OPTIMAL CONTROL APPLICATIONS AND METHODS Optim Control Appl Meth 2010; 31:581–591 Published online 11 November 2010 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/oca.971 Optimal regional pole placement for sun tracking control of high-concentration photovoltaic (HCPV) systems: case study Chee-Fai Yung1 , Hong-YihYeh2 , Cheng-Dar Lee2 , Jenq-Lang Wu1, ∗, † , Pei-Chang Zhou1 , Jia-Cian Feng1 , Hong-Xun Wang1 and Sheng-Jin Peng1 Department of Electrical Engineering, National Taiwan Ocean University, Keelung 202, Taiwan of Nuclear Energy Research, Longtan 325, Taoyuan, Taiwan Institute SUMMARY This paper proposes an optimal regional pole placement approach for sun tracking control of high-concentration photovoltaic systems A static output feedback controller is designed to minimize an LQG cost function with a sector region pole constraint The problem cannot be solved by LMI approach since it is a non-convex optimization problem Based on the barrier method, we instead solve an auxiliary minimization problem to obtain an approximate solution Simulation results show the benefit of our approach Copyright q 2010 John Wiley & Sons, Ltd Received 26 May 2010; Revised 27 August 2010; Accepted October 2010 KEY WORDS: regional pole placement; LQG optimal control; barrier method; Lagrange multiplier method; sun tracker INTRODUCTION Recently, the problems of shortage of fossil fuel source and global warming effects have become more and more severe People begin to seek various possible solutions to those problems One of the potential options is the use of the sun energy, which not only provides an alternative energy source but also improves environmental pollution Therefore, sun tracking systems have attracted much attention in recent years In the literature, common sun tracking systems consist of open-loop and closed-loop types In [1], a lookup table was pre-established to obtain the position of the sun at any time and then the direction of tracking mechanism is adjusted to point the direction of the sun In [2] a type of on–off control was utilized for two-axis sun tracking On the other hand, common closed-loop control methodologies include robust proportional (P) control, proportional–integral (PI) control, derivative (D)-like control, proportional–integral–derivative (PID) control [3–8], fuzzy control [9–12], LQG control [13] or H ∞ control [14, 15] Various controllers have individual advantages and disadvantages [14] For instance, PID control and fuzzy control could be good options when accurate model of tracking system is absent, while LQG or H ∞ control are preferred if higher accuracy tracking performance or tracking robustness against to exogenous disturbance, like wind gusts or cloud effects, is the main concern [16–18] ∗ Correspondence † to: Jenq-Lang Wu, Department of Electrical Engineering, National Taiwan Ocean University, Keelung 202, Taiwan E-mail: wujl@mail.ntou.edu.tw Copyright q 2010 John Wiley & Sons, Ltd 582 C.-F YUNG ET AL Most recently, the Institute of Nuclear Energy Research (INER) has developed a high concentration photovoltaic (HCPV) high power generation system with III-V solar cells, as an alternative source to the application of solar PV and as a dependable energy source to the mankind [19, 20] The main purpose of the present paper is to develop an accurate sun tracking control strategy for the HCPV power generation system implemented and installed at INER An optimal regional pole placement approach is proposed for designing static output feedback sun tracking controllers The minimization of quadratic cost functions can improve systems’ static responses but cannot guarantee good transient responses Good transient responses can be ensured by properly assigning closed-loop poles to some particular regions In [21–25], the authors determined feedback controllers to assign the closed-loop poles to some particular regions Moreover, a quadratic cost function being minimized by the resultant controller is found Nevertheless, for a given cost function, how to find the optimal controller subject to the regional pole’s constraint was not discussed In [26], the authors solved a modified Lyapunov equation to obtain a controller which minimizes a function, which is an upper bound of the original cost function, and guarantees that the resultant closed-loop poles lie in a desired region Recently, based on the barrier method, Wu and Lee [27–29] have developed a novel approach for solving optimal regional pole placement problems Wu and Lee [27, 28] considered the state feedback case and [29] considered the output feedback case Different to the approach in [26], in barrier method a solution arbitrarily close to the infimal solution of the constraint optimization problem can be obtained In this paper we employ a similar approach to solve static output feedback optimal regional pole placement problem for sun tracking systems The considered cost function is quadratic and the closed-loop poles are required to locate on a sector region For sector constraint region, in [29] three constraint matrix equations must be included but in this paper only two constraint matrix equations should be considered This is a constrained optimization problem and its minimum point may not exist It often happens that its infimum point lies on the boundary of the admissible solution set and it is not a stationary point Therefore, the Lagrange multiplier method cannot be employed to derive the necessary conditions for optimum Moreover, this problem cannot be solved via linear matrix inequality (LMI) approach since the admissible solution set may be non-convex It is known that static output feedback control problems are difficult to solve [30] In this paper, based on the barrier method (see [31]), we instead solve an auxiliary minimization problem to obtain an approximate solution of the original problem The new cost function is the sum of the actual cost function of the original problem and a weighted ‘barrier function’ If the admissible solution set is non-empty, the minimal solution of the auxiliary minimization problem exists and is a stationary point Therefore, the Lagrange multiplier method can be used to derive the necessary conditions for optimum The minimal solution of the auxiliary minimization problem converges to the infimal solution of the original problem if the weighting factor of the barrier function approaches zero Notations In this paper, E(.) denotes the expected value, (M) is the spectrum of matrix M, Tr(M) means the trace of matrix M, MT (M∗ ) is the (conjugate) transpose of matrix M, M>0( 0) means that the matrix M is positive (semi)definite, and ¯ is the complex conjugate of ∈ C PROBLEM FORMULATION AND PRELIMINARIES Based on the technology of the semiconductor radiation detector, Institute of Nuclear Energy Research (INER) of Atomic Energy Council (AEC), Executive Yuan in Taiwan, has started the R&D projects on the 100 kW HCPV systems The detailed architecture of the HCPV power system is shown in Figure [19] The HCPV system is composed of the III–V solar cell, concentrating solar module, solar tracker, inverter, and the tracking control system (Figure 2) This tracker, an azimuth-elevation tracker, consists of two axes One axis is a vertical pivot shaft that allows the device to be swung to a compass point The other axis is a horizontal elevation pivot mounted upon the Copyright q 2010 John Wiley & Sons, Ltd Optim Control Appl Meth 2010; 31:581–591 DOI: 10.1002/oca OPTIMAL REGIONAL POLE PLACEMENT FOR SUN TRACKING CONTROL 583 Figure 100 kW HCPV Power System Architecture u Controller Motors Tracker y y + - Sun Sensor Figure The sun tracking control system azimuth platform A photo sensor mechanism oriented to sun direction is mainly composed of four photo detectors, located at 90 degrees apart from each other and oriented to the cardinal points Two differential signals between east and west detectors, and south and north detectors are sent to the tracking controller A CCTV system was implemented for remote monitoring of the tracking system The DC power output was connected to the charge controller, which tracks the maximum peak power point to keep the HCPV modules output power in the maximum condition There are two kinds of power measurement design implementation in the system One is the measurement of DC current and voltage for HCPV modules output power, and the other is the measurement of AC current and voltage for consumption power of the load The PC controller collects those signals from power measurement devices through interface modules and Ethernet network The control function of the tracker is implemented into PC-based controller by high-level programming language The controller collects signals from the photo-sensor mechanism, and sends commands to control the tracker motion The main purpose of the present paper is to develop an optimal regional pole placement method for designing sun tracking control strategy (see Figure 2) for the HCPV power generation system at INER Suppose that the considered azimuth/elevation tracking system is modeled as x˙ (t) = Ax+Bu y(t) = Cx Copyright q 2010 John Wiley & Sons, Ltd (1) Optim Control Appl Meth 2010; 31:581–591 DOI: 10.1002/oca 584 C.-F YUNG ET AL Im s-plane π/4 α Ω β Figure The constraint region Re where x ∈ R n is the state, u ∈ R is the control input (the voltage to the DC motor), and y ∈ R is the measured output (the tracking error); A, B, and C are constant matrices of appropriate dimensions Define H ( , ) ≡ {s ∈ C| Re[ei (s − )]0 Two useful lemmas are introduced in the following Lemma (Wu and Lee [29]) ˆ lie in the region W ( ) if and only if for any given positive definite matrix All the eigenvalues of a real matrix A ˆ Q, the Lyapunov equation ˆ ˆ ˆ =0 ˆ P( ˆ A− (A− I)T P+ I)+ Q (7) ˆ has a unique solution P>0 Lemma ˆ lie in the region H ( , /4)∪ H ( , /4) if and only if for any given positive All the eigenvalues of a real matrix A ˆ the matrix equation definite matrix Q, T ˆ ˆ =0 ˆ ˆ P( ˆ A− −(A− I)2 P− I)2 + Q (8) ˆ has a unique solution P>0 Proof ˆ and v is an associated eigenvector, i.e Av ˆ = v Pre- and post-multiplying (8) by Sufficiency: Suppose that ∈ (A) ∗ v and v respectively yields T ∗ˆ ˆ ˆ =0 ˆ ˆ −v∗ (A− I)2 Pv−v P(A− I)2 v+v∗ Qv That is, ˆ ˆ ˆ = v∗ Qv ( ¯ − )2 v∗ Pv+( − )2 v∗ Pv ˆ ˆ By the fact that Q>0 and P>0, we have ( ¯ − )2 +( − )2 = ˆ v∗ Qv >0 ∗ ˆ v Pv This is equivalent to ∈ H ( , /4)∪ H ( , /4) since ( ¯ − )2 +( − )2 = 2((Re( )− )2 −(Im( ))2 ) ˆ and let − be expressed as − = | − |ei Then, ∈ H ( , /4)∪ H ( , /4) is equivNecessary: Let ∈ ( A) ˆ alent to − /4 /4 or /4 /4 It is clear that −| − |2 ei2 = | − |2 ei2( + /2) ∈ (−(A− I)2 ) By the Copyright q 2010 John Wiley & Sons, Ltd Optim Control Appl Meth 2010; 31:581–591 DOI: 10.1002/oca 586 C.-F YUNG ET AL ˆ fact that − /4 /4 or /4 /4, we have /2 2( + /2) /2 This implies that (−(A− I)2 ) ⊂ W (0) ˆ ˆ the following equation Since −(A− I) is Hurwitz, for any positive definite matrix Q T ˆ ˆ ˆ =0 ˆ P( ˆ A− −(A− I)2 P− I)2 + Q ˆ This completes the proof has a unique positive definite solution P The result in Lemma is a generalization of some results in [27] THE AUXILIARY MINIMIZATION PROBLEM The problem under consideration is a constrained optimization problem To solve this problem analytically is difficult since its minimal solution may not exist In fact, its infimal solution may lie on the boundary of the set r ; and furthermore, it may not be a stationary point In this paper, motivated by the barrier method (Luenberger [31]), we instead solve an auxiliary minimization problem to obtain an approximate solution of the original problem The auxiliary cost function Jaux (F) is the sum of the actual cost function J (F) and an additional barrier function Jpole (F) The auxiliary minimization problem is formulated as follows: Find F, over r , to minimize the auxiliary cost function Jaux (F) = J (F)+ · Jpole (F) where the term J (F) is defined in (3), is a weighting factor, and Jpole (F) = Tr(P1 )+Tr(P2 ) if F ∈ ∞ r (9) otherwise with matrices P1 >0 and P2 >0 being the solutions of (A+BFC− I)T P1 +P1 (A+BFC− I)+Q1 = (10) and T −(A+BFC− I)2 P2 −P2 (A+BFC− I)2 +Q2 = (11) for given positive definite matrices Q1 and Q2 As shown in [31], a barrier function must satisfy: (1) it is continuous, (2) it is nonnegative over the set r , and (3) it will approach infinity as F approaches the boundary of the set r Now we will show that the function Jpole (F) satisfies these three conditions Lemma The function Jpole (F) defined in (9) satisfies (1) Jpole (F) is continuous in the set r , (2) Jpole (F)>0 over the set r , and (3) Jpole (F) approaches infinity as F approaches the boundary of the set r Proof The proof is similar to that in [29] and therefore is omitted here Copyright q 2010 John Wiley & Sons, Ltd Optim Control Appl Meth 2010; 31:581–591 DOI: 10.1002/oca OPTIMAL REGIONAL POLE PLACEMENT FOR SUN TRACKING CONTROL 587 Although the auxiliary minimization problem is, from a formal viewpoint, a minimization problem with inequality constraints; from a computational viewpoint it is unconstrained [31] The advantage of the auxiliary minimization problem is that it can be solved by unconstrained search techniques Remark It is shown in [31] that the optimal solution of the auxiliary minimization problem converges to the solution of the original problem as the weighting factor → 0+ This suggests a way to approximate the infimal solution of the original problem in our approach As we have shown in [29], if the set r is non-empty, the auxiliary cost function Jaux (F) has a minimum point in the set r Since the minimum point of the auxiliary cost function Jaux (F) lies in the interior of the admissible solution set, it must be a stationary point The Lagrange multiplier method can be employed to derive the necessary conditions for local optimum of cost function Jaux (F) Theorem Let F ∈ r minimize Jaux (F) Then there exist P 0, P1 >0, P2 >0, L>0, L1 >0, and L2 >0 satisfying (A+BFC)T P+P(A+BFC)+CT FT RFC+Q+CT SC = (12) (A+BFC)L+L(A+BFC)T +X0 = (13) (A+BFC− I)T P1 +P1 (A+BFC− I)+Q1 = (14) (A+BFC− I)L1 +L1 (A+BFC− I)T + I = (15) 2T −(A+BFC− I) P2 −P2 (A+BFC− I)2 +Q2 = T −(A+BFC− I)2 L2 −L2 (A+BFC− I)2 + I = (16) (17) and Fgrad (U) ≡ 2(BT PL+RFCL+BT P1 L1 −BT AT P2 L2 −BT P2 L2 AT −BT P2 L2 CT FT BT −BT CT FT BT P2 L2 +2 BT P2 L2 )CT = (18) Proof The Lagragian Ham is defined as Ham = Tr(PX0 )+ ·(Tr(P1 )+Tr(P2 )) +Tr(L((A+BFC)T P+P(A+BFC)+CT FT RFC+Q+CT SC)) +Tr(L1 ((A+BFC− I)T P1 +P1 (A+BFC− I)+Q1 )) T +Tr(L2 (−(A+BFC− I)2 P2 −P2 (A+BFC− I)2 +Q2 )) The necessary conditions for local optimum are *Ham /*F = 0, *Ham /*L = 0, *Ham /*P = 0, *Ham /*L1 = 0, *Ham /*P1 = 0, *Ham /*L2 = 0, and *Ham /*P2 = After some manipulations, we have (12)–(18) The above theorem provides not only a necessary conditions for optimum, but also a method to calculate the gradient direction of Jaux (F) at a given point F The gradient of Jaux (F) at a fixed point F is Fgrad (F) In the solution algorithm, this gradient direction is used as the searching direction Copyright q 2010 John Wiley & Sons, Ltd Optim Control Appl Meth 2010; 31:581–591 DOI: 10.1002/oca 588 C.-F YUNG ET AL AN ILLUSTRATIVE EXAMPLE In order to aim a collector aperture toward the sun during daytime, the two-axis movement of sun tracker is always required Two independent tracking systems (azimuth and elevation) are designed Case 1: For azimuth tracking system, by applying system identification method on practical experimental data of the tracking system, we have the following dynamic equation: ⎡ ⎤ ⎤ −10.65 −15.63 −8.938 −7.602 −1.294 ⎢ ⎥ ⎥ ⎢ 0 0 ⎥ ⎢ 0⎥ ⎢ 16 ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ 0 ⎥ x+ ⎢ 0⎥ u x˙ = ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ 0⎥ ⎢ 0 0 ⎥ ⎣ ⎦ ⎦ ⎣ ⎡ 0 0 y = [0.2533 0.2054 0.2534 0.1888 0.5181]x Suppose that E{x(0)xT (0)} = X0 = I5×5 The design goal is to find a static output feedback gain F such that the controller u = Fy achieves the infimum of the cost function ∞ J (F) = E (yT Sy+xT Qx+uT Ru) dt subject to the constraints that (A+BFC) ∈ ≡ {s ∈ C|s ∈ H (7, /4)∩W (−1)}, where S = 1, Q = I5×5 , and R = Choosing different values for parameters and will lead to different responses In general the tracker will have faster response for small However, for sun tracking systems, very fast response is not necessary and therefore we let = −1 We find that, for the azimuth tracking system with = −1, no solutions can be found if