Robust disturbance rejection method for uncertain ... - Springer Link

Nonlinear Dyn (2009) 55: 329–336 DOI 10.1007/s11071-008-9365-z

O R I G I N A L PA P E R

Robust disturbance rejection method for uncertain system with disturbances of unknown frequencies Yi-De Chen · Pi-Cheng Tung · Chyun-Chau Fuh

Received: 8 December 2007 / Accepted: 1 May 2008 / Published online: 3 June 2008 © Springer Science+Business Media B.V. 2008

Abstract In this paper, we discuss a robust disturbance rejection method for dealing with disturbances of unknown frequencies. Unlike many other approaches, the method proposed here does not require the disturbance frequencies of the separate harmonics to be estimated. The current approach is based on disturbance reduction and disturbance suppression. This novel disturbance reduction controller consists of an inverse of the nominal model with an input deduction and a high gain integral term. The proposed controller can reduce both periodic and nonperiodic unknown disturbances with uncertainties in both stable and unstable systems. In addition, undesired responses caused by residual disturbances and residual modeling uncertainties are suppressed by combining the novel disturbance reduction controller with a sliding mode controller. The simulation results demonstrate that the proposed disturbance rejection method performs well under different disturbance inputs including random signals. Y.-D. Chen · P.-C. Tung () Department of Mechanical Engineering, National Central University, No. 300, Jhongda Rd., Jhongli City, Taoyuan County 32001, Taiwan, ROC e-mail: [email protected] C.-C. Fuh Department of Mechanical and Mechatronic Engineering, National Taiwan Ocean University, No. 2, Pei-Ning Rd., Keelung 20224, Taiwan, ROC e-mail: [email protected]

Keywords Robust · Disturbance rejection · Sliding mode control

1 Introduction Periodic disturbances can occur in many different engineering control applications, particularly in those involving rotating mechanical systems. For example, in active magnetic bearings, mass imbalances, and sensor runout generate disturbances at the first harmonic and multiple harmonics of rotation, respectively. Clearly, these disturbances will degrade system performance. Furthermore, the vibrations caused by a rotating imbalance affect both the precision and the reliability of the system. Similar disturbance phenomena are also apparent in electric motors, where they manifest in the form of torque disturbances caused by cogging torque and detent torque. In disk drive systems, internal and external disturbances act to reduce the storage capacity of the drive. Since these types of disturbances cannot simply be ignored, many proposals for disturbance rejection in active vibration and noise control applications have found their way into the published literature. One of the most common disturbance rejection techniques based on the internal model principle (IMP), was proposed by Francis and Wonham [1]. The IMP method states that a model of the disturbance generation system must be included in the feedback

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system to enable a complete rejection of the disturbance. For sinusoidal disturbances, this implies that the controller must have a pair of poles on the j ω-axis at a location corresponding to the frequency of the disturbances. The exact frequency estimation is essential in the IMP method if the periodic disturbance is to be completely canceled. For multiple-frequency disturbances, the stability problem becomes increasingly complex as poles are added on the j ω-axis. However, if the frequencies of the disturbances are known and are invariant, the disturbances can be effectively canceled by adding a disturbance observer into the controller [2]. Another common disturbance rejection technique is the adaptive feedforward cancellation (AFC) method where a disturbance at the input of the plant can be canceled by constantly adding the negative of the estimated disturbance. In the case where the frequency of the periodic disturbance is known, the standard AFC method provides an effective means for rejecting the disturbances by estimating their unknown magnitude and phase [3–5]. In 1994, Bodson, Sacks and Khosla [6, 7] proposed an equivalent relationship between the IMP and the AFC, which can be used to design a suitable adaptive gain by a linear analysis of the equivalent system. Messner and Bodson [8] further described an AFC algorithm with sinusoidal regressors for repetitive control which is also equivalent to a linear controller based on the IMP. However, the analysis of the equivalent system is dependent on the accuracy of the identified model and requires the frequencies of disturbances to be known. For the rejection of sinusoidal disturbances with unknown frequencies, Bodson and Douglas [9] proposed an indirect algorithm that combined an adaptive notch filter [10] with a direct algorithm using a phaselocked loop for frequency estimation. This direct algorithm was later extended to be used for the noise control of periodic disturbances with multiple harmonics [11, 12]. Other researchers [13–15] have proposed different approaches for dealing with periodic disturbances with unknown frequencies. We now discuss a novel method for the rejection of disturbances with unknown and time-varying frequencies or with random signals. The proposed method can be extended to disturbances with an arbitrary number of harmonic components over the frequencies of interest. This approach also has the advantage that the disturbance frequencies need not be estimated. The proposed disturbance rejection scheme is a combination

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of a novel disturbance reduction controller and a sliding mode controller. The disturbance reduction controller consists of an inverse of the nominal plant with input deduction and a high gain integral term. Even if the real unknown system is unstable, it can be modeled as the nominal plant, which can be chosen to be Hurwitz and minimum phase, with modeling uncertainties. If the relative degree of the chosen nominal plant is greater than one, adding an appropriate all pole filter in series with the integral term can let the proposed controller become proper, [16] thereby avoid in sensitivity to measurement noise such as occurs in practical applications. This approach provides good low-band disturbance reduction and is combination with a sliding mode controller can deal with matching condition uncertainties. The remainder of this paper is organized as follows. In Sect. 2.1, the problem statement is presented, while in Sect. 2.2, the novel disturbance reduction controller is introduced. Section 2.3 describes the integration of the sliding mode controller with the disturbance reduction controller in order to deal with residual disturbances and modeling uncertainties under matching conditions. In Sect. 3, the simulation results obtained under different conditions are discussed to illustrate the performance of the proposed method when applied to a stable and an unstable system. Finally, Sect. 4 presents some brief conclusions.

2 Disturbance rejection scheme Disturbance rejection problems in active noise and vibration control applications are frequently encountered. In a linear system, irregardless of where the disturbances act on the plant, the problem can be transformed into one in which equivalent disturbances are applied as control input. 2.1 Problem statement Consider an unknown plant, P (s), which can be modeled as a nominal plant, Pˆ (s), and which can be chosen to be Hurwitz and minimum phase. It is assumed that the system with uncertainties is subjected to a finite and differentiable disturbance input d(t). The perfect disturbance cancellation objective is satisfied when the control input becomes equivalent to the disturbance input. In other words, the controller is designed to generate a control input u(t) such that output y(t) → 0 as t → ∞.

Robust disturbance rejection method for uncertain system with disturbances of unknown frequencies

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The system in Fig. 1 can be described by y(s) = Pˆ (s) d(s) + ρ(s) − u1 (s) − u2 (s) = Pˆ (s)e(s),

(4)

where the control force u1 (s) is given by: u1 (s) =

k e(s) + u2 (s) , s

(5)

and k is the nonnegative gain of the integral part. From (4), we get e(s) = d(s) + ρ(s) − u1 (s) − u2 (s), and (5) can be written as Fig. 1 Block diagram of the disturbance reduction controller and sliding mode controller system

The disturbance rejection scheme, which consists of the proposed novel disturbance reduction controller with a sliding mode controller, designed to achieve this goal of disturbance rejection, is shown in Fig. 1. The aforementioned plant, P (s), is modeled by x˙ (t) = Ax(t) + b b−1 Ax(t) + d(t) − bu(t), u(t) = u1 (t) + u2 (t),

(1)

y(t) = gx(t), where A and A ∈ Rn×n ; b ∈ Rn×1 ; g ∈ R1×n ; d(t), u1 (t), u2 (t), and y(t) ∈ R; A denotes the modeling uncertainty and is assumed to satisfy the matching condition; d(t) is the unknown disturbance input; u1 (t) is the control force for disturbance reduction and u2 (t) is the control force of the sliding mode controller; x(t) ∈ Rn×1 is the observable state. Considering the uncertainty of modeling, the nominal model is assumed to have the form. Pˆ (s) = g(sI − A)−1 b.

(2)

2.2 Novel disturbance reduction controller Even if the real unknown system is unstable, it can be modeled as a nominal plant Pˆ (s) which can be chosen to be Hurwitz and minimum phase, with the modeling uncertainty A. The proposed disturbance reduction controller is shown in Fig. 1, where the unknown parametric error function ρ(s), which is due to modeling uncertainties, is defined as ρ(s) = b−1 Ag−1 e(s)Pˆ (s).

(3)

k d(s) + ρ(s) − u1 (s) s k d(s) + ρ(s) . = s +k

u1 (s) =

(6)

Hence, the control force u1 (s) tracks the sum of the unknown disturbance input and the unknown parametric error function due to modeling uncertainties. The performance of the tracking effect improves when pole −k is further away from the imaginary axis. In other words, the control force u1 (s) can cancel both the unknown disturbance input and the modeling uncertainty. The relationship between them is similar to the transient response of a first-order system. When an appropriate sampling frequency is specified, disturbances with an arbitrary number of harmonic components and modeling uncertainties can be successfully tracked by the control input. To ensure that this control force can be effectively used to reduce the influence of disturbances and the modeling uncertainty acting on the plant, the error input signal e(s) can be written as e(s) = d(s) + ρ(s) − u1 (s) − u2 (s) s s = d(s) + ρ(s) − u2 (s) s+k s+k = v1 (s) + v2 (s) − u2 (s),

(7)

where the signal v1 (s) can be considered as the residual disturbances of the disturbance reduction controller, and the signal v2 (s) is the residual uncertainty term arising from the modeling uncertainty. A block diagram of the control system shown in Fig. 1 can be simplified into the form shown in Fig. 2. The advantage of the proposed disturbance reduction controller is that it can be applied to disturbances with unknown frequencies, amplitudes, and phases. If the relative degree of the chosen nominal plant is greater than one,

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vector c to ensure system stability on the sliding surface [17]. For Assumption 1, there exists a positivedefinite matrix P that solves the Lyapunov equation, i.e., AT P + PA = −Q,

where Q is a positive-definite matrix. If we choose the Lyapunov function

Fig. 2 Equivalent block diagram of the control system

V (t) = xT (t)Px(t) > 0, adding an appropriate all pole filter in series with the integral term can let the proposed controller be proper [16] to avoid sensitivity to measurement noise that is a problem in practical applications. Since residual signals cannot be canceled out by the proposed disturbance reduction controller, a robust sliding mode controller must be employed with the disturbance reduction controller. Its development is discussed in the next sector. 2.3 Combined disturbance reduction controller and sliding mode controller After adding the disturbance reduction controller into the system, the model in (1) can now be described by x˙ (t) = Ax(t) + b v1 (t) + v2 (t) − bu2 (t), y(t) = gx(t).

(11)

(8)

(12)

then V˙ (t) = −xT (t)Qx(t) + 2xT (t)Pb v1 (t) + v2 (t) − u2 (t) .

(13)

If the vector c chosen is S(t) = cx(t) = bT Px(t),

(14)

then on the sliding surface in (10), where S(t) = 0, it can be shown that V˙ (t) = −xT (t)Qx(t) ≤ 0.

(15)

Hence, when the sliding mode controller is applied, the system is stable on the sliding surface. The following lemma is used to design a switching controller to establish the reaching condition of the sliding mode:

The following assumptions are taken for this system Assumption 1 For this nominal system, the matrix pair (A, b) is controllable and the matrix A is stable. Assumption 2 When the appropriate gain k of the integral part is chosen, there exists known nonnegative constants δ1 and δ2 for uncertainties v1 (t) and v2 (t), such that v1 (t) < δ1

and v2 (t) < δ2 x(t).

(9)

Lemma 1 If the following condition holds, the motion of the sliding mode in (10) is asymptotically stable: ˙ < 0, S(t)S(t)

∀t ≥ 0.

(16)

Proof Let the Lyapunov function candidate of the system given in (8) be 1 V (t) = S 2 (t). 2

(17)

Now condition in (16) ensures that

A switching surface s(t) is defined in the state space as

˙ < 0. V˙ (t) = S(t)S(t)

S(t) = cx(t) = 0,

To meet condition (16) declared in Lemma 1, the switching controller is defined as (19) u2 (t) = (cb)−1 cAx(t) + β x(t) sat(S, ε) ,

(10)

where S ∈ R and c ∈ R1×n such that cb is invertible. The Lyapunov approach is used to choose a proper

(18)


3 Simulation results

where sat(S, ε) and β(x(t)) are defined as ⎧ ⎨ 1, sat(S, ε) = S(t)/ε, ⎩ −1,

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S(t) > ε |S(t)| ≤ ε S(t) < −ε.

(20)

β x(t) > cb δ1 + δ2 x(t) .

(21)

In the control law (19), the saturation function sat(S, ε) [18] is employed to eliminate chattering. This can be achieved by smoothing out the control discontinuity in the thin boundary layer neighboring the switching surface defined in (10). The following theorem proves that the proposed control law is capable of driving an uncertain system trajectory into the boundary layer. Theorem 1 Consider the system in (8) with a disturbance input subjected to Assumptions 1 and 2. If the sliding mode controller (19) is applied to the system, then the global reaching condition (18) for the boundary layer |s(t)| ≤ ε neighboring the switching surface is satisfied. Proof Substituting (8) and (10) into the time derivative of V (t), gives

˙ = S(t)c Ax(t) + b v1 (t) + v2 (t) − u2 (t) . S(t)S(t) (22) Using the property AB ≤ AB, and control law (19), it can be shown that

In this section, some simulation examples and the proposed controllers are discussed. In the following, the initial condition is assumed to be zero and the sampling time is 1 ms. First, in order to illustrate the performance of the proposed novel disturbance reduction controller, a second order system [6] is employed, and the plant is assumed to be known. The parameters of this second order system [6] are P (s) = (s + 2)/(s + 1)(s + 3), d(t) = sin(0.1t) − 0.2 sin(0.3t). The proposed disturbance reduction controller is added into the second order system to cancel the disturbance input. The nonnegative gain of the proposed controller is set to be k = 50. M(s) represents the transfer function of the disturbance input and the system output. The bode diagram of M(s), shown in Fig. 3 illustrates the good low-band disturbance reduction obtained by adding the disturbance reduction controller into the second order system. Figure 4a shows the disturbance input and Fig. 4b shows the output of the disturbance reduction controller. As seen in Fig. 4b, the output of the controller u1 (t) is almost similar to the disturbance input d(t) as shown in Fig. 4a. Figure 4c shows the time response of a second order system with the proposed disturbance reduction controller. A comparison of the system responses [6, Figs. 3 and 5] shows that the proposed disturbance reduction controller was effectively able to reduce the time response due to the disturbance.

˙ S(t)S(t)

= S(t) cb v1 (t) + v2 (t) − β x(t) sat(S, ε) ≤ S(t) cb v1 (t) + v2 (t) − β x(t) (23) ∀ S(t) > ε.

Substituting (9) and (21) into (23), gives S(t)S(t) < S(t) cb δ1 + δ2 x(t) − β x(t) (24) < 0, ∀ S(t) > ε. This confirms that the trajectory is bound and convergent to the sliding surface.

Fig. 3 Bode diagram of M(s): solid line indicates the system; dotted line indicates the system with disturbance reduction controller

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can be added in series with the integral term, thereby letting the proposed controller be proper. Hence, the control force u1 (s) in (6) can be derived as u1 (s) =

Fig. 4 (a) Disturbance input; (b) output of the disturbance reduction controller; and (c) time response of system with proposed disturbance reduction controller

Furthermore, consider the case where the actual system is unstable, but the nominal plant chosen is stable with modeling uncertainties. The state space equation is

0 1 0 0 A= , A = , −3 −4 4 1

0 b= , and g = 3 0 . 1 Hence, the unstable system, P (s), can be expressed as P (s) =

3 , 2 s + 3s − 1

and the transfer function of the nominal plant can be expressed as Pˆ (s) =

3 . s 2 + 4s + 3

A disturbance reduction controller is incorporated to suppress the disturbances that are applied to the unstable system. The nonnegative gain of the proposed controller is set to be k = 900. A high frequency measurement noise

s2

900 d(s) + ρ(s) . + 6s + 900

A sliding mode controller and the proposed disturbance reduction controller combined with a sliding mode controller are, respectively, applied to eliminate the disturbance. To design an appropriate sliding mode controller, the positive-definite matrix Q is first defined to be

18 0 Q= , 0 2 and is then used to solve the Lyapunov equation given in (14). The positive-definite matrix P is calculated:

15 3 P= . 3 1 The values of the sliding controller used in the present simulations are: ε = 0.1, δ1 = 1, c = bT P = 3 1 , δ2 = 1, and β x(t) = cb δ1 + δ2 x(t) . Three different disturbance inputs, as shown in Fig. 7, are considered in the present simulations, namely: (1) One harmonic signal: d1 (t) = 5 sin(0.5t), (2) Three harmonic signals: d2 (t) = 5 sin(0.5t) + 2 sin(5t) − 3 sin(50t), (3) Random signals: d3 (t) = 5 sin(0.5t + r), where r is a normally distributed random function.

dm (s) = 0.001 sin(300t), is considered in the unknown system. Now since the relative degree of the chosen nominal plant is greater than one, an all pole filter FP (s) =

1 , s+6

Figure 6a shows the output of the system controlled by the proposed disturbance reduction controller alone without being combined with a sliding mode controller, with one harmonic disturbance d1 (t) applied to the unstable system. The result for taking the proposed disturbance reduction controller with a sliding mode controller to reject one harmonic disturbance is


Fig. 5 (a) One harmonic disturbance input d1 (t) = 5 sin(0.5t); (b) three harmonic disturbance input d2 (t) = 5 sin(0.5t) + 2 sin(5t) − 3 sin(50t); and (c) random disturbance input d3 (t) = 5 sin(0.5t + r)

Fig. 6 Disturbance reduction effect of one harmonic signal, d1 (t) = 5 sin(0.5t) achieved using the (a) proposed disturbance reduction controller, and (b) proposed disturbance reduction controller with sliding mode controller

shown in Fig. 6b. Consider the case where the disturbance is assumed to be the application of three harmonic signals d2 (t) to the system. The effects of disturbance reduction obtained using the proposed disturbance reduction controller, without the sliding mode controller, and with the proposed disturbance rejection scheme, are shown in Fig. 7a, b. The disturbance is also assumed to consist of random signals d3 (t). To generate a random signal as the disturbance input, a normally distributed random function r is obtained using the MATLAB “randn” instruction. The elements of this function are normally distributed with a zeromean, a variance of σ 2 = 1, and a standard deviation of σ = 1. The random disturbance reduction results controlled by the proposed disturbance reduction con-

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Fig. 7 Disturbance reduction effect of three harmonic signals, d2 (t) = 5 sin(0.5t) + 2 sin(5t) − 3 sin(50t) achieved using the (a) proposed disturbance reduction controller, and (b) proposed disturbance reduction controller with sliding mode controller

Fig. 8 Disturbance reduction effect of random signals, d3 (t) = 5 sin(0.5t + r) achieved using the (a) proposed disturbance reduction controller, and (b) proposed disturbance reduction controller with sliding mode controller

troller, without the sliding mode controller, and with the proposed disturbance rejection scheme, are illustrated in Fig. 8a, b. Figures 6, 7, and 8 show the simulation results given three different disturbance inputs. They demonstrate that the effectiveness of the disturbance rejection achieved using the proposed disturbance reduction controller and the proposed disturbance rejection scheme.

4 Conclusions This paper described a robust disturbance rejection method to deal with disturbances of unknown frequencies. The method successfully reduced both peri-

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Table 1 Nomenclature Symbol

Description

P (s) Pˆ (s)

Transfer function of unknown plant

Pˆ −1 (s)

Inverse of nominal plant

k

Nonnegative gain of the integral part

Transfer function of nominal plant

A

Uncertainty of matrix A

ρ(s)

Parametric error function

v1 (s)

Residual disturbances of the disturbance reduction scheme

v2 (s)

Uncertainty from the modeling imprecision

δ1 , δ2

Known nonnegative constants

sat(·)

Saturation function of (·)

ε

Thickness of the boundary layer

·

2-norm of (·)

r

Normally distributed random function

σ

Standard deviation of a random function r

odic and nonperiodic unknown disturbances with uncertainties in both a stable system and in an unstable system. A distinguishing feature of the proposed method is that the frequencies of the disturbance need not be estimated. The proposed disturbance reduction controller, which can be used to reduce the influence of disturbances and the modeling uncertainties, is incorporated with a sliding mode controller, so as to suppress any undesired response caused by residual disturbances and residual modeling uncertainties. The simulation results confirm that the proposed disturbance rejection scheme rejects disturbances effectively and rapidly.

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