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Design and experimental validation of robust controllers for machine tool drives with linear motor
Mechatronics 11 (2001) 545±562
Design and experimental validation of robust controllers for machine tool drives with linear motor P. Van den Braembussche, J. Swevers*, H. Van Brussel K.U. Leuven, Department of Mechanical Engineering, Division PMA, Celestijnenlaan 300 B, 3001 Heverlee, Belgium Received 15 March 1999; accepted 11 November 1999
Abstract This paper discusses the design and experimental validation of two controllers for a prototype machine tool axis, actuated by a linear motor: an H1 controller and a discretetime sliding mode controller. Robustness of the tracking performance with respect to load changes up to 300% is considered. Experiments show that the discrete-time sliding mode controller is robustly performant in this load range. The H1 controller based on standard weighing functions is not robustly performant for the considered range of load changes. This paper proposes an alternative performance weighing function, inspired by the sliding mode controller, which yields an H1 controller with improved robust performance, comparable to that of the sliding mode controller. 7 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction Direct drive design using linear motors is widely used in high-precision component placement machines, e.g. in semiconductor industry where only pointto-point movements have to be considered. The use of linear motors is also very * Corresponding author. Fax: +32-16-322987. E-mail address: [email protected] (J. Swevers). 0957-4158/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 5 7 - 4 1 5 8 ( 0 0 ) 0 0 0 1 3 - 1
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promising for tracking applications, like laser cutting and high-speed milling [2,3,22]. Due to the lack of a transmission unit, the tracking behaviour of a direct drive design is very prone to disturbances and model parameter variations [24]. Therefore, much attention must be paid to the controller design. Beside the inherent model inaccuracies due to approximation of a physical system by a reduced-order model, the model of the machine drive can also have an uncertain gain factor due to an uncertain load of the workpiece. The mass to be moved by the drive motor not only changes with the mass of the workpiece, but can also depend on the position of the machine table in its workspace. For example, in a gantry machine with two parallel X-axis motors, the load distribution between the two motors depends on the position of the Y-axis carriage. Moreover, there is a trend towards modular machine design, where modular axes, containing all the mechanics, electronics and software, are used as building blocks and combined into various machine con®gurations. Since the axis module can be used as X, Y or Z-axis, the load is not known in advance and the motion controller of such modular machines has to be made robustly performant for a broad range of loads. Alter and Tsao [2] have studied the control of a linear motor axis taking into account a model of the cutting process, in order to investigate the chatter behaviour. They have shown that minimising chatter is equivalent to maximising servo-dynamic stiness. Therefore, design for good nominal performance (good disturbance rejection) leads to reduced chatter. In [3], an H1 controller is designed for a linear motor axis. There, the authors focus on nominal performance, not on robustness. In this paper, the design of an H1 and a sliding mode controller for a prototype machine tool axis with a linear motor is discussed and the robustness against load variations is compared experimentally. The performance and robustness of an H1 controller depends on the selection of the weighing functions. Generally, the performance is obtained by shaping the performance weighing function, such that the sensitivity function is small in the low-frequency range. Robustness is then obtained by selecting the robustness weighing function in order to make the complementary sensitivity function small enough in the highfrequency range to allow for the expected model uncertainty. LundstroÈm et al. [20] and Postlethwaite et al. [21] give guidelines for the selection of these weighing functions. In these guidelines, the mid-frequency range, which is the range around the cross-over frequency, is often not considered. This mid-frequency range is however very important for robustness [19,21]. In this mid-frequency range, neither the sensitivity nor the complementary sensitivity function is small. Here, the closed-loop system behaviour is determined by a combination of the performance weight and robustness weight. This interaction between both weights is generally neglected in H1 design. For an H2 controller, this interaction has been considered by Jovik and Lennartson [17], where they gain robustness, introducing loop transfer recovery (LTR), by a proper choice of the robustness weighing function. This paper considers the interaction between performance and robustness weights in H1 control design for the above mentioned application, by
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proposing an alternative approach for the selection of the weighing functions, yielding improved robust performance. Other techniques to select the performance weighing function are given by Franchek [14] and Yang et al. [28]. Franchek [14] describes a method that allows to extract the performance weights from certain speci®c time domain constraints, such as the maximum value of the response to a step disturbance, and actuator eort constraints. Yang et al. [28] use matrix experiments as used in quality control to determine the eect of the various parameters in the weighing function and to ®nd the optimal parameter settings. However, neither method leads to generally applicable design guidelines. In a continuous-time sliding mode controller, the system is requested to move on a sliding surface [25]. A discontinuous controller forces the system on this sliding surface. The parameters of the sliding surface de®ne the system's closedloop dynamics. Drazenovic [12] showed that the system in the sliding mode is invariant against a certain class of parameter uncertainty and disturbances. However, this invariance property only holds when the system is moving exactly on the sliding surface. In practical applications, the theoretical invariance property is lost, due to actuator limitations, system delays or the use of observers. Also, for discrete-time systems, this theoretical invariance is lost. A sampleddata system will seldom be able to move exactly on the sliding surface, it will rather move zigzag around it and towards the origin. Sarpturk et al. [23] design a controller, based on a discrete-time equivalent of the continuous-time reaching condition. They show that this imposes an upper and a lower bound on the control signal. However, these bounds converge, when approaching the sliding surface, which would require a priori knowledge of the uncertainties and disturbances [18]. Furuta [15] and Wang et al. [27] propose a method that relaxes the reaching conditions in a neighbourhood of the switching surface. The reaching law method of Gao et al. [16] starts from the desired dynamics of the switching variable (the distance from the switching surface) in the discrete-time domain. This method allows to analyse the robustness of the motion around the switching surface [5]. Utkin [26] designs a discrete-time sliding mode controller, based on a new de®nition of the discrete-time sliding mode. In contrast to the earlier methods, this control algorithm excludes discontinuities in the control law, and thus chattering. In milling applications, chattering leads to high-frequency errors on the surface, which are clearly visible and not acceptable.Therefore, the sliding mode controller presented in this paper, uses a discrete-time design approach. The designed controllers are experimentally tested on a prototype machine tool axis actuated by a linear synchronous motor. The aim is to achieve a maximum tracking error below 10 mm for a broad range of loads. This desired accuracy is typical for many applications, e.g. high-speed milling. Sections 2 and 3 brie¯y review the H1 control and the discrete-time sliding mode control design approaches, respectively. Section 4 presents the experimental test set-up and discusses the design of two dierent controllers: an H1 controller, and a discrete-time sliding mode controller. Section 5 discusses the experimental
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performance validation of these controllers in the presence of load variations. Section 6 proposes a novel sliding-mode-inspired selection of weighing functions for the H1 design, which yield better robustness against load changes. 2. H1 control design Fig. 1 shows a typical H1 design scheme, where e, u, y, yref, d and di represent the tracking error, the controlled system input, the system output, the reference trajectory, the output disturbance and the input disturbance, respectively. G(s ) and C(s ) represent the system and controller, respectively, with s the Laplace variable. Wp(s ), Wu(s ) and Wr(s ) are H1 design weighing functions which will be explained below. y1, y2, and y3 are the weighed tracking error, control input, and undistorted system output, respectively. Minimizing the H1 norm of the transfer functions of these weighed signals is the goal of the H1 control design. The H1 norm of the system sensitivity function S 1= 1 GC , represented by kSk1 , is a measure for the tracking and disturbance rejection performance of the closed-loop system, since S relates the disturbance d to the system output, y, as well as the reference input, yref, to the tracking error, e. This means that a small kSk1 is desirable for good disturbance compensation and tracking behaviour. It can be easily derived from Fig. 1 that the in¯uence of the input disturbance, di, is measured by kGSk1 : Usually, a controller that minimises kSk1 for good rejection of output disturbances, d, will also minimise kGSk1 , since G is constant. The robust stability for unstructured multiplicative uncertainties can be measured by the H1 norm of the system complementary sensitivity function T GC= 1 GC : The Small Gain Theorem yields that for multiplicative uncertainties Dm , the perturbed system, G 1 Dm , is stable if kDm k1 < 1=kTk1 [8]. This yields that large multiplicative uncertainty, kDm k1 , is allowed, when kTk1 is small. For additive uncertainties, Da , the perturbed system, G Da , is stable if kDa k1 < 1=kRk1 , with R C= 1 GC : Since S + T = 1, it is not possible to make kSk1 and kTk1 both arbitrarily small for good nominal performance and for robust stability, simultaneously.
Fig. 1. H1 design scheme.
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Therefore, the H1 control synthesis procedure usually minimises a weighed sensitivity, kWp Sk1 , and weighed complementary sensitivity function, kWr Tk1 , where Wpÿ1 represents the desired frequency-dependent disturbance attenuation factor and Wr an upper bound on the expected multiplicative uncertainties. Typically, Wpÿ1 is small at low frequencies, where good nominal performance is desired, and Wrÿ1 is small at high frequencies, where large model uncertainties are expected. kWu Rk1 should be considered in the control design objectives, when additive uncertainties are expected. Since R is also the transfer function from the reference trajectory and the output disturbance to the control input signal, u, its norm measures the required control eort. Therefore, Wu represents the expected additive uncertainties or a weight allowing to account for system input limitations. For robust performance, the weighed mixed-sensitivity design approach [6] can be used, where the control synthesis procedure aims at simultaneously optimising nominal performance and robust stability, by minimising the following norm of stacked weighed transfer functions:
Wp S
Wu R : 1
Wr T 1
Theoretically, robust performance is only guaranteed when Eq. (1) is smaller than p 1= 2: The robustness of the H1 controller is not inherent to the synthesis design itself. The robustness, as well as the performance, result from the weighing functions chosen in the synthesis. The diculty of designing a robust controller with H1 synthesis lies therefore in the selection of the weighing functions. This will be discussed in Section 4.
3. Sliding mode control design The discrete-time sliding mode control method of Utkin [26] has been used here. This method is based on the de®nition of discrete-time sliding mode [11]. Consider a linear discrete-time SISO system, described by an nth order state space model: x k 1 A x k B u k ,
2
with x k, x k 1 2 Rn the system states at time instant k and k 1: u k 2 R is the system input, and A and B are the system and input matrices of the state space model, respectively. For a SISO system, the sliding surface is reduced to a line, which can be described as ÿ 3 s k Csl x k ÿ x ref k 0, where x ref k 2 Rn are the reference states and the scalar s k is a measure of the
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distance from the sliding line. The slope of the sliding line, Csl 2 Rn , is designed such that the system has the desired closed-loop dynamics, when moving on the sliding line. The control input u k that will bring the system to the sliding line at moment k 1 is found as the solution of the equation s k 1 0:
4
Combining Eqs. (2)±(4) yields: u k
Csl Csl A x ref k 1 ÿ x k Csl B Csl B
5
Combining Eqs. (2) and (5) yields the state matrix of the closed-loop dynamics: Acl A ÿ B
Csl A : Csl B
6
The sliding manifold (4) de®nes 1 pole of the closed loop system at z = 0. This can easily be seen as follows. If the matrix A is singular, it is obvious from Eq. (6) that Acl will also be singular, thus having an eigenvalue at zero, i.e. a pole at z = 0. If A is not singular, the vector: X A
ÿ1
B,
7
is an eigenvector of Acl with corresponding eigenvalue l 0: Acl X lX 0X:
8
The other n ÿ 1 poles in Eq. (6) are de®ned by the selection of the slope of the sliding line, i.e. Csl : The method can also be seen as a special case of the reaching law method [16], where a controller is designed that results in the reaching law s k 1 1 ÿ qTs s k ÿ ETs sign s k ,
9
with Ts the sampling period. Choosing qTs 1 and ETs 0, yields Eq. (4). Fig. 2 shows the resulting controller structure. This ®gure shows that the states
Fig. 2. Controller scheme of sliding mode controller.
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are estimated by means of a state observer, which is necessary when the state variables are not directly measurable. This is the case for the experimental system described below. It should be remarked that state observers can cause loss of robustness [9]. However, the experiments presented in Section 5, give no indication of this. 4. Controller design for a machine tool axis with linear motor 4.1. Experimental set-up Fig. 3 shows the experimental set-up, representing a prototype machine axis. It consists of a linear synchronous motor operating in current control mode. The linear motor consists of a series of magnets ®xed to the machine base, and a moving part which contains the coils. The carriage connected to the moving part of the motor is supported on two rolling element slideways. The position of the carriage is measured by means of a magnetic linear scale on one of the slideways with an incremental measurement system with a resolution of 1 mm. The total mass of moving part is around 12 kg. The control unit is a dSPACE digital signal processing (DSP) board using a TMS320-C31 processor, equipped with a 12-bit D/A converter (settling time is 4 ms) and a 24-bit incremental encoder input [1]. The sampling frequency is selected to be 2200 Hz. The dynamics of the system, relating system input signal, u [Volt], which is proportional to the motor current, to system output, y [mm], have been identi®ed experimentally using broad-band multisine excitation up to 220 Hz, and a nonlinear least-squares frequency-domain identi®cation method [29] (Schoukens and Pintelon, 1991). The obtained discrete-time model equals: G z
ÿ1:7429 : z ÿ 1 z ÿ 0:9981
10
It describes the system dynamics, including the zero-order-hold eect of the digital control hardware, accurately (Fig. 4). This model is a compromise between accuracy (the model is accurate only up to 220 Hz) and complexity. The ®rst
Fig. 3. Experimental test set-up.
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Fig. 4. Measured frequency response function and the frequency characteristic of the identi®ed model (a) amplitude, (b) phase.
structural resonance frequency is at 300 Hz and therefore, not included in the model. The model, (10), contains a pole at z = 1, due to integration in the relation between force (input) and position (output). Robustness of the controllers is checked against load variation, which is accomplished by attaching extra masses of 7 kg to the carriage. The masses are added while the system is not operational. 4.2. Design of the H1 controller The design of the H1 controller is an iterative procedure in the continuous-time domain, i.e. based on the continuous-time equivalent of G(z), cfr. Eq. (10). The obtained controller will then be transformed to the discrete-time domain using a bilinear transformation. This transformation maps the closed unit disk in the zplane onto the closed left-half s-plane. The obtained performance is not lost, since the bilinear transformation is norm-preserving. The design of the H1 controllers does not take into account any uncertainty model. The only uncertainty that is considered in this paper is the mass of the load carried by the motor. Other system uncertainties are negligible with respect to the large mass variations considered (up to 300%). The mass variations results in an uncertainty on the system gain. The corresponding uncertainty model for the H-in®nity control design would therefore be a constant but complex number over all frequencies, which would lead to a very conservative controller with very bad performance, since the actual uncertainty is real, not complex. As a result, it is not appropriate to include such an uncertainty model in the control design. Therefore, the weighing functions have been tuned experimentally on the nominal model, until a sucient tracking performance is obtained. Fig. 5 shows the resulting third-order weighing function Wp : ÿ 1:2 s 350 s 2 2 0:7 350 s 350 2 : 11 Wp s 2:5 s 2 2 0:7 2:5 s 2:5 2
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Fig. 5. Wp (solid) and S (dashed) obtained with the H1 controller.
Fig. 6 shows the robustness weighing function Wr , which consists of a ®rst-order roll-o at high frequencies: Wr
s : 2200
12
Wu is chosen to be a constant; small enough not to in¯uence the design, e.g. 0.5. In case actuator saturation is observed during the experiments, Wu should be increased. These weighing function shapes are common in H1 control design. S is shaped by Wp resulting in low sensitivity at low frequencies. As a result, T will be very close to one at these frequencies. The weighing function Wr will force T to have a ÿ20 dB/decade roll-o above 350 Hz. The continuous-time equivalent of Eq. (10), which is used for control synthesis, contains a pole at the origin. However, the standard H1 synthesis procedure of Doyle et al. [10] does not allow this. The use of a bilinear transformation, as
Fig. 6. Wr (solid) and T (dashed) obtained with the H1 controller.
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proposed by [6], avoids this problem: s~
ÿs p1 : s=p2 ÿ 1
13
This method ®rst shifts the poles of the system to the left (e.g. p1 = ÿ1 and p2 = 1), then calculates the controller and shifts the resulting controller poles back to the right. This controller will then compensate this pole at the origin with a zero. However, a pole at the origin represents an integration in the system, which is required in most motion control applications to remove steady-state errors. The following two properties [6] lead to an H1 synthesis procedure that can handle system poles at the origin without removing them by pole-zero cancellation. The resulting controller contains integration and thus yields perfect compensation of steady state errors. Property 1: In the weighed mixed-sensitivity formulation, any unstable pole of the plant inside the speci®ed control bandwidth will be shifted approximately to its jo-axis mirror image, once the feedback loop is closed with an H1 controller. Property 2: If the controller C(s ) for G(s ) is obtained from the transformation of C(sÄ ), using the inverse of Eq. (13), where C(sÄ ) is synthesised for the bilinear transformation G ~s of the system model G(s ) using Eq. (13), it is guaranteed that the following inequality holds:
ÿ
T jo R ~ ,
T~ jo 14 1 1
with T the complementary transfer function of the system, G(s ), and TÄ the complementary transfer function of the bilinearly transformed system G(sÄ). This ensures that the ®nal s-plane controller C s will at least meet the robustness speci®cations requested from C ~s and will probably exceed it. These properties yield the following design procedure: A.1. The continuous-time equivalent G(s ) of the plant (10) is transformed using Eq. (13) with p1 = 1 and p2 = 1 towards G(sÄ). This transformation shifts the system poles to the right over 1 rad/s, yielding that the pole in the origin shifts to a pole at 1. A.2. The controller is designed using Wp, Wu and Wr using the mixedsensitivity approach (1). The pole of G(sÄ) at 1 is compensated by a zero of the controller C(sÄ) at ÿ1. A.3. The poles and zeros of the controller are shifted to the left over 1 rad/s, yielding a zero at ÿ2. A.4. The closed-loop system includes integration (due to the pole of G(s ) at s = 0) until 2 rad/s (due to the zero of C(s ) at s = ÿ2), as can be seen in Fig. 5: the sensitivity function S has a slope of +20 dB at low frequencies. A.5. The obtained controller is transformed to the discrete-time domain for implementation.
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Although H1 control design can also be accomplished directly in discrete-time, the described continuous-time design was preferred, since this can easily be implemented using commercial software packages, such as the MATLAB m-Tools [4] or Robust Control Toolbox [7]. Figs. 5 and 6 show the sensitivity and complementary sensitivity function of the resulting closed-loop system. Fig. 5 shows that this H1 controller yields a sensitivity function, S, with zero DC gain. This means that the closed loop system has a perfect asymptotic compensation of step disturbances at the system output, d, and perfect asymptotic tracking of step trajectories. Eq. (18) in the appendix of this paper shows the discrete-time transfer function of the H1 controller.
4.3. Design of the sliding mode controller The sliding mode controller selected for the test set-up is based on Eq. (4), which yields equivalent control law Eq. (5) [15]. It is a special case of the reaching law method [16] (Eq. (9)) as explained in Section 3. Its design consists of the selection of the slope of the sliding line Eq. (3), which determines along with the choice of the reaching law Eq. (9), the poles of the closed-loop dynamics Eq. (6). First, the system model (Eq. (10)) is transformed into a state space model (the controller canonical form) and extended with a third state variable, which corresponds to the (discrete) integration of the output. This ensures integral feedback of the position and therefore, removes steady-state errors. The resulting third order model equals: 2
1:9981 x k 1 4 1 0
y k 0
ÿ 1:7429
ÿ0:9981 0 ÿ1:7429
3 2 3 0 1 0 5x k 4 0 5u k 1 0
0 x k
15
Reaching law (4) de®nes one closed-loop pole at z = 0 (18). The remaining twoclosed loop poles are determined by selecting the coecients of Csl : using a regular form based approach [13]. This approach is based on the canonical form of the state space model, also called the regular form, calculates the last two coecients of Csl using any common pole placement algorithm, and selects the ®rst coecient of Csl equal to one [13]. These two closed-loop poles are selected experimentally at 10 Hz and 100 Hz, yielding: Csl 1
ÿ 0:7234
ÿ 0:0040
The precise choice of these poles is not critical.
16
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5. Robustness against load variation Both the sliding mode and the H1 controller have been implemented on the linear motor axis, described in Section 4.1. The moving mass of the nominal system is around 12 kg. The robustness of the controllers is tested for increasing mass up to 35 kg, by incrementally adding masses of 7 kg, while the system is not operational. The linear synchronous motor of the test set-up has a signi®cant motor ripple resulting in a deterministic and periodic tracking error. In the experiments, this has been compensated by an appropriate ripple model based feedforward [24], in order to give a realistic representation of the tracking accuracies achievable for this test set-up. Fig. 7 shows the tracking error for the H1 controller for the nominal system, i.e. without extra mass. The reference trajectory is a 9th-order polynomial. The maximum velocity is 0.5 m/s and the maximum acceleration is 2 m/s2. The maximal acceleration of the trajectory is rather low compared to the maximum achievable acceleration (40m/s2 for the nominal system). This is to avoid actuator saturation when the extra masses are added, and to better demonstrate the robust performance of the feedback. If a trajectory with higher acceleration would be chosen, e.g. 5 m/s2, the maximum tracking error is determined by the tracking error during acceleration and deceleration, which is de®ned by the feedback and the feedforward. The feedforward becomes very inaccurate for a large amount of extra mass, since it is based on the nominal system. This shows that for good tracking behaviour for all cases, an adaptive feedforward is required. Since this paper focuses on the feedback controller, a slower trajectory is chosen, which shows better feedback properties of the overall system, like disturbance compensation (ripple, friction, etc.). Fig. 8 shows the maximum tracking error for the dierent controllers for the load increasing in steps of 7 kg until 35 kg. The sliding mode controllers and the H1 controller have the same nominal performance. The maximum tracking error
Fig. 7. Nominal tracking error for the sliding mode controller.
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Fig. 8. Comparison of robustness: maximum tracking error for dierent load.
for the nominal system, i.e. for zero extra load, is 7 mm. The sliding mode controller is stable for all extra masses, without increase in maximum tracking error for increasing load. The H1 controller reaches the edge of stability for 35 kg added. The tracking error shows an undamped oscillation for this load. Unlike most sliding mode controllers which have a non-linear term, the applied sliding mode controller is a linear controller. The only dierence with a standard state space controller lies in the selection of the poles. This allows to use standard techniques to compare the control performance, like Bode and Nyquist plots. Fig. 9 shows the Nyquist plot for the original system without extra mass, connected to the H1 controller and to the sliding mode controller. Both open loop systems have approximately the same phase margin. However, the sliding mode controller appears to allow much more decrease in system gain, i.e., has a larger gain margin, therefore allowing much more increase in system mass. The relatively poor robust performance of the H1 controller against load changes, is a result of the selected weighing functions. Section 6 proposes an adapted selection of the weighing functions leading to better robust performance for the considered application.
Fig. 9. Nyquist plots for the nominal system (solid line: sliding mode controller, dashed line: H1).
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6. Adaptation of weighing functions for robust control against load changes The preceding section shows the dierence between both controllers by means of Nyquist plots. This dierence is also visible in the frequency response of the sensitivity and complementary sensitivity function (Fig. 10). Both sensitivity functions seem to have a comparable frequency response. Both have a slope of 60 dB/dec at low frequencies. However, close inspection at the crossing of the 0 dB line shows clear dierences: the slope of the H1 sensitivity function changes quite abruptly from 60 dB/dec to 0 dB/dec, while the slope of the sliding mode sensitivity function decreases ®rst to 20 dB/dec before reaching the 0 dB line, after which it reduces further to 0 dB/dec. This dierence in shape of the sensitivity functions is used to select an adapted weighing function Wp for the H1 synthesis. The weighing function Wp, is adapted such that its slope increases ®rst from ÿ60 dB/dec to ÿ20 dB/dec before crossing the 0 dB line, after which it further increases to 0 dB/dec. This shape is accomplished by the following transfer function: Wp
1:2 s 75 s 500 s 880 s 3:5 s 2 2 0:7 3:5 s 3:5 2
17
Fig. 11 shows the frequency response of the old and the adapted performance weighing function, Wp : Using the adapted Wp, (17), a new H1 controller can be designed, following the same procedure as in Section 4.3. Eq. (19) in the appendix of this paper shows the discrete-time transfer function of this new H1 controller. Fig. 12 shows the Nyquist plot of the loop transfer function of the nominal system with the sliding mode controller and with the new H1 controller. As expected, the Nyquist plots shows that the new H1 controller and the sliding mode controller have a comparable robustness.
Fig. 10. Sensitivity (left) and complementary sensitivity (right) for sliding mode controller (solid line), H1 controller (dashed line), weighing function, resp. Wp and Wr (dash-dotted line).
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Fig. 11. Old (dashed line) and new (solid line) weighing function Wp :
Fig. 12. Nyquist plots for the nominal system for the sliding mode (solid line) and new H1 controller (dashed line).
Fig. 13. Comparison of robustness: maximum tracking error for dierent load with adapted weighing function Wp.
Fig. 13 shows the experimental robustness results of the adapted H1 controller together with the results of the sliding mode controller. As in Fig. 8, it shows the tracking error for the given trajectory for increasing load.
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The H1 controller with adapted weighing function Wp is much more robust than the one with the standard weighing functions: the tracking error increases from 7mm for the nominal system to only 10 mm for 35 kg added. This is much better than the ®rst H1 controller, but not as good as the sliding mode controller. This dierence is caused by the dierence in their controller structure. The H1 controller has a more complex structure: its order is higher, which leads to extra zeros in the loop, which are unfavourable for the performance. The simplicity of the sliding mode controller results in better performance.
7. Conclusion Experiments on a prototype machine tool axis, actuated by a linear motor, show a discrete-time sliding mode controller is robustly performant with respect to load variations up to 300%. An H1 controller design using standard weighing functions is unstable for this load increase. This poor robustness can be improved signi®cantly by selecting dierent weighing functions. The proposed adapted performance weighing function is based on the shape of the sensitvity function of the sliding mode controller, and yields robustness comparable to that of the sliding mode controller without loss of nominal performance. This approach of selecting weighing functions (although probably not generally applicable) can inspire control engineers in applying H1 control theory successfully in practical situations, since this cannot be learned from control theory only.
Acknowledgements This research is carried out in the framework of BE-programme of the European Union, under the project KERNEL2, BE-7423. This text also presents research results of the Belgian Programme on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister's Oce, Science Policy Programming. The scienti®c responsibility is assumed by its authors.
Appendix A The discrete-time transfer function of the H1 controller discussed in Section 4.2 equals: ÿ0:2118z5 0:6071z4 ÿ 0:3704z3 ÿ 0:4170z 2 0:5756z ÿ 0:1836 z5 ÿ 1:7363z4 ÿ 0:4726z3 1:8462z 2 ÿ 0:3293z ÿ 0:3080
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The discrete-time transfer function of the H1 controller discussed in Section 6 equals: ÿ0:2358z5 0:6785z4 ÿ 0:4153z3 ÿ 0:4685z 2 0:6480z ÿ 0:2069 z5 ÿ 1:7883z4 ÿ 0:3618z3 1:8237z 2 ÿ 0:4087z ÿ 0:2649
19
References [1] DS1102 User's guide, dSpace digital signal processing and control engineering, Paderborn, Germany, 1992. [2] Alter DM, Tsao T-C. Stability of turning processes with actively controlled linear motor feed drives. Transactions of the ASME, J of Engineering for Industry 1994;116:298±307. [3] Alter DM, Tsao T-C. Control of linear motors for machine tool feed drives: design and implementation of H1 optimal feedback control''. Transactions of the ASME, J of Dynamic Systems, Measurement and Control 1996;118:649±56. [4] Balas GJ, Doyle JC, Glover K, Packard A, Smith R. m-Analysis and synthesis toolbox. Natick, Mass., USA: MUSYN and The MathWorks, 1991. [5] Bartoszewicz A. Remarks on Discrete-time variable structure control systems. IEEE Transactions on Industrial Electronics 1996;31(1):235±8. [6] Chiang RY, Safonov MG. H1 Synthesis using a bilinear pole shifting transform. Journal of Guidance, Control and Dynamics 1992;15(5):1111±7. [7] Chiang RY, Safonov MG. Robust control toolbox. Natick, MA, USA: The Mathworks, 1992. [8] Dailey RL. Lecture notes for the Workshop on H1 and m Methods for Robust Control. In: IEEE Conf. on Decision and Control, Dec. 9±10, Brighton, England. 1991. [9] Doyle JC, Stein G. Robustness with observers. IEEE Transactions on Automatic Control 1979;AC-24(4):607±11. [10] Doyle JC, Glover K, Khargonekar PP, Francis BA. State space solutions to standard H2 and H1 control problems. IEEE Trans on Automatic Control 1989;34(8):831±47. [11] Drakunov SV, Utkin VI. Sliding mode control in dynamic systems. Int J of Control 1992;55(4):1029±37. [12] Drazenovic B. The invariance conditions in variable structure systems. Automatica 1969;5:287±95. [13] Edwards C, Spurgeon SK. Sliding model control: theory and applications. London: Taylor and Francis Systems and Control Book Series, 1998. [14] Franchek MA. Selecting the performance weights for the m and H1 synthesis methods for SISO regulating systems. Transactions of the ASME, J of Dynamic Systems, Measurement and Control 1996;118:126±31. [15] Furuta K. Sliding mode control of a discrete system. System and Control Letters 1990;14:145±52. [16] Gao W, Wang Y, Homaifa A. Discrete-time variable sructure control systems. IEEE Trans on Industrial Electronics 1995;42(2):117±22. [17] Jovik I, Lennartson B. On the choice of criteria and weighing functions of an H2 =H1 -design problem. In: IFAC 13th Triennial World Congress, San Fransisco, USA. 1996. p. 373±8. [18] Kotta U. Comments on On the stability of discrete-time sliding mode control systems. IEEE Trans on Automatic Control 1989;34(9):1021±2. [19] Kwakernaak H. Robust control and H1 optimization Ð Tutorial Paper. Automatica 1993;29(2):255±73. [20] LundstroÈm P, Skogestad S, Wang Z-Q. Performance weight selection for H-in®nity and m-control methods. Trans Inst MC 1991;13(5):241±52. [21] Postlethwaite I, Lin J-L, Gu D-W. A loop-shaping approach to robust performance for SISO systems. Trans Inst MC 1991;13(5):262±8.
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[22] Pritschow G. Increasing accuracy through utilisation of linear direct drives. Manufacturing Systems 1996;25(2):117±20. [23] Sarpturk SZ, Istefanopulos Y, Kaynak O. On the stability of discrete-time sliding mode control systems. IEEE Trans on Automatic Control 1987;32(10):930±2. [24] Van den Braembussche P, Swevers J, Van Brussel H, Vanherck P. Accurate tracking control of linear synchronous motor machine tool axes. Mechatronics 1996;6(5):507±21. [25] Utkin VI. Variable structure systems with sliding modes. IEEE Trans on Automatic Control 1977;22(2):212±22. [26] Utkin VI. Sliding mode control in discrete-time and dierence systems. In: Zinober ASI, editor. Variable structure and Lyapunov control. Berlin: Springer±Verlag, 1994. p. 87±107. [27] Wang W-J, Wu G-H, Yang D-C. Variable structure control design for uncertain discrete-time systems. IEEE Transactions on Automatic Control 1994;AC-39(1):99±102. [28] Yang C-D, Tai H-C, Lee CC. Experimental approach to selecting H1 Weighing functions for DC servos. Transactions of the ASME, Journal of Dynamic Systems, Measurement and Control 1997;119:101±5. [29] Schoukens J, Pintelon R. Identi®cation of linear systems: a practical guideline to accurate modeling. Oxford: Pergamon Press, 1991.
Design and experimental validation of robust controllers for machine tool drives with linear motor P. Van den Braembussche, J. Swevers*, H. Van Brussel K.U. Leuven, Department of Mechanical Engineering, Division PMA, Celestijnenlaan 300 B, 3001 Heverlee, Belgium Received 15 March 1999; accepted 11 November 1999
Abstract This paper discusses the design and experimental validation of two controllers for a prototype machine tool axis, actuated by a linear motor: an H1 controller and a discretetime sliding mode controller. Robustness of the tracking performance with respect to load changes up to 300% is considered. Experiments show that the discrete-time sliding mode controller is robustly performant in this load range. The H1 controller based on standard weighing functions is not robustly performant for the considered range of load changes. This paper proposes an alternative performance weighing function, inspired by the sliding mode controller, which yields an H1 controller with improved robust performance, comparable to that of the sliding mode controller. 7 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction Direct drive design using linear motors is widely used in high-precision component placement machines, e.g. in semiconductor industry where only pointto-point movements have to be considered. The use of linear motors is also very * Corresponding author. Fax: +32-16-322987. E-mail address: [email protected] (J. Swevers). 0957-4158/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 5 7 - 4 1 5 8 ( 0 0 ) 0 0 0 1 3 - 1
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promising for tracking applications, like laser cutting and high-speed milling [2,3,22]. Due to the lack of a transmission unit, the tracking behaviour of a direct drive design is very prone to disturbances and model parameter variations [24]. Therefore, much attention must be paid to the controller design. Beside the inherent model inaccuracies due to approximation of a physical system by a reduced-order model, the model of the machine drive can also have an uncertain gain factor due to an uncertain load of the workpiece. The mass to be moved by the drive motor not only changes with the mass of the workpiece, but can also depend on the position of the machine table in its workspace. For example, in a gantry machine with two parallel X-axis motors, the load distribution between the two motors depends on the position of the Y-axis carriage. Moreover, there is a trend towards modular machine design, where modular axes, containing all the mechanics, electronics and software, are used as building blocks and combined into various machine con®gurations. Since the axis module can be used as X, Y or Z-axis, the load is not known in advance and the motion controller of such modular machines has to be made robustly performant for a broad range of loads. Alter and Tsao [2] have studied the control of a linear motor axis taking into account a model of the cutting process, in order to investigate the chatter behaviour. They have shown that minimising chatter is equivalent to maximising servo-dynamic stiness. Therefore, design for good nominal performance (good disturbance rejection) leads to reduced chatter. In [3], an H1 controller is designed for a linear motor axis. There, the authors focus on nominal performance, not on robustness. In this paper, the design of an H1 and a sliding mode controller for a prototype machine tool axis with a linear motor is discussed and the robustness against load variations is compared experimentally. The performance and robustness of an H1 controller depends on the selection of the weighing functions. Generally, the performance is obtained by shaping the performance weighing function, such that the sensitivity function is small in the low-frequency range. Robustness is then obtained by selecting the robustness weighing function in order to make the complementary sensitivity function small enough in the highfrequency range to allow for the expected model uncertainty. LundstroÈm et al. [20] and Postlethwaite et al. [21] give guidelines for the selection of these weighing functions. In these guidelines, the mid-frequency range, which is the range around the cross-over frequency, is often not considered. This mid-frequency range is however very important for robustness [19,21]. In this mid-frequency range, neither the sensitivity nor the complementary sensitivity function is small. Here, the closed-loop system behaviour is determined by a combination of the performance weight and robustness weight. This interaction between both weights is generally neglected in H1 design. For an H2 controller, this interaction has been considered by Jovik and Lennartson [17], where they gain robustness, introducing loop transfer recovery (LTR), by a proper choice of the robustness weighing function. This paper considers the interaction between performance and robustness weights in H1 control design for the above mentioned application, by
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proposing an alternative approach for the selection of the weighing functions, yielding improved robust performance. Other techniques to select the performance weighing function are given by Franchek [14] and Yang et al. [28]. Franchek [14] describes a method that allows to extract the performance weights from certain speci®c time domain constraints, such as the maximum value of the response to a step disturbance, and actuator eort constraints. Yang et al. [28] use matrix experiments as used in quality control to determine the eect of the various parameters in the weighing function and to ®nd the optimal parameter settings. However, neither method leads to generally applicable design guidelines. In a continuous-time sliding mode controller, the system is requested to move on a sliding surface [25]. A discontinuous controller forces the system on this sliding surface. The parameters of the sliding surface de®ne the system's closedloop dynamics. Drazenovic [12] showed that the system in the sliding mode is invariant against a certain class of parameter uncertainty and disturbances. However, this invariance property only holds when the system is moving exactly on the sliding surface. In practical applications, the theoretical invariance property is lost, due to actuator limitations, system delays or the use of observers. Also, for discrete-time systems, this theoretical invariance is lost. A sampleddata system will seldom be able to move exactly on the sliding surface, it will rather move zigzag around it and towards the origin. Sarpturk et al. [23] design a controller, based on a discrete-time equivalent of the continuous-time reaching condition. They show that this imposes an upper and a lower bound on the control signal. However, these bounds converge, when approaching the sliding surface, which would require a priori knowledge of the uncertainties and disturbances [18]. Furuta [15] and Wang et al. [27] propose a method that relaxes the reaching conditions in a neighbourhood of the switching surface. The reaching law method of Gao et al. [16] starts from the desired dynamics of the switching variable (the distance from the switching surface) in the discrete-time domain. This method allows to analyse the robustness of the motion around the switching surface [5]. Utkin [26] designs a discrete-time sliding mode controller, based on a new de®nition of the discrete-time sliding mode. In contrast to the earlier methods, this control algorithm excludes discontinuities in the control law, and thus chattering. In milling applications, chattering leads to high-frequency errors on the surface, which are clearly visible and not acceptable.Therefore, the sliding mode controller presented in this paper, uses a discrete-time design approach. The designed controllers are experimentally tested on a prototype machine tool axis actuated by a linear synchronous motor. The aim is to achieve a maximum tracking error below 10 mm for a broad range of loads. This desired accuracy is typical for many applications, e.g. high-speed milling. Sections 2 and 3 brie¯y review the H1 control and the discrete-time sliding mode control design approaches, respectively. Section 4 presents the experimental test set-up and discusses the design of two dierent controllers: an H1 controller, and a discrete-time sliding mode controller. Section 5 discusses the experimental
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performance validation of these controllers in the presence of load variations. Section 6 proposes a novel sliding-mode-inspired selection of weighing functions for the H1 design, which yield better robustness against load changes. 2. H1 control design Fig. 1 shows a typical H1 design scheme, where e, u, y, yref, d and di represent the tracking error, the controlled system input, the system output, the reference trajectory, the output disturbance and the input disturbance, respectively. G(s ) and C(s ) represent the system and controller, respectively, with s the Laplace variable. Wp(s ), Wu(s ) and Wr(s ) are H1 design weighing functions which will be explained below. y1, y2, and y3 are the weighed tracking error, control input, and undistorted system output, respectively. Minimizing the H1 norm of the transfer functions of these weighed signals is the goal of the H1 control design. The H1 norm of the system sensitivity function S 1= 1 GC , represented by kSk1 , is a measure for the tracking and disturbance rejection performance of the closed-loop system, since S relates the disturbance d to the system output, y, as well as the reference input, yref, to the tracking error, e. This means that a small kSk1 is desirable for good disturbance compensation and tracking behaviour. It can be easily derived from Fig. 1 that the in¯uence of the input disturbance, di, is measured by kGSk1 : Usually, a controller that minimises kSk1 for good rejection of output disturbances, d, will also minimise kGSk1 , since G is constant. The robust stability for unstructured multiplicative uncertainties can be measured by the H1 norm of the system complementary sensitivity function T GC= 1 GC : The Small Gain Theorem yields that for multiplicative uncertainties Dm , the perturbed system, G 1 Dm , is stable if kDm k1 < 1=kTk1 [8]. This yields that large multiplicative uncertainty, kDm k1 , is allowed, when kTk1 is small. For additive uncertainties, Da , the perturbed system, G Da , is stable if kDa k1 < 1=kRk1 , with R C= 1 GC : Since S + T = 1, it is not possible to make kSk1 and kTk1 both arbitrarily small for good nominal performance and for robust stability, simultaneously.
Fig. 1. H1 design scheme.
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Therefore, the H1 control synthesis procedure usually minimises a weighed sensitivity, kWp Sk1 , and weighed complementary sensitivity function, kWr Tk1 , where Wpÿ1 represents the desired frequency-dependent disturbance attenuation factor and Wr an upper bound on the expected multiplicative uncertainties. Typically, Wpÿ1 is small at low frequencies, where good nominal performance is desired, and Wrÿ1 is small at high frequencies, where large model uncertainties are expected. kWu Rk1 should be considered in the control design objectives, when additive uncertainties are expected. Since R is also the transfer function from the reference trajectory and the output disturbance to the control input signal, u, its norm measures the required control eort. Therefore, Wu represents the expected additive uncertainties or a weight allowing to account for system input limitations. For robust performance, the weighed mixed-sensitivity design approach [6] can be used, where the control synthesis procedure aims at simultaneously optimising nominal performance and robust stability, by minimising the following norm of stacked weighed transfer functions:
Wp S
Wu R : 1
Wr T 1
Theoretically, robust performance is only guaranteed when Eq. (1) is smaller than p 1= 2: The robustness of the H1 controller is not inherent to the synthesis design itself. The robustness, as well as the performance, result from the weighing functions chosen in the synthesis. The diculty of designing a robust controller with H1 synthesis lies therefore in the selection of the weighing functions. This will be discussed in Section 4.
3. Sliding mode control design The discrete-time sliding mode control method of Utkin [26] has been used here. This method is based on the de®nition of discrete-time sliding mode [11]. Consider a linear discrete-time SISO system, described by an nth order state space model: x k 1 A x k B u k ,
2
with x k, x k 1 2 Rn the system states at time instant k and k 1: u k 2 R is the system input, and A and B are the system and input matrices of the state space model, respectively. For a SISO system, the sliding surface is reduced to a line, which can be described as ÿ 3 s k Csl x k ÿ x ref k 0, where x ref k 2 Rn are the reference states and the scalar s k is a measure of the
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distance from the sliding line. The slope of the sliding line, Csl 2 Rn , is designed such that the system has the desired closed-loop dynamics, when moving on the sliding line. The control input u k that will bring the system to the sliding line at moment k 1 is found as the solution of the equation s k 1 0:
4
Combining Eqs. (2)±(4) yields: u k
Csl Csl A x ref k 1 ÿ x k Csl B Csl B
5
Combining Eqs. (2) and (5) yields the state matrix of the closed-loop dynamics: Acl A ÿ B
Csl A : Csl B
6
The sliding manifold (4) de®nes 1 pole of the closed loop system at z = 0. This can easily be seen as follows. If the matrix A is singular, it is obvious from Eq. (6) that Acl will also be singular, thus having an eigenvalue at zero, i.e. a pole at z = 0. If A is not singular, the vector: X A
ÿ1
B,
7
is an eigenvector of Acl with corresponding eigenvalue l 0: Acl X lX 0X:
8
The other n ÿ 1 poles in Eq. (6) are de®ned by the selection of the slope of the sliding line, i.e. Csl : The method can also be seen as a special case of the reaching law method [16], where a controller is designed that results in the reaching law s k 1 1 ÿ qTs s k ÿ ETs sign s k ,
9
with Ts the sampling period. Choosing qTs 1 and ETs 0, yields Eq. (4). Fig. 2 shows the resulting controller structure. This ®gure shows that the states
Fig. 2. Controller scheme of sliding mode controller.
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are estimated by means of a state observer, which is necessary when the state variables are not directly measurable. This is the case for the experimental system described below. It should be remarked that state observers can cause loss of robustness [9]. However, the experiments presented in Section 5, give no indication of this. 4. Controller design for a machine tool axis with linear motor 4.1. Experimental set-up Fig. 3 shows the experimental set-up, representing a prototype machine axis. It consists of a linear synchronous motor operating in current control mode. The linear motor consists of a series of magnets ®xed to the machine base, and a moving part which contains the coils. The carriage connected to the moving part of the motor is supported on two rolling element slideways. The position of the carriage is measured by means of a magnetic linear scale on one of the slideways with an incremental measurement system with a resolution of 1 mm. The total mass of moving part is around 12 kg. The control unit is a dSPACE digital signal processing (DSP) board using a TMS320-C31 processor, equipped with a 12-bit D/A converter (settling time is 4 ms) and a 24-bit incremental encoder input [1]. The sampling frequency is selected to be 2200 Hz. The dynamics of the system, relating system input signal, u [Volt], which is proportional to the motor current, to system output, y [mm], have been identi®ed experimentally using broad-band multisine excitation up to 220 Hz, and a nonlinear least-squares frequency-domain identi®cation method [29] (Schoukens and Pintelon, 1991). The obtained discrete-time model equals: G z
ÿ1:7429 : z ÿ 1 z ÿ 0:9981
10
It describes the system dynamics, including the zero-order-hold eect of the digital control hardware, accurately (Fig. 4). This model is a compromise between accuracy (the model is accurate only up to 220 Hz) and complexity. The ®rst
Fig. 3. Experimental test set-up.
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Fig. 4. Measured frequency response function and the frequency characteristic of the identi®ed model (a) amplitude, (b) phase.
structural resonance frequency is at 300 Hz and therefore, not included in the model. The model, (10), contains a pole at z = 1, due to integration in the relation between force (input) and position (output). Robustness of the controllers is checked against load variation, which is accomplished by attaching extra masses of 7 kg to the carriage. The masses are added while the system is not operational. 4.2. Design of the H1 controller The design of the H1 controller is an iterative procedure in the continuous-time domain, i.e. based on the continuous-time equivalent of G(z), cfr. Eq. (10). The obtained controller will then be transformed to the discrete-time domain using a bilinear transformation. This transformation maps the closed unit disk in the zplane onto the closed left-half s-plane. The obtained performance is not lost, since the bilinear transformation is norm-preserving. The design of the H1 controllers does not take into account any uncertainty model. The only uncertainty that is considered in this paper is the mass of the load carried by the motor. Other system uncertainties are negligible with respect to the large mass variations considered (up to 300%). The mass variations results in an uncertainty on the system gain. The corresponding uncertainty model for the H-in®nity control design would therefore be a constant but complex number over all frequencies, which would lead to a very conservative controller with very bad performance, since the actual uncertainty is real, not complex. As a result, it is not appropriate to include such an uncertainty model in the control design. Therefore, the weighing functions have been tuned experimentally on the nominal model, until a sucient tracking performance is obtained. Fig. 5 shows the resulting third-order weighing function Wp : ÿ 1:2 s 350 s 2 2 0:7 350 s 350 2 : 11 Wp s 2:5 s 2 2 0:7 2:5 s 2:5 2
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Fig. 5. Wp (solid) and S (dashed) obtained with the H1 controller.
Fig. 6 shows the robustness weighing function Wr , which consists of a ®rst-order roll-o at high frequencies: Wr
s : 2200
12
Wu is chosen to be a constant; small enough not to in¯uence the design, e.g. 0.5. In case actuator saturation is observed during the experiments, Wu should be increased. These weighing function shapes are common in H1 control design. S is shaped by Wp resulting in low sensitivity at low frequencies. As a result, T will be very close to one at these frequencies. The weighing function Wr will force T to have a ÿ20 dB/decade roll-o above 350 Hz. The continuous-time equivalent of Eq. (10), which is used for control synthesis, contains a pole at the origin. However, the standard H1 synthesis procedure of Doyle et al. [10] does not allow this. The use of a bilinear transformation, as
Fig. 6. Wr (solid) and T (dashed) obtained with the H1 controller.
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proposed by [6], avoids this problem: s~
ÿs p1 : s=p2 ÿ 1
13
This method ®rst shifts the poles of the system to the left (e.g. p1 = ÿ1 and p2 = 1), then calculates the controller and shifts the resulting controller poles back to the right. This controller will then compensate this pole at the origin with a zero. However, a pole at the origin represents an integration in the system, which is required in most motion control applications to remove steady-state errors. The following two properties [6] lead to an H1 synthesis procedure that can handle system poles at the origin without removing them by pole-zero cancellation. The resulting controller contains integration and thus yields perfect compensation of steady state errors. Property 1: In the weighed mixed-sensitivity formulation, any unstable pole of the plant inside the speci®ed control bandwidth will be shifted approximately to its jo-axis mirror image, once the feedback loop is closed with an H1 controller. Property 2: If the controller C(s ) for G(s ) is obtained from the transformation of C(sÄ ), using the inverse of Eq. (13), where C(sÄ ) is synthesised for the bilinear transformation G ~s of the system model G(s ) using Eq. (13), it is guaranteed that the following inequality holds:
ÿ
T jo R ~ ,
T~ jo 14 1 1
with T the complementary transfer function of the system, G(s ), and TÄ the complementary transfer function of the bilinearly transformed system G(sÄ). This ensures that the ®nal s-plane controller C s will at least meet the robustness speci®cations requested from C ~s and will probably exceed it. These properties yield the following design procedure: A.1. The continuous-time equivalent G(s ) of the plant (10) is transformed using Eq. (13) with p1 = 1 and p2 = 1 towards G(sÄ). This transformation shifts the system poles to the right over 1 rad/s, yielding that the pole in the origin shifts to a pole at 1. A.2. The controller is designed using Wp, Wu and Wr using the mixedsensitivity approach (1). The pole of G(sÄ) at 1 is compensated by a zero of the controller C(sÄ) at ÿ1. A.3. The poles and zeros of the controller are shifted to the left over 1 rad/s, yielding a zero at ÿ2. A.4. The closed-loop system includes integration (due to the pole of G(s ) at s = 0) until 2 rad/s (due to the zero of C(s ) at s = ÿ2), as can be seen in Fig. 5: the sensitivity function S has a slope of +20 dB at low frequencies. A.5. The obtained controller is transformed to the discrete-time domain for implementation.
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Although H1 control design can also be accomplished directly in discrete-time, the described continuous-time design was preferred, since this can easily be implemented using commercial software packages, such as the MATLAB m-Tools [4] or Robust Control Toolbox [7]. Figs. 5 and 6 show the sensitivity and complementary sensitivity function of the resulting closed-loop system. Fig. 5 shows that this H1 controller yields a sensitivity function, S, with zero DC gain. This means that the closed loop system has a perfect asymptotic compensation of step disturbances at the system output, d, and perfect asymptotic tracking of step trajectories. Eq. (18) in the appendix of this paper shows the discrete-time transfer function of the H1 controller.
4.3. Design of the sliding mode controller The sliding mode controller selected for the test set-up is based on Eq. (4), which yields equivalent control law Eq. (5) [15]. It is a special case of the reaching law method [16] (Eq. (9)) as explained in Section 3. Its design consists of the selection of the slope of the sliding line Eq. (3), which determines along with the choice of the reaching law Eq. (9), the poles of the closed-loop dynamics Eq. (6). First, the system model (Eq. (10)) is transformed into a state space model (the controller canonical form) and extended with a third state variable, which corresponds to the (discrete) integration of the output. This ensures integral feedback of the position and therefore, removes steady-state errors. The resulting third order model equals: 2
1:9981 x k 1 4 1 0
y k 0
ÿ 1:7429
ÿ0:9981 0 ÿ1:7429
3 2 3 0 1 0 5x k 4 0 5u k 1 0
0 x k
15
Reaching law (4) de®nes one closed-loop pole at z = 0 (18). The remaining twoclosed loop poles are determined by selecting the coecients of Csl : using a regular form based approach [13]. This approach is based on the canonical form of the state space model, also called the regular form, calculates the last two coecients of Csl using any common pole placement algorithm, and selects the ®rst coecient of Csl equal to one [13]. These two closed-loop poles are selected experimentally at 10 Hz and 100 Hz, yielding: Csl 1
ÿ 0:7234
ÿ 0:0040
The precise choice of these poles is not critical.
16
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5. Robustness against load variation Both the sliding mode and the H1 controller have been implemented on the linear motor axis, described in Section 4.1. The moving mass of the nominal system is around 12 kg. The robustness of the controllers is tested for increasing mass up to 35 kg, by incrementally adding masses of 7 kg, while the system is not operational. The linear synchronous motor of the test set-up has a signi®cant motor ripple resulting in a deterministic and periodic tracking error. In the experiments, this has been compensated by an appropriate ripple model based feedforward [24], in order to give a realistic representation of the tracking accuracies achievable for this test set-up. Fig. 7 shows the tracking error for the H1 controller for the nominal system, i.e. without extra mass. The reference trajectory is a 9th-order polynomial. The maximum velocity is 0.5 m/s and the maximum acceleration is 2 m/s2. The maximal acceleration of the trajectory is rather low compared to the maximum achievable acceleration (40m/s2 for the nominal system). This is to avoid actuator saturation when the extra masses are added, and to better demonstrate the robust performance of the feedback. If a trajectory with higher acceleration would be chosen, e.g. 5 m/s2, the maximum tracking error is determined by the tracking error during acceleration and deceleration, which is de®ned by the feedback and the feedforward. The feedforward becomes very inaccurate for a large amount of extra mass, since it is based on the nominal system. This shows that for good tracking behaviour for all cases, an adaptive feedforward is required. Since this paper focuses on the feedback controller, a slower trajectory is chosen, which shows better feedback properties of the overall system, like disturbance compensation (ripple, friction, etc.). Fig. 8 shows the maximum tracking error for the dierent controllers for the load increasing in steps of 7 kg until 35 kg. The sliding mode controllers and the H1 controller have the same nominal performance. The maximum tracking error
Fig. 7. Nominal tracking error for the sliding mode controller.
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Fig. 8. Comparison of robustness: maximum tracking error for dierent load.
for the nominal system, i.e. for zero extra load, is 7 mm. The sliding mode controller is stable for all extra masses, without increase in maximum tracking error for increasing load. The H1 controller reaches the edge of stability for 35 kg added. The tracking error shows an undamped oscillation for this load. Unlike most sliding mode controllers which have a non-linear term, the applied sliding mode controller is a linear controller. The only dierence with a standard state space controller lies in the selection of the poles. This allows to use standard techniques to compare the control performance, like Bode and Nyquist plots. Fig. 9 shows the Nyquist plot for the original system without extra mass, connected to the H1 controller and to the sliding mode controller. Both open loop systems have approximately the same phase margin. However, the sliding mode controller appears to allow much more decrease in system gain, i.e., has a larger gain margin, therefore allowing much more increase in system mass. The relatively poor robust performance of the H1 controller against load changes, is a result of the selected weighing functions. Section 6 proposes an adapted selection of the weighing functions leading to better robust performance for the considered application.
Fig. 9. Nyquist plots for the nominal system (solid line: sliding mode controller, dashed line: H1).
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6. Adaptation of weighing functions for robust control against load changes The preceding section shows the dierence between both controllers by means of Nyquist plots. This dierence is also visible in the frequency response of the sensitivity and complementary sensitivity function (Fig. 10). Both sensitivity functions seem to have a comparable frequency response. Both have a slope of 60 dB/dec at low frequencies. However, close inspection at the crossing of the 0 dB line shows clear dierences: the slope of the H1 sensitivity function changes quite abruptly from 60 dB/dec to 0 dB/dec, while the slope of the sliding mode sensitivity function decreases ®rst to 20 dB/dec before reaching the 0 dB line, after which it reduces further to 0 dB/dec. This dierence in shape of the sensitivity functions is used to select an adapted weighing function Wp for the H1 synthesis. The weighing function Wp, is adapted such that its slope increases ®rst from ÿ60 dB/dec to ÿ20 dB/dec before crossing the 0 dB line, after which it further increases to 0 dB/dec. This shape is accomplished by the following transfer function: Wp
1:2 s 75 s 500 s 880 s 3:5 s 2 2 0:7 3:5 s 3:5 2
17
Fig. 11 shows the frequency response of the old and the adapted performance weighing function, Wp : Using the adapted Wp, (17), a new H1 controller can be designed, following the same procedure as in Section 4.3. Eq. (19) in the appendix of this paper shows the discrete-time transfer function of this new H1 controller. Fig. 12 shows the Nyquist plot of the loop transfer function of the nominal system with the sliding mode controller and with the new H1 controller. As expected, the Nyquist plots shows that the new H1 controller and the sliding mode controller have a comparable robustness.
Fig. 10. Sensitivity (left) and complementary sensitivity (right) for sliding mode controller (solid line), H1 controller (dashed line), weighing function, resp. Wp and Wr (dash-dotted line).
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Fig. 11. Old (dashed line) and new (solid line) weighing function Wp :
Fig. 12. Nyquist plots for the nominal system for the sliding mode (solid line) and new H1 controller (dashed line).
Fig. 13. Comparison of robustness: maximum tracking error for dierent load with adapted weighing function Wp.
Fig. 13 shows the experimental robustness results of the adapted H1 controller together with the results of the sliding mode controller. As in Fig. 8, it shows the tracking error for the given trajectory for increasing load.
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The H1 controller with adapted weighing function Wp is much more robust than the one with the standard weighing functions: the tracking error increases from 7mm for the nominal system to only 10 mm for 35 kg added. This is much better than the ®rst H1 controller, but not as good as the sliding mode controller. This dierence is caused by the dierence in their controller structure. The H1 controller has a more complex structure: its order is higher, which leads to extra zeros in the loop, which are unfavourable for the performance. The simplicity of the sliding mode controller results in better performance.
7. Conclusion Experiments on a prototype machine tool axis, actuated by a linear motor, show a discrete-time sliding mode controller is robustly performant with respect to load variations up to 300%. An H1 controller design using standard weighing functions is unstable for this load increase. This poor robustness can be improved signi®cantly by selecting dierent weighing functions. The proposed adapted performance weighing function is based on the shape of the sensitvity function of the sliding mode controller, and yields robustness comparable to that of the sliding mode controller without loss of nominal performance. This approach of selecting weighing functions (although probably not generally applicable) can inspire control engineers in applying H1 control theory successfully in practical situations, since this cannot be learned from control theory only.
Acknowledgements This research is carried out in the framework of BE-programme of the European Union, under the project KERNEL2, BE-7423. This text also presents research results of the Belgian Programme on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister's Oce, Science Policy Programming. The scienti®c responsibility is assumed by its authors.
Appendix A The discrete-time transfer function of the H1 controller discussed in Section 4.2 equals: ÿ0:2118z5 0:6071z4 ÿ 0:3704z3 ÿ 0:4170z 2 0:5756z ÿ 0:1836 z5 ÿ 1:7363z4 ÿ 0:4726z3 1:8462z 2 ÿ 0:3293z ÿ 0:3080
18
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The discrete-time transfer function of the H1 controller discussed in Section 6 equals: ÿ0:2358z5 0:6785z4 ÿ 0:4153z3 ÿ 0:4685z 2 0:6480z ÿ 0:2069 z5 ÿ 1:7883z4 ÿ 0:3618z3 1:8237z 2 ÿ 0:4087z ÿ 0:2649
19
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