Model-free adaptive optimal controller design for aeroelastic system with input constraints

Abstract

In this thesis, an innovative model-free adaptive control strategy based on a multi-observer technique that takes advantage of input/output measurement data is proposed for the aeroelastic system of the two degree of freedom pitch-plunge wing, and this unknown complicated nonlinear system is a general multi-input multi-output plant with input constraints. In this algorithm, the multi-observer technique is applied to estimate the value of the pseudopartial derivative parameter matrix in the approach of the compact form dynamic linearization designed to linearize the model of the two-dimensional wing-flap system with input constraints. At the same time, this model-free adaptive control method consists of the approximate internal model and the optimal controller. Moreover, this control scheme is based on the linear matrix inequalities, which is a kind of real-time computation. In the design process for controlling this two-dimensional wing-flap system in the condition that the control inputs are subjected to amplitude and change rate limits, the problem of the dynamic control is transformed into the optimization problem, which can minimize the performance index. Finally, simulation results for the two-dimensional wing-flap system with input constraints can demonstrate the availability and potential of the presented multi-observer-based model-free optimal control strategy for unknown nonlinear multi-input multi-output system with input saturation.

Keywords

Model-free adaptive control method multi-observer MIMO nonlinear system optimal control input constraints

Introduction

Model-based control strategies can be used in a variety of domains when the superior identification of plant dynamics and the knowledge of the processes can be obtained. However, due to the shortage of the adequate priori physical models, these methods cannot get the good performance and strong robust.^1
–3 Unfortunately, it is the common case that we cannot obtain the precise matching models of control systems with complex dynamics and highly nonlinear characteristics in the modern industry. For solving this intractable and unavoidable problem, the adaptive techniques for the general nonlinear systems on the basis of neural network and fuzzy logic come into being during the last several decades.^4

–7 However, there are still some disadvantages for the methods of fuzzy logic and neural network, such as no assurance of high convergence speed and so on. Furthermore, it is very difficult to select the amount of the fuzzy control rule base in fuzzy logic. Similar to the condition of fuzzy control, it is very hard to do the decision for the amount of the hidden units in the general neural network control.

The data-driven control (DDC) serves as the effective approach when the indispensable control information is obtained immediately from a lot of process data. This term of data driven was first proposed in computer science, and many researchers in the control society pay attention to this method in recently years.^8,9 The controller of the DDC strategy is only based on input/output measurements in the process dynamics. In the design procedure, the unmodelled dynamics are not necessary. In addition, DDC technique makes good use of a plenty of measurable control data to get the adequate information of the control plants.¹⁰ In recent years, the methods of data driven can be used to many aspects, such as fault diagnosis,¹¹ robot control,¹² and aeronautical and aerospace area.¹³ With the development of the control theory, there are several typical DDC strategies, for example, model-free adaptive control (MFAC),¹⁴ lazy learning approach,¹⁵ and dynamic optimization strategy.¹⁶

MFAC is one of the most popular DDC methods used in many fields. Hou and Jin^17,18 and Xu et al.¹⁹ have taken use of compact form dynamic linearization (CFDL) to design MFAC algorithm for discrete-time systems. Others also have utilized the techniques of partial form dynamic linearization or full form dynamic linearization to achieve the same goal.²⁰ But the vast majority of results for MFAC is aimed at the systems of single-input single-output (SISO). Many control laws for SISO systems cannot be applied to the multi-input multi-output (MIMO) systems due to unavoidable existence of the input coupling.²¹ Moreover, in many control design for MIMO systems, the decoupling matrix must be calculable for the convenience to decouple the control inputs. In some cases, the decoupling control is very difficult to be realized, and the performance of the control is not satisfactory.

Under the background that modern science and technology have been developed at top speed, there has been a noticeable increase in methods to control the aeroelastic system, which is a typical nonlinear MIMO system. However, most of the strategies have been adopted on the basis of accurate mathematical models for aeroelastic plants. According to inherent characteristics of the structural and aerodynamic nonlinearities for the aeroelastic system, the precise model of the aeroelastic system cannot be easy to get.²² When those nonlinearities exist in the system, the aeroelastic system may present several undesired responses such as limit cycle oscillation or flutter.²³ Under the condition of considering those nonlinear characteristics in the aeroelastic system, the general model-based methods are difficult to obtain the satisfactory control performance. In actual control process of aeroelastic system, magnitude and rate constraints of control inputs are known as the most important nonlinearity in the aeroelastic system although many researchers have ignored these serious problems.^24,25 If the constraint problems cannot be considered, the performance of the aeroelastic system may decrease in some extent. In more serious condition, the system can become unstable. Optimal control, which has the capability of handling multiple constraints, is the common systematic method. A model predictive control method with the capability of prediction serves as an effective way to solve the problems of various constraints, including amplitude, rate, and state constraints.

In this thesis, we present a kind of innovative MFAC method for aeroelastic systems with input constraints, which are MIMO nonlinear control systems with unmatched model and varied disturbances. From some research achievements in dynamic linearization,¹⁸ we propose an innovative control strategy based on the multi-observer for the two degree of freedom (2-DOF) pitch-plunge wing, which is the typical aeroelastic system. In this algorithm, the pseudopartial derivative (PPD) can be adopted to linearize this complex aeroelastic wing with nonlinear characteristics. Due to the inevitable existence of the control input constraints in this dynamic process, we must design a unique constrained controller from online identified multi-observer model and the internal model optimal control. Simulation results for the 2-DOF pitch-plunge wing are shown to prove the high efficiency of this presented MAFC strategy. The main innovative points in our paper are described as the following details: (1) a novel multi-observer-based MFAC scheme is presented for such MIMO aeroelastic system with uncertainties; (2) in the design process of controller, magnitude and rate limits of control inputs are considered in this MIMO nonlinear aeroelastic system; and (3) the optimal control problem subjected to the magnitude and rate constraints of control inputs is transformed as a problem of real-time computation based on the linear matrix inequalities (LMIs).

The remaining parts of the presented article can be organized as follows. System description for 2-DOF pitch-plunge wing subjected to magnitude and rate constraints of control inputs is shown in ‘System description for 2-DOF pitch-plunge wing’ section. In ‘Main results’ section, CFDL for this MIMO aeroelastic system and MFAC strategy based on multi-observer is presented. In ‘Simulation results’ section, simulation results for 2-DOF pitch-plunge wing are given to demonstrate the availability and potential of this presented method. The conclusions are provided at the end of the thesis.

System description for 2-DOF pitch-plunge wing

In this section, we shortly introduce the 2-DOF pitch-plunge wing, which is the typical aeroelastic system, and the structure of the 2-DOF pitch-plunge wing is depicted in Figure 1. The general governing equation of this aeroelastic system with external disturbances, which is developed from the previous research results in,^13,26,27 is as follows

Figure 1.

The constructional drawing of 2-DOF pitch-plunge wing. DOF: degree of freedom.

[\begin{matrix} m_{T} & m_{ω} x_{α} b \\ m_{ω} x_{α} b & I_{α} \end{matrix}] [\begin{matrix} \ddot{h} \\ \ddot{α} \end{matrix}] + [\begin{matrix} c_{h} & 0 \\ 0 & c_{α} \end{matrix}] [\begin{matrix} \dot{h} \\ \dot{α} \end{matrix}] + [\begin{matrix} h k_{h} \\ α k_{α} (α) \end{matrix}] = [\begin{matrix} - L - L_{g} \\ M + M_{g} \end{matrix}]

where α denotes the distance from midchord to elastic axis, b is the semichord, and c_h and k_h represent the values of structural damping coefficients and structural spring stiffness in plunging, respectively. c_α and k_α are similar to c_h and k_h in pitching, respectively. I_α denotes the elastic axis inertia, L and M represent aerodynamic lift and moment, L_g and M_g are aerodynamic lift and moment from external disturbances, m_T and m_ω are mass of wing and pitch-plunge system, x_α denotes distance from elastic axis to midchord, and α and h are pitching and plunging, respectively. In equation (1), the quasi-steady lift L and aerodynamic moment M can be written as

\begin{array}{l} L = ρ U_{\infty}^{2} b s C_{l α} (α + \frac{\dot{h}}{U_{\infty}} + (\frac{1}{2} - a) b \frac{\dot{α}}{U_{\infty}}) \\ + ρ U_{\infty}^{2} b s C_{l β} β + ρ U_{\infty}^{2} b s C_{l γ} γ \\ M = ρ U_{\infty}^{2} b^{2} s C_{m α - eff} (α + \frac{\dot{h}}{U_{\infty}} + (\frac{1}{2} - a) b \frac{\dot{α}}{U_{\infty}}) \\ + ρ U_{\infty}^{2} b^{2} s C_{m β - eff} β + ρ U_{\infty}^{2} b^{2} s C_{m γ - eff} γ \end{array}

where C_lα, C_mα, and C_mα−eff are change rate of lift, moment, and effective moment with respect to angle of attack, respectively. C_lβ, C_mβ, and C_mβ−eff denote change rate of lift, moment, and effective moment with respect to trailing-edge control surface deflections, respectively. C_lγ, C_mγ, and C_mγ−eff are change rate of lift, moment, and effective moment with respect to leading-edge control surface deflections, respectively. U_∞ and ρ are freestream velocity and air density, respectively. β and γ represent trailing-edge flap and leading-edge flap, respectively. s denotes wing section span. C_mα−eff, C_mβ−eff, and C_mγ−eff are written as follows

\begin{array}{l} C_{m α - eff} = (\frac{1}{2} + a) C_{l α} + 2 C_{m α} \\ C_{m β - eff} = (\frac{1}{2} + a) C_{l β} + 2 C_{m β} \\ C_{m γ - eff} = (\frac{1}{2} + a) C_{l γ} + 2 C_{m γ} \end{array}

From the above equation, we define the aerodynamic loads which have the bounded external disturbance as

\begin{matrix} L_{g} = ρ U_{\infty}^{2} b s C_{l α} ω_{G} (τ) / U_{\infty} = ρ U_{\infty} b s C_{l α} ω_{G} (τ) \\ M_{g} = (\frac{1}{2} - a) b L_{g} \end{matrix}

where ω_G(τ) represents the disturbance velocity, therein τ means the dimensionless time variable, which can be written as τ = U_∞t/b. From the above formulas, we can get the following equation

\begin{array}{l} [\begin{matrix} - L - L_{g} \\ M + M_{g} \end{matrix}] = [\begin{matrix} - ρ U_{\infty}^{2} b s C_{l α} (α + \frac{\dot{h}}{U_{\infty}} + (\frac{1}{2} - a) b \frac{\dot{α}}{U_{\infty}}) \\ - ρ U_{\infty}^{2} b^{2} s C_{m α - eff} (α + \frac{\dot{h}}{U_{\infty}} + (\frac{1}{2} - a) b \frac{\dot{α}}{U_{\infty}}) \end{matrix}] + [\begin{matrix} - ρ U_{\infty}^{2} b s C_{l β} & - ρ U_{\infty}^{2} b s C_{l γ} \\ ρ U_{\infty}^{2} b^{2} s C_{m β - eff} & ρ U_{\infty}^{2} b^{2} s C_{m γ - eff} \end{matrix}] \cdot [\begin{matrix} β \\ γ \end{matrix}] + [\begin{matrix} - L_{g} \\ M_{g} \end{matrix}] \\ = A + H \cdot [\begin{matrix} β \\ γ \end{matrix}] + C \end{array}

where

\begin{array}{l} A = [\begin{matrix} - ρ U_{\infty}^{2} b s C_{l α} (α + \frac{\dot{h}}{U_{\infty}} + (\frac{1}{2} - a) b \frac{\dot{α}}{U_{\infty}}) \\ - ρ U_{\infty}^{2} b^{2} s C_{m α - eff} (α + \frac{\dot{h}}{U_{\infty}} + (\frac{1}{2} - a) b \frac{\dot{α}}{U_{\infty}}) \end{matrix}], H = [\begin{matrix} - ρ U_{\infty}^{2} b s C_{l β} & - ρ U_{\infty}^{2} b s C_{l γ} \\ ρ U_{\infty}^{2} b^{2} s C_{m β - eff} & ρ U_{\infty}^{2} b^{2} s C_{m γ - eff} \end{matrix}], C = [\begin{matrix} - L_{g} \\ M_{g} \end{matrix}] \end{array}

In dynamic modeling, the 2-DOF pitch-plunge wing can be redefined in the following form

\begin{array}{l} ϖ = F (p) \ddot{p} + Q (p) \dot{p} + N (p) \end{array}

therein

\begin{array}{l} ϖ = [\begin{matrix} - L - L_{g} \\ M + M_{g} \end{matrix}] = A + H \cdot [\begin{matrix} β \\ γ \end{matrix}] + C, F = [\begin{matrix} m_{T} & m_{ω} x_{α} b \\ m_{ω} x_{α} b & I_{α} \end{matrix}], Q = [\begin{matrix} c_{h} & 0 \\ 0 & c_{α} \end{matrix}], N = [\begin{matrix} h k_{h} \\ α k_{α} (α) \end{matrix}], p = [\begin{matrix} h \\ α \end{matrix}] \end{array}

p(k) and $\dot{p} (k)$ denote the specified values at the kth moment. So F(p), Q(p), and N(p) must be the exact matrice at the k th moment. Therefore, this dynamic equation for 2-DOF pitch-plunge wing is rewritten as the following form

\begin{array}{l} ϖ (k) = F (k) \ddot{p} (k) + Q (k) \dot{p} (k) + N (k) \end{array}

The input of this system is defined as $u (k) = ϖ (k) = [u_{1}, u_{2}]^{T}$ , and the corresponding output can be written as $y (k) = \dot{p} (k) = [y_{1}, y_{2}]^{T}$ . F(k) denotes the symmetric positive-definite matrix. Therefore, equation (6) is redefined as the following form

\begin{array}{l} u (k) = F (k) \dot{y} (k) + Q (k) y (k) + N (k) \end{array}

\begin{array}{l} \dot{y} (k) = \frac{u (k)}{F (k)} - \frac{Q (k) y (k)}{F (k)} - \frac{N (k)}{F (k)} \end{array}

Due to $\dot{y} (k) = \frac{y (k + 1) - y (k)}{T}$ in the interval time T, equation (8) should be redefined the expression

\begin{array}{l} y (k + 1) = (I - \frac{T Q (k)}{F (k)}) y (k) + \frac{T}{F (k)} u (k) - \frac{T}{F (k)} N (k) \end{array}

Now, we consider the condition of control input saturation in this system of the 2-DOF pitch-plunge wing. The amplitude constraint of control input is shown as

\begin{array}{l} u_{min} (k) \leq u (k) \leq u_{max} (k) \end{array}

where u_min(k) and u_max(k) are the lower and upper bounds for amplitude constraints, respectively. The change rate for control input is another constraint, which is easy to be ignored in many actual systems. Due to the existence of inertia in the most of the actual systems, states and outputs (actuators) are unable to change quickly in the short time interval. So this change rate limit, which is a physical restriction, must be considered in the design of the controller. In this system, the constraint of the change rate $\dot{u} (k)$ can be defined as

\begin{array}{l} \underline{\dot{u}} (k) \leq \dot{u} (k) \leq \bar{\dot{u}} (k) \end{array}

where $\underline{\dot{u}} (k)$ and $\bar{\dot{u}} (k)$ denote the lower and upper bounds for rate constraints, respectively. In this discrete time algorithm, the change rate of control input is described as the following form used the first-order Euler approximation

\begin{array}{l} \dot{u} (k) = \frac{u (k + 1) - u (k)}{T_{s}} \end{array}

where T_s denotes the sampling time. Considering the actual amplitude and the change rate constraints from the above analysis, for the next interval, two restrictive constraints can be concluded as

\begin{array}{l} \bar{u} (k + 1) = min [u_{max} (k), u (k) + \bar{\dot{u}} (k) T_{s}] \\ \underline{u} (k + 1) = max [u_{min} (k), u (k) + \underline{\dot{u}} (k) T_{s}] \end{array}

Main results

Dynamics transformation and linearization

In this section, a presented innovative constrained method is designed with the dynamic linearization. We transform the general system model of the 2-DOF pitch-plunge wing to the MIMO CFDL model. Furthermore, we use the method based on multi-observer to estimate the values of PPD parameters. Main contribution in this section is that we adopt a new PPD estimation strategy based on multi-observer.

This CFDL of the system for 2-DOF pitch-plunge wing is based on two necessary conditions.¹⁹ First, the partial derivative of the output in the system with respect to u(k) must be continuous. Second, this aeroelastic system (1) must be generalized Lipschitz, which can be satisfying Δy(k + 1) ≤ c₁|Δu(k)| when ∀k and ∥ Δu(k)∥ ≠ 0, where Δy(k + 1) = y(k + 1) − y(k), Δu(k) = u(k) − u(k − 1), where c₁ denotes the positive constant.

From equation (9), we can realize that partial derivatives of y(k + 1) with regard to output and input can be continuous for 2-DOF pitch-plunge wing. At the same time, we find that this system has the characteristic of generalized Lipschitz. According to equation (9), in the condition of $∥ {[Δ y^{T} (k), Δ u^{T} (k)]}^{T} ∥ \neq 0$ at each moment, there can get the PPD matrix Φ(k). Based on previous analysis, the dynamic linearization model of the 2-DOF pitch-plunge wing is rewritten as the following form.^12,19

\begin{array}{l} Δ y (k + 1) = Φ (k) {[Δ y^{T} (k), Δ u^{T} (k)]}^{T} \end{array}

where $Φ (k) = [χ_{1} (k), χ_{2} (k)]$ , χ₁(k) and χ₂(k) ∈ R^2×2. Moreover, ‖Φ(k)‖ < c₂, where c₂ is a positive constant. χ₁(k), χ₂(k) and Φ(k) are written as

χ_{1} (k) = [\begin{matrix} Φ_{11} (k) \\ Φ_{12} (k) \end{matrix}] = [\begin{matrix} φ_{11} (k) & φ_{12} (k) \\ φ_{21} (k) & φ_{22} (k) \end{matrix}]

χ_{2} (k) = [\begin{matrix} Φ_{12} (k) \\ Φ_{22} (k) \end{matrix}] = [\begin{matrix} φ_{31} (k) & φ_{32} (k) \\ φ_{41} (k) & φ_{42} (k) \end{matrix}]

Φ (k) = [\begin{matrix} Φ_{11} (k) & Φ_{12} (k) \\ Φ_{21} (k) & Φ_{22} (k) \end{matrix}]

The CFDL form of equation (14) can be divided into two MISO equations, and the two MISO equations based on dynamic linearization are defined as

\begin{array}{l} Δ y_{1} (k + 1) = Φ_{1} (k) S (k) = S^{T} (k) Φ_{1}^{T} (k) \\ Δ y_{2} (k + 1) = Φ_{2} (k) S (k) = S^{T} (k) Φ_{2}^{T} (k) \end{array}

where Φ₁(k) = [Φ₁₁(k), Φ₁₂(k)], Φ₂(k) = [Φ₂₁(k), Φ₂₂(k)], and S(k) = [Δy^T(k), Δu^T(k)]^T.

Then, we design an observer to estimate the parameter vector Φ₁(k) of PPD matrix for the first MISO function, and the structure of the first observer is given the definition that

\begin{array}{l} {\hat{y}}_{1} (k + 1) = {\hat{y}}_{1} (k) + S^{T} (k) {\hat{Φ}}_{1}^{T} (k) + k_{1} {\tilde{y}}_{1} (k) \end{array}

where ${\tilde{y}}_{1} (k) = y_{1} (k) - {\hat{y}}_{1} (k)$ represents an output estimation error, and ${\hat{Φ}}_{1} (k)$ denotes the estimated value for the first row vector in PPD matrix. The parameter k₁ must be satisfied with R₁ = 1 − k₁ < 1.

Therefore, due to equations (14) and (16), the output estimation error dynamic can be described as

\begin{array}{l} {\tilde{y}}_{1} (k + 1) = R_{1} {\tilde{y}}_{1} (k) + S^{T} (k) {\tilde{Φ}}_{1}^{T} (k) \end{array}

where ${\tilde{Φ}}_{1} (k) = Φ_{1} (k) - {\hat{Φ}}_{1} (k)$ denotes a parameter estimation error of PPD. An adaptive law of parameter vector Φ₁(k) in PPD matrix is defined in the following form

\begin{array}{l} {\hat{Φ}}_{1}^{T} (k + 1) = {\hat{Φ}}_{1}^{T} (k) + S (k) Ψ_{1} (k) ({\tilde{y}}_{1} (k + 1) - R_{1} {\tilde{y}}_{1} (k)) \end{array}

The gain Ψ₁(k) can be defined that

\begin{array}{l} Ψ_{1} (k) = 2 {(∥ S (k) ∥^{2} + s_{1})}^{- 1} \end{array}

where s₁ denotes the positive constant. Therefore, Ψ₁(k) can be positive definite at every moment.

According to equations (16) and (17) in previous analysis, the error dynamics is described that

\begin{array}{l} {\tilde{y}}_{1} (k + 1) = R_{1} {\tilde{y}}_{1} (k) + S^{T} (k) {\tilde{Φ}}_{1}^{T} (k) \\ {\tilde{Φ}}_{1} (k + 1) = Ω_{1} {\tilde{Φ}}_{1} (k) \end{array}

where Ω₁ is given by

Ω_{1} = I_{1} - S (k) Ψ_{1} (k) S^{T} (k)

where I₁ represents the identity matrix. The major property for estimation strategy can be summarized in a conclusion: assume that ∥Δu(k)∥ ≤ ∊ where the constant ∊ > 0, the equilibrium ${[{\tilde{y}}_{1}, {\tilde{Φ}}_{1}^{T}]}^{T} {= [0, 0}_{2 \times 4}^{T}]^{T}$ in equation (19), this system is globally uniformly stable and ${\tilde{y}}_{1} (k)$ can achieve the asymptotic convergence to 0. This detailed process of proof can be found in the study by Xu et al.¹⁹ and we don’t have detailed description in this section.

Following the same steps as in the first observer, we design the second observer to estimate the parameter vector Φ₂(k) of PPD matrix for the second MISO function. The structure of the second observer is the same as the first observer, and the definitions of parameters in the the second observer are similar to the first one. Then, the following relation equation can be easily proved. For this aeroelastic system of 2-DOF pitch-plunge wing (equation (14)), a group of observers have the description of the following form

\begin{array}{l} Observer 1 {\begin{array}{l} {\hat{y}}_{1} (k + 1) = {\hat{y}}_{1} (k) + S^{T} (k) {\hat{Φ}}_{1}^{T} (k) + k_{1} {\tilde{y}}_{1} (k) \\ Δ {\hat{Φ}}_{1}^{T} (k + 1) = S (k) Ψ_{1} (k) ({\tilde{y}}_{1} (k + 1) - R_{1} {\tilde{y}}_{1} (k)) \\ Ψ_{1} (k) = 2 {(∥ S (k) ∥^{2} + s_{1})}^{- 1} \end{array} \end{array}

\begin{array}{l} Observer 2 {\begin{array}{l} {\hat{y}}_{2} (k + 1) = {\hat{y}}_{2} (k) + S^{T} (k) {\hat{Φ}}_{2}^{T} (k) + k_{2} {\tilde{y}}_{2} (k) \\ Δ {\hat{Φ}}_{2}^{T} (k + 1) = S (k) Ψ_{2} (k) ({\tilde{y}}_{2} (k + 1) - R_{2} {\tilde{y}}_{2} (k)) \\ Ψ_{2} (k) = 2 {(∥ S (k) ∥^{2} + s_{2})}^{- 1} \end{array} \end{array}

where $\lim_{k \to \infty} {\tilde{y}}_{i} (k) = 0$ , therein ${\tilde{y}}_{i} (k) = y_{i} (k) - {\hat{y}}_{i} (k)$ , $i = 1, 2 $, $ \tilde{y} (k) = {[{\tilde{y}}_{1} (k), {\tilde{y}}_{2} (k)]}^{T}$ .

For ascertaining research topic for the next content, two observers can be jointed as

\hat{y} (k + 1) = \hat{y} (k) + \hat{Φ} (k) S (k) + Y \tilde{y} (k) = \hat{y} (k) + {\hat{χ}}_{1} (k) Δ y (k) + {\hat{χ}}_{2} (k) Δ u (k) + Y \tilde{y} (k) = \hat{y} (k) + {\hat{χ}}_{1} (k) Δ y (k) + Y \tilde{y} (k) + {\hat{χ}}_{2} (k) Δ u (k)

where Y = diag(k₁, k₂), ${\hat{χ}}_{1} (k)$ , and ${\hat{χ}}_{2} (k)$ mean the estimation values related to χ₁ and χ₂.

Adaptive optimal controller design based on multi-observer

From this part, we show the adaptive optimal control method, which is a new strategy of internal model optimal control based on the multi-observer for the two-input and two-output system of the 2-DOF pitch-plunge wing.

Based on the above section, equation (21) can be rewritten as the following new form

\begin{array}{l} \hat{y} (k + 1) = G_{1} (k) + G_{2} (k) Δ u (k) \end{array}

where $G_{1} (k) = \hat{y} (k) + {\hat{χ}}_{1} (k) Δ y (k) + Y \tilde{y} (k)$ and $G_{2} (k) = {\hat{χ}}_{2} (k)$ .

It is easy to obtain that equation (22) can be redefined as

\begin{array}{l} \hat{y} (k + 1) & = G_{1} (k) - G_{2} (k) u (k - 1) + G_{2} (k) u (k) \\ = G_{3} (k) + G_{2} (k) u (k) \end{array}

where G₃(k) = G₁(k) − G₂(k)u(k − 1). Then, the nonlinear control law for this aeroelastic system can be easily determined. According to this system model (23), a nonlinear adaptive controller under the condition of input saturation is designed. In general design process of control, the dynamic control, which does not take into account the control input constraints, usually has the description as the optimization problem. The performance index is written as

\begin{array}{l} J_{1} = u {(k)}^{T} B^{2} u (k) \end{array}

where B = diag(B₁, B₂) is positive definite, and the parameters B₁ and B₂ in this matrix denote the relative weight. In the actual system of 2-DOF pitch-plunge wing, J₁ is minimized and subjected to the constraints in equation (13). This performance index may lead to a serious problem under the condition that amplitude or rate saturation of the control input exists in the system. In that undesirable condition, the control demand is unattainable, and the system may become unstable. To avoid this situation, a novel performance index J₂, which adds a term related to moment error, is chosen as the following form

J_{2} = u {(k)}^{T} B^{2} u (k) + {[y^{*} (k + 1) - y (k + 1)]}^{T} D^{2} [y^{*} (k + 1) - y (k + 1)] = u {(k)}^{T} B^{2} u (k) + [y^{*} (k + 1) - G_{3} (k) - G_{2} (k) u (k)]^{T} D^{2} [y^{*} (k + 1) - G_{3} (k) - G_{2} (k) u (k)]

where D = diag(D₁, D₂) is positive definite, therein D₁ and D₂ represent the relative weight. Components of D must be choosen some higher values compared with the components of B, which can eliminate the allocation error. ι > 0 denotes the minimum value of this performance, and we have the relationship that

ι - J_{2} > 0

and

ι - u {(k)}^{T} B^{2} u (k) - [y^{*} (k + 1) - G_{3} (k) - G_{2} (k) u (k {)]}^{T} D^{2} [y^{*} (k + 1) - G_{3} (k) - G_{2} (k) u (k)] > 0

Schur complements. Define the diagonal matrices A_i, LMI A(x) > 0 can be realized as the set. Nonlinear inequalities can be transformed into LMI with the method of Schur complements. The description of this theorem can be written that

[\begin{matrix} P (x) & S (x) \\ S {(x)}^{T} & W (x) \end{matrix}] > 0

where P(x) = P(x)^T, W(x) = W(x)^T, and S(x) is the function related to x. This expression used Schur complements can be written as

W (x) > 0, P (x) - S (x) W {(x)}^{- 1} S {(x)}^{T} > 0

Based on Schur complements, equation (26) leads to the matrix inequality

[\begin{matrix} ι & u {(k)}^{T} B & {[y^{*} (k + 1) - G_{3} (k) - G_{2} (k) u (k)]}^{T} D \\ B u (k) & I & 0 \\ D [y^{*} (k + 1) - G_{3} (k) - G_{2} (k) u (k)] & 0 & I \end{matrix}] > 0

The control input is subjected to equation (13), and the constraint conditions are redefined that

e_{i} u (k) - e_{i} \underline{u} (k) > 0, e_{j} \bar{u} (k) - e_{j} u (k) > 0, i, j = 1, 2

where e_i and e_j denote unit vectors along u_i and u_j. Based on equations (29) and (30), we convert the control problem into an optimization problem, which is a solution for LMI constrained minimization problem

\begin{array}{l} \lim_{u (k)} ι \\ [\begin{matrix} ι & u {(k)}^{T} B & {[y^{*} (k + 1) - G_{3} (k) - G_{2} (k) u (k)]}^{T} D \\ B u (k) & I & 0 \\ D [y^{*} (k + 1) - G_{3} (k) - G_{2} (k) u (k)] & 0 & I \end{matrix}] > 0 \\ ι > 0 \\ e_{i} u (k) - e_{i} \underline{u} (k) > 0, i = 1, 2 \\ e_{j} \bar{u} (k) - e_{j} u (k) > 0, j = 1, 2 \end{array}

where

\begin{array}{l} \bar{u} (k + 1) = min [u_{max} (k), u (k) + \bar{\dot{u}} (k) T_{s}] \\ \underline{u} (k + 1) = max [u_{min} (k), u (k) + \underline{\dot{u}} (k) T_{s}] \end{array}

This LMI formulation in equation (31) can be solved by standard algorithms. The controller of this aeroelastic system uses the reference signal y*(k + 1), past control input u(k − 1) to determine control input. This control optimization problem can be handled iteratively. Based on the condition that a feasible solution of this function exists, the convergence of this method can be guaranteed. The proposed strategy, in this article, can be verified under the help of LMI toolbox in MATLAB.

Internal gmodel control (IMC) method has the characteristic that its control device consists of a controller and a predictive model for control plant,^28,29 and this predictive model is known as the internal model. The internal model loop can obtain the error of actual outputs and IMC outputs. Figure 2 depicts the structure of MFAC scheme based on multi-observer for aeroelastic system with input constraints proposed in this article.

Figure 2.

Model-free adaptive optimal controller design based on multi-observer for aeroelastic system with input constraints.

Some uncertainties including external disturbances can be eliminated to some extent filters, which can be properly chosen, like F₁(z) and F₂(z). The methods of design process for filter F₁(z) and F₂(z) can be found in the study by Nahas et al.³⁰

Because of that modeling error will be very large at high frequencies, we infer that the low-pass filter should increase the robustness of 2-DOF pitch-plunge wing.³⁰ In this section, the filter is chosen as F₁ = (1 − κ₁)/(1 − κ₁z⁻¹). The filter should be tuned, so as to have 0 < κ₁ < 1 to guarantee stability of this aeroelastic system.

Remark 1

In many practical systems like many aeroelastic systems, the amplitude and the change rate of the control inputs in the system are not unrestricted. So the control saturation problems must be considered in the process of the controller design. However, a few researchers make substantial results in such problem for this nonaffine nonlinear systems. In this article, the amplitude and rate constraints of control input u(k) are considered in the typical aeroelastic system, which is a MIMO nonlinear system.

Simulation results

This part shows some simulation results for proposed multi-observer-based model-free optimal control method for the 2-D wing-flap system with input constraints, showing that the tracking error of this aeroelastic system can be made arbitrarily small, and the proposed model-free optimal control strategy can eliminate the negative influence of the input saturation.

This 2-DOF pitch-plunge wing is directly impacted with variables γ and β and external disturbances. In this aeroelastic system (1), the parameter values applied for the simulation may be similar to the values used by Wang et al.¹³ and Platanitis and Strganac.³¹ The parameters, values, and units are summarized in Table 1. It is worth mentioning that all of the parameter values are applied to simulate this system of 2-DOF pitch-plunge wing. However, these parameters are realized as uncertain for the controller design. The initial conditions for α and h can be selected as 4.612° and 0 m. The initial values of $\dot{α}$ , $\dot{h}$ , $\ddot{α}$ , and $\ddot{h}$ are all 0. According to the simple static exploration of this aeroelastic system, we can find the relationship between sustained external disturbance, leading edge γ, and trailing edge β. We assume that the 2-D wing-flap system can meet the expected condition of balance points (h, α, $\dot{α}$ , $\dot{h}$ , $\ddot{α}$ , and $\ddot{h}$ are unified 0) with the influence of external disturbances. In that condition, the left-hand side of equation (1) must be 0, and all parts related to h, α, $\dot{α}$ , and $\dot{h}$ on another side of this equation can be 0. Then, we can get the function that

\begin{array}{l} ρ U_{\infty} 2 b s C_{l β} β + ρ U_{\infty} 2 b s C_{l γ} γ + ρ U_{\infty} b s C_{l α} ω_{G} (τ) = 0 \\ ρ U_{\infty} 2 b^{2} s C_{m β - eff} β + ρ U_{\infty} 2 b^{2} s C_{m γ - eff} γ + (\frac{1}{2} - a) b L_{g} = 0 \end{array}

Table 1.

Parameters in the model of 2-DOF pitch-plunge wing.

Parameters	Values	Units
a	−0.6634	—
b	0.2014	m
s	0.6116	m
ρ	1.304	kg·m³
r _cg	−b(0.0998 + a)	m
x_α	r_cg/b	—
c_h	26.12	kg s⁻¹
c_α	0.0351	kg·m² s⁻¹
k_h	2921	N m⁻¹
m _wing	4.340	kg
m_ω	5.23	kg
m_T	15.57	kg
I _cgω	0.04342	kg·m ²
I_cam	0.04697	kg·m ²
C_lα	6.438	rad ⁻¹
C_mα	0	rad ⁻¹
C_lβ	3.685	rad ⁻¹
C_mβ	−0.6719	rad ⁻¹
C_lγ	−0.1624	rad ⁻¹
C_mγ	−0.0984	rad ⁻¹
U_∞	13.28	m s⁻¹
k_α(α)	12.77 + 53.47α + 1003α³	N·m
I_α	I_cam + I_cgω + m_wingr_cg²	kg·m ²

DOF: degree of freedom.

when the matrix in equation (32) is nonsingular, we can get the relationship as follows

\begin{array}{l} [\begin{matrix} β \\ γ \end{matrix}] = - \frac{ω_{G} (τ)}{U_{\infty}} {[\begin{matrix} C_{l β} & C_{l γ} \\ C_{m β - eff} & C_{m γ - eff} \end{matrix}]}^{- 1} [\begin{matrix} C_{l α} \\ (\frac{1}{2} - a) C_{l α} \end{matrix}] \end{array}

The upper bound of ω_G(τ) are related to the value of U_∞. When ω_G(τ) becomes quite larger than constrained control signals, equation (32) can be nonexistent. Moreover, model variables α and h in this 2-DOF pitch-plunge wing are unable to reach the origin. For this simulation, all values of external signals are all less than 0.077 m s⁻¹, corresponding to 0.58% of the velocity value U_∞. In this article, ω_G(τ) = V(τ)ω₀ sin_ωτ, where V(⋅) is a unit step function, and ω₀ is selected as 0.4 rad s⁻¹.

In the design of the multi-observer and the adaptive optimal controller, Y = diag(0.7, 0.7) and s₁ = s₂ = 0.2. B = diag(1, 1), D = diag(1000, 1000), F₂ = 1, and κ₁ = 0.85. We use the proposed control laws in equation (31) to adjust y(k) to track the reference signals, which are $y_{1}^{*} = y_{2}^{*} = 0$ . Amplitude limit of control inputs is −6 ≤ u_i ≤ 6 N·m, and the rate limit of control inputs is $- 2 \leq {\dot{u}}_{i} \leq 2$ N·m s⁻¹, i = 1, 2. Figure 3 shows the estimated values of the PPD parameters in multi-observer. Figure 4 describes the dynamic responses for aeroelastic system used the presented MFAC scheme, where ϖ₁ = u₁ and ϖ₂ = u₂. In Figure 4, we can find that the tracking ability of the multi-observer is very good, and the control inputs u₁ and u₂ meet preset requirements for amplitude and rate limits. Therefore, these simulation results show that the negative effects of inexact model or external disturbances on the traceability are eliminated efficiently by the proposed MFAC method, and the 2-D wing-flap system has the good performance under the condition that the control inputs of this system meet the requirements of magnitude and rate limits.

Figure 3.

Estimations of the PPD parameters in multi-observer. PPD: pseudopartial derivative.

Figure 4.

Dynamic responses for aeroelastic system based on the proposed model-free adaptive optimal control method.

Conclusions

A novel multiple adaptive observer-based model-free optimal control strategy is proposed for the 2-D wing-flap system with input constraints, which is a MIMO nonlinear aeroelastic system under model mismatch and disturbances. According to CFDL technique, we use an innovative MFAC scheme combined with the adaptive multi-observer for nonlinear MIMO aeroelastic system. And the PPD was used to dynamically linearize this nonlinear aeroelastic system. Moreover, the constraints of control inputs are considered in the design process of this controller. In addition, that the presented control method is based on real-time computation of LMIs, which is an optimization problem to minimize the performance index. Finally, simulation results for the 2-D wing-flap system achieve well tracking ability and satisfactory robustness, which can certify the high efficient and potential of this creative control strategy. In the future, we will improve the robust to optimize this algorithm and improve the accuracy of measured data. Furthermore, we will use this method to nonlinear aeroelastic systems with more complex characteristics. We hope that we provide a new alternative method to deal with saturation problems for control inputs in MIMO nonlinear system for the other researchers.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was partially supported by the National Natural Science Foundation of China (61503156, 61473250, 61403161, and 51405198), the Fundamental Research Funds for the Central Universities (JUSRP11562 and NJ20150011), and the Natural Science Foundation of Jiangsu Higher Education Institution (14KJB120013).

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