning the magnitude of these uncertainties. A second order sliding mode control method is used to establish the desired output tracking controller. It is shown that the outputs of the closed-loop system asymptotically track given output trajectories despite the uncertainties while maintaining the boundedness of all signals inside the loop. From the computer simulation results for the numerical example, we prove the validity of the proposed control algorithm.

Problem of tracking in linear systems with parametric uncertainties under unstable zero dynamics

Proposed was a solution to the problem of tracking in linear SISO systems with unstable zero dynamics in the conditions of parametric uncertainty of the models of plant and reference generator under incomplete information about the state vectors. The approach developed employs the methods of the sliding mode theory in the problems of real-time feedback design, observation, and parameter identification.

Zero dynamics of sampled-data models for nonlinear multivariable systems in fractional-order hold case

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).

Zero dynamics and funnel control for linear electrical circuits

We consider electrical circuits containing linear resistances, capacitances and inductances. The circuits can be described by differential-algebraic input–output systems, where the input consists of voltages of voltage sources and currents of current sources and the output consists of currents of voltage sources and voltages of current sources. We generalize a characterization of asymptotic stability of the circuit and give sufficient topological criteria for its invariant zeros being located in the open left half-plane. We show that asymptotic stability of the zero dynamics can be characterized by means of the interconnectivity of the circuit and that it implies that the circuit is high-gain stabilizable with any positive high-gain factor. Thereafter we consider the output regulation problem for electrical circuits by funnel control. We show that for circuits with asymptotically stable zero dynamics, the funnel controller achieves tracking of a class of reference signals within a pre-specified funnel; this means in particular that the transient behavior of the output error can be prescribed and the funnel controller does neither incorporate any internal model for the reference signals nor any identification mechanism, it is simple in its design. The results are illustrated by a simulation of a discretized transmission line.

Zero Dynamics and Stabilization for Analytic Linear Systems

We combine different system theoretic concepts to solve the feedback stabilization problem for linear time-varying systems with real analytic coefficients. The algebraic concept of the skew polynomial ring with meromorphic coefficients and the geometric concept of (A,B)-invariant time-varying subspaces are invoked. They are exploited for a description of the zero dynamics and to derive the zero dynamics form. The latter is essential for stabilization by state feedback: the subsystem describing the zero dynamics are decoupled from the remaining system which is controllable and observable. The zero dynamics form requires an assumption close to autonomous zero dynamics; this in some sense resembles the Byrnes-Isidori form for systems with strict relative degree. Some aspects of the latter are also proved. Finally, using the zero dynamics form, we show for square systems with autonomous zero dynamics that there exists a linear state feedback such that the Lyapunov exponent of the closed-loop system equals the Lyapunov exponent of the zero dynamics; some boundedness conditions are required, too. If the zero dynamics are exponentially stable this implies that the system can be exponentially stabilized. These results are to some extent also new for time-invariant systems.

The effects of foot geometric properties on the gait of planar bipeds walking under HZD-based control

It has been hypothesized by many that foot design can influence gait. This idea was investigated in both simulation and hardware for the five-link, planar biped ERNIE controlled under the Hybrid Zero Dynamics paradigm. The effects of walking speed, foot radius, and foot center of curvature location on gait efficiency and kinematics were investigated in a full factorial study of gaits optimized using a work-based objective function. In most cases, the simulation correctly predicted the trends observed in hardware, indicating that simulation can be used for foot design. As expected, increasing walking speed decreased the energetic efficiency. The dominant effect of speed on joint kinematics was to alter the timing of the peak hip flexion. Increasing foot radius up to the length of the shank improved the energetic efficiency and increased the range of motion of the hip and knee joints. Shifting the foot center of curvature location forward altered the energetic efficiency in a manner that interacted with changes in foot radius. The energetically optimal foot center of curvature location was coincident with the shank for a large foot radius and shifted far in front of the shank for a small foot radius. In all cases, the forward shift increased the range of motion of the hip and knee joints. Therefore, a robot designer can achieve similar energetic benefits across a range of speeds with either a larger radius foot or a smaller radius foot whose center of curvature is located forward of the shank.

Decoupling control and zero dynamics stabilization for shunt hybrid active power filter

An inverse system method based optimal control strategy was proposed for the shunt hybrid active power filter (SHAPF) to enhance its harmonic elimination performance. Based on the inverse system method, the d-axis and q-axis current dynamics of the SHAPF system were decoupled and linearized into two pseudolinear subsystems. Then, an optimal feedback controller was designed for the pseudolinear system, and the stability condition of the resulting zero dynamics was presented. Under the control strategy, the current dynamics can asymptotically converge to their reference states and the zero dynamics can be bounded. Simulation results show that the proposed control strategy is robust against load variations and system parameter mismatches, its steady-state performance is better than that of the traditional linear control strategy.

Rapidly Exponentially Stabilizing Control Lyapunov Functions and Hybrid Zero Dynamics

This paper addresses the problem of exponentially stabilizing periodic orbits in a special class of hybrid models-systems with impulse effects-through control Lyapunov functions. The periodic orbit is assumed to lie in a C1 submanifold Z that is contained in the zero set of an output function and is invariant under both the continuous and discrete dynamics; the associated restriction dynamics are termed the hybrid zero dynamics. The orbit is furthermore assumed to be exponentially stable within the hybrid zero dynamics. Prior results on the stabilization of such periodic orbits with respect to the full-order dynamics of the system with impulse effects have relied on input-output linearization of the dynamics transverse to the zero dynamics manifold. The principal result of this paper demonstrates that a variant of control Lyapunov functions that enforce rapid exponential convergence to the zero dynamics surface, Z, can be used to achieve exponential stability of the periodic orbit in the full-order dynamics, thereby significantly extending the class of stabilizing controllers. The main result is illustrated on a hybrid model of a bipedal walking robot through simulations and is utilized to experimentally achieve bipedal locomotion via control Lyapunov functions.

On Passivity of a Class of Discrete-Time Switched Nonlinear Systems

This paper analyzes the passivity and feedback passivity of discrete-time-switched nonlinear systems with passive and nonpassive modes that are affine in the control input. When a nonpassive mode is active, the increase in storage function is not necessarily bounded by the energy supplied to the switched system at every time step. Therefore, a switched system with at least one nonpassive mode is defined to be nonpassive in the classical passivity theory. In this paper, we propose a framework to analyze the passivity of such switched systems in a more general sense. We consider switched nonlinear systems which are affine in the control input and may consist of passive, feedback passive modes, and modes which cannot be rendered passive using feedback. In the proposed framework, we prove that a switched nonlinear system is locally feedback passive if and only if its zero dynamics are locally passive. A lower bound on the ratio of total activation time between (feedback) passive and nonfeedback passive modes is obtained to guarantee passive zero dynamics. Finally, we prove that two important properties of classical passivity still hold for the proposed passivity definition, that is: 1) output feedback control can be used to stabilize the switched system, and 2) parallel and negative feedback interconnections of two such passive systems are also passive.

High-Sampling Rate Dynamic Inversion—Filter Realization and Applications in Digital Control

This paper presents a stable inversion of nonminimum phase systems with highly efficient computation for high-sampling rate applications. The stable filter that inverts the dynamics of a nonminimum system is based on cascading a stable pole-zero cancellation infinite impulse response (IIR) filter with a high-order finite impulse response (FIR) filter which inverts the unstable zero dynamics. The high-order FIR filter is realized based on efficient IIR filter implementation first introduced by Powell and Chau then later modified by Kurosu. As a demonstrative example, the inversion filters are applied to feedforward tracking and repetitive control algorithms and realized by a field programmable gate array. The controllers are implemented at 100-kHz sampling rate to control the motion of a 4 degrees-of-freedom magnetically levitated shaft in experiment.

Discrepancy based control of particulate processes

This article deals with a new approach to particulate process control. The model system under investigation, the continuous fluidized bed spray granulation with external product classification, is described by a nonlinear partial integro-differential equation, the population balance equation for the particle size distribution. This process exhibits interesting dynamical behavior, i.e. a change of the stability behavior and the occurrence of limit cycles. In addition, the zero dynamics with respect to moment measurements frequently used in practice are unstable in certain parameter regions. In order to stabilize these types of systems in this contribution the use of a generalized distance measure, the discrepancy, is proposed. Applying, the associated stability theory, i.e. stability theory with respect to two discrepancies, a stabilizing control law can be derived. One of the main advantages of the proposed discrepancy based control method is that no model reduction is required.

Zero Dynamics Analysis for Inverse Decoupling Control of Asynchronous Traction Motor

Considering the problem for inverse system method in EMU AC induction traction motor linear decoupling, the zero dynamics subsystem will be separated from the original dynamic system through coordinate transformation. Firstly, a getting method for zero dynamics of the multiple input multiple output nonlinear system is discussed when γ＜ n. Second, the zero dynamics analysis for five order nonlinear model of asynchronous traction motor which base on the stationary coordinate system is given by using inverse decoupling method. The analysis results show that if the stability of the zero dynamics can be ensured, then the entire linearization of original nonlinear system is not necessary, need only partial linearization which effect on the external dynamic portion. The inverse decoupling process of asynchronous traction motor can be simplified by this conclusion. DOI : http://dx.doi.org/10.11591/telkomnika.v12i1.3354

Improvement of the Asymptotic Properties of Zero Dynamics for Sampled-Data Systems in the Case of a Time Delay

It is well known that the existence of unstable zero dynamics is recognized as a major barrier in many control systems, and deeply limits the achievable control performance. When a continuous-time system with relative degree greater than or equal to three is discretized using a zero-order hold (ZOH), at least one of the zero dynamics of the resulting sampled-data model is obviously unstable for sufficiently small sampling periods, irrespective of whether they involve time delay or not. Thus, attention is here focused on continuous-time systems with time delay and relative degree two. This paper analyzes the asymptotic behavior of zero dynamics for the sampled-data models corresponding to the continuous-time systems mentioned above, and further gives an approximate expression of the zero dynamics in the form of a power series expansion up to the third order term of sampling period. Meanwhile, the stability of the zero dynamics is discussed for sufficiently small sampling periods and a new stability condition is also derived. The ideas presented here generalize well-known results from the delay-free control system to time-delay case.

Minimalphasigkeit – keine relevante Eigenschaft für die Regelungstechnik!

Zusammenfassung Ursprünglich wurde die Minimalphasigkeit von H. W. Bode eingeführt, um den eineindeutigen Zusammenhang zwischen Amplitude und Phase des Frequenzgangs zu definieren. Beim Entwurf von digitalen Filtern wird eine andere Definition der Minimalphasigkeit verwendet und ein asymptotisch stabiles Filter als minimalphasig bezeichnet, wenn auch das inverse Filter asymptotisch stabil ist. Schließlich werden von C. I. Byrnes und A. Isidori Systeme mit einer asymptotisch stabilen Nulldynamik als minimalphasig bezeichnet. Wegen der nicht konsistenten Definitionen wird dafür plädiert, den Minimalphasigkeits-Begriff in der Regelungstechnik zu vermeiden und die Stabilität von Filtern und der Nulldynamik mit den wohl definierten Kriterien von Hurwitz oder Ljapunow zu beschreiben.

A New Dual Lagrangian Model and Input/Output Feedback Linearization Control of 3-Phase/Level NPC Voltage-Source Rectifier

An input/output feedback linearization control strategy is developed for control of three-level three-phase neutral-point-clamped rectifier using its dual Lagrangian modeling. The load current of the rectifier can be given in two forms: (i) the load current involving the current of capacitor C1 (ii) the load current involving the current of capacitor C2. Applying the obtained load current to the Euler-Lagrange parameters of the rectifier, two nonlinear models of the system are derived. The models are based on the superposition law of the load current and the Euler-Lagrange description of the rectifier. Also, the two power-balance equations between the input and output sides are obtained by considering the effects of two load currents separately. Then, the two nonlinear models and power-balance equations are linearized using input-output feedback linearization, and the state feedback law controlled inputs are completed by pole placement. With the definition of outputs of the feedback linearization law, the zero dynamics of the system are avoided and a fast output voltages control scheme is designed. Finally, some integrators are added to robust control against parasitic elements. The proposed nonlinear controller is compared to the PI controller and the MATLAB/SIMULINK results verify the proposed control strategy.

Tracking Problem for Linear Systems with Parametric Uncertainties and Unstable Zero Dynamics

The paper deals with tracking problem for single-input single-output linear systems with unstable zero dynamics under parametric and signal uncertainty of control plant and reference model within the framework of the sliding mode techniques for feedback design, state observation and parameters identification in real time.

Graph-theoretic characterisations of zeros for the input–output dynamics of complex network processes

We examine linear network dynamics in which an input can be applied at only one network component and measurements can only be made at one (in general different) component. The infinite-zero and finite-zero structures of these dynamics are characterised explicitly in terms of the network's graph matrix, using the special coordinate basis for linear system. These algebraic characterisations are then used to identify relationships between features of the network's graph topology and the state matrix of its finite zero dynamics.