Improvement of stability of zeros in discrete-time multivariable systems using fractional-order hold

This paper is concerned with the stability of zeros of the discrete-time multivariable systems composed of a fractional-order hold (FROH), a continuous -time plant and a sampler in cascade. The properties of the limiting zeros are studied and a condition for obtaining stable zeros for sufficiently small sampling periods is derived. An approximate fractional-order hold (AFROH) is proposed as an implementation of FROH. It is shown that the AFROH as well as the FROH can locate the zeros of discrete-time multivariable systems inside the stable region and improve stability properties when the zero-order hold (ZOH) cannot. Experimental results of adaptive control with AFROH demonstrate a significant improvement of control performance compared with the use of a ZOH.     

On the stabilization of discrete system zeros

The stabilization of discrete system zeros is commonly achieved by allowing the input pattern to deviate from zero-order hold (ZOH). In this paper, we show that this design technique may not be practical at small sampling times when the continuous system has unstable zero(s). Deviation of the input pattern from ZOH may also cause undesirable fluctuation in the inter-sampling-time output. In order to reduce this effect, we present a method which uses linear quadratic optimization for the determination of suitable zero locations.     

Two-delay robust digital control and its applications-avoiding theproblem on unstable limiting zeros

In the design of linear control systems, the existence of unstable zeros makes it difficult to construct many control systems. When the usual digital control scheme is used, unstable zeros appear in the discrete-time model due to the existence of limiting zeros even though the continuous-time plant is minimal phase. To avoid this unstable zero problem, two new digital control schemes, called two-delay output control and two-delay input control, are proposed. These control systems are proved to have no finite zeros. Asymptotic inverse systems, pole assignable unknown input observers, and output feedback controllers having the LTR (loop transfer recovery) property are developed using the two-delay output control. Zero assignable control systems and model matching control systems are developed using the two-delay input control     

Improving the Stability Behavior of Limiting Zeros for Multivariable Systems Based on Multirate Sampling

It is well-known that existence of unstable sampling zeros is recognized as a major barrier in many control problems, and stability of sampling zeros, in general, depends on the type of hold circuit used to generate the continuous-time system input. This paper is concerned with stability of limiting zeros, as the sampling period tends to zero, of multivariable sampled-data models composed of a generalized sample hold function (GSHF), a continuous-time plant with the relative degrees being two and three, and a sampler in cascade. In particular, the main focus of the paper is how to preserve the stability of limiting zeros when at least one of the relative degrees of a multivariable system is more than two. In this case, the asymptotic properties of the limiting zeros on the basis of normal form representation of continuous-time systems are analyzed and approximate expressions for their stability are discussed as power series expansions with respect to a sufficiently small sampling period. More importantly, unstable sampling zeros of the sampled-data models mentioned above can be avoided successfully through the contribution of this paper, whereas a zero-order hold (ZOH) or a fractional-order hold (FROH) fails to do so. It is a further extension of previous results for single-input single-output cases to multivariable systems.     

Asymptotic behaviours and stability of zero dynamics in nonlinear discrete-time systems using Taylor approach and multirate input

It is well known that the existence of unstable sampled zero dynamics is recognised as a major barrier in many control problems. When the usual digital control with zero-order hold (ZOH) or fractional-order hold (FROH) input is used, unstable sampled zero dynamics inevitably appear in the discrete-time model even though the continuous-time system with relative degree more than or equal to three is of minimum phase. In this paper, we show how an approximate sampled-data model can be obtained for nonlinear systems by the use of multirate input and hold such as a generalised sample hold function (GSHF) in order that discrete zero dynamics of the resulting model can be arbitrarily placed. Furthermore, the properties of sampled zero dynamics are studied and the conditions for ensuring the stability of sampling zero dynamics of the desired model are derived. The results presented here generalise well-known notion of sampling zero dynamics from the linear case to nonlinear systems, and GSHF can provide some advantages over ZOH or FROH in terms of stability of discrete system zero dynamics.     

The relationship between real poles and real zeros in SISO sampleddata systems

A relationship between real poles and real zeros of SISO (single-input, single-output) sampled data systems is presented, which is independent of the sampling period. The relation is stated in terms of a parity property involving the number of real zeros between any two real poles. This property allows the investigation of conditions for the preservation of stable or unstable zeros under sampling. A sufficient condition for stability of a sampled system with a continuous-time plant having all real zeros, and a sufficient condition for instability of a two-sampled system with an unstable plant is developed     

Compensation of uncertain continuous systems by using the internal model control principle

In this paper the design of compensators for uncertain continuous plants is investigated. The standard derived compensators are based on the application of the internal model control (IMC) method. The required a priori knowledge on the plant is rather weak, namely, an upper bound of the plant relative order, the numbers of the strictly unstable and critically unstable plant poles being integrators and upper and lower bounds of the amplitude-versus-frequency plot over the low frequency band in the case of minimum-phase open-loop systems. If the open-loop system has unstable zeros and/or poles then the above bounds are required to be known for a modified magnitude plot which substitutes the unstable zeros (poles) by stable poles (zeros) which are their complex-conjugate reflections on the left-hand plane. An absolute upper bound of the open-loop phase plot is obtained on a finite frequency interval which allows the closed-loop system to guarantee a prescribed relative stability in many practical situations. The method is dependent on the alternative design of phase lead/lag classical compensators and to indirect adaptive control situations where the adaptive identifier is used for the parametrization of the adaptive controller.     

Digital controller design for multivariable systems with structural closed-loop performance specifications

The problem of the direct design of the closed-loop transfer function matrix is addressed for multivariable discrete systems. The limitations imposed by unstable zeros, time delays and the structure associated with these are quantified. A design procedure is formulated that provides the designer with quantitative measures for evaluating the tradeoffs between different closed-loop interaction structures and durations. The problem of intersample rippling is also considered. The procedure requires only linear-algebra operations, includes the eventual construction of the feedback controller in state space, and is presented in a way that allows its straightforward computer implementation.     

Imperfect model reference adaptive control: deadbeat output error approach

It is well known that while the perfect model matching condition (i.e. unstable plant zeros must be zeros of the reference model) is not met, the model reference adaptive control cannot easily be implemented. In adaptive control systems, since the plant is assumed to be unknown previously, it is a difficult task to choose an adequate reference model such that the perfect model matching condition is guaranteed in every adaptive step. In this paper, a new design algorithm for model reference adaptive control systems is proposed to synthesize an adaptive controller such that the error between the reference model output and the plant output can vanish in a deadbeat manner. In this situation, the stringent matching condition can be relaxed in every adaptive step, so our design algorithm is also suitable for unstable or non-minimum phase systems. Several simulation results are presented to illustrate the good behaviour of our design algorithm.     

A note on the Youla-Bongiorno-Lu condition

Youla, Bongiorno and Lu have presented a condition for the stabilizability using stable controllers of an unstable plant. We show this condition may be reformulated in terms of Cauchy indices, allowing its checking in a finite number of rational operations, should the plant not be given in factored form, i.e. should its poles and zeros be unknown. The results applies to discrete and continuous-time, and scalar and multivariable plants.     

Stabilization for a class of nonlinear systems: A fuzzy logic approach

In this paper, the problem of stabilization for the class of continuous time nonlinear systems which are discretized in closed form is addressed. By using the Takagi–Sugeno model approach, a discrete controller capable of stabilizing the discrete Takagi–Sugeno model and the continuous model as well, is obtained. This scheme allows using a digital controller for stabilizing an analog plant.     

Virtual Reference Feedback Tuning for non-minimum phase plants

Model reference control design methods fail when the plant has one or more non-minimum phase zeros that are not included in the reference model, leading possibly to an unstable closed loop. This is a very serious problem for data-based control design methods, where the plant is typically unknown. In this paper, we extend the Virtual Reference Feedback Tuning method to non-minimum phase plants. This extension is based on the idea proposed in Lecchini and Gevers (2002) for Iterative Feedback Tuning. We present a simple two-step procedure that can cope with the situation where the unknown plant may or may not have non-minimum phase zeros.     

AN ALGORITHM FOR DIGITAL IMPLEMENTATION OF TRAJECTORY TRACKING CONTROLLERS

In this work, we study the problem of the digital implementation of continuous‐time solutions to the trajectory tracking control problem for nonlinear systems. We exhibit an example that shows that, in general, no proper behavior of the tracking error can be expected when these solutions are implemented via the Sampling and Zero Order Hold technique. Inspired in some constructions developed in the context of Positional Games, we present a sampled‐data controller that, based on a continuous‐time solution of the tracking problem, assures semiglobal practical stability of the tracking error, with final error arbitrarily small if we choose a suitable sampling period. The controller is robust with respect to small external disturbances and small errors in the measurements.     

Limiting zeros of decouplable MIMO systems

For sufficiently rapid zero-order hold (ZOH) sampling, it is known that the zeros of single-input/single-output discrete-time systems expressed in the forward shift operator converge to values determined only by the degrees of the finite and infinite zeros of the underlying continuous-time system. In this paper, it is shown how this result can be generalized to multi-input/multi-output (MIMO) systems decouplable by static-state feedback (equivalently, having a diagonal interactor matrix). In the fast sampling limit, the authors show how invariant and infinite zero structure is mapped under ZOH sampling for discrete-time systems expressed in either the delta (forward difference) or forward shift operators     

具有随机延时的网络控制系统的均方可镇定性分析

针对网络系统的可镇定性问题,研究整数步随机延时离散时间线性系统的均方可镇定性.利用Youla参数化与内外分解方法,结合均方小增益定理得到系统输出反馈均方可镇定的充分必要条件.该条件明确给出系统可镇定性与被控对象特性(不稳定极点、非最小相位零点、相对阶)和信道特性(频域信噪比函数)的关系,其中频域信噪比函数在被控对象不稳定极点的取值对可镇定影响甚大.利用仿真算例量化被控对象的非最小相位零点及相对阶对可镇定性的影响,验证可镇定性条件的正确性.     

Conditions for stable zeros of sampled systems

A criterion for there being no unstable discrete zeros is derived in terms of coefficients of a continuous-time transfer function and of a sampling period when a continuous-time, stable, and strictly proper plant is sampled. It is also shown that the shortest sampling period ensuring minimum-phase behavior can be calculated from a simple equation     

Stabilization of discrete system zeros: An improved design

Stabilization of discrete system zeros has been previously achieved by providing a fixed pattern of inter-sampling time input which deviates from zero-order-hold. However, a fixed non-ZOH input creates several problems: indefinite fluctuation of the system output which causes a deterioration in performance, highly fluctuating input which accelerates actuator fatigue, and counter-acting control which increases demand for input energy. In this work, zero placement is achieved with a temporary non-ZOH input. This temporary input pattern, which converges asymptotically toward a ZOH input pattern, removes the problems associated with a fixed non-ZOH input design.     

Signal-to-noise ratio fundamental limitations in the discrete-time domain

Fundamental limitations in feedback control are a well established area of research. In recent years it has been extended to the study of limitations imposed by the consideration of a communication channel in the control loop. Previous results characterised these limitations in terms of a minimal data transmission rate necessary for stabilisation. In this paper a signal-to-noise ratio (SNR) approach is used to obtain a tight condition for the linear time invariant output feedback stabilisation of a discrete-time, single-input-single-output (SISO) unstable, non-minimum phase (NMP) plant with arbitrary relative degree over an additive Gaussian coloured noise (ACGN) communication channel with memory. The obtained result gives a guideline in estimating the severity of the fundamental SNR limitation imposed by the plant unstable poles, NMP zeros and relative degree as well as the channel NMP zeros, bandwidth, and noise colouring. We then characterise the output feedback sensitivity function for the infimal SNR solution and follow up by quantifying the extra SNR imposed by suboptimal solutions (for example due to plant modelling errors).     

Two-degree-of-freedom H-infinity control of combustion in diesel engine using a discrete dynamics model

This paper proposes an H-infinity combustion control method for diesel engines. The plant model is the discrete dynamics model developed by Yasuda et al., which is implementable on a real engine control unit. We introduce a two-degree-of-freedom control scheme with a feedback controller and a feedforward controller. This scheme achieves both good feedback properties, such as disturbance suppression and robust stability, and a good transient response. The feedforward controller is designed by taking the inverse of the static plant model, and the feedback controller is designed by the H-infinity control method, which reduces the effect of the trubocharger lag. The effectiveness of the proposed method is evaluated in simulations using the nonlinear discrete dynamics model.     

Feedback, minimax sensitivity, and optimal robustness

In this paper, we look for feedbacks that minimize the sensitivity function of a linear single-variable feedback system represented by its frequency responses. Sensitivity to disturbances and robustness under plant perturbations are measured in a weightedH^{infty}norm. In an earlier paper, Zames proposed an approach to feedback design involving the measurement of sensitivity by "multiplicative seminorms," which have certain advantages over the widely used quadratic norm in problems where there is plant uncertainty, or where signal power-spectra are not fixed, but belong to sets. The problem was studied in a general setting, and someH^{infty}examples were solved. Here, a detailed study of the single-variable case is undertaken. The results are extended to unstable plants, and explicit formulas for the general situation of a finite number of right half-plane (RHP) plant zeros or poles are provided. TheQor "approximate-inverse" parametrization of feedbacks that maintain closed-loop stability is extended to the ease of unstable plants. TheH^{infty}and Wiener-Hopf approaches are compared.