Pulse-Width and VSS-Modulated Controllers in Motion Systems

Copyrigth © IFAC Motion Control for Intclligcnt Automation Pcrugia. Italy. October 27 ·29. 1992

PULSE·\VIDTH AND VSS-MODULATED CONTROLLERS IN MOTION SYSTEl\1S P.P.J. van den 1l0SCH, A.E. van den (; ROEI' and H.R. VISSER

Delft UniversilY of Technology, ElccLIical Engineering Dcpc P.O.Box 5031,2600 GA Dclfl, The NcLhcrlands Abstract. Discrete pulse modulation is necessary to convert a continuous control signal into a binary signal to control switches. This paper compares state feedback modulated with PWM with a controller based on variable structure systems (VSS) equipped with hysteresis. Simulations with an AC-DC converter are used to calculate the frequency spectra. It is shown that VSS with hysteresis reduces the peak hannonic values in the frequency spectra considerably. Compared with other peak-value reducing measures, VSS yields attractive results during transients and in ACoperation. Keywords. Discrete pulse modulation, PWM, Pulse width modulation, VSS, Variable structure systems, boost buck converter. This paper illustrates the differences in using PWM or VSS with respect to ripple , thermal stress, audible noise and control perfonnance . It is shown that PWM yields a sharp peak-like spectrum and VSS a flat broad spectrum. VSS limits the ripple and is relatively insensitive for parameter variations. In contrast, the spectrum and so the audible noise and the thennal stress can not always be guaranteed.

INTRODUCTION In motion control systems electronic converters are used for controlling the power flow between power source and electrical machine. To reduce losses, power electronic switches are used. Control actions are based on measurements of some or of all states (currents and voltages) of the converter. Once an appropriate feedback signal is calculated, the switches are activated. The control signal is implemented, i.e. modulated into an binary on- or off-signal. Two different approaches can be distinguished to realize this modulation, namely: - time driven, e.g. PWM or PFM modulation (Pulse Width or Pulse Frequency Modulation) or - state driven, via a test on the value of some signal, e.g. VSS (Variable Structure System). Both PWM and VSS have to deal with a number of mutually-conflicting requirements: - limitations on ripple of currents and voltages; - limitation of acoustical and audible noise; - legal requirements pose constraints on the conduc ted and radiated electromagnetic interference (EMI); - limitations on the thermal stress owing to the switching losses of the semiconductor switches; These last three requirements pose constraints on the frequency spectrum of the currents. To satisfy these requirements, different approaches are followed: - The designer of PWM selects the switching frequency and the values of the components, which yield the ripple. - The designer of VSS selects the ripple and the values of the components which determine the spectrum of the signals.

In an example the characteristics of PWM and VSS are illustrated with respect to spectrum, ripple, parameter sensitivity and control performance.

DISCRETE PULSE l\fODULATION Signals in a control system are in general continuous in amplitude and either continuous (analogous) or discrete (sampled date) with respect to their time behaviour. As soon as switches are introduced to control the power flow in a converter, a large amplitude quantisation is introduced. The continuous control signal uc(t) is converted into a binary signal lI.t(t) , representing whether the switch is open or closed. Although switches are to preferred for their efficiency, they introduce a severe nonlinearity. By utilizing a high switching frequency, the effects of this nonlinearity on the performance are reduced. However, switched signals introduce also high frequencies, which introduce audible noise and EMI , and thermal stress for the switch devices. Consequently, a compromise has to be made between the degradation of the control perfonnance and the introduction of high frequency components in the currents and voltages of a converter. The basic modulation scheme to convert a continuous control signal uc(t) into a binary signal ud(t) is using pulse-

265

VA~ DE~

width modulation (PWM). illustrated in figure 1.

BOSCIII'.PJ .• VA~ DE." GROEF A.E.. VISSER ilK

A PWM

signal is

u (t)

Fig. 1 PWM with fixed To

The pulse frequency fo= lrro determines the basic frequency component. The duty cycle d is proportional to uc(t), so d(t)=a.u,,(t). Consequently, the average value n.t'" of ud(t) for fixed To, amplitude 1 and constant uc' yields

As long as the bandwidth fc ofuc(t) is small compared with fo' fc
= L 2·c.·cos(n·2·lt·fJ .-0

With

c.

CD

=

the nib harmonic: sin(n·lt·d) n·lt

n>O

The spectrum Pit) ofn.t(t) is determined by poet) and fc' It is much broader than the spectrum Pe(t) of u,,(t), as illustrated in figure 2.

f

Fig. 2 Spectra P jt) and Pd(t)

To reduce acoustical noise and EMI, the peak: harmonic values in the spectrum Pit) of n.t(t) or of some current in the converter have to decrease. Especially, these currents are expected to generate the largest contributions to acoustical noise and/or EM!.

266

Much research has been and is devoted to reduce these peak: harmonics or to influence the shape of the frequency spectrum Pit). Enjeti et al. (1988) have given a survey of these techniques. These techniques use different approaches to modify the fixed basic frequency fo of the PWM. - Some techniques are based on extensive offiine calculations to reduce some harmonics or the peak: values. For example, Wang and Sanders (1992) have proposed an optimally programmed PWM, such that the peak: harmonics are reduced. They combine several periods (about 32) and determine for each period k an appropriate value for both ~ and T k. The average value of Tk equals To' These vales are stored in an EPROM and retrieved by the modulator. Using such an approach reduces the peak: harmonic with about 65 % at the cost of an increase at lower and higher frequencies. The scheme is effective in steady state operations. - Another approach adds some uncorrelated signal in determining the switching intervals T k' so T k = f(To' ucCt), 1/(t», where 1/(t) is an independent noise source satisfying a required probability distribution. Tanaka and Ninomiya (1992) give a theoretical analysis of the influence of noise 1/(t) on the spectrum Pit). They show that a reduction of more than 50% of the peak harmonic value can be achieved. - Venkataramanan and Divan (1992) describe and analyse the use of the Delta (current regulated delta modulator) and the Sigma-delta modulator for obtaining a PWM like signal ud(t) . These modulators are synchronously sampled comparators of the error signal in the current (delta) or voltage (sigma delta) control loop. Their profit arises from using the values of n.t(t) of all three control loops active in controlling a three-phase AC motor. Again some harmonics and the total harmonic distortion (THD) are reduced. These approaches influence the timing instances in such a way that the peak harmonics of PWM with a fixed frequency fo are reduced. In general, the proposed techniques achieve this goal. The price to be paid is an increase in both lower and higher frequencies compared with PWM with fixed frequency fo ' In this paper we propose a modulation scheme arising from the application of VSS (Variable structure systems). VSS has been proposed in the sixties for improving the robustness of existing linear feedback controllers. Indeed, it can be shown that a VSScontrolled system exhibits perfect parameter insensitivity and disturbance rejection for some classes of variations and disturbances (Utkin, 1977). VSS allows a complete theoretical analysis and design of systems equipped with actively controlled switches.

PULSE-WID'nI A:'-TI VSS-\10J)CLATED CO\"IlWLI n{S ,'\ WnlO:\ SYS1D1S

VSS is state driven, so an asynchronous pulse pattern is expected. In sliding mode, a(e) = 0 and o{e) = 0, equivalent system behaviour is:

the

VARIABLE STRUCTURE SYSTEMS VSS is used to design variable structure controllers (VSC), e.g. Utkin (1977). A short review explains the basics of VSC. Although VSS can be applied to nonlinear systems and may yield a nonlinear control law u=G(x,t), now only a linear model and a linear control law is evaluated. An nib order switched system is described as:

x = A·x + x E RD E Rm A E RD.D B E Rn.m DE RD U

When a(e)=O the controlloo system exhibits a reduced-order behaviour. The states x(t) have to statisfy the n model equations (i = Ax + bu + d) and the the m switching surface constraints (a(e) =0). Consequently, the order is reduced from n to n-m. Several techniques exist to derive an appropriate feedback matrix G, e.g. via LQ-like function minimization or via pole placement of the resulting nm poles of the closed loop system. (Utkin, 1977).

B·u + D state vector input vector ui E {O,l} state transition matrix input transition matrix constant vector

VSS FOR PULSE GENERATION In the previous section VSS is introduced as an analysis and design method for continuous systems equipped with actively controlled switches. VSScontrolled systems yield a good dynamic performance, a good robustness against parameter variations and unmodelled disturbances. These nice characteristics are only obtained if the switch is allowed to operate at high frequencies. In practical applications, the dynamic switching losses of the semiconductors prohibit high switch frequencies. Constraints are imposed on the average switch time T. and the minimum on TOIl and off Toff times. With these constraints imbedded in the problem formulation. ' no analytical solution exists of the VSS problem. Still. the basic theory is applied with some modification to satisfy the stated timing constraints. Simulation is used to verify the results.

VSC defines a switching function a (E Rm) in statespace, defined by a feedback matrix G (E Rm.D) and the error e between a setpoint xoc• and x: a(e) = G.e = G·(x - x...)

o(e) - 0

o(e)>O u-

y ,

'.

o(e)
Fig. 3 Switching surface in state-space.

The switching surface a(e)=O is determined by the design variable G and elucidated in figure 3. It defines the behaviour of the controlled system. Each switch i, i = I,m, is defined via: ai(e) ai(e)

< >

0 then u i = 0 then u i =

~+ ~-

A necessary condition for VSC is that all trajectories will move towards the switching surface a(e)=O and, subsequently, will move along a(e) =0 to the required setpoint x... or mathematically formulated:

A pulse modulator for the ideal VSS controller is proposed equipped with hysteresis and a timing mechanism to avoid too fast or too slow switching. I Hysteresis Hysteresis is added to the system to avoid too fast switching, as elucidated in figure 4. The switching function a(e) = G(e,t) calculated with VSS, is used as input of the hysteresis, yielding a uit) . Because the switch is restricted, the switching surface a(e)=O is no longer followed accurately. The deviation from the surface a(e)=O is determined by the value E of the hysteresis. The signal a(t) will bounce between +E and -E. as illustrated in figure 4. 2 Timing mechanism A timing mechanism is implemented in the pulse modulator that forces the size of the switch intervals to satisfy the minimum-on (ToJ and

267

VA'I. DEi'< BOSClIl'.I'J ., VA,\ DE" GROEF A.E., VISSER II.R.

The current through L. is called i. and the current through Lz is i2 • The switches s. and ~ are the control inputs ud(t).

e(t)

e

-e

n[

Fig. 5 Boost-buck converter

Fig. 4 VSC with hysteresis

It is assumed that the currents are continuous. Then,

the state description of the boost-buck converter, with x=(i .. u.' i2 , uo)T yields: minimum-off (Toff) times. Moreover, a maximum 1"""" is imposed on the values of Tk • Suppose Ton = Toff = Toof' then the timing constraint for the start It of a new pulse becomes with ~_. the end of the previous pulse:

i.

0

li.

~ lio

1

r..

0

0

i.

1 C

0

0

0

u.

0

0

0

0

0

1

~

Lz

1

- -I -

Co

Ro' Co

Uo

SIMULATION EXPERIMENTS In this section a number of simulation experiments are executed. As a consequence of the many pulse modulation processes, the simulation environment should support several timing mechanisms to ensure accurate simulation results. For example, if a PWM has to be realized with T=50 Ils and d has to be realized with an accuracy of e.g. 1 %, then time steps down to 0.5 IlS have to be taken. With a fixed-step numerical integration method this requires an integration step of 0.5 Ils during the whole simulation, requiring excessive amounts of calculation time. Variable-step integration methods are of no use because they cannot accurately detect the switching instants. Consequently, some timing mechanism is required, called state-event detection and solution (Van den Bosch and Visser, 1990). Then, a relatively large integration interval is selected while still the required accuracy in determining and realizing short, asynchronous pulses is achieved. Moreover, if signal post-processing is used, e.g. calculation of spectra with MATLAB, equidistant samples of the signal are needed. Only few simulation programs support these timing facilities, e.g. ACSL and PSI. The proposed pulse modulators are implemented on an AC-DC boost-buck converter of 1000 W (Spies and Bergsma, 1991). In figure 5 this converter is elucidated with the wave patterns of u., u. and U o' The voltage source u. is a rectified 50 Hz, 220 V AC voltage with T.... = 10 ms. The output is the voltage Uo'

26X

U.

L. +

-i.

-~

C

C

0 0

S. , S2

u. 0

u.

Lz 0

L.

[J

0 0

0

E {O, l} 0: open/OFF t: closed/ON

Besides frequency requirements, this converter has to fulfil the following constraints: - power factor: pf ~ 0.98 - maximum ripple on the input current: 1 % of its maximum value - output voltage, without load variation: 199 Volt ~ U o ~ 201 Volt - output voltage, with load variation: 196 Volt ~ U o ~ 204 Volt - maximum output power: P o,mox = 1000 Watt - maximum allowed load variation: maximum .:1io = 4.5 Ampere (Po: 100 - 1000 Watt) Based on these requirements and a selected switch frequency fo of 40 kHz, the passive components of the converter are selected as: L. = 40.5 mH, C = 470 IlF, Lz = 5 mH and Co = 84.5 ~F The following control strategies and modulators are

PCLSE·wnm 1 A\J) VSS·\10DCLATED CO\'IlWLLmS 1\ \10TIO\ SYSTE\'IS

compared: a Linear state feedback with a fixed frequency PWM, with fo=4O kRz and 0.02 < =d< =0.98; b VSS with hysteresis I u(e) I < 1 and timing constraints T m(= 2 p.s and T"'"" = 50p.s.

i]d ' o

Although the mathematical model represents a MIMO system with 2 inputs and 4 states, good performance is achieved in using 2 separate SISO controllers. The first control loop takes care of the power factor by controlling i l in the boost part with switch SI' The second control loop controls the output voltage U o in the buck part with switch ~. The reference values i l. and u... of controller 1 are calculated. E.g. i l..." = Kpf. u.(t) with Kpf calculated according to energy conservation laws.

i

I,)OO~

001

':W o

0 ,0 05

0

:::t_~ ··[5j"" · "< ,oo~ "1 138

~

I

196 L_.

o.

(I

_ 0 ,00 5

0 .0 1

, IOC

For both controllers the feedback matrix G is calculated with

~::brJ-n-, 0.00

0.002

0.004

0.006

0.008

From a control point of view, e.g. accuracy, disturbance rejections and speea of responses, both controllers yield a comparable result. This is not a surprising result, since both controllers utilize a comparable control law. The major difference arises as a consequence of the pulse modulation process, either being a PWM with fixed frequency fo or a hysteresis with free frequencies. In the figures 8-11 the frequency spectra of the currents il and i2 are shown both for VSS and state feedback with PWM (fixed fJ. In figure 8 the influence of the timing constraint T""" is visible. This constraint avoids the introduction of frequencies below 20 kHz.

0.01

==C> time

O--J:~

0.00

0.002

0.004

0.006

0.008

0.01

==:> time H,

Fig. 6 Load variation.

The values of Gij are calculated separately for state feedback and for VSS, using the appropriate algorithms and the linear model. These calculated values are further optimized in a simulation program (PSI) to take care of the nonlinearities. The controllers are compared after the converter has reached the steady state, starting at the beginning of a new period T.,... (il = 0 Ampere, U c = 376 Volt, i2 = 4.75 Ampere and Uo = 200 Volt). The load is changed between 10 and 100% according to figure 6. Both control schemes can control the converter and satisfy the stated control requirements. In figure 7 signals are illustrated if VSS with hysteresis is used.

Fig. 8 il with VSS

In using VSS, the peak harmonic values of both il and i2 owing to fo are reduced considerably, namely with about a factor 20 (to 5%) for i l and about a factor 3 (to 30%) for i2. The price to be paid is a much broader spectrum, as elucidated in figure 8 and 9. The spectrum of il also reflects the rectified sine, visible at f = 100Hz and it higher harmonics. These components are much larger than the harmonics introduced by the switches, because they reflect the energy flow through the converter. The filters of the converter are optimized for one single frequency, namely fo' In using VSS, also other frequencies are active. So, a redesign can yield better results for VSS.

V,\:\ DE:" Bosell P.P.J , V,\:" DE:" «ROE!' ,\ E. VISSER ilK

10-

1

1

! ! ! !:! !!

ta-'

f ....... ,.... ,....... ..

10- l

I ~.

10- ·

~ ·· · ·;-·-i--;· ; ·~~·r~i

10- '

l

·+.!+~i·~~~··

...:.. "-'--."'-" " .....

..~-~ ..:.~.~!~!

· · ·t··;,

~ ~

. ....... ..

. :: ... "'

These harmonics are determined by the operating points. In contrast, adding noise to the pulse modulator, guarantees that peak harmonics are reduced in steady state, but unpredictable results occur during a transient. Consequently, VSS with hysteresis is more appropriate for motion control systems with changing setpoints or for AC converters, while adding noise might be an alternative for DC converters operating in steady state.

CONCLUSIONS

Fig. 9 i2 with VSS

It has been shown that VSS can be profitably used to control systems equipped with switches. If VSS is P o ... , spec tr um 0 1 cvrr .. "t "

extended with a hysteresis and timing constraints, an attractive discrete pulse modulator arises. This controller yields flat frequency spectra. Compared with PWM with a fixed frequency fo' the peak harmonic values can be reduced considerably, while the control performance is maintained.

ACKNOWLEDGEl\fENTS We want to acknowledge the fruitful cooperation with Klaassens, Spies and Bergsma to realize this study and the converter in hardware.

H,

Fig. 10 i, with state feedback

+

PWM

REFERENCES Po.... r spectrwm o f cur,."t i2

10-)

' 0-'

.

N 10-'

Ai 10---

I

10-'

H,

Fig. 11 i2 with state feedback

+ PWM

VSS works most attractively during transients. Then, the goal is to reduce ripple and maintain good dynamic behaviour. This situation occurs in the boost part as illustrated for i, (figure 8 and 10), because i,.1ICt is a rectified sine wave. The buck part of the converter, so i2 , is almost constant apart from some transients, as indicated in figure 7. Then the VSS controller tends to realize a fixed duty cycle. In steady state, VSS displays a periodic behaviour, yielding higher peak values for some harmonics. Then, VSS with hysteresis is comparable with PWM.

-

-

-

270

Bosch, P.P.I. van den and H.R. Visser (1990). Simulation of state-events in power electronic devices, Proceedings 4th International Conference on Power Electronics and Variable Speed Drives, London, (184-189) . Enjeti, P.N., P.D. Ziogas and I.F. Lindsay (1988). Programmed PWM techniques to eliminate harmonics: A critical evaluation. IEEE IAS conference record (419-430). Spies, R. en G. Bergsma (1991). A 1000 Watt Boost-huck converter. A91-493, Control Laboratory, Delft University of technology. Tanaka, T. and T. Ninomiya (1992). Random switching control for DC-to-DC converter: analysis of noise spectrum. IEEE PESC conference record (579-586). Utkin, V.1. (1977). Variable structure systems with sliding modes. IEEE AC-22 (212-222). Venkatarmanan, G. and D.M. Divan (1992). Improved performance voltage and current regulators using discrete pulse modulation. IEEE PESC conference record (601-606). Wang, A. and S.R. Sanders (1992). On optimal programmed PWM waveforms for DC-DC converters. IEEE PESC conference record (571578).