How to Solve Modern Control Theory Questions ?

Introduction

Jiwei Wen , ... Dhafer J. Almakhles , in Non-Monotonic Approach to Robust H-infinity Control of Multi-model Systems, 2019

1.1.1 Nonmonotonic Approach

Modern control theory involves many research fields with a set of rigorous analysis and synthesis methods. In control systems theory, stability analysis is the foundation of almost all approaches. By taking the system uncertainties into consideration, the so-called robust stability is also a hot research topic in the last twenty years. The concepts of the asymptotic stability [1], global uniform asymptotic stability (GUAS) [2], stochastic stability [3], etc. usually aim at the equilibrium points of the dynamical systems. The depth and breadth of their theoretical developments are far beyond solving specific problems of certain control systems. As a sufficient condition, Lyapunov stability is a simple and straightforward approach to address the stability analysis problem by properly choosing or constructing a Lyapunov function (LF). However, from an engineering point of view, it inevitably introduces conservativeness to some extent. Therefore, many research efforts are devoted to conservatism reduction problem, for example, designing parameter-dependent LF [4], discussing the necessity condition [5], etc. Some of these approaches successfully reduced the conservatism to a certain extent; other approaches got the analysis results but brought severe difficulties to the controller synthesis. However, none of these approaches fundamentally solved all the analysis and synthesis problems of the controlled systems. Generally, it is believed that if the framework of Lyapunov stability based on the equilibrium point is not breached, then the excessive search for the necessary conditions will gradually remove the research from the engineering background. As a matter of fact, many engineering problems can be solved only by sufficient conditions.

The nonmonotonic Lyapunov function (NLF) is a relaxed version of LF and is based on such a simple idea: consider a proper LF $V>0$ and $V\to 0$ eventually; a natural question arises: Is the monotonic decreasing of V a necessary condition to guarantee $V\to 0$ as $t\to 0$ ?. Actually, some pioneering work already proposed NLFs in the continuous-time domain when Butz [6] tried to replace $\dot{V}<0$ by $\ddot{V}<0$ to obtain a sufficient condition of GUAS. The obtained condition in [6], however, cannot be transformed into a convex optimization problem. Subsequently, Yorke [7] gave convex conditions to address the Lyapunov stability, which cannot guarantee the GUAS. Based on the fundamental work of [6,7], Aeyels and Peuteman [8] developed an efficient approach, i.e., allowing the LF to occasionally increase on several small intervals. However, such an approach cannot achieve controller synthesis, and also the paper has not given detailed solutions in the discrete-time domain.

Since 2008, the NLF approach, which is aiming at the discrete-time nonlinear system, starts to penetrate into the research front line [9]. The main idea is to obtain a GUAS criteria with less conservatism by allowing the LF to increase locally within several sampling period. For the T-S fuzzy model, Derakhshan et al. employed NLF using only 2-samples variations $V(k+2)-V(k)<0$ . This approach allowed LF to increase locally at $k+1$ , which enlarges the stability region for a class of nonlinear systems. The state feedback synthesis [10], observer-based synthesis [11], and robust $H_2$ synthesis [12] have also been achieved. The authors in [13–16] extended the results to the general case $V(k+N)-V(k)<0$ , which allowed LF to increase in several sampling periods between the sampling point k and $k+N$ , to further reduce the conservativeness when obtaining the stable region of nonlinear systems. The corresponding nonquadratic state feedback stabilization [13], robust $H_{\infty }$ state feedback control [14], guaranteed cost control [15], and robust dynamic output feedback control [16] have been intensively studied using the general case of NLF with N-sample variations. The studies of NLF approach for T-S fuzzy model is also an important part of this book.

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Controllability and Observability of an LSS

Tong Zhou , ... Tao Li , in Estimation and Control of Large Scale Networked Systems, 2018

3.1 Introduction

A prominent characteristic of modern control theory is the adoption of a state space model, which uses a set of first-order differential equations in the description of system input–output dynamics. The states of a plant might be either its actual physical variables or some of their linear combinations. Usually, not all of these states can be directly manipulated and/or measured. An essential issue is therefore whether it is possible to maneuver a plant state to a desirable value using feasible inputs and to estimate a plant state through measurements of accessible variables. The former is usually called controllability of the plant, whereas the latter its observability [1,2].

It is now extensively known that controllability and observability are closely related to other important characteristics of a plant. For example, controllability is required to locate eigenvalues of a linear system into an arbitrary desirable area through state feedbacks, whereas to guarantee the existence of a linear control law that makes the $H_2$ or $H_{\infty }$ norm smaller than a prescribed value, the plant must be both controllable and observable. In addition to these, convergence of a state estimator is also closely related to the observability of a plant.

Controllability and observability are primarily formulated and investigated by Kalman [3] in his pursuit of system analysis and synthesis using a state space model. Through extensive pursuits of many researchers, various results have been obtained for the verification of the controllability and observability of a system. Originally, this problem is investigated for a linear dynamic system without any restrictions on plant inputs, states, and outputs. Afterward, these results are extended to more practical situations in which plant inputs and/or states and/or outputs are restricted to some prescribed sets and to nonlinear dynamic systems with the help of Lie brackets and Lie algebras. There are also studies on relations between controllability/observability of a system and its structure, which are usually called structural controllability and structural observability, respectively. Rather than system parameters, the corresponding results depend only on the positions of plant inputs and outputs, directed connections among plant states, way that the plant states are connected to its inputs, and the way that the plant outputs are connected to its states.

Although issues related to system controllability/observability have been investigated for more than half a century, it is still an active research topic. Especially, stimulated by the development of sensor technologies, communication technologies, computer technologies, and so on, network technologies are widely recognized to be very helpful in the sense of providing more structural flexibilities, reducing greatly hardware investments, and so on in the construction of a control system. With the dream of making these advantages realistic, recently, there emerged extensive interests in the verification of controllability and observability of a networked system.

In this chapter, we aim at developing a numerically stable and computationally feasible method for checking these properties of a system constituted from a great number of subsystems. Verification of controllability and observability of a linear time-invariant (LTI) system is investigated without any constraints on its inputs, outputs, and states. At first, we summarize major associated results for a lumped LTI system. A model for spatially connected systems is introduced afterward, which is more convenient than the available descriptions and can represent the dynamics of a larger plant class. Necessary and sufficient conditions are given in Section 4.3 for the controllability and observability of a spatially connected system, which independently depends on parameters of each of its subsystems and its subsystem connection matrix. As a model is no longer required for the associated lumped system, this property is quite attractive in large-scale system analysis and synthesis.

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Industrial control engineering

Peng Zhang , in Advanced Industrial Control Technology, 2010

(6) Optimum control

Optimal control is an important component of modern control theory. In principle, optimal control problems belong to the calculus of variations. Pontryagin's maximum principle and Bellman's dynamic programming are two powerful tools that are used to solve closed-set constrained variation problems, which are related to most optimal control problems. The statement of a typical optimal control problem can be expressed as follows: The state equation and initial condition of the system to be controlled are given. The objective set is also provided. Find a feasible control, such that the system that starts from the given initial condition transfers its state to the objective set, and in so doing minimizes a performance index.

In industrial systems, there are some situations where optimal control can be applied, such as the control of bacterial content in a bioengineering system. However, most process control problems are related to the control of flow, pressure, temperature, and level. They are not well suited to the application of optimal control techniques.

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Nonlinear Systems; Stability and Control†

JEFFERY LEWINS PhD (Cantab), PhD (MIT) , in Nuclear Reactor Kinetics and Control, 1978

Conclusion

There are several other interesting developments in modern control theory which space precludes. Methods of optimisation are well reviewed by Terney and Wade ⁽¹¹⁾ in a general context. Control theory based on the extension of classical mechanics in the state space admits the treatment of uncertainties in some of the state vectors (e.g. refs. 12 and 13), while an interesting development suitable for direct digital control (DDC) lies in the region of adaptive control. ⁽¹⁴⁾

It is fair criticism of these methods that when they are applied to problems of any complexity instead of the specially simple examples used here, then a considerable burden of computation arises. It is likely, therefore, that their use in practice will not arise until DDC itself is accepted as safe and reliable in the operation of nuclear reactors. If we assume that this very reasonable expectation will be met, then we can foresee a developing use of state space methods in DDC since they provide a more logical and consistent way to optimise behaviour than the empirical optimisation of analogue control elements.

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Robust H∞ Filtering of Nonhomogeneous Markovian Jump Delay Systems via N-Step Ahead Lyapunov–Krasovskii Function Approach

Jiwei Wen , ... Dhafer J. Almakhles , in Non-Monotonic Approach to Robust H-infinity Control of Multi-model Systems, 2019

10.1 Introduction

Filtering is one of the most fundamental problems in modern control theory and application fields, such as control system synthesis, state estimation, information fusion, etc. From the signal-processing point of view, useful signals are inevitably contaminated by exogenous noise, which leads to the discrepancy between obtained signals and original signals during the measurement and transmission. It is thus necessary to minimize inaccuracies caused by noise and to estimate the signal close to the original one [1]. Filtering methods are rightly aimed at this purpose. The goal of filtering is estimating signals that are corrupted by noises or unmeasurable or technically difficult to be measured.

With the development of Kalman filtering theory [2] for stochastic systems, numerous extended results of such an optimal filtering have been reported with burgeoning research interest. To name a few, a robust Kalman filter has been designed for continuous-time delay systems with norm bounded uncertainties [3] and discrete-time uncertain systems [4] by fully making use of the mean value and the variance knowledge of Gaussian noise. Nowadays, a new research tendency is studying the Kalman filtering problem under the networked environment. The distributed Kalman filtering with effects of network structures was introduced by a thorough bibliographic review in [5]. Recently, a distributed extended Kalman filter with nonlinear consensus estimate [6] or with event-triggered scheme and stability guarantee [7] has been developed. As a typical application example, formation of autonomous multivehicles was achieved by employing distributed Kalman filter design. Each vehicle aims to estimate its own state, such as position, velocity, and acceleration, by locally available measurements and limited communication with its neighbors [8]. Fading measurements are also noteworthy phenomena induced by the wireless network, where fading rates during the communication are described by random variables with known statistical properties dependent or independent on the system mode. It usually results in unpredictable performance degradation for the filters [9]. The Tobit Kalman filter [10] and modified extended Kalman filter [11] have been further investigated over fading channels with transmission failure or signal fluctuation.

A successful application of Kalman filter in the aerospace and aviation industry has led applications in ordinary industry in the 1970s. However, these attempts have shown that there was a serious mismatch between the underlying assumptions of Kalman filtering and the industrial state estimation problem. In terms of engineering applications, it is quite costly or difficult to get an accurate system model, and engineers are seldom aware of the statistical characteristics of noise that affect the industrial processes. A useful scheme to deal with modeling uncertainty and non-Gaussian noise is $H_{\infty }$ filtering. The objective is to minimize the $l_2$ gain of the filtering error system from noise inputs to filtering errors. $H_{\infty }$ filtering can be effective in the case that noise inputs are energy-bounded signals, without any statistical knowledge of the noise, rather than Gaussian white noise [12]. Moreover, it is robust against uncertainties both in the system modeling and exogenous noise.

The development of $H_{\infty }$ filtering is reviewed in [13–16] and references therein. It is worth mentioning that $H_{\infty }$ filtering for a class of special switched systems, i.e., Markovian jump systems (MJSs), attracted much attention of the control community. MJSs can well depict various physical phenomena, such as solar thermal receiver, Samuelson's multiplier-accelerator model, NASA F-8 test aircraft, and networked control systems [17]. Such systems may run with external environmental changes, actuator fault, communication time delay, data packet loss, and so on. These random factors often cause a jumping phenomenon of system structure or parameters. Aiming at the $H_{\infty }$ filtering for MJSs, a great deal of research work has emerged (see [17–19] and references therein). The essential difference between MJSs and linear systems is the modes jumping character, which is governed by the transition probabilities (TPs). Therefore, the $H_{\infty }$ filtering for nonhomogeneous Markovian jump systems (NMJSs), that is, MJSs with time-varying TPs, starts to penetrate into the research frontline of the filtering. Filtering for discrete-time uncertain NMJSs [20], robust $H_{\infty }$ filtering for continuous-time nonlinear NMJSs with randomly occurring uncertainties [21], and finite-time $H_{\infty }$ filtering for nonlinear singular NMJSs [22] have been extensively studied. Moreover, the $H_{\infty }$ filtering for time-delayed systems has also attracted great research interests (see [38–40] and references therein). Many effective approaches, such as the famous Lyapunov–Krasovskii function (LKF) approach and Lyapunov–Razumikhin function approach are developed independently for handling time delay [41]. Specifically, robust filtering performance can be achieved by constructing a proper LKF for time-delay systems.

However, filter design approaches developed for NMJSs in the above-mentioned work are conservative. To reduce the design conservatism, an interesting idea is to construct nonmonotonic Lyapunov function (NLF) [23]. It has been successfully used for T-S fuzzy models to relax the monotonicity requirement of LF and further reduce the conservatism of the stability criteria, i.e., allowing the LF to increase locally during several sampling periods. Two-samples variation, i.e., $V(x_{k+2})<V(x_k)$ [24–26], and N-sample variation, i.e., $V(x_{k+N})<V(x_k)$ [27–33], were fully developed for the T-S fuzzy model. Stability analysis and synthesis, robust $H_{\infty }$ controller design, observer-based fuzzy controller design, and output feedback stabilization have been intensively studied. Some preliminary results for switched systems [34,35] and NMJSs [36] have also been reported.

Based on these observations, we can conclude that the NLF approach has not been fully developed for filtering of time-delayed systems. The bigger challenge is how to deal with noise of future time and delay intervals according to the predictive horizon N. In this study, we aim to investigate the $H_{\infty }$ filtering problem for a class of nonhomogeneous Markovian jump delay systems (NMJDSs) via an N-step ahead Lyapunov–Krasovskii function (NALKF) approach. The main contributions and novelties of this chapter are summarized as follows:

• The NALKF approach is developed to reduce the conservatism of the filtering design by properly constructing an LKF and allowing the underlying LKF to increase during the period of N sampling time steps ahead of the current time within each jump mode.

• The LMI formulation of sufficient conditions for filtering is obtained by moving the horizon from $k+N-1$ to $k+1$ step by step. Due to the predictive horizon, the derivation is not trivial.

• For all possible time-varying TPs and all admissible parameter uncertainties and time delays, the filtering error system is mean-square stable with smaller estimated error and lower dissipative level.

Notations: ${\mathbb{R}}^n$ and ${\mathbb{R}}^{n\times m}$ denote the n-dimensional Euclidean space and the set of all $n\times m$ matrices, respectively. ${\Vert \cdot \Vert }_2$ refers to the Euclidean vector norm; $\mathcal{E}\{\cdot |{\mathcal{F}}_k\}$ stands for the conditional mathematical expectation, where ${\mathcal{F}}_k=\sigma \{(x_0,r_0),\dots ,(x_k,r_k)\}$ is the σ-algebra; ${\lambda }_{\min }(\mathrm{\Theta })$ represents the minimum eigenvalue of matrix Θ; and ${\mathrm{diag}}_n(\cdots )$ stands for the block-diagonal matrix with n blocks given by matrices in $(\cdots )$ . In symmetric matrices, we use ⁎ as an ellipsis for the symmetric terms above or below the diagonal. The identity and zero matrices of appropriate dimensions are denoted by I and 0, respectively; ${\mathbb{Z}}_{[s_1,s_2]}\triangleq \{l|l\in \mathbb{Z},s_1⩽l⩽s_2\}$ , where $\mathbb{Z}$ is the set of integers; and $l_{2[0,S]}$ is the space of summable sequences on ${\mathbb{Z}}_{[0,S]}$ , where S may be finite or infinite.

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Application and potential of the artificial intelligence technology

Yu Luo , ... Ningsheng Cai , in Hybrid Systems and Multi-energy Networks for the Future Energy Internet, 2021

9.3 Control and optimization based on artificial algorithm

Traditional control methods are based on classical control and modern control theory, both of which require an explicit mathematical description of the controlled object, in the form of transfer function or state-space equation. These approaches have been of great importance in the control of conventional power and thermal system. However, these methods highly rely on the mathematical model of controlled object, either in the form of function or equation group. What makes these not easy to be used is the growing complexity of modern energy system. It is believed that building a perfect model for a system, even for a simple small energy network, would be extremely challenging. That is why we need artificial intelligence to deal with this task.

Control approach based on artificial intelligence, also known as intelligent control, is a promising direction of modern control strategy. It consists of fuzzy control, ANN control, and so on. The common ground of these control methods is the imitation of natural intelligence. They adopt different ideas to simulate the process how human beings make control of a system.

Fuzzy control is a method based on fuzzy math. Its aim is to "translate" the human prior knowledge into a form that can be understood by computer using the theory of fuzzy set and fuzzy relation. Human beings have accumulated a large number of knowledge and experience. For example, if the room temperature is too high, we might need to improve the power of air conditioner. We can also say that if the temperature is extremely higher than what expected, we need to improve the power of air conditioner as much as possible. When it comes to solid oxide fuel cell, if the current increases, the hydrogen flow rate should also be improved. These rules can easily be understood by engineers or scientists, however, they are not able to be implemented by computers.

Fig. 9.5 shows a typical process of fuzzy control, which covers several steps, mainly fuzzification, fuzzy inference, and defuzzification. The continuous input signal, generally the system error or the error variation, which is collected from real physical terminal, is first mapped to a discrete one by using quantization factor. The discrete signals are further mapped to fuzzy sets by fuzzification method. These fuzzy sets qualitatively describe the input signal in the form of NB (negative big), NS (negative small), Z (Zero), PS (positive small), and PB (positive big). By fuzzy inference, which is actually a calculation between fuzzy set and fuzzy relation, computer can acquire the output fuzzy set, in the same form with the input sets. The next step is the inverse process of fuzzification. Output fuzzy sets are mapped to a series of discrete data and further to the control signal that can be detected by the real system. Fuzzy control is widely implemented in hybrid system. Such applications cover wind energy conversion system, battery management system, heating energy supply system, and energy storage system.

Figure 9.5. Fuzzy controller.

The main idea of control method based on ANN is to replace the physical model of the controlled object with its ANN model. It is known to us that ANN model is a "black box" where given an input; the output will be calculated without solving physical equations. It uses the historical data to train the model to get the most suitable parameters, as mentioned before. ANN control can be utilized when the physical mechanism is too complex to be modeled.

The optimization of energy system is another important topic, especially when more and more renewable energy and distributed subsystems are integrated. The optimization involves how to improve the efficiency of the system, how to maximize the profits and how to minimize the costs. Some of them are in accordance with economy and some are regarding physical performance. Different from general optimization methods that implement optimization algorithm to find the optimal solution of problem, due to the complexity of the system, intelligent algorithm relies on heuristic algorithm that is focused on finding a feasible solution. Heuristic algorithm derives from intuitive thinking and experience, which may try to imitate some natural phenomenon like the evolutionary of creatures, the migration of a bird flock. There are several types of heuristic algorithms including genetic algorithm, ant algorithm, particle swarm optimization, simulated anneal algorithm, etc [9].

In the ant algorithm, the idea of finding the optimal solution is based on foraging activities of ants. Provided that there is a food near ants' nest and between them are some obstacles, the ant colony will find a feasible and even the shortest path by self-organization, as shown in Fig. 9.6. This intelligent behavior lies in two factors. The first one is whenever an individual ant passes through a certain path; pheromones will be remained on this path. Assuming that the speed of ant is constant, it can be inferred that the shorter the path connecting food and nest is, the more pheromones will exist on this path. The second rule is that ants tend to follow a path that has more pheromones. It is also possible that an ant will go to a path with less pheromones. However, the probability could be significantly low. These two rules actually form a positive-feedback cycle, which means the more pheromones a path has, the more ants pass through this path, vice versa. Eventually, the paths of ant will converge to a specific path that may be the shortest one with a high probability.

Figure 9.6. Foraging activities of ant colony.

Ant algorithm can solve the NP problems in combinatorial optimization and has been used in a variety of fields. When it is used in energy system, there appear a few valuable researches. Using multi-layer ant colony optimization, Mousa Marzbanda etc. figure out the optimum operation of micro-sources for decreasing the electricity production cost by hourly day-ahead and real time scheduling. On mass rapid transit system, Bwo-Ren Ke, etc. propose a method for block-layout design to save energy. They used a Max-Min Ant System of ant colony algorithm, which can also reduce the computational time [10].

Genetic algorithm, also widely used, imitates the process of evolutionary of creatures [11]. It is well known that creatures are evolving to adapt to the environment by inheritance and variation. The individuals with high adaptation tend to be alive during the natural selection while the ones with bad traits will be eliminated. Fig. 9.7 gives the flowchart of genetic algorithm. The first step is to map the possible value of variables in continuous space to a chromosome. For example, assuming that the independent variable $x_1$ and $x_2$ are both in the range (0, 1), we can map them to a binary string in the form of 0010101010001 010. The first eight bits represent $x_1$ and the latter bits represent $x_2$ . It can be seen that the higher the precision is, the more bits are needed to discretize independent variable space. The objective function of optimization is further used as a criterion of adaptation for the individuals from genetic model. A higher adaption means that this individual has a larger probability to be chosen to crossover with other individuals and its gene is more probably inherited. Mutation of genes in chromosome, which can be considered as a change of a bit in the binary string, is also crucial because the initial population is selected stochastically thus without mutation, the model may jump into local optimization quickly. Choosing a suitable possibility for mutation plays a major part in the calculation. Too large possibility means more iterations, slower computational speed, and bad convergence performance. On the other hand, small possibility leads to local optimization.

Figure 9.7. Flowchart for genetic algorithm.

A research from Jonathan Reynolds, which is focused on district heat supply system, utilized genetic algorithm to manage. The authors first built a system model with ANN method and then got the optimal operating schedules of the heat generation equipment, thermal storage, and the heating set point temperature using genetic algorithm. Abbassi Abdelkader used a multi-objective genetic algorithm to optimize the sizing of a stand-alone PV and wind power supply system, where all storage dynamics are considered. The objective is to minimize the total cost electricity and the loss of power supply probability of the load. Results demonstrated renewable energy play an important role in promoting the energy sector.

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Digital Control Systems Implementation and Computational Techniques

Guang-Qian Xing , Peter M. Bainum , in Control and Dynamic Systems, 1996

I Introduction

The concept of controllability is one of the cornerstones of modern control theory. Yet despite its fundamental importance from a theoretical point of view, its practical utility for control system evaluation is limited due to its binary nature. That is, a system is either controllable or it is uncontrollable. There is no provision for consideration of the more subtle question: how controllable is the system?

The desirability of a degree of controllability (earlier it was called the controllability index) concept has been recognized in the literature since 1961 when Kalman, Ho and Narendra [1] discussed it. Early papers in the area [2–4] concentrated on particular properties of either the controllability Grammian matrix or the controllability matrix itself in developing definitions of the degree of controllability. It is natural to try to connect the degree of controllability to properties of the standard controllability matrix P_c = (B:AB: …: A^n-1B), and define the degree of controllability as the square root of the minimum eigenvalue of ${\text{P}}_{\text{c}}{\text{P}}_{\text{c}}^{\text{T}}$ . In 1979 Viswanathan, Longman and Likins considered the "shortcomings" of this definition [5]. Four apparent difficulties with this definition must somehow be handled before the definition becomes "viable"; therefore, they presented a new definition of the degree of controllability of a control system by means of a scalar measure based on the concept of the "recovery region".

In this paper three candidate definitions of degree of controllability, which were presented first by Muller and Weber [4] for linear continuous systems, are presented for linear discrete-time systems based on the scalar measure of the Grammian matrix. The three candidates for the degree of observability of linear discrete-time systems are also presented. Because some difficulties with this definition (as pointed out by Viswanathan [5]) have been handled here, and the degree of controllability (observability) based on a scalar measure of the Grammian matrix can be readily calculated, the three candidates for the degree of controllability (observability) are viable for practical engineering design.

The emphasis of this paper is in showing the physical and geometrical meanings and the general properties of the three candidate definitions of degree of controllability (observability) as physically meaningful measures. The concepts of degree of controllability are applied to the actuator placement problem for the orbiting shallow spherical shell control system. The degree of controllability for seven cases in which the placement of actuators are all different are compared. The LQG transient responses for several typical systems with different controllability are also shown.

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Modal analysis methods – time domain

Jimin He , Zhi-Fang Fu , in Modal Analysis, 2001

9.6 Summary of time domain modal analysis

The development of time domain modal analysis methods relies upon the progress of modern control theory. Theoretical and mathematical aspects of these methods are similar to that in system identification developed for modern control engineering. When applying modern control engineering to structural dynamics and modal analysis, additional requirements are placed on vibration testing, data acquisition and signal processing, especially when dealing with large structures.

Traditional frequency domain modal analysis is based on frequency analysis theory. The physical interpretation of this modal analysis is usually evident. For example, an FRF exhibits resonances and anti-resonances of a test structure and the amount of damping for each vibration mode is evidenced by the sharpness of resonance peaks. There are developed techniques to combat measurement noise and other errors. For example, the average of records and application of windows are effective means of reducing random and systematic errors in FRF estimation. To obtain the response measurement for the estimation of FRF data, a structure normally needs to undergo excitations which are accurately measured. This type of measurement may require costly measurement equipment and a laboratory environment. It is not suitable for online measurement and analysis.

Time domain modal analysis is able to make use of vibration responses due to ambient excitations. This is a clear advantage over frequency domain modal analysis. The time responses obviously do not exhibit clear modal information such as resonances as FRF data do. Noise experienced in measurement is more a problem in time domain than in frequency domain because we do not have the luxury to utilize FRF data around resonances for better signal to noise ratio. Modes with less vibration energy at a measurement location can be inundated by noise and therefore disappear from analysis radar screen. This poses great demands on modal analysis methods to improve their algorithms in order to deliver accurate modal analysis results.

Ideally, time domain modal analysis requires data measured from all measurement points on a structure simultaneously. This avoids unnecessary phase difference among responses and ensures that all responses come from the same initial forcing conditions. In reality, it may not be practical to measure many channels of responses at the same time due to hardware limitations. One solution is to divide measurement points into groups. Each group can be analysed separately using the time domain modal analysis methods. By having one or more overlapping points between groups, it is possible later to scale all the mode shape data consistently.

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Overview

Qinglei Hu , ... Youmin Zhang , in Fault-Tolerant Attitude Control of Spacecraft, 2021

1.6.1 Attitude FTC design using adaptive control

Adaptive control technique Krstic et al. (1995), as one of modern control theories, has been widely applied to handling uncertain system parameters, that is, owing to its capability to estimate those parameters online. Taking this advantage, many investigations on spacecraft have applied adaptive control to designing attitude FTC law. In Boskovic et al. (2005) an adaptive controller was designed for flight control system. An adaptive attitude FTC stabilization control law was proposed for a tethered satellite. The work in Cai et al. (2008) showed that adaptive control can be successfully applied to performing an attitude tracking maneuver for a rigid spacecraft with thrusters failures. Of particular interest to this paper, input constraint, uncertain inertia parameters and even external disturbance were explicitly addressed. The author in Jiang et al. (2010) derived an FTC strategy to follow the desired attitude for flexible spacecraft. The proposed scheme estimated the upper bounds on disturbances and model parameters online and used there estimates in the control law; more specifically, two types of reaction wheel faults were explicitly compensated by using adaptive technique. An adaptive control-based intelligent FTC law was derived in Chandrasekar et al. (2010) to achieve attitude control with high pointing accuracy, and actuator fault and misalignment were adaptively handled. In a recent work Zou and Kumar (2011), combing adaptive control with fuzzy control, an adaptive attitude controller was synthesized. System uncertainties, disturbances, and actuator faults were addressed, and two adaptive updating laws were presented to estimate uncertain parameters. For the spacecraft formation flying control subjecting to actuator fault, a passive FTC controller by using adaptive control was developed in Zou and Kumar (2012). The implementation of this controller did not require any FDD approach. To compensate for disturbances and actuator faults, an adaptive FTC law was designed to perform attitude tracking maneuver. For more recent development of FTC law design by using adaptive control technique, the works Zhang et al. (2004); Ma et al. (2014); Qiao et al. (2008); Bustan et al. (2013) can be further referred.

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FISSION SOURCES & ENERGY PRODUCTION

Smithsonian Science Information Exchange, Inc. , in Summary of International Energy Research and Development Activities 1974–1976, 2013

3.0537 EXPERIMENTAL DIRECT DIGITAL CONTROL OF THE POWER PLANT A 1 REACTOR BASED ON A MODERN CONTROL THEORY APPROACH

C. KARPETA, Power Research Institute, Prague 7, Czechoslovakia The research project represents an application of the modern control theory to a heavy-water gas-cooled nuclear power reactor. A computerized control system is to be developed to control the coolant output temperature at the nominal power level. The design procedure will be based on the solution of the so-called linear-quadratic gaussian problem.

Scheduled topics for the current year are as follows: (a) linearization of the model of the process, (b) sensitivity analysis of the linearized model with respect to the process parameters variation, (c) design of an optimum digital estimator and controller, (d) simulation studies of the closed control loop on a hybrid computer.

SUPPORTED BY International Atomic Energy Agency - Vienna, Austria

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How to Solve Modern Control Theory Questions ?

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