Optimal Control of Piecewise Affine Systems

Majid akbarianDepartment of Electrical EngineeringFerdowsi university of mashhad

Iran,mashhadakbarian@stu.um.ac.ir

Najmeh Eghbal Department of Electrical Engineeringsadjjad university of technology

Iran,mashhadnajmeh.eghbal@sadjad.ac.ir

Naser ParizDepartment of Electrical EngineeringFerdowsi university of mashhad

n-pariz@um.ac.irIran,mashhad

Abstract—In this paper we discuss the issue of optimal control of piecewise affine systems based on discontinuous quadratic Lyapunov function, given the stability of the said system, first we calculate the upper bound of the cost function, then we consider the control input as the state feedback and finally, by minimizing the upper bound of the cost function, the state feedback coefficients are calculated. Note that if we minimize upper bound the cost function is minimized. It is worth noting that the optimization problem is turned into a semidefinite programming problem with bilinear constraints. These problems can be solved using numerical or any other methods. Finally, to check the effectiveness of the method, we have listed a numerical example.

I. INTRODUCTION

Hybrid systems are dynamical systems with continuous time and discrete components. These systems can be used for modeling industrial processes. Piecewise affine systems are a subset of hybrid systems and their equivalence with other classes of the hybrid systems are shown [1]. Piecewise affine systems are an important category for modeling of complex and nonlinear systems, because many of such nonlinear systems such as saturated and etc. are inherently modeled in form of piecewise affine systems or can be approximated as a piecewise affine system [2]. Therefore, piecewise affine systems are an important tool and a starting point for studying nonlinear systems. These systems are defined by a series of linear systems that are defined by polyhedral in each region. In references [3] and [4] you can see the application of these systems in power electronics and process control. Furthermore, a wide range of nonlinear systems in engineering applications can be modeled using a piecewise affine system [5]. On the other hand,, the stability and performance analysis of the piecewise affine systems can be formulated and easily solved in the form on convex problems that are easily solved by numerical methods and this is an advantage for these systems. Analysis and controller design problems for piecewise affine systems has been a subject of debate over recent years and has been one of the most challenging issues in the field. Controllability and observability of these systems are discussed in reference [6]. Stability and synthesis analysis of piecewise affine systems are discussed in references [7] and [8]. In these cases stability is expressed in the form of linear matrix inequalities. In reference [9] controller design is done based on the output feedback and controller is obtained by solving a series of bilinear matrix inequalities that are solved using numerical algorithms. Quadratic controls of piecewise affine systems are discussed in reference [10]. Gain calculation of L2 for piecewise affine systems is done in [11] and [12].Optimal control of switch affine systems discussed in [15] and [18].Optimal control

of switch Hybrid systems discussed in reference [16].In reference [17] and [19] and [21]optimal control of discrete piecwise affine systems obtained.In reference [20] optimal control of switch affine systems with dynamic programming obtained.But optimal control of these systems using the state feedback is not performed and this prompted us to perform this research. Here, the cost function is considered of Linear Quadratic Regulator (LQR) for a closed-loop system. The goal is controlling the state feedback in such a way that it stabilizes the closed-loop system and reduces the cost function to a minimum. Initially we discuss the conditions of stability in a piecewise affine systems, then we generalize these conditions to a closed-loop system based on discontinuous Lyapunov function. These conditions are expressed in terms of linear matrix inequalities. Then we prove a theorem and we use it to calculate the upper bound of the cost function. In the end based on the discontinuous Lyapunov function, the optimal controller design problem will become a semi-definite programming optimization problem with bilinear constraint which can be solved by genetic algorithm or numerical algorithms.All the previos works in fact designed optimal control for switch-affine systems or for discrete piecwise affine systems or the way they are used to design optimal control is limited and we cant increase the number of optimal control designe constraints but the method mentioned in this paper is not limited.

II. DEFINITIONS,STABILITY ANALYIS AND UPPER BOUND

In this section, the definitions and backgrounds necessary for studying the next sections are described. Then we discuss the necessary conditions for stability analysis and , we obtain an upper bound for the cost function introduced A. Positive Matrix

The A=[aij ] ,i , j=1 ,…, n matrix is called a positive matrix when ∀ i , j aij>0 is similarly defined as negative matrix.B. Definite positive matrix

Matrix A=[aij ] ,i , j=1 ,…,n is called definite positive if

∀ xϵ Rn , xT Ax>0 and the notation A>0 is used to show it. In the same way, definite negative matrix is also defined.C. piecewise affine systemsThe mathematical description of PWA class in general is

{ x=ai+ A i x+B iuy=c i+C i x+Diu

for xϵ X i(1)

In which X i shapes the designated areas and their collection is a partitioned space of the state. According to the type of partitioning systems of the PWA, there are two types of PWA, multi-faceted and oval.

X i={xϵ R2 , Ei x≥ e i }iϵI (2)

In (1) and (2) Ei and e i of the matrix and vector are appropriate to the size.It’s worth noting that system (1) assuming Bi=0 turns into a piecewise linear system as follows:

x=a i+ A i x for xϵ X i(3)

The border of neighboring areas X i and X j is defined asif X i∩ X j≠∅ then X i ∩ X j⊆{x|x=F ij s+f ij , sϵR } (4)

Using the notation x=(x1) and Ei=[ Ei−e i ] we’ll have

x= Ai x for x ϵ X i (5)

X i={x∈ R2 ,E i x≥ 0 } (6)

X i ∩ X j⊆{x|x=F ij s , s ϵR , s=(s1)}

(7)

F ij=(F ij f ij

0 1 ) (8)

D. Stability Analysis Thorem 1:Suppose U i and W i are unknown matrices with non-negative elements and appropriate dimensions and (k=1,2) ωij

k are unknown vectors with appropriate dimensions and

non-negative elements and (i∈I) pi ϵ R3× 3 is a symmetric matrix, then we define following variables [13]

H ij=EiT ωij

1 Cij A i+E jT ωij

2C ij A j

Li=EiT U i Ei

M i=EiT W i Ei

(9)

If there is a choice between Pi and U i and W i matrices and (k=1,2)ωij

kvectors that can establish the following restrictions, then for system defined by equation (5) all the trajectory starting at X will exponentially converges to origin.

Pi=( pi 00 o)>0 for iϵ I 0

(10)

Pi−Li>0 for iϵ I 1 i∉ I 0 (11)

( I n 0 ) ( Pi−Li )(I n

0 )>0∀ i∈ I 0(12)

AiT Pi+Pi A i+ Mi<0∀ i∈ I , i∉ I0

(13)

( I n 0 ) ( AiT Pi+Pi A i+ Mi ) (I n

0 )<0 ∀ i∈ I 0(14)

F ijT ( Pi−P j ) Fij=F ij

T (H ij+H ijT ) Fij∀ i∈ I , j∈N i , where Fij ≠ 0(15

)

Where N i={k∈ I , k≠ i , X i⋂X j≠∅ } (16)

E. Calculating the upper boundThorem 2:For the system (5) if the conditions of therom (1) is met and the inequality is established, then the upper bound for cost function

j=∫0

∞

x (t)T Qi x (t )dt Qi>0 is calculated [14]:

i∈ I 0 Pi A i+ A iT Pi+Qi+ Ei

T W i E i<0 (17)

i∈ I1 Pi A i+ A iT P i+Q i+E i

T W i E i<0 (18)

Proof: Suppose that we prove i∈ I1 for the therom, another proof is the same. We multiply the said inequality in X from left and right and remove of non-negative terms. Then we take the integral of the expression that we desire in the interval [0 , ∞ ]:

xT Pi A i x+xT A iT Pi x+xT Qi x+xT Ei

T W i Ei x<0

xT Pi x+ xT Pi x+xT Qi x+xT EiT W i Ei x<0

ddt (xT Pi x )+ xT Qi x+ xT E i

T W i E i x<0

¿ddt (xT Pi x )+ xT Qi x≤ 0

∫0

∞

[ ddt ( xT Pi x )+xT Qi x ]dt ≤ 0

[ xT Pi 0 x ]|∞0 + j≤ 0

0−x (0 )T P i 0 x (0 )+ j ≤ 0

j ≤ x (0 )T Pi 0 x (0 )

E ( j ) ≤ E (tr (P i 0 x0 x0T ))=∑

i∈Iα i tr (Pi 0 Li)

(19)

Li={E (x0 x0T ) x0∈ X i ,i∈ I 0

E (x0 x0T ) x0∈ X i ,i∈ I 1

(20)

You may notice in the above inequalities that all of them are a series of linear matrix inequalities in relation to the variables Pi and ( Pi ). Therefore, stability conditions for a closed-loop system are a series of linear matrix inequalities in relation to Pi and ( Pi ) and they are convex optimization problems that can be solved using numerical methods. Pay attention that because the cost function is dependent on the initial point, and this point is unknown and a random variable, we assume that the initial point of a random variable is monotonous with distribution, so that the dependency is eliminated. Operator E expresses the expected value and α i represents the probability that x0 belongs to area X i. Since we considered the initial state as a monotonous random variable, therefor the probability of α i and the covariance matrix Li can be determined using the desired area’s information and the partition X i.

III. OPTIMAL CONTROL DESIGN

In this section we describe the optimal controller design issues for piecewise affine systems using the state feedback. We assume that the designated system balance point is the initial point. Consider the system described with equations (1), in this case assume that state feedback controller is u ( t )=K i x ( t ). The closed loop system takes the form below:

{x=( Ai+Bi K i ) x ( t )+ai

x (t 0 )=x0(21)

We consider the cost function as:

j ( x0 ,u )=∫0

∞

[x ( t )T Qi x ( t )+u ( t )T Ri u(t )]dt(22)

With the consideration of the appropriate state feedback, the cost functions comes in the form of:

j=∫0

∞

[ x ( t )T (Qi+K iT R i K i)x ( t )¿]dt ¿

(23)

j=∫0

∞

(x ( t )T 1)(Qi+K iT Ri K i 0

0 0)(x ( t )1 )dt

(24)

j=∫0

∞

x (t)T Qi x (t )dt (25)

Using the notations of equation (21), it gives:˙x (t )=(Ai+Bi K i ai

0 0 ) x( t)(26)

Ai=(Ai+Bi K i a i

0 0 ) s(27)

By applying the said changes in the form of the equations, therom 2 for the system (26) is rewritten as:Therom 3:For the system (26) with (assuming that the system is stable) if the equations (30-41) are met, then the upper bound for the equation (22) is obtained:i∈ I 0 Pi ( A i+B i K i )+( A i+Bi K i )

T Pi+Qi+K iT Ri K i+ Ei

T W i E i<0 (28)

i∈ I1 Pi A i+ A iT P i+Q i+E i

T W i E i<0 (29)

In this case we’ll have: j ≤ x (0 )T Pi 0 x (0 )Proof: To prove this therom in therom 2, we convert Ai to Ai+Bi K i. Now we can merge therom 1 and 3 and generally express the result in terms of therom 4: Therom 4:For the system defined by equations (26) if the following conditions are met, then the system for each respective system trajectory exponentially convergent origin and j ≤ x (0 )T Pi 0 x (0 )

i∈ I 0 Pi=( pi 00 o)>0

(30)

( Pi−Li )>0 (31)

( A iT P i+P i Ai+M i )<0 (32)

Ai=(Ai+Bi K i 00 0) (33)

Pi ( A i+B i K i )+( A i+Bi K i )T Pi+Qi+K i

T Ri K i

+EiT W i Ei<0

(34)

i∈ I1 Pi−Li>0 (35)

AiT Pi+Pi A i+ Mi<0 (36)

Pi A i+ A iT Pi+Qi+ Ei

T W i E i<0 (37)

Ai=( Ai+Bi K i a i

0 0 ) (38)

for x∈ X i∩ X j we haveF ij

T ( Pi−P j ) Fij=F ijT (H ij+H ij

T ) Fij∀ i∈ I , j∈N i , where Fij ≠ 0(39)

Where N i= {k∈ I , k ≠ i , X i⋂X j≠∅ } (40)

( Ai+Bi K i ) x (t )+ai=( A j+B j K j ) x ( t )+a j⟹

(Ai+Bi K i a i

0 0 )(x1)=(A j+B j K j a j

0 0 )(x1)

X i ∩ X j⊆{x|x=F ij s , s ϵR , s=(s1)}

F ij=(F ij f ij

0 1 )⟹Ai Fij=A j F ij (41)

j ≤ x (0 )T Pi 0 x (0 )⇒E ( j ) ≤ E (tr (P i 0 x0 x0

T ))=∑i∈ I

α itr (Pi 0 Li) (42)

Li={E (x0 x0T ) x0∈ X i ,i∈ I 0

E (x0 x0T ) x0∈ X i ,i∈ I 1

(43)

Finally, for optimal control design, the coefficient K i must be calculated. To calculate these coefficients we consider a

controller that minimizes the upper bound of cost function ∑i∈ I

αi tr(Pi 0 Li). So the desired optimization problem that

leads to the controller design is as follows:

min(∑i∈Iαi tr ( Pi 0 Li ))subject

¿{ K i ϵK

(30 )−(41)¿

To solve optimization problem we use genetic algorithm for selecting the variable Ki. After defining Ki, the optimization problem is converted to a linear matrix inequality form which can be solved as a semi-definite programming problem

IV. NUMERICAL EXAMPLE

Consider System (1) with grade 2 and i=1,2,3 and the following matrices:

A1=(0 10 −0.1) , A2=(0 1

1 −0.1) , A3=(0 10 −0.1)

a1=−a3=( 0−1) , a2=(00) , B1=B2=B3=(1 0

0 1)X1={x|x1 ϵ [−2 ,−1 ]} , X2={x|x1 ϵ [−1,1 ] } ,X3={x ⌊ x1¿¿

Suppose that the initial state x1(0) is a random variable with monotonous distribution in the interval [−2,2 ] . We assume the cost function as equation (22) and assume Ri=1 and Qi=1. We consider the control coefficient in interval [−5,5 ]. It becomes clear that the source is located in region X2 and the closed loop system is unstable. Matrices E1 and E2 and E3 and e1 and e2 and e3are calculated as:

E1=( 1 0−1 0), E2=( 1 0

−1 0) ,E3=( 1 0−1 0) ,

e1=(−21 ) , e2=(−1

−1) , e3=( 1−2)

Also, the parameters required to analyze the stability using

therom (1) are:

C12=(10 ),C23=(1 0),F12=(01),F23=(01)c12=(1 );c23=(−1),f 12=(−1

0 ),f 23=(10)

After the simulation, the appropriate control coefficients are obtained as follows. Also, the appropriate Lyapunov function for region X2is demonstrated in figure (1):

K1=( 0.6882 3.1029−0.3061 −0.0499)

K2=(−4.8810 −3.3435−1.6288 1.0198 )

K3=(−3.3782 −3.37822.9428 2.9428 )Joptimal=0.1698

-2 -1 0 1 2-2

02

-40

-30

-20

-10

0

10

20

30

x1

Figure 1: lyapunov for region X2

V. CONCLUSION

In this paper, we introduced a class of hybrid systems that are able to model a wide range of practical systems, then we discussed the mathematical description of affine linear systems and stability conditions of piece wise affine systems in form of terms of linear matrix inequalities and then calculated the upper bound of the cost function. In fact, theroms 3 and 4 are one the innovations of this article. We proved that the problem of optimal control of piece wise affine systems leads to solving an optimization problem with bilinear constraints. Then, by minimizing the upper bound and use of genetic algorithms and semi-definite programming we can calculate the controller coefficients. In the end, we demonstrated the effectiveness of the methods listed by using a numerical example.

REFERENCES

[1] W. P. M. H. Heemels, B. de. Schutter, and A. Bemporad. Equivalence of hybrid dynamical models. Automatica, 37:1085–1091, 2001.

[2] R. Lin, J.-N.; Unbehauen. Canonical piecewise-linear approximations. IEEE Trans. on Circuit and Systems I: Fundamental theory and applicatons, 39(8):697 – 699, Aug 1992

[3] M.senesky,G.Eirea,and T.John Koo,2003,”Hybrid modeling and control of power electronics”,Hybrid systems:computer Science,Springer-Verlag:Berlin,in the proceeding of 6th International Workshop,pp.450-465

[4] Panagiotis D.Christofides and Nael H.El-Farra,Control of Nonlinear and Hybrid Process systems Design for Uncertainty ,Constraints and Time-Delays(Lecture Notes in control and Imformation Sciences 324),Springer –Verlag Berlin Heidelberg,2005.

[5] A.Bempporad,A.Garulli,S.Paoletti,and A.Vicino.A boundederror approach to piecewise affine system identification.IEEE Trans.on Automatic Control,50(10):1567-1580,2005

[6] A.Bempporad,G.Ferrari-Trecate,M.Morari”Observability and controllability of piecewise affine and hybrid systems.”IEEE Transactions on Automatic control 2000;45(10):1864-1876.

[7] M.Johansson and A.Rantzer.computation of piecewise quadratic Lyapunov functions for hybrid systems.IEEE Transactions on Automatic Control,43(4):555-559,April 1998

[8] S.Pettersson.Modelling control and stability Analysis of Hybrid systems.Licentiate thesis,Chalmers University of Technology,November 1996.

[9] J.Zhang and W.Tang.”Output feedback optimal guaranteed cost control of uncertain piecewise linear systems ”INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL Int.J.Robust Nonlinear Control 2009;19:569-590 Published online 4 June 2008 in Wiley InterScience

[10] S.P.Banks and S.A.Khathur.Structure and control of piecewise linear systems.International Journal of control,50(2):667-686,1989

[11] M.R.James and S.Yuliar.Numerical approximation of the H∞ norm for nonlinear systems.Automatica,31:1075-1086,1995.

[12] A.J.vander schaft L2-gain analysis of nonlinear systems and nonlinear state feedback H∞ control.IEEE Transactions Aumatica Control,37(6770-784),1992

[13] N.Eghbal,N.Pariz and A.Karimpour.”Discontinuous piecewise quadratic Lyapunov function for planar piecewise affine systems”Journal of Mathematical Analysis and Applications.399(2013)586-593

[14] A.Rantzer,M.Johansson.”Piecewise linear quadratic optimal conrol” IEEE Transactions on Aumatica Control 2000;45(4):629-637.

[15] H.W.J.Lee,K.L.Teo,V.Rehbock,L.S.Jennings”Control parametrization enhancing technique for optimal discrete-valued control problems”Elsevier Science on Aumatica 35(1999)1401-1407

[16] F.Zhu,P.J.Antsaklis”Optimal Control of Switch Hybrid Systems”Interdisciplinary studies in Intelligent systems July 2013

[17] M.Baric,P.Grieder,M.Baotic,M.Morari,”An efficient algoritm foroptimal control of PWA systems with polyhedral performanceindices”Science Direct on Automatica(2007).

[18] J.Imura”Optimal control of sampled-data piecwise affine systems” Elsevier Science on Aumatica 40(2004)661-669

[19] S.V.Rakovic,E.C.Kerrigan,D.Q.Mayne”Optimal control of constrained piecwise affine systems with stste-and input-dependent disturbances”Research supported by the enginearing and physical sciences research council and the royal academy of enginearing UK

[20] P.Riedinger,C.Iung,F.Kratz”An Optimal Control Approach for Hybrid Systems”European Journal of Control(2003)9:449-458

[21] “Optimal Control of Piecewise Affine Systems “Doctor of Sciences presented by MATO BAOTIC