Max-Plus Methods for Nonlinear Control and Estimation (Systems & Control: Foundations & Applications)

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What is MDS? LibraryThing's MDS system is based on the classification work of libraries around the world, whose assignments are not copyrightable. MDS "scheduldes" the words that describe the numbers are user-added, and based on public domain editions of the system.

Journal of Formalized Reasoning

Wordings, which are entered by members, can only come from public domain sources. Where useful or necessary, wording comes from the edition of the Dewey Decimal System. Language and concepts may be changed to fit modern tastes, or to better describe books cataloged. Based on equations 6 and 33 , the stimulation control input is designed as. Note that equation 35 is valid only for the case where the control signal in equation 34 is not saturated i. In other words, equation 35 is valid for. The controller given in equation 34 is the actual stimulation input to the muscle.

The variables v max and v min are saturation and threshold stimulation inputs, respectively and can be selected appropriately, depending on the type of modulation employed. For example, the controller can be implemented as a voltage modulation or current modulation or pulsewidth modulation controller.

A block diagram describing the controller is shown in Figure 2. Theorem 1. The controller given in equations 25 and 34 ensures that all system signals are bounded under closed-loop operation and that the position tracking error is uniformly ultimately bounded in the sense that. Proof : See Appendix.

The controller given in equation 34 is a robust and adaptive controller; i. The controller, however, uses estimates of the muscle activation dynamics in equation 5 , the muscle recruitment threshold and saturation in equation 6 , and the fatigue variable in equation 7. This section describes the procedures that were used to estimate these parameters. The parameter estimation was performed on the right and left legs of three able-bodied subjects.

The parameters estimated for all participants can be found in the following section. During the parameter estimation procedures the participants were asked to relax and avoid any voluntary contractions that might influence the results during electrical stimulation. The isometric joint torque was computed from the force measured by the load cell Omega Engineering Inc. Isometric tests were used because it can be shown that in an isometric contraction the joint torque normalized by the maximum torque that can be produced at that joint angle is equal to the product of the muscle activation and the fatigue state i.

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A note on Ultra-discrete equations

The FNS was modified from the standard model to increase the amplitude range from 20 to mA, as NMES of human quadriceps muscles typically require current amplitudes in the range of 20— mA to achieve a significant muscle contraction. The stimulation train was delivered to the quadriceps muscles through 2. The muscle activation and fatigue parameters estimated in this section are dependent on the stimulation parameters used during identification.

Therefore, it is important to note that the stimulation parameters frequency and pulse width used during the parameter estimation procedures are same as the stimulation parameters during the controller validation experiments. It is also important to note that the parameters can change with placement of the electrodes.


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Figure 3. Test setup for estimating fatigue and activation dynamics. The test setup was also used to validate the developed controller for NMES knee extension tracking experiments. A rotary encoder was used to measure the knee-joint angle.

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The depicted individual provided written and informed consent for the publication of this image. Pulse trains of increasing current amplitude were used to determine the saturation and threshold current amplitudes. The threshold current amplitude is the amplitude that produces the first noticeable isometric joint torque, and the saturation current amplitude is the minimum amplitude that produces no further significant increase in joint torque. One-second long stimulation pulse trains were used to minimize muscle fatigue that may occur during the procedure.

From the results shown in Figure 4 A, the threshold amplitude was determined to be 33 mA. The increase in the isometric torque was found to be an insignificant much less than 1 Nm at stimulation amplitudes beyond 77 mA. Therefore, the saturation current amplitude was chosen as 77 mA. Figure 4. A The threshold and saturation current amplitudes were determined from this plot. The threshold amplitude was determined to be 33 mA, and the saturation amplitude was determined to be 77 mA. B This plot shows the measured muscle activation and the best fit first-order response for a unit step input, which has an RMS error of 0.

The first-order response has a time constant of 0. This procedure was conducted 3 min after the previous procedure to ensure that the subject was not fatigued. This procedure was only conducted once, since further trials would induce fatigue. Any induced muscle fatigue would cause the isometric contraction to not reach a normalized activation of one, which would affect the parameter estimation.

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An optimization method was then used to solve for the first-order system with a time constant that best matches the response measured by the load cell. The normalized load cell measurement and the first-order response that best fits the measured data are shown in Figure 4 B.

A first-order system with a time constant of 0. To generate an estimate of the fatigue state, an estimated model of the fatigue dynamics, described in equation 15 , was used; i.


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This procedure was conducted on a separate day from the previous procedures muscle activation estimation to ensure that the muscle was fully rested. First, the muscle was potentiated using 10, 1-s long pulse trains with 10 s between the trains. The current stimulation amplitude was set at the saturation level. This was done to warm up the quadriceps muscles to electrical stimulation, and the duration of the potentiation was short enough to prevent the muscles from fatiguing.

After the potentiation sequence, a constant stimulation at the saturation amplitude was used for 3 min. Immediately after the fatiguing protocol, 1-s long pulse trains of stimulation were used every 10 s to see the rate at which the joint torque magnitude recovered. The measured load cell data during the potentiation, fatigue, and recovery processes can be seen in Figure 5. The steady decrease in the measured joint torque during the fatigue process and the steady increase in the joint torque produced during the recovery process illustrate that fatigue and recovery are occurring as expected.

These measurements were then used to estimate the parameters of muscle fatigue dynamics in equation Figure 5. Results of the experiments to determine the parameters of the muscle fatigue parameters. These three plots show the torque measured during the potentiation, fatigue, and recovery segments of the procedure. It was assumed that the muscle was fully rested at the beginning of the fatigue process. Also, because the participant was being stimulated at the saturation amplitude for the duration of the fatigue procedure, the muscle activation variable was assumed to be 1 throughout the fatigue process this is because the duration of the procedure is significantly longer than the muscle activation time constant that was previously determined.

This approximation is necessary, as it allows us to measure the rate at which the muscle recovers without unnecessarily fatiguing the muscle. This procedure is identical to the potentiation procedure, where it can be observed that no noticeable muscle fatigue occurs. The normalized load cell data and the plot of the fatigue state that best fits the measured data are shown in Figure 6 A.

The resulting fit has an RMS error of 0. Figure 6. A The fatigue time constant and minimum fatigue states were determined by fitting the solution of the differential equation of the fatigue state to the normalized load cell data.