International Journal of Automation and Computing DOI: 10.1007/s11633-016-0954-x
Model Predictive Control of Resonant Systems Using Kautz Model Shamik Misra
Rajasekhara Reddy
Prabirkumar Saha
Department of Chemical Engineering, Indian Institute of Technology Guwahati, Assam 781039, India
Abstract: The scope of this paper broadly spans in two areas: system identification of resonant system and design of an efficient control scheme suitable for resonant systems. Use of filters based on orthogonal basis functions (OBF) have been advocated for modelling of resonant process. Kautz filter has been identified as best suited OBF for this purpose. A state space based system identification technique using Kautz filters, viz. Kautz model, has been demonstrated. Model based controllers are believed to be more efficient than classical controllers because explicit use of process model is essential with these modelling techniques. Extensive literature search concludes that very few reports are available which explore use of the model based control studies on resonant system. Two such model based controllers are considered in this work, viz. model predictive controller and internal model controller. A model predictive control algorithm has been developed using the Kautz model. The efficacy of the model and the controller has been verified by two case studies, viz. linear second order underdamped process and a mildly nonlinear magnetic ball suspension system. Comparative assessment of performances of these controllers in those case studies have been carried out. Keywords:
1
Model predictive control, resonant systems, Kautz model, orthonormal basis function, internal model control.
Introduction
There are several mechanical and electrical systems which show resonating characteristics. Existence of one or more pairs of complex poles in those systems yields oscillatory behaviour in their output profiles. Such kind of systems require an efficient and adequate control strategy that can offer tight and stable closed loop control. Some examples of such resonating systems can be found in robotics, power system electronics, mechanical systems like crane, etc. Even in large scale chemical processes some oscillatory behaviour may be observed in the process outputs, especially where multiple recycle loops exist in the process network or the case of a cascade control with the primary loop cut off[1] . PID controller, the most widely used controller over the decades due to its robustness and simplicity, is however not well understood for plants with resonating response. Some tuning techniques based on heuristic knowledge have been proposed by ˚ Astr¨ om and H¨ agglund[2] , nevertheless it has been observed that such controllers seldom need human intervention[3] , and one may have to tune the PID controller manually through trial and error while implementing it in practice. One of the reasons of failure of PID controllers for resonating systems may be attributed to its classical design procedure where no scope exists for explicit use of exact model of the process. Model based control strategies, such as internal model control (IMC) and model predictive control (MPC), might be good alternatives for classical conResearch Article Manuscript received December 9, 2013; accepted February 15, 2015 Recommended by Associate Editor Dong-Hua Zhou c Institute of Automation, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2016
trol strategies as these control systems embed the process model in the control algorithm. They may always not be the substitute of the conventional control schemes rather they would act as an aid to improve the traditional control strategies[4] . In fact Riveara et al.[5] has proved that the IMC design procedure, in certain situations, leads to traditional PID controller with feedback loop. Design method of IMC based PID controller is quite well-established and is available even in textbooks[6] . On the other hand, though originally developed to meet the specialized control needs of power plant & petroleum refineries, MPC technology can now be found in a wide variety of application areas, including chemicals, food processing, automotive & aerospace applications. MPC uses an explicit dynamic model of the plant in order to predict the future output by simulating input manipulation and thereby optimizes an appropriate objective function to calculate the best control action (i.e., the best set of input manipulations) for the actual process. Process/model mismatch, arising out of the actual implementation of control action, is fed back to the MPC algorithm before calculating the next set of best control actions. Due to its immense prospect, researchers of both academic and industrial fields show great interest in MPC that resulted in various developments in its techniques over the years[7] . Nevertheless, there hardly exists any research report that explores the applicability of MPC on resonant systems. As the name suggests, modelling is a very important part of a model predictive control scheme. It is perhaps the absence of appropriate modelling techniques for resonant systems, that has barred the researchers to study MPC for resonant systems. This work is mainly focussed on application of MPC for resonant systems.
2 Modelling of a process consists of formulating a set of mathematical equations which describes dynamic input/output behavior of the process[8] . Modelling can either be knowledge based (mechanistic modelling) or experience based (black-box modelling). A mechanistic model needs physical insight of the system that includes differential equations of balance of states (mass, energy or momentum) and algebraic equations of thermodynamic and/or chemical equilibrium of a process. However, in most of the real cases, it is difficult to obtain complete knowledge of the system, therefore people resort to black-box model which consists of a recursive filter whose present values of output variables are expressed as functions of past values of outputs and inputs. These functions (linear, polynomial functions, etc.) are associated with a set of parameters. The input/output model structure (i.e., the filter function) is fixed apriori and parameters of the model are extracted through optimization using available input/output data. Simplicity in model structure and parsimony of the model (few numbers of parameters to be evaluated) are keys to success in efficient black-box modelling of a process. In recent years, the use of orthogonal basis functions (OBF) in system identification of dynamic processes has been increased appreciably[9] . The main reason of using OBF in such areas is that the corresponding models usually are parsimonious in nature and thereby have simpler solutions. A model based on OBFs incorporates an approximate knowledge about the dominant dynamics of the process into the procedure of system identification. With the help of this knowledge, the number of free design parameters can be set and thus the variance of their estimates gets reduced. This results in an increase in robustness and accuracy in the model. The simplest structure based upon OBF, which is also most popular is the finite impulse response (FIR) model. FIR modelling corresponds to the estimation of coefficients of a partial expansion in terms of standard OBFs z −k . The main advantage of FIR modelling is its parameters which appear linearly in the model structure. So the system identification problem simplifies to a linear regression estimation problem. However, when FIR is used to approximate a system with long impulse response the minimum number of delays required to provide an acceptable approximation is quite high. In other words, the parsimony of this OBF is lost for such case. This is due to the fact that the time domain equivalents of the basis function z −k are the pulse function δ (t − k). But in general, impulse response of a system shows exponential decay. To overcome this problem of non-parsimony, a special type of model structure viz. Takenaka-Malmquist basis[10] , is introduced which consists of the sum of orthonormal basis functions that also have exponential delay. Nevertheless, TakenakaMalmquist constructions are not much popular due to their complex structure. Various researches have been done till now on a specific case of this generalized structure, best known as Laguerre functions. Classical orthonormal Laguerre functions, as explained in [11], were originally introduced by French mathematician E. Laguerre way back in
International Journal of Automation and Computing
1879. The recursive nature of Laguerre construction makes it easy to compute. The reason of popularity of Laguerre filter is its simplicity as it can be parameterized by single real-valued pole. The Laguerre basis is preferable for representing well damped dynamic system. Systems with poorly damped dynamics, however cannot be accurately described by Laguerre functions, i.e., these functions are not appropriate for approximating signals having strong oscillatory behavior. This drawback has led to an increasing interest in the Kautz functions. Kautz filters are more generalized structure of Laguerre filters and a model developed with these filters deals with complex poles; thus facilitates an efficient modelling of resonant systems. Though Kautz functions were introduced by Kautz in 1954[13] , very few works have been done using it because of its complexity. The aim of this paper is two fold, firstly to establish the efficacy of Kautz modelling for resonant system having linear and mildly nonlinear characteristics and secondly develop an MPC based on Kautz model that can successfully be used with those resonant systems. This paper proposes a suitable state space form of Kautz filter on the lines of [12] and thereby develops an MPC algorithm using this state space model. To understand the efficacy of this control scheme, two case studies, viz. a linear second order underdamped process and a mildly nonlinear magnetic ball suspension system, have been carried out. Results are compared with conventional control schemes viz. IMC or IMC based PID controller.
2
Theoretical development
The problem of orthogonalizing a set of discrete time exponential functions can be summarized[14] as following: Theorem 1. The sequence of funcions Ψj (z)
(n) 1 − a1 z Γ(n) (z) (n) (n) Ψ2n (z) =C2 1 − a2 z Γ(n) (z) (n)
Ψ2n−1 (z) =C1
(1) (2)
for ∀n = 1, 2, · · · , where n−1
(1 − βj z) 1 − βj∗ z
Γ(n) (z) = n j=1
j=1
(3)
(z − βj ) z − βj∗
(n) (n) {1 + βn βn∗ } − 0 = 1 + a1 a 2 (n) (n) {βn + βn∗ } a1 + a2 ⎡
(n)
C1
⎢ ⎢ ⎢ 1 − β 2 1 − β ∗2 (1 − β β ∗ ) n n n n ⎢ =⎢
2 ⎢ (n) ⎢ 1 + a1 {1 + βn βn∗ } − ⎣ (n) 2a1 {βn + βn∗ }
(4) ⎤1 2
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(5)
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S. Misra et al. / Model Predictive Control of Resonant Systems Using Kautz Model
⎡
(n)
C2
⎤1 2
⎢ ⎢ ⎢ 1 − β 2 1 − β ∗2 (1 − β β ∗ ) n n n n ⎢ =⎢
2 ⎢ (n) ∗ ⎢ 1 + a2 {1 + βn βn } − ⎣ (n) 2a2 {βn + βn∗ }
form an orthonormal set, i.e., dz 1 δjl = Ψj (z) Ψl z −1 2πi z
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(6)
and q is the shift operator. The parameter vector Θ can be estimated using least squares approach. In this paper, we develop a model only for an SISO process, nevertheless the technique can be extended for MIMO process too. The state space representation of Kautz model is given as follows. Define the following states xodd,1 × u (k) , for n = 1 (11) x2n−1 (k) = xodd,n × x2n−3 (k) , for n ≥ 2
(7) x2n (k) = x2n−1 (k − 1) ; for n ≥ 1 q xodd,1 = (1) (1) 2 q + h1 q + h2
where δjl is the Kronecker delta function and {βn , βn∗ } are pairs of complex numbers in the region |βn | < 1. The functions {Ψj (z) , ∀j = 1, 2, · · · } are called discrete Kautz functions.
2.1
Kautz model and its state space representation
(n−1) 2
xodd,n =
2N n=1
where input-output relations can be written as
(n)
(8)
where
where ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ Φ (k) = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ Θ =⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
ϕ1 (k) ϕ3 (k) ··· ϕ2N−1 (k) ϕ2 (k) ϕ4 (k) ··· ϕ2N (k) Ψ1 (q) Ψ3 (q) ··· Ψ2N−1 (q) Ψ2 (q) Ψ4 (q) ··· Ψ2N (q) ⎤T θ1 ⎥ θ3 ⎥ ⎥ ⎥ ··· ⎥ θ2N−1 ⎥ ⎥ ⎥ ⎥ θ2 ⎥ ⎥ θ4 ⎥ ⎥ ··· ⎦ θ2N
⎤
(k)
C1
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥= ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(k)
C2
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ × u (k) ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
q+1
(n)
+ h2
h1
= − (βn + βn∗ )
(13)
(n) h2
=βn βn∗ .
(14)
The components of regression vector in (9) are expressed as (n) (n) (15) x2n (k) − a1 x2n−1 (k) ϕ2n−1 (k) =C1 (n) (n) x2n (k) − a2 x2n−1 (k) ϕ2n (k) =C2 (16)
θn Ψn (z)
y (k) = Θ × Φ (k)
q2 +
(n) h1 q
where
A stable transfer function based on the linear combination of discrete Kautz functions can be expressed as[14] G (z) =
(n−1)
q + h1
h2
(12)
(9)
(k)2 (k) 1 − h(k)2 + 3h 1 − h 1 2 2 = (k)2 (k) (k) (k) 1 + a1 1 + h2 + 2a1 h1 (k)2 (k) 1 − h(k)2 + 3h2 1 − h2 1 = . (k)2 (k) (k) (k) 1 + a2 1 + h2 + 2a2 h1
(18)
The states in (11) and (12) can further be arranged in a vector form as ⎡ ⎤ x1 (k) ⎢ ⎥ ⎢ x3 (k) ⎥ ⎢ ⎥ ⎢ ⎥ (19) xodd (k) = ⎢ x5 (k) ⎥ ⎢ ⎥ .. ⎢ ⎥ . ⎣ ⎦ x2n−1 (k) ⎡ ⎤ ⎡ ⎤ x1 (k − 1) x2 (k) ⎢ ⎥ ⎢ ⎥ ⎢ x4 (k) ⎥ ⎢ x3 (k − 1) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ xeven (k) = ⎢ x6 (k) ⎥ = ⎢ x5 (k − 1) ⎥ = ⎢ ⎥ ⎢ ⎥ . . ⎢ ⎥ ⎢ ⎥ .. .. ⎣ ⎦ ⎣ ⎦ x2n (k) x2n−1 (k − 1) xodd (k − 1)
(10)
(17)
(20)
which would yield the complete state space representation as xodd (k) = A1 xodd (k − 1) + A2 xodd (k − 2) + Bu (k − 1) (21)
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International Journal of Automation and Computing
⎡
where A1 =
A2 =
a12
a13
···
a1n
a21
a22
a23
···
a2n
⎡
a11
⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎣ ⎡
a12
⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎣
(1)
a13
⎡
a15
⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣ ⎡
a21
⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎣ ⎡
a22
⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎣ ⎡
a23
⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣
(1) −h1
(1)
(2) (3)
(n−1)
h1 h2 · · · h2
(3) h1
0 0 (3) −h1 .. .
(n−1) · · · h2
0 0 0 .. . (n) −h1
⎤
(22)
1 − h2 (1) (2) (1) h1 h2 1 − h2 .. . (1) (2) (3) (n−1) (1) 1 − h2 h1 h2 h2 · · · h2 ⎤ 0 ⎥ (2) ⎥ −h1 ⎥ (2) (2) ⎥ h1 1 − h2 ⎥ ⎥ .. ⎥ ⎥ . ⎦ h1
⎡ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣
a11
(23) ⎤
⎡
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
⎢ ⎢ ⎢ ⎢ B =⎢ ⎢ ⎢ ⎣
(2)
⎤
1−
1 (1) h2 (1) (2) h2 h2 .. . (1) (2) (3) (4) (n−1) h2 h2 h2 h2 · · · h2
⎤ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦
(24)
ϕodd (k) =C 1 ◦ [xeven (k) − a1 ◦ xodd (k)]
(25)
ϕeven (k) =C 2 ◦ [xeven (k) − a2 ◦ xodd (k)]
(26)
⎡ ⎢ ⎢ ⎢ ϕodd (k) = ⎢ ⎢ ⎢ ⎣
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
⎡
(1)
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
where ◦ denotes the Schur product and
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
−h2 2(1) 1 − h2 (2) 2(1) h2 1 − h2 .. . (2) (3) (4) (n−1) 2(1) 1 − h2 h2 h2 h2 · · · h2 ⎤ 0 ⎥ (2) ⎥ −h2 ⎥ 2(2) ⎥ 1 − h2 ⎥ ⎥ .. ⎥ ⎥ . ⎦ (3) (4) (n−1) 2(2) h2 h2 · · · h2 1 − h2 ⎤ 0 ⎥ 0 ⎥ ⎥ (3) ⎥ −h2 ⎥ ⎥ .. ⎥ . ⎦ (4) (n−1) 2(3) h2 · · · h2 1 − h2
⎤
The regressors in (15) and (16) can be written as
1 − h2
(3) h2
a25
0 0 0 .. . (n) −h2
⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
⎢ ⎢ ⎢ ϕeven (k) = ⎢ ⎢ ⎢ ⎣ C1 =
(1)
C1
C2 = a1 =
(1)
a1
a2 =
(1)
C2
(1)
a2
ϕ1 (k) ϕ3 (k) ϕ5 (k) ··· ϕ(2n−1) (k)
(2)
(2)
C2 (2)
a1
(2)
a2
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(27)
⎤
ϕ2 (k) ϕ4 (k) ϕ6 (k) ··· ϕ(2n) (k) C1
⎤
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(28)
(3)
···
C1
(3)
···
C2
C1 C2
(n)
(n)
(3)
···
a1
(3)
···
a2
a1 a2
(n)
(n)
T T
T T
(29) (30) (31)
.
(32)
Detailed derivation of (11)−(32) have been given in the Appendix A. The equations (8), (9), (20), (21), (25) and (26) can be written in the incremental forms as
δy (k) =δΦ (k) × Θ
(33)
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S. Misra et al. / Model Predictive Control of Resonant Systems Using Kautz Model
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ δΦ (k) = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
δϕ1 (k) δϕ3 (k) ··· δϕ2n−1 (k) δϕ2 (k) δϕ4 (k) ··· δϕ2n (k)
⎤T
where
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
⎡ (34)
δϕodd (k) =C 1 ◦ [δxeven (k) − a1 ◦ δxodd (k)]
(35)
⎡
δϕeven (k) =C 2 ◦ [δxeven (k) − a2 ◦ δxodd (k)]
(36)
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
δxodd (k) =A1 δxodd (k − 1) + A2 δxodd (k − 2) + Bδu (k − 1)
(37)
δxeven (k) =δxodd (k − 1)
(38)
⎡
where δy (k) =y (k) − y (k − 1) δxi (k) =xi (k) − xi (k − 1) ,
(39) i is an integer
δu (k) =u (k) − u (k − 1) .
(40) (41)
Now, (33)−(41) may be used to develop a Kautz model of a resonant system and thereby used in the formulation of an MPC algorithm.
2.2
⎢ ⎢ ⎢ ⎢ Qi = ⎢ ⎢ ⎢ ⎢ ⎣
Formulation of model predictive control law
Let us consider the following equalities which will be useful for derivation of predictors. Qa1 Qa2 Qa3 .. . Qai .. .
= = = .. . = .. .
A1 A1 Qa1 + A2 A1 Qa2 + A2 Qa1 .. . A1 Qa(i−1) + A2 Qa(i−2) .. .
Qb1 Qb2 Qb3 .. . Qbi .. .
= = = .. . = .. .
A2 A1 Qb1 + A2 A1 Qb2 + A2 Qb1 .. . A1 Qb(i−1) + A2 Qb(i−2) .. .
⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
⎥ ⎥ ⎥ ⎥ ⎥ × B, ⎥ ⎥ ⎥ ⎦
Qa(i−1) Qa(i−2) Qa(i−3) ··· Qa(i−M +1) Qa(i−M )
Then it is easy to derive an i-step ahead prediction of states, as in (21), over a prediction horizon of P and control horizon M as δxodd (k + i) = Qai δxodd (k) + Qbi δxodd (k − 1) + Qi δUi (44)
⎥ ⎥ ⎥ ⎥ ⎥ × B, ⎥ ⎥ ⎥ ⎦
(45)
for
i>M
(46)
⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
δu (k) δu (k + 1) δu (k + 2) ··· δu (k + M − 1)
for
i≤M
(47)
⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
for
i>M
(48)
and I in (45) is an identity matrix. The reader may refer to the Appendix B for complete derivation of (44)−(52). And (44) can be used in (34)−(36) in order to obtain the following prediction of regressors
(43)
i≤M
for
⎤
δu (k) δu (k + 1) δu (k + 2) ··· δu (k + i − 1)
δϕodd (k + i) =C 1 ◦
⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
⎡
⎤
(42)
and
.
⎢ ⎢ ⎢ δUi = ⎢ ⎢ ⎢ ⎣
Qa(i−1) Qa(i−2) Qa(i−3) ··· Qa1 I
δxeven (k + i) − a1 ◦ δxodd (k + i)
δxeven (k + i) − δϕeven (k + i) =C 2 ◦ a2 ◦ δxodd (k + i) ⎤T ⎡ δϕ1 (k + i) ⎥ ⎢ ⎢ δϕ3 (k + i) ⎥ ⎥ ⎢ ⎥ ⎢ ··· ⎥ ⎢ ⎥ ⎢ δϕ 2n−1 (k + i) ⎥ ⎢ δΦ (k + i) = ⎢ ⎥ ⎢ δϕ2 (k + i) ⎥ ⎥ ⎢ ⎢ δϕ4 (k + i) ⎥ ⎥ ⎢ ⎥ ⎢ ··· ⎦ ⎣ δϕ2n (k + i)
(49) (50)
(51)
the algebraic manipulation of which would lead to
δΦ (k + i) =μ1i δxodd (k) + μ2i δxodd (k − 1) + μ3i δU(i−1) + μ4i δUi
(52)
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International Journal of Automation and Computing
where μ1i , μ2i , μ3i and μ4i are 2N ×N, 2N ×N, 2N ×(i − 1) and 2N × i matrices.
μ1i μ2i μ3i μ4i
C 1 ◦ Qa(i−1) − a1 ◦ Qai = C 2 ◦ Qa(i−1) − a2 ◦ Qai C 1 ◦ Qb(i−1) − a1 ◦ Qbi = C 2 ◦ Qb(i−1) − a2 ◦ Qbi C 1 ◦ Q(i−1) = C 2 ◦ Q(i−1) C 1 ◦ a1 ◦ Qi = C 2 ◦ a2 ◦ Qi
where ym (k) is the process measurement at k-th instant. Equation (60) can further be simplified as
⎡ (53)
(54)
⎧ ⎪ ⎪ ⎪ ⎨
⎢ P ⎢ ⎢ ⎢ J= ⎢ i=1 ⎢ ⎣
(55)
ysp (k + i) − yp (k) + % & $i μxj + + Θ j=1 μuj δUM [ym (k) − yp (k)]
⎪ ⎪ ⎪ ⎩
⎤2 ⎫ ⎪ ⎪ ⎪ ⎬
⎥ ⎥ ⎥ ⎥ + ⎥ ⎪ ⎥ ⎦ ⎪ ⎪ ⎭
T RδUM = δUM
⎡
(56)
P ⎢ ⎢ ⎣ i=1
ysp (k + i) − ym (k) − $ Θ i μ $ j=1 xj − Θ ij=1 μuj δUM
' ⎤2 ⎥ ⎥ + ⎦
and (52) can be re-written as T δUM RδUM = P
δΦ (k + i) =μ1i δxodd (k) + μ2i δxodd (k − 1) + μui δUM =
T [ji − μU i δUM ]2 + δUM RδUM
(62)
i=1
μxi + μui δUM
(57) where
where μxi is a 2N × 1 matrix ji =ysp (k + i) − ym (k) − Θ μxi = μ1i δxodd (k) + μ2i δxodd (k − 1)
(58)
and μui is a 2N ×M matrix, composed of μ3i and μ4i in such a manner that (57) holds. From (8) and (39) the (k + i)-th prediction of output is
yp (k + i) =yp (k) +
⎡
i
⎢ ⎢ ⎢ ⎢ R =⎢ ⎢ ⎢ ⎣
δyp (k + j) =
i
(μxj + μuj δUM ) .
(59)
j=1
J= J=
(63)
(64)
λ1 0 0 .. . 0
0 λ2 0 .. . 0
0 0 λ3 .. . 0
··· ··· ··· ··· ···
0 0 0 .. . λM
⎤ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦
(65)
For an SISO process, ji is a scalar quantity, μU i is a row vector with M elements and R is an M × M matrix. Continuing with (63)
The predictive control law is in general obtained by minimization of the following criterion: P
μuj
j=1
j=1
yp (k) + Θ
μxj
j=1
μU i =Θ
i
i
P
T T μU i μU i δUM − 2ji μU i δUM + ji2 + δUM
i=1 2
[ysp (k + i) − {yp (k + i) + d (k + i)}] +
T RδUM = δUM
i=1 M −1
% λi+1 [δu (k + i)]
2
(60)
i=0
P
% +
T δUM
R+
i=1
% where d (k + i) is the process/model mismatch at the (k + i)-th prediction. It is customary to assume
d (k + i) = d (k) = ym (k) − yp (k)
& ji2
(61)
2
P
P
& μT U i μU i
δUM −
i=1
& ji μU i
δUM .
(66)
i=1
Without constraints, the optimal solution of the cost function (66) is given by
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S. Misra et al. / Model Predictive Control of Resonant Systems Using Kautz Model
∂J =0 = ∂ (δUM ) % R+ %
P
& μT U i μU i
i=1 P
loop, IMC can also be written in terms of regular feedback control structure as δUM −
&
Gf eedback =
T μT U i ji
(67)
i=1
or δUM = R +
P i=1
−1 μT U i μU i
P
T ji μU i
.
(68)
i=1
Equation (68) can be solved to compute the incremental control move which can then be used to compute the actual control move using (41).
2.3
Design of internal model controller
The most desirable features of tuning a controller are its simplicity and optimality. In most of the cases, first a simple control structure is fixed and then optimality is [5]
extracted out of it . In other words, the structure of the model (along with all the poles and zeros of the process) is not explicitly used while designing a classical controller; the controller parameters are rather tuned as a function of a simplified approximation of the “more accurate” model of the process. Moreover, classical PID controllers are not designed to handle constraints. IMC strategy, on the other hand, is based on the concept of perfect control. For an open loop process y (s) = Gp (s) Gc (s) ysp (s)
(69)
where Gp (s) and Gc (s) represent the process and controller transfer functions and y (s) and ysp (s) are the output and setpoint trajectories respectively. Now from (70) it is ev1 ident that if Gc (s) = then output trajectory will Gp (s) be the same as the reference (setpoint) one. This holds the key idea of IMC control, i.e. output of the system resembles the reference signal if the controller transfer function is just the reciprocal of the system s transfer function. The control scheme is given elaborately in [6]. At first, the ( process model should be divided in invertible Gp− and ( p+ parts. The non-invertible part connon-invertible G tains right half plane (RHP) zeros and time delay. The controller constitutes the reciprocal of the invertible part. To make the controller proper, a filter f (s) is added. ( −1 GIM C = G p− (s) f (s) .
(70)
Controller calculates the desired control move; the process and model are excited by the same control move (manipulation of input). Difference in their outputs, i.e. the process/model mismatch, is fed back to the controller. Controller then recalculates the future control move and procedure repeats itself. By structural re-formatting of control
GIM C . 1 − Gp (s) GIM C
(71)
By appropriate choice of filter transfer function it is possible, at least for some cases, to obtain Gf eedback in the form of regular PID controller. Unlike other tuning techniques, IMC based tuning can handle underdamped systems. IMC works perfectly if the model is perfect. But perfect model is a utopian concept. The order of the filter is chosen to make the controller proper. The filter parameter is the main tuning parameter of IMC. Large value of filter ensures robustness whereas smaller value yields faster response. There is always a tradeoff between robustness and speed of response[6, 15] .
3
Simulation case studies
The efficacy of the model predictive controller based on Kautz model has been tested on the following two case studies. In the first case, a linear second order underdamped system, originally proposed by [14], has been considered. A time series analysis has been done for modelling as opposed to frequency domain analysis in [14]. In the second case, a magnetic suspension system has been considered which is mildly nonlinear as well as resonating in nature. A comparative study of performance of the two controllers, viz. Kautz-MPC and PID-IMC, has been presented subsequently. All simulations were carried out using Matlab (Version 7, 64-bit) under Windows 7 (64-bit) operating system on a PC with Intel CoreTM 2 Duo processor @2.4 GHz speed.
3.1
Case study I: Linear second order underdamped system
Consider this continuous time transfer function of a linear second order underdamped system G (s) =
1 s2 + 0.2s + 1
(72)
with a resonant frequency ω = 1 and damping coefficient ξ = 0.1. This system is sampled using a zero order hold circuit with sampling period, T = 0.5. Reference [14] studied the efficacy of the Kautz model by analyzing its steady state characteristics. A relevant Bode diagram was generated and the validity of the Kautz model was established within a certain frequency range. 3.1.1 Development of a Kautz model The system has been subjected to an input perturbation with mean 1.921 and variance 0.072. A total of 1000 data points were collected for both input and output. 75% of the data set was used to train the Kautz model while 25% data were used to test it. A Kautz model of order 3 has been developed using (33)−(41) that yields a very good match with the process. The results are shown in Fig. 1. In fact the
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International Journal of Automation and Computing
process/model mismatch is so low that the profiles almost overlap with each other. The root mean square (RMS) value of the process/model mismatch is 0.018 3.
2ξ ω 1 = 2ξω
τI =
(75)
τD
(76)
(a) Event-triggered instants (a) Event-triggered instants
(b) The successful broadcast release instants
Fig. 1 System identification of a linear second order process. (a) Series of random perturbation introduced to the input of the process; (b) Profile of output of the process as a result of random perturbation in its input
3.1.2
Closed loop control study with MPC using Kautz model
An unconstrained MPC law has been formulated using (69). The prediction and control horizons are chosen as 1000 and 1 respectively. Since the process is highly oscillatory, a longer prediction horizon is needed for the MPC law to calculate an appropriate control action. The closed loop response using the above MPC-Kautz controller is shown in Fig. 2. 3.1.3
Closed loop control study with IMC
The tuning parameters for PID controller based on IMC techniques for a second order underdamped system having a transfer function Gp (s) = [6]
are as follows
Kp s2 + 2ξωs + ω 2
(73)
: Kc =
2ξω λKp
(74)
(b) The successful broadcast release instants
Fig. 2 The performance of two controllers, viz. MPC using Kautz model and PID with IMC tuning, in controlling the linear second order process. (a) Profile of output; (b) Profile of input
The parameter λ is a user specified filter tuning parameter which helps to adjust the speed of the closed loop response. In the present case, the value of the filter tuning parameter has been taken as λ = 0.9. Comparing (73) with (72) one obtains, Kp = 1; ω = 1; ξ = 0.1 and hence using these values in (74)−(76) one obtains, Kc = 0.222; τI = 0.2; τD = 5. The closed loop response using the above IMC-PID controller is shown in Fig. 2. 3.1.4
Comparative study of the controller performance
It is observed in Fig. 2 that for a 40% change in the setpoint of the process, the MPC-Kautz is able to guide the process to its new setpoint in 18 whereas IMC-PID controller fails to achieve it even after 125. Although minor oscillation is observed in the output profile under MPCKautz controlled process, the output remains within ±5% of the final setpoint after 18 of the simulation run. An overshoot of 31.4% is observed in the output profile, however an initial decay ratio of 16.69% is good enough to arrest the oscillation in the controlled output. On the other hand, the IMC-PID controller yields higher overshoot in the con-
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S. Misra et al. / Model Predictive Control of Resonant Systems Using Kautz Model
trolled output and its insufficient decay ratio fails the output to sustain within acceptable limit of oscillation near the setpoint. This justifies the superiority of the MPC-Kautz controller over IMC-PID controller in controlling a linear second order process.
are es = 50 and is = 0.5 respectively. The nominal value of (78) indicates that the process is nonlinear in nature. It is worth to examine how efficiently a linear Kautz model can approximate this process.
3.2
For modelling purpose, the system is subjected to a series of random changes in the input voltage. The change in the voltage causes a change in the current passing through the system and subsequently the steel ball s position changes. The process has been subjected to an input perturbation with mean 49.98 and variance 0.084. A total of 1000 data points were collected for both input and output. Similar to the Case Study I, 75% of the data set was used to train the Kautz model while 25% data were used to test it. A Kautz model of order 5 has been developed using (33)−(41) that yields a very good match with the process. The results are shown in Fig. 4.
Case study II: Magnetic ball suspension system
Fig. 3 shows a schematic of magnetic ball suspension system (MBSS). It consists of an electromagnet firmly placed at the ceiling of an encloser while an iron ball is suspended over the floor by means of a spring. The electric coil that winds the electromagnet has a resistor (R) and an inductor (L) in series. The voltage, e, supplied to the electromagnet yields a current, i, which in turn generates the magnetic field sufficient to pull the iron ball upwards. The mass of the ball is M and the spring constant is k. The distance between the ball and the electromagnet is denoted by y. The objective of the system is to control the position of the ball (y) by adjusting the input voltage (e).
3.2.1
Development of a Kautz model
(a) Event-triggered instants
Fig. 3
The schematic of a magnetic ball suspension system (b) The successful broadcast release instants
The differential equations of the system are given by d2 y (t) dy (t) i2 (t) M +k + Mg = 2 dt dt y (t) di (t) L + Ri (t) = e (t) dt
(77)
Fig. 4 System identification of a magnetic ball suspension system. (a) Series of random perturbation introduced to the input of the process; (b) Profile of output of the process as a result of random perturbation in its input.
(78)
where g is the accelaration due to gravity. For all simulation studies in this paper, the following numerical values have been considered, g = 9.8; M = 0.01; L = 10; R = 100; k = 0.01. The nominal steady state position of the ball is 2.551, (i.e., ys = 2.551) away from the magnet; and the corresponding nominal value of input voltage and current
The process/model mismatch is quite low and the profiles of process and model overlap with each other at most of the places. The root mean square (RMS) value of the process/model mismatch is 0.015 7. The above observation indicates that the linear Kautz model is capable of capturing the process dynamics quite well despite the fact that the process is inherently nonlinear in nature. It is perhaps
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International Journal of Automation and Computing
due to the fact that the quadratic nonlinearity which is evident in the mechanistic model, is mild enough in the region of nominal steady state; and a linear Kautz model is good enough for this mild nonlinearity to be captured. 3.2.2 Closed loop control study with MPC using Kautz model An unconstrained MPC law has been formulated using (68). The prediction and control horizons are chosen as 10 and 1 respectively. The closed loop response using the above MPC-Kautz controller is shown in Fig. 5.
Using (79) in (77), one obtains * ) 2* ) 2is dy (t) d2 y (t) is + {i (t) − is } − + k M + M g = dt2 dt ys ys ) 2* is {y (t) − ys } . (80) ys2 Taking deviation form of the variables and subsequently converting in Laplace domain, (80) and (78) take the form * ) 2* ) is 2is i (s) − y (s) (81) M s2 y (s) + ksy (s) = ys ys2 Lsi (s) + Ri (s) =e (s) .
(82)
Algebraic rearrangement of (81) and (82) yield y (s) =
)
* 2is e (s) LM ys * * ) *. ) ) 2 Ri2s k is kR R s2 + s + s3 + + + M L M ys2 LM LM ys2 (83)
(a) Event-triggered instants
And putting the values of the coefficient terms, one obtains the linearized transfer function model of the MBSS process G (s) =
y (s) 3.92 = 3 . e (s) s + 11s2 + 13.84s + 38.42
(84)
This transfer function may be used to construct the IMC. As it does not contain any positive zeros or dead time element the entire transfer function is invertible. A low pass filter of 3rd order is used to make the controller proper. So the controller transfer function takes the form[6] GIM C (s) =
(b) The successful broadcast release instants
Fig. 5 The performance of three controllers, viz. MPC using Kautz model, MPC using transfer function model and PID with IMC tuning, in controlling the magnetic ball suspension system. (a) Profile of distance of steel ball from ground; (b) Profile of input voltage to the electromagnet
3.2.3 Closed loop control study with IMC In order to design an IMC for the MBSS system one needs to linearize the nonlinear system. The only nonlinear term i2 (t) in the process is in (77). Applying Taylor s series y (t) expansion (upto first order only) in the nonlinear term, one obtains ) 2* ) * i 2is i2 (t) = s + {i (t) − is } − y (t) ys ys ) 2* is (79) {y (t) − ys } . ys2
s3 + 11s2 + 13.84s + 38.42 . 3.92 (λs + 1)3
(85)
The parameter λ is a user specified filter tuning parameter which helps to adjust the speed of the closed loop response. In the present case, the value of the filter tuning parameter has been taken as λ = 0.9. The closed loop response using the above IMC controller is shown in Fig. 5. 3.2.4 Comparative study of the controller performance It is observed in the Fig. 5 that for a 40% change in the setpoint of the process, the MPC-Kautz is able to guide the process to its new setpoint in 120 time units while IMC controller is also able to achieve it in same period of time. Although minor oscillation is observed in the output profile under MPC-Kautz controlled process, the output remains within ±5% of the final setpoint after 40 time units of the simulation run. An overshoot of 16% is observed in the output profile, however an initial decay ratio of 34.72% is good enough to arrest the oscillation in the controlled output. It is also observed that the MPC based on Kautz model performs better than MPC based on transfer function model. The ISE value of MPC-Kautz is 12.02 whereas that of MPC-TF is 14.57. On the other hand, the IMC
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S. Misra et al. / Model Predictive Control of Resonant Systems Using Kautz Model
controller yields higher overshoot in the controlled output. While the performances of IMC and MPC-Kautz are comparable for smaller changes in setpoint, they grossly differ at large change in setpoint. Interestingly, large offset (24%) is observed for IMC controller when it is used for large setpoint changes. This further justifies the superiority of the MPC-Kautz controller over IMC controller in controlling a MBSS process.
4
Using (A4) and (A5) in (A1) one obtains (1) (1) ϕ1 (k) =C1 q −1 − a1 x1 (k) = (1) (1) C1 x1 (k − 1) − a1 x1 (k) = (1) (1) C1 x2 (k) − a1 x1 (k) . Similarly
Conclusions
The Kautz model has been proved to be an efficient modelling technique for resonant systems. The case studies with both linear and mildly nonlinear processes support this fact. The process/model mismatch has been extremely less in both the cases. The MPC developed on the basis of Kautz model turns out to be a better controller than IMC and/or IMC based PID controller. The extent of overshoot and the duration of unacceptable oscillation are less for MPC controlled processes. Decay ratio of MPC controlled output is stronger than IMC controlled output. The IMC generates offset in case of mildly nonlinear process whereas MPC yields offset free response. For all practical purposes, MPC with Kautz model stands out to be a far better option for modelling and controlling a resonant system.
ϕ2 (k) =Ψ2 (q) u (k) = (1) (1) 1 − a2 q Γ(1) (q) u (k) = C2 (1) (1) C2 q −1 − a2 × q (1) h1 q
q2
+
(1)
C2
(1) x2 (k) − a2 x1 (k) .
(1 − β1 q) (1 − β1∗ q) u (k) = (q − β1 ) (q − β1∗ ) (q − β2 ) (q − β2∗ ) (2) (2) q −1 − a1 × C1 (1)
(1)
h2 q 2 + h1 q + 1
Using Theorem 1 the Kautz model in (9) can further be represented as the following state space realization (with shift operator q):
q2
q2 +
(2) h1 q
(2)
+ h2
q
q (1)
u (k)
(1) h1 q
+
(2)
C1
(1)
+ h2
(2)
q −1 − a1
q2 +
(2)
(2) h2
(A1)
(A2)
(1) h2
(A3)
Define the following states q (1)
(1)
q 2 + h1 q + h2
x2 (k) =x1 (k − 1) .
× u (k)
(A8)
(A9)
=β2 β2∗ .
(A10)
Define the following states
h1 = − (β1 + β1∗ )
x1 (k) =
× x1 (k)
h1 = − (β2 + β2∗ )
(1)
x3 (k) =
(1)
h2 q 2 + h1 q + 1 (2)
(A4) (A5)
(2)
q 2 + h1 q + h2
x4 (k) =x3 (k − 1) .
=β1 β1∗ .
×
(1) + h1 q + 1 (2) (2) h1 q + h2
where
where (1)
×
u (k) =
(1) h2 q 2
1 u (k) = (q − β1 ) (q − β1∗ ) (1) (1) C1 q −1 − a1 × (1)
(A7)
ϕ3 (k) =Ψ3 (q) u (k) = (2) (2) C1 1 − a1 q Γ(2) (q) u (k) = (2) (2) 1 − a1 q × C1
Detailed derivation of state space Kautz model
q 2 + h1 q + h2
× u (k) =
(1)
+ h2
Further
Appendix A
ϕ1 (k) =Ψ1 (q) u (k) = (1) (1) 1 − a1 q Γ(1) (q) u (k) = C1 (1) (1) C1 1 − a1 q ×
(A6)
× x1 (k)
(A11) (A12)
Using (A11) and (A12) in (A8), one obtains the regressors (similar to A6 and A7) as (2) (2) (A13) ϕ3 (k) =C1 x4 (k) − a1 x3 (k) (2) (2) ϕ4 (k) =C2 x4 (k) − a2 x3 (k) . (A14) The derivation of regressors can further be continued and a generalized expression[14] can be given as in (11)−(16).
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International Journal of Automation and Computing
For n = 1, in (11) ⎡ x1 (k) =
q (1)
(1)
q 2 + h1 q + h2 q
× u (k) =
−1
(1) 1 + h1 q −1 (1) − h1 x1 (k
(1)
+ h2 q −2
× u (k) =
(1)
− 1) − h2 x1 (k − 2) +
u (k − 1) .
(A15)
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ x2n−1 (k) = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
(1)
x3 (k) =
(1)
h2 q 2 + h1 q + 1
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
× x (k) =
1 (2) (2) q 2 + h1 q + h2 (1) (1) h2 + h1 q −1 + q −2 × x1 (k) = (2) (2) 1 + h1 q −1 + h2 q −2 (2) (2) − h1 x3 (k − 1) − h2 x3 (k − 2) + (1)
⎡
(1)
h2 x1 (k) + h1 x1 (k − 1) + x1 (k − 2) .
(A16)
(n−1)
(1)
1 − h2 (2) (3) (4) (n−1) (2) 1 − h2 h1 h2 h2 · · · h2 (3) (4) (n−1) (3) h1 h2 · · · h2 1 − h2 .. . (n−2) (n−1) (n−2) h2 1 − h2 h1 (n−1) (n−1) 1 − h2 h1
⎤T ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ × ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(n)
⎡ For n = 2, in (11)
(1) (2) (3) (4)
h1 h2 h2 h2 · · · h2
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
−h1 ⎤
x1 (k − 1) x3 (k − 1) x5 (k − 1) .. . x2n−1 (k − 1)
⎥ ⎥ ⎥ ⎥ ⎥+ ⎥ ⎥ ⎦
(2) (3) (4)
(n−1)
h2 h2 h2 · · · h2 (3) (4) (4)
(n−1)
h2 h2 · · · h2
(n−1)
h2 · · · h2
2(1)
1 − h2
2(2)
1 − h2
2(3)
1 − h2
.. . (n−1) 2(n−2) h2 1 − h2 2(n−1) 1 − h2
⎤T ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ × ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(n)
Using (A15) in (A16) and simplifying one gets
(1)
(1)
⎡
(2)
x3 (k) =h1 1 − h2 x1 (k − 1) − h1 x3 (k − 1) + 2(1) (2) x1 (k − 2) − h2 x3 (k − 2) + 1 − h2 (1)
h2 u (k − 1) .
(A17)
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
x1 (k − 2) x3 (k − 2) x5 (k − 2) .. . x2n−1 (k − 2)
−h2 ⎤
(1) (2) (3) (4)
⎥ ⎥ ⎥ ⎥ ⎥+ ⎥ ⎥ ⎦ (n−1)
h2 h2 h2 h2 · · · h2
u (k − 1) . (A19)
(k)
Similarly continuing with n = 3 and 4 one obtains the following state space representation ⎡ ⎢ ⎢ ⎢ ⎣
x1 (k) x3 (k) x5 (k) x7 (k)
⎤
⎡
⎢ ⎥ ⎢ ⎥ ⎥ =A1 ⎢ ⎣ ⎦ ⎡ ⎢ ⎢ A2 ⎢ ⎣
x1 (k − 1) x3 (k − 1) x5 (k − 1) x7 (k − 1) x1 (k − 2) x3 (k − 2) x5 (k − 2) x7 (k − 2)
Bu (k − 1)
⎤ ⎥ ⎥ ⎥+ ⎦ ⎤ ⎥ ⎥ ⎥+ ⎦ (A18)
where A1 , A2 and B are obtained from (22), (23) and (24). The complete state space representation can be given as (21) whose (2n − 1)-th row will be
The value of C2 in (18) can be simplified as 1 − βk2 1 − βk∗2 (1 − βk βk∗ ) (k) C1 = = (k)2 (k) 1 + a1 (1 + βk βk∗ ) − 2a1 (βk + βk∗ ) 1 − βk2 − βk∗2 + βk2 βk∗2 (1 − βk βk∗ ) = (k)2 (k) 1 + a1 (1 + βk βk∗ ) − 2a1 (βk + βk∗ ) 2 ∗2 2 ∗2 2 ∗2 ∗ 1 − βk − βk − 2βk βk + 3βk βk (1 − βk βk ) = (k)2 (k) 1 + a1 (1 + βk βk∗ ) − 2a1 (βk + βk∗ ) 2 1 − (βk + βk∗ ) + 3βk2 βk∗2 (1 − βk βk∗ ) = (k)2 (k) 1 + a1 (1 + βk βk∗ ) − 2a1 (βk + βk∗ ) (k)2 (k) 1 − h(k)2 + 3h 1 − h 1 2 2 . (A20) (k)2 (k) (k) (k) 1 + a1 1 + h2 + 2a1 h1 (k)
(k)
(k)
(k)
The value of C2 in terms of a2 , h1 and h2 can be derived in the similar fashion. Hence (17) and (18) can be obtained.
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S. Misra et al. / Model Predictive Control of Resonant Systems Using Kautz Model
Appendix B
and the (M + 1)-th prediction is
Detailed derivation of model predictive control law
δxodd (k + M + 1) =A1 δxodd (k + M ) + A2 δxodd (k + M − 1) = ⎧ ⎫ ⎪ ⎨ QaM δxodd (k) + ⎪ ⎬ A1 + QbM δxodd (k − 1) + ⎪ ⎪ ⎩ ⎭ QM δUM ⎧ ⎫ ⎪ ⎨ Qa(M −1) δxodd (k) + ⎪ ⎬ = A2 Qb(M −1) δxodd (k − 1) + ⎪ ⎪ ⎩ ⎭ QM −1 δUM −1 ' A1 QaM + δxodd (k) + A2 Qa(M −1) ' A1 QbM + δxodd (k − 1) + A2 Qb(M −1)
Using equalities in (42), (37) can be written as δxodd (k + 1) = Qa1 δxodd (k) + Qb1 δxodd (k − 1) + Bδu (k) .
(B1)
Further the i-step ahead incremental predictor can be written as δxodd (k + 2) =A1 δxodd (k + 1) + A2 δxodd (k) + Bδu (k + 1) = ⎧ ⎪ ⎨ Qa1 δxodd (k) + A1 Qb1 δxodd (k − 1) + ⎪ ⎩ Bδu (k)
⎫ ⎪ ⎬ ⎪ ⎭
+
A1 QM δUM + A2 QM −1 δUM −1 =
A2 δxodd (k) + Bδu (k + 1) =
Qa(M +1) δxodd (k) +
(A1 Qa1 + A2 ) δxodd (k) +
Qb(M +1) δxodd (k − 1) +
A1 Qb1 δxodd (k − 1) +
A1 QM δUM +
A1 Bδu (k) + Bδu (k + 1) =
A2 QM −1 δUM −1 .
Qa2 δxodd (k) + Qb2 δxodd (k − 1) + Qa1 Bδu (k) + Bδu (k + 1)
(B2)
δxodd (k + 3) =A1 δxodd (k + 2) + A2 δxodd (k + 1) + Bδu (k + 2) = ⎧ Qa2 δxodd (k) + ⎪ ⎪ ⎪ ⎨ Q δx (k − 1) + b2 odd A1 ⎪ Q Bδu (k) + a1 ⎪ ⎪ ⎩ Bδu (k + 1) ⎧ ⎪ ⎨ Qa1 δxodd (k) + A2 Qb1 δxodd (k − 1) + ⎪ ⎩ Bδu (k)
Now A1 QM δUM =A1 Qa(M −1) Bδu (k) + A1 Qa(M −2) Bδu (k + 1) +
⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭
A1 Qa(M −3) Bδu (k + 2) + · · · + A1 Qa1 Bδu (k + M − 2) +
+
A1 Bδu (k + M − 1)
⎫ ⎪ ⎬ ⎪ ⎭
(B6)
(B7)
A2 QM −1 δUM −1 =A2 Qa(M −2) Bδu (k) + A2 Qa(M −3) Bδu (k + 1) +
+
A2 Qa(M −4) Bδu (k + 2) + · · · + A2 Bδu (k + M − 2) .
Bδu (k + 2) = {A1 Qa2 + A2 Qa1 } δxodd (k) +
Adding (B7) and (B8) one obtains
{A1 Qb2 + A2 Qb1 } δxodd (k − 1) + {A1 Qa1 + A2 } Bδu (k) +
SU M =A1 QM δUM + A2 QM −1 δUM −1 = A1 Qa(M −1) + A2 Qa(M −2) Bδu (k) + A1 Qa(M −2) + A2 Qa(M −3) Bδu (k + 1) + A1 Qa(M −3) + A2 Qa(M −4) Bδu (k + 2) + · · · +
Qa1 Bδu (k + 1) + Bδu (k + 2) = Qa3 δxodd (k) + Qb3 δxodd (k − 1) + Qa2 Bδu (k) + Qa1 Bδu (k + 1) + Bδu (k + 2)
(B3)
QaM Bδu (k) + Qa(M −1) Bδu (k + 1) +
δxodd (k + i) =Qai δxodd (k) + Qbi δxodd (k − 1) + (B4)
where Qi and δUi are given in (45) and (47) respectively. Again the M -th prediction is
Qa(M −2) Bδu (k + 2) + · · · + Qa2 Bδu (k + M − 2) + Qa1 Bδu (k + M − 1) = QM +1 δUM
δxodd (k + M ) = QaM δxodd (k) + QbM δxodd (k − 1) + QM δUM
{A1 Qa1 + A2 } Bδu (k + M − 2) + A1 Bδu (k + M − 1) =
.. . Qi δUi
(B8)
(B5)
(B9)
where QM +1 can be denoted by (46). Hence, (B6) takes the
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International Journal of Automation and Computing
Similarly it can be proved that
form of
δxodd (k + M + 3) =Qa(M +3) δxodd (k) +
δxodd (k + M + 1) =Qa(M +1) δxodd (k) +
Qb(M +3) δxodd (k − 1) +
Qb(M +1) δxodd (k − 1) + QM +1 δUM .
(B10)
QM +3 δUM
(B14)
δxodd (k + M + 4) =Qa(M +4) δxodd (k) + Qb(M +4) δxodd (k − 1) +
Again,
QM +4 δUM .. .
δxodd (k + M + 2) =A1 δxodd (k + M + 1) + A2 δxodd (k + M ) = ⎧ ⎫ ⎪ ⎨ Qa(M +1) δxodd (k) + ⎪ ⎬ A1 + Qb(M +1) δxodd (k − 1) + ⎪ ⎪ ⎩ ⎭ QM +1 δUM ⎧ ⎫ ⎪ ⎨ QaM δxodd (k) + ⎪ ⎬ A2 = QbM δxodd (k − 1) + ⎪ ⎪ ⎩ ⎭ QM δUM ' A1 Q+ δxodd (k) + A2 QaM ' A1 Qb(M +1) + δxodd (k − 1) + A2 QaM
δxodd (k + M + i) =Qa(M +i) δxodd (k) + Qb(M +i) δxodd (k − 1) + QM +i δUM
δxodd (k + P ) =QaP δxodd (k) + QbP δxodd (k − 1) + QP δUM
⎡
Qa(M +2) δxodd (k) +
(B11)
A1 QM +1 + A2 QM =A1 QaM B + A1 Qa(M −1) B+ A1 Qa(M −2) B + · · · + A1 Qa2 B + A1 Qa1 B+ A2 Qa(M −1) B+ A2 Qa(M −2) B+ A2 Qa(M −3) B + · · · + A2 Qa1 B + A2 B = Qa(M +1) B + QaM B+ Qa(M −1) B + · · · + Qa3 B + Qa2 B = QM +2
(B12)
Qa(P −1) Qa(P −2) Qa(P −3) ··· Qa(P −M +1) Qa(P −M )
Qb(M +2) δxodd (k − 1) + QM +2 δUM .
(B13)
⎥ ⎥ ⎥ ⎥ ⎥ × B. ⎥ ⎥ ⎥ ⎦
(B18)
Similarly
δϕeven (k + i) = C 2 ·
Qa(i−1) − a2 · Qai
' δxodd (k) +
' Qb(i−1) − δxodd (k − 1) + a2 · Qbi , + C 2 · Q(i−1) δU(i−1) − , + C 2 · a2 · Qi δUi . (B20)
C2 ·
δxodd (k + M + 2) =Qa(M +2) δxodd (k) +
⎤T
Hence the above derivation leads to (44)−(48). The i-th prediction of the regressors are δxeven (k + i) − = δϕodd (k + i) =C 1 · a1 · δxodd (k + i) δxodd (k + i − 1) − = C1 · a1 · δxodd (k + i) ' Qa(i−1) − C1 · δxodd (k) + a1 · Qai ' Qb(i−1) − C1 · δxodd (k − 1) + a1 · Qbi , + C 1 · Q(i−1) δU(i−1) − , + C 1 · a1 · Qi δUi . (B19)
where QM +2 can also be denoted by (46). Hence (B11) can be re-written as
(B17)
where QM +3 , QM +4 , · · · , QM +i can be denoted by (46) and QP can be represented by ⎢ ⎢ ⎢ ⎢ QP = ⎢ ⎢ ⎢ ⎢ ⎣
Now,
(B16)
.. .
{A1 QM +1 + A2 QM } δUM = Qb(M +2) δxodd (k − 1) + ' A1 QM +1 + δUM . A2 QM
(B15)
S. Misra et al. / Model Predictive Control of Resonant Systems Using Kautz Model
Hence, from (9), ⎡
' ⎤
Qa(i−1) − ⎥ ⎢ C1 · ⎥ ⎢ a · Qai ' ⎥ δxodd (k) + 1 δΦ (k + i) = ⎢ ⎥ ⎢ Qa(i−1) − ⎦ ⎣ C2 · a2 · Qai ' ⎤ ⎡ Qb(i−1) − ⎥ ⎢ C1 · ⎥ ⎢ a · Qbi ⎢ 1 ' ⎥ δxodd (k − 1) + ⎥ ⎢ Q − b(i−1) ⎦ ⎣ C2 · a2 · Qbi C 1 · Q(i−1) δU(i−1) + C 2 · Q(i−1) C 1 · a1 · Qi δUi . (B21) C 2 · a2 ◦ Qi
which is same as (52).
References [1] C. T. Huang, C. J. Chou. Estimation of the under damped second-order parameters from the system transient. Industrial and Engineering Chemistry Research, vol. 33, no. 1, pp. 174–176, 1994. ˚str¨ [2] K. J. A om, T. H¨ agglund. PID Controllers: Theory, Design, and Tuning, New York, USA: The International Society of Automation, 1995. [3] W. K. Ho, C. C. Hang, J. H. Zhou. Self tuning PID control of a plant with under damped response with specifications on gain and phase margins. IEEE Transactions on Control System Technology, vol. 5, no. 4, pp. 446–452, 1997. [4] C. Brosilow, B. Joseph. Techniques of Model-based Control, New Jersey, USA: Prentice-Hall, 2002. [5] D. E. Riveara, M. Morari, S. Skogestad. Internal model control: PID controller design. Industrial and Engineering Chemistry Process Design and Development, vol. 25, no. 1, pp. 252–265, 1986. [6] B. W. Bequette. Process Control: Modeling, Design, and Simulation, New Jersey, USA: Prentice Hall PTR, 2003. [7] K. S. Holkar, L. M. Waghmare. An overview of model predictive control. International Journal of Control and Automation, vol. 3, no. 4, pp. 47–63, 2010. [8] I. Rivals, L. Personnaz. Black-box modeling with statespace neural networks. Neural Adaptive Control Technology I, pp. 237–264, 1996. [9] A. da Rosa, R. J. G. B. Campello, W. C. Amral. Exact search directions for optimization of linear and nonlinear models based on generalized orthonormal functions. IEEE Transactions on Automatic Control, vol. 54, no. 12, pp. 2757–2772, 2009. [10] W. Mi, T. Qian, F. Wan. A fast adaptive model reduction method based on Takenaka-Malmquist systems. Systems & Control Letters, vol. 61, no. 1, pp. 223–230, 2012.
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[11] P. Saha, S. C. Patwardhan, V. S. R. Rao. Maximizing productivity of a continuous feremeter using nonlinear adaptive control. Bioprocess Engineering, vol. 20, no. 1, pp. 15– 20, 1999. [12] B. Wahlberg, P. M. M¨ akil¨ a. On approximation of stable linear dynamical systems using Laguerre and Kautz functions. Automatica, vol. 32, no. 5, pp. 693–708, 1996. [13] W. H. Kautz. Transient synthesis in the time domain. Transactions of the IRE Professional Group on Circuit Theory, vol. CT-1, no. 3, pp. 29–39, 1954. [14] B. Wahlberg. Identification of resonant systems using Kautz filters. In Proceedings of the 30th IEEE Conference on Decision and Control, IEEE, Brighton, UK, pp. 2005–2010, 1991. [15] P. S. Agachi, Z. K. Nagy, M. V. Cristea, A. I. Lucaci. Model Based Control, New York, USA: John Wiley & Sons Inc., 2007.
Shamik Misra received the bachelor degree from Heritage Institute of Technology, Kolkata and Masters degree from IIT Guwahati, India. He is now working as senior research fellow at IIT Guwahati, India. His research interests include modelling and control of chemical and biochemical processes. E-mail:
[email protected] ORCID iD: 0000-0002-1684-4174 Rajasekhara Reddy received the bachelor degree from Kakatiya Institute of Technology and Science, Warangal and master degree from Anna University, India. Currently he is a Ph. D. degree candidate at IIT Guwahati, India. His research interests include nonlinear system identification and predictive control. E-mail:
[email protected] Prabirkumar Saha received the B. Eng. degree in chemical engineering from Jadavpur University, India in 1992 and the M. Tech. and Ph. D. degrees in chemical engineering from the Indian Institute of Technology Madras, in 1994 and 1998, respectively. He has got 15 years of post-Ph. D. experience both in industry and academia. Prior to joining his present job, he had undertaken professional responsibilities at the National University of Singapore, General Electric (USA) and Cranfield University (England). Currently, he is a professor in the Department of Chemical Engineering at Indian Institute of Technology Guwahati. He has published about 70 refereed journal and conference papers. He is a member of American Institute of Chemical Engineers. He is a recipient of Fulbright-Nehru Award for International Education Administrators. His research interest include process control and liquid membrane based separation process. E-mail:
[email protected] (Corresponding author) ORCID iD: 0000-0002-1121-1829
