On the Parameter-dependent H ∞ control for MEMS gyroscopes: synthesis and analysis

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On the Parameter-dependent H ∞ control for MEMS gyroscopes: synthesis and analysis

Fabricio Saggin, Jorge Ayala-Cuevas, Anton Korniienko, Gérard Scorletti

To cite this version:

Fabricio Saggin, Jorge Ayala-Cuevas, Anton Korniienko, Gérard Scorletti. On the Parameter-

dependent H

control for MEMS gyroscopes: synthesis and analysis. [Research Report] Ecole

Centrale Lyon; Laboratoire Ampère. 2020. �hal-02505581v3�

On the Parameter-dependent H _∞ control for MEMS gyroscopes: synthesis and analysis

Fabr´ıcio Saggin, Jorge Ayala-Cuevas, Anton Korniienko, G´ erard Scorletti

Amp`ere laboratory UMR CNRS 5005 Ecole Centrale de Lyon, Ecully, France

Abstract

This document provides further details on the paper “Parameter- dependentH∞control for MEMS gyroscopes: synthesis and analysis”.

These details concern the choice of the weighting function parameters and the system model in pseudo-continuous time (PCT).

1 Background and objectives

The to-be-controlled system is the drive mode of a MEMS, whose model, in continuous-time (CT), is given by:

G

(s) = y(s)

u(s) = k

(s/ω

)

+ (s/ω

) /Q + 1 , (1) where y is the displacement of the drive mode, u is the input force, k is the static gain, Q is the quality factor, and ω

is the resonance frequency (in rad/s), which slowly ranges [ω

, ω

].

The control objectives are:

• tracking of a sinusoidal reference signal y

of frequency ω

;

• minimization of the control effort u;

• robust stability.

Moreover, to ensure high performance, the controller depends on ω

.

In Saggin et al. (2020), this problem is solved either for an analog or for

a digital implementation of the ω

-dependent controller. The solutions are based on time/frequency normalization and on the H

synthesis.

In this report, we detail the choice of the H

criterion, more specifically, the choice of the weighting function parameters, which is presented in Sec- tion 2. For the specific problem of a digital implementation of the controller, its design is based on the gyroscope model in the pseudo-continuous time (PCT). This model is developed in Section 3.

2 Choice of the weighting function parameters

In the H

synthesis, the control specifications are expressed through the choice of the weighting functions and of the weighted closed-loop transfer functions. We consider the criterion presented in Fig. 1, where we include an input disturbance d, a measurement noise n and weighting functions W

, and we define ε = y

− y

and y

= y + n.

Kω0

W_ω^r₀ W_ω^d₀ Wωⁿ0

Gω0

w1 yr

u + y+ ym

w2

d + w3

n +

−

+ Wω^ε0

W_ω^u₀ z2

ε z1

Figure 1: H

criterion.

The H

problem is then: given a performance level γ > 0, compute a controller K

, if there is any, such that k P

? K

k

< γ. If this problem has a solution for γ = 1, then the following H

criterion is also ensured:

W

T

W

W

T

W

W

T

W

W

T

W

W

T

W

W

T

W

<1. (2) Then, with the following weighting functions

W

(s) = 1 M

(s/ω

)

+ (s/ω

) α

+ 1 (s/ω

)

+ (s/ω

) α

A

/M

+ 1 , W

(s) = M

(s/ω

)

+ (s/ω

) α

A

/M

+ 1

(s/ω

)

+ (s/ω

) α

+ 1 ,

W

(s) = k

, W

(s) = k

and W

(s) = k

,

the choice of the parameters A

≤ 1, M

≥ 1, α

, A

≤ 1, M

≥ 1, α

, k

, k

and k

ensures the desired specifications, as follows Skogestad, Postlethwaite (2001):

• Reference tracking: (2) implies that

∀ ω, | T

(jω) | ≤ 1

| W

(jω)W

(jω) | , (3) which ensures the tracking of the sinusoidal reference signal y

by y

with a frequency equal to ω

, error bounded by A

/k

and convergence speed constrained by α

.

• Control limitation: (2) implies that

∀ ω, | T

(jω) | ≤ 1

| W

(jω)W

(jω) | , (4)

∀ ω, | T

(jω) | ≤ 1

| W

(jω)W

(jω) | , (5)

∀ ω, | T

(jω) | ≤ 1

| W

(jω)W

(jω) | , (6) which constrains by α

the bandwidth of the controller and by A

(for frequencies close to ω

) and M

(for low and high frequencies) the control signal amplitude and, therefore, its power.

• Robust stability: finally, (2) also implies

∀ ω, | T

(jω) | ≤ 1

| W

(jω)W

(jω) | (7) and ∀ ω, | T

(jω) | ≤ 1

| W

(jω)W

(jω) | ,

which are respectively used to avoid pole-zero compensations and to enforce a lower bound on the modulus margin M , which is defined as M

1/ k T

k

. Moreover,

k T

k

< M

/k

. (8)

Hence, (2) also implies M > k

/M

.

Please note that the weighting functions are parameterized by ω

, express- ing the control specifications in continuous-time. Nevertheless, the above discussion also holds for the weighting functions in a normalized or pseudo- continuous space.

In (Saggin et al., 2020, Section 6), the control specification are:

1. track a reference signal y

(t) = Y

sin (ω

t) with an error ε(t) = y

(t) − y

(t) such that | ε(t) | < 10

· Y

in steady-state;

2. the control signal amplitude is less than 0.02 · Y

in steady-state;

3. the closed-loop system is stable and has a modulus margin M > 1/2.

The first control specification demands | T

(jω

) | < 10

, which is bounded by A

/k

, see (3). Then, we choose k

= 1 and A

= 5 · 10

.

The third control specification demands k T

k

< 2, which is bounded by M

/k

, see (8). Then, we choose M

= 2 and k

= 1.

To avoid pole-zero compensation, | T

| has to be bounded where the plant presents a resonance peak. Then, we choose k

= 0.05, see (7).

We choose M

= 400 to minimize the control signal amplitude at low and high frequencies, see (4), (5) and (6). Moreover, we choose A

= 0.004 to allow high controller gains around ω

. Note that the control signal amplitude at ω

(second specification) is structurally given by Y

/ | G(jω

) | and cannot be modified by the choice of the weighting functions.

Finally, α

and α

are tuned (α

= 0.2 and α

= 1632) to select adequate response times.

3 Details on the PCT model

Here, we intend to provide further details on G

, see (1), when analyzing its equivalent discrete-time (DT) model, G

. We highlight that the latter one takes the presence of the zero-order holder into account.

For Q > 1, the equivalent DT system G

, with sampling period T

, is given by

G

(z) = k b

z + b

z

+ a

z + a

with

a

= − 2e

cos (ω

T

) a

= e

b

= 1 − e

sin (ω

T

)

p

4Q

− 1 + cos (ω

T

)

!

b

= e

+ e

sin (ω

T

)

p

4Q

− 1 − cos (ω

T

)

!

Then, by applying the bilinear transform, we obtain G

(s

) = k (1 − s

T

/2) (1 + s

z

)

s

+ $

s

/ Q + $

with

$

= 4 T

1 − 2e

cos (ω

T

) + e

1 + 2e

cos (ω

T

) + e

$

Q = 2

T

1 − e

1 + 2e

cos (ω

T

) + e

z

= T

2

1 − 2e

√

4Q²−1

sin (ω

T

) − e

1 − 2e

cos (ω

T

) + e

For Q 1, we may approximate e

≈ 1 and ξ

≈ 0. Thus,

$

≈ 4 T

1 − cos (ω

T

)

1 + cos (ω

T

) = ⇒ $

≈ 2 T

tan ω

T

2

,

$

Q ≈ ω

(2Q) (1 + cos (ω

T

)) = ⇒ Q ≈ Q sinc (ω

T

) and z

≈ T

2

− sin (ω

T

)

(2Q) (1 − cos (ω

T

)) = 1 2Q$

.

References

Saggin F., Ayala-Cuevas J., Korniienko A., Scorletti G. Parameter- dependent H

control for MEMS gyroscopes: synthesis and analysis // IFAC World Conference. 2020.

Skogestad S., Postlethwaite I. Multivariable feedback control - analisys and

design. 2001. Second. 585.