Linear Control Theory with an ℋ∞ Optimality Criterion

This expository paper sets out the principal results in ℋ∞ control theory in the context of continuous-time linear systems. The focus is on the mathematical theory rather than computational methods

Decentralized Robust Capacity Control of Job Shop Systems with Reconfigurable Machine Tools

Manufacturing companies are confronted with various challenges from the perspective of customers individual requirements concerning variations of types of products, quantities and delivery dates. This renders the manufacturing process to be more dynamic and complex, which may result in bottlenecks and unbalanced capacity distributions. To cope with these problems, capacity adjustment is an effective approach to balance capacity and load for short or medium term fluctuations on the operational layer. Particularly, new technologies and algorithms need to be developed for the implementation of capacity adjustment. Reconfigurable machine tools (RMTs) and operator-based robust right coprime factorization (RRCF) provide an opportunity for a new capacity control strategy. Therefore, the main purpose of the research is to develop an effective machinery-oriented capacity control strategy by incorporating RMTs and RRCF for a job shop system to deal with volatile customer demands

Optimization methods for deadbeat control design: a state space approach

This thesis addresses the synthesis problem of state deadbeat regulator using state space techniques. Deadbeat control is a linear control strategy in discrete time systems and consists of driving the system from any arbitrary initial state to a desired final state infinite number of time steps. Having described the framework for development of the thesis which is in the form of a lower linear-fractional transformation (LFT), the conditions for internal stability based on the notion of coprime factorization over the set of proper and stable transfer matrices, namely RH, is discussed. This leads to the derivation of the class of all stabilizing linear controllers, which are parameterized affinely in terms of a stable but otherwise free parameter Q, usually known as the Q-parameterization. In this work, the classical Q- parameterization is generalized to deliver a parameterization for the family of deadbeat regulators. Time response characteristics of the deadbeat system are investigated. In particular, the deadbeat regulator design problem in which the system must satisfy time domain specifications and minimize a quadratic (LQG-type) performance criterion is examined. It is shown that the attained parameterization for deadbeat controllers leads to the formulation of the synthesis problem in a quadratic programming framework with Q regarded as the design variable. The equivalent formulation of this objective as a quadratic integral in the frequency domain provides the means for shaping the frequency response characteristics of the system. Using the LMI characterization of the standard H problem, a new scheme for shaping the system frequency response characteristics by minimizing the infinity norm of an appropriate closed-loop transfer function is introduced. As shown, the derived parameterization of deadbeat compensators simplifies considerably the formulation and solution of this problem. The last part of the work described in this thesis is devoted to addressing the synthesis problem of deadbeat regulators in a robust way, when the plant is subject to structured norm-bounded parametric uncertainties. A novel approach which is expressed as an LMI feasibility condition has been proposed and analysed

Guaranteed safe switching for switching adaptive control

Adaptive control algorithms may not behave well in practice due to discrepancies between the theory and actual practice. The proposed results in this manuscript constitute an effort in providing algorithms which assure more reliable operation in practice. Our emphasis is on algorithms that will be safe in the sense of not permitting destabilizing controllers to be switched in the closed-loop and to prevent wild signal fluctuations to occur. Coping with the connection or possible connection of destabilizing controllers is indeed a daunting task. One of the most intuitive forms of adaptive control, gain scheduling, is an approach to control of non-linear systems which utilizes a family of linear controllers, each of which provides satisfactory control for a different operating point of the system. We provide a mechanism for guaranteeing closed-loop stability over rapid switching between controllers. Our proposed design provides a simplification using only finite number of pre-determined values for the controller gain, where the observer gain is computed via a table look-up method. In comparison to the original gain scheduling design which our procedure builds on, our design achieves similar performance but with much less computational burden. Many multi-controller adaptive switching algorithms do not explicitly rule out the possibility of switching a destabilizing controller into the closed-loop. Even if the new controller is ensured to be stabilizing, performance verification with the new controller is not straightforward. The importance of this arises in iterative identification and control algorithms and multiple model adaptive control (MMAC). We utilize a limited amount of experimental and possibly noisy data obtained from a closed-loop consisting of an existing known stabilizing controller connected to an unknown plant-to infer if the introduction of a prospective controller will stabilize the unknown plant. We propose analysis results in a nonlinear setting and provide data-based tests for verifying the closed-loop stability with the introduction of a new nonlinear controller to replace a linear controller. We also propose verification tools for the closed-loop performance with the introduction of a new stabilizing controller using a limited amount of data obtained from the existing stable closed-loop. The simulation results in different practical scenarios demonstrate efficacy and versatility of our results, and illustrate practicality of our novel data-based tests in addressing an instability problem in adaptive control algorithms

Finite settling time stabilization for linear multivariable time-invariant discrete-time systems: An algebraic approach

The problem of Total Finite Settling Time Stabilization of linear time-invariant discrete-time systems is investigated in this thesis. This problem falls within the same area of the well-known deadbeat (time-optimal) control and in particular, constitutes a generalization of this problem. That is, instead of seeking time-optimum performance, it is required that all internal and external variables (signals) of the closed-loop system settle to a new steady state after a finite time from the application of a step change to any of its inputs and for every initial condition. The state/output deadbeat control is a special case of the Total FSTS problem. Using a mathematical and system theory framework based on sequences and the polynomial equation (algebraic) approach, we are able to tackle the FSTS problem in a unifying manner. The one-parameter (unity) feedback configuration is mainly used for the solution of the FSTS problem and FSTS related control strategies. The whole problem is reduced to the solution of a polynomial matrix Diophantine equation which guarantees not only internal stability but also internal FSTS and is further reduced to the solution of a linear algebra problem over R. This approach enables the parametrizat ion of the family of all FSTS controllers, as well as those which are causal, in a Youla-Bongiorno-Kucera type parametrization. The minimal McMillan degree FSTS problem is completely solved for vector plants and a parametrization of the FSTS controllers according to their McMillan degree is obtained. In the MIMO case bounds of the minimum McMillan degree controllers are derived and families of FSTS controllers with given lower/upper McMillan degree bounds are provided in parametric form. Having parametrized the family of all FSTS controllers, the state deadbeat regulation is treated as a special case of FSTS and complete parametrization of all the deadbeat regulators is presented. In addition, further performance criteria, or design constraints are imposed such as, FSTS tracking and/or disturbance rejection, partial assignment of controller dynamics, l1-, l∞-optimization and robustness to plant parameter variations. Finally, the Simultaneous-FSTS problem is formulated, and necessary as well as sufficient conditions for its solution are derived. Also, a two-parameter control scheme is introduced to alleviate some of the drawbacks of the one-parameter control. A parametrization of the family of FSTS controllers as well as the FSTS controllers for tracking and/or disturbance rejection is given as an illustration of the particular advantages of the two-parameter FSTS controllers

Algebraic geometric methods for the stabilizability and reliability of multivariable and of multimode systems

The extent to which feedback can alter the dynamic characteristics (e.g., instability, oscillations) of a control system, possibly operating in one or more modes (e.g., failure versus nonfailure of one or more components) is examined

Study on Spectrum Estimation in Biophoton Emission Signal Analysis of Wheat Varieties

The photon emission signal in visible range (380 nm–630 nm) was measured from various wheat kernels by means of a low noise photomultiplier system. To study the features of the photon emission signal, the spectrum estimation method of the photon emission signal is described for the first time. The biophoton emission signal, belonging to four varieties of wheat, is analyzed in time domain and frequency domain. It shows that the intensity of the biophoton emission signal for four varieties of wheat kernels is relatively weak and has dramatic changes over time. Mean and mean square value are obviously different in four varieties; the range was, respectively, 3.7837 and 74.8819. The difference of variance is not significant. The range is 1.1764. The results of power spectrum estimation deduced that the biophoton emission signal is a low frequency signal, and its power spectrum is mostly distributed in the frequency less than 0.1 Hz. Then three parameters, which are spectral edge frequency, spectral gravity frequency, and power spectral entropy, are adopted to explain the features of the kernels’ spontaneous biophoton emission signal. It shows that the parameters of the spontaneous biophoton emission signal for different varieties of wheat are similar

데이터 기반 궤적 최적화를 결합한 계산 효율적인 다중선형 모델 기반 제어

학위논문 (박사) -- 서울대학교 대학원 : 공과대학 화학생물공학부, 2021. 2. 이종민.Model predictive control (MPC) is a widely used advanced control strategy applied in the process industry due to its capability to handle multivariate systems and constraints. When applied to nonlinear processes, linear MPC (LMPC) is limited to a relatively small operating region. On the other hand, nonlinear MPC (NMPC) is challenging due to the need for a nonlinear model with a large domain of validity and the computational load to solve nonlinear optimization problems. Multilinear MPC (MLMPC) or linear time-varying MPC (LTVMPC) complements the limitations, employing multiple linear models to predict dynamic behavior in a wide operating range. However, the main issue is obtaining the linear models, which is difficult to obtain without the nonlinear model and a trajectory from an initial condition to a set-point. Differential dynamic programming (DDP) can help to get the linear models and the suboptimal trajectory simultaneously. DDP iteratively improves the trajectory with the linear models of the previous trajectory, which can be identified by excitation of input around the trajectory. We propose four novel methodologies in the thesis. First, we propose a scheme to design MLMPC based on gap metric, which achieves convergence to LMPC and offset-free tracking. Second, we propose a switching strategy of MLMPC. It consists of a design of the subregions from an initial point to a set-point and LMPC for each subregion. Next, we develop a scheme that combines constrained differential dynamic programming (CDDP) and MLMPC, starting without any models. Finally, we developed an algorithm that combines LTVMPC and LMPC based on the models from CDDP. It exploits the suboptimal trajectory from CDDP and achieves offset-free tracking. We apply developed MPC algorithms to an illustrative example for validation. It also supports that multiple linear models are appropriate to control nonlinear processes with or without the nonlinear models.모델예측제어 (model predictive control) 는 산업에서 널리 쓰이는 고급 공정 제어 기법으로, 다변수 시스템의 동역학과 제약 조건을 고려하여 실시간으로 현재 상태에 대해 최적해를 도출해낸다. 선형 모델을 이용하는 선형 모델예측제어 (linear model predictive control) 가 가장 간단하고 많은 이론들이 정립되어 있으나, 실제 비선형 공정에서는 선형 모델이 근사할 수 있는 좁은 운전 조건에서만 사용할 수 있다는 한계가 있다. 비선형 모델예측제어 (nonlinear model predictive control) 는 비선형 모델을 이용하여 넓은 운전 조건에서도 최적해를 제공할 수 있지만, 비선형 최적화 문제를 실시간으로 풀어야 하기 때문에 샘플링 타임이 작을 경우, 적용하기 어렵다는 한계가 있다. 다중 선형 모델예측제어 (multilinear model predictive control) 또는 선형 시변 모델예측제어 (linear time-varying model predictive control) 는 여러 개의 선형 모델을 이용하여 넓은 운전 범위에서 공정의 거동을 표현하고 최적에 가까운 해를 제공할 수 있기 때문에 앞선 두 가지 제어 기법의 한계를 보완할 수 있다. 모델예측제어 기법을 실제 비선형 공정에 적용하기에 또 다른 어려운 점은 실제 공정의 비선형 모델을 얻기가 힘들다는 것이다. 미분동적계획법 (differential dynamic programming) 은 이러한 상황에서 현재 공정 운전 데이터에 기반해 동적 최적화 (dynamic optimization) 를 수행하여 초기 조건에서 설정점까지의 최적에 가까운 경로를 찾아 모델예측제어 기법을 적용할 수 있도록 도움을 줄 수 있다. 구체적으로, 미분동적계획법은 현재 공정 운전 데이터를 이용해 운전 데이터 근처의 거동을 묘사하는 선형 모델들을 얻고, 이를 이용하여 반복적으로 다음 운전에서의 최적해를 제공하여 운전 궤적을 개선한다. 본 학위 논문에서는 넓은 운전 범위를 가진 공정에서 운전 조건을 변경하기에 적합한 다중 선형 모델예측제어와 선형 시변 모델예측제어 전략을 제시한다. 첫번째로, gap metric을 이용하여 설정점에서 모델예측제어를 적용한 시스템의 안정성을 보장하고 잔류 편차를 제거하는 다중 선형 모델예측제어 기법을 제시한다. 두번째로, 다중 선형 모델예측제어기의 진동 가능성을 막기 위해, 초기 조건에서 설정점까지의 구간을 gap metric에 기반하여 나누고, 각각의 구간에서의 하위 설정점들을 정하여 초기 조건에서 설정점까지의 경로를 하위 설정점들의 그래프로 표현하고, 각 하위 설정점까지 각각 배정된 선형 모델예측제어기를 이용하여 설정점까지 도달하게 하는 제어 전략을 제시한다. 다음으로는 공정의 모델이 없을 때, 공정의 입력 제약 조건을 고려하는 미분동적계획법을 이용하여 최적에 가까운 개루프 (open-loop) 제어 입력과 해당 운전 데이터 근방을 근사하는 선형 모델들을 얻어, 다중 선형 모델예측제어를 적용하여 설정점까지 도달하고 잔류 편차를 제거하는 제어 전략을 제시한다. 마지막으로, 미분동적계획법이 제공하는 준최적 (suboptimal) 운전 데이터를 활용하기 위해, 선형 시변 모델예측제어 기법과 잔류편차-제거 모델예측제어 기법을 이어서 사용하는 전략이 제시되었다. 구체적으로, 제공된 준최적 운전데이터를 과도 응답 (transient response) 과 정상 상태 응답 (steady-state response) 이 나타나는 구간으로 나누고, 선형 시변 모델예측제어를 통해 과도 응답에서의 준최적 궤적을 추적하고 상태 변수가 정상 상태에 가까워지면 잔류편차-제거 모델예측제어를 적용해 설정점에 도달하도록 한다. 제안된 기법들을 공정 예제에 적용하여 공정의 모델 유무와 관계없이 선형 모델들을 이용한 모델예측제어 기법이 넓은 운전 조건을 가진 비선형 공정의 공정 조건을 이동하기에 적합한 방법론임을 검증하였다.Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Motivation and previous work . . . . . . . . . . . . . 1 1.2 Statement of contributions . . . . . . . . . . . . . . . 4 1.3 Outline of the thesis . . . . . . . . . . . . . . . . . . 7 2. Background and preliminaries . . . . . . . . . . . . . 8 2.1 Offset-free linear model predictive control . . . . . . 8 2.2 Gap metric and stability margin . . . . . . . . . . . . 12 2.3 Multilinear model predictive control . . . . . . . . . 19 2.4 Linear time-varying model predictive control . . . . . 22 2.5 Differential dynamic programming . . . . . . . . . . 24 3. Offset-free multilinear model predictive control based on gap metric . . . . . . . . . . . . . . . . . . . . . . . 28 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 28 3.2 Local linear MPC design . . . . . . . . . . . . . . . . 31 3.3 Gap metric-based multilinear MPC . . . . . . . . . . 35 3.3.1 Gap metric-based gridding algorithm . . . . . 36 3.3.2 Gap metric-based K-medoids clustering . . . . 39 3.3.3 MLMPC design . . . . . . . . . . . . . . . . 42 3.4 Results and discussions . . . . . . . . . . . . . . . . 50 3.4.1 Example 1 (SISO CSTR) . . . . . . . . . . . 50 3.4.2 Example 2 (MIMO CSTR) . . . . . . . . . . . 62 4. Switching multilinear model predictive control based on gap metric . . . . . . . . . . . . . . . . . . . . . . . 75 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 75 4.2 Shortest path problem . . . . . . . . . . . . . . . . . 76 4.3 Switching Multilinear Model Predictive Control . . . 80 4.3.1 Local MPC design . . . . . . . . . . . . . . . 80 4.3.2 Path design based on gap metric . . . . . . . . 84 4.3.3 Global MPC design . . . . . . . . . . . . . . 90 4.4 Results and discussions . . . . . . . . . . . . . . . . 96 5. Design of data-driven multilinear model predictive control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 112 5.2 Data-driven trajectory optimization . . . . . . . . . . 114 5.2.1 Constrained differential dynamic programming 114 5.2.2 Model identification around a trajectory . . . . 123 5.3 Data-driven offset-free MLMPC . . . . . . . . . . . . 126 5.3.1 Gap metric-based clustering algorithm . . . . 126 5.3.2 Prediction-based MLMPC . . . . . . . . . . . 129 5.4 Results and discussions . . . . . . . . . . . . . . . . 134 6. Design of data-driven linear time-varying model predictive control . . . . . . . . . . . . . . . . . . . . . . 153 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 153 6.2 Design of data-driven linear time-varying model predictive control . . . . . . . . . . . . . . . . . . . . . 154 6.2.1 Gap metric-based model selection . . . . . . . 154 6.2.2 Offset-free linear time-varying model predictive control . . . . . . . . . . . . . . . . . . . 158 6.3 Results and discussions . . . . . . . . . . . . . . . . 167 7. Conclusions and future works . . . . . . . . . . . . . . 187 7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . 187 7.2 Future works . . . . . . . . . . . . . . . . . . . . . . 188 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 190Docto