Delayed-error Equations for Controller Design - IEEE Xplore

Delayed-error Equations for Controller Design A. G´arate Garc´ıa1 , S. V´azquez Vall´ın1 , L.A. M´arquez Mart´ınez2 1

Master of Science in Engineering. Universidad Polit´ecnica de Aguascalientes, Calle Paseo San Gerardo No. 207, Fracc. San Gerardo, C.P. 20342, Aguascalientes, Ags., MEXICO. E-mail:{araceli.garate; susana.vazquez}@upa.edu.mx 2 Department of Applied Physics. CICESE, Carr. Ensenada-Tijuana No. 3918, Zona Playitas, C.P. 22860, Ensenada B.C. MEXICO. E-mail: [email protected]

Abstract—A methodology to solve the trajectory tracking problem for systems with a noisy position signal is proposed in this paper. Such signal is obtained from a direct measurement, and must be filtered to reduce unwanted noise, and to estimate its time-derivative. It is shown that the use of delay-differential error equations is instrumental to achieve the stability despite a constant delay, which is inherent to the filtering process.

II. P RELIMINARIES Concepts and mathematical background which are fundamental to understand the proposed approach are stated in this section. First, the filtering process will be explained. A. Filtering process

Keywords: Trajectory tracking, delayed PD, noise. I. I NTRODUCTION Real processes with output signals of excellent quality, free from noise, are extremely rare. Therefore, it is not strange that several works on the literature are devoted to the analysis of systems where the signal position is subject to noise and to signal reconstruction [1], [2]. There are different techniques to circumvent this problem: different kind of filters are used to estimate velocity from noisy output signals [3], Luenberger observers are implemented because noisy position signals cannot be easily used to obtain good approximations for controller design [4], [5]. Delays are known to have complex effects on stability [6], [7]. However, there are works that take advantage of the properties of delay-differential equations (DDE). Some examples are [8], [9], [10] where DDE are used to stabilize particular systems and [11], [12] where they are used to solve the master-slave synchronization problem. This paper concerns the trajectory tracking problem in systems where the output signal is subject to noise. The use of filters is essential to assure the signal quality estimation and a constant delay, which depends on the filter order, is introduced. The main contribution is the stabilization of systems with noisy position signal, using delay-differential equations, and in some particular cases, even the use of the derivative is not necessary. Previous results were shown on [13]. Basic assumptions on the systems under interest, the problem statement and some preliminaries are given in Section II. The proposed approach to solve the trajectory tracking problem is explained in Section III. Finally, some concluding remarks are stated in Section IV.

Fig. 1.

Filtering process

Filtering approaches can be used to alleviate the difficulties associated with the noise amplification resulting from the differentiation of a noisy signal [14]. However, a constant time-delay is induced depending of the filters and its order. Figure 1 show a filtering process scheme under the following assumptions: 1) The noise amplitude yields a poor performance using the position signal without filtering. 2) The velocity is required, then a derivative filter is introduced. Fixed low-pass filters can be applied to improve the estimate obtained by difference methods. For instance, antialiasing analog filters can be applied to the position signal before it is sampled and quantized [14]. A Bessel filter was chosen because they have been optimized to obtain a maximally flat delay response; although this is payed by a low selectivity in the frequency response. It is conveyed with a low-pass filtering stage, included in the finite-impulse response (FIR) differentiator to get e(t ˙ − τ ). This double function FIR design was taken from [15].

The low-pass FIR digital filter is used to reduce the unwanted noise from eˆ (t). In all cases, regardless of the noise source, the idea is to assume that the noisy position signal can be separated into spectral components: a low frequency component from which a velocity estimate can be reliably derived and a noisy component which must be filtered out [14].

2) η does not have strong components in the system’s operation range. The complete system is shown on Fig. 2.

Now, some mathematical background about the stability of DDE will be recalled. The equations were adapted to the notation of this paper. B. Delay-differential equations Let consider the general second order DDE: Fig. 2.

x ¨(t) = a2 x(t) ˙ + a1 x(t) + b2 x(t ˙ − τ ) + b1 x(t − τ )

Considered system

(1)

where aj , bj for j = {1, 2}, and τ > 0 are constants. The quasipolynomial ∆(z) corresponding to this equation is ∆(z) = P (z) + Q(z)e−τ z with P (z) = z 2 − a2 z − a1 and Q(z) = −b2 z − b1 . Lemma 1: [16] Assume that |b1 | < |a1 |. Then the system (1) is asymptotically stable for a delay τ ≥ 0 if and only if the polynomial P (z) is asymptotically stable and |Q(iω)| < |P (iω)|, i2 = −1, ω > 0.

D. Problem statement Given a C ∞ output reference yr (t), t ≥ 0, a noisy output signal y(t) = x1 (t) + η and a constant delay τ induced from the filters used to reduce the unwanted noise η and to get the velocity signal. Find a suitable control input u(t) for system (2) such that the trajectory tracking error e(t) = yr (t) − x1 (t)

Lemma 2: [17] Suppose that C = 0, A 6= 0, BD < 0. Necessary conditions for the zero solution of (1) to be asymptotically stable are A < 0 and B + D < 0, where A = a2 τ , B = a1 τ 2 , C = b2 τ and D = b1 τ 2 . Lemma 3: [17] Assume A < 0, B > 0, C < 0, D < 0, and −C = −A. Then necessary conditions for the zero solution of (1) to be asymptotically stable are B + D < 0 and B < −A − C, where A = a2 τ , B = a1 τ 2 , C = b2 τ and D = b1 τ 2 . Note that Lemmas 2 and 3 are delay-dependent stability criteria. C. Considered systems A second order system in the controller canonical form will be considered. x˙ 1 x˙ 2 y

= x2 (t) = a2 x2 (t) + a1 x1 (t) + u(t) = x1 (t),

(3)

satisfies limt→∞ e(t) = 0. Remark 1: Note in Fig. 1 that the system’s error eˆ (t)=yr (t) − y(t) include noise because y(t) = x1 (t) + η. Therefore, the challenge is that x1 (t) is not available from measurement to have equation (3). III. M AIN RESULTS In this section is shown that trajectory tracking for systems with a noisy output signal can be solved thanks to a linear difference-differential equation, which defines the dynamics of the tracking error. Consider the error equation (3). Differentiating it up to the relative degree of system (2) we have the following control input: u(t) = y¨r − a1 x1 (t) − a2 x2 (t) − e¨(t).

(4)

(2)

where x(t) ∈ IR2 is the state vector, u(t) ∈ IR is the system input, y(t) ∈ IR the output and a1 , a2 are constants. The following assumptions are given for system 2: 1) The output signal y(t) can be expressed as x(t) + η, where η represents a noisy undesired signal.

The choice of the tracking error dynamics e¨(t) is fundamental. Regarding equation (4) the signals x1 (t) and x2 (t) = x˙ 1 (t) are not available directly. The output signal for control design is y(t) = x1 (t)+η, and differentiating y(t), we get x1 (t − τ ) for the delay induced by the filtering process. The usual solution is to design an observer to reconstruct x1 (t).

response is shown in Figure 3.

A. Proposed approach The classical solution to the trajectory tracking problem was given in the previously section. It was shown that the output signal approximation is corrupted by noise. Now, the key to derive a controller will be the use of delay-differential equations for the trajectory tracking error.

•

1) Scheme 1. Let consider the following dynamics for e¨(t): e¨(t) = a2 e(t) ˙ + a1 e(t) + b2 e(t ˙ − 1) + b1 e(t − 1)

(5)

where aj and bj for j = {1, 2} are constants such that the stability conditions given in Lemma 1 or given by [17] are fulfilled. This is the case of Fig. 1.

η provided by Simulink’s limited bandwith noise generator considered a noise power of 0.00001 and sampling time of 0.001. Some other kind of noise can be used if assumption 2 is satisfied. The total delay is τ = τBessel + τF IR = 112 ms.. •

From (4) the proposed controller is u(t) = y¨r − a2 y˙ r − a1 yr − b2 e(t ˙ − τ ) − b1 e(t − τ ).

The FIR discrete filter is a combined low-pass and time-differentiator filter design, with sampling rate of 100 Hz., frequency response of s from 0 to 10 Hz. and zero from 10 to 50 Hz. The FIR transfer fuction is (9), the phase is a linear function of frequency representing a constant time delay of MT seconds. The 13-order FIR filter has the following numerator coefficients: [1.3353140743 2.4534859947 3.1983929114 3.3267967565 2.7467372783 1.551489954 0 -1.551489954 -2.7467372783 3.3267967565 -3.1983929114 -2.4534859947 1.3353140743] and it induces a constant delay of τF IR = M T = (6)(0.01) = 60 ms.. The frequency response is shown in Figure 4.

(6)

The coefficients aj are chosen according to the constants of system (2), and bj are chosen s.t. (5) is stable. Note that x1 at time t is not necessary to implement input (6). Remark 2: Delay-independent stability conditions of Lemma 1 are for aj , bj < 0, with j = {1, 2}, i.e., the stable ordinary differential equation (ODE) associated e¨(t) = a2 e(t) ˙ + a1 e(t)

(7)

is stable. The work of Cahlon and Schmidt in [17] is a complete study of this equation, it consider the cases where (7) is unstable and it gives delay-dependent stability conditions.

Fig. 3.

Bessel frequency response.

Example 1. Let consider the following system x˙ 1 (t) = x2 (t) x˙ 2 (t) = 0.8x1 (t) − 3x2 (t) y(t) = x1 (t).

(8)

Find, if possible, a control law u(t) such that the system follows a C ∞ reference trajectory yr (t) = sin(3t), t ≥ 0 despite the noisy output signal y(t) and the constant delay τ induced by the filters. c R2010a, Simulations were done using Matlab/Simulink running on windows 7, 64 bits with the following characteristics: • A sampling frequency of 400 Hz. •

The low-pass antialias Bessel filter was design of order 20 and a 50 Hz. cutoff frequency with the Simulink’s analog filter. The delay was measured directly in simulation and τBessel = 52 ms.. The frequency

Fig. 4.

FIR frequency response.

H(z) =

2M X

ai z −i

(9)

i=0

ai = cM −i where 2M is the order of the filter and ai are the numerator coefficients which are computed according to the coefficients of the Fourier series (further information in [15]).

Equation (5) is asymptotically stable for a1 = 0.8, a2 = −3, b1 = −3 and b2 = −6 from (3), given by [17].

low-pass filters, the combined delay is again τ = 0.112. The coefficients of equation (10) are a1 = 0.8, a2 = −3 and b1 = −5. Simulation results are shown on Fig. 7

Fig. 5 show the results for the trajectory tracking considering the input control (6).

Fig. 7. Fig. 5.

Performance of scheme 1. Controller (6).

Performance of scheme 2. Controller (11).

1) Scheme 3: Let consider the following dynamics for e¨(t):

2) Scheme 2.

e¨(t) = a2 e(t) ˙ + a1 e(t) + b2 e(t ˙ − 1)

Let consider the following dynamics for e¨(t): e¨(t) = a2 e(t) ˙ + a1 e(t) + b1 e(t − 1)

(10)

(12)

where aj for j = {1, 2} and b2 are constants such that the stability conditions given in Lemma 1 or given by [17] are fulfilled. The proposed scheme is shown on Fig. 8.

where aj for j = {1, 2} and b1 are constants such that the stability conditions in Lemma 2 are fulfilled. Fig. 6 shows the proposed scheme. From (4) the controller is u(t) = y¨r − a2 y˙ r − a1 yr − b1 e(t − τ ).

(11)

The coefficients aj are given by system (2) and b1 is chosen s.t. (10) is asymptotically stable. Note again that x1 at time t is not necessary to implement the proposed solution (11). Example 1 (Continuation). Let consider the example 1, with the same filter characteristics for the Bessel antialising and the FIR

Fig. 8.

Configurationof scheme 3

From (4) the controller is u(t) = y¨r − a2 y˙ r − a1 yr − b2 e(t ˙ − τ ).

(13)

The constants aj are given by system (2). Example 2. Let consider system (2) x˙ 1 (t) = x2 (t) x˙ 2 (t) = −2x1 (t) − 10x2 (t) y(t) = x1 (t).

Fig. 6.

Configuration of scheme 2.

(14)

Find, if possible, a control law u(t) such that the system follows a C ∞ reference trajectory yr (t) = sin(3t), t ≥ 0 despite the noisy output signal y(t) and the constant delay τ

induced by the filters, without compute the velocity. The filter characteristics of example 1 are considered for the Bessel antialising and the FIR derivative filters, the combined delay is again τ = 0.112. The coefficients of equation (10) are a1 = −2, a2 = −10 and b2 = 2. Simulation results are shown on Fig. 9.

Fig. 10.

Dynamics of the delay-differential error equations

R EFERENCES

Fig. 9.

Performance of scheme 3. Controller (13).

Remark 3: Note that (7) stable corresponds to a stable open-loop system (2). Therefore if the open-loop system is unstable, the conditions in [17] must be used. B. Discussion The response of the system controller is closely related to the stability of the delayed error equations. The coefficients of (5), (10) and (12) were chosen looking for an acceptable asymptotically stable response, satisfying conditions given by [16] or [17]. A further characterization of the transient response parameters is in process to optimize the closed loop performance. Their transient response is shown on Fig. 10, with initial conditions x = 10 and x˙ = 1. IV. C ONCLUSIONS In this work a methodology to solve the trajectory tracking problem of systems with a noisy position signal is proposed. The novelty with respect to standard methodologies is the inclusion of the delay introduced by the measured signal filtering process. The use of delay-differential error equation allow us to guarantee the convergence of the proposed controllers. The use of a simple FIR filter as a combination of a low-pass filter and differentiator displayed a good performance in simulation examples. This is a first step towards the general case, where more complex delay-differential error equations are required.

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