Predictive Functional Control for Linear Systems under Partial Actuator

Article pubs.acs.org/IECR

Predictive Functional Control for Linear Systems under Partial Actuator Faults and Application on an Injection Molding Batch Process Ridong Zhang,†,‡ Renquan Lu,† Anke Xue,† and Furong Gao*,‡ †

Information and Control Institute, Hangzhou Dianzi University, Hangzhou 310018, People’s Republic of China Department of Chemical and Biomolecular Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

‡

ABSTRACT: Actuator faults commonly exist in process control systems. Due to these faults, the controller may not achieve the required target, so the control performance may degrade. In this paper, the fault-tolerant control is designed through a predictive functional control framework to deal with partial actuator faults and unknown disturbances, which exist widely in process control systems. Based on a new state space formulation of the process models, an improved predictive functional control scheme is proposed, where satisfactory closed-loop control performance is achieved even with unknown disturbances and actuator faults. With the actuator faults, the system becomes a process with parameter uncertainties. Hence, a sufficient condition that guarantees closed-loop robust stability is presented. Simulations are given to illustrate the feasibility and effectiveness of the proposed scheme.

1. INTRODUCTION Batch processing technologies have achieved significant progress in the past 10 years due to their advantages of manufacturing lowvolume and high-value products. At the same time, progress in the corresponding control technologies has also been reported, driven by the business of manufacturing.1 However, batch processes are often encountering strict operation conditions required by high productivity and safety issues, which may lead to system failures. In practice, it is often difficult for an actuator to achieve the desired position due to physical limitations such as friction, oil pollution, saturation, etc., which cause actuator faults. A fault in a chemical process system can cause significant damage or performance deterioration if it is not detected and corrected immediately. In view of this, fault-tolerant control (FTC) is an important issue to maintain closed-loop control performance. Recently, there have been quite a lot of studies of FTC on actuator faults. In ref 2, an estimation method is proposed for the backlash phenomenon of industrial actuator in automotive powertrains. Tao and Kokotovic also proposed an adaptive faulttolerant-control strategy for such actuator faults.3 In ref 4, Mosemann et al. showed the stability analysis of friction-type actuator faults. For the cogging and coulomb friction in permanent-magnet linear motors, Ahn et al. proposed a stateperiodic adaptive compensation method.5 To deal with the actuator saturation, Wu et al. proposed a theoretic feedback approach of stabilizing linear systems.6 In ref 7, an iterative learning control strategy is presented to cope with the actuator dead zone. There are mainly three cases of actuator faults in batch processes, i.e., partial degradation of the actuator, the outage case, and the stuck fault. Under the latter two cases, the control system can no longer be controlled and thus it is meaningless to design any controllers. Therefore, finding suitable controllers to cope with the first case is a meaningful issue among researchers. However, reported work on control system design for it is really © 2013 American Chemical Society

scarce. The only results focusing on actuator fault diagnosis and control available up to now are as follows. In ref 8, issues of fault diagnosis of batch processes are discussed. Wang et al. proposed a two-dimensional (2D) iterative learning reliable controller for batch processes with actuator faults using a particular 2D Fornasini−Marchsini model.9 In ref 10, a methodology of robust detection, isolation, and compensation of control actuator faults for particulate processes is proposed using robust feedback control. In ref 11, by assuming a delay dependent situation, a robust iterative learning fault-tolerant control is proposed to guarantee the stability of a closed-loop fault system in terms of linear matrix inequality (LMI). The same idea has also been extended to the fault-tolerant guaranteed cost control in ref 12. As the actual fault is not generally known, the design of a controller is based on the ideal assumption of perfect actuator action in response to the controller output. This causes model/ process mismatch for the designed controller.13 Though there have been some results on fault diagnosis and control of different kinds of processes,14−17 the issue of enhancing control performance under model/process mismatch is still an active topic. Iterative learning control has been widely used in batch processes; however, it is confined to the repetitive nature and is only suitable for improving tracking performance. In practice, many batch processes are slowly time-varying with nonrepetitive behavior and disturbances, which poses great difficulty for iterative learning methods. Recently, model predictive control (MPC) has been another choice for improving process control performance.18−20 However, improving MPC performance under model/process mismatch and actuator faults for batch processes to achieve the desired product quality is still an open issue.21−23 Also, to the best of the authors’ knowledge, few MPC Received: Revised: Accepted: Published: 723

April 26, 2013 November 28, 2013 December 19, 2013 December 19, 2013 dx.doi.org/10.1021/ie401329x | Ind. Eng. Chem. Res. 2014, 53, 723−731

Industrial & Engineering Chemistry Research

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For the controller output u(k) that will be fed to the actuator, let uF(k) denote the signal from the actuator that has failed. Then, the following failure model is adopted

results for batch processes with actuator failures have appeared up to the present. In this paper, a predictive functional control for batch processes with actuator faults is proposed. The batch process is first transformed into a new nonminimal state space transformation, and then the subsequent predictive functional control is designed to improve the output tracking performance under unknown disturbances for both normal operation and admissible faults. The advantages of the proposed strategy lie in the fact that the controller design can regulate both the system state and error variables, which cannot be achieved by traditional predictive functional control methods. The differences between the proposed control strategy and traditional robust design methods lie in the following. Traditional robust designs require that the information of the upper/lower limits of the actuator parameters faults should be known; thus subsequent robust designs can be adopted to yield a conservative control law that can guarantee robust stability under both nominal and uncertain situations. This strategy is passive because the design should consider both the nominal and the mismatch cases simultaneously, which in fact results in suboptimal control parameters for the closed-loop control systems. However, the proposed does not need the fault information and it guarantees robust stability through an active mechanism by regulating both the state and the output error dynamics in its controller design. In other words, the proposed control is a straightforward design of controllers to cope with process uncertainties. If the upper/lower limits of fault information are known, the proposed control strategy can also incorporate this information and design an active controller based on the robust criterion in this article, where both the state and the output error dynamics in its controller design can still be considered and tuned to result in an active controller. An injection velocity control is illustrated to show the feasibility and effectiveness of the proposed. The remainder of this paper is organized as follows. Section 2 deals with the problem formulation. In section 3, traditional FTC strategy is introduced. The main strategy of the proposed model transformation and controller design are detailed in section 4. In section 5, the effectiveness of the proposed is demonstrated through injection velocity control. Section 6 concludes the paper.

u F(k) = αu(k)

where (3) 0 < α̲ ≤ α ≤ α̅ The terms α̲ (α̲ ≤ 1) and α̅ (α̅ ≥ 1) are known scalars. Remark 2. The failure model described by eq 2 is widely adopted; see refs 25 and 26. Note that there are mainly three actuator failure situations: partial failure case, i.e., partial degradation of the actuator, the outage case, and the stuck fault case. Since the system is no longer controllable under the latter two failures, the first case is considered in this article. Remark 3. Since α > 0 denotes the partial failure case and α = 0 denotes the outage case, it can be seen that α > 0 is used in this paper. It is also noted that the parameter α is assumed to vary within a known range described by eq 3. Thus α̲ = α̅ or α = 1 corresponds to the normal case. Hence, a batch process with such actuator failures can be described by

x(k + 1) = Ax ̅ (k) + B̅ αu(k − d) y(k) = Cx̅ (k) + w(k)

(4)

The control objective is now to design a fault-tolerant control such that the output of the batch process tracks the set point as closely as possible under actuator failures and unknown disturbances.

3. TRADITIONAL FAULT-TOLERANT CONTROL In order to make the subsequent comparisons more clear, the idea of traditional FTC is first shown here. To facilitate the controller design, the process model is based on the nominal state space model. x(k + 1) = Ax ̅ (k) + B̅ u(k − d) y(k) = Cx̅ (k)

(5)

It is well-known that a tracking error integral action can eliminate the steady-state error. To obtain this, traditional FTC first defines the following tracking error:

2. PROBLEM FORMULATION For simplicity, it is assumed that the underlying process is singleinput single-output (SISO). When considering the operation around a set point, the batch process can be described through linearization as

e(k) = r(k) − y(k)

(6)

where r(k) is the expected output. Then the following dynamic integral of error model is introduced:

x(k + 1) = Ax ̅ (k) + B̅ u(k − d) y(k) = Cx̅ (k) + w(k)

(2)

(1)

xe(k + 1) = xe(k) + e(k)

where k denotes the time instant. x(k) ∈ Rn, y(k) ∈ R, and u(k) ∈ R represent the state, output, and input of the process at time instant k, respectively. d is the dead time and w(k) ∈ R is the measurement noise. {A̅ ,B̅ ,C̅ } are the process matrices of appropriate dimensions. Remark 1. Since {A̅ ,B̅ ,C̅ } are constant matrices, eq 1 is a description of linear processes. However, batch processes are generally nonlinear in many cases. In ref 24, a linear deviation model is considered to describe nonlinear batch processes. Similarly, if nonlinear batch processes are considered being operated around a set point, the proposed method in this paper is also applicable for nonlinear processes.

= xe(k) + r(k) − y(k) = xe(k) + r(k) − Cx̅ (k)

(7)

where xe(0) ≡ 0

and xe(k) denotes the integral of the tracking error. By combination of eq 7 with eq 5, the augmented model is derived as xI(k + 1) = AI x(k) + BI αu(k − d) + R(k) 724

(8)

dx.doi.org/10.1021/ie401329x | Ind. Eng. Chem. Res. 2014, 53, 723−731


Article

⎡ Δx (k)⎤ m ⎥ z(k ) = ⎢ ⎢⎣ e(k) ⎥⎦

where ⎡ x(k) ⎤ ⎥ xI(k + 1) = ⎢ ⎢⎣ xe(k)⎥⎦

(9)

In eq 17, 0 is a zero vector with appropriate dimension. Remark 4. Equation 16 is the newly derived state space model that will be used to design the corresponding predictive functional controller. This treatment facilitates the controller design to regulate both the process output error and the state changes, leading to improved control performance. 4.2. Predictive Functional Control. 4.2.1. Cost Function. Following the general idea of predictive functional control, the cost function of the proposed is

with ⎡ A̅ 0 ⎤ AI = ⎢ ⎥, ⎣−C̅ I ⎦

⎡ B̅ ⎤ BI = ⎢ ⎥ , ⎣0 ⎦

⎡0 ⎤ R (k ) = ⎢ ⎥ ⎣ r(k)⎦ (10)

For the above-described augmented system eq 9, the corresponding FTC can be designed to yield tracking performance under both normal and admissible faults.

P

J=

4. IMPROVED STATE SPACE PREDICTIVE FUNCTIONAL CONTROL 4.1. New State Space Model. For the nominal batch process described by eq 5, introduce a new state variable as

Q j = diag{qjx , qjx , ..., qjx , qju , qju , ..., qju , qje} 1

n

1

2

d

(20)

Remark 5. The variables qjx1, qjx2, ..., qjxn and qju1, qju2, ..., qjud are associated with the regulations of the process states and the process input increments, while qje is associated with the regulation of the process output tracking error. 4.2.2. Controller Derivation. As is known,27 predictive functional control action is associated with the process character and the base functions of the set point as follows:

(12)

with 0̅ 0̅ ··· 0̅ B̅ ⎤ ⎥ 0 0 ··· 0 0 ⎥ 1 0 ··· ⋮ 0 ⎥ ⎥ 0 1 0 0 ⋮⎥ ⎥ ⋮ ⋱ ⋱ ⋱ ⋮⎥ 0 ··· 0 1 0 ⎦⎥

N

u(k + i) =

∑ μj f j (i)

(21)

j=1

where u(k+i) is the input signal at time instant k + i, μj is the weight coefficient, f j(i) is the value of the base functions at sampling time k + i, and N is the degree of the base functions. Denote

Bm = [0 1 0 ··· 0]T

Cm = [C̅ 0 0 ··· 0 ]

(13)

Ti = [f1 (i), f2 (i), ..., fN (i)]

where 0̅ and 0̲ are the zero vectors with appropriate dimensions. Since the expected output is r(k), the output tracking error is therefore formulated as

e(k) = y(k) − r(k)

where i = 0, 1, ..., P − 1. Then eq 21 can be further expressed as

The formulation of the dynamics of this error variable can then be derived as

u(k + i) = Ti Υ

(15)

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

The combination of eqs 12 and 15 further leads to an augmented model as z(k + 1) = Az(k) + BΔu(k) + C Δr(k + 1)

(23)

The future predictions of state variables from sampling instant k are based on eq 16 and denote

e(k + 1) = e(k) + CmA mΔxm(k) + CmBmΔu(k) − Δr(k + 1)

(22)

Υ = [μ1 , μ2 , ..., μN ]T

(14)

(16)

where ⎡CmB ⎤ m ⎥, B=⎢ ⎢⎣ Bm ⎥⎦

2

1≤j≤P

Δxm(k + 1) = A mΔxm(k) + BmΔu(k)

⎡ Am 0⎤ ⎥, A=⎢ ⎢⎣Cm A m 1 ⎥⎦

(19)

where P is the prediction horizon; Qj is the symmetrical weight matrix with appropriate dimension and is specified as

Then the corresponding state space model is derived as

⎡ A̅ ⎢ ⎢ 0̲ ⎢ 0̲ Am = ⎢ ⎢ 0̲ ⎢ ⎢⋮ ⎣⎢ 0̲

∑ z T(k + j) Q jz(k + j) j=1

Δxm(k) = [Δx(k) Δu(k − 1) Δu(k − 2) ··· Δu(k − d)]T (11)

Δy(k) = CmΔxm(k)

(18)

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

(24)

Then the future state vector Z is related to the current state z(k) and the future control vector Υ via the following equation:

⎡0 ⎤ C=⎢ ⎥ ⎣−1⎦

Z = Fz(k) − Gu(k − 1) + ΦΥ + SΔR

(17)

and

(25)

with 725


Industrial & Engineering Chemistry Research ⎡B ⎤ ⎥ ⎢ ⎢ AB ⎥ G = ⎢ A2 B ⎥ ⎥ ⎢ ⎢⋮ ⎥ ⎢ P ⎥ ⎣ A B⎦

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

Article N

H=

∑ f j (0)hj j=1 N

Hu =

∑ f j (0)hu

(31)

j

j=1

(26a)

N

⎡C ⎢ ⎢ AC S = ⎢ A2 C ⎢ ⎢⋮ ⎢ P−1 ⎣A C

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

M=

0 0

⎡ BT0 ⎤ ⎢ ⎥ ⎢(AB − B)T0 + BT1 ⎥ ⎢ ⎥ 2 ⎢(A B − AB)T0 + (AB − B)T1 + BT2 ⎥ Φ=⎢ ⎥ ⎢⋮ ⎥ ⎢ P−1 ⎥ ⎢ ∑ (Ak B − Ak − 1B)T ⎥ + BT P−1−k P−1 ⎥ ⎢⎣ ⎦ k=1

∑ f j (0)mj j=1

Remark 6. Note that, although the proposed predictive functional control design follows the general idea of ref 27, the derived control law is a totally different one. This is because the proposed design adopts a newly developed state space model (see eq 16) that incorporates both the process state and error dynamics, which is different from traditional state space models. The resulting control is of course a new one that can facilitate the designer to consider both the process state and the error regulation. Remark 7. The optimal control vector shown in eq 28 does not contain any uncertain parameters. However, control performance enhancement is done through the choices of the parameters in Qj shown in eq 20 to allow the designer to consider regulating the process states, the process input increments, and the process output tracking error, which cannot be achieved by traditional predictive functional control due to its model limitations.

(26b)

(26c)

5. ROBUST STABILITY CRITERION FOR CLOSED-LOOP CONTROL SYSTEM It is noted that the fault-tolerant-control system design is actually the issue of controller design under system uncertainties. Thus a robust stability criterion will facilitate the designer to select appropriate control parameters to ensure a stable system. A robust stability condition for traditional state space predictive control has been proposed in ref 28. In this section, a robust criterion is also provided for the proposed control, which is summarized as follows: Theorem. For the batch process with uncertain actuator faults described by eq 4, if a fault-tolerant controller is designed using the nominal model eq 1 such that the following holds:

The cost function eq 19 can be expressed in vector form as J = ZTQZ

(27)

with Q = block diag{Q1, Q2, ..., QP}. The optimal control vector can be directly obtained by substituting eq 25 into eq 27 as Υ = −(ΦTQ Φ)−1ΦTQ (Fz(k) − Gu(k − 1) + SΔR ) (28)

Note that if we define μ1 = −(1, 0, ..., 0)(ΦTQ Φ)−1ΦT × Q (Fz(k) − Gu(k − 1) + SΔR )

σmax(ΔA) < −σmax(A − BKs)

= −h1z(k) + hu1u(k − 1) − m1ΔR

+

μ2 = −(0, 1, ..., 0)(ΦTQ Φ)−1ΦT × Q (Fz(k) − Gu(k − 1) + SΔR ) = −h2z(k) + hu2u(k − 1) − m2ΔR

λmin(W ) λmax (P)

(32)

where σmax(χ), λmin(χ), and λmax(χ) denote the maximum singular value and the minimum/maximum eigenvalues of χ, respectively. The matrices P and W are symmetric positive matrices defined in

(29)

⋮

(A − BKs)T P(A − BKs) − P = −W

μN = −(0, 0, ..., 1)(ΦTQ Φ)−1ΦT × Q (Fz(k) − Gu(k − 1) + SΔR )

(33)

and the matrices ΔA and Ks are given as ⎡ 0̃ ⎢ ⎢ 0̲ ⎢ 0̲ ⎢ ΔA = ⎢⋮ ⎢ ⎢⋮ ⎢ 0̲ ⎢ ⎣ 0̲

= −hN z(k) + huN u(k − 1) − mN ΔR

Then the control signal at current time instant is N

u(k) =

σmax 2(A − BKs) +

∑ μj f j (0) = −Hz(k) + Huu(k − 1) − M ΔR j=1

(30)

where 726

0̅ ⎤ ⎥ 0⎥ 0 0⎥ ⎥ ⋮ ⋮⎥ ⎥ ⋮ ⋮⎥ 0 0⎥ ⎥ CB ̅ ̅ α − CB ̅ ̅ 0⎦

0̅ 0̅ ··· 0̅ B̅ α − B̅ 0 0 ··· 0 0 0 0 ··· 0 ⋮ ⋮

··· ⋮

⋮ ⋮

··· ⋮

0 ⋮ ⋮ ⋮ 0 ··· ··· 0

(34)


Industrial & Engineering Chemistry Research Ks = (1, 0, ..., 0)(ΨTΦΦTQ Ψ)−1ΨTΦΦTQF

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then the proposed control holds robust stability for the uncertain system. Proof. This robust stability criterion follows the general idea of Lyapunov theory and we will demonstrate it as follows. We first derive the increment control action as follows. Equation 28 can be rewritten as

(35)

where 0̃ is a zero matrix of dimension n × n and ⎡B ⎢ ⎢ AB Ψ = ⎢ A2 B ⎢ ⎢⋮ ⎢ P−1 ⎣A B

··· 0 ⎤ ⎥ ··· 0 ⎥ ··· 0 ⎥ AB B ⎥ ⋮ ⋮ ⋱ ⋮⎥ ⎥ AP − 2 B AP − 3B ··· B ⎦ 0 B

0 0

ΦTQ [ΦΥ − Gu(k − 1)] = −ΦTQ (Fz(k) + SΔR ) (36)

(37)

From eqs 22, 23, and 26a−26c, it can be derived that

⎡ BT0 ⎤ ⎢ ⎥ ⎡B ⎤ ⎢(AB − B)T0 + BT1 ⎥ ⎢ ⎥ ⎢ ⎥ AB ⎥ ⎢ 2 ⎢(A B − AB)T0 + (AB − B)T + BT2 ⎥ 2 ΦΥ − Gu(k − 1) = ⎢ ⎥Υ − ⎢⎢ A B ⎥⎥u(k − 1) ⋮ ⎢ ⎥ ⎢⋮ ⎥ ⎢ P−1 ⎥ ⎢ P ⎥ ⎢ ∑ (Ak B − Ak − 1B)T ⎣ A B⎦ + BTP − 1 ⎥ P − 1 − k ⎢⎣ ⎥⎦ k=1

(38)

= ΨΔU

Substituting eq 45 into eq 46 and rearranging terms, we have

where

ΔV (z(k)) = z T(k)(A − BKs)T P(A − BKs)z(k)

ΔU = [Δu(k) Δu(k + 1) ··· Δu(k + P − 1)]T

+ z T(k)(A − BKs)T P ΔAz(k)

This shows that the following holds: ΦTQ ΨΔU = −ΦTQ (Fz(k) + SΔR )

+ z T(k)ΔATP(A − BKs)z(k)

(39)

+ z T(k)ΔATP ΔAz(k) − z T(k)Pz(k)

Premultiply both sides of eq 22 with ΨTΦ to get ΔU = −(ΨTΦΦTQ Ψ)−1ΨTΦΦTQ (Fz(k) + SΔR )

From eq 33, the first term and the last term on the righ-hand side of eq 47 say that

(40)

Let

z T(k)[(A − BKs)T P(A − BKs) − P]z(k)

Ks = (1, 0, ..., 0)(ΨTΦΦTQ Ψ)−1ΨTΦΦTQF T

T

−1 T

T

KR = (1, 0, ..., 0)(Ψ ΦΦ Q Ψ) Ψ ΦΦ QS

≤ −λmin(W ) z(k)

(41)

(48)

z T(k)(A − BKs)T P ΔAz(k) + z T(k)ΔATP(A − BKs)z(k)

(42)

≤ 2σmax(A − BKs)λmax (P) ΔA

When closed-loop stability is considered, the set point can be chosen to be ΔR = 0 without loss of generality. This shows that the proposed control law is Δu(k) = −Ksz(k)

2

The second and third terms yield the following equation:

Then the incremental control vector at time instant k is Δu(k) = −Ksz(k) − KR ΔR

z (k )

z T(k)ΔATP ΔAz(k) ≤ λmax (P) ΔA

(43)

(49)

2

z (k )

2

(50)

We can now see that ΔV (z(k)) ≤ z(k) 2 (− λmin(W ) + 2σmax(A − BKs)λmax (P) ΔA + λmax (P) ΔA 2 )

z(k + 1) = (A + ΔA)z(k) + BΔu(k) + C Δr(k + 1)

(51)

It is not difficult to see that if the following is satisfied (44)

−σmax(A − BKs) −

Substituting eq 43 into eq 44, we are now in a position to check the stability of the following closed-loop uncertain system:


λmin(W ) λmax (P)

< ΔA

(45)

Consider the Lyapunov function of the closed-loop system between the time instants k and k + 1.

< −σmax(A − BKs) +


λmin(W ) λmax (P) (52)

ΔV (z(k)) = V (z(k + 1)) − V (z(k)) = z T(k + 1)Pz(k + 1) − z T(k)Pz(k)

2

The fourth term will lead to the following result:

Using the same idea that is shown in section 4.1, it is directly seen that the uncertain system eq 4 is related to the nominal system eq 5 through the following:

z(k + 1) = (A − BKs)z(k) + ΔAz(k)

(47)

where the first term and last term in eq 52 are the two solutions to the following equation

(46) 727



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Figure 1. (a) Schematic diagram of injection molding machine. (b) Injection molding process: (a) filling, (b) packing/holding, (c) cooling, (d) mold open and part ejection.

6. ILLUSTRATION This section illustrates a case study of the proposed control of injection velocity control in injection molding. 6.1. Injection Molding Process.29−31 Figure 1 shows a schematic diagram of the injection molding machine and a corresponding injection molding process. The job of this process is to transform plastic granules into various products, which is operated through batch mode and mainly consists of three stages: filling, packing/holding, and cooling. The whole process is as follows. During the filling stage, high pressure in the hydraulic cylinder is exerted on the injection screw to force the plastic to be melted and pushed into the mold cavity. When the mold is completely or almost completely filled (see Figure 1b (subpanel a)), the packing/holding stage starts, where additional material is “packed” into the mold cavity to compensate for the shrinkage associated with cooling and solidification (see Figure 1b (subpanel b)). This stage will stop if the material at the mold gate is frozen and the material inside the mold is no longer influenced by the injection nozzle. The next stage is the cooling stage, where the material inside the mold is cooled until it is rigid enough to be ejected. At the same time, plastication is taking place inside the barrel with the material solidification, which causes the polymer to melt to the screw tip under the screw rotation (see Figure 1b (subpanel c)). The screw rotation will stop after a sufficient amount of melted polymer is generated

−λmin(W ) + 2σmax(A − BKs)λmax (P) ΔA + λmax (P) ΔA

2

=0

(53)

ΔV(z(k)) < 0 will be guaranteed, indicating the robust stability of the closed-loop control system. Note that the first inequality is always true because the lefthand side is always negative and ∥ΔA∥ ≥ 0. Thus the condition is reduced to ΔA = σmax(ΔA) < − σmax(A − BKs) +


λmin(W ) λmax (P) (54)

This completes the proof. Remark 8. It can be seen that if eq 54 is satisfied, the closedloop system will be stable. In practice, since the relationship between the control parameters and eq 33 is not clear, it is difficult to find any analytical selection rule of the P and W matrices. At this stage, this robust stability criterion can be used to test whether a designed control is robust or not. Because ΔA is known, and in combination of (A − BKs), if P and W satisfying eq 33 can be found to let eq 54 hold, then the closed-loop system is robust. 728



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on eq 5 in section 3. The control parameters of the two methods are the same with P = 4. The weighting elements on the output error are both chosen as 1. From Remark 5, it is seen that the proposed control can regulate the state changes in the controller design by choosing the appropriate weights, which is the advantage that TFTPFC cannot achieve. By observing the specific batch process model described by eq 55, it can be seen that the process output changes Δy(k) and Δy(k−1) and the process input change Δu(k − 1) can be weighted. In this simulation, the weights are chosen as 1, 0, and 0, respectively. 6.2. Constant Fault and Nonrepetitive Unknown Disturbance. In this case, different constant values of α are studied to value the performance of the proposed control scheme. A random white noise sequence with a standard deviation of 0.2 is added to the process output as the unknown disturbance. The following three cases are given:

in front of the screw. The mold opens until the material in it becomes rigid enough, and then the material is ejected (see Figure 1b (subpanel d)). Thus the current cycle is over and the machine is then ready for the next cycle. For high product quality, the critical process variables in each stage, such as the injection speed in the injection stage and packing pressure in the pressure stage, should be controlled with high precision during the batch process. Here, the injection velocity in the filling stage is considered since it has a significant impact on the quality of the final product associated with mechanical strength, deformation, and accuracy. This variable should be controlled to follow a given set point to maintain product quality. The injection velocity response to the proportional valve has been identified as an autoregressive model in refs 9 and 32. G (z ) =

1.69z + 1.419 z − 1.582z + 0.5916 2

(55)

case 1. α = 0.65 case 2. α = 0.45 case 3. α = 0.25

Assume there exists an unknown actuator fault α. In this study, the set-point profile takes the following forms: r(k) = 15 (for 1 ≤ k < 40) r(k) = 30 (for 51 ≤ k < 120)

The simulation results are shown in Figures 2−4. It is clear that, as the fault becomes severe, the performance of both methods deteriorates. However, the process output given by the proposed control scheme can still track the set point as quickly and smoothly as possible. TFTPFC gives the oscillatory responses. The input signals are shown in Figures 2b−4b, where the drastic input signals of TFTPFC are witnessed. It is demonstrated that, even in case of a relatively severe fault, the proposed can also give admissible control results.

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r(k) = 15 (for 120 ≤ k < 200)

Different cases of faults and unknown disturbance are illustrated and comparison is made with traditional fault-tolerant predictive functional control (TFTPFC) that is designed based

Figure 2. (a) Output responses under case 1. (b) Input signals under case 1.

Figure 3. (a) Output responses under case 2. (b) Input signals under case 2. 729



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Figure 4. Input signals under case 3.



Figure 7. (a) Output responses under case 6. (b) Input signals under case 6. 730



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6.3. Time-Varying Fault and Nonrepetitive Unknown Disturbance. This section follows the previous researchers that consider time-varying faults. In this case, different time-varying values of α are studied to value the performance of the proposed control scheme. The white noise sequence remains the same as that in section 6.2. The following three cases are considered: case 4. α = 0.5 + 0.4 sin(k) case 5. α = 0.5 + 0.1 sin(k) case 6. α = 0.5 + 0.01 sin(k) The control performance comparisons can be seen in Figures 5−7. Again, they show that the proposed control scheme provides improved control results even under the time-varying actuator faults. In summary, the proposed predictive functional control selects a new state space representation of the batch process and considers both the tracking error and the process state regulations in the design, which further improves control performance.

7. CONCLUSION In this work, a predictive functional control is designed for batch processes with partial actuator faults. The process has been transformed into a new state space model for further predictive functional control. Comparison results on injection velocity control with traditional predictive functional control shows that the control performance has been improved under both the constant and time-varying actuator faults.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +852-2358-7139. Fax: +852-2358-0054. 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 Hong Kong, Macao, and Taiwan Science & Technology Cooperation Program of China (Grant 2013DFH10120), National Natural Science Foundation of China (Grant 61273101, 61104058), and Guangzhou scientific and technological project (2012J5100032).

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REFERENCES

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Predictive Functional Control for Linear Systems under Partial Actuator

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