- Email: [email protected]
Robust Position and Speed Control of Variable Displacement Hydraulic Motors
Copyright © IFAC Robust Control Design, Budapest, Hungary, 1997
ROBUST POSITION AND SPEED CONTROL OF VARIABLE DISPLACEMENT HYDRAULIC MOTORS H. Berg Department of Measurement and Control (Prof. Dr.-Ing. II. Schwarz, Prof· Dr.-Ing. M. Ivantysynova), Faculty of Mechanical Engineering, University of Duisburg, D-47048 Duisburg, Germany Phone: +49 (203) 379 - 1576; Fax: +49 (203) 379 - 3027, email: [email protected]
Abstract: This paper addresses the position and speed control of a variable displacement hydraulic motor (VDHM) at a constant pressure net (also referred to as "secondary control") . The system specifications relate to the application in future large aircraft, where this modern type of hydraulic motors can help to reduce weight and energy consumption of flight control actuators. LQG/LTR and Hoo design approaches are considered, emphasizing the problem of sufficiently precise position control in the face of non linear friction. Simulation results are compared with experiments performed a VDHM test rig. Finally, the use of J..I analysis techniques is briefly outlined. Keywords: Hydraulic motors, LQG control, H 00 control, J..I analysis, position control
1. INTRODUCTION
(1983) , adaptive and fuzzy based designs have been investigated by Weishaupt and Backe (1993) . Guo and Schwarz (1989) have developed a nonlinear approach to speed control using a bilinear system description.
The use of energy and weight saving technologies for hydraulic drives is of increasing importance in large aircraft design as well as other industrial applications. Variable displacement hydraulic motors (VDHM) can help to significantly reduce the energy consumption of aircraft actuation systems. Examples are the power drive units of high lift systems and the horizontal stabilizer actuator. Both are presently driven by valve operated fixed displacement motors with poor efficiency due high pressure losses in the control valves. VDHMs automatically adapt the hydraulic input power to the required mechanical output power. Moreover, they are able to deliver hydraulic power back to the supply system in case of aiding loads. While speed and torque control of VDHMs are used in numerous industrial applications, position control has received only few attention up to now. Both speed and position control have been subject to university research since the early eighties. Classical linear design methods have been employed by Haas (1989), Metzner (1985) and Murrenhoff
In this paper, emphasis is given to some aspects which have not been considered so far. For the use in an aircraft actuator a solution must be found that meets all functional and safety requirements on the one hand and on the other hand provides a control structure that is robust in the face of plant uncertainties and of low order to minimize the computational effort. A means is presented by which the problem of limit cycles in the position control loop can be handled without the use of switching integrators or other nonlinear techniques. There is a number of industrial applications apart from aircraft systems where the issues discussed here are relevant.
473
2. THE VARlABLE DISPLACEMENT HYDRAULIC MOTOR (VDHM)
• The nonlinear equations of flow are linearized about the operating point p* ~ (Po + Pr) , Y;v = 0 :
Bs v is the flow coefficient of the servovalve. • The "hydraulic dynamics" of the transfer function between Q and xp (pressure buildup, piston dynamics) may be approximated by a second-order linear model with the parameters Wh and D h : (3) • In the equation of motion for the motor only linear viscous friction is modeled (coefficient dm ):
Figure 1. Variable displacement hydraulic motor Fig. 1 shows a scheme of the VDHM at a constant pressure supply. For the test rig experiments a cam-type axial piston motor "Rexroth A10VSO" with a maximum motor displacement of Vm max = 28 cm3 /rev is used. The swash plate of the motor (i.e. the motor displacement) is controlled by a single-acting cylinder. The flow to the cylinder is controlled by an electrohydraulic servovalve. Depending on the sw ash plate position and the torque at the output shaft , the unit works either as a pump or a motor. In case of aiding loads, the system delivers hydraulic power to the supply system. A digital controller implemented on a PC calculates the input signal for the servovalve using motor shaft speed, motor shaft position and cylinder stroke feedback. The controller limits the maximum motor output speed . A sampling frequency of 1 kHz allows continuous time techniques to be used for design and analysis.
Vm I inr-
max (Po
- Pr)
Xp
_ d
.
(4)
m
21rXp max
The nominal values of the parameters PO,Pr, V.v,Bsv , Ap,xpmax, Vmmax , I and dm must be determined for controller synthesis, see Table 1. The servovalve dynamics and the hydraulic dynamics can be covered by a complex uncertainty in the j.J. analysis of the controlled system and it suffices to obtain rough estimates of the parameters. The servovalve dynamics are assumed to act in a frequency region around 80 Hz (440 rad/s)' the hydraulic dynamics lie approximately one decade higher. Due to the relatively low bandwidth requirements of an aircraft actuator, both are neglected in the linear model for controller synthesis. ~ ···· · ·······················se;v~;;i';;m:i ·s:';~Pb~·';~CYiindOt \
3. MATHEMATICAL MODEL ~
In industrial applications it is usually desired to find the simplest mathematical system description that is still fully sufficient for controller design and system analysis under given specifications. This keeps the effort for system identification and simulation on a minimum level. Moreover, the controller order is related to the order of the (possibly augmented) plant model, if LQG/LTR or H00 synthesis methods are used. The following linear model of the VDHM can be derived for controller design:
~ .
--. ----. ------------------------------------------------- --------
--~
Motor ;
'--- ---- --- --- ------- --- ---- ----- ---- ---------- --- ------------------ ----
Figure 2. Linear model for controller synthesis The exact values of Vsv and the geometrical quantities piston stroke xp max , piston area Ap and maximum motor displacement Vm max are given by the manufacturer. All other parameters can take values within certain intervals, either because of imprecise identification or because they vary with the operating conditions. More information about system analysis is given in Section 5. For technical reasons , the system pressure amounts to less than
• The servovalve dynamics may be approximated by a second-order linear model with the parameters Wsv and Dsv :
(1)
474
half of the nominal value for this type of motor. This makes the system more susceptible to saturations of Ysv or xp . The linear model from Fig. 2 is controllable, observable and minimum phase.
(4) Rejection of disturbances caused by load, friction and moment pulsation of the motor: Z = cp/ M/ < 0 dB for all frequencies . (5) No overshoot of the command and load step responses of the position. (6) No integral action of the speed controller. (7) Control effort due to a speed command step of 150 rad/s without external loads: xp < 0.5·
Table 1. Nominal parameters of the linear model System pressure Po Return pressure Pr Static valve gain v'v Flow coefficient Bsv Piston area Ap Maximum piston stroke
llO .10 5 Pa 3 .105 Pa 6 .10- 5 m/V 3.05 .10- 4 m3 /s/VN 4.05 .10- 4 m 2 17.72 .10- 3 m
X pmax ·
(8) Stability margin of the nominal system: ISI < 1/0.8 (= 2 dB) for position and speed control. (9) A peak value of the complementary sensitivity T of less than 0.1 dB for speed control. (10) A high roll-off rate of the complementary sensitivity T shortly after the crossover frequency to ensure good rejection of measurement noise and good robustness against unstructured multiplicative uncertainty such as neglected dynamics . (ll) A low sampling frequency and a simple controller structure.
Xpmax
Maximum motor displacement Vm max Moment of inertia I Viscous friction coefficient dm
28 .10- 6 m 3 /rev 7.85 .10- 3 kgm 2 2.9 .10- 2 Nms/rad
Requirement 6 is necessary for position control , because · integrating behaviour of the controller causes undesired control action trying to achieve zero steady-state position error of the motor shaft in the face of nonlinear friction (limit cycles) . Alternatively, an integrator can be used which is switched off for small position errors. Possible implications of this approach such as cyclic action and oscillations are described in (Weishaupt, 1995). If the bound from requirement 3 turns out to be insufficient, prefilters can be introduced to further reduce the steady-state errors.
4. ROBUST CONTROLLER DESIGN The main intention of this work is to design a control system for the motor shaft position. An important constraint is given by the maximum allowed motor shaft speed which is chosen here as CPc max = 150 rad/s . Moreover, it is demanded that the swash plate position can be controlled independently. On the one hand this is necessary for safety reasons , on the other hand it prevents the control system from undesired action due to leakage of the swash plate operating cylinder. Fig. 3 shows a scheme of the entire position control system. Fij denotes the transfer function from signal i to signal j. A saturation element is used to limit the positioning speed by bounding the command input CPc for the speed control loop. For large values of the tracking error e, the saturation element gives the constant speed command CPc max . For values of e beneath a defined threshold, it works as a linear gain ks in the position control loop.
In the following the main design steps are presented . The speed controllers are calculated using a state-space representation of the linear plant model from Fig. 2 with the nominal parameters from Table 1. As shown in Fig. 3 , a PI compensator is selected to control the piston position Xp:
C Xp
= 300 (s + 5;ad/s) .
(5)
The integrator provides good rejection of disturbances that are caused by a servovalve offset and by the swash plate operating force. Without this , especially the servovalve offset , which usually occurs after some operating time, has a strong impact on the tracking error of the motor speed and position . The swash plate operating force is a speed dependent , oscillating force which is caused by internal mechanisms of the hydrauli~ motor. It has a mean absolute value larger than zero (Ivantysyn and I vantysynova, 1993) .
Figure 3. Block diagram of the position control system The following design specifications relate to the application in an aircraft actuator: (1) A bandwidth of the position control loop of 2-4 rad/s . (2) A bandwidth of the speed control loop of 2040 rad/s. (3) A steady-state error in the face of position and speed command steps of less than 5 %.
After the controller for the swash plate operating cylinder has been designed, the next step is to 475
find a control law for the motor shaft speed . The plant is a series connection of the controlled operating cylinder and the motor dynamics , represented by a first-order transfer function (Eqn. (4)) . The LQG/LTR method is used to design the speed control system based on open loop considerations, see (Maciejowski, 1989) or (Geering, 1994). It requires only standard software for linear-quadratic regulator and Kalman filter design. A loop gain roll-off of -60 dB/decade shortly after the crossover frequency yields good robustness against multiplicative uncertainty (e.g. neglected dynamics) , minimum control action and good attenuation of measurement noise.
(7)
must remain below a certain bandwidth to keep the whole system stable. To maintain the good performance and stability margins of the LQG/LTR design, a value of Ta 0.2 s is selected . Fig. 8 shows calculated and measured command step responses of the speed control system with and without augmentation. The tracking error of the system without ·augmentation is caused by the command step and the coulombic friction moment acting as a load disturbance of M/ ~ 3 Nm. Slight oscillations due to moment pulsations of the motor can be seen in the step responses from Fig. 8. Their frequency of approximately" 15 rad/s corresponds to the peak of Z in the crossover region (Fig. 7) . Better rejection of these disturbances can only be obtained by raising the bandwidth.
=
Ib) 151 (-). In 1--)
la) IU 1-). ILFI 1--)
SO
------
iD ~
.0
c;
iD
~-20
0
.0
c;
O·r-----~------~----~----~
-40
-SO
10':
10' Frequency [rad's]
10'
10':
10' Frequency [rad's]
10'
-50
Figure 4. (a) Loop gains of the Kalman filter and the whole system; (b) Sensitivity and complementary sensitivity The controller transfer function (after one polezero cancellation) is
G . _ 0.0128s + 0.6646 '" - s2 + 165.24s + 12451·
-200
-2~~----~------~----~----~ 1~2 1~ 1~ 1~ 1i Frequency [radial
(6)
Figure 6. Influence of ka on the tranfer function xp (s) /I{;e (s) for the open speed control loop, Ta = 0.2 s
It may be further reduced to a PD controller which has significantly worse characteristics due to a loop gain roll-off of only -20 dB/decade. The damping can be increased by shifting the gain of Gzp from 300 to 350. The low bandwidth requirement results in a large steady-state error and poor disturbance rejection properties. Significantly improved results can be obtained by augmenting the structure with piston position feedback (Fig. 5) . The choice ka 1/G.,o(0) leads to integrating behaviour of the system xp (s) /I{;e (s) for the open speed control loop. This must be avoided to prevent limit cycles of the position control. In Fig. 6 gain plots of xp (s) Ne (s) for different values of ka are shown. With the value ka 0.9· l/Go, sufficiently small steady-state errors can be achieved without causing limit cycles. The first order transfer function
=
=
Figure 7. Load sensitivity Z = I{; / M/
Hoc loop shaping techniques allow to design a controller directly for the augmented plant. Fig. 9 shows the speed control system with weighting filters. Appropriate weights can be derived from the experience with the LQG /LTR design . It turns out that the loop shaping configuration from Fig . 9 is useful to investigate the trade-off between the "amount of integral behaviour" of the system
Figure 5. Block diagram of the augmented speed control system 476
measured (-), linear simulation (--)
Closed speed control loop T.
Open position control loop
dB
dB A
+
J~radlS ~ c) oL ~,o i ... I
4
10
k,
I
I
~radls
-t
Figure 10. Transition from
... .PiSton positiOn .. . -- - -- - --:---
"
0.1
0.2
0.4
0.6
0.8
1.2
Time[s]
command step responses,
Figure 11. Normalized measured command step responses, t.pc = 300 rad, LQG/LTR controller
I~L < 1
certainty descriptions can be found in (Balas et al., 1995) and (Gahinet et al., 1995). The goal of system analysis is to check whether the desired performance in terms of bandwidth, disturbance attenuation and stability margins can be sustained in the face of plant uncertainties. First of all, only speed control is considered where the uncertainties can be categorized in two groups:
Figure 8.
Normali~ed
subopt. H_ - Problem:
• Dynamical uncertainty due to the neglected servovalve dynamics W sv , Dsv and "hydraulic dynamics" Wh, Dh which can be together taken into account in a complex, frequency dependent, multiplicative uncertainty ~ (s) . • Uncertainty of physical parameters either because of imprecise knowledge of their values or because their values change during operation. The parameters (Po - Pr), Bsv, I and dm can take values within a certain range and are modeled as real, multiplicative uncertainties
Figure 9. Block diagram with weights for Hoc design
xp (s) /
8i . All uncertainties are assumed to be linear and time-invariant which represents a strong simplification of the real situation. A frequency dependent upper bound is given on I~I, constant upper bounds are given on 8i . Fig. 12 shows a linear fractional interconnection of the nominal plant P and the structured uncertainty description represented by the block-diagonal matrix
Appropriate scalings to normalize the uncertainty bounds are inserted into P such that 11 A 1100 < 1 for all admissible perturbations. The system
5. J.L ANALYSIS OF THE CONTROLLED SYSTEM
In this Section, the procedure for J.L analysis of the controlled system is briefly outlined. A tutorial exposition of the J.L approach and the required un477
at a constant pressure net. The specifications relate to a low-bandwidth application in an aircraft actuator. An LQG/LTR controller design is discussed which is based on a simple mathematical plant description with both dynamical and parametric uncertainty. A control structure is proposed that allows sufficiently precise control of the motor shaft position avoiding limit cycles. Simulations and experiments show that satisfactory performance can be achieved with this design. An Hoc loop shaping approach is better suited to determine the design limits, but involves more iterations and leads to more complex controllers. fJ. analysis techniques which use a structured uncertainty description can be employed to establish a link between the accuracy of the mathematical model and the achievable performance and stability margins of the actual control system. An important future topic is the investigation of high-bandwidth applications, where the servovalve dynamics must be incorporated into controller synthesis.
(9) for all perturbations satisfying lIalloo < 1, an equivalent robust stability problem can be formulated. In Fig. 12 the weights of the Hoc design are used to bound the gains of T and Z . It is an important objective to investigate how
much the neglected high frequency dynamics and real parametric uncertainty constrain the achievable performance. System analysis is used to keep the effort for parameter identification on a minimum level under given performance specifications. As a prerequisite for analysis, the bounds on the different kinds of uncertainty must be found . In the considered application of an aircraft actuator, the maximum and minimum values of system and return pressure are given by the aircraft manufacturer. The parameters B. v , I and dm can be obtained relatively easy by measurement and calculation with reasonable uncertainty ranges. An upper bound on the dynamical uncertainty ~ can be calculated from worst-case estimates of the parameters w. v , D. v , Wh and D h . Calculations show that the dynamical uncertainty suffices to slightly violate the performance bounds imposed by Wz and WT . Simulations and test rig experiments are presently prepared to find out how accurate fJ. analysis allows to determine the limits of linear controller design which is based on the simple plant description presented in this paper.
REFERENCES Balas, G. J. , J . C. Doyle, K. Glover, A. Packard and R. Smith (1995). fJ.Analysis and Synthesis Toolbox User's Guide. The Math Works. Gahinet, P., A. Nemirovski, A. J . Laub and M. Chilali (1995) . LMI Control Toolbox User 's Guide. The MathWorks. Geering, H. P. (1994) . Regelungstechnik. Springer Verlag, Berlin, 3rd edition. Guo, L. and H. Schwarz (1989). A control scheme for bilinear systems and application to a secondary controlled hydraulic rotary drive. In: Proc. 28th IEEE Conference on Decision and Control. Tampa, Florida. Haas, H.-J . (1989) . Sekundargeregelte hydrostatische Antriebe im Drehzahl- und Drehwinkelregelkreis. PhD Thesis. RWTH Aachen . Ivantysyn, J . and M. Ivantysynova (1993) . Hydrostatische Pumpen und Motoren . Vogel Verlag, Wiirzburg, Germany. Maciejowski, J . M. (1989) . Multivariable Feeback Design. Addison-Wesley, Reading , Mass. Metzner, F. (1985) . Kennwerte der Dynamik sekundargeregelter Axialkolbeneinheiten. PhD Thesis. HsBw Hamburg. Murrenhoff, H. (1983) . Regelung von verstellbaren Verdrangereinheiten am J{onstantdrucknetz. PhD Thesis. RWTH Aachen . Weishaupt, E. (1995). Adaptive Regelungskonzepte fur eine hydraulische Verstelleinheit am Netz mit aufgepragtem Versorgungsdruck im Drehzahl- und Drehwinkelregelkreis. PhD Thesis. RWTH Aachen .
If position control is considered , the saturation
nonlinearity must be incorporated into stability analysis to detect a possible danger of limit cycles. After normalizing it, the saturation element may be modeled as a linear, time-invariant uncertainty ~. with I~.I < 0.5. This makes it tractable within the fJ. framework together with other uncertainties. Further study is in progress to evaluate the conservativeness of this approach .
p
L _ _
..J----;~
Figure 12. Block diagram for fJ. analysis of performance robustness
6. CONCLUSIONS Presented above are theoretical and experimental aspects of the robust position and speed control of a variable displacement hydraulic motor operated 478
ROBUST POSITION AND SPEED CONTROL OF VARIABLE DISPLACEMENT HYDRAULIC MOTORS H. Berg Department of Measurement and Control (Prof. Dr.-Ing. II. Schwarz, Prof· Dr.-Ing. M. Ivantysynova), Faculty of Mechanical Engineering, University of Duisburg, D-47048 Duisburg, Germany Phone: +49 (203) 379 - 1576; Fax: +49 (203) 379 - 3027, email: [email protected]
Abstract: This paper addresses the position and speed control of a variable displacement hydraulic motor (VDHM) at a constant pressure net (also referred to as "secondary control") . The system specifications relate to the application in future large aircraft, where this modern type of hydraulic motors can help to reduce weight and energy consumption of flight control actuators. LQG/LTR and Hoo design approaches are considered, emphasizing the problem of sufficiently precise position control in the face of non linear friction. Simulation results are compared with experiments performed a VDHM test rig. Finally, the use of J..I analysis techniques is briefly outlined. Keywords: Hydraulic motors, LQG control, H 00 control, J..I analysis, position control
1. INTRODUCTION
(1983) , adaptive and fuzzy based designs have been investigated by Weishaupt and Backe (1993) . Guo and Schwarz (1989) have developed a nonlinear approach to speed control using a bilinear system description.
The use of energy and weight saving technologies for hydraulic drives is of increasing importance in large aircraft design as well as other industrial applications. Variable displacement hydraulic motors (VDHM) can help to significantly reduce the energy consumption of aircraft actuation systems. Examples are the power drive units of high lift systems and the horizontal stabilizer actuator. Both are presently driven by valve operated fixed displacement motors with poor efficiency due high pressure losses in the control valves. VDHMs automatically adapt the hydraulic input power to the required mechanical output power. Moreover, they are able to deliver hydraulic power back to the supply system in case of aiding loads. While speed and torque control of VDHMs are used in numerous industrial applications, position control has received only few attention up to now. Both speed and position control have been subject to university research since the early eighties. Classical linear design methods have been employed by Haas (1989), Metzner (1985) and Murrenhoff
In this paper, emphasis is given to some aspects which have not been considered so far. For the use in an aircraft actuator a solution must be found that meets all functional and safety requirements on the one hand and on the other hand provides a control structure that is robust in the face of plant uncertainties and of low order to minimize the computational effort. A means is presented by which the problem of limit cycles in the position control loop can be handled without the use of switching integrators or other nonlinear techniques. There is a number of industrial applications apart from aircraft systems where the issues discussed here are relevant.
473
2. THE VARlABLE DISPLACEMENT HYDRAULIC MOTOR (VDHM)
• The nonlinear equations of flow are linearized about the operating point p* ~ (Po + Pr) , Y;v = 0 :
Bs v is the flow coefficient of the servovalve. • The "hydraulic dynamics" of the transfer function between Q and xp (pressure buildup, piston dynamics) may be approximated by a second-order linear model with the parameters Wh and D h : (3) • In the equation of motion for the motor only linear viscous friction is modeled (coefficient dm ):
Figure 1. Variable displacement hydraulic motor Fig. 1 shows a scheme of the VDHM at a constant pressure supply. For the test rig experiments a cam-type axial piston motor "Rexroth A10VSO" with a maximum motor displacement of Vm max = 28 cm3 /rev is used. The swash plate of the motor (i.e. the motor displacement) is controlled by a single-acting cylinder. The flow to the cylinder is controlled by an electrohydraulic servovalve. Depending on the sw ash plate position and the torque at the output shaft , the unit works either as a pump or a motor. In case of aiding loads, the system delivers hydraulic power to the supply system. A digital controller implemented on a PC calculates the input signal for the servovalve using motor shaft speed, motor shaft position and cylinder stroke feedback. The controller limits the maximum motor output speed . A sampling frequency of 1 kHz allows continuous time techniques to be used for design and analysis.
Vm I inr-
max (Po
- Pr)
Xp
_ d
.
(4)
m
21rXp max
The nominal values of the parameters PO,Pr, V.v,Bsv , Ap,xpmax, Vmmax , I and dm must be determined for controller synthesis, see Table 1. The servovalve dynamics and the hydraulic dynamics can be covered by a complex uncertainty in the j.J. analysis of the controlled system and it suffices to obtain rough estimates of the parameters. The servovalve dynamics are assumed to act in a frequency region around 80 Hz (440 rad/s)' the hydraulic dynamics lie approximately one decade higher. Due to the relatively low bandwidth requirements of an aircraft actuator, both are neglected in the linear model for controller synthesis. ~ ···· · ·······················se;v~;;i';;m:i ·s:';~Pb~·';~CYiindOt \
3. MATHEMATICAL MODEL ~
In industrial applications it is usually desired to find the simplest mathematical system description that is still fully sufficient for controller design and system analysis under given specifications. This keeps the effort for system identification and simulation on a minimum level. Moreover, the controller order is related to the order of the (possibly augmented) plant model, if LQG/LTR or H00 synthesis methods are used. The following linear model of the VDHM can be derived for controller design:
~ .
--. ----. ------------------------------------------------- --------
--~
Motor ;
'--- ---- --- --- ------- --- ---- ----- ---- ---------- --- ------------------ ----
Figure 2. Linear model for controller synthesis The exact values of Vsv and the geometrical quantities piston stroke xp max , piston area Ap and maximum motor displacement Vm max are given by the manufacturer. All other parameters can take values within certain intervals, either because of imprecise identification or because they vary with the operating conditions. More information about system analysis is given in Section 5. For technical reasons , the system pressure amounts to less than
• The servovalve dynamics may be approximated by a second-order linear model with the parameters Wsv and Dsv :
(1)
474
half of the nominal value for this type of motor. This makes the system more susceptible to saturations of Ysv or xp . The linear model from Fig. 2 is controllable, observable and minimum phase.
(4) Rejection of disturbances caused by load, friction and moment pulsation of the motor: Z = cp/ M/ < 0 dB for all frequencies . (5) No overshoot of the command and load step responses of the position. (6) No integral action of the speed controller. (7) Control effort due to a speed command step of 150 rad/s without external loads: xp < 0.5·
Table 1. Nominal parameters of the linear model System pressure Po Return pressure Pr Static valve gain v'v Flow coefficient Bsv Piston area Ap Maximum piston stroke
llO .10 5 Pa 3 .105 Pa 6 .10- 5 m/V 3.05 .10- 4 m3 /s/VN 4.05 .10- 4 m 2 17.72 .10- 3 m
X pmax ·
(8) Stability margin of the nominal system: ISI < 1/0.8 (= 2 dB) for position and speed control. (9) A peak value of the complementary sensitivity T of less than 0.1 dB for speed control. (10) A high roll-off rate of the complementary sensitivity T shortly after the crossover frequency to ensure good rejection of measurement noise and good robustness against unstructured multiplicative uncertainty such as neglected dynamics . (ll) A low sampling frequency and a simple controller structure.
Xpmax
Maximum motor displacement Vm max Moment of inertia I Viscous friction coefficient dm
28 .10- 6 m 3 /rev 7.85 .10- 3 kgm 2 2.9 .10- 2 Nms/rad
Requirement 6 is necessary for position control , because · integrating behaviour of the controller causes undesired control action trying to achieve zero steady-state position error of the motor shaft in the face of nonlinear friction (limit cycles) . Alternatively, an integrator can be used which is switched off for small position errors. Possible implications of this approach such as cyclic action and oscillations are described in (Weishaupt, 1995). If the bound from requirement 3 turns out to be insufficient, prefilters can be introduced to further reduce the steady-state errors.
4. ROBUST CONTROLLER DESIGN The main intention of this work is to design a control system for the motor shaft position. An important constraint is given by the maximum allowed motor shaft speed which is chosen here as CPc max = 150 rad/s . Moreover, it is demanded that the swash plate position can be controlled independently. On the one hand this is necessary for safety reasons , on the other hand it prevents the control system from undesired action due to leakage of the swash plate operating cylinder. Fig. 3 shows a scheme of the entire position control system. Fij denotes the transfer function from signal i to signal j. A saturation element is used to limit the positioning speed by bounding the command input CPc for the speed control loop. For large values of the tracking error e, the saturation element gives the constant speed command CPc max . For values of e beneath a defined threshold, it works as a linear gain ks in the position control loop.
In the following the main design steps are presented . The speed controllers are calculated using a state-space representation of the linear plant model from Fig. 2 with the nominal parameters from Table 1. As shown in Fig. 3 , a PI compensator is selected to control the piston position Xp:
C Xp
= 300 (s + 5;ad/s) .
(5)
The integrator provides good rejection of disturbances that are caused by a servovalve offset and by the swash plate operating force. Without this , especially the servovalve offset , which usually occurs after some operating time, has a strong impact on the tracking error of the motor speed and position . The swash plate operating force is a speed dependent , oscillating force which is caused by internal mechanisms of the hydrauli~ motor. It has a mean absolute value larger than zero (Ivantysyn and I vantysynova, 1993) .
Figure 3. Block diagram of the position control system The following design specifications relate to the application in an aircraft actuator: (1) A bandwidth of the position control loop of 2-4 rad/s . (2) A bandwidth of the speed control loop of 2040 rad/s. (3) A steady-state error in the face of position and speed command steps of less than 5 %.
After the controller for the swash plate operating cylinder has been designed, the next step is to 475
find a control law for the motor shaft speed . The plant is a series connection of the controlled operating cylinder and the motor dynamics , represented by a first-order transfer function (Eqn. (4)) . The LQG/LTR method is used to design the speed control system based on open loop considerations, see (Maciejowski, 1989) or (Geering, 1994). It requires only standard software for linear-quadratic regulator and Kalman filter design. A loop gain roll-off of -60 dB/decade shortly after the crossover frequency yields good robustness against multiplicative uncertainty (e.g. neglected dynamics) , minimum control action and good attenuation of measurement noise.
(7)
must remain below a certain bandwidth to keep the whole system stable. To maintain the good performance and stability margins of the LQG/LTR design, a value of Ta 0.2 s is selected . Fig. 8 shows calculated and measured command step responses of the speed control system with and without augmentation. The tracking error of the system without ·augmentation is caused by the command step and the coulombic friction moment acting as a load disturbance of M/ ~ 3 Nm. Slight oscillations due to moment pulsations of the motor can be seen in the step responses from Fig. 8. Their frequency of approximately" 15 rad/s corresponds to the peak of Z in the crossover region (Fig. 7) . Better rejection of these disturbances can only be obtained by raising the bandwidth.
=
Ib) 151 (-). In 1--)
la) IU 1-). ILFI 1--)
SO
------
iD ~
.0
c;
iD
~-20
0
.0
c;
O·r-----~------~----~----~
-40
-SO
10':
10' Frequency [rad's]
10'
10':
10' Frequency [rad's]
10'
-50
Figure 4. (a) Loop gains of the Kalman filter and the whole system; (b) Sensitivity and complementary sensitivity The controller transfer function (after one polezero cancellation) is
G . _ 0.0128s + 0.6646 '" - s2 + 165.24s + 12451·
-200
-2~~----~------~----~----~ 1~2 1~ 1~ 1~ 1i Frequency [radial
(6)
Figure 6. Influence of ka on the tranfer function xp (s) /I{;e (s) for the open speed control loop, Ta = 0.2 s
It may be further reduced to a PD controller which has significantly worse characteristics due to a loop gain roll-off of only -20 dB/decade. The damping can be increased by shifting the gain of Gzp from 300 to 350. The low bandwidth requirement results in a large steady-state error and poor disturbance rejection properties. Significantly improved results can be obtained by augmenting the structure with piston position feedback (Fig. 5) . The choice ka 1/G.,o(0) leads to integrating behaviour of the system xp (s) /I{;e (s) for the open speed control loop. This must be avoided to prevent limit cycles of the position control. In Fig. 6 gain plots of xp (s) Ne (s) for different values of ka are shown. With the value ka 0.9· l/Go, sufficiently small steady-state errors can be achieved without causing limit cycles. The first order transfer function
=
=
Figure 7. Load sensitivity Z = I{; / M/
Hoc loop shaping techniques allow to design a controller directly for the augmented plant. Fig. 9 shows the speed control system with weighting filters. Appropriate weights can be derived from the experience with the LQG /LTR design . It turns out that the loop shaping configuration from Fig . 9 is useful to investigate the trade-off between the "amount of integral behaviour" of the system
Figure 5. Block diagram of the augmented speed control system 476
measured (-), linear simulation (--)
Closed speed control loop T.
Open position control loop
dB
dB A
+
J~radlS ~ c) oL ~,o i ... I
4
10
k,
I
I
~radls
-t
Figure 10. Transition from
... .PiSton positiOn .. . -- - -- - --:---
"
0.1
0.2
0.4
0.6
0.8
1.2
Time[s]
command step responses,
Figure 11. Normalized measured command step responses, t.pc = 300 rad, LQG/LTR controller
I~L < 1
certainty descriptions can be found in (Balas et al., 1995) and (Gahinet et al., 1995). The goal of system analysis is to check whether the desired performance in terms of bandwidth, disturbance attenuation and stability margins can be sustained in the face of plant uncertainties. First of all, only speed control is considered where the uncertainties can be categorized in two groups:
Figure 8.
Normali~ed
subopt. H_ - Problem:
• Dynamical uncertainty due to the neglected servovalve dynamics W sv , Dsv and "hydraulic dynamics" Wh, Dh which can be together taken into account in a complex, frequency dependent, multiplicative uncertainty ~ (s) . • Uncertainty of physical parameters either because of imprecise knowledge of their values or because their values change during operation. The parameters (Po - Pr), Bsv, I and dm can take values within a certain range and are modeled as real, multiplicative uncertainties
Figure 9. Block diagram with weights for Hoc design
xp (s) /
8i . All uncertainties are assumed to be linear and time-invariant which represents a strong simplification of the real situation. A frequency dependent upper bound is given on I~I, constant upper bounds are given on 8i . Fig. 12 shows a linear fractional interconnection of the nominal plant P and the structured uncertainty description represented by the block-diagonal matrix
Appropriate scalings to normalize the uncertainty bounds are inserted into P such that 11 A 1100 < 1 for all admissible perturbations. The system
5. J.L ANALYSIS OF THE CONTROLLED SYSTEM
In this Section, the procedure for J.L analysis of the controlled system is briefly outlined. A tutorial exposition of the J.L approach and the required un477
at a constant pressure net. The specifications relate to a low-bandwidth application in an aircraft actuator. An LQG/LTR controller design is discussed which is based on a simple mathematical plant description with both dynamical and parametric uncertainty. A control structure is proposed that allows sufficiently precise control of the motor shaft position avoiding limit cycles. Simulations and experiments show that satisfactory performance can be achieved with this design. An Hoc loop shaping approach is better suited to determine the design limits, but involves more iterations and leads to more complex controllers. fJ. analysis techniques which use a structured uncertainty description can be employed to establish a link between the accuracy of the mathematical model and the achievable performance and stability margins of the actual control system. An important future topic is the investigation of high-bandwidth applications, where the servovalve dynamics must be incorporated into controller synthesis.
(9) for all perturbations satisfying lIalloo < 1, an equivalent robust stability problem can be formulated. In Fig. 12 the weights of the Hoc design are used to bound the gains of T and Z . It is an important objective to investigate how
much the neglected high frequency dynamics and real parametric uncertainty constrain the achievable performance. System analysis is used to keep the effort for parameter identification on a minimum level under given performance specifications. As a prerequisite for analysis, the bounds on the different kinds of uncertainty must be found . In the considered application of an aircraft actuator, the maximum and minimum values of system and return pressure are given by the aircraft manufacturer. The parameters B. v , I and dm can be obtained relatively easy by measurement and calculation with reasonable uncertainty ranges. An upper bound on the dynamical uncertainty ~ can be calculated from worst-case estimates of the parameters w. v , D. v , Wh and D h . Calculations show that the dynamical uncertainty suffices to slightly violate the performance bounds imposed by Wz and WT . Simulations and test rig experiments are presently prepared to find out how accurate fJ. analysis allows to determine the limits of linear controller design which is based on the simple plant description presented in this paper.
REFERENCES Balas, G. J. , J . C. Doyle, K. Glover, A. Packard and R. Smith (1995). fJ.Analysis and Synthesis Toolbox User's Guide. The Math Works. Gahinet, P., A. Nemirovski, A. J . Laub and M. Chilali (1995) . LMI Control Toolbox User 's Guide. The MathWorks. Geering, H. P. (1994) . Regelungstechnik. Springer Verlag, Berlin, 3rd edition. Guo, L. and H. Schwarz (1989). A control scheme for bilinear systems and application to a secondary controlled hydraulic rotary drive. In: Proc. 28th IEEE Conference on Decision and Control. Tampa, Florida. Haas, H.-J . (1989) . Sekundargeregelte hydrostatische Antriebe im Drehzahl- und Drehwinkelregelkreis. PhD Thesis. RWTH Aachen . Ivantysyn, J . and M. Ivantysynova (1993) . Hydrostatische Pumpen und Motoren . Vogel Verlag, Wiirzburg, Germany. Maciejowski, J . M. (1989) . Multivariable Feeback Design. Addison-Wesley, Reading , Mass. Metzner, F. (1985) . Kennwerte der Dynamik sekundargeregelter Axialkolbeneinheiten. PhD Thesis. HsBw Hamburg. Murrenhoff, H. (1983) . Regelung von verstellbaren Verdrangereinheiten am J{onstantdrucknetz. PhD Thesis. RWTH Aachen . Weishaupt, E. (1995). Adaptive Regelungskonzepte fur eine hydraulische Verstelleinheit am Netz mit aufgepragtem Versorgungsdruck im Drehzahl- und Drehwinkelregelkreis. PhD Thesis. RWTH Aachen .
If position control is considered , the saturation
nonlinearity must be incorporated into stability analysis to detect a possible danger of limit cycles. After normalizing it, the saturation element may be modeled as a linear, time-invariant uncertainty ~. with I~.I < 0.5. This makes it tractable within the fJ. framework together with other uncertainties. Further study is in progress to evaluate the conservativeness of this approach .
p
L _ _
..J----;~
Figure 12. Block diagram for fJ. analysis of performance robustness
6. CONCLUSIONS Presented above are theoretical and experimental aspects of the robust position and speed control of a variable displacement hydraulic motor operated 478