Improving an adaptive controller for non-minimum phase plants

This study improves an adaptive controller for linear discrete-time plants in two aspects: (1) a new adaptation law is proposed to reduce the residual and the parameter errors; (2) a new method is proposed to modify the parameter estimates if the estimated model is not controllable. The second improvement is required when the plant is non-minimum phase. It obtains a controllable model from available parameter estimates while minimizing the modification offset. These features enhance closed-loop tracking performance of the adaptive controller when it is applied to non-minimum phase plants.     

Robust adaptive control of nonlinear non-minimum phase systems with uncertainties

This paper, presents a robust adaptive control method for a class of nonlinear non-minimum phase systems with uncertainties. The development of the control method comprises two steps. First, stabilization of the system is considered based on the availability of the output and internal dynamics of the system. The reference signal is designed to stabilize the internal dynamics with respect to the output tracking error. Moreover, a combined neuro-adaptive controller is proposed to guarantee asymptotic stability of the tracking error. Then, the overall stability is achieved using the small gain theorem. Next, the availability of internal dynamics is relaxed by using a linear error observer. The unmatched uncertainty is compensated using a suitable reference signal. The ultimate boundedness of the reconstruction error signals is analytically shown using an extension of the Lyapunov theory. The theoretical results are applied to a translational oscillator/rotational actuator model to illustrate the effectiveness of the proposed scheme.     

Dynamic non-minimum phase compensation for SISO nonlinear, affine in the input systems

Non-minimum phase difficulties prevent the use of input-output feedback linearization. To avoid this problem earlier work proposed the use of synthetic outputs in order to place zeros in prescribed locations under a condition of static equivalence. The objective here is also to introduce a synthetic output but this is defined through the use of perturbations aimed at yielding maximum relative degree. Resetting of the initial conditions allows design freedom for closing the gap between the behaviour of the synthetic and actual output.     

Comments on a robust model reference adaptive control for non-minimum phase systems with unknown or time-varying delay

In the above paper Makoudi and Radouane, 2000 [A robust model reference adaptive control for non-minimum phase systems with unknown or time-varying delay. Automatica, 36, 1057-1065.], a model reference adaptive control (MRAC) algorithm is presented for non-minimum phase systems with unknown or time-varying delays. Unfortunately, this paper contains several mathematical mistakes that render the proofs of the claimed results erroneous. Furthermore, there are no obvious ways to correct these errors, so that the results presented in this paper are questionable.     

A minimal k-step delay controller for robust tracking of non-minimum phase systems

This paper considers the problem of look-ahead robust tracking for a non-minimum phase system under a sensitivity constraint. Because of non-minimum phase zeros, the exact tracking of arbitrary reference input is impossible. However, with an introduction of some delay, tracking is possible within any given “tolerance” ?₁ > 0. It is shown in the paper that the minimum delay required to satisfy the tracking performance is decoupled from a sensitivity/disturbance rejection constraint. Moreover, it is shown that calculation of the minimum delay reduces to a binary search problem.     

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.     

Stability conditions for a class of time-varying discrete-time systems

This paper deals with the stability conditions for a class of dynamical discrete-time systems, called ‘P-invariants’ and ‘not P-invariants’. Stability conditions are given in terms of polyhedral convex cones and are obtained by using some extensions on M-matrices.     

An adaptive method for consistent estimation of real-valued non-minimum phase zeros in stable LTI systems

An adaptive algorithm, consisting of a recursive estimator for a finite impulse response model having two non-zero lags only, and an adaptive input are presented. The model is parametrized in terms of the first impulse response coefficient and the model zero. For linear time-invariant single-input single-output systems with real rational transfer functions possessing at least one real-valued non-minimum phase zero of multiplicity one, it is shown that the model zero converges to such a zero of the true system. In the case of multiple non-minimum phase zeros, the algorithm can be tailored to converge to a particular zero. The result is shown to hold for systems and noise spectra of arbitrary degree. The algorithm requires prior knowledge of the sign of the high frequency gain of the system as well as an interval to which the non-minimum phase zero of interest belongs.     

A note on the differential regulator equation for non-minimum phase linear systems with time-varying exosystems

Some control problems in practice are often formulated as a linear output regulation problem with time-varying exosystems. Although this problem has been studied recently, an explicit and constructive solution has been given only for minimum phase systems. This paper presents a solution for non-minimum phase systems whose zero dynamics is hyperbolic (or has an exponentially dichotomic split in the case of time-varying zero dynamics). The idea is inspired by the non-causal stable inversion.     

输出重定义下的非线性非最小相位系统迭代学习控制

讨论非线性非最小相位系统实现完全跟踪的迭代学习控制方法, 适于在有限作业区间上重复运行的受控系统. 在控制器设计时, 通过输出重定义以使非最小相位系统的零动态变成渐近稳定特性. 分别采用部分限幅和完全限幅两种学习算法设计控制器, 理论分析表明两种算法能够保证学习系统中所有变量的有界性和跟踪误差在整个作业区间上渐近收敛于零. 数值仿真验证了两种迭代学习控制系统的跟踪性能.     

Modelling of non-minimum phase effects in discrete-time norm optimal iterative learning control

The subject of this article is the modelling of the influence of non-minimum phase discrete-time system dynamics on the performance of norm optimal iterative learning control (NOILC) algorithms with the intent of explaining the observed phenomenon and predicting its primary characteristics. It is established that performance in the presence of one or more non-minimum phase plant zeros typically has two phases. These consist of an initial fast monotonic reduction of the L ₂ error norm (mean square error) followed by a very slow asymptotic convergence. Although the norm of the tracking error does eventually converge to zero, the practical implications over a finite number of trials is apparent convergence to a non-zero error. The source of this slow convergence is identified using the singular value distribution of the system's all pass component. A predictive model of the onset of slow convergence behaviour is developed as a set of linear constraints and shown to be valid when the iteration time interval is sufficiently long. The results provide a good prediction of the magnitude of error norm where slow convergence begins. Formulae for this norm and associated error time series are obtained for single-input single-output systems with several non-minimum phase zeros outside the unit circle using Lagrangian techniques. Numerical simulations are given to confirm the validity of the analysis.     

Asymptotic output tracking in a class of non-minimum phase nonlinear systems via learning-based inversion

Asymptotic output tracking of non-minimum phase (NMP) nonlinear systems has been a popular topic in control theory and applications. Many approaches have focused on finding solutions under minimal assumptions either in the target system or desired trajectories, as there is no general solution available. In this article, we propose a practical and simple solution for cases where the reference trajectory is periodic in time. Our approach employs a learning-based scheme to iteratively determine the desired feedforward input. Unlike previous learning-based frameworks, our method only requires the output tracking error to update the feedforward input iteratively and can be applicable to NMP systems. Our method retains the key advantages of the learning-based framework, including robustness to parameter uncertainties and periodic disturbances. We evaluate the effectiveness of our algorithm using simulation results with an inverted pendulum on a cart, a typical NMP nonlinear system.     

LPV decoupling for control of multivariable systems

This article investigates methods for decoupling multivariable linear parameter varying (LPV) systems and proposes a new interaction measure for decoupled proportional-integral (PI) feedback control design in LPV systems. The proposed approach seeks to benefit the multivariable control of multi-input multi-output (MIMO) systems with variable operating conditions, variable parameters or nonlinear behaviour. This method can improve the tracking performance and reduce the operating conditions variability of such systems with significant coupling in the system dynamics. We design MIMO decoupling feedback LPV controllers to address loop interaction effects. The proposed method uses a parameter-dependent static inversion or SVD decomposition of the system to minimise the effects of the off-diagonal terms in the MIMO system transfer function matrix. A new parameter-dependent interaction measure is introduced based on the SVD decomposition and static inversion which is subsequently utilised for tuning multi-loop PI controller gains. Numerical examples are presented to illustrate the validity of the proposed LPV decoupling methods, as well as the use of the proposed interaction measures for a decoupled multi-loop PI control design.     

Integral controller design based on disturbance cancellation: Partial LTR approach for non-minimum phase plants

For non-minimum phase plants, the loop transfer recovery (LTR) design of integral controllers based on the disturbance cancellation is considered. Since the design requires an unstabilizable extended plant, the standard LTR method cannot be applied. A new partial LTR method is proposed to overcome the difficulty. The target of the proposed design is a fictitious controller including a disturbance estimator based on the measurement of the minimum phase state. It is shown that the Riccati equation including the gain matrix of the disturbance estimator in the target can be used to recover the target feedback property in the output feedback controller. A simple design example is presented to show the effectiveness of the proposed method.     

一类非最小相位系统的多变量多模型解耦控制器

为解决系统暂态响应变差问题，提出一种基于多模型切换的多变量直接自适应控制器，通过加权多项式矩阵的选择，可消除稳态误差，实现静态解耦控制，该控制器由多个参数已知固定模型和两个自适应模型构成，多个固定参数控制器模型可由系统参数模型通过映射直接得到，并与邻城一起完全覆盖控制器参数模型集，仿真结果表明，对于非最小相位系统，暂态响应可得到明显改善。     

基于Kp = LDU分解的多变量离散时间系统鲁棒模型参考自适应控制

针对具有未建模动态且相对阶大于1的一类多变量离散时间系统, 利用其高频增益矩阵分解建立新的参数模型, 在较弱假设下, 进一步研究了直接型鲁棒模型参考自适应控制问题. 由离散时间系统的交换引理, 建立了闭环系统的所有信号与规范化信号的联系. 以一种系统化的方法, 严格地分析了闭环系统的稳定性与鲁棒性.