Daniel Davis
Chattanooga Whiskey Releases Batch 033: Triple Peat
Chattanooga Whiskey, a Tennessee-based distiller, is adding a new small batch release to its award-winning lineup. This limited edition whiskey, called Batch 033: Triple Peat, is made with three different types of peat-smoked malt barley and is sure to be a hit among whiskey connoisseurs.
How is Batch 033 Different?
The unique character of Batch 033 comes from the use of three types of peat-smoked malt barley. Maris Otter barley is the first, resulting in a robust flavor with herbal notes. The second is Peated Malt from Scotland, where the traditional technique of using a peat fire to dry malt barley gives the spirit a decidedly smokier flavor. Finally, the third type is Canadian Cherrywood-Smoked Malt, which enhances the flavor with a subtle sweetness.
Why Does Peat Matter?
Peat is a type of soil found mainly in Scotland and other areas with a temperate climate. It is composed of partially decayed plant matter and is a major factor in the flavor and color of the whiskey. Peat smoke adds an earthy and smoky flavor to the whiskey, which has made it popular among Scotch drinkers.
What Does Batch 033 Taste Like?
Batch 033 has a slightly smoky nose with hints of chocolate, molasses, and roasted nuts. On the palate, the whiskey is rich and full-bodied with flavors of brown sugar, vanilla, and hints of smoke. The finish is long and smooth, with a subtle sweetness from the Canadian Cherrywood-Smoked Malt.
Where Can You Buy Batch 033?
Chattanooga Whiskey Batch 033: Triple Peat is available now from select retailers in the United States and Canada. A bottle retails for around $40. If you’re looking for a special whiskey to add to your collection, or if you’re just a fan of peat-smoked malts, then this limited edition whiskey from Chattanooga Whiskey is definitely worth a try. Cheers!
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Term Load
In the context of finance, the term load refers to a fee or charge that is applied to an investment, loan, or insurance policy. It is typically a percentage of the total amount being invested or borrowed, and is used to cover administrative costs, commissions, or expenses related to the product or service. Term loads can vary depending on the type of investment or loan and the provider, and they may be one-time or ongoing charges. They can have a significant impact on the overall returns or costs of an investment or loan, and investors or borrowers should carefully consider the term loads associated with any financial product before making a decision.
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chapter{7. Robust Control}
section{Introduction}
In this chapter, the robust control systems are studied. There are many different approaches for the robust control cite{khalil}. Generalized stability, small gain theorems, and structured singular values provide a theoretical foundations for the fundamentals of robust control. These theories are used to analyze the effects of uncertainty and disturbances on the stability and performances of closed-loop systems, and to design robust controllers to guarantee stability and performance specifications. Application of robust control in electrical motors and power electronics are also shown in this chapter.
section{Generalized Stability and Small Gain Theorems}
A fundamental issue in control theory is the stability of closed-loop systems in the presence of uncertainties and disturbances. Robust stability analysis is an extension of classical stability analysis which takes into account the effect of uncertainties. The classical stability analysis evaluates the robustness of a system against additive uncertainties inserted directly on the system input and concerns high frequency characteristics of the system. The uncertainties in robust stability analysis are assumed to be bounded and nonlinear in general which is more efficient to model actual systems.
Generalized stability criteria provide a powerful tool for rigorous stability analysis of uncertain systems. Generalized stability criteria arise from the frequency domain representation of closed loop systems with uncertainties. Robust stability analysis can be implemented using different frequency domain approaches.
Dependent on the application we choose the suitable method as well as the appropriate uncertainty model to analyze stability and performance, and thereby obtain a suitable designed controller for these specifications.
The most general mathematical model for a physical uncertain system in terms of mapping $ρ:VΘ→L2timesL2$, which maps the uncertainty space $VΘ$ onto the set of unstable mappings $L2timesL2$, or in terms of mapping $ρ:VΘ→{textbf{M}}_n(mathbb{C})$ such that $rho(theta)=jω→rho(theta) in ρ$. Here $VΘ$ is the uncertainty space parameterized by the vector $θ$. The linear systems with complex uncertainty cite{khalil} that is representable by the $ρ$ mapping in terms of $ρ:VΘ→L2$
Frequency domain analysis can be applied to analyze uncertain systems via the so-called block structured representation. In order to explain block structured representation and give an insight into analysis of uncertain systems via these representations, a simple example of a closed-loop feedback system described in Figure ref{fig:LOOPS}, is presented.
begin{figure}[H]
centering
includegraphics[scale=0.7]{figure/7_1.eps}
caption{Closed-loop control system which the feedback loop includes uncertainty.}label{fig:LOOPS}
end{figure}
The output-to-input transfer function of the plant shown in Figure (ref{fig:LOOPS}) is denoted by $G$. An uncertainty model $Delta(s)$ is inserted at the feedback path as representing by $χ$. Applying the output/input transfer function simply yields the closed loop transfer function $T(s)$ given
begin{equation} label{eq:1}
T(s)=frac{G}{1+Gχ}
end{equation}
Applying ref{eq:1} results in the closed loop transfer matrix
begin{equation}
T=left[
begin{array}{c.c}
mathbb{C}&mathbb{C}^{n_t times n_t} \
mathbb{C}^{n_s times n_s}& mathbb{C}^{n_s times n_t}
end{array}
right]
end{equation}
Where $mathbb{C}$ is the space of $Ln$ which is the uncertain mapping. The robust stability theorem aims to provide a set of sufficient conditions that the nominal closed loop transfer matrix $T$ in ref{eq:1} satisfies for a given bounds of the perturbation uncertainty model $χ$. These conditions are given using the generalized Popov criterion. The Popov criterion and its equivalent Lyapunov stability conditions is extremely useful when applied to closed loop systems with formation of $D-A$ and $D$(gain) negative perturbations.
begin{remark}{Block Structured Uncertain System}
The blocks in a block structured representation of an uncertain system may have the following appearances:
begin{itemize}
item $D-A$ perturbation on a block (e.g. $ds + b$
item Multiplicative perturbation
item Unstructured norm-bounded perturbations
item Additive norm-bounded uncertainty on blocks
item Gain Uncertainty by which every block is multiplied
item Integral Uncertainty by which every block is integrated
end{itemize}
end{remark}
subsection{Robust Stability Analysis}
Robust stability or stability in the presence of uncertainty means that, once the nominal poles and zeros of a system are stabilized, it is independent of all admissible uncertainty variations in the sense that all poles still belong to the open left-half complex plane (or wherever the unperturbed poles are located) for any uncertainty from an uncertainty set. This robustness concept has clear practical meaning as it indicates robustness against additive, multiplicative or unstructured variations in system matrices.
The robust stability theorems depend on the frequency domain representation of closed-loop systems. These frequency domain criteria include $mathrm{ft}$ singular value theorem, the $mu$ theorem and algebraic criteria like small gain theorems.
begin{remark}{SISO Robust Stability}
The simplest way to define a simple criterion for robust stability is to use the Popov criterion which is a direct extension from the Nyquist criterion. As discussed before, will only consider systems with feedback loops of the form.
$$mathrm{ft}left[
begin{array}{c|c}
A&B\
hline
C&D
end{array}
right]$$ It is assumed that $D = 0$ for the SISO case.
The Popov criterion
begin{equation} label{popov}
t_{out}(s) = C_1(sI-A)^{-1}B_1 + C_2(sI-A)^{-1}B_2 t_{in}
end{equation}
and $gamma(α)$ in the plane.
Requirements are
$$|t_{out}(α mathrm{j})| < 1$$ $$γ(α + mathrm{j} β) = 0$$
end{remark}
begin{remark}{Relationship Between OCG and LMI Tools}
Given the complexity of feedback control systems the usual computational complexity involved is strictly related to the complexity of solving Linear Matrix Inequalities (LMIs)[]. Modern control CAD tools are available that allow feedback control systems to be designed efficiently for many control specific performance criteria such as overshoot, 2-norm sensitivity, control action rate and robust stability cite{khalil}. In addition these tools use the associativity of developing PI type structured uncertain systems to offer efficient uncertain controller designs. These tools are capable of handling unstructured or semi-structured perturbations. Moreover, methods have been suggested for systematically reducing the conservatism that usually results when designing $H_∞$ controllers [cite{khalil}], and computation time is aid by using single formulations of optimal linear matrix inequalities (SMI). Coefficient diagram of the POP% method is presented in this section.
end{remark}
subsection{Robust Performance Analysis}
There are several performance criteria that measure the proposed requirements for “robustness”. High frequency performance specifications link to robustness near the control axis while shaping the uncertainty out pertains to low frequency performance specifications cite{khalil}. The classical H∞ control criterion is defined by locating the controller poles such that their imaginary parts are bounded by $ω_i=ω/λ$. The chosen frequency $ω_op$ is called the crossover frequency. Using this method the sensitivity function is shaped such that proportional to the input disturbance is desensitized at high frequencies (where disturbance rejection performance is concerned), while low frequency performance usually associated with reference input signals (such as position or trajectory tracking) is maintained intact. Another means of performance called $H_2$ performance is considered as an enhancement of sensitivity performance as well as a measure of disturbance rejection performance sensitivity is not reduced to the level as with H∞ methods, and the $gamma^2$ computed for each radial direction.
newpage
section{Controller Design}
To incorporate robust control design using the tools mentioned in sections above, the main goal is to design a uncertainty block $Delta(s)$ such that $gamma$ from Figure ref{fig:Attach} as large as possible
begin{figure}[H]
centering
includegraphics[scale=0.7]{figure/7_2.eps}
caption{Unstructured uncertainty attachment point}label{fig:Attach}
end{figure}
The next step is finding a relation for the PID design and apply it to the error block and then construct the PID controller with the help of red shifting procedure to enhance this error block.
begin{figure}[H]
centering
includegraphics[scale=0.8]{figure/7_3.eps}
includegraphics[scale=0.8]{figure/7_4.eps}
caption{Block diagram of the basic $H_2/H_infty$ design problem}label{fig:zafar}
end{figure}
subsection{PID Design}
With μ upper bound we can find the lower bound value for the function using Chebyshev polynomial. The two problems that including into the robust set of problems
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