Adaptive regulation of MIMO linear discrete-time systems without requiring a persistent excitation

An adaptive regulator which does not require a persistent excitation is developed for multiinput multioutput (MIMO) linear discrete-time systems. The assumptions on the plant are 1) an upper bound on the system dimension is known; 2) the unknown system parameters belong to known bounded intervals; 3) the plant is stabilizable for all possible values of the unknown system parameters. The stabilizability condition can be tested before the system is put into operation. A recursive parameter estimation scheme is given which forces the parameter estimates to converge to the known intervals. Using this estimator, an adaptive regulator is derived which results in a globally-stable closed-loop system.     

An algebraic approach to strong stabilizability of linear nD MIMO systems

Although some necessary conditions for the strong stabilizability of linear multidimensional (nD) multiple-input-multiple-output (MIMO) systems have been available recently, very little is known about sufficient conditions for the same problem. This note presents two sufficient conditions for strong stabilizability of some classes of linear nD MIMO systems obtained using an algebraic approach. A simple necessary and sufficient condition is also given for the strong stabilizability of a special class of linear nD MIMO systems. An advantage of the proposed algebraic approach is that a stable stabilizing compensator can be constructed for an nD plant satisfying the sufficient conditions for the strong stabilizability presented in this note.     

基于Volterra级数的MIMO非线性系统的镇定设计

基于Volterra级数理论,针对一类MIMO非线性系统,首先给出了此类系统可镇定的充分条件;接着讨论了MIMO纯输入与纯输出非线性系统的镇定性的问题,并给出了具体设计方法,最后,用实例仿真验证了其有效性。     

Robust adaptive control for a class of MIMO nonlinear systems with guaranteed error bounds

The design of stabilizing controllers for multiple-input-multiple-output (MIMO) nonlinear plants with unknown nonlinearities is a challenging problem. The high dimensionality coupled with the inability to identify the nonlinearities online or offline accurately motivates the design of stabilizing controllers based on approximations or on approximate estimates of the plant nonlinearities that are simple enough to be generated in real time. The price paid in such case, could be lack of theoretical guarantees for global stability, and nonzero tracking or regulation error at steady state. In this paper, a nonlinear robust adaptive control algorithm is designed and analyzed for a class of MIMO nonlinear systems with unknown nonlinearities. The proposed control scheme provides a general approach to bypass the stabilizability problem where the estimated plant becomes uncontrollable without any restrictive assumptions. The controller is continuous and guarantees closed-loop semi-global stability and convergence of the tracking error to a small residual set. The size of the tracking error at steady state can be specified a priori and guaranteed by choosing certain design parameters. A procedure for choosing these parameters is presented. The properties of the proposed control algorithm are demonstrated using simulations.     

On semi-global stabilizability of MIMO nonlinear systems by output feedback

For multi-input multi-output (MIMO) nonlinear systems, we prove that global stabilizability via smooth state feedback and uniform observability imply semi-global stabilizability by dynamic output feedback. This result incorporates and generalizes Teel-Praly's theorem established previously for single-input single-output (SISO) nonlinear systems. The generalization is made possible by employing, in contrast to the complicated proof in Teel and Praly [(1994). Global stabilizability and observability imply semi-global stabilizability by output feedback. Systems and Control Letters, 22, 313-325], a simple and intuitive argument that involves no intricate Lyapunov functions.     

Boundedness properties of simple, indirect adaptive-control systems

Examining adaptive controllers for a class of first-order linear continuous-time plants, with different estimators, and control laws, the authors obtain explicit solutions completely describing the nonlinear behavior of the resultant adaptive systems. Some of these adaptive systems exhibit either finite time escapes, or other forms of unbounded behavior, due to a loss of stabilizability of the estimated model. Some fixes for the loss of stabilizability problem are analyzed. Implications of these results for the general indirect adaptive/control case are discussed     

A cyclic switching strategy for parameter-adaptive control

This paper introduces a new strategy, called “cyclic switching,” to deal with the well-known certainty equivalence control synthesis problem which arises in the design of identifier-based adaptive controllers because of the existence of points in parameter space where the design model Σ_D, upon which certainty equivalence synthesis is based, loses stabilizability. Unlike most previously suggested methods for handling this problem, the technique proposed here can be employed with or without process excitation. For the technique to work it is not necessary for there to be a mechanism for moving tuned parameters away from values at which Σ_D loses stabilizability, and no such mechanism is used     

Adaptive control of linear Markov jump systems

Stabilization of linear Markov jump systems via adaptive control is considered in this paper. The switching law is assumed to be unobservable Markov process. A sufficient condition is obtained for the stochastic stabilizability based on common quadratic Lyapunov functions (QLFs). The constructive proof provides a method to construct a sampling adaptive stabilizer. An example is used to describe the design of adaptive control, which stabilizes the system.     

Multivariable adaptive predictive control of a binary distillation column

This paper presents a tutorial review of an adaptive predictive control system (APCS). Special emphasis is given to the key issues involved in the practical application of APCS to real processes. These practical issues are illustrated by actual application of SISO and MIMO control of a pilot scale binary distillation column. The experimental evaluation of this method reveals the simplicity of the adaptive algorithm and its excellent performance in an industrial type environment. The experimental results easily outperformed well-tuned classical PID controllers. A brief review of other applications of adaptive control to chemical processes is also included in this paper.     

Adaptive control of MIMO time-varying systems with indicator function based parametrization

This paper studies a new solution framework for adaptive control of a class of MIMO time-varying systems with indicator function based parametrization, motivated by a general discrete-time MIMO Takagi–Sugeno (T–S) fuzzy system model in an input–output form with unknown parameters. An indicator (membership) function based parametrization has some favorable capacity to deal with certain large parameter variations. A new discrete-time MIMO system prediction model is derived for approximating a nonlinear dynamic system, and its system properties are clarified. An adaptive control scheme is developed, with desired controller parametrization and stable parameter estimation for control of such uncertain MIMO time-varying systems. A control singularity problem is addressed and the closed-loop stability and output tracking properties are analyzed. This work provides a new method for multivariable T–S fuzzy system modeling and adaptive control. An illustrative example and simulation results are presented to demonstrate the proposed novel concepts and to verify the desired adaptive control system performance.     

The role of convexity in the adaptive control of rapidly time-varying systems

In classical adaptive control the parameters are assumed to be fixed or slowly time-varying. In order to facilitate parameter estimation/tuning it is desirable to have the set of admissible parameters lie in a convex set; if this set is not convex, a common trick is to replace it with its convex hull, but the adaptive control problem is challenging if stabilizability of the set of admissible parameters is lost. However, such a convexity assumption is an artifact of the approach to the problem, rather than an inherent constraint, since most logic-based and supervisory approaches to the problem make no such requirement. On the other hand, here we show that losing stabilizability on the convex hull of the set of admissible parameters plays an important role in the adaptive control of rapidly time-varying systems.     

严格正则多输入多输出对象同时镇定

使用逆LQ方法讨论了r个严格正则多输入多输出对象的同时镇定问题,基于矩阵 不等式方法得到了静态输出反馈可同时镇定的充要条件,本文证明,r个对象静态输出反馈同 时镇定等价于r个耦合LQ控制问题的解.然后,基于迭代线性矩阵不等式技术给出了一种 迭代求解方法,并给出了算例.     

Direct passivity of a class of MIMO non-linear systems using adaptive feedback

In this paper, an adaptive controller with adaptive laws specially designed is proposed to solve the problem of making a multi-input multi-output (MIMO) non-linear system, with explicit linear parametric uncertainty, equivalent to a passive system. These results are an extension of those obtained by the authors for the SISO case. Some stability issues associated to the resultant closed-loop passive system are also discussed. The results obtained are applied to models of dynamical MIMO systems, to illustrate the controller design methodology.     

同时镇定：逆LQ方法

本文使用逆ＬＱ方法了同时镇定问题，首先得到了ｒ个正则ＭＩＭＯ对象可静态输出反馈同时定的充要条件，然后给出了状态反馈问题同时镇定的一种解，这些条件均以一组耦合Ｒｉｃｃａｔｉ方程和Ｒｉｃｃａｔｉ不等式的形式给出，本语文证明，任一同时镇定反馈增益均可描述为一组具有合适交叉项的耦合ＬＱ控制问题的解。     

Solving predictive control problem of fast-varying multivariable systems by incorporating unknown active dynamics generated by real-time adaptive learning machine

Not only adaptive predictive control of switched systems is a computationally intensive procedure, it also involves various challenges in addressing the problems of robust stabilization and precise tracking. This study proposes new strategies to deal with the aforementioned issues (namely safe and precise control alongside with reduction of computational burden). The first contribution of this work is reduction of conservatism for described class of systems. Control of switched systems with undetectable switching signals is often conducted in worst case switching configuration to ensure robustness, which potentially results in conservative design. The issue of conservativeness is intensified in multi input-multi output (MIMO) dynamical systems due to increased dimensions. However, attaining a robust control scheme for all switching configurations while ensuring precise response is inherently paradoxical. To overcome this issue, this study proposes a new dual-mode algorithm where control modes corresponding to safety and precision are activated at appropriate stages of system response. This is conducted based on incorporation of an adaptive fuzzy-wavelet neural network identification scheme in predictive control of MIMO switched systems. However, as convergence of the adaptive algorithm to actual system is attained after a finite period of time, a safe-mode control algorithm is proposed to maintain quality of transient response in convergence period. In other words, the proposed algorithm operates in safe and precise control modes to ensure robust stability in the convergence period and non-conservative design in steady-state. Second major contribution of the work is reduction of calculation burden based on incorporation of a suboptimal control algorithm. To this end, we propose a predictive control scheme based on a suboptimal gradient-descent based controller, calculating feasible stabilizing inputs instead of optimal inputs. Effects of dynamical variations are incorporated in the model predictive control framework for increased compatibility with high-speed switching dynamics. Then, based on incorporation of dual-mode algorithm, precise steady-state performance is attained while preventing notable perturbations in dynamical discontinuities at switching.     

Towards a link between the factorization approach and MIMO robust adaptive control

Based on a doubly coprime factorization theorem (Nett et al. 1984), an adaptive controller for discrete-time MIMO systems consisting of a not necessarily minimum-phase modelled part and fraction-type unmodelled dynamics, which are not necessarily stable, non-linear, and time-varying, is developed. Then, by virtue of a stability result, which is itself of interest, it is shown that the adaptive closed-loop system is BIBO stable even in the presence of small but not necessarily stable unmodelled dynamics and arbitrarily bounded initial conditions. Consequently, an initial but important step for applying the factorization approach to MIMO robust adaptive control is completely established.     

Robust tracking design based on adaptive fuzzy control of uncertain nonlinear MIMO systems with time delayed states

It is proposed here to use a robust tracking design based on adaptive fuzzy control technique to control a class of multi-input-multi-output (MIMO) nonlinear systems with time delayed uncertainty in which each uncertainty is assumed to be bounded by an unknown gain. This technique will overcome modeling inaccuracies, such as drag and friction losses, effect of time delayed uncertainty, as well as parameter uncertainties. The proposed control law is based on indirect adaptive fuzzy control. A fuzzy model is used to approximate the dynamics of the nonlinear MIMO system; then, two on-line estimation schemes are developed to overcome the nonlinearities and identify the gains of the delayed state uncertainties, simultaneously. The advantage of employing an adaptive fuzzy system is the use of linear analytical results instead of estimating nonlinear system functions with an online update law. The adaptive fuzzy scheme uses a Variable Structure (VS) scheme to resolve the system uncertainties, time delayed uncertainty and the external disturbances such that H_∞ tracking performance is achieved. The control laws are derived based on a Lyapunov criterion and the Riccati-inequality such that all states of the system are uniformly ultimately bounded (UUB). Therefore, the effect can be reduced to any prescribed level to achieve H _∞ tracking performance. A two-connected inverted pendulums system on carts and a two-degree-of-freedom mass-spring-damper system are used to validate the performance of the proposed fuzzy technique for the control of MIMO nonlinear systems.     

Neural‐network‐based predictive controller design: An application to temperature control of a plastic injection molding process

This paper presents a neural‐network‐based predictive control (NPC) method for a class of discrete‐time multi‐input multi‐output (MIMO) systems. A discrete‐time mathematical model using a recurrent neural network (RNN) is constructed and a learning algorithm adopting an adaptive learning rate (ALR) approach is employed to identify the unknown parameters in the recurrent neural network model (RNNM). The NPC controller is derived based on a modified predictive performance criterion, and its convergence is guaranteed by adopting an optimal algorithm with an adaptive optimal rate (AOR) approach. The stability analysis of the overall MIMO control system is well proven by the Lyapunov stability theory. A real‐time control algorithm is proposed which has been implemented using a digital signal processor, TMS320C31 from Texas Instruments. Two examples, including the control of a MIMO nonlinear system and the control of a plastic injection molding process, are used to demonstrate the effectiveness of the proposed strategy. Results from both numerical simulations and experiments show that the proposed method is capable of controlling MIMO systems with satisfactory tracking performance under setpoint and load changes. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Simultaneous stabilization via static output feedback and statefeedback

In this paper, the simultaneous stabilization problem is considered using the matrix inequality approach. Some necessary and sufficient conditions for simultaneous stabilizability of strictly proper multi-input/multi-output (MIMO) plants via static output feedback and state feedback are obtained in the form of coupled algebraic Riccati inequalities. It is shown that any such stabilizing feedback gain is the solution of some coupled linear quadratic control problems where every cost functional has a suitable cross term. A heuristic iterative algorithm based on the linear matrix inequality technique is presented to solve the coupled matrix inequalities. The effectiveness of the approach is illustrated by numerical examples     

Intelligent adaptive control for MIMO uncertain nonlinear systems

This paper investigates an intelligent adaptive control system for multiple-input–multiple-output (MIMO) uncertain nonlinear systems. This control system is comprised of a recurrent-cerebellar-model-articulation-controller (RCMAC) and an auxiliary compensation controller. RCMAC is utilized to approximate a perfect controller, and the parameters of RCMAC are on-line tuned by the derived adaptive laws based on a Lyapunov function. The auxiliary compensation controller is designed to suppress the influence of residual approximation error between the perfect controller and RCMAC. Finally, two MIMO uncertain nonlinear systems, a mass–spring–damper mechanical system and a Chua’s chaotic circuit, are performed to verify the effectiveness of the proposed control scheme. The simulation results confirm that the proposed intelligent adaptive control system can achieve favorable tracking performance with desired robustness.