Gain-scheduling control for mechatronic systems ... - Semantic Scholar

Gain-scheduling control for mechatronic systems with position dependent dynamics Bart Paijmans1,2 , Wim Symens2 , Hendrik Van Brussel1 and Jan Swevers1 (1) K.U.Leuven, Department of Mechanical Engineering, Celestijnenlaan 300 B, B-3001, Heverlee, Belgium (2) Flanders Mechatronics Technology Center, Celestijnenlaan 300 D, B-3001, Heverlee, Belgium e-mail: [email protected]

Abstract This paper presents a gain-scheduling-control technique for mechatronic systems with position-dependent dynamics. The proposed method fits in the framework of traditional gain scheduling, where several controllers designed for fixed operating points are interpolated to construct a global gain-scheduling controller. A new interpolation approach is proposed starting from an affine interpolation between the poles, zeros and gains of the local controllers as a function of the varying parameter, resulting in a polynomial state-space representation. The presented method is applied on an industrial pick-and-place machine which has positiondependent dynamics. Experimental results show the benefit of the proposed method.

1 Introduction High-speed, high-accuracy and low-cost operation are the main thrusts behind the design of light weight mechatronic structures such as robot manipulators, pick-and- place machines, etc. Light-weight structures operating at high speeds, however, may lead to significant vibration problems, that degrade the positioning accuracy. An additional problem is that the dynamic behavior of a machine tool can depend on the position of the tool in its workspace. As this position is measured, the changes in dynamic behavior can be modeled exactly in a deterministic way. High-performance motion controllers that take into account these varying structural resonances are needed. Gain-scheduling control is an appropriate control solution for systems with position-dependent dynamics and will be adopted in this paper. Gain scheduling is a controller implementation where the controller coefficients are changed according to the current value of scheduling signals, which may be signals external and/or internal to the plant [1]. Two main approaches can be distinguished [2], namely traditional gain-scheduling control and Linear Parameter Varying (LPV) control. LPV control is a design method that guarantees the closed loop to be stable for all possible time-varying parameter trajectories. Only the first approach, traditional gain scheduling, will be considered in this paper. A new method is presented to systematically interpolate between local LTI controllers. The proposed gain-scheduling technique is tested on an industrial pick-and-place machine. First, linear models are identified in different operating points, then linear optimal controllers are designed using H∞ techniques and finally an interpolation is made between these controllers. Section 2 explains the interpolation technique that will be used to design a gain-scheduling controller for the setup. Section 3 first gives a description of the layout of the setup and the identification of linear models. Then the design of controllers based on these models is shortly commented. Section 4 discusses the interpolation between these controllers and the experimental results. Section 5, gives the main conclusions of this paper and gives some directions for further research. 93

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2 Interpolation between controllers: Theory 2.1 Introduction Gain scheduling is a common engineering method that can be used to control LPV systems in a variety of applications such as flight control and process control. An LPV system is defined as a linear system whose describing matrices in state-space form depend on a vector of time-varying parameters, which can be measured in real-time [3]:

x˙ = A(l)x + B(l)u y = C(l)x + D(l)u

(1)

with x the vector of states of the system, u and y respectively the input and output and l a vector of varying parameters. This paper focuses on traditional gain scheduling. The main idea of traditional gain scheduling is to break the control design into two parts. First, local linear controllers are designed based on linearizations of the plant for several fixed values of the varying parameter. Second, a global parameter-dependent controller for the system is obtained by interpolating, or "scheduling" the local controllers. Interpolating between controllers is expanding the dynamic behaviour of local controllers to a continuous varying controller. This leads to the following three requirements: 1. stability during interpolation must be preserved: the controller must stabilize the plant for all intermediate operating points; 2. performance during interpolation must be preserved: the system should satisfy the performance specifications for all intermediate operating points; 3. the parameters of the interpolated controllers must be smooth and continuous functions of the varying parameter to avoid discontinuous controller states and output signals as a function of time. This last condition is especially important if fast parameter variations can occur. An extra requirement is to have reliable techniques that can interpolate between high-order controllers, since modern controller techniques applied on complex mechatronic systems, such as H∞ -algorithms, often result in high-order controllers. Many interpolation methods have been successfully applied by control engineers (e.g.[4],[5],[6]). These methods are either ad-hoc methods which require a lot of trial and error, or they fail when the order of the system is too high. The method proposed in this paper is an attempt to be a systematic interpolation approach that can handle complex controllers. The idea is to first make an interpolation between the poles and zeros of the local controllers. Such an interpolation clearly expands the dynamic behaviour of the local controllers to a global varying controller, since the dynamic behaviour of a controller is fully determined by the location of the poles and zeros in the complex plane. In a second step a global varying state-space controller is constructed with a polynomial dependence on the varying parameter, which makes it easy to implement. The third subsection describes the characteristiques of the proposed interpolation technique and highlights some differences with existing techniques.

2.2 Interpolation between poles, zeros and gains A first requirement that has to be fulfilled for a correct interpolation between poles and zeros, is that all the local controllers have the same number of poles and zeros. A second requirement is that the operating points

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for which the local controllers are designed should be sufficiently close such that migration of poles and zeros from one to the next is recognizable. The technique proposed in this paper interpolates between the discrete-time poles and zeros, to make a discrete-time implementation possible. The variation of the vectors of poles and zeros and the variation of the gain each has to be described by an affine function, containing a sum of a constant term and one varying term. The varying term consists of a vector of coefficients multiplied with a scalar real analytical function of the scheduling parameter. This is shown in (2) for the vector of varying poles:

    

p1 (l) p2 (l) .. . pn (l)





    =  

p0,1 p0,2 .. . p0,n





    +  

p1,1 p1,2 .. . p1,n



   · f1 (l) 

(2)

with p1 till pn the poles of the controller, p0,1 till p0,n and p1,1 till p1,n the complex coefficients and f1 (l) a real scalar analytical function of the scheduling parameter l. Similar affine functions have to be made to describe the varying zeros and gain, but they have to contain the same analytical function. Notice that he proposed modeling of the poles and zeros implies that the same analytical function is used in the description of the real part and the imaginary part of the poles and zeros, resulting in linear variations of the poles and zeros in the complex plane. The number of operating points should be high enough sucht that an analytical function can be found that describes the variation of the poles and zeros correctly.

2.3 Transformation to state-space The next step is to construct varying state-space matrices. The resulting state-space matrices (A,B,C,D) are each polynomial functions of the varying parameter: they consist of a sum of a real constant matrix plus two real varying matrices. A varying matrix consists of a constant matrix of coefficients multiplied with an analytical function of the scheduling parameter. This is shown in (3) for the A-matrix :

A(l) = A0 + f1 (l) · A1 + f12 (l) · A2

(3)

The same expressions can be constructed for the B,C and D matrices. The first analytical function f1 (l) is the same as the one used to describe the varying poles, zeros and gains, the second analytical function is the square of this function, f12 (l). The key idea in the algorithm is that each system defined by a gain and a number of poles and zeros can be written as a series concatenation of first-order subsystems containing a real pole and possibly a real zero, and second order subsystems containing a complex pole-pair and possibly a real zero or a complex zero-pair. So there are five types of different subsystems to consider. Each subsystem is created using the control canonical state-space form [12], since for first order and second order subsystems, the control canonical form can be explicitly written in function of the poles and zeros. By means of a concatenation of the states of each subsystem, the global system is obtained. Equation (4) illustrates how a subsystem defined by one pair of complex conjugated poles (pi and pi+1 ) and one pair of complex conjugated zeros (zi and zi+1 ) is transformed to state-space using the control canonical form:

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pi (l) + pi+1 (l) −pi (l) · pi+1 (l) Asub (l) = Re 1 0 1 Bsub = 0 T −zi (l) − zi+1 (l) + pi (l) + pi+1 (l) Csub (l) = Re zi (l) · zi+1 (l) − pi (l) · pi+1 (l) Dsub = [1]

(4)

Filling in the affine functions of poles and zeros in each subsystem, results in polynomial varying state-space matrices. Equation (5) illustrates this by filling in the affine functions of poles and zeros (2) in the A -matrix of subsystem (4) :

p0,i + p0,i+1 −p0,i · p0,i+1 + Asub (l) = Re 1 0 p + p1,i+1 −p1,i · p0,i+1 − p0,i · p1,i+1 f1 (l) · Re 1,i + 0 0 0 −p1,i · p1,i+1 2 f1 (l) · Re 0 0

(5)

Only additions and multiplications of maximum two poles and/or zeros are needed to calculate the elements of the A,B,C,D matrices of the subsystems. Because there are multiplications of two poles or zeros, products of the analytical function in the expression of the varying state-space matrices occur, as can be seen in (5). Since the concatenation of two subsystems can be written as :

0 Asub1 (l) A(l) = Bsub2 · Csub1 (l) Asub2 (l) Bsub1 B= Bsub2 · Dsub1 C(l) = Dsub2 · Csub1 (l) Csub2 (l)

(6)

D = [Dsub1 · Dsub2 ]

there are no multiplications of two parameter-dependent matrices, so the resulting varying controller will have the same parameter dependence as the subsystems. Each matrix contains only real elements. The number of columns and rows of the A-matrix is the number of complex conjugated poles of the system times two plus the number of real poles. The coefficients of the varying terms of the B- and D-matrices will always be zero, as in (4). This means that eight constant matrices together with one analytical function are sufficient to implement the gain-scheduling controller.


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2.4 Characteristics of the proposed technique compared to other existing techniques This method fulfils the three requirements formulated in the beginning of this section. Because smooth scheduling is done on the poles and zeros, only stable poles of the controller can be imposed. No approximation errors are made to convert the pole-zero representation to state-space, hence there will be no risk of loss of stability or performance in intermediate operating points. Only basic calculations are needed for the digital implementation of the gain-scheduling controller, since a simple polynomial parameter-dependent state-space representation is proposed. Being able to observe the variation of the poles is a benefit in comparison with a direct interpolation between state-space matrices of controllers designed for fixed operating points. Directly interpolating between the state-space matrices can lead to unstable controllers for intermediate operating points, due to the fact that there is usually no direct control over the variation of the poles and zeros. The interpolation between controllers then boils down to a trial-and-error procedure of finding the state-space representation of the controllers that have the best behavior for intermediate operating points. In [7] for example, it is observed that an interpolation between the parameters of controllers in observable canonical form leads to instability in certain operating points, but an interpolation between the controllers in delta-operating form [10] results in a stable system in each operating point. The proposed interpolation technique puts no restriction on the number of poles and zeros of the local controllers, so that the method can be applied to complex mechatronic systems. This is in contrast to methods that use the transfer function form [4], [13], which can be numerically ill conditioned [11] for higher-order LTI-systems. Small errors on the coefficients of the denominator polynomial, can result in large differences of the pole locations. The sensitivity grows if higher-order systems are used and particularly when multiple poles occur. This procedure also has its restrictions. The algorithm is only valid for SISO-systems. MIMO-systems will have to be split up in SISO-systems from each input to each output. Another situation that cannot be handled yet is where complex poles become real or vice versa. Further research on this topic is needed. When the variation of the poles, zeros and gains cannot be described accurately enough by an affine function with only one varying term, then the algorithm can be expanded with more varying terms. This way the variation of the real parts of the poles and zeros can be modeled differently then the variation of the imaginary part of the poles and zeros, which leads to arbitrary variations of the poles and zeros in the complex plane. The number of constant matrices then grows rapidly. The same expansion can be used if there is not one but two scheduling signals. Then the first analytical functions describes the variation as a function of the first scheduling signal and the second analytical function describes the variation as a function of the second parameter.

3 Control design for a practical setup The first part of this section describes the layout and modelling of a pick-and-place machine. Then controllers are derived based on this model for different operating points.

3.1 Description of the setup The considered test-case is an industrial 3-axis pick-and place machine shown in Fig. 1 . The Y-motion is gantry driven by two linear motors and the X-motion of the carriage over the gantry is also driven by a linear motor. The vertical Z-motion is actuated by a rotary brushless DC-motor which drives a vertical beam by a ball screw/nut combination. The position of the linear motors and the length of the beam are measured with

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Figure 1: Picture of the setup optical encoders and the acceleration of the end point of the beam in the X-direction, representing the tool tip in this setup, is measured with an accelerometer. The objective is to move the end point of the beam as accurate and fast as possible along a prescribed trajectory in the X-Y-Z plane. Fast movements of the linear motor will excite the eigenfrequencies of the flexible beam and during motion, the length of the beam is continuously changed, giving rise to varying resonance frequencies. This paper focuses on motion control in one direction, namely in the X-direction, for different fixed values of the length of the beam in Z-direction.

3.2 Modelling of the setup The setup is identified for 4 different lengths of the beam. SIMO-models are derived in the X-direction. The model has one input, the force of the motor, and two outputs: the motor position, which is a collocated measurement,is the first output and the position of the end-point of the beam, which is a non-collocated measurement, is the second output. On the setup, the position of the end-point of the beam is obtained by integrating the accelerometer twice. The integrator is combined with a high-pass filter with a cut-off frequency of 3 Hz, because no reliable signal can be obtained under that frequency. The modelling is based on frequency response function measurements using multi-sine excitation [8]. The details of the modelling are not presented in this paper. The four SIMO-models are each of tenth order. Fig. 2 shows the FRF’s of the identified models for four different beam lengths. A mass-line characteristic (-40dB/decade) is recognised at low frequencies and a varying resonance at higher frequencies, which is the first eigenfrequency of the beam. At the bottom position of the beam this eigenfrequency is around 30 Hz and in the highest position it is around 70 Hz. There is also a fixed resonance around 125 Hz, which is most likely a machine-frame resonance.

3.3 Control design Both the collocated and the non-collocated measurement in the X-direction are taken as input for the controller. The resulting MISO controller will therefore have two inputs and one output. This controller is then split up in two SISO controllers, so it can be used to make two seperate gain-scheduling controllers by applying the proposed interpolation technique. Finally, the output of both gain-scheduling controllers have to be summed up.


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Bode Diagram 60

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Magnitude (dB)

Magnitude (dB)

Bode Diagram

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20 0 −20 −40 l=8 l=10 l=15 l=20

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Phase (deg)

0

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−90 −135 2

−270 −360 −450 2

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10 Frequency (rad/sec)

Frequency (rad/sec)

Figure 2: FRF’s of the identified state-space models for four different values of the length of the beam; left: from motor force (N) to encoder position (m); right: from motor force (N) to the position of the end point of the beam (m) The goal of this design is to obtain a stable controller that has good tracking performance of the end-point of the beam and good disturbance rejection with respect to external forces at the end up to a frequency of 20 Hz. H∞ -techniques are used to design the MISO-controllers. The method that is used to choose the weighing functions and the inputs and outputs of the general control configuration is described in [9]. Weighting functions are proposed for which only a few design parameters have to be specified for each control loop: 1. the target bandwidth, which is defined as the frequency where the open-loop gain first crosses unity from above; 2. the frequency below which the controller must have integral action to suppress low-frequency and constant disturbances; 3. the frequency beyond which the controller must roll-off, to suppress noise and to achieve robustness against high-frequency model uncertainty. These design parameters have a clear interpretation, which greatly facilitates their tuning. The left part of Fig. 3 shows the first set of controllers, which has the position error of the motor encoder as input and the force of the motor as output. The right part of Fig. 3 shows the second set of controllers, which has the position error at the end-point of the beam as input and the force of the motor as output. The single SISO-controllers for different lengths of the beam have each 12 states. The number of poles and zeros are reduced manually for each SISO-controller to 9 poles and 8 zeros by deleting the higher order poles and zeros and near pole-zero cancellations. The derivation of the global gain-scheduling controller by interpolating between the local controllers for this setup is presented next.

4 Application of the interpolation method 4.1 Derivation of the gain-scheduling controller The interpolation method explained in section 2 is applied to schedule the four controllers created in the previous section. The first step is to fit affine functions through the poles, zeros and gains of these controllers.

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Bode Diagram

Bode Diagram Magnitude (dB)

Magnitude (dB)

−40 −45 −50 −55 −60

−45 −50 −55 −60 −65 −70 180

Phase (deg)

Phase (deg)

90 0 −90 2


l=20 l=15 l=10 l=8

0 −180 −360 −540

2


Figure 3: FRF’s of controllers for four different values of the length of the beam; left: from position error of the motor encoder (meter) to the motor force (N); right: from position error at the end-point of the beam (m) to the motor force (N)

Figure 4: Pole-zero-map of some of the varying poles and zeros of the controller + the interpolating functions An affine interpolation with f (l) = l gives a good fit. Fig. 4 shows a part of the pole-zero-map of the four fixed controllers and the resulting interpolation. It can be seen that the limitation of linear variations in the complex plane is a good approximation in this case. To evaluate the proposed interpolation, fixed controllers are designed for intermediate operating points using this linear interpolation. The combination of these controllers with the local models still fulfill the design specifications. Therefore the proposed linear interpolation is accurate enough. The transformation to state-space representation yields 16 constant matrices, 8 for each varying SISOcontroller: the A- and the C-matrices each consist of a sum of three terms (see (3)) and the B- and D-matrices are parameter independent. The controller parameters are now directly adapted by varying the length of the beam l in (3). Fig. 5 shows the FRF’s obtained by evaluating the gain-scheduling controllers for 10 different values of the scheduling parameter, covering the whole parameter range.


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Bode Diagram Magnitude (dB)

Magnitude (dB)

−40 −45 −50 −55 −60

−50 −60 −70

−65 −80 90 Phase (deg)

Phase (deg)

45 0 −45 −90 −135 2

0 −90 −180 −270 −360 −450 −540


0


Figure 5: Range of fixed controllers obtained by evaluating the gain-scheduling controller for 10 different values of the scheduling parameter step response of the motor: experiment (blue) versus simulation (red)

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0.5 0.4 0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4 0.5 time [s]

0.6

0.7

0.8

Figure 6: Response of the motor on a step in the setpoint: experiment versus simulation (l = 20 cm)

4.2 Experimental results To evaluate the gain-scheduling controller, two types of experiments are made. First the dynamic behaviour of the system is analysed in several intermediate operating points. A good gain-scheduling controller preserves the performance and stability of the closed loop in all intermediate operating points. Secondly the controller is analysed during trajectories of the parameter, which will only be briefly discussed. The first experiment shows the performance of the controller designed for a fixed length of the beam (l = 20 cm). The response on a step command can be seen in Fig. 6. The corresponding acceleration signal can be seen in Fig. 7. In both figures also the response is shown of the simulated closed-loop in red. There is a good correspondence between the simulations and the experiments. In Fig. 8 the disturbance rejection at the end-point of the beam can be seen. A comparison is made between the time-constants of the exponential decay of the vibrarion at the end-point of the beam with and without control. Two exponential functions are fitted on this data. Without control the time constant of the exponential decay is 0.15 seconds, with control this is reduced to 0.075 seconds. There is an important trade-off in the design of the controller: good command following behaviour without

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0.15

0.2

0.25

0.3 0.35 time [s]

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Figure 7: Response of the end-point of the beam on a step in the setpoint: experiment versus simulation (l = 20 cm) Acceleration during a force disturbance 60

acceleration [m/s2]

40 20 0 −20 −40 −60 3.6

3.8

4

4.2 time [s]

4.4

4.6

Figure 8: Acceleration on the setup of the end-point of the beam after a force disturbance: with control (red) versus without control (blue) overshoot is contrary to disturbance rejection with respect to external forces at the end-point of the beam. In this case, more weight is put on the disturbance rejection at the end-point of the beam, resulting in a rather large overshoot of the motor when a step is applied. Next, the gain-scheduling controller is analysed in intermediate operating points. As can be seen in Fig. 11 the performance of the gain-scheduling controller (expressed as the time-constant of the exponential decay of the vibrations) remains constant for different lengths of the beam. In the same figure, also the performance of a LTI-controller designed for a fixed length (l = 16 cm) is visualised for different lengths of the beam, showing that the closed-loop becomes unstable outside a certain interval. Also the natural exponential decay without control is depicted. It can be seen that the natural damping increases for higher values of the length of the beam. This can also be seen in the transfer functions of the models (Fig. 2). The second type of experiments evaluates the performance and stability of the closed-loop during trajectories of the parameter. In Fig. 9 the motor position and the acceleration of the end-point of the beam can be seen as a response on an agressive setpoint (with continuous djerk). During the motion in the X-direction there is no motion in the Z-direction. The same responses can be seen in Fig. 10, where there is simulanuous motion


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470 460 450 440 430 420 410 400

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Figure 9: Left: acceleration of the end-point of the beam; right: (blue) motor position [µ], (red) length of the beam (is scaled to fit the picture). 480 motor position [mm] (blue), Z−position (red)

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Figure 10: Left: acceleration of the end-point of the beam; right: (blue) motor position [µ], (red) length of the beam (is scaled to fit the picture). in X- and Z-direction. Other experiments with simultanous movements in the X- and Z-direction confirm that the performance remains constant during variation of the parameter. Simulations have shown that the closed-loop becomes unstable only for sinusoidal variations of the parameter with a frequency around 88 Hz. These reults clearly show that for this application traditional gain-scheduling is an appropriate solution. First results with LPV controllers designed for the same setup show a substantial amount of conservatism.

5 CONCLUSIONS This paper presents an interpolation method for the gain-scheduling control of mechatronic systems with position-dependent dynamics. The proposed interpolation method starts with fitting affine functions on the variation of the poles, zeros and gains of controllers designed in fixed operating points. The next step is an exact transformation of these functions to a polynomial state-space representation. This is a smooth interpolation method that can be applied to schedule complex linear controllers while conserving stability and performance in intermediate operating points.


time constant of the exponential decay (seconds)

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0.18 0.16 0.14 0.12 0.1 0.08 0.06

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14 15 16 17 18 length of the beam (centimeter)

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Figure 11: Time constants of the exponential decay after an excitation at the end-point of the beam for different lengths of the beam This technique is applied on an industrial pick-and-place machine that has position-dependent dynamics. The experimental results confirm that the performance obtained in fixed working points can be expanded to the complete range of operating points. In further research, an extensive comparison will be made with a LPV controller. The derived interpolation method is also extended so it can be used for LPV identification of models with an affine state-space dependence.

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