Nonlinear model predictive control using parameter varying BP-ARX combination model

A novel back-propagation AutoRegressive with eXternal input (BP-ARX) combination model is constructed for model predictive control (MPC) of MIMO nonlinear systems, whose steady-state relation between inputs and outputs can be obtained. The BP neural network represents the steady-state relation, and the ARX model represents the linear dynamic relation between inputs and outputs of the nonlinear systems. The BP-ARX model is a global model and is identified offline, while the parameters of the ARX model are rescaled online according to BP neural network and operating data. Sequential quadratic programming is employed to solve the quadratic objective function online, and a shift coefficient is defined to constrain the effect time of the recursive least-squares algorithm. Thus, a parameter varying nonlinear MPC (PVNMPC) algorithm that responds quickly to large changes in system set-points and shows good dynamic performance when system outputs approach set-points is proposed. Simulation results in a multivariable stirred tank and a multivariable pH neutralisation process illustrate the applicability of the proposed method and comparisons of the control effect between PVNMPC and multivariable recursive generalised predictive controller are also performed.     

一类MIMO系统连续状态空间模型的参数辨识频域方法

在连续时间状态空间模型的参数辨识中,针对系统状态微分项获取困难这一问题,对输入、状态及输出序列应用离散傅里叶变换,得到复数域线性回归方程,并给出了不同形式的最小二乘解估计式.以飞行器多输入多输出(Multiple-input multiple-output, MIMO)状态空间模型为例,设计正交多正弦信号对系统进行多通道同时激励,在一次激励的情况下就可以辨识出所有模型参数,从而提高辨识实验效率.仿真实验证明了方法的有效性和结果的准确性.     

小型无人直升机纵横角动态耦合辨识建模

针对小型无人直升机耦合建模问题提出了一种频域解耦辨识建模方法,该方法通过处理针对耦合辨识的实验数据得到指定频域范围内被辨识耦合的频域特性,对频域特性进行拟合从而获得耦合模型.提出了适用于多输入输出(MIMO)系统的频域特性计算方法,定义了一种复合相干函数并证明其能够用于表达在耦合通道辨识中输入输出的相关性.基于该方法,对一种小型无人直升机在悬停状态的纵横角动态耦合模型进行了辨识,并将耦合模型加入到直升机仿真模型中考察其对模型预测精度的影响.模型预测输出与实际输出的比较表明,相较于普通模型,考虑了耦合动态的仿真模型能够更为精确地预测实际输出.     

Identification of noncontrollable systems from impulse response measurements

We analyze the problem of modeling an observed impulse response by means of a finite-dimensional, linear, time-invariant system. Our approach differs from classical realization theory in the following respects. The modeling problem is split in two steps, namely, identification for determining a model for the observations, and realization for determining parameters which describe the model. Systems are considered as sets of time series, not as input-output maps. In particular, the partitioning of variables into inputs and outputs need not be known, and it is not required that there exist a causal relationship between inputs and outputs. Further, we make no assumptions concerning initial conditions, which in particular may be nonzero. Determination of initial conditions is part of the modeling problem. A final significant distinction from classical realization theory is that the systems need not be controllable.We characterize the class of systems which can be identified from impulse response measurements. Necessary and sufficient conditions are formulated in terms of state-space realizations. It turns out that noncontrollable systems are also identifiable. For causal systems, the condition is that the state transition matrix, restricted to the noncontrollable states, has sufficiently small cyclic index. For noncausal systems, the condition is expressed in terms of the rank of the (singular) state evolution equation.     

Left invertibility of discrete systems with finite inputs and quantised output

The aim of this article is to address left invertibility for dynamical systems with inputs and outputs in discrete sets. We study systems which evolve in discrete time within a continuous state-space. Quantised outputs are generated by the system according to a given partition of the state-space, while inputs are arbitrary sequences of symbols in a finite alphabet, which are associated to specific actions on the system. Our main results are obtained under some contractivity hypotheses. The problem of left invertibility, i.e. recovering an unknown input sequence from the knowledge of the corresponding output string, is addressed using the theory of iterated function systems (IFS), a tool developed for the study of fractals. We show how the IFS naturally associated to a system and the geometric properties of its attractor are linked to the invertibility property of the system. Our main result is a necessary and sufficient condition for left invertibility and uniform left invertibility for joint contractive systems. In addition, an algorithm is proposed to recover inputs from output strings. A few examples are presented to illustrate the application of the proposed method.     

多自由度非线性系统动态参数化模型建模方法研究

针对多自由度非线性系统的动态模型辨识问题,基于NARX(Non-linear Autoregressive with Exogenous inputs)模型的建模方法,考虑系统的物理设计参数,建立非线性系统动态参数化模型.首先,根据系统输入、输出数据建立系统不同参数下的NARX模型,并通过EFOR(Extended Forward Orthogonal Regression)算法对不同参数下NARX模型进行修正,以统一辨识得到的系统模型结构.随后,建立NARX模型系数与物理设计参数间的函数关系,得到多自由度非线性系统的动态参数化模型.以单输入、单输出两自由度非线性系统为例,根据数值仿真结果,对系统的动态参数化模型建模过程进行说明.最后,以带非线性涂层阻尼的悬臂梁作为试验对象,建立其动态参数化模型以反映其动力学特性.试验结果表明,非线性系统动态参数化模型能准确预测多自由度非线性系统的输出响应,为非线性系统的分析与优化设计提供了理论基础.     

Oscillation control in non-linear systems using a first-order filter

In this paper we shall address the oscillation control problems in certain classes of non-linear systems whose outputs are required to follow their inputs. It is assumed that the non-linear systems can be well represented by a set of state-space equations and undergo Hopf bifurcation at some particular value of their parameters or their inputs. A simple first-order output feedback controller is proposed for oscillation control in these non-linear systems. First it is shown that in most cases the first-order controller is effective in locally stabilizing a second-order non-linear system which is undergoing Hopf bifurcation. Then a state separation method based on the solution of the associated Riccati equation is applied to the oscillation control of higher-order non-linear systems and a second-order approximated model is developed for the purpose of designing an oscillation controller. The closed-loop stability of the reduced-order model based design is analysed and some sufficient stability conditions are provided. Finally, a detailed application example of a stepper motor is given to show how the controller design method developed in this paper is applied to practical oscillation control problems.     

Output feedback adaptive stabilization and command following for minimum phase dynamical systems with unmatched uncertainties and disturbances

In this article, we develop an output feedback adaptive control framework for continuous-time minimum phase multivariable dynamical systems for output stabilisation and command following. The approach is based on a nonminimal state-space realisation that generates an expanded set of states using the filtered inputs and filtered outputs and their derivatives of the original system. Specifically, a direct adaptive controller for the nonminimal state-space model is constructed using the expanded states of the nonminimal realisation and is shown to be effective for multi-input, multi-output linear dynamical systems with unmatched disturbances, unmatched uncertainties and unstable dynamics. The proposed adaptive control architecture requires only knowledge of the open-loop system's relative degree as well as a bound on the system's order. Several illustrative numerical examples are provided to demonstrate the efficacy of the proposed approach.     

Stable recursive canonical variate state space modeling for time-varying processes

An adaptive recursive process modeling approach is developed to improve the accuracy of modeling time-varying processes. We adopt the exponential weighted moving average approach to update the covariance and cross-covariance of past and future observation vectors. Forgetting factors are adjusted in the recursive modeling process based on the residual of model outputs. To ensure the stability of the identified model, we introduce a constrained nonlinear optimization approach and propose a stable recursive canonical variate state space modeling (SRCVSS) method. The performance of the proposed method is illustrated with an open-loop numerical example and simulation with the closed-loop data from a continuous stirred tank heater (CSTH) system. The results indicate that the accuracy of proposed SRCVSS modeling method is higher than that of state space modeling with traditional canonical variate analysis.     

Kernel-Based State-Space Kriging for Predictive Control

In this paper, we extend the state-space kriging (SSK) modeling technique presented in a previous work by the authors in order to consider non-autonomous systems. SSK is a data-driven method that computes predictions as linear combinations of past outputs. To model the nonlinear dynamics of the system, we propose the kernel-based state-space kriging (K-SSK), a new version of the SSK where kernel functions are used instead of resorting to considerations about the locality of the data. Also, a Kalman filter can be used to improve the predictions at each time step in the case of noisy measurements. A constrained tracking nonlinear model predictive control (NMPC) scheme using the black-box input-output model obtained by means of the K-SSK prediction method is proposed. Finally, a simulation example and a real experiment are provided in order to assess the performance of the proposed controller.      

Experimental Study of Dynamic Based Feedback Linearization for Trajectory Tracking of a Four-Wheel Autonomous Ground Vehicle

In this paper, experimental study of dynamic based trajectory tracking of an autonomous ground vehicle is presented.The vehicle with two front (steering) and two rear (driving) wheels and also an on-board computer, two DC motors, two batteries and two measurement systems is a good example of an autonomous ground vehicle. The dynamic model of this vehicle is presented in the state-space form with steering and driving torques as inputs; kinematic and dynamic parameters of the model and also electrical parameters of the motors are identified, measurement systems are calibrated and the simulation of controlling this model by feedback linearization method is compared with the experiments.The results of simulations and experiments for the feedback linearization technique are compared with those of a simple PID controller and also the results for sharp turn trajectory tracking illustrate the validity of the method used and the usefulness of the built autonomous ground vehicle.     

Approximated modeling and minimal realization of transfer function matrices with multiple time delays

The modeling and minimal realization techniques for a specific multiple time-delay continuous-time transfer function matrix with a delay-free denominator and a multiple (integer/fractional) time-delay numerator matrix have been developed in the literature. However, this is not the case for a general multiple time-delay continuous-time transfer function matrix with multiple (integer/fractional) time delays in both the denominator and the numerator matrix. This paper presents a new approximated modeling and minimal realization technique for the general multiple time-delay transfer function matrices. According to the proposed technique, an approximated discrete-time state-space model and its corresponding discrete-time transfer function matrix are first determined, by utilizing the balanced realization and model reduction methods with the sampled unit-step response data of the afore-mentioned multiple time-delay (known/unknown) continuous-time systems. Then, the modified Z-transform method is applied to the obtained discrete-time transfer function matrix to find an equivalent specific multiple time-delay continuous-time transfer function matrix with multiple time delays in only the inputs and outputs, for which the existing control and design methodologies and minimal realization techniques can be effectively applied. Illustrative examples are given to demonstrate the effectiveness of the proposed method.     

Discrete state-space modeling for linearnth-order constant coefficient distributed-parameter systems

A systematic procedure is developed for state-space modeling and solving the dynamic behavior of any linearn order constant coefficient distributed-parameter system with two or more independent variables. The state-space model is a set of first-order linear difference equations and is also referred to as a discrete multidimensional state-space model. Transformation of a continuous distributed-parameter system into a discrete state-space model is based on the multidimensional Laplace-bilinear mapping technique. A procedure is outlined for converting the initial and boundary conditions of the system into a set of discrete conditions appropriate for the statespace model. Convergence of the state-space model's solution to the exact solution depends on the sampling rates of the independent variables and the ratio of increments. A few examples when state-space modeling of a distributed-parameter system is useful are: to estimate optimal feedback or optimal feedforward gains in active control applications; model reference optimal-distributed tracking systems; optimal tracking of desired trajectories; realtime system identification.     

Explicit input and output feedback control for discrete-time systems

A new form of output feedback control, referred herein as explicit input and output feedback control (EIOC), is proposed for linear discrete-time systems. Unlike the conventional dynamic output feedback control described by a state-space model, the proposed EIOC has a batch form, where current control is explicitly expressed using current and past system outputs and past control inputs over a recent time horizon. The paper formulates the EIOC law and discusses its features and desirable characteristics. The EIOC is shown to be equivalent to static output feedback control for an augmented system. The coefficients of the EIOC are obtained to achieve the H _∞ performance criterion. Finally, numerical examples are presented to illustrate the effectiveness of the EIOC.     

高炉炼铁过程多元铁水质量非线性子空间建模及应用

高炉炼铁是一个物理化学反应复杂、多相多场耦合的大滞后、非线性动态系统,其关键工艺指标——铁水质量参数的检测、建模和控制一直是冶金工程和自动控制领域的难题.本文提出一种面向控制的数据驱动高炉炼铁多元铁水质量非线性子空间建模方法.首先,为了提高建模效率和降低计算复杂度,采用数据驱动典型相关性分析与相关性分析相结合的方法提取与铁水质量相关性最强的关键可控变量作为建模的输入变量;同时,为了更好地反映高炉非线性动态特性,将相关输入输出变量的时序和时滞关系在建模过程进行考虑;最后,采用基于最小二乘支持向量机（Least square support vector machine,LS-SVM）的非线性Hammerstein系统子空间辨识方法建立数据驱动的多元铁水质量非线性状态空间模型.同时,将核函数表示的模型非线性特性用多项式函数拟合,在仅损失很小模型精度的前提下大大降低模型的计算复杂度.基于实际数据的工业试验验证了所提建模方法的准确性、有效性和先进性.     

Order estimation of multivariable ill-conditioned processes based on PCA method

Ill-conditioned multivariable processes exhibit significantly strong interactions among system variables and large gain directions from the system inputs to the outputs, which makes the identification and control a challenging task. The objective of this paper is to develop an order estimation algorithm for model identification of ill-conditioned processes using subspace methods. In this paper, the order is determined from noise-corrupted samples with high accuracy based on the principal component analysis (PCA) method. To excite each direction in the ill-conditioned process, test signals are designed carefully based on the system characteristics. Using the PCA modeling, the model prediction error is first reconstructed, and the Akaike Information Criterion (AIC) is then used to examine the modeling error bound and thus to determine the process order. A new weighted direction variable is proposed to strengthen the interactions along the small gain directions, thus improving the identifiability and accuracy of the ill-conditioned model. The effectiveness of the proposed method is confirmed by an application study on a high purity distillation column process under noise conditions.     

State-space realizations of linear differential-algebraic-equationsystems with control-dependent state space

This note addresses the derivation of state-space realizations for the feedback control of linear, high-index differential-algebraic-equation systems that are not controllable at infinity. In particular, a class of systems is considered for which the underlying algebraic constraints involve the control inputs, and thus a state-space realization cannot be derived independently of the feedback controller. The proposed methodology involves the design of a dynamic state feedback compensator such that the underlying algebraic constraints in the resulting modified system are independent of the new inputs. A state-space realization of the feedback-modified system is then derived that can be used as the basis for controller synthesis     

Constrained state-space system identification with application to structural dynamics

Constrained identification of state-space models representing structural dynamic systems is addressed. Based on physical insight, transfer function constraints are formulated in terms of the state-space parametrization. A simple example shows that a method tailored for this application, which utilizes the non-uniqueness of a state-space model, outperforms the classic sequential quadratic programming method in terms of robustness and convergence properties. The method is also successfully applied to real experimental data of a plane frame structure.     

EFFECTIVE DECENTRALIZED TITO PROCESS IDENTIFICATION FROM CLOSED‐LOOP STEP RESPONSES

In this paper, a new identification method performed in the time domain based on the decentralized step‐test is proposed for two inputs and two outputs (TITO) processes with significant interactions. In terms of parameter identification, the coupled closed‐loop TITO system is decoupled to obtain four individual single open‐loop systems with the same input signal. As in the SISO case, new linear regression equations are derived, from which the parameters of a first‐ or second‐order plus dead‐time model can be obtained directly. The proposed method outperforms the existing estimation methods for multivariable control systems that use step‐test responses. Furthermore, the method is robust in the presence of high levels of measurement noise. Simulation examples are given to show both effectiveness and practicality of the identification method for a wide range of multivariable processes. The usefulness of the identified method in multivariable process modeling and controller design is demonstrated.