www.ietdl.org Published in IET Control Theory and Applications Received on 22nd August 2012 Revised on 4th March 2013 Accepted on 17th March 2013 doi: 10.1049/iet-cta.2013.0054
ISSN 1751-8644
Robust disturbance rejection based on equivalent-input-disturbance approach Rui-Juan Liu1,2,3 , Guo-Ping Liu2,4 , Min Wu1,3 , Fang-Chun Xiao1,3 , Jinhua She5 1 School
of Information Science and Engineering, Central South University, Changsha 410083, People’s Republic of China of AdvancedTechnology, University of South Wales, Pontypridd CF37 1DL, UK 3 Hunan Engineering Laboratory for Advanced Control and Intelligent Automation, Changsha 410083, People’s Republic of China 4 CTGT Center, Harbin Institute of Technology, Harbin 150001, People’s Republic of China 5 School of Computer Science,Tokyo University of Technology, Hachioji,Tokyo 192-0982, Japan E-mail:
[email protected] 2 Faculty
Abstract: A robust disturbance rejection method for uncertain systems is presented in this study. It effectively rejects both matched and unmatched disturbances and guarantees robust stability of the system when there exist modelling uncertainties. First, the authors show the configuration of the system that is based on the idea of the equivalent input disturbance (EID). An EID estimator is designed to compensate disturbances without requiring their prior knowledge. Then, a robust stability condition and the method of the controller design are presented using a linear-matrix-inequality. Finally, the validity of the authors method and its superiority over a conventional control method and the sliding-mode control method are demonstrated through the simulations of a numerical example and experiments of a rotational-speed control system.
1
Introduction
One important objective in the design of a control system is to achieve satisfactory disturbance rejection performance. It is well known that disturbance rejection performance is closely related to the sensitivity function in a one-degree-offreedom control system. On the other hand, robust stability for modelling error and parameter perturbation is related to the complementary sensitivity function of the control system. Since, the sum of the sensitivity and complementary sensitivity functions is a constant, satisfactory control performance of a one-degree-of-freedom control system relies on a delicate adjustment of those two functions. Over the last decades, the performance of disturbance rejection and robust stability have fully been taken into account, and many design methodologies have been proposed. For example, a fuzzy-logic-based PID controller was presented to improve the robustness of the control system [1]; adaptive robust control methods, such as predictive model control [2] and fuzzy observer control [3, 4], have been used to obtain satisfactory system performance without requiring a precise system model; an H2 control method was proposed to handle uncertain linear continuous-time systems under matched disturbances [5]. Sliding mode control (SMC) is a widely used robust technique to handle disturbances and modelling uncertainties [6–9]. However, the trade-off still exists in the system and may impose a limit to the effect of disturbance rejection. Active disturbance rejection (ADR) methods [10–16], in contrast, have two-degrees-of-freedom. Since these methods IET Control Theory Appl., 2013, Vol. 7, Iss. 9, pp. 1261–1268 doi: 10.1049/iet-cta.2013.0054
actively process the information of the disturbances and construct a compensation signal for them, they result in good disturbance rejection performance. Disturbance observer (DOB) is a common ADR method [11–14]. It has been used to solve many problems in control engineering, such as disturbance rejection in an optical disk drive system [15] and positioning control for a magnetic suspension rotor [16]. Some design methods have been proposed in the frequency domain to handle the uncertainties of a plant [17, 18]. However, they require a plant to be a minimum-phase one. A DOB was constructed in the state space to remove this restriction [19], but the order of the system is high because it needs to construct two dynamics: a state observer of the plant and a state-estimator-based DOB, to carry out disturbance estimation. The equivalent-input-disturbance (EID) approach was devised to overcome the drawbacks in the above methods. This method rejects both matched and unmatched disturbances effectively [20–22]. It does not require an inverse model of the plant or prior information on the disturbances [20]. So, it avoids the cancellation of unstable poles and/or zeros. An EID is estimated by making the best use of the state observer of the plant. However, the design method shown in [20] did not consider uncertainties in a plant. In fact, the existence of uncertainties makes the separation theorem no longer applicable. So, the state observer and the state-feedback controller cannot be designed independently. To solve this problem, we need to build a new framework to design the gains of the state observer and state-feedback controller simultaneously. In this study, we present a robust 1261 © The Institution of Engineering and Technology 2013
www.ietdl.org EID-based control method using the linear-matrix-inequality (LMI) technique to solve the design problem. This method not only inherits the advantages of the EID approach, but also effectively handles uncertainties of a plant. The rest of this paper is organised as follows. First, the configuration of a robust EID-based control system is constructed in Section 2. Then, Section 3 gives a robust stability condition and a design method for a state observer and a state-feedback controller in terms of an LMI. Simulations and experimental results are given in Section 4 to compare our method with a conventional control method and the SMC method, and to demonstrate the superiority of our method over them. Finally, some conclusion remarks are made in Section 5.
2 Configuration of robust EID-based control system Consider the following linear uncertain plant x˙ (t) = [A + A(t)]x(t) + [B + B(t)]u(t) + Bd d(t) y(t) = Cx(t) (1) where x(t) ∈ Rn is the state of the plant; u(t) ∈ Rp is the control input; y(t) ∈ Rq is the output; and d(t) ∈ Rnd is a disturbance; A, B, Bd and C are the constant matrices of appropriate dimensions. A(t) and B(t) are the time-varying structured uncertainties with the general form A(t) B(t) = ME(t) N0 N1 (2) where M , N0 and N1 are known constant matrices of appropriate dimensions; and E(t) ∈ Rn×n is an unknown matrix function satisfying E T (t)E(t) ≤ I ,
∀t > 0
(3)
d (t) r (t )
Internal model
−
u f (t)
KR State feedback KP
u (t)
− d (t) Low-pass filter dˆ (t)
Uncertain Plant
y (t )
EID estimator
− State observer
Fig. 1
Configuration of robust EID-based control system
In the plant (1), let the input u(t) be zero. A signal, de (t), on the control input channel is called an EID of the disturbance d(t), if it produces the same effect on the output as the disturbance d(t) does for all t ≥ 0 [20, 23]. As shown in [22, 23], there always exists an EID for both matched and unmatched disturbances. On the other hand, we write the plant (1) as x˙ (t) = Ax(t) + Bu(t) + [A(t)x(t) + B(t)u(t) + Bd d(t)] y(t) = Cx(t) (7) The uncertainties of the plant can be regarded as a stateand input-dependent disturbance [24]. Under Assumptions 1 and 2, there always exists a signal, de (t), on the control input channel that produces the same effect on the output as A(t)x(t) + B(t)u(t) + Bd d(t) does. So, we write the plant (7) in the state space as x˙ (t) = Ax(t) + B[u(t) + de (t)] (8) y(t) = Cx(t) As explained in [20], by making the full use of the state observer, we obtain an estimate of the EID ˆ = B+ LC[x(t) − xˆ (t)] + uf (t) − u(t) d(t)
with its elements being Lebesgue measurable. The following assumptions are made for the plant:
(9)
where Assumption 1: Plant (1) is controllable and observable.
B+ = (BT B)−1 BT
Assumption 2: Plant (1) has no zeros on the imaginary axis. The configuration of the robust EID-based control system is shown in Fig. 1. It contains five parts: the plant, an internal model, a state observer, a state-feedback controller and an EID estimator. It can be interpreted as a conventional control system (CCS) combined with the EID estimator. The following internal model x˙ R (t) = AR xR (t) + BR [r(t) − y(t)]
(4)
is employed to guarantee perfect tracking for a reference input, r(t). Since the reference input is known, AR and BR are easily determined. A full-state Luenberger observer of the plant x˙ˆ (t) = Aˆx(t) + Buf (t) + L[y(t) − yˆ (t)] (5) yˆ (t) = C xˆ (t)
Since the output, y(t), contains a measurement noise, we use a low-pass filter, F(s), to select the angular frequency band for the estimate. It satisfies |F(jω)| 1,
∀ω ∈ [0, ωr ]
(10)
where ωr is the highest angular frequency selected for disturbance estimation. A suitable filter has its cutoff angular frequency being 5−10 times larger than ωr , and a first-order low-pass filter is good enough [20]. A state-space form of F(s) is given by ˆ x˙ F (t) = AF xF (t) + BF d(t) (11) ˜d(t) = CF xF (t) ˜ where d(t) is the filtered disturbance estimate. ˜ in the state-feedback control law yields Incorporating d(t) an imposed control law
is used to reproduce the state of the plant. The state-feedback control law is
˜ u(t) = uf (t) − d(t)
uf (t) = KR xR (t) + KP xˆ (t)
The EID estimator in Fig. 1 plays a key role in improving the control performance. Incorporating the EID estimate actively
1262 © The Institution of Engineering and Technology 2013
(6)
(12)
IET Control Theory Appl., 2013, Vol. 7, Iss. 9, pp. 1261–1268 doi: 10.1049/iet-cta.2013.0054
www.ietdl.org suppresses the uncertainties and disturbances. To apply the EID-based method to an uncertain plant, we present an algorithm of designing a robust EID-based control system in the next section.
Substituting (15) into (5) yields x˙ˆ (t) = Aˆx(t) + Buf (t) + LCx(t)
(17)
From (5), (12), (14) and (15), we have
3 Analysis and design of robust EID-based control system Since exogenous signals do not influence the stability of the system, we set the input and disturbance to be zero, that is, r(t) = 0,
d(t) = 0
˙x(t) = [A + A(t) − LC]x(t) + A(t)ˆx(t) + B(t)uf (t) − [B + B(t)]CF xF (t)
(18)
Combining (9), (11) and (12) yields x˙ F (t) = (AF + BF CF )xF (t) + BF B+ LCx(t)
(13)
(19)
Then the plant (1) becomes
x˙ (t) = [A + A(t)]x(t) + [B + B(t)]u(t) y(t) = Cx(t)
(14)
for the filter. It follows from (13) and (15) that the internal model (4) becomes x˙ R (t) = −BR C xˆ (t) − BR Cx(t) + AR xR (t)
The gains of the state observer and the state-feedback controller were designed separately in [20]. However, the separation theorem cannot be used for their design in this study because the uncertainties of the plant mix these design together. Moreover, the gain of the observer was designed based on the concept of perfect regulation, which can only handle a minimum-phase plant. To overcome these drawbacks, we derive an LMI-based method for the stability analysis and the design of the state observer and the state-feedback controller in this study. First, we recall the following lemmas. Lemma 1 Schur complement [25]: For a given symmetric matrix 11 12 = T 12 22
(20)
(17)–(20) yield the state-space representation of the closedloop system in Fig. 1 ¯ ¯ f (t) ϕ(t) ˙ = Aϕ(t) + Bu where ⎡
A A(t) ⎢ A¯ = ⎣ 0 −BR C
LC A + A(t) − LC BF B+ LC −BR C B¯ = BT
(21)
0 −BCF − B(t)CF AF + BF CF 0
BT (t)
0
T 0
⎤ 0 0⎥ 0⎦ AR (22) (23)
The state-feedback control law is
the following statements are equivalent: ¯ uf (t) = Kϕ(t)
1. < 0, −1 T 11 12 < 0, and 2. 11 < 0 and 22 − 12 −1 T 3. 22 < 0 and 11 − 12 22 12 < 0.
where
Lemma 2 [26]: For a given matrix ∈ Rp×n with rank() = p, there exists a matrix X¯ ∈ Rp×p such that X = X¯ holds for any X ∈ Rn×n if and only if X can be decomposed as X = W X¯ W T ,
X¯ = diag{X¯ 11 , X¯ 22 }
Lemma 3 [27]: Let 0 (x) and 1 (x) be quadratic matrix functions over Rn and 1 (x) ≤ 0 for all x ∈ Rn − {0}. Then 0 (x) < 0 for all x ∈ Rn − {0} if and only if there exists an ε ≥ 0 such that 0 (x) − ε1 (x) < 0 holds. Since the states in the control system in Fig. 1 are x(t), xˆ (t), xF (t), and xR (t), we define x(t) = x(t) − xˆ (t) ϕ(t) = xˆ T (t) xT (t)
xFT (t)
(15)
xRT (t)
0
0
KR
(25)
Substituting (24) into (21), and separating the certain and uncertain items give ˆ ˆ ϕ(t) ˙ = Aϕ(t) + B (t)
(26)
where
where W ∈ Rn×n is a unitary matrix, X¯ 11 ∈ Rp×p , and X¯ 22 ∈ R(n−p)×(n−p) .
and
K¯ = KP
(24)
T
and use ϕ(t) to describe the closed-loop system. IET Control Theory Appl., 2013, Vol. 7, Iss. 9, pp. 1261–1268 doi: 10.1049/iet-cta.2013.0054
(16)
(t) = E(t) ϕ(t)
= N0 + N1 KP N0 −N1 CF N1 KR ⎡ ⎤ LC 0 BKR A + BKP 0 A − LC −BCF 0 ⎥ ⎢ Aˆ = ⎣ 0 BF B+ LC AF + BF CF 0 ⎦ −BR C −BR C 0 AR T Bˆ = 0 M T 0 0
(27) (28) (29)
(30)
Assume that the singular-value decomposition of the output matrix is C = U [S 0]V T (31) where S is a semi-positive definite matrix and U and V are unitary matrices. Letting V = V1 V2 , we have the following theorem. 1263 © The Institution of Engineering and Technology 2013
www.ietdl.org H22 = P2 A + AT P2 − P2 LC − C T LT P2
Theorem 1: For given parameters α and β, the system (20) is robustly stable under the control law (24), if there exist symmetric positive-definite matrices X1 , X11 , X22 , X3 and X4 , and appropriate matrices W1 , W2 and W3 such that the following LMI is feasible ⎡
11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗
W2 C
22 ∗ ∗ ∗ ∗
0
23
33 ∗ ∗ ∗
14 −X2 C T BRT 0
44 ∗ ∗
⎤
16 X2 N0T ⎥ ⎥ −X3 N1T ⎥ ⎥ ⎥ < 0 (32) βW3T N1T ⎥ ⎥ 0 ⎦ −I
0 M 0 0 −I ∗
H23 = −P2 BCF + C T LT B+T BFT P3 1 H24 = − C T BRT P4 β H33 = P3 (AF + BF CF ) + (AF + BF CF )T P3 1 H44 = (P4 AR + ATR P4 ) β So, V˙ (t) − [ T (t) (t) − ϕ T (t) T ϕ(t)] T ϕ(t) T ϕ (t) (t) = (t)
where
11 = αAX1 + αX1 AT + αBW1 + αW1T BT
14 = βBW3 − αX1 C
T
where ⎡
BRT
22 = AX2 + X2 AT − W2 C − C T W2T
23 = −BCF X3 + C T W2T B+T BFT
33 = (AF + BF CF )X3 + X3 (AF + BF CF )T
V2
X11 0
0 H23 H33 ∗ ∗
H14 H24 0 H44 ∗
⎤ 0 P2 M ⎥ ⎥ T
⎥ 0 ⎥+
0 ⎥ 0 ⎦ −I
0
(36)
Using Lemma 1, we write < 0 as
βX4 ATR
⎡
and the singular-value decomposition of X2 is X2 = V1
H12 H22 ∗ ∗ ∗
H11 ⎢ ∗ ⎢ ⎢ =⎢ ∗ ⎢ ⎣ ∗ ∗
16 = αX1 N0T + αW1T N1T
44 = βAR X4 +
(35)
0 X22
V1T V2T
Moreover, the gains of the state-feedback controller and the observer are
H11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗
H12 H22 ∗ ∗ ∗ ∗
0 H23 H33 ∗ ∗ ∗
H14 H24 0 H44 ∗ ∗
0 P2 M 0 0 −I ∗
⎤ N0T + KPT N1T ⎥ N0T ⎥ −CF N1T ⎥ ⎥ ⎥ < 0 (37) KRT N1T ⎥ ⎥ ⎦ 0 −I
Pre- and post-multiplying (37) by KP = W1 X1−1 ,
KR = W3 X4−1 ,
L = W2 USX11−1 S −1 U T (33)
= diag{αP1−1 , P2−1 , P3−1 , βP4−1 , I , I } = diag{αX1 , X2 , X3 , βX4 , I , I }
(38)
Proof: Choose a Lyapunov functional candidate to be yields V (t) = ϕ (t)Pϕ(t) T
(34)
where P = diag{ α1 P1 P2 P3 β1 P4 } and P1 , P2 , P3 and P4 are positive-definite matrices to be determined. The derivative of V (t) along (26) is V˙ (t) = ϕ T (t)P ϕ(t) ˙ + ϕ˙ T (t)Pϕ(t) ⎤ ⎡ 0 H14 H11 H12 ⎢ ∗ H22 H23 H24 ⎥ = ϕ T (t)⎣ ϕ(t) + 2ϕ T (t)P2 M (t) ∗ ∗ H33 0 ⎦ ∗ ∗ ∗ H44
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
˜ 11
∗
LCX2 ˜ 22
0 ˜ 23
∗
∗
˜ 33
∗ ∗ ∗
∗ ∗ ∗
∗ ∗ ∗
˜ 14
0
−X2 C T BRT
M
0 ˜
44 ∗ ∗
0 0 −I ∗
˜ 16
⎤
⎥ ⎥ ⎥ T⎥ −X3 CF N1 ⎥ ⎥
