ROBUSTNESS ANALYSIS OF SUPER-ROBUST AND ADAPTIVE ALGORITHM OF INPUT SHAPING

_______________________________________

* Poznan University of Technology.

Maciej GNIADEK*

ROBUSTNESS ANALYSIS OF SUPER-ROBUST AND ADAPTIVE ALGORITHM OF INPUT SHAPING

Many mechanical systems are characterized by flexible joints. One of the main problems in control of this kind of devices are the mechanical oscillations. To avoid the problem of vibrations the input shaping method might be used. For the systems with variable parameters the standard methods of simple input shaping are insufficient and non-robust. More complicated algorithms, exemplary super-robust, can be used, but have negative influence on the regulation time. The alternative method is an adaptive algorithm of input shaping. Both methods are compared on the crane system with variable load mass and line length. The algorithms of system parameter changing are executed with variable speed. The research was made in Matlab/Simulink environment.

KEYWORDS: adaptive input shaping, super-robust input shaping, crane positioning, oscillation avoidance

1. INTRODUCTION

Multiple mechanical systems contain flexible joints between elements. This kind of systems are represented by robotic arms, multi-mass systems, conveyers, cranes and other similar systems. The common feature of all of those objects is susceptibility for oscillations. One of the method of system control without of oscillations excitation is input shaping.

The method of input shaping is very effective for the objects with fixed natural resonant frequencies. If this parameter is varying the efficiency of method is decreasing. To solve this problem robust algorithm of input shaping may be used.

The greatest disadvantage of described algorithm is extension of control time. The alternative method is connected with gain scheduling or, in general, adaptation of parameters in input shaper.

Ideal example of objects with flexible joint and variable resonant frequency are the cranes. Because of this feature the research is conducted on simulation model of crane. All the research was made using Matlab/Simulink environment.

2. INPUT SHAPING

The Input Shaping method is one of possible algorithms of system’s control without of excitation of oscillations. Described method should me extinguished as one of the simplest and most effective algorithms for selected task.

In general – the method of input shaping is based on very simple observation (confirmed by mathematical equations), that for linear objects superposition principle is valid. As the consequence – the sum of outputs coming from two different signals is equal to the output achieved due to excitation of object by sum of those inputs. If the object is vulnerable for oscillations the input reference signal might be convolved with a series of Dirac impulses applied in specified moments of times and characterized by specific amplitudes. The situation is presented in Figures 1 [5] and 2 [2].

Fig. 1. Input shaping basis – object response

Fig. 2. Input shaping basis – signal convolution

According to the Figures 1 and 2 and previous information is obvious, that the Dirac impulses cannot be applied randomly. According to the theory

presented in [7], the amplitudes A_i and moments of application t_i should fulfil following dependency:

1

1 1

0 0.5

Ai K K K

ti Td

 

   

   

   

 

 

  

where

1 2 K e





 

 

 

 

 



 

(2) Td is the period of object natural frequency and ϛ is the damping coefficient of this frequency. The tuning of input shaping is based only on two simple parameters for each natural resonant frequencies. All the needed information might be also simply calculated form the system transfer function (see:[3]).

The most simple version of the algorithm (presented in previous equations) will work with high efficiency only if the object parameters are not varying. If the natural resonant frequency is being changed the method might be even harmful for the system – the pulsation amplitude might be even (in the worst case) magnified and the object might resonate. Due to this case the robust algorithm of input shaping was designed.

2.2. Robust algorithm of input shaping

The robust and super robust algorithms of input shaping are quite similar, in theory, to the simple algorithm. The main difference is in the number of Dirac impulses convolved with the baseline command. For the robust algorithm the reference is convolved with three, and for super robust with four impulses. The parameters Ai and ti might be calculated from dependency (3):

0.25 0.25 0.25 0.25 0 0.5

1.5

Ai

ti T T T

   

   

   

 

 



where all of variables are the same, as the described in previous chapter.

The period of natural oscillations and dampening in the previous example should be adjusted to the expected mean value in the whole work range.

This algorithm will perfectly reduce the oscillations with expected resonant frequencies, but the other oscillations in the proximity of selected point will also be highly reduced. The biggest disadvantage of the method is extension of the control time and, in general, limitations in system dynamics. The comparison of the algorithms was presented in [1].

3. GAIN SCHEDULING AND ADAPTAION

The adaptive control algorithm is one of the greatest achievements in control theory since PID controllers. The main idea of this algorithm is based on two basic elements – object identification and tuneable controller.

In most of the control algorithms the object transfer function has to be identified. In normal case the Least Square Method may be used. The object rank has to be defined. If the object is treated as the “black box” the Akaike criteria may be used for rank identification. The controller parameters are defined by parameters of estimated transfer function. The controller parameters definition depends from selected control strategy.

The classical adaptive algorithm of adaptation presented above will be used to tune the parameters of input shaping. The natural oscillations periods and their dampening are strongly connected with the object transfer function and might be used to tune the whole system.

In various practical situations the whole adaptation is not needed. When the object structure is not changing and the significant for oscillations parameters are measureable gain scheduling method might be used.

Gain scheduling is usually based on simple connection between the measurable parameter of the system and controller parameter. If the mathematical formula for optimal settings of controller may be determined using only physical signals the gain scheduling will be more effective and faster than adaptation.

The general structure of adaptive controller is presented in Figure 3.

Fig. 3. Adaptive control – general structure [4]

4. CRANE MODEL

As was mentioned in the first chapter the research was made on the crane model. Used example is presented in the base of Simulink examples [6]. The usage of selected model has some positive aspects - all the mechanics is

modelled good, usage of ready model projected with control system has more in common with the practical situations – usually the control system is ready and additional element (in this case input shaping) is simply added. Additionally a good visualization of the system is prepared. Some elements are easier to observe on the visualization instead of single figures. Finally the optimization for computing of pre-prepared model is much better than will be in self- prepared model.

All these reasons finally leaded to usage of this model. The model is built from 3 main parts – dynamics in X and Z axes – orthogonal to the line, and line dynamics in Y axis.

The dynamics of X and Z axes is the same and independent. The dynamics of line includes the limitation of length changes speed. The physical structure id described in [6].

Fig. 4. Crane model visualization

The visualization of the model is well-optimized and does not significantly change the operational time.

5. SIMULATION AND RESULTS

The model described in previous chapter was modified in the structure. The Input Shaper block was implemented in position presented in Figure 5.

The simulation was conducted for two different strategies of Input Shaping tuning. First strategy was based on gain scheduling (adaptation) and the second on super-robust input shaping algorithm. The task was the same in both cases – set trajectories of movement are presented in Figure 6. The dynamics should be not limited and the deviation of the load from vertical should be small for the optimal control strategy.

Fig. 5. Input shaping (IS) position in the control structure (R – controller, Obj – object (crane))

Fig. 6. Set trajectories for crane movement

As was presented in figure (6) the control task was set to move the crane by 15 meters in both axes with line in upper position without of load and go back to 0,0 point with load of 2 tones. The task was selected to achieve variable resonant frequencies during the simulations. The X axis is shaped and Z axis is left without of shaping to illustrate the profit coming from input shaping usage.

5.1. Robust input shaping

The period of load oscillations depends (in general) only form the length of cable with load. The medium length was set and oscillations period was measured. Input shaper was tuned according to the dependency (3). The results are presented in Figure 7.

As was presented in Figure 7 the usage of shaper strongly improves the results of pre-prepared control strategy. The object oscillations are strongly reduced. Dynamics of movement is limited.

0 10 20 30 40 50 60 -5

0 5 10 15

Time [s]

Head position [m] Reference

X Axis Z Axis

0 10 20 30 40 50 60

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3

Time [s]

Angle [rad]

X Axis Z Axis

Fig. 7. Position and angle courses – robust input shaping

5.2. Gain scheduling

Second simulation was conducted using gain scheduling for Input shaping tuning. The test is similar to the presented in previous point. The gain scheduling was based on observation, that the load oscillations period might be treated as mathematical pendulum. The dampening of oscillations is not varying in wide range for the whole operation point. The results are presented in Figure 8.

0 10 20 30 40 50 60

-5 0 5 10 15

Time [s]

Head position [m] Reference

X Axis Z Axis

0 10 20 30 40 50 60

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3

Time [s]

Angle [rad]

X Axis Z Axis

Fig. 8. Position and angle courses – gain scheduling of input shaping

As was presented in Figure 8 the oscillations are also strongly reduced. The limitation of dynamics is very insignificant because of usage of the simplest algorithm if input shaping with adaptation.

6. CONCLUSION

Two algorithms of input shaping were tested. Both algorithms are working properly and very significantly improve the results achieved without of shaping.

The oscillations are damped very well in both cases. The maximal level of oscillations is comparable. The dynamics of movement is much better when adaptive algorithm is used. Because of the usage of less-complicated algorithm of input shaping the limitations of dynamics are insignificant. This leads to the conclusion, that gain scheduling gave much better results than robust algorithm of input shaping.

REFERENCES

[1] S. Brock and M. Gniadek, “Robust input shaping for two mass system with variable parameters.,” presented at the ZKwE, Poznań, Poland, 2014.

[2] M. Gniadek, “Analysis of input shaping robustness for two-mass system.” WPP, 2014.

[3] M. Gniadek, “Auto tuning of systems with input shaping,” presented at the 16th Mechatronika 2014, Brno, Czech Republic, 2014.

[4] D. Horla, Sterowanie adaptacyjne. Cwiczenia laboratoryjne, 3rd ed. Poznań, Poland: Wydawnictwo Politechniki Poznańskiej, 2010.

[5] J. R. Huey, “The intelligent combination of input shaping and PID feedback control,” 2006.

[6] Mathworks, “Portal Crane with Predefined Trajectory,” 11-Jan-2015. [Online].

Available: http://www.mathworks.com/help/sl3d/examples/portal-crane-with- predefined-trajectory-and-sound-support.html?refresh=true. [Accessed: 11-Jan- 2015].

[7] W. E. Singhose, “Command generation for flexible system.” MIT, 1997.