J. Cent. South Univ. (2018) 25: 2615-2625

DOI: https://doi.org/10.1007/s11771-018-3940-3

Robust control with compensation of adaptive model for dual-stage inertially stabilized platform

SONG Jiang-peng(宋江鹏)^{1, 2}, ZHOU Di(周荻)¹, SUN Guang-li(孙广利)², QI Zhi-hui(祁智辉)²

1. Department of Control Science and Engineering, Harbin Institute of Technology, Harbin 150001, China;

2. Tianjin Jinhang Institute of Technical Physics, Tianjin 300192, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract: To achieve excellent tracking accuracy, a coarse-fine dual-stage control system is chosen for inertially stabilized platform. The coarse stage is a conventional inertially stabilized platform, and the fine stage is a secondary servo mechanism to control lens motion in the imaging optical path. Firstly, the dual-stage dynamics is mathematically modeled as a coupling multi-input multi-output (MIMO) control system. Then, by incorporating compensation of adaptive model to deal with parameter variations and nonlinearity, a systematic robust H_∞ control scheme is designed, which can achieve good tracking performance, as well as improve system robustness against model uncertainties. Lyapunov stability analysis confirmed the stability of the overall control system. Finally, simulation and experiment results are provided to demonstrate the feasibility and effectiveness of the proposed control design method.

Key words: dual-stage control; inertially stabilized platform; robust H_∞ control; adaptive model

Cite this article as: SONG Jiang-peng, ZHOU Di, SUN Guang-li, QI Zhi-hui. Robust control with compensation of adaptive model for dual-stage inertially stabilized platform [J]. Journal of Central South University, 2018, 25(11): 2615–2625. DOI: https://doi.org/10.1007/s11771-018-3940-3.

1 Introduction

Inertially stabilized platforms are used to achieve the line of sight (LOS) of the optical sensors controlled motion on the moving platforms or ground-mounted devices. Typically, a conventional inertially stabilized platform consists of two orthogonal gimbals to control the inertial orientation of the optical sensors [1, 2].

In the past decade, the dual-stage control system has been successfully applied to improving control performance in the field of hard disk drives [3–7]. Enlightened by these applications, a dual- stage stabilized platform has been paid attention recently, which can provide similar control capability without increasing the shape and size [8].

There are many research works for the design of dual-stage controller. RYOO et al [9] proposed a dual-stage PQ controller to simplify the controller design into two single-input single-output (SISO) control problems. KOGANEZAWA et al [10] used the master slave controller structure in the dual-stage system to deal with the coupling disturbance. RAYMOND et al [11] summarized the application of modern control techniques such as adaptive control, LQG/LTR technique and robust control technique in the dual-stage control system. CHAO et al [12] successfully applied robust H_∞ technique to achieve high precision control and robustness for a dual- stage control system. HUANG et al [13] proposed a robust mixed H₂/H_∞ controller to enhance the tracking performance and stability robustness of a dual-stage servo system.CHEN et al [14] proposed an adaptive fuzzy H_∞ control method to reduce the effect of the model error and external disturbance.

Note that, robust H_∞ controller is usually conservative due to model uncertainties and other nonlinearities. It is beneficial to combine with some nonlinear compensation technique, such as friction disturbance compensation, to further improve the control performance of H_∞ control [15–20]. For example, a robust μ-synthesis controller combined with adaptive model compensation technique has been used in the linear motor systems to overcome the effect of nonlinearities [21].

In this paper, a robust H_∞ MIMO controller with compensation of adaptive model is proposed for the dual-stage stabilized platform. It can effectively improve the tracking performance and guarantee overall system stability. Firstly, the dynamic mode of the dual-stage control system is derived. Then a robust H_∞ control scheme with compensation of adaptive model is designed. By incorporating compensation of adaptive model, the proposed controller can effectively attenuate the effect of model errors and system nonlinearities, such as friction nonlinearity. Finally, stability of the proposed dual-stage robust controller is proved by means of Lyapunov theory.

The remainder of this paper is organized as follows. In Section 2, the dynamic model is derived and the system parameters are given by system identification experiment. In Section 3, a systematic control scheme using a robust H_∞ controller with compensation of adaptive model is developed. In Section 4, simulation and experiment are carried out to verify the effectiveness of the proposed controller. Finally, some conclusions are summarized in Section 5.

2 Plant model and identification

The dual-stage stabilized platform studied in this paper is composed with the coarse stage and the fine stage. The fine stage is placed in the imaging optical path to control lens motion as the micro- scan lens (MSL).

2.1 MSL model

MSL mechanical structure is shown in Figure 1. It can be seen that the flexure hinge is used to amplify and transmit the voice coil motor (VCM) displacement without the bearing [22].

Figure 1 Mechanical structure of MSL

Therefore, the dynamic model of the MSL is simplified as a mass–spring–damper system model, as shown in Figure 2.

Figure 2 Simplified linear dynamic model of MSL

The linear dynamic model of MSL can be expressed as

                  (1)

where u_f is the input voltage; η is the translation displacement; m_f, c_f and k_f represent the equivalent mass, damping coefficient and stiffness respectively; L is the distance between the gimbal rotation axis and the VCM actuator position; θ and represent the gimbal angle and gimbal velocity respectively; d_f is the lumped external disturbances in the MSL.

In order to improve the damping effect of the MSL, the inner loop damping controller of MSL is designed by using the embedded displacement transducer before the dual-stage controller design. The damping closed-loop dynamics of MSL can be expressed as

                     (2)

where ω_f, δ_f and A_f are the natural frequency, the damping ratio and the equivalent gain, respectively; is the equivalent lumped external disturbances in the MSL.

The system parameters ω_f, δ_f and A_f of the MSL can be obtained from the frequency response of the damping MSL as

         (3)

The high-frequency model is susceptible to the product manufacture tolerance and measurement error, since it is treated as unstructured uncertainty. The friction of the MSL is assumed to have slight influence because of the flexure hinge mechanism.

2.2 Two-axis gimbal platform model

A two-axis inertially stabilized platform is shown in Figure 3.

Figure 3 Structure of a two-axis gimbal configuration

It can be seen that this two-axis gimbal platform mainly consists of DC torque motor, encoder, optical sensor and a two-axis rate gyro, which is used to measure the inertial angular rates.

The gimbal dynamics can be represented in the following form:

          (4)

whereis the inertial angular rate; M_my is gimbal motor torque; d_fy is gimbal nonlinear friction torque; d_uy is the mass imbalance torquel; d_cry is the cable restraint torque; d_{y _cross} is the cross coupling torque and J_s is the equivalent moment of inertia.

Then, the corresponding outer gimbal state space model can be written as

                   (5)

where u_my denotes control voltage input; R denotes the resistance of motor armature windings; d_ly denotes other lumped unknown disturbances including d_uy, d_cry, d_{y_cross}; the scalars K_e and K_t are the back EMF constant and the motor force constant, respectively.

From Eq. (5), d_fy is a nonlinear friction torque which can be described by a nonlinear static coulomb friction model, i.e.,

                     (6)

where B_fy is the viscous friction coefficient; A_fy is the coulomb friction coefficient;is the gimbal relative angular velocity;is the signum function about.

The system parameters J_s, K_t, K_e, R and the friction parameters A_fy, B_fy can be obtained by the system identification technique as

                (7)

Similar to the MSL, the high-frequency gimbal model is treated as unstructured uncertainty.

The gimbal is susceptible to nonlinear friction and other nonlinearities. If these nonlinearities are treated as unstructured uncertainty, the controller will be too conservative, so the compensation of adaptive model is introduced to deal with these nonlinearities.

2.3 Dual-stage platform model

In the dual-stage platform, MSL is mounted on the two-axis gimbal platform. Based on the dynamic model mentioned above, the dual-stage platform dynamic model can be derived as follows, ignoring unstructured uncertainty.

                 (8)

From Eq. (8), it can be seen that the gimbal dynamics will affect the MSL dynamics through the coupling term

The structure of the dual-stage system is illustrated in Figure 4.

Figure 4 Structure of dual-stage control system

From Figure 4, the angular position θ is the LOS angular output caused by gimbal motion and θ_f is the LOS angular output caused by MSL translation displacement; A_g is the relationship between the LOS angular output and MSL translation displacement, which is determined by imaging optical properties. The output angular y_T is the combination of the gimbal output θ and the VCM output θ_f.

Based on the above description, this dual-stage control system can be regarded as a coupled MIMO system with nonlinearities and external disturbance. It is natural to employ a multi-objective H_∞ optimal approach to design a dual-stage controller.

Meanwhile, the compensation of adaptive model is suitable for this control system to deal with model uncertainties, such as model errors, parameter variations, nonlinear friction and other nonlinearities.

In the controller design, the coupling term can be considered an external disturbance.

Define variation parameter sets as θ_yu=k_t/J_SR, A_fy/J_s,, in which can be considered the constant nominal value of the lumped uncertainties d_ly,

Then Eq. (8) can be rewritten as

 (9)

where the time-varying portions of are denoted as

2.4 Problem formulation

The following reasonable and practical assumptions are made.

Assumption 1: The variation parameter sets and θf_u are within a known bounded convex set, and the boundary set is assumed to be known from experiment results.

Assumption 2: The disturbance terms are bounded and satisfied where δ_dy and δ_df are the known bounding function.

Assumption 3: All the states and commands of system are bounded and second-order differentiable for the controller design.

Therefore, under assumptions 1–3, the control problem for this dual-stage control system can be described as follows.

1) Design a dual-stage robust H_∞ controller to achieve the specified system performance and guaranteed robustness.

2) The compensation of adaptive model is introduced into the controller design to reduce the effect of model uncertainties and system nonlinearity.

3) The stability of the control scheme, integrating robust H_∞ technique and compensation of adaptive model can be proved by means of Lyapunov method.

3 Controller design and stability analysis

A basic control architecture for dual-stage servo control system shown in Figure 5 is employed in this paper. From this figure, it can be seen that the output of the VCM fine stage is added to the input of the gimbal coarse stage, which restricts the VCM channel motion and guarantees the output of the VCM on the center of its stroke.

Figure 5 Block diagram of dual-stage platform

where e is the angular position tracking error of the overall system; e_c is the angular position tracking error of gimbal coarse stage; y_d is reference angular position input of the system; P_gimb and P_vcm are the nominal plants of the coarse gimbal stage and fine VCM stage, respectively; K_gimb and K_vcm are the designed controllers respectively. The notations, such as y_T, θ and θ_f, are used in Figure 4.

Based on the control architecture shown in Figure 5, a robust H_∞ controller with compensation of adaptive model will be developed in this section.

3.1 Adaptive model compensation

1) Define tracking errors as

                    (10)

                   (11)

2) Then define the state vector as

       (12)

3) With Eqs. (10), Eq. (11) and Eq. (12), the state space equation of tracking error x can be written as

 (13)

where

4) The controller is designed as

,              (14)

where

         (15)

             (16)

are the compensation term of adaptive model; andare the adaptive estimation parameters. And u_mys, u_fs are the robust control law to be designed using robust H_∞ design approach.

5) The adaptation laws are designed as

     (17)

        (18)

where Γ_y, Γ_yu, Γ_f and Γ_fu are the positive definite matrices, and τ_y, τ_yu, τ_f and τ_fu are adaptive estimation functions. The projection mapping proj(·) is defined as

            (19)

where θ_max and θ_min represent the maximum and minimum boundaries of the adaptive estimation parameters respectively.

5) Combining Eq. (13) with Eq. (14), the state space equation becomes

                      (20)

where

and

3.2 Robust H_∞ controller design

With compensation of adaptive model, the control problem for this dual-stage control system is converted into the standard robust H_∞ design problem based on the state space Eq. (20). The H_∞ control structure is given in Figure 6.

The augmentation state space equation can be represented in the following form:

                  (21)

where

The weighting transfer function W₁ is chosen based on the unstructured additive uncertainty; W₂ is chosen to shape the sensitivity function in order to achieve the specified control accuracy and disturbance attenuation ability. The choice of these weighting transfer functions is detailed as follows.

The model uncertainty such as the high- frequency dynamics is treated as unstructured uncertainty, which can be represented by an additive weighting transfer function W₁.

The weighting transfer function W₁ consists of gimbal weighting transfer function W_G and MSL weighting transfer function W_m. W_G mainly reflects the neglected and unmodeled gimbal high- frequency dynamics, and W_m mainly reflects the neglected and unmodeled MSL high-frequency dynamics. W_G and W_m can be obtained by the following equations.

            (22)

where G_Gnom(s) and G_Mnom(s) are the nominal reduced order models of the gimbal and MSL, respectively. G_Gnom(s) and G_Mnom(s) can be obtained by Eq. (3) and Eq. (7). G_G(s) and G_M(s) represent the actual system models without neglecting high- frequency dynamics, which can be obtained through system frequency identification results. Δ_G(s) and Δ_M(s) are normalized such that ||Δ_G(s)||≤1, ||Δ_M(s)||≤1.

Figure 6 H_∞ control structure of dual-stage platform

Thus, based on Eq. (22), the weighting transfer functions W_G and W_M can be captured as

The performance weighting transfer function W₂ is usually designed to meet system sensitivity function requirements. In order to guarantee good tracking performance at the concerned low frequencies, a simple low pass filter is chosen to obtain the desired performance weighting transfer function W₂ [23]:

                        (23)

where M_s is the upper bound on the magnitude of sensitivity function in all frequencies; ω_b is chosen to achieve approximately the desired crossover frequency of sensitivity function; A is a non-zero small number to approximate the integral part of the filter.

Based on Eq. (23) and practical control requirements of this dual-stage control system, W₂ is chosen as

where M_s=1.5, ω_b=50.24 rad/s, A=0.01.

In general, H_∞ control problem requires the solutions of two Riccati equations, the state equation and the observer equations [24]. This general H_∞ control problem is often called H_∞ output feedback control problem.

In this paper, due to the robust H_∞ control law u_s=Kx_e, the general H_∞ control problem can be converted into H_∞ stage feedback control problem, and the full state feedback is available. This special H_∞ state feedback controller K can be derived by solving a single Riccati equation rather than two Riccati equations [25]:

   (24)

whereis the solution of the above algebraic Riccati equation.

With the algebraic Riccati solution P, the state feedback controller K is written as

                          (25)

3.3 Stability analysis by means of Lyapunov function

Theorem 1: In the state space Eq. (21), the adaptive estimation functions are chosen as . Then under Assumptions 1–3, the following results can be obtained:

1) The H_∞ tracking performance is guaranteed as follows:

                (26)

where and

2) All state vectors in the state space Eq. (21) are bounded. Furthermore, the Lyapunov function V_s is bounded by

    (27)

Proof: The Lyapunov function is defined as

Its derivative with respect to time is given by

                      (28)

The derivative of V_s is rewritten with the Riccati Eq. (24) as

                  (29)

where

Integratingwith time t=0 to T yields

    (30)

Since the above inequation becomes

             (31)

This proves the situation 1) of Theorem 1.

Now consider the situation 2) of Theorem 1.

Assumption 4: The estimation errors and are bounded within known boundaries by projection mapping proj(·). Thus, it is reasonable to assume M_f= and

Then, Eq. (29) can be derived as

                          (32)

whereλ_max(·) and λ_min(·) are the maximum and minimum eigenvalue of the matrix, respectively.

Thus by comparison lemma, part B is proved.

4 Simulation and experiment analysis

4.1 Simulation analysis and results

The simulation parameters of the dual-stage platform are obtained from a prototype identified results, as listed in Table 1.

Table 1 System parameters of real platform

The frequency response of the closed-loop control system is given in Figure 7. The results in Figure 7 show that the coarse stage has been limited in the low frequency region below 27 Hz. The overall closed-loop bandwidth of dual-stage system can reach up to 104 Hz.

Here, the proposed robust H_∞ controller with compensation of adaptive model is abbreviated as AH_∞ controller. For performance comparison, a conventional H_∞ controller is also designed to have similar closed loop bandwidth around 104 Hz with AH_∞ controller.

In order to verify the control performance of the proposed controller, three typical simulation conditions are introduced to compare the tracking performances of the two controllers mentioned above.

Case 1: To verify the trajectory tracking performance of this dual-stage platform, the reference input of system is chosen as the sinusoidal motion with 5°/0.3 Hz.

Case 2: To verify the performance under external disturbance, a sinusoidal motion with 2°/2 Hz is introduced as external disturbance and the dual-stage platform is controlled by the proposed controller.

Figure 7 Simulated bode diagram of dual-stage control system

Case 3: To verify the adaptive estimation performance of the controlled system to parameter variations, set the coulomb friction coefficient fluctuation to be 10%, while the external disturbance is chosen as the sinusoidal motion with 2°/2 Hz.

The simulation results under Case 1 are shown in Figure 8. The results show that the maximum tracking error of AH_∞ is 0.011° while the maximum tracking error of conventional H_∞ is 0.041°.

The simulation results under Case 2 are shown in Figure 9. The maximum position tracking error of AH_∞ is 0.012° while the maximum position tracking error of conventional H_∞ is 0.042°. These results reflect the effectiveness of the proposed controller to overcome external disturbance.

The simulation results under Case 3 are shown in Figures 10 and 11. It is obvious that the tracking error becomes smaller after several periods due to the compensation of adaptive model with converging online coulomb friction coefficient estimate.

Figure 8 Position tracking errors with AH_∞ and H_∞ under Case1

Figure 9 Position tracking errors with AH_∞ and H_∞ under Case 2

Figure 10 Parameter estimation of coulomb friction coefficient

4.2 Experiment analysis and results

The proposed controller has been also verified in the prototype to further assess its feasibility. The hardware setup is shown in Figure 12.

In the experiment, this dual-stage platform is mounted on a flight simulation table which is controlled by the sinusoidal motion with 2°/2 Hz as external disturbance caused by the base motions.

Figure 11 Position tracking errors under Case 3

Figure 12 Structure of mirror stabilization platforms

The results in Figure 13 show that the maximum tracking error of AH_∞ is 0.012°while the maximum tracking error of conventional H_∞ is 0.041°.

It is clear that the simulation and experiment results are very similar. The slight difference is mainly due to some model errors, such as the friction model.

5 Conclusions

In this paper, a dual-stage H_∞ controller with compensation of adaptive model is designed. The compensation of adaptive model is used to deal with the nonlinearity dynamics, such as friction nonlinearity, and the H_∞ state feedback controller is developed to achieve the specified system performance, such as the tracking accuracy and robustness. Stability of the closed-loop system is proved by means of Lyapunov analysis.

Figure 13 Position tracking error in experiment under external disturbance

Simulation and experiment results show that the proposed controller can obtain satisfactory tracking performance and disturbance rejection performance, which is superior to the conventional H_∞ controller. With the proposed controller，the maximum position tracking error is reduced to 0.012° under the condition of external disturbance.

Future work will include experiment verification of the proposed controller in handling the effects of parameter variations and model uncertainties.

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(Edited by YANG Hua)

中文导读

基于自适应模型补偿的双级惯性稳定平台鲁棒控制

摘要：为了实现更加优越的跟踪精度，光电惯性稳定平台采取粗、精双级控制系统形式。其中，粗级控制系统为常规的陀螺稳定平台，而精级控制系统由二级微动伺服控制系统组成，用来控制成像光路中透镜的精密运动。首先，本文对双级控制系统动力学模型进行建模，将其表示为多输入、多输出（MIMO）的耦合控制系统。然后，针对双级控制系统动力学模型的特点，推导了基于自适应模型前馈补偿的综合鲁棒H_∞控制策略，保证了双级控制系统的整体稳定性，实现良好的跟踪性能和鲁棒性；然后，通过Lyapunov方法证明了双级控制系统的稳定性。最后，仿真和实验结果证明了所提控制器的有效性和可行性。

关键词：双级控制；惯性稳定平台；鲁棒H_∞控制；自适应模型

Foundation item: Project(61174203) supported by the National Natural Science Foundation of China

Received date: 2017-05-30; Accepted date: 2018-03-29

Corresponding author: SONG Jiang-peng, PhD Candidate; Tel: +86-13702140809; E-mail: jjh_sjp@163.com; ORCID: 0000-0001- 8274-3745