State Space Model Predictive Control Using Partial Decoupling and

Article pubs.acs.org/IECR

State Space Model Predictive Control Using Partial Decoupling and Output Weighting for Improved Model/Plant Mismatch Performance Ridong Zhang†,‡,§ and Furong Gao*,‡ †

Information and Control Institute, Hangzhou Dianzi University, Hangzhou 310018, P R China Department of Chemical and Biomolecular Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong § National Key Laboratory of Industrial Control Technology, Institute of Cyber-Systems & Control, Zhejiang University, Hangzhou 310027, P R China ‡

ABSTRACT: Focusing on multivariable control, this paper presents a design method of model predictive control that enjoys the benefits of both the partial decoupling and the state space design. The multivariable process is first decoupled into a set of multiinput single-output (MISO) structures and then transformed into an extended state space model (partial decoupling extended state space model, PD-ESS). Consequently, a systematic design of model predictive control is proposed. The proposed controller is tested on three typical cases for comparison with previous controllers. Results show that control performance is improved. In addition, a closed-form of transfer function representation that facilitates frequency analysis of the control system is provided for further insight into the proposed method.

1. INTRODUCTION Model predictive control (MPC) has been widely practiced both in control theory and in control engineering since its first appearance in the 1970s. The three general MPC design methods are typically focused: finite impulse response (FIR) or step response models, transfer function models and state space models. Earlier FIR model based MPC can be represented by dynamic matrix control (DMC)1 and quadratic DMC,2 which is limited to stable processes and the design model order is usually high. Transfer function based MPC can be seen in generalized predictive control (GPC)3 by Clarke et al. It can deal with both stable and unstable processes but is considered to be less effective for multivariable processes and real-time application. In recent years, MPC based on state space models has been an area of intensive research. A number of state space model based predictive control methods have been developed.4−16 Some surveys on the state space predictive control are also available.17−22 Some state space MPC designs need to assume that the observer dynamics are faster than that of the state feedback controller. However, this assumption will cause a major problem of numerical difficulty.23 The new idea of incorporating the measured process input, output, and their past values into a nonminimal state space model (NMSS) can overcome the observerbased problems. Results on the NMSS model based control have been reported in many different areas.24−35 Recently, Wang and Young36 proposed a multivariable NMSS model based MPC scheme. By augmenting the state variable to include the input−output variables, this strategy eliminates the need of an observer and improves control performance compared with existing methods. For model/plant mismatch cases, however, this scheme might not always guarantee satisfactory control performance due to the limitation of the design procedure. In this work, a partial decoupling scheme is first given and further treated to an extended state space formulation (PD-ESS). © 2012 American Chemical Society

Then a new MPC design is presented to show improved control performance. The main contributions of this work include the following: (1) A partial decoupling strategy based extended state space model is derived that directly uses the measured input and output variables which avoids the inclusion of any observer and provides a balance between controller design and control performance improvement. (2) A subsequent MPC scheme will enable the controller design to be based on the regulations of both the output errors and the process states, thus control performance is improved. (3) A closed-form of transfer function representation that facilitates frequency analysis of the control system will give further insight into the proposed method. The remainder of this paper is organized as follows. Section 2 presents the new partial decoupling extended state space model structure. In section 3, an MPC design based on the new model is presented, where integral action and set-point tracking are naturally incorporated in the algorithm. Section 4 presents the transfer function analysis of the proposed control algorithm. Section 5 discusses the comparisons of three typical processes, the 24-plate bubble-cup distillation column process,37 the glasshouse process,36 and a double integrating plant.36 Conclusion is in section 6.

2. PARTIAL DECOUPLING STRUCTURE BASED EXTENDED STATE SPACE MODEL 2.1. Discussion. This section is the direct extension of the authors’ previous work to the general case where the input number and the output number are not equal.38 We consider a plant Received: Revised: Accepted: Published: 817

March 29, 2012 September 16, 2012 November 26, 2012 November 26, 2012 dx.doi.org/10.1021/ie300836m | Ind. Eng. Chem. Res. 2013, 52, 817−829

Industrial & Engineering Chemistry Research

Article

Di (4a) D where D = det(F̅) and Di is the determinant obtained by substituting for the i column of F̅ with H̅ . In eq 4a, the division does not need to be done, it is further treated as

with p outputs and q inputs and is described by the following difference equation model

yi (k) =

F11̅ (z −1)y1(k) + F12̅ (z −1)y2 (k) + ··· + F1̅ p(z −1)yp (k) −1 −1 −1 = H11 ̅ (z )u1(k) + H12 ̅ (z )u 2(k) + ··· + H1̅ q(z )uq(k)

F21̅ (z −1)y1(k) + F22̅ (z −1)y2 (k) + ··· + F2̅ p(z −1)yp (k) −1

−1

Dyi (k) = Di

(4b)

where Di = Di1 × u1(k)+Di2 × u2(k) + ··· + Diq × uq(k) (i = 1, 2, ..., p). The treatment will lead to the following form of D and Dij

−1

= H̅ 21(z )u1(k) + H̅ 22(z )u 2(k) + ··· + H̅ 2q(z )uq(k)

⋮

D = 1 + f1 z −1 + f2 z −2 + ··· + fn z −n

Fp̅ 1(z −1)y1(k) + Fp̅ 2(z −1)y2 (k) + ··· + Fpp̅ (z −1)yp (k) −1

Dij = hij1z −1 + hij2z −2 + ··· + hijsz −s

−1

= H̅ p1(z )u1(k) + H̅ p2(z )u 2(k) + ···

(i = 1, 2, ..., p; j = 1, 2, ..., q)

+ H̅ pq(z −1)uq(k) −1

−1

(1) −1

−1

where n and s are the orders of D and Dij, respectively. Now further denote

−1

where F̅11(z ), F̅12(z ), ..., F̅pp(z ) and H̅ 11(z ), H̅ 12(z ), ..., H̅ pq(z−1) are the polynomials and yi(k) and uj(k) (i = 1, 2, ..., p; j = 1, 2, ..., q) are the output and input variables, respectively. Denote ⎡ F ̅ (z −1) ⎢ 11 ⎢ F21̅ (z −1) F̅ = ⎢ ⎢⋮ ⎢ ⎢ F ̅ (z −1) ⎣ p1

F(z −1) = I + F1z −1 + F2z −2 + ··· + Fnz −n H(z −1) = H1z −1 + H 2z −2 + ··· + Hsz −s

F12̅ (z −1) ··· F1̅ p(z −1) ⎤ ⎥ ⎥ F22̅ (z −1) ··· F2̅ p(z −1) ⎥ ⎥ ⋮ ⋱ ⋮ ⎥ Fp̅ 2(z −1) ··· Fpp̅ (z −1)⎥⎦

where ⎡f ⎢ k ⎢0 Fk = ⎢ ⎢⋮ ⎢ ⎣0

⎡ y (k ) ⎤ ⎢ 1 ⎥ ⎢ y (k ) ⎥ 2 ⎥ Y=⎢ ⎥ ⎢⋮ ⎥ ⎢ ⎢⎣ yp (k)⎥⎦

⎡ h11i ⎢ ⎢ h21i Hi = ⎢ ⎢⋮ ⎢ ⎢⎣ hp1i

⎡ H (z −1)u (k) + H (z −1)u (k) + ··· ⎤ ̅ ̅ 1 12 2 ⎢ 11 ⎥ −1 ⎢ + H1̅ q(z )uq(k) ⎥ ⎢ ⎥ ⎢ H̅ 21(z −1)u1(k) + H̅ 22(z −1)u 2(k) + ···⎥ ⎢ ⎥ −1 H̅ = ⎢ + H̅ 2q(z )uq(k) ⎥ ⎢ ⎥ ⋮ ⎢ ⎥ ⎢ ⎥ −1 −1 ⎢ H̅ p1(z )u1(k) + H̅ p2(z )u 2(k) + ··· ⎥ ⎢⎣ + H̅ pq(z −1)uq(k) ⎥⎦

(4c)

0 ··· 0 ⎤ ⎥ fk ··· 0 ⎥ ⎥ 0 ⋱ 0⎥ ⎥ 0 ··· fk ⎦ h12i ··· h1qi ⎤ ⎥ h22i ··· h2qi ⎥ ⎥ ⋮ ⋱ ⋮ ⎥ ⎥ hp2i ··· hpqi ⎥⎦

k = 1, 2, ..., n; i = 1, 2, ..., s

Then eq 4b is expressed as a new MIMO structure, in which the relationship of MISO holds for each output variable F(z −1)y(k) = H(z −1)u(k)

(5) T

where y(k) = [y1(k), y2(k), ..., yp(k)] and u(k) = [u1(k), u2(k), ..., uq(k)]T are output and input vectors of the process, respectively. Add the difference operator Δ to eq 5 and rewrite it in the form of differenced input and output variables as (2)

Δy(k+1) + F1Δy(k) + F2Δy(k−1) + ··· + FnΔy(k−n+1)

Then eq 1 can be expressed as linear equations

= H1Δu(k) + H 2Δu(k−1) + ··· + HsΔu(k−s+1)

(3) FY ̅ = H̅ The well-known Cramer’s Rule for solving the above linear equations gives the following solution

(6)

Define the input-output variable included state space vector Δxm(k) as

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

Δx m(k+1) = A mΔx m(k) + BmΔu(k)

where m = dim(Δxm) = p × (n + q) × (s − 1).

Δy(k+1) = CmΔx m(k+1)

Then the corresponding state space model is 818

(7)

dx.doi.org/10.1021/ie300836m | Ind. Eng. Chem. Res. 2013, 52, 817−829


Article

3. PREDICTIVE CONTROL DESIGN BASED ON PD-ESS MODEL 3.1. Cost Function.

where ⎡−F1 ⎢ ⎢ Ip ⎢ ⎢0 ⎢ ⎢⋮ A m = ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢⋮ ⎢ ⎢⎣ 0

Bm =

[H1T

−F2 ··· −Fn − 1 −Fn H 2 ··· Hs − 1 0

··· 0

0

0

··· 0

Ip

··· 0

0

0

··· 0

⋮

··· ⋮

⋮

⋮

··· ⋮

0

··· Ip

0

0

··· 0

0 0

··· 0 ··· 0

0 0

0 Iq

··· 0 ··· 0

⋮

··· ⋮

⋮

···

⋮ ⋮

0

··· 0

0

0

··· Iq

Hs ⎤ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ ⋮ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⋮ ⎥⎥ 0 ⎥⎦

P

J=

j=1

Q j = diag{qjy1, qjy2 , ..., qjynp , qju1, qju2 , ..., qju(n − 1)q , qje1, qje2 , ..., qjep}

e(k) = y(k) − r(k)

(9)

Thus by combining eq 7 and eq 9, the formulation of e(k+1) is derived as e(k+1) = e(k) + CmA mΔx m(k) + CmBmΔu(k) (10)

Define a new state variable as

(11)

Then we get a partial decoupling extended state space model (PD-ESS) (12)

⎡ z(k+1) ⎤ ⎢ ⎥ ⎢ z(k+2) ⎥ Z=⎢ ⎥ ⎢⋮ ⎥ ⎢ z(k+P)⎥ ⎣ ⎦

where ⎡Am 0⎤ ⎥ A=⎢ ⎢⎣ CmA m Ip ⎥⎦

⎡ Bm ⎤ ⎥ B=⎢ ⎣ CmBm ⎦

(15)

Discussion. • The variables qjy1, qjy2, ..., qjynp and qju1, qju2, ..., qju(n−1)q are associated with the regulation of the process states (including output changes and the input changes), whereas qje1, qje2, ..., qjep are associated with the regulation of the process output tracking errors. • If Qj = 0, j < P1 < P is adopted, where P1 is a value ranging from 1 to P, this is to show that the optimization starts from the sampling instant k + P1. Hence, time delay d and single point prediction can be implemented conveniently by setting P1 = d + 1 and P1 = P. Remarks 2. • The selection of the cost function eq 14 enables the incorporation of both the output error vector and the process state dynamics and avoids the shortcomings of that of Wang’s, which, due to its model structure treatment, can only incorporate the output error vector in it. 3.2. State Prediction and Controller Design. On the basis of the prediction model eq 12 and with the following

(8)

Define the expected output vector as r(k), and then the output tracking error vector as

z(k+1) = Az(k) + BΔu(k) + CΔr(k+1)

j≥M

1≤j≤P

Cm = [Ip 0 0 ··· 0 0 0 0]

(14)

where P is the maximum prediction horizon, M is the control horizon, Qj is the symmetrical weight matrix with dimension (m + p) × (m + p), and Lj ≥ 0 is the weight factor of control input increments. Generally, Qj is taken as

0 0 ··· 0 Iq 0 0]

⎡ Δx (k)⎤ m ⎥ z (k ) = ⎢ ⎢⎣ e(k) ⎥⎦

j=1

s.t. Δu(k+j) = 0

T

− Δr(k+1)

M

∑ z T(k+j)Q j z(k+j) + ∑ Δu T(k+j−1)LjΔu(k+j−1)

⎡0 ⎤ C=⎢ ⎥ ⎣− I p ⎦ (13)

In eq 13, 0 is a zero matrix with dimension m × p and Ip is a unit matrix with dimension p. 2.2. Remarks 1. • The above eq 12 is the derived new state space model, which will show later that this treatment facilitates the controller design to regulate both the process output error and the state changes, leading to improved control performance. • This treatment avoids the two main drawbacks of those of Astrom and Wittenmark,23 which uses the observability matrix and linear transformation of original state space structure to reconstruct state variables: (1) sensitivity to measurement noise; (2) numerical stability of observability matrix in case of large number of state variables and unstable plants. • Time delay can be incorporated in the above model by letting the coefficients (or the coefficients matrices) H1 = H2 = ... = Hd = 0.

⎡ Δu(k) ⎤ ⎢ ⎥ ⎢ Δu(k+1) ⎥ ΔU = ⎢ ⎥ ⎢⋮ ⎥ ⎢ Δu(k+M −1)⎥ ⎣ ⎦

⎡ Δr(k+1) ⎤ ⎢ ⎥ ⎢ Δr(k+2) ⎥ ΔR = ⎢ ⎥ ⎢⋮ ⎥ ⎢ Δr(k+P)⎥ ⎣ ⎦

(16)

the future state vector Z is then predicted as Z = Fz(k) + ΦΔU + SΔR

(17)

where

⎡A ⎤ ⎢ 2⎥ ⎢A ⎥ F=⎢ ⎥ ⋮ ⎢ ⎥ ⎢⎣ AP ⎥⎦ 819


Industrial & Engineering Chemistry Research ⎡B ⎢ ⎢ AB Φ = ⎢ A2B ⎢ ⎢⋮ ⎢ P−1 ⎣A B ⎡C ⎢ ⎢ AC S = ⎢ A2C ⎢ ⎢⋮ ⎢ P−1 ⎣A C

Article

⎤ ⎥ ⎥ ⎥ AB B ··· 0 ⎥ ⋮ ⋮ ⋱ ⋮ ⎥ P−2 P−3 P−M−1 ⎥ A B A B ··· A B⎦ 0 B

0 0⎤ ⎥ 0 0⎥ AC C 0 0⎥ ⎥ ⋮ ⋮ ⋱ ⋮⎥ ⎥ AP − 2C AP − 3C ··· C ⎦ 0 C

Define two polynomial functions as following

··· 0 ··· 0

0 0

P(z) = k1 + k 2z −1 + k 3z −2 + ··· + knz −(n − 1) L(z) = 1 + kn + 1z −1 + kn + 2z −2 + ··· + k 2n − 1z −(n − 1) (25)

Note that in model predictive control, the reference trajectory associated with the smoothing factor μ is set as

0 0

r(k) = y(k) r(k+i) = μi y(k) + (1 − μi )ys

T

J = Z Q Z + ΔU LΔU

r(k) = y(k) Δr(k+1) = r(k+1) − r(k) = (1 − μ)(ys − y(k))

(19)

⋮

where Q = block diag{Q1,Q2,...,QP} and L = block diag{L1,L2,...,LM}. Substituting eq 17 into eq 19, the optimal control vector is derived as T

−1

T

ΔU = −(Φ Q Φ + L) Φ Q (Fz(k) + SΔR)

Δr(k + P) = r(k+P) − r(k+P−1) = μ P − 1(1 − μ)(ys − y(k))

(20)

⎡(1 − μ)(y − y(k)) ⎤ s ⎢ ⎥ ⎢ μ(1 − μ)(y − y(k)) ⎥ s ⎥ ΔR = ⎢ ⎢⋮ ⎥ ⎢ ⎥ P−1 ⎢⎣ μ (1 − μ)(ys − y(k))⎥⎦

K s = (ΦTQ Φ + L)−1ΦTQ F (21)

Then the incremental control vector at time k is Δu(k) = −k sz(k) − kRΔR

k′R = kR1(1−μ) + kR 2μ(1−μ) + ··· + kRPμ P − 1(1−μ) (29)

(23)

The proposed MPC control law is expressed in the following polynomial form

4. CONTROL PERFORMANCE INTERPRETATIONS This section will further cast the above MPC design in terms of transfer functions. This will give the convenience for using frequency response tools to value the control performances. For notational simplicity, this article will show the single input and single output case only. 4.1. Proposition. For the process that is described by eq 1, if it is treated into the form of eq 6 and the subsequent predictive control law is designed as eq 22, then the closed-loop MPC law will yield the following results: it tracks the constant set-point without steady error and rejects constant output disturbances and the constant input disturbance totally without steady errors. 4.2. Proof. The transfer function is assumed to be Bn(z)/ Ad(z), and the model order for the input and output transfer function is n. Then the dimensionality of z(k) is 2n. The feedback control gain vector ks and kR in eq 22 can be further expressed as

(1 − z −1) L(z)U(z) = −(1 − z −1) P(z)Y(z) + k′R(Ys(z) − Y(z))

(30)

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

k′RBn(z) −1

(1 − z )( L(z)Ad (z) + P(z)Bn(z)) + k′RBn(z) (31)

It can be seen from eq 31 that lim T̅(z) = 1

(32)

z→1

which proves that the closed-loop system tracks the constant setpoint without steady error. The closed-loop transfer function from the output disturbance to output response is shown as

k s = [k1 k 2 ··· kn kn + 1 kn + 2 ··· k 2n] kR = [kR1 kR 2 ··· kRP]

(28)

Then from the control law eq 22, together with eqs 11, 24, and 28, denotes

(22)

where ks and kR are the first P rows of the matrices Ks and KR, respectively. The control input is expressed as u(k) = u(k − 1) + Δu(k)

(27)

In view of eq 27, the incremental reference vector in eq 16 can be rewritten as

Let

KR = (ΦTQ Φ + L)−1ΦTQ S

(26)

where ys is the set point vector. Then by iteration of eq 26, the followings hold

(18)

The vector form of the cost function in eq 14 can be further expressed as T

i = 1, 2, ..., P

S ̅ (z ) = (24)

(1 − z −1) L(z)Ad (z) −1

(1 − z )( L(z)Ad (z) + P(z)Bn(z)) + k′RBn(z) (33)

820



Article

It can be seen that

x(k) = [Δy T(k) Δy T(k−1) Δu T(k−1) Δu T(k−2) Δu T(k−3) e T(k)]T

lim S ̅ (z) = 0

(34)

z→1

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

Δu(k) = [Δu1(k) Δu 2(k)]T

y(k) = [ y1(k) y2 (k)]T

(1 − z −1) L(z)Bn(z)

e(k) = [ e1(k) e 2(k)]T

(1 − z −1)( L(z)Ad (z) + P(z)Bn(z)) + k′RBn(z) (35)

And the corresponding state space model is

Note that

x(k+1) = Ax(k) + BΔu(k) + CΔr(k+1)

lim Si̅ (z) = 0

z→1

(36)

where A, B, and C can be derived through eq 13. Then the corresponding predictive control can be designed as described in sections 2 and 3. On the basis of Wang’s strategy, the state space variables are

which means that the constant input disturbance can also be rejected totally without steady error.

5. CASE STUDIES Three examples are given to illustrate the performance of the proposed method. The representative state space model based MPC proposed by Wang and Young36 is adopted here for comparison. Note that in Wang’s method, the design procedure does not mention the decoupling structure. Hence, to show a clear and fair comparison, Wang’s method can also be extended to this partial decoupling structure for comparison. The tests on the three examples are as follows. In the first and second examples, the tracking performance and disturbance rejection performance under model/plant mismatch are tested. In the third example, the constrained optimization under both model/plant match and model/plant mismatch are tested. 5.1. Unconstrained Optimization Case. Example 1. Consider a 24-plate bubble-cup distillation column process given by Luyben,37 the corresponding discrete model with sample time 1 min is

x(k) = [Δy T(k) Δy T(k−1) Δu T(k−1) Δu T(k−2) Δu T(k−3) y T(k)]T

Δu(k) = [Δu1(k) Δu 2(k)]T

y(k) = [ y1(k) y2 (k)]T

And the corresponding model is then derived as x(k+1) = Ax(k) + BΔu(k) y(k) = Cx(k)

where A, B, and C can be derived by substituting eq 8 into the following equation as described by Wang and Young36 ⎡Am 0⎤ ⎥ A=⎢ ⎢⎣ Cm A m Ip ⎥⎦

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

⎡ Bm ⎤ ⎥ B=⎢ ⎣ CmBm ⎦

C = [ 0 Ip ]

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 amplitudes of −0.1 and −0.1 are added to each of the process output at time instant k = 1000 and k = 1800, respectively. In the two controller designs, the control parameters are the same with P = 20, M = 3, μ = 0.95. Model/Plant Parameter Mismatch Test. Suppose some model/ plant parameter mismatch is introduced into the process, which causes the real process to change to

The partial decoupling input-output state space model is then ⎡ y (k) ⎤ ⎡ ⎤⎡⎢ y1(k−1) ⎤⎥ ⎡ 0 0 ⎤ ⎢ 1 ⎥ = ⎢ 0.8669 0 +⎢ ⎥⎢ ⎥ ⎢ y (k)⎥ ⎣ 0 ⎥ ⎣ ⎦ 0 −0.8073897 ⎦ 1.7971 ⎣ y (k−1)⎦ ⎣ 2 ⎦ 2 ⎡ y (k−2) ⎤ ⎡ u (k−1) ⎤ 1 ⎥ + ⎡ 0 0.1237 ⎤⎢ 1 ⎥ ×⎢ ⎥ ⎢ ⎢ y (k−2)⎥ ⎣ 0 0.2933 ⎦⎢⎣ u (k−1)⎥⎦ 2 ⎣ 2 ⎦

⎡ y (k) ⎤ ⎢ 1 ⎥ ⎢ y (k)⎥ ⎣ 2 ⎦

⎡−0.2929 0.04935 ⎤⎡ u1(k−2) ⎤ ⎥ +⎢ ⎥⎢ ⎣ 0.05833 −0.11439933⎦⎢⎣ u 2(k−2)⎥⎦

⎡ − 0.2529z−2 0.1137z−1 + 0.06935z−2 ⎤ ⎥⎡ ⎢ −1 ⎤ 1 − 0.8969z−1 ⎥⎢ u1(k) ⎥ ⎢ 1 − 0.7669z =⎢ −2 −3 −1 −2 ⎥⎢ u (k)⎥ 0.2633z + 0.1096z ⎥⎣ 2 ⎦ ⎢ − 0.07833z − 0.1914z ⎥⎦ ⎢⎣ 1 − 0.9251z−1 1 − 0.867z−1

⎡0 ⎤⎡ u1(k−3) ⎤ 0 ⎥ +⎢ ⎥⎢ ⎣−0.16907799 −0.13465496 ⎦⎢⎣ u 2(k−3)⎥⎦ ⎡0 0 ⎤⎡⎢ u1(k−4) ⎤⎥ +⎢ ⎣ 0.1985958 0 ⎥⎦⎢⎣ u (k−4)⎥⎦ 2

The designs of the proposed and Wang’s method are still based on the original plant model. The weighting parameters for the proposed MPC are chosen as

On the basis of the proposed design procedure, the state space model variables for the proposed are 821



Article

However, the two controllers are still designed on the basis of the original plant model mentioned at the beginning of this section. The simulation parameters remain the same as those in the model/plant parameter mismatch test. This time Wang’s method results in the unstable responses. The closed-loop responses of the proposed can be seen in Figure 2. It can be seen that the responses are still acceptable.

Q j = diag{10 10 5 5 0 0 0 0 0 0 1 1} (j = 1, 2, ..., 20) Q = block diag{Q 1, Q 2 , ..., Q 20} Lj = diag{1 1}

(j = 1, 2, 3)

L = block diag{L1, L 2 , L3}

The limitation of Wang’s method shows that it cannot consider the state variables in its controller design, which means that weighting parameters for Wang’s method are still Q j = diag{1 1}

(j = 1, 2, ..., 20)

Q = diag{Q 1, Q 2 , ..., Q 20} Lj = diag{1 1}

(j = 1, 2, 3)

L = block diag{L1, L 2 , L3}

The comparison of closed-loop responses can be seen in Figure 1. It is clearly seen that improved control performance is obtained.

Figure 2. Closed-loop responses of proposed under structure mismatch for example 1.

Though we cannot generalize the results, we can conclude that the proposed is to some degree more robust to model/plant mismatch than conventional state space predictive control. Example 2. Consider the glasshouse process given by Wang and Young36 ⎡ 0.015z −1 ⎤ −0.077z −1⎥⎡ ⎡ y (k ) ⎤ ⎢ −1 ⎤ ⎥⎢ u1(k) ⎥ ⎢ 1 ⎥ = ⎢ 1 − 0.905z ⎥⎢ ⎢ y (k)⎥ ⎢ −0.058z −1 ⎥ ⎣ 2 ⎦ ⎢ −1 ⎥⎣ u 2(k)⎦ 0.753 z ⎢⎣ 1 − 0.793z −1 ⎥⎦ Figure 1. Closed-loop responses under parameter mismatch for example 1. Dotted line: Wang. Solid line: proposed.

where y1(k) is the air temperature, y2(k) is the relative humidity of the air, u1(k) is the fractional valve aperture of the boiler, and u2(k) is the mist spraying system input. The input−output state space model is

The proposed algorithm provides satisfactory performance of both set-point tracking and disturbance rejection. This is because the proposed MPC enables the controller to regulate the dynamics of the process states, which can cope with the model/plant mismatch by placing weights on the process states to improve their responses. Model/Plant Structure Mismatch Test. Suppose there is also kind of structure mismatch. Now arbitrarily change the orders of the four transfer functions of the real plant to

⎤ ⎡ y (k ) ⎤ ⎡ ⎤⎢ y1(k−1) ⎥ ⎢ 1 ⎥ + ⎡−0.905 0 ⎢ ⎥ ⎢ y (k)⎥ ⎣ 0 −0.793⎦⎢⎣ y (k−1)⎥⎦ ⎣ 2 ⎦ 2 ⎡ 0.015 −0.077 ⎤⎡ u1(k−1) ⎤ ⎢ ⎥ =⎢ ⎣−0.058 0.753 ⎥⎦⎢⎣ u 2(k−1)⎥⎦ ⎡ 0.0 0.07 ⎤⎡ u1(k−2) ⎤ ⎢ ⎥ +⎢ ⎣ 0.0 − 0.597 ⎥⎦⎢⎣ u 2(k−2)⎥⎦

⎤ ⎡ 0.06935z−2 − 0.2529z−2 ⎥⎡ ⎡ y (k) ⎤ ⎢ −1 −1 ⎤ 1 0.8669 0.11 1 0.8669 0.11 z z − + − + 1 ⎥⎢ u1(k) ⎥ ⎢ ⎢ ⎥= ⎥ ⎢ ⎢ y (k)⎥ 0.2633z−1 + 0.1096z−2 ⎥⎢⎣ u 2(k)⎥⎦ − 0.1914z−3 ⎣ 2 ⎦ ⎢ ⎢⎣ 1 − 0.9251z−1 − 0.03 1 − 0.867z−1 − 0.05 ⎥⎦

On the basis of the proposed design procedure, the state space model variables for the proposed are x(k) = [ Δy1(k) Δy2 (k) Δu1(k − 1) Δu 2(k − 1) e1(k) e 2(k)]T

It can be seen that kinds of both parameter mismatch and structure mismatch are encountered. Besides the above real plant model also includes kind of dead time mismatch (see the (1,2) and (2,1) elements, the real dead time values are both increased compared with the original plant model).

Δu(k) = [ Δu1(k) Δu 2(k)]

y(k) = [ y1(k) y2 (k)] 822



Article

The two control designs are still based on the original model. The weighting parameters for the proposed are

And the corresponding state space model is then derived as x(k+1) = Ax(k) + BΔu(k) + CΔr(k+1)

Q j = diag{120 120 0 0 1 1}

with ⎡ 0.905 ⎢ ⎢0 ⎢0 A=⎢ ⎢0 ⎢ 0.905 ⎢ ⎣0

0 0.793 0 0 0 0.793

⎡ 0.015 ⎢ ⎢−0.058 ⎢1 B=⎢ ⎢0 ⎢ 0.015 ⎢⎣ −0.058

0.0 0 0 0 0.0 0

0.07 − 0.597 0 0 0.07 − 0.597

−0.077 ⎤ ⎥ 0.753 ⎥ ⎥ 0 ⎥ 1 ⎥ −0.077 ⎥ ⎥ 0.753 ⎦

0 0 0 0 1 0

0⎤ ⎥ 0⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ 1⎦

⎡0 ⎢ ⎢0 ⎢0 C=⎢ ⎢0 ⎢− 1 ⎢⎣ 0

(j = 1, 2, ..., 15)

Q = block diag{Q 1, Q 2 , ..., Q 15} Lj = diag{1 1}

(j = 1, 2, ..., 4)

L = block diag{L1, L 2 , ..., L4}

Qj and Q for Wang’s method are Q j = diag{1 1} 0 ⎤ ⎥ 0 ⎥ 0 ⎥ ⎥ 0 ⎥ 0 ⎥ −1⎥⎦

(j = 1, 2, ..., 15)

Q = block diag{Q 1, Q 2 , ..., Q 15} Lj = diag{1 1}

(j = 1, 2, ..., 4)

L = block diag{L1, L 2 , ..., L4}

Figure 3 shows the control results. It is obvious that when model/plant mismatch is encountered, Wang’s method exhibits

On the basis of Wang’s strategy, the state space variables are x(k) = [Δy1(k) Δy2 (k) Δu1(k−1) Δu 2(k−1) y1(k) y2 (k)]T

Δu(k) = [Δu1(k) Δu 2(k)] y(k) = [ y1(k) y2 (k)]

And the corresponding model is then derived as x(k+1) = Ax(k) + BΔu(k) y(k) = Cx(k)

with ⎡ 0.905 ⎢ ⎢0 ⎢0 A=⎢ ⎢0 ⎢ 0.905 ⎢ ⎣0

⎡ 0.015 ⎢ ⎢−0.058 ⎢1 B=⎢ ⎢0 ⎢ 0.015 ⎢⎣ −0.058

0 0.793 0 0 0 0.793

0.0 0 0 0 0.0 0

−0.077 ⎤ ⎥ 0.753 ⎥ ⎥ 0 ⎥ 1 ⎥ −0.077 ⎥ ⎥ 0.753 ⎦

0.07 − 0.597 0 0 0.07 − 0.597

0 0 0 0 1 0

0⎤ ⎥ 0⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ 1⎦

Figure 3. Closed-loop responses under model mismatch for example 2. Dotted line: Wang. Solid line: proposed.

large oscillation both in the output responses and in the input signals. It is seen that when the weights on the process states are chosen, this kind of oscillation can be reduced a lot. 5.2. Effects of Weight Parameters. This section will give a brief illustration of the effects of the weight parameters on the closedloop control performance. Generally speaking, the weights Q and L for MPC are considered as “design” parameters rather than “tuning” parameters because they are associated with the compromise between the reference tracking errors and the control actions. However, different choices of Q and L will indeed result in different control performances. Thus we will value whether the proposed still provide improved control performance under these conditions. To this end and in view of the fact that the values for the reference tracking errors and the control actions in both Q and L are chosen as 1 in the previous study. We will further illustrate some results based on different combinations of Q and L. For the first example, we give the control results based on the following four cases: Case 1: the weights on the reference tracking errors are 10 whereas the weights on the control actions are 1, i.e., Qj = diag{10 10}, Lj = diag{1 1} for Wang’s method.

⎡0 0 0 0 1 0⎤ C=⎢ ⎣ 0 0 0 0 0 1 ⎥⎦

Model/Plant Mismatch Case Test. In this section, the model/ plant mismatch case is studied. The real process is supposed to be ⎡ 0.021z −1 ⎤ −1 ⎥⎡ − 0.027 z ⎡ y (k ) ⎤ ⎢ ⎤ −1 ⎥⎢ u1(k) ⎥ ⎢ 1 ⎥ = ⎢ 1 − 1.055z ⎥⎢ ⎢ y (k)⎥ ⎢ −0.098z −1 ⎥ ⎣ 2 ⎦ ⎢ −1 ⎥⎣ u 2(k)⎦ 1.653 z ⎢⎣ 1 − 0.293z −1 ⎥⎦

The model/plant mismatch will have an impact on the process’s gain and time constant. The simulation parameters are P = 15, M = 4, μ = 0.95. 823



Article

Case 2: the weights on the reference tracking errors are 0.1 whereas the weights on the control actions are 1, i.e., Qj = diag{0.1 0.1}, Lj = diag{1 1} for Wang’s method. Case 3: the weights on the reference tracking errors are 1 whereas the weights on the control actions are 10, i.e., Qj = diag{1 1}, Lj = diag{10 10} for Wang’s method. Case 4: the weights on the reference tracking errors are 1 whereas the weights on the control actions are 0.1, i.e., Qj = diag{1 1}, Lj = diag{0.1 0.1} for Wang’s method. Note that for standard MPC performance index that only has the reference tracking errors and the control actions parts, such as NMSSMPC, case 1 will be almost the same with case 4 because the weights on the reference tracking errors are both 10 times the weights on the control actions. For the ideal value of min J = 0, the effects can be seen clearly because the right-hand side of J = 0 is zero. We can simply interpret this as we pay 10 times of attention to the reference tracking errors than that of the control actions. And the same applies for case 2 and case 3. However, it is not the case for the proposed because besides the reference tracking errors and the control actions, its performance index includes the weights on the changes of process outputs and inputs signals. This also tells us that if we increase the weight values that correspond to the reference tracking errors in Q, we should also increase the weight values that correspond to the process output changes in Q because they are also associated with the regulation of the reference tracking errors. Large values of weights on the reference tracking errors inQ and small values of weights on the process output changes in Q will contradict with each other because the former would like the reference tracking errors to decrease quickly but the latter is doing the opposite action. The control results under the above four conditions are illustrated in Figures 4−7, respectively, with the

Figure 5. Closed-loop control responses under parameter mismatch for Example 1 under case 2. Dotted line: Wang with Qj = diag{0.1 0.1}, Lj = diag{1 1}. Solid line: proposed with Qj = diag{0.5 0.5 0.3 0.3 0 0 0 0 0 0 0.1 0.1}, Lj = diag{1 1}.

Figure 6. Closed-loop control responses under parameter mismatch for example 1. under case 3. Dotted line: Wang with Qj = diag{1 1}, Lj = diag{10 10}. Solid line: proposed with Qj = diag{5 5 2 2 0 0 0 0 0 0 1 1}, Lj = diag{10 10}.

It can be seen that when the weights on the Qi matrices are smaller, Wang’s method gives better performance. However, the proposed also shows improved performance due to the fact that the process state variables are considered in the controller design. It is also interesting to see that the effect of the weight pair Qj = diag{10 10}, Lj = diag{1 1} is almost the same with that of Qj = diag{1 1}, Lj = diag{0.1 0.1} and Qj = diag{0.1 0.1}, Lj = diag{1 1} with that of Qj = diag{1 1}, Lj = diag{10 10} for Wang’s method, as previously mentioned. For example 2, we will try to give an extreme case of the simulation on the weights. Prior simulations show that too small or large weights on the reference tracking errors or the control actions will result in unstable performance for Wang’s method. Thus we give the control results based on the following four cases (the values are all chosen to be their limit values; i.e., any further increase or decrease will lead to unstable performance):

Figure 4. Closed-loop control responses under parameter mismatch for example 1 under case 1. Dotted line: Wang with Qj = diag{10 10}, Lj = diag{1 1}. Solid line: proposed with Qj = diag{50 50 20 20 0 0 0 0 0 0 10 10}, Lj = diag{1 1}.

corresponding Qi and Li matrices for the proposed chosen as the following: Case 1: Qj = diag{50 50 20 20 0 0 0 0 0 0 10 10}, Lj = diag{1 1} Case 2: Qj = diag{0.5 0.5 0.3 0.3 0 0 0 0 0 0 0.1 0.1}, Lj = diag{1 1} Case 3: Qj = diag{5 5 2 2 0 0 0 0 0 0 1 1}, Lj = diag{10 10} Case 4: Qj = diag{10 10 5 5 0 0 0 0 0 0 1 1}, Lj = diag{0.1 0.1} 824



Article

Figure 9. Closed-loop control responses under parameter mismatch for example 2 under case 6. Dotted line: Wang with Qj = diag{0.6 0.6}, Lj = diag{1 1}. Solid line: proposed with Qj = diag{120 100 0 0 0.6 0.6}, Lj = diag{1 1}.

Figure 7. Closed-loop control responses under parameter mismatch for example 1 under case 4. Dotted line: Wang with Qj = diag{1 1}, Lj = diag{0.1 0.1}. Solid line: proposed with Qj = diag{10 10 5 5 0 0 0 0 0 0 1 1}, Lj = diag{0.1 0.1}.

Case 5: the weights on the reference tracking errors are 100 whereas the weights on the control actions are 1, i.e., Qj = diag{100 100}, Lj = diag{1 1} for Wang’s method. Case 6: the weights on the reference tracking errors are 0.6 whereas the weights on the control actions are 1, i.e., Qj = diag{0.6 0.6}, Lj = diag{1 1} for Wang’s method. Case 7: the weights on the reference tracking errors are 1 whereas the weights on the control actions are 1.5, i.e., Qj = diag{1 1}, Lj = diag{1.5 1.5} for Wang’s method. Case 8: the weights on the reference tracking errors are 1 whereas the weights on the control actions are 0.1, i.e., Qj = diag{1 1}, Lj = diag{0.1 0.1} for Wang’s method. It is again seen from Figures 8−11 that the ensemble performance of the proposed is improved. The corresponding Qi and Li matrices for the proposed are as the following:

Figure 10. Closed-loop control responses under parameter mismatch for example 2 under case 7. Dotted line: Wang with Qj = diag{1 1}, Lj = diag{1.5 1.5}. Solid line: proposed with Qj = diag{120 100 0 0 1 1}, Lj = diag{1.5 1.5}.

Case 5: Qj = diag{1200 100 0 0 100 100}, Lj = diag{1 1} Case 6: Qj = diag{120 100 0 0 0.6 0.6}, Lj = diag{1 1} Case 7: Qj = diag{120 100 0 0 1 1}, Lj = diag{1.5 1.5} Case 8: Qj = diag{10 0 0 0 1 1}, Lj = diag{0.1 0.1} 5.3. Constrained Optimization Case. Example 3. For this example, we will briefly discuss the constraints dealing in the proposed MPC framework. The idea of dealing constraints is transformed into a simplex method for linear programming problem, which is previously mentioned for a linearized case by the authors.39 Consider the constrained optimization of a double integrating plant given by Wang and Young,36 the corresponding discrete-time transfer function of the double integrating plant is

Figure 8. Closed-loop control responses under parameter mismatch for example 2 under case 5. Dotted line: Wang with Qj = diag{100 100}, Lj = diag{1 1}. Solid line: proposed with Qj = diag{1200 100 0 0 100 100}, Lj = diag{1 1}.

y(k+1) = 2y(k) − y(k−1) + 0.5u(k) + 0.5u(k−1) 825



Article

simulation study, a step input disturbance with amplitude −10 is added to the process at time instant k = 20. The two control designs are still based on the original model. The weighting parameters for the proposed are Q = diag{0.4 0 0 1} L=1

The weighting parameters for Wang’s method are

Q=1 L=1 This shows that the proposed has other degrees of weighting on the changes of the state variables for improved performance, whereas Wang’s method cannot achieve this goal due to the design limitation. Figure 12 shows when the output constraints

Figure 11. Closed-loop control responses under parameter mismatch for example 2 under case 8. Dotted line: Wang with Qj = diag{1 1}, Lj = diag{0.1 0.1}. Solid line: proposed with Qj = diag{10 0 0 0 1 1}, Lj = diag{0.1 0.1}.

And the state space model is given by Δx m(k + 1) = A mΔx m(k) + BmΔu(k) Δy(k + 1) = CmΔx m(k + 1)

where ⎡ 2 −1 0.5⎤ ⎢ ⎥ A m = ⎢1 0 0 ⎥ ⎣0 0 0 ⎦

⎡ 0.5⎤ ⎢ ⎥ Bm = ⎢ 0 ⎥ ⎣1 ⎦

Cm = [1 0 0]

And the NMSS model based on it is Figure 12. Closed-loop responses with output constraints for example 3 under model/plant match case. Dotted line: Wang. Solid line: proposed.

x(k+1) = Ax(k) + BΔu(k) y(k) = Cx(k)

are considered. It is clearly seen that both methods can deal well with the required constraints. However, the proposed gives the improved performance because it considers the regulation of the process state variables (the element 0.4 in Q). Figure 13 shows

with x(k) = [Δy(k) Δy(k−1) Δu(k−1) y(k)]T and ⎡ 2 − 1 0.5 0 ⎤ ⎢ ⎥ 1 0 0 0⎥ A=⎢ ⎢0 0 0 0⎥ ⎢⎣ ⎥ 2 − 1 0.5 1 ⎦

⎡ 0.5⎤ ⎢ ⎥ 0 B=⎢ ⎥ ⎢1 ⎥ ⎢⎣ ⎥⎦ 0.5

C = [0 0 0 1]

The state space model based on the proposed structure is x(k+1) = Ax(k) + BΔu(k) + CΔr(k+1)

with x(k) = [Δy(k) Δy(k−1) Δu(k−1) e(k)]T and ⎡ 2 −1 0.5 0 ⎤ ⎢ ⎥ 1 0 0 0⎥ A=⎢ ⎢0 0 0 0⎥ ⎢⎣ ⎥ 2 −1 0.5 1 ⎦

⎡ 0.5⎤ ⎢ ⎥ 0 B=⎢ ⎥ ⎢1 ⎥ ⎢⎣ ⎥⎦ 0.5

⎡0⎤ ⎢ ⎥ 0 C=⎢ ⎥ ⎢0⎥ ⎢⎣ ⎥⎦ −1

Constrained Optimization under No Model/Plant Mismatch. As has been stated by Wang and Young36 that this “toy” example is chosen with the intention of making the comparisons more transparent. To alleviate the computation burden, we give a onestep optimization of the two methods, that is, the prediction horizon P = 1 and the control horizon M = 1 are for the two methods. The smoothing factor is chosen to be 0.95. In the

Figure 13. Closed-loop responses with input constraints for example 3 under model/plant match case. Dotted line: Wang. Solid line: proposed. 826



Article

Figure 14 shows the responses of output and input for case 1. It is seen that both output and input signals of Wang’s method oscillate a lot; in addition, the required upper and lower limits cannot be guaranteed. In contrast, the responses of the proposed are steadier and the output does not violate the two limits. In Figure 15, it is clearly seen that Wang’s method results in the unacceptable responses. The output response and the corresponding input signal oscillate all the time, together with the two output limits being continuously broken. Figure 16 shows the

Figure 14. Closed-loop responses with output constraints for example 3 under model/plant mismatch case 1. Dotted line: Wang. Solid line: proposed.



the input constraints case. It is seen that by considering the impact of the process state variables, the performance can also be improved. Generally, the output response will tend to be more steadier if the weights are considered. Constrained Optimization under Model/Plant Mismatch. In this section, model/plant mismatch cases are studied. The real process is supposed to be the following three cases, respectively.

Figure 17. Closed-loop responses with output constraints (−15 ≤ y ≥ +1) for example 3 under model/plant mismatch case 1. Dotted line: Wang with Q = 1, L = 0.1. Solid line: proposed with Q = diag{0.4 0 0 1}, L = 0.1.

Case 1: y(k+1) = 2.04y(k) − 1.02y(k−1) + 0.4u(k) + 0.416u(k−1) Case 2: y(k+1) = 2.1y(k) − 1.05y(k−1) + 0.4u(k) + 0.416u(k−1) Case 3: y(k+1) = 1.96y(k) − 0.98y(k−1) + 0.8u(k) + 0.808u(k−1) The three comparisons are all studied in the case of output constraints, where the target is to maintain the output within the limits of −40 and +32. The control designs of the two methods are still based on the original process model. The prediction horizon, control horizon, smoothing factor, and the weighting parameters remain the same as for the model/plant match case.

responses for case 3. From this figure, we draw the conclusion that when the limits are not violated, placing weight on the state variables during the controller design can smooth the closed-loop responses, which in turn results in more acceptable performance. Effects of Weight Parameters under Model/Plant Mismatch. In this section, we will also illustrate the effect of the weights on the control performance through the constrained optimization. 827



Article

violates the upper limit when it tries to regulate the output to its steady value. In the case of the proposed, it successfully maintains its output within the two limits after breaking the lower limit for the first time. Figure 19 shows the control results when the limits are changed to −3 and +1, respectively. It can be seen that Wang’s method violates the limits three times (two for the lower limit and one for the upper limit) whereas the proposed violates the limits twice (one for the lower limit and one for the upper limit). If we change the weighting parameter for the proposed as Q = diag{0.7 0 0 1}, we will get the response shown in Figure 20,

This time we keep the weight on the reference tracking error unchanged and adjust the weight on the control action for better performance. Here we take the model/plant mismatch case 1 from the above constrained optimization as the illustration. When the output limit range is shortened, the control performance tends to become unstable. Thus for better and stable control performance, we select a new value of L = 0.1. Through the following simulations, the weighting parameters for the proposed are all chosen as Q = diag{0.4 0 0 1}. Figure 17 shows the performances of the two methods when the two output limits are −15 and +1, respectively. It can be seen that the output of Wang’s method violates the lower limit whereas the proposed successfully maintains its output within the limits. Figure 18

Figure 20. Closed-loop responses with output constraints (−3 ≤ y ≥ +1) for example 3 under model/plant mismatch case 1. Dotted line: Wang with Q = 1, L = 0.1. Solid line: proposed with Q = diag{0.7 0 0 1}, L = 0.1. Figure 18. Closed-loop responses with output constraints (−4 ≤ y ≥ +1) for example 3 under model/plant mismatch case 1. Dotted line: Wang with Q = 1, L = 0.1. Solid line: proposed with Q = diag{0.4 0 0 1}, L = 0.1.

in which the limits are only violated once. This shows that the proposed provides improved control performance.

6. CONCLUSION In this work, a MPC design based on a partial decoupling extended state space model (PD-ESS) structure is proposed, in which the measured input and output variables, their past values together with the defined output errors are chosen as the state variables. The design offers the advantage of incorporating its cost function to include the changes of the process state variables. As a result, closed-loop performance for tracking performance and disturbance rejection under model/plant mismatch is improved as demonstrated by three examples.

■

AUTHOR INFORMATION

Corresponding Author

*Tel: +852 2358 7139. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

Figure 19. Closed-loop responses with output constraints (−3 ≤ y ≥ +1) for example 3 under model/plant mismatch case 1. Dotted line: Wang with Q = 1, L = 0.1. Solid line: proposed with Q = diag{0.4 0 0 1}, L = 0.1.

ACKNOWLEDGMENTS Part of this project was supported by State Key Program of National Natural Science of China (Grant No. 61134007), the Special Fund for Basic Research on Scientific Instruments of National Natural Science of China (Grant No. 61227005), National Natural Science Foundation of China (Grant Nos. 61273101, 61203025, 61074030, 61104058), National 973 Program (Grant Nos. 2009CB320602, 2012CB821204), key innovation team program of Zhejiang Province of China: (Grant No. 2009R50019), and China

shows the results when the limits are changed to −4 and +1, respectively. It can be seen that when the limit range is shortened, performances of both methods deteriorate. By observing the detailed responses of the two methods, one sees that both methods violate the lower limit. However, Wang’s method also 828



Article

(24) Chotai, A.; Young, P. C.; Mckenna, P. G.; Tych, W. Proportionalintegral-plus (PIP) design for delta operator systems: Parts 1 and 2. Int. J. Control, 1998, 70, 123−147, 149−168. (25) Exadaktylos, V.; Taylor, C. J. Multi-objective performance optimization for model predictive control by goal attainment. Int. J. Control 2010, 83, 1374−1386. (26) Gonzalez, A. H.; Adam, E. J.; Marcovecchio, M. G.; Odloak, D. Application of an extended IHMPC to an unstable reactor system: Study of feasibility and performance. J. Process Control 2011, 21 (10), 1493− 1503. (27) Zhang, R. D.; Xue, A. K.; Wang, S. Q.; Ren, Z. Y. An improved model predictive control approach based on extended non-minimal state space formulation. J.Process Control 2011, 21 (8), 1183−1192. (28) Pannocchia, G.; Rawlings, J. Disturbance Models for offset-free Model Predictive Control. AIChE J. 2003, 39, 262−287. (29) Taylor, C. J.; Chotai, A.; Young, P. C. State space control system design based on non-minimal state variable feedback: further generalization and unification results. Int. J. Control 2000, 73, 1329−1345. (30) Taylor, C. J.; Chotai, A.; Young, P. C. Design and application of PIP controllers: robust control of the IFAC93 benchmark. Trans. Inst. Measurement Control 2001, 23, 183−200. (31) Taylor, C. J.; McCabe, A. P.; Young, P. C.; Chotai, A. Proportional-integral-plus (PIP) control of the gasifier benchmark system, Proceedings of the Institute of Mechanical Engineers Part I. J. Syst. Control Eng. 2000, 214, 469−480. (32) Grossmann, C.; Ströhlein, G.; Morari, M.; Morbidelli, M. Optimizing model predictive control of the chromatographic multicolumn solvent gradient purification (MCSGP) process. J. Process Control 2010, 20 (5), 618−629. (33) Young, P. C.; Lee, M. J.; Chotai, A.; Tych, W.; Chalabi, Z. S. Modelling and PIP control of a glasshouse micro-climate. Control Eng. Practice 1994, 2, 591−604. (34) Young, P. C.; Willems, J. C. An approach to the linear multivariable servomechanism problem. Int. J. Control 1972, 15, 961− 975. (35) Wang, C.; Young, P. C. Direct digital control by input-output state variable feedback: theoretical background. Int. J. Control 1988, 747, 97− 109. (36) Wang, L.; Young, P. C. An improved structure for model predictive control using non-minimal state space realization. J. Process Control 2006, 16, 355−371. (37) Lubyben, W. L. Simple method for tuning SISO controllers in multivariable system. Ind. Eng. Chem. Process Des. Dev. 1986, 25, 654− 660. (38) Zhang, R. D.; Xue, A. K.; Wang, S. Q.; Zhang, J. M.; Gao, F. R. Partially decoupled approach of extended state space predictive functional control for MIMO processes. J. Process Control 2012, 22, 837−851. (39) Zhang, R. D.; Xue, A. K.; Wang, S. Q. Dynamic modeling and nonlinear predictive control based on partitioned model and nonlinear optimization. Ind. Eng. Chem. Res. 2011, 50 (13), 8110−8121.

Postdoctoral Science Foundation (Grant Nos. 2012M511367, 2012M511368).

■

REFERENCES

(1) Cutler, C. R.; Ramaker, B. L. Dynamic matrix controla computer control algorithm. Proceedings of the Joint Automatic Control Conference, San Francisco; University of Michigan: Ann Arbor, MI, 1980; wp5-B, pp 1−6. (2) Garcia, C. E.; Morshedi, A. M. Quadratic programming solution to dynamic matrix control (QDMC). Chem. Eng. Commun. 1986, 46, 73− 87. (3) Clarke, D. W.; Mohtadi, C.; Tuffs, P. S. Generalized predictive control. Part 1: The basic algorithm. Part 2: Extensions and interpretations. Automatica 1987, 23, 137−160. (4) Ricker, N. L. Model predictive control with state estimation. Ind. Eng. Chem. Res. 1990, 29 (3), 569−577. (5) Balchen, J. G.; Ljungguist, D.; Strand, S. State-space predictive control. Chem. Eng. Sci. 1992, 47 (4), 787−807. (6) Muske, K.; Rawlings, J. B. Linear model predictive control of unstable processes. J. Process Control 1993, 3, 85−96. (7) Scokaert, P. O.; Rawlings, J. B. Constrained linear quadratic regulation. IEEE Trans. Automatic Control 1998, 43, 1163−1169. (8) Vogel, E. F.; Downs, J. J. Industrial experience with state-space model predictive control. In Proceedings of the sixth international conference on chemical process control; Rawlings, J. B., Ogunnaike, B. A., Eaton, J. W., Eds.; Elsevier: Amsterdam, 2001; Vol. 98 pp , 438−442. (9) Kadali, R.; Huang, B.; Rossiter, A. A data driven subspace approach to predictive controller design. Control Engineering Practice 2003, 11, 261−278. (10) Prakash, J.; Patwardhan, S. C.; Shah, S. L. State estimation and nonlinear predictive control of autonomous hybrid system using derivative free state estimators. J. Process Control 2010, 20 (7), 787−799. (11) Scheu, H.; Marquardt, W. Sensitivity-based coordination in distributed model predictive control. J. Process Control 2011, 21 (5), 715−728. (12) Lauri, D.; Rossiter, J. A.; Sanchis, J.; Martínez, M. Data-driven latent-variable model-based predictive control for continuous processes. J. Process Control 2010, 20 (10), 1207−1219. (13) Oblak, S.; Skrjanc, I. Continuous-time Wiener-model predictive control of a pH process based on a PWL approximation. Chem. Eng. Sci. 2010, 65 (5), 1720−1728. (14) Dubljevic, S. Model predictive control of Kuramoto−Sivashinsky equation with state and input constraints. Chem. Eng. Sci. 2010, 65 (15), 4388−4396. (15) Jacob, N. C.; Dhib, R. Unscented Kalman filter based nonlinear model predictive control of a LDPE autoclave reactor. J. Process Control 2011, 21 (9), 1332−1344. (16) Shead, L. R. E.; Muske, K. R.; Rossiter, J. A. Conditions for which linear MPC converges to the correct target. J. Process Control 2010, 20 (10), 1243−1251. (17) Brian Froisy, J. Model predictive control-Building a bridge between theory and practice. Comput. Chem. Eng. 2006, 30 (12), 1426− 1435. (18) Mayne, D. Q.; Rawlings, J. B.; Rao, C. V.; Scokaert, P.O. M. Constrained model predictive control: stability and optimality. Automatica 2000, 36, 789−814. (19) Qin, S. J.; Badgwell, T. A. A survey of industrial model predictive control technology. Control Eng. Practice 2003, 11, 733−764. (20) Rawlings, J. B. Tutorial overview of model predictive control. IEEE Control Systems Mag. 2000, 20, 38−52. (21) Ricker, N. L. Model-predictive control: state of the art. Proceedings of Fourth International Conference on Chemical Process Control, Padre Island, TX; Elsevier: Amsterdam, 1991; pp 271−296. (22) Tenny, M. J.; Rawlings, J. B.; Wright, S. J. Closed-loop behavior of nonlinear model predictive control. AIChE J. 2004, 50, 2142−2154. (23) Astrom, K. J.; Wittenmark, B. Computer Controlled Systems: Theory and Design, 3rd ed.; Prentice-Hall: Englewood Cliffs, NJ, 1997. 829


State Space Model Predictive Control Using Partial Decoupling and

Recommend Documents