14-02-2011, 05:02 PM
The author is from CET, Trivandrum
ABSTRACT
Traditionally, input shaping has been used outside the feedback control loop. Utilizing input shaping in this way prevents it from directly addressing some important vibration control issues such as disturbances, nonzero initial conditions, actuator saturation, etc., These potential applications, however, will never be fully exploited unless a good understanding exists of how to create stable CLSS controllers.
Study of closed-loop stability of feedback loops containing input shapers by analyzing their dynamics via root locus plots is done.. Despite the fact that CLSS controllers add partial time delays to the feedback loop, these controllers can be stabilized even when modeling errors occur.
In fact, they rely upon accurate models more than standard PID feedback control. The study presented provides a basic approach to creating stable CLSS controllers. Experimental results were given to verify the predictions made via the root locus, the stability properties of a crane under CLSS control were experimentally demonstrated.
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LIST OF CONTENTS
1.Introduction……………………………………………………….….…..….(3)
2.Review of Outside-the-Loop Input Shaping…………………………….….. (4)
3.Advantages of Closed-Loop Signal Shaping…………………………….…...(6)
4. General approach…………………………………………………..……..(9)
4.1. First-Order Effect of Shaper Zeros…………………………….…….…..(9)
4.2 Root Loci Proof for Closed-Loop Signal Shaping……………………..(10)
4.3 Stability Analysis of a Damped Second-Order System………………...(12)
4.3.1 Root Locus Analysis of an Undamped Second-Order
System……………………………………………………….……(13)
4.3.2 Parameter Influence on Closed-Loop Stability…………………..….(16)
4.3.3 Influence of Gain, K…………………………………………..……..(16)
4.3.4 Influence of Damping Ratio……………………………………..…..(17)
4.3.5 Influence of Lead Compensator…………………………….……….(18)
4.3.6 General Design Approach………………………………….………..(19)
5. Stability Analysis Of a Mass-Spring-Damper-Mass Plant………………………..(19)
5.1 Root Locus Analysis……………………………………………………..(21)
5.2. Effect of Modeling Errors on Stability…………………………………..(22)
5.3 Results From the Mass-Spring-Damper-Mass Study……………………..(23)
6. Experimental Verification…………………………………………………………(24)
7.0 Disturbance rejection in CLSS Controllers………………………………………(29)
8.0 Conclusion………………………………………………………………………..(30
1. INTRODUCTION
The control of flexible systems is an immense field of research. Consequently, there are many types of vibration control, including feedback control, optimal control, filters, etc. These control strategies are utilized in a variety of control architectures, such as open-loop control, feedback control and feedforward control. One particularly effective form of vibration control is input shaping. This type of control has been widely used on cranes coordinate measuring machines flexible spacecraft, and long-reach robots. Input shaping has also been extended for use in more complicated applications, including multi-input systems multi-mode systems adaptive control and digital control .
Traditionally, input shaping has been used outside the feedback control loop. Outside-the-loop input prevents it from directly addressing some important vibration control issues such as disturbances, nonzero initial conditions, actuator saturation, etc..
Considering the overall simplicity of input shaping, from a theoretical and practical perspective, it is certainly desirable to investigate whether input shaping can address these phenomena by including it inside feedback loops.
2. Review of Outside-the-Loop Input Shaping
Figure1: Input shaping procedure.
Traditional outside-the-loop input shaping works by convolving a reference command with a series of impulses specifically designed to eliminate unwanted vibratory modes. Figure 1 depicts this process by showing a step command convolved with a two-impulse shaper, which produces a staircase command that results in zero vibration. A useful way to understand input shaping is to analyze it in the Laplace domain. An input shaper is designed to cancel a system’s stable poles by generating zeros that occur at the same place in the s-plane as the vibratory poles. For example, a “Zero Vibration” (ZV) shaper, like the one shown in figure 1, contains two impulses. The first impulse occurs at time t=0 with a magnitude of A1. The second impulse, which has a magnitude of A2, occurs at some delayed time.
In the Laplace domain, the equation for a ZV shaper is
ISZV(s) = A1+A2e-st2 (1)
Using, s=σ+jω, the zeros of ISZV can be determined. This leads to two separate equations that must be simultaneously satisfied.
(2)
-1= e-ωt2 (3)
From (2) and (3), we find that, ω= π/t2 and one solution is, σ= -(1)/(t2)ln(A1/A2).
If the shaper is to cancel a set of damped poles, then ω must be set to ωd , the system’s damped natural frequency. In other words, t2=(π)/(ωd). Also, and must be set so that, where and are the damping ratio and natural frequency of the system.
Figure 2: Zero Vibration shaper cancelling flexible poles.
One interesting note that becomes important when input shapers are included within feedback loops is that a ZV shaper has an infinite number of zeros. From (3) it is evident that ωt2 can be any odd multiple of π.Therefore, ω= +(nπ)/(t2), (where n=1,2,3) will satisfy (3). This means that an input shaper produces an infinite column of zeros, with the first set of zeros (n=1) usually set to cancel the poles of the flexible system. This column of zeros can be seen in figure 2, which shows the zeros of an input shaper canceling the oscillatory poles of a flexible system. It is also important to note that a ZV shaper has no finite, open-loop poles, because ISZV(s) goes to infinity only if the real part of s equals
3. Advantages of Closed-Loop Signal Shaping
As noted in the previous section, input shapers have been widely studied as an open-loop command shaper, filtering out system natural frequencies from the system’s reference command. The system can be an open or a closed-loop system, but usually, the input shaper is left outside of any feedback loops. However, as will be detailed in the following section, some work has studied the effectiveness of placing input shapers inside the feedback loop. Here, any control scheme that includes an input shaper within a feedback loop is referred to as a closed-loop signal shaping (CLSS) controller.
Figure 3: Block diagram of CLSS controller
One obvious potential for closed-loop signal shaping (placing input shapers inside the feedback loop) is disturbance rejection. With traditional input shaping acting as open-loop control, its vibration-reducing capabilities cannot address disturbances. Typically, disturbance rejection for a low-damped, oscillatory system is accomplished with some form of derivative feedback control. This derivative action creates a closed- loop system that is more damped, improving the closed-loop system’s response to disturbances. Closed-loop signal shaping provides a possible alternative to derivative control. By placing the shaper inside the loop, it has the potential to eliminate the vibration induced by disturbances without overly slowing the closed-loop system by increasing the closed-loop damping ratio.
Another possible use for closed-loop signal shaping is non-collocated control. Due to the inherent difficulty of stabilizing non-collocated control systems (even with PID control), some researchers have intentionally introduced time delays into the closed-loop controller to enhance stability . By placing input shapers (which are created with time delays) inside the feedback loop, it is possible to create stable, non-collocated controllers that also result in rapid, low-vibration motion.
Figure 4: Open loop input shaping control system with saturation.
A third application for closed-loop signal shaping involves systems with hard nonlinearities. In some physical systems, successfully implementing open-loop input shaping is not possible because of the presence of a hard nonlinearity, such as a slewrate limiter or a saturator inside the feedback controller. A block diagram of such a control scheme with a saturator is shown in figure4. When a hard nonlinearity modifies an input-shaped command, the vibration-reducing properties of the shaped command entering the oscillatory plant can be significantly degraded . Therefore, one solution would be to design the baseline reference command so that the input-shaped reference command would result in zero saturation of the actuator input to the plant. While possible in some cases, this is clearly a difficult endeavor in many cases. However, by placing an input shaper with all positive impulses inside the loop, one can guarantee (for any possible reference command) that the input-shaped command entering the plant is unmodified by the hard nonlinearity. This allows the input shaper to fully employ its vibration-reducing capabilities.
To understand this result, one must first realize that the output of the input-shaping process is never greater in magnitude than the input to the input-shaping block. So, if instead of placing the input shaper outside the loop, it was placed inside the loop (acting on the error signal) , then one only needs to constrain the input to the input shaper (the error signal) to be within the bounds of the “Actual” saturator. This can easily be done by preconditioning the error signal with a model of the saturation limits. By placing the input shaper inside the loop and using a model of the saturation to condition the error signal, one is able to fully utilize the vibration-reducing characteristics of input shaping.
Finally, CLSS controllers formed by placing input shapers inside feedback loops formed by a human operator (such as in crane control) have proven to greatly improve performance .The potential applications of CLSS controllers however, will never be fully exploited unless a good understanding exists of how to create stable CLSS controllers.
4. General approach
This section will establish the general design approach for creating stable CLSS controllers. The plants studied in this paper will be limited to linear systems, meaning that their transfer functions can be written as fractions of polynomials in “s”. However, the input shaper (in the Laplace domain) contains exponentials of “s”, due to the presence of time delays. Therefore, analytical solutions to the “s” values that result in instability are not generally possible. Note that this is also true when high order (fourth-order or above) linear systems are studied. Therefore, the approach here will be to use a graphical
tool to illustrate several key concepts of creating stable CLSS controllers. These concepts can then be used to create new CLSS controllers for any order plant. The graphical tool
can be used to verify stability and to adjust gains for optimal performance of any specific system.
Figure 5: Block diagram of CLSS controller.
The graphical tool used is Root Locus because it most easily demonstrates several key characteristics of CLSS controllers. Certainly, the Bode, Nyquist, or any number of other methods could also be used.
4.1. First-Order Effect of Shaper Zeros
The closed-loop signal shaping control scheme considered here is shown in figure5. Here, “IS” is an input shaper and “C” is some other controller. To begin the stability study, “C” is restricted to be a proportional gain, K. If the plant is assumed to be unity, then the closed-loop system is only a function of the input shaper and the proportional gain . The system’s root locus, for the case of a ZV shaper, C=K and Plant=1, is shown in figure 5.
Figure 6:ZV shaper inside a feedback loop.
The root locus in figure 6 shows that the input shaper’s infinite number of open-loop poles, located at negative infinity, create an infinite number of root locus branches when a shaper is included within the feedback loop. Note that the closed-loop poles arising from the input shaper tend to form a column that extends infinitely away from the real axis. This occurs because of the infinite column of open-loop zeros, to which the closed-loop poles approach. As discussed in the next section, the closed-loop poles do not generally form a vertical column, but one that tilts toward negative infinity as it extends away from the real axis. Note that, in the general case, the input shaper’s open-loop zeros do not lie on the imaginary axis. They can lie anywhere in the left-half plane, depending upon the oscillatory poles they are designed to cancel.
4.2 Root Loci Proof for Closed-Loop Signal Shaping
Constructing continuous domain root loci of CLSS controllers presents a few interesting challenges. First, the closed-loop characteristic equation is not a polynomial in “s”. This means that numerical methods must be employed to construct the root locus. This paper uses a numerical root locus drawing technique created by Nishioka . In addition, the input shaper contains an infinite column of open-loop zeros, resulting in an infinite number of closed-loop poles. These poles are not necessarily in a straight, vertical column, although they do extend vertically away from the real axis as described in the previous paragraph and illustrated in figure 6.
This ambiguity as to the location of an infinite number of closed-loop poles can make the use of the continuous domain root locus tool difficult, as it is not obvious what axes range should be drawn to include all significant closed-loop poles. However, it is shown below that beyond a sufficiently large radial distance from the origin, the closed-loop poles arising from the input shaper move left as they get farther from the real axis. This means that the most significant closed-loop poles arising from an input shaper are those closest to the real axis. Therefore, a controls engineer can reasonably ignore all but a small, finite number of these closed-loop root loci branches.
To show this, a simple proof is presented. The characteristic equation of the controller shown in figure 3 is assumed to be of the form,
1 + K*IS*F=0 (4)
Where K*IS*F is the open-loop transfer function of the closed-loop system. Furthermore, K is a proportional gain, IS is the input shaper, and F is the portion of the open-loop transfer function that can be written as a rational transfer function (a fraction of polynomials in s). Note that F incorporates the plant and any other desired controller. One further assumption is that the denominator of F is of higher order than the numerator.
For some value of K, there is an infinite number of values that will satisfy the characteristic equation. In addition, the closed-loop poles arising from the input shaper dynamics tend to form a vertical column as they approach the column of open-loop zeros established by the input shaper. By moving away from the real axis, the magnitude of these ‘s’ values tends toward infinity. As , │s│ tends to ∞ it follows from the strictly proper nature of F that │F(s)│tends to zero. However, because K*IS*F (with K being a finite constant) must always equal -1, it follows that │IS│tends to ∞ as ‘s’ tends to zero.
An input shaper with n impulses will have the form
n n
IS = A1 + ∑Aie-sti=A1 + ∑Aie-σtiAie-jωti
i=2 i=2 (5)
Where Ai and ti are the magnitude and time, respectively, of the impulse. Note that every input shaper can be written in this form, as they are all a summation of time-delayed impulses (ti=0 , following standard practice). In order for (5) to approach infinity as s tends towards infinity, some term in (5) must approach infinity along with s . The Ai terms are constant and usually less than or equal to one. The term e-jωti has a constant unity magnitude. Therefore, the term e-σti must be the term whose magnitude approaches infinity. Because ti is fixed and finite, σ must approach ∞ as │s│ tends to ∞.This means that as the closed-loop poles for a single K value lie farther from the real axis, they also lie farther to the left of the imaginary axis. Therefore, at some point, they become insignificant and only a finite portion of the real-imaginary plane is needed for sufficient system identification and control.
One point of caution is the fact that the trend established above is only valid s as approaches infinity. There is currently no established method for determining the exact value for which the trend begins. For small ‘s’ values, where the closed-loop poles from the input shaper are close to the dynamics arising from F, no specific trend exists. The controls engineer must establish the suitable real-imaginary plane area outside of which the above mentioned trend holds and where the closed-loop poles are insignificant.
The controls engineer can also avoid this difficulty inherent to the continuous time root locus tool by utilizing frequency domain tools (like Bode plots ) or by utilizing the digital root locus .
4.3 Stability Analysis of a Damped Second-Order System
Given the insights gained by the previous analysis, a damped second-order plant can be considered.
G(s) = ωa2
s2+2ξωa+ωa2 (6)
To begin, the undamped version is analyzed via the root locus, then the additional elements of damping and lead compensators are added. The goal of this section is to understand the most basic stability issues inherent to CLSS control. For example, the following sections will reveal that high gains and modeling errors lead to instability, whereas system damping and lead compensators improve stability.
4.3.1 Root Locus Analysis of an Undamped Second-Order System
Given the control system depicted in figure 3, the controller C, is first defined as a proportional controller(C=K). If the input shaper is tuned perfectly to the plant frequency, then the root locus will be similar to that shown in figure 7. Here, the plant parameters are ωa=2π and ξ =0.
As shown in figure 6 and reiterated in figure 7, placing an input shaper within the loop will result in an infinite number of closed loop poles. Their presence indicates another important result. Using an input shaper within a feedback loop, as done within the control scheme depicted in figure 3, will result in additional oscillatory dynamics arising from the input shaper. Note that many of the root locus figures in this paper stop at a finite value of to illustrate some feature of CLSS controllers. For example, figure 7 illustrates how the real part of closed-loop poles tend toward the left as they lie further from the real axis, as was proven in Section 4.2.
Figure 7: Root locus with pole/zero cancellation.
If modeling errors occur, then the plant poles are not completely cancelled by the shaper because the modeled frequency used to design the shaper(ωm) does not equal the actual frequency(ωa) of the plant. If ωa<(ωm), then the root locus is as shown in figure 8. The root locus branch extending from the plant poles immediately goes unstable. However, if ωa>ωm, then this branch remains stable, as shown in figure 9. This is the pattern usually seen with CLSS controllers when the plant G, is a lightly damped second-order system.
Figure 8: Root locus where ωa<ωm Figure 9: Root locus where ωa>ωm
If the plant pole is below the shaper zero to which it is closest, then the root locus branch extending from that plant pole bends to the right, making the system unstable. On the other hand, if the pole is above the zero to which it is closest, then the root locus branch bends left, creating a stable system. Under such modeling errors, closed-loop stability depends heavily on the relationship between ωa and ωm. Staehlin and Singh demonstrated similar stability results for their CLSS controller. Note that the simple shapes of the branches shown in figures 8 and 9 occur when the plant pole is close to one of the shaper zeros. If the pole is near the equidistant point between two shaper zeros, then the bend-left/bend-right characteristics will hold in general, but the branch may initially depart in a direction that is not directly left or right.
These departure angle observations can be explicitly shown for the case discussed here: a ZV shaper combined with an undamped second-order plant. The open-loop transfer function of this CLSS controller is
GC = (0.5+0.5e-sП/ωm) ωa2
s2+ ωa2 (7)
The departure angle from the plant pole can be determined by testing an value arbitrarily close to the plant pole. For example, set s=reθi+ωai where r establishes the radial distance of the test point from the plant pole and θ establishes the angle of the test point from the positive real axis.
The open-loop transfer function evaluated at this test point is
(0.5ωa2)(1+e-Л/ωm(ωa+rsinθ)i)
(r2cos2θ-(ωa+rsinθ)2+ωa2)+((ωa+rsinθ)(rcosθ))i (8)
As ‘r’ tends to zero, angle of the numerator is
α = tan-1 (sin(-Лωa/ωm)
(1+cos(-Лωa/ωm) (9)
In addition, as , the angle of the denominator (with the help of L’Hospital’s Rule) can be shown to be
β= tan-1(0/0) (10)
tan-1(cosθ/-2sinθ) (11)
Figure 10: Departure angle versus modeling error.
In order for this test point to lie on the root locus, α-ɤ=Л.This equation can be rearranged via trigonometric identities to yield
tan α = tan β (12)
Substituting (10) and (11) into (12) yields
tan θ=1+cos(-лωa/ωm)
-2sin(-лωa/ωm) (13)
Solving for θ gives the departure angle from the plant pole s=ωai . Varying ωa/ωm relationship reveals how the departure angle relates to modeling errors, as was depicted in figures 8 and 9. Note that a ZV shaper puts zeros at odd multiples of the plant pole . Therefore, when ωa/ωm<1, the plant pole is below the first shaper zero, as shown in figure 8. If 1<ωa/ωm<3 , then the plant pole is between the first and second shaper zero, as shown in figure 9. This pattern is repeated as ωa/ωm crosses 1,3,5….
Figure 9 shows a generalized representation of what is depicted for specific cases in figures 8 and 9. When the plant pole is below the shaper zero to which it is closest (i.e.ωa/ωm, is close to but less than 1,3,5.. ), then the departure angle is between zero and 900, making the root locus branch immediately unstable. If the plant pole is above the shaper zero to which it is closest (i.e ωa/ωm, is close to but greater than 1,3,5,7.. ), then the departure angle is between 900 and 1800 ,making the root locus branch initially stable.
4.3.2.Parameter Influence on Closed-Loop Stability
The previous sections established the basic form of the root locus for closed-loop signal shaping controllers. Also, the previous sections revealed that modeling errors are a primary source of instability for CLSS controllers. Here, the influence of other parameters (namely K and ξ) that are present in the control system of figure 3 will be analyzed. Also, the addition of a lead compensator is investigated. The lead compensator is studied here because it is known to increase stability margins. Because small modeling errors can result in unstable, CLSS controllers, it is important to show that a control block C can be added to the block diagram shown in figure 3 so as to increase stability margins even when modeling errors occur.
4.3.3 Influence of Gain, K
By increasing the gain K , the system shown in figure 3 will eventually be driven unstable, even with a favorable modeling error condition, figure 11 shows how an initially stable system can be driven unstable by making K too high.
Figure 11: Root locus with high K value.
This figure indicates that the dynamics arising from the inclusion of input shaping filters inside of feedback loops can, themselves, be the cause of instability. That is, the root locus branch emanating from the plant pole remains stable, while the branch emanating from one of the input shaper open-loop poles is what eventually causes instability.
4.3.4 Influence of Damping Ratio
When the second-order plant has damping, this increases the regions of stability. figsures 10 and 11 show this effect for each modeling error condition. The modeling error used here is +20 % error in the frequency. By increasing , the root locus branches are shifted to the left. This effect enables higher proportional gain(K) values to result in stable closed-loop poles.
Figure 12: Influence of ξ when ωa< ωm Figure 13: Influence of ξ when ωa >ωm
4.3.5 Influence of Lead Compensator
Figure 14: Root locus with lead compensator
Because of the stability issues associated with closed-loop signal shaping, it is desirable to study the effect of stability-enhancing controllers. One of the most effective and practical controllers used to increase stability margins is the lead compensator. The lead compensator is implemented here by setting C=K(S+Z)/(S+P). Figure 14 shows how the root locus branches are pulled to the left by the lead compensator.
The “No Lead Comp” root locus shown in figure 14 is a good example of what happens when the plant pole is nearly equidistant from two shaper zeros. The branch does not look exactly like those in figures 12 or 13. It follows the departure angle plot shown in figure 10, having an angle of approximately 90 degrees. However, because it is closer to the shaper zero above it, this branch eventually behaves like those shown in figure 12, tending toward instability.
While there are many other types of controllers (PID, lag compensator, etc.) that could have been used as the C block in figure 3, the main goal of this section is to show that reasonable stability margins can be achieved in CLSS systems. The lead compensator sufficiently demonstrates this point.
4.3.6 General Design Approach
This section revealed several key insights into the best way to create a stable CLSS controller. First, it is important to note that adding an input shaper into a closed-loop controller will add higher-order dynamics. As with any system that contains high modes that are either unmodeled or not actively controlled, these higher-order dynamics can cause instability. Therefore, as shown earlier in this section, the gains of CLSS controllers will always be limited by these higher modes. Note that this influence of high-order modes is true whether they derive from unmodeled dynamics of the plant or if they arise from the modes added by the input shaper.
Another key insight was the impact of modeling errors. Modeling errors in CLSS controllers can quickly cause instability. Therefore, CLSS controllers will need to be limited to usage with plants that are well defined. This section also showed that if the plant contains some amount of damping, then small modeling errors can be tolerated. So, CLSS controllers should also be limited to use with systems that have some reasonable amount of damping. Finally, if higher-order modes, modeling errors, or a lack of a minimal amount of system damping is causing stability problems, then the CLSS controller can be quickly stabilized with more traditional control schemes. For example, this section showed how a lead controller significantly increased the stability of one type of CLSS controller.
5. Stability Analysis Of a Mass-Spring-Damper-Mass Plant
Figure 15: Mass-spring-mass system.
In order to demonstrate the effects of CLSS on more complicated systems, the mass-spring-damper-mass system shown in figure 15 is now considered. A block diagram of a collocated feedback controller for the system is shown in figure 15. Analyzing the system shown in figure 16, (X)/(F) can be shown to be
Figure 16: Diagram of collocated controller
X 1 ( s2 + αs + ß)
F M1 S2(S2+2ξ1ω1 +ω12) (12)
where
ω1 = ksp M1 + M2 , ß = ksp
M1M2 M2
2ξ1ω1 = b( M1 +M2)
M1M2 (13)
Figure 15 represents a large number of systems such as satellites and cranes, which have a rigid body connected to a lightweight, flexible appendage. These flexible systems often present a more difficult control challenge, and the study of this system will reveal additional insights into CLSS control.
Following the lessons learned in the previous section, the CLSS controller will maintain limited gains so as not to drive the higher modes introduced by the input shaper unstable. Also, the plant studied here will have a small, but nonzero, damping coefficient,b. This will ensure that the CLSS controller can tolerate some level of modeling errors. In addition to a nonzero damping coefficient, all of the plant parameters will be assumed to be reasonably well known. This section will, however, investigate the effect of modeling errors. Last, the presence of a rigid-body mode in the plant will present a stability problem. However, as shown in the previous sections, the use of a lead compensator in addition to the input shaper will greatly increase the range of stable controllers.
For this example, the following values are used : M1=100 , M2 = 1, ksp= 20, and b = 0.2. These values lead to ω1 = 4.69 rad/sec, ß = 19.98, ξ1 = 0.024, α = 0.18.
5.1 Root Locus Analysis
The root locus of the closed-loop system shown in figure 16, without the input shaper, is shown in figure 17. This system has one rigid-body mode, one flexible mode and one set of complex zeros. This system is stable for any proportional gain(K) value. However, if the input shaper is added to the closed-loop system, then the root locus is as shown by the two, dark, thick lines in figure 17.
Figure 17: Fourth-order system under control—Without CLSS
The input shaper is designed without modeling error, so the input shaper zero exactly cancels the plant’s oscillatory pole. However, even with no modeling errors, this system starts out (with small K values) as unstable. And, even when large K values bring the closed-loop poles from the first root locus branch back to the left-half plane, higher modes introduced by the shaper have already gone unstable. The general instability of this control scheme is reiterated by the six, thinner root loci shown in figure 18. These root loci were obtained by varying the ω1 and ß between +30 % of their original values.
Figure 18: CLSS of fourth-order plant.
According to this root locus analysis, CLSS control of a mass spring- damper-mass system is unstable for a significant range of parameter values. That is, this system is unstable when the controller, , is just proportional control. This is an important reminder of the difficulty of using controllers that contain time delays within a feedback loop.
One solution is to use a lead compensator, in addition to proportional control (i.e. C=K(S+Z)/(S+P)). If the lead compensator pole is chosen to be at-20 while the zero is set at -1, then the system root locus will look like the one depicted by the two, dark, thick lines in figure 19. Even with extremely high K values, the system remains stable. Under the same parameter variations as in figure 19, the resulting root loci are significantly more stable. In conclusion, by adding a lead compensator, closed-loop signal shaping of a fourth-order system can be stabilized.
Figure 19: Addition of lead compensator
5.2. Effect of Modeling Errors on Stability
Given the sensitivity to modeling errors of CLSS on the second-order system shown in Section 4.3.1 it is important to investigate the general stability of CLSS on a fourth-order system with a lead compensator. Here, modeling errors will be examined by using input shapers designed for frequencies 30% above and below the actual frequency of the plant’s oscillatory poles. If the shaper design frequency is smaller than the plant frequency, then a root locus like the one shown in figure 20 results.
Figure 20: Root locus with ωa< ω1
Notice how the input shaper zero no longer cancels the plant’s oscillatory pole. As expected from the earlier study of second-order systems, the root locus branch from the oscillatory plant pole bends to the left. However, if the shaper frequency is larger than the plant frequency, then the root locus branch bends to the right, as shown in figure 21.
Figure 21: Root locus with ωa>ω1
With second-order systems, this usually causes significant stability problems. However, in this fourth-order system, the plant pole is now located between two zeros, one from the input shaper and one from the plant’s numerator dynamics. Because it lies between two zeros, the plant pole does not adversely affect stability as much as in the second-order case. As shown in figure 21, the root locus branch extending from the plant pole remains stable for a significant range of K values.
5.3 Results From the Mass-Spring-Damper-Mass Study
The study of this mass-spring-damper-mass plant showed can be used to create a stable, CLSS controller for a fourth-order plant that contains numerator dynamics. The system was assumed to be well known, although the CLSS controller with the lead compensator did allow for significant modeling errors. The system was also damped, although the damping ratio was low. More traditional controllers (in this case a lead compensator) were effectively utilized to improve stability.
6. Experimental Verification
In order to verify the stability characteristics of CLSS controllers predicted by the root locus analysis technique, experiments were conducted on an industrial crane located at the Georgia Institute of Technology . A sketch of the 10 ton bridge crane is shown in figure 22. This bridge crane has at least four notable nonlinearities: a velocity limit, an acceleration limit, a built-in velocity smoothing algorithm that prevents sudden sign changes in velocity, and a velocity dead zone.
Figure 22: Crane used for stability experiments.
Obviously, the linear-based root locus analysis tool utilized in this paper cannot account for the nonlinear behavior of this real world system. However, it will be shown here that a good linear model is sufficient to explain the dynamic behavior and approximately predict the gain value that induces instability. The motion of the crane trolley is controlled by a CLSS controller of the form shown in figure 5 where
C = K0(Kp + Kds)
IS = A1 +A2e-st2
G = ωn2
s(s2 + 2ξΩns+ ωn2) (14)
Figure 23. Root locus of crane controller.
This closed-loop system consists of a PD feedback controller, a two impulse (ZV) shaper, and a third-order plant (second-order oscillator plus an integrator). The plant contains an integrator because it is driven by velocity commands, while its output is position. For these experiments, values within the closed-loop system were held constant, and K0 was varied to show its effect on stability. The theoretical root locus of this closed-loop system is shown in figure 23. Zooming in closer to the real axis, figure 23 shows that this closed-loop system will go unstable for some finite value of gain K0. The root locus predicts that a K0 value of 10.5 will create a marginally stable system.
Figure 24: Controller stability limit.
In order to verify these predictions, the crane trolley was commanded to move 1 meter. As an initial test, the experiments were conducted without an input shaper in the loop. That is, IS was set to 1 and C and G were as given in (14). This was done to ensure that the system would not go unstable purely due to its nonlinearities and PD controller. As seen in the experimental results shown in figure 25, no tested value of K0 resulted in instability.
Figure 25: Trolley response without CLSS.
However, when the input shaper was included within the feedback loop, increasing K0 from 0.5 to 10 caused a limit cycle, as shown in figure 26. The proportional and derivative gains used in this experiment were Kp = 0.8 and Kd = 0.5 . Although the response is not exponentially growing, this limit cycle response is unstable for practical purposes. Because the linear dynamics of the bridge crane dominate its response, the experimentally determined critical gain K0 =10 of is very close to K0 =10.5 the theoretical value of predicted by the root locus.
Figure 26: Trolley response with CLSS
It should be noted that the oscillatory response depicted in the K0 = 10 plot has a period of oscillation of approximately four seconds, corresponding to a frequency of approximately 1.6 radians per second. Looking at figure 24, the root locus branch which goes unstable at K0 =10.5 would indicate an oscillation of more than 10 radians per second. However, figure 24 shows that between Ko = 9 and Ko = 11 and , several closed-loop poles are close to instability. While the pole which first goes unstable does not dominate the time-domain response, the critical value does correctly predict when the system will exhibit steady oscillations.
Figure 27: Root locus of second system.
A second set of experiments were conducted using a different Kd gain( Kd = 0.1)in the controller block “C” defined by (14). The root locus of this system is shown in figure 26. In this case, the root locus predicts that Ko = 12 will result in instability. The 1 m experimental step responses shown in figure 27 again demonstrate that Ko= 10 results in a limit cycle response. However, this response is different from that shown in figure 26, exhibiting a significantly higher amplitude.
Figure 28: CLSS response with new PD gains.
These experimental results match fairly well with the theoretically predicted effects of utilizing input shapers within feedback loops. The difference between the predicted and actual critical value of K0 can be attributed to nonlinearities that are not accounted for in the linear model.
It should be noted that the range of Ko values experimentally tested on the bridge crane was rather course. The only value tested between Ko = 5 and Ko = 10 was Ko = 7 This value did not result in sustained oscillations for either set of Kp and Kd gains. Therefore, the actual critical Ko value that results in instability (for both experiments) lies somewhere between seven and ten. However, the precise, critical Ko value is not extremely important, as the root locus prediction will always be somewhat inaccurate due to the nonlinear nature of any system. The important point is that the root locus technique utilized here gives a quick, reasonable estimate of Ko value that will result in instability. This will always need to be verified experimentally if the precise value is needed. Also, the root locus technique shown here gives the engineer a good understanding of the closed-loop system’s secondary dynamics, even when the overall system is stable.
7. Disturbance rejection in CLSS Controllers
One potential application of closed-loop signal shaping controllers is disturbance rejection. If the disturbance enters the closed-loop system just before the plant (called an actuator disturbance), then the traditional use of input shaping outside of the loop performs better. Figure 29 shows that for a second-order plant, an impulse disturbance at the actuator is better handled by the outside-the-loop input shaper and PD closed-loop controller combination (OLIS/PD control).
Figure 29: Actuator disturbance response. Figure 30: Sensor disturbance response
However, if the disturbance enters the system at the sensor (known as a sensor disturbance), then a closed-loop input-shaping controller can often outperform a controller that relies solely on PD control to attenuate the disturbance. This can be seen in figure 30 with a second-order plant.
8. Conclusion
This paper utilized the root locus to develop a design process intended to ensure the creation of a stable CLSS controller. Despite the fact that CLSS controllers add partial time delays to the feedback loop, these controllers can be stabilized even when modeling errors occur. But, they rely upon accurate models more than standard PID feedback control. Experimental results were given to verify the predictions made via the root locus. These experimental results were performed on a 10 ton bridge crane operating under a CLSS controller that combined input shaping and PD control. The several possible applications of CLSS Controllers motivated the need for a design process that would create a stable CLSS controller. The design process proposed here shows that stable CLSS controllers are practical and implementable.
Therefore, the next logical step is to compare the performance of CLSS controllers to current, state-of-the-art control schemes. The most logical state-of-the-art control scheme is the traditional combination of input shaping and feedback control .This controller utilizes some form of PID control (to create a stable closed-loop system) combined with an input shaper (completely outside of any feedback loops) to filter the reference command of any frequencies that would incite vibration in the closed-loop system. One potential application of closed-loop signal shaping controllers is disturbance rejection.The examples demonstrate how CLSS controllers can provide better control than current, state-of-the-art control schemes.
A study of CLSS controllers with the development of a basic approach for creating stable CLSS controllers was done. The developed design scheme was then utilized to create a stable, CLSS controller for a mass-spring-damper-mass plant. Finally, the stability properties of a crane under CLSS control were experimentally demonstrated.
References
[1] John R. Huey and William Singhose, “Trends in the Stability Properties of CLSS
Controllers: A Root-Locus Analysis,” IEEE Transactions on Control Systems
Technology, Volume 18, No. 5, September 2010.
[2] N. C. Singer and W. P. Seering, “Preshaping command inputs to reduce system
vibration,” Journal of Dynamics of Systems, Measurement and Control, volume 112, pp. 76–82, March 1990.
[3] D. Lewis, G. G. Parker, B. Driessen, and R. D. Robinett, “Command shaping control
of an operator-in-the-loop boom crane,” in Proceedings of American Control
Conference, Philadelphia, PA , pp. 2643–2647,March 1998.
[4] K. T. Hong and K. S. Hong, “Input shaping and VSC of container cranes,” presented
at the IEEE International Conference on Control Applications, Taipei, Taiwan, 2004.
[5] K. Zuo and D. Wang, “Closed loop shaped-input control of a class of manipulators
with a single flexible link,” in Proceedings of IEEE International Conference on
Robotic Automation, Nice, France, pp. 782–787,1992.
[6] T. Singh and S. R. Vadali, “Robust time-delay control,” ASME Journal of Dynamic
Systems, Measurement and Control, volume 115, pp. 303–306, June 1993.
[7] D S. Kwon, D H Hwang, S. M. Babcock, and B. L. Burks, “Input shaping filter
methods for the control of structurally flexible, longreach manipulators,” in
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4, pp. 3259–3264, 1994.
ABSTRACT
Traditionally, input shaping has been used outside the feedback control loop. Utilizing input shaping in this way prevents it from directly addressing some important vibration control issues such as disturbances, nonzero initial conditions, actuator saturation, etc., These potential applications, however, will never be fully exploited unless a good understanding exists of how to create stable CLSS controllers.
Study of closed-loop stability of feedback loops containing input shapers by analyzing their dynamics via root locus plots is done.. Despite the fact that CLSS controllers add partial time delays to the feedback loop, these controllers can be stabilized even when modeling errors occur.
In fact, they rely upon accurate models more than standard PID feedback control. The study presented provides a basic approach to creating stable CLSS controllers. Experimental results were given to verify the predictions made via the root locus, the stability properties of a crane under CLSS control were experimentally demonstrated.
[attachment=8648]
LIST OF CONTENTS
1.Introduction……………………………………………………….….…..….(3)
2.Review of Outside-the-Loop Input Shaping…………………………….….. (4)
3.Advantages of Closed-Loop Signal Shaping…………………………….…...(6)
4. General approach…………………………………………………..……..(9)
4.1. First-Order Effect of Shaper Zeros…………………………….…….…..(9)
4.2 Root Loci Proof for Closed-Loop Signal Shaping……………………..(10)
4.3 Stability Analysis of a Damped Second-Order System………………...(12)
4.3.1 Root Locus Analysis of an Undamped Second-Order
System……………………………………………………….……(13)
4.3.2 Parameter Influence on Closed-Loop Stability…………………..….(16)
4.3.3 Influence of Gain, K…………………………………………..……..(16)
4.3.4 Influence of Damping Ratio……………………………………..…..(17)
4.3.5 Influence of Lead Compensator…………………………….……….(18)
4.3.6 General Design Approach………………………………….………..(19)
5. Stability Analysis Of a Mass-Spring-Damper-Mass Plant………………………..(19)
5.1 Root Locus Analysis……………………………………………………..(21)
5.2. Effect of Modeling Errors on Stability…………………………………..(22)
5.3 Results From the Mass-Spring-Damper-Mass Study……………………..(23)
6. Experimental Verification…………………………………………………………(24)
7.0 Disturbance rejection in CLSS Controllers………………………………………(29)
8.0 Conclusion………………………………………………………………………..(30
1. INTRODUCTION
The control of flexible systems is an immense field of research. Consequently, there are many types of vibration control, including feedback control, optimal control, filters, etc. These control strategies are utilized in a variety of control architectures, such as open-loop control, feedback control and feedforward control. One particularly effective form of vibration control is input shaping. This type of control has been widely used on cranes coordinate measuring machines flexible spacecraft, and long-reach robots. Input shaping has also been extended for use in more complicated applications, including multi-input systems multi-mode systems adaptive control and digital control .
Traditionally, input shaping has been used outside the feedback control loop. Outside-the-loop input prevents it from directly addressing some important vibration control issues such as disturbances, nonzero initial conditions, actuator saturation, etc..
Considering the overall simplicity of input shaping, from a theoretical and practical perspective, it is certainly desirable to investigate whether input shaping can address these phenomena by including it inside feedback loops.
2. Review of Outside-the-Loop Input Shaping
Figure1: Input shaping procedure.
Traditional outside-the-loop input shaping works by convolving a reference command with a series of impulses specifically designed to eliminate unwanted vibratory modes. Figure 1 depicts this process by showing a step command convolved with a two-impulse shaper, which produces a staircase command that results in zero vibration. A useful way to understand input shaping is to analyze it in the Laplace domain. An input shaper is designed to cancel a system’s stable poles by generating zeros that occur at the same place in the s-plane as the vibratory poles. For example, a “Zero Vibration” (ZV) shaper, like the one shown in figure 1, contains two impulses. The first impulse occurs at time t=0 with a magnitude of A1. The second impulse, which has a magnitude of A2, occurs at some delayed time.
In the Laplace domain, the equation for a ZV shaper is
ISZV(s) = A1+A2e-st2 (1)
Using, s=σ+jω, the zeros of ISZV can be determined. This leads to two separate equations that must be simultaneously satisfied.
(2)
-1= e-ωt2 (3)
From (2) and (3), we find that, ω= π/t2 and one solution is, σ= -(1)/(t2)ln(A1/A2).
If the shaper is to cancel a set of damped poles, then ω must be set to ωd , the system’s damped natural frequency. In other words, t2=(π)/(ωd). Also, and must be set so that, where and are the damping ratio and natural frequency of the system.
Figure 2: Zero Vibration shaper cancelling flexible poles.
One interesting note that becomes important when input shapers are included within feedback loops is that a ZV shaper has an infinite number of zeros. From (3) it is evident that ωt2 can be any odd multiple of π.Therefore, ω= +(nπ)/(t2), (where n=1,2,3) will satisfy (3). This means that an input shaper produces an infinite column of zeros, with the first set of zeros (n=1) usually set to cancel the poles of the flexible system. This column of zeros can be seen in figure 2, which shows the zeros of an input shaper canceling the oscillatory poles of a flexible system. It is also important to note that a ZV shaper has no finite, open-loop poles, because ISZV(s) goes to infinity only if the real part of s equals
3. Advantages of Closed-Loop Signal Shaping
As noted in the previous section, input shapers have been widely studied as an open-loop command shaper, filtering out system natural frequencies from the system’s reference command. The system can be an open or a closed-loop system, but usually, the input shaper is left outside of any feedback loops. However, as will be detailed in the following section, some work has studied the effectiveness of placing input shapers inside the feedback loop. Here, any control scheme that includes an input shaper within a feedback loop is referred to as a closed-loop signal shaping (CLSS) controller.
Figure 3: Block diagram of CLSS controller
One obvious potential for closed-loop signal shaping (placing input shapers inside the feedback loop) is disturbance rejection. With traditional input shaping acting as open-loop control, its vibration-reducing capabilities cannot address disturbances. Typically, disturbance rejection for a low-damped, oscillatory system is accomplished with some form of derivative feedback control. This derivative action creates a closed- loop system that is more damped, improving the closed-loop system’s response to disturbances. Closed-loop signal shaping provides a possible alternative to derivative control. By placing the shaper inside the loop, it has the potential to eliminate the vibration induced by disturbances without overly slowing the closed-loop system by increasing the closed-loop damping ratio.
Another possible use for closed-loop signal shaping is non-collocated control. Due to the inherent difficulty of stabilizing non-collocated control systems (even with PID control), some researchers have intentionally introduced time delays into the closed-loop controller to enhance stability . By placing input shapers (which are created with time delays) inside the feedback loop, it is possible to create stable, non-collocated controllers that also result in rapid, low-vibration motion.
Figure 4: Open loop input shaping control system with saturation.
A third application for closed-loop signal shaping involves systems with hard nonlinearities. In some physical systems, successfully implementing open-loop input shaping is not possible because of the presence of a hard nonlinearity, such as a slewrate limiter or a saturator inside the feedback controller. A block diagram of such a control scheme with a saturator is shown in figure4. When a hard nonlinearity modifies an input-shaped command, the vibration-reducing properties of the shaped command entering the oscillatory plant can be significantly degraded . Therefore, one solution would be to design the baseline reference command so that the input-shaped reference command would result in zero saturation of the actuator input to the plant. While possible in some cases, this is clearly a difficult endeavor in many cases. However, by placing an input shaper with all positive impulses inside the loop, one can guarantee (for any possible reference command) that the input-shaped command entering the plant is unmodified by the hard nonlinearity. This allows the input shaper to fully employ its vibration-reducing capabilities.
To understand this result, one must first realize that the output of the input-shaping process is never greater in magnitude than the input to the input-shaping block. So, if instead of placing the input shaper outside the loop, it was placed inside the loop (acting on the error signal) , then one only needs to constrain the input to the input shaper (the error signal) to be within the bounds of the “Actual” saturator. This can easily be done by preconditioning the error signal with a model of the saturation limits. By placing the input shaper inside the loop and using a model of the saturation to condition the error signal, one is able to fully utilize the vibration-reducing characteristics of input shaping.
Finally, CLSS controllers formed by placing input shapers inside feedback loops formed by a human operator (such as in crane control) have proven to greatly improve performance .The potential applications of CLSS controllers however, will never be fully exploited unless a good understanding exists of how to create stable CLSS controllers.
4. General approach
This section will establish the general design approach for creating stable CLSS controllers. The plants studied in this paper will be limited to linear systems, meaning that their transfer functions can be written as fractions of polynomials in “s”. However, the input shaper (in the Laplace domain) contains exponentials of “s”, due to the presence of time delays. Therefore, analytical solutions to the “s” values that result in instability are not generally possible. Note that this is also true when high order (fourth-order or above) linear systems are studied. Therefore, the approach here will be to use a graphical
tool to illustrate several key concepts of creating stable CLSS controllers. These concepts can then be used to create new CLSS controllers for any order plant. The graphical tool
can be used to verify stability and to adjust gains for optimal performance of any specific system.
Figure 5: Block diagram of CLSS controller.
The graphical tool used is Root Locus because it most easily demonstrates several key characteristics of CLSS controllers. Certainly, the Bode, Nyquist, or any number of other methods could also be used.
4.1. First-Order Effect of Shaper Zeros
The closed-loop signal shaping control scheme considered here is shown in figure5. Here, “IS” is an input shaper and “C” is some other controller. To begin the stability study, “C” is restricted to be a proportional gain, K. If the plant is assumed to be unity, then the closed-loop system is only a function of the input shaper and the proportional gain . The system’s root locus, for the case of a ZV shaper, C=K and Plant=1, is shown in figure 5.
Figure 6:ZV shaper inside a feedback loop.
The root locus in figure 6 shows that the input shaper’s infinite number of open-loop poles, located at negative infinity, create an infinite number of root locus branches when a shaper is included within the feedback loop. Note that the closed-loop poles arising from the input shaper tend to form a column that extends infinitely away from the real axis. This occurs because of the infinite column of open-loop zeros, to which the closed-loop poles approach. As discussed in the next section, the closed-loop poles do not generally form a vertical column, but one that tilts toward negative infinity as it extends away from the real axis. Note that, in the general case, the input shaper’s open-loop zeros do not lie on the imaginary axis. They can lie anywhere in the left-half plane, depending upon the oscillatory poles they are designed to cancel.
4.2 Root Loci Proof for Closed-Loop Signal Shaping
Constructing continuous domain root loci of CLSS controllers presents a few interesting challenges. First, the closed-loop characteristic equation is not a polynomial in “s”. This means that numerical methods must be employed to construct the root locus. This paper uses a numerical root locus drawing technique created by Nishioka . In addition, the input shaper contains an infinite column of open-loop zeros, resulting in an infinite number of closed-loop poles. These poles are not necessarily in a straight, vertical column, although they do extend vertically away from the real axis as described in the previous paragraph and illustrated in figure 6.
This ambiguity as to the location of an infinite number of closed-loop poles can make the use of the continuous domain root locus tool difficult, as it is not obvious what axes range should be drawn to include all significant closed-loop poles. However, it is shown below that beyond a sufficiently large radial distance from the origin, the closed-loop poles arising from the input shaper move left as they get farther from the real axis. This means that the most significant closed-loop poles arising from an input shaper are those closest to the real axis. Therefore, a controls engineer can reasonably ignore all but a small, finite number of these closed-loop root loci branches.
To show this, a simple proof is presented. The characteristic equation of the controller shown in figure 3 is assumed to be of the form,
1 + K*IS*F=0 (4)
Where K*IS*F is the open-loop transfer function of the closed-loop system. Furthermore, K is a proportional gain, IS is the input shaper, and F is the portion of the open-loop transfer function that can be written as a rational transfer function (a fraction of polynomials in s). Note that F incorporates the plant and any other desired controller. One further assumption is that the denominator of F is of higher order than the numerator.
For some value of K, there is an infinite number of values that will satisfy the characteristic equation. In addition, the closed-loop poles arising from the input shaper dynamics tend to form a vertical column as they approach the column of open-loop zeros established by the input shaper. By moving away from the real axis, the magnitude of these ‘s’ values tends toward infinity. As , │s│ tends to ∞ it follows from the strictly proper nature of F that │F(s)│tends to zero. However, because K*IS*F (with K being a finite constant) must always equal -1, it follows that │IS│tends to ∞ as ‘s’ tends to zero.
An input shaper with n impulses will have the form
n n
IS = A1 + ∑Aie-sti=A1 + ∑Aie-σtiAie-jωti
i=2 i=2 (5)
Where Ai and ti are the magnitude and time, respectively, of the impulse. Note that every input shaper can be written in this form, as they are all a summation of time-delayed impulses (ti=0 , following standard practice). In order for (5) to approach infinity as s tends towards infinity, some term in (5) must approach infinity along with s . The Ai terms are constant and usually less than or equal to one. The term e-jωti has a constant unity magnitude. Therefore, the term e-σti must be the term whose magnitude approaches infinity. Because ti is fixed and finite, σ must approach ∞ as │s│ tends to ∞.This means that as the closed-loop poles for a single K value lie farther from the real axis, they also lie farther to the left of the imaginary axis. Therefore, at some point, they become insignificant and only a finite portion of the real-imaginary plane is needed for sufficient system identification and control.
One point of caution is the fact that the trend established above is only valid s as approaches infinity. There is currently no established method for determining the exact value for which the trend begins. For small ‘s’ values, where the closed-loop poles from the input shaper are close to the dynamics arising from F, no specific trend exists. The controls engineer must establish the suitable real-imaginary plane area outside of which the above mentioned trend holds and where the closed-loop poles are insignificant.
The controls engineer can also avoid this difficulty inherent to the continuous time root locus tool by utilizing frequency domain tools (like Bode plots ) or by utilizing the digital root locus .
4.3 Stability Analysis of a Damped Second-Order System
Given the insights gained by the previous analysis, a damped second-order plant can be considered.
G(s) = ωa2
s2+2ξωa+ωa2 (6)
To begin, the undamped version is analyzed via the root locus, then the additional elements of damping and lead compensators are added. The goal of this section is to understand the most basic stability issues inherent to CLSS control. For example, the following sections will reveal that high gains and modeling errors lead to instability, whereas system damping and lead compensators improve stability.
4.3.1 Root Locus Analysis of an Undamped Second-Order System
Given the control system depicted in figure 3, the controller C, is first defined as a proportional controller(C=K). If the input shaper is tuned perfectly to the plant frequency, then the root locus will be similar to that shown in figure 7. Here, the plant parameters are ωa=2π and ξ =0.
As shown in figure 6 and reiterated in figure 7, placing an input shaper within the loop will result in an infinite number of closed loop poles. Their presence indicates another important result. Using an input shaper within a feedback loop, as done within the control scheme depicted in figure 3, will result in additional oscillatory dynamics arising from the input shaper. Note that many of the root locus figures in this paper stop at a finite value of to illustrate some feature of CLSS controllers. For example, figure 7 illustrates how the real part of closed-loop poles tend toward the left as they lie further from the real axis, as was proven in Section 4.2.
Figure 7: Root locus with pole/zero cancellation.
If modeling errors occur, then the plant poles are not completely cancelled by the shaper because the modeled frequency used to design the shaper(ωm) does not equal the actual frequency(ωa) of the plant. If ωa<(ωm), then the root locus is as shown in figure 8. The root locus branch extending from the plant poles immediately goes unstable. However, if ωa>ωm, then this branch remains stable, as shown in figure 9. This is the pattern usually seen with CLSS controllers when the plant G, is a lightly damped second-order system.
Figure 8: Root locus where ωa<ωm Figure 9: Root locus where ωa>ωm
If the plant pole is below the shaper zero to which it is closest, then the root locus branch extending from that plant pole bends to the right, making the system unstable. On the other hand, if the pole is above the zero to which it is closest, then the root locus branch bends left, creating a stable system. Under such modeling errors, closed-loop stability depends heavily on the relationship between ωa and ωm. Staehlin and Singh demonstrated similar stability results for their CLSS controller. Note that the simple shapes of the branches shown in figures 8 and 9 occur when the plant pole is close to one of the shaper zeros. If the pole is near the equidistant point between two shaper zeros, then the bend-left/bend-right characteristics will hold in general, but the branch may initially depart in a direction that is not directly left or right.
These departure angle observations can be explicitly shown for the case discussed here: a ZV shaper combined with an undamped second-order plant. The open-loop transfer function of this CLSS controller is
GC = (0.5+0.5e-sП/ωm) ωa2
s2+ ωa2 (7)
The departure angle from the plant pole can be determined by testing an value arbitrarily close to the plant pole. For example, set s=reθi+ωai where r establishes the radial distance of the test point from the plant pole and θ establishes the angle of the test point from the positive real axis.
The open-loop transfer function evaluated at this test point is
(0.5ωa2)(1+e-Л/ωm(ωa+rsinθ)i)
(r2cos2θ-(ωa+rsinθ)2+ωa2)+((ωa+rsinθ)(rcosθ))i (8)
As ‘r’ tends to zero, angle of the numerator is
α = tan-1 (sin(-Лωa/ωm)
(1+cos(-Лωa/ωm) (9)
In addition, as , the angle of the denominator (with the help of L’Hospital’s Rule) can be shown to be
β= tan-1(0/0) (10)
tan-1(cosθ/-2sinθ) (11)
Figure 10: Departure angle versus modeling error.
In order for this test point to lie on the root locus, α-ɤ=Л.This equation can be rearranged via trigonometric identities to yield
tan α = tan β (12)
Substituting (10) and (11) into (12) yields
tan θ=1+cos(-лωa/ωm)
-2sin(-лωa/ωm) (13)
Solving for θ gives the departure angle from the plant pole s=ωai . Varying ωa/ωm relationship reveals how the departure angle relates to modeling errors, as was depicted in figures 8 and 9. Note that a ZV shaper puts zeros at odd multiples of the plant pole . Therefore, when ωa/ωm<1, the plant pole is below the first shaper zero, as shown in figure 8. If 1<ωa/ωm<3 , then the plant pole is between the first and second shaper zero, as shown in figure 9. This pattern is repeated as ωa/ωm crosses 1,3,5….
Figure 9 shows a generalized representation of what is depicted for specific cases in figures 8 and 9. When the plant pole is below the shaper zero to which it is closest (i.e.ωa/ωm, is close to but less than 1,3,5.. ), then the departure angle is between zero and 900, making the root locus branch immediately unstable. If the plant pole is above the shaper zero to which it is closest (i.e ωa/ωm, is close to but greater than 1,3,5,7.. ), then the departure angle is between 900 and 1800 ,making the root locus branch initially stable.
4.3.2.Parameter Influence on Closed-Loop Stability
The previous sections established the basic form of the root locus for closed-loop signal shaping controllers. Also, the previous sections revealed that modeling errors are a primary source of instability for CLSS controllers. Here, the influence of other parameters (namely K and ξ) that are present in the control system of figure 3 will be analyzed. Also, the addition of a lead compensator is investigated. The lead compensator is studied here because it is known to increase stability margins. Because small modeling errors can result in unstable, CLSS controllers, it is important to show that a control block C can be added to the block diagram shown in figure 3 so as to increase stability margins even when modeling errors occur.
4.3.3 Influence of Gain, K
By increasing the gain K , the system shown in figure 3 will eventually be driven unstable, even with a favorable modeling error condition, figure 11 shows how an initially stable system can be driven unstable by making K too high.
Figure 11: Root locus with high K value.
This figure indicates that the dynamics arising from the inclusion of input shaping filters inside of feedback loops can, themselves, be the cause of instability. That is, the root locus branch emanating from the plant pole remains stable, while the branch emanating from one of the input shaper open-loop poles is what eventually causes instability.
4.3.4 Influence of Damping Ratio
When the second-order plant has damping, this increases the regions of stability. figsures 10 and 11 show this effect for each modeling error condition. The modeling error used here is +20 % error in the frequency. By increasing , the root locus branches are shifted to the left. This effect enables higher proportional gain(K) values to result in stable closed-loop poles.
Figure 12: Influence of ξ when ωa< ωm Figure 13: Influence of ξ when ωa >ωm
4.3.5 Influence of Lead Compensator
Figure 14: Root locus with lead compensator
Because of the stability issues associated with closed-loop signal shaping, it is desirable to study the effect of stability-enhancing controllers. One of the most effective and practical controllers used to increase stability margins is the lead compensator. The lead compensator is implemented here by setting C=K(S+Z)/(S+P). Figure 14 shows how the root locus branches are pulled to the left by the lead compensator.
The “No Lead Comp” root locus shown in figure 14 is a good example of what happens when the plant pole is nearly equidistant from two shaper zeros. The branch does not look exactly like those in figures 12 or 13. It follows the departure angle plot shown in figure 10, having an angle of approximately 90 degrees. However, because it is closer to the shaper zero above it, this branch eventually behaves like those shown in figure 12, tending toward instability.
While there are many other types of controllers (PID, lag compensator, etc.) that could have been used as the C block in figure 3, the main goal of this section is to show that reasonable stability margins can be achieved in CLSS systems. The lead compensator sufficiently demonstrates this point.
4.3.6 General Design Approach
This section revealed several key insights into the best way to create a stable CLSS controller. First, it is important to note that adding an input shaper into a closed-loop controller will add higher-order dynamics. As with any system that contains high modes that are either unmodeled or not actively controlled, these higher-order dynamics can cause instability. Therefore, as shown earlier in this section, the gains of CLSS controllers will always be limited by these higher modes. Note that this influence of high-order modes is true whether they derive from unmodeled dynamics of the plant or if they arise from the modes added by the input shaper.
Another key insight was the impact of modeling errors. Modeling errors in CLSS controllers can quickly cause instability. Therefore, CLSS controllers will need to be limited to usage with plants that are well defined. This section also showed that if the plant contains some amount of damping, then small modeling errors can be tolerated. So, CLSS controllers should also be limited to use with systems that have some reasonable amount of damping. Finally, if higher-order modes, modeling errors, or a lack of a minimal amount of system damping is causing stability problems, then the CLSS controller can be quickly stabilized with more traditional control schemes. For example, this section showed how a lead controller significantly increased the stability of one type of CLSS controller.
5. Stability Analysis Of a Mass-Spring-Damper-Mass Plant
Figure 15: Mass-spring-mass system.
In order to demonstrate the effects of CLSS on more complicated systems, the mass-spring-damper-mass system shown in figure 15 is now considered. A block diagram of a collocated feedback controller for the system is shown in figure 15. Analyzing the system shown in figure 16, (X)/(F) can be shown to be
Figure 16: Diagram of collocated controller
X 1 ( s2 + αs + ß)
F M1 S2(S2+2ξ1ω1 +ω12) (12)
where
ω1 = ksp M1 + M2 , ß = ksp
M1M2 M2
2ξ1ω1 = b( M1 +M2)
M1M2 (13)
Figure 15 represents a large number of systems such as satellites and cranes, which have a rigid body connected to a lightweight, flexible appendage. These flexible systems often present a more difficult control challenge, and the study of this system will reveal additional insights into CLSS control.
Following the lessons learned in the previous section, the CLSS controller will maintain limited gains so as not to drive the higher modes introduced by the input shaper unstable. Also, the plant studied here will have a small, but nonzero, damping coefficient,b. This will ensure that the CLSS controller can tolerate some level of modeling errors. In addition to a nonzero damping coefficient, all of the plant parameters will be assumed to be reasonably well known. This section will, however, investigate the effect of modeling errors. Last, the presence of a rigid-body mode in the plant will present a stability problem. However, as shown in the previous sections, the use of a lead compensator in addition to the input shaper will greatly increase the range of stable controllers.
For this example, the following values are used : M1=100 , M2 = 1, ksp= 20, and b = 0.2. These values lead to ω1 = 4.69 rad/sec, ß = 19.98, ξ1 = 0.024, α = 0.18.
5.1 Root Locus Analysis
The root locus of the closed-loop system shown in figure 16, without the input shaper, is shown in figure 17. This system has one rigid-body mode, one flexible mode and one set of complex zeros. This system is stable for any proportional gain(K) value. However, if the input shaper is added to the closed-loop system, then the root locus is as shown by the two, dark, thick lines in figure 17.
Figure 17: Fourth-order system under control—Without CLSS
The input shaper is designed without modeling error, so the input shaper zero exactly cancels the plant’s oscillatory pole. However, even with no modeling errors, this system starts out (with small K values) as unstable. And, even when large K values bring the closed-loop poles from the first root locus branch back to the left-half plane, higher modes introduced by the shaper have already gone unstable. The general instability of this control scheme is reiterated by the six, thinner root loci shown in figure 18. These root loci were obtained by varying the ω1 and ß between +30 % of their original values.
Figure 18: CLSS of fourth-order plant.
According to this root locus analysis, CLSS control of a mass spring- damper-mass system is unstable for a significant range of parameter values. That is, this system is unstable when the controller, , is just proportional control. This is an important reminder of the difficulty of using controllers that contain time delays within a feedback loop.
One solution is to use a lead compensator, in addition to proportional control (i.e. C=K(S+Z)/(S+P)). If the lead compensator pole is chosen to be at-20 while the zero is set at -1, then the system root locus will look like the one depicted by the two, dark, thick lines in figure 19. Even with extremely high K values, the system remains stable. Under the same parameter variations as in figure 19, the resulting root loci are significantly more stable. In conclusion, by adding a lead compensator, closed-loop signal shaping of a fourth-order system can be stabilized.
Figure 19: Addition of lead compensator
5.2. Effect of Modeling Errors on Stability
Given the sensitivity to modeling errors of CLSS on the second-order system shown in Section 4.3.1 it is important to investigate the general stability of CLSS on a fourth-order system with a lead compensator. Here, modeling errors will be examined by using input shapers designed for frequencies 30% above and below the actual frequency of the plant’s oscillatory poles. If the shaper design frequency is smaller than the plant frequency, then a root locus like the one shown in figure 20 results.
Figure 20: Root locus with ωa< ω1
Notice how the input shaper zero no longer cancels the plant’s oscillatory pole. As expected from the earlier study of second-order systems, the root locus branch from the oscillatory plant pole bends to the left. However, if the shaper frequency is larger than the plant frequency, then the root locus branch bends to the right, as shown in figure 21.
Figure 21: Root locus with ωa>ω1
With second-order systems, this usually causes significant stability problems. However, in this fourth-order system, the plant pole is now located between two zeros, one from the input shaper and one from the plant’s numerator dynamics. Because it lies between two zeros, the plant pole does not adversely affect stability as much as in the second-order case. As shown in figure 21, the root locus branch extending from the plant pole remains stable for a significant range of K values.
5.3 Results From the Mass-Spring-Damper-Mass Study
The study of this mass-spring-damper-mass plant showed can be used to create a stable, CLSS controller for a fourth-order plant that contains numerator dynamics. The system was assumed to be well known, although the CLSS controller with the lead compensator did allow for significant modeling errors. The system was also damped, although the damping ratio was low. More traditional controllers (in this case a lead compensator) were effectively utilized to improve stability.
6. Experimental Verification
In order to verify the stability characteristics of CLSS controllers predicted by the root locus analysis technique, experiments were conducted on an industrial crane located at the Georgia Institute of Technology . A sketch of the 10 ton bridge crane is shown in figure 22. This bridge crane has at least four notable nonlinearities: a velocity limit, an acceleration limit, a built-in velocity smoothing algorithm that prevents sudden sign changes in velocity, and a velocity dead zone.
Figure 22: Crane used for stability experiments.
Obviously, the linear-based root locus analysis tool utilized in this paper cannot account for the nonlinear behavior of this real world system. However, it will be shown here that a good linear model is sufficient to explain the dynamic behavior and approximately predict the gain value that induces instability. The motion of the crane trolley is controlled by a CLSS controller of the form shown in figure 5 where
C = K0(Kp + Kds)
IS = A1 +A2e-st2
G = ωn2
s(s2 + 2ξΩns+ ωn2) (14)
Figure 23. Root locus of crane controller.
This closed-loop system consists of a PD feedback controller, a two impulse (ZV) shaper, and a third-order plant (second-order oscillator plus an integrator). The plant contains an integrator because it is driven by velocity commands, while its output is position. For these experiments, values within the closed-loop system were held constant, and K0 was varied to show its effect on stability. The theoretical root locus of this closed-loop system is shown in figure 23. Zooming in closer to the real axis, figure 23 shows that this closed-loop system will go unstable for some finite value of gain K0. The root locus predicts that a K0 value of 10.5 will create a marginally stable system.
Figure 24: Controller stability limit.
In order to verify these predictions, the crane trolley was commanded to move 1 meter. As an initial test, the experiments were conducted without an input shaper in the loop. That is, IS was set to 1 and C and G were as given in (14). This was done to ensure that the system would not go unstable purely due to its nonlinearities and PD controller. As seen in the experimental results shown in figure 25, no tested value of K0 resulted in instability.
Figure 25: Trolley response without CLSS.
However, when the input shaper was included within the feedback loop, increasing K0 from 0.5 to 10 caused a limit cycle, as shown in figure 26. The proportional and derivative gains used in this experiment were Kp = 0.8 and Kd = 0.5 . Although the response is not exponentially growing, this limit cycle response is unstable for practical purposes. Because the linear dynamics of the bridge crane dominate its response, the experimentally determined critical gain K0 =10 of is very close to K0 =10.5 the theoretical value of predicted by the root locus.
Figure 26: Trolley response with CLSS
It should be noted that the oscillatory response depicted in the K0 = 10 plot has a period of oscillation of approximately four seconds, corresponding to a frequency of approximately 1.6 radians per second. Looking at figure 24, the root locus branch which goes unstable at K0 =10.5 would indicate an oscillation of more than 10 radians per second. However, figure 24 shows that between Ko = 9 and Ko = 11 and , several closed-loop poles are close to instability. While the pole which first goes unstable does not dominate the time-domain response, the critical value does correctly predict when the system will exhibit steady oscillations.
Figure 27: Root locus of second system.
A second set of experiments were conducted using a different Kd gain( Kd = 0.1)in the controller block “C” defined by (14). The root locus of this system is shown in figure 26. In this case, the root locus predicts that Ko = 12 will result in instability. The 1 m experimental step responses shown in figure 27 again demonstrate that Ko= 10 results in a limit cycle response. However, this response is different from that shown in figure 26, exhibiting a significantly higher amplitude.
Figure 28: CLSS response with new PD gains.
These experimental results match fairly well with the theoretically predicted effects of utilizing input shapers within feedback loops. The difference between the predicted and actual critical value of K0 can be attributed to nonlinearities that are not accounted for in the linear model.
It should be noted that the range of Ko values experimentally tested on the bridge crane was rather course. The only value tested between Ko = 5 and Ko = 10 was Ko = 7 This value did not result in sustained oscillations for either set of Kp and Kd gains. Therefore, the actual critical Ko value that results in instability (for both experiments) lies somewhere between seven and ten. However, the precise, critical Ko value is not extremely important, as the root locus prediction will always be somewhat inaccurate due to the nonlinear nature of any system. The important point is that the root locus technique utilized here gives a quick, reasonable estimate of Ko value that will result in instability. This will always need to be verified experimentally if the precise value is needed. Also, the root locus technique shown here gives the engineer a good understanding of the closed-loop system’s secondary dynamics, even when the overall system is stable.
7. Disturbance rejection in CLSS Controllers
One potential application of closed-loop signal shaping controllers is disturbance rejection. If the disturbance enters the closed-loop system just before the plant (called an actuator disturbance), then the traditional use of input shaping outside of the loop performs better. Figure 29 shows that for a second-order plant, an impulse disturbance at the actuator is better handled by the outside-the-loop input shaper and PD closed-loop controller combination (OLIS/PD control).
Figure 29: Actuator disturbance response. Figure 30: Sensor disturbance response
However, if the disturbance enters the system at the sensor (known as a sensor disturbance), then a closed-loop input-shaping controller can often outperform a controller that relies solely on PD control to attenuate the disturbance. This can be seen in figure 30 with a second-order plant.
8. Conclusion
This paper utilized the root locus to develop a design process intended to ensure the creation of a stable CLSS controller. Despite the fact that CLSS controllers add partial time delays to the feedback loop, these controllers can be stabilized even when modeling errors occur. But, they rely upon accurate models more than standard PID feedback control. Experimental results were given to verify the predictions made via the root locus. These experimental results were performed on a 10 ton bridge crane operating under a CLSS controller that combined input shaping and PD control. The several possible applications of CLSS Controllers motivated the need for a design process that would create a stable CLSS controller. The design process proposed here shows that stable CLSS controllers are practical and implementable.
Therefore, the next logical step is to compare the performance of CLSS controllers to current, state-of-the-art control schemes. The most logical state-of-the-art control scheme is the traditional combination of input shaping and feedback control .This controller utilizes some form of PID control (to create a stable closed-loop system) combined with an input shaper (completely outside of any feedback loops) to filter the reference command of any frequencies that would incite vibration in the closed-loop system. One potential application of closed-loop signal shaping controllers is disturbance rejection.The examples demonstrate how CLSS controllers can provide better control than current, state-of-the-art control schemes.
A study of CLSS controllers with the development of a basic approach for creating stable CLSS controllers was done. The developed design scheme was then utilized to create a stable, CLSS controller for a mass-spring-damper-mass plant. Finally, the stability properties of a crane under CLSS control were experimentally demonstrated.
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