Data-driven asymptotic stabilization for discrete-time nonlinear systems

In this paper, we propose a data-driven feedback controller design method based on Lyapunov approach, which can guarantee the asymptotic stability of the closed-loop and enlarge the estimate of domain of attraction (DOA) for the closed-loop. First, sufficient conditions for a feedback controller asymptotically stabilizing the discrete-time nonlinear plant are proposed. That is, if a feedback controller belongs to an open set consisting of pairs of control input and state, whose elements can make the difference of a control Lyapunov function (CLF) to be negative-definite, then the controller asymptotically stabilizes the plant. Then, for a given CLF candidate, an algorithm, to estimate the open set only using data, is proposed. With the estimate, it is checked whether the candidate is or is not a CLF. If it is, a feedback controller is designed just using data, which satisfies sufficient conditions mentioned above. Finally, the estimate of DOA for closed-loop is enlarged by finding an appropriate CLF from a CLF candidate set based on data. Because the controller is designed directly from data, complexity in building the model and modeling error are avoided.     

Constructing modal control systems for plants with meromorphic transfer function

A problem of constructing an infinite-dimensional controller that ensures the given desired distribution of poles and a part of zeros of the transfer function of the closed-loop system is solved for the plant with distributed parameters. Transfer functions of the plant, controller and closed-loop system are considered in the class of meromorphic functions. The modal controller is synthesized directly based on the desired transfer function of the closed-loop system. The method of synthesizing the controller is reduced to searching an interpolation series. An example is given.     

数据驱动闭环子空间预测控制方法研究与应用

提出一种完全数据驱动的闭环子空间辨识及预测控制器设计方法. 该方法完全由闭环系统的输入输出数据辨识子空间矩阵, 通过子空间矩阵的拆分, 排除了与扰动相关的模型输入, 进而获取子空间矩阵参数的无偏估计; 将辨识得到的闭环系统子空间矩阵描述直接作为预测模型, 设计预测控制器; 将其应用于某钢铁集团焦炉炭化室压力控制系统, 取得了良好的控制效果.

     

Controller discretization: a gap metric framework for analysis and synthesis

Although techniques for directly synthesising sampled-data (SD) compensators are available in the literature, feedback controller design is perhaps best understood in a purely continuous-time setting. As such, a feedback controller is often designed in the continuous-time domain and then discretized for digital implementation. It is important for the discretization step involved to yield an SD approximation which captures the essential features of the original controller from the perspective of closed-loop behavior. In this note, a gap metric framework is developed for studying the controller discretization problem for linear time-invariant (LTI) plants and controllers. Importantly, knowledge of a gap metric distance between an LTI controller and an SD approximation permits explicit characterization of the possible difference in closed-loop performance, with any LTI plant for which the LTI controller is known to work well, accounting for intersample behavior. The central result of the new framework gives rise to an algorithm for computing a gap metric measure of the distance between an LTI controller and a given discretization, and a technique for synthesising a SD approximation which is optimal with respect to this metric.     

Input-state model matching and ripple-free response for dual-rate systems

In this paper, we address the model matching problem for dual-rate systems where the controller output is generated at a faster rate than the measurement update rate. The model matching problem that has been studied in the literature requires the input-output properties of the closed-loop multirate system to match those of a desired single-rate linear time-invariant (LTI) system. In this paper, we consider the model matching problem from the input-state viewpoint: given a desired LTI system, find conditions and provide a controller design procedure to achieve matching between the closed-loop system and the desired system state variables at the measurement update rate. We provide a solution to this problem using a particular time-varying controller structure. In addition, we give conditions to avoid ripples in the steady-state output of the continuous-time plant; in particular, we show that some constraints on the input matrix of the desired system have to be posed. Numerical examples are given to illustrate the proposed method.     

A discrete-time internal model-based controller and its application to a binary distillation column

Explicit state-space formulae for an ^∞ two degrees-of-freedom (2-DOF) Internal Model Control (IMC)-based controller are derived. The IMC controller solves a special ^∞ control problem directly in discrete-time, providing robust stability against coprime factor uncertainty, and a degree of robust performance in the sense of making the closed-loop system match a pre-specified reference model. The IMC controller structure is shown to comprise a plant state observer, an on-line step response model, and a generalised state-feedback law associated with the plant and model states. A design for a binary (methanol/water) distillation column is presented as an illustrative example.     

Monitoring control performance via structured closed-loop response subject to output variance/covariance upper bound

Owing to the process time delays, the closed-loop response can be divided into feedback control invariant part and feedback controller dependent part. If the latter part is replaced by a user specified response trajectory, we refer to the resultant closed-loop response as structured closed-loop response. The user specified structured closed-loop response has been used as an achievable control against which one can assess performance of control loops. In the control performance monitoring literature, the user specified response is often given as a first-order transfer function with some specified performance requirement, such as time constant. In this paper, we solve this problem from a systematic approach, i.e., in viewpoint of a variance/covariance upper bound on the outputs. With available closed-loop routine operating output data and process time delay/interactor matrix, the desired structured closed-loop response can be obtained directly via estimated closed-loop time series model. A significant feature is that the output variance/covariance upper bound constraint can be explicitly specified according to the product specifications and is always satisfied when the problem is feasible. This desired structured closed-loop response can thus be served as a benchmark against which the existing controller performance can be compared. We also show that two approaches, linearizing change of variables and Frank and Wolfe algorithm, are suitable for solving this problem, which result in a full order and a reduced order structured closed-loop response, respectively. Both approaches are illustrated by two case studies.     

基于多控制器的直接自适应控制

为解决多模型自适应控制可能造成的模型失配不稳定性问题,利用非伪控制理论,提出了一种直接辨识控制器的方法。由一步超前控制器构造控制器集,通过一个具有一定属性的恰当选择的代价函数,只要控制器集中至少存在一个镇定的控制器,即自适应控制问题是可行的,滞后切换逻辑总能快速将镇定控制器切换到控制回路中,保证闭环系统稳定和输出误差渐近趋于零。给出了闭环系统在L2e意义下输入输出稳定性结论,并给出仿真实例验证其有效性。     

Safe Adaptive Switching Control: Stability and Convergence

A formal theoretical explanation of the model-mismatch instability problem associated with certain adaptive control design schemes is proposed, and a solution is provided. To address the model-mismatch problem, a primary task of adaptive control is formulated as finding an asymptotically optimal, stabilizing controller, given the feasibility of adaptive control problem. A class of data-driven cost functions called cost-detectable is introduced that detect evidence of instability without reference to prior plant models or plant assumptions. The problem of designing adaptive systems that are robustly immune to mismatch instability problems is thus placed in a setting of a standard optimization problem. We call the result safe adaptive control because it robustly achieves adaptive stabilization goals whenever feasible, without prior assumptions on the plant model and, hence, without the risk of model-mismatch instability. The result improves the robustness of previous results in hysteresis switching control, both for discrete and for continuously-parameterized candidate controller sets. Examples are provided.     

Data-driven criteria synthesis of system with two-degree-of-freedom controller

This paper is concerned with the problem of designing a two-degree-of-freedom (2-DOF) controller based on data. Without requiring the specific mathematical model of a plant, the parameters of stabilising controller are calculated directly from data for a linear time-invariant system. In fact, using the traditional unity feedback control, the existing results which focus on calculating stabilising PID or first-order controllers have been reported in the literature. However, there exists certain steady-state error in these approaches if the integrators are not added into the controller. In this paper, a 2-DOF controller is designed to ensure no changes in characteristic polynomial of the closed-loop systems, furthermore, in terms of a pre-specified criterion function, by applying the iterative feedback tuning technique, a local optimal performance is achieved. Especially, for the step input, through tuning the parameters of controller, the steady-state error can be reduced to zero. Finally, a numerical example is given to illustrate the effectiveness of this method.     

Multivariable closed-loop deadbeat control: a polynomial-matrix approach

The closed-loop deadbeat servo problem (CDSP), considered in this paper, consists of the synthesis of a linear, output feedback controller such that the control signal and tracking error both vanish, after a finite period of time, for every reference sequence from a prespecified class and for every initial state of a plant and the controller. The closed-loop structure is determined by studying necessary and sufficient conditions for deadbeat tracking performance. A new theorem asserts that if an open-loop deadbeat control strategy exists for every initial state of the plant and every reference function from a given class, then CDSP is solvable and all desired control laws are found in an explicit parametric form by solving simple, unilateral, linear equations in polynomial matrices. On the basis of this theorem a design algorithm is developed. Asymptotic stability of the closed-loop system exhibiting deadbeat properties is demonstrated. A numerical example is given to illustrate the usefulness and computational efficiency of the new design algorithm presented.     

Solution to the positive real control problem for lineartime-invariant systems

In this paper we study the problem of synthesizing an internally stabilizing linear time-invariant controller for a linear time-invariant plant such that a given closed-loop transfer function is extended strictly positive real. Necessary and sufficient conditions for the existence of a controller are obtained. State-space formulas for the controller design are given in terms of solutions to algebraic Riccati equations (or inequalities). The order of the constructed controller does not exceed that of the plant     

Robust fuzzy stabilisation of interval plants

The paper deals with the problem of stabilisation of interval systems. To this end, by using Takagi–Sugeno fuzzy mechanism, a Mamdani-type PID-like fuzzy controller is modified and extended to develop a new PID-like Takagi–Sugeno fuzzy stabilising controller for the plant described by an interval system. Indeed, a PID-like Takagi–Sugeno fuzzy controller and an interval plant are considered in the forward path of a unity feedback system, and parameters in Takagi–Sugeno fuzzy controller are determined so that the stability of the closed-loop system is assured. The closed-loop system has a multilinear uncertainty structure. Therefore, based on the Zero Exclusion Condition for multilinear uncertain systems, a new theorem presenting sufficient conditions for the Takagi–Sugeno fuzzy controller to be robust stability guaranteed is also derived. An example is given to illustrate the application and the effectiveness of the proposed controller.     

Self-tuning in master–slave synchronization of high-precision stage systems

For synchronization of high-precision stage systems, in particular the synchronization between a wafer and a reticle stage system of a wafer scanner, a master–slave controller design is presented. The design consists of a synchronization controller based on FIR filters and a data-driven self-tuning approach is used to find the coefficients of these filters. In the context of Lur'e systems, i.e. the reticle stage slave system has a variable gain controller with saturation nonlinearity, a part of the gradients needed for self-tuning is obtained from reconstruction using closed-loop nonlinear models. The remaining part is given by sampled data obtained primarily from time-series measurements. Performance with the synchronization controller is shown to be bounded by a waterbed effect: low-frequency suppression comes at the price of high-frequency amplification. For the considered Lur'e stage systems the ability of the self-tuning to induce improved tracking is discussed in view of this waterbed effect for either simulation results or experimental results.     

On validating closed-loop behaviour from noisy frequency-response measurements

It is shown how noisy closed-loop frequency-response measurements can be used to obtain pointwise in frequency bounds on the possible difference between the actual closed-loop system and the closed-loop comprising a nominal model of the plant and the stabilising controller. To this end, Vinnicombe's gap metric framework for robustness analysis plays a central role. Indeed, an optimisation problem and corresponding algorithm are proposed for estimating the chordal distance between the frequency responses of the nominal plant model and a plant that is consistent with the closed-loop data and a priori information, when projected onto the Riemann sphere.     

二阶加纯滞后过程的非脆弱PID稳定化控制器设计

根据Hermite-Biehler定理在准多项式稳定性问题上的推广，研究用PID控制器镇定二阶加纯滞后过程的问题，给出了使闭环系统稳定的PID参数区域，并以稳定的PID参数区域内切圆半径为目标函数，寻优得到非脆弱PID控制器参数，最后通过线性规划给出了非脆弱PID稳定化控制器的设计方法。     

Controller Design Under Fuzzy Pole-Placement Specifications: An Interval Arithmetic Approach

This paper discusses fuzzy specifications for robust controller design, as a way to define different specification levels for different plants in a family and allow the control of performance degradation. Controller synthesis will be understood as mapping a fuzzy plant onto a desired fuzzy set of closed-loop specifications. In this context, a fuzzy plant is considered as a possibility distribution on a given plant space. In particular, pole placement in linear plants with fuzzy parametric uncertainty is discussed, although the basic idea is general and could be applied to other settings. In the case under consideration, the controller coefficients are the solution of a fuzzy linear system of equations with a particular semantics. Modal interval arithmetic is used to solve the system for each alpha-cut. The intersection of the solutions, if not empty, constitutes the solution to the robust control problem     

Virtual reference direct design method: an off-line approach todata-based control system design

This paper presents a direct model-reference approach to the off-line design of linear controllers, suited to deal with plants described by a single set of open-loop I/O measurements only. The method is direct inasmuch as the controller parameters are directly estimated with no preliminary identification of any model to describe the plant. The design can be carried out off-line and, in the present formulation, leads to a nonadaptive controller. The basic idea is that of interpreting the open-loop I/O measurements of the plant as closed-loop data produced by a “virtual” reference signal that can be computed by backpropagating the measured output of the plant through the reference model; thus, the controller design reduces to a standard identification problem, in which the “output” signal to be matched is the measured input of the plant. Both a deterministic (noise-free) and a stochastic setting, are considered     

基于观测器的受扰多项式系统H∞输出跟踪控制

考虑一类受到外部扰动影响的多项式系统在状态不完全可测情况下的H_∞输出跟踪控制问题.首先,综合前馈-反馈复合控制思想,设计基于观测器的输出跟踪控制器,其中反馈镇定控制器用于保证闭环系统稳定,前馈补偿控制器用以实现对参考模型输出信号的跟踪;然后,提出具有输出反馈结构的跟踪控制方法,其优势在于实现了分离原则,可单独设计观测器和控制器,降低计算复杂度;接着,利用依赖全状态的齐次多项式Lyapunov函数导出使得闭环系统渐近稳定且满足H_∞跟踪性能的充分条件,借助多项式平方和凸优化技术可直接求得相应的观测器和控制器;最后,通过数值仿真实例验证所提出设计方法的有效性和优越性.     

Simultaneous controller design for linear time-invariant systems

The use of generalized sampled-data hold functions (GSHF) in the problem of simultaneous controller design for linear time-invariant plants is discussed. This problem can be stated as follows: given plants P₁, P₂, . . ., P_N , find a controller C which achieves not only simultaneous stability, but also simultaneous optimal performance in the N given systems. By this, it is meant that C must optimize an overall cost function reflecting the closed-loop performance of each plant when it is regulated by C. The problem is solved in three aspects: simultaneous stabilization, simultaneous optimal quadratic performance, and simultaneous pole assignment in combination with simultaneous intersampling performance