U-model based predictive control for nonlinear processes with

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U-model based predictive control for nonlinear processes with

Transcript Of U-model based predictive control for nonlinear processes with

U-model based predictive control for nonlinear processes with input delay
Xinpeng Geng a, Quanmin Zhu b,*, Tao Liu a, *, Jing Na c
a Institute of Advanced Control Technology, Dalian University of Technology, Dalian, 116024, P. R. China b Department of Engineering Design and Mathematics, University of the West of England, Frenchy Campus
Coldharbour Lane, Bristol, BS16 1QY, UK c Faculty of Mechanical & Electrical Engineering, Kunming University of Science & Technology, Kunming,
650500, P.R. China * Corresponding authors. Tel: +86-411-84706465; Fax: +86-411-84706706 Emails: [email protected] (X. Geng); [email protected] (Q. Zhu); [email protected] (T. Liu);
[email protected] (J. Na)
Abstract: In this paper, a general control scheme is proposed for nonlinear dynamic processes with input delay described by different models, including polynomial models, state-space models, nonlinear autoregressive moving average with eXogenous inputs (NARMAX) models, Hammerstein or Wiener type models. To tackle the input delay and nonlinear dynamics involved with the control system design, it integrates the classical Smith predictor and a U-model based controller into a U-model based predictive control scheme, which gives a general solution of two-degree-of-freedom (2DOF) control for the set-point tracking and disturbance rejection, respectively. Both controllers are analytically designed by proposing the desired transfer functions for the above objectives in terms of a linear system expression with the U-model, and therefor are independent of the process model for implementation. Meanwhile, the control system robust stability is analyzed in the presence of process uncertainties. To demonstrate the control performance and advantage, three examples from the literature are conducted with a user-friendly step by step procedure for the ease of understanding by readers.
Keywords: nonlinear dynamic control, input delay, Smith Predictor, U-model, U-Smith predictor controller, robust stability

1. Introduction
Time delay appears in various industrial operations associated with transmissions of material or energy between interconnected systems, data transmission in communication systems and networked control systems. The presence of time delay may lead to a sluggish response, limit the achievable control performance, or even provoke instability of the closed-loop systems. The main challenge for controlling such a process is to avoid overshoot in tracking a desired set-point profile and to accommodate stability and robustness against process uncertainties [1]. In particular, the input delay is a notorious barrier for the control of industrial processes with nonlinear dynamics [2, 3].
For a general review of the existing references contributing to nonlinear systems with input delay, the research development is divided in terms of continuous-time domain and discrete-time domain. In continuous-time domain, backstepping based designs were developed to deal with time-invariant input delay for control implementation [4-6]. The backstepping strategy was further extended to stabilize nonlinear systems with time-varying input delay [7], by tuning the controllers with respect to the output bound. In contrast, the finite spectrum assignment (FSA) approach was explored for retarded nonlinear systems by transforming such a nonlinear system into a delay-free form for control design [8]. A truncated predictor feedback control design based on state estimation or delay-free output estimation were developed for nonlinear processes with input delay [9, 10]. By using a high-gain output predictor, a feedback linearization design was proposed to maintain asymptotic stability of the closed-loop control system for a nonlinear process with time-varying input delay and output delay [11]. By comparison, approximate predictors and high-gain observers were adopted for estimating the delay-free state and output for control design by means of numerical computation on the ordinary differential equations of nonlinear systems [12]. To reduce the high gain of such an observer, a chain of observation algorithms was proposed to reconstruct the system state and output at different delayed time instants [13]. In contrast, a nonlinear filtered Smith predictor structure [14] was proposed for predicting the delay-free output of a nonlinear process, based on the Hammerstein type or Volterra model for describing the system dynamics. For sampled control systems implemented in
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discrete-time domain, the approximated implementation of continuous-time control design may affect or even destroy the system stability [1, 15]. Only a few papers, however, presented discrete-time control designs for nonlinear systems with input delay. Based on the delay-free output prediction, an adaptive neural network control scheme was presented to approximate the desired control input for implementation [16]. By representing a nonlinear time-delay system into a T–S fuzzy system comprised of linear delay difference inclusions, a nonlinear model predictive control (MPC) method [17] was proposed to improve system performance for disturbance rejection. A dynamic output feedback linearization control algorithm was developed by using the additive nonlinear autoregressive moving average with eXogenous inputs (NARMAX) models established by training a neural network with a specific connectivity structure [18]. Note that the robust stability under process uncertainties was left open in the above references. To deal with input delay uncertainty, a predictor based robust control design was given which could allow the input delay to be varying in a range [19]. Another predictor based control design was proposed to deal with state-dependent input delay [20]. To improve disturbance rejection performance in the presence of input delay uncertainty, a modified active disturbance rejection control (ADRC) design based on the extended state observer (ESO) was proposed [21], which could be applied to linear or nonlinear systems based on a low-order model description of the fundamental dynamics of such a system. This approach was recently extended to systems with longer input delay, by using a filtered SP in combination with ESO [22].
It should be noted that most of the existing references for nonlinear systems included the aforementioned almost gave up the Smith Predictor (SP) based control structure [23] that had been successfully used for linear systems with input delay [24]-[30], probably due to the difficulty in dealing with the related nonlinear models. It was noted [31] that SP could not be directly used for nonlinear control design because nonlinear systems generally could not be expressed by linear transfer function models. Besides, it had been recognized that the SP could not be directly used in terms of a state-space description of the system under control.
To deal with the above issues, a universal framework is proposed in this paper for controlling nonlinear processes with input delay described by different models like polynomial models,
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state-space models, NARMAX models, Hammerstein or Wiener type models, by extending the U model enabled design for delay-free nonlinear systems [32]. This paper will concentrate on the control design in the presence of time-invariant input delay, which is the foundation for solving the next step challenge of time-varying input delay. By integrating the SP with the delay-free U-model based control structure, a two-degree-of-freedom (2DOF) control scheme with output prediction is proposed for application to various nonlinear systems, based on a general U-model representation of different plant models for control design. By specifying the desired transfer functions for the set-point tracking and disturbance rejection, both controllers are analytically derived, respectively. They can be separately tuned for control performance optimization. A notable advantage of the proposed control design is that both controllers could be tuned relatively independent of the plant model or its variation, therefore facilitating practical applications. Moreover, the system performance could be monotonically tuned by the single adjustable parameter of each controller. For clarity, the paper is organised as follows. Section 2 presents the U-model representations of different classical models of nonlinear processes with input delay, which lays a foundation for the general framework of U-model based control design. Section 3 proposes a general U-model based control system design with a step by step implementation procedure. Section 4 presents robust stability analysis of the proposed control scheme. Section 5 shows three illustrative examples to demonstrate the performance of the proposed control scheme. Section 6 draws some conclusions along with potential research topics for study in the future.
Notations: Throughout the study, a series of conventional notations are used, R , Rn , and Rnm for sets of real number, n-dimension real factor, and n×m real matrix. Denote by t 1, 2, for sampling time instance.
2. U-model description of nonlinear processes with input delay
Consider a general single-input single-output (SISO) nonlinear process with input delay described by the discrete-time domain state-space model,
X (t  1)  F ( X (t),u(t  )) (1)
y(t)  h( X (t))
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where X  Rn is the state vector, u  R is the control input,   0 is time delay in the control input, y  R is the system output. F  Rn is a smooth vector function describing the model

dynamics and h  R is a smooth function relating the system states to the output. Throughout

the study, assume the system relative degree r equals to the system order n and has stable

zero dynamics (i.e., the system model has a stable inverse), and the state vector X is available

by measurement or observer.

Equivalently, the corresponding input-output polynomial model of (1) is written by

y(t)  f ( y(t 1) ,..., y(t  n), u(t 1 ) ,..., u(t  n  ))


where y(t)  R and u(t 1)  R are the output and input (also known as the controller output in control system design) signals of the plant, respectively, at the discrete-time instance t 1, 2, . f  R is a smooth linear or nonlinear function.
For delay-free nonlinear systems, it has been explored that the U-model based control strategy could establish a universal framework to design a control system based on the plant polynomial or state-space model [32, 33]. Figure 1 shows the U-model based control structure, where Gp denotes the plant, Gp1(U- model) an inverse of the plant model represented by the U-model, Gc the closed-loop controller. The U-model is defined as a polynomial of u(t) with time-varying coefficients. With the control oriented model structure, the U-model transforms a smooth (polynomial) nonlinear model of the plant into a class of polynomials, so as to make a dynamic inversion of the plant for deriving the control implementation, which is resolved by finding one of the roots of the U-model. Given a state-space model of the plant, the classical backstepping algorithm could be expanded to recursively resolve the multi-layer roots. In this way, the U-model based control scheme is capable of radically relieving the dependence on the plant model that has been the foundation of classical control system designs in the literature. The role of plant model is therefore reduced as a reference for converting to U-model and determining the next step controller output. Note that the U-model based control design takes a hypothesis on the feasibility using linear methodologies to directly design nonlinear control systems within a universal framework.
To extend the U-model based control strategy for nonlinear processes with input delay, the
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first step is to transform the plant model of polynomial, state-space, or else with time delay into the U-model, as detailed in the following subsections.
2.1 Single-layer U-model in polynomial
2.1.1 Polynomial U-model set The general polynomial model in (2) can be alternatively written in terms of a parametric
form, y(t)  f ( y(t 1) ,..., y(t  n), u(t 1 ) ,..., u(t  n  ), ) (3)  f ( p(t), )

where   [0

L ] RL1 is the associated parameter vector, L is the number of regression

terms, p  RL1 is a regression term composed of the past inputs and outputs.

Definition [34]: Assign a triplet ( X , f , h) , X is an irreducible real affine variety, ( f , h)

are mapping functions. A discrete-time process model  with delayed input space

U (t  )  Rm,t,   and output Y (t)  Rr is defined as polynomial/rational, while the
functions f   f  U and h : X  Rr both on X are mappings from input space to state

space and from state space to output space polynomial/rational, respectively. That is, for

polynomial systems, hi  A for all i  1, , r where A is the algebra of all polynomials on

the variety X , and for rational systems, hi Q for all i  1, , r where Q is the algebra of

all rational functions on the variety X .

Assume a mapping  : RL1  RM 1 and its inverse  1 exist, that is

f ( p ,  )  f (u j ,  ) , it defines the corresponding U-model structure of (3) as an input



oriented polynomial,


 y(t)  j (t)u j (t 1 )



where M is the degree of model input, the time-varying parameter vector

(t)  0(t)

M (t) RM1 is a function of past inputs and outputs,

u t  2, ,u t  n, y t 1, , y t  n, and the parameters vector  .

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Property 1: For the above manifolds f p ( pi , i ) , fu (u j , j ) , the differentiable

f ( p ,  )  f (u j ,  ) is diffeomorphism, which is bijection and its




(u j ,

 1 ) 





is differentiable as well [32].



mapping inverse

For illustration, consider a nonlinear process described by a polynomial model,

y(t)  0.5y(t 1) y(t  2)  0.7u(t  3  )u(t 1 )  0.9y(t  2)u2(t 1 )  0.1u3(t 1 ) (5)

With the above U mapping, the corresponding U-model is given by

y(t)  0(t)  1(t)u(t 1  )  2(t)u2(t 1  )  3(t)u3(t 1  )


where 0 (t)  0.5y(t 1) y(t  2) , 1(t)  0.7u(t  3  ) , 2 (t)  0.9y(t  2) , and 3(t)  0.1.

Clearly, j (t) is a time-varying function, absorbing the past variables and parameters of the

original polynomial in (5) associated with u j (t 1 ) .

Remark 1: Concerning the U-model in (4), an input oriented prototype, there is no change of

model properties compared with the classical polynomial models in representation of (3).

Therefore, the U-model and classical polynomial models are equivalent to each other. However,

the U-model is linear-like with time-varying parameters (t) as shown in (4) and therefore, is

convenient for applying the developed linear control methods based on real-time estimation of

(t) , compared to the nonlinear polynomial model in (3) used for exploring specific nonlinear

control designs in the literature (e.g. [4-8, 10-12]). Hence, the U-model can bridge various types

of polynomial models for nonlinear systems with the developed linear control system designs

within a universal framework.

2.1.2 Rational U-model set

This model set is defined as

y(t)Dp (t)  Np (t)


where Dp (t) and Np (t) are two U-polynomials, as expressed in the parametric form of



  y(t) i (t)ui (t 1 )  j (t)u j (t 1 )




This is a representation of the classical rational model set, which is expressed as a ratio of the numerator to denominator polynomials. For illustration, a rational model

y(t)  y(t 1) 1 0.8u(t 1 ) 1 0.5u2 (t 1 )

may be converted into a U-model

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y(t)[ 0   2(t)u2(t 1  )]  0(t)  1(t)u(t 1  ) 0 (t)  y(t 1) , 1(t)  0.8y(t 1) .


 0 (t)  1 ,  2 (t)  0.5 ,

2.1.3 Extended U-model set

This model set is defined as

y(k) fb (u(t 1 ))  fa (u(t 1 ))


where fb (u(t 1 ))  R and fa (u(t 1 ))  R are smooth functions that may be generally
expressed by
fb (u(t 1 ))   fbj (u(t 1 )) j (10)
fa (u(t 1 ))  faj (u(t 1 ))

For instance, consider a nonlinear process, y(t)  y(t 1)sin(u(t 1 )) , using the 1 0.5cos2 (u(t 1 ))








fa (u(t 1 ))  1(t)sin(u(t 1 )) , fb (u(t 1 ))   0 (t)  1(t) cos2(u(t 1 )) , where

 0 (t)  1, 1(t)  0.5 , 1(t)  y(t 1) .

2.2 Multi-layer U-model in state space

With reference to the U model in (4), the state-space model in (1) can be converted into a

multi-layer U model as

 











 


 

xn 1 (t


n 1
n1 j (t) xn j (t)



 


 xn (t  1)  nj (t)u j (t  )


 y(t)  h( X (t))

where, for each line, M j is the degree of next state variable x j1(t) , the time-varying parameter

vector i (t)  j0(t)






 






i 1

n is a function of the other state variables.

In the penultimate line, Mn is the degree of the model input (controller output) u(t 1) , the

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time-varying parameter vector states.

n (t)  n0(t)

nMn (t)  RMn 1 is a function of all system

Remark 2: In the multi-layer U-model shown in (11), the controller output u(t  ) can be

derived by using a backstepping routine as long as x (t 1) is determined.


2.3 U-model representations for Hammerstein and Wiener models

Considering that Hammerstein type models, Wiener type models, and NAMAX models have

also been widely used for describing various nonlinear system dynamics, the corresponding

U-model representations are given blow for the convenience of control design.

Hammerstein type models

The Hammerstein type model is a cascade structure of a nonlinear static input block and a

linear dynamic block. For describing a nonlinear process with input delay, it is in the form of

X H (t 1)  fH (u(t 1 ))

  m



y(t)  bi (t) X H (t 1 i)  ai (t) y(t  i)


i 1

where y(t) and u(t 1  ) are the output and input of the plant at the discrete-time instant

t  1, 2, , respectively, X H (t) is the output of the nonlinear input block, and fH () is a nonlinear function of the input u(t 1  ) .

The corresponding equivalent U-model can be expressed as

y(t)  0 (t)  1(t)XH (t 1)  0(t)  1(t) fH (u(t 1 ))





  0 (t)  bi (t) X H (t 1 i)  ai (t) y(t  i) , 1(t)  b0


i 1

i 1

Wiener type models

The Wiener type model is a cascade structure of a linear dynamic block and a nonlinear static

output block. For a nonlinear process with input delay, it has the form of



  X W (t)  i0 bi (t)u(t 1  i)  i1 ai (t) X W (t  i) (15)

y(t)  fW ( X W (t))

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where y(t) and u(t 1 ) denote the output and input of the plant at the discrete-time instant t  1, 2, , respectively, XW (t) is the output of the linear dynamics, and fW () is a nonlinear function of XW (t) .
The corresponding equivalent U-model can be expressed as X W (t)  0 (t)  1u(t 1 ) (16) y(t)  fW ( X W (t))




  0 (t)  bi (t) X H (t 1 i)  ai (t) y(t  i) , 1(t)  b0


i 1

i 1

NAMAX models

Generally, an NARMAX model has the form,


 y(t)  pl (t)l



where the regression terms pl (t) are the products of past inputs and outputs such as u(t 1 ) y(t  3) , u(t 1 )u(t  2  ) , y2 (t 1) , and l ( l  0,1, , L. ) are the associated


The corresponding equivalent U-model can be straightforwardly expressed as


y(t)  j (t)u j (t 1 )



where  j (t) can be viewed as a time-varying parameter absorbing pl (t) and l associated with u j (t 1  ) in (18).

3. U-model based predictive control design

3.1 Framework of U-model based control design

Without loss of generality, consider a single input ( u  R ) and single output ( y  R ) linear

feedback and delay-free control system structured with a triplet

   Flfbc Gc1 Gip 


where the linear invariant controller Gc1 : y  u and the constant unit Gip  GP1Gp  1: u  y are both within a linear feedback control framework, Flfbc .

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Input DelayOutputControl DesignNonlinear SystemsControl