Disturbance observer control for AC speed servo with improved noise attenuation J. Vonkomer ∗ I. B´ elai ∗∗ M. Huba ∗∗ ∗∗
∗ Software, R&D, VONSCH s.r.o. (e-mail:
[email protected]) STU FEI Bratislava (e-mail: mikulas.huba,
[email protected])
Abstract: The paper deals with a noise attenuation design of a modification of the disturbance observer (DO) based PI control for an AC servo drive with the dominant first-order dynamics. The achieved results are compared with the traditional PI control. The DO based filtered PI (DO-FPI) controller consists of the P control with adjustable filter. It is augmented with an integral action based on load reconstruction using equally filtered plant dynamics inversion. The availability of broadly scalable filters requires a simple one-parameter based tuning. In looking for a possibly good noise attenuation, it allows changing the filter order and time constants by keeping nearly the same speed of disturbance responses as in the PI control. An application to the speed servo control with an incremental position sensor brings to light interesting properties of the compared solutions. The DO FPI and PI control give nearly equally robust solutions. However, in case of PI control it is at the expense of significantly increased torque ripple. Keywords: PI control, disturbance observer, AC speed servo drive, performance, robustness, noise attenuation 1. INTRODUCTION Recently, numerous papers have appeared dealing with a noise attenuation motivated control (see e.g. Segovia et al. (2014); Huba (2015) and the references therein). For many years the PI control has represented the highest percentage of all applications being used. Its main application domain corresponds with the first order plant control. Despite its maturity, one may still find here some aspects insufficiently explored. This contribution focuses on impact of typical servo nonlinearities as the position quantization in incremental sensors, or the PWM in the torque generator resulting in an increased noise level for decreasing sampling periods. The traditional PI control is compared with a disturbance observer (DO) based filtered PI control (DO-FPI) with scalable filtering properties. The DO-FPI control (Huba, 2013a; Huba and B´elai, 2014) enhanced the DO based control (Ohishi et al., 1987; Schrijver and van Dijk, 2002) by a filter also introduced to the stabilizing controller, which has brought about high tuning flexibility with significant noise attenuation increase with respect to traditional PI control (Huba and B´elai, 2014). One of the key aspects of the paper lies in a modified cost function used in the controller tuning. Instead of full plant input and output trajectories, used in such optimal problems like the LQR design, the new approach evaluates deviations from ideal shapes of transient responses achieved on the plant input and output. These, together with the traditionally used IAE (Integral of Absolute Error), are used to formulate a new tuning scenario. Another new key point is a modified design paradigm. In the traditional two-loop structure based paradigm (Schrijver and van Dijk, 2002; Yun et al., 2013), design of an internalloop DO based compensator is carried out in an openloop framework for robustness to force a current system
to become the given nominal model. Then the externalloop controller design may be carried out for the nominal model. In the DO-FPI tuning, modularity is imposed by designing firstly a stabilizing P controller combined with an optimal low-pass filter design. After that the loop is augmented by a DO based disturbance compensation. This, in the nominal case, does not change the closed loop stability or performance. The DO base controllers establish alternatives to such approaches as the Active Disturbance Rejection Control (ADRC) Han (2009), or to the intelligent PID (iPID) (Fliess and Join, 2013) and GPI observer based control (Sira-Ramirez et al., 2013). These are appropriate especially for control of systems under measurement and quantization noise and high performance requirements. Whereas in a nonlinear observer based disturbance estimation in ADRC an increased noise attenuation is achieved by a differentiator consisting of two lags with different time constants. This finally corresponds to a second order filter. Integral filters used in iPID control may be shown to offer better noise attenuation than DO with a simple first order filter. In this paper all filtered structures may work with an arbitrary filter order. As in Huba and B´elai (2014), a significant improvement is already achieved by the 2nd order filters. The paper is structured as follows: Section 2 discusses use of 2DOF P-control for the first order plants and the relevant time and shape related performance measures. It should be noted that P control is the basis for disturbance observer based control with a filtered feedback signal analyzed in the sections 3 and 5. Section 3 deals with the optimal P-controller tuning considering intentionally introduced filter dynamics. The controller is tuned optimally in terms of the fastest response without any
overshoot. The P-control is later (in section 5) supplemented by the disturbance observer. Section 4 gives a short overview of basic methods used for compensation of acting disturbances. Two methods are considered: a) Disturbance elimination by an integral action of a 2DOF PI control; b) disturbance reconstruction and compensation by a disturbance observer. For the 2DOF PI control, controller tuning yielding a required closed loop performance is considered. Section 5 analyzes the nominal optimal tuning of the DOFPI control. For a chosen low-pass binomial filter with the degree n its time constant is expressed as a function of the required closed loop performance. Subsequently, the proposed filter dynamics is taken into account in calculating the stabilizing P controller gain. New alternative closed loop model based structures with a disturbance feedforward are treated in Section 6. Properties of both noiseattenuation motivated controllers are then demonstrated by an illustrative example in Section 7 and summarized in Conclusions.
smoothness result into a piecewise monotonicity summarized by the theorem: Theorem 1. MO transients y(t) at the output of linear first order plants running from a steady state y0 to a new steady state y∞ correspond to a one-pulse (1P) input u(t) consisting of two MO intervals. For stable plants (a > 0) MO output responses may also be achieved by MO input.
2. CONTROL OF NOMINAL FIRST ORDER PLANTS
As the performance indexes for evaluating deviations from required shapes, we use normed Total Variance (TV) measures proposed by TV (Skogestad, 2003) modification Z ∞ X du dt ≈ TV = |ui+1 − ui | (4) dt 0 i
Controller structures will be derived for the dominant firstorder plant with the input disturbance di dy/dt = Ks (u + di ) − ay (1) u being the controller output, y being the plant output. The “pole-zero form“ transfer function Y (s) Ks S(s) = = (2) U (s) di =do =0 s+a allows a uniform consideration of all stable, unstable and integral plants and an easy comparison with the model free control corresponding to a = 0. 2.1 2DOF pole assignment P-controller Compensation of measurable disturbances with a Pcontroller Huba (2013a) yields u = KP e + u∞ ; e=r−y (3) KP = (1/Tr − a) /Ks ; u∞ = a(r − do )/Ks − di Thereby u∞ corresponds with the static feedforward control keeping y∞ = r in steady states. The closed loop with plant (2) becomes stable with the time constant Tr > 0 (a closed loop pole α = −1/Tr < 0). Such a pole assignment control aim may be defined as ”to bring the output y from an initial value y0 to a new setpoint reference value r with the shortest possible time constant Tr ”. This, however, gives no information about the optimal controller tuning. Since the output and the disturbances are usually not measured, the plant parameters not exactly known and the output measurements corrupted by measurement noise and nonmodelled dynamics, the control problem is far from being completely solved yet. 2.2 Expected control performance After setpoint steps, the simplest possible output ys (t) is a monotonic (MO) one Huba (2013b). Monotonicity plays a central role in mathematics, as well as in physical and technological requirements. It also includes such requirements as no or limited output overshooting, or non-oscillatory transients. Restrictions on the corresponding plant input
For a single integrator, the one-pulse (1P) input corresponding to the required S-shaped MO output may be simply found by the output derivation (inversion of its dynamics). Still, when requiring a relatively fast closed loop dynamics, the 1P control is typical for all the first order plants. When evaluating a disturbance step response yd (t) one has to note that a disturbance step makes the plant output rise (fall) and the controller needs some time for balancing its effect and to reverse the output to move back to the reference value. Thus the evaluation of MO output increase (decrease) starts right after its turnover, which in total corresponds with a 1P shape.
Contribution of the superimposed oscillation in 1P dominant control with um = max{u} may be expressed by X uT V1 = |ui+1 − ui | − |2um − u∞ − u0 | (5) i
uT V1 = 0 just for strictly 1P response, else uT V1 > 0. Low uT V1 means a low additional control effort, high frequency oscillations, noise impact, actuator wear, etc. This measure will be used for quantifying overall excessive control effort appearing in motion control in the form of torque ripples. Speed of transients at the plant output will be evaluated by the IAE (Integral of Absolute Error) index Z ∞ |e(t)| dt (6) IAE = 0
3. FILTERED P-CONTROL A first order plant approximation (1) used for the Pcontroller structure design is insufficient for its reliable tuning and use. As in Rivera et al. (1986); Huba (2013a); Huba and B´elai (2014), because of different filtering properties achievement, the plant model (2) will be extended to Sn (s) by a binomial low-pass filter Qn (s) Ks 1 Sn (s) = n ; Qn (s) = n (7) (s + a) (Tf s + 1) (Tf s + 1) 0 < Tf
