MIMO nonlinear PID predictive controller

A class of nonlinear generalised predictive controllers (NGPC) is derived for multi-input multi-output (MIMO) nonlinear systems with offset or steady-state response error. The MIMO composite controller consists of an optimal NGPC and a nonlinear disturbance observer. The design of the nonlinear disturbance observer to estimate the offset is particularly simple, as is the associated proof of overall nonlinear closed-loop system stability. Moreover, the transient error response of the disturbance observer can be arbitrarily specified by simple design parameters. Very satisfactory performance of the proposed MIMO nonlinear predictive controller is demonstrated for a three-link nonlinear robotic manipulator example

Lyapunov-based Control Design For Uncertain Mimo Systems

In this dissertation. we document the progress in the control design for a class of MIMO nonlinear uncertain system from five papers. In the first part, we address the problem of adaptive control design for a class of multi-input multi-output (MIMO) nonlinear systems. A Lypaunov based singularity free control law, which compensates for parametric uncertainty in both the drift vector and the input gain matrix, is proposed under the mild assumption that the signs of the leading minors of the control input gain matrix are known. Lyapunov analysis shows global uniform ultimate boundedness (GUUB) result for the tracking error under full state feedback (FSFB). Under the restriction that only the output vector is available for measurement, an output feedback (OFB) controller is designed based on a standard high gain observer (HGO) stability under OFB is fostered by the uniformity of the FSFB solution. Simulation results for both FSFB and OFB controllers demonstrate the efcacy of the MIMO control design in the classical 2-DOF robot manipulator model. In the second part, an adaptive feedback control is designed for a class of MIMO nonlinear systems containing parametric uncertainty in both the drift vector and the input gain matrix, which is assumed to be full-rank and non-symmetric in general. Based on an SDU decomposition of the gain matrix, a singularity-free adaptive tracking control law is proposed that is shown to be globally asymptotically stable (GAS) under full-state feedback. iii Output feedback results are facilitated via the use of a high-gain observer (HGO). Under output feedback control, ultimate boundedness of the error signals is obtained the size of the bound is related to the size of the uncertainty in the parameters. An explicit upper bound is also provided on the size of the HGO gain constant. In third part, a class of aeroelastic systems with an unmodeled nonlinearity and external disturbance is considered. By using leading- and trailing-edge control surface actuations, a full-state feedforward/feedback controller is designed to suppress the aeroelastic vibrations of a nonlinear wing section subject to external disturbance. The full-state feedback control yields a uniformly ultimately bounded result for two-axis vibration suppression. With the restriction that only pitching and plunging displacements are measurable while their rates are not, a high-gain observer is used to modify the full-state feedback control design to an output feedback design. Simulation results demonstrate the ef cacy of the multi-input multioutput control toward suppressing aeroelastic vibration and limit cycle oscillations occurring in pre and post utter velocity regimes when the system is subjected to a variety of external disturbance signals. Comparisons are drawn with a previously designed adaptive multi-input multi-output controller. In the fourth part, a continuous robust feedback control is designed for a class of high-order multi-input multi-output (MIMO) nonlinear systems with two degrees of freedom containing unstructured nonlinear uncertainties in the drift vector and parametric uncertainties in the high frequency gain matrix, which is allowed to be non-symmetric in general. Given some mild assumptions on the system model, a singularity-free continuous robust tracking coniv trol law is designed that is shown to be semi-globally asymptotically stable under full-state feedback through a Lyapunov stability analysis. The performance of the proposed algorithm have been verified on a two-link robot manipulator model and 2-DOF aeroelastic model

Reinforcement Learning-Based Output Feedback Control of Nonlinear Systems with Input Constraints

A novel neural network (NN) -based output feedback controller with magnitude constraints is designed to deliver a desired tracking performance for a class of multi-input-multi-output (MIMO) discrete-time strict feedback nonlinear systems. Reinforcement learning in discrete time is proposed for the output feedback controller, which uses three NN: 1) a NN observer to estimate the system states with the input-output data; 2) a critic NN to approximate certain strategic utility function; and 3) an action NN to minimize both the strategic utility function and the unknown dynamics estimation errors. The magnitude constraints are manifested as saturation nonlinearities in the output feedback controller design. Using the Lyapunov approach, the uniformly ultimate boundedness (UUB) of the state estimation errors, the tracking errors and weight estimates is shown

Adaptive Output Feedback Based on Closed-Loop Reference Models for Hypersonic Vehicles

This paper presents a new method of synthesizing an output feedback adaptive controller for a class of uncertain, non-square, multi-input multi-output systems that often occur in hypersonic vehicle models. The main challenge that needs to be addressed is the determination of a corresponding square and strictly positive real transfer function. This paper proposes a new procedure to synthesize two gain matrices that allows the realization of such a transfer function, thereby allowing a globally stable adaptive output feedback law to be generated. The unique features of this output feedback adaptive controller are a baseline controller that uses a Luenberger observer, a closed-loop reference model, manipulations of a bilinear matrix inequality, and the Kalman-Yakubovich Lemma. Using these features, a simple design procedure is proposed for the adaptive controller, and the corresponding stability property is established. The proposed adaptive controller is compared to the classical multi-input multi-output adaptive controller. A numerical example based on a scramjet powered, blended wing-body generic hypersonic vehicle model is presented. The 6 degree-of-freedom nonlinear vehicle model is linearized, giving the design model for which the controller is synthesized. The adaptive output feedback controller is then applied to an evaluation model, which is nonlinear, coupled, and includes actuator dynamics, and is shown to result in stable tracking in the presence of uncertainties that destabilize the baseline linear output feedback controller.This research is funded by the Air Force Research Laboratory/Aerospace Systems Directorate grant FA 8650-07-2-3744 for the Michigan/MIT/AFRL Collaborative Center in Control Sciences and the Boeing Strategic University Initiative. Approved for Public Release; Distribution Unlimited. Case Number 88ABW- 2014-2551

Estimation and Control of Dynamical Systems with Applications to Multi-Processor Systems

System and control theory is playing an increasingly important role in the design and analysis of computing systems. This thesis investigates a set of estimation and control problems that are driven by new challenges presented by next-generation Multi-Processor Systems on Chips (MPSoCs). Specifically, we consider problems related to state norm estimation, state estimation for positive systems, sensor selection, and nonlinear output tracking. Although these problems are motivated by applications to multi-processor systems, the corresponding theory and algorithms are developed for general dynamical systems. We first study state norm estimation for linear systems with unknown inputs. Specifically, we consider a formulation where the unknown inputs and initial condition of the system are bounded in magnitude, and the objective is to construct an unknown input norm-observer which estimates an upper bound for the norm of the states. This class of problems is motivated by the need to estimate the maximum temperature across a multi-core processor, based on a given model of the thermal dynamics. In order to characterize the existence of the norm observer, we propose a notion of bounded-input-bounded-output-bounded-state (BIBOBS) stability; this concept supplements various system properties, including bounded-input-bounded-output (BIBO) stability, bounded-input-bounded-state (BIBS) stability, and input-output-to-state stability (IOSS).We provide necessary and sufficient conditions on the system matrices under which a linear system is BIBOBS stable, and show that the set of modes of the system with magnitude 1 plays a key role. A construction for the unknown input norm-observer follows as a byproduct. Then we investigate the state estimation problem for positive linear systems with unknown inputs. This problem is also motivated by the need to monitor the temperature of a multi-processor system and the property of positivity arises due to the physical nature of the thermal model. We extend the concept of strong observability to positive systems and as a negative result, we show that the additional information on positivity does not help in state estimation. Since the states of the system are always positive, negative state estimates are meaningless and the positivity of the observers themselves may be desirable in certain applications. Moreover, positive systems possess certain desired robustness properties. Thus, for positive systems where state estimation with unknown inputs is possible, we provide a linear programming based design procedure for delayed positive observers. Next we consider the problem of selecting an optimal set of sensors to estimate the states of linear dynamical systems; in the context of multi-core processors, this problem arises due to the need to place thermal sensors in order to perform state estimation. The goal is to choose (at design-time) a subset of sensors (satisfying certain budget constraints) from a given set in order to minimize the trace of the steady state a priori or a posteriori error covariance produced by a Kalman filter. We show that the a priori and a posteriori error covariance-based sensor selection problems are both NP-hard, even under the additional assumption that the system is stable. We then provide bounds on the worst-case performance of sensor selection algorithms based on the system dynamics, and show that certain greedy algorithms are optimal for two classes of systems. However, as a negative result, we show that certain typical objective functions are not submodular or supermodular in general. While this makes it difficult to evaluate the performance of greedy algorithms for sensor selection (outside of certain special cases), we show via simulations that these greedy algorithms perform well in practice. Finally, we study the output tracking problem for nonlinear systems with constraints. This class of problems arises due to the need to optimize the energy consumption of the CPU-GPU subsystem in multi-processor systems while satisfying certain Quality of Service (QoS) requirements. In order for the system output to track a class of bounded reference signals with limited online computational resources, we propose a sampling-based explicit nonlinear model predictive control (ENMPC) approach, where only a bound on the admissible references is known to the designer a priori. The basic idea of sampling-based ENMPC is to sample the state and reference signal space using deterministic sampling and construct the ENMPC by using regression methods. The proposed approach guarantees feasibility and stability for all admissible references and ensures asymptotic convergence to the set-point. Furthermore, robustness through the use of an ancillary controller is added to the nominal ENMPC for a class of nonlinear systems with additive disturbances, where the robust controller keeps the system output close to the desired nominal trajectory

축소 차원 동적 관측기 오차 선형화와 확장된 비선형 관측기 정준형을 통한 비선형 관측기 설계

학위논문 (박사)-- 서울대학교 대학원 : 전기·컴퓨터공학부, 2014. 8. 서진헌.본 논문은 비선형 시스템에 대한 관측기 설계 문제를 다루고 있다. 관측기 설계 문제란 주어진 시스템의 입력과 출력 정보만을 활용하여 대상 시스템의 상태 변수를 추정할 수 있는 시스템을 설계하는 것이다. 선형 시스템의 경우에는 루엔버거 관측기(Luenberger observer)로 알려진 일반적인 해법이 존재하는 반면, 일반적인 비선형 시스템에 대해 관측기를 설계하는 방법에 대한 연구 결과는 현재까지 보고된 바가 없다. 다만, 특정한 형태의 비선형 시스템에 대해 관측기를 설계하는 문제에 대한 연구는 활발하게 진행되어 오고 있다. 관측기 오차 선형화(observer error linearization) 기법은 이 문제에 대한 가장 잘 알려진 방법론 중의 하나로서, 주어진 비선형 시스템을 좌표 변환을 통해 관측 가능한 선형 시스템과 출력주입(output injection) 부분들로 구성된 비선형 관측기 정준형(nonlinear observer canonical form)으로 변환시키는 문제이다. 비선형 관측기 정준형으로 변환 가능한 좌표계에서는 시스템의 모든 비선형성이 시스템의 입력과 출력의 함수로 이루어진 출력 주입 부분에 제한되므로, 이를 상쇄시킴으로써 선형 시스템의 경우와 비슷한 형태의 루엔버거형의 관측기(Luenberger-type observer)를 설계하는 것이 가능하고, 이에 따라 선형화된 관측기 오차 동역학(observer error dynamics)을 얻을 수 있다. 관측기 오차 선형화 기법의 출현 이래로, 이를 적용할 수 있는 시스템의 범위를 확장시키기 위한 여러 연구가 진행되어 왔다. 그 중 하나는 주어진 시스템을 보다 높은 차수의 비선형 관측기 정준형으로 변환시키는 방법이다. 이러한 방식에는 시스템 이머젼 기법과 동적 관측기 오차 선형화(dynamic observer error linearization) 기법이 있는데, 그 중에서도 동적 관측기 오차 선형화 기법의 특징은 다음과 같이 크게 두 가지로 요약될 수 있다. 첫째는 대상 시스템의 출력을 입력으로 하는 보조 동역학(auxiliary dynamics)을 설계하는 것이고, 둘째는 보조 동역학을 포함하는 확장된 시스템을 대상 시스템보다 높은 차수의 일반화된 비선형 관측기 정준형(generalized nonlinear observer canonical form)으로 변환하는 것이다. 동적 관측기 오차 선형화 기법에서 제안된 일반화된 비선형 관측기 정준형은 관측 가능한 선형 시스템과 일반화된 출력 주입(generalized output injection)으로 구성되어 있고, 일반화된 출력 주입은 대상 시스템의 출력 뿐만 아니라 보조동역학의 상태 변수에 대한 함수로 이루어져 있다는 차이점이 있다. 하지만, 이 방법론은 관측기의 차수가 대상 시스템의 차수보다 크다는 단점을 가지고 있다. 이러한 문제를 해결하기 위해, 최근에는 동적 관측기 오차 선형화의 변형된 기법으로서 축소 차원 동적 관측기 오차 선형화(reduced-order dynamic observer error linearization)란 기법이 단일 출력 시스템에 대해 새롭게 제안되었다. 축소 차원 동적 관측기 오차 선형화 기법 역시 보조 동역학을 설계하여 확장된 시스템을 일반화된 비선형 관측기 정준형으로 변환시킨다는 점에서 동적 관측기 오차 선형화 기법과 공통점을 갖지만, 변환된 일반화된 비선형 관측기 정준형의 차수가 대상 시스템의 차수와 같다는 차이점이 있다. 비록 축소 차원 동적 관측기 오차 선형화 기법이 적용 가능한 시스템의 범주는 동적 관측기 선형화 기법이 적용 가능한 시스템 범주를 벗어날 수는 없지만, 축소 차원 동적 관측기 오차 선형화 기법은 동적 관측기 선형화 기법에 비해 더 작은 차수의 관측기를 설계할 수 있다는 이점이 있고, 보조 동역학의 개념을 도입함으로써 관측기 오차 선형화 기법에 비해 더 넓은 범주의 시스템에 적용 가능하다는 장점을 지닌다. 뿐만 아니라, 축소 차원 동적 관측기 오차 선형화 기법의 개념 자체가 관측기 오차 선형화 기법의 개념과 매우 흡사하기 때문에 (보조 동역학을 고려하지 않은 축소 차원 동적 관측기 오차 선형화 문제는 관측기 오차 선형화 문제와 일치한다.) 축소 차원 동적 관측기 오차 선형화 기법에 대한 연구를 통해 기존의 관측기 오차 선형화 기법을 해석할 수도 있다. 이에 따라, 본 논문에서는 축소 차원 동적 관측기 오차 선형화 기법을 다중 출력 시스템에 대해 확장시키고, 이에 대한 연구를 수행하여 궁극적으로는 주어진 다중 출력 시스템이 이 기법에 의해 일반화된 비선형 관측기 정준형으로 변환될 수 있는 필요충분 조건을 제시한다. 이 결과는 현재까지 확립되지 않았던 일반적인 형태의 출력 변환까지 고려하였을 경우의 다중 출력 시스템에 대한 관측기 오차 선형화 문제의 필요충분 조건을 내포하고 있다. 또한, 본 논문에서는 비선형 관측기 정준형의 선형 부분 또한 시스템의 출력과 보조 동역학의 상태 변수에 의해 결정되는 확장된 비선형 관측기 정준형(extended nonlinear observer canonical form)을 제안하고, 축소 차원 동적 관측기 오차 선형화의 확장된 기법으로서 주어진 단일 출력 시스템을 보조 동역학을 설계하여 확장된 비선형 관측기 정준형으로 변환하는 문제를 제안하고 이에 대한 필요충분 조건을 제시한다. 또한 이 결과를 뢰슬러 시스템(Rossler system)에 적용시켜봄으로써 새롭게 제안된 방법론이 축소 차원 동적 관측기 오차 선형화에 비해 더 넓은 범주의 시스템에 적용될 수 있음을 예증한다.This dissertation contributes to the observer design problem for some classes of nonlinear systems. The observer design problem is to construct a dynamic system (called observer) that can estimate the state of a given dynamic system by using available signals which are commonly the input and the output of the given system. While a standard solution (called Luenberger observer) to the problem was solved for linear systems, there has not been a unified solution for general nonlinear systems. However, there have been significant research efforts on the problem of designing observers for special classes of nonlinear systems. Observer error linearization (OEL) is one of the well-known methods, and it is the problem of transforming a nonlinear system into a nonlinear observer canonical form (NOCF) that is an observable linear system modulo output injection. If a nonlinear system can be transformed into the NOCF, then all the nonlinearities of the system are restricted to the output injection term which is a vector-valued function of the system input and the system output. As a result, we can design a Luenberger-type observer that cancels out the output injection and thus has a linear observer error dynamics in the transformed coordinates. In order to extend the class of systems to which the OEL approach is applicable, a lot of attempts have been made in the past three decades. One of them is to transform a nonlinear system into a higher-dimensional NOCF: system immersion and dynamic observer error linearization (DOEL). In particular, the main idea of DOEL is twofold: the first is to introduce an auxiliary dynamics whose input is system output, and the second is to transform the extended system into a generalized nonlinear observer canonical form (GNOCF) that is an observable linear system modulo generalized output injection depending not only on the system output but also on the state of auxiliary dynamics. By introducing such an auxiliary dynamics, the DOEL problem can be solved for a larger class of systems compared with the (conventional) OEL problem. However, it has a drawback on the dimension of observer. That is, the dimension of observer designed by the DOEL approach is larger than that of the given system, because the dimension of GNOCF equals to the sum of dimensions of the given system and the auxiliary dynamics. Recently, inspired by this fact, a new approach called reduced-order dynamic observer error linearization (RDOEL) was proposed for single output nonlinear systems. In the framework of RDOEL, we also introduce an auxiliary dynamics and transform the extended system into GNOCF in a similar fashion to DOEL, but the coordinate transformation preserves the coordinates corresponding to the state of auxiliary dynamics so that the dimension of GNOCF equals to that of the given system. Although RDOEL is a special case of DOEL (that is, the class of systems to which the RDOEL approach can be applied is a subset of that of DOEL), the RDOEL approach offers a lower-dimensional observer compared to the DOEL approach, and it is also applicable to a larger class of systems compared to the (conventional) OEL approach. In addition, since the framework of RDOEL is coterminous with that of OEL (in fact, the OEL problem is identical to the RDOEL problem with no auxiliary dynamics), most of results for the RDOEL problem can be also used to analyze the OEL problem by slight modification. In this respect, one of the topics of this dissertation is to deal with the RDOEL problem for multi-output systems. We first formulate the framework of RDOEL for multi-output nonlinear systems and provide three necessary conditions. And then, by means of the necessary conditions, we derive a geometric necessary and sufficient condition in terms of Lie algebras of vector fields. Since the proposed RDOEL problem is a natural extension of the (conventional) OEL problem, the result can be easily translated into a geometric necessary and sufficient condition for the OEL problem, which has not yet been completely established in the case where an output transformation of general form is considered. The other topic of the dissertation is to introduce an extended nonlinear observer canonical form (ENOCF) whose linear part also depends on the system output and the state of auxiliary dynamics, and to deal with the problem of transforming a single output nonlinear system with an auxiliary dynamics into the ENOCF as an extension of the RDOEL problem. Since the proposed ENOCF admits a kind of high-gain observers, the solvability of the problem allows us to design observers for a class of single output nonlinear systems. We also first present two necessary conditions, and then derive a geometric necessary and sufficient condition for the problem. Furthermore, as a case study, we apply the results to the Rӧssler system in order to show that the proposed method enlarges the class of applicable systems compared with the RDOEL approach.ABSTRACT i List of Figures ix Notation and Acronyms x 1 Introduction 1 1.1 Research Background 1 1.2 Organization and Contributions of the Dissertation 5 2 Mathematical Preliminaries 7 2.1 Manifolds and Differentiable Structures 7 2.2 Vector Fields and Covector Fields 10 2.3 Lie Derivatives and Lie Brackets 13 2.4 Distributions and Codistributions 16 3 Review of Related Previous Works 21 3.1 Observability of Multi-Output Nonlinear Systems 21 3.2 Observer Error Linearization (OEL) 23 3.3 System Immersion 28 3.4 Dynamic Observer Error Linearization (DOEL) 30 3.5 Reduced-Order Dynamic Observer Error Linearization (RDOEL) for Single Output Systems 36 3.6 Inclusion Relation among OEL, System Immersion, DOEL, and RDOEL 39 4 Reduced-Order Dynamic Observer Error Linearization (RDOEL) for Multi-Output Systems 43 4.1 Problem Statement 43 4.2 Necessary Conditions 47 4.2.1 Observability 47 4.2.2 Inverse Output Transformation 52 4.2.3 System Dynamics 61 4.3 Necessary and Sufficient Conditions 65 4.3.1 Necessary and Sufficient Condition for RDOEL 65 4.3.2 Necessary and Sufficient Condition for OEL 80 4.3.3 Procedure to Solve OEL and RDOEL 81 4.4 Illustrative Examples 85 5 Extension of RDOEL: System into Extended Nonlinear Observer Canonical Form (ENOCF) 97 5.1 Problem Statement 99 5.2 Necessary Conditions 102 5.2.1 Output Transformation and Observability 102 5.2.2 System Dynamics 105 5.3 Necessary and Sufficient Condition 109 5.4 Case Study: Rӧssler System into ENOCF 117 6 Conclusions 125 BIBLIOGRAPHY 129 국문초록 139 감사의 글 143Docto

High gain observer for structured multi-output nonlinear systems

In this note, we present two system structures that characterize classes of multi-input multi-output uniformly observable systems. The first structure is decomposable into a linear and a nonlinear part while the second takes a more general form. It is shown that the second system structure, being more general, contains several system structures that are available in the literature. Two high gain observer design methodologies are presented for both structures and their distinct features are highlighted

Time-and event-driven communication process for networked control systems: A survey

Copyright © 2014 Lei Zou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.In recent years, theoretical and practical research topics on networked control systems (NCSs) have gained an increasing interest from many researchers in a variety of disciplines owing to the extensive applications of NCSs in practice. In particular, an urgent need has arisen to understand the effects of communication processes on system performances. Sampling and protocol are two fundamental aspects of a communication process which have attracted a great deal of research attention. Most research focus has been on the analysis and control of dynamical behaviors under certain sampling procedures and communication protocols. In this paper, we aim to survey some recent advances on the analysis and synthesis issues of NCSs with different sampling procedures (time-and event-driven sampling) and protocols (static and dynamic protocols). First, these sampling procedures and protocols are introduced in detail according to their engineering backgrounds as well as dynamic natures. Then, the developments of the stabilization, control, and filtering problems are systematically reviewed and discussed in great detail. Finally, we conclude the paper by outlining future research challenges for analysis and synthesis problems of NCSs with different communication processes.This work was supported in part by the National Natural Science Foundation of China under Grants 61329301, 61374127, and 61374010, the Royal Society of the UK, and the Alexander von Humboldt Foundation of Germany