Robust control for line-of-sight stabilization of a two-axis gimbal system

Turkish Journal of Electrical Engineering & Computer Sciences http://journals.tubitak.gov.tr/elektrik/ Turk J Elec Eng & Comp Sci (2017) 25: 3839 – 3853 c TÜBİTAK ⃝ doi:10.3906/elk-1606-435 Research Article Robust control for line-of-sight stabilization of a two-axis gimbal system Mehmet BASKIN1,∗, Kemal LEBLEBİCİOĞLU2 Control Systems Design, MGEO, Aselsan Inc., Ankara, Turkey 2 Department of Electrical & Electronics Engineering, Faculty of Engineering, Middle East Technical University, Ankara, Turkey 1 Received: 26.06.2016 • Accepted/Published Online: 16.06.2017 • Final Version: 05.10.2017 Abstract: Line-of-sight stabilization against various disturbances is an essential property of gimbaled imaging systems mounted on mobile platforms. In recent years, the importance of target detection from higher distances has increased. This has raised the need for better stabilization performance. For that reason, stabilization loops are designed such that they have higher gains and larger bandwidths. As these are required for good disturbance attenuation, sufficient loop stability is also needed. However, model uncertainties around structural resonances impose strict restrictions on sufficient loop stability. Therefore, to satisfy high stabilization performance in the presence of model uncertainties, robust control methods are required. In this paper, a robust controller design in LQG/LTR, H ∞ , and µ -synthesis framework is described for a two-axis gimbal. First, the performance criteria and weights are determined to minimize the stabilization error with moderate control effort under known platform disturbance profile. Second, model uncertainties are determined by considering locally linearized models at different operating points. Next, robust LQG/LTR, H ∞ , and µ controllers are designed. Robust stability and performance of the three designs are investigated and compared. The paper finishes with the experimental performances to validate the designed robust controllers. Key words: LQG/LTR, H ∞ , µ -synthesis, two-axis gimbal, robust multivariable control 1. Introduction For precise pointing and tracking performance, line-of-sight (LOS) stabilization against various disturbances is essential for imaging systems. To obtain better performance, bandwidth and gain of stabilization loops need to be increased while sufficient loop stability is maintained. For gimbaled imaging systems, the main difficulties in satisfying sufficient loop stability and good performance at the same time are model uncertainties around structural resonances. Therefore, robust control methods are needed to maintain high stabilization performance under model uncertainties. In this aspect, this paper deals with the design of a stabilization loop for a two-axis gimbal. Classical control methods were used for stabilization loops in the past [1,2]. However, finding a classical controller that satisfies both stability and performance criteria is a time-consuming iterative procedure. Moreover, this method suffers from lack of optimality. Over the past decade, different methods have been used to obtain good stability and performance properties. Linear quadratic methods [3–5], H ∞ control methods [6-8], and µ -synthesis [9] are applied to the LOS control problem. However, in most of these reports the performance is evaluated only for nominal models. In other words, the stability and performance change due to model ∗ Correspondence: 3839 BASKIN and LEBLEBİCİOĞLU/Turk J Elec Eng & Comp Sci uncertainty are not considered. In addition, although some designers conduct an analysis for robustness, there are not clear experimental data to validate the robustness of the stabilization loops. Therefore, in this paper the authors try to fully support all theoretical findings with experimental data. In the next section, three controller design methods are reviewed. Firstly, LQG/LTR design is discussed. In traditional LQG method, the desired loop shape is obtained by adjusting weighting matrices or intensities of process and measurement noises. However, in this paper a different approach is followed to shape the loop easily [10,11]. By using this modified technique, the sensitivity is successfully shaped for good disturbance rejection. Next, H ∞ and µ -synthesis design in mixed sensitivity framework are investigated. In previous mixed sensitivity designs, performance and uncertainty weights are determined by using general rules. However, in this paper all weights are determined by using experimental data. After designing three controllers, the performance and stability of the three designs are investigated and compared. Firstly, the comparison is made by investigating theoretical results. Next, experimental findings are obtained, and they are compared with theoretical results. Both theoretical and experimental results show that the stabilization loop has robust stability and robust performance properties for each of the three design methods. 2. Design methods 2.1. LQG/LTR design The traditional LQG method uses a linear time invariant plant, and it assumes that the uncertainty in the states and measurements are additive [10]. The state space form of the plant is represented as in (1), where wd andwn are uncorrelated zero mean white noise processes having constant power spectral densities W and V as illustrated in (2). ẋ = Ax + Bu + Γwd y = Cx + wn (1) { } E wd}(t) wdT (τ ) = W δ (t { { − τ) , } E wn (t) wnT (τ ) = V δ (t − τ ) , E wd (t) wnT (τ ) = 0 (2) The aim of the LQG theory is to find a feedback control law to minimize the cost (3), where Q = QT ≥ 0 and R = RT > 0 are weighting matrices. { J = lim E T→∞ T( ) ∫ xT Qx + uT Ru dt } (3) 0 The solution turns out to be a cascade connection of Kalman filter and LQ regulator, each of which can tolerate gain variation between (1/2, ∞) and phase variation less than 60 ◦ in each channel [10]. However, the cascaded form, LQG regulator, does not have guaranteed stability margins, and the closed loop may suffer from poor stability [11]. If one applies loop transfer recovery (LTR), the closed loop recovers the good stability properties of the Kalman filter [12]. Since the overall loop approaches the Kalman filter, good Kalman filter shape is essential for good disturbance rejection. In most of the reported designs, the desired Kalman filter shape is obtained by iteratively changing covariance matrices W and V . On the other hand, if frequency dependent weighting matrices W (s) and V (s) are used, to obtain a good Kalman filter is simpler [11,13]. As given in Figure 1, assume that instead of state disturbances the plant has output disturbance d and measurement noise v , which have power spectral density D(s) and V (s) , respectively. An augmented system can be obtained 3840 BASKIN and LEBLEBİCİOĞLU/Turk J Elec Eng & Comp Sci Figure 1. Plant augmentation. if the states of the plant and frequency dependent weights are combined. If one assumes that d˜, ṽ , and θ are un (...truncated)