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Improved Nonminimal State Space Model Predictive Control for Multivariable Processes Using a Non-Zero−Pole Decoupling Formulation Jianming Zhang* State Key Laboratory of Industrial Control Technology, Institute of Cyber-Systems & Control, Zhejiang University, Hangzhou 310027, P. R. China ABSTRACT: In this article, a decoupling approach is first formulated, and then a corresponding nonminimal state space predictive control is proposed. The proposed decoupling structure can avoid zero−pole cancellations between the decoupler and the process, and thus, realization is guaranteed. Consequently, a systematic design of nonminimal state space predictive control can be designed in terms of a SISO procedure. Simulation results of a typical multivariable process are provided to demonstrate the effectiveness of the proposed method. In addition, a closed-form transfer function representation that facilitates frequency analysis of the control system is also provided to give further insight into the proposed strategy.

1. INTRODUCTION Model predictive control (MPC) has been intensively studied in control theory and engineering since the 1970s.1 Thus far, MPC has been extended to several categories including finite impulse response (FIR) or step response model-based dynamic matrix control (DMC) or quadratic DMC;2,3 transfer function model-based linear or nonlinear generalized predictive control (GPC);4−6 state space model-based MPC;7−25 and most recently, nonminimal state space (NMSS) model-based MPC.26−29 MPC strategies are often directly designed through the multivariable description of a process without considering input/output decoupling. However, as demonstrated later in this article, MPC performance can be further improved by decoupling design. Toward this end, decoupling MPC is also an important issue, where the problems of theoretical feasibility and realization need to be addressed. Currently, ideal decoupling, simplified decoupling, and inverted decoupling represent three typical methods.30,31 However, some inconveniences are still encountered. For example, the major problems associated with ideal decoupling are the complex decoupler structure and practical realizability. Simplified decoupling, however, results in complex decoupled transfer function matrices. Inverted decoupling, in fact, is difficult to implement and sensitive to modeling errors.32 Moreover, a major problem with these three decoupling strategies is that zero−pole cancellations might exist between the decoupler and the process, causing realizability obstacles. This article first presents a new decoupling idea that avoids zero− pole cancellations and can be realized in practice, where feedforward compensation through the adjugate matrix of the process is used. Then, the decoupling strategy is tested on the nonminimal state space model predictive control (NMSSMPC). The main contributions of this work include the following: (1) A non-zero−pole cancellation decoupling strategy is proposed, which enables the realization of the decoupler. (2) A corresponding single-input− single-output (SISO) NMSSMPC is proposed. (3) A closedform evaluation of the transfer function of control performance is given to provide further insight into the proposed method. © 2013 American Chemical Society

This article is organized as follows: Section 2 details the new decoupling structure. In section 3, a SISO predictive control design is presented based on the new decoupling model, and some of its features are highlighted. Section 4 presents the transfer function analysis of the proposed control algorithm. Section 5 discusses the simulation results on a typical 24-plate bubble-cup distillation column process given by Luyben.33 Conclusions are presented in section 6.

2. FEED-FORWARD NON-ZERO−POLE CANCELLATION DECOUPLING 2.1. Model Structure. In this case, the design of a decoupler network for an n × n process, Gp(z), is obtained as N (z) = adj[Gp(z)]

(1)

where N(z) is the decoupler. Note that Gp−1(z) =

adj[Gp(z)] det[Gp(z)]

(2)

This shows that W (z) = Gp(z) ·N (z) = Gp(z) ·adj[Gp(z)] = Gp(z) ·Gp−1(z) ·det[Gp(z)] = diag{det[Gp(z)]}

(3)

where W(z) is the derived decoupled transfer function matrix. 2.2. Properties of the Proposed Decoupler. Theorem 1: For the decoupling strategy described in the preceding section, the zeros in the original transfer function matrix of the process Received: Revised: Accepted: Published: 4874

December 21, 2012 March 2, 2013 March 6, 2013 March 6, 2013 dx.doi.org/10.1021/ie303558f | Ind. Eng. Chem. Res. 2013, 52, 4874−4880

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3. SISO DESIGN OF NMSSMPC Because the multivariable process under consideration is decomposed into a number of independent single loops, simple SISO controllers can be designed instead of multi-input−multioutput (MIMO) controllers. Thus, the following sections provide the detailed SISO nonminimal state space model predictive control (NMSSMPC) design. 3.1. SISO Nonminimal State Space Model. Consider the general discrete input−output difference equation model of a process

still remain in the decoupled open-loop system, resulting in no zero−pole cancellations. Proof: Rewrite the process transfer function matrix as Gp(z) = Nr(z) ·Dr −1(z)

(4)

where Nr(z) and Dr(z) are the matrix numerator and denominator, respectively. Obviously, the zeros and poles of Gp(z) are the zeros of Nr(z) and Dr(z), respectively. Equation 4 can be further formulated as Gp(z) =

Nr(z) ·adj[Dr (z)] Λ (z )

y(k + 1) + Fy 1 (k) + F2y(k − 1) + ··· + Fny(k − n + 1)

(5)

= H1u(k) + H2u(k − 1) + ··· + Hmu(k − m + 1)

where

Λ(z) = det[Dr (z)]

Equation 15 can be transformed into a NMSS form by first adding the difference operator Δ and then selecting a state vector Δxm(k) as follows29

(6)

From eq 6, it is directly obtained that

det[Dr −1(z)] =

1 Λ (z )

Δxm(k + 1) = A mΔxm(k) + BmΔu(k) (7)

Δy(k + 1) = CmΔxm(k + 1)

Note that ⎧ adj[Dr (z)] ⎫ ⎬ det[Dr (z)] = det⎨ ⎩ Λ (z ) ⎭

(16)

where

−1

Δxm(k)

(8)

= [Δy(k) Δy(k − 1) ··· Δy(k − n + 1) Δu(k − 1) Δu(k − 2) ··· Δu(k − m + 1)]T

(17)

Thus, we obtain ⎧ adj[Dr (z)] ⎫ 1 ⎬= det⎨ Λ(z ) ⎩ Λ(z ) ⎭

and the dimension of Δxm(k) is m1 = dim(Δxm) = (m − 1) + n. The corresponding matrices are as follows (9)

⎡− F1 ⎢ ⎢1 ⎢0 ⎢ ⎢⋮ A m= ⎢ 0 ⎢ ⎢0 ⎢0 ⎢ ⎢⋮ ⎢⎣ 0

which indicates that det{adj[Dr (z)]} 1 = n Λ (z ) Λ (z )

(10)

Equation 10 is further written as det{adj[Dr (z)]} = Λn − 1(z)

(11)

From eq 5, it is seen that ⎧ N (z) ·adj[Dr (z)] ⎫ ⎬ det[Gp(z)] = det⎨ r Λ (z ) ⎭ ⎩ det[Nr(z)]·det{adj[Dr (z)]} = Λn(z) det[Nr(z)] = Λ (z )

⎧ det[Nr(z)] ⎫ ⎬ W (z) = diag det[Gp(z)] = diag⎨ ⎩ Λ (z ) ⎭

}

Cm= [1 0 0 ··· 0 0 0 0] (18)

From eq 16, it is obtained that

(12)

Δy(k + 1) = CmA mΔxm(k) + CmBmΔu(k)

(19)

Then, y(k + 1) is expressed as y(k + 1) = y(k) + CmA mΔxm(k) + CmBmΔu(k)

(13)

(20)

Thus, by augmenting the state variable xm(k) to include the output y(k) and defining

Note that, for the unity feedback structure of the control system, the closed-loop transfer function matrix is ⎧ ⎫ det[Nr(z)] ⎬ Φ(z) = diag⎨ ⎩ Λ(z) + det[Nr(z)] ⎭

− F2 ··· − Fn − 1 − Fn H2 ··· Hm − 1 Hm ⎤ ⎥ ··· 0 0 0 0 ··· 0 0 ⎥ ··· 0 1 0 0 ··· 0 0 ⎥ ⎥ ⋮ ⋱ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⎥ ··· 1 0 0 0 ··· 0 0 ⎥ ⎥ ··· 0 0 0 0 ··· 0 0 ⎥ ··· 0 0 0 1 ··· 0 0 ⎥ ⎥ ⋮ ⋱ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⎥ ··· 0 0 0 0 ··· 1 0 ⎥⎦

Bm= [ H1 0 0 ··· 0 1 0 0 ]T

Thus, by combining eqs 3 and 12, the transfer function matrix of the decoupler is

{

(15)

⎡ Δx (k)⎤ m ⎥ z(k ) = ⎢ ⎢⎣ y(k) ⎥⎦

(14)

It can be seen that, when Λ(z) and det[Nr(z)] have no common factors, all of the zeros of the process remain in the closed-loop transfer function matrix, causing no zero−pole cancellations in the decoupling design. This completes the proof. □

(21)

we obtain the nonminimal state space model (NMSS) z(k + 1) = Az(k) + BΔu(k) y(k) = Cz(k) 4875

(22)

dx.doi.org/10.1021/ie303558f | Ind. Eng. Chem. Res. 2013, 52, 4874−4880


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where

The cost function in eq 24a can be further expressed as

⎡ Am 01⎤ ⎥, A=⎢ ⎢⎣Cm A m 1 ⎥⎦

⎡ Bm ⎤ ⎥, B=⎢ ⎢⎣CmBm ⎥⎦

J = (S − Y )T Q (S − Y ) + ΔUTLΔU

C = [ 02 1]

where Q = block diag{Q1 Q2 ··· QP} and L = block diag{L1 L2 ··· LM}. By substituting eq 26 into eq 28, the optimal control vector can be derived as

(23)

In eq 23, 01 is a zero vector with dimensions m1 × 1, and 02 is a zero vector with dimensions 1 × m1. 3.2. Predictive Control Design. 3.2.1. Cost Function. The cost function is given by

ΔU = −(GTQG + L)−1GQ [Tz(k) − S]

∑ [s(k + j) − y(k + j)]T Q j[s(k + j) − y(k + j)]

KF = (GTQG + L)−1GQT

j=1

K s = (GTQG + L)−1GQ

M

+

∑ Δu

(29)

Let

P

J=

(28)

T

(k + j − 1)LjΔu(k + j − 1)

Then, the control vector at time instant k is

j=1

s.t. Δu(k + j) = 0, j ≥ M

Δu(k) = −kFz(k) + ksS

(24a)

4. CONTROL PERFORMANCE INTERPRETATIONS To obtain further insight into the MPC design described in the preceding section, the interpretation of the transfer function is addressed. This will allow the convenience of using frequency response tools to evaluate the control performance. Theorem 2: For the process described by eq 15, if it is treated in the form of eq 20 and the subsequent predictive control law is designed using eq 31, then the closed-loop control law will track the constant set point without steady error and reject constant output disturbances and the constant input disturbance completely without steady error. Proof: Denote the transfer function of eq 15 as Bm(z)/An(z) and the model orders for the input and output as m and n, respectively. Then, the dimensionality of z(k) is m + n + 1. Write the feedback control gain vectors kF and ks in eq 31 as

s(k) = y(k) (24b)

where ys is the set point. 3.2.2. State Prediction and Controller Design. The future state variables from sampling instant k are based on eq 22 and are denoted as ⎡ z(k + 1) ⎤ ⎢ ⎥ ⎢ z(k + 2) ⎥ Z=⎢ ⎥, ⎢⋮ ⎥ ⎢ z(k + P)⎥ ⎣ ⎦ ⎡ y(k + 1) ⎤ ⎢ ⎥ ⎢ y(k + 2) ⎥ Y=⎢ ⎥, ⎢⋮ ⎥ ⎢ ⎥ ⎣ y(k + P)⎦

⎡ Δu(k) ⎤ ⎢ ⎥ ⎢ Δu(k + 1) ⎥ ΔU = ⎢ ⎥, ⎢⋮ ⎥ ⎢ Δu(k + M − 1)⎥ ⎣ ⎦ ⎡ s(k + 1) ⎤ ⎢ ⎥ ⎢ s(k + 2) ⎥ S=⎢ ⎥ ⎢⋮ ⎥ ⎢ s(k + P)⎥ ⎣ ⎦

kF = [ k1 k 2 ··· kn kn + 1 kn + 2 ··· km + n + 1] ks = [ ks1 ks2 ··· ksP ]

(32)

Define two polynomial functions P̲ (z) = k1 + k 2z −1 + k 3z −2 + ··· + knz −(n − 1) L̲ (z) = 1 + kn + 1z −1 + kn + 2z −2 + ··· + km + nz −(m − 1)

(25)

(33)

Then future state vector Z can be expressed as

Then, from the control law in eq 31, together with eqs 21, 32, and 33 and the definitions

Z = Fz(k) + ΦΔU Y = Tz(k) + GΔU

(31)

where kF and ks are the first rows of KF and Ks, respectively.

where P is the maximum prediction horizon; M is the control horizon; and s(k + j) and y(k + j), j = 1, 2, ..., P, are the future set-point trajectories and the predicted outputs, respectively. In predictive control principle, when the smoothing factor μ is introduced, the reference trajectory is set as

s(k + i) = μi y(k) + (1 − μi )ys i = 1, 2, ..., P

(30)

kx1 = ks1λ + ks2λ 2 + ··· + ksPλ P

(26)

kx 2 = ks1(1 − λ) + ks2(1 − λ 2) + ··· + ksP(1 − λ P )

where

(34)

⎤ ⎡B 0 0 ··· 0 ⎥ ⎢ AB B 0 ··· 0 ⎥ ⎢ ⎥, Φ = ⎢ A2B AB B ··· 0 ⎥ ⎢ ⋮ ⋮ ⋱ ⋮ ⎥ ⎢⋮ ⎥ ⎢ P−1 ⎣ A B AP − 2B AP − 3B ··· AP − M − 1B ⎦ ⎡CB ⎤ 0 0 ··· 0 ⎡CA ⎤ ⎢ ⎥ CAB CB 0 ··· 0 ⎢ 2⎥ ⎢ ⎥ ⎢CA ⎥ ⎥ T=⎢ , G = ⎢CA2B CAB CB ··· 0 ⎥ ⎢ ⎥ ⋮ ⎥ ⎢ ⋮ ⋮ ⋱ ⋮ ⎢⋮ ⎥ ⎢⎣CAP ⎥⎦ ⎢ P−1 ⎥ 2 3 1 P P P M − − − − ⎣CA B CA B CA B ··· CA B⎦

⎡A ⎤ ⎢ 2⎥ ⎢A ⎥ F = ⎢ ⎥, ⋮ ⎢ ⎥ ⎢⎣ AP ⎥⎦

the control law can be derived in the following polynomial form (1 − z −1) L̲ (z) U (z) = −(1 − z −1)P̲ (z) Y (z) − km + n + 1Y (z) + kx1Y (z) + kx 2Ys(z)

(35)

where Y(z) and Ys(z) are the z transforms of y(k) and ys, respectively. Then, the transfer function from the set point to the output is T̅ (z) =

(27)

kx2Bm(z) (1 − z −1)[ L̲ (z) A n(z) + P̲ (z) Bm(z)] + (km + n + 1 − kx1)Bm(z)

(36) 4876



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From eq 27, it is verified that ⎡CA ⎤ ⎡*1 ⎢ 2⎥ ⎢ ⎢CA ⎥ ⎢*2 T=⎢ =⎢ ⋮ ⎥ ⎢⋮ ⎢ ⎥ ⎢⎣CAP ⎥⎦ ⎢⎣* P

Si̅ (z) =

1⎤ ⎥ 1⎥ ⎥ 1⎥ 1⎥⎦

(42)

It can be seen that lim Si̅ (z) = 0

(37)

z→1

where *1, *2, ..., *P are matrices that are irrelevant for the proof. From eqs 30 and 32 and the fact that the last column of T has 1 for all elements, it is easily seen that km+n+1 satisfies km + n + 1 = ks1 + ks2 + ··· + ksP

lim T̅(z) = 1

5. CASE STUDIES In this section, an example is employed to illustrate the ensemble performance of the proposed method. The performance is highlighted by comparisons of decoupling NMSSMPC with nondecoupling NMSSMPC26−28 and the recently proposed partial decoupling MPC.29 The distillation column model proposed by Lubyben33 that has been used for study for many years is employed here for comparisons. The effects of decoupling NMSSMPC will be tested on both model/process match and mismatch cases. Consider the 24-plate bubble-cup distillation column process given by Luyben.33 The corresponding discrete model with a sample time of 1 min is

(38)

(39)

This shows that the closed-loop system will track the constant set point without any steady error. The closed-loop transfer function from the output disturbance to the output response is derived as S ̅(z) =

kx2(1 − z −1) L̲ (z) A n(z) (1 − z )[ L̲ (z) A n(z) + P̲ (z) Bm(z)] + (km + n + 1 − kx1)Bm(z) −1

(40)

with lim S ̅ (z) = 0

⎡ − 0.2929z −2 0.1237z −1 + 0.04935z −2 ⎤ ⎥⎡ ⎡ y (k) ⎤ ⎢ −1 ⎤ 1 − 0.8669z −1 ⎥⎢ u1(k) ⎥ ⎢ 1 ⎥ = ⎢ 1 − 0.8669z ⎢ ⎥ −2 −3 ⎢ y (k)⎥ 0.2933z −1 + 0.1496z −2 ⎥⎢⎣ u 2(k)⎥⎦ ⎣ 2 ⎦ ⎢ − 0.05833z − 0.2214z ⎢⎣ ⎥⎦ 1 − 0.9001z −1 1 − 0.897z −1

(41)

z→1

indicating that the closed-loop control system can reject constant output disturbances completely. The closed-loop transfer function from the input disturbance to the output response is derived as G(z) =

(43)

showing that the constant input disturbance can also be rejected completely without steady error. □

resulting in z→1

kx2(1 − z−1) L̲ (z) Bm(z) (1 − z )[ L̲ (z) A n(z) + P̲ (z) Bm(z)] + (km + n + 1 − kx1)Bm(z) −1

For the proposed decoupling case, this process model is first divided into two SISO processes with the same transfer function model

−0.078692149z −3 + 0.05730109662z −4 + 0.0232181361305z −5 − 0.00980070273z −6 1 − 2.664z −1 + 2.36529569z −2 − 0.69992613093z −3

The relationship between [u1(k) u2(k)]T and [ud1(k) ud2(k)]T can be further calculated as

⎡ − 0.2829z −2 0.1337z −1 + 0.05935z −2 ⎤ ⎥⎡ ⎡ y (k) ⎤ ⎢ −1 ⎤ 1 − 0.8969z −1 ⎥⎢ u1(k) ⎥ ⎢ 1 ⎥ = ⎢ 1 − 0.8569z ⎢ y (k)⎥ ⎢ − 0.06833z −2 − 0.2114z −3 0.2833z −1 + 0.1096z −2 ⎥⎢⎣ u (k)⎥⎦ ⎣ 2 ⎦ ⎢ ⎥ 2 ⎢⎣ ⎥⎦ 1 − 0.9101z −1 1 − 0.867z −1

Then, the corresponding NMSSMPC can be designed for the decoupled process model. The simulation procedure is as follows: A unit step change is added to the set points at time instant k = 0, and step changes of load disturbance with an amplitude of −0.1 are added to each of the process outputs at time instants k = 1000 and k = 1800. In the controller designs, the control parameters are the same with the prediction horizon chosen to be 20, the control horizon chosen to be 3, and the smoothing factor chosen to be 0.95. The weighting elements on the output errors are all set to 1, and the weighting elements on the control signal are 1. 5.1. Comparison of the Proposed with Nondecoupling NMSSMPC. The detailed design of the NMSSMPC based on the example model can be seen in Wang and Young.26 This section provides comparisons under three cases: Case 1 represents the case of model/process match. Case 2 is the case of model/ process mismatch where the real process is supposed to be

However, the designs of the decoupling and nondecoupling strategies are still based on the original process model. Case 3 is a Monte Carlo experiment. The process parameters (process gains, poles, and zeros) are changed randomly, and the upper and lower ranges of the parameter changes are restricted to be within the interval [−0.05 0.05], that is, the mismatched process is estimated as ⎡ − 0.2798z−2 0.143z−1 + 0.078z−2 ⎤ ⎥⎡ ⎡ y (k) ⎤ ⎢ ⎤ −1 1 − 0.8715z−1 ⎥⎢ u1(k) ⎥ ⎢ 1 ⎥ = ⎢ 1 − 0.8451z ⎢ y (k)⎥ ⎢ − 0.0641z−2 − 0.2361z−3 0.3109z−1 + 0.1695z−2 ⎥⎢⎣ u (k)⎥⎦ ⎣ 2 ⎦ ⎢ ⎥ 2 ⎥⎦ ⎢⎣ 1 − 0.9347z−1 1 − 0.8742z−1 4877



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However, the designs of the decoupling and nondecoupling strategies are still based on the original process model. Figures 1 and 2 show the output responses and control signals, respectively, for case 1. It can be seen that, under the proposed strategy, the original multivariable process is completely decoupled into two SISO processes. In the nondecoupling case, when one of the output disturbances is encountered, both of the process outputs

are influenced. In addition, it can be seen that the overall set-point tracking and disturbance rejection of the proposed are also acceptable. The proposed method successfully provides decoupled responses and improved control performance. The model/process mismatch studies under case 2 are illustrated in Figures 3 and 4. Again, it can be seen that improved results are obtained for both outputs. In the two figures, the decoupling

Figure 1. Closed-loop responses of [y1(k) u1(k)] under case 1.





Figure 6. Closed-loop responses of [y2(k) u2(k)] under case 3. 4878



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NMSSMPC produces faster disturbance rejection with smaller magnitudes. The model/process mismatch case studies for case 3 are illustrated in Figures 5 and 6. It can be seen that improved results are obtained for both outputs. The initial tracking performance of the nondecoupling control might be slightly faster, but large overshoot results. In terms of disturbance rejection, the responses of both y1(k) and y2(k) are improved through the proposed method. 5.2. Comparison of the Proposed with Partial Decoupling MPC.29 The detailed design of partial decoupling state space MPC can be seen in Zhang and Gao.29 This section provides comparisons for a Monte Carlo experiment in which relatively severe model/process mismatch is generated. In this case, the process parameters (process gains, poles, and zeros) are simultaneously assumed to vary by ±30%, which introduces severe model/process mismatch into the process. In case 4, the mismatched process is estimated as

The responses of the Monte Carlo experiment can be seen in Figures 7 and 8. It can be clearly seen from these figures that improved control performance can be obtained through the proposed decoupling MPC. When the model/process mismatch becomes more severe, the responses of partialdecoupling-based MPC tends to deteriorate. However, the responses of the proposed decoupling MPC remain smooth, showing improved performance. All in all, the Monte Carlo experiment shows that the proposed MPC can further improve control performance.

6. CONCLUSIONS In this work, a decoupling design is first proposed that offers the advantage of no zero−pole cancellations and enables realization in practice. A subsequent SISO NMSSMPC controller can then be designed for the multivariable process. The NMSSMPC design is tested on both model/process match and mismatch. As a result, improved control performance results are obtained, as demonstrated by an example of a 24plate bubble-cup distillation column.

⎡ − 0.3163z−2 0.1472z−1 + 0.0564z−2 ⎤ ⎥⎡ ⎡ y (k) ⎤ ⎢ ⎤ −1 1 − 0.939z−1 ⎥⎢ u1(k) ⎥ ⎢ 1 ⎥ = ⎢ 1 − 0.9556z ⎥ ⎢ −2 −3 ⎢ y (k)⎥ 0.2397z−1 + 0.1484z−2 ⎥⎢⎣ u 2(k)⎥⎦ ⎣ 2 ⎦ ⎢ − 0.0636z − 0.2568z ⎥⎦ ⎢⎣ 1 − 0.8515z−1 1 − 0.7513z−1

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +86-0571-87952233. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Part of this project was supported by the National Natural Science Foundation of China (Grants 61273101 and 61104058), China Postdoctoral Science Foundation (Grants 2012M511367 and 2012M511368), and Doctor Scientific Research Foundation of Liaoning Province (20121046).

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REFERENCES

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Figure 8. Closed-loop responses of [y2(k) u2(k)] under case 4. 4879



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