A constructive approach for an optimal control applied to a class of nonlinear time delay systems

In this contribution, we obtain a nonlinear controller for a class of nonlinear time delay systems, by using the inverse optimality approach. We avoid the solution of the Hamilton Jacobi Bellman type equation and the determination of the Bellman's functional by extending the inverse optimality approach for delay free nonlinear systems to time delay nonlinear systems. This is achieved by combining the Control Lyapunov Function framework and Lyapunov-Krasovskii functionals of complete type. Explicit formulas for an optimal control are obtained. The efficiency of the proposed method is illustrated via experimental results applied to a dehydration process whose model includes a delayed state linear part and a delayed nonlinear part. To give evidence of the good performance of the proposed control law, experimental comparison against an industrial Proportional Integral Derivative controller and optimal linear controller. Additionally experimental robustness tests are presented.     

Wiener sliding-mode control for artificial pancreas: a new nonlinear approach to glucose regulation

Type 1 diabetic patients need insulin therapy to keep their blood glucose close to normal. In this paper an attempt is made to show how nonlinear control-oriented model may be used to improve the performance of closed-loop control of blood glucose in diabetic patients. The nonlinear Wiener model is used as a novel modeling approach to be applied to the glucose control problem. The identified Wiener model is used in the design of a robust nonlinear sliding mode control strategy. Two configurations of the nonlinear controller are tested and compared to a controller designed with a linear model. The controllers are designed in a Smith predictor structure to reduce the effect of system time delay. To improve the meal compensation features, the controllers are provided with a simple feedforward controller to inject an insulin bolus at meal time. Different simulation scenarios have been used to evaluate the proposed controllers. The obtained results show that the new approach outperforms the linear control scheme, and regulates the glucose level within safe limits in the presence of measurement and modeling errors, meal uncertainty and patient variations.     

A new constructive approach to constraint-based geometric design

In the paper, a new constructive approach to solving geometric constraints in 2-D space is presented. Constraints are employed on lines and points only, but more sophisticated geometric elements like Bézier curves and ellipses can also be constrained by mapping them onto auxiliary lines and points. The algorithm is based on local propagation, but first, the problem is transformed into a form that guarantees success of employing this simple technique. The most important steps are substitution of complex constraints with sets of simpler ones and insertion of redundant constraints by solving triangles and determining sums and differences of adjacent angles. In this way, various well-constrained problems with a few exceptions are solved, over-constrained scenes and input data contradictory to some well-known mathematical theorems are detected, and the algorithm is proved successful in many under-constrained cases as well.     

On adaptive control of nonlinearly parameterized nonlinear systems: Towards a constructive procedure

A new framework to design immersion and invariance adaptive controllers for nonlinearly parameterized, nonlinear systems was recently proposed by the authors. The key step is the construction of a monotone mapping, via a suitable selection of a controller tuning function, which has to satisfy some integrability conditions—this translates into the need to solve a partial differential equation (PDE). In this paper this result is extended providing some answers to the questions of characterization of “monotonizable” systems and solvability of the PDE. First, adding to the design a nonlinear dynamic scaling, we obviate the need to solve the PDE. Second, for the case of factorizable nonlinearities, the following results are established. (i) It is shown that the monotonicity condition is satisfied if a linear matrix inequality is feasible. (ii) Directly verifiable involutivity conditions that ensure the solution of the PDE are presented. (iii) An explicit formula for the required tuning function is given, provided the regressor matrix satisfies some rank conditions. Hence, adding a dynamic scaling, this yields a constructive solution to the problem.     

A new approach to inversion-based feedforward control design for nonlinear systems

The finite-time transition between stationary setpoints of nonlinear SISO systems is considered as a scenario for the presentation of a new design approach for inversion-based feedforward control. Design techniques which are based on a stable system inversion result in input trajectories with pre- and/or post-actuation intervals. The presented approach treats the considered transition task as a two-point boundary value problem (BVP) and yields causal feedforward trajectories, which are constant outside the transition interval. The main idea of this approach is to provide free parameters in the desired output trajectory to solve the BVP of the internal dynamics. Thereby, a standard MATLAB function can be used for the numerical solution of the BVP. Feedforward control design techniques are illustrated by simulation results for a simple example.     

A constructive approach for nonlinear system identification usingmultilayer perceptrons

This paper combines a conventional method of multivariable system identification with a dynamic multi-layer perceptron (MLP) to achieve a constructive method of nonlinear system identification. The class of nonlinear systems is assumed to operate nominally around an equilibrium point in the neighborhood of which a linearized model exists to represent the system, although normal operation is not limited to the linear region. The result is an accurate discrete-time nonlinear model, extended from a MIMO linear model, which captures the nonlinear behavior of the system.     

Optimal control of nonlinear systems: a predictive control approach

A new nonlinear predictive control law for a class of multivariable nonlinear systems is presented in this paper. It is shown that the closed-loop dynamics under this nonlinear predictive controller explicitly depend on design parameters (prediction time and control order). The main features of this result are that an explicitly analytical form of the optimal predictive controller is given, on-line optimisation is not required, stability of the closed-loop system is guaranteed, the whole design procedure is transparent to designers and the resultant controller is easy to implement. By establishing the relationship between the design parameters and time-domain transient, it is shown that the design of an optimal generalised predictive controller to achieve desired time-domain specifications for nonlinear systems can be performed by looking up tables. The design procedure is illustrated by designing an autopilot for a missile.     

A constructive approach to local stabilization of nonlinear systems by dynamic output feedback

The note considers the problem of local stabilization of nonlinear systems by dynamic output feedback. A new concept, namely, local uniform observability of feedback control law, is introduced. The main result is that if a nonlinear system is Nth-order approximately stabilizable by a locally uniformly observable state feedback, then it is stabilizable by dynamic output feedback. Based on the approximate stability, a constructive method for designing dynamic compensators is presented. The design of the dynamic compensators is beyond the separation principle and can handle systems whose linearization might be uncontrollable and/or unobservable. An example of nonminimum phase nonlinear systems is presented to illustrate the utility of the results.     

Adaptive control of parametric nonlinear autoregressive models via a new martingale approach

The purpose of this note is to investigate the stability and the optimality of the adaptive tracking for a wide class of parametric nonlinear autoregressive models, via a new martingale approach. Several asymptotic results for the standard least squares estimator of the unknown model parameter, such as a central limit theorem, a law of iterated logarithm, and strong laws of large numbers are also provided.     

A new approach to adaptive control design without overparametrization for a class of uncertain nonlinear systems

This paper considers the globally stabilizing adaptive controller design for a class of more general uncertain high-order nonlinear systems with unknown control coefficients.Although the existing literature has solved the problem,for n-dimensional systems,the existing methods need at least n + 1 dynamic updating laws for the unknown parameters to construct the stabilizing adaptive controller;that is,the dimension of the dynamic compensator is not less than n + 1,and therefore,there exists serious overparame...     

一类不确定非线性系统无过参数自适应控制设计新方法

文中研究了一类控制系数未知的更一般高阶不确定非线性系统的全局稳定自适应控制设计问题.尽管现有文献已解决了该问题,但对于n维系统,现有构造稳定自适应控制的方法至少需要设计n+1个未知参数的动态调节律,即动态补偿器的维数至少为n+1,存在较为严重的过参数问题.文中通过定义新的需动态调节的未知参数,运用增加幂积分和有关自适应技术,成功地解决了已有方法中的过参数问题,给出了构造只含有一个参数调节律的稳定自适应控制设计新方法.仿真算例验证了文中所给方法的有效性.     

Global numerical approach to nonlinear discrete-time control

Interval analysis is used to characterize the set of all input sequences with a given length that drive a nonlinear discrete-time state-space system from a given initial state to a given set of terminal states. No requirement other than computability (i.e. ability to be evaluated by a finite algorithm) is put on the nature of the state equations. The method is based on an algorithm for set inversion and approximates the solution set in a guaranteed way     

Invertibility of affine nonlinear control systems: a geometric approach

The paper deals with the invertibility of multivariable non-linear control systems. By using the recently developed theory on controlled invariant and controllability distributions necessary and sufficient conditions for invertibility are derived.     

Suboptimal control for nonlinear systems: a successive approximation approach

This paper presents a successive approximation approach (SAA) designing optimal controllers for a class of nonlinear systems with a quadratic performance index. By using the SAA, the nonlinear optimal control problem is transformed into a sequence of nonhomogeneous linear two-point boundary value (TPBV) problems. The optimal control law obtained consists of an accurate linear feedback term and a nonlinear compensation term which is the limit of an adjoint vector sequence. By using the finite-step iteration of the nonlinear compensation sequence, we can obtain a suboptimal control law. Simulation examples are employed to test the validity of the SAA.     

A new approach to model reduction of nonlinear control systems using smooth orthogonal decomposition

A new approach to model order reduction of nonlinear control systems is aimed at developing persistent reduced order models (ROMs) that are robust to the changes in system's energy level. A multivariate analysis method called smooth orthogonal decomposition (SOD) is used to identify the dynamically relevant modal structures of the control system. The identified SOD subspaces are used to develop persistent ROMs. Performance of the resultant SOD‐based ROM is compared with proper orthogonal decomposition (POD)–based ROM by evaluating their robustness to the changes in system's energy level. Results show that SOD‐based ROMs are valid for a relatively wider range of the nonlinear control system's energy when compared with POD‐based models. In addition, the SOD‐based ROMs show considerably faster computations compared to the POD‐based ROMs of same order. For the considered dynamic system, SOD provides more effective reduction in dimension and complexity compared to POD.     

A new approach to nonlinear optimal feedback law

For a fixed-time free endpoint optimal control problem it is shown that the optimal feedback control satisfies a system of ordinary differential equations. They are obtained using an elimination procedure of the adjoint vector which appears linearly in a set of differential equations. These equations, involving Lie brackets of vector fields, are derived from the Maximum Principle. An application of this approach to robotics is given.     

A constructive approach to sequential Nash equilibria

We present a Coq-formalized proof that all non-cooperative, sequential games have a Nash equilibrium point. Our proof methodology follows the style advocated by LCF-style theorem provers, i.e., it is based on inductive definitions and is computational in nature. The proof (i) uses simple computational means, only, (ii) basically is by construction, and (iii) reaches a constructively stronger conclusion than informal efforts. We believe the development is a first as far as formalized game theory goes.     

A new approach to fuzzy modeling and control for nonlinear dynamic systems: Neuro-fuzzy dynamic characteristic modeling and adaptive control mechanism

The study on nonlinear control system has received great interest from the international research field of automatic engineering. There are currently some alternative and complementary methods used to predict the behavior of nonlinear systems and design nonlinear control systems. Among them, characteristic modeling (CM) and fuzzy dynamic modeling are two effective methods. However, there are also some deficiencies in dealing with complex nonlinear system. In order to overcome the deficiencies, a novel intelligent modeling method is proposed by combining fuzzy dynamic modeling and characteristic modeling methods. Meanwhile, the proposed method also introduces the low-level learning power of neural network into the fuzzy logic system to implement parameters identification. This novel method is called neuro-fuzzy dynamic characteristic modeling (NFDCM). The neuro-fuzzy dynamic characteristic model based overall fuzzy control law is also discussed. Meanwhile the local adaptive controller is designed through the golden section adaptive control law and feedforward control law. In addition, the stability condition for the proposed closed-loop control system is briefly analyzed. The proposed approach has been shown to be effective via an example. Recommended by Editor Young-Hoon Joo. This work was jointly supported by National Natural Science Foundation of China under Grant 60604010, 90716021, and 90405017 and Foundation of National Laboratory of Space Intelligent Control of China under Grant SIC07010202. Xiong Luo received the Ph.D. degree from Central South University, Changsha, China, in 2004. From 2005 to 2006, he was a Postdoctoral Fellow in the Department of Computer Science and Technology at Tsinghua University. He currently works as an Associate Professor in the Department of Computer Science and Technology, University of Science and Technology Beijing. His research interests include intelligent control for spacecraft, intelligent optimization algorithms, and intelligent robot system. Zengqi Sun received the bachelor degree from Tsinghua University, Beijing, China, in 1966, and the Ph.D. degree from Chalmers University of the Technology, Gothenburg, Sweden, in 1981. He currently works as a Professor in the Department of Computer Science and Technology, Tsinghua University. His research interests include intelligent control of robotics, fuzzy neural networks, and intelligent flight control. Fuchun Sun received the Ph.D. degree from Tsinghua University, Beijing, China, in 1998. From 1998 to 2000, he was a Postdoctoral Fellow in the Department of Automation at Tsinghua University, where he is currently a Professor in the Department of Computer Science and Technology. His research interests include neural-fuzzy systems, variable structure control, networked control systems, and robotics.