New results on general observers for nonlinear systems

In this paper, we establish that detectability is a necessary condition for the existence ofgeneral observers (asymptotic or exponential) for nonlinear systems. Using this necessary condition, we show that there does not exist any general observer (asymptotic or exponential), for nonlinear systems with real parametric uncertainty, if the state equilibrium does not change with the parameter values and if the plant output function is purely a function of the state. Next, using center manifold theory, we derive necessary and sufficient conditions for the existence of general exponential observers for Lyapunov stable nonlinear systems. As an application of this result, we show that for the existence of general exponential observers for Lyapunov stable nonlinear systems, the dimension of the state of the general exponential observer should not be less than the number of critical eigenvalues of the linearization matrix of the state dynamics of the plant.     

Analytical failure-detection using adpative,secondary observers

The problem of designing analytical failure-detection systems, using adaptive observers, is addressed in this paper. Failure-detection systems can be applied to linear multi-input, multi-output systems and are related to the examination of then-dimensional observer error vector which carries the necessary information on possible failures. This approach leads toward the design of highly sensitive failure detection systems, obtaining a unique fingerprint for every possible failure (abrupt or soft). In order to keep the observer's false-alarm rate under a certain specified value, it is necessary to have an acceptable matching between the observer model and the system parameters. It is shown here that properly designed adaptive observers are able to detect abrupt changes in the system (actuator, sensor failures, etc.) with adequate reliability. Conditions for convergence for the adaptive observer algorithm are obtained. Good tracking performance with small observer output errors, coupled with accurate and fast parameter identification in both deterministic and stochastic cases, is obtained.Dedicated to G. LeitmannThis research was supported by a National Research Council Associateship at NASA Ames Research Center. The author is indebted to both NRC and NASA Ames Research Center.     

New results on general observers for discrete-time nonlinear systems

In this paper, we establish that detectability is a necessary condition for the existence of general observers (asymptotic or exponential) for discrete-time nonlinear systems. Using this necessary condition, we show that there does not exist any general observer (asymptotic or exponential) for discrete-time nonlinear systems with real parametric uncertainty, if the state equilibrium does not change with the parameter values and if the plant output function is purely a function of the state. Next, using center manifold theory for maps, we derive necessary and sufficient conditions for the existence of general exponential observers for Lyapunov stable discrete-time nonlinear systems. As an application of this result, we show that for the existence of general exponential observers for Lyapunov stable discrete-time nonlinear systems, the dimension of the state of the general exponential observer should not be less than the number of critical eigenvalues of the linearization matrix of the state dynamics of the plant.     

Nonlinear model-predictive control with hysteresis compensation of an electro-pneumatic clutch for truck applications

This contribution presents two real-time capable nonlinear model-predictive control (NMPC) approaches for an electro-pneumatic clutch for heavy trucks: a centralized control approach and a cascaded one. A clutch is necessary at start-up or during gear shifts to connect or disconnect the combustion engine and the gear box. This automated actuator disburdens the driver and provides the necessary actuation force according to the large torque typically transmitted through the drive train. The force characteristic of the clutch, however, is subject to hysteresis, which is described by a generalized Bouc–Wen model and used for a feedforward hysteresis compensation in the control algorithm. The proposed NMPC-algorithm involves (i) a minimization of the difference between the desired and predicted state vector at the end of the prediction horizon and (ii) flatness-based techniques to compute desired trajectories for the complete state vector as well as the control input. The optimal control is given by an additional, minimum-norm control input that minimizes the difference between the predicted state vector and the desired one at the end of the prediction horizon. Thereby, the computation effort of the NMPC approaches can be kept relatively small, and a real-time evaluation becomes possible. A reduced-order observer estimates an effective pressure in the clutch that also accounts for an uncertain disturbance force. Thereby, a disturbance compensation and a high tracking accuracy is achievable for the piston position as controlled variable. The efficiency of the two proposed control structures is emphasized by experimental results from a dedicated test rig.     

Optimal estimation and control of discrete multiplicative systems with unknown second-order statistics

The problem of estimation and control for discrete-time systems with multiplicative noise is examined. Such systems occur naturally in the modeling of stochastic systems with random or unknown coefficients and appear to be robust in contrast to LQG regulators which are sensitive to errors in the coefficients.The statistics of the white sequences of the system are unknown. The problem of stochastic estimation and control of such a system is difficult not only because of the unknown statistics but also because the state is not Gaussian.The approach of this work is to convert the stochastic problem to a deterministic game-theoretic one. We find the estimator and controller so as to minimize a suitable performance measure assuming the worst behavior of nature.A set of necessary and sufficient conditions is developed for the existence of a saddle-point estimator. When both estimation and control are considered, two difficulties appear: the optimality conditions are only necessary and the separation principle collapses. As a result, the saddle-point conditions are only necessary. If the covariances belong to sets with maximal points, then the necessary conditions are satisfied at these points. If, on the other hand, they belong to convex and compact sets and the system has a steady state, then the estimation problem alone has always a saddle-point solution.     

Designing SISO observers with unknown input

In this paper we investigate the observer design for Single-InputSingle-Output (SISO) systems with unknown input. We find thatthe design conditions proposed by several authors in the pastlead to unity relative degree of the plant and zeros of theplant in the open left half of the complex plane, for SISO systems.In other words an observer for a SISO plant with unknown inputcan be designed if and only if the plant is of unity relativedegree and the numerator of the transfer function is Hurwitz.These conditions have an interesting consequence in the contextof model reference adaptive control (MRAC) for the case of relativedegree one, in the sense that strictly real positive (SPR) conditionfor the model reference is not needed.     

Regularity of solutions for an optimal control problem with mixed control-state constraints

A class of optimal control problems for a semilinear elliptic partial differential equation with mixed control-state constraints is considered. Existence results of an optimal control and necessary optimality conditions are stated. Moreover, a projection formula is derived that is equivalent to the necessary optimality conditions. As main result, the Lipschitz continuity of the optimal control is obtained.     

具终端观测且与年龄相关的非线性种群扩散系统的最优控制

讨论了一类具终端观测且与年龄相关的非线性时变种群扩散系统的最优分布控制问题利用偏微控制理论和先验估计,证明了系统最优分布控制的存在性,得到了控制为最优的一阶必要条件,并进而讨论了系统的最优反馈控制问题.     

Necessary conditions for optimal singular stochastic control problems

In this paper, necessary conditions of optimality, in the form of a maximum principle, are obtained for singular stochastic control problems. This maximum principle is derived for a state process satisfying a general stochastic differential equation where the coefficient associated to the control process can be dependent on the state, extending earlier results of the literature.     

Investigation of regularity conditions in optimal control problems with geometric mixed constraints

We investigate regularity conditions in optimal control problems with mixed constraints of a general geometric type, in which a closed non-convex constraint set appears. A closely related question to this issue concerns the derivation of necessary optimality conditions under some regularity conditions on the constraints. By imposing strong and weak regularity condition on the constraints, we provide necessary optimality conditions in the form of Pontryagin maximum principle for the control problem with mixed constraints. The optimality conditions obtained here turn out to be more general than earlier results even in the case when the constraint set is convex. The proofs of our main results are based on a series of technical lemmas which are gathered in the Appendix.     

一类改进的非线性种群系统的最优控制

研究某一类改进的关于年龄结构的非线性种群系统的最优控制问题.首先利用常用的极小化序列方法证明最优控制问题的最优解存在;其次定义原系统对应的共轭系统,借助于法锥定义,得到最优解所要满足的必要条件.     

Dynamic Robust Output Min–Max Control for Discrete Uncertain Systems

Min–max control is a robust control, which guarantees stability in the presence of matched uncertainties. The basic min–max control is a static state feedback law. Recently, the applicability conditions of discrete static min–max control through the output have been derived. In this paper, the results for output static min–max control are further extended to a class of output dynamic min–max controllers, and a general parametrization of all such controllers is derived. The dynamic output min–max control is shown to exist in many circumstances under which the output static min–max control does not exist, and usually allows for broader bounds on uncertainties. Another family of robust output min–max controllers, constructed from an asymptotic observer which is insensitive to uncertainties and a state min–max control, is derived. The latter is shown to be a particular case of the dynamic min–max control when the nominal system has no zeros at the origin. In the case where the insensitive observer exists, it is shown that the observer-controller has the same stability properties as those of the full state feedback min–max control.     

Robin-type boundary control problems for the nonlinear Boussinesq type equations

In this paper, we study a linear and a nonlinear boundary control problems arising from viscous flows. The equations are of nonlinear Navier-Stokes type for the velocity and pressure, of transport-diffusion type for the temperature and the salinity. The essential difficulties are due to the nonlinear nature of a part of the boundary conditions and to the nature of the equations: time-dependent, coupled and nonlinear. The existence and the conditions of the uniqueness of the solution, for the variational problem, are studied. The control is of linear or nonlinear Robin-type and acts on a part of the boundary during a time T. The cost function measures the distance between the observed and the computed vorticity. The existence of an optimal control in the admissible set of states and controls is proved. A first order necessary conditions of optimality are obtained.     

Improved Performance of Universal Integral Regulators

The paper is concerned with the design of an integral control for minimum-phase nonlinear systems that have a well-defined relative degree. Earlier work by the author and coworkers has shown how integral control can be combined with robust control techniques to achieve nonlocal and semiglobal regulation results. The results are applicable to output feedback control problems through the use of high-gain observers and saturation nonlinearities. When used with sliding-mode control, the technique produces universal integral regulators that have been shown to coincide with PI and PID controllers, followed by limiters, for relative-degree-one systems and relative-degree-two systems. The use of integral action improves the steady-state response at the expense of degrading the transient response. The present paper summarizes two recent techniques to modify the universal integral regulator so as to improve its transient performance. One technique uses antiwindup integration, while the other uses nonlinear integrators.     

Optimal control in nonlinear infinite-dimensional systems with nondifferentiability of two types

We consider the optimal control problem for systems described by nonlinear equations of elliptic type. If the nonlinear term in the equation is smooth and the nonlinearity increases at a comparatively low rate of growth, then necessary conditions for optimality can be obtained by well-known methods. For small values of the nonlinearity exponent in the smooth case, we propose to approximate the state operator by a certain differentiable operator. We show that the solution of the approximate problem obtained by standard methods ensures that the optimality criterion for the initial problem is close to its minimal value. For sufficiently large values of the nonlinearity exponent, the dependence of the state function on the control is nondifferentiable even under smoothness conditions for the operator. But this dependence becomes differentiable in a certain extended sense, which is sufficient for obtaining necessary conditions for optimality. Finally, if there is no smoothness and no restrictions are imposed on the nonlinearity exponent of the equation, then a smooth approximation of the state operator is possible. Next, we obtain necessary conditions for optimality of the approximate problem using the notion of extended differentiability of the solution of the equation approximated with respect to the control, and then we show that the optimal control of the approximated extremum problem minimizes the original functional with arbitrary accuracy.     

A dynamic parameter estimator to control chaos with distinct feedback schemes

A dynamic strategy is proposed to estimate parameters of chaotic systems. The dynamic estimator of parameters can be used with diverse control functions; for example, those based on: (i) Lie algebra, (ii) backstepping, or (iii) variable feedback structure (sliding-mode). The proposal has adaptive structure because of interaction between dynamic estimation of parameters and a feedback control function. Without lost of generality, a class of dynamical systems with chaotic behavior is considered as benchmark. The proposed scheme is compared with a previous low-parameterized robust adaptive feedback in terms of execution and performance. The comparison is motivated to ask: What is the suitable adaptive scheme to suppress chaos in an specific implementation? Experimental results of proposed scheme are discussed in terms of control execution and performance and are relevant in specific implementations; for example, in order to induce synchrony in complex networks.     

Output tracking control of Boolean control networks with impulsive effects

This paper studies the output tracking problem of Boolean control networks (BCNs) with impulsive effects via the algebraic state‐space representation approach. The dynamics of BCNs with impulsive effects is converted to an algebraic form. Based on the algebraic form, some necessary and sufficient conditions are presented for the feedback output tracking control of BCNs with impulsive effects. These conditions contain constant reference signal case and time‐varying reference signal case. The study of an illustrative example shows that the obtained new results are effective.     

Image Space Analysis for Vector Variational Inequalities with Matrix Inequality Constraints and Applications

In this paper, vector variational inequalities (VVI) with matrix inequality constraints are investigated by using the image space analysis. Linear separation for VVI with matrix inequality constraints is characterized by using the saddle-point conditions of the Lagrangian function. Lagrangian-type necessary and sufficient optimality conditions for VVI with matrix inequality constraints are derived by utilizing the separation theorem. Gap functions for VVI with matrix inequality constraints and weak sharp minimum property for the solutions set of VVI with matrix inequality constraints are also considered. The results obtained above are applied to investigate the Lagrangian-type necessary and sufficient optimality conditions for vector linear semidefinite programming problems as well as VVI with convex quadratic inequality constraints.     

Optimal Control of the Instationary Three Dimensional Navier-Stokes-Voigt Equations

We study an optimal control problem with quadratic objective functional for the three dimensional Navier-Stokes-Voigt equations in bounded domains. We show the existence of optimal solutions, the necessary optimality conditions and the sufficient optimality conditions. The second-order optimality conditions obtained in the article seem to be optimal.     

On Necessary Optimality Conditions for Singular Controls in Dynamical Systems with a Delay in Control

Abstract

In this article, an optimal control problem with a delay in control is considered. The second-order necessary condition is obtained for the optimality of singular (in the sense of the maximum principle) control. Also, the notion of degenerate singular control of order k (k?≥?1) is introduced and for optimality of this, the high-order necessary condition is obtained. Moreover, while studying the problem, one of the strengthened version of an analog of the maximum principle is shown. Finally, the rich content of the obtained results is illustrated by specific examples.