Feedback Control of Dynamic Systems Solutions Chapter 3

ROBUSTNESS OF MULTIVARIABLE NON-LINEAR ADAPTIVE FEEDBACK STABILIZATION

A. Hmamed , L. Radouane , in Adaptive Systems in Control and Signal Processing 1983, 1984

Abstract

Here considered is the stabilazing property of the adaptive nonlinear dynamic feedback control proposed in ( Hmamed and Radouane, 1983) for multivariable systems in the presence of uncertainty. A multivariable system is decomposed into a set of single-input subsystems and the second Luenberger canonical form is obtained. This representation is suitable for non-linear adaptive control of unknown systems. The measure of robustness of the proposed scheme is defined in terms of bounds of allowable perturbations such that the stability is preserved. It is shown that the system can always remain stable for a large class of perturbations.

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Introduction to Modeling and Design of Feedback Control Systems

Nicolae Lobontiu , in System Dynamics for Engineering Students, 2010

Summary

This chapter introduces the main notions, concepts, and modeling tools of feedback control dynamic systems. Using transfer functions interconnected in negative-feedback block diagrams, methods are presented for analyzing and designing dynamic systems with feedback control included. Methods focusing on system stability are the Routh-Hurwitz criterion and the closed-pole position, whereas the root locus technique allows studying both the stability and the systems' characteristics. Nyquist plots and Bode diagrams are utilized to analyze and design feedback control systems in the frequency domain. MATLAB built-in commands are utilized throughout this chapter to aid in the solution of almost all the problems, whereas Simulink application is exemplified in the time-domain modeling of control systems. Studied also are the steady-state response of feedback control systems and their sensitivity to parameter variation.

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Block Diagrams and Feedback Control System Modeling

Nicolae Lobontiu , in System Dynamics for Engineering Students (Second Edition), 2018

Summary

This chapter introduces the main notions, concepts, and modeling tools of feedback control dynamic systems. Using transfer functions interconnected in negative-feedback block diagrams, models are presented for SISO systems. The closed-loop transfer function synthesizes the added actions of the controller and feedback and directly connects the input (reference) signal to the output (controlled) signal. Studied is also the sensitivity of SISO systems to parameter variation. MIMO feedback systems are also studied in this chapter and are modeled by means of transfer function matrices connected in block diagrams. Feedback systems with disturbances are analyzed as a specific case of MIMO systems. SISO systems are also modeled by using the state-space approach. The chapter illustrates and models numerous examples of actual physical systems that can operate as individual components in or full feedback control systems.

Problems

11.1

Convert the control system of Example 11.3 and shown in the block diagram of Figure 11.14(a) into a basic unity-feedback control system as the one shown in Figure 11.12.

11.2

Convert the control system of block diagram of Example 11.2 and Figure 11.13 into a basic nonunity-feedback control system.

11.3

A MIMO feedback control system is connected as in Figure 11.13 of Example 11.2 where [G ₁(s)], [G ₂(s)], [G ₃(s)] are matrices, while {R(s)}, {C(s)} are vectors. Convert this system into the regular feedback system of Figure 11.35(b).

11.4

Convert the system of Problem 11.3 into a unity-feedback system where [H(s)] is the unity matrix in Figure 11.35(b).

11.5

Convert the control system described by the block diagram of Figure 11.40 into a unity-feedback control system as the one illustrated in Figure 11.12.

Figure 11.40. Block Diagram of a Control System With Embedded Feedback Loop and Prefilter

11.6

Convert the control system of Figure 11.40 into a basic nonunity control system as the generic one of Figure 11.11(a).

11.7

Convert the control system described by the block diagram of Figure 11.41 into a unity-feedback control system as the generic one shown in Figure 11.12.

Figure 11.41. Block Diagram of a Control System With Two Feedback Loops and Prefilter.

11.8

Convert the control system of Figure 11.41 into a basic nonunity control system as the generic one of Figure 11.11(a).

11.9

Calculate the controlled output C(s) resulting from the feedback system of Figure 11.42 when R(s)   =   1/s.

Figure 11.42. Block Diagram With Two Feedback Loops.

11.10

Using only basic P, D, and I controllers, as well as summing and pick off points, design the block diagrams to model the controllers defined by G _c1(s)   =   1/(a  − b·s) and G _c2(s)   =   (c− d·s)/(a  − b·s) where a, b, c, and d are real numbers.

11.11

A unity-feedback control system is defined by the closed-loop transfer function $G_{CL}\left(s\right)=\left(s+1\right)/\left(s+4\right)$ . Design a block diagram formed of only proportional gains, integrators, derivation operators, summers, and pick off points to represent this transfer function.

11.12

Solve Problem 11.11 for the closed-loop transfer function: $G_{CL}\left(s\right)=\left(s+2\right)/\left(s^2+s-3\right)$ .

11.13

Solve Problem 11.11 for the closed-loop transfer function: $G_{CL}\left(s\right)=s/\left(s^3+4s^2+5s+2\right)$ .

11.14

Solve Problem 11.11 for the closed-loop transfer function: $G_{CL}\left(s\right)=\left(s^2+s+1\right)/\left(s^3+2s^2+3s\right)$ .

11.15

Demonstrate that electrical circuit of Figure 11.43 with three cascading operational-amplifier stages and seven identical electrical elements can operate as a summing point in a basic control system when considering the three signals are the voltages V _s (s), V _r (s), and V _e (s).

Figure 11.43. Electrical System as a Summing Point in a Feedback Control System.

11.16

Disposing of three identical pulleys, two identical translatory-motion springs, and some rigid rods, design a rotary mechanical system to operate as a summing point when considering small rotations. Hint: Use pulley rotation angles as the signals of the summing point.

11.17

Design an active electrical system to operate as P  + I controller. The system comprises two cascading operational-amplifier stages, one inductor, and four identical resistors. Knowing the resistance ranges in the 200–300   Ω interval and the inductance is within the 1–5   H range, find the electrical components that minimize the controller's integral time T _I .

11.18

Design an active electrical system to operate as a P  + D controller. The system comprises two cascading operational-amplifier stages, one capacitor, and four identical resistors. Knowing the resistance ranges in the 150–250   Ω interval and the capacitance is within the 3–5   μF range, find the electrical components that minimize the controller's derivative time T _d .

11.19

Derive the transfer function of the fluid system of Example 7.3 and shown in Figure 7.4 when considering that the flow rate q _i is the input and the flow rate q _o is the output. Using basic P, D, I controllers together with summing and junction points, propose a block diagram to model the system's transfer function.

11.20

Design a P  + I  + D controller formed of the following rotary-motion mechanical elements: a disk of mass moment of inertia J, a damper of damping coefficient c _r , and a spring of stiffness k _r by defining a transfer function that connects two rotation angles.

11.21

Design a P  + I  + D controller with two operational amplifiers, three identical resistors, and three identical inductors. Determine the values of R and L that will result in a derivative time T _D   =   0.05   s and an integral time T _I   =   0.1   s.

11.22

Disposing of two operational amplifiers, four resistors (two of resistance R ₁ and two of resistance R ₂), and four inductors (two of inductance L ₁ and two of inductance L ₂), design an active lag–lead compensator. Express the gain K, the time constants τ₁, τ₂ and the constants α₁, α₂, as well as the relationships between the four electrical elements properties resulting in a lag–lead compensator.

11.23

Design a unity-feedback control system entirely composed of mechanical elements where the controller is a proportional one. As shown in Figure 11.44, the plant contains a rotary (torsional) spring of stiffness k_r and a cylinder of mass moment of inertia J. The plant's input and output are the angles θ _i and θ _o .

Figure 11.44. Rotary Mechanical Plant With Angular Input and Output.

11.24

Consider the plant is the circuit of Figure 7.7(a). Design a unity-feedback control system entirely composed of electrical elements resulting in a P  + I controller.

11.25

Demonstrate that the passive circuit of Figure 11.45 can be utilized as a lag compensator.

Figure 11.45. Passive Electrical Network as a Lag Compensator.

11.26

Decide whether the electrical circuit of Figure 7.2 in Example 7.1 can operate as a lead or a lag compensator in terms of its four electrical elements.

11.27

Can the subsystem defined by the transfer function G _c (s)   =   (s ²  + a·s  + b)/(s ²  + c·s  + d) operate as a lag–lead compensator where a, b, c, d are real parameters?

11.28

A mechanical second-order microplant is controlled by a P  + D unit, resulting in the feed-forward transfer function shown in Figure 11.46. Study the sensitivity of the open-loop transfer function and of the closed-loop transfer function to the gain K.

Figure 11.46. Block Diagram of Nonunity-Feedback Control System.

11.29

Calculate and analyze the sensitivity of the closed-loop transfer function of the control system shown in Figure 11.13 of Example 11.2 in terms of K when G ₁(s)   = G ₃(s)   = K and G ₂(s)   =   1/(s ²  + s  +   1).

11.30

Calculate and analyze the sensitivity of the closed-loop transfer function of the control system sketched in Figure 11.40 of Problem 11.5 in terms of K ₁, K ₂, and K ₃ for G ₁(s)   = K ₁, G ₂(s)   = K ₂/[s·(s  +   1)·(s  +   2)²] and G ₃(s)   = K ₃/s.

11.31

Obtain a state-space model for the control system of Problem 11.28 for K  =   0.5 and plot its time response c(t) when r(t)   =   5.

11.32

Considering K ₁  =   10, K ₂  =   2, and K ₃  =   120 in Problem 11.30, find a state-space model for the control system and plot its time response c(t) for r(t)   =   40·sin(110t).

11.33

Compare the time responses of the control systems sketched in Figures 11.34 and 11.47. Use Simulink to plot the time response c(t) for both systems considering that r(t)   =   1 and d(t)   =   1/(t ²  +   1). Known are G _p (s)   =   100/(s ²  +   5s  +   100), G _c (s)   =   0.02/s, and H(s)   =   1.

Figure 11.47. Block Diagram of a Nonunity-Feedback Control System With Disturbance.

11.34

A two-input, two-output unity-feedback control system similar to the one depicted in Figure 11.39 is defined by the following control and plant transfer function matrices:

$\left[G_c\left(s\right)\right]=\left[\begin{array}{cc}20+\frac{100}{s}& 0\\ 0& 10+0.02s\end{array}\right];\left[G_p\left(s\right)\right]=\left[\begin{array}{cc}\frac{s}{s^2+2s+10}& \frac{2}{s^2+2s+10}\\ \frac{2}{s^2+2s+10}& \frac{s+1}{s^2+2s+10}\end{array}\right];$

Calculate the closed-loop transfer function matrix and plot the system response for a reference input {r(t)}   =   {1 1}^T.

11.35

The control and plant transfer functions of Problem 11.34 are combined in a nonunity-feedback control system as the general one sketched in Figure 11.36. For identical prefilter and feedback transfer function matrices $\left[P\left(s\right)\right]=\left[H\left(s\right)\right]=\left[\begin{array}{cc}1200& 0\\ 0& 1200\end{array}\right]$ , calculate the closed-loop transfer function matrix of this system and plot the time response for {r(t)}   =   {1 1/(t  +   1)}^T.

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Simplification of Buffer Formulation and Improvement of Buffer Control with In-Line Conditioning (IC)

Enrique N. Carredano , ... Günter Jagschies , in Biopharmaceutical Processing, 2018

27.4 In-Line Conditioning—Controlled Production of Any Buffer

In-line conditioning (IC) is a concept for in-line buffer formulation at large scale using stock solutions and water for injection (WFI) in combination with a purification step such as chromatography or filtration (i.e., directly delivering the buffer to the process without intermediate storage in bags or tanks; Fig. 27.2B). As with ILD, but more pronounced, significant floor space and tank volume reductions are possible. Stock solutions contain only one buffer component and can thus usually be much more concentrated than a prepared concentrated buffer (see Table 27.3). With IC, it is possible to prepare buffers of different strength, pH and salt concentration from these component stock solutions kept in separate bags or tanks, independent on effects like CIE or the least soluble component (i.e., without reworking diluted buffers; Table 27.2).

Table 27.3. Concentration Factors for Different Buffers and Buffer Components

Buffer/salt	ILD Conc. Factor	IC Conc. Factor	Comment
20   mM Phosphate pH   7.2, 50   mM NaCl	10,2	16,7	Equilibration buffer in mAb purification
35   mM Phosphate pH   7.2, 500   mM NaCl	2,1	4,0	Wash buffer in mAb purification
20   mM Phosphate pH   7.2	18,3	24,4	Load buffer in mAb purification

27.4.1 IC System Layout

An IC system has at least three inlets for buffer preparation: acid component, base component, and water for injection (WFI). A fourth inlet is provided for salt solution if needed. The system has multiple sensors for monitoring flow, pH, and conductivity, which may be used for dynamic feedback control. A typical flow diagram enabling the IC buffer preparation approach is shown in Fig. 27.3.

Fig. 27.3. Typical IC system flow scheme with inlets for acid and base components of the buffer and WFI. Optional inlets for salt and additives can be included in the buffer recipe. IC can either be installed as a central buffer preparation station replacing the current, largely manual processes but still storing the buffers prior to use, or as an integrated part of chromatography and filtration skids delivering buffers directly to columns and filter holders.

27.4.2 IC Control Modes

There are different possibilities for preparing the buffer by using three different types of control: (a) flow feedback with recipe, (b) pH-Flow feedback, or (c) pH-Conductivity feedback. They have different benefits and drawbacks and can all be used with advantage, dependent on the scenario to be addressed.

All modes of control include flow feedback to maintain the total flow rate, approximated as the sum of all flow rates, constant by adjusting the flow of WFI. It is also possible to reduce the risk of bias on the pH sensors by appropriate choice of monitor for control.

In flow feedback mode , the system is able to use pre-defined recipes to determine the flow set point for each pump, where the ratios among the four different pumps can be set in percentage together with a system total flow rate set point in L/h. This corresponds to a pure in-line dilution procedure where several concentrated solutions are diluted with water according to a recipe tested to yield the correct buffer. The pump percentages can be changed in steps or linearly to obtain a gradient. Flow feedback makes sure that the correct flows from the concentrated stock solutions of acid, base, salt and water are combined. In-line conductivity and pH measurements can be used for monitoring of the buffer properties and for release. For many processes, flow feedback with recipe is a robust alternative, especially if the temperature is well controlled. On the other hand, temperature variations may lead to variations in pH and conductivity. Accurate stock solutions are required (there is lower risk relative to preparing accurate buffer concentrate mixtures) as there is no possibility for dynamic feedback control correcting pH and conductivity.

When using pH-flow feedback the pH of the buffer is compared with a target value to determine whether the base or the acid percentage should increase or decrease. Flow feedback is used to make sure that the buffer concentration is kept constant, and in the case where there is salt, to keep the salt concentration at the desired level, either in a gradient or at a constant concentration. The percentage of water added is always regulated to keep a constant overall flow rate. Using this control mode, the user needs to specify the buffer concentration and the stock solution concentrations for the acid and base.

A combination of pH and conductivity feedback requires the operator to specify both a target pH and target conductivity. If there is salt, it requires the specification of both the stock solution concentration for the acid and base and also the buffer concentration. On the other hand, if there is no salt, the stock solutions can vary. This mode of control is useful whenever there is a set point for the conductivity, which can be reached by adjusting the flow of the salt (if there is salt) or of the buffer (when there is no salt).

Whenever using pH feedback it is then possible to choose between the three different types of buffer mixing: weak acid and weak base, strong acid and weak base, and weak acid and strong base.

The use of flow feedback mode requires a recipe. A recipe can be either calculated or empirically determined. It is possible to use the IC system itself to determine the recipe by running it in pH-flow mode, in which case the system will work as an "analog machine" solving the equations of the buffer equilibrium. After the run, it is possible to read the percentage (%) of each flow in the result file to obtain a recipe. This approach will always work independently of the buffer system or salts and additives added.

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Raw Gas Transmission

Saeid Mokhatab , ... John Y. Mak , in Handbook of Natural Gas Transmission and Processing (Fourth Edition), 2019

3.8.4.3.3.3 Control Methods

Control methods (feed forward control, slug choking, and active feedback control) for slug handling are characterized by the use of process and/or pipeline information to adjust available degrees of freedom (pipeline chokes, pressure, and levels) to reduce or eliminate the effect of slugs in the downstream separation and compression unit. Control based strategies are designed based on simulations using rigorous multiphase simulators, process knowledge, and iterative procedures. To design efficient control systems, it is therefore advantageous to have an accurate model of the process (Bjune et al., 2002).

The feed-forwarded control aims to detect the buildup of slugs and, accordingly, prepares the separators to receive them, e.g., via feed-forwarded control to the separator level and pressure control loops. The aim of slug choking is to avoid overloading the process facilities with liquid or gas. This method makes use of a topside pipeline choke by reducing its opening in the presence of a slug, and thereby protecting the downstream equipment (Courbot, 1996 ). Like slug choking, active feedback control makes use of a topside choke. However, with dynamic feedback control, the approach is to solve the slug problem by stabilizing the multiphase flow. Using feedback control to prevent severe slugging has been proposed by Hedne and Linga (1990), and by other researchers (Molyneux et al., 2000; Havre and Dalsmo, 2001; Bjune et al., 2002). The use of feedback control to stabilize an unstable operating point has several advantages. Most importantly, one is able to operate with even, nonoscillatory flow at a pressure drop that would otherwise give severe slugging. Fig. 3.31 shows a typical application of an active feedback control approach on a production flow line/pipeline system, and illustrates how the system uses pressure and temperature measurements (PT and TT) at the pipeline inlet and outlet to adjust the choke valve. If the pipeline flow measurements (FT) are also available, these can be used to adjust the nominal operating point and tuning parameters of the controller.

Figure 3.31. Typical configuration of a feedback control technique in flow line/riser systems (Bjune et al., 2002).

Note, the response times of large multiphase chokes are usually too long for such a system to be practical. The slug suppression system (S3) developed by Shell has avoided this problem by separating the fluids into a gas and liquid stream, controlling the liquid level in the separator by throttling the liquid stream and controlling the total volumetric flow rate by throttling the gas stream. Hence, the gas control valve back pressures the separator to suppress surges and as it is a gas choke, it is smaller and therefore more responsive than a multiphase choke.

The S3 is a small separator with dynamically controlled valves at the gas and liquid outlets, positioned between the pipeline outlet and the production separator. The outlet valves are regulated by the control system using signals calculated from locally measured parameters, including pressure and liquid level in the S3 vessel and gas and liquid flow rates. The objective is to maintain constant total volumetric outflow. The system is designed to suppress severe slugging and decelerate transient slugs so that associated fluids can be produced at controlled rates. In fact, implementation of the S3 results in a stabilized gas and liquid production approximating the ideal production system. Installing S3 is a cost effective modification and has lower capital costs than other slug catchers on production platforms. The slug suppression technology also has two advantages over other slug-mitigation solutions, where unlike a topside choke, the S3 does not cause production deferment and controls gas production, and the S3 controller uses locally measured variables as input variables and is independent of downstream facilities (Kovalev et al., 2003).

The design of stable pipeline-riser systems is particularly important in deepwater fields, since the propensity towards severe slugging is likely to be greater and the associated surges more pronounced at greater water depths. Therefore, system design and methodology used to control or eliminate severe slugging phenomenon become very crucial when considering the safety of the operation and the limited available space on the platform. Currently, there are three basic elimination methods that have been already proposed. However, the applicability of current elimination methods to deepwater systems is very much in question. Anticipating this problem, different techniques should be developed to be suitable for different types of problems and production systems (Mokhatab et al., 2007a).

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On a quasioptimal spectrum assignment for delayed systems

Libor Pekař , in Control Strategy for Time-Delay Systems, 2021

4.5.1 Problem statement

Let us first provide the reader with motivation and the problem statement. Recall that the subject of the study of this chapter is an infinite spectrum of control feedback characteristic values, resulting from a control design (e.g., using the $R_{QM}$ ring). It is worth noting that zeros in LTI TDSs have the same impact on the system dynamics as for finite-dimensional systems (it implies, e.g., that zeros located right from ${\alpha }_P$ may cause a control response overshoot even for aperiodic control systems). Hence it is desirable to adjust the spectrum ${\mathrm{\Sigma }}_Z$ of zeros as well, which enables us to adjust the feedback control dynamics more comprehensively. The goal is to place a selected subset of the rightmost poles and zeros to desired loci and, simultaneously, to minimize both abscissae of the remaining spectra.

A sketch of the algorithm framework is as follows. The initial step consists in the direct zero/pole placement (analogously to step (1) of Algorithm 4.1). Whenever the placed roots are not dominant, a subset of the rightmost roots is quasicontinuously shifted to the prescribed loci (via the QCSA of an iterative optimization algorithm), whereas the rests of both spectra are attempted to be pushed to the left. The optimization procedure should take into account the distance of dominant adjusted roots from the prescribed loci and abscissae of rests of the spectra. However, the task is rather tricky because the controller parameters are generally simultaneously included in the numerator and denominator of the closed-loop transfer function. In addition, the use of habitual optimization techniques may go wrong due to the properties introduced in item (6) of Proposition 4.2.

Denote by ${\mathrm{\Delta }}_Z\left(s\right)$ the numerator (quasi)polynomial of $G_{RY}\left(s\right)=Y\left(s\right)/R\left(s\right)$ . Assume, moreover, that there is no s such that it is a common root of ${\mathrm{\Delta }}_P\left(s\right)$ and ${\mathrm{\Delta }}_Z\left(s\right)$ , that is, ${\mathrm{\Sigma }}_{ZP}:=\{s:{\mathrm{\Delta }}_Z\left(s\right)={\mathrm{\Delta }}_P\left(s\right)=0\}=\varnothing$ . The matching model transfer function $G_{RY,m}\left(s\right)={\mathrm{\Delta }}_{Z,m}\left(s\right)/{\mathrm{\Delta }}_{P,m}\left(s\right)$ gives rise to the desired zero and pole loci ${\zeta }_i$ , $i=1,2,\dots ,n_Z=\mathrm{deg}{\mathrm{\Delta }}_{Z,m}\left(s\right)$ and ${\sigma }_i$ , $i=1,2,\dots ,n_P=\mathrm{deg}{\mathrm{\Delta }}_{P,m}\left(s\right)$ , respectively. The values $n_Z$ , $n_P$ include possible root multiplicities. Let selectable (tunable) parameters in the feedback system numerator and denominator, respectively, be elements of vectors ${\mathbf{p}}_Z\in {\mathbb{R}}^{r_Z}$ and ${\mathbf{p}}_P\in {\mathbb{R}}^{r_P}$ , $r_P\ge n_P$ . The numbers of currently shifting zeros and poles are $m_Z$ , $m_P$ , respectively.

Assuming that the sensitivity matrix $\mathbf{S}\in {\mathbb{C}}^{(m_Z+m_P)\times (r_Z+r_P)}$ has the full rank, the natural requirement is that

(4.34) $r_Z\ge n_Z,r_P\ge n_P$

to get a solvable model matching problem. In addition, some tunable parameters must serve to the minimization of abscissae of the remaining spectra, not to the approaching of the dominant roots to the desired ones, that is, a stricter condition is to be met:

(4.35) $0<n_P<r_P,0\le n_Z<r_Z.$

Condition (4.35) is separately applied to the numerator and denominator. Moreover, there must be taken into consideration the overall number of prescribed roots and free parameters, $n=n_Z+n_P$ and $r=r_Z+r_P$ , respectively, under the restriction

(4.36) $n<r.$

Regarding the matching model itself, it must be strictly proper (i.e., strongly feasible):

(4.37) $n_Z<n_P.$

Conditions (4.34)–(4.37) must be considered when selecting $G_{RY,m}\left(s\right)$ . The particular desired zero/pole loci can be chosen based on the required model dynamics. A concise analysis of simple-model dynamics is elaborated is Section 4.5.2.

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Finite-time extended state observer based fault tolerant output feedback control for UAV attitude stabilization under actuator failures and disturbances

Bo Li , ... Yongsheng Yang , in Fundamental Design and Automation Technologies in Offshore Robotics, 2020

13.3.3 Finite-time attitude dynamic feedback control algorithm

In order to achieve fast convergence performance of the control system, terminal sliding mode control has attracted significant interest [40]. However, it has two noteworthy disadvantages, that is, the singularity problem and chattering phenomenon. To this end, a supertwisting algorithm based continuous homogeneous sliding mode control scheme was proposed for a perturbed second-order system in [41]. It has ensured the finite-time convergence of the system states and chattering reducing performance. And that, it is worth mentioning that the supertwisting algorithm based sliding mode control laws can actuate continuous control signals to restrain the chattering phenomenon and obtain the finite-time convergence [42]. Inspired by these approaches, a novel continuous finite-time control law is investigated by using the reconstructed angular velocity and system failure information deriving from the proposed FTESO. The proposed control law is considered as a combination of supertwisting and nonsingular terminal sliding mode control, with the superiorities of chattering attenuating and nonsingularity.

Consider the attitude dynamics in the form of the second-order plant as (13.9) by denoting ${\mathit{x}}_1=\mathit{\Theta }$ and ${\mathit{x}}_2=\dot{\mathit{\Theta }}$ . The attitude control system can be rewritten as

(13.26a) ${\dot{\mathit{x}}}_1={\mathit{x}}_2,$

(13.26b) ${\dot{\mathit{x}}}_2=-{\mathit{M}}^{-1}\mathit{C}({\mathit{x}}_1,{\mathit{x}}_2){\mathit{x}}_2+\bar{\mathit{B}}\mathit{\tau }+{\mathit{x}}_3,$

with $\bar{\mathit{B}}={\mathit{M}}^{-1}{\mathit{F}}^{\mathrm{T}}$ and ${\mathit{x}}_3={\mathit{M}}^{-1}\bar{\mathit{d}}$ . Then, a nonsingular terminal sliding mode surface (NTSMS) [40] can be defined as

(13.27) ${\mathit{S}}_L={\mathit{x}}_1+\frac{{\alpha }_{31}}{L^{\frac{1}{2}}}{\mathrm{sig}}^{\frac{3}{2}}\left({\hat{\mathit{x}}}_2\right)$

where ${\mathit{S}}_L={[s_{L1},s_{L2},s_{L3}]}^{\mathrm{T}}$ , $L>0$ and ${\alpha }_{31}>0$ are positive gains to be designed. It should be noted that (13.27) is introduced as a real sliding mode surface. When $L=1$ , we define ${\mathit{S}}_L$ simply as S . For the attitude control system in (13.26) and the NTSMS in (13.27), a novel simple FTDFC is developed in the form of

(13.28) $\mathit{\tau }={\bar{\mathit{B}}}^{-1}(-k_1{L}^{\frac{2}{3}}{\mathrm{sig}}^{\frac{1}{3}}\left({\mathit{S}}_L\right)+{\mathit{M}}^{-1}\mathit{C}({\mathit{x}}_1,{\hat{\mathit{x}}}_2){\hat{\mathit{x}}}_2-{\hat{\mathit{x}}}_3+{\mathit{z}}_1),$

(13.29) ${\dot{\mathit{z}}}_1=-k_{2}L\mathrm{sign}\left({\mathit{S}}_L\right)$

where ${\hat{\mathit{x}}}_2$ and ${\hat{\mathit{x}}}_3$ are the observation values of the unknown attitude angular velocity and synthetic failure deriving from the proposed FTESO in Sect. 13.3.2, whereas $k_1$ and $k_2$ are positive gains to be designed. Submitting the FTDFC in (13.28)–(13.29) into the attitude dynamics in (13.26), it follows that

(13.30a) ${\dot{\mathit{x}}}_1={\mathit{x}}_2,$

(13.30b) ${\dot{\mathit{x}}}_2=-k_1{L}^{\frac{2}{3}}{\mathrm{sig}}^{\frac{1}{3}}\left({\mathit{S}}_L\right)-\tilde{\zeta }-{\mathit{e}}_3+{\mathit{z}}_1.$

Defining $\mu =-\tilde{\zeta }-{\mathit{e}}_3$ and ${\mathit{z}}_3\triangleq -\tilde{\zeta }-{\mathit{e}}_3+{\mathit{z}}_1$ , the closed-loop attitude control system in (13.26)–(13.29) yields the following third order plant:

(13.31a) ${\dot{\mathit{x}}}_1={\mathit{x}}_2,$

(13.31b) ${\dot{\mathit{x}}}_2=-k_1{L}^{\frac{2}{3}}{\mathrm{sig}}^{\frac{1}{3}}\left({\mathit{S}}_L\right)+{\mathit{z}}_3,$

(13.31c) ${\dot{\mathit{z}}}_3=-k_{2}L\mathrm{sign}\left({\mathit{S}}_L\right)+\dot{\mu }$

where ${\mathit{x}}_1$ , ${\mathit{x}}_2$ , and ${\mathit{z}}_3$ are the states of the above closed-loop plant. According to the results in [41], the solutions of the above third order system will be found in the sense of Filippov. It is easily obtained that the Filippov differential inclusion corresponding to the closed-loop third-order attitude stabilization control system in (13.31) is homogeneous of degree (scaled to) $\delta =-1$ with weights $(r_1=3,r_2=2,r_3=1)$ . Then, the other key contribution of this paper is presented as the following theorem.

Theorem 13.2

Consider the simplified attitude stabilization control system in (13.9) with the assumption that $\left|{\dot{\mu }}_i\right|\le \bar{\mu }$ , $i=1,2,3$ , $\bar{\mu }>0$ . The attitudes are globally and uniformly finite-time stable, if the NTSMS based attitude control scheme FTDFC is developed in the form of (13.28) – (13.29) .

To prove and analyze the above Theorem 13.2, the following definition will be introduced and used [41,43].

Definition 13.2

The origin $x=0$ of a differential inclusion $\dot{x}\in F(x)$ (a differential equation $\dot{x}=f(x)$ ) is called globally uniformly finite-time stable, if it is Lyapunov stable and there exists $T>0$ such that the trajectory with initial condition $\Vert x_0\Vert <\varsigma (\varsigma >0)$ will converge to $x=0$ at time T.

Proof of Theorem 13.2

Firstly, select the candidate Lyapunov function as follows:

(13.32) $V_{2i}=\frac{2}{5}{\alpha }_{31}{\left|x_{2i}\right|}^{\frac{5}{2}}+{\beta }_{31}{\left|x_{1i}\right|}^{\frac{5}{3}}+x_{1i}{x}_{2i}-\frac{1}{k_1^3}{x}_{2i}{z}_{3i}^3+k_3{\left|z_{3i}\right|}^5,i=1,2,3$

where $k_3>0$ and ${\beta }_{31}>0$ . For the above candidate Lyapunov function $V_{2i}$ , one can obtain easily that $V_{2i}$ is homogeneous with the degree ${\delta }_{V_{2i}}=5$ , and also continuously differentiable. Utilizing the Young's inequality, $V_{2i}$ can be calculated to be positive definite and bounded, if the control gains are selected as ${\beta }_{31}>\frac{3}{5}{\left(\frac{4}{{\alpha }_{31}}\right)}^{\frac{2}{3}}$ and $k_1^5{k}_3>\frac{3}{5}{\left(\frac{4}{{\alpha }_{31}}\right)}^{\frac{2}{3}}$ .

Taking the derivative of $V_{2i}$ with respect to time, it follows that

(13.33) $\begin{array}{cc}\hfill {\dot{V}}_{2i}=& -({\alpha }_{31}{\mathrm{sig}}^{\frac{3}{2}}\left(s_{Li}\right)-\frac{1}{k_1^3}{z}_{3i}^3)(k_1{\mathrm{sig}}^{\frac{1}{3}}\left(s_{Li}\right)-z_{3i})\hfill \\ \hfill & +(\frac{5}{3}{\beta }_{31}{\mathrm{sig}}^{\frac{2}{3}}\left(x_{1i}\right)+x_{2i})x_{2i}\hfill \\ \hfill & -{\left|z_{3i}\right|}^2(5k_3{\left|z_{3i}\right|}^2-\frac{3}{k_1^3}{x}_{2i})(k_2\mathrm{sign}\left(s_{Li}\right)+\frac{{\dot{\mu }}_i}{L}).\hfill \end{array}$

Utilizing Lemma 2 and Procedure 1 in [41], one can get that ${\dot{V}}_{2i}$ is continuous and negative definite. And that ${\dot{V}}_{2i}$ is homogeneous with the degree ${\delta }_{{\dot{V}}_{2i}}=4$ according to the theory of homogeneous functions. Then, one can see that

(13.34) ${\dot{V}}_{2i}\le -{\eta }_i{V}_{2i}^{\frac{4}{5}}$

where ${\eta }_i$ is a positive constant. Thus, the attitude stabilization control system in (13.9) is uniformly globally asymptotically stable according to the basic Lyapunov theory for differential inclusions. Furthermore, it is globally uniformly finite-time stable since the closed-loop system in (13.31) is homogeneous of degree $\delta =-1$ . Certainly, the results in Theorem 13.2 could be achieved by using Lemma 13.1 as well.   □

Remark 13.2

The control scheme proposed in (13.28)–(13.29) is inspired by [41]. This work applies the control methods in [41] to the quadrotor UAV attitude stabilization problem even though there exist model uncertainty, actuator failures, and disturbances. As for the specific analysis of Theorem 13.2, it is similar to [41] and the detailed analysis is omitted here for space limitation. Nevertheless, it should be noted that the assumption $\left|{\dot{\mu }}_i\right|\le \bar{\mu }$ is reasonable. According to Theorem 13.1, one can easily obtain that ${\mathit{e}}_2$ , ${\dot{\mathit{e}}}_2$ , ${\mathit{e}}_3$ , and ${\dot{\mathit{e}}}_3$ are all continuous and bounded under the assumption that $\mathit{g}(t)$ is bounded. And $\tilde{\mathit{\zeta }}({\mathit{x}}_1,{\mathit{x}}_2)={\mathit{M}}^{-1}\mathit{C}({\mathit{x}}_1,{\mathit{x}}_2){\mathit{e}}_2-{\mathit{M}}^{-1}\mathit{C}({\mathit{x}}_1,{\mathit{e}}_2)({\mathit{x}}_2+{\mathit{e}}_2)$ is also bounded deriving from (13.15), and so is $\mathit{\mu }=-\tilde{\mathit{\zeta }}-{\mathit{e}}_3$ . Thus, it is also reasonable to suppose that there exists a positive and sufficiently large constant $\bar{\mu }$ such that $\left|{\dot{\mu }}_i\right|\le \bar{\mu }$ on the basis of Assumption 13.2. Although the assumption may appear restrictive, it readily provides some help for the finite-time stability analysis of the proposed control law. And also this bound $\bar{\mu }$ could be an arbitrary positive value, because it has nothing to do with the dynamic process of the attitude control. Therefore, Theorem 13.2 will be applicable with the only requirement that $\left|{\dot{\mu }}_i\right|$ is bounded.

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Active control of flow-induced cavity oscillations

Louis N. CattafestaIII, ... Farrukh S. Alvi , in Progress in Aerospace Sciences, 2008

A further non-standard but useful classification of closed-loop flow control is that of quasi-static vs. dynamic feedback control. The quasi-static case corresponds to slow tuning of an open-loop control approach and occurs when the time scales of feedback are large compared to the time scales of the plant (i.e., flow). This approach is particularly relevant in nonlinear fluid dynamic systems, where the fundamental notion of frequency preservation in a linear system does not hold. As discussed in Section 5, the quasi-static approach was successfully used by Shaw and Northcraft [27]. The usual dynamic compensation case corresponds to the situation when the above time scales are commensurate. This can be implemented using an analog (see, for example, Williams et al. [28]) or "real-time" digital control systems [29]. In this context, "real time" refers to the situation in which the control signal is updated at the sampling rate of the data system, and the actuator response to the flow state changes at the time scales of the dynamics.

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Supporting disassembly processes through simulation tools: A systematic literature review with a focus on printed circuit boards

Claudio Sassanelli , ... Sergio Terzi , in Journal of Manufacturing Systems, 2021

4.1.7 Benefits and business impacts

Ref. [98] used a dynamic business model simulation to estimate CE business impacts. Ref. [99] conducted a simulation, based on SD modelling, whose results stated that manufacturing transition towards CE can foster coal power and cement companies to decrease waste emission and improve economic profits. Ref. [100 ] focused on the metallurgical industry. Given that: i) all metals have strong intrinsic recycling potentials and ii) a digital integration of metallurgical reactor technologies and systems can support dynamic feedback control loops, they used modelling, simulation, and optimization to perform real-time measurement of ore and scrap properties in intelligent plant structures, by enabling CE-oriented big data analysis and process control of industrial metallurgical systems. Results were used to elaborate in an easy way the resource efficiency of the CE system. Ref. [ 101] proposed a similar SD model, but applied on coal resource utilization systems with a full lifecycle perspective. Thirteen development projects divided in two types of scenarios were run on the model. Simulation results were analysed through the efficacy coefficient method to determine the best project of coal resource utilization system and demonstrate the benefits coming from CE adoption. Ref. [102] used SD methods to enlighten the benefits deriving from CE in countries without resource shortage issues. The simulation demonstrated that, despite the investment costs, countries can obtain economic benefits through a lower raw material cost in a long run.

Table 9 describes the "benefits and business impacts" category. Only SD simulation method is used to support either strategic analysis about raw material costs and sale volume or specific product lifecycle phases (from product design, through manufacturing up to disassembly), attempting to optimize waste emission, economic profits, resource utilization or the materials recovery process.

Table 9. Benefits and business impacts category.

Authors	What is proposed	Lifecycle phase improved through simulation	Technologies/tools involved	Type of simulation			Improved human machine interaction	Variables optimized
				Physics-based	VR	AR
[98]	- Dynamic business model simulation using the SD methodology, - CE design framework to support circular product design development	Product design	–	SD	–	–	–	CE Business model
[99]	SD modelling to infer that the transformation of manufacturing towards a CE system can prominently help coal power and cement enterprises reduce waste emission and increase economic profits	Manufacturing	–	SD	–	–	–	- waste emission, - increase economic profits
[101]	Coal resource utilization SD model based on full life cycle	Manufacturing	–	SD	–	–	–	Coal resource utilization
[100]	Integration of metallurgical reactor technology and systems digitally, not only on one site but linking different sites globally via hardware, for describing CE systems as dynamic feedback control loops, i.e., the metallurgical IoT	Disassembly	Connection via digital technologies	SD	–	–	–	Recovery of all minor and technology elements in the refining metallurgical infrastructure of the global carrier
[102]	Economic benefits of a CE in countries without resource scarcity problem. SD method was used to overcome the data limitation problem	Strategic level	–	SD	–	–	–	- raw material costs, - raw material sale volumes
TOTAL				5	0	0	0

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Feedback Control of Dynamic Systems Solutions Chapter 3

Source: https://www.sciencedirect.com/topics/engineering/dynamic-feedback-control

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