Interpolation-based Off-line Robust MPC for Uncertain Polytopic Discrete-time Systems

In this paper, interpolation-based off-line robust MPC for uncertain polytopic discrete-time systems is presented. Instead of solving an on-line optimization problem at each sampling time to find a state feedback gain, a sequence of state feedback gains is pre-computed off-line in order to reduce the on-line computational time. At each sampling time, the real-time state feedback gain is calculated by linear interpolation between the pre-computed state feedback gains. Three interpolation techniques are proposed. In the first technique, the smallest ellipsoids containing the measured state are approximated and the corresponding real-time state feedback gain is calculated. In the second technique, the pre-computed state feedback gains are interpolated in order to get the largest possible real-time state feedback gain while robust stability is still guaranteed. In the last technique, the real-time state feedback gain is calculated by minimizing the violation of the constraints of the adjacent inner ellipsoids so the real-time state feedback gain calculated has to regulate the state from the current ellipsoids to the adjacent inner ellipsoids as fast as possible. As compared to on-line robust MPC, the proposed techniques can significantly reduce on-line computational time while the same level of control performance is still ensured

이동블록 및 잔류편차 제거 모델예측제어 기법의 최적성 향상

학위논문(박사)--서울대학교 대학원 :공과대학 화학생물공학부,2020. 2. 이종민.Model predictive control (MPC) is a receding horizon control which derives finite-horizon optimal solution for current state on-line by solving an optimal control problem. MPC has had a tremendous impact on both industrial and control research areas. There are several outstanding issues in MPC. MPC has to solve the optimization problem within a sampling period so that the reduction of on-line computational complexity is a one of the main research subject in MPC. Another major issue is model-plant mismatch due to the model based predictive approach so that offset-free tracking schemes by compensating model-plant mismatch or unmeasured disturbance has been developed. In this thesis, we focused on the optimality performance of move blocking which fixes the decision variables over arbitrary time intervals to reduce computational load for on-line optimization in MPC and disturbance estimator approach based offset-free MPC which is the most standardly used method to accomplish offset-free tracking in MPC. We improve the optimality performance of move blocked MPC in two ways. The first scheme provides a superior base sequence by linearly interpolating complementary base sequences, and the second scheme provides a proper time-varying blocking structure with semi-explicit approach. Moreover, we improve the optimality performance of offset-free MPC by exploiting learned model-plant mismatch compensating signal from estimated disturbance data. With the proposed schemes, we efficiently improve the optimality performance while guaranteeing the recursive feasibility and closed-loop stability.모델예측제어는 현재 시스템 상태에 대한 유한 구간 최적해를 도출하는 온라인 이동 구간 제어 방식이다. 모델예측제어는 피드백을 통한 공정 동특성과 제약 조건을 효과적으로 반영하는 장점으로 인해 산업 및 제어 연구 분야에 큰 영향을 미쳤다. 이러한 모델예측제어에는 몇 가지 해결되어야 할 문제가 있다. 모델예측제어에서는 샘플링 기간 내에 최적화 문제를 풀어내야 하기 때문에, 온라인 계산 복잡성의 감소가 주요 연구 주제 중 하나로 활발히 연구되고 있다. 또 다른 주요 문제는 모델에 기반한 예측을 이용하는 접근 방식으로 인해 모델-플랜트 불일치로 인한 오차를 해결해야 한다는 점이며, 모델 플랜트 불일치 또는 측정되지 않은 외란을 보상하여 잔류편차 없이 참조신호를 추적하는 연구가 활발히 이루어지고 있다. 이 논문에서는 모델예측제어에서의 온라인 최적화를 위한 계산 부하를 줄이기 위해 임의의 시간 간격에 걸쳐 결정 변수를 고정시키는 이동 블록 전략의 최적성 향상에 중점을 두었으며, 또한 잔류편차를 제거하기 위해 가장 표준적으로 사용되는 외란 추정기를 이용한 잔류편차-제거 모델예측제어 기법의 최적성 향상에 중점을 두었다. 이 논문에서는 이동 블록 모델예측제어의 최적 성능을 향상시키기 위한 두 가지 전략을 제시한다. 첫 번째 전략은 이동 블록 전략에서 일반적으로 고정된 채로 사용되는 기반 시퀀스를 상호 보완적인 두 기반 시퀀스의 선형 보간으로 대체함으로써 보다 우수한 기반 시퀀스를 제공하며, 두 번째 전략은 준-명시적 접근법을 활용하여 현재 시스템 상태에 적절한 시변 블록 구조를 온라인에서 제공한다. 또한, 잔류편차-제거 모델예측제어 기법의 최적 성능을 향상시키기 위해 추정 외란 데이터로부터 학습된 모델-플랜트 불일치 보상 신호를 온라인에서 이용하는 전략을 제안하였다. 제안된 세 가지 기법을 통해 모델예측제어의 반복적 실현가능성과 폐쇄-루프 안정성을 보장하면서 최적 성능을 효율적으로 개선 하였다.1. Introduction 1 2. Move-blocked model predictive control with linear interpolation of base sequences 5 2.1 Introduction 5 2.2 Preliminaries 9 2.2.1 MPC formulation 9 2.2.2 Move blocking 12 2.2.3 Move blocked MPC (MBMPC) 15 2.3 Move blocking schemes 16 2.3.1 Previous solution based offset blocking 17 2.3.2 LQR solution based offset blocking 18 2.4 Interpolated solution based move blocking 20 2.4.1 Interpolated solution based MBMPC 20 2.4.2 QP formulation 26 2.5 Numerical examples 29 2.5.1 Example 1 (Feasible region) 30 2.5.2 Example 2 (Performance in regulation problem) 33 2.5.3 Example 3 (Performance in tracking problem) 36 3. Move-blocked model predictive control with time-varying blocking structure by semi-explicit approach 43 3.1 Introduction 43 3.2 Problem formulation 46 3.3 Move blocked MPC 48 3.3.1 Move blocking scheme 48 3.3.2 Implementation of move blocking 51 3.4 Semi-explicit approach for move blocked MPC 53 3.4.1 Off-line generation of critical region 56 3.4.2 On-line MPC scheme with critical region search 60 3.4.3 Property of semi-explicit move blocked MPC 62 3.5 Numerical examples 70 3.5.1 Example 1 (Regulation problem) 71 3.5.2 Example 2 (Tracking problem) 77 4. Model-plant mismatch learning offset-free model predictive control 83 4.1 Introduction 83 4.2 Offset-free MPC: Disturbance estimator approach 86 4.2.1 Preliminaries 86 4.2.2 Disturbance estimator and controller design 87 4.2.3 Offset-free tracking condition 89 4.3 Model-plant mismatch learning offset-free MPC 91 4.3.1 Model-plant mismatch learning 92 4.3.2 Application of learned model-plant mismatch 97 4.3.3 Robust asymptotic stability of model-plant mismatch learning offset-free MPC 102 4.4 Numerical example 117 4.4.1 System with random set-point 120 4.4.2 Transformed system 125 4.4.3 System with multiple random set-points 128 5. Concluding remarks 134 5.1 Move-blocked model predictive control with linear interpolation of base sequences 134 5.2 Move-blocked model predictive control with time-varying blocking structure by semi-explicit approach 135 5.3 Model-plant mismatch learning offset-free model predictive control 136 5.4 Conclusions 138 5.5 Future work 139 Bibliography 145Docto

Interpolating Control with Periodic Invariant Sets

This paper presents a novel low-complexity interpolating control scheme involving periodic invariance or vertex reachability of target sets for the constrained control of LTI systems. Periodic invariance relaxes the strict one-step positively invariant set notion, by allowing the state trajectory to leave the set temporarily but return into the set in a finite number of steps. To reduce the complexity of the representation of the required controllable invariant set, a periodic invariant set is employed. This set should be defined within the controllable stabilising region, which is considered unknown during the design process. Since periodic invariant sets are not traditional invariant sets, a reachability problem can be solved off-line for each vertex of the outer set to provide an admissible control sequence that steers the system state back into the original target set after a finite number of steps. This work develops a periodic interpolating control (pIC) scheme between such periodic invariant sets and a maximal admissible inner set by means of an inexpensive linear programming problem, solved on-line at the beginning of each periodic control sequence. Theorems on recursive feasibility and asymptotic stability of the pIC are given. A numerical example demonstrates that pIC provides similar performance compared to more expensive optimization-based schemes previously proposed in the literature, though it employs a naive representation of the controllable invariant set

Polytopic Approximation of Explicit Model Predictive Controllers

A model predictive control law (MPC) is given by the solution to a parametric optimization problem that can be pre-computed offline, which provides an explicit map from state to input that can be rapidly evaluated online. However, the primary limitations of these optimal explicit solutions are that they are applicable to only a restricted set of systems and that the complexity can grow quickly with problem size. In this paper we compute approximate explicit control laws that trade-off complexity against approximation error for MPC controllers that give rise to convex parametric optimization problems. The algorithm is based on the classic double- description method and returns a polyhedral approx- imation to the optimal cost function. The proposed method has three main advantages from a control point of view: it is an incremental approach, meaning that an approximation of any specified complexity can be produced, it operates on implicitly-defined convex sets, meaning that the prohibitively complex optimal explicit solution is not required and finally it can be applied to any convex parametric optimization problem. A sub-optimal controller based on barycentric in- terpolation is then generated from this approximate polyhedral cost function that is feasible and stabiliz- ing. The resulting control law is continuous, although non-linear and defined over a non-simplical polytopic partition of the state space. The non-simplical nature of the partition generates significantly simpler approx- imate control laws, which is demonstrated on several examples

Stochastic model predictive control of LPV systems via scenario optimization

A stochastic receding-horizon control approach for constrained Linear Parameter Varying discrete-time systems is proposed in this paper. It is assumed that the time-varying parameters have stochastic nature and that the system's matrices are bounded but otherwise arbitrary nonlinear functions of these parameters. No specific assumption on the statistics of the parameters is required. By using a randomization approach, a scenario-based finite-horizon optimal control problem is formulated, where only a finite number M of sampled predicted parameter trajectories (‘scenarios') are considered. This problem is convex and its solution is a priori guaranteed to be probabilistically robust, up to a user-defined probability level p. The p level is linked to M by an analytic relationship, which establishes a tradeoff between computational complexity and robustness of the solution. Then, a receding horizon strategy is presented, involving the iterated solution of a scenario-based finite-horizon control problem at each time step. Our key result is to show that the state trajectories of the controlled system reach a terminal positively invariant set in finite time, either deterministically, or with probability no smaller than p. The features of the approach are illustrated by a numerical example

Interpolating Control Toolbox (ICT)

Interpolating control toolbox (ICT) is a free and open-source MATLAB toolbox that implements interpolation-based control (IC) for time-invariant and uncertain time-varying linear discrete-time systems with local state and control constraints. The toolbox combines geometrical features to compute robust invariant sets offline and solves a linear programming problem to compute the required IC online. This paper provides an overview on interpolating control and shows how to use ICT to robustly control input centralised and input decentralised interconnected systems. ICT includes some demo files to compare the performance of centralised versus decentralised IC

Interpolating Control Toolbox (ICT)

Interpolating control toolbox (ICT) is a free and open-source MATLAB toolbox that implements interpolation-based control (IC) for time-invariant and uncertain time-varying linear discrete-time systems with local state and control constraints. The toolbox combines geometrical features to compute robust invariant sets offline and solves a linear programming problem to compute the required IC online. This paper provides an overview on interpolating control and shows how to use ICT to robustly control input centralised and input decentralised interconnected systems. ICT includes some demo files to compare the performance of centralised versus decentralised IC

Explicit feedback synthesis for nonlinear robust model predictive control driven by quasi-interpolation

We present QuIFS (Quasi-Interpolation driven Feedback Synthesis): an offline feedback synthesis algorithm for explicit nonlinear robust minmax model predictive control (MPC) problems with guaranteed quality of approximation. The underlying technique is driven by a particular type of grid-based quasi-interpolation scheme. The QuIFS algorithm departs drastically from conventional approximation algorithms that are employed in the MPC industry (in particular, it is neither based on multi-parametric programming tools and nor does it involve kernel methods), and the essence of its point of departure is encoded in the following challenge-answer approach: Given an error margin $\varepsilon>0$, compute in a single stroke a feasible feedback policy that is uniformly $\varepsilon$-close to the optimal MPC feedback policy for a given nonlinear system subjected to constraints and bounded uncertainties. Closed-loop stability and recursive feasibility under the approximate feedback policy are also established. We provide a library of numerical examples to illustrate our results.Comment: 31 Page

Robust model predictive control: robust control invariant sets and efficient implementation

Robust model predictive control (RMPC) is widely used in industry. However, the online computational burden of this algorithm restricts its development and application to systems with relatively slow dynamics. We investigate this problem in this thesis with the overall aim of reducing the online computational burden and improving the online efficiency. In RMPC schemes, robust control invariant (RCI) sets are vitally important in dealing with constraints and providing stability. They can be used as terminal (invariant) sets in RMPC schemes to reduce the online computational burden and ensure stability simultaneously. To this end, we present a novel algorithm for the computation of full-complexity polytopic RCI sets, and the corresponding feedback control law, for linear discrete-time systems subject to output and initial state constraints, performance bounds, and bounded additive disturbances. Two types of uncertainty, structured norm-bounded and polytopic uncertainty, are considered. These algorithms are then extended to deal with systems subject to asymmetric initial state and output constraints. Furthermore, the concept of RCI sets can be extended to invariant tubes, which are fundamental elements in tube based RMPC scheme. The online computational burden of tube based RMPC schemes is largely reduced to the same level as model predictive control for nominal systems. However, it is important that the constraint tightening that is needed is not excessive, otherwise the performance of the MPC design may deteriorate, and there may even not exist a feasible control law. Here, the algorithms we proposed for RCI set approximations are extended and applied to the problem of reducing the constraint tightening in tube based RMPC schemes. In order to ameliorate the computational complexity of the online RMPC algorithms, we propose an online-offline RMPC method, where a causal state feedback structure on the controller is considered. In order to improve the efficiency of the online computation, we calculate the state feedback gain offline using a semi-definite program (SDP). Then we propose a novel method to compute the control perturbation component online. The online optimization problem is derived using Farkas' Theorem, and then approximated by a quadratic program (QP) to reduce the online computational burden. A further approximation is made to derive a simplified online optimization problem, which results in a large reduction in the number of variables. Numerical examples are provided that demonstrate the advantages of all our proposed algorithms over current schemes.Open Acces