Design of stabilized digital inverse systems

As is well known, due to the existence of unstable limiting zeros and/or the minimum phase property of continuous-time systems, there are many cases that the discretized systems are of non-minimum phase. In this case, it is impossible to construct a stable digital inverse by usual methods. However, in this paper, we will show that if the inter-sample output is used in the design, one can always construct a stable digital inverse system.     

Stabilization mechanism underlying passive dynamic running

In this study, we discuss the stabilization mechanism underlying passive dynamic running (PDR), focusing on the feedback structure in an analytical Poincaré map. To this end, we derived a linearized analytical Poincaré map for PDR and analyzed its stability in terms of linear control theory. Through our theoretical analysis, we found that an ‘implicit two-delay feedback structure,’ which can be seen as a certain type of two-delay input digital feedback control developed as an artificial control structure for digital control, is an inherent stabilization mechanism in PDR appearing from a minimalistic biped model with elastic elements. To the best of our knowledge, this has yet to be addressed and studied. Our results shed new light on the principles underlying bipedal locomotion as ‘morphological computation.’     

Feedforward digital tracking controllers for motion control applications

This paper reviews digital tracking control algorithms for motion control applications. In tracking control, the control objective is to steer the control object along the time-varying desired output. Two design approaches are presented for the case where the desired signal is known in advance, i.e. previewable. One approach is based on the mathematical inverse of a closed-loop system consisting of a controlled plant and a feedback controller. If the mathematical inverse is asymptotically stable, i.e. the closed-loop system does not possess zeros outside the unit circle (unstable zeros), it is an ideal feedforward controller for achieving perfect tracking under the preview assumption. For closed-loop systems with unstable zeros, a cancellation technique for the phase shift induced by unstable zeros is introduced. Another approach is based on linear quadratic optimal control and is known as finite optimal preview control. In this approach, the feedback controller and feedforward controller are determined simultaneously by minimizing a quadratic performance index which involves a tracking error term and a term related to the control effort. Applications of these tracking control algorithms to mechanical systems control are described.     

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.     

Simultaneous Stabilization and Constant Reference Tracking of LTI,MIMO Systems

It is shown that any finite number of plants that belong to certain classes of multi‐input multi‐output systems with no zeros in the region of instability can be simultaneously stabilized using linear, time‐invariant integral‐action controllers. These plants may be stable or unstable and their poles are not restricted; they may also have any number of zeros in the stable region of the complex plane. The classes of systems under consideration include plants with blocking or transmission zeros at infinity. The common controller achieves asymptotic tracking of step‐input references with zero steady‐state error and has a low order transfer‐function. Systematic synthesis methods are presented, and a parametrization of all simultaneously stabilizing controllers with integral‐action is also provided.     

Inversion of sampled-data system approximates the continuous-time counterpart in a noncausal framework

Although correspondence between the poles of a continuous-time and sampled-data system with a piecewise constant input is simple and desirable from the stability viewpoint, the relationship between zeros is intricate. Inversion of a sampled-data system is mostly unstable irrespective of the stability of the continuous-time counterpart. This makes it difficult to apply inversion-based control techniques such as perfect tracking, transient response shaping or iterative learning control to sampled-data systems. Although recently developed noncausal inversion techniques help us to circumvent unboundedness of the inversion caused by unstable zeros, whether the inversion of sampled-data systems approximates the continuous-time counterpart or not as the sample period is shortened is still to be determined. This article gives a positive conclusion to this problem.     

一种新的非最小相位系统的控制方法

非最小相位系统的控制中，需要抑制由不稳定零点引起的负调并同时缩短系统的调节时间．针对非最小相位系统负调与调节时间的相互影响及负调与不稳定零点的相互关系，提出将控制过程分为抑制负调阶段和跟踪输入阶段，并适时改变系统不稳定零点数，用遗传算法统一优化各阶段的控制器参数．仿真结果表明该控制方法大大减弱了负调，并同时缩短了调节时间，达到了良好的控制效果．     

Discrete-time output feedback sliding-mode control design for uncertain systems using linear matrix inequalities

An output feedback-based sliding-mode control design methodology for discrete-time systems is considered in this article. In previous work, it has been shown that by identifying a minimal set of current and past outputs, an augmented system can be obtained which permits the design of a sliding surface based upon output information only, if the invariant zeros of this augmented system are stable. In this work, a procedure for realising discrete-time controllers via a particular set of extended outputs is presented for non-square systems with uncertainties. This method is applicable when unstable invariant zeros are present in the original system. The conditions for existence of a sliding manifold guaranteeing a stable sliding motion are given. A procedure to obtain a Lyapunov matrix, which simultaneously satisfies both a Riccati inequality and a structural constraint, is used to formulate the corresponding control to solve the reachability problem. A numerical method using linear matrix inequalities is suggested to obtain the Lyapunov matrix. Finally, the design approach given in this article is applied to an aircraft problem and the use of the method as a reconfigurable control strategy in the presence of sensor failure is demonstrated.     

Robust digital control of a DC–DC buck converter with low frequency samplings

A robust DC?CDC converter which can covers extensive load change and also input voltage changes with one controller is needed. Then the demand to suppress output voltage change becomes still severer. We propose an approximate 2-degree-of-freedom (2DOF) digital controller which realized start-up response and dynamic load response independently. The controller makes a control bandwidth wider, and at the same time makes variations of the output voltage small at sudden changes of a load and an input voltage. In this paper, a new approximate 2DOF digital control system with additional zeros is proposed. Using the additional zeros, the second-order differential transfer characteristics between equivalent disturbances and a output voltage are realized. Therefore, the new controller makes variations of the output voltage smaller and the sudden changes of the load and the input voltage. This controller is actually implement on a DSP and is connected to the DC?CDC converter. Experimental results demonstrate that this type of digital controller can satisfy given severe specifications with low frequency sampling.     

多采样率数字控制系统的零点与无纹波渐近跟踪

采用函数空间的方法,利用先后两次的提升技术,构造出多采样率数字控制系统的函数空间模型和相应的脉冲传递函数矩阵模型.在此基础上,给出了多采样率数字控制系统零点和零方向的定义及其计算方法.并利用零点和零方向的概念,得出了多采样率数字控制系统实现包括采样点上和采样点之间的整个连续时间区间内无纹波渐近跟踪的条件.     

Digital sliding mode controller design for multiple time-delay continuous-time transfer function matrices with a long input–output delay

This paper extends the dominant eigenvector-based sliding mode control (SMC) design methodology, which was originally developed for delay-free continuous-time processes with known parameters, to the case of multiple time-delay continuous-time processes with known/unknown parameters. In addition, this paper presents a new prediction-based Chebyshev quadrature digital redesign methodology for indirect design of the digital counterpart of the analog sliding mode controller (ASMC) for multiple time-delay continuous-time transfer function matrices with either a long input delay or a long output delay. An approximated discrete-time model and its corresponding continuous-time model are constructed for multiple time-delay continuous-time stable/unstable dynamical processes with known/unknown parameters, using first the conventional observer/Kalman filter identification (OKID) method. Then, an optimal ASMC is developed using the linear quadratic regulator (LQR) approach, in which the corresponding sliding surface is designed using the user-specified eigenvectors and the scalar sign function. For digital implementation of the proposed non-augmented low-dimensional ASMC, a digital counterpart is designed based on the existing prediction-based digital redesign method and the newly developed prediction-based Chebyshev quadrature digital redesign method. Finally, a non-augmented low dimensional digital observer with a long input or output dead time is constructed for the implementation of the digitally redesigned sliding mode controller, to improve the performances of multiple time-delay dynamical processes. The effectiveness of the proposed method has been verified by means of two illustrative examples.     

An approximated scalar sign function approach to optimal anti-windup digital controller design for continuous-time nonlinear systems with input constraints

This article presents an approximated scalar sign function-based digital design methodology to develop an optimal anti-windup digital controller for analogue nonlinear systems with input constraints. The approximated scalar sign function, a mathematically smooth nonlinear function, is utilised to represent the constrained input functions, which are often expressed by mathematically non-smooth nonlinear functions. Then, an optimal linearisation technique is applied to the resulting nonlinear system (with smooth nonlinear input functions) for finding an optimal linear model, which has the exact dynamics of the original nonlinear system at the operating point of interest. This optimal linear model is used to design an optimal anti-windup LQR, and an iterative procedure is developed to systematically adjust the weighting matrices in the performance index as the actuator saturation occurs. Hence, the designed optimal anti-windup controller would lie within the desired saturation range. In addition, the designed optimal analogue controller is digitally implemented using the prediction-based digital redesign technique for the effective digital control of stable and unstable multivariable nonlinear systems with input constraints.     

Adaptive tracking control of MIMO nonlinear nonminimum phase system with unknown input nonlinearity

This paper proposes a nonlinear adaptive control for output tracking of multi‐input multi‐output nonlinear nonminimum phase system with input nonlinearity. The parameters of the input nonlinearity are assumed to be unknown. This problem is challenging, not only because of the unstable internal dynamics of nonminimum phase system, but also the existence of the unknown input nonlinearity. The partially linearized model of the original system is obtained through input/output linearization, and a states tracking model is constructed based on the computed ideal internal dynamics. A nonlinear adaptive controller, which can guarantee the bounded of output tracking error in the existence of unknown input nonlinearity, is proposed. Finally, a numerical simulation on vertical takeoff and landing aircraft is given to show the effectiveness of the proposed control methods.     

Digital inverse model control using Generalised holds with extensions to the adaptive case

In this paper, a digital implementation of an inverse-model based control scheme is proposed using Generalised Sampling and Hold Functions. The implementation of the controller using this kind of holds allows overcoming the difficulties related to the presence of unstable zeros in the continuous-time model and the usual appearance of unstable discretisation zeros in the discrete model when a ZOH is applied. The Generalised Sampling and Hold Functions allows obtaining a discrete model of the plant with all its zeros stable which allows realizing an exact inverse model of the plant in comparison to the use of a classical ZOH which only allows, in general, an approximate inversion of the plant. The proposed approach is then extended to the adaptive case where the stability and tracking properties of the general scheme are fully proved. Simulation examples showing the scope and application of the method are also presented.     

A generalised optimal linear quadratic tracker with universal applications. Part 2: discrete-time systems

Contrastive to Part 1, Part 2 presents a generalised optimal linear quadratic digital tracker (LQDT) with universal applications for the discrete-time (DT) systems. This includes (1) a generalised optimal LQDT design for the system with the pre-specified trajectories of the output and the control input and additionally with both the input-to-output direct-feedthrough term and known/estimated system disturbances or extra input/output signals; (2) a new optimal filter-shaped proportional plus integral state-feedback LQDT design for non-square non-minimum phase DT systems to achieve a minimum-phase-like tracking performance; (3) a new approach for computing the control zeros of the given non-square DT systems; and (4) a one-learning-epoch input-constrained iterative learning LQDT design for the repetitive DT systems.     

Iterative learning control for linear discrete time nonminimum phase systems

This paper investigates iterative learning control for linear discrete time nonminimum phase systems. First, iterative learning control with advanced output data is considered for maximum phase systems. Next, the results are extended to nonminimum phase systems. The stability of the inverse mapping from the desired output to the input is proven based on the results for maximum phase systems. The input should be updated with the output which is more advanced than the input by the sum of the relative degree of the system and the number of nonminimum phase zeros. An example is given to indicate the importance of proper advances of output in the input update law.     

Adaptive output feedback control methodology applicable to non-minimum phase nonlinear systems

An adaptive output feedback control methodology is developed for a class of uncertain multi-input multi-output nonlinear systems using linearly parameterized neural networks. The methodology can be applied to non-minimum phase systems if the non-minimum phase zeros are modeled to a sufficient accuracy. The control architecture is comprised of a linear controller and a neural network. The neural network operates over a tapped delay line of memory units, comprised of the system's input/output signals. The adaptive laws for the neural-network weights employ a linear observer of the nominal system's error dynamics. Ultimate boundedness of the error signals is shown through Lyapunov's direct method. Simulations of an inverted pendulum on a cart illustrate the theoretical results.     

Reduced-order controllers for the H∞ control problem with unstable invariant zeros

This paper addresses the existence and design methods of reduced-order controllers for the H_∞ control problem with unstable invariant zeros in the state-space realization of the transfer function matrix from the control input to the controlled error or from the exogenous input to the observation output, where the realization is induced from a stabilizable and detectable realization of the generalized plant. This paper presents a new controller degree bound for the H_∞ control problem in terms of the minimal rank of the system matrix pencils of these two transfer function matrices in the unstable region. When the unstable invariant zero exists, this paper shows that reduced-order controllers with orders strictly less than that of the generalized plant exist if the H_∞ control problem is solvable. Moreover, this paper shows that the computational problem of finding the controllers with the new degree bound is convex by providing two linear matrix inequality-based design methods (algorithms) for constructing the reduced-order controllers. The results developed in this paper are valid both for the continuous- and discrete-time H_∞ control problems.     

Discrete model reference adaptive control with an augmented error signal

A method is developed for designing discrete model reference adaptive control systems when one has access to only the plant's input and output signals. Controllers for single-input, single-output, nonlinear, nonautonomous plants are developed via Lyapunov's second method. The augmented error signal method is employed to ensure that the normally used true error signal approaches zero asymptotically without requiring anticipative values of the plant output signal. Such anticipative signals are replaced by others easily obtained from low pass digital filters operating on the plant's input and output signals.     

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.