On dissipativity, passivity and dynamic operability of nonlinear processes

Passivity and dissipativity have long being important tools in nonlinear systems analysis and control design. In this article, connections between dissipativity and the dynamic operability of nonlinear processes are explored. In particular, we study nonlinear processes that are dissipative with respect to a supply rate quadratic in the inputs and outputs. These type of dissipative systems include processes that can be unstable and can have unstable zero dynamics. Two methods to quantify operability are proposed. The first method analyses the speed of response of the closed-loop with no exogenous signals. The second method studies the output disturbance attenuation problem. The proposed framework reveals important insights into those nonlinear process characteristics that can limit dynamic operability.     

On stabilization for a class of nonlinear stochastic time-delay systems： a matrix inequality approach

This paper treats the feedback stabilization of nonlinear stochastic time-delay systems with state and control-dependent noise. Some locally （globally） robustly stabilizable conditions are given in terms of matrix inequalities that are independent of the delay size. When it is applied to linear stochastic time-delay systems, sufficient conditions for the state-feedback stabilization are presented via linear matrix inequalities. Several previous results are extended to more general systems with both state and control-dependent noise, and easy computation algorithms are also given.     

On stabilization for a class of nonlinear stochastic time-delay systems:a matrix inequality approach

This paper treats the feedback stabilization of nonlinear stochastic time-delay systems with state and control-dependent noise. Some locally (globally) robustly stabilizable conditions are given in terms of matrix inequalities that are independent of the delay size. When it is applied to linear stochastic time-delay systems, sufficient conditions for the state-feedback stabilization are presented via linear matrix inequalities. Several previous results are extended to more general systems with both state and control-dependent noise, and easy computation algorithms are also eiven.     

A new delay-dependent absolute stability criterion for a class of nonlinear neutral systems

This paper deals with absolute stability for a class of nonlinear neutral systems using a discretized Lyapunov functional approach. A delay-dependent absolute stability criterion is obtained and formulated in the form of linear matrix inequalities (LMIs). The criterion is valid not only for systems with small delay, but also for systems with non-small delay. Neither model transformation nor bounding technique for cross terms, nor free weighting matrix method is involved through derivation of the stability criterion. Numerical examples show that for small delay case, the results obtained in this paper significantly improve the estimate of the stability limit over some existing result in the literature; for non-small delay case, the ideal results can also be achieved.     

基于矩阵测度的非线性广义时滞系统鲁棒模糊控制

研究T-S模糊广义时滞系统的鲁棒控制问题.不同于传统的寻求公共正定矩阵的方法,基于矩阵测度给出保证系统鲁棒稳定的充分条件,并将此条件进一步转化为线性矩阵不等式.通过求解线性矩阵不等式,得到状态反馈控制器和静态输出反馈控制器.最后通过算例仿真验证了方法的有效性.     

Stabilization for nonlinear systems via a limited capacity communication channel with data packet dropout

This paper addresses the stabilization problem for a class of nonlinear systems. It is assumed that the controller can only receive the transmitted sequence of finite coded signals via a limited digital communication channel. Both state and output feedback coder-decoder-controller procedures are proposed. Stabilization conditions involving the size of coding alphabet, the sampling period, system state growth rate and data packet dropout rate are obtained. Finally, an example is given to illustrate the design procedures and effectiveness of the proposed results.     

Ultimate boundedness control of linear systems with band-bounded nonlinear actuators and additive measurement noise

For linear systems driven by band-bounded nonlinear actuators, a set of linear matrix inequality (LMI) based sufficient conditions are derived for finding state feedback controllers which assure ultimate boundedness of state trajectories. Besides actuator nonlinearity, it is assumed that additive noise exists when state variables are measured for feedback. The purpose is to minimize the ultimate boundedness region while tolerating noise of the largest magnitude. When a state feedback controller is determined for a given system by solving the LMI conditions or by any other means, a less conservative LMI condition is given for further examination of the resultant ultimate boundedness region and tolerable noise magnitude.     

一类非线性时滞广义系统的状态反馈控制

利用线性矩阵不等式(LMI),讨论了一类非线性时滞广义系统的状态反馈控制问题.通过对非线性映射Frechet导数的逆矩阵的估计,给出了系统渐近稳定的充分条件.这一条件与时滞相关,可归结为一个线性矩阵不等式的可解性.在LMI有解的条件下,可得到系统状态反馈控制器的参数表示.数值实例表明该方法是有效的.     

Exponential stability analysis of time-delay systems based on Taylor expansion-based weighted integral inequality

This paper investigates the problem of exponential stability analysis of linear time-delay systems. First, based on the Gram-Schmidt-based integral inequality and the Taylor expansion of exponential function, we develop a new weighted multiple integral inequality called Taylor expansion-based weighted integral inequality. Second, a new Lyapunov-krasovskii functional is constructed, and then, with the help of the Taylor expansion-based inequality, a new exponential stability criterion is established in terms of linear matrix inequality. Finally, numerical examples are presented to show the effectiveness of the newly established stability criterion.     

基于输出严格无源的切换非线性系统的H∞ 分析

针对切换非线性系统,考虑了基于输出严格无源性的 ∞问题.利用每个子系统的输出严格无源性(这里并不要求每个子系统都在整个时间域上满足输出严格无源性,只要求子系统在其激活时间段内满足输出严格无源性),同时借助切换系统理论中的多 Lyapunov 函数方法,用输出严格无源性中的存储函数作为备选的多 Lyapunov 函数,设计了依赖存储函数的最小切换规则,研究了一类切换非线性系统的 ∞问题,给出了存在 ∞性能指标的充分条件.此外,这种基于输出严格无源性的方法也可用于分析切换级联系统的 ∞问题.最后,通过仿真算例验证了这种基于输出严格无源性的方法的有效性.     

Stabilisation of nonlinear chemical processes via dynamic power-shaping passivity-based control

It is well known that energy-balancing passivity-based control is stymied by the presence of pervasive dissipation. To overcome this problem in electrical circuits, some authors have used power-shaping techniques, where stabilisation is achieved by shaping a function akin to power instead of energy. Some extensions of the techniques to general nonlinear systems, yielding static state-feedback control laws, have also been reported. In this article, we extend these techniques to dynamic feedback control and apply them to nonlinear chemical processes. The stability analysis is carried out using the shaped power function as Lyapunov function. The proposed technique is illustrated with two nonlinear chemical process examples.     

Input-output finite-time stability of time-varying linear singular systems

This paper studies the input-output finite-time stabilization problem for time-varying linear singular systems. The output and the input refer to the controlled output and the disturbance input,respectively.Two classes of disturbance inputs are considered,which belong to L-two and L-infinity.Sufficient conditions are firstly provided which guarantee the input-output finite-time stability.Based on this,state feedback controllers are designed such that the resultant closed-loop systems are input-output finite-time stable.The conditions are presented in terms of differential linear matrix inequalities.Finally,an example is presented to show the validity of the proposed results.     

Repetitive process based design and experimental verification of a dynamic iterative learning control law

This paper gives new results on iterative learning control (ILC) design and experimental verification using the stability theory of linear repetitive processes. Using this theory a control law can be designed in one step to force error convergence and produce acceptable transient dynamics. Previous research developed algorithms for the design of a static control law with supporting experimental verification. Should a static law not give the required levels of performance one option is to allow the control law to have internal dynamics. This paper develops a procedure for the design of such a control law with supporting experimental verification on a gantry robot, including a comparative performance against a static law applied to the same robot. The resulting ILC design is an efficient combination of linear matrix inequalities and optimization algorithms.     

Closed-form solutions of singular KYP lemma: strongly passive systems,and fast lossless trajectories

In this paper, we deal with a special class of passive systems, which possess the characteristic property of having no finite spectral zeros. We call these systems strongly passive. It is well known that, for these systems, storage functions, i.e. solutions to the linear matrix inequality (LMI) arising from the Kalman–Yakubovich–Popov (KYP) lemma, cannot be obtained by the conventional approach of algebraic Riccati equations (AREs) and Hamiltonian matrices. In this paper, we first show that a strongly passive system always admits a unique storage function. We then provide a closed-form expression for this unique storage function. Using the closed-form formula of the unique storage function we characterise the ‘lossless’ trajectories of strongly passive systems and show that such systems admit impulsive lossless trajectories on the half-line; we call them fast lossless trajectories. This adds to the existing notion that such systems do not admit any ‘slow’ lossless trajectories.     

Stability analysis of nonlinear processes using empirical state affine models and LMIs

A novel methodology is proposed for the analysis of robust stability of a nonlinear process under PI (Proportional-Integral) control. The analysis is based on state-affine empirical models regressed from input-output data. The state model is represented by a set of polynomial matrices nonlinear with respect to the manipulated variables. This model in combination with a linear PI controller results in a closed loop model that can be shown to lie in a polytope of matrices. This allows for the formulation of a Lyapunov stability test in terms of a simple set of LMIs (linear matrix inequalities). This set of inequalities can be also expanded to account for input saturation. The stability analysis produces regions of stability, in terms of the PI controller parameters, that are significantly larger than the regions previously calculated by a singular value test. The issues of saturation and modeling error are also incorporated into the analysis. The technique permits also to test the stability of the closed loop system with a gain scheduling PI controller. The conservativeness of the analysis is assessed by comparison to closed loop simulations of a highly nonlinear CSTR (continuous stirred tank reactor) under PI control.     

Dissipativity and stabilization of nonlinear repetitive processes

Repetitive processes are characterized by repeated executions of a task defined over a finite duration with resetting after each execution is complete. Also the output from any execution directly influences the output produced on the next execution. The repetitive process model structure arises in the modeling of physical processes and can also be used to effect in the control of other systems, the design of iterative learning control laws where experimental verification of designs has been reported. The existing systems theory for them is, in the main, linear model based. This paper considers nonlinear repetitive processes using a dissipative setting and develops a stabilizing control law with the required conditions expressed in terms of vector storage functions. This design is then extended to stabilization plus disturbance attenuation.     

Delay-dependent stabilization of stochastic interval delay systems with nonlinear disturbances

In this paper, a delay-dependent approach is developed to deal with the robust stabilization problem for a class of stochastic time-delay interval systems with nonlinear disturbances. The system matrices are assumed to be uncertain within given intervals, the time delays appear in both the system states and the nonlinear disturbances, and the stochastic perturbation is in the form of a Brownian motion. The purpose of the addressed stochastic stabilization problem is to design a memoryless state feedback controller such that, for all admissible interval uncertainties and nonlinear disturbances, the closed-loop system is asymptotically stable in the mean square, where the stability criteria are dependent on the length of the time delay and therefore less conservative. By using Itô's differential formula and the Lyapunov stability theory, sufficient conditions are first derived for ensuring the stability of the stochastic interval delay systems. Then, the controller gain is characterized in terms of the solution to a delay-dependent linear matrix inequality (LMI), which can be easily solved by using available software packages. A numerical example is exploited to demonstrate the effectiveness of the proposed design procedure.     

Contraction theory on Riemannian manifolds

Contraction theory is a methodology for assessing the stability of trajectories of a dynamical system with respect to one another. In this work, we present the fundamental results of contraction theory in an intrinsic, coordinate-free setting, with the presentation highlighting the underlying geometric foundation of contraction theory and the resulting stability properties. We provide coordinate-free proofs of the main results for autonomous vector fields, and clarify the assumptions under which the results hold. We state and prove several interesting corollaries to the main result, study cascade and feedback interconnections of contracting systems, study some simple examples, and highlight how contraction theory has arisen independently in other scientific disciplines. We conclude by illustrating the developed theory for the case of gradient dynamics.     

Output tracking of constrained nonlinear processes with offset-free input-to-state stable fuzzy predictive control

This paper develops an efficient offset-free output feedback predictive control approach to nonlinear processes based on their approximate fuzzy models as well as an integrating disturbance model. The estimated disturbance signals account for all the plant-model mismatch and unmodeled plant disturbances. An augmented piecewise observer, constructed by solving some linear matrix inequalities, is used to estimate the system states and the lumped disturbances. Based on the reference from an online constrained target generator, the fuzzy model predictive control law can be easily obtained by solving a convex semi-definite programming optimization problem subject to several linear matrix inequalities. The resulting closed-loop system is guaranteed to be input-to-state stable even in the presence of observer estimation error. The zero offset output tracking property of the proposed control approach is proved, and subsequently demonstrated by the simulation results on a strongly nonlinear benchmark plant.