Survey Paper:Optimization in Model Predictive Control

M. A. AbbasFEAS, UOIT, Canada.

Abstract— This paper presents a comprehensive survey ofdifferent optimization methods used in Model Predictive Con-trol (MPC) systems. We discuss optimization methods forboth linear and non-linear systems and describe advan-tages/disadvantages for each method and also the specificconditions in which each method should be used. State of theArt in this field is described together with current and futureresearch trends.

I. INTRODUCTION

Control theory is a branch of engineering and mathematicswhich deals with the behaviour of dynamical systems[1].There are many control strategies in use today like intelligentcontrol, adaptive control, stochastic control, optimal controletc. Optimal control is such a control technique in which weminimize certain cost index to achieve desired performance.The two types of optimal control techniques are• Linear Quadratic Gaussian (LQG)• Model Predictive Control (MPC)In this survey we consider optimization problems in Model

Predictive Control technique only because it is most widelyused in industry as opposed to LQG which was termed asfailure. The reasons cited for this failure are [2], [3]:• constraints• process nonlinearities• model uncertainty (robustness)• unique performance criteria• cultural reasons (people, education, etc.)

A. Brief History

History of optimal control can be traced back to 1960swhen two ground breaking paper by Kalman appeared[4], [5]. These papers were in fact the first to introducethe algorithm for computing the state feedback gain ofthe optimal controller for a linear system with a quadraticperformance criterion. Kalman introduced the notions ofcontrollability and observability, and their exploitation inthe regulator problem, which is considered as principalcontribution of the paper. These papers had a significanteffect on researchers working in the field of optimalcontrols. It was this development of of Linear QuadraticGaussian (LQG) controller that later led to the developmentof Model Predictive Control theory. MPC is basically aform of LQG controller with added finite prediction horizonand constraints handling.

Unconst. infinite horizon Linear MPC = simple LQG

Until 70s MPC was only used for plants with slow dy-namics. It was widely applied in petro-chemical and relatedindustries where satisfaction of constraints was particularlyimportant because efficiency demands operating points on orclose to the boundary of the set of admissible states andcontrol. One of the primary advantages of this techniqueis its explicit capability to handle constraints. However, thefact that the optimisation procedure is to be repeated everytime-step, is the reason that the application of MPC hasbeen limited to the slow dynamics of systems in the processindustry until recently. The boom in MPC started in 1990swhen faster computer became available together with rapiddevelopment of optimization algorithms. These days MPC isapplied to various types of plants with fast dynamics suchas airplanes, satellites, robotics, automotives etc.

B. Purpose of this survey

With our discussion above, many theoretical issues arisein MPC by application of control law. One of the majorissues in model predictive control is finding the appropriateoptimization algorithm to be employed in order to reducefuture errors. In our survey, we focus on these varieties ofoptimization methods. This is, however, a general surveycovering common optimization techniques used in MPC.Due to space limitation, we do not go into detail of eachalgorithm, rather touch many of them with little detail andfocus on general trends and methodology employed.

C. Organization of paper

This paper is organized in 6 sections. Section I providesintroduction to the problem in hand, whereas Section IIformulates the problem mathematically. In Section III andIV we survey various optimization methods for Linear andNonlinear MPC respectively. Section V details some practicalimplementations of MPC in industrial processes. In sectionIV we conclude our discussion and predict some futureresearch trends.

II. PROBLEM FORMULATION

Model predictive control (MPC) algorithms utilize anexplicit process model to predict the future response of aplant. At each control interval an MPC algorithm attemptsto optimize future plant behaviour by computing a sequenceof future manipulated variable adjustments. The first inputin the optimal sequence is then sent into the plant, and theentire calculation is repeated at subsequent control intervals.Thus optimizer solves an optimization problem during each

sampling interval of the controller.Figure 1 shows the oper-ation of MPC

Fig. 1. MPC operation concept

Mathematically, if we have Y = process output and u =controller input

Ydesired(t) = a1u(t− h) + a2u(t− 2h) + ......

+b1y(t− h) + b2y(t− 2h) + .... (1)

Ypredicted(t) = a′

1u(t− h) + a′

2u(t− 2h) + ......

+b′

1y(t− h) + b′

2y(t− 2h) + .... (2)

where t = t − h is the time step. Equation (1) describessystem desirable input/output behaviour of our system toachieve goals within its operational limits. Meanwhile equa-tion (2) empirically predicts system’s future behaviour basedon system’s past input u(t− h) and output y(t− h).

The error between predicted and desired output can becalculated as

e(t) = Ypredicted(t)− Ydesired (3)

Uoptimized(t) = ce(t) (4)

Equation (4) scales the error by c and predicts the cal-culated optimized input Uoptimized(t) based on previouscalculations. Equations (1),(2),(3) and (4) describe systembehaviour for one future time step. We now iterate the fourequations for time t + h,t + 2h,.... to predict system futureI/O profile similar to that shown in Figure 1. By this iterationwe also get associated errors e(t), e(t + h), e(t + 2h) + ...for each time step. To optimize the system i.e., minimizethe deviation between predicted output of the system from

desired behaviour we need to define the system’s futureresidual function.

Residual = e2(t+ ih), i = 0, 1, .., n (5)

Our objective is to use appropriate optimization algorithmsand find future inputs that minimize this residual i.e., findMin(Residual). With the value of c well defined, ouroptimized input Uoptimized is now well defined. The wholeoptimization procedure is depicted in Figure 2

Fig. 2. MPC Optimization Flow

To probe further, we divide our discussion into two sec-tions• Linear Model Predictive Control• Non-Linear Model Predictive ControlBoth linear and nonlinear systems have specific problem

statements and utilize different optimization methods. It isimportant to describe each of them separately. Below is thebrief discussion and optimization methods used for each ofthese types of MPC.

III. LINEAR MPC

A. Problem Formulation

Consider a linear time-invariant and discrete-time systemdescribed by following set of equations:

xt+1 = Axt +But (6)yt = Cxt (7)

subject to the following constraints

ymin ≤ yt ≤ ymax (8)umin ≤ ut ≤ umax (9)

where xt ∈ Rn,ut ∈ Rm and yt ∈ Rp are state, input andoutput vectors respectively. Subscripts min and max denote

lower and upper bounds respectively. Generally our objectivein linear MPC is minimization of cost function of the form

J = xtQx+ utRu (10)

Majority of industrial MPC applications use linear empiricalmodels, therefore most of MPC products and optimizationalgorithms are based on this model type.

B. Optimization Methods

Main challenge in MPC is to find fastest ways of optimiza-tion as time required for solving on-optimization is very lim-ited. Thus we need real-time optimal solution. Sometimes wefind a trade off by looking for a suboptimal solution which isless complex. Constraints are linear. Exact solution methodsare well studied for linear MPC. The optimal solution relieson a linear dynamic model of the process, respects all inputand output constraints, and minimizes a performance figure.This is usually expressed as a quadratic or a linear criterion,so that the resulting optimization problem can be cast as aquadratic program (QP) or linear program (LP), respectively,for which a rich variety of efficient active-set and interior-point solvers are available[10].

1) Linear Programming (LP): A few authors have inves-tigated MPC optimization based on linear programming [6],[7], [8]. If we have objective function of the form

minJ = min ||PxN ||∞+

N−1∑k=0

||Qxk||∞ + ||Ruk||∞

(11)

subject to constraints

Gz ≤W + Sx(t)

then MPC law can be defined by the solution of a LinearProgram[9]. Schechter[9] proved that this is true for anysum of convex piecewise affine costs.

2) Algebraic Method: If we consider the following objec-tive function:

minJ = x′t+Ny|tPxt+Ny|t+

Ny−1∑k=0

x′t+k|tQxt+k|t

+u′t+kRut+k (12)

subject to constraints

ymin ≤ yt ≤ ymax, k = 1, 2, ..., Nc

umin ≤ ut ≤ umax, k = 0, 1, ..., Nc

and system dynamics

xt+k+1 = Axt+k|t +But+k, k ≥ 0,

yt+k|t + Cxt+k|t, k ≥ 0,

ut+k = Kxt+k|t, Nu ≤ k ≤ Ny,

where matrices Q = Q′ ≥ 0, R = R′ ≥ 0 and P ≥ 0.Nu, Ny, Nc are input horizon, output horizon and constrainthorizon respectively such that Ny ≥ Nu and Nc ≤ Ny −

1. We solve this problem (12) repeatedly at eact time tfor current measurement xt and predicted state variable,xt+1|t, ..., xt+k|t at time steps t + 1, ..., t + k and corre-sponding optimal control actions U∗ = {u∗t , ..., u∗t+k−1} isobtained. The first predicted input is applied to the system asfirst control action i.e., ut = u∗t . This procedure is repeatedat time t+ 1 based on new state xt+1.

The the tuning cost function matrix P and state feedbackgain K is generally used to guarantee closed loop stability ofststem (12). Algebraic solution of this system depends uponfinding the values of P and Q matrices. P is found by thesolution of discrete Lyapunov equation

P = A′PA+Q.

Assuming the problem is unconstrained, infinite horizoni.e., Nc = Nu = Ny = ∞ we can find the state feedbackgain K by solving the Algebraic Ricatti equation:

K = −(R+B′PB)−1B′PA,

P = (A+BK)′P (A+BK) +K ′RK +Q

Solution of Lyapunov and Algebraic Ricatti equations isthe most popular method to find the values of K and Pmatrices[13], [14] thus solving the problem algebraically.

3) Quadratic Programming (QP): Rawlins and Morari[10], [11], [12] proved that linear MPC can be posed asQuadratic Programming (QP) problem. If we incorporate thefollowing realtion

xt+k|t = Akxt+

k−1∑j=0

AjBut+k−1kj

into system represented by set of equations (12) then it givesus the following quadratic programming (QP) optimizationproblem [18]

J∗(xt) = min1

2U ′HU + x′tFU +

1

2x′tY x(t) (13)

subject to constraints

Gz ≤W + Sx(t)

where U , [u′t, ..., u′t+Nu−1]′ ∈ Rs, and s , mNu, is

a vector of optimization variables, H = H ′ � 0, andH,F, Y,G,W,E matrices are obtained from state constraintmatrix S and input matrix R. MPC is applied by solvingQP problem (13) repeatedly at each time t ≥ 0 for xt, thecurrent state value .Despite the fact that efficient QP solversare available to solve , computing the input ut online mayrequire significant computational effort [15].

4) Multi Parametric Quadratic Programming (mp-QP):In MPC our goal is to reduce online optimization timebecause system operates in real-time. These days, substantialresearch is being carried out to fiend more efficient opti-mization algorithms. Bemporad et. al. [13], [16], [17] solvedthe problem (12) by multiparametric quadratic programming

TABLE IADVANTAGES / DISADVANTAGES OF OPTIMIZATION METHODS USED IN

LINEAR MPC

Algebraic LP QP Constrained QPDifficulty Small Medium Large Larger

Optimization No Yes Better BetterConstraints No Yes No Yes

(mp-QP) which avoids repetitive optimization. They trans-formed the QP problem (13) into multiparametric optimiza-tion problem by using the following linear transformation:

z , U +H−1F ′xt

where z ∈ Rs is optimization variable parameter. Then QPproblem (13) can be written as following (mp-QP) problem:

Vz(xt) = min1

2z′Hz (14)

subject to constraints

Gz ≤W + Sxt

where xt is vector of parameters and

S = E +GH−1F ′

The advantage obtained from such a formulation is that xtonly appears in right hand side of constraints and not inobjective function. As opposed to Equations (13) where statevector xt appears on right hand sides of both constraints andobjective function. Thus in Equation (14) z can be obtainedas affine function of x for complete feasible space of x [15].Vassilis et.al [15] proved that the set of feasible parametersXf ⊆ X is convex, the optimal solution, z(x) : Xf 7−→Rs is continuous and piecewise affine, and the optimzationobjective function Vz(x) : Xf 7−→ R is continuous, convex,and piecewise quadratic.

According to our survey the optimization methods can betabulated according to Table I.

Normally, limits on the storage space or the computationtime restrict the applicability of model predictive controllers(MPC) in many real problems. Morari et. al. [19] introduceda new approach combining the two paradigms of explicitand online MPC to overcome their individual limitations.His developed algorithm computed a piecewise affine ap-proximation of the optimal solution that warm-started anactive set linear programming procedure. A pre-processingmethod was introduced that provided hard realtime, stabilityand performance guarantees for controller. Doing so, theresearcher was able to trade off some optimization perfor-mance in-order to find faster processing time. Advantagewith this method is that its easy to implement and makes on-line evaluation faster. Disadvantage is that, with increasingstates of system, the controller regions may become largethus making algorithm difficult to implement.

IV. NONLINEAR MPC

Model predictive control (MPC), also referred to as mov-ing horizon control or receding horizon control, has becomean attractive feedback strategy, especially for linear processes[21]. Many systems are, however, in general inherentlynonlinear. This, together with higher product quality specifi-cations and increasing productivity demands, tighter environ-mental regulations and demanding economical considerationsin the process industry require to operate systems closer tothe boundary of the admissible operating region. In thesecases, linear models are often not good enough to describethe process dynamics and nonlinear models have to be used.Fortunately, considerable progress has been achieved in thelast decade that allows to reduce both, computational delaysand approximation errors. This progress is possible by thedevelopment of dedicated real-time optimization algorithmsfor NMPC and moving horizon estimation (MHE) that allowsnowadays applying NMPC to plants with tens of thousandsof states or to mechatronic applications. By now linear MPCis widely used in industrial applications (Qin and Badgwell;Garca et al; Morari and Lee; Froisy)[22]. The basic structureof nonlinear MPC is depicted in Figure 3

Fig. 3. Basic NMPC control loop [21]

The basic NMPC scheme works as follows:1) Obtain measurements/estimates of the states of the

system2) Compute an optimal input signal by minimizing a

given cost function over a certain prediction horizonin the future using a model of the system

3) Implement the first part of the optimal input signal untilnew measurements/estimates of the state are available

4) Continue with 1

A. Problem Formulation

Consider a nonlinear discrete time dynamic system de-scribed by following set of equations[23]

xk = fts(xk−1, uk−1)

yk|k−1 = g(xk)

subject to the following constraints

U = u ∈ Rm|umin ≤ u ≤ umax

X = x ∈ Rn|xmin ≤ x ≤ xmax

Where xk is a vector of states, uk is the vector ofmanipulated inputs, and yk is a vector of outputs.ts is sampletime; the k|k − 1 subscript notation is used to indicate theprediction at step k based on measurements at step k-1. Hereumin, umax and xmin , xmax are given constant vectors. Theerror in the model is calculated by this equation

dk = yk − yk|k−1

so the objective is to minimize the error as much as possibleto get the optimal output which is given by this equation

xk+1 = fts(xk), uk

yk+1|k = g(xk+1) + dk

Based on this dynamic system form, the following sim-plified optimal control problem in discrete time is given bythese set of equations

minimizex,z,u

N−1∑i=0

Li(xi, zi, ui) + E(xN ) (15)

subject to x0 − x̄o = 0 (16)xi+1 − fi(xi, zi, ui) = 0, i = 0, . . . , N − 1, (17)

gi(xi, zi, ui) = 0, i = 0, . . . , N − 1, (18)hi(xi, zi, ui) ≤= 0, i = 0, . . . , N − 1, (19)

r(xN ) ≤ 0. (20)

B. Optimization Methods for Nonlinear MPC

There are many methods to solve these problems, in thissurvey we cover two methods to solve these NMPC problems• Newton Type Optimization• Numerical Method1) Newton type optimization: Newton’s method for solu-

tion of a nonlinear equation R(W) =0 starts with an initialguess that W 0 and generates a series of iterates W k that eachsolves a linerazation of the system at the previous iterates,i.e, for given W k the next iterate W k+1 shall satisfy

R(W k) +5(W k)T (W k+1 −W k) = 0.

Newtons method has locally a quadratic convergance ratewhics is as fast as making any numerical analyst happy[22].Ifthe jacobian 5R(W k)T is not computed or inverted exactly,this leads to slower convergance rates , but cheaper iteration,and gives rise to the larger class of ”Newton type methods”.Agood overview of the field is given in [24].

The NMPC problem as stated above is the speciallystructured form of a generic nonlinear program that has theform

minimizeX

F (X) s.t

{G(X) = 0

H(X) ≤ 0

for the optimal solution X* must satisfy the famous Karush-Kuhn-Tucker(KKT) conditions which are:

5x(X∗, λ∗, µ∗) = 0 (21)G(X∗) = 0 (22)

0 ≥ H(X∗) ⊥ µ∗ ≥ 0 (23)

Here we have used the defination of Lagrange function

L(X,λ, µ) = F (X) +G(X)Tλ+H(X)Tµ

and the symbol ⊥ between the two vector valued inequal-ities in (23) that also the complementarity condition shouldhold.

All the Newton type optimization try to linearize theproblem functions and for this they use Sequential QuadraticProgramming (SQP) type method.

a) Sequential Quadratic Programming: First step tosolve the KKT system is to linearize all nonlinear functionsappearing in (21)-(23) by using the conditions of quadraticprgramming(QP)

minimizeX

F kQP (X)s.t

{G(Xk) +5G(Xk)T (X −Xk) = 0

H(Xk) +5H(Xk)T (X −Xk) ≤ 0

with objective function

F kQP (X) = 5H(Xk)T (X) +

1

2(X −Xk)T (24)

52xL(Xk, λk, µk)(X −Xk)

52xL(Xk, λk, µk) is called the Hessian matrix it is positive

semidefinite, this QP is convex so that global solution canbe found reliably. This general approach to address the non-linear optimization problem is called Sequential QuadraticProgramming(SQP).

b) Powell’s Classical SQP Method: One of the mostsuccessfully used SQP variants is due to Powell [25]. Ituses exact constraint Jacobians, but replaces the Hessianmatrix 52

xL(Xk, λk, µk)by an approximation Ak. Each newHessian approximation Ak+1 is obtained from the previousapproximation Ak by an update formula that uses the differ-ence of the Lagrange gradients,

γ = 5xL(Xk+1, λk+1, µk+1)−5xL(Xk, λk+1, µk+1)(25)

Aim of these Quasi-Newton or Variable-Metric methodsis to collect second order information in Ak+1 by satisfyingthe secant equation

Ak+1σ = γ.

The most widely used update formula is the Broyden-Fletcher-Goldfarb-Shanno (BFGS) update[26]

Ak+1 =Ak + γγT

(γTσ)− Akσσ

TAk

(σTAkσ)

Quasi-Newton methods converge very linearly under mildconditions, and had a tremendous impact in the field of

nonlinear optimization. Successful implementations are thepackages NPSOL and SNOPT for general NLPs [27], andMUSCOD-II [28] for optimal control.

c) Constrained Gauss-Newton Method: Another partic-ularly successful SQP variant the Constrained (or General-ized) Gauss-Newton method is also based on approximationsof the Hessian. It is applicable when the objective functionis a sum of squares:

F (X) =1

2||R(X)||22

For this case Hessian is defined as

Ak = 5R(Xk)5R(Xk)T

the corresponding QP objective is defined as:

F kQP (X) =

1

2||R(Xk) +5R(k)T (X −Xk)||22

The constrained Gauss-Newton method has only linearconvergance but often with a surprisingly fast contractionrate. Newton type SQP methods use approximation of Hes-sain, as well as the constrained jacobians which was analysedin [31],[29] It uses approximations Ak, bk, ck of the Hessianmatrix whic is already defined in the SQP and also called”modified gradient”.

ak = 5xL(Xk, λk, µk)−Bkλk − Ckµ

k (26)

Now QP objective is defined as

F kadjQP (X) = aTkX +

1

2(X −Xk)TAk(X −Xk)

and this is the equation of QP which is solved in eachiteration:

minimizeX

F kadjQP (X)s.t

{G(Xk) +BT

k (X −Xk) = 0

H(Xk) + CTk (X −Xk) ≤ 0

In this equation it is shown that by using a modifiedgradient ak allows to locally converge to solutions of theoriginal nonliner NLP even in the presence of inequalityconstraint Jacobians [29], [30], [31].

2) Numerical Method: When Newton type optimizationis applied to the optimal control problem (15)-(20). Byusing sequential approach, where all state variables x, z areeliminated and the optimization routine only sees the controlvariables u, the specific optimal control problem structureplays a minor role. Thus, often an off-the-shelf code fornonlinear optimization can be used. This makes practicalimplementation very easy and is a major reason why thesequential approach is used by many practitioners [32].

a) The Linearized Optimal Control Problem: Let usregard the linearization of the optimal control problem (15)-(20) within an SQP method, which is a structured QP. It turnsout that due to the dynamic system structure the Hessian ofthe Lagrangian function has the same separable structure asthe Hessian of the original objective function (26), so thatthe quadratic QP objective is still representable as a sum oflinearquadratic stage costs, which was first observed by Bock

and Plitt [33]. Thus, the QP subproblem has the followingform:

minimizex,z,u

N−1∑i=0

LQP,i(xi, zi, ui) + EQP (xN ) (27)

subject to x0 − x̄0 = 0(28)

xi+1 − f′

i − F xi xi − F z

i zi − Fui u = 0, i = 0, . . . , N − 1,

(29)

g′

i +Gxi xi +Gz

i zi +Gui ui = 0 i = 0 . . . , N − 1,

(30)

h′

i +Hxi xi +Hz

i zi +Hui ui ≤ 0, i = 0, . . . , N − 1,

(31)

r′+RxN

≤ 0.

This partially reduced QP can be post-processed either bya condensing or a band structure exploiting strategy[32].

C. Advantages and Disadvantages of NMPC

In general one would like to use an infinite predictionand control horizon, to minimize the performance objectivedetermined by the cost. However, solving a nonlinear optimalcontrol problem over an infinite horizon is often computa-tionally not feasible. Thus typically a finite prediction hori-zon is used. In this case the actual closed-loop input and statetrajectories differ from the predicted open-loop trajectories,even if no model plant mismatch and no disturbances arepresent. This can be explained considering somebody hikingin the mountains without a map. The goal of the hiker is totake the shortest route to his goal. Since he is not able tosee infinitely far (or up to his goal), the only thing he cando is to plan a certain route based on the current information(skyline/horizon) and then follow this route. After some timethe hiker re evaluates his route based on the fact that hemight be able to see further. The new route obtained mightbe significantly different from the previous route and he willchange his route, even though he has not yet reached the endof the previous considered route [21]. Basically, the sameapproach is employed in a finite horizon NMPC strategy.At a recalculation instant the future is only predicted overthe prediction horizon. At the next recalculation instant theprediction horizon moves further, thus allowing obtainingmore information and re-planning.

V. MPC IN INDUSTRY

For complex constrained multivariable control problems,model predictive control (MPC) has become the acceptedstandard in the process industries [20]. So we found it worth-while to extend our survey to different practical applicationsand technologies being used in the industry.

A. Applications

According to Badgwell [40] there are 4500+ succesfulindustrial applications of linear MPC and 50+ applicationsof nonlinear MPC. In Table II some applications of MPC are

TABLE IIMPC APPLICATIONS

Application Sampling Rate Company

Integrated room automation [34] 0.002 Siemens

Adaptive cruise control [35] 2 Chrysler

Mechanical systems with backlash [36] 25 –

Car automatic steering [37] 30 Ford

Automotive hybrid traction control [38] 50 Ford

Electronic throttle control [39] 200 Ford

DC-DC voltage Inverters [41] 10× 103 –

Induction motors torque control [42] 40× 103 ABB

DC-DC converters / power balance [43] 50× 103 STM

tabulated with emphasis on controller operating frequency,which is the most critical constraint in the development ofindustrial MPC.

B. Commercial Technologies and Softwares

There are many versions of MPC software dependingupon developer group. These different versions are similarin principles but differ in implementation procedure, modeltype, objective function and optimization method used.Below we list some of the most popular MPC technologiesused in industry and optimization procedure used in each ofthem:

• IDCOM (Identification and Command) [44], 1987– Company: Set point, Inc. USA– Methodology: Model algorithmic control– Model Type: Impulse response, linear in inputs or

internal variables.– Optimization method: QP method– Other Features: Direct interface to Honeywell MPC

systems, input and output constraints included inthe formulation, 1st generation technology.

• DMC (Dynamic Matrix Control) [45], 1985– Company: Shell Co.– Methodology: Dynamic Matrix control– Model Type: Step response– Optimization method: LP method– Other Features: Direct interface to Honeywell MPC

systems, 1st generation technology.• OPC (Optimum Predictive Control), 1987

– Company: Treiber Controls, Inc.– Model Type: Step response– Optimization method: LP method– Other Features: Controller design and simulation

can be performed on personal computers• PCT (Predictive Control Technology), 1994

– Company: Profimatics, Inc.– Model Type: Combines aspects of IDCOM and

DMC– Optimization method: Solves optimization for one

control move only.

– Other Features: 3rd generation technology.• HMPC (Horizon multivariable Predictive Control), RM-

PCT (Robust Model Predictive Control Technology),1991

– Company: Honeywell.– Other Features: Different to any other scheme, no

data available due to proprietary reasons.• DMC-plus and RMPCT, 2000

– Company: Honeywell + Profimatics + Treiber Con-trols

– Model Type: Linear and non linear– Optimization method: Multiple optimization meth-

ods depending on control objective.– Other Features: 4th generation cutting edge tech-

nology in use today.

VI. CONCLUSION AND FUTURE RESEARCH TRENDS

In this paper, we presented a survey of different opti-mization methods use in linear / non-linear MPC. MPCtechnology has progressed steadily since its conception. Withthe availability of faster computing power, it has now becomepossible to implement better optimization methods (requiringimmense computing power) for control systems. There isplenty of room to develop new algorithms and prove newfacts in MPC domain. Also, there is a need to extendcurrent MPC implementations to new areas because existingimplementation domains are uneven. Many researchers areworking to reduce time and find better solutions to opti-mization problem. MPC is finding new application on small-scale, fast loops as well as large-scale, networked systems.Specifically researchers are working in following fields:• Reducing complexity of online optimization [47]• Reduce complexity of explicit solution (reduce number

of regions) [48]• Combination of online and explicit off-line optimization• MPC Optimization for non-linear plant models• Robustness of MPC• MPC for stochastic systems [49]• Adaptive MPC• MPC for switched / hybrid systems• MPC for hierarchical / decentralized structures

In future we hope to see amazing developments to fillin the vacuum in the field of optimal control systems.

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