外文翻譯---基于模型預(yù)測(cè)控制利用不確定集方法的魯棒優(yōu)化（英文）

1、The Robust Optimization Based Model Predictive Control using Box Uncertainty Set Endra Joelianto*, Dwita Rismayasari* and Diah Chaerani# *Bandung Institute of Technology Jalan Ganesha 10, Bandung 40132, INDONESIA E-m

2、ail: ejoel@tf.itb.ac.id #Padjajaran University, INDONESIA E-mail: d_chaerani@yahoo.com Abstract-The paper considers the application of Robust Optimization (RO) to Model Predictive Control (MPC). This optimization met

3、hodology incorporates the uncertain data, which means the data of an optimization problem is not known exactly at the time when its solution has to be determined. The robust optimization has been expanded and applied

4、to various kinds of application, in this paper, it is shown the application in MPC. The RO based MPC is simulated to a waste heat boiler control. Keyword: robust counterpart, robust optimization, model predictive con

5、trol, interior point method, quadratic cost. I. INTRODUCTION Model Predictive Control (MPC) is a kind of control algorithms which explicitly uses a model of the process to obtain the control signal by minimizing an o

6、bjective function. The model is used to predict the process output at future time instant (horizon). It has been commonly known that MPC handles constrained multivariable control problem in process industry. Knowing

7、the process output, a control sequence can be calculated to minimize the designed objective function. However, only the first element of the control signal is applied at each step to the plant, this is known as recedi

8、ng horizon strategy. The calculation is then repeated at the next sampling time using the last measurements. In the optimization process, MPC uses a linear dynamic model of the process, linear constraints of input, o

9、utput, and input increment, resulting in a linear program (LP) or quadratic program (QP) where an optimal solution can be obtained [1]. In the case where the plant dynamics are uncertain, robust MPC has been develope

10、d to tackle this problem [2-3] which applies general approach by describing uncertainties in the mathematical model of the plant using various available frameworks that available in the literatures. Next, a performan

11、ce index with respect to robustness type of the closed loop system is chosen. The robust MPC is then obtained by solving the robust optimal control sequence at each sampling interval. The receding horizon strategy is

12、applied at each sampling interval to complete the MPC algorithm. This approach limits the capability and the power of the intensive computational used in the optimization program that can handle uncertainty problems.

13、 Recently, a methodology called Robust Optimisation (RO) has been extensively studies in the area of mathematical programming and operations research. The RO methodology is designed to solve optimization problems whe

14、re data are uncertain and are only known to belong to some uncertainty sets. This method is pioneered by Ben-Tal and Nemirovski [4- 6]. In [7], the RO has been used to develop a new class of robust MPC using ellipsoid

15、al uncertainty type in the optimization. It has been shown that the proposed robust MPC has been effective in handling disturbances caused by uncertainties in the plant. Using RO, the uncertainties need not to be def

16、ined completely in the linear plant model. Instead, it can be relaxed and incorporated later in the optimization problem formulation when developing the optimization model. This paper considers a robust MPC formulated

17、 as a RO by using box uncertainty type in the robust optimization. This new class of robust MPC gives a different point of view in the designing of a robust MPC where there present uncertainties in the plant. II. PR

18、OBLEM FORMULATION Constrained MPC can be formulated in a linear, discrete- time, state-space model of the plant [8], in the form ) ( ) ( ) 1 ( k u B k x A k x d d + = +(1a) ) ( ) ( k x C k y d =(1b) ) ( ) ( k x C k z z

19、=(1c) where y(k), z(k), u(k), and x(k) represent its system outputs, controlled outputs, inputs, and states respectively. In this paper, it is assumed that the system outputs are equal to controlled outputs. The pred

20、icted outputs is obtained by iterating the model ∑ =? ? + + =+ = +ij d j d d i d ddk j i k u B A C k x A Ck i k x C k i k y11 ) | ( ? ) () | ( ? ) | ( ?(2) By collecting the predicted output into a vector, then the pre

21、dicted outputs are obtained in a vector form The cost function subjects to linear inequality in the inputs, input increments, and outputs 1 ,..., 1 , 0 , max min ? = ≤ ≤ + u i k H i u u u1 ,..., 1 , 0 , max min ? = Δ ≤

22、 Δ ≤ Δ + u i k H i u u u(11) p i k H i y y y ,..., 2 , 1 , max min = ≤ ≤ +To make it more explicit, all inequalities can be combined together into a single set k k k k D L Mx Fu U ? ? 1 + + + ≤ ΩΔ ? β(12) The inequalit

23、y matrix needs to be composed once and use it for every optimization since constraints are constant. III. MPC WITH ROBUST COUNTERPART This section discusses the robust counterpart methodology as proposed by Ben-Tal an

24、 Nemirovski [4-6] and extended by Chaerani [9]. The robust counterpart is one of the existing methodologies for handling uncertainty in the data of an optimization problem. The most important thing in this methodolog

25、y is how and when the robust counterpart of the uncertain problem can be reformulated as a computationally tractable optimization problem. Consequently, the robust counterpart highly depends on the selection of the un

26、certainty set. One of the models of uncertainty set proposed by Ben-Tal and Nemirovski [4-6] is the box uncertainty set. The advantage of this methodology is that the resulting of this optimization problem belongs to

27、 the class of Conic Optimization (CO) that is Linear Optimization (LO), Conic Quadratic Optimization (CQO), or Semi Definite Optimization (SDO) which can be solved by the interior point methods. In the following, it

28、will be explained how to obtain the RC by assuming that there is an uncertainty on the objective function and the uncertainty is modeled as a box uncertainty set. Recall the objective function of MPC in (10) and (11)

29、can be written as ( )ω ≤ ΩΔΔ ? Δ Δ = Δ ΔUU G U H U J T TU U ? . s.t? ? ? min min ? ?(13) and assume that the uncertainty is in G from sensor errors, measurement noise and disturbance. The uncertain MPC problem is the

30、n given by ( ) { } U ∈ ? ≤ ΩΔ Δ ? Δ Δ Δ G U U G U H U T TU , ? : ? ? ? min ? ωRemove the uncertainty from the objective function, then the uncertain MPC problem becomes τ min, ? ? ? . . U G U H U t s T T Δ ? Δ Δ ≥ τ, ?

31、 ω ≤ ΩΔU U ∈ ?G(14) In this paper, the uncertainty set is defined as a box uncertainty set as follows ( ) { } n n n G G G G ) 1 ( 1 : γ γ + ≤ ≤ ? = U(15) where γ is the uncertainty value given 0 ≥ γ , Gn is nominal

32、vector of G. The feasibility of (τ, U ? Δ ) on the constraints set guarantees that the vector U ? Δexists and satisfies the constraints. Using the worst case approach principal, it can then be obtained ( ) ( ) [ ] {

33、} [ ] U G G G G U H U T n nGT ? 1 , 1 : max ? ? Δ + ? ∈ ? Δ Δ ≥ ∈ γ γ τ U(16) The minimum value of the right hand side of the equation (16) will be determined when G is maximum, i.e. ( ) n G G γ + = 1Thus, the robust

34、counterpart of constrained MPC problem is τ min( ) [ ] 0 ? 1 ? ? s.t. = Δ + ? Δ Δ + ? U G U H U T n T γ τ(17) ω ≤ ΩΔU ?The above equation can be obtained as a conic quadratic problem; with H is a positive semi definit

35、e symmetric matrix. H can be decomposed as H = STS, such that U S U H U T ? ? ? Δ = Δ Δ . Assume that β ≤ ΔU S ? , then the MPC with Robust Counterpart can be reformulated as ( ) [ ]βωγ β ττ≤ Δ≤ ΩΔ= Δ + ? + ?U

36、 SUU G T n??0 ? 1 s.t.min(18) The robust optimization now is in a class of conic optimization problems. The next section will discuss about how to solve the problem (18) using a software package, called SeDuMi which

37、stand for self-dual minimization that was developed by J.F. Sturm [10]. IV. AN ILLUSTRATIVE EXAMPLE The model predictive control with robust optimization is implemented to a waste heat boiler having five outputs, thre

38、e inputs, and one measured disturbance. The outputs are steam pressure, furnace temperature, HP drum level, steam flow rate and steam temperature, while the inputs are fuel flow rate, boiler feed water flow rate to H

39、P drum and boiler feed water flow rate to a super-heater, and the measured disturbance is exhaust gas temperature. The state space equation in discrete- form [11] is given by the following equation ) (0073 . 00005 . 00

40、014 . 00007 . 00046 . 0) (2405 . 0 0107 . 0 0011 . 00408 . 0 0102 . 0 0000 . 00488 . 0 0052 . 0 0001 . 01127 . 0 0114 . 0 0000 . 02594 . 0 0064 . 0 0006 . 0) (9559 . 0 0057 . 0 0216 . 0 0048 . 0 0154 . 00024 . 0 9922 . 0

41、 0125 . 0 0002 . 0 0007 . 00013 . 0 0024 . 0 9441 . 0 0021 . 0 0009 . 00189 . 0 0028 . 0 4013 . 0 0072 . 1 0347 . 00564 . 0 0104 . 0 1790 . 0 0038 . 0 9792 . 0) 1 (k d k uk x k x? ? ? ? ? ???? ? ? ? ? ??????+? ? ? ? ? ??