Least-restrictive robust periodic model predictive control applied to room temperature regulation

State-feedback model predictive control (MPC) of constrained discrete-time periodic affine systems is considered. The periodic systems’ states and inputs are subject to periodically time-dependent, hard, polyhedral constraints. Disturbances are additive, bounded and subject to periodically time-dependent bounds. The objective is to design MPC laws that robustly enforce constraint satisfaction in a manner that is least-restrictive, i.e., have the largest possible domain. The proposed design method is demonstrated on a building climate control example. The proposed method is directly applicable to time-invariant MPC.     

Receding horizon output feedback control for constrained uncertain systems using periodic invariance

In this article, we consider a receding horizon output feedback control (RHOC) method for linear discrete-time systems with polytopic model uncertainties and input constraints. First, we derive a set of estimator gains and then we obtain, on the basis of the periodic invariance, a series of state feedback gains stabilising the augmented output feedback system with these estimator gains. These procedures are formulated as linear matrix inequalities. An RHOC strategy is proposed based on these state feedback and state estimator gains in conjunction with their corresponding periodically invariant sets. The proposed RHOC strategy enhances the performance in comparison with the case in which static periodic gains are used, and increases the size of the stabilisable region by introducing a degree of freedom to steer the augmented state into periodically invariant sets.     

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.     

Pole-assignment problem for discrete-time linear periodic systems

This paper considers the pole-assignment problem for discrete-time linear periodic systems through the use of linear periodic state-variable feedback control. It is shown that if the N-periodic system with m inputs and n states is completely reachable then the problem can be reduced to the pole-assignment problem for a discrete-time linear invariant system with Nm inputs and n states.     

Compensating periodic descriptor systems

The compensating problem for forward–backward periodic systems in discrete-time has been studied. A normal compensator for this kind of systems is proposed using a suitable observer. In addition, some properties to construct the periodic observer and compensator can be obtained from the associated invariant systems.     

On a general optimal algorithm for multirate output feedbackcontrollers for linear stochastic periodic systems

A modified optimal algorithm for multirate output feedback controllers of linear stochastic periodic systems is developed. By combining the discrete-time linear quadratic regulation (LQR) control problem and the discrete-time stochastic linear quadratic regulation (SLQR) control problem to obtain an extended linear quadratic regulation (ELQR) control problem, one derives a general optimal algorithm to balance the advantages of the optimal transient response of the LQR control problem and the optimal steady-state regulation of the SLQR control problem. In general, the solution of this algorithm is obtained by solving a set of coupled matrix equations. Special cases for which the coupled matrix equations can be reduced to a discrete-time algebraic Riccati equation are discussed. A reducable case is the optimal algorithm derived by H.M. Al-Rahmani and G.F. Franklin (1990), where the system has complete state information and the discrete-time quadratic performance index is transformed from a continuous-time one     

On input-to-state stability of min–max nonlinear model predictive control

In this paper we consider discrete-time nonlinear systems that are affected, possibly simultaneously, by parametric uncertainties and other disturbance inputs. The min–max model predictive control (MPC) methodology is employed to obtain a controller that robustly steers the state of the system towards a desired equilibrium. The aim is to provide a priori sufficient conditions for robust stability of the resulting closed-loop system using the input-to-state stability (ISS) framework. First, we show that only input-to-state practical stability can be ensured in general for closed-loop min–max MPC systems; and we provide explicit bounds on the evolution of the closed-loop system state. Then, we derive new conditions for guaranteeing ISS of min–max MPC closed-loop systems, using a dual-mode approach. An example illustrates the presented theory.     

连续非线性系统的滑模鲁棒正不变集控制

针对一类具有控制和状态有界约束的连续非线性系统, 提出了一种基于单向辅助面滑模控制的正不变集设计方法. 该方法通过将约束条件引入单向辅助面的设计中, 利用单向辅助面构造系统状态的正不变集, 以保证系统状态和控制输入在整个过程中都能满足约束条件. 同时, 滑模控制器设计不再受到切换面的限制, 一些不稳定的超平面也可以作为单向辅助面以设计控制器. 随后给出该方法的稳定性分析以及正不变集的理论证明, 并且通过仿真验证了设计方法的有效性.     

The periodic optimality of LQ controllers satisfying strong stabilization

An LQ strong stabilization problem is proposed. To determine when a controller with periodic gains is locally superior to a linear time invariant compensator for this problem, a Π test is presented. For systems with strictly proper transfer functions, it is proven that the frequency range where stable periodic controllers based on weak variations about the LTI case can give better performance than stable LTI compensators is finite. In the development, a means to evaluate the second partials of functions with respect to matrix-valued parameters is introduced. For those systems where periodic control is warranted, techniques for designing optimal periodic strongly stabilizing controllers are presented. Two examples detailing the application of the Π test are provided, as well as an optimal periodic controller design example.     

Constrained linear quadratic regulation

The paper is a contribution to the theory of the infinite-horizon linear quadratic regulator (LQR) problem subject to inequality constraints on the inputs and states, extending an approach first proposed by Sznaier and Damborg (1987). A solution algorithm is presented, which requires solving a finite number of finite-dimensional positive definite quadratic programs. The constrained LQR outlined does not feature the undesirable mismatch between open-loop and closed-loop nominal system trajectories, which is present in the other popular forms of model predictive control (MPC) that can be implemented with a finite quadratic programming algorithm. The constrained LQR is shown to be both optimal and stabilizing. The solution algorithm is guaranteed to terminate in finite time with a computational cost that has a reasonable upper bound compared to the minimal cost for computing the optimal solution. Inherent to the approach is the removal of a tuning parameter, the control horizon, which is present in other MPC approaches and for which no reliable tuning guidelines are available. Two examples are presented that compare constrained LQR and two other popular forms of MPC. The examples demonstrate that constrained LQR achieves significantly better performance than the other forms of MPC on some plants, and the computational cost is not prohibitive for online implementation     

Control synthesis for invariant sets of families of linear and nonlinear discrete systems with bounded disturbances

Two sets of controlled linear and nonlinear discrete-time systems under bounded disturbances are analyzed. The control synthesis problem for these two sets of systems, including systems with nonlinear functions of a certain class, is constructively solved. The optimal control minimizes the radius of the invariant set. An illustrative example is presented.     

基于鲁棒一步集的Tube不变集鲁棒模型预测控制

针对一类干扰有界的输入和状态受约束线性离散系统,提出了一种基于鲁棒一步集的Tube不 变集鲁棒模型预测控制方法.首先采用多面体不变集离线设计方法得到基于多面体不变集序列的扩 展终端约束集;然后为了扩大鲁棒模型预测控制的初始状态允许区域,并提高系统的鲁棒性,在扩展终端约束集的基础上,通过引入鲁棒一步集并借助Tube不变集控制策略,设计了基于鲁棒一步集的鲁棒模型预测控制方法,并给出了算法的存在性和稳定性证明. 该方法不仅极大地扩大了初始状态允许区域,而且对有界干扰具有有效的抑制作用,使得受扰系统收敛到以原点为中心的最小鲁棒正不变集内.最后仿真验证了算法的有效性.     

A note on robust pole assignment for periodic systems

In this note a robust pole assignment algorithm is proposed for linear periodic discrete-time systems with time-varying dimensions of the state and/or input spaces. The algorithm deduces a periodic state feedback law by the minimization of the condition numbers of the eigenvector matrices of the closed-loop system. Numerical examples are provided to show the performances of the algorithm     

Feedback limitations of linear sampled‐data periodic digital control

We present a set of feedback limitations for linear time‐invariant systems controlled by periodic digital controllers based upon an analysis of the inter‐sample response of the closed‐loop system to sinusoidal inputs. Fundamental sensitivity and complementary sensitivity functions govern the fundamental and harmonic components of the continuous closed‐loop response. The continuous and discrete response of the system is sensitive to variations in the analog plant at frequencies integer multiples of ω_s/N away from the excitation frequency, where ω_s is the sampling frequency and N is the period of the controller. These functions satisfy interpolation and integral constraints due to open‐loop non‐minimum phase zeros and unstable poles. In addition, the use of periodic digital control may result in a reduction in closed‐loop bandwidth. Copyright © 2000 John Wiley & Sons, Ltd.     

Fault detection of linear discrete-time periodic systems

In this note, an approach to the design of optimal fault detection systems for linear discrete-time periodic systems is proposed, which leads to an optimized compromise between robustness to unknown disturbances and sensitivity to faults. The needed computation mainly consists in solving a difference periodic Riccati system. The proposed approach is finally illustrated by a numerical example.     

Stability and stabilization of discrete-time periodic linear systems with actuator saturation

This paper is concerned with the problems of stability and stabilization for discrete-time periodic linear systems subject to input saturation. Both local results and global results are obtained. For local stability and stabilization, the so-called periodic invariant set is used to estimate the domain of attraction. The conditions for periodic invariance of an ellipsoid can be expressed as linear matrix inequalities (LMIs) which can be used for both enlarging the domain of attraction with a given controller and synthesizing controllers. The periodic enhancement technique is introduced to reduce the conservatism in the methods. As a by-product, less conservative results for controller analysis and design for discrete-time time-invariant systems with input saturation are obtained. For global stability, by utilizing the special properties of the saturation function, a saturation dependent periodic Lyapunov function is constructed to derive sufficient conditions for guaranteeing the global stability of the system. The corresponding conditions are expressed in the form of LMIs and can be efficiently solved. Several numerical and practical examples are given to illustrate the theoretical results proposed in the paper.     

Robust model predictive control of constrained linear systems with bounded disturbances

This paper provides a novel solution to the problem of robust model predictive control of constrained, linear, discrete-time systems in the presence of bounded disturbances. The optimal control problem that is solved online includes, uniquely, the initial state of the model employed in the problem as a decision variable. The associated value function is zero in a disturbance invariant set that serves as the ‘origin’ when bounded disturbances are present, and permits a strong stability result, namely robust exponential stability of the disturbance invariant set for the controlled system with bounded disturbances, to be obtained. The resultant online algorithm is a quadratic program of similar complexity to that required in conventional model predictive control.     

Positively invariant polyhedral sets of discrete-time linear systems

The problem of the existence of positively invariant polyhedral sets for linear discrete-time dynamical systems is studied. In the first part of the paper, necessary and sufficient conditions for a given polyhedral set to be a positively invariant set of a linear system are obtained. Then, the spectral properties of systems possessing this kind of invariant set are established. Finally the class of systems possessing positively invariant polyhedral cones is studied.     

Robust invariant sets for constrained storage systems

Several items are produced and stored into n buffers in order to supply an external demand without interruptions. We consider the classical problem of determining control laws and smallest buffer levels guaranteeing that an unknown bounded demand is always satisfied. A simple model with n decoupled integrators and n additive bounded disturbances is employed. The coupling arises from bounds on the total production capacity and on the total demand. Invariant set theory for linear and switched linear systems is exploited to compute robust positive invariant sets and controlled robust invariant sets for two commonly adopted scheduling policies. This paper provides the explicit expression of the invariant sets for any arbitrary n.