Solving a class of linear and non-linear optimal control problems by homotopy perturbation method

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1 IMA Journal of Mathematical Control and Information (2011) 28, doi:101093/imamci/dnr018 Solving a class of linear and non-linear optimal control problems by homotopy perturbation method S EFFATI AND H SABERI NIK Department of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran Corresponding author: effati911@yahoocom saberi hssn@yahoocom [Received on 5 December 2010; revised on 6 May 2011; accepted on 4 June 2011] In this paper, we give an analytical approximate solution for non-linear quadratic optimal control problems and optimal control of linear systems using the homotopy perturbation method (HPM) Applying the HPM, the non-linear two-point boundary-value problem (TPBVP) and linear systems, derived from the Pontryagin s maximum principle, are transformed into a sequence of linear time-invariant TPBVPs Solving the proposed linear TPBVP sequence in a recursive manner leads to the optimal control law and the optimal trajectory in the form of rapid convergent series Finally, a non-linear example and several linear examples are given to verify the reliability and efficiency of the proposed method Keywords: homotopy perturbation method; optimal control problems; Pontryagin s maximum principle; Hamiltonian system 1 Introduction The investigation of the optimal control is of importance in modern control theory One of the most active subjects in control theory is the optimal control which has a wide range of practical applications not only in all areas of physics but also in economy, aerospace, chemical engineering, robotic, etc (Garrard & Jordan, 1997; Manousiouthakis & Chmielewski, 2002; Tang, 2005; Notsu et al, 2008) The theory and the application of optimal control for linear time-invariant systems have been developed perfectly However, as for non-linear systems, synthesis problems that are solved by classic control theory lead to difficult computations It is well-known that the non-linear optimal control problem (OCP) can be reduced to a two-point boundary-value problem (TPBVP), implementing the Pontryagin s maximum principle (PMP) This TPBVP cannot be solved analytically, in general, and most researches have been devoted to find an approximate solution, for non-linear TPBVP s (Ascher et al, 1995) In order to find an analytical approximate solution, some successive methods have been proposed, eg Tang (2005) One familiar scheme is to determine the optimal control law using dynamic programming This approach leads to the Hamilton Jacobi Bellman equation that is hard to solve in most cases Yousefi et al (2010) used the original or basic Variational Iteration Method (VIM) for linear quadratic OCPs They transfer the linear TPBVP obtained from PMP to an initial value problem (IVP) and then implement the basic VIM to get a feedback controller In He (2008), optimal problems are solved in a more attractive way In recent years, the homotopy perturbation method (HPM), first proposed by He (1999, 2003b), has successfully been applied to solve many types of linear and non-linear functional equations This method, which is a combination of homotopy in topology and classic perturbation techniques, provides us with a convenient way to obtain analytic or approximate solutions for a wide variety of problems arising in different fields He used HPM to solve the Lighthill equation (He, 1999), the Duffing equation (He, 2003b) and the Blasius equation (He, 2003a), and then the idea found its way in sciences and c Crown copyright 2011

2 540 S EFFATI AND H SABERI NIK has been used to solve non-linear wave equations (He, 2005), boundary-value problems (He, 2006), quadratic Riccati differential equations (Abbasbandy, 2006), integral equations (Golbabai & Keramati, 2008), Klein Gordon and sine Gordon equations (Odibat & Momani, 2007), initial value problems (Chowdhury & Hashim, 2007), non-linear evolution equations (Ganji et al, 2007) and many other problems Biazar & Alizadeh (2010) performed decomposition of source terms in HPM Hesameddini & Latifizadeh (2009) used an optimal choice of initial solutions in the HPM and Mohyud-Din et al (2010) used coupling of He s polynomials and Laplace transformation The main advantage of applying HPM is that the results are readily obtained and a few iterations are used The significant merit of the analytic approach is to provide scientists with the general parametric relation between the dependent and independent variables, namely, displacement and time, respectively Therefore, the related equations can be simply obtained, giving one the opportunity for further studies, for different cases and thereby different parameters This paper concerns with a class of nonlinear quadratic OCPs and linear OCP First, HPM is employed for finding the optimal control of linear systems As discussed earlier, non-linear quadratic OCPs, using PMP, leads to a nonlinear Two-Point Boundary Value Problem (TPBVP) Then applying a new method as shooting method for selection of the initial approximations, we solve this TPBVP by the HPM We will apply He s polynomials in order to make the solution procedure easier, more effective and more straightforward 2 Optimality conditions for linear optimal control system In the present work, we consider the following linear OCP: ẋ = Ax(t) + Bu(t), x(t 0 ) = x 0, J = 1 2 x(t f) Sx(t f ) tf t 0 (x Px + 2x Qu + u Ru)dt, (1) where x R n, u R m, A R n n and B R m n The control u(t) is an admissible control if it is piecewise continuous in t for t [t 0, t f ] Its values belong to a given closed subset U of R + The input u(t) is derived by minimizing the quadratic performance index J, where S R n n, P R n n and Q R n m are positive semi-definite matrices and R R m m is a positive definite matrix Consider the Hamiltonian for system (1) as H(x, u, λ, t) = 1 2 (x Px + 2x Qu + u Ru) + λ (Ax + Bu), (2) where λ R n is a co-state vector According to the PMP (Datta and Mohan, 1995), we can obtain the following optimal control law u = R 1 Q x R 1 B λ (3) and the Hamiltonian system ẋ = [A B R 1 Q ]x B R 1 B λ, λ = [ P + Q R 1 Q ]x + [Q R 1 B A ]λ, (4) with the condition x(t 0 ) = x 0 Since x(t f ) is indeterminate, then λ(t f ) = Sx(t f ) (5)

3 SOLVING LINEAR AND NON-LINEAR OCPS 541 By applying the terminal condition (5), we can write the solution of system (4) in the form x(t) = (F + GS)x(t f ), where F, G, L and M are n n matrices By taking Y (t, t f ) = (L + M S)(F + GS) 1, we have Differentiating (7) with respect to t and summarizing lead to λ(t) = (L + M S)x(t f ), (6) λ(t) = Y (t, t f )x(t) (7) Ẏ = (Y B + Q)R 1 (B Y + Q ) Y A A Y P (8) By considering (3) and (7), we can see the optimal control law as We now introduce the following variables: u (t) = R 1 Q T x R 1 B T Y (t, t f )x(t) (9) V (t) = F(t, t f ) + G(t, t f )S, W (t) = L(t, t f ) + M(t, t f )S (10) Substituting (10) into (6) and then into (4), we obtain the following system: V (t) = [A B R 1 Q ]V (t) B R 1 B W (t), Ẇ (t) = [ P + Q R 1 Q ]V (t) + [Q R 1 B A ]W (t), (11) with conditions V (t f ) = I and W (t f ) = S 3 Non-linear quadratic OCPs and solution guidelines Consider the non-linear dynamical system ẋ(t) = f (t, x(t)) + g(t, x(t))u(t), t [t 0, t f ], x(t 0 ) = x 0, x(t f ) = x f, (12) with x(t) R n denoting the state variable, u(t) R m the control variable and x 0 and x f the given initial and final states at t 0 and t f, respectively Moreover, f (t, x(t)) R n and g(t, x(t)) R n m are two continuously differentiable functions in all arguments Our aim is to minimize the quadratic objective functional J[x, u] = 1 2 tf t 0 (x (t)qx(t) + u (t)ru(t))dt, (13)

4 542 S EFFATI AND H SABERI NIK subject to the non-linear system (12), for Q R n n and R R m m, positive semi-definite and positive definite matrices, respectively Since the performance index (13) is convex, the following extreme necessary conditions are also sufficient for optimality (Geering, 2007): ẋ = f (t, x) + g(t, x)u, λ = H x (x, u, λ), u = argmin u H(x, u, λ), x(t 0 ) = x 0, x(t f ) = x f, (14) where H(x, u, λ) = 1 2 [x Qx + u Ru] + λ [ f (t, x) + g(t, x)u] is referred to as the Hamiltonian Equivalently, (14) can be written in the form ẋ = f (t, x) + g(t, x)[ R 1 g (t, x)λ], ( ( ) ) f (t, x) n λ = Qx + λ + λ i [ R 1 g (t, x)λ] g i(t, x), x x i=1 x(t 0 ) = x 0, x(t f ) = x f, (15) where λ(t) R n is the co-state vector with the ith component λ i (t), i = 1,, n, and g(t, x) = [g 1 (t, x),, g n (t, x)] with g i (t, x) R m, i = 1,, n Also the optimal control law is obtained by u = R 1 g (t, x)λ (16) For solving such a TPBVP in (15), we use a shooting-method-like procedure combined with the HPM So, first we apply HPM to solve the following IVP: ẋ = f (t, x) + g(t, x)[ R 1 g (t, x)λ], ( ( ) ) f (t, x) n λ = Qx + λ + λ i [ R 1 g (t, x)λ] g i(t, x), x x i=1 x(t 0 ) = x 0, λ(t 0 ) = α, (17) where α R is an unknown parameter Then it will be identified after sufficient iterations of HPM, as discussed hereinafter 4 Basic idea of HPM The principles of the HPM can be described as follows (He, 1999) Consider the following non-linear differential equation L(u) + N(u) f (r) = 0, r Ω, (18) subject to the boundary condition ( B u, u ) = 0, r Γ, (19) n

5 SOLVING LINEAR AND NON-LINEAR OCPS 543 where L is a linear operator, while N is a non-linear operator, B is a boundary operator, f (r) is a known analytical function and Γ is the boundary of domain Ω By the homotopy technique, He constructed a homotopy v(r, p): Ω [0, 1], which satisfies H(v, p) = (1 p)[l(v) L(u 0 )] + p[a(v) f (r)] = 0, p [0, 1], r Ω, (20) where p [0, 1] is an embedding parameter and u 0 is an initial guess approximation of (18), which satisfies the boundary conditions It follows from (20) that H(v, 0) = L(v) L(u 0 ) = 0, H(v, 1) = A(v) f (r) = 0 Here we assume that the solution of (20) is a power series in p: Setting p = 1, we obtain the approximate solution of (18), The convergence of series (22) has been proved in He (2003b) v = v 0 + pv 1 + p 2 v 2 + (21) u = lim p 1 v = v 0 + v 1 + v 2 + (22) THEOREM 1 Suppose N(v) is a non-linear function and v = p k v k, then we have ( n ) ( p n N(v) p=0 = n p n N n ) p k v k = n p n N p k v k p=0 p=0 Proof For more details, see Ghorbani (2009) THEOREM 2 The approximate solution of (18) obtained by the HPM can be expressed in He s polynomials as u(r) = f (r) + H 0 (v 0 ) + H 1 (v 0, v 1 ) + + H n (v 0, v 1,, v n ), where He s polynomials are defined as follows: H j (v 0, v 1, v 2,, v j ) = L 1 1 j j! p j N p k v k, j = 0,, n Proof For more details, see Ghorbani (2009) 5 Application of the HPM for non-linear quadratic OCPs In this section, we shall introduce a new reliable procedure for choosing the initial approximations in HPM to solve non-linear quadratic OCPs, which reduce to a TPBVP

6 544 S EFFATI AND H SABERI NIK 51 Selection of the initial approximations As mentioned in He (1999, 2003a), n k=1 x k (t) and n k=1 λ k (t) tend to the exact solutions of (17), say ˆx(t, α) and ˆλ(t, α), as n Thus, for sufficiently large number of HPM iterations, N, we can write Nk=1 x k (t) = ˆx(t, α) and N k=1 λ k (t) = ˆλ(t, α) or, more precisely, since N k=1 x k and N k=1 λ k are functions of both t and α, we can write N k=1 x k (t, α) = ˆx(t, α) and N k=1 λ k (t, α) = ˆλ(t, α) Note that we did not use the final-state condition x(t f ) = x f until now Considering this condition, α should be identified such that N k=1 x k (t f, α) = ˆx(t f, α) = x f That is, α should be a real root of Nk=1 x k (t f, α) x f = 0, which can be easily approximated by numerical methods such as Newton or Secant method Let us denote the approximated α by ˆα Therefore, the analytic approximate solution of (15) is x(t) = N x k (t, ˆα), λ(t) = k=1 and the optimal control law can be obtained by (16) N λ k (t, ˆα), (23) k=1 52 Suboptimal control design Consider the OCP of the non-linear system (12) with the quadratic cost function (13) Then, the Nthorder suboptimal trajectory control pair is obtained as follows: x (N) (t) = N i=0 x i (t), u (N) (t) = R 1 g (t, x) N i=0 λ i (t) The integer N in (24) is generally determined according to a concrete control precision As we will present in the next subsection, every time x i (t) and λ i (t) are obtained from the linear TPBVP sequence, we let N = i and calculate x (N) (t) and λ (N) (t) from (24) Then the following quadratic performance index (QPI) can be calculated as J (N) = 1 2 tf (24) t 0 ((x (N) ) (t)q(x (N) )(t) + (u (N) ) (t)r(u (N) )(t))dt (25) The N th-order suboptimal trajectory control pair in (24) has desirable accuracy if for two given positive constants ɛ 1 > 0 and ɛ 2 > 0, the following conditions hold jointly: J (N) J (N 1) < ɛ 1, (26) J (N) x(t f ) x f < ɛ 2, (27) where is a suitable norm on R n and x(t f ) is the value of the corresponding state trajectory at the final time t f If the tolerance error bounds ɛ 1 > 0 and ɛ 2 > 0 be chosen small enough, the Nth-order suboptimal control law will be very close to the optimal control law u (t), the value of QPI in (24) will be very close to its optimal value J and the boundary condition will be satisfied tightly

7 SOLVING LINEAR AND NON-LINEAR OCPS An illustrative example Consider the following non-linear OCP: min J = 1 0 u 2 (t)dt st ẋ = 1 2 x2 (t) sin x(t) + u(t), t [0, 1], x(0) = 0, x(1) = 05 (28) According to (12, 13), we have f (t, x(t)) = 1 2 x2 (t) sin x(t), g(t, x(t)) = 1, Q = 0, R = 1, t 0 = 0 and t f = 1 As mentioned in Section 51, we solve the following IVP: ẋ = 1 2 x2 (t) sin x(t) 1 2 λ(t), λ = λ(t)x(t) sin x(t) 1 2 λ(t)x2 (t) cos x(t), t [0, 1], x(0) = 0, λ(0) = α, (29) where α R is an unknown parameter Also the optimal control law is given by u (t) = 1 λ(t) (30) 2 To solve system (29) by HMP, we construct the following homotopy: (1 p)( v(t) ẋ 0 (t)) + p ( v(t) 1 2 v2 (t) sin v(t) w(t)) = 0, (1 p)(ẇ(t) λ 0 (t)) + p ( ẇ(t) + w(t)v(t) sin v(t) w(t)v2 (t) cos v(t) ) = 0, (31) where p [0, 1] is an embedding parameter and x 0 and λ 0 are initial approximations that satisfy the initial conditions Suppose the solution of (29) has the form v = x 0 + px 1 + p 2 x 2 +, w = λ 0 + pλ 1 + p 2 λ 2 + (32) Substituting (32) into (31) and comparing coefficients of the terms with the identical powers of p lead to

8 546 S EFFATI AND H SABERI NIK p 0 : v 0 (t) ẋ 0 (t) = 0, v 0 (0) = 0, (33) ẇ 0 (t) λ 0 (t) = 0, w 0 (0) = α, p 1 : v 1 (t) + ẋ 0 (t) 1 2 v 0 2 (t) sin v 0 (t) w 0(t) = 0, v 1 (0) = 0, (34) ẇ 1 (t) + λ 0 (t) + w 0 (t)v 0 (t) sin v 0 (t) w 0(t)v 0 2 (t) cos v 0 (t) = 0, w 1 (0) = 0, p 2 : ( 1 v 2 (t) 1 2 p N 1 p k v k, ( 1 ẇ 2 (t) + p N 2 ( p N 3 p k v k, p k v k, ) 1 p k v k ) 1 p k w k ) 1 p k w k p=0 p=0 p= v 1(t) = 0, v 2 (0) = 0, = 0, w 2 (0) = 0, (35) p j : v j (t) 1 1 j 1 2 ( j 1)! p j 1 N j 1 j 1 1 p k v k, p k w k 1 j 1 ẇ j (t) + ( j 1)! p j 1 N j 1 j 1 2 p k v k, p k w k ( j 1)! p 2 N j 1 3 p k v k, p k w k p=0 p=0 p= w j 1(t) = 0, v j (0) = 0, = 0, w j (0) = 0, (36) where n! 1 ( p n n N n i p k v k, n p k w k, i = 1, 2, 3, are He s polynomials relative to non-linear )p=0 terms

9 SOLVING LINEAR AND NON-LINEAR OCPS 547 FIG 1 The optimal control Solving the presented sequence of linear time-invariant TPBVPs in (33 36) with the initial approximations of x 0 and λ 0 leads to x 0 (t) = 0, λ 0 (t) = α, λ 4 (t) = 0, x 1 (t) = 1 2 αt, λ 1(t) = 0, λ 5 (t) = α5 t 5, x n (t) = 0, n 2, λ 2 (t) = 0, λ 3 (t) = 1 8 α3 t 3 By considering the final-state condition, we should have i x k (1) = 05, for i = 1,, 5 That is, 1 2α = 05 or, equivalently, α = 1 Therefore, we have x 0 (t) = 0, λ 0 (t) = 1, λ 3 (t) = 1 8 t3, and the associated optimal control is x 1 (t) = 1 2 t, λ 1(t) = 0, λ 4 (t) = 0, x n (t) = 0, n 2, λ 2 (t) = 0, λ 5 (t) = t5

10 548 S EFFATI AND H SABERI NIK FIG 2 The optimal state trajectory TABLE 1 Simulation results of example at different iteration times i J(i) J (i) J (i 1) J (i) x(t f ) x f u(t) i=0 λ i (t) = t t5 Simulation curves of u(t) and x(t) for n = 5 are shown in Figs 1 and 2, respectively Also, we compared the results of HPM with the solutions obtained using the collocation method (Ascher et al, 1995) and modal series (Jajarmi et al, 2011) Results of both methods are very close to each other as shown by Figs 1 and 2 This confirms that the proposed method yields excellent results In order to obtain an accurate enough suboptimal trajectory control pair, we applied the strategy proposed in Section 52, with the tolerance error bounds ɛ 1 = and ɛ 2 = In this case, convergence is achieved after five iterations, ie J (5) J (4) J = < and (5) x(1) 05 = < A minimum of J (5) = is obtained Problem (28) has also been solved by Rubio (1986) via the measure theory in which to find an acceptable solution, a linear programming problem with 1000 variables and 20 constraints should be solved Results are listed in Tables 1 and 2

11 SOLVING LINEAR AND NON-LINEAR OCPS 549 TABLE 2 Results of the proposed method and measure theoretical method Method Performance index value Final-state error HPM Measure theory method Modal series method Application of the HPM for linear optimal control system In this section, to illustrate the effectiveness of the HPM we shall consider two examples of optimal control of linear systems In the following examples, we assume, for the sake of simplicity, Q = 0 and k(t) = R 1 B Y (t), and according to (9), we have u (t) = k(t)x(t) Other numerical methods for approximating k(t) based on orthogonal functions are available in Datta and Mohan (1995) EXAMPLE 3 Consider a single-input scalar system as follows: ẋ = 2x(t) + u(t), (37) J = 1 2 x2 (1) (x 2 (t) + u 2 (t))dt (38) According to system (1), we have A = 2, B = 1, S = 1, P = 1, Q = 0, R = 1 and t 1 = 1 By using system (11), we have The exact solution of k(t) is k(t) = V (t) = 2V (t) W (t), Ẇ (t) = V (t) + 2W (t) (39) 5 cosh 5(1 t) sinh 5(1 t) 5 cosh 5(1 t) + 3 sinh 5(1 t) To solve system (39) by HPM, we construct the following homotopy: (1 p)( V (t) V 0 (t)) + p( V (t) + 2V (t) + W (t)) = 0, (1 p)(ẇ (t) Ẇ 0 (t)) + p(ẇ (t) + V (t) 2W (t)) = 0 (40) In Fig 3, the approximate value for k(t), obtained from HPM with n = 15, and the following exact value are plotted Table 3 gives the results from the HPM and exact solution of Example 61 and illustrates the absolute error

12 550 S EFFATI AND H SABERI NIK FIG 3 Comparison of the exact solution with the HPM solution TABLE 3 Comparison of the error of the 15th-order HPM, exact solution t HPM Exact solution Absolute error EXAMPLE 4 Consider a single-input scalar system as follows: x = x(t) + u(t), (41) J = (x 2 (t) + u 2 (t))dt (42) According to system (1), we have A = 1, B = 1, S = 0, Q = 1, R = 1 and t f = 1, then V (t) = V (t) W (t), W (t) = V (t) + W (t) (43)

13 SOLVING LINEAR AND NON-LINEAR OCPS 551 The exact solution of k(t) is where k(t) = (1 + 2β) cosh( 2t) + ( 2 + β) sinh( 2t) cosh( 2t) + β sinh(, 2t) β = cosh( 2) + 2 sinh( 2) 2 cosh( 2) + sinh( 2) In Fig 4, the approximate value for k(t), obtained from HPM with n = 15, and the following exact value are plotted Table 4 gives the results from the HPM and exact solution of Example 62 and illustrates the absolute error FIG 4 Comparison of the exact solution with the HPM solution TABLE 4 Comparison of the error of the 15th-order HPM approximate solution with exact solution t HPM Exact solution Absolute error

14 552 S EFFATI AND H SABERI NIK 7 Conclusions In this paper, we have successfully developed HPM and He s polynomials for solving non-linear quadratic OCPs Then we employed HPM to finding optimal control of linear systems Applying the HPM, the optimal control law and the optimal trajectory are determined in the form of rapid convergent series with easy computable terms The proposed method avoids directly solving the non-linear TPBVP Therefore, in view of computational complexity, the proposed method is more practical than the other approximate methods Matlab has been used for computations in this paper REFERENCES ABBASBANDY, S (2006) Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian s decomposition method Appl Math Comput, 172, ASCHER, U M, MATTHEIJ, R M M & Russel, R D (1995) Numerical Solution of Boundary Value Problems for Ordinary Differential Equations Philadelphia, PA: SIAM BIAZAR, J & ALIZADEH, S (2010) Decomposition of source terms in homotopy perturbation method Nonlinear Sci Lett A, 1, CHOWDHURY, M S H & HASHIM, I (2007) Solutions of class of singular second-order IVPs by homotopy perturbation method Phys Lett A, 365, DATTA, K B & MOHAN, B M (1995) Orthogonal Functions in Systems and Control Singapore: World Scientific GANJI, D D, TARI, H & BAKHSHI JOOYBARI, M (2007) Variational iteration method and homotopy perturbation method for nonlinear evolution equations Int J Comput Math Appl, 54, GARRARD, W L & JORDAN, J M (1997) Design of nonlinear automatic flight control systems Automatica, 13, GEERING, H P (2007) Optimal Control with Engineering Applications Berlin: Springer GOLBABAI, A & KERAMATI, B (2008) Modified homotopy perturbation method for solving Fredholm integral equations Chaos Solitons Fractals, 37, GHORBANI, A (2009) Beyond Adomian polynomials: He polynomials Chaos Solitons Fractals 39, HESAMEDDINI, E & LATIFIZADEH, H (2009) An optimal choice of initial solutions in the homotopy perturbation method Int J Nonlinear Sci Numer, 10, HE, J H (1999) Homotopy perturbation technique Comput Methods Appl Mech Eng, 178, HE, J H (2003) A simple perturbation approach to Blasius equation Appl Math Comput, 140, HE, J H (2003) Homotopy perturbation method: a new nonlinear analytical technique Appl Math Comput, 135, HE, J H (2005) Application of homotopy perturbation method to nonlinear wave equations Chaos Solitons Fractals, 26, HE, J H (2006) Homotopy perturbation method for solving boundary value problems Phys Lett A, 350, HE, J H (2008) An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering Int J Mod Phys B, 22, JAJARMI, A, PARIZ, N, VAHIDIAN, A & EFFATI, S (2011) A novel modal series representation approach to solve a class of nonlinear optimal control problems Int J Innov Comput Inf Control, 7, MANOUSIOUTHAKIS, V & CHMIELEWSKI, D J (2002) On constrained infinite-time nonlinear optimal control Chem Eng Sci, 57, MOHYUD-DIN, S T, NOOR, M A, NOOR, K I HOSSEINI, M M (2010) On the coupling of He s polynomials and Laplace transformation Int J Nonlinear Sci Numer, 11, 93 96

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