Suboptimal Control of Singularly Perturbed Two Parameter Discrete Control System

International Electrical Engineering Journal (IEEJ) Vol. 5 (2014) No.11, pp. 1594-1604 ISSN 2078-2365 http://www.ieejournal.com/

Suboptimal Control of Singularly Perturbed Two Parameter Discrete Control System G. KISHORE BABU * and M. S. KRISHNARAYALU ** * Department of Electrical Engineering, PSCMR Engineering College, Vijayawada, A. P. 520002, India ** Department of Electrical and Electronics Engineering, VR Siddhartha Engineering Vijayawada, A. P. 520007, India. [email protected]

Abstract— A singularly perturbed discrete power system with two parameters (three-time-scales) is considered. An optimal controller using Pontriagin's minimum principle (PMP) is designed for this stiff system. But the solution for control law of this type of system requires special numerical techniques such as shooting methods. Hence it is better to go for a suboptimal controller that eliminates this problem. Singular perturbation method (SPM) that reduces the order, removes the stiffness of the system and with approximate solutions closer to optimal solution is an ideal choice for this problem. The SPM consists of an outer, two initial boundary layer corrections (BLC) and two final BLC solutions for this problem. The SPM is applied to a fifth order power system model as a case study. The results of case study validate the proposed suboptimal controller. Key words— Two parameter discrete system, Open-loop optimal control, Stiff two-point boundary value problem, Suboptimal control, Singular perturbation method, Outer solution, Boundary layer corrections.

I. INTRODUCTION The dynamics of many continuous-time and digital systems is described by high order differential and difference equations respectively. Frequently, the presence of small parameter such as time constants, masses, moments of inertia, inductances and capacitances is the source for increased order of the system. A system in which the suppression of a small parameter is responsible for the degeneration of the dimension of the system is called a singularly perturbed system. Such a system possesses widely separated groups of eigenvalues exhibiting slow, medium fast and fast phenomena or time-scale behavior resulting in computationally stiff. These stiff systems require the use of extensive numerical routines. We frequently encounter two-point boundary value problems in optimal control. The solution of two-point boundary value problems of stiff systems requires special methods such as shooting techniques [20]. Even these special

College,

methods may end as trial and error methods if the problem is too stiff. The singular perturbation methods, which are not trial and error methods, remove stiffness, reduce the order of the system and satisfy the specified boundary conditions of the system. The essence of singular perturbation methodology is as follows. The degenerate system, obtained by suppressing the small parameters is of reduced order and can satisfy the specified boundary conditions of the slow variables only. The rest of the specified boundary conditions of the fast variables are lost in the process of stiffness removal (degeneration). Boundary layers are formed due to non-uniform convergence of exact solution to the degenerate solution. Now boundary layer corrections have to be added to recover the lost boundary conditions and to improve the degenerate solution. Also boundary layer corrections should ensure that solution is unique. Singularly perturbed and time-scale systems are one and the same [10]. Power system control results in multi-time-scale systems as number of time constants of different magnitudes (governor, turbine, generator & load, amplifier, exciter) are involved [5]. Thus modeling single area and multi-area power systems, yields time scale systems. Also multi-time-scale (MTS) systems exhibit the chaotic characteristic [21]. Hence obtaining the exact solution of MTS is difficult and special numerical methods are required [20]. Singular perturbation theory in continuous-time systems is well developed [1-8], whereas singular perturbation theory of discrete control systems is still in developing stage [9-19]. However applications are also lacking. Hence in the present paper, the open-loop optimal control of a linear, time-invariant singularly perturbed three-time-scale discrete power system with two parameters is considered. A SPM is presented up to second-order approximation for this two parameter problem, using asymptotic expansions and transformations required to get a unique solution. Suboptimal control is obtained using the SPM and compared with optimal control. 1594

Kishore and Krishnarayalu. Suboptimal Control of Singularly Perturbed Two Parameter Discrete Control System

International Electrical Engineering Journal (IEEJ) Vol. 5 (2014) No.11, pp. 1594-1604 ISSN 2078-2365 http://www.ieejournal.com/ II. STATEMENT OF THE PROBLEM Consider the linear, time-invariant, completely controllable singularly perturbed two-parameter discrete system. x0 (k + 1) x0 (k) [x1 (k + 1)] =A [ ε1 x1 (k) ] + B u(k) 1(a) x2 (k + 1) ε1 ε2 x2 (k) with initial conditions xi (k = 0) = xi (0) , i=0,1,2. 1(b) A11 A12 A13 B1 where A=[A21 A22 A23 ] ; B = [B2 ] ; B3 A31 A32 A33 and xj−1 (k) ∈ Rnj , j=1,2,3. Sub matrices Aij and Bi are matrices of appropriate dimensionality and the control vector u(k) ∈ Rr is independent of the small parameters ε1 and ε2 . The performance index to be minimized is ′ ′ J=1/2∑N−1 (2) k=0 [w (k)Dw(k) + u (k)Ru(k)] where w(k) = [x0 (k) ε1 x1 (k) ε1 ε2 x2 (k)]. ′ indicates transpose. D is a real positive-semidefinite symmetric matrix of order (n1+n2+n3) x (n1+n2+n3). R is real positive-definite symmetric matrix of order (r x r) and N is a fixed integer indicating the terminal (final) time. Here note that the states are incorporated in an appropriate manner to bring the resulting TPBVP into singularly perturbed form. The Hamiltonian of the problem is 1 1 H(k) = w ′ (k)Dw(k) + u′ (k)Ru(k) + p′ (k + 1)[A w(k) + 2 2 B u(k)] (3) p0 (k) where the co-state vector p(k) = [ ε1 p1 (k) ] ε1 ε2 p2 (k) Using the results of digital optimal control theory ∂H(k) = x0 (k + 1) (4a) ∂p0 (k + 1) ∂H(k) = x1 (k + 1) ∂ε1 p1 (k + 1)

(4b)

∂H(k) = x2 (k + 1) ∂ε1 ε2 p2 (k + 1)

(4c)

∂H(k) ∂x0 (k) ∂H(k) ∂x1 (k)

= p0 (k)

(4d)

= ε1 p1 (k)

(4e)

∂H(k) ∂x2 (k)

= ε1 ε2 p2 (k)

∂H(k) ∂u(k)

(4f)

=0

(4g)

Form (3) and (4), the states and co-states are obtained as x0 (k + 1 ) x0 (k ) x1 (k + 1)(1a) ε1 x1 (k) x2 (k + 1) ε1 ε2 x2 (k) =C p0 (k) p0 (k + 1) p1 (k) ε1 p1 (k + 1) [ p2 (k) ] [ε1 ε2 p2 (k + 1)] A13 E11 A11 A12 A21 A22 A23 E12 ′ A A32 A33 E13 ′ where C = 31 D11 D12 D13 A11 ′ D12 ′ D22 D23 A12 ′ [D13 ′ D23 ′ D33 A13 ′

(5a)

(1b)

E12 E22 E23 ′ A21 ′ A22 ′ A23 ′

E13 E23 E33 A31 ′ A32 ′ A33 ′]

Eij = −Bi R−1 Bj′ , i = 1,2,3; j = 1,2,3. The final conditions of the system (5) are pi (N) = 0, i = 0,1,2. (5b) The optimal control is obtained as u(k)= -R-1[B1’p0(k+1)+ ε1 B2’ p1(k+1)+ ε1 ε2 B3’p2(k+1)] (6) The set of equations (5) – (6) constitutes the open-loop optimal control problem. The 2(n1+n2+n3)th order discrete TPBVP represented by (5) is in the singularly perturbed form in the sense that the degenerate TPBVP x000 (k + 1) x000 (k) x100 (k + 1) 0 x200 (k + 1) = C 00 0 (7) p00 p0 (k + 1) 0 (k) 0 p100 (k) [ ] 00 0 [ p2 (k) ] obtained by suppressing the small parameters ε1 and ε2 in (5a) is of order 2n1 and can satisfy the boundary conditions x000 (k = 0) = x0 (0) and p00 (8) 0 (k = N) = p0 (N) The other boundary conditions, in general, are xi00 (k = 0) ≠ xi (0) and p00 i = 1,2 . i (k = N) ≠ pi (N), (9) The boundary conditions x10 (0), x20 (0), p10 (N) and p02 (N) are lost in the process of degeneration and the loss of these boundary conditions contributes to the existence of boundary layers at initial and terminal points. The 2(n2+n3) boundary conditions lost in the process of degeneration are recovered by the following singular method which gives an approximate 1595

Kishore and Krishnarayalu. Suboptimal Control of Singularly Perturbed Two Parameter Discrete Control System

International Electrical Engineering Journal (IEEJ) Vol. 5 (2014) No.11, pp. 1594-1604 ISSN 2078-2365 http://www.ieejournal.com/ solution to the stiff TPBVP represented by (5) and (1b). Consequently it results in a suboptimal control. III. SINGULAR PERTURBATION METHOD 3.1 Outer solution Let us assume asymptotic expansions for the outer solution in the small parameters as 𝑞 𝑖𝑗 𝑖𝑗 𝑗 {xv,o (k), pv,o (k)}=∑𝑖,𝑗≥0[𝑥𝑣 (𝑘), 𝑝𝑣 (𝑘)]𝜀1𝑖 𝜀2 ; v=0, 1, 2. (10) where q is the desired order of approximation. Substituting (10) in (5a) and equating like powers of the small parameter a set of equations may be obtained. The zero-order equation is the same as that given by (7). For first- order approximation x10 0 (k + 1) x10 0 (k) 10 x1 (k + 1) x100 (k) 10 x (k + 1) 𝜀11 ∶ 2 10 = C 10 0 (11a) p0 (k) p0 (k + 1) p10 p100 (k + 1) 1 (k) 10 [ ] 0 [ p2 (k) ] 01 x0 (k + 1) x001 (k) x101 (k + 1) 0 x201 (k + 1) 1 𝜀2 ∶ = C 01 0 (11b) p01 p0 (k + 1) 0 (k) 0 p101 (k) [ ] 0 [ p01 (k) ] 2 For second- order approximation x11 x11 0 (k) 0 (k + 1) 11 x101 (k) x1 (k + 1) 11 x (k + 1) x 00 (k) 𝜀11 𝜀21 ∶ 2 11 = C 11 2 (11c) p0 (k) p0 (k + 1) p11 p101 (k + 1) 1 (k) 11 [ p2 (k) ] [p00 2 (k + 1)] 20 x0 (k + 1) x020 (k) 20 x1 (k + 1) x101 (k) 20 x (k + 1) 𝜀12 ∶ 2 20 = C 20 0 (11d) p0 (k) p0 (k + 1) p120 (k) p101 (k + 1) 20 [ ] 0 [ p2 (k) ]

𝜀22

x002 (k + 1) x002 (k) x102 (k + 1) 0 x202 (k + 1) ∶ = C 02 0 p02 (k) p0 (k + 1) 0 02 0 p1 (k) [ ] 02 0 [ p2 (k) ]

(11e)

3.2 Boundary layer correction (BLC) solutions In order to recover the boundary conditions lost in the process of degeneration and to provide the necessary data for solving the outer equations, we introduce the following transformations corresponding to the four boundary layer corrections (two initial and two final) that give unique solution as

x0ci (k) = x0 (k)/(ε1 . . . εi )k+1 , i=1,2 . x0fi (k) = x0 (k)/(ε1 . . . εi )N−k+1 , i=1,2. x1c1 (k) = x1 (k)/(ε1 )k x1c2 (k) = x1 (k)/(𝜀1𝑘 𝜀2𝑘+1 ) x1fi (k) = x1 (k)/(ε1 … εi )N−k+1 , i=1,2. x2ci (k) = x2 (k)/(ε1 … εi )k , i=1,2. x2fi (k) = x2 (k)/(ε1 . . . εi )N−k+1 , i=1,2. p0ci (k) = p0 (k)/(ε1 . . . εi )k+1 , i = 1,2. p0fi (k) = p0 (k)/(ε1 . . . εi )N−k+1, i=1,2. p1ci (k) = p1 (k)/(ε1 . . . εi )k+1 , i = 1,2. p1f1 (k) = p1 (k)/(ε1 )N−k p1f2 (k) = p1 (k)/(𝜀1𝑁−𝑘 𝜀2𝑁−𝑘+1 ) p2ci (k) = p2 (k)/(ε1 . . . εi )k+1 , i = 1,2. p2fi (k) = p2 (k)/(ε1 . . . εi )N−k, i=1,2.

(12a) (12b) (12c) (12d) (12e) (12f) (12g) (12h) (12i) (12j) (12k) (12l) (12m) (12n)

Here suffixes c and f refer to initial and terminal boundary layer corrections respectively. Inserting (12) in (5a), the following sets of subsystems are obtained. For the initial BLC 1 x0c1 (k ) ε1 x0c1 (k + 1 ) x1c1 (k) x1c1 (k + 1) ε2 x2c1 (k) x2c1 (k + 1) =C (13a) ε1 p0c1 (k + 1) p0c1 (k) 2 p1c1 (k) 𝜀1 p1c1 (k + 1) [ p2c1 (k) ] [𝜀12 ε2 p2c1 (k + 1)] and the subsystem for the initial BLC 2 is given by

1596 Kishore and Krishnarayalu. Suboptimal Control of Singularly Perturbed Two Parameter Discrete Control System

International Electrical Engineering Journal (IEEJ) Vol. 5 (2014) No.11, pp. 1594-1604 ISSN 2078-2365 http://www.ieejournal.com/ x0c2 (k ) ε1 ε2 x0c2 (k + 1 ) x1c2 (k) ε2 x1c2 (k + 1) x2c2 (k) x2c2 (k + 1) =C (13b) ε1 p0c2 (k + 1) p0c2 (k) p1c2 (k) 𝜀12 ε2 p1c2 (k + 1) [ p2c2 (k) ] [𝜀12 𝜀22 p2c2 (k + 1)] For the final BLC 1 ε1 x0f1 (k ) x0f1 (k + 1 ) x1f1 (k + 1) 𝜀12 x1f1 (k) x2f1 (k + 1) 𝜀 2 ε x (k) = C 1 2 2f1 (13c) ε1 p0f1 (k) p0f1 (k + 1) p1f1 (k) p1f1 (k + 1) [ p2f1 (k) ] [ε2 p2f1 (k + 1)] and for the final BLC 2 ε1 ε2 x0f2 (k ) x0f2 (k + 1 ) x1f2 (k + 1) 𝜀12 ε2 x1f2 (k) x2f2 (k + 1) 𝜀 2 𝜀 2 x (k) = C 1 2 2f2 (13d) ε1 ε2 p0f2 (k) p0f2 (k + 1) ε2 p1f2 (k) p1f2 (k + 1) [ p2f2 (k) ] [ p2f2 (k + 1)] Seeking asymptotic expansions as ij {xvcs (k), pvfs (k)}= ∑qij≥0{xvcs (k), pijvfs (k)} 𝜀1𝑖 𝜀2𝑗 ; v = 0,1,2; s = 1,2. (14) where q is the desired order of approximation. Substituting (14) in (5a) and collecting the coefficients of like powers of the small parameter ε, we get for the initial BLC 1 Zero- order: 0 00 x0c1 (k) 00 x1c1 (k + 1) 00 x1c1 (k) 00 x2c1 (k + 1) 0 𝜀10 𝜀20 ∶ =C (15a) p00 0c1 (k) 0 00 0 p1c1 (k) [ 00 0 ] [ p2c1 (k) ] First-order: 00 x0c1 (k + 1) x10 0c1 (k) 10 x1c1 (k + 1) 10 x1c1 (k) 10 x (k + 1) 2c1 1 0 𝜀1 ∶ = C 00 (15b) p10 p0c1 (k + 1) 0c1 (k) 0 p10 1c1 (k) [ ] 10 0 [ p2c1 (k) ]

0 01 x0c1 (k) 01 x1c1 (k + 1) 01 x1c1 (k) 01 x2c1 (k + 1) 0 𝜀21 ∶ =C 01 p0c1 (k) 0 01 0 p1c1 (k) [ 0 ] (k) [ p01 ] 2c1 Second-order: 01 x0c1 (k + 1) x11 0c1 (k) 11 x1c1 (k + 1) 11 x1c1 (k) 11 x (k + 1) 10 2c1 1 1 x 𝜀1 𝜀2 ∶ = C 2c1 (k) p11 0c1 (k) 0 0 p11 1c1 (k) [ 0 ] [ p11 (k) ] 2c1 x10 20 0c1 (k + 1) x0c1 (k) 20 x1c1 (k + 1) 20 x1c1 (k) 20 x2c1 (k + 1) 2 𝜀1 ∶ = C 10 0 p20 p0c1 (k + 1) 0c1 (k) 20 0 p1c1 (k) [ ] 0 (k) [ p20 ] 2c1 02 0 x0c1 (k) 02 x1c1 (k + 1) 02 x1c1 (k) 02 x (k + 1) 01 2c1 x 2 2c1 (k) 𝜀2 ∶ =C p02 01 0c1 (k) p0c1 (k + 1) 02 p1c1 (k) 0 [ ] [ p02 0 2c1 (k) ] and for the initial BLC 2 Zero-order: 00 0 x0c2 (k) 0 00 x1c2 (k) 00 x2c2 (k + 1) 00 0 0 x 𝜀1 𝜀2 ∶ = C 2c2 (k) p00 0c2 (k) 0 00 p1c2 (k) 0 [ 0 ] [ p00 2c2 (k) ] First-order: 0 x10 0c2 (k) 0 10 x1c2 (k) x10 2c2 (k + 1) 10 1 x 𝜀1 ∶ = C 2c2 (k) p10 0c2 (k) 0 p10 1c2 (k) 0 [ 0 ] [ p10 2c2 (k) ]

(15c)

(15d)

(15e)

(15f)

(15g)

(15h)

1597 Kishore and Krishnarayalu. Suboptimal Control of Singularly Perturbed Two Parameter Discrete Control System

International Electrical Engineering Journal (IEEJ) Vol. 5 (2014) No.11, pp. 1594-1604 ISSN 2078-2365 http://www.ieejournal.com/ 0

𝜀21 ∶

00 x1c2 (k + 1) 01 x2c2 (k + 1) p01 0c2 (k) 01 p1c2 (k) 01 [ p2c2 (k) ]

01 x0c2 (k) 01 x1c2 (k) 01 = C x2c2 (k) 0 0 [ 0 ]

Second-order: 00 x0c2 (k + 1) x11 0c2 (k) 10 x1c2 (k + 1) 11 x1c2 (k) 11 x (k + 1) 11 2c2 1 1 x 𝜀1 𝜀2 : = C 2c2 (k) p11 0c2 (k) 0 0 p11 1c2 (k) [ 0 ] (k) [ p11 ] 2c2 20 0 x0c2 (k) 0 20 x1c2 (k) 20 x2c2 (k + 1) 20 x 2 2c2 (k) 𝜀1 ∶ =C p20 0c2 (k) 10 p0c2 (k + 1) 20 p1c2 (k) 0 [ p20 [ ] 2c2 (k) ] 0 0 02 x (k) 0c2 01 x1c2 (k + 1) 02 x (k) 1c2 02 x (k + 1) 2c2 02 2 x 𝜀2 ∶ = C 2c2 (k) p02 0c1 (k) 0 02 p1c1 (k) 0 [ 0 ] [ p02 2c1 (k) ] For the final BLC 1 Zero-order: 00 x0f1 (k + 1) 0 00 0 x1f1 (k + 1) 00 0 𝜀10 𝜀20 ∶ x1f2 (k + 1) = C p00 (k + 1) 0f1 0 00 00 p1f1 (k + 1) p1f1 (k) [ ] 00 0 [ p2f1 (k) ] First-order: x10 00 0f1 (k + 1) x0f1 (k + 1) 10 x1f1 (k + 1) 0 x10 0 2f1 (k + 1) 1 𝜀1 ∶ = C 10 p (k + 1) p00 (k) 0f1 0f1 10 10 p (k + 1) p1f1 (k) 1f1 [ ] 10 0 [ p2f1 (k) ]

(15i)

(15j)

(15k)

(15l)

(15m)

(15n)

01 x0f1 (k + 1) 0 01 0 x1f1 (k + 1) 01 0 x (k + 1) 𝜀21 ∶ 2f1 = C p01 (k + 1) 0f1 0 01 01 p1f1 (k + 1) p1f1 (k) [ ] 01 0 [ p2f1 (k) ] Second-order: 01 x11 0f1 (k + 1) x0f1 (k) 11 x1f1 (k + 1) 0 0 x11 (k + 1) 𝜀11 𝜀21 ∶ 2f101 = C p11 (k + 1) 0f1 p0f1 (k) p11 1f1 (k + 1) p11 1f1 (k) 10 11 [p2f1 (k + 1)] [ p2f1 (k) ] 20 x0f1 (k + 1) x10 0f1 (k) 20 x1f1 (k + 1) 00 x1f1 (k) 20 x2f1 (k + 1) 0 2 𝜀1 ∶ = C 20 p10 p0f1 (k + 1) 0f1 (k) 20 20 p1f1 (k) p1f1 (k + 1) 20 [ ] 0 [ p2f1 (k) ] 02 x0f1 (k + 1) 0 02 0 x1f1 (k + 1) 0 02 𝜀22 ∶ x2f1 (k + 1) = C p02 (k + 1) 0f1 0 02 02 p1f1 (k + 1) p1f1 (k) 01 02 p [ 2f1 (k + 1)] [ p2f1 (k) ] and the final BLC 2 Zero-order: 00 0 x0f2 (k + 1) 0 00 x1f2 (k + 1) 0 00 𝜀10 𝜀20 ∶ x2f2 (k + 1) = C p00 (k + 1) 0f2 0 00 p1f2 (k + 1) 0 00 00 p [ [ p2f2 (k) ] 2f2 (k + 1)] First-order: 0 x10 0f2 (k + 1) 0 10 x1f2 (k + 1) 0 10 𝜀11 ∶ x2f2 (k + 1) = C p10 0f2 (k + 1) 0 p10 1f2 (k + 1) 0 10 10 [p2f2 (k + 1)] [ p2f2 (k) ]

(15o)

(15p)

(15q)

(15r)

(15s)

(15t)

1598 Kishore and Krishnarayalu. Suboptimal Control of Singularly Perturbed Two Parameter Discrete Control System

International Electrical Engineering Journal (IEEJ) Vol. 5 (2014) No.11, pp. 1594-1604 ISSN 2078-2365 http://www.ieejournal.com/ 01 x0f2 (k + 1) 0 01 0 x1f2 (k + 1) 0 01 𝜀21 ∶ x2f2 (k + 1) = C p01 (k + 1) 0f2 0 01 00 p1f2 (k + 1) p1f2 (k) 01 01 p [ 2f2 (k + 1)] [ p2f2 (k) ] Second-order: 00 x11 2f2 (k + 1) x0f2 (k) 11 x1f2 (k + 1) 0 0 x11 (k + 1) 𝜀11 𝜀21 : 2f200 = C p11 (k + 1) 0f2 p0f2 (k) p11 1f2 (k + 1) p10 1f2 (k) 11 11 [p2f2 (k + 1)] [ p2f2 (k) ] 20 0 x2f2 (k + 1) 0 20 x1f2 (k + 1) 0 20 𝜀12 ∶ x2f2 (k + 1) = C p20 0f2 (k + 1) 0 20 p1f2 (k + 1) 0 20 20 [p2f2 (k + 1)] [ p2f2 (k) ] 02 x2f2 (k + 1) 0 02 0 x1f2 (k + 1) 0 02 𝜀22 ∶ x2f2 (k + 1) = C p02 (k + 1) 0f2 0 02 10 p1f2 (k + 1) p1f2 (k) 02 02 p [ 2f2 (k + 1)] [ p (k) ]

q

ij

(16e)

ij

(15u)

(15w)

(15x)

ij

ij

𝑗

(16a)

ij

ij

ij

𝑗

ij

𝑗

𝜀1𝑁−𝑘+1 x2f1 (k)+( ε1 ε2 )N−k+1 x2f2 (k)} 𝜀1𝑖 𝜀2 =

𝑗

ij

𝑗

𝜀1𝑁−𝑘 𝑝2f1 (k)+( ε1 ε2 )N−k p2f2 (k)} 𝜀1𝑖 𝜀2

𝜀1𝑁−𝑘+1 x1f1 (k)+( ε1 ε2 )N−k+1 x1f2 (k)} 𝜀1𝑖 𝜀2 (16b) ij ij ij q q x2T (k) = ∑ij≥0{x2 (k) + 𝜀1𝑘 x2c1 (k)+( ε1 ε2 )k x2c2 (k) +

q p0T (k)

ij

ij ij ij q q p2T (k) = ∑ij≥0{p02 (k) + 𝜀1𝑘+1 𝑝2c1 (k) + ( ε1 ε2 )k+1 p2c2 (k) +

x1T (k) = ∑ij≥0{x1 (k) + 𝜀1𝑘 x1c1 (k) + 𝜀1𝑘 𝜀2𝑘+1 x1c2 (k) +

ij

ij

(15v)

𝜀1𝑁−𝑘+1 x0f1 (k)+( ε1 ε2 )N−k+1 x0f2 (k)} 𝜀1𝑖 𝜀2 q

ij

ij

3.3 Total series solution: The total series solution of states x(k), co-states p(k) and control law u(k) may be obtained as ij ij ij q q x0T (k) = ∑ij≥0{x0 (k) + 𝜀1𝑘+1 x0c1 (k)+( ε1 ε2 )k+1 x0c2 (k) +

q

ij

𝜀1𝑁−𝑘 𝑝1f1 (k) + 𝜀1𝑁−𝑘 𝜀2𝑁−𝑘+1 p1f2 (k)}𝜀1𝑖 𝜀2

2f2

ij

q

p1T (k) = ∑ij≥0{p1 (k) + 𝜀1𝑘+1 𝑝1c1 (k) + ( ε1 ε2 )k+1 p1c2 (k) +

(16c) +

ij ∑qij≥0{pij0 (k) + 𝜀1𝑘+1 𝑝0c1 (k) + ( ε1 ε2 )k+1 pij0c2 (k) ij ij 𝑗 𝜀1𝑁−𝑘+1 𝑝0f1 (k)+( ε1 ε2 )N−k+1 p0f2 (k)} 𝜀1𝑖 𝜀2

(16d)

(16f)

uq(k)= -R-1[B1’p0q(k+1)+ ε1 B2’ p1q-1(k+1)+ ε1 ε2 B2’ p2q-2(k+1)] (17) here q is the desired order of approximation. 3.4 Boundary conditions: 𝜀10 𝜀20 ∶ x000 (0) = x0 (0) 00 (0) x1c1 = x1 (0) - x100 (0) 00 (0) 00 (0) x2c2 = x2 (0) - x200 (0) - x2c1 00 (0) 1 10 (0) 𝜀1 : x0 = -x0c1 10 (0) x1c1 = -x110 (0) 10 (0) x10 = -x10 2c2 2 (0) - x 2c1 (0) 𝜀21 : x001 (0) = 0 01 (0) 00 (0) x1c1 = -x101 (0) - x1c2 01 (0) 01 (0) 01 (0) x2c2 = -x2 - x2c1 and 𝜀10 𝜀20 ∶ p00 0 (N) = p0 (N) 00 (N) = p1 (N) - p100 (N) p1f1 00 00 p2f2 (N) = p2 (N) - p00 2 (N) - p2f1 (N) 00 𝜀11 : p10 0 (N) = -p0f1 (N) 10 p1f1 (N) = -p10 1 (N) 10 10 p10 2f2 (N) = − p2 (N) - p2f1 (N) 01 (N) 1 𝜀2 : p0 =0 01 00 (N) = - p101 (N) - p1f2 (N) p1f1 01 01 p2f2 (N) = − p2 (N) - p01 2f1 (N) Second-order 01 00 𝜀11 𝜀21 ∶ x11 0 (0) = − x 0c1 (0) − x 0c2 (0) 10 (0) 11 (0) 11 (0) x1c1 = - x1 - x1c2 11 11 x11 2c2 (0) = - x 2c1 (0) - x 2 (0) 10 (0) 2 20 (0) 𝜀1 : x0 = -x0c1 20 (0) x1c1 = -x120 (0) 20 (0) 20 (0) x2c2 = -x220 (0) - x2c1 2 02 (0) 𝜀2 : x0 =0 02 (0) 01 (0) x1c1 = -x102 (0) - x1c2 02 (0) 02 (0) 02 (0) x2c2 = -x2 - x2c1 and 01 00 𝜀11 𝜀21 ∶ p11 0 (N) = − p0f1 (N) − p0f2 (N) 11 10 11 p1f1 (N) = − p1 (N) − p1f2 (N) 11 11 p11 2f2 (N) = - p2 (N) - p2f1 (N) 10 20 (N) 2 𝜀1 : p0 = -p0f1 (N)

(18a) (18b) (18c) (18d) (18e) (18f) (18g) (18h) (18i) (18j) (18k) (18l) (18m) (18n) (18o) (18p) (18q) (18r) (19a) (19b) (19c) (19d) (19e) (19f) (19g) (19h) (19i) (19j) (19k) (19l) (19m) 1599

Kishore and Krishnarayalu. Suboptimal Control of Singularly Perturbed Two Parameter Discrete Control System

International Electrical Engineering Journal (IEEJ) Vol. 5 (2014) No.11, pp. 1594-1604 ISSN 2078-2365 http://www.ieejournal.com/ 20 (N) = -p120 (N) p1f1 20 20 p2f2 (N) = − p20 2 (N) - p2f1 (N) 02 2 𝜀2 : p0 (N) = 0 02 01 (N) = -p102 (N) - p1f2 (N) p1f1 02 02 (N) p2f2 (N) = − p02 p 2 2f1 (N)

(19n) (19o) (19p) (19q) (19r)

3.5 Algorithm: For a particular order of approximate solution, first find the outer solution for states and co-states. Then add the BLC corresponding to the least singular transformation. Continue this process and finally add the BLC corresponding to the most singular transformation. Once a particular order of solution is obtained for states and co-states using (16), then obtain the corresponding suboptimal control using (17). IV. ILLUSTRATIVE EXAMPLE Consider a fifth order power system model sampled with 0.7s [17]. The resulting system is given by 𝑥01 (𝑘 + 1) 𝑥01 (𝑘) 0.0097 0.9147 0.0253 0.0125 0.0075 0.0051 𝑥02 (𝑘 + 1) 𝑥02 (𝑘) 0.0616 −0.0602 0.8893 −0.0003 0.0456 0.0295 𝑥11 (𝑘 + 1) = −0.0195 0.7016 0.2465 0.0209 0.0192 𝑥11 (𝑘) + 0.0198 u(k) 𝑥12 (𝑘 + 1) −1.4300 −0.0219 −0.0138 0.2399 −0.0063 𝑥12 (𝑘) 1.5006 [ ] [ 𝑥2 (𝑘 + 1) ] −1.1124 −0.0125 −0.0089 0.3388 0.0259 [ 𝑥2 (𝑘) ] [1.1545]

(19a) The eigenspectrum of this system (0.8928 ± 0.0937i, 0.2506 ± 0.0251i, 0.0296) clearly indicates three-time-scale nature. Hence it is represented as a two-parameter system as shown below. 𝑥01 (𝑘 + 1) 𝑥01 (𝑘) 0.9147 0.0253 0.0625 0.0375 0.2550 0.0097 𝑥02 (𝑘 + 1) 𝑥02 (𝑘) −0.0602 0.8893 −0.0015 0.2280 1.4750 0.0616 (𝑘 (𝑘) + 0.0198 u(k) 𝑥11 + 1) = −0.0195 0.7016 1.2325 0.1045 0.9600 𝜀1 𝑥11 1.5006 𝑥12 (𝑘 + 1) −1.4300 −0.0219 −0.0690 1.1995 −0.3150 𝜀1 𝑥12 (𝑘) [ 𝑥2 (𝑘 + 1) ] [−1.1124 −0.0125 −0.0445 1.6940 1.2950 ] [𝜀1 𝜀2 𝑥2 (𝑘)] [1.1545]

where ε1 = 0.2; ε2 = 0.1.

(19b) The initial conditions are given as x01(0) = 2; x02(0) = -1; x10(0) = 1; x12(0) = 5; x2(0) = -4 The performance index is T T J = 1/2∑N−1 (21a) k=0 [w (k)Dw(k) + u (k)Ru(k)] , where N =6; R = 1; 0.25 0 D= 0 0 [ 0

0 0.25 0 0 0

0 0 0.75 0 0

0 0 0 0.75 0

𝑥01 (𝑘) 0 𝑥02 (𝑘) 0 0 ; w(k) = 𝜀1 𝑥11 (𝑘) 0 𝜀1 𝑥12 (𝑘) 1] [𝜀1 𝜀2 𝑥2 (𝑘)]

(21b)

The singularly perturbed TPBVP of tenth-order corresponding to (5a) is

𝑥01 (𝑘 + 1) 𝑥02 (𝑘 + 1) 𝑥11 (𝑘 + 1) 𝑥12 (𝑘 + 1) 𝑥2 (𝑘 + 1) = 𝑝01 (𝑘) 𝑝02 (𝑘) 𝑝11 (𝑘) 𝑝12 (𝑘) [ 𝑝2 (𝑘) ] 0.9147 0.0253 0.0625 0.0375 0.2550 −0.0000 −0.0602 0.8893 −0.0015 0.2280 1.4750 −0.0005 −0.0195 0.7016 1.2325 0.1045 0.9600 −0.0001 −1.4300 −0.0219 −0.0690 1.1995 −0.3150 −0.0145 −0.0111 −1.1124 −0.0125 −0.0445 1.6940 1.2950 0.9147 0.2500 0.0000 0.0000 0.0000 0.0000 0.0253 0.0000 0.2500 0.0000 0.0000 0.0000 0.0000 0.0000 0.7500 0.0000 0.0000 0.0625 0.0000 0.0000 0.0000 0.7500 0.0000 0.0375 [ 0.0000 0.0000 0.0000 0.0000 1.0000 0.2550 𝑥01 (𝑘) 𝑥02 (𝑘) 𝜀1 𝑥11 (𝑘) 𝜀1 𝑥12 (𝑘) 𝜀1 𝜀2 𝑥2 (𝑘) 𝑝01 (𝑘 + 1) 𝑝02 (𝑘 + 1) 𝜀1 𝑝11 (𝑘 + 1) 𝜀1 𝑝12 (𝑘 + 1) [𝜀1 𝜀2 𝑝2 (𝑘 + 1)]

−0.0005 −0.0001 −0.0037 −0.0012 −0.0003 −0.0012 −0.0924 −0.0297 −0.0711 −0.0228 −0.0602 −0.0195 0.8893 0.7016 −0.0015 1.2325 0.2280 0.1045 1.47500 0.9600

−0.0145 −0.0924 −0.0297 −2.2518 −1.7324 −1.4300 −0.0219 −0.0690 1.1995 −0.3150

−0.0111 −0.0711 −0.0228 −1.7324 −1.3328 −1.1124 −0.0125 −0.0445 1.6940 1.2950 ]

(22)

Using the singular perturbation method developed in the previous section, the degenerate, zero, first and second-order solutions are evaluated and compared with the optimal solution in Table 1. Observations from this table are  The degenerate solution, obtained by making ε1 and ε2 equal to zero in (22), is unable to satisfy the boundary conditions of fast states x11(0), x12(0) and faster state x2(0) and their co-states specified at p11(6), p12(6) and p2(6).  The zero-order solution, obtained from (16), incorporates BLCs and recovers these boundary conditions x11(0), x12(0), x2(0), p11(6), p12(6) and p2(6).  The first-order solution improves the zero-order solution.  The second-order solution improves the firstorder solution and is much closer to the optimal (exact) solution. The optimal solution of the tenth-order singularly perturbed discrete TPBVP given by (22) is obtained by the method of complementary functions. This necessitates a special numerical algorithm to be implemented on a digital computer. On the other hand, by using the present SPM, the various series solutions are easily obtained as the stiffness is removed and at the same time are very close to the optimal solution. Thus it is seen that the singular perturbation method not only reduces the order but also removes the ‘stiffness’ of the problem. This can be evidenced from the eigenvalues of full and degenerate optimal control systems. Eigen values of full optimal control system are {33.84855; 4.08196 ± 0.40125i; 1.13125 ± 0.10143i; 0.87692 ± 0.07862i; 0.242635 ± 0.02385i; 0.029543} Eigen values of degenerate optimal control system are {0.90007 ± 0.03575i; 1.10927 ±0.04406i} 1600

Kishore and Krishnarayalu. Suboptimal Control of Singularly Perturbed Two Parameter Discrete Control System

International Electrical Engineering Journal (IEEJ) Vol. 5 (2014) No.11, pp. 1594-1604 ISSN 2078-2365 http://www.ieejournal.com/ Table 1 – Comparison of suboptimal and optimal solutions

x(k) & p(k) x01(0) x02(0) x11(0) x12(0) x2(0) p01(0) p02(0) p11(0) p12(0) p2(0) u(0)

Degenera te Solution 2.00000 -1.00000 -0.72820 -3.07979 -2.40075 2.10992 -1.01957 0.10761 -0.14458 -0.91421 0.03980

Zero Order Solution 2.00000 -1.00000 1.00000 5.00000 -4.00000 2.10992 -1.01957 0.10761 -0.14458 -0.91421 0.03980

First Order Solution 2.00000 -1.00000 1.00000 5.00000 -4.00000 2.21378 0.98447 0.28458 0.58441 -0.83619 0.07598

Second Order Solution 2.00000 -1.00000 1.00000 5.00000 -4.00000 2.41855 -1.10976 0.29984 0.52040 -1.00021 0.24019

2.00000 -1.00000 1.00000 5.00000 -4.00000 2.86755 -1.11978 0.31332 0.43092 -1.02475 0.23163

x01(1) x02(1) x11(1) x12(1) x2(1) p01(1) p02(1) p11(1) p12(1) p2(1) u(1)

1.80448 -1.00725 -0.73981 -2.77830 -2.16635 1.69991 -0.91373 0.08328 -0.12895 -0.81793 0.0354

1.80448 -1.00725 -0.73981 -2.77830 -2.16635 1.69991 -0.91373 0.08328 -0.12895 -0.81793 0.0354

1.85483 -0.77732 -0.38809 -1.53828 -0.43946 1.77503 -0.89017 -0.00926 -0.58003 -0.86389 0.06990

1.82376 -0.93199 -0.59403 -1.22298 -0.39563 1.94814 -1.04133 0.03404 -0.32797 -1.00091 0.21798

1.83594 -0.88573 -0.46181 -1.27960 -0.36337 2.32156 -1.05230 0.03291 -0.55425 -1.00144 0.27842

x01(2) x02(2) x11(2) x12(2) x2(2) p01(2) p02(2) p11(2) p12(2)

1.62542 -1.00219 -0.74117 -2.50522 -1.95384 1.31380 -0.78169 0.06037 -0.10668

1.62542 -1.00219 -0.74117 -2.50522 -1.95384 1.31380 -0.78169 0.06037 -0.10668

1.64755 -0.92508 -0.82057 -3.18683 -2.90763 1.34584 -0.83080 -0.04007 -0.51936

1.62345 -1.03047 -0.82008 -2.60544 -2.21276 1.50018 -0.95619 -0.04337 -0.56352

1.64240 -0.94998 -0.79927 -2.48652 -2.14862 1.72688 -0.97576 -0.07234 -0.70535

Optimal Solution

1601 Kishore and Krishnarayalu. Suboptimal Control of Singularly Perturbed Two Parameter Discrete Control System

International Electrical Engineering Journal (IEEJ) Vol. 5 (2014) No.11, pp. 1594-1604 ISSN 2078-2365 http://www.ieejournal.com/ p2(2) u(2)

-0.67833 0.02923

-0.67833 0.02923

-0.78915 0.05758

-1.01889 0.18735

-0.99976 0.25132

x01(3) x02(3) x11(3) x12(3) x2(3) p01(3) p02(3) p11(3) p12(3) p2(3) u(3)

1.46170 -0.98730 -0.73425 -2.25853 -1.76184 0.95097 -0.62432 0.03885 -0.07783 -0.49591 0.02129

1.46170 -0.98730 -0.73425 -2.25853 -1.76184 0.95097 -0.62432 0.03885 -0.07783 -0.49591 0.02129

1.45612 -1.03233 -0.91508 -2.84010 -2.59689 0.95029 -0.70772 -0.06708 -0.44614 -0.61778 0.03925

1.42163 -1.13605 -1.02569 -2.76278 -2.59680 1.00768 -0.81745 -0.07115 -0.47610 -0.79554 0.14706

1.44111 -1.10474 -0.98380 -2.52265 -2.41595 1.16381 -0.84362 -0.14593 -0.63653 -0.82417 0.19339

x01(4) x02(4) x11(4) x12(4) x2(4) p01(4) p02(4) p11(4) p12(4) p2(4) u(4)

1.31224 -0.96469 -0.72077 -2.03666 -1.58907 0.61107 -0.44187 0.01872 -0.04233 -0.27023 0.01156

1.31224 -0.96469 -0.72077 -2.03666 -1.58907 0.61107 -0.44187 0.01872 -0.04233 -0.27023 0.01156

1.28005 -1.10606 -0.98009 -2.53242 -2.32021 0.59318 -0.52564 -0.09020 -0.36069 -0.35361 0.01503

1.23586 -1.28751 -1.10173 -2.45098 -2.30195 0.60342 -0.60527 -0.09347 -0.37482 -0.45698 0.05077

1.24857 -1.24329 -1.14097 -2.32278 -2.27450 0.66903 -0.62869 -0.19641 -0.49655 -0.49148 0.11497

x01(5) x02(5) x11(5) x12(5) x2(5) p01(5) p02(5) p11(5) p12(5) p2(5) u(5)

1.17601 -0.93613 -0.70218 -1.83803 -1.43433 0.29400 -0.23404 0.00000 0.00000 0.00000 0.00000

1.17601 -0.93613 -0.70218 -1.83803 -1.43433 0.29400 -0.23404 0.00000 0.00000 0.00000 0.00000

1.11874 -1.15240 -1.02090 -2.26235 -2.07636 0.27968 -0.28809 -0.10532 -0.27570 0.00000 0.00000

1.06569 -1.35756 -1.17772 -2.17501 -2.06827 0.27900 -0.32527 -0.25523 -0.30392 -0.02695 0.00000

1.06844 -1.34642 -1.26783 -2.11286 -2.07634 0.26711 -0.33660 -0.19017 -0.31693 -0.04152 0.00000

x01(6) x02(6) x11(6) x12(6) x2(6) p01(6) p02(6) p11(6) p12(6) p2(6)

1.05202 -0.90334 -0.67975 -1.66120 -1.29650 0.00000 0.00000 0.01728

1.05202 -0.90334 -0.67975 -1.66120 -1.29650 0.00000 0.00000 0.00000 0.00000 0.00000

0.97160 -1.17578 -1.04184 -2.00582 -1.84656 0.00000 0.00000 0.00000 0.00000 0.00000

0.901045 -1.40805 -1.28261 -1.99829 -1.91126 0.00000 0.00000 0.00000 0.00000 0.00000

0.90095 -1.41891 -1.36202 -1.97468 -1.93003 0.00000 0.00000 0.00000 0.00000 0.00000

0.04951 0.31697

1602 Kishore and Krishnarayalu. Suboptimal Control of Singularly Perturbed Two Parameter Discrete Control System

International Electrical Engineering Journal (IEEJ) Vol. 5 (2014) No.11, pp. 1594-1604 ISSN 2078-2365 http://www.ieejournal.com/ V. CONCLUSIONS A singularly perturbed discrete power system with two small parameters (exhibiting three-time-scales) is considered. An open-loop optimal controller using PMP is designed for this system. However the solution for control law of this type of system requires special numerical techniques, such as shooting method, which may end up as trial and error method. Hence it is advised to go for a suboptimal controller using SPM that eliminates this problem. SPM that reduces the order and removes the stiffness of the system is an ideal choice for this problem. Here the SPM consists of an outer, two initial BLC and two final BLC solutions. The method is applied to the power system problem up to second order approximation. The results validate the proposed suboptimal controller. The salient features of the present paper are:  The characteristics of singular perturbations such as stiffness, order reduction, loss of boundary conditions and boundary layer corrections are clearly demonstrated.  A method is presented to treat singularly perturbed, discrete open-loop optimal control problems with two small parameters up to second-order approximation in a concise form  The suboptimal control using SPM is applied to a power system problem Hence a suboptimal controller is advised based on SPM for singularly perturbed discrete open-loop optimal control problems, which are notorious for their prohibitive numerical computation. In the future, suboptimal controllers will be developed to multi-parameter discrete control systems. Acknowledgements We greatly acknowledge Siddhartha Academy of General and Technical Education, Vijayawada for providing the facilities to carry out this research. VI. REFERENCES [1] Yuan Yuan, Fuchun Sun and Yenan Hu (2012), Decentralized multi-objective robust control of interconnected fuzzy singular perturbed model with multiple perturbation parameters. Fuzzy Systems (FUZZ-IEEE), IEEE International Conference, pp 1-8. [2] Xin, H., Gan, D. ,Huang, M. and Wang, K.(2010), Estimating the stability region of singular perturbation power systems with saturation nonlinearities: an linear matrix

inequality based method. Control Theory & Applications, IET ,Vol. 4 , Issue 3, pp 351 – 361. [3] Baolin Zhang and MingQu Fan (2008), Near optimal control for singularly perturbed systems with small time-delay. Intelligent Control and Automation, WCICA 2008. 7th World Congress, pp 7212 – 7216. [4] Dmitriev M. and Kurina G. (2006), Singular perturbations in control problems. Automation and Remote Control, 67, 1, 143. [5] Yong Chen and Yongqiang Liu (2005), Summary of Singular Perturbation Modeling of Multi-time Scale Power Systems. Transmission and Distribution Conference and Exhibition: Asia and Pacific, IEEE/PES, pp 1-4. [6] Naidu D. S. (2002), Singular Perturbations and Time Scales in Control Theory and Applications: An Overview. Dynamics of Continuous, Discrete & Impulsive Systems, 9, 2, 233-278. [7] Kevorkian J. K. and Cole J. D.(1996). Multiple Scale and Singular Perturbation Methods. Springer-Verlag, New York. [8] Saksena V. R., O’Reiley J. and Kokotovic P. V. (1984), Singular Perturbations and Time-Scale Methods in Control Theory: Survey 1976-1983. Automatica, 20, 3, 273-293. [9] Naidu, D.S and Rao, A.K. (1985), Singular perturbation analysis of discrete control systems. Volume 1154 of Lecture Nots in Mathematics, A.Dold and B. Eckmann, eds, SpringerVerlag [10] Naidu, D.S and D.B Price (1988), Singular perturbations and time scales in the design of digital flight control systems. NASA Technical paper 2844. [11] Krishnarayalu M. S. (1989), Singular perturbation method applied to the open-loop discrete optimal control problem with two small parameters. Int. J. Systems Science, 20, 5, 793-809. [12] Krishnarayalu M. S. (1990), Singular perturbation method applied to the closed-loop discrete optimal control problem. Optimal Control Applications & Methods, 11, 1, 75-83. [13] Krishnarayalu M. S. (1994), Singular perturbation analysis of a class of initial and boundary value problems in multiparameter digital control systems. Control- Theory and Advanced Technology, 10, 3, 465-477. [14] Krishnarayalu M. S. (1999), Singular perturbation methods for one-point, two-point and multi-point boundary value problems in multiparameter digital control systems. Journal of Electrical and Electronics Engineering, Australia, 19, 3, 97-110. [15] Krishnarayalu M. S. (2004), Singular perturbation methods for a class of initial and boundary value problems in multi-parameter classical digital control systems. ANZIAM J., 46, 67-77. [16] Krishnarayalu M. S. (2008), Singular perturbation method applied to the discrete Euler-Lagrange free-endpoint optimal 1603

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International Electrical Engineering Journal (IEEJ) Vol. 5 (2014) No.11, pp. 1594-1604 ISSN 2078-2365 http://www.ieejournal.com/ control problem. Automatic Control (theory and applications) AMSE journal, 63, 3, 16-29. [17] Kishore Babu G. and Krishnarayalu M. S. (2012), An Application Of Discrete Two Parameter Singular Perturbation Method, IJERT, Vol. 1 Issue 10. [18] Kishore Babu G. and Krishnarayalu M. S.(2014) Some Applications of Discrete One Parameter Singular Perturbation Method. JCET Vol. 4 Iss.1, PP. 76-81. [19] Kishore Babu G. and Krishnarayalu M. S.(2014) Suboptimal Control of Singularly Perturbed Discrete Control

System. International Electrical Engineering Journal (IEEJ) Vol. 5 (2014) No.5, pp. 1413-1419 [20] Roberts S.M. and Shipman J.S. (1972) Two-point Boundary Value Problems: Shooting Methods. Elsevier, New York. [21] Koichi F. and Kunihiko K. (2003). Bifurcation cascade as chaotic itinerancy with multiple time scales. Chaos: An Interdisciplinary Journal of Nonlinear Science, 13, 1041-1056.

1604 Kishore and Krishnarayalu. Suboptimal Control of Singularly Perturbed Two Parameter Discrete Control System