Feedback gain

In equation (8.93), r(t) is a vector of desired state variables and K is referred to as the state feedback gain matrix. Equations (8.92) and (8.93) are represented in state variable block diagram form in Figure 8.7. 

In reverse-time, starting with P(A ) = 0 at NT = 20 seconds, compute the state feedback gain matrix K(kT) and Riccati matrix P(kT) using equations (9.29) and (9.30). Aiso in reverse time, use the desired state vector r(/c7 ) to drive the tracking equation (9.53) with the boundary condition s(N) = 0 and hence compute the command vector y kT). 

Calculate feedback gain matrix using Ackermann s formula K=acker(A,B,desiredpoles)... 

Discrete-time steady-state feedback gain... 

Thus in general, we can calculate all the state feedback gains in Kby... 

There are other methods in pole-placement design. One of them is the Ackermann s formula. The derivation of Eq. (9-21) predicates that we have put (9-13) in the controllable canonical form. Ackermann s formula only requires that the system (9-13) be completely state controllable. If so, we can evaluate the state feedback gain as 1... 

You may notice that nothing that we have covered so far does integral control as in a PID controller. To implement integral action, we need to add one state variable as in Fig. 9.2. Here, we integrate the error [r(t) -, (t) to generate the new variable xn+1. This quantity is multiplied by the additional feedback gain Kn+1 before being added to the rest of the feedback data. 

Example 9.2 Consider the second order model in Example 9.1. What are the state feedback gains if we specify that the closed-loop poles are to be at -3 3j and -6 ... 

To obtain the state feedback gains with Eq. (9-21), we should subtract the coefficients of the polynomial pi from p2, starting with the last constant coefficient. The result is, indeed,... 

Example 4.7B Let us revisit the two CSTR-in-series problem in Example 4.7 (p. 4-5). Use the inlet concentration as the input variable and check that the system is controllable and observable. Find the state feedback gain such that the reactor system is very slightly underdamped with a damping ratio of 0.8, which is equivalent to about a 1.5% overshoot. 

The state space state feedback gain (K2) related to the output variable C2 is the same as the proportional gain obtained with root locus. Given any set of closed-loop poles, we can find the state feedback gain of a controllable system using state-space pole placement methods. The use of root locus is not necessary, but it is a handy tool that we can take advantage of. 

To find the new state feedback gain is a matter of applying Eq. (9-29) and the Ackermann s formula. The hard part is to make an intelligent decision on the choice of closed-loop poles. Following the lead of Example 4.7B, we use root locus plots to help us. With the understanding that we have two open-loop poles at -4 and -5, a reasonable choice of the integral time constant is 1/3 min. With the open-loop zero at -3, the reactor system is always stable, and the dominant closed-loop pole is real and the reactor system will not suffer from excessive oscillation. 

The state feedback gain including integral control K is [0 1.66 -4.99], Unlike the simple proportional gain, we cannot expect that Kn+1 = 4.99 would resemble the integral time constant in classical PI control. To do the time domain simulation, the task is similar to the hints that we provide for Example 7.5B in the Review Problems. The actual statements will also be provided on our Web Support. 

Define the closed-loop poles %Calculate the feedback gains... 

Do the time response simulation in Example 7.5B. We found that the state space system has a steady state error. Implement integral control and find the new state feedback gain vector. Perform a time response simulation to confirm the result. 

We would like to see how the output step response changes for different feedback gains. Presently the feedback gain is 1E3 or 1000. We would like to vary the feedback gain to see how the feedback affects the operation of the system. We will use a parameter to change the value of the feedback gain. Double-click on the text PARAMETERS to obtain the parts spreadsheet ... 

We would like to observe the step response for several different values of the feedback gain. We will use a Parametric Sweep to vary the value of the parameter FB gain. Click the LEFT mouse button on the square O next to the text Parametric Sweep to enable the sweep and display its options. Fill in the Parametric dialog box as shown ... 

We will be running the Transient Analysis 11 times. The first time the feedback gain will be zero, the second time it will be 0.1, the third time it will be 0.2, and so on. Run the simulation and plot the output ... 

Jot s method of incorporating absorptive filters into a lossless prototype yields a system whose poles lie on a curve specified by the reverberation time. An alternative method to obtain the same pole locus is to combine a bank of bandpass filters with a bank of comb filters, such that each comb filter processes a different frequency range. The feedback gain of each comb filter then determines the reverberation time for the corresponding frequency band. 

Biological control systems are often regarded as some sloppy variants of the more precise engineering control systems. Classic control theory considers linear, stable and stationary systems [1-3]. To this could be added well defined. Biological systems are nonlinear, often unstable, and never stationary. They work with small feedback gains, typically less than 10 [4—6] they are interwoven, so completely different systems share common routes (hormones, nerves, etc.) and their properties vary from person to person, even in healthy people. 

Here, max and jrm n denote, respectively, the maximum and the minimum values of the muscular activation, a determines the slope of the feedback curve, S is the displacement of the curve along the flow axis, and Fneno is a normalization value for the Henle flow. The relation between the glomerular filtration and the flow into the loop of Henle can be obtained from open-loop experiments in which a paraffin block is inserted into the proximal tubule and the rate of glomerular filtration (or, alternatively, the so-called tubular stop pressure at which the filtration ceases) is measured as a function of an externally forced rate of flow of artificial tubular fluid into the loop of Henle. Translation of the experimental results into a relation between muscular activation and Henle flow is performed by means of the model, i.e., the relation is adjusted such that it can reproduce the experimentally observed steady state relation. We have previously discussed the significance of the feedback gain a in controlling the dynamics of the system, a is one of the parameters that differ between hypertensive and normotensive rats, and a will also be one of the control parameters in our analysis of the simulation results. 

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