In the system shown below find the range of k for closed loop stability Search MathWorks. MathWorks Answers Support. Open Mobile Search. Trial software. You are now following this question You will see updates in your activity feed. You may receive emails, depending on your notification preferences. Vote 0. Commented: Fatima Mohamad Hussein on 17 Apr Fatima Mohamad Hussein on 17 Apr Cancel Copy to Clipboard. Find minimum K that makes the system stable. Find K at the break in point. Find the break in point.

Find K that makes -3 one pf the system poles.This is equivalent to asking whether the denominator of the transfer function which is the characteristic equation of the system. What we want, though, is Z, the number of zeros in the right half plane. Before continuing we make one small change. In the previous section, we specified that the path of s should enclose the entire right half plane. This is shown below.

Control Systems/Stability

Let's examine this procedure with a couple of examples followed by a video of the two examples. A variety of examples follow on the next page Examples. If we map this function from "s" to "L s " with the variable s following the Nyquist path we get the following image note: the image on the left is the "Nyquist path" the image on the right is called the "Nyquist plot". This tells us that the system is stable. And, if we close the loop, we find that the characteristic equation of the closed loop transfer function is.

Recall that the roots of the characteristic equation are the poles of the transfer function. However, during this segment of the plot, the path in L s will not move, assuming L s is a proper transfer function.

Determining Stability using the Nyquist Plot

The order of the numerator polynomial of a proper transfer function is less than or equal to that of the denominator. Let's consider, first, the case when the order of both polynomials is equal to n:.

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Note: In the Nyquist diagrams, since there are several arrowheads on the path in "s" as it makes it excursion at infinity, there are also several arrowheads at this single point in "L s.

If the loop gain, L sis strictly proper i. If we map this function from "s" to "L s " with the variable s following the Nyquist path we get the following image.

We can check this by closing the loop to get the characteristic equation of the closed loop transfer function:. To handle that situation we make small "detours" around the poles that are on the axis. For instance if we have. However, because the path is so close to the pole, the magnitude of the path in "L s " is at infinity. And because the path is going around the pole in the counterclockwise direction in "s" the path in "L s " is in the clockwise direction.

This is shown below in this diagram the radius of the detours is exaggerated so they can be seen on the graph. There are two ways to do this. The first is easier conceptually, the second is easier practically. For example we can start at the point near the origin and call this angle zero Note: since we will be traversing the entire path the place where we start on the graph is arbitrary.

As we do so we keep track of the total angle that has been swept out by the arrow, as shown below. However, with a slightly different Nyquist diagram we get a different result.Documentation Help Center. The root locus returns the closed-loop pole trajectories as a function of the feedback gain k assuming negative feedback.

Root loci are used to study the effects of varying feedback gains on closed-loop pole locations.

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In turn, these locations provide indirect information on the time and frequency responses. You can use rlocus to plot the root locus diagram of any of the following negative feedback loops by setting sys as shown below:. The root locus plot depicts the trajectories of closed-loop poles when the feedback-gain k varies from 0 to infinity. The poles on the root locus plot are denoted by x and the zeros are denoted by o.

You can specify a color, line style, and marker for each model. For even more plot customization options, see rlocusplot. The poles of the system are denoted by xwhile the zeros are denoted by o on the root locus plot. You can use the menu within the generated root locus plot to add grid lines, zoom in or out, and also invoke the Property Editor to customize the plot.

For more plot customization options, use rlocusplot. For this example, consider sisoModels. Create the root locus plot using rlocus and specify the color for each system. Also add a legend to the root locus plot. The figure contains root locus diagrams for all three systems in the same plot. For more plot customization, see rlocusplot. Use the above transfer function model with rlocus to extract the closed-loop poles and associated feedback gain values. Since sys contains 3 poles, the size of the resultant array of poles r is 3x Each column in r corresponds to a gain value from vector k.

For this example, rlocus automatically chose 53 values of k from zero to infinity to obtain a smooth trajectory for the three closed-loop poles.

For instance, r :,39 contains the above closed-loop poles for a feedback gain value of Define the transfer function model and required vector of feedback gain values.

For this example, consider a set of gain values varying from 1 to 8 with increments of 0. Since sys contains 4 closed-loop poles, the size of the resultant array of closed-pole locations r is 4x9 where the 9 columns correspond to the 9 specific gain values defined in k.

You can also visualize the trajectory of the closed-loop poles for the specific gain values in k on the root locus plot. Continuous-time or discrete-time numeric LTI models which include tfzpkor ss models. Generalized or uncertain LTI models such as genss or uss models. Identified LTI models, such as idtfidssidprocidpolyand idgrey models.

Feedback gain values that pertain to pole locations, specified as a vector. The feedback gains define the trajectory of the poles thereby affecting the shape of the root locus plot. Feedback gain values that pertain to pole locations, returned as a vector.Documentation Help Center. You can plot the step and impulse responses of this system using the step and impulse commands:.

You can also simulate the response to an arbitrary signal, for example, a sine wave, using the lsim command. The input signal appears in gray and the system's response in blue. For state-space models, you can also plot the unforced response from some given initial state, for example:. Frequency-domain analysis is key to understanding stability and performance properties of control systems. Bode plots, Nyquist plots, and Nichols chart are three standard ways to plot and analyze the frequency response of a linear system. You can create these plots using the bodenicholsand nyquist commands. For example:. The poles and zeros of a system contain valuable information about its dynamics, stability, and limits of performance.

For example, consider the feedback loop in Figure 1 where. The closed-loop poles marked by blue x's lie in the left half-plane so the feedback loop is stable for this choice of gain k. You can read the damping ratio of the closed-loop poles from this chart see labels on the radial lines.

Here the damping ratio is about 0. To further understand how the loop gain k affects closed-loop stability, you can plot the locus of the closed-loop poles as a function of k :. So the loop gain should remain smaller than 1. Right-clicking on response plots gives access to a variety of options and annotations.

In particular, the Characteristics menu lets you display standard metrics such as rise time and settling time for step responses, or peak gain and stability margins for frequency response plots. Now, right-click on the plot to display the Peak Response and Settling Time Characteristics, and click on the blue dots to read the corresponding overshoot and settling time values:.

All commands mentioned so far fully support multi-input multi-output MIMO systems. In the MIMO case, these commands produce arrays of plots. For example, the step response of the two-input, two-output system. The resulting plot is shown below. Singular value plot sigmawhich shows the principal gains of the frequency response. You can plot multiple systems at once using any of the response plot commands.

You can assign a specific color, marker, or line style to each system for easy comparison. Using the feedback example above, plot the closed-loop step response for three values of the loop gain k in three different colors:.

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Search MathWorks. Open Mobile Search.Several root locus examples are provided. If you click on the link in each column labelled "New" it will take you to a page I have recently written that demonstrates the rules for drawing the root locus for an arbitrary transfer function. I would love to get comments links at bottom of page about how well it worked for you.

If all goes well I will replace this page with the new one. The root locus is obviously a very powerful technique for design and analysis of control systems, but it must be used with some care, and results obtained with it should always be checked.

To show potential pitfalls of this method, consider the two systems G1 s and G2 s. If we control these systems with a simple proportional controller, as shown.

However, if we plot the two responses, we get something quite different. T1 s resembles somewhat a first order system, and has no overshoot, and its settling time is almost exactly 4 seconds, as predicted. However, T2 s behaves very differently, it is much faster and more oscillatory than expected.

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How can we explain this? If we look more closely at T1 s and T2 swe can understand what happened. In particular, lets look at pole-zero plots of both closed loop transfer functions. The lesson here is that while the poles of a system the roots of the denominator polynomial are very important in determining the behavior of a system, the zeros of the system the roots of the numerator polynomial can also be important.

After performing a root-locus design, it is critical to go back and test the closed loop system to ensure that it behaves as expected. A Weakness of the root locus.When a system is unstable, the output of the system may be infinite even though the input to the system was finite. This causes a number of practical problems.

For instance, a robot arm controller that is unstable may cause the robot to move dangerously. Also, systems that are unstable often incur a certain amount of physical damage, which can become costly. Nonetheless, many systems are inherently unstable - a fighter jet, for instance, or a rocket at liftoff, are examples of naturally unstable systems. Although we can design controllers that stabilize the system, it is first important to understand what stability is, how it is determined, and why it matters.

The chapters in this section are heavily mathematical, and many require a background in linear differential equations. Readers without a strong mathematical background might want to review the necessary chapters in the Calculus and Ordinary Differential Equations books or equivalent before reading this material. For most of this chapter we will be assuming that the system is linear, and can be represented either by a set of transfer functions or in state space.

Linear systems have an associated characteristic polynomial, and this polynomial tells us a great deal about the stability of the system.

Negativeness of any coefficient of a characteristic polynomial indicates that the system is either unstable or at most marginally stable. It is important to note, though, that even if all of the coefficients of the characteristic polynomial are positive the system may still be unstable. We will look into this in more detail below. This must hold for all initial times t o. So long as we don't input infinity to our system, we won't get infinity output.

A system is defined to be uniformly BIBO Stable if there exists a positive constant k that is independent of t 0 such that for all t 0 the following conditions:. There are a number of different types of stability, and keywords that are used with the topic of stability.

All of these words mean slightly different things. We can prove mathematically that a system f is BIBO stable if an arbitrary input x is bounded by two finite but large arbitrary constants M and -M:.

We apply the input x, and the arbitrary boundaries M and -M to the system to produce three outputs:. Now, all three outputs should be finite for all possible values of M and x, and they should satisfy the following relationship:. And this inequality should be satisfied for all possible values of x. However, we can see that when x is zero, we have the following:.

Which means that x is between -M and M, but the value y x is not between y -M and y M. Therefore, this system is not stable. When the poles of the closed-loop transfer function of a given system are located in the right-half of the S-plane RHPthe system becomes unstable. When the poles of the system are located in the left-half plane LHP and the system is not improper, the system is shown to be stable.Consider the closed-loop system shown in Figure 2.

After solving the characteristic equation, we'll get a polynomial equation. We'll arrange the coefficients of the polynomial according to Routh's criterion. Routh's criterion states that the number of sign changes in the first column is equal to the number of roots in the right-half plane. Become a Study.