Output feedback robust MPC for LPV system with polytopic model parametric uncertainty and bounded disturbance

The output feedback robust model predictive control (MPC), for the linear parameter varying (LPV) system with norm-bounded disturbance, is addressed, where the model parametric matrices are only known to be bounded within a polytope. The previous techniques of norm-bounding technique, quadratic boundedness (QB), dynamic output feedback, and ellipsoid (true-state bound; TSB) refreshment formula for guaranteeing recursive feasibility, are fused into the newly proposed approaches. In the notion of QB, the full Lyapunov matrix is applied for the first time in this context. The single-step dynamic output feedback robust MPC, where the infinite-horizon control moves are parameterised as a dynamic output feedback law, is the main topic of this paper, while the multi-step method is also suggested. In order to strictly guarantee the physical constraints, the outer bound of the true state replaces the true state itself, so tightness of this bound has a major effect on the control performance. In order to tighten the TSB, a procedure for refreshing the real-time ellipsoid based on that of the last sampling instant is given. This paper is conclusive for the past results and far-reaching for the future researches. Two benchmark examples are given to show the effectiveness of the novel results.     

Robust sampled‐data model predictive control for networked systems with time‐varying delay

In this paper, the problem of sampled‐data model predictive control (MPC) is investigated for linear networked control systems with both input delay and input saturation. The delay‐induced nonlinearity is overapproximatively modeled as a polytopic inclusion. The nonlinear behavior of input saturation is expressed as a convex polytope. The resulting closed‐loop systems are represented as linear systems with polytopic and additive norm‐bounded uncertainties. The aim is to determine a robust MPC controller that asymptotically stabilizes the uncertain system at the origin with a certain level of quadratic performance. The effectiveness of the proposed algorithm is demonstrated by a numerical example.     

鲁棒模型预测控制在变风量空调系统中的应用

在变风量空调系统中二次泵压差控制可以有效地减少空调能耗,为克服二次泵模型的不确定性,提高二次泵变频调速控制的响应速度和精度,采用基于线性矩阵不等式的鲁棒预测控制策略。算法分为离线和在线两个部分,离线时首先用传统算法得出目标函数上界,以此为已知量重新优化得到一系列较大的渐近稳定的不变椭圆集。在线时,每个采样周期用三个相邻的椭圆集优化来对状态变量进行精确定位,并给出控制量。给出在线优化的理论证明。通过和传统算法的仿真比较,表明该算法的有效性。二次泵压差控制的实验表明该算法可得到较大的可行域,系统响应快,控制效果好。     

基于终端凸集约束的新MPC 控制器

针对一类离散系统,研究了带有终端约束凸集的MPC控制问题.通过离线设计一组椭圆不变集,并将其组合成一个终端约束凸集,其中凸集参数作为在线优化变量.在线运算时,根据实际的终端状态即时地选择合适的终端不变集,从而有效地扩大了系统的可行域.分别给出了设计MPC控制器的离线和在线算法,仿真实例说明了该方法的有效性.     

基于可达集的鲁棒模型预测控制

设计了一种基于可达集的鲁棒模型预测控制算法.首先确定了一个鲁棒不变集,并将此不变集用作模型预测控制的终端约束集;接着采用终端约束集对可达集的包含度作为优化指标;最后,采用预测时域逐渐减小的控制策略以保证在线优化存在可行解.从理论上证明了吸引域内的任意点在有限时域内都会被引导至终端约束集并始终停留在此集之内,并由仿真算例验证了本文所设计鲁棒模型预测控制算法的可行性.     

Enlarging the domain of attraction of MPC controllers

This paper presents a method for enlarging the domain of attraction of nonlinear model predictive control (MPC). The usual way of guaranteeing stability of nonlinear MPC is to add a terminal constraint and a terminal cost to the optimization problem such that the terminal region is a positively invariant set for the system and the terminal cost is an associated Lyapunov function. The domain of attraction of the controller depends on the size of the terminal region and the control horizon. By increasing the control horizon, the domain of attraction is enlarged but at the expense of a greater computational burden, while increasing the terminal region produces an enlargement without an extra cost.In this paper, the MPC formulation with terminal cost and constraint is modified, replacing the terminal constraint by a contractive terminal constraint. This constraint is given by a sequence of sets computed off-line that is based on the positively invariant set. Each set of this sequence does not need to be an invariant set and can be computed by a procedure which provides an inner approximation to the one-step set. This property allows us to use one-step approximations with a trade off between accuracy and computational burden for the computation of the sequence. This strategy guarantees closed loop-stability ensuring the enlargement of the domain of attraction and the local optimality of the controller. Moreover, this idea can be directly translated to robust MPC.     

MPC for tracking piecewise constant references for constrained linear systems

In this paper, a novel model predictive control (MPC) for constrained (non-square) linear systems to track piecewise constant references is presented. This controller ensures constraint satisfaction and asymptotic evolution of the system to any target which is an admissible steady-state. Therefore, any sequence of piecewise admissible setpoints can be tracked without error. If the target steady state is not admissible, the controller steers the system to the closest admissible steady state.These objectives are achieved by: (i) adding an artificial steady state and input as decision variables, (ii) using a modified cost function to penalize the distance from the artificial to the target steady state (iii) considering an extended terminal constraint based on the notion of invariant set for tracking. The control law is derived from the solution of a single quadratic programming problem which is feasible for any target. Furthermore, the proposed controller provides a larger domain of attraction (for a given control horizon) than the standard MPC and can be explicitly computed by means of multiparametric programming tools. On the other hand, the extra degrees of freedom added to the MPC may cause a loss of optimality that can be arbitrarily reduced by an appropriate weighting of the offset cost term.     

可变多胞体结构的鲁棒模型预测控制

针对一类具有输入输出约束的线性变参数系统,提出了一种可变多胞体结构的多面体不变集鲁棒模型预测控制算法,算法分为在线和离线两个部分。离线算法构建系统的未知但有界误差描述模型,根据状态的变化得到可变参数的多胞体结构,并转化为状态空间模型的多胞体形式,然后采用线性矩阵不等式的优化方式得到一系列离线嵌套多面体不变集。在线算法根据状态变量在嵌套多面体不变集中的位置,结合可变多胞体结构,通过线性插值的优化方式得到系统的实际控制律。给出算法的详细步骤和系统闭环稳定性证明。仿真结果验证了该算法的有效性,表明该算法使系统的闭环响应更为快速和稳定。     

Recursive algorithms for inner ellipsoidal approximation of convex polytopes

In this paper, fast recursive algorithms for the approximation of an n-dimensional convex polytope by means of an inscribed ellipsoid are presented. These algorithms consider at each step a single inequality describing the polytope and, under mild assumptions, they are guaranteed to converge in a finite number of steps. For their recursive nature, the proposed algorithms are better suited to treat a quite large number of constraints than standard off-line solutions, and have their natural application to problems where the set of constraints is iteratively updated, as on-line estimation problems, nonlinear convex optimization procedures and set membership identification.     

输入状态稳定的鲁棒预测控制

以有界干扰非线性系统为研究对象,设计一种基于近似可达集的鲁棒预测控制方法。该方法以鲁棒控制不变集作为终端约束集,采用一种简单的三次多项式逼近预测控制的待优化控制律,通过在线优化求解三次多项式的各项系数,并从理论上证明了所设计的鲁棒预测控制律可以使系统输入状态稳定。最后通过仿真实例验证了所提出的鲁棒预测控制方法的可行性和有效性。     

Computation of Feasible and Invariant Sets for Interpolation-based MPC

The terminal invariant set plays a key role in the stabilizing MPC (Model Predictive Control) formulation. When control gains of the terminal local control laws and corresponding feasible and invariant sets are given, the existing interpolation methods unite them to enlarge the stabilizable region and enhance performance. In this paper, when an invariant set is given, an algorithm is proposed to find another invariant set such that their convex hull is maximized and also invariant. Numerical examples show that the set of the stabilizable initial state of the MPC is enlarged by the terminal constraint set computed by an interpolation-based approach.

     

WinDICOM: A program for determining inclusion shape and orientation

Digital images obtained from X-ray computed tomography scans are analysed for the estimation of inclusion shape and orientation. Three-dimensional computer imagery and segmentation algorithms are used to visualise and isolate the regions of interest. These regions are then approximated by best-fit ellipsoids and the mean best-fit ellipsoid is used as a measure of preferred inclusion orientation. A Windows program is developed to implement these procedures and results found from both manufactured and natural data are presented. These results show that the radiodensity contrast plays a major role in the ability of the software to isolate inclusions from their matrix and hence determine rock fabric.     

Constrained linear MPC with time-varying terminal cost using convex combinations

Recent papers (IEEE Transactions on Automatic Control 48(6) (2003) 1092-1096, Automatica 38 (2002) 1061-1068, Systems and Control Letters 48 (2003) 375-383) have introduced dual-mode MPC algorithms using a time-varying terminal cost and/or constraint. The advantage of these methods is the enlargement of the admissible set of initial states without sacrificing local optimality of the controller, but this comes at the cost of a higher computational complexity. This paper delivers two main contributions in this area. First, a new MPC algorithm with a time-varying terminal cost and constraint is introduced. The algorithm uses convex combinations of off-line computed ellipsoidal terminal constraint sets and uses the associated cost as a terminal cost. In this way, a significant on-line computational advantage is obtained. The second main contribution is the introduction of a general stability theorem, proving stability of both the new MPC algorithm and several existing MPC schemes (IEEE Transactions on Automatic Control 48(6) (2003) 1092-1096, Automatica 38 (2002) 1061-1068). This allows a theoretical comparison to be made between the different algorithms. The new algorithm using convex combinations is illustrated and compared with other methods on the example of an inverted pendulum.     

离线椭圆集的非线性模型预测控制算法研究

针对一类带有约束的非线性系统,提出一种非线性时间最优模型预测控制算法。这种方法首先基于Jacobian线性化将非线性进行线性化,能够推导一系列凸优化问题,而且产生的线性化误差在Lipschitz条件下确定上边界范围。然后采用双重模式策略,在离线情况下构造一系列椭圆集来描述[t]步可行区域,每个椭圆集的平衡点根据上一个椭圆来选取,最后再根据在线计算合适的输入使系统稳定。采用逐步倒退计算的方法能够确保迭代的可行性和稳定性,大大减少了计算负担。数值例子证明了算法的有效性。     

On IQC approach to the analysis and design of linear systems subject to actuator saturation

This paper establishes IQC-based (Integral Quadratic Constraints) conditions under which an ellipsoid is contractively invariant for a single input linear system under a saturated linear feedback law. Based on these set invariance conditions, the determination of the largest such ellipsoid, for use as an estimate of the domain of attraction, can be formulated and solved as an LMI optimization problem. Such an LMI problem can also be readily adapted for the design of the feedback gain that achieves the largest contractively invariant ellipsoid. While the advantages of the proposed IQC approach remain to be explored, it is shown in this paper that the largest contractively invariant ellipsoid determined by this approach is the same as the one determined by the existing approach based on expressing the saturated linear feedback as a linear differential inclusion (LDI), which is known to lead to non-conservative result in determining the largest contractively invariant ellipsoid for single input systems.     

Operations Research Techniques in Manufacturing Planning and Control Systems

This paper investigates the possible uses for operations research (OR) models and techniques in manufacturing planning and control (MPC) systems. We discuss various MPC problems where OR can be applied and the potential inclusion of such models in computer-aided MPC systems. It is found that even though the use of OR techniques in current standard MPC packages is rather restricted, there is a potential for utilizing more OR techniques in computer-aided MPC systems, at least in theory. In practice, OR-related techniques are more often developed as stand-alone systems for decision support as a complement to the MPC system. In this paper, specific OR techniques are discussed for specific MPC problems in terms of applicability and the appropriate type of software solution; none at all, stand-alone system or integrated part of the MPC system.     

Extended invariance and its use in model predictive control

The use of time invariant linear state feedback control laws for the definition of terminal invariant regions can be conservative, thereby reducing the efficacy of predictive control in terms of size of stabilisable sets and closed-loop performance. This difficulty, which is particularly pronounced in the case of nonlinear and/or uncertain dynamics, can be remedied through the use of time-varying control laws and terminal invariant sets. In existing MPC schemes, however, these have to be computed online thereby rendering implementation impracticable for anything other than low-dimensional systems. Here, the definition of invariance is extended to apply over ν predicted control moves, thereby enabling the use of pre-determined (offline) time-varying state feedback gains. More importantly, this extension allows for the use of local uncertainty or linear difference inclusion sets, and thus affords significant improvements, e.g. in terms of the size of terminal regions.     

Controlled invariant feasibility — A general approach to enforcing strong feasibility in MPC applied to move-blocking

Strong feasibility of MPC problems is usually enforced by constraining the state at the final prediction step to a controlled invariant set. However, such terminal constraints fail to enforce strong feasibility in a rich class of MPC problems, for example when employing move-blocking. In this paper a generalized, least restrictive approach for enforcing strong feasibility of MPC problems is proposed and applied to move-blocking MPC. The approach hinges on the novel concept of controlled invariant feasibility. Instead of a terminal constraint, the state of an earlier prediction step is constrained to a controlled invariant feasible set. Controlled invariant feasibility is a generalization of controlled invariance. The convergence of well-known approaches for determining maximum controlled invariant sets, and j-step admissible sets, is formally proved. Thus an algorithm for rigorously approximating maximum controlled invariant feasible sets is developed for situations where the exact maximum cannot be determined.     

A fast procedure for optimizing dynamic force distribution in multifingered grasping.

This correspondence deals with the dynamic force distribution (DFD) problem, i.e., computing the contact forces to equilibrate a dynamic external wrench on the grasped object. The sum of the normal force components is minimized for enhancing safety and saving energy. By this optimality criterion, the DFD problem can be transformed into a linear programming (LP) problem. Its objective function is the inner product of the dynamic external wrench and a vector, and the constraints on the vector, given by a set of linear inequalities, define a polytope. The solution to the LP problem can always be attained at the vertex of the polytope called the solution vertex. We notice that the polytope is determined by the grasp configuration. Along with the direction change of the dynamic external wrench, only the solution vertex moves to an adjacent vertex sequentially, whereas the polytope with all its vertices remains unchanged. Therefore, the polytope and the adjacencies of each vertex can be computed in the offline phase. Then, in the online phase, simply search the adjacencies of the old solution vertex for the new one. Without lost of optimality, such a DFD algorithm runs a thousandfold faster than solving the LP problem by the simplex method in real time.     

MPC for continuous piecewise-affine systems

A large class of hybrid systems can be described by a max–min-plus-scaling (MMPS) model (i.e., using the operations maximization, minimization, addition and scalar multiplication). First, we show that continuous piecewise-affine systems are equivalent to MMPS systems. Next, we consider model predictive control (MPC) for these systems. In general, this leads to nonlinear, nonconvex optimization problems. We present a new MPC method for MMPS systems that is based on canonical forms for MMPS functions. In case the MPC constraints are linear constraints in the inputs only, this results in a sequence of linear optimization problems such that the MPC control can often be computed in a much more efficient way than by just applying nonlinear optimization as was done in previous work.