DOI 10.1007/s12555-011-0505-8
Proportional-Derivative Unknown Input Observer Design Using
Descriptor System Approach for Non-Minimum Phase Systems
Huan-Chan Ting, Jeang-Lin Chang, and Yon-Ping Chen
Abstract: This paper considers the problem of estimating the state of an MIMO linear system with un-known inputs in the state and output. Through a series of linear transformations in the state and output equations, the original system can be transformed into a descriptor system form. The proposed propor-tional derivative observer can accurately estimate the system state and avoid the peaking phenomenon. Moreover, the approach developed in this paper does not require the derivatives of the output and can be applied to the system with unstable zeros (with respect to the relation between the output and the unknown input). Finally, our algorithm can prove the valid feasibility and the property of disturbance attenuation through demonstrating a simulation-base example.
Keywords: Descriptor system, non-minimum phase, state estimation, unknown input observer.
1. INTRODUCTION
External disturbances and coefficient variations of a plant bring difficulties designing the controller. These unknown perturbations are incorporated into the system model, as unknown inputs. When the system is subject to the unknown inputs, the standard Luenberger observer can not obtain the perfect estimation. As a result, the design of state observer for systems with unknown inputs, called the unknown input observer (UIO) design, is an important topic in several control applications [1-19]. However, these papers [1-4] only consider the unknown input in the state equation. In this regard, Darouach et al. [2] presented a full-order UIO, and Hou and Muller [3] established a reduced-order UIO. UIO can also apply to the fault detection and isolation problems [5-8]. When there exists unknown inputs in state and output equations of the system model, these papers [9-14] developed different design methods under some constraints of system matrices. Alwi et al. [13] proposed an effective sliding mode observer to estimate states and noises for LTI systems with disturbances and measurement noises. Sharma and Aldeen [15,16] decoupled the unknown inputs from the rest of the system through a series of coordinate transformations. There are two necessary and sufficient conditions to check the existence of a stable UIO. These two important conditions are that the transfer
function matrix between the unknown input and the system output must be minimum phase and with relative degree one. If one of the conditions mentioned above is not satisfied, how to design a stable UIO is a difficult research problem. Releasing the minimum phase condition, this paper proposes a reduced-order UIO design method to effectively estimate the state in which there exists unknown inputs in the state and output equations of the system with unstable zeros. In contrast with [13], they presented a specific linear matrix inequality (LMI) to design parameters of the observer and guarantee the estimation convergence simultane-ously. In this paper, the matrix stabilizing the estimated error dynamics is determined by a pole-placement method, and then another independent matrix is used to suppress the effect of unknown inputs.
On the other hand, Boutayeb et al. [17] involved the descriptor system approach to design a nonlinear observer for simultaneously estimating the system state and unknown inputs. Based on the same method, Fernando and Trinh [18] designed a reduced-order functional observer of the linear systems where both input and output disturbances are presented. Gao and Wang [20] developed the descriptor system observer to reduce the effect of measurement noises. They constructed a modified proportional derivative observer to asymptotically estimate the system state and output noise at the same time.
In this paper, we introduce the descriptor system approach, developed by Gao and Wang [20], into the observer design for linear systems with unknown inputs in the state and output. A series of similar transform-ations is first proposed to transform the original system into a descriptor system form. Then a proportional derivative observer is designed to estimate the system state in which the derivative gain and the proportional gain can adjust the unknown input amplification and ensure the robust stability of the estimation error dynamics, respectively. Although the perfect estimation © ICROS, KIEE and Springer 2011
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Manuscript received July 26, 2010; revised January 18, 2011; accepted June 17, 2011. Recommended by Editorial Board mem-ber Young Soo Suh under the direction of Editor Young Il Lee. This work was supported by the National Science Council under grant NSC-99-2221-E-161-607.
Huan-Chan Ting and Yon-Ping Chen are with the Institute of Electrical Control Engineering, National Chiao-Tung University, 1001 University Rd., Hsinchu 300, Taiwan (e-mails: hcting. ece92g@nctu.edu.tw, ypchen@mail.nctu.edu.tw).
Jeang-Lin Chang is with the Department of Electrical Engi-neering, Oriental Institute of Technology, 58, Sec. 2, Sichuan Rd., Banqiao Dist., New Taipei City 220, Taiwan (e-mail: jlchang@ ee.oit.edu.tw).
is impossible, the proposed algorithm can be successfully implemented in the systems with unstable invariant zeros and the estimation error is finally bounded in a small region around zero. Moreover, the observer in this paper does not require the derivative of the output and is capable of avoiding the peaking phenomenon.
In the next section, a class of controlled systems is first introduced with some important assumptions in relation to the system matrices. Sections 3 and 4 describe some coordinate transformation and the unknown input observer design. To verify the developed observer, a numeric example is shown in section 5. Finally, section 6 gives concluding remarks.
2. PROBLEM FORMULATION
Consider the following MIMO linear system with the unknown input as ( ) ( ) ( ) ( ), ( ) ( ) ( ), t t t t t t t = + + = + x Ax Bu Dd y Cx Fd (1)
where x »∈ n, u »∈ m, d » and ∈ l, y »∈ p are the system state vector, the control input vector, the unknown input vector, and the system output vector, respectively. Suppose that system (1) is detectable and the system matrices with appropriate dimensions are known. Without loss of generality, we assume that rank( )F =k and rank( ) pC = where p l≥ and 0 k<
.l
≤ Although the input d(t) is unknown (cannot be measured), the target is to design a UIO which can accurately estimate the system state x(t) for any instant of time. Using the standard Luenberger observer to estimate the state of system (1) precisely is difficult due to the existence of unknown inputs. Busawon and Kabore [21] have shown that the conventional Luenberger observer is not adequate for handling measurement noises. Several authors [1-4,9-16] have solved this problem and proposed different UIO design methods. There exists two conditions checking the existence of a stable UIO [1-4,9-16]. The first is that the system with respect to the relation between the output and the unknown input must be minimum phase, i.e. the invariant zeros of the system are located on the left-half open plane. The second is that the relative degree of the transfer matrix function from the unknown input to the system output is one. For a linear system with the unknown inputs in the state and output equations and unstable invariant zeros, a UIO design using the descriptor system approach is proposed in this paper.
3. COORDINATE TRANSFORMATION AND SYSTEM ANALYSIS
Since rank( )F =k, applying the singular value decomposition [22] can obtain
, k = 0 0 UFV 0 I (2)
where the matrices U »∈ p p× and V »∈ l l× are nonsingular. Define the following vectors
1 2 ( ) ( ) ( ) t t t = d d V d and y( )t =Uy( ),t (3) where d1∈»l k− and d2∈»k. As a result, system (1)
can be rewritten as 1 1 2 2 2 ( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ), k t t t t t t t t = + + + = + x Ax Bu D d D d 0 y Cx d I (4)
where C UC= and DV=[D D Assuming rank 1 2].
1 1
(CD) rank( )= D = −l k, there exists a transformation
1∈ n n×
T » such that the matrix D can be partitioned as 1 1 1 1= , D T D 0 (5)
where D1∈»( ) ( )l k− × −l k is invertible. Let [C C1 2] = 1
1−
CT where C1∈»p l k× −( ) and C2∈»p n l k× − +( ), we can obtain that
1 1 1 1 1 1 1 2 1 1 rank( ) rank( ) rank rank( ) l k. − = = = = − CD CT T D D C C C D 0 (6)
From rank( ) l kD1 = − and (6) we have rank( ) =C1 .
l k− Define the transformation 1 1 1 p p + × = ∈ Φ C U »
where Φ ∈ »1T p p l k× − +( ). The matrix C1+ =(C C1 1T )−1C1T is a null space of C1T. Multiplying U1 and CT1−1 attains 12 1 1 1 1 1 2 22 , l k− − = = I C U CT U C C 0 C (7)
where C12 =C C1+ 2 and C22= Φ1 2C Furthermore, . we introduce a transformation 2 l k 12 n l k − − + − = I C T 0 I and define 1 21 1 2 − = = z z T T x z where z1∈»l k− and z2∈ . n l k− + Notice that 11 12 1 1 1 1 2 1 1 2 2 1 21 22 2 12 12 1 2 1 2 1 22 22 , , , and . k − − − − = = = = A A B T T AT T T T B A A B 0 D F T T D U I D F (8)
Based on the mentioned transformations above, system (4) can be transformed into the following form:
1( )t = 11 1( )t + 12 2( )t + 1 ( )t + 12 2( )t + 1 1( ),t z A z A z B u D d D d 2 21 1 22 2 2 22 2 1 1 12 2 2 22 2 22 2 ( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ), ( ) ( ) ( ), t t t t t t t t t t t = + + + = + = + z A z A z B u D d y z F d y C z F d (9)
where y1∈»l k− and y2∈»p l k− + . Observing (9) can obtain z1=y1−F d12 2. Substitute z1 into (9) to obtain
(
)
2 22 2 2 21 1 22 21 12 2 22 2 2 21 1 2 2 2 22 2 22 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ), t t t t t t t t t t t t = + + + − = + + + = + z A z B u A y D A F d A z B u A y D d y C z F d (10)where D2 =D22−A F21 12. In the following, a reduced-order observer algorithm is proposed by the descriptor system form of estimating simultaneously the system state z2 and the unknown input d2. Define a vector
2 n l k− + ∈ w » as 2 2 = z w d and 22 2 2 21 1 , , , , and . n l k k k − + = = − = = = A D I 0 E H 0 I 0 0 0 u B A G v M I y 0 0 (11)
Then the descriptor system form of system (10) is given by 22 2 2 2 2 2 2 21 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), n l k k k t t t t t t t t t t − + = − + + = + + A D z z I 0 0 I d d 0 0 0 u B A d I y 0 0 Hw Gv Md
[
]
2 2 22 22 1 2 ( ) ( ) ( ), ( ) t t t t = = z y C F C w d (12)where C1=[C22 F22]. The detectability of (12) is proven
in the following lemmas.
Lemma 1: If the following conditions hold:
1 ranks n− − = + −n l k ∀ ∈s + I A D C 0 » and 1 1 rank(CD) rank( )= D = −l k, then the pair (A22, C22) is detectable.
Proof: According to rank(CD1) rank( ) l k= D1 = − and (8), we have 1 11 12 1 21 22 22 rank rank n l k n l k l k s s s − − + − − − − − − − − = I A D C 0 I A A D A I A 0 I 0 0 0 C 0
(
)
22 22 ranks n l k− + − 2 l k n l k. = + − = + − I A C Since ranks n− − 1= + −n l k ∀ ∈s +, I A D C 0 » wecan ensure the following relationship,
22 22 ranks n l k− + − = − +n l k ∀ ∈s + I A C »
and complete the proof.
Lemma 2: The descriptor system (12) is completely detectable, i.e., (i) 1 ranks − = − +n l 2 , k ∀ ∈s +, E H C » (ii) 1 rank = − +n l 2 .k E C
The completely detectable system represents that it has neither unstable finite nor infinite output decoupling zeros [22,23].
Proof: From rank(F22)=k and Lemma 1, we have
22 2 1 22 22 22 22 rank rank rank 2 n k k n k s s s k n l k s − − + − − − = − = + = − + ∀ ∈ I A D E H 0 I C C F I A C » and 1 22 22 22 rank rank rank( ) 2 . n l k n l k n l k − + = = − + + = − + I 0 E 0 0 C C F F
Hence, system (12) is completely detectable. The proof of the lemma is finished.
Design 1 ( 2 ) ( ) 12 22 n l k− + × − +p l k + = ∈ 0 K K F » where the
gain matrix K12∈»k k× is invertible and F22+ = 1 22 22 22 (F FT )− FT. Note that
(
)
[
]
1 1 1 1 22 22 12 22 1 1 22 22 12 12 . n l k n k k − − − + + − + − − + = + =− = E K C M 0 I 0 C F M 0 0 K F I 0 0 0 I F C K K (13)4. OBSERVER FORMULATION AND DESIGN The proportional derivative observer for system (12) is designed as
(
1 1 1 1)
1 1 1 1 2 1 1 1 1 2 ( ) ( ) ( ) ( ) ( )( ) ( ) ( ), ˆ ( ) ( ) ( ) ( ), t t t t t t t − − + = + + + + − + = + + E K C H LC H LC E K C K L y Gv w E K C K y η η η (14)where η∈ »n l k− +2 is an auxiliary state of the observer
and 2 2 2 ˆ ˆ = ˆ ∈ n l k− + z w
d » denotes the estimation of w. Adding K y1 2 into both sides of the first equation in (12) obtains 1 1 1 1 2 2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). t t t t t t + = + + + − + E K C w H LC w Gv K y Ly Md (15) Substitute η= −w E K Cˆ
(
+ 1 1)
−1K y1 2 into (14) to obtain 1 1 1 2 1 2 ˆ ˆ ( ) ( ) ( ) ( ) ( ) ( ) ( ). E K C w H LC w Ly K y Gv t t t t t + = + − + + (16) Define 2 2 ˆ = − = z w w w das the estimation error of w. It follows from (15) and (16) that the dynamics of w can be expressed as
(
)
1 1 1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 12 2 ( ) ( ) ( ) ( ) ( ) (( ) ) ( ) ( ) ( ) (( ) ) ( ) ( ), t t t t t t t − − − − − = + + + = + − + + = + − + w E K C H LC w Md E K C H NC w E K C Md E K C H NC w K d (17)where N= − +(E K C1 1)−1L. As a result, the design rule is to choose the gain K12 reducing the effect of d2, and
the matrix N ensuring the stability of the error dynamics. Define the estimation state 11 2 1 12 2
2 ˆ ˆ ˆ − − = y F d x T T z and
the estimation error
12 1 1 2 ( ) ( ), n l k t − t − + − = 0 F x T T w I 0 (18)
where the dynamics of w is given by
1 1
1 1 1 12 2
( ) ((t = + )− − ) ( )t + − ( )t
w E K C H NC w K d .
The estimation performance of the developed observer is shown in the following theorem.
Theorem 1: Consider system (4) which satisfies the following conditions 1 ranks n− − = + −n l k ∀ ∈s + I A D C 0 » and 1 1 rank(CD) rank( )= D = −l k,
and the observer (14). If the unknown input d2 is
bounded and the matrix ((E K C+ 1 1)−1H NC− 1) is stabilizing by N, then the estimation error is finally bounded in a small region.
Proof: Since 11 2 1 11 2 1 12 2 2 2 , − − − = = z y F d x T T T T z z
the estimation error is
1 12 2 1 2 2 12 1 1 2 ˆ ( ) ( ) ( ) ( ). n l k t t t t − − − + − = − = − = F d x x x T T z 0 F T T w I 0
Hence, the term w( )t dominates the estimation performance. In the following, we show that the pair
1 1 1 1
((E K C+ )− H C, ) in (17) is detectable. From Lemma 1 and
(
)
(
)
1 2 1 1 1 1 1 1 2 1 1 1 rank rank rank rank , n l k n l k p l k s s s s s − − + − + − + − + + − = − = − = I E K C H C E K C H C I K E H 0 I C E H C it follows that(
)
1 2 1 1 1 1 rank rank 2 , . n l k s s n l k s − − + + − + − = = − + ∀ ∈ I E K C H C E H C »Hence, the pair ((E K C+ 1 1)−1H C, )1 is detectable and there exists the gain matrix N stabilizing ((E K C+ 1 1)−1⋅
1).
−
H NC Furthermore, in order to satisfy the property of disturbance attenuation, choosing a high gain K12 to
reduce the effect of d2 is recommended. The proof is
completed.
Remark 1: When the matrix F is full rank, system (1) can directly transfer to the descriptor system form (12) without the coordinate transformation (8). In this special case, the descriptor system form for system (1) is described as 1 1 1 1 1 1 2 1 ( ) ( ) ( ) ( ), ( ) ( ), t t t t t t = + + = E w H w G u M d y C w where
1= , x w d ( ) ( ) 1= n ∈ n l+ × +n l , I 0 E 0 0 1 , l = − A D H 0 I 1 ( ) , n l m+ × = ∈ B G 0 » ( ) 1 n l l, l + × = ∈ 0 M I and C2= C F .
Follow the same rule mentioned above to design the gain of observer K12 reducing the effect of the unknown input.
Then design another parameter matrix N to place the desired eigenvalues of estimated error dynamics and avoid the peaking phenomenon.
Remark 2: For system (1), the conventional UIO observer designs [1-4,9-16] can obtain the perfect estimation if the following condition is satisfied:
ranks n− − = + ∀ ∈n l s +. I A D C F »
The above condition implies that the system with respect to the relation between the output and the unknown input must be minimum phase. This condition has been altered that the pair ( , )A C is detectable in our proposed observer.
5. NUMERICAL EXAMPLE
To demonstrate the designed observer, we consider the following system: 1 2 1 2 0 1 1 3 ( ) 1 2 1 0 1 1 ( ) ( ) , ( ) 1 0 3 3 1 1 1 2 0 2 1 1 d t t t d t − − − − − = + − − − − − − x x 1 1 2 2 ( ) 1 0 1 0 0 0 ( ) ( ) . ( ) 0 1 1 1 0 1 ( ) y t d t t y t d t = + − x
The invariant zeros of the system between the output and the unknown input are 1.7890 and −4.1445 0.3897 .i± Due to the unstable invariant zero 1.7890, the conventional UIO methods [1-4,9-16] can not be implemented in the system. Using the linear transformations proposed in Section 3, the above system can be transformed into the following form:
1( ) 4 ( ) [0.8 0.8 1.8] ( ) 0.6 ( )1 2 2 1( ), z t = − z t + − z t + d t +d t 2 1 2 2 1 1 2 2 2 2 2 0.4 0.2 0.6 ( ) 3 ( ) 2.6 1.2 4.6 ( ) 3 3.4 1.8 2.4 [4 4 4] ( ), ( ) ( ) 0.2 ( ), ( ) [0.8944 0.4472 0.8944] ( ) 0.8944 ( ), T t z t t d t y t z t d t y t t d t − − = − + − − − − − − + − − = − = − + z z z
where the unknown inputs are set as d t1( ) 2cos( )= t and d t2( ) 0.2sin(5 ) 0.1cos(20 ).= πt + t Design K =12 3
which places the desired eigenvalues of the observer at { 4, 3, 2 }.− − − ±i The proposed observer is given by
1 2 1 33.25 58.5 61.5 2 17.75 24.5 24.5 ( ) ( ) 2 14.25 29.5 27.5 3 21.0417 37.5833 35.25 [ 5 3 1 2.5] ( ) [ 1.1180 5.1430 0.2236 3.1678] ( ), T T t t y t y t − − − − = − − − − + − − + − − η η 2 2 2 ˆ ( ) ( ) [0 0 0 1.118] ( ), ˆ ( ) T t t y t d t = + z η
and the estimation state can be attained by
1 2 2 1 0 0.6 0.2 ˆ 1 1 0.4 0.2 ( ) 0.2 ( ) ˆ( ) . 1 0 0.4 0.2 ˆ ( ) 1 1 0.4 1.2 y t d t t t − − + = − − − x z
The noise attenuation observer [20] is simultaneously simulated. For the same system, the observer [20] can be designed as 316.0110 238.9735 74.0375 223.7300 500.9847 279.2548 31.1998 278.7039 251.5041 ( ) 339.2302 889.4069 547.1768 265.1693 38.9530 305.1223 143.8446 639.4346 494.5901 239.9735 315.0110 240. t ξ − − − − − = − − − − − − − 1 2 9735 502.9847 224.7300 502.9847 275.7039 30.7998 278.7039 ( ) 889.4069 340.2302 887.4069 34.9530 264.8360 36.9530 634.4346 142.8446 635.1013 ( ) 0 0 0 0 0.3333 0 ( ) 0 0 0 0 0 0.3333 T t y t y t ξ − − − − − − + − 1 1 1 1 2 1 ˆ ( ), 3 1 1 1 2 1 T t − − + − − − − d 398.1891 983.7861 585.5971 ˆ( ) 103.1360 32.3694 135.5054 983.7861 398.1891 983.7861 ( ), 32.3694 103.1360 32.3694 t t ξ − − = − − − − − − d ˆ( ) 0 0 0 0 1 0 ( ) ( ), ˆ( ) 0 0 0 0 0 1 T t t t t ξ ω = + x y
where ωˆ( )t is the estimation of the noise in output signals. The eigenvalues of observer are placed at { 15, 12, 10, 8, 4, 3, 2 },− − − − − − − ±i and the gain
suppress-ing the disturbances is also three as K12=3 in the
proposed method above. Using the initial state (0) [ 0.2 2= − −1 −1]T
x and η(0)=ξ(0)= 0in both observers, Figs. 1-4 illustrate the responses of the real and estimated states in two methods. From these figures, the proposed method can obtain better estimation responses in the steady state and the disturbance attenuation property is evident. Hence, the proposed observer is capable of estimating the state when the
underlying system has the unstable zero and does not suffer from the peaking phenomenon.
6. CONCLUSIONS
With respect to an MIMO linear system with the unknown input in the state and output equations, this paper have developed an observer design method using the descriptor system approach. It is shown that the proposed proportional derivative observer is able to reconstruct the system state under the system with respect to the relation between the output and the unknown input is nonminimum phase. The robust stability of the estimation dynamics can be guaranteed and the estimation error is bounded in a small region. Simulation results demonstrate that the present observer scheme exhibits reasonably good estimation performance and avoids the peaking phenomenon.
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Huan-Chan Ting received his B.S. degree in Electrical Engineering from National Taipei University of Technol-ogy, Taipei, Taiwan in 2003, and his Ph.D. degree in Electrical and Control Engineering from National Chiao-Tung University, Hsinchu, Taiwan in 2011. His major fields of research are sliding mode control theorem, robust control, and power electronics.
Jeang-Lin Chang received his B.S., M.S., and Ph.D. degrees in Electrical and Control Engineering from National Chiao-Tung University, Hsinchu, Taiwan. He is a professor in the Department of Electrical Engineering, Orient Institute of Technology. His researches include variable structure control, robust control, and discrete signal processing.
Yon-Ping Chen received his B.S. degree in Electrical Engineering from National Taiwan University in 1981, and his M.S. and Ph.D. degrees in Electrical Engineer-ing from University of Texas at Arlington, U.S.A. He is a professor in the Institute of Electrical Control Engineer-ing, National Chiao-Tung University. His researches include variable structure control, intelligent control, and image processing.
