Constrained quadratic stabilization of discrete-time uncertain non-linear multi-model systems using piecewise affine state-feedback

In this paper a method for non-linear robust stabilization based on solving a bilinear matrix inequality (BMI) feasibility problem is developed. Robustness against model uncertainty is handled. In different non-overlapping regions of the statespace known as clusters the plant is assumed to be an element in a polytope whose vertices (local models) are affine systems. In the clusters containing the origin in their closure, the local models are restricted to being linear systems. The clusters cover the region of interest in the state-space. A n affine state-feedback is associated with each cluster. By utilizing the affinity of the local models and the state-feedback, a set of linear matrix inequalities (LMIs) combined with a single non-convex BMI are obtained which, if feasible, guarantee quadratic stability of the origin of the closed loop. The feasibility problem is attacked by a branch-and-bound-based global approach. If the feasibility check is successful, the Lyapunov matrix and the piecewise affine state-feedback are given directly by the feasible solution. Control constraints are shown to be representable by LMIs or BMIs, and an application of the control design method to robustify constrained non-linear model predictive control is presented. In addition, the control design method is applied to a simple example.     

Robust constrained control of piecewise affine systems through set‐based reachability computations

This article addresses finite‐horizon robust control of a piecewise affine system affected by uncertainty and characterized by different affine dynamics (modes) associated with a polyhedral partition of the state space. The goal is to design a static state‐feedback control law that maintains the state of the system within given—possibly time‐varying—sets, subject to actuation constraints. The proposed approach rests on two phases: a reference mode sequence with a sufficiently large robustness level is determined first, and then a tracking state‐feedback control law defined on the reach sets of the controlled system is designed to counteract uncertainty and maintain the reach sets within the reference sequence. If this is not possible and the reach sets split over different modes, then, further reference mode sequences and tracking controllers are computed. The designed state‐feedback control law is represented through a collection of controllers defined on precomputed reach sets of the closed‐loop control system. Performance of the approach is shown on some numerical examples.     

Optimal input design for model discrimination using Pontryagin’s maximum principle: Application to kinetic model structures

The paper presents a methodology for an optimal input design for model discrimination. To allow analytical solutions, the method, using Pontryagin’s maximum principle, is developed for non-linear single-state systems that are affine in their joint input. The method is demonstrated on a fed-batch reactor case study with first-order and Monod kinetics.     

On persistently exciting observers and a non-linear separation principle: Application to the stabilization of a generator

We provide a separation principle for a class of non-linear time-varying systems. The only assumption we make on the state feedback control law is that it does not vanish as the states grow unboundedly. For the observer, we propose a design method based on the assumption that the system is transformable into a form affine in the unmeasured variable and a property of persistency of excitation. This assumption covers but is not restricted to the common observability condition that the vector field multiplying the unmeasured variable is separated from zero for all state values. Our separation principle relies on stability results for cascades systems.     

Ellipsoidal parameter or state estimation under model uncertainty

Ellipsoidal outer-bounding of the set of all feasible state vectors under model uncertainty is a natural extension of state estimation for deterministic models with unknown-but-bounded state perturbations and measurement noise. The technique described in this paper applies to linear discrete-time dynamic systems; it can also be applied to weakly non-linear systems if non-linearity is replaced by uncertainty. Many difficulties arise because of the non-convexity of feasible sets. Combined quadratic constraints on model uncertainty and additive disturbances are considered in order to simplify the analysis. Analytical optimal or suboptimal solutions of the basic problems involved in parameter or state estimation are presented, which are counterparts in this context of uncertain models to classical approximations of the sum and intersection of ellipsoids. The results obtained for combined quadratic constraints are extended to other types of model uncertainty.     

Non-linear output feedback stabilization on a bounded region of attraction

In this paper it is shown that, for multi-input single-output non-affine non-linear systems, when a state feedback control stabilizes an equilibrium point of a plant with a certain bounded region of attraction, it is also stabilized by an output feedback controller with arbitrarily small loss of the region. Moreover, the proposed output feedback controller has the dynamic order n which is the same as the order of the plant. From any given state feedback, an explicit form of the overall controller is provided. A sufficient condition presented for the result is shown to be necessary and sufficient for regional uniform observability when the system is input affine. Thus, the result can be regarded as a regional separation principle for affine non-linear systems.     

Parameterization and Stabilization of the Equilibrium Points of Affine Non-linear Control Systems

The paper investigates the equilibrium points of affine non-linear control systems and constructs a scheduled control law composed of a feedforward control and a family of state feedbacks. When the design of a non-linear system is decomposed into the design of a family of linear time-invariant systems, it is required to generate parameterized linear models for the plant and to develop a scheduling scheme guaranteeing stability. To solve these difficulties, we parameterized the equilibrium points of the system by constructing a coordinate transformation into a new coordinate system with the parameter coordinates. Then, the system is represented by a parameterized family of linear models along its equilibrium manifold. With these parameterized linear families, we designed a scheduled control law with a feedforward control and local linear robust controllers so that the overall feedback stabilizes the plant about the equilibrium points. The approach is illustrated by applying it to the control of an arm-driven inverted pendulum.     

Synthesis of controllers for target problems of hybrid systems using approximate computation

This paper presents methodologies based on approximate computations for the target control problem of hybrid systems modelled by hybrid automata. The problem of backward reachability and its relation to the control synthesis is studied using approximate analysis techniques. The reachability operators, considering non-linear and linear dynamics with affine disturbances, are under-approximated using state space discretization that involves hyper-cubes. The timing information provided by the backward reachability computation is used in order to design a sub-optimal controller. The computational techniques are applied to the batch evaporator benchmark process which has practical interest.     

Robust H ∞ filtering for a class of non-linear systems with state delay and parameter uncertainty

This paper deals with the problem of robust H X filtering for a class of state-delayed non-linear systems with normbounded parameter uncertainty appearing in all the matrices of the linear part of the system model. The non-linearities are assumed to satisfy the global Lipschitz conditions and appear in both the state and measured output equations. Attention is focused on the design of a non-linear filter which ensures both the robust stability and a prescribed H X performance of the filtering error dynamics for all admissible uncertainties. A sufficient condition for the existence of such a filter is given in terms of a linear matrix inequality (LMI). When this LMI is feasible, the expression of a desired H X filter is also presented. A numerical example is provided to demonstrate the applicability of the proposed approach.     

Finite Volterra-series realizations and input-output approximations of non-linear discrete-time systems

It is shown that a separability property of a given finite family of stationary kernels turns out to be necessary and sufficient for the realization of the associated functional by means of a polynomial affine system (i.e. a system that is polynomial in the input and affine in the state). Moreover, the discrete Volterra kernels associated with the input-output map of a non-linear analytic discrete-lime system, initialized at an equilibrium point, are shown to possess a separability property. On this basis, we state an approximation result for the given input-output map by considering the first kernels of the discrete Volterra series. Two explicit constructions of the approximating polynomial affine systems are proposed.     

An optimal control approach to robust control design

We propose an optimal control approach to robust control design. Our goal is to design a state feedback to stabilize a system under uncertainty. We translate this robust control problem into an optimal control problem of minimizing a cost. Because the uncertainty bound is reflected in the cost, the solution to the optimal control problem is a solution to the robust control problem. Our approach can deal with both linear and non-linear systems. Furthermore it can handle both matched and unmatched uncertainties. It can also handle uncertainty in the control input matrix.     

Non-linear control system design via fuzzy modelling and LMIs

The method of fuzzy-model-based control has emerged as an alternative approach to the solution of analysis and synthesis problems associated with plants that exhibit complex non-linear behaviour. At present, the literature in this field has addressed the control design problem related to the stabilization of state-space fuzzy models. In practical situations, however, where perturbations exist in the state-space model, the problem becomes one of robust stabilization that has yet to be posed and solved. The present paper contributes in this direction through the development of a framework that exploits the distinctive property of the fuzzy model as the convex hull of linear system matrices. Using such a quasi-linear model structure, the robust stabilization of complex non-linear systems, against modelling error and parametric uncertainty, based on static state or dynamic output feedback, is reduced to a linear matrix inequality (LMI) problem.     

基于Backstepping设计的不确定非线性系统的预测控制

本文的目的是针对一类带有不确定性的单输入单输出的仿射非线性系统,设计一种非线性预测控制器.用反步设计思想获得具有待定参数的控制器表达式,然后用预测控制在线优化获得控制器的参数.用这种方法设计的控制器更易使闭环系统稳定,且闭环系统具有良好的动态特性.连续发酵过程的仿真结果也验证了控制器是有效的.     

Predictive control based on neural network models with I/O feedback linearization

This paper presents an approach for the constrained non-linear predictive control problem based on the input-output feedback linearization (IOFL) of a general non-linear system modelled by a discrete-time affine neural network model. Using the resulting linear system in the formulation of the original non-linear predictive control problem enables to restate the optimization problem as the minimization of a quadratic function, which solution can be found using reliable and fast quadratic programming (QP) routines. However, the presence of a non-linear feedback linearizing controller maps the original linear input constraints onto non-linear and state dependent constraints on the controller output, which invalidates a direct application of QP routines. In order to cope with this problem and still be able to use QP routines, an approximate method is proposed which simultaneously guarantees a feasible solution without constraints violation over the complete prediction horizon within a finite number of steps, while allowing only for a small performance degradation.     

基于输入输出线性化方法的一类多输入多输出非线性系统的观测器设计

研究一类多输入多输出仿射非线性系统的状态观测器设计问题. 基于输入输出线性化方法提出了一类多输入多输出仿射非线性系统的状态观测器设计的新方法, 并给出保证状态估计误差渐近趋于零的充分条件. 算例表明了所得结果的有效性.     

Robust observer design for non-linear interconnected systems using structural characteristics

In this paper, a kind of robust observer design scheme based on a constrained Lyapunov equation is presented for a class of non-linear interconnected systems using structural characteristics. The non-linear interconnected system is not required to be affine or linearizable. Both matched and unmatched uncertainties are considered, and a method to deal with uncertain interconnections is proposed. The bounding functions of all the uncertainties may be arbitrarily large since non-linear gain functions are introduced to eliminate the effects of the uncertainties by exploiting their structural characteristics. Simulation is given to demonstrate the effectiveness of the proposed scheme.     

Feedback stabilization of strongly non-linear systems using the CBH formula

This paper presents an approach to the construction of stabilizing feedback for strongly non-linear systems. The class of systems of interest includes systems with drift which are affine in the controls and which cannot be stabilized by continuous state feedback. The approach is independent of the selection of a Lyapunov function, but requires the on-line solution of a non-linear programming satisficing problem stated in terms of expressions resulting from the composition of flows via the Campbell–Baker–Hausdorff formula.     

Robust near-optimal output feedback control of non-linear systems

This work proposes a robust near-optimal non-linear output feedback controller design for a broad class of non-linear systems with time-varying bounded uncertain variables. Both vanishing and non-vanishing uncertainties are considered. Under the assumptions of input-to-state stable (ISS) inverse dynamics and vanishing uncertainty, a robust dynamic output feedback controller is constructed through combination of a high-gain observer with a robust optimal state feedback controller synthesized via Lyapunov's direct method and the inverse optimal approach. The controller enforces exponential stability and robust asymptotic output tracking with arbitrary degree of attenuation of the effect of the uncertain variables on the output of the closed-loop system, for initial conditions and uncertainty in arbitrarily large compact sets, provided that the observer gain is sufficiently large. Utilizing the inverse optimal control approach and singular perturbation techniques, the controller is shown to be near-optimal in the sense that its performance can be made arbitrarily close to the optimal performance of the robust optimal state feedback controller on the infinite time-interval by selecting the observer gain to be sufficiently large. For systems with non-vanishing uncertainties, the same controller is shown to ensure boundedness of the states, uncertainty attenuation and near-optimality on a finite time-interval. The developed controller is successfully applied to a chemical reactor example.     

Dynamic state feedback control of two cooperative manipulators

Dynamic coordinated control of two robot manipulators that rigidly grasp a common object is studied. A dynamic coordinated control model for the two manipulators is derived that is suitable for system analysis and design in state space. The model takes into account kinematic and dynamic constraints between the two manipulators, and is explicitly described by non-linear state equtions and non-linear output equations in the state space. Since coordinated control requires the control of forces applied to the object by manipulators, the output equations include both position components and force components. While robotic systems with position outputs can be linearized using a static state feedback, systems with force outputs, such as the present two robot system, require a dynamic non-linear state feedback for exact linearization. By using dynamic non-linear feedback, coordinated control of two robotic manipulators is converted into a control problem of linear systems.