Adaptive feedback suppression of unknown periodic components of acoustic noises with time-varying characteristics

Abstract

Effective attenuation of the noise level is an important problem in acoustic systems. In this paper, we propose a robust adaptive output feedback control scheme that can considerably attenuate narrow-band noises made up of periodic signals mixed with random noise in the presence of modeling uncertainties. The amplitude, phase and frequencies as well as the number of periodic terms are unknown and could vary with time. The performance and robustness of the proposed scheme with respect to unstructured modeling uncertainties are analyzed for continuous-time single-input, single-output systems; the results, however, are extendable to multi-channel systems. The successful attenuation of the unknown periodic components of the disturbance despite the time variations, modeling errors, and random noise is demonstrated using simulations. In addition, guidelines how to choose certain design parameters for performance improvement have been presented.

Keywords

Periodic noise suppression unknown time-varying frequencies robust adaptive control

1. Introduction

The control of unwanted sound and vibrations in a wide class of systems has been the subject of engineering research over the recent decades. In many industrial applications, undesired sound and vibrations can be classified as periodic or quasi-periodic disturbances which are mainly caused by rotary or reciprocating components such as electric motors, compressors, engines, cooling systems, fans, propellers, air-conditioning systems, or can be generated by resonance coils in magnetic resonance imaging (MRI) system (Nelson and Elliott, 1992; Benesty et al., 2008; Rudd et al., 2012). In active noise control, the main objective is to reduce the noise level over a useful volume of space in an environment by producing anti-noise, i.e. generating noise from some speakers (control actuators) to interfere with the sound field such that noise level at microphones (error sensors) position is made as small as possible. Due to spatial constrains, the active noise control primarily deals with low-frequency acoustical noise (typically less than 500 Hz); high-frequency components can be effectively suppressed by using passive sound absorbers (Elliott and Nelson, 1990, 1993). Some successful applications including reduction of propeller-induced cabin noise in aircraft, engine and road noise in cars, and noise in acoustic ducts can be found in Elliott et al. (1990), Sutton et al. (1994), and Amara et al. (1999).

Several control schemes for noise and vibration reduction have been developed and empirically tested (Bodson, 2005; Patt et al., 2005; Lau et al., 2006; Perez-Arancibia et al., 2009; Pigg and Bodson, 2010; Kim et al., 2011; Landau et al., 2011a,b; Chen and Tomizuka, 2012; Isidori et al., 2012; Aranovskiy and Freidovich, 2013; Marino and Tomei, 2013). Basturk and Krstic (2012) studied the problem of adaptive rejection of unknown periodic components of a disturbance using state derivative feedback. A repetitive control method has been proposed in Houtzager et al. (2013) for rejection of periodic wind disturbances in wind turbines. In Duan et al. (2013, 2014a,b), active noise control algorithms have been developed for the reduction of road noise inside a vehicle cabin.

In almost all past work on noise attenuation, the controller is designed based on a simplified model of the plant. The subsequent analysis, in most cases, assumes that there is no discrepancy between the plant model and the actual plant which is not true in practice. A practical control scheme for noise attenuation has to be designed to tolerate the inevitable modeling errors and be able to handle time variations in the unknown characteristics of the noise components.

This paper presents the design and analysis of an adaptive feedback control scheme for attenuation of unknown unmeasurable acoustical noise dominated by periodic terms and corrupted by broadband random noise. The robustness of the proposed scheme with respect to plant modeling uncertainties and variations of disturbance parameters, i.e. frequencies, amplitudes, and phases is analyzed. The effect of various design parameters on stability and performance is investigated.

The paper is organized as follows. We introduce the notation and preliminaries in Section 2. The problem formulation and the underlying assumptions are described in Section 3. In Section 4, the architecture of the controller is presented and the conditions for existence of a problem-solving controller is examined. The adaptive controller is designed and analyzed in Section 5. The numerical simulations demonstrating the proposed scheme are presented in Section 6. The concluding remarks are given in Section 7.

2. Notation and preliminaries

In this section we present the notation and basic definitions used throughout the paper. The superscript (ċ) ^T denotes the transpose operator. For a signal x(t), the truncated signal x_t for any t ∈ [0,∞) is defined as x_t(τ) = x(τ), for 0 ≤ τ ≤ t, and x_t(τ) = 0, for τ > t (Ioannou and Sun, 1996). A finite-dimensional linear time-invariant (LTI) system with transfer function G(s), input u(t) and output y(t) may be written as $y (t) = G (s) [u (t)] = L - 1 {G (s) U (s)}$ , where U(s) is the Laplace transform of u(t), $L - 1 (\cdot)$ denotes the inverse Laplace transform, and s is the Laplace variable. The $H \infty$ -norm of a system is defined as $| |$ G(s) $| |$ _∞ = max _ω |G(jω)| and the $L 1$ -norm is $| | G (s) | | 1 = \int_{0}^{\infty} | g (τ) | 1 mu d τ$ , where g(t) is the impulse response of the system. For a system with minimal realization of degree n, the two norms are related as $| |$ G(s) $| |$ _∞ ≤ $| |$ G(s) $| |$ ₁ ≤ (2n + 1) $| |$ G(s) $| |$ _∞ (Dahleh and Diaz-Bobillo, 1995; Ioannou and Sun, 1996; Skogestad and Postlethwaite, 2005). We say that a function x: [0, ∞) → $ℝ n$ is μ-small in the mean square sense and write $x \in S (μ)$ , if for a given constant μ ≥ 0, we have $\int_{t}^{t + T} x T (τ) x (τ) d τ \leq c 0 μ T + c 1$ , for any t, T ≥ 0 and some finite constants c₀, c₁ ≥ 0 independent of μ (Ioannou and Sun, 1996).

Swapping Lemma (Ioannou and Sun, 1996). Let x(t), y(t): $ℝ +$ → $ℝ n$ and x(t) be differentiable. Let W(s) be a proper stable rational transfer function with a minimal realization (A,B,C,D), i.e. W(s) = C^T (sI − A)⁻¹B + D. Then, W(s)[x^T(t)y(t)] = x^T (t)W(s)[y(t)] + W_c(s)[(W_b(s)[y(t)]) ^T $x \cdot$ (t)], where W_c(s) = −C^T (sI − A)⁻¹ and W_b(s) = (sI − A)⁻¹B.

3. Problem formulation

We consider an acoustic feedback configuration with a single loadspeaker (actuator) and a single microphone (error sensor) as shown in Figure 1, where y(t) is the output signal from the microphone and u(t) is the input signal to the loadspeaker.

The objective is to produce a control signal u(t) to minimize the noise level at the microphone position. The block diagram of the system is shown in Figure 2, where d(t) is an unknown acoustical disturbance which is not directly observable whose characteristics may be time-varying, and G(s) is the transfer function from u(t) to y(t) which is typically not perfectly known consisting of speaker, acoustic path, and microphone dynamics. We consider G₀(s) as the known modeled part of the plant with the multiplicative uncertainty term Δ _m (s) representing modeling errors, time delays, etc. (Ioannou and Sun, 1996). Since in real physical systems a very accurate linear model over a wide range of frequencies is usually unavailable especially in the high-frequency range, then any model must include such sources of uncertainties, and its effect on stability and performance must be carefully investigated.

Based on Figure 2, the plant dynamics are expressed as

G (s) = G 0 (s) (1 + Δ m (s))

(1)

where G₀(s) is assumed to have stable poles, i.e. poles in the open left half-plane with no restriction on the location of its zeros. The modeling uncertainty term Δ _m (s) accounts for neglected dynamics and is assumed to be small in some norm sense in the low-frequency range which is the range of interest, and may be large at high frequencies. It may represent unmodeled input delay, fast actuator dynamics and other modeling errors. An example of such uncertainty is Δ _m (s) = −μs, where 0 < μ ≪ 1 is a small constant, which may represent the approximation of a time delay μ in the input, i.e. u(t − μ) whose Laplace is e⁻ ^μs [u(t)] ≈ (1 − μs)[u(t)]. The parameters and structure of Δ _m (s) are unknown and it is just assumed that G₀(s)Δ _m (s) is proper and bounded-input, bounded-output (BIBO) stable.

In many applications, the unknown narrow-band acoustical disturbance d(t) is dominated by periodic unknown signals and can be approximated as sum of a finite number of sinusoidal terms corrupted with low-amplitude broadband random noise as

d (t) = \sum_{i = 1}^{n f (t)} a i (t) \sin (ω i (t) t + ϕ i (t)) + v (t)

(2)

where the parameters a_i, ω_i, ϕ_i, the number of distinct frequencies n_f are unknown and may vary with time and v is a broadband random noise with amplitude much smaller than that of the periodic terms. We assume that an upper bound

n ¯ max

for the maximum number of distinct frequencies of the periodic components of the disturbance and an upper bound

ω ¯ max

for its largest frequency are known, i.e. n_f(t) ≤

n ¯ max

and ω_i(t) ≤

ω ¯ max

, for any t. If no knowledge about these parameters is available, one may choose conservative upper bounds.

In active acoustic noise control problems, we often deal with disturbances at low-frequency ranges, because passive sound absorbers can efficiently suppress disturbances with high-frequency components (Elliott and Nelson, 1993). The proposed control scheme in this paper, however, can be used for any range of frequencies over which the size of modeling uncertainties is sufficiently small.

The control law must produce the control signal u(t) by using the knowledge of the output signal y(t) and nominal plant model G₀(s) to counteract the effect of the unknown disturbance d(t) on y(t) as much as possible. It is obvious from Figure 2 that if G(s)[u(t)] = −d(t), then y(t) = 0. This choice of u(t), however, is not acceptable as this control law cannot be implemented in practice, because G(s), even if known exactly, may have unstable zeros and its inverse is not causal. In the following section, we present a feedback adaptive control that can meet the objective of attenuating the unknown output disturbance despite the presence of random noise and modeling uncertainties.

4. Controller architecture

Since the characteristics of the disturbance are unknown and may vary with time, an adaptive control law is designed to achieve the control objective. We consider the controller structure shown in Figure 3, where F(s) is a fixed-parameter BIBO stable filter and K(s, θ) is an adaptive filter whose parameter vector θ(t) is to be generated at each time t.

We assume the adaptive filter K has the following form

K (s, θ (t)) = \sum_{k = 0}^{N - 1} θ k (t) \frac{λ N - k}{(s + λ) N - k}

(3)

where

θ (t) = [θ 0 (t), θ 1 (t), \dots, θ N - 1 (t)] T

is the controller parameter vector, the scalar λ > 0, and integer N are design parameters. The parameter vector θ(t) is to be adjusted at each time t by an adaptive law to be designed.

Before designing the adaptive law we examine whether there exists a desired vector θ* (which θ(t) is trying to find), which meets the control objective.

From Figures 2 and 3, the transfer function from d to y is given by

y (t) = S (s, θ) [d (t)] = \frac{1 - G' 0 (s) K (s, θ)}{1 + Δ m (s) G' 0 (s) K (s, θ)} [d (t)]

(4)

where

G 0'

(s) = G₀(s)F(s) and the filter F(s) (to be designed) is such that

G 0'

(s) is BIBO stable and strictly proper. The effect of the periodic components of d on y is rejected if the zeros of S(s) include zeros at s = ± jω_i, for

i = 1, \dots, n f

. In other words, the periodic components of d(t) can be rejected exactly if S(s) contains their internal model. This can be achieved if the filters F(s) and K(s,θ) are chosen so that S(s) contains the internal model of all the periodic terms in d(t). Such choice, however, should be such that the poles of S(s) remain in the stable region.

Assuming that we have perfect knowledge of the frequencies in d(t), is the structure of the controller given in Figure 3 able to meet the control objective for some choice of the filters F(s) and K(s,θ)? In particular, before we proceed with the case of the unknown disturbance we address the following questions: do there exist filters F(s) and K(s,θ) that reject the periodic terms in d(t)? If such filters exist, how do they affect stability and performance? How does the plant uncertainty Δ _m (s) affects stability and performance? The following theorem provides answers to the above questions and becomes the reference of the best performance that can be achieved in the case of unknown disturbance frequencies.

Theorem 1

Consider the closed-loop system shown in Figures 2 and 3 with a LTI filter of the form (3). Let $ω 1, \dots, ω n f$ be the distinct frequencies of the disturbance. There exists a $θ *$ such that $K (s, θ *)$ completely rejects the periodic components of the disturbances if and only if $G' 0 (s) = G 0 (s) F (s)$ has no zero at $s = \pm j ω i$ , for $i = 1, \dots, n f$ , and $N \geq 2 n f$ , provided the stability condition $| | Δ m (s) G' 0 (s) K (s, θ *) | | \infty < 1$ is satisfied. The choice of θ* is unique if $N = 2 n f$ . In addition, all signals in the closed-loop system are guaranteed to be uniformly bounded and the plant output satisfies

lim t \to \infty sup τ \geq t | y (τ) | \leq | | S (s, θ *) | | 1 v 0 \leq c (1 + | | G' 0 (s) K (s, θ *) | | 1) v 0

(5)

where

v 0 = sup t | v (t) |

and

c = ‖ \frac{1}{1 + Δ m (s) G' 0 (s) K (s, θ *)} ‖ 1

is a finite positive constant. Moreover, in the absence of broadband noise (

v (t) = 0), y (t)

converges to zero exponentially fast.

Proof

The proof is given in Appendix A.

Theorem 1 gives conditions under which we can completely reject the periodic components of the disturbance when the frequencies of the disturbance are known and fixed with time. It shows that if the plant uncertainty satisfies an upper bound, the output is of the order of the broadband random noise level. When this noise is absent, the output converges to zero exponentially fast, demonstrating that the periodic terms of the disturbance are completely rejected. It should be noted that with N = 2n_f, there exists a unique θ* for which complete rejection of the period part of the disturbance is possible. Such a unique value, however, does not provide any flexibility to improve performance and/or robust stability margins.

As shown in the proof of Theorem 1, for N > 2n_f there is an infinite number of vectors θ* that guarantee the results of Theorem 1. In such case, one may choose the θ* that minimizes $| | G 0'$ (s)K(s,θ*) $| |$ _∞. The minimization of $| | G 0'$ (s)K(s,θ*) $| |$ _∞, while maintaining rejection of the periodic terms and stability, guarantees that the norm sensitivity function from the output to the random noise v(t) is also minimized over the set of all possible θ*. If the frequencies of the disturbance are varying with time in some sense, then θ*(t) is a function of time. It can be easily shown that if the time derivatives of the unknown frequencies of the disturbance are μ-small in the mean square sense then $θ \cdot * (t)$ is also μ-small in the mean square sense, i.e. for some constants $c 0, μ 0 \geq 0, \int_{t}^{t + T} ‖ θ \cdot * (τ) ‖ 2 d τ \leq μ 0 T + c 0$ (Ioannou and Sun, 1996; Rugh, 1996).

5. Adaptive control design

From Figure 3, the control input signal is given by

u (t) = - F (s) [K (s, θ (t)) [z (t)]] z (t) = y (t) - G 0 (s) [u (t)]

(6)

where

θ (t) = [θ 0 (t), \dots, θ N - 1 (t)] T

is the estimate of a desired parameter vector θ*(t) at each time, and

K (s, θ (t)) = (θ 0 (t) λ N + θ 1 (t) λ N - 1 (s + λ) + \dots + θ N - 1 (t) λ (s + λ) N - 1) (s + λ) - N

(7)

In order to estimate θ* online, we express it in the form of a parametric model as follows (Tsakalis and Ioannou, 1993; Ioannou and Sun, 1996). We define

w (t) = Δ G' 0 (s) [K (s, θ * (t)) [z (t)]] = G' 0 (s) [θ * T (t) Λ (s) [z (t)]]

(8)

where

Λ (s) = [\frac{λ N}{(s + λ) N}, \dots, \frac{λ}{s + λ}] T

is a BIBO stable transfer function matrix and

G' 0 (s) = Δ G 0 (s) F (s)

. Since

G' 0 (s) K (s, θ * (t))

at each frozen time t is stable, then from Theorem 3.4.11 in Ioannou and Sun (1996), there exists a

μ_{0}^{*} > 0

such that if

| | θ \cdot * (t) | | \in S (μ 0)

for any

μ 0 \in [0, μ_{0}^{*})

, the linear time-varying system from z(t) to w(t) is BIBO stable.

The vector $θ * (t)$ is such that at each time t, $S (s, θ * (t)) = Δ 1 - G' 0 (s) K (s, θ * (t))$ has zero gain at the frequencies of the disturbance ω_i, $i = 1, \dots, n f$ , then

e (t) = Δ z (t) - w (t) = S (s, θ * (t)) [z (t)] = S (s, θ * (t)) [d (t) + Δ m (s) G 0 (s) [u (t)]] = S (s, θ * (t)) [v (t) + Δ m (s) G 0 (s) [u (t)]]

(9)

where the possible exponentially decaying terms are neglected as they do not affect the analysis.

From the Swapping Lemma (see Section 2),

w (t) = G' 0 (s) [θ * T (t) Λ (s) [z (t)]] = θ * T (t) G' 0 (s) Λ (s) [z (t)] + f (θ \cdot * (t))

(10)

where

f (θ \cdot * (t)) = G' 0 c (s) [(G' 0 b (s) [(Λ (s) [z (t)]) T]) θ \cdot * (t)] G' 0 b (s) = (sI - A g) - 1 B g G' 0 c (s) = - C_{g}^{T} (sI - A g) - 1

(11)

and the triple (A_g, B_g, C_g) is a minimal state-space realization of

G 0' (s)

. The last term in the right-hand side of (10) is due to the time variations in the parameters of the disturbance, that acts as a disturbance term. By adding z(t) to both sides of (10) we obtain

z (t) = θ * T (t) G' 0 (s) Λ (s) [z (t)] + (z (t) - w (t)) + f (θ \cdot * (t))

Then, a parametric model is obtained as

z (t) = θ * T (t) φ (t) + η (t) φ (t) = G' 0 (s) Λ (s) [z (t)] = F (s) G 0 (s) Λ (s) [z (t)] η (t) = e (t) + f (θ \cdot * (t))

(12)

where

z (t) = y (t) - G 0 (s) [u (t)]

and

φ (t)

are available for measurement, the error term η(t) is unknown and goes to zero if v(t) = 0, Δ _m (s) = 0, and

θ \cdot * (t) = 0

(i.e. in the absence of broadband noise, modeling error, and if disturbance characteristics do not change with time).

Using parametric model (12), a robust adaptive law can be designed to estimate θ*(t) and achieve the control objective. The amplitude of the regressor vector φ(t) at the frequencies of the disturbance is an important factor contributing to the rate of adaptation. In (12), the regressor φ(t) is obtained by passing z(t) through W(s) = F(s)G₀(s)Λ(s). If W(s) severely suppresses some of the frequency components of z(t), there will not be adequate information about those frequencies in the regressor and this will lead to a very small or zero learning rate of the unknown frequencies. As a result, the feedback controller will have difficulty attenuating the respective disturbance periodic terms associated with these frequencies. In order to remedy or improve this situation we design the LTI filter F(s) such that F(s)G₀(s)Λ(s) has a large enough gain over the frequency range of 0 ≤ ω ≤ $ω ¯ max$ . It is to be noted that λ is chosen such that $Λ (s) = [λ N (s + λ) N, \dots, λ s + λ] T$ has a large enough gain over the frequency range of interest.

5.1. Design of the LTI filter F(s)

The objective of introducing the filter F(s) is to flatten the spectrum of all elements of W(s) = F(s)G₀(s)Λ(s) over the frequency range ω ∈ [0, $ω ¯ max$ ] to the extent possible. Since all elements of Λ(s) are chosen to be low-pass filters with DC gain one and bandwidth greater than $ω ¯ max$ , we just need to choose F(s) such that G₀′(s) = F(s)G₀(s) has a sufficiently large gain at almost all frequencies in the frequency range of ω ≤ $ω ¯ max$ .

Let $G ~ 0 (s)$ be a rational transfer function obtained from G₀(s) by modifying the numerator of G₀(s) as follows.

Let $G ~ 0 (s) = G 0 (s)$ .

If $G ~ 0 (s)$ has a zero on the jω-axis at s = jω₀, change it to s = −δ₀ + jω₀, where δ₀ > 0 is a small positive constant.

If $G ~ 0 (s)$ has an unstable zero at s = σ₀ ± jω₀ with σ₀ > 0, change it to s = −σ₀ ± jω₀, i.e. reflect it across the jω-axis and make it stable.

Now, we let

F (s) = κ 0 \frac{α m}{(s + α) m} {G ~}_{0}^{- 1} (s) = Δ κ 0 F ¯ (s)

(13)

where m > 0 is an integer greater than the relative degree of G₀(s) and α > 0 is a design constant chosen such that the low-pass filter

\frac{α m}{(s + α) m}

has a large enough gain over the frequency range of interest. Then

G 0'

(s) = F(s)G₀(s) is a BIBO stable strictly proper transfer function with gain about κ₀ at most frequencies over ω ≤

ω ¯ max

except at and around the frequencies where G₀(s) has zeros on the jω-axis.

Example

Assume that the nominal plant model is given by

G 0 (s) = \frac{s (s 2 + 4) (s - 0.8) (s + 1.4)}{(s + 0.5) 3 (s + 2) 2 (s + 3)}

Then the filter F(s) can be designed as

F (s) = \frac{κ 0 α 2 (s + 0.5) 3 (s + 2) 2 (s + 3)}{(s + α) 2 (s + δ 0) ((s + δ 0) 2 + 4) (s + 0.8) (s + 1.4)}

where δ₀ is a positive small constant, e.g. δ₀ = 0.01. Let κ₀ = 1 and assume

ω ¯ max

= 300 rad/sec, then we choose α = 300. Figure 4 shows the magnitude plot of G₀(s) and

G 0'

(s) = F(s)G₀(s) which has gain of κ₀ = 1 over the frequency range of interest except at and around ω = 0 and 2 rad/s at which G₀(s) has zeros on the jω axis. It should be noted that from Theorem 1, for this plant model, rejection of disturbances with frequencies ω = 0 and 2 rad/s is impossible as the plant has zero gain at these frequencies.

The above procedure increases the gain of the nominal plant by introducing a filter F(s) and flattening the magnitude of its frequency response over the frequency range of interest. However, some other considerations must be taken into account in order to ensure the stability of the closed-loop system in the presence of unmodeled dynamics. We will show below that the stability of the adaptive closed-loop system is guaranteed if the norm of F(s)G₀(s)Δ _m (s) is sufficiently small. With filter F(s) obtained from the above procedure, the gain of F(s)G₀(s) at low frequency range ω < α is less than or equal to κ₀ and at very high frequencies ω ≫ α is very small. Since the size of the modeling error term Δ _m (s) is usually small at low frequencies and is possibly large at high frequencies, then if the magnitude of Δ _m (s) at low frequencies is small enough and the roll-off rate of F(s)G₀(s) for ω ≫ α is high enough, the size of F(s)G₀(s)Δ _m (s) is small in the low- and high-frequency range; however, in the moderate-frequency range, especially when ω_max is large, F(s)G₀(s)Δ _m (s) may have a large peak magnitude which may lead to instability of the closed-loop system. Since the frequency range where the unmodeled dynamics term Δ _m (s) is dominant is unknown, we may need to choose conservative parameters for the filter F(s).

In (13), the parameters κ₀ and α must be chosen carefully to have a good stability margin and a satisfactory performance. For example, increasing the value of κ₀ can speed up adaptation and improve performance, but it may reduce the margin of stability in the presence of modeling error. Also, decreasing α reduces the bandwidth of F(s)G₀(s) and compensates the high-frequency unmodeled dynamics; hence, improves the stability margin, but adversely affects the performance if the disturbance has dominant terms with frequencies beyond the bandwidth of F(s)G₀(s). These trade offs are well known in robust control and robust adaptive control and need to be kept in mind in choosing the various design parameters (Ioannou and Sun, 1996). In Silva et al. (2014) a conceptually similar approach has been proposed to improve the gain of G₀(s) by using the Youla–Kučera parametrization and an Fourier infrared (FIR) filter in the feedback loop.

5.2. Design of the adaptive filter K(s,θ(t))

Using the parametric model (12) and employing the techniques discussed in Ioannou and Sun (1996); Ioannou and Fidan (2006) we design a robust adaptive law to estimate the unknown parameter vector θ*(t).

The estimated model of (12) is obtained by replacing θ*(t) by its estimate θ(t) at time t and ignoring the unknown error term η(t) as

z \land (t) = θ T (t) φ (t)

(14)

where

z \land

(t) is the predicted value of z(t) based on θ(t) at time t. In order to measure the difference between θ(t) and θ*(t) at each time, we define the estimation error as

ɛ (t) = \frac{z (t) - z \land (t)}{m_{s}^{2} (t)} m_{s}^{2} (t) = 1 + γ 0 φ T (t) φ (t)

(15)

where

m_{s}^{2} (t)

is a normalizing signal and γ₀ > 0 is a design constant. Following the design procedure of Ioannou and Sun (1996); Ioannou and Fidan (2006) we consider the robust least-squares algorithm to generate the estimates of θ(t):

P \cdot (t) = - P (t) \frac{φ (t) φ T (t)}{m_{s}^{2} (t)} P (t)

(16a)

θ \cdot (t) = proj (P (t) ɛ (t) φ (t))

(16b)

where θ(0) = 0,

P (0) = P 0 = P_{0}^{T} > 0

is a design constant matrix. To prevent P(t) in (16a) from becoming singular, we can use covariance resetting modification (Ioannou and Sun, 1996), i.e. set P(

t_{r}^{+}

) = ρ₀I, where t_r is the time at each λ_min(P) ≤ ρ₁, and ρ₀ > ρ₁ > 0 are some design constants (where λ_min denotes the minimum eigenvalue). This modification guarantees that P(t) for all time is positive definite and never becomes close to singularity. The projection operator proj(ċ) in (16b) is used to ensure that the estimated vector θ(t) belongs to a known compact set defined as

= {θ ∈

ℝ N

|g(θ)

= Δ

θ^Tθ −

θ_{\max}^{2}

≤ 0}, where θ_max > 0 is chosen such that θ*(t) ∈

, ∀t. Since no a priori knowledge on the norm of θ* is available, θ_max must be chosen sufficiently large. Some alternatives to projection and other robust modifications can be found in Ioannou and Sun (1996). The projection operator in (16b) may be implemented as

where

and

denote the interior and the boundary of

The following theorem summarizes the properties of the adaptive closed-loop system shown in Figures 2 and 3.

Theorem 2

Consider the closed-loop system shown in Figures 2 and 3 with an adaptive filter of the form (3), and choose $N >> 2 n ¯ max$ . If the unmodeled dynamics satisfy

κ 0 θ ¯ M | | F ¯ (s) G 0 (s) Δ m (s) | | 1 < 1

(18)

where

κ 0 > 0

is a design constant, and

F ¯ (s)

is defined in (13), then all signals in the closed-loop system are uniformly bounded and the plant output satisfies

lim T \to \infty sup \frac{1}{T} \int_{t}^{t + T} | y (τ) | 21 mu d τ \leq c (v_{0}^{2} + μ_{Δ m}^{2} + μ_{θ \cdot *}^{2})

(19)

for any t and some finite constant c > 0 independent of T,t, where

v 0 = sup_{t} v (t)

μ Δ m > 0

is a constant proportional to the size of the unmodeled dynamics, and

μ θ \cdot * > 0

is a constant proportional to the size of

| | θ \cdot * | |

. In addition, in the absence of broadband noise and modeling error and when the disturbance frequencies do not vary with time (i.e. when

v (t) = 0

Δ m (s) = 0

, and

θ \cdot * = 0

), the plant output y(t) converges to zero.

Proof

The proof is given in Appendix B.

Theorem 2 states that if the size of modeling uncertainty is small enough, the energy of the output signal will be of the order of broadband noise level and the size of unmodeled dynamics. It is to be noted that (18) is a sufficient condition for the stability of the closed-loop system and states that the stability is ensured if FG₀ is sufficiently small over the frequency range where Δ _m is large. Since Δ _m is typically negligible over the frequency range of active noise cancellation (low-frequency range) we can choose the filter F to make FG₀ very small over medium and high frequencies.

Even though the bounds in (18) and (19) cannot be explicitly calculated as they depend on the unknown parameter θ* and Δ _m (s), they offer some indication how to choose the filter F(s) and other design parameters. The performance bound (19) is typical in robust adaptive control. It does not prevent the output y(t) to assume large values over short interval of times a phenomenon known as “bursting” (Ioannou and Sun, 1996). This phenomenon can be prevented by further modifying the adaptive law using a dead-zone in addition to projection.

6. Numerical simulation

Consider the system shown in Figure 2. The actual transfer function from u to y is given by G(s) = G₀(s)(1 + Δ _m (s)), where

G 0 (s) = \frac{0.5 (s - 0.2)}{s 2 + s + 1.25}

is the known modeled part of the plant and Δ _m (s) = −μs, with μ = 0.001 (for simulation purposes), is the unknown unmodeled dynamics. The magnitude of Δ _m (s) is very small at low frequencies and is large at high frequencies. Assume the number of distinct frequencies of the periodic terms of the disturbance is at most

n ¯ max

= 5 and the largest frequency of interest of the disturbance is less than

ω ¯ max

= 600 rad/s. The unknown disturbance signal is given by

d (t) = 0.6 \sin (ω 1 t + \frac{π}{4}) + 0.7 \sin (ω 2 t + \frac{π}{2}) + v (t)

(20)

where ω₁ = 70 and ω₂ = 187 rad/s, v(t) is a zero mean Gaussian noise with standard deviation 0.02 and |v(t)| ≤ 0.1. The objective is to design the feedback control signal u to minimize the effect of d on y.

As shown in Figure 5, the magnitude bode plot of G₀(s) is very small at the expected frequency range of the disturbance. This small plant gain may drastically slow down the adaptation and adversely affect the performance of the proposed control scheme. In order to increase the plant gain over the frequency range of interest of the disturbance, we use the procedure proposed in Section 5.1 and design a stable filter F(s) such that the compensated plant model $G 0'$ (s) = F(s)G₀(s) has a large enough gain over the frequency range of interest. The filter F(s) designed for this plant is given by

F (s) = κ 0 \frac{2 α 2 (s 2 + s + 1.25)}{(s + α) 2 (s + 0.2)}

Figure 5 shows the magnitude bode plot of

G 0'

(s) = F(s)G₀(s) for α = 500 and κ₀ = 0.5 = −6 dB.

For a fixed value of α, increasing κ₀ increases the gain of the plant as well as the level of the regressor signal vector φ(t) over the frequency range of interest; however, from the bound in (18), in the presence of unmodeled dynamics, increasing κ₀ reduces the margin of stability and that may lead to instability. By choosing α within the expected frequency range of disturbance 0 < α ≤ $ω ¯ max$ , the gain or magnitude of $G 0'$ (s) = F(s)G₀(s) for s = jω with ω < α is about κ₀. Then, by choosing reasonable values of κ₀, the gain of $G 0'$ (s) is high enough in the frequency range of interest and low in the range where the unmodeled dynamics become important which is outside the frequency range of interest. If, however, the frequency range of interest is expanded to come close to the range where the unmodeled dynamics are dominant we may run into instability problems. This phenomenon suggests the classical trade-off between robustness and performance. Therefore, a proper choice of filter F(s) can provide an appropriate trade-off between performance and robust stability.

Consider the adaptive control scheme proposed in Section 5.2 with P(0) = 500I, θ(0) = 0, N = 20, κ₀ = 0.5, α = 500, and λ = 500. We assume the sampling period for simulations is T_s = 10⁻⁴ s. Figure 6 shows the plant output y(t), where the control input u(t) is applied at t = 4 s. Figure 6(a) is the performance of the proposed scheme without filter F(s) in the loop (i.e. F(s) = 1). The rate of adaptation in this case is very small as the plant model G₀(s) has a very small gain at the frequencies of the disturbance ω₁ = 70 and ω₂ = 187 rad/s (see Figure 5). It is clear that the feedback adaptive controller with F(s) = 1 and N = 20 failed to improve performance by rejecting the periodic disturbances at least during the first 20 seconds. We should note that by increasing the size of the adaptive filter, i.e. N, the performance shown in Figure 6(a) can be improved but not as effectively as using the filter F(s) above with κ₀ = 0.5, α = 500 for the same N = 20. As shown in Figure 6(b) the periodic terms have been rejected very fast right after the control input is switched on at t = 4 s. The output is driven by mostly the output broadband random noise.

To show the performance of the proposed scheme in case of change in the characteristics of the disturbance, we assume from t = 6 to t = 16 sec, the frequency ω₁ in (20) changes from 70 to 100 rad/s linearly with time, i.e.

ω 1 (t) = {70 if t \leq 6 3 (t - 6) + 70 if 6 < t \leq 16 100 if t > 16

As shown in Figure 7, the adaptive controller readjusts its parameters and rejects the new disturbance periodic terms. As predicted by analysis, during the time the frequency varies with time the output error is proportional to the speed of variation, i.e. the derivative of the frequency which for this example is rather high, i.e. 3 for 6 < t ≤ 16. Despite the time variation the adaptive scheme is still able to maintain small regulation error which reduces further when the frequency of the disturbance becomes constant.

7. Conclusions

We have considered the problem of attenuation of unknown unobservable acoustical noise in an environment. The acoustical noise is assumed to be dominated by an unknown number of sinusoidal terms with unknown amplitudes, phases and frequencies all of which can change over time, mixed with lower-amplitude broadband random noise. A robust adaptive control scheme has been proposed to attenuate the noise level. The effect of plant unmodeled dynamics, variations of disturbance characteristics, and broadband noise on the stability and performance have been investigated. It has been shown that a proper shaping of the magnitude frequency response of the plant model can significantly improve robustness and performance of the adaptive closed-loop system. By making an appropriate selection of the design parameters, the proposed scheme provides a suitable compromise between robustness and performance. The approach can be generalized to multi-input multi-output continuous or discrete-time systems.

Footnotes

Funding

This work has been supported by Northrop Grumman Aerospace Systems.

Appendix A: proof of Theorem 1

Appendix B: proof of Theorem 2

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