Effects of finite switching frequency and delay on PWM ... - IEEE Xplore

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 47, NO. 4, APRIL 2000

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Effects of Finite Switching Frequency and Delay on PWM Controlled Systems Timoor A. Sakharuk, Brad Lehman, Member, IEEE, Aleksandar M. Stankovic´, Member, IEEE, and Gilead Tadmor, Senior Member, IEEE

Abstract—The paper examines the effects of (low) switching frequency and delays on a closed loop system with pulse-width modulated (PWM) components. A modeling procedure is described together with analytical formulas. Deviations from standard (idealized) models which replace a PWM unit with a fixed gain are quantified. Low switching frequency reduces the effective PWM gain, and delays make the gain larger than the idealized one at low voltages. The necessary and sufficient condition for existence of a -periodic stable equilibrium in the closed loop linear system with a PWM is obtained. This result is based on an analysis of a piecewise affine discrete-time map under the assumption of linear-ripple approximation. Experimental results are provided for a PWM controlled dc servo motor, while analytical results are presented for certain examples from the power electronics literature. Index Terms—Delay effects, gain control, PWM, voltage control.

I. INTRODUCTION A. Background

T

HE common modeling approach to linear systems with power switches [1] is to replace the pulse-width modulated (PWM) block unit with a fixed gain. To justify this idealization, the PWM input signal should be slowly varying in comparison to the switching frequency, and the switching frequency should be fast enough, so that ripple effects on the feedback signal can be neglected. There are applications, however, in which characteristics of employed semiconductor devices, or difficulties in heat removal lead to fairly low switching frequencies. This slow switching frequency causes the ripple (i.e., the deviation of a waveform from its cycle-by-cycle mean) to become considerable. This paper deals with two major aspects associated with high ripple magnitude. First, a new equivalent PWM gain in a steady-state regime is derived as a nonlinear function of the operating point. Second, it is shown that a system may have a nonconstant duty ratio, even when input signals are constant, which may affect performance in several ways. It is shown in this paper that delays (e.g., due to digital implementation) can substantially influence both system performance and ripple patterns. When switching is slow, the delay should not be neglected. In fact, depending on system conditions, a delay can either decrease or increase the equivalent PWM gain. The Manuscript received March 11, 1998; revised October 19, 1998 and December 9, 1998. This work was supported in part by the Office of Naval Research (ONR) under Grant N14-95-1-0723 and in part by the ONR under grant N14-97-1-0704 and the National Science Foundation under Grants ECS-9410354, ECS-9502636, and CMS-9596268. This work was recommended by Associate Editor M. K. Kazimierczkuk. The authors are with the Department of Electrical and Computer Engineering, Northeastern University, Boston, MA 02115 USA. Publisher Item Identifier S 1057-7122(00)01828-6.

introduction of delay into the control loop also makes the system more prone to PWM switching aperiodicity. While the difference in the effective gain can be compensated in the closed-loop controller design, PWM aperiodicity, such as period doubling, may lead to higher magnitude and lower frequency of the ripple, with implications on performance such as low power efficiency and saturation. Subharmonic oscillations in classes of PWM power converters have been well documented, of which [2]–[4] are a few examples. Period doubling and bifurcations have been detected in current controlled converters, and must always be avoided in practice. This paper attempts to analytically generalize (sometimes heuristic) results from power converters to arbitrary piece-wise linear systems and to systems with delay. When the delay is set to zero and a specific circuit topology is assumed, then known stability results are recovered. A suitable approximate model is needed to analyze the complicated behavior of the time-varying PWM system. A method introduced in [5] for power converters yields averaged models where the PWM is approximated as a nonlinear gain. These models are obtained by using periodic ripple functions to improve averaging approximations. Compared to conventional averaging [6] which approximates the PWM as a constant gain and neglects ripple, this approach provides a better approximation of the system state trajectory, eliminates a dc offset error in steady state, and yields an accurate estimate of the ripple magnitude. This paper generalizes the averaging method of [5] to linear systems with PWM and delay, and then this new result is used to obtain a piecewise affine discrete time model and to derive a criterion for PWM periodic operation. B. Outline of the Paper Fig. 1 depicts a block diagram of the closed loop system, where the plant and the controller are linear time invariant (LTI). is compared with a sawThe output signal of the controller, tri of period and maximum tooth function This generates the on-off driving signal for the switch. Specifically, the switch is turned “on” at the beginning of the cycle while the switch is “on,” the controller output i.e., at decreases; at the switch is turned “off” and remains so until the next cycle begins, as shown in Fig. 2. (The meaning of “ ” and “ψ” in Fig. 2, which correspond to the average state and the ripple function, respectively, will be clarified shortly.) tri The PWM output voltage is where is the Heaviside unit step function, i.e., for and for If were a constant reference, the “classical” averaged output voltage of the PWM

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Fig. 1.

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 47, NO. 4, APRIL 2000

Block diagram of a PWM actuated system.

II. AVERAGING THEORY The method of averaging is a very commonly used technique in the analysis of periodic differential equations. The purpose of this section is to briefly review some pertinent background on averaging theory. A rigorous mathematical justification of the basic theory can be found in [9], [10]; extensions to switching systems are discussed in [11], [12]. Consider a feedback controlled pulse-width modulated system with constant input signals where the plant and the controller are linear time invariant. It is straightforward to check that the closed loop system can be written in the form tri

(1) (2)

Fig. 2. (a) PWM input and (b) output voltages.

is whereby is commonly referred to as the nominal gain of the PWM block [1]. In practice, however, PWM implementation introduces ripple (higher harmonics) into the plant state trajectory. This ripple is fed back through the control loop and causes ripple in the traas illustrated by the (schematic) curve in Fig. 2(a). jectory As a result, the classical averaging assumption that remains constant over a period is invalid. As noted in [5] for dc-dc converters, one effect of ripple in is a change in the equivalent effective gain when the PWM is replaced by such a gain in an averaged model, a phenomenon that becomes more pronounced at lower switching frequencies. The starting point of this paper is the analytical framework of [5]. Section III quantifies deviations from standard (idealized) models, illustrated by an example of a fractional horse power dc drive. Section IV extends the averaging technique of [5] to include delay effects, and presents experimental results with the dc servo motor to support analysis. The effects of low switching frequency is to reduce the effective PWM gain by up to 20% at low voltages. Delays make the gain larger than the idealized gain at low voltages. The relative simplicity of the gain calculation formulas makes them candidates for design toolboxes for engineers involved in control synthesis of high-performance PWM systems. A necessary and sufficient condition for existence of a -periodic steady-state equilibrium for the approximate system is developed in Section V. It is shown that the obtained criterion simplifies to the well known results [7], [8], when there is no delay, or the delay is small.

where is the input to the PWM, is the combined state of the plant and the controller, and is a vector of the reference and disturbance signals. In the example of the system shown in Fig. 1 This matrix is nonzero when switching the matrix occurs between two LTI systems, such as in PWM boost dc/dc power converters. From here to the end of this section assume that the reference and disturbances are constant (or vary slowly). Then (1) fits into the general form (3) is -periodic and where where is the time constant of the dominant pole of the open loop transfer function from the output of the PWM to its input. If ε is small, the solution will vary slowly relative to the period The aim of averaging is to approximate the solution of (3) by the solution of an averaged time invariant system (4) where the function

is given by the time average Define (5)

chosen so that is zero mean. Since is zero mean and -periodic, the function is -periodic as well. Consider the change of variables

where

(6)

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where takes the role of an averaged, slow trajectory and is a ripple function (Fig. 3). Differentiating this expression with respect to time and substituting for in (3) leads to the equation

Fig. 3. Averaged trajectory and ripple function.

(7) This equation is a perturbation of the averaged system (4). The by idenerror is introduced by approximating by As tity matrix and by approximating this error becomes small and can be neglected. In practice but the error due to the approximation of by can be eliminated by a technique proposed in [5]. The and to replace idea is to define new functions and so that they will simultaneously satisfy the following equations:

(8) (9) It can be a difficult task to solve these equations in general, but using the special properties of (1) it is possible to obtain a good be the instant of time when the approximate solution. Let unit step function switches from one to zero, or equivalently tri Then and equals the duty when ratio. To derive a closed form solution the following properties of the vector function ψ will be postuof the components lated: is zero average over both and 1. 2. is a triangle wave (see Fig. 3); Assumption 2 introduces some error due to replacing exponential curves by straight lines. The accuracy of this approximation is dependent on the decay rate of solutions of (1). For experiments show that the approximaexample, if tion error will be less than 1%. Under the above assumptions can be explicitly evaluated by noting the following facts: • • tri •

tri

on the PWM input signal. To calculate τ using the modified method, first, find the function ψ explicitly. Substituting explicit expressions for and into (9) and using the equality tri tri obtain

tri

(11)

When integrating (11), it is assumed that is a constant, which is justified when an averaged trajectory is slowly varying; this equation then can be solved explicitly. The solution defines a triangle-shaped ripple function as shown in Fig. 3. Because the ripple function achieves its minimum at switching instances, the ripple magnitude is (12) Then, to solve for τ, note that the switching occurs when the (see Fig. 2). Since PWM input voltage is and this gives rise to the equawhich leads to the quadratic tion equation in τ (13) and the relevant where The function is solution is in the closed interval interpreted as the averaged value of the PWM input. If or (open loop) then which is the and then the effective conventional solution. If value of the input PWM voltage is less than the averaged value due to the ripple in III. BASIC PWM CONTROL OF THE DC SERVO MOTOR

Combining these facts the averaged model becomes

(10) The difference between this formula and the conventional averaged model (without the modification from [5]) is only in the way τ is calculated. In the conventional method which does not take into consideration the influence of ripple

The techniques of Section II are now applied to a dc-servo motor. Fig. 4 shows a block diagram of a dc servo motor with a nested two-loop control structure [1]: the electrical (current) subsystem in the inner loop and the mechanical (speed) subsystem in the outer loop, with their respective PI controllers. The parameters of the motor and the constant load in the experiments, as well as the selection of controller parameters, are is assumed to given in Appendix I. In this section the delay, be zero.

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Fig. 4. DC servo motor block diagram.

For the purpose of controller design, a linear model of the PWM may be useful. To derive a linearized small deviation model of the PWM, assume that the system is in quasi steady state and Then is known, and

A state space model for the closed loop system is

(16) can be solved for

The small deviation model of the PWM is (17)

(14)

which will be written concisely in the form (1), (2) where the “ ” matrix in (14)

is

Results of this section show that slow switching frequency may have pronounced effects on accuracy of PWM models. The analytical expressions (15), (17) are quite effective in quantifying the deviations from idealized (standard) models. IV. DELAY EFFECTS

and and

The averaged model (10) is defined accordingly. The average duty ratio τ can be found from (13) with Due to the intrinsic nonlinearity of the PWM, the average PWM gain will depend on the operating point. The magnitude of the ripple function is given by (12) (with One can now compute the magnitude of the current and reference voltage ripple as and (15) where the magnitude of the ripple function is defined as a difference between maximum and an average value. is compared with In Fig. 5 the ideal PWM gain (with the effective gain in an averaged model. Evidently, ripple effects reduce the effective PWM gain.

In this section the effects of delays in the PWM input signal are studied, such as those caused by digital implementation of the controller. Delay effects require modifications in the previously presented computations. Let the delay be With a delay, the expression tri in (1) tri The effect of is replaced by this change on the PWM switching is depicted in Fig. 6. Since is approximately the basic assumption in averaging is that the delay in can be ignored. constant for The ripple function , however, represents fast zero average harmonics, where a small delay may have significant effects. That is

(18) Justification of ignoring the delay in is given in the classical averaging papers [13], [14], [9], where the change of variables (18) is used to transform time-varying delay systems to corresponding approximate averaged ODE’s. Additional discussion by is of the error introduced by approximating given in [15] and [16]. Now, to find τ, solve (19) The formula for the magnitude of the ripple in

as a function

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Fig. 5. Comparison of the ideal and effective averaged PWM gain.

where and Next, we substitute (21) into (19) to obtain four equations, to be solved Then τ turns out to be the solution to one of the with following two equations (22)

(23) or explicitly, using (20) (24) Fig. 6. (a) PWM input, and (b) output voltages.

(25)

of τ remains the same as in (12) (20) Using piecewise linear approxiwhere in terms of mations, as before, the explicit expression for and τ follow from a simple geometrical observation

if

if (21)

where the smallest positive solution, τ, of each quadratic equation is selected. It is evident that (24) becomes equivalent to (13) if Fig. 7 depicts the absolute deviation of the averaged normalized PWM gain from the idealized model, for different time delays in the example of the dc motor. (Maximum relative deviafor s at input voltage ) With tion no delay, the average PWM gain is always less than ideal; the presence of delay may increase the average gain to a level higher than the ideal, as verified in the experiments. Fig. 7 presents for different delays the theoretical predictions of (solid lines), and experimental measurements. Since the noise is present, approximately 15 experiments were performed to obtain data. Average values of data points are represented by ’*’

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Fig. 7.

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 47, NO. 4, APRIL 2000

Absolute deviations of the averaged normalized PWM gain from ideal 

and the upper and lower bounds of the experiments are given by and respectively, in the figure. In this section tools for analysis of combined effects of finite switching frequency and delay have been developed. Resulting expressions (24), (25) are of modest complexity, while their predictive power is considerable, as evidenced by our experiments.

V. CONDITIONS FOR EXISTENCE OF A -PERIODIC STEADY STATE FOR APPROXIMATE SYSTEM Equations (24)–(25) were obtained under the assumption of existence of a -periodic steady state, and using a piecewise linear approximation of the steady-state averaged current trajectory. In this section necessary and sufficient conditions for the existence of a -periodic solution are explored. Note that a deviation from a -periodic regime may result in an increase of ripple magnitude, and in creation of low frequency harmonics. To illustrate this point, note that, from (12) the maximum ripple magnitude is achieved if τ = 0.5. If the system is in a 2 -periodic regime the maximum ripple magnitude will exist if the switch is “on” during one cycle and “off,” during the next one. In this case, the magnitude of the ripple function will be twice larger, and its frequency will be twice lower. Higher magnitude and lower frequency components may have adverse influence on the performance of the system, because low frequency components are less attenuated by the plant, and input and output filters. All existence conditions developed in this Section are based on the analysis of the averaged system, and therefore are approximate. The conditions derived are very simple to apply; however, the derivations and proofs to verify the conditions are, at times,

0 m(z)=V (2100) as a function of m(z)=V

for different time delays.

involved. Therefore, we present a summary of the results obtained in this section. In Section V-A a necessary condition will be derived for existence of a -periodic steady state that involves only basic system parameters and does not depend on the system operating point, (i.e., on the input signals). The meaning of the obtained is larger than condition is that, if normalized delay then for the given system parameters an operating point exists for which system will not have a -periodic steady state. The converse statement is not necessary true. In Section V-B, stronger conditions are given for the existence of a -periodic steady state. These conditions are given ) in the terms of normalized (by the slope of the sawtooth rising β and falling α slopes of the PWM input signal These slopes depend not only on system parameters, but also on the operating point of the system. Specifically, suppose that is stable operating of averaged equation (10), then (26) (27) and (Note, that this will lead to steady state exists if and only if then (a) and

) Then a -periodic

(28)

then

(b) if or

(29)

Furthermore, the average duty ratio given by does not depend on the particular slope within which the intersection occurs.

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=T < =( 0 ) < 1 < t >; (b) 1 > t =T > =( 0 ); (c) uniform sampling.

Fig. 8.

Stability regions on (α, β) plane: (a) t

Fig. 9.

Different types of intersections for the time interval T

< t  2T:

Fig. 8(a) and (b) show these regions graphically; from these plots it is clear that the “rule of thumb” often used in practice does not guarantee the existence of a -periodic that steady state if the delay is sufficiently large [7]. Section V-B also gives conditions for the existence of a -periodic steady state of a PWM system subject to uniform sampling. The stability region for this case, presented on Fig. 8(c), does not depend on sampling instance.

(31)

(32) A. A Necessary Condition for Existence of a Steady State of the Approximate System

-Periodic

We assume existence of a -periodic steady states with duty ratio τ, and use a piecewise linear approximation of the steadystate trajectory. Then the input of the PWM is given by (21), is a function of time alone, as τ is where the trajectory for σ, may give rise to now fixed. Solving = 1, 2, 3 within one up to three possible solutions PWM period. Those points are given by

(30)

and correspond to the intersections with where the slopes 1, 2 and 3, as marked on Fig. 9. Slopes 2 and 3 correspond to the ripple function up- and down-slopes of the current switching period, respectively, while slope 1 corresponds to the down-slope of the previous switching period. Following from the PWM switching rule, in a -periodic steady state, τ must coincide with the smallest of these solutions, similarly (when all three exist). creates a contradicThe existence of an intersection at tion that precludes the assumed existence of a -periodic trais the smallest possible solution, its jectory. Indeed, since but the situation is imexistence requires contradicts (30). Therefore in order possible because to derive a necessary condition of a -periodic steady state, an algebraic condition for (30) to have no solution will be derived.

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If exists, it requires that Substituting into this inequality the right-hand side of (30) for denoting (33) and after a few algebraic manipulations, it is straightforward to does not exist if conclude that (34) is dropped, as it is always (the condition false). To make the condition (34) practically useful, an expresin terms of system parameters is now derived. sion for Invoking (20) and (24)–(25) (35) (36) (for dc servo motor case, described where ). Substituting this in the previous sections into (34) the following equations are obtained: (37) (38) The left-hand side of both inequalities achieves its minimum at (as ). It means that if the inequality (39) is satisfied then (34) will be true for any τ. This quadratic equahas roots at and 1. Since, is always tion in smaller then 1, a necessary condition for the existence of an equilibrium point is (40) requires that does not Indeed, the possibility of then the existence of exist. It is easy to see that if implies also the existence of The condition (40) is conservative, but has the advantage that it involves only basic system parameters. Example (DC servo motor). For the dc servo motor a stanthen dard design procedure leads to the selection of but if the inequality (40) is true for any is selected to make the controller more aggressive, then and a necessary condition for the existence of a -periodic equilibrium is that B. Necessary and Sufficient Conditions for Existence of a -Periodic Stable Equilibrium (Based on the Approximate System) In this section necessary and sufficient conditions for existence of a -periodic equilibrium point are derived, and uniqueness and local stability of such point is established. Although we were unable to provide a rigorous proof for global stability, extensive simulations suggest that, under the local stability con-

dition, the equilibrium point is globally stable. The main tool used in this section is a discrete piecewise affine map, which describes the evolution of the closed loop system duty ratio from one time interval to the next. The system operates cyclically taking a succession of two configurations in each cycle. The state vector is governed by the LTI equations (41) (42) The trajectory of the system state space vector and switching are completely determined instant in the time interval and This observation may be used to obtain an by iterative map for τ, but this map will be nonlinear. In this section an approximate affine map is derived. First, an explicit equation for as a function of time will be derived. As before, assume that the reference and disturbance signals are constant, but that the system is not necessarily in a -periodic steady state, and assume a piecewise linear approximation To define as a piecewise linear function it is sufficient for to find the derivative of in every switching interval. By defiand therefore the derivative of denition pends on the state vector as in (41)–(42). The assumption that is piecewise linear within every time interval implies in the right-hand side of (41)–(42) is approximated by that equal to the time averaged value of over that a constant is neglected in (6). In interval; equivalently, the term ε in a -periodic steady-state regime, remains constant from one time interval to another. When the system is not in a -periodic may vary. To determine a coarse apsteady state, will be used. The validity of proximation of by a constant has this approximation is justified if the error for = 0, 1. the same order as The approximate value of the state space vector, for the case will be found as a steady-state solution of the conventional averaged model (43) (44) Using the modified method described in where Section II, it is possible to get a better approximation, but due to the nonlinearity of the equation for τ it is difficult to derive a simple close form expression for Substituting for τ and letting from (43)–(44) the steady-state averaged closed loop vector is obtained (45) Now equations for the straight-line segments of the function will be derived. From (41)–(42) it follows that in steady is approximated by ( = 0, 1), then state, if is (46) (47)

SAKHARUK et al.: EFFECTS OF FINITE SWITCHING FREQUENCY AND DELAY ON PWM CONTROLLED SYSTEMS

With the notation for one period

for the function

and during

is

was shown that a -periodic steady-state equilibrium may not Thus the two cases are left: exist if Intersection within slope 2

if if (48) and where (Explicit formulas for α and β in terms of the system parameters in the dc servo motor example are given in Appendix II.) is derived. In the preNext, an affine iterative map for vious section we distinguished between three cases where the tri intersects with the function at sawtooth function different slopes of (enumerated from 1 to 3). To completely determine the behavior of the system, two more cases must be considered. There are two possibilities that an absence of an intri tersection occurs due to saturation: one when and and the other when tri and for all (these cases are denoted by 4 and 5, reshould spectively). In general, one more state, for example, well posed (this be added to make the discrete evolution of is not necessary if only cases 2 and 3 are considered). to In Appendix III the transition from state is derived as a continuous, piecewise affine map of the form (49) where inequalities

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Using the explicit formulas from Appendix III, condition (52) becomes

(53) and The first two inequalities are trivially satisfied, as The third one can be simplified to Since the matrix has only one non zero eigenvalue, stability of the solution is equivalent to (54) and which is further equivalent to The coordinates of the equilibrium point are given by (51) (55)

Intersection within slope 3 A similar analysis, applied to the slope 3, yields the intersection condition

is determined by the validity of one of five (56) (50)

into five which divide the region of all possible pairs nonintersecting subregions (excluding boundary intervals). The and depend only on α and β; and matrices depend also on The following conditions can now be derived from (49)–(50): A necessary and sufficient condition for existence of a -periodic steady state. By (49), a -periodic steady state exists within the region if and only if (51) and if the inequality (50) is satisfied, meaning (52) A sufficient condition for local stability of a -periodic is stable, i.e., all its steady state. If and only if the matrix eigenvalues are inside the unit circle in the complex plane, then the equilibrium point given by (51) is locally stable. Now explicit expressions for the necessary and sufficient conditions for existence of a stable -periodic equilibrium within different regions will be obtained from (51)–(52). It easy to see that a -periodic equilibrium within the saturation region may exist if and only if and similarly, a -periodic equilibrium exists within saturation region and These two simple cases will if and only if be excluded from the remaining discussion. In Section V-A it

(the second inequality which is true if has only one non zero always holds). Since the matrix eigenvalue, stability of the solution is equivalent to (57) or which is true when the equilibrium point are given by (51)

The coordinates of

(58) Uniqueness of a -periodic stable equilibrium. The conditions (54) and (57) for existence of an equilibrium point for and (and the conditions for cases 4 and 5) are mutually exclusive. Therefore a -periodic equilibrium, should it exist, is unique. A few examples will be considered next, to demonstrate the broad applications of the results in this section. 1) Examples: DC servo motor. For the benchmark PWM and formulas dc servo motor previously discussed, for α and β are given in Appendix II. Using the motors values from Appendix I, and This leads to the existence of a stable -periodic orbit of the system if and (28) is satisfied or and (29) is satisfied, where Finally, note that in absence of a stable steady-state -periodic -periodic equilibrium or equilibrium, the system may have

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(a) Fig. 10.

(b)

3T -periodic stable equilibrium: (a) the PWM input voltage and sawtooth; (b) the evolution of the state vector ( ; m ):

(a) Fig. 11.

(b)

Aperiodic behavior: (a) the PWM input voltage and sawtooth; (b) the evolution of the state vector ( ; m ):

may even display an aperiodic (chaotic) behavior. This will be illustrated by the following examples: and then a) If and In this case inequality (52) is but the matrix is unstable. satisfied only for It was observed in the simulations (see Fig. 10) that the -periodic equilibrium. system has and then b) If and In this case again inequality (52) but the matrix is unstable. is satisfied only for The system exhibits an aperiodic behavior, as shown in Fig. 11.

challenging, however, to detect precisely the crossing instant of two time-variable signals when a microprocessor is used. This difficulty is often bypassed with the help of sampling and hold techniques. The PWM input signal is sampled only once within the switching period . The switching instants are computed on-line as the respective points where the triangular reference reaches the sampled value. where Assume that sampling instants are and Then and can be rewritten as iterative map (59) Stability of the solution is equivalent to

While it may be feasible to obtain analytical conditions for -periodic solution, these formulas will likely the existence of be far more complicated than the expressions for the -periodic case. PWM with uniform sampling. The PWM technique discussed in the Introduction for the generation of switching instants is simple to implement using analog comparators. It is

(60) (see Fig. 8(c)). which is further equivalent to Current programmed DC-DC converter [7]. In a current programmed buck dc-dc converter inductor current is sensed, A clock signal initiates and compared to a control current each switching cycle at a constant frequency. Duty ratio is

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determined implicitly by the instant when inductor current reaches a certain peak value set by For stability reasons it is (sawtooth a common practice to add a compensation ramp Thus, the current programmed signal in our terminology) to converters fall into the class of closed loop linear systems with considered in Introduction. The condition for PWM and existence of a -periodic steady state (57), developed in this paper, is identical to the stability criterion available in the literand ature (e.g., (17) in [8]) after the substitution . Fig. 12.

PWM input voltage and a sawtooth in the presence of a delay.

VI. CONCLUSIONS In this paper analytical and experimental study of effects of finite switching frequency and delays on linear systems with pulse-width modulated components are presented. In the first part of the paper deviations from standard (idealized) fixed-gain models have been quantified and experimentally verified. In the second part, a discrete-time piecewise affine map which describes the behavior of the closed loop system is derived, and conditions for existence of a -periodic stable equilibrium are presented. APPENDIX I PARAMETERS OF THE DC SERVO MOTOR SETUP The motor and load parameters are: = 0.92 , = 4.9 mH, = 3.9 10 kg-m = 1.2 10 Nms, = 0.296 = 0.294 Nm/A and = 0.44 Nm. V/(rad/s), Following the standard procedure [1], the PI control param= 0.1, = 47.283, = 2.513 eters were selected as = 118.416. These selections provides for a damping and ” in both loops, with the respective desired natural of “ frequencies of 10 rad/sec (current loop) and 10 rad/sec (speed loop), when the PWM block is replaced by a nominal gain. The switching frequency of the PWM is 1.5 kHz, = 1.56 V = 60 V; the compensator sampling frequency in our and experiments is 10 kHz.

APPENDIX III DERIVATION OF THE PIECEWISE AFFINE MAP In this section, with the help of simple geometrical observations (see Fig. 12), explicit expressions used in (49)–(50) for will be derived. First, equamatricses tions for the straight line segments which correspond to slopes will be obtained. 1, 2 and 3 for the time interval Then conditions for existence of the intersection of the specific with the sawtooth function will yield inequalities slope of for the given slope, will be obtained (50). An equation for as a solution of the following equation (65) as a function of and is straightAn equation for forward, and does not depend on the intersected slope (66) Then (65) and (66) yield (49); to solve (65), observe that along the slopes 1, 2, 3 for the interval is

(67)

APPENDIX II THE RIPPLE FUNCTION COEFFICIENTS The coefficients of the ripple function slopes in terms of system parameters

(68)

(61)

(69)

(62) where (63)

(70) To simplify derivations more notation is needed (see Fig. 12) (71)

(64)

(72)

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Case 4—No intersection

Case 1—Intersection within slope 1

tri

Intersection of the slope 1 with the sawtooth function may occur if and only if the inequalities and

(84)

(73)

are true, or equivalently, if (85)

(74) (75) If an intersection occurs, then the next

can be found from (65)

Case 5—No intersection

tri

(76) (86)

is computed from (66)

and the next ripple peak magnitude

(77) Finally we obtain (87) (78) REFERENCES (79) Analogous observations lead to the following results: Case 2—Intersection within slope 2

(80)

sig

sig

(81)

Case 3—Intersection within slope 3

(82)

(83)

[1] N. Mohan, T. M. Undeland, and W. P. Robbins, Power Electronics. New York: Wiley, 1995, ch. 2nd ed.. [2] L. Resnick, S. Hsu, A. R. Brown, and R. D. Middlebrook, “Modeling and analysis of switch dc-to-dc converters in constant-frequency currentprogrammed mode,” IEEE PESC Rec., pp. 284–301, 1979. [3] J. H. B. Deane and D. C. Hamill, “Instability, subharmonics, and chaos in power electronics systems,” IEEE Trans. Power Electron., vol. 5, pp. 260–268, 1990. [4] I. Zafrany and S. Ben-Yaakov, “A chaos model of subharmonic oscillations in current mode pwm boost converters,” IEEE PESC Rec., pp. 1111–1117, 1995. [5] B. Lehman and R. M. Bass, “Switching frequency dependent averaged models for PWM dc-dc converters,” IEEE Trans. Power Electron., vol. 11, pp. 89–98, 1996. ´ uk, “A general unified approach to modeling [6] R. D. Middlebrook and S. C switching power stages,” in IEEE Power Electronics Special (Specialists?) Conf. Rec., 1976, pp. 18–34. [7] F. D. Tan and R. D. Middlebrook, “A unified model for current-programmed converters,” IEEE Trans. Power Electron., vol. 10, pp. 397–408, 1995. [8] R. Tymerski and D. Li, “State-space models for current programmed pulsewidth-modulated converters,” IEEE Trans. Power Electron., vol. 8, pp. 271–278, 1993. [9] N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic Methods in the Theory of Non-Linear Oscillators. New York: Gordon and Breach, 1961. [10] H. K. Khalil, Nonlinear Systems. Englewood Cliffs, NJ: Prentice-Hall, ch. 2nd ed. 1996. [11] B. Lehman and R. M. Bass, “Extensions of averaging theory for power electronic systems,” IEEE Trans. Power Electron., vol. 11, pp. 542–553, 1996. [12] P. T. Krein, J. Bentsman, R. M. Bass, and B. C. Lesieutre, “On the use of averaging for the analysis of power electronic systems,” IEEE Trans. Power Electron., vol. 5, pp. 182–190, 1990. [13] A. Halanay, “Averaging methnods for differential equations with retarded arguments with a small parameter,” J. Differential Equations, vol. 2, pp. 57–73, 1966. [14] A. Hale, “On the method of averaging for differential equations with retarded arguments,” J. Mathemat. Anal. Appl., vol. 14, pp. 70–76, 1966. [15] B. Lehman and V. M. Kolmanovskii, “Extensions of classical averaging techniques to delay differential equations,” in Proc. IEEE Conf. Decision Control, 1994, pp. 411–416.

SAKHARUK et al.: EFFECTS OF FINITE SWITCHING FREQUENCY AND DELAY ON PWM CONTROLLED SYSTEMS

[16] B. Lehman and S. Weibel, “Fundamental averaging theorems for functional differential equations,” in Proc. American Control Conf., vol. 5, 1997, pp. 3215–3219.

Timoor A. Sakharuk was born in Leningrad, Russia, in 1965. He received the Diploma in Electrical Engineering from Leningrad Institute of Fine Mechanics and Optics and Diploma in Mathematics from Leningrad State University in 1988 and 1992, respectively. He received the Ph.D. degree in Electrical Engineering from Northeastern University, Boston, MA, in 1999. From 1988 to 1994, he was employed by the Institute of Analytical Instrumentation, USSR Academy of Sciences as a Software Engineer. While there, he participated in the development of signal processing algorithms for analytical instruments. During 1994–1998, he was with the Department of Electrical and Computer Engineering, Northeastern University as a Research Assistant. At present, he is a DSP Engineer at the R&D Carrier Systems Group of the 3Com Corporation. His current research interests are in signal processing and communication.

Brad Lehman (S’92–M’92) received the B.E.E. from Georgia Institute of Technology, Atlanta, the M.S.E.E. from University of Illinois at Champaign-Urbana, and the Ph.D. E.E. from Georgia Institute of Technology in 1987, 1988, and 1992, respectively. He is presently an Associate Professor in the Department of Electrical and Computer Engineering at Northeastern University, Boston, MA, and previously was a Hearin Hess Distinguished Assistant Professor at Mississippi State University. He actively performs research in the areas of dc-dc converters, averaging methods, electric motor drives, open loop oscillatory control, and functional differential equations. In 1994, Dr. Lehman was awarded an NSF Presidential Faculty Fellow. He is also a recipient of an Alcoa Science Foundation Fellowship and previously was a visiting scientist at MIT. He has previously served as an Associate Editor of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL.

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Aleksandar M. Stankovic´ (S’88–M’91) received the Dipl. Ing. and M.Sc. degrees in electrical engineering from the University of Belgrade, Belgrade, Yugoslavia, in 1982 and 1987, respectively, and the Ph.D. degree in electrical engineering from the Massachusetts Institute of Technology, Cambridge, MA, in 1993. He has been with the Department of Electrical and Computer Engineering, Northeastern University since 1993, presently as an Associate Professor. His research interests are in modeling, analysis and control of power electronic converters, electric drives and power systems. Dr. Stankovic´ is a member of the IEEE Power Electronics, Control Systems, Power Systems, Industry Applications and Industrial Electronics Societies, and presently serves as an Associate Editor for the IEEE TRANSACTION ON CONTROL SYSTEM TECHNOLOGY.

Gilead Tadmor (M’87–SM’92) received the B.Sc in mathematics from Tel Aviv University, Tel Aviv, Israel, and the M.Sc. and Ph.D. degrees in mathematics from the Weizmann Institute of Science, Israel, in 1977, 1979, and 1984, respectively. He joined the Department of Electrical and Computer Engineering, Northeastern University, Boston, MA, in 1989 where he is currently an Associate Professor. Previously he held research and faculty positions at Tel Aviv University, Brown University, Providence, Rhode Island, the University of Texas, Dallas, and the Laboratory for Information and Decision Systems, at M.I.T. During 1998–1999 he visited SatCon Technology Co., Cambridge, MA. His background is in the areas of robust ( ) and optimal control, distributed parameter systems, and systems theory. His most recent active interest has been in nonlinear dissipative control, with applications in power electronics, power systems, electric motor drives, and mechanical systems. Dr. Tadmor is currently a member of the IEEE Control Systems Society Conference Editorial Board.

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