A Robust Algorithm for a Class of Optimal Control Problems Subject to Regional Stability Constraints and Disturbances

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1 A Robust Algorithm for a Class of Optimal Control Problems Subject to Regional Stability Constraints and Disturbances Saeid Rastegar, and Rui Araújo Institute for Systems and Robotics (ISR-UC), and Department of Electrical and Computer Engineering (DEEC-UC), University of Coimbra, Pólo II, PT Coimbra srastegar@isr.uc.pt, rui@isr.uc.pt December 4, 2012 Abstract A robust globally convergent algorithm for solving the optimization control problem (OCP) in both state feedback controller and observation control system is investigated. Finding the OCP adjoint parameter for computing the optimal observer gain and feedback gain vectors are desired. First, the optimal control problem considering stability of degree constrains and disturbance that affects the dynamics of system is converted into a two-point boundary value problem (TPBVP). Then, we apply the He s polynomials based on homotopy perturbation method (HPM) as an efficient method to find both optimal gains. The algorithm will be modified do decrease the number of iterations required to attain a desired control problem cost function. As a result lower computational complexity is required when compared with other state of the art methods. Applying the HPM makes the solution procedure become easier, simpler and more straightforward. In the proposed method the control problem can be solved with lower amplitudes of the input signal (control effort), comparing with analytical method. Lower control efforts may also help to avoid saturation effects, and to restrain the system to work within linear operating areas of the state space. On the other hand, there is a tradeoff between control effort and the degree of optimality obtained. For demonstrating the simplicity and efficiency of the proposed optimal control method, the algorithm is compared with a control architecture using the Kalman filter estimator and a controller gain designed by the Lyapunov s second method. Keywords: Optimization control problem (OCP), Observer gain vector, State gain matrix, Two-point boundary value problem (TPBVP), stability degree constrains, Homotopy perturbation method (HPM), Disturbances, Kalman filter. 1 Introduction Multi-variable feedback control systems are important in many areas of science and engineering. In the last decade, for example, many researches and progresses have been developed in structural control [1], [2]. Among the various methods for control of structure vibration, feedback active control is an efficient and robust method. Many practical examples can be found in building industry, where most structural control investigation focus on using state feedback control methods. In such systems, high dimensional models are needed to describe the dynamics states. This means that the use of full state feedback control systems may be require a large number of sensors and actuators which probably may not be realized in practical implementations. Thus, replacing full-state measurement vector by a state estimation derived from incomplete state measurements is inevitable. The state estimator can be designed as a Luenberger observer [3] or as a Kalman Bucy filter [4], [5]. These observers use a linear feedback of the difference between the measurement and the computed output to reduce the estimation error. Fast observer response is generally achieved by locating the observer poles deep enough in the left complex plane, which implies 1

2 a large observer gain matrix. However, using a large observer gain matrix in practice is difficult or not possible due to the observer sensitivity to sensor noise, calculation error or time delay. Recently, in [6] has been tried to use fuzzy rule-based logic to overcome this problem. An optimal observer like the Kalman filter makes an unbalanced estimation with minimum of variance which leads to optimal estimation of state vectors. The combination of this optimal observability design and optimal state gain feedback results in a complete optimal closed-loop control systems. These designs are considered independently, according to the separation principle [7]. The procedure required to design an optimal observer or feedback controller involves the non-linear the Hamiltonian-Jacobi-Bellman (HJB) equations which satisfies the two-point boundary value problem (TPBVP) which in turn can be solved by the Pontryagin s maximum principle. By solving the HJB equations, the optimal adjoint parameter for state space estimator is obtained. In general, the HJB equations are difficult to solve nonlinear partial differential equations. The problem is more complicated especially when disturbances that affect the dynamics of the system states, and regional stability constrains are considered. Besides, a nonlinear TP- BVP has no analytical solution except for a few simple cases. Thus, many researches have been devoted to find an approximate solution for the nonlinear TPBVP s. In [8], a successive Galerkin approximation (SGA) method is introduced. In the SGA method, a sequence of generalized HJB equations is solved in a iterative procedure to obtain a sequence of approximations to the solution of the HJB equations. However, the proposed sequence may converge very slowly or may even be diverge. In another work the measure theory approach was applied to solve the problem [9]. This converts the nonlinear OCP to a linear programming and makes a piecewise constant control rule. In other two researches a successive approximation approach (SAA) and sensitivity approach have been introduced in [10] and [11], respectively. In both cases, instead of directly solving the nonlinear TPBVP, a sequence of non homogeneous linear time-varying TPBVPs are solved. However, simulation results show that solving time-invariant equations is simpler when comparing to solving time-varying equations. This motivates to solve HJB equation by an approximate-analytic method. Recently, the HJB equations have been solved by HPM in simple mathematical view witout considering control constraints [12], [13], [14], while the problem will be more complicated in OCP when considering the control constrains that affect behaviour of the dynamic state vector. The homotopy perturbation method (HPM) was initially proposed by He [15]. He has shown that HPM, as well as some other semi-analitical methods such as variational iteration method (VIM) [16], [17], can be successfully applied to solve many types of linear and nonlinear functional equations. HPM is based on the combination of the homotopy deformations in topology and classic perturbation techniques. It provides us such a convenient method to find analytic or approximate solutions for a various of problems which are arising in different fields. Initially, HPM has been applied to solve the Lighthill equation [15], Duffing equation [18], Blasius equation [19] and electric potential equation [20]. Then, the idea was developed into various areas. It has been used in boundary value problems [21], integral equations [22], Klein Gordon and sine Gordon equations [23], initial value problems [24], nonlinear evolution equations [25], heat equations [26], [27], linear and nonlinear optimal control problems [28] and many other practical problems. Recently application of HPM on unsteady motion of a spherical and non-spherical particles in incompressible Newtonian environment was investigated in [29],[30] and [31] respectively. A wide variety of uses and applications of the HPM technique has shown the efficiency of the method in practice. In this work, a globally convergent algorithm is proposed to solve the OCP. First, the optimal control problem with stability of degree constrains and environmental disturbances is transformed into a TPBVP. Then, the algorithm uses HPM to find the optimal observer gain vector. Integrating the optimal state vector estimates obtained with HPM and the optimal state feedback controller obtained by TPBVP results in a complete optimal closed-loop control design. An atractive characteristic of the proposed methods is its simplicity. Moreover, the proposed methods have lower computational costs when compared to other state of the art methods discussed above. Furthermore, in the proposed method the control problem can be solved with lower amplitudes of the input signal (control effort), comparing with analytical method. Lower control efforts may also help to avoid saturation effects, and to restrain the system to work within linear operating areas of the state space. On the other hand, there is a tradeoff between control effort and the degree of optimality obtained. The paper is organized as follows. In section 2 the statement of problem is discussed. In Section 3 the proposed algorithm is presented and then extended. Section 4 contains a numerical example to show the effectiveness and simplicity of proposed algorithm comparing with conventional HJB solution. Finally, section 5 presents conclusions. 2

3 2 Problem Statement Consider the following dynamic control system: ẋ(t) = Ax(t)+Bu(t), x(t 0 ) = x 0, (1) y(t) = Cx(t)+Du(t), (2) where A and B are real constant matrices, x(t) R n is the state space vector and u(t) R m is the input control vector, x 0 R n is the initial value of state, and t it the time. The objective is to find the optimal control u (t) that minimizes the following finite horizon quadratic integral criterion: J = t1 t 0 [x T (t)qx(t)+u T (t)ru(t)]dt+x T (t 1 )Sx(t 1 ), (3) where Q,S R n n and R R m m are non-negative definite weighting matrices. This is called a linear quadratic regulator (LQR) problem. Many numerical and analytic methods have been proposed to solve this optimal control problem [32], [33]. An optimal control law that minimizes the cost criterion results the following optimal feedback gain matrix K [34]: u (t) = Kx(t), K = R 1 B T P(t), (4) which results in the following optimal close-loop state matrix: A c = A BK. (5) In (4), P(t) is a symmetric positive semi-definite matrix called the adjoint variable which satisfies following matrix Riccati differential equation: P(t) = Q P(t)BR 1 B T P(t)+P(t)A+A T P(t), P(t 1 ) = S. (6) In most cases in control design, one goal is that α degree region stability constraints are meet, i.e. the real parts of system poles never exceed some α. In order to meet this constaint on problem (1)-(3), the state space equation (1) and optimal quadratic regulator criterion (3) are respectively re-arranged as follows [35]: ẋ(t) = (A+αI n )x(t)+bu(t), x(t 0 ) = x 0, (7) J = t1 t 0 e 2αt [x T (t)qx(t)+u T (t)ru(t)]dt+x T (t 1 )Sx(t 1 ). (8) Comparing (7) with (1) and (6) results in the following matrix Riccati equation: P α (t) = Q P α (t)br 1 B T P α (t)+p α (t)(a+αi n )+(A+αI n ) T P α (t), P α (t 1 ) = S. (9) After having introduced the stability degree constraints, now disturbances will be inserted in the formulation of the control problem. Specifically, consider the following stochastic system: ẋ(t) = Ax(t)+Bu(t)+W(t), x(t 0 ) = x 0, (10) y(t) = Cx(t)+Du(t)+V(t), (11) where x(t) represents the dynamic state, u(t) is the deterministic input, W(t) represents environment disturbance that affects the dynamics of the system, and V(t) represents output measurement noise. Assume that the disturbance W(t) and noise V(t) are zero-mean, Gaussian white noise but not necessarily stationary, with probability density functions p(w), and p(v), respectively, given by: p(w) = 1 n e 1 2 w T R 1 w w, EW(t+τ)W T (t)} = Qδ(τ) = R w (t)δ(τ), (12) 2π det Rw p(v) = 1 n e 1 2 v T R 1 v v, EV(t+τ)V T (t)} = Rδ(τ) = R v (t)δ(τ), (13) 2π det Rv 3

4 Figure 1: Diagram of control design system via kalman filter. where R w R m m and R v R p p. It is also assumed that there is no cross correlation between W and V, EW(t + τ)v T (t)} = R vw (t) = 0. In the scalar case R w = σ 2 w, and R v = σ 2 v. In general the optimal estimation problem is formulated by finding the estimate ˆx(t) that minimizes the mean square error E[x(t) ˆx(t)][x(t) ˆx(t)] T } given Y(t ) : 0 <= t <= t}. It can be shown that this is equivalent to finding the estimated value of x(t) respecting the constraint given by the all the previous measurements, thus ˆx(t) = Ex(t) y(t ),t <= t}. In fact this is the way that Kalman originally formulated the problem. According to the Kalman-Bucy theorem [4], the optimal estimator has the form of the following linear observer: ˆx(t) = Aˆx(t)+Bu(t)+L obs (t)[y(t) Du(t) Cˆx(t)], (14) where L obs (t) = P(t)C T R 1 v = R 1 CP(t), and P(t) = E[x(t) ˆx(t)][x(t) ˆx(t)] T }, which satisfies P(t) = AP(t)+P(t)A T P(t)C T R 1 v CP(t)+FR w (t)f T, P(t 0 ) = Ex(t 0 )x T (t 0 )}. (15) It has been shown that the Kalman filter extracts the maximum possible information about output data. Considering the innovations process, µ = Y C ˆX. The correlation matrix of µ is given by µ µ (t,t+τ) = Eµ(t+τ)µ T (t)} = W(t)δ(τ). This means that another property of Kalman filter is that the residuals are a white noise process. Thus, the output prediction error has no remaining dynamic information content. Figure 1 shows the optimal control design integrating a Kalman-Bucy filter. In (2), (7)-(8) an LQR problem with regional stability constraints has been considered. Now consider stochastic system (10)-(11) with regional stability constrains. Having optimal closed-loop control with feedback gain subject to stability degree constrains is desired. In order to simultaneously meet stability degree constraints and cope to disturbances, equations (9) and (15) are considered, and the following Riccati equation should be solved: P ˆ α (t) = (A+αI n ) P ˆ α (t)+ P ˆ α (t)(a+αi n ) T P ˆ α (t)c T Rv 1 CP ˆ α (t)+fr w (t)f T, (16) Pˆ α (t 0 ) = Ex(t 0 )x T (t 0 )}, (17) 4

5 where P ˆ α (t) is the optimal observer matrix with respect to the constrains for the stochastic system. Equation (16) satisfies the following two-point boundary value problem (TPBVP) [36]: ˆx(t) = (A+αIn ) Tˆx(t)+C T Rv 1 CP ˆ α (t), ˆx(t 0 ) = x 0, P ˆ α (t) = FR w (t)f Tˆx(t)+(A+αI n ) P ˆ (18) α (t), Pα ˆ (0) = Ex(t 0 )x T (t 0 )}. The estimated observer vector ˆx(t) obtained from (18) can then be used to generate the optimal control input as u (t) = Kˆx(t). 3 Proposed Method The solution of the TPBVP problem (18) is not easy in most cases. This paper proposes the HPM as an efficient method to solve the problem. 3.1 Review of the Homotopy Perturbation Method Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem. Perturbation theory is applicable if the problem at hand can be formulated by introducing a small parameter, p, to the mathematical description of the exactly solvable problem. To illustrate the basic ideas of HPM [15], consider the following nonlinear differential equation: A(u) f(r) = 0, r Ω, (19) subject to the following boundary condition: ( B u, u ) = 0, r Γ, (20) n where A is a general differential operator, B a boundary operator, f(r) a known analytical function and Γ is the boundary of domain Ω. Operator A can be divided into two parts: a linear part L, and a nonlinear part N. Then, (19) can be rewritten as follows: By the homotopy technique, we construct a homotopy H(v,p): where p [0,1] is an embedding parameter, L(u)+N(u) f(r) = 0, r Ω. (21) H(v,p) = (1 p)[l(v) L(u 0 )]+p[a(v) f(r)] = 0, (22) v(r,p) : Ω [0,1] R, (23) and u 0 is the first approximation that satisfies the boundary condition. From (22), H(v,0) = L(v) L(u 0 ) = 0 and H(v,1) = A(v) f(r) = 0. The solution of (22) can be written as a taylor expansion in p as follows: v = p 0 v 0 +p 1 v 1 +p 2 v 2 +p 3 v = p i v i. (24) Setting p = 1 results in the following approximate solution of (19): i=0 u = v(r,p 1) = limv p 1 = v 0 +v 1 +v 2 +v = v i (25) The series (25) is convergent for most cases. However, the convergence rate depends on the nonlinear operator A(v). After the initial HPM proposal [15], some approaches have been proposed by He in [21], [37] to attain better results: The second derivative of N(v) with respect to v must be small because the parameter p may be relatively large. The norm of L 1 N/ v must be smaller than one until the series (25) converges. i=0 5

6 3.2 Proposed Method The proposed method transforms the nonlinear TPBVP (18) into a sequence of linear time invariant TPBVP s. To do so, from (18) operators Z 1 ( ˆX(t), ˆP α (t)) and Z 2 ( ˆX(t), ˆP α (t)) are defined as follows: Z1 (ˆx(t), ˆP α (t)) = ˆx(t)+(A+αI n ) Tˆx(t) C T Rv 1 C ˆP α (t) = 0, Z 2 (ˆx(t), ˆP α (t)) = ˆPα (t) FR w (t)f Tˆx(t) (A+αI (26) n )ˆP α (t) = 0. Operators Z 1 and Z 2 can generally be divided into two separate parts, a linear part and nonlinear part, as follows: L1 (ˆx(t), ˆP α (t))+pn 1 (ˆx(t), ˆP α (t)) = 0, L 2 (ˆx(t), ˆP α (t))+pn 2 (ˆx(t), ˆP (27) α (t)) = 0, or (1 p)l1 (ˆx(t), P ˆ α (t))+pz 1 (ˆx(t), P ˆ α (t)) = 0, (1 p)l 2 (ˆx(t), P ˆ α (t))+pz 2 (ˆx(t), P ˆ α (t)) = 0, where L i and N i are respectively the linear and nonlinear parts of Z i for i = 1,2, subject to boundary conditions: ˆx(t 0 ) = x 0, Pα ˆ (t 0 ) = Ex(t 0 )x T (t 0 )}. (29) L must be chosen in such a way that one has no difficulty in subsequently solving systems of resulting equations. It should be mentioned that, although according to the He s analysis L i, i = 1,2, are not restricted to be linear, it is strongly recommended for simplicity and/or as a first approach to select a linear operator for L i. In most cases, in previous HPM works some part of the original equation has been selected as for the linear part L (L i in this case). The linear part wil be defined as follows: (28) L1 (ˆx(t), P ˆ α (t)) = ˆx(t)/ t, L 2 (ˆx(t), P ˆ α (t)) = P ˆ α (t))/ t. (30) Next consider v = [ˆP α (t) ˆx(t)] T, L(u 0 ) = 0, and A(v) f(r) = [Z 1 (ˆx(t), ˆP α (t)) Z 2 (ˆx(t), ˆP α (t))] T in (22). Thus, from (28), (30), a homotopy-perturbation equation is reconstructed for (26) as follows: [ (1 p) ˆx(t)/ t+p ˆx(t)/ t+(a+αin ) Tˆx(t) C T Rv 1 CP ˆ α (t) ] = 0, (1 p) P ˆ α (t))/ t+p [ P ˆ α (t))/ t FR w (t)f Tˆx(t) (A+αI n ) P ˆ α (t) ] = 0. (31) Likewise (24), expanding ˆx(t) and ˆP α (t) in Taylor series in both equations of (31), and by equating the coefficients of the terms of equal powers of p it results: ˆP 0 : ˆx0 (t)/ t = 0, ˆx 0 (t 0 ) = x 0, ˆP α0 (t))/ t = 0, ˆPα0 (t 0 ) = E(x(t 0 )x T (t 0 )), (32) ˆP 1 : ˆx1 (t)/ t+(a+αi n ) Tˆx 0 (t) C T Rv 1 C ˆP α0 (t) = 0, ˆx 1 (t 0 ) = 0, ˆP α1 (t)/ t (A+αI n )ˆP α0 (t) FR w F Tˆx 0 (t) = 0, ˆPα1 (t 0 ) = 0, (33) ˆP N : ˆP 2 : ˆx2 (t)/ t+(a+αi n ) Tˆx 1 (t) C T R 1 v C ˆP α1 (t) = 0, ˆx 2 (t 0 ) = 0, ˆP α2 (t)/ t (A+αI n )ˆP α1 (t) FR w F Tˆx 1 (t) = 0, ˆPα2 (t 0 ) = 0,. ˆxN (t)/ t+(a+αi n ) Tˆx N 1 (t) C T Rv 1 C ˆP αn 1 (t) = 0, ˆx N (t 0 ) = 0, ˆP αn (t)/ t (A+αI n )ˆP αn 1 (t) FR w F Tˆx N 1 (t) = 0, ˆPαN (t 0 ) = 0. (34) (35) After solving TPBVPs (32)-(35) for ˆP α0 (t), ˆP α1 (t), ˆP α2 (t),..., ˆP αn (t), and ˆx 0 (t),ˆx 1 (t),ˆx 2 (t),...,ˆx N (t), then, the final result considering (25) is obtained as follows: ˆPα (t) = ˆP α0 (t)+ ˆP α1 (t)+ ˆP α2 (t)+...+ ˆP αn (t), (36) ˆx(t) = ˆx 0 (t)+ ˆx 1 (t)+ ˆx 2 (t)+...+ ˆx N (t). 6

7 Substituting ˆx(t) from (36) into (4), the optimal control law is obtained as follows: u (t) = K(t)ˆx(t). (37) After having obtained the optimal state estimate ˆx(t) from the observer, the value of the feedback controller gain K(t) remains to be obtained. For this purpose, equation (9) is used instead of (16). Equation (9) satisfies the following two-point boundary value problem (TPBVP) [36]: ẋ(t) = (A+αIn )x(t) BR 1B T P(t), ˆx(t 0 ) = x 0, P(t) = Qx(t) (A+αI n ) T (38) P(t), P(t 1 ) = Sx(t 1 ). A procedure similar to the one used in (26)-(36) to solve (18) will now be used to to solve (38). Specifically, consider the following linear parts L i, i = 1,2, for the HPM: L1 (x(t),p(t)) = x(t)/ t, (39) L 2 (x(t),p(t)) = P(t))/ t. For (38), a homotopy-perturbation equation is reconstructed as follows: (1 p) x(t)/ t+p [ x(t)/ t (A+αIn )x(t)+br 1B T P(t) ] = 0, (1 p) P(t)/ t+p [ P(t)/ t+qx(t)+(a+αi n ) T P(t) ] = 0. (40) Likewise (31)-(35), expanding x(t) and P(t) in Taylor series in both equations of (40), and by equating the coefficients of the terms of equal powers of p it results: P 0 : x0 (t)/ t = 0, x 0 (t 0 ) = x 0, P 0 (t)/ t = 0, P(t 1 ) = Sx(t 1 ), (41) P 1 : x1 (t)/ t (A+αI n )x 0 (t)+br 1 B T P 0 (t) = 0, x 1 (t 0 ) = 0, P 1 (t)/ t+(a+αi n ) T P 0 (t)+qx 0 (t) = 0, P 1 (t 0 ) = 0, P 2 : x2 (t)/ t (A+αI n )x 1 (t)+br 1 B T P 1 (t) = 0, x 2 (t 0 ) = 0, P 2 (t)/ t+(a+αi n ) T P 1 (t)+qx 1 (t) = 0, P 2 (t 0 ) = 0,. P N : xn (t)/ t (A+αI n )x N 1 (t)+br 1 B T P N 1 (t) = 0, x N (t 0 ) = 0, P N (t)/ t+(a+αi n ) T P N 1 (t)+qx N 1 (t) = 0, P N (t 0 ) = 0. Thus, similarly to (36), the P(t) and x(t) results are obtained in (45). Finally, K(t) can be obtained from P(t), and the controller is completed. P(t) = P0 (t)+p 1 (t)+p 2 (t)+...+p N (t), (45) x(t) = x 0 (t)+x 1 (t)+x 2 (t)+...+x N (t). 3.3 Algorithm Modification In experimental applications, having optimal an control solution using an infinite series implementation is not practical or even possible. Therefore, we consider an N-th order suboptimal control as follows: where, the LQR criterion (3) can be rearranged as follows: J (N) = t1 (42) (43) (44) N u(t) = u N(t) = R 1 B T P n (t)ˆx(t), (46) n=0 [ˆx T (t)qˆx(t)+u T N (t)ru N(t)]dt+ ˆx T (t 1 )Sˆx(t 1 ). (47) t 0 Integer N in (46) is generally determined by the designer according to the desired output control precision, or by considering the computional complexity, control cost, etc. The accuracy of the solution of the problem is defined to be ǫ > 0 if the following condition holds: J (N) J (N 1) < ǫ, (48) 7

8 System States x LK(t) 1 x LK(t) 2 x 1 PM(t) x PM(t) Time(sec) Figure 2: Comparing of the state space control performance results obtainned by the proposed methods (PM) and by the Lyapunov-Kalman (LK) control architecture. where ǫ > 0 is a positive constant. If the tolerance error margin ǫ is chosen to be small enough, then the value of the performance index in (47) would be very close to its optimal value J. Thus, in order to obtain an accurate enough suboptimal control, we propose an iterative algorithm as follows: Step 1: Let N 1; Step 2: Calculate the P n (t) (n = 1,...,N) and ˆx(t) using the approach presented in Section 3.2; Step 3: Calculate the N-th order control command approximation u N (t) in (46); Step 4: Calculate N-th approximation of J (N) using (47); Step 5: If (48) holds for a given small enough constant ǫ > 0, go to Step 6; else N N +1 and go to Step 2; Step 6: Stop the algorithm; u N (t) is the suboptimal control command result and N indicates the approximation order. 4 Simulation Results To illustrate the feasibility and effectiveness of the proposed method, this section applies it to design an optimal control system for the following stochastic system example: [ ] [ ] [ ] ẋ(t) = x(t)+ [u(t)+w(t)], x T (0) = [0 1], ẏ(t) = x(t)+v(t), (49) where noises w(t) and v(t) are stationary Gaussian Processes, so that: E[ww T (t)] = Qδ(t) = [1]δ(t) = δ(t), (50) [ ] 2 0 E[vv T (t)] = Rδ(t) = δ(t). (51) 0 1 The degree of stability is considered to be α = 2, and the desired precision is ǫ = To show the effectiveness of the proposed method, the algorithm performance is compared with a Lyapunov-Kalman (LK) control architecture. The LK architecture integrates (i) a complete-precision LQR controller solved by Lyapunov methods [38], and (ii) a Kalman filter observer. Figure 2 shows x 1 (t) and x 2 (t) obtained in both methods. Figures 3 and 4, present the K(t) and u(t) results for both the 8

9 FeedBack Gain K LK(t) 2 K LK(t) 1 K PM(t) 2 K PM(t) Time [sec] Figure 3: Feedback gain obtained with the proposed methods (PM) and the LK method. Table 1: Comparison of the exact solution and the proposed HPM solution of u(t), for N = 5. t(s) HPM solution Analitical solution Absolute error e e e-06 proposed approximation-based method and the LK method, respectively. The simulation shows that after five iterations the proposed method has converged, with J 5 = In this case, the obtained adjoint variable was P = [ ; ], and the obtained observer gain was L = [ ; ]. The closed-loop gain matrix of the optimal system is: [ ] 0 1 A cl = A BK α =, (52) d T = eig(a cl ) = [ ]. (53) As shown in (53), the poles of A cl are mapped into the left part of α = 2. Table 1 shows u(t) performance in some points in both the LK analytical method and the proposed method. 5 Conclusion In this work, presents the application of the HPM as a semi-analitical iterative method to solve a class of nonlinear optimal control problem. The controller integrates the optimal state vector estimates obtained with HPM and the optimal state feedback controller obtained by TPBVP results in a complete optimal closed-loop control design. Simulation results were presented. The results show that the algorithm has good performance even in the presence of disturbences and stability degree constrains. The proposed method is not more complicated when compared with other methods. The results show that having more iterations has resulted in more similarity in system gain and input signals, when compared to 9

10 0.2 Input Control Signal U PM Time(sec) U LK Figure 4: Input control command to the plant which results from the proposed method (PM) and and the LK method. the complete-precision Lyapunov-based analytical method. it also has lower computational costs when compared to other state of the art methods. Furthermore, in the proposed method the control problem can be solved with lower amplitudes of the input signal (control effort), comparing with analytical method. Lower control efforts may also help to avoid saturation effects, and to restrain the system to work within linear operating areas of the state space. On the other hand, there is a tradeoff between control effort and the degree of optimality obtained. Unlike the analitical methods, the application of the proposed algorithm gives the designer important parameters, such as N in (46) in order to reach a solution with performance index J N close enough to the desired J. The main disadvantage of HPM is that we should suitably choose an initial guess. However, in most of the cases in OCP suitable initial boundary values are available. Acknowledgments This work was supported by Project SCIAD/2011/21531 co-financed by QREN, in the framework of the Mais Centro - Regional Operational Program of the Centro, and by the European Union through the European Regional Development Fund (ERDF). References [1] B. F. and M. K. Sain Spencer, Jr. Controlling buildings: A new frontier in feedback. IEEE Control Systems Magazine on Emerging Technology, 17(6):19 35, [2] G. W. Housner and M. K. Sain. Structural control: Past, present and future. Journal of Engineering Mechanics, 123(9): , [3] D. J. Luenberger. New results in linear filtering and prediction theory. IEEE Transactions on Automatic Control, 11(2):95 108, [4] R. E. Kalman and R. S. Bucy. Observers for multivariable systems. Journal of Basic Engineering, 83:95 108, [5] H. Kwakernaak and R. Sivan. Linear Optimal Control Systems. Wiley-Interscience, first edition, October

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