High order sigma-delta modulator design via sliding mode control

High Order Sigma-Delta Modulator Design via Sliding Mode Control Sergey Plekhanov, Ilya A. Shkolnikov, Yuri B. Shtessel Department of Electrical and Computer Engineering, University of Alabama in Huntsville, AL 35899

Abstract- In this paper a Sliding Mode Approach to stability analysis and design of Sigma-Delta modulators is introduced. It is shown that a continuous-time sigma-delta modulator can be studied as a nonlinear dynamical system with feedback control. Representing a modulator as ”plant” and “controller” transforms a modulator design problem into a control design problem. Moreover, using advanced sliding mode techniques; new structures of sigma-delta modulator can be obtained. A non-linear second order modulator is proposed and analyzed as a part of high order non-linear modulators family.

filters [1]. This structure allows creating inexpensive, simple, low-distortion A/D converters with high resolution. 1.1 First Order Sigma-Delta Modulator

Simple, first-order Sigma-Delta modulator is presented in fig. 2, where m(t ) - analog signal (message), u[n] - digital, discrete-time signal {-1,+1},

m[n] = m(n ⋅ T ) - sampled analog input signal; Tos = 1 / f os sampling time for oversampling frequency.

1. Introduction The rapid growing market of electronic devices, availability of inexpensive digital integrated circuits and increased demands for higher precision analog to digital converters (A/D) attracts designer's attention to the Sigma-Delta modulators. Their popularity comes from less analog signals processing and relaxed requirements for analog components. These advantages come at expense of large amount of digital signal processing at relatively high clock speed. Sigma-Delta data conversion has been known for several decades but only with resent advances of VLSI technology they became truly popular. Before applying any Sliding Mode Control techniques for analyzing Sigma-Delta modulators we introduce a concept of such modulators, principle of operation, advantages and downsides. Then a method for designing stable modulators will be described. In contrary to conventional Pulse Code Modulator (PCM), which employs a lot of analog comparators and samples at Nyquist rate, Sigma-Delta modulator uses only 1 comparator but samples the analog signal at much higher rate. General form of an oversampling A/D converter (Sigma-Delta Modulator) is shown in fig 1.

Fig.1. Oversampling A/D converter Since the original signal is sampled at very high frequency, it has very relaxed requirements for the anti-aliasing prefilter. After sampling a digital low pass filter-downsampler is applied. (Usually it is implemented by nondistortion FIR

Fig 2a. continuous-time sigma-delta modulator.

Fig 2b. discrete-time implementation. Since the output of the quantizer u (t ) has only two levels and it can be directly represented as a series of binary numbers, the terms Delta modulation and Sigma-Delta nodulation could refer to both digital signals, consisting of +1 and –1, and analog waveforms consisting of pulses with amplitude +1, -1 and duration of Ts = 1 / f os (binary waveform). In this work, the terms Delta and Sigma-Delta modulation will denote the binary waveforms. In discrete-time system the integrator is replaced by a delay element (built with switched capacitors) and the comparator is presented as an additive noise with assumption that it is white noise with variance σ

2

=

∆2 , where ∆ is a quantization 12

step. Such transformation models continuous-time, inherently nonlinear system as a discrete-time linear system and standard discrete-time analysis can be applied. The overall transfer function of this discrete-time system is the following: Y ( z) = H x ( z) ⋅ X ( z) + H e ( z) ⋅ E ( z) , where H x ( z ) = z −1 is a transfer function of an input signal pure delay that doesn't change the form of the signal, and

H e ( z ) = (1 − z −1 ) is a transfer function of quantization noise, shown in fig 4 [1]. 1.2 Second Order Sigma-Delta Modulator Increasing number of integrals in the analog part of the modulator will improve the noise shaping performance and, consequently, a higher resolution of overall system. One of the 2-nd order modulators is shown in fig.3 with

H e ( z ) = (1 − z −1 ) 2

Fig. 3 Second-order Discrete-time Sigma-delta modulator The noise transfer function in frequency domain for 1-st and 2-nd order systems is shown in fig. 4.

2-order

simplicity L=1, m(t ) is normalized) Usually, the sampling frequency is much higher than the highest frequency in the signal spectrum, and there is no need to make a pass-band of a system greater than a signal band so the first assumption is not far from reality. The second one is just a matter of scaling.

3

He

The objective is to develop algorithms for design and analysis of continuous-time Sigma-Delta Modulators of Nth order using sliding mode control theory (SMC) [6]. The algorithm is developed under the following assumption: 1. Sampling rate is infinite. 2. The input signal is bounded: | m(t ) |≤ L , (for

Transfer Function of noise 5

4

higher signal to noise ratio and better resolution of the modulator [1,2]. Due to simplicity of analysis the discrete-time Sigma-Delta modulators on switched capacitors are the most widespread, but they have certain limitation in maximal sampling frequency. For the high frequencies the analog, continuoustime modulator are more preferable. As it was said earlier, modern Sigma-Delta modulators can be divided into 2 major groups: 1.Continuous-time implementation, based on operational amplifiers 2.Analog, discrete-time implementation, based on switched capacitors. The second group is easier to analyze and design but it is less tolerant to the element imperfection and for very high sampling rates the transient of the capacitors become an issue. Continuous-time systems are more difficult to analyze due to non-linearity of the quantizer, but they are simpler to implement in analog circuit and they are more suitable for higher frequency signals and some special applications. In order to provide the higher resolution, the modulator should contain several integrals ( 4-7 in latest implementation). The main problem of the high order SigmaDelta modulator design is in providing for its stability. Although there is a well-developed analysis method for discrete-time systems, survey shows that there don’t exist any methods for designing continuous-time Sigma-Delta modulators [3,4,5].

1-order 2

2.

1

0

0

0.1

0.2

0.3

f / fs

0.4

0.5

Fig 4. H e in frequency domain. Vertical line shows a band limit of a signal, where

f B = 0.02 ⋅ f s . It can be noticed that the most part of the noise is out of band. Only a quantization noise to the left of this line will appear after a digital low-pass filter at the end of modulator. Thus the Sigma-Delta modulator also called a "noise shaping" system. The second- and higher-order systems amplify the noise in the out of band region even more and have higher in-band noise attenuation. That provides

Sliding Mode Control Design For The High Order Sigma-Delta Modulator

In this section we will demonstrate analogy between a sliding mode observer and a Sigma Delta modulator. Then two sliding mode controllers will be designed to synthesize a family of high order Sigma-Delta modulators. First, a formal definition of sigma-delta modulation is introduced: Sigma-Delta Modulation u (t ) of the signal m(t ) is a Delta Modulation of the integral of the input signal m(t ) . In other words, an ideal Sigma-Delta modulation of a signal m(t ) is a binary waveform u (t ) such that: t

t

∫0u (τ ) ⋅ dτ = ∫0m(τ ) ⋅ dτ

(1)

The system in fig. 2a can be described by the differential equations:

xD1 (t ) = m(t ) − u (t )

(2)

u = sign( x1 )

that can be treated as a first order sliding mode disturbance estimator [6,7,10]. | m(t ) ≤ 1 , the sliding mode exists and Since

u eq = m(t ) ]. Writing σ dynamics in terms of

u

and

u eq

one can get:

∂σ ≠ 0 [6]. ∂x n Considering that B is defined as in (5) and x = 0 .

Integrating both equations results in t

∫0 u ⋅ dτ =∫0 u eq ⋅ dτ = ∫0 m ⋅ dτ

(3)

According to the definition of sigma-delta modulation (1), discontinuous signal u (t ) = ρ ⋅ sign( x) (4) is an ideal sigma-delta modulation of the input signal m(t ) .

Integrating (6) results in:

σ =∫

As it was stated in the introduction, the high-order Sigma-Delta modulators have better signal resolution and higher quantization noise attenuation. In this section a sliding mode control theory will be used in studying the N-th order system. Consider an n-th order SISO dynamical system in the following format:

 xD (t ) = A ⋅ x(t ) + B ⋅ (m(t ) − u (t ))   y (t ) = x1 (t ) + f ⋅ m(t )

(5)

where

arbitrary

⋅ ⋅ ⋅ x n ]T

state

vector,

B = [b1 b2

matrix,

(b1 , b2 ,..., bn −1 ) - arbitrary constants,

y (t )

∂σ ⋅ (m(τ ) − u (τ )) ⋅ dτ = 0 ∂x n

∂σ ⋅ (m(τ ) − u eq (τ )) ⋅ dτ = 0 0 ∂x n

σ =∫

t

and, finally, t

x2

t

0

m(t ) = u eq (t )

2.1 N-th Order Modulator

x(t ) = [ x1

A ∈ R n×n

⋅ ⋅ ⋅ bn −1 1] , u (t ) scalar control, T

scalar output, f = 0 ( f ≠ 0 in a delta modulator).

Note, that signal m(t ) acts as a disturbance, and this “disturbance” is matched. Theorem 1. If the state vector x(t ) in (5) is stabilized at zero by means of sliding mode control u (t ) , then u (t ) is a sigma-delta

(7) t

∫0u (τ ) ⋅ dτ = ∫0u eq (τ ) ⋅ dτ

(8)

According to the definition of sigma-delta modulation, one can conclude that a sliding mode discontinuous control signal u (t ) = ρ ⋅ sign( x) is a Sigma-Delta modulation of the equivalent control u eq (t ) .

■

Remark. It can be shown, that this result is valid for system (5) with unmatched disturbance m(t ) if the theorem 1 conditions hold. Thus, the system (5) with sliding mode control, in fact, a sigma-delta modulator. On the other hand, any continuoustime sigma delta modulator can be represented as “plant” and a “controller” that stabilizes the “plant”. Based on this result a following strategy for analysis of given sigma-delta modulator is proposed • Divide the modulator into “plant” and “controller”, • Determine the conditions for the existence of sliding mode, (this part was published in [8]) • Determine the value of u eq (t ) . The general form of these modulators is shown in fig.5

modulation of m(t ) and m(t ) = u eq (t ) . Proof: Note that u (t ) = ρ ⋅ sign(σ ( x)) is a binary waveform. If

x = 0 then xD n = m − u . Integrating this equation, one gets t

(6)

part is made stable by the choice of σ (x) ) and

σ = m − u t

σ = σD = 0 ∂σ ∂σ ⋅ xD = ⋅ ( A ⋅ x + B ⋅ (m − ⋅u ) σD = ∂x ∂x ∂σ ⋅ ( A ⋅ x + B ⋅ (m − u eq ) σD = ∂x

Note, that the system (6) is controllable, (uncontrollable

σ = m − u eq

t

u eq , by definition, is a continuous control that provides the same σ dynamics in sliding mode as a discontinuous, sliding mode control. In sliding mode An equivalent control

t

∫0u (τ ) ⋅ dτ = ∫0m(τ ) ⋅ dτ and u (t ) is a sigma-delta modulation of m(t ) . Fig. 5. General form of Sigma-Delta modulator

Another interesting result can be obtained from the fact that stabilizing a dynamical system one creates a sigma-delta modulator as a side effect. Deploying this approach and using non-trivial sliding mode controller one can obtain new structures of sigma-delta modulators. An example of new modulator with non-linear sliding surface, and nonlinear element in the modulator structure is presented later. First, an example of interpolative modulator is studied thoroughly. 2.1 Interpolative modulator from sliding mode control point of view. Consider a particular case of the system (5) as follows:

 xD1 = x 2 D  x 2 = x 3 ...  xD = m(t ) − u (t )  N  y = x1

(9)

from any finite initial point, σ will reach zero in finite time and stay there. At the same time eq.(10) becomes a homogeneous differential equation and if its solution is stable, then all the states will tend towards origin accordingly to its eigenvalues. System dynamics in sliding is defined by

σ = 0 , and

n −1

∑ ci ⋅ x1(i) = 0 i=0

or

c 0 ⋅ x1 + c1 ⋅ xD1 + c 2 ⋅ DxD1 + ... + c n −1 ⋅ x1( n −1) = 0 , (12) which becomes a sliding surface. For this equation to have stable solution, all its eigenvalues must have negative real parts. Once in Sliding, the equivalent control u eq (t ) can be derived: n−2

σC = ∑ c i ⋅ x1(i +1) + c n −1 (m − u eq ) = 0 i=0

and a control task of stabilizing the states of this system at zero by means of sliding mode control. There exist several solutions for σ (x) that will stabilize the system. In this section, a relative degree approach is used [7]. Take

σ = c 0 ⋅ y + c1 ⋅ y (1) + c 2 ⋅ y (2) + ... + c n −1 ⋅ y ( n −1)

u eq = m +

∑ ci ⋅ x1(i+1) i =0

c N −1 n− 2

u eq = m + ∑ c i ⋅ x1(i +1)

(13)

i =0

n −1

σ = ∑ c i ⋅ y (i )

n− 2

is a n-th time derivative

According to (12), x1 and all its derivatives asymptotically go to zero, so the residual in (13) will tend to zero as well. Consequently, while sliding mode exists u eq (t ) → m(t ) as time increases •

of an output, and a control function u = ρ ⋅ sign(σ ) This architecture is known as an interpolative modulator [8]. It is known that most sigma-delta modulators can be transformed to this form.

• becomes a Sigma-Delta modulation of m(t ) . Thus, it is shown that if (11) satisfies, then sliding mode exists and the whole modulator is stable. In interpolative modulator u eq (t ) converges to m(t )

It can be seen from (10) that y (i ) = xi +1 . Using sliding mode existing condition σ ⋅ σ ≤ 0 [7] we obtain from (6)

asymptotically. In order to provide finite time convergence, a non-linear modulator is proposed.

or

(10)

i=0

where c n −1 = 1 and y ( n) =

d n ( y) tn

n −1

σ = ∑ ci ⋅ y (i +1)

3. Non-linear modulator.

or

In this section, a second order example of the system (9) is studied.

i =0

σ = c 0⋅ ⋅ x1 + c1⋅ ⋅ x 2 + c 2 ⋅ x 3 + ... + c n −1 ⋅ x n x n = m(t ) − u (t ) and

σ ⋅ σ = σ ⋅ (c 0⋅ ⋅ x1 + ... + c n −1 ⋅ m(t ) − σ ⋅ c n −1 ⋅ ρ ⋅ sign(σ ) | c 0⋅ ⋅ x1 + c1⋅ ⋅ x 2 + ... + c n −1 ⋅ m(t ) | −c n −1 ⋅ ρ < 0 And, finally,

| c ⋅ xD + c ⋅ xD + ... + c n − 2 ⋅ xD n −1 | ρ >| m(t ) | + 0⋅ 1 1⋅ 2 c n −1 or

(11)

ρ >| m(t ) | + | c~0⋅ ⋅ x 2 + ~ c1⋅ ⋅ x 3 + ... + c~n − 2 ⋅ x n |

Eq. (11) declares a criterion for existence of Sliding Mode of the system. If the coefficients satisfy this condition, then,

 xD1 = x 2   xD 2 = m(t ) − u (t ) y = x 1 

σ is chosen as σ = x1 + c⋅ | xD1 | ⋅ xD1

(14)

(15)

In contrary to the interpolative architecture, in this case the sliding surface σ = 0 is not linear (fig.7). The solution of (15) defines the system dynamics in sliding. When σ = 0 , eq.(15) transforms to the

| x1 |= −

| x1 | c

(16)

With solution:

| x1 |= (| x1 (0) | −

1

2

⋅ t) (17) 2⋅c for 0 < t < t * , where t * denotes time at which x1 reaches zero. ( x1 (t*) = 0 and x1 (t ) ≠ 0 for t < t * )

In order to define u eq , consider the original system (14) with equivalent control (19) applied. The system exibits the following dynamics:

 xD1 = x 2 = 0   xD = m(t ) − u = − 1 ⋅ sign( x ) eq 2  2 2⋅c

(20)

The last eq. can be presented in the form: x 2 = v ,

v=−

1 ⋅ sign( s ) , 2⋅c

(21)

s = x2

At s = x 2 = 0 , sign(s) is not defined, but the system (21) is nothing more then another first order system with a sliding mode control v(t ) . So, applying SMC analysis one gets:

s ⋅ sD = x 2 (−

1 1 ⋅ sign( x 2 )) = − | x 2 |< 0 2⋅c 2⋅c

(22)

and the system exibits another sliding mode on s = x 2 = 0 , for which v eq is defined as follows: Fig.7 Non-linear sliding surface. After that, x1 and x1 stay at zero. Eq.(17) implies that, in sliding, the system will reach the origin in finite time. Note that in case of a linear Sliding Surface in section 2.1 the system motion to the origin is asymptotical. Existence of the sliding mode is proved by Lyapunov function

V=

1 ⋅ sign( x 2 )) eq = 0 (23) 2⋅c From (23) and (19) follows that u eq = m . It means, that the sD = 0 and sD = xD 2 = (−

system (14) is stabiblized with finite time by the controller u (t ) = ρ ⋅ sign( x) is a sigma-delta (15), (4); and modulation of m(t ) .

1 2 σ > 0 , VD = σ ⋅ σD < 0 2

Now ρ should be defined such that σ ⋅ σD < 0

σD = xD1 + c ⋅

d (| xD1 | ⋅xD1 ) ⋅ DxD1 dxD1

σD = xD1 + 2 ⋅ c⋅ | xD1 | ⋅DxD1 = x 2 + 2 ⋅ c⋅ | xD1 | ⋅(m − ρ ⋅ sign(σ )) σ ⋅ σD = σ ⋅ ( x 2 + 2 ⋅ c⋅ | xD1 | ⋅m) − σ ⋅ ρ ⋅ sign(σ )) σ ⋅ σD ≤| σ | ⋅ | x 2 + 2 ⋅ c⋅ | xD1 | ⋅m | − | σ | ⋅ρ σ ⋅ σD ≤| σ | ⋅(| x 2 + 2 ⋅ c⋅ | xD1 | ⋅m | − ρ ) Existence condition now can be written as

ρ ≥| x 2 + 2 ⋅ c⋅ | xD1 | ⋅m ≥| x 2 | ⋅ | 1 + 2 ⋅ c ⋅ m |

(18) If (18) is satisfied, the system reaches sliding surface from any initial point in finite time. After that, all the states x1 , x 2 will go to zero in finite time as well. The equivalent control, u eq (t ) is defined as follows:

σ = x 2 + 2 ⋅ c⋅ | x 2 | ⋅(m − u eq ) and

1 ⋅ sign( x 2 ) 2⋅c But at x1 = x 2 = 0 , u eq still undefined. u eq = m +

(19)

Fig. 8. Non-linear second order modulator 3.1 Simulations Simulation shows that non-linear modulator can provide competitive resolution. 2 systems were simulated. Simple, second order interpolative modulator of the structure in fig. 6, had coefficients C0=100 and C1=0.01. (These coefficients show the best results) Non-linear modulator shown in fig. 8 had the coefficient C0=0.5. Analog input sine-wave of 100Hz was represented digitally with a sampling frequency 384.61kHz and double precision. Then it was digitized with Sigma-Delta modulators with sampling rate fs=38,461Hz and a FFT of 131072 output points were taken. Results (enlarged) are presented in fig.9 and fig.10.

is very promising to use an adaptive gain based on the estimations of states. Non-linear Sliding Mode Controller presented in this paper demonstrates the opportunity of using all the family of different controllers for the purpose of Sigma-Delta modulator design. More detailed quantitative analysis of nonlinear modulators along with an adaptation in quantizer gain is a subject for following research. High order sliding mode controllers with better σ stabilization, could increase the resolution of sigma-delta modulators, and they are the subject of the following work.

FFT of the output of interpolative modulator 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03

References

0.02 0.01 0

1. 0

100

200

300 frequency (Hz)

400

500

600

Fig. 9 interpolative modulator output spectrum FFT of the output of non-linear modulator 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

0

100

200

300 frequency (Hz)

400

500

600

Fig. 10 non-linear modulator output spectrum It can be seen from the graphs that in case of non-linear modulator the level of quantization noise is lower. This means that non-linear modulator will provide a better resolution. This gives us a ground for further analysis and optimization of such modulators 4. Conclusion Given analysis and examples demonstrate the application of Sliding Mode methods for designing and analysis of oversampled Sigma-Delta modulators. Dividing modulators into 2 parts (plant and controller) enable us to treat a modulator design as a control design problem. It is shown that solving the problem of stabilization; one gets a stable sigmadelta modulator. Given stability criteria allow to choose the right quantizer gain. Notice that this gain ρ depends on current states x ,and those states change relatively slow, so it

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