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Research Article
Model predictive control for a class of systems with isolated nonlinearity Jili Tao n, Yong Zhu, Qinru Fan Ningbo Institute of Technology, Zhejiang University, Ningbo 315100, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 24 December 2013 Received in revised form 20 March 2014 Accepted 26 March 2014 This paper was recommended for publication by Dr. Dong Lili
The paper is concerned with an overall convergent nonlinear model predictive control design for a kind of nonlinear mechatronic drive systems. The proposed nonlinear model predictive control results in the improvement of regulatory capacity for reference tracking and load disturbance rejection. The design of the nonlinear model predictive controller consists of two steps: the first step is to design a linear model predictive controller based on the linear part of the system at each sample instant, then an overall convergent nonlinear part is added to the linear model predictive controller to combine a nonlinear controller using error driven. The structure of the proposed controller is similar to that of classical PI optimal regulator but it also bears a set-point feed forward control loop, thus tracking ability and disturbance rejection are improved. The proposed method is compared with the results from recent literature, where control performance under both model match and mismatch cases are enlightened. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Overall convergent control strategy Model predictive control Mechatronic drive systems State space control
1. Introduction Mechanical engineering, electronics and intelligent computer control in the design and manufacturing of industrial products and processes lead to the field of mechatronics [1,2]. In many cases, the mechatronic system is a coupled and complex one. This is because the mechanical part of the system is often coupled with the electrical, thermodynamical, chemical or information processing part, which causes the difficulty of designing suitable control systems for the nonlinear characteristics. To overcome the bad impact of these nonlinearities, different control methods, for the control of position, speed or force of various mechatronic systems are provided based on various control design theories. These methods are generally designed to compensate for the system nonlinearities. However, they are either computationally demanding or requiring high processing capability of CPUs and the control performance greatly relies on the accuracy of the compensators. Typical examples are as follows. Sun proposed an optimal linear quadratic controller (LQ) [3], however, the output response is rather unsatisfactory since it oscillates a lot with undesirable overshoot/undershoot due to nonlinear coupling. Isermann and Raab proposed a compensation method for nonlinear static characteristics, however, it needs to design an inverse function compensator and suppose that the
n
Corresponding author. Tel.: þ 86 574 88130021. E-mail address: [email protected] (J. Tao).
nonlinear function has an inverse function [4]. In view of the above shortcomings, several model predictive control (MPC) techniques have been proposed. Rau and Schroder [5] proposed a linearized process model along the reference trajectory to cope with the nonlinearity for further predictive control design. However, the shortcomings lie in the fact that undesirable dynamic output response and control signal oscillations are resulted. Zhao et al. [6] proposed a nonlinear recursive predictive functional control (PFC), however, the control system structure is complex and time-consuming computation is resulted. The neural network iterative MPC proposed by Zhang et al. [7] further improves the control performance; however, it also faces the problems of complex structure and excessive computation. The most recent method can be seen in [8], where Zhang et al. assumed that the nonlinear coupling can be treated into a time-varying part and then designed a nonlinear adaptive extended state space model based predictive control method (NAESSPC). However, it is still a kind of approximation and may not be general in practice. It is known that state space model based predictive control can improve performance since the state information can be considered when designing the control systems. Recently, MPC based on state space models has attracted a lot of interest from researchers [9–15]. However, the existence of observers may cause numerical difficulty for traditional state space MPC. For nonlinear processes, a simple linear design of controller may not achieve the desirable control performance. On the other hand, for the implementation of practical controllers, linear design is indeed convenient. To facilitate the issues of nonlinear design and easy implementation,
http://dx.doi.org/10.1016/j.isatra.2014.03.019 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Tao J, et al. Model predictive control for a class of systems with isolated nonlinearity. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.019i
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iterative learning is a good choice. A lot of literature has shown that iterative learning MPC can provide improved performance. Following the iterative learning strategy [16–18], an overall convergent iterative learning state space MPC is proposed for the mechatronic drive systems [3–8]. Generally, the procedure of the proposed method consists of two steps: (i) The nonlinear part of the process is first ignored and a linear predictive controller is designed based on the linear part that is treated in to a nonminimal-like structure. (ii) Secondly, a nonlinear control part is added to the linear predictive controller to form an overall convergent nonlinear predictive control law based on the error between the real process and the linear system. A positive scalar decreasing sequence is introduced to ensure the convergence of the nonlinear predictive control law. The paper is organized as follows. Section 2 gives a brief description of the nonlinear process and its new state space model. Section 3 deals with the proposed MPC controller. The closed-loop control performance analysis is presented in Section 4. Section 5 details a comparison case study. Conclusion is in Section 6.
2. Process description and its presentation
Define " # eðkÞ zðkÞ ¼ ΔxðkÞ
ð6Þ
and combine Eqs. (3) and (5), a new process model is derived ~ ~ ~ ~ zðk þ 1Þ ¼ AzðkÞ þ bΔuðkÞ þ cΔrðk þ 1Þ þ DΔNLðyðkÞÞ where 1 A~ ¼ o~
" 0# 1 cb ck ~ ~ ~ ; b¼ ; c¼ ; D¼ 0 A o~ b k
cA
The process under study is taken from previous articles [6–8] and described by the following nonlinear state space model
ð7bÞ
are matrices of appropriate dimensions and o~ is a zero vector of appropriate dimension. The next step is to design a nonlinear predictive control method. Here the linear part of the derived model (see Eq. (7c)) is first used to design a linear predictive controller. Then a nonlinear part based on the error driven is added to the linear predictive controller. This two-step design method does not need the knowledge of the nonlinear part of the process. Thus the nonlinear function observer [5] is not needed and the structure of the control system is simpler. The linear part of the derived model (Eq. (7a)) will be used to design a linear predictive controller first. ~ ~ ~ zðk þ 1Þ ¼ AzðkÞ þ bΔuðkÞ þ cΔrðk þ 1Þ
2.1. Process description
ð7aÞ
ð7cÞ
3. Predictive controller design
xðk þ 1Þ ¼ AxðkÞ þbuðk dÞ þ k UNLðyðkÞÞ ð1Þ yðkÞ ¼ cT xðkÞ þ huðk dÞ T is the state vector with where xðkÞ ¼ x1 ðkÞ; x2 ðkÞ; ⋯; xn ðkÞ dimension n 1, A; b; cT ; h; k are matrices or constants of appropriate dimensions, k describes the coupling of the nonlinearity into the system. NL is a nonlinear function. 2.2. The treatment of the derived model Take uðk 1Þ; …; uðk dÞ as the system state variables and uðkÞ as the input only 0
xðk þ 1Þ ¼ AxðkÞ þbuðkÞ þ k NLðyðkÞÞ yðkÞ ¼ cxðkÞ where, xðkÞ ¼ ½xðkÞ ; uðk dÞ; uðk d þ1Þ; …; uðk 1Þ state variable, 2 3 A b o o UUU o 6 7 6o 0 1 0 UUU 07 6 7 6 A¼6⋮ ⋮ ⋱ ⋮7 7 6 7 4o 0 5 0 0
P
M
j¼1
j¼1
J ¼ ∑ zT ðk þ jÞQ j zðk þ jÞ þ ∑ λj ½Δuðk þ j 1Þ2 s:t: Δuðk þ jÞ ¼ 0 j Z M
ð8Þ
where P is the maximum prediction horizon. M is the control horizon, Q j is the symmetrical weighted matrix with dimension ðn þ d þ 1Þ ðn þ d þ 1Þ, λj is the weighted factor of control input increments, generally Q j is taken as n o Q j ¼ diag qj1 ; qj2;…; qjðn þ 1Þ ; 0; …; 0
1rjrP
ð2aÞ T
o
3.1. Cost function
UUU
b ¼ ½o T ; 0; …; 0; 1T
T
3.2. State prediction and controller design is the new
0 0
T
T
k ¼ ½k ; 0; 0; …; 0T
c ¼ ½c ; h; 0; …; 0
ð2bÞ
o and o are matrices with appropriate dimensions. Add the back shift operator to Eq. (2), then 0
Δxðk þ 1Þ ¼ AΔxðkÞ þ bΔuðkÞ þ k ΔNLðyðkÞÞ
ð3Þ
Define the expected output as rðkÞ, and the output tracking error is eðkÞ ¼ yðkÞ rðkÞ
ð4Þ
Combine Eqs. (2) and (4)
Based on Eq. (7) and define 2 3 2 3 2 3 zðk þ 1Þ ΔuðkÞ Δrðk þ 1Þ 6 7 6 7 6 7 6 zðk þ 2Þ 7 6 Δuðk þ 1Þ 7 6 Δrðk þ 2Þ 7 7; Δu ¼ 6 7; Δr ¼ 6 7 z¼6 6 7 6 7 6 7 ⋮ ⋮ ⋮ 4 5 4 5 4 5 zðk þ PÞ Δuðk þ M 1Þ Δrðk þ PÞ 2 2 3 b~ 0 6 6 7 6 A~ 6 A~ b~ 7 b~ 6 7 ; G ¼ F¼6 6 ~2 ~P 6 ⋮ 7 ⋮ ⋱ ⋮ 4 A ⋮A 4 5 P 1 P 2 PM ~ ~ ~ b A~ b ⋯ A~ b A~ 2 3 c~ 6 ~~ 7 c~ 07 6 Ac 7 S¼6 6 ⋮ ⋮ ⋱ ⋮7 4 5 P 1 P2 c~ A~ c~ ⋯ c~ A~ ~ ~ ~ Δε1 ¼ DΔNLðyðkÞÞ; Δε2 ¼ A~ DΔNLðyðkÞÞ þ DΔNLðyðk þ 1ÞÞ; …; P 1 P 2 ~ ~ ~ DΔNLðyðkÞÞ þ A~ DΔNLðyðk þ 1ÞÞ þ ⋯ þ DΔNLðyðk þ P 1ÞÞ ΔεP ¼ A~
0
eðk þ 1Þ ¼ eðkÞ þ cAΔxðkÞ þcbΔuðkÞ þ ck ΔNLðyðkÞÞ Δrðk þ 1Þ
ð5Þ
Δε ¼ ½Δε1T ; Δε2T ; …; ΔεP T T
Please cite this article as: Tao J, et al. Model predictive control for a class of systems with isolated nonlinearity. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.019i
ð9Þ
J. Tao et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Q ¼ Block diagfQ 1 ; Q 2 ; …; Q P g
ð10Þ
3.3. Remarks
L ¼ diagfλ1 ; λ2 ; …; λM g
ð11Þ
3
z ¼ FzðkÞ þ GΔu þ SΔr þ Δε
ð12Þ
The proposed controller needs the error between the actual process output and the model output to trim the action of the linear controller, and by introducing the convergence factor, the modified controller action will eventually converge to the desired one provided the actual process output can be measured, which is reasonable.
J ¼ zT Q z þ ΔuT LΔu
ð13Þ
4. Convergence analysis
Then the vector form of the output prediction and the cost function are
From ð∂J=∂ΔuÞ ¼ 0, the control law is Δu ¼ ðGT Q G þ LÞ 1 GT Q ½FzðkÞ þ SΔr þΔε
ð14Þ
Denote qT as the first row of ðGT Q G þ LÞ 1 GT Q , then the control increment is ΔuðkÞ ¼ ð1; 0; …; 0ÞΔu ¼ hzðkÞ fΔr qT Δε
ð15aÞ
where
From Eq. (18), it is seen that when k is becoming larger and larger, δðkÞ is becoming smaller and smaller, and the changes of Δε and Δu are becoming smaller and smaller. When Δεk -Δε, where Δε is the true value of Δεk , zk -zkm is derived, thus the optimal control law is derived. So the next step is to analyze the convergence of Δεk , if Δεk -Δε, then the control law is convergent. Lemma. [19] Consider the following recurrence algorithm
h ¼ ð1; 0; …; 0ÞðGT Q G þ LÞ 1 GT Q F ¼ ðK e ; K x1 ; …; K xn ; K d ; …; K 1 Þ f ¼ ð1; 0; …; 0ÞðGT Q G þ LÞ1 GT Q S ¼ ðf 1 ; f 2 ; …; f p Þ
αðkÞ ¼ αðk 1Þ þ δðkÞQ ðk; αðk 1Þ; ΦðkÞÞ ð15bÞ
Combine Eqs. (6) with (15) n
d
P
j¼1
j¼1
j¼1
ΔuðkÞ ¼ K e eðkÞ ∑ K xj Δxj ðkÞ ∑ K j Δuðk jÞ ∑ f j Δrðk þjÞ qT Δε
ð16Þ
k
n
d
P
j¼0
j¼1
j¼1
j¼1
uðkÞ ¼ K e ∑ eðjÞ ∑ K xj xj ðkÞ ∑ K j uðk jÞ ∑ f j rðk þ jÞ qT ε
ð17Þ From the above formulae (Eqs. (16) and (17)), it is seen that the control action includes the integration of output tracking error, the feedback of the states, past d-step control actions, and the feed forward of future P-step set-points. The integral part is to eliminate the static tracking error, the feedback of the states and past d-step control actions improve the dynamic performance of the system and the feed forward of future P-step set-points improves the system tracking performance. However, the system is nonlinear, thus Δε is not known, the following nonlinear part is added to the linear predictive controller Δε0 ¼ 0 Δu0 ¼ ðGT Q G þ LÞ 1 GT Q ½FzðkÞ þ SΔr þ Δε0
ΦðkÞ ¼ Aðαðk 1ÞÞΦðk 1Þ þ Bðαðk 1ÞÞeðkÞ
ð19aÞ
where αðkÞ is the variable to be estimated, define Ds ¼ fαjall the eigenvalues of AðαÞ are within the unit circleg:, DR is a connected open subset of Ds . InDR , the functions of Eq. (24) satisfy the following regular conditions C1–C5: C1: Q ðk; α; ΦÞ is Lipschitz continuous of α and Φ near ðα; ΦÞ, where α A DR ; Φ is arbitrary; and Q ðk; α; ΦÞ is also a continuous differentiable function of α and Φ. C2: For α A DR , functions AðαÞ; BðαÞ are Lipschitz continuous of α. C3: feðkÞg is the independent random vector sequence, which makes n
lim EfQ ðk; α; Φðk; αÞÞg-f ðαÞ; 8 α A DR :
k-1
where Φðk; αÞ is defined as follows ( Φðk; αÞ ¼ AðαÞΦðk 1; αÞ þ BðαÞeðkÞ Φð0; αÞ ¼ 0
ð19bÞ
p 1 C4: ∑1 k ¼ 1 δðkÞ ¼ 1; ∑k ¼ 1 ½δðkÞ o 1 ðp is a positive number; p 4 1Þ C5:δðkÞ is a positive scalar decreasing sequence, with 1 1 o1 lim δðkÞ ¼ 0; lim sup δðk 1Þ δðkÞ k-1 k-1
Δεk þ 1 ¼ Δεk þ δðkÞðzkm zk Þ Δuk þ 1 ¼ ðGT Q G þ LÞ 1 GT Q ½FzðkÞ þSΔr þ Δεk
ð18Þ
Where, Δε0 is the initial value of Δε and it can be seen that it is initially set to be zero, Δu0 is the initial value of Δu, superscript k denotes the value of k step.zkm is derived by substituting Δuk into Eq. 7a, zk is derived by substituting Δuk into Eq. (7c), δðkÞ is the convergent factor, it is a positive scalar decreasing sequence, δðkÞ A ð0Þ1, lim δðkÞ ¼ 0. When δðkÞ is properly chosen, the above k-1 recurrence Eq. (18) is convergent, here δðkÞ is chosen as δðkÞ ¼ 1=k to ensure the convergence of the control law. When zkm ¼ zk , the control law Δuk is the optimal one, actually, if jjΔεk þ 1 Δεk jj o a desired tolerance Δuk can be thought as the optimal control law. The algorithm is summarized as follows: (1) (2) (3) (4) (5)
Get the output of the system yðkÞ. Describe the system using the method of Section 2. Compute the prediction z using Eq. (12). Get the control action using Eq. (18). Return to step (1).
For any α A DR , define a stationary random function Q ðk; α; Φðk; αÞ, then the adjoint differential equation of Eq. (19) is d n αD ðτÞ ¼ f ðαD ðτÞÞ dτ
ð20Þ
where τ is the pseudo-time, τ ¼ ∑ki ¼ 1 δðkÞ, and f ¼ lim EfQ ðk; δ; k-1 Φðk; αÞÞg If a positive function of Eq. (20) VðαD Þ exists, with (d VðαD Þ r 0; 8 αD A DA dτ ð21Þ d VðαD Þ ¼ 0; 8 αD A Dc ; Dc A DA dτ D . Meantime, if αn is the overall Then, when k-1, αðkÞ-w:p:1 k-1 c asymptotic stable equilibruim point, thus when k-1, αðkÞ-αn with 1 probability, that is αðkÞ-w:p:1 αn . k-1 Theorem. For nonlinear mechatronic system (1), if its model is treated into the form of Eq. (7), the predictive control law is designed by Eq. (18), and the controller parameters satisfy the following conditions: δðkÞ is a positive scalar decreasing sequence, δðkÞ A ð0; 1Þ,
Please cite this article as: Tao J, et al. Model predictive control for a class of systems with isolated nonlinearity. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.019i
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lim δðkÞ ¼ 0, then the nonlinear predictive control law (18) is k-1 convergent.
While zkm ¼ FzðkÞ þ GΔuk þ SΔr þ Δε, then substitute it into Eq. (24)
Proof. From Eqs. (18) and (12)
f ðΔεðτÞÞ ¼ Δε ΔεðτÞ
Δεk þ 1 ¼ Δεk þ δðkÞðzkm zk Þ ¼ Δεk þ δðkÞðzkm FzðkÞ GΔuk SΔr Δεk Þ
where, Δε is the true value of Δε.
n
~ k þD ¼ FΔε
ð22aÞ
where, D;F~ are constant matrices at each recurrence step D ¼ δðkÞ½zkm FzðkÞ GΔuk SΔr F~ ¼ diagð1 δðkÞ; 1 δðkÞ; …; 1 δðkÞÞ
ð22bÞ
Define ~ ~ the eigenvalues of Fare within the unit circleg:. It is Ds ¼ fFjall seen that ~ ¼ 1 δðkÞ o 1 λk ðFÞ
k ¼ 1; …; p
ð23Þ n
which means DS is the whole plane, that is, DR ¼ DS ¼ R . From Eq. (22) and δðkÞ ¼ 1=k, all conditions C1–C5 can be satisfied. Thus the adjoint differential equation of Eq. (22) is n
o dΔεðτÞ n ¼ f ðΔεðτÞÞ ¼ lim E zkm FzðkÞ GΔuk SΔr ΔεðτÞ dτ k-1
ð24Þ
Define Δεn as the equilibruim Eq. (25), then it can be seen that
ð25Þ
point
of
differencing
Δεn ¼ Δε
ð26Þ
From Eq. (26), it is known that the equilibruim point of Eq. (25) is at the true value. The Lyapunov function of Eq. (25) is 1 ð27Þ V ΔεÞ ¼ ðΔε ΔεÞT ðΔε Δε 4 0 2 then d d VðΔεÞ ¼ ðΔε ΔεÞT ðΔε ΔεÞ ¼ ðΔε ΔεÞT ðΔε ΔεÞ r0 8 Δε dτ dτ ð28Þ where d VðΔεÞ ¼ 0; Δε ¼ Δεn ¼ Δε dτ
ð29Þ
Fig. 1. (a) Results for set-point tracking without model mismatch: output, (b) results for set-point tracking without model mismatch: state, (c) results for set-point tracking without model mismatch: control action, and (d) error convergence for set-point tracking without model mismatch.
Please cite this article as: Tao J, et al. Model predictive control for a class of systems with isolated nonlinearity. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.019i
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which means that there exists an invariant set DC ¼ fΔεg for the differential equation, and the attraction domain is the whole plane. Thus from the lemma, Δε-Δε.
5. Example In this section, comparison results are given to show the effectiveness of the proposed method. Note that the nonlinear adaptive state space predictive control (NAESSPC) method in Zhang, et al. [8] is the latest one and had demonstrated its superiority over many existing methods, thus is adopted here for comparison. The comparisons are based on the following two conditions: (1) there exists no model mismatch; (2) there exists model mismatch. Both tracking performance and disturbance rejection performance are studied. Consider the following open-loop unstable and non-minimum phase system with dead time [8]. " # # " x1 ðk þ 1Þ x1 ðkÞ 1:1053 0 ¼ x2 ðk þ 1Þ 0:01 0:8186 x2 ðkÞ 1 0 þ uðk 2Þ 2:3 arctan ð10yðkÞÞ 0:0858 1 " # x1 ðkÞ ð30Þ yðkÞ ¼ x2 ðkÞ ¼ ½0 1 x2 ðkÞ
5
5.1. No model mismatch 5.1.1. Tracking performance The above two controllers are compared under the same conditions. The control parameters are P ¼ 21; M ¼ 1; λ1 ¼ 0:001. The smoothing factor of reference trajectory for the two predictive controllers is μ ¼ 0:9. Fig. 1a–c gives the tracking performance of the above two controllers. It can be seen from Fig. 1 that Zhang's controller gives a relatively fast but drastic output tracking performance. The responses of the proposed are steadier and fast when approaching the steady state. The proposed controller provides reasonably satisfactory performance because of its unique design method. Fig. 1b–c shows the state and control action of the two controllers. It is shown that the state and control action of Zhang's controller are not satisfactory. The system states and control action of the proposed act more steadily.
5.1.2. Disturbance rejection The disturbance rejection is tested on the above two controllers. At time t ¼ 150 s, a step disturbance ε ¼ 10 is introduced to the process. The control parameters remain the same for the two controllers, as can be seen in Section 5.1.1. Fig. 1a–c shows the disturbance rejection of the two controllers. It is seen that each method has its own characteristics for disturbance rejection. The response of Zhang's controller is slower than the proposed. For the responses of state and control action, Zhang's controller gives the
Fig. 2. (a) Results for set-point tracking with model mismatch: output, (b) results for set-point tracking with model mismatch: state, and (c) results for set-point tracking with model mismatch: control action.
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drastic one while the proposed yields the smoother one. Fig. 1d shows the algorithm convergence, where the output error is eventually converging to zero.
5.2. Model mismatch case To further compare the control performance of the two methods, the model mismatch is introduced by changing some elements of the process matrix. In this case, one of the poles is changed from 0.8186 to 0.8586, resulting in a kind of model uncertainty. The control parameters remain the same as those in the model match case.
5.2.1. Tracking performance Fig. 2a–c shows the comparison results. It can be seen from Fig. 2a that Zhang's controller gives a relatively faster but later a slower and drastic response. The proposed controller gives the overall smoothing one.
5.2.2. Disturbance rejection Disturbance rejection is tested on the above two controllers. At time t ¼ 150 s, a step disturbance ε ¼ 10 is introduced to the process. Fig. 2a–c shows the responses of the two controllers. The proposed controller displays a satisfactory response. Zhang's controller gives a relatively slower output.
5.3. General cases of pole mismatch Here, a more general situation of the model/plant mismatch is studied. The two poles of the process are changed randomly. Two cases are generated as follows. Case 1. the two poles are changed from 1.1053 to 0.9913 and from 0.8186 to 0.8386, respectively. Case 2. the two poles are changed from 1.1053 to 1.2246 and from 0.8186 to 0.8167, respectively. The results are seen in Figs. 3a–4c. From these results, it is seen that the ensemble performance of both the set point tracking and disturbance rejection of the process is improved by the proposed MPC. The mean tracking error and the control input are also illustrated in Table 1. In all cases, the proposed controller yields the improvement of performance concerning tracking performance and disturbance rejection due to its anticipated capacity. 6. Conclusion An overall convergent model predictive control for nonlinear mechatronic drive systems is considered in this study. Control performance is compared for both model match and mismatch cases. In the case of the proposed control scheme, a two-step method of designing controller has been performed instead of the linearization method. Results have shown that the proposed method yields improved performance.
Fig. 3. (a) Results for set-point tracking under case 1: output, (b) results for set-point tracking under case 1: state, and (c) results for set-point tracking under case 1: input.
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Fig. 4. (a) Results for set-point tracking under case 2: output, (b) results for set-point tracking under case 2: state, and (c) results for set-point tracking under case 2: input.
Table 1 Statistical results under cases 1 and 2. Case
Item
NAESSPC
Proposed
Case 1
Mean absolute error Mean input
28.72 0.9
28.77 1.2
Case 2
Mean absolute error Mean input
6.15 6.03
7.27 8.41
Acknowledgment The work is supported by Zhejiang Provincial Natural Science Foundation of China (Q13F030023). References [1] Isermann R. Mechatronic systems: concepts and applications. Trans Inst Meas Control 2000;22(1):29–55. [2] Harashima F, Tomizuka M, Fukuda T. Mechatronics – What is it, why and how? An editorial IEEE/ASME Trans Mech 1996;1:1–4. [3] Sun ZQ. Computer control theory and applications. Beijing: Tsinghua University Press; 1989. [4] Isermann R, Raab U. Intelligent actuators. Automatica 1993;29:1315–31. [5] Rau M, Schroder D. Model predictive control with nonlinear state space models. In: IEEE international workshop on advanced motion control, Slovenia; 2002 p. 136–141. [6] Zhao HQ, Cao J, Li ZW, Liu YQ. Nonlinear predictive functional control of recursive subspace model using support vector machine. In: IEEE Proceedings of the Chinese conference on control and decision; 2008, p. 4909–4913.
[7] Zhang RD, Xue AK, Wang JZ, Wang SQ, Ren ZY. Neural network based iterative learning predictive control design for mechatronic systems with isolated nonlinearity. J Process Control 2009;19(1):68–74. [8] Zhang RD, Wang SQ, Xue AK, Ren ZY, Li P. Adaptive extended state space predictive control for a kind of nonlinear systems. ISA Trans 2009;48(4): 491–6. [9] Shakouri P, Ordys A, Askari MR. Adaptive cruise control with stop&go function using the state-dependent nonlinear model predictive control approach. ISA Trans 2012;51(5):622–31. [10] Zhang RD, Xue AK, Wang SQ, Zhang JM. An improved state space model structure and a corresponding predictive functional control design with improved control performance. Int J Control 2012;85(8):1146–61. [11] Das S, Das S, Pan I. Multi-objective optimization framework for networked predictive controller design. ISA Trans 2013;52(1):56–77. [12] Karlgaard CD, Shen HJ. Robust state estimation using desensitized divided difference filter. ISA Trans 2013;52(5):629–37. [13] Zhang RD, Gao FR. State space model predictive control using partial decoupling and output weighting for improved model/plant mismatch performance. Ind Eng Chem Res 2013;52:817–29. [14] Wang L, Young PC. An improved structure for model predictive control using non-minimal state space realization. J Process Control 2006;16:355–71. [15] Zhang RD, Xue AK, Wang SQ, Ren ZY. An improved model predictive control approach based on extended non-minimal state space formulation. J Process Control 2011;21(8):1183–92. [16] Zhang R, Wang S. Support vector machine based predictive functional control design for output temperature of coking furnace. J Process Control 2008;18(5): 439–48. [17] Elaissi I, Jaffel I, Taouali O, Messaoud H. Online prediction model based on the SVD–KPCA method. ISA Trans 2013;52(1):96–104. [18] Zhang RD, Xue AK, Wang SQ. Dynamic modeling and nonlinear predictive control based on partitioned model and nonlinear optimization. Ind Eng Chem Res 2011;50:8110–21. [19] Zhang R, Li P, Xue A, Jiang A, Wang S. A simplified linear iterative predictive functional control approach for chamber pressure of industrial coke furnace. J Process Control 2010;20(4):464–71.
Please cite this article as: Tao J, et al. Model predictive control for a class of systems with isolated nonlinearity. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.019i
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Research Article
Model predictive control for a class of systems with isolated nonlinearity Jili Tao n, Yong Zhu, Qinru Fan Ningbo Institute of Technology, Zhejiang University, Ningbo 315100, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 24 December 2013 Received in revised form 20 March 2014 Accepted 26 March 2014 This paper was recommended for publication by Dr. Dong Lili
The paper is concerned with an overall convergent nonlinear model predictive control design for a kind of nonlinear mechatronic drive systems. The proposed nonlinear model predictive control results in the improvement of regulatory capacity for reference tracking and load disturbance rejection. The design of the nonlinear model predictive controller consists of two steps: the first step is to design a linear model predictive controller based on the linear part of the system at each sample instant, then an overall convergent nonlinear part is added to the linear model predictive controller to combine a nonlinear controller using error driven. The structure of the proposed controller is similar to that of classical PI optimal regulator but it also bears a set-point feed forward control loop, thus tracking ability and disturbance rejection are improved. The proposed method is compared with the results from recent literature, where control performance under both model match and mismatch cases are enlightened. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Overall convergent control strategy Model predictive control Mechatronic drive systems State space control
1. Introduction Mechanical engineering, electronics and intelligent computer control in the design and manufacturing of industrial products and processes lead to the field of mechatronics [1,2]. In many cases, the mechatronic system is a coupled and complex one. This is because the mechanical part of the system is often coupled with the electrical, thermodynamical, chemical or information processing part, which causes the difficulty of designing suitable control systems for the nonlinear characteristics. To overcome the bad impact of these nonlinearities, different control methods, for the control of position, speed or force of various mechatronic systems are provided based on various control design theories. These methods are generally designed to compensate for the system nonlinearities. However, they are either computationally demanding or requiring high processing capability of CPUs and the control performance greatly relies on the accuracy of the compensators. Typical examples are as follows. Sun proposed an optimal linear quadratic controller (LQ) [3], however, the output response is rather unsatisfactory since it oscillates a lot with undesirable overshoot/undershoot due to nonlinear coupling. Isermann and Raab proposed a compensation method for nonlinear static characteristics, however, it needs to design an inverse function compensator and suppose that the
n
Corresponding author. Tel.: þ 86 574 88130021. E-mail address: [email protected] (J. Tao).
nonlinear function has an inverse function [4]. In view of the above shortcomings, several model predictive control (MPC) techniques have been proposed. Rau and Schroder [5] proposed a linearized process model along the reference trajectory to cope with the nonlinearity for further predictive control design. However, the shortcomings lie in the fact that undesirable dynamic output response and control signal oscillations are resulted. Zhao et al. [6] proposed a nonlinear recursive predictive functional control (PFC), however, the control system structure is complex and time-consuming computation is resulted. The neural network iterative MPC proposed by Zhang et al. [7] further improves the control performance; however, it also faces the problems of complex structure and excessive computation. The most recent method can be seen in [8], where Zhang et al. assumed that the nonlinear coupling can be treated into a time-varying part and then designed a nonlinear adaptive extended state space model based predictive control method (NAESSPC). However, it is still a kind of approximation and may not be general in practice. It is known that state space model based predictive control can improve performance since the state information can be considered when designing the control systems. Recently, MPC based on state space models has attracted a lot of interest from researchers [9–15]. However, the existence of observers may cause numerical difficulty for traditional state space MPC. For nonlinear processes, a simple linear design of controller may not achieve the desirable control performance. On the other hand, for the implementation of practical controllers, linear design is indeed convenient. To facilitate the issues of nonlinear design and easy implementation,
http://dx.doi.org/10.1016/j.isatra.2014.03.019 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Tao J, et al. Model predictive control for a class of systems with isolated nonlinearity. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.019i
J. Tao et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
iterative learning is a good choice. A lot of literature has shown that iterative learning MPC can provide improved performance. Following the iterative learning strategy [16–18], an overall convergent iterative learning state space MPC is proposed for the mechatronic drive systems [3–8]. Generally, the procedure of the proposed method consists of two steps: (i) The nonlinear part of the process is first ignored and a linear predictive controller is designed based on the linear part that is treated in to a nonminimal-like structure. (ii) Secondly, a nonlinear control part is added to the linear predictive controller to form an overall convergent nonlinear predictive control law based on the error between the real process and the linear system. A positive scalar decreasing sequence is introduced to ensure the convergence of the nonlinear predictive control law. The paper is organized as follows. Section 2 gives a brief description of the nonlinear process and its new state space model. Section 3 deals with the proposed MPC controller. The closed-loop control performance analysis is presented in Section 4. Section 5 details a comparison case study. Conclusion is in Section 6.
2. Process description and its presentation
Define " # eðkÞ zðkÞ ¼ ΔxðkÞ
ð6Þ
and combine Eqs. (3) and (5), a new process model is derived ~ ~ ~ ~ zðk þ 1Þ ¼ AzðkÞ þ bΔuðkÞ þ cΔrðk þ 1Þ þ DΔNLðyðkÞÞ where 1 A~ ¼ o~
" 0# 1 cb ck ~ ~ ~ ; b¼ ; c¼ ; D¼ 0 A o~ b k
cA
The process under study is taken from previous articles [6–8] and described by the following nonlinear state space model
ð7bÞ
are matrices of appropriate dimensions and o~ is a zero vector of appropriate dimension. The next step is to design a nonlinear predictive control method. Here the linear part of the derived model (see Eq. (7c)) is first used to design a linear predictive controller. Then a nonlinear part based on the error driven is added to the linear predictive controller. This two-step design method does not need the knowledge of the nonlinear part of the process. Thus the nonlinear function observer [5] is not needed and the structure of the control system is simpler. The linear part of the derived model (Eq. (7a)) will be used to design a linear predictive controller first. ~ ~ ~ zðk þ 1Þ ¼ AzðkÞ þ bΔuðkÞ þ cΔrðk þ 1Þ
2.1. Process description
ð7aÞ
ð7cÞ
3. Predictive controller design
xðk þ 1Þ ¼ AxðkÞ þbuðk dÞ þ k UNLðyðkÞÞ ð1Þ yðkÞ ¼ cT xðkÞ þ huðk dÞ T is the state vector with where xðkÞ ¼ x1 ðkÞ; x2 ðkÞ; ⋯; xn ðkÞ dimension n 1, A; b; cT ; h; k are matrices or constants of appropriate dimensions, k describes the coupling of the nonlinearity into the system. NL is a nonlinear function. 2.2. The treatment of the derived model Take uðk 1Þ; …; uðk dÞ as the system state variables and uðkÞ as the input only 0
xðk þ 1Þ ¼ AxðkÞ þbuðkÞ þ k NLðyðkÞÞ yðkÞ ¼ cxðkÞ where, xðkÞ ¼ ½xðkÞ ; uðk dÞ; uðk d þ1Þ; …; uðk 1Þ state variable, 2 3 A b o o UUU o 6 7 6o 0 1 0 UUU 07 6 7 6 A¼6⋮ ⋮ ⋱ ⋮7 7 6 7 4o 0 5 0 0
P
M
j¼1
j¼1
J ¼ ∑ zT ðk þ jÞQ j zðk þ jÞ þ ∑ λj ½Δuðk þ j 1Þ2 s:t: Δuðk þ jÞ ¼ 0 j Z M
ð8Þ
where P is the maximum prediction horizon. M is the control horizon, Q j is the symmetrical weighted matrix with dimension ðn þ d þ 1Þ ðn þ d þ 1Þ, λj is the weighted factor of control input increments, generally Q j is taken as n o Q j ¼ diag qj1 ; qj2;…; qjðn þ 1Þ ; 0; …; 0
1rjrP
ð2aÞ T
o
3.1. Cost function
UUU
b ¼ ½o T ; 0; …; 0; 1T
T
3.2. State prediction and controller design is the new
0 0
T
T
k ¼ ½k ; 0; 0; …; 0T
c ¼ ½c ; h; 0; …; 0
ð2bÞ
o and o are matrices with appropriate dimensions. Add the back shift operator to Eq. (2), then 0
Δxðk þ 1Þ ¼ AΔxðkÞ þ bΔuðkÞ þ k ΔNLðyðkÞÞ
ð3Þ
Define the expected output as rðkÞ, and the output tracking error is eðkÞ ¼ yðkÞ rðkÞ
ð4Þ
Combine Eqs. (2) and (4)
Based on Eq. (7) and define 2 3 2 3 2 3 zðk þ 1Þ ΔuðkÞ Δrðk þ 1Þ 6 7 6 7 6 7 6 zðk þ 2Þ 7 6 Δuðk þ 1Þ 7 6 Δrðk þ 2Þ 7 7; Δu ¼ 6 7; Δr ¼ 6 7 z¼6 6 7 6 7 6 7 ⋮ ⋮ ⋮ 4 5 4 5 4 5 zðk þ PÞ Δuðk þ M 1Þ Δrðk þ PÞ 2 2 3 b~ 0 6 6 7 6 A~ 6 A~ b~ 7 b~ 6 7 ; G ¼ F¼6 6 ~2 ~P 6 ⋮ 7 ⋮ ⋱ ⋮ 4 A ⋮A 4 5 P 1 P 2 PM ~ ~ ~ b A~ b ⋯ A~ b A~ 2 3 c~ 6 ~~ 7 c~ 07 6 Ac 7 S¼6 6 ⋮ ⋮ ⋱ ⋮7 4 5 P 1 P2 c~ A~ c~ ⋯ c~ A~ ~ ~ ~ Δε1 ¼ DΔNLðyðkÞÞ; Δε2 ¼ A~ DΔNLðyðkÞÞ þ DΔNLðyðk þ 1ÞÞ; …; P 1 P 2 ~ ~ ~ DΔNLðyðkÞÞ þ A~ DΔNLðyðk þ 1ÞÞ þ ⋯ þ DΔNLðyðk þ P 1ÞÞ ΔεP ¼ A~
0
eðk þ 1Þ ¼ eðkÞ þ cAΔxðkÞ þcbΔuðkÞ þ ck ΔNLðyðkÞÞ Δrðk þ 1Þ
ð5Þ
Δε ¼ ½Δε1T ; Δε2T ; …; ΔεP T T
Please cite this article as: Tao J, et al. Model predictive control for a class of systems with isolated nonlinearity. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.019i
ð9Þ
J. Tao et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Q ¼ Block diagfQ 1 ; Q 2 ; …; Q P g
ð10Þ
3.3. Remarks
L ¼ diagfλ1 ; λ2 ; …; λM g
ð11Þ
3
z ¼ FzðkÞ þ GΔu þ SΔr þ Δε
ð12Þ
The proposed controller needs the error between the actual process output and the model output to trim the action of the linear controller, and by introducing the convergence factor, the modified controller action will eventually converge to the desired one provided the actual process output can be measured, which is reasonable.
J ¼ zT Q z þ ΔuT LΔu
ð13Þ
4. Convergence analysis
Then the vector form of the output prediction and the cost function are
From ð∂J=∂ΔuÞ ¼ 0, the control law is Δu ¼ ðGT Q G þ LÞ 1 GT Q ½FzðkÞ þ SΔr þΔε
ð14Þ
Denote qT as the first row of ðGT Q G þ LÞ 1 GT Q , then the control increment is ΔuðkÞ ¼ ð1; 0; …; 0ÞΔu ¼ hzðkÞ fΔr qT Δε
ð15aÞ
where
From Eq. (18), it is seen that when k is becoming larger and larger, δðkÞ is becoming smaller and smaller, and the changes of Δε and Δu are becoming smaller and smaller. When Δεk -Δε, where Δε is the true value of Δεk , zk -zkm is derived, thus the optimal control law is derived. So the next step is to analyze the convergence of Δεk , if Δεk -Δε, then the control law is convergent. Lemma. [19] Consider the following recurrence algorithm
h ¼ ð1; 0; …; 0ÞðGT Q G þ LÞ 1 GT Q F ¼ ðK e ; K x1 ; …; K xn ; K d ; …; K 1 Þ f ¼ ð1; 0; …; 0ÞðGT Q G þ LÞ1 GT Q S ¼ ðf 1 ; f 2 ; …; f p Þ
αðkÞ ¼ αðk 1Þ þ δðkÞQ ðk; αðk 1Þ; ΦðkÞÞ ð15bÞ
Combine Eqs. (6) with (15) n
d
P
j¼1
j¼1
j¼1
ΔuðkÞ ¼ K e eðkÞ ∑ K xj Δxj ðkÞ ∑ K j Δuðk jÞ ∑ f j Δrðk þjÞ qT Δε
ð16Þ
k
n
d
P
j¼0
j¼1
j¼1
j¼1
uðkÞ ¼ K e ∑ eðjÞ ∑ K xj xj ðkÞ ∑ K j uðk jÞ ∑ f j rðk þ jÞ qT ε
ð17Þ From the above formulae (Eqs. (16) and (17)), it is seen that the control action includes the integration of output tracking error, the feedback of the states, past d-step control actions, and the feed forward of future P-step set-points. The integral part is to eliminate the static tracking error, the feedback of the states and past d-step control actions improve the dynamic performance of the system and the feed forward of future P-step set-points improves the system tracking performance. However, the system is nonlinear, thus Δε is not known, the following nonlinear part is added to the linear predictive controller Δε0 ¼ 0 Δu0 ¼ ðGT Q G þ LÞ 1 GT Q ½FzðkÞ þ SΔr þ Δε0
ΦðkÞ ¼ Aðαðk 1ÞÞΦðk 1Þ þ Bðαðk 1ÞÞeðkÞ
ð19aÞ
where αðkÞ is the variable to be estimated, define Ds ¼ fαjall the eigenvalues of AðαÞ are within the unit circleg:, DR is a connected open subset of Ds . InDR , the functions of Eq. (24) satisfy the following regular conditions C1–C5: C1: Q ðk; α; ΦÞ is Lipschitz continuous of α and Φ near ðα; ΦÞ, where α A DR ; Φ is arbitrary; and Q ðk; α; ΦÞ is also a continuous differentiable function of α and Φ. C2: For α A DR , functions AðαÞ; BðαÞ are Lipschitz continuous of α. C3: feðkÞg is the independent random vector sequence, which makes n
lim EfQ ðk; α; Φðk; αÞÞg-f ðαÞ; 8 α A DR :
k-1
where Φðk; αÞ is defined as follows ( Φðk; αÞ ¼ AðαÞΦðk 1; αÞ þ BðαÞeðkÞ Φð0; αÞ ¼ 0
ð19bÞ
p 1 C4: ∑1 k ¼ 1 δðkÞ ¼ 1; ∑k ¼ 1 ½δðkÞ o 1 ðp is a positive number; p 4 1Þ C5:δðkÞ is a positive scalar decreasing sequence, with 1 1 o1 lim δðkÞ ¼ 0; lim sup δðk 1Þ δðkÞ k-1 k-1
Δεk þ 1 ¼ Δεk þ δðkÞðzkm zk Þ Δuk þ 1 ¼ ðGT Q G þ LÞ 1 GT Q ½FzðkÞ þSΔr þ Δεk
ð18Þ
Where, Δε0 is the initial value of Δε and it can be seen that it is initially set to be zero, Δu0 is the initial value of Δu, superscript k denotes the value of k step.zkm is derived by substituting Δuk into Eq. 7a, zk is derived by substituting Δuk into Eq. (7c), δðkÞ is the convergent factor, it is a positive scalar decreasing sequence, δðkÞ A ð0Þ1, lim δðkÞ ¼ 0. When δðkÞ is properly chosen, the above k-1 recurrence Eq. (18) is convergent, here δðkÞ is chosen as δðkÞ ¼ 1=k to ensure the convergence of the control law. When zkm ¼ zk , the control law Δuk is the optimal one, actually, if jjΔεk þ 1 Δεk jj o a desired tolerance Δuk can be thought as the optimal control law. The algorithm is summarized as follows: (1) (2) (3) (4) (5)
Get the output of the system yðkÞ. Describe the system using the method of Section 2. Compute the prediction z using Eq. (12). Get the control action using Eq. (18). Return to step (1).
For any α A DR , define a stationary random function Q ðk; α; Φðk; αÞ, then the adjoint differential equation of Eq. (19) is d n αD ðτÞ ¼ f ðαD ðτÞÞ dτ
ð20Þ
where τ is the pseudo-time, τ ¼ ∑ki ¼ 1 δðkÞ, and f ¼ lim EfQ ðk; δ; k-1 Φðk; αÞÞg If a positive function of Eq. (20) VðαD Þ exists, with (d VðαD Þ r 0; 8 αD A DA dτ ð21Þ d VðαD Þ ¼ 0; 8 αD A Dc ; Dc A DA dτ D . Meantime, if αn is the overall Then, when k-1, αðkÞ-w:p:1 k-1 c asymptotic stable equilibruim point, thus when k-1, αðkÞ-αn with 1 probability, that is αðkÞ-w:p:1 αn . k-1 Theorem. For nonlinear mechatronic system (1), if its model is treated into the form of Eq. (7), the predictive control law is designed by Eq. (18), and the controller parameters satisfy the following conditions: δðkÞ is a positive scalar decreasing sequence, δðkÞ A ð0; 1Þ,
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lim δðkÞ ¼ 0, then the nonlinear predictive control law (18) is k-1 convergent.
While zkm ¼ FzðkÞ þ GΔuk þ SΔr þ Δε, then substitute it into Eq. (24)
Proof. From Eqs. (18) and (12)
f ðΔεðτÞÞ ¼ Δε ΔεðτÞ
Δεk þ 1 ¼ Δεk þ δðkÞðzkm zk Þ ¼ Δεk þ δðkÞðzkm FzðkÞ GΔuk SΔr Δεk Þ
where, Δε is the true value of Δε.
n
~ k þD ¼ FΔε
ð22aÞ
where, D;F~ are constant matrices at each recurrence step D ¼ δðkÞ½zkm FzðkÞ GΔuk SΔr F~ ¼ diagð1 δðkÞ; 1 δðkÞ; …; 1 δðkÞÞ
ð22bÞ
Define ~ ~ the eigenvalues of Fare within the unit circleg:. It is Ds ¼ fFjall seen that ~ ¼ 1 δðkÞ o 1 λk ðFÞ
k ¼ 1; …; p
ð23Þ n
which means DS is the whole plane, that is, DR ¼ DS ¼ R . From Eq. (22) and δðkÞ ¼ 1=k, all conditions C1–C5 can be satisfied. Thus the adjoint differential equation of Eq. (22) is n
o dΔεðτÞ n ¼ f ðΔεðτÞÞ ¼ lim E zkm FzðkÞ GΔuk SΔr ΔεðτÞ dτ k-1
ð24Þ
Define Δεn as the equilibruim Eq. (25), then it can be seen that
ð25Þ
point
of
differencing
Δεn ¼ Δε
ð26Þ
From Eq. (26), it is known that the equilibruim point of Eq. (25) is at the true value. The Lyapunov function of Eq. (25) is 1 ð27Þ V ΔεÞ ¼ ðΔε ΔεÞT ðΔε Δε 4 0 2 then d d VðΔεÞ ¼ ðΔε ΔεÞT ðΔε ΔεÞ ¼ ðΔε ΔεÞT ðΔε ΔεÞ r0 8 Δε dτ dτ ð28Þ where d VðΔεÞ ¼ 0; Δε ¼ Δεn ¼ Δε dτ
ð29Þ
Fig. 1. (a) Results for set-point tracking without model mismatch: output, (b) results for set-point tracking without model mismatch: state, (c) results for set-point tracking without model mismatch: control action, and (d) error convergence for set-point tracking without model mismatch.
Please cite this article as: Tao J, et al. Model predictive control for a class of systems with isolated nonlinearity. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.019i
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which means that there exists an invariant set DC ¼ fΔεg for the differential equation, and the attraction domain is the whole plane. Thus from the lemma, Δε-Δε.
5. Example In this section, comparison results are given to show the effectiveness of the proposed method. Note that the nonlinear adaptive state space predictive control (NAESSPC) method in Zhang, et al. [8] is the latest one and had demonstrated its superiority over many existing methods, thus is adopted here for comparison. The comparisons are based on the following two conditions: (1) there exists no model mismatch; (2) there exists model mismatch. Both tracking performance and disturbance rejection performance are studied. Consider the following open-loop unstable and non-minimum phase system with dead time [8]. " # # " x1 ðk þ 1Þ x1 ðkÞ 1:1053 0 ¼ x2 ðk þ 1Þ 0:01 0:8186 x2 ðkÞ 1 0 þ uðk 2Þ 2:3 arctan ð10yðkÞÞ 0:0858 1 " # x1 ðkÞ ð30Þ yðkÞ ¼ x2 ðkÞ ¼ ½0 1 x2 ðkÞ
5
5.1. No model mismatch 5.1.1. Tracking performance The above two controllers are compared under the same conditions. The control parameters are P ¼ 21; M ¼ 1; λ1 ¼ 0:001. The smoothing factor of reference trajectory for the two predictive controllers is μ ¼ 0:9. Fig. 1a–c gives the tracking performance of the above two controllers. It can be seen from Fig. 1 that Zhang's controller gives a relatively fast but drastic output tracking performance. The responses of the proposed are steadier and fast when approaching the steady state. The proposed controller provides reasonably satisfactory performance because of its unique design method. Fig. 1b–c shows the state and control action of the two controllers. It is shown that the state and control action of Zhang's controller are not satisfactory. The system states and control action of the proposed act more steadily.
5.1.2. Disturbance rejection The disturbance rejection is tested on the above two controllers. At time t ¼ 150 s, a step disturbance ε ¼ 10 is introduced to the process. The control parameters remain the same for the two controllers, as can be seen in Section 5.1.1. Fig. 1a–c shows the disturbance rejection of the two controllers. It is seen that each method has its own characteristics for disturbance rejection. The response of Zhang's controller is slower than the proposed. For the responses of state and control action, Zhang's controller gives the
Fig. 2. (a) Results for set-point tracking with model mismatch: output, (b) results for set-point tracking with model mismatch: state, and (c) results for set-point tracking with model mismatch: control action.
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drastic one while the proposed yields the smoother one. Fig. 1d shows the algorithm convergence, where the output error is eventually converging to zero.
5.2. Model mismatch case To further compare the control performance of the two methods, the model mismatch is introduced by changing some elements of the process matrix. In this case, one of the poles is changed from 0.8186 to 0.8586, resulting in a kind of model uncertainty. The control parameters remain the same as those in the model match case.
5.2.1. Tracking performance Fig. 2a–c shows the comparison results. It can be seen from Fig. 2a that Zhang's controller gives a relatively faster but later a slower and drastic response. The proposed controller gives the overall smoothing one.
5.2.2. Disturbance rejection Disturbance rejection is tested on the above two controllers. At time t ¼ 150 s, a step disturbance ε ¼ 10 is introduced to the process. Fig. 2a–c shows the responses of the two controllers. The proposed controller displays a satisfactory response. Zhang's controller gives a relatively slower output.
5.3. General cases of pole mismatch Here, a more general situation of the model/plant mismatch is studied. The two poles of the process are changed randomly. Two cases are generated as follows. Case 1. the two poles are changed from 1.1053 to 0.9913 and from 0.8186 to 0.8386, respectively. Case 2. the two poles are changed from 1.1053 to 1.2246 and from 0.8186 to 0.8167, respectively. The results are seen in Figs. 3a–4c. From these results, it is seen that the ensemble performance of both the set point tracking and disturbance rejection of the process is improved by the proposed MPC. The mean tracking error and the control input are also illustrated in Table 1. In all cases, the proposed controller yields the improvement of performance concerning tracking performance and disturbance rejection due to its anticipated capacity. 6. Conclusion An overall convergent model predictive control for nonlinear mechatronic drive systems is considered in this study. Control performance is compared for both model match and mismatch cases. In the case of the proposed control scheme, a two-step method of designing controller has been performed instead of the linearization method. Results have shown that the proposed method yields improved performance.
Fig. 3. (a) Results for set-point tracking under case 1: output, (b) results for set-point tracking under case 1: state, and (c) results for set-point tracking under case 1: input.
Please cite this article as: Tao J, et al. Model predictive control for a class of systems with isolated nonlinearity. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.019i
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Fig. 4. (a) Results for set-point tracking under case 2: output, (b) results for set-point tracking under case 2: state, and (c) results for set-point tracking under case 2: input.
Table 1 Statistical results under cases 1 and 2. Case
Item
NAESSPC
Proposed
Case 1
Mean absolute error Mean input
28.72 0.9
28.77 1.2
Case 2
Mean absolute error Mean input
6.15 6.03
7.27 8.41
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Please cite this article as: Tao J, et al. Model predictive control for a class of systems with isolated nonlinearity. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.019i
