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Medical Instructor, Edward Via College of Osteopathic Medicine
We do not insist erectile dysfunction pills cvs cheap kamagra gold 100mg overnight delivery, however erectile dysfunction without drugs buy 100 mg kamagra gold amex, that the state is the least such information although this is often a convenient assumption impotence remedy discount 100 mg kamagra gold free shipping. The concept of reachability is introduced and used to investigate how to design the dynamics of a system through assignment of its eigenvalues erectile dysfunction instrumental buy kamagra gold 100mg lowest price. In particular, it will be shown that under certain conditions it is possible to assign the system eigenvalues arbitrarily by appropriate feedback of the system state. It turns out that the property of reachability is also fundamental in understanding the extent to which feedback can be used to design the dynamics of a system. Definition of Reachability We begin by disregarding the output measurements of the system and focusing on the evolution of the state, given by dx = Ax + Bu, (6. A fundamental question is whether it is possible to find control signals so that any point in the state space can be reached through some choice of input. To study this, we define the reachable set R(x0, T) as the set of all points x f such that there exists an input u(t), 0 t T that steers the system from x(0) = x0 to x(T) = x f, as illustrated in Figure 6. A linear system is reachable if for any x0, x f Rn there exists a T > 0 and u: [0, T] R such that the corresponding solution satisfies x(0) = x0 and x(T) = x f. The set R(x0, T) shown in (a) is the set of points reachable from x0 in time less than T. The phase portrait in (b) shows the dynamics for a double integrator, with the natural dynamics drawn as horizontal arrows and the control inputs drawn as vertical arrows. By setting the control inputs as a function of the state, it is possible to steer the system to the origin, as shown on the sample path. The definition of reachability addresses whether it is possible to reach all points in the state space in a transient fashion. In many applications, the set of points that we are most interested in reaching is the set of equilibrium points of the system (since we can remain at those points once we get there). The set of all possible equilibria for constant controls is given by E = {xe: Axe + Bu e = 0 for some u e R}. This means that possible equilibria lie in a one- (or possibly higher) dimensional subspace. The open loop dynamics (u = 0) are shown as horizontal arrows pointed to the right for x2 > 0 and to the left for x2 < 0. The control input is represented by a double-headed arrow in the vertical direction, corresponding to our ability to set the value of x2. We can directly move the state up and down in the phase plane, but we must rely on the natural dynamics to control the motion to the left and right. If a > 0, we can move the origin by first setting u < 0, which will cause x2 to become negative. After a while, we can set u 2 to be positive, moving x2 back toward zero and slowing the motion in the x1 direction. Note that if we steer the system to an equilibrium point, it is possible to remain there indefinitely (since x1 = 0 when x2 = 0), but if we go to any other point in the state space, we can pass through the point only in a transient fashion. To find general conditions under which a linear system is reachable, we will first give a heuristic argument based on formal calculations with impulse functions. We note that if we can reach all points in the state space through some choice of input, then we can also reach all equilibrium points. Testing for Reachability When the initial state is zero, the response of the system to an input u(t) is given by t x(t) = e A(t-) Bu() d. To reach an arbitrary point in the state space, we thus require that there are n linear independent columns of the matrix Wr. An input consisting of a sum of impulse functions and their derivatives is a very violent signal. Assuming that the initial condition is zero, the state of a linear system is given by t t x(t) = 0 e A(t-) Bu()d = 0 e A Bu(t -)d. It follows from the theory of matrix functions, specifically the CayleyHamilton theorem (see Exercise 6. Again we observe that the right-hand side is a linear combination of the columns of the reachability matrix Wr given by equation (6. A linear system is reachable if and only if the reachability matrix Wr is invertible. Recall that this system is a model for a class of examples in which the center of mass is balanced above a pivot point.
If a hospital performs better than an average hospital that admitted similar patients (that is erectile dysfunction recreational drugs best 100mg kamagra gold, patients with the same risk factors for readmission such as age and comorbidities) erectile dysfunction pills pictures discount 100mg kamagra gold with visa, the ratio will be less than one low libido erectile dysfunction treatment buy 100 mg kamagra gold visa. Hospitals with a ratio greater than one have excess readmissions relative to average quality hospitals with similar types of patients erectile dysfunction drugs used discount 100mg kamagra gold fast delivery. The riskstandardized ratio is the unique result produced by the measures for each hospital for each condition to assess relative hospital performance. Multiplying by a constant transforms the ratio into a rate (the riskstandardized readmission rate) that is better understood by the public. The hospital-specific effect will be negative for a hospital above the national average (that is, with lower than average adjusted rates of readmissions), positive for a hospital below the national average (that is, with higher than average adjusted rates of readmissions), and close to zero for an average hospital. If there are no quality differences resulting in excess readmissions among hospitals (if all hospitals had the same readmission rates relative to hospitals with similar patients), the hospital-specific effects for all hospitals will be zero and the ratio for all hospitals will be one. Comment: One commenter expressed concern that multiplying the ratio by the national raw rate of readmissions could inflate the readmission rate for a given hospital. Response: As discussed above, the Excess Readmission Ratio is calculated using hierarchical logistic regression which produces an adjusted actual (or ``predicted') number in the numerator and an ``expected' number in the denominator. This serves to standardize all hospitals rates to the national rate but should not be interpreted as the unadjusted rate for a given hospitals. The Hospital Readmissions Reduction Program uses the Excess Readmission Ratio rather than the raw readmission rate. This estimated probability of readmission for each patient is calculated using: · the hospital-specific effect (probability of readmission relative to the probability of readmission at an average hospital); · the intercept term for the model (this is the average hospital-specific effect and is the same for all hospitals and for both numerator and denominator equations). The specific approach and variables used in the risk adjustment are discussed below. This approach to calculating the numerator, although more complex than that used for logistic regression, is the method traditionally used in hierarchical regression modeling and is statistically more accurate given the type of data being used. Hospitals with more adjusted actual readmissions than expected readmissions will have a riskstandardized ratio (Excess Readmission Ratio) greater than one. Because the ratio is risk-adjusted, a hospital may have high crude readmission rates (number of 30-day readmissions among patients with the applicable condition divided by number of admissions for patients with the applicable condition) yet have a riskstandardized ratio (Excess Readmission Ratio) less than one. For example, if a hospital with a higher than average raw readmission rate cares for very sick patients, the ratio may show that the adjusted actual number of readmissions (the numerator), which accounts for the case-mix, is actually lower than what would be expected for an average hospital caring for these patients (denominator) and therefore the Excess Readmission Ratio, as proposed, will be less than one, demonstrating that this hospital performs better than average, despite having a high crude readmission rate. Similarly, if a hospital has a seemingly low unadjusted readmission rate but cares for a very low risk population of patients, it may be found to have an adjusted actual number of readmissions that is higher than the expected number of readmissions, and therefore a ratio greater than one. If a hospital performs better than an average hospital that admitted similar patients (that is, patients with the same risk factors for readmission such as age and comorbidities), the ratio will be less than 1. We welcomed public comment on our proposal to use this methodology for 51676 Federal Register / Vol. Finally, within hierarchical models, we can account for both differences in case mix and sample size to more fairly profile hospital performance. Some commenters compared the hierarchical modeling approach to the logistic regression model, which produces an expected rate for the denominator and uses the observed (raw count of readmission) for the numerator. Consistent with the statutory requirement that the Secretary must develop a risk-adjusted Excess Readmission Ratio that is the ratio of ``the risk adjusted readmissions based on actual readmission, as determined consistent with a readmission measure methodology that has been endorsed under paragraph (5)(A)(ii)(I) * * * to the risk adjusted expected readmissions,' we proposed to calculate the Excess Readmission Ratio using hierarchical modeling (rather than logistic regression, which produces an observed over expected ratio). We believe that hierarchical modeling is a more appropriate statistical approach for hospital outcomes measures than the calculation of observed over expected ratio using the logistic regression model for various reasons. Second, we believe that hierarchical modeling is a more appropriate statistical approach given the structure of the data and the underlying assumption of such measures which is that hospital quality of care influences 30-day readmission rates. The advantage of using the hierarchical modeling is that it accounts for the clustering of patients within hospitals. As specified in section 1886(d)(5)(C)(ii) of the Act, the national standard is set at 5,000 discharges. We note that the median number of discharges for hospitals in each census region is greater than the national standard of 5,000 discharges. Therefore, 5,000 discharges is the minimum criterion for all hospitals under this final rule. The additional payment adjustment to a low-volume hospital provided for under section 1886(d)(12) of the Act is ``in addition to any payment calculated under this section.
The basic concepts that we describe hold more generally and can be used to understand dynamical behavior in higher dimensions impotence herbal medicine purchase kamagra gold 100 mg without a prescription. Phase Portraits A convenient way to understand the behavior of dynamical systems with state x R2 is to plot the phase portrait of the system erectile dysfunction medication side effects generic kamagra gold 100 mg free shipping, briefly introduced in Chapter 2 antihypertensive that causes erectile dysfunction generic 100 mg kamagra gold otc. For a system of ordinary differential equations dx = F(x) erectile dysfunction exercise video kamagra gold 100 mg free shipping, dt the right-hand side of the differential equation defines at every x Rn a velocity F(x) Rn. For planar dynamical systems, each state corresponds to a point in the plane and F(x) is a vector representing the velocity of that state. We can plot these vectors on a grid of points in the plane and obtain a visual image of the dynamics of the system, as shown in Figure 4. The points where the velocities are zero are of particular interest since they define stationary points of the flow: if we start at such a state, we stay at that state. A phase portrait is constructed by plotting the flow of the vector field corresponding to the planar dynamical system. That is, for a set of initial conditions, we plot the solution of the differential equation in the plane R2. This corresponds to following the arrows at each point in the phase plane and drawing the resulting trajectory. By plotting the solutions for several different initial conditions, we obtain a phase portrait, as show in Figure 4. Phase portraits give insight into the dynamics of the system by showing the solutions plotted in the (two-dimensional) state space of the system. For example, we can see whether all trajectories tend to a single point as time increases or whether there are more complicated behaviors. An inverted pendulum is a model for a class of balance systems in which we wish to keep a system upright, such as a rocket (a). Using a simplified model of an inverted pendulum (b), we can develop a phase portrait that shows the dynamics of the system (c). The system has multiple equilibrium points, marked by the solid dots along the x2 = 0 line. Equilibrium Points and Limit Cycles An equilibrium point of a dynamical system represents a stationary condition for the dynamics. We say that a state xe is an equilibrium point for a dynamical system dx = F(x) dt if F(xe) = 0. If a dynamical system has an initial condition x(0) = xe, then it will stay at the equilibrium point: x(t) = xe for all t 0, where we have taken t0 = 0. Equilibrium points are one of the most important features of a dynamical system since they define the states corresponding to constant operating conditions. The inverted pendulum is a simplified version of the problem of stabilizing a rocket: by applying forces at the base of the rocket, we seek to keep the rocket stabilized in the upright position. The state variables are the angle = x1 and the angular velocity d/dt = x2, the control variable is the acceleration u of the pivot and the output is the angle. For simplicity we assume that mgl/Jt = 1 and ml/Jt = 1, so that the dynamics (equation (2. This same set of equations can also be obtained by appropriate normalization of the system dynamics as illustrated in Example 2. The phase portrait (a) shows the states of the solution plotted for different initial conditions. The simulation (b) shows a single solution plotted as a function of time, with the limit cycle corresponding to a steady oscillation of fixed amplitude. The equilibrium points for n even correspond to the pendulum pointing up and those for n odd correspond to the pendulum hanging down. A phase portrait for this system (without corrective inputs) is shown in Figure 4. The equilibrium points for the system are given by ±n, xe = 0 Nonlinear systems can exhibit rich behavior. This is of great practical value in generating sinusoidally varying voltages in power systems or in generating periodic signals for animal locomotion. A normalized model of the oscillator is given by the equation d x2 d x1 2 2 2 2 (4. The figure shows that the solutions in the phase plane converge to a circular trajectory. More formally, we call an isolated solution x(t) a limit cycle of period T > 0 if x(t + T) = x(t) for all t R.
The condition that V (x) is negative simply means that the vector field points toward lower-level contours erectile dysfunction doctors jacksonville fl kamagra gold 100 mg online. This means that the trajectories move to smaller and smaller values of V and if V is negative definite then x must approach 0 erectile dysfunction treatment cincinnati cheap 100 mg kamagra gold amex. We consider the equilibrium point at x = 1 and rewrite the dynamics using z = x - 1: 2 dz = - z - 1 erectile dysfunction natural remedies over the counter herbs buy cheap kamagra gold 100 mg on-line, dt 2+z which has an equilibrium point at z = 0 impotence after prostatectomy cheap 100mg kamagra gold with mastercard. The derivative of V along trajectories of the system is given by 2z V (z) = z z = - z 2 - z. In this case it is possible that V (x) = 0 when x = 0, and hence x could stop decreasing in value. The equation has an equilibrium x1 = x2 = 0, which corresponds to the pendulum hanging straight down. To explore the stability of this equilibrium we choose the total energy as a Lyapunov function: 1 2 V (x) = 1 - cos x1 + x2 2 the Taylor series approximation shows that the small x. When perturbed, the pendulum actually moves in a trajectory that corresponds to constant energy. In many cases energy functions can be used as a starting point, as was done in Example 4. It turns out that Lyapunov functions can always be found for any stable system (under certain conditions), and hence one knows that if a system is stable, a Lyapunov function exists (and vice versa). Sum-ofsquares techniques can be applied to a broad variety of systems, including systems whose dynamics are described by polynomial equations, as well as hybrid systems, which can have different models for different regions of state space. To do so, we consider quadratic functions of the form V (x) = x T P x, where P RnЧn is a symmetric matrix (P = P T). The condition that V be positive definite is equivalent to the condition that P be a positive definite matrix: x T P x > 0, for all x = 0, which we write as P > 0. It can be shown that if P is symmetric, then P is positive definite if and only if all of its eigenvalues are real and positive. Given a candidate Lyapunov function V (x) = x T P x, we can now compute its derivative along flows of the system: V dx V = = x T (A T P + P A)x =: -x T Qx. Thus, to find a Lyapunov function for a linear system it is sufficient to choose a Q > 0 and solve the Lyapunov equation: A T P + P A = -Q. It can be shown that the equation always has a solution if all of the eigenvalues of the matrix A are in the left half-plane. It is thus always possible to find a quadratic Lyapunov function for a stable linear system. We will defer a proof of this until Chapter 5, where more tools for analysis of linear systems will be developed. Knowing that we have a direct method to find Lyapunov functions for linear systems, we can now investigate the stability of nonlinear systems. The function Ax is an approximation of F(x) near the origin, and we can determine the Lyapunov function for the linear approximation and investigate if it is also a Lyapunov function for the full nonlinear system. The circuit diagram in (a) represents two proteins that are each repressing the production of the other. The inputs u 1 and u 2 interfere with this repression, allowing the circuit dynamics to be modified. The equilibrium points for this circuit can be determined by the intersection of the two curves shown in (b). The equilibrium points for the system are found by equating the time derivatives to zero. We define f (u) = µ, 1 + un f (u) = df -µnu n-1 =, du (1 + u n)2 and the equilibrium points are defined as the solutions of the equations z 1 = f (z 2), z 2 = f (z 1). If we plot the curves (z 1, f (z 1)) and (f (z 2), z 2) on a graph, then these equations will have a solution when the curves intersect, as shown in Figure 4. Because of the shape of the curves, it can be shown that there will always be three solutions: one at z 1e = z 2e, one with z 1e < z 2e and one with z 1e > z 2e.
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