Design of Fixed Structure Preview Controller with Constraints

Nidhika Birla Research Scholar, Department of Electrical Engineering,

National Institute of Technology, Kurukshetra Haryana, India

nidhikabirla@gmail.com

Akhilesh Swarup Professor, Department of Electrical Engineering,

National Institute of Technology, Kurukshetra Haryana, India

akhilesh.swarup@gmail.com

Abstract—This paper presents the design of fixed structure Preview Controller with multiple objectives in constrained environment as min-max problem. The paper explains the novel MOCC-PSO (Multi-Objective Constrained Co-evolutionary Particle Swarm Optimization) procedure to solve this optimization problem, with techniques to handle infeasible solution. The proposed procedure is tested and verified for control problems using MATLAB platform. The system models are taken from the literature. The results validate the ability of the procedure in terms of the quality of solution obtained in the constrained environment and the ease to implement the non-conventional objectives and constraints.

Keywords-Preview Control, Discrete – time System, State Feedback Control, Co-Evolutionary PSO

I. INTRODUCTION The field of Preview Control primarily deals with fixed-

structure controllers as the length of preview is fixed depending on the available resources. Also, the availability of the computer resource and inevitability of on-site controller tuning in practical engineering problems confirm the significance of fixed-structure controllers.

The preview controllers are mostly designed using robust control design techniques that include H∞ performance criteria. The classical solution approach to Preview Control problem is given using H∞ control and state augmentation, solved using Algebraic Riccati Equation. The mathematical formulation and solution of the H∞ Preview Control problem is given by A. Kojima, et. al. and G. Tadmor, et. al., for preview compensation, output feedback setting and fixed lag smoothing [1-4]. The discrete version of the preview control problem and its various issues are studied with numerical examples by Polyakov, et. al. [5]. Y. Kuroiwa, et. al. have analysed the H∞ Preview Control problem for the systems with delay [6]. M. M. Negm, et. al. have synthesized Optimal Preview Control for three-phase induction motor [7]. Analysis and Design of H∞ Preview Tracking Control.

Systems and its various variations using state augmentation have also been studied [8-10]. The solutions to all the problems are given using Algebraic Riccati Equation for the continuous and Discrete Algebraic Riccati Equation for the discrete – time systems. Other techniques proposed include H2/H∞ criteria. The solution approaches also includes linear matrix inequalities

(LMI) formulae. Robust H2 and/or H∞ tracking problems for linear time invariant systems have been studied using LMIs [11-12]. A design method of a state feedback controller with integral and preview actions to achieve the robust LQ tracking performance by using LMI approach for various systems is also available in the literature [13-14].

A study about the criterion reveals that H∞ performance criteria guarantees the robustness aspects while the H2 performance criteria LQG or noise insensitivity behaviour of the system. Thus, even with a mixed H2/H∞ design the transient behaviour is a secondary feature in the optimization problem. Also, design of controller with multiple specifications is a difficult task and requires in-depth knowledge of robust control theory and semi-definite programming (SDP) that may not be an easy task for most design engineers. Thus, it generates the requirement of a technique that can solve the fixed-structure Preview Control optimization problem with multiple objective functions and constraints on transient behaviour for practical purposes.

Particle Swarm Optimization (PSO), a heuristic technique for automated optimization of multi-dimensional problems, introduced by Kennedy and Eberhart [15-16] has attracted many researchers working in the field of optimization. It is inspired from the birds’ flocking behaviour. Instead of using evolutionary operators to manipulate the individuals, each individual in PSO flies in the search space with a velocity which is dynamically adjusted according to its own flying experience and its companions’ flying experience. Many improvements in the basic algorithm are proposed to improve the performance of the PSO algorithm. The integration of concept of multiple parallel populations of Genetic Algorithms has given rise to co-evolutionary PSO algorithm [17]. The concept of co-evolutionary PSO can easily optimize the fixed-structure Preview Control problem, for multiple objectives and constraints, with multiple parallel populations.

Thus, a novel method for the design of fixed structure preview controller satisfying multiple objective functions and constraints using Co-Evolutionary PSO approach is proposed in this paper. The paper is organised into five sections. The first section introduces the literature and the objective of the paper. The problem of fixed structure preview controller synthesis is formulated as min-max problem in the second section. The next section explains the solution methodology for the problem

using Multi-Objective Constrained Co-evolutionary Particle Swarm Optimization (MOCC-PSO) approach. The fourth section presents the results of application of the methodology to benchmark control systems and the conclusion is drawn in the last section.

II. PROBLEM FORMULATION A simple SISO Preview Tracking problem is shown in

figure 1. The Preview Control System has its control signal dependent on the present error (that is augmented to be a part of system’s state) between reference signal and system output and the future information available for the reference signal or the disturbance and the present condition of system states.

Figure 1. Structure of SISO Preview Tracking System

The general discrete-time system is described by

)()()()1( tEdtButAxtx ++=+ (1-a)

)()()( tDutCxtz += (1-b)

where nRtx ∈)( and mRtu ∈)( are the state vector and

control input, respectively. The signal pRtz ∈)( denotes the

controlled output or the tracking error. Moreover, lRtd ∈)( denotes the exogenous signal which can be considered as the reference signal or the disturbance.

The following assumptions are made for the system:

H1. (A, B) is stabilizable.

H2. ⎥⎥⎦

⎤

⎢⎢⎣

⎡ −DCBIeA jθ

has full column rank for any

)2,0[ πθ ∈ .

H3. The values of )(),...,1(),( htdtdtd ++ are available for control, where h is a nonnegative constant that is called preview length.

The preview controller to be designed is of the form [10],

∑ =++=

h

i dix itdKtxKtu0

)()()( (2)

or, [ ]dhdx KKKzK ...)( 1= .

The first and second terms on the right-hand-side of the controller equation (2) represent the state feedback and preview compensation, respectively.

The optimization based controller synthesis problem considered in this paper is to design a preview controller of the form shown in equation (2) such that the multiple objective functions are optimized along with no violation of the constraints. Some of the various objective functions can be minimize cost, maximize performance, maximize reliability, etc. The objective functions and constraints considered in this paper are ;)(;)( 221 zGJzGJ == ∞ ∫= );(3 teJ

);(max4 tyJ = )(min5 tyJ = etc.

The constrained optimization problem is expressed as

)(min KJnK ℜ∈ (3)

subject to

miKgi ,...,2,1,0)( =≤ (4)

where the vector K consists of n preview controller gains [ ]dhdx KKKzK ...)( 1= .

Using the Lagrangian formulation, the dual problem associated with the primal problem (3) can be written as

),(max λλ KL (5)

subject to

mii ,...,2,1,0 =≥λ (6)

with )()(),( KgKJKL Tλλ += and λ is 1×m multiplier vector for the inequality constraints. Thus, the problem can be expressed as min-max problem, defined as below,

),(maxmin λλ KLK (7)

provides the optimal K that minimizes )(KJ and optimizes λ the multiplier. The next section explains the MOCC-PSO technique based on optimizing both the preview controller gains and the Lagrangian multiplier using proposed min-max problem.

III. METHODOLOGY

A. Standard PSO Particle Swarm Optimization (PSO), a heuristic technique

for automated optimization of multi-dimensional problems, was introduced by Kennedy and Eberhart [15-16]. It is inspired from the birds’ flocking behaviour. Instead of using evolutionary operators to manipulate the individuals, each individual in PSO flies in the search space with a velocity which is dynamically adjusted according to its own flying experience and its companions’ flying experience. Each individual is treated as a volume-less particle (a point) in the D-dimensional search space. The ith particle is represented as

),...,,( 21 iDiii xxxX = . The best previous position (the position giving the best fitness value) of the ith particle is recorded and represented as ),...,,( 21 iDiii pppP = . The index of the best particle among all the particles in the population is represented

Future Information

+ +

Output Signal

Disturbance Signal

Reference Signal

Ke +

P -

+ +

Kd

Kx

System States

+

by the symbol g. The rate of position change (velocity) for particle i is represented as ),...,,( 21 iDiii vvvV = . The particles are manipulated according to the following equation:

+−⋅⋅+⋅=+ )(() )(1

)()1( tid

tid

tid xpbestrandcvwv

)(() )(2

tidxgbestrandc −⋅⋅ (8-a)

)1()()1( ++ += t

idt

idt

id vxx (8-b) The equation (8-a) is used to calculate the new velocity

according to its previous velocity and distance of its current position from both its best historical position and its neighbours’ best position. Then, the particle flies towards the new position according to equation (8-b). The process is repeated until a user-defined stopping criterion is reached.

B. Multi-Objective Constrained Co-Evolutionary PSO (MOCC-PSO) The concept of co-evolution was proposed and incorporated

into GA to solve constrained optimization problems by Coello [17]. In this paper, the notion of co-evolution is incorporated into PSO for solving multi-objective multi-constraint optimization problems. The co-evolution model is developed based on the fact that for a preview control system, the system’s stability is dependent on the preview controller gains while optimal satisfaction of the constraints depends on the weight associated with the constraints.

The principle of co-evolution model in MOCC-PSO is shown figure 2. In this MOCC-PSO, two types of swarms are used. In particular, one kind of swarm (denoted by Swarm2) with size N2 is used to adapt the constraint weight, another kind of multiple swarms (denoted by Swarm1,1, Swarm1,2,…, Swarm1,N2) each with size N1 are used in parallel to search good decision solutions for static preview controller gains. Each particle KCj in Swarm2 represents a set of constraints’ weights for particles in Swarm1,j, where each particle KXj represents a set of preview controller gains.

Figure 2. Graphical Representation of the Concept of Co-evolution.

For every generation of co-evolution process, each Swarm1,j will evolve by using PSO for a certain number of generations (G1) with particle KCj in Swarm2 as constraints’ weights for every set of preview controller gains in Swarm1,j. After obtaining the best result KXj from this evolution of Swarm1,j, the fitness of particle KCj of Swarm2 is evaluated. After all the particles in Swarm2 are evaluated, Swarm2 will evolve by using PSO with one generation to get a new Swarm2 with modified preview controller gains. The above process is repeated until a pre-defined stopping criterion is satisfied (e.g. a maximum number of co-evolution generations G2 is reached or the fitness of the best solution obtained lies in the desired range).

Thus, the two swarms evolve interactively, with Swarm1,j is used to evolve preview controller gains and Swarm2 used to modify constraints’ weights. Consequently the controller gains are adjusted to satisfy the multiple objectives and multiple constraints by a self-tuning approach.

C. Multiple Objectives The synthesis procedure requires a definition of the

objectives that specify the design requirements. As the numerical values of the objectives will be in different ranges, so the values of these objectives are normalized with respect to the maximum value of the respective objective function that provides the fitness of the solution on a normalized range. The normalized objective functions are then combined using weighted sum approach to deal with the multiple-objectives and the objective function is represented as

)(...)()()( ''22

'11 KzwKzwKzwKf nn+++= (9)

where )(' kzi is the normalized objective function. The weights iw are assigned values depending on the design preferences, for equal preference all the weights are kept as 1. The advantage of this approach is its easy implementation and computational efficiency that allows the optimization algorithm to search multiple solutions simultaneously.

D. Handling Constraints The presence of constraints may significantly affect the

optimization performances of any optimization algorithms for unconstrained problems. So far, penalty function methods have been the most popular methods for constrained optimization problem. The penalty term varies with the quality of solution but does not discard the infeasible solutions. Thus, in this paper, barrier function approach is used to handle the constraints. The solutions that are on or beyond the boundary are heavily penalized, and so, infeasible solutions are removed from the search space. In this paper a logarithmic barrier function is assumed, defined as below,

( )( )( ) miKgKci ,...,2,1,,0maxlog)(' =−= (10)

The overall constraint term can now be expressed as,

∑=

=m

iii KcKc

1

' )()( λ (11)

KC1

KC2

.

.

.

KCN2

KX1 KX2 …… KXN1

Swarm1,1

KX1 KX2 …… KXN1

Swarm1,2

KX1 KX2 …… KXN1

Swarm1,N2

where the adaptation gains iλ are adjusted by the MOCC-PSO mechanism for optimal results.

E. Infeasible Solution As the barrier approach to handle constraints is used, so the

infeasible solutions in the population are required to be replaced by the feasible solutions. The Marriage in Honey Bees Optimization (MBO) approach is used to produce new feasible solutions. The local best and global best solutions are assumed as set of queens while set of drones are randomly generated. The broods are then generated probabilistically, using crossover and mutation operator. The broods are tested for their feasibility. The feasible broods are, then, used to replace the infeasible solutions.

F. MOCC-PSO Algorithm The The lagrangian formulation of the constrained Preview

Controller optimization problem is expressed as,

),(maxmin λλ KLK (12)

where )()(),( KcKfKL +=λ .

The design procedure for the synthesis of Fixed Structure Preview Controller using Multi-Objective Co-Evolutionary PSO algorithm is presented as below:

Step 1: Initialize Swarm1 with N1 particles and Swarm2 with N2 particles. Create N2 replica of Swarm1 as Swarm1,1, Swarm1,2,…, Swarm1,N2.

Step 2: Repeat the steps 2.1, 2.2 1nd 2.3 for fixed number of generations i.e. G2

Step 2.1: Repeat step 2.1.1 for j = 1 to N2

Step 2.1.1: Repeat the steps 2.1.1.1 to 2.1.1.3 for fixed number of generations i.e. G1

Step 2.1.1.1: Evaluate the fitness of each particle KXi (where i varies from 1 to N1) in Swarm1,j with KCj.

Step 2.1.1.2: Replace the infeasible solutions with feasible solutions by using the MBO procedure as explained in the previous section.

Step 2.1.1.3: Evolve Swarm1,j using PSO with KCj as maximization problem.

Step 2.2: Evaluate the fitness of all the particles KCj in Swarm2.

Step 2.3: Evolve Swarm2 using PSO as minimization problem.

Step 3: The particle of Swarm2 i.e. KCj along with the best particle of Swarm1,j i.e. KXj that attains the maximum fitness for all the objective functions is the optimized solution.

The verification of the procedure is done for the linear discrete control systems and the results are presented in the next section.

IV. SIMULATION AND RESULTS The proposed procedure is tested on two linear discrete

systems borrowed from the literature, servo mechanism [10] and missile trajectory following problem [9]. The results are presented as below:

A. Servo Mechanism The discrete state-variable model of the 4th-order Servo

Mechanism is taken from the literature [10].

The values of the parameters chosen are 1,5 == RQ and preview length is assumed to be h=4. The optimization problem can be expressed as,

( )ηγ +min (13)

subject to

∫ ≤ 9)(te (14)

where γ is the H∞ norm and η is the H2 norm of the system.

The system is solved for an Optimal Preview Controller using the proposed procedure on MATLAB platform and the LMI based Mixed H2-H∞ Design method [13] with the help of YALMIP toolbox.

Figure 3 shows a comparison of the time-response Servo Mechanism for a unit-step input.

The optimal value of constraint weight obtained using proposed procedure is 4683.8=λ . A comparison of the results obtained is summed up in table I.

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

1.2

1.4

LMI based SolutionSolution using Proposed Methodology

Figure 3. Transient Response of Servo Mechanism with LMI based and proposed MOCC-PSO based Methodology.

TABLE I. CHARACTERISTICS OF SOLURION OBTAINED FOR SERVO MECHANISM

System Characteristics

LMI based Mixed H2-H∞ Design method

Proposed MOCC-PSO Algorithm

γ

(H∞ Criteria) 14.2554 16.0405

η

(H2 Criteria) 5.3357 5.2951

IAE 11.0065 8.9934

Objective Function Value ∞ 39.8048

B. Missile Trajectory Following Problem The discrete state-variable model of the Missile for

trajectory following is acquired from the literature [9].

The above model is converted to discrete-domain using sampling time ts = 1 seconds.

The values of the parameters chosen are 50=Q , 10=R and preview length is assumed to be h=3. The optimization problem can be expressed as,

( )ηγ +min (15)

subject to

( )( ) 01.1max ≤ty (16)

∫ ≤ 1)(te (17)

where γ is the H∞ norm and η is the H2 norm of the system.

The system is solved for an Optimal Preview Controller using the proposed procedure on MATLAB platform and the LMI based Mixed H2-H∞ Design method [13] with the help of YALMIP toolbox.

Figure 4 shows a comparison of the time-response of missile model following problem for a unit-step input.

The optimal value of constraints’ weights obtained using proposed procedure is 1016.31 =λ and 6632.32 =λ . A comparison of the results obtained is summed up in table II.

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time Step

Out

put

Am

plitu

de

LMI based Solution

Solution using Proposed Methodology

Figure 4. Transient Response of Missile Model with LMI based and proposed MOCC-PSO based Methodology.

TABLE II. CHARACTERISTICS OF SOLURION OBTAINED FOR MISSILE MODEL

System Characteristics

LMI based Mixed H2-H∞ Design method

Proposed MOCC-PSO Algorithm

γ

(H∞ Criteria) 4.5870 8.4375

η

(H2 Criteria) 3.4375 6.6067

IAE 2.4379 0.5206

Max (y(t)) 1.5070 1.0090

Objective Function Value ∞ 27.9468

The results show that the optimal solutions are strongly affected by the constraints imposed. The classical solution methods do not provide a flexible approach to present the non-classical constraints. Also, the classical solution approach does not provide the ease to include non-conventional objective functions. The proposed MOCC-PSO approach is simple in implementation and straight forward to include the conventional and non-conventional forms of objective functions and constraints. The results obtained confirm the superiority of the proposed algorithm to obtain the optimal solution in the constrained environment.

V. CONCLUSION This paper presents MOCC-PSO algorithm to design fixed

structure Preview Controller as min-max optimization problem. The algorithm presents a novel technique to handle multiple objectives and constraints simultaneously. Also, the procedure is able to implement the solution replacement for infeasible solutions and also to keep a check that the particles do not fall into local optima, using the brood generation method of MBO algorithm. The algorithm is tested for control problems borrowed from the literature. The results verify the ability of the MOCC-PSO algorithm in terms of quality of solution obtained in the constrained environment with multiple objectives. In addition, the procedure is flexible enough to include the non-conventional objectives and constraints that is not an easy task with the classical techniques. The results for implementation of procedure for non-linear and uncertain systems shall be reported in the further communications.

REFERENCES [1] A. Kojima and S. Ishijima, “H∞ Control with Preview Compensation”,

Proceedings of American Control Conference, New Mexico, pp. 1692-1697, June 1997.

[2] A. Kojima and S. Ishijima, “H∞ Preview Tracking in Output Feedback Setting”, Proceedings of Conference on Decision and Control, Arizona, USA, pp. 3162-3164, December 1999.

[3] G. Tadmor and L. Mirkinn, “H∞ Preview Control and Fixed- Lag Smoothing I: Matrix ARE Solutions in Continuous-Time Systems”, Proceedings of IEEE Conference on Decision and Control, Hawaii, USA, pp. 6515-6520, December 2003.

[4] G. Tadmor and L. Mirkinn, “H∞ Preview Control and Fixed- Lag Smoothing II: Matrix ARE Solutions in Discrete-Time Systems”,

Proceedings of IEEE Conference on Decision and Control, Hawaii, USA, pp. 6521-6526, December 2003.

[5] K. Y. Polyakov, E. N. Rosenwasser and B. P. Lampe, “Optimal Open-Loop Tracking using Sampled Data System with Preview”, Proceedings of 11th IEEE Mediterranean Conference on Control and Automation, Greece, CD-IV04-03, 2003.

[6] Y. Kuroiwa and H. Kimura, “H∞ Control with Preview and Delay”, Proceedings of American Control Conference, Boston, pp. 4794-4799, June 2004.

[7] M. M. Negm, J. M. Bakhashwain and M. H. Shwehdi, “Speed Control of Three Phase Induction Motor based on Robust Optimal Preview Control Theory”, IEEE Transactions on Energy Conversion, Vol. 21, No. 1, pp. 77-84, March 2006.

[8] A. Moran, Y. Mikami and M. Hayase, “Analysis and Design of H∞ Preview Tracking Control Systems”, Proceedings of 4th International Workshop on Advanced Motion Control (AMC’96- MIE), Japan, Vol. 2, pp. 482-487, 1996.

[9] A. Farooq and D. J. N. Limebeer, “Path Following of Optimal Trajectories using Preview Control”, Proceedings of IEEE Conference on Decision and Control and the European Control Conference, Spain, pp. 2787-2792, December 2005.

[10] K. Takaba, “A Tutorial on Preview Control Systems”, SICE Annual Conference, Fukui, pp. 1388-1393, August 2003.

[11] K. M. Grigoriadis, J. T. Watson, “Reduced Order H∞ and L2 - L∞ Filtering via Linear Matrix Inequalities”, IEEE Transactions on Aerospace and Electronic Systems, Vol. 33, pp. 1326-1338, 1997.

[12] H. D. Tuan, P. Apkarian and T. Q. Nguyen, “Robust and Reduced Order Filtering: New Characterizations and Methhods”, Proceedings of American Control Conference, Chicago, USA, pp. 1327-1331, 2000.

[13] K. Takaba, “Robust Preview Tracking Control for Polytopic Uncertain Systems”, Proceedings of 37th IEEE Conference on Decision and Control, Tampa, Florida, pp. 1765-1770, 1998.

[14] F. Liao, J. L. Wan and G. H. Yang, “LMI-Based Reliable Robust Preview Tracking Control Against Actuator Faults”, Proceedings of the American Control Conference, Arlington, VA, pp. 1047-1052, 2001.

[15] J. Kennedy and R. Eberhart, “Particle Swarm Optimization”, Proceedings of IEEE International Conference on Neural Network, Vol. IV, Perth, Australia, 1995, pp. 1942-1948.

[16] Y. Shi and R. C. Eberhart, “Empirical Study of Particle Swarm Optimization”, Proceedings of the 1999 Congress on Evolutionary Computation (CEC 99), U.S.A., Vol. 3, pp. 1945-1950, 1999.

[17] C.A.C. Coello, “Use of a self-adaptive penalty approach for engineering optimization problems”, Computers in Industry, Vol. 41, 2000, pp. 113-127.