Simultaneous design and control of processes under uncertainty: A robust modelling approach

This paper presents a new methodology to integrate process design and control. The key idea in this method is to represent the system’s closed-loop nonlinear behaviour as a linear state space model complemented with uncertain model parameters. Then, robust control tools are applied to calculate bounds on the process stability, the process feasibility and the worst-case scenario. The new methodology was applied to the simultaneous design and control of a mixing tank process. The resulting design avoids the solution of computationally intensive dynamic optimizations since the integration of design and control problem is reduced to a nonlinear constrained optimization problem.     

Uncertainty propagation in dynamical systems

Dynamical propagation of parametric and initial condition uncertainty is studied. The notion of input measure of an observable is defined and its propagation to output measure of the observable is studied by means of transfer operators. Uncertainty of these measures is defined in terms of their cumulative probability distributions. Comparison with alternative uncertainty metrics such as variance and entropy is pursued. The developed formalism is illustrated through an analysis of the effect of pitchfork bifurcation on uncertainty. Finally, the implications of these concepts in the design of nonlinear systems are discussed.     

Decoupled approach to integrated optimum design of structures and robust control systems

Integrated optimum design of structures and control systems is studied by using H ₂ and robust control formulations. It is derived that conventional simultaneous optimization approach by using these robust control laws can be approximated by a decoupled optimization approach in which the structures are optimized by shaping the structural singular values and then the controllers can be designed, namely, decoupled, sequential or successive design approach. It is shown that the proposed decoupled optimization approach can be used to design optimum robust structures and has certain advantages over the conventional simultaneous optimization procedures such as it avoids the drawbacks of pure robust control laws and faster, especially if the number of degrees of freedom (DOF) of the associated structure is large. The bounds for achievable robustness measures are also obtained. Following, simultaneous and decoupled optimization approaches are applied to active control of two structures. The optimization results are presented, and it is concluded that the proposed decoupled optimization approach yields the achieved global minimum much faster than the simultaneous optimization approach.     

一类不确定离散时间系统的鲁棒故障诊断滤波器优化设计方法

研究线性不确定离散时间系统的鲁棒故障诊断滤波器设计问题。基于新提出的性能指标函数，将不确定离散时间系统的故障诊断漶波器设计问题归结为肌优化问题，并通过选择适当的后滤波器和观测器增益矩阵，得到未知输入和模型不确定性鲁棒性的最优解。     

A dynamical systems approach to weighted graph matching

Graph matching is a fundamental problem that arises frequently in the areas of distributed control, computer vision, and facility allocation. In this paper, we consider the optimal graph matching problem for weighted graphs, which is computationally challenging due the combinatorial nature of the set of permutations. Contrary to optimization-based relaxations to this problem, in this paper we develop a novel relaxation by constructing dynamical systems on the manifold of orthogonal matrices. In particular, since permutation matrices are orthogonal matrices with nonnegative elements, we define two gradient flows in the space of orthogonal matrices. The first minimizes the cost of weighted graph matching over orthogonal matrices, whereas the second minimizes the distance of an orthogonal matrix from the finite set of all permutations. The combination of the two dynamical systems converges to a permutation matrix, which provides a suboptimal solution to the weighted graph matching problem. Finally, our approach is shown to be promising by illustrating it on nontrivial problems.     

Modelling of uncertain systems with application to robust process control

A method for black-box identification of uncertain systems is presented. The method identifies a nominal model and an uncertainty model set, consisting of unfalsified uncertainty models. Minimisation of a Chebyshev criterion leads to computationally favourable linear programming problems and allows the possibility to include a priori information in the form of linear constraints without making the computations more complex. Using data compression via correlation computations solves the computation problem associated with identifying unfalsified uncertainty models. The application of set-valued uncertainty models to robust process control is illustrated in a simulation study of robust model predictive control of a distillation column.     

On robust stability of neutral systems with time-varying discrete delay and norm-bounded uncertainty

This paper investigates the robust stability of uncertain linear neutral systems with time-varying discrete delay. A delay-dependent stability criterion is obtained and formulated in the form of a linear matrix inequality. Two numerical examples are given to indicate significant improvements over some existing results.     

Aggregation of a class of interconnected, linear dynamical systems

In this paper, we introduce the notion of a “meaningful” average of a collection of dynamical systems as distinct from an “ensemble” average. Such a notion is useful for the study of a variety of dynamical systems such as traffic flow, power systems, and econometric systems. We also address the associated issue of the existence and computation of such an average for a class of interconnected, linear, time invariant dynamical systems. Such an “average” dynamical system is not only attractive from a computational perspective, but also represents the average behavior of the interconnected dynamical systems. The problem of analysis and control of heirarchical, large scale control systems can be simplified by approximating the lower level dynamics of such systems with such an average dynamical system.     

The challenge of equality constraints in robust design optimization: examination and new approach

In recent years, robust design optimization (RDO) has emerged as a significant area of research. The focus of RDO is to obtain a design that minimizes the effects of uncertainty on product reliability and performance. The effectiveness of the resulting solution in RDO highly depends on how the objective function and the constraints are formulated to account for uncertainties. Inequality constraint and objective function formulations under uncertainty have been studied extensively in the literature. However, the approaches for formulating equality constraints in the RDO literature are in a state of disharmony. Moreover, we observe that these approaches are generally applicable only to certain special cases of equality constraints. There is a need for a systematic approach for handling equality constraints in RDO, which is the motivation for this research. In this paper, we examine critical issues pertinent to formulating equality constraints in RDO. Equality constraints in RDO can be classified as belonging to two classes: (1) those that cannot be satisfied, because of the uncertainty inherently present in the RDO problem, and (2) those that must be satisfied, regardless of the uncertainty present in the problem. In this paper, we propose a linearization- based approach to classify equality constraints into the above two classes, and propose respective formulation methods. The theoretical developments presented in this paper are illustrated with the help of two numerical examples.     

A GUI platform for uncertainty quantification of complex dynamical models

Uncertainty quantification (UQ) refers to quantitative characterization and reduction of uncertainties present in computer model simulations. It is widely used in engineering and geophysics fields to assess and predict the likelihood of various outcomes. This paper describes a UQ platform called UQ-PyL (Uncertainty Quantification Python Laboratory), a flexible software platform designed to quantify uncertainty of complex dynamical models. UQ-PyL integrates different kinds of UQ methods, including experimental design, statistical analysis, sensitivity analysis, surrogate modeling and parameter optimization. It is written in Python language and runs on all common operating systems. UQ-PyL has a graphical user interface that allows users to enter commands via pull-down menus. It is equipped with a model driver generator that allows any computer model to be linked with the software. We illustrate the different functions of UQ-PyL by applying it to the uncertainty analysis of the Sacramento Soil Moisture Accounting Model. We will also demonstrate that UQ-PyL can be applied to a wide range of applications.     

From curve design algorithms to the design of rigid body motions

We discuss the following problem, which arises in computer animation and robot motion planning: given are N positions or keyframes (t_i) of a moving body ³ at time instances t_i. Compute a smooth rigid body motion (t) that interpolates or approximates the given positions (t_i) such that chosen feature points of the moving system run on smooth paths. We present an algorithm that can be considered as a transfer principle from curve design algorithms to motion design. The algorithm relies on known curve design algorithms and on registration techniques from computer vision. We prove that the motion generated in this way is of the same smoothness as the curve design algorithm employed. Supplementary material is available in the online version of this article at http://dx.doi.org/10.1007/s00371-003-0221-3To the memory of Peter Steiner     

Stabilization of interconnected dynamical systems by online convex optimization

The problem of stabilizing networks of interconnected dynamical systems (NDS) in a scalable fashion is considered. As the first contribution, a generalized lemma and example network are provided to demonstrate that state‐of‐the‐art, tractable dissipation‐based NDS stabilization methods can fail even for simple unconstrained, linear, and time‐invariant dynamics. Then, a solution to this issue is proposed, in which controller synthesis is decentralized via a set of parameterized storage functions. The corresponding stability conditions allow for max‐type construction of a trajectory‐specific Lyapunov function for the full closed‐loop network, whereas neither of the local storage functions is required to be monotonically converging. The provided approach is indicated to be nonconservative in the sense that it can generate converging closed‐loop trajectories for the motivating example network and a prescribed set of initial conditions. For input‐affine NDS and quadratic parameterized storage functions, the synthesis scheme can be formulated as a set of low‐complexity semidefinite programs that are solved online, in a receding horizon fashion. Moreover, for linear and time‐invariant networks, an even simpler, explicit control scheme is derived by interpolating a collection of a priori generated converging state and control trajectories in a distributed fashion. Copyright © 2012 John Wiley & Sons, Ltd.     

Non-linear dynamical system approach to behavior modeling

angular velocity control dynamic system guides the agent's direction angle, while another dynamic system selects the environmental input that will be used in the control system. The agent interacts with the environment through its knowledge of the position of stationary and moving objects. In our system agents automatically avoid stationary and moving obstacles to reach the desired target(s). This approach allows us to prove the stability conditions that result in a principled methodology for the computation of the system's dynamic parameters. We present a variety of real-time simulations that illustrate the power of our approach.     

A robust two degree-of-freedom controller for systems with both model and measurement uncertainty

This paper presents a design method for robust two degree-of-freedom (DOF) controllers that optimize the control performance with respect to both model uncertainty and signal measurement uncertainty. In many situations, non-causal feedforward is a welcome control addition when closed loop feedback bandwidth limitations exist due to plant dynamics such as: delays, non-minimum phase zeros, poorly placed zeros and poles (Xie, Alleyne, Greer, and Deneault (2013); Xie (2013), etc. However, feedforward control is sensitive to both model uncertainty and signal measurement uncertainty. The latter is particularly true when the feedforward is responding to pre-measured disturbance signals. The combined sensitivity will deteriorate the feedforward controller performance if care is not taken in design. In this paper a two DOF design is introduced which optimizes the performance based on a given estimate of uncertainties. The controller design uses H_∞ tools to balance the controlled system bandwidth with increased sensitivity to signal measurement uncertainties. A successful case study on an experimental header height control system for a combine harvester is shown as an example of the approach.     

Adaptive robust stabilisation of uncertain nonlinear dynamical systems: an improved backstepping approach

The main purpose of this paper is to propose a design approach by which some simple adaptive robust controllers can be synthesised for a class of uncertain nonlinear dynamical systems which can be transformed into uncertain strict-feedback nonlinear systems. In this paper, an improved backstepping design approach is presented to synthesising a class of continuous adaptive robust state-feedback controllers with a rather simple structure. The improved backstepping design approach can avoid the repeated differentiation problem which appears in using the conventional backstepping algorithm. In particular, it is not required to know the nonlinear upper bound functions of uncertainties. In the light of the presented approach, the state-feedback controllers can be constructed to be linear in the state, with the time-varying control gains which can be self-tuned by the adaptive laws. Similar to the conventional backstepping algorithm, the improved backstepping approach can be extended to a rather large class of uncertain nonlinear systems, and by combining the improved backstepping approach with other control methods, it may be expected to obtain a number of interesting results.     

Decentralized global robust stabilization of a class of interconnected minimum-phase nonlinear systems

This paper focuses on a class of large-scale interconnected minimum-phase nonlinear systems with parameter uncertainty and nonlinear interconnections. The uncertain parameters are allowed to be time-varying and enter the systems nonlinearly. The interconnections are bounded by nonlinear functions of states. The problem we address is to design a decentralized robust controller such that the closed-loop large-scale interconnected nonlinear system is globally asymptotically stable for all admissible uncertain parameters and interconnections. It is shown that decentralized global robust stabilization of the system can be achieved using a control law obtained by a recursive design method together with an appropriate Lyapunov function.     

Characterization of robust stability of a class of interconnected systems

We consider robust stability analysis of a class of large scale interconnected systems. The individual subsystems may be different but they are assumed to share a property that characterizes a set of interconnection matrices which lead to stable overall systems. The main contribution of the paper is to show that, for the case where the network interconnection matrix is normal, (robust) stability verification can be simplified to a low complexity problem of checking whether the frequency response of the individual subsystems and the eigenvalues of the interconnection matrix can be mutually separated using a class of quadratic forms. Most interestingly, it is shown that this criterion is also necessary, in the sense that if the criterion is violated, an interconnection matrix of the same eigenvalue distribution can be found to make the overall system unstable.     

A top-down approach to calibration, validation, uncertainty quantification and predictive accuracy assessment

This paper describes a “top-down” uncertainty quantification (UQ) approach for calibration, validation and predictive accuracy assessment of the SNL Validation Workshop Structural Dynamics Challenge Problem. The top-down UQ approach differs from the more conventional (“bottom-up”) approach in that correlated statistical analysis is performed directly with the modal characteristics (frequencies, mode shapes and damping ratios) rather than using the modal characteristics to derive the statistics of physical model parameters (springs, masses and viscous damping elements in the present application). In this application, a stochastic subsystem model is coupled with a deterministic subsystem model to analyze stochastic system response to stochastic forcing functions. The weak nonlinearity of the stochastic subsystem was characterized by testing it at three different input levels, low, medium and high. The calibrated subsystem models were validated with additional test data using published NASA and Air Force validation criteria. The validated subsystem models were first installed in the accreditation test bed where system response simulations involving stochastic shock-type force inputs were conducted. The validated stochastic subsystem model was then installed in the target application and simulations involving limited duration segments of stationary random vibration excitation were conducted.