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Abstract

For a class of multi-input multi-output nonlinear systems, a disturbance observer-based control is proposed to solve the tracking problem in the presence of mismatched disturbances. By designing a novel compensation gain matrix, the disturbances can be removed from the output channel completely as well as retaining the nominal performance. Compared with the state of the art, the gain matrix reduces to be constant; therefore, the complexity of the controller is simplified greatly. This method is applied to the control of hypersonic flight vehicles to demonstrate its effectiveness.

Keywords

Feedback linearization disturbance observer disturbance compensation hypersonic flight vehicle

Introduction

In modern practical control systems, unexpected disturbances occur inevitably due to the external disturbances, unmodelled dynamics and parameter perturbations.^1
–3 Efficient control techniques have been extensively investigated in the literature, such as H₂ control,⁴ H_∞ control,⁵ sliding control,⁶ adaptive control^7,8 and so on.

However, it should be pointed out that mismatched disturbances (the disturbances are not in the input channel) are more universal in practice. Considering hypersonic flight vehicles (HFVs) for example, the mismatched disturbances caused by the dramatic changes in flight environments may influence the states directly.⁹ Recently, the mismatched disturbances problem has been a hot topic. In the study by Zheng et al.,¹⁰ a robust tracking controller was proposed for multi-input multi-output (MIMO) nonlinear systems using a variable structure control approach with the mismatched disturbances. In the study by Lin et al.,¹¹ a robust controller was constructed using the sliding mode control (SMC) technique, while the norms of the mismatched disturbances were confined to be bounded. Some researchers attenuated mismatched disturbances by incorporating an integral item into the controller.¹² However, the chattering would increase as the gain of the controller increases.

Most of the previous controllers were robust, focusing on the system stability at the price of control performance. Recently, disturbance observer-based control (DOBC)¹³ was introduced to solve the disturbances problem in another way. The core of DOBC is to amend disturbances due to the estimation of the disturbance observer without destroying the original performance. DOBC will work to maintain the nominal performance, when there is no disturbance. The initial applications of DOBC focused on the matched disturbances, which could be eliminated directly by the state feedback. The mismatched disturbances attenuation approaches were proposed for linear systems.^14,15 Some results about sliding control made advances to eliminate the disturbances effectively by incorporating the disturbances estimation into dynamic surfaces^16
–18 for nonlinear systems, but the researches concentrated on single input single output nonlinear systems. However, the mismatched disturbances attenuation of MIMO nonlinear systems is still a challenge in present research. Literature¹⁹ proposed a feedback control law to remove the disturbances by solving ordinary constant equations, but singularity of the solution was inevitable yet. Literature²⁰ investigated the fault-tolerant control problem for a class of continuous-time Markovian jump systems with the mismatched disturbances. In the study by Yang et al.,²¹ disturbances were totally removed from output by designing the decoupling gain matrix. In the study by Han et al.,²² a back-stepping controller was designed to acquire exact tracking for a class of uncertain systems in the presence of the mismatched disturbances. Continuous-time model predictive control combined with DOBC was proposed to guarantee the desired offset-free tracking performance for a class of nonlinear systems.²³

Inspired by the previous works, a tracking controller is proposed to counteract the mismatched disturbances for feedback linearizable nonlinear systems. By designing a proper compensation gain matrix, the disturbances can be removed in the output channel completely. The disturbance observers are designed for every subsystem of the original system, such that the gain compensating matrix reduces to a constant one. The gain matrix of the previous result²¹ was more complicated, which involved Lie derivative computation of the nonlinear system dynamics. Therefore, less information of the nonlinear system is required and the controller complexity is reduced subsequently. In addition, the proposed method keeps the nominal performance recovery while it can compensate for the effect of the disturbances at the same time.

The proposed method is applied to velocity and altitude tracking of an HFV model undergoing mismatched disturbances. The HFV has been widely applied to the commercial use and military demands for its outstanding flight performance. However, for its complicated dynamics, the longitudinal motion of HFV is the only mature part of HFV.²⁴ Many powerful controllers have been studied for models of HFV, such as adaptive control,²⁵ sliding control,²⁶ back-stepping control²⁷ and intelligent control.²⁸ The emphasis of the research is to weaken the uncertainty and disturbances of the HFV system. Literature²⁹ summarized the powerful techniques and challenges for the common models of HFVs. A novel dynamic surface control was applied to develop two global neural controllers for HFVs with unknown dynamics.³⁰ A robust control was proposed for a generic hypersonic aircraft using command filter to simplify the back-stepping design in the study by Xu et al.³¹ An et al.³² designed an auxiliary system to counteract the mismatched disturbances for regulating control of air-breathing hypersonic vehicles subject to constrained inputs. Besides, the large flight envelope and the tremendous changes of flight environment lead to the external disturbances. Thus, it is very necessary to design the controllers which can decrease the effect of the disturbances and retain the nominal stability.

The main contributions of our work are as follows:

The compensation gain matrix is a constant one, which simplifies the structure of the controller.

The proposed controller possesses the ability to recover the nominal performance.

The influence of mismatched disturbances can be completely compensated in the output channel.

The rest of the article is organized as follows. In ‘Problem formulation and preliminaries’ section, the main problem and some preliminaries are stated. In ‘The tracking control design’ section, the tracking controller is provided along with its stability analysis. In ‘Application to the control of HFVs’ section, the proposed method is applied to the HFVs and simulation results are presented. This article is finally concluded in the last section.

Problem formulation and preliminaries

Problem formulation

Consider an MIMO feedback linearizable nonlinear system with lumped disturbance represented as

{\begin{cases} \dot{x} = f (x) + g (x) u + q (x) d \\ y_{1} = h_{1} (x) \\ y_{2} = h_{2} (x) \\ \dots \\ y_{m} = h_{m} (x) \end{cases}

where $x (t) \in R^{n}, u (t) = {[u_{1} u_{2} \dots u_{m}]}^{T} \in R^{m} and {[h_{1}, \dots, h_{m}]}^{T} \in R^{m}$ are the state, input and output, respectively. $d = {[d_{1} d_{2} \dots d_{n}]}^{T} \in R^{n}$ is the external unknown disturbance.

$f (x), g (x) = [g_{1} (x) g_{2} (x) \dots g_{n} (x)] and q (x) = [q_{1} (x), q_{2} (x), ..., q_{n} (x)]$ are smooth vector or matrix fields on Rⁿ.

For functions $h_{i} (x), i = 1, 2, \dots, m$ and vector field $f (x) = [f_{1} (x) f_{2} (x) \dots f_{n} (x)]$ , the Lie derivative is defined as $L_{f} h_{i} (x) = \sum_{j = 1}^{n} \frac{\partial h_{i} (x)}{\partial x_{j}} f_{j} (x) and L_{f}^{k} h_{i} (x) = \sum_{j = 1}^{n} \frac{\partial L_{f}^{k - 1} h_{i} (x)}{\partial x_{j}} f_{j} (x) .$

Assumption 1

The nominal system of equation (1) is satisfied with exact linearization condition^33,34 that the sum of the relative degree of $h_{i} (x)$ is n. This assumption is necessary for MIMO feedback linearizable nonlinear systems. The relative degree is defined as follows: if there is a positive integer λ, such that

\begin{array}{l} L_{g_{k}} L_{f}^{λ} h_{i} (x) = 0, λ < r_{i} - 1 \\ L_{g_{k}} L_{f}^{λ - 1} h_{i} (x) \neq 0, λ = r_{i} \end{array}

then, the relative degree of output h_i is r_i.

By the definition of the relative degree of system (1), a new vector can be designed as

h_{*} = {[h_{1}^{(r_{1})} h_{2}^{(r_{2})} \dots h_{m}^{(r_{m})}]}^{T} = b (x) + A (x) u + d^{*}

where $b (x) = {[L_{f}^{r_{1}} (h_{1} (x)) \dots L_{f}^{r_{m}} (h_{m} (x))]}^{T},$

A (x) = [\begin{matrix} L_{g_{1}} L_{f}^{r_{1} - 1} (h_{1} (x)) & L_{g_{2}} L_{f}^{r_{1} - 1} (h_{1} (x)) & \dots & L_{g_{m}} L_{f}^{r_{1} - 1} (h_{1} (x)) \\ L_{g_{1}} L_{f}^{r_{2} - 1} (h_{2} (x)) & L_{g_{2}} L_{f}^{r_{2} - 1} (h_{2} (x)) & \dots & L_{g_{m}} L_{f}^{r_{2} - 1} (h_{2} (x)) \\ ⋮ & ⋮ & ⋮ \\ L_{g_{1}} L_{f}^{r_{m} - 1} (h_{m} (x)) & L_{g_{2}} L_{f}^{r_{m} - 1} (h_{m} (x)) & \dots & L_{g_{m}} L_{f}^{r_{m} - 1} (h_{m} (x)) \end{matrix}], \begin{matrix}  \end{matrix} d^{*} = [\begin{array}{l} \sum_{k = 1}^{n} L_{q_{k}}^{r_{1} - 1} h_{1} d_{k} \\ \sum_{k = 1}^{n} L_{q_{k}}^{r_{2} - 1} h_{2} d_{k} \\ ⋮ \\ \sum_{k = 1}^{n} L_{q_{k}}^{r_{m} - 1} h_{m} d_{k} \end{array}] .

To be specific, output h_i corresponds to a linear subsystem Ξ_i.³³ The coordinate transformations are defined as

ξ^{i} = {[ξ^{i}_{1} ξ^{i}_{2} \dots ξ^{i}_{r_{i}}]}^{T} = {[ϕ^{i}_{1} (x) ϕ^{i}_{2} (x) \dots ϕ^{i}_{r_{i}} (x)]}^{T}

where $ϕ^{i}_{j} = L_{f}^{j - 1} h_{i} (x), i = 1, 2, \dots, m$ , $j = 1, 2, \dots r_{i} .$

Under coordinates (3), the dynamics of subsystem Ξ_i is described as

{\begin{cases} {\dot{ξ}}^{i}_{1} = ξ^{i}_{2} + d_{1}^{i} \\ {\dot{ξ}}^{i}_{2} = ξ^{i}_{3} + d_{2}^{i} \\ \dots \\ {\dot{ξ}}^{i}_{r_{i}} = b_{i} (x) + A_{i} (x) u + d_{r_{i}}^{i} \\ h_{i} = ξ_{1}^{i} \end{cases}

where $A_{i} (x) = [L_{g_{1}} L_{f}^{r_{i} - 1} (h_{i} (x)) L_{g_{2}} L_{f}^{r_{i} - 1} (h_{i} (x)) \dots L_{g_{m}} L_{f}^{r_{i} - 1} (h_{i} (x))], b_{i} (x) = L_{f}^{r_{i}} (h_{i} (x)) and d_{j}^{i} = \sum_{k = 1}^{n} L_{q_{k}}^{j - 1} h_{i} d_{k}$

Moreover, the subsystem disturbance vector ${[d_{1}^{i} d_{2}^{i} \dots d_{r_{i}}^{i}]}^{T}$ is denoted by dⁱ for briefness.

Assumption 2

The full states are available for measurement or known. This assumption is necessary for the state feedback control. The states can be acquired by the sensors or observers.

Assumption 3

The derivatives of the disturbances are bounded. This assumption is a general assumption for the disturbance estimation.

Assumption 4

The disturbances in equation (1) are satisfied with $\lim_{t \to \infty} {\dot{d}}_{i} = 0$ . This assumption is made for the steady-state analysis of disturbance estimation and compensation. However, the disturbances are just satisfied with assumption 3; the disturbances would be cancelled by the proposed controller in this article.

The nonlinear disturbance observer

The nonlinear disturbance observer for the disturbance in equation (1) was introduced by Chen et al.,³⁵ designed as

{\begin{cases} \hat{d} = ρ + λ (x), \\ \dot{ρ} = - l (x) q (x) ρ - l (x) (q (x) λ (x) + f (x) + g (x) u) \end{cases}

where $\hat{d}$ and ρ are the estimation of the disturbance d and the internal states of the disturbance observer (5), respectively. $λ (x)$ is a nonlinear vector-valued function to be designed. The gain $l (x)$ is designed as

l (x) = \frac{\partial λ (x)}{\partial x}

Applying equation (1) into equation (5), dynamics of estimation error $e (t) = \hat{d} - d$ is acquired as

\dot{e} (t) = - l (x) p (x) e (t) + \dot{d}

Lemma 1 (Chen and Guo³⁶)

Suppose that the disturbance d is bounded, the disturbance estimation error system of (5) is locally input-to-state stable, if the appropriate observer gain $l (x)$ can be chosen, such that

\dot{e} (t) + l (x) q (x) e (t) = 0

is asymptotically stable for all $x \in R^{n}$ .

In this article, we propose an output tracking control scheme for equation (1) in the presence of the mismatched disturbances via feedback linearization and nonlinear disturbance observer.

The tracking control design

The proposed tracking control scheme for system (1) will compensate for the mismatched disturbances in the output channel by designing a proper gain matrix. The specific controller is described in theorem 1.

Theorem 1

Consider MIMO nonlinear control system (1) with assumptions 1–4 satisfied. The disturbance observer (5) is assumed to guarantee the tracking error asymptotically stable. The proposed control scheme which compensates effect of the disturbances in the output and tracks the desired trajectory $h_{i d} (1 \leq i \leq m)$ is

$u = A^{- 1} [- b (x) + v + d_{comp}]$
6

where $A (x) and b (x)$ are defined as equation (2). Additionally, the input of the nominal system v is designed as

$\begin{array}{l} v = {[v_{1} v_{2} \dots v_{m}]}^{T}, \\ v_{i} = - c^{i}_{0} [h_{i} (x) - h_{i d}] - c^{i}_{1} [L_{f} h_{i} (x) - {\dot{h}}_{i d}] - \dots \\ \begin{matrix} - c^{i}_{r_{i} - 1} [L_{f}^{r_{i} - 1} h_{i} (x) - h_{i d}^{(r_{i} - 1)}] + h_{i d}^{(r_{i})} \end{matrix} \end{array}$

where h_id is the desired output of h_i and parameter $c_{j}^{i}$ is given to make the polynomials

$p_{0}^{i} (s) = c_{0}^{i} + c_{1}^{i} s \dots + c_{r_{i} - 1}^{i} s^{r_{i} - 1} + s^{r_{i} - 1}$

Hurwitz stable. The disturbances compensation $d_{comp} = {[d_{1} d_{2} \dots d_{m}]}^{T}$ is designed as $d_{i} = - \sum_{j = 1}^{r_{i - 1}} c_{j}^{i} {\hat{d}}_{j}^{i} - {\hat{d}}_{r_{i}}^{i}$ , where $c_{j}^{i}$ is the same as that of v_i, and ${\hat{d}}_{j}^{i}$ is acquired from the disturbance observer for subsystem Ξ_i.

Proof

A new vector $ξ^{} = {[ξ_{r_{1}}^{1} ξ_{r_{2}}^{2} \dots ξ_{r_{m}}^{m}]}^{T}$ is formulated by selecting each last equation of equation (4). Thus, equation (2) can be reformulated as

${\dot{ξ}}^{} = b (x) + A (x) u + d^{}$
7

Substituting equation (6) to equation (7), the closed-loop system equation is obtained

${\dot{ξ}}^{} = v + d^{} + d_{comp}$
8

Correspondently, subsystem Ξ_i* is depicted as

${\begin{cases} {\dot{ξ}}^{i}_{1} = ξ^{i}_{2} + d_{1}^{i} \\ {\dot{ξ}}^{i}_{2} = ξ^{i}_{3} + d_{2}^{i} \\ \dots \\ ξ^{i}_{r_{i}} = v_{i} + d_{r_{i}}^{i} \\ h_{i} = ξ_{1}^{i} \end{cases}$
9

By translational transformation, equation (9) is described as

${\begin{cases} {\dot{ξ}}^{i}_{1} - {\dot{h}}_{i d} = ξ^{i}_{2} - {\dot{h}}_{i d} + d_{1}^{i} \\ {\dot{ξ}}^{i}_{2} - {\ddot{h}}_{i d} = ξ^{i}_{3} - {\ddot{h}}_{i d} + d_{2}^{i} \\ \dots \\ {\dot{ξ}}^{i}_{r_{i}} - h_{i d}^{(r_{i})} = v_{i} - h_{i d}^{(r_{i})} + d_{r_{i}}^{i} \end{cases}$
10

Equation (10) is rewritten in the matrix form

${\begin{cases} {\dot{ξ}}^{i} - {\dot{h}}^{i d} = {\tilde{A}}_{i} (ξ^{i} - h^{i d}) + D_{i} d^{i} + Ψ_{i} e^{i} \\ h_{i} - h_{i d} = C_{i} (ξ^{i} - h^{i d}) \end{cases}$
11

where $e^{i} = {[{\hat{d}}_{1}^{i} - d_{1}^{i} {\hat{d}}_{2}^{i} - d_{2}^{i} \dots {\hat{d}}_{r_{i}}^{i} - d_{r_{i}}^{i}]}^{T}, h^{i d} = {[h_{i d} {\dot{h}}_{i d} \dots h_{i d}^{(r_{i} - 1)}]}^{T}, {\tilde{A}}_{i} = [\begin{array}{l} \begin{matrix} \begin{matrix} 0 & 1 \end{matrix} & 0 & \dots & 0 \end{matrix} \\ \begin{matrix} \begin{matrix} 0 & 0 \end{matrix} & 1 & \dots & 0 \end{matrix} \\ \begin{matrix} \begin{matrix} ⋮ & ⋮ \end{matrix} & ⋮ & ⋱ & ⋮ \end{matrix} \\ \begin{matrix} \begin{matrix} 0 & 0 \end{matrix} & 0 & \dots & 1 \end{matrix} \\ - c_{0}^{i} - c_{1}^{i} \dots - c_{r_{i} - 1}^{i} \end{array}], C_{i} = [1 0 \dots 0], D_{i} = [\begin{array}{l} \begin{matrix} \begin{matrix} 1 & 0 \end{matrix} & 0 & \dots \dots & 0 \end{matrix} \\ \begin{matrix} \begin{matrix} 0 & 1 \end{matrix} & 0 & \dots \dots & 0 \end{matrix} \\ \begin{matrix} \begin{matrix} ⋮ & ⋮ \end{matrix} & ⋮ & ⋱ & ⋮ \end{matrix} \\ \begin{matrix} \begin{matrix} 0 & 0 \end{matrix} & 0 & \dots \dots 1 & 0 \end{matrix} \\ - c_{1}^{i} - c_{2}^{i} \dots - c_{r_{i} - 1}^{i} 0 \end{array}] and Ψ_{i} = [\begin{array}{l} \begin{matrix} \begin{matrix} 0 & 0 \end{matrix} & 0 & \dots \dots & 0 \end{matrix} \\ \begin{matrix} \begin{matrix} 0 & 0 \end{matrix} & 0 & \dots \dots & 0 \end{matrix} \\ \begin{matrix} \begin{matrix} ⋮ & ⋮ \end{matrix} & ⋮ & ⋱ & ⋮ \end{matrix} \\ \begin{matrix} \begin{matrix} 0 & 0 \end{matrix} & 0 & \dots \dots 0 & 0 \end{matrix} \\ - c_{1}^{i} - c_{2}^{i} \dots - c_{r_{i} - 1}^{i} - 1 \end{array}] .$

By simple calculation, it can be concluded from equation (11) that

$ξ^{i} - h^{i d} = {({\tilde{A}}_{i})}^{- 1} [{\dot{ξ}}^{i} - {\dot{h}}^{i d} - D_{i} d^{i} - Ψ_{i} e^{i}]$
12

Employing $C_{i} {({\tilde{A}}_{i})}^{- 1} D^{i} = 0$ , equation (12) is written as

$h_{i} = C_{i} {({\tilde{A}}_{i})}^{- 1} [{\dot{ξ}}^{i} - {\dot{h}}^{i d} - Ψ_{i} e^{i}] + h_{i d}$
13

Equation (13) demonstrates that the disturbances are removed in the output channel by the gain matrix D_i. The tracking error system (11) has a steady state that is ${\dot{ξ}}^{i} - {\dot{h}}_{i d} = 0$ . Applying it to equation (13), we have

$| h_{i} - h_{i d} | = | | C_{i} {({\tilde{A}}_{i})}^{- 1} | | ‖ Ψ_{i} ‖ | e^{i} |$
14

By lemma 1 and assumption 4, the observer gain $l (x)$ can be assumed to make the tracking error asymptotically stable; therefore, $\lim_{t \to \infty} | e^{i} (t) | = 0$ , $i = 1, 2, \dots, m$ is obvious. Additionally, $‖ Ψ_{i} ‖$ is bounded, because Ψ_i is composed of constants $c_{j}^{i}$ . From the previous conclusions, we have

$\lim_{t \to \infty} | h_{i} - h_{i d} | = | | C_{i} {({\tilde{A}}_{i})}^{- 1} | | ‖ Ψ_{i} ‖ \lim_{t \to \infty} | e^{i} | = 0$

We can conclude that output h_i will track the desired trajectory h_id locally in the sense of input state stability. □

Remark 1

The superiority of gain matrix D_i here is constant, which reduces the overall design complexity greatly and enhances the robustness.

Remark 2

The gain matrix D_i is determined by the nominal controller. Moreover, the separation of gain matrix and state is beneficial to the steady state.

Application to the control of HFVs

This section takes the velocity and attitude tracking of HFVs as an example to show the effectiveness of our proposed method. Based on the model of the study by Marrison and Stengel,³⁷ the longitudinal motion of HFVs with the mismatched disturbances can be written in the form of equation (1) as follows

${\begin{matrix} \dot{x} = f (x) + \sum_{i = 1}^{2} g_{i} (x) u_{i} + d \\ y = {[\begin{matrix} V & h \end{matrix}]}^{T} \end{matrix}$
15

with

$f (x) = [\begin{array}{l} \frac{T \cos α - D}{m} - \frac{μ sin γ}{r^{2}} \\ \frac{L + T sin α}{m V} - \frac{(μ - V^{2} r) \cos γ}{V r^{2}} \\ V sin γ \\ q - \dot{γ} \\ \frac{\bar{q} S \bar{c} [C_{M} (α) + C_{M} (q) - 0.0292 α]}{I_{y y}} \\ \dot{β} \\ - 2 ζ ω_{n} η - ω_{n}^{2} β \end{array}] and [g_{1}, g_{2}] = [\begin{array}{l} 0 0 \\ 0 0 \\ 0 0 \\ 0 0 \\ 0 \frac{0.0292 \bar{q} S \bar{c}}{I_{y y}} \\ 0 0 \\ \begin{matrix} ω_{n}^{2} & 0 \end{matrix} \end{array}]$

where $x = {[V γ h α q β \dot{β}]}^{T}$ is the vector of system states. System input $u = {[β_{c} δ_{e}]}^{T}$ and system output $y = {[V, h]}^{T}$ are chosen to satisfy the conditions of feedback linearization. V is the velocity, h is the attitude, γ is the flight-path angle, α is the angle of attack, q is the pitch rate and β is the engine throttle setting. $d = {[d_{1} d_{2} \dots d_{7}]}^{T}$ is the unknown vector of the mismatched disturbance. $q (x) = [q_{1} q_{2} \dots q_{7}]$ is a seventh-order identity matrix with unit vectors $q_{i}, i = 1, 2, \dots, 7$ . $f (x)$ is written as $f (x) = {[f_{1} (x) f_{2} (x) f_{3} (x) f_{4} (x) f_{5} (x) f_{6} (x) f_{7} (x)]}^{T}$ for computation of Lie derivatives. β_c is the command of engine throttle and δ_e is the elevator deflection. The values of lift, drag and thrust are modelled as

$\begin{array}{l} L = \bar{q} S C_{L} (α, M) \\ D = \bar{q} S C_{D} (α, M) \\ T = \bar{q} S C_{T} (α, M, β) \end{array}$

where $\bar{q}$ is the dynamic pressure, S is the reference area and M is the Mach number. The aerodynamic coefficients are acquired by the following fitting functions

$\begin{array}{l} C_{L} = α (0.493 + \frac{1.91}{M}) \\ C_{D} = 0.0082 (171 α^{2} + 1.15 α + 1) \times (0.0012 M^{2} - 0.054 M + 1) \\ C_{T} = {\begin{matrix} 0.0105 [1 - 164 {(α - α_{0})}^{2}] (1 + \frac{17}{M}) 1.15 β if β < 1 \\ 0.0105 [1 - 164 {(α - α_{0})}^{2}] (1 + \frac{17}{M}) (1 + 0.15 β) if β \geq 1 \end{matrix} \\ C_{M} (α) = 10^{- 4} \times (0.06 - e^{\frac{M}{3}}) (- 6565 α^{2} + 6875 α + 1) \\ C_{M} (q) = \frac{\bar{c}}{2 V} q (- 0.025 M + 1.37) (- 6.83 α^{2} + 0.303 α - 0.23) \end{array}$

The details and meanings of other notations can be found in the study by Marrison and Stengel.³⁷

By calculations, the relative degrees of output V and h are three and four, separately. We differentiate V three times and h four times. Therefore, the HFV model (15) can be separated into two subsystems. Subsystem Ξ₁ is related to the output V and subsystem Ξ₂ is related to the output h. Correspondingly, the coordinate transformations of two subsystems are described as

$ξ^{1} = {[ξ_{1}^{1} ξ_{2}^{1} ξ_{3}^{1}]}^{T}, ξ^{2} = {[ξ_{1}^{2} ξ_{2}^{2} ξ_{3}^{2} ξ_{4}^{2}]}^{T}$

which are defined as follows

$\begin{array}{l} ξ_{1}^{1} = V, \begin{matrix} ξ_{2}^{1} = L_{f} V = \frac{T \cos α - D}{m} - \frac{μ sin γ}{r^{2}} \end{matrix} \\ ξ_{3}^{1} = L_{f}^{2} V = f_{1} (\frac{T_{V} \cos α - D_{V}}{m}) + f_{2} (- \frac{μ \cos γ}{r^{2}}) + f_{3} (\frac{T_{h} \cos α - D_{h}}{m} + \frac{2 μ sin γ}{r^{3}} r_{h}) \\ + f_{4} (\frac{T_{α} \cos α - T sin α - D_{α}}{m}) + f_{6} (\frac{T_{β} \cos α}{m}) \\ ξ_{1}^{2} = h, \begin{matrix} ξ_{2}^{2} = L_{f} h = V sin γ \end{matrix} \\ ξ_{3}^{2} = L_{f}^{2} h = f_{1} \cdot (sin γ) + f_{2} \cdot (V \cos γ) \\ ξ_{4}^{2} = L_{f}^{3} h = (L_{f}^{2} V) sin γ + f_{1} \cdot f_{2} \cdot \cos γ + L_{f} (\frac{\partial f_{2}}{\partial x} \cdot V \cdot \cos γ) \\ - L_{f} (f_{2} \cdot V \cdot sin γ) + L_{f} (f_{2} \cdot \cos γ) \end{array}$
16

Under the coordinate transformations (16), subsystems Ξ₁ and Ξ₂ can be represented as

$Ξ_{1} : {\begin{cases} {\dot{ξ}}^{1}_{1} = ξ_{2}^{1} + L_{d} V \\ {\dot{ξ}}^{1}_{2} = ξ_{3}^{1} + L_{d} L_{f} V \\ {\dot{ξ}}^{1}_{3} = L_{f}^{3} V + \sum_{k = 1}^{2} (L_{g_{k}} L_{_{f}}^{2} V) \cdot u_{k} + L_{d} L_{_{f}}^{2} V \end{cases} and Ξ_{2} : {\begin{cases} {\dot{ξ}}^{2}_{1} = ξ_{2}^{2} + L_{d} h \\ {\dot{ξ}}^{2}_{2} = ξ_{3}^{2} + L_{d} L_{f} h \\ {\dot{ξ}}^{2}_{3} = ξ_{4}^{2} + L_{d} L_{_{f}}^{2} h \\ {\dot{ξ}}^{2}_{4} = L_{f}^{4} h + \sum_{k = 1}^{2} (L_{g_{k}} L_{_{f}}^{2} h) \cdot u_{k} + L_{d} L_{_{f}}^{3} h \end{cases}$

According to equation (6), the controller for equation (15) is designed as

$u = A^{- 1} [- b (x) + v + d_{comp}]$

where $A (x) = [\begin{matrix} L_{g_{1}} L_{f}^{2} V & L_{g_{2}} L_{f}^{2} V \\ L_{g_{1}} L_{f}^{3} h & L_{g_{2}} L_{f}^{3} h \end{matrix}], b (x) = {[L_{f}^{3} V L_{f}^{4} h]}^{T} and d^{} = {[\begin{matrix} \sum_{k = 1}^{2} L_{q_{k}}^{2} V d_{k} & \sum_{k = 1}^{2} L_{q_{k}}^{2} h d_{k} \end{matrix}]}^{T} .$

By coordinate transformations (16), we obtain the nominal control $v = {[\begin{matrix} v_{1} & v_{2} \end{matrix}]}^{T}$ as follows

$\begin{array}{l} v_{1} = - c_{0}^{1} [ξ_{1}^{1} - V_{d}] - c_{1}^{1} [ξ_{2}^{1} - {\dot{V}}_{d}] - c_{2}^{1} [ξ_{3}^{1} - {\ddot{V}}_{d}] + {\overset{⃛}{V}}_{d} \\ v_{2} = - c_{0}^{2} [ξ_{1}^{2} - h_{d}] - c_{1}^{2} [ξ_{2}^{2} - {\dot{h}}_{d}] - c_{2}^{2} [ξ_{3}^{2} - {\ddot{h}}_{d}] - c_{3}^{2} [ξ_{4}^{2} - {\overset{⃛}{h}}_{d}] + h_{d^{(4)}} \end{array}$

where

$c_{0}^{1} = 0.625, c_{1}^{1} = 0.004378, c_{2}^{1} = 0.003772, c_{0}^{2} = 0.0224, c_{1}^{2} = 164, c_{2}^{2} = 1.15, c_{3}^{2} = 0.15.$

The disturbances compensation $d_{comp} = {[d_{1} d_{2}]}^{T}$ is designed as $d_{i} = - \sum_{j = 1}^{r_{i - 1}} c_{j}^{i} {\hat{d}}_{j}^{i} - {\hat{d}}_{r_{i}}^{i}$ , $i = 1, 2$ , $r_{1} = 3, r_{2} = 4.$ where $c_{j}^{i}$ is the same as v₁ and v₂.

Disturbances estimation ${\hat{d}}_{j}^{i}$ , $i = 1, 2, j = 1, 2 \dots, r_{i} - 1$ is acquired from the disturbance observers for subsystems Ξ₁ and Ξ₂. The nonlinear vector-valued functions in disturbance observer (5) for Ξ₁ and Ξ₂ are chosen as

$λ (ξ^{1}) = {[10 ξ_{_{1}}^{1} 10 ξ_{2}^{1} 10 ξ_{3}^{1}]}^{T}, λ (ξ^{2}) = {[10 ξ_{1}^{2} 10 ξ_{2}^{2} 10 ξ_{3}^{2} 10 ξ_{4}^{2}]}^{T}$

Supposing the HFV is cruising at h _{0 =} 110,000 ft and V ₀ = 15,060 ft/s. The control objective is to let the vehicle climb Δh = 1000 ft with the velocity increasing about ΔV = 400 ft/s. The altitude and velocity references, that is, h_d* and V_d, are generated by the following command filter

$[\begin{matrix} h_{d} \\ V_{d} \end{matrix}] = \frac{ω^{2}}{s^{2} + 2 ξ ω s + ω^{2}} [\begin{matrix} Δ h \\ Δ V \end{matrix}]$

where ξ = 0.9 and ω = 0.1 rad/s. The parameters in aerodynamic coefficients of lift, drag, thrust and pitch moment are randomly selected with relative uncertainties up to 20%. That is to say, for a parameter with a nominal value P₀, the actual value in simulation is selected as $(P_{0} + Δ P)$ with $| Δ P | / | P_{0} | \leq 0.2$ . In this case, the difference between the nominal and actual dynamics of the HFV model can be lumped as the additive disturbance in equation (15). Consider the velocity dynamics, for example. The actual dynamics of velocity can be written as

$\dot{V} = \frac{(T + Δ T) \cos α - (D + Δ D)}{m} - \frac{μ sin γ}{r^{2}} = \frac{T \cos α - D}{m} - \frac{μ sin γ}{r^{2}} + d_{1}$

where T and D are the thrust and drag calculated according to the nominal parameters; ΔT and ΔD are the differences between the actual and nominal values of thrust and drag; d₁ is the resulting disturbance with the expression of $d_{1} = (Δ T \cos α - Δ D) / m$ . Additionally, we also suppose that the 30% decrease of lift occurs at 100 s of the simulation, which can be regarded as the mismatched disturbances applied on the nominal system.

Simulation results are shown in Figures 1 –8. We can see that our proposed controller (referred to as ‘NDOBC’ in Figures 1 –4) can well track the references. However, the so-called baseline controller $u = A^{- 1} [- b (x) + v]$ , which does not have the mechanism of disturbance compensation (referred to as ‘baseline control’ in Figures 1 –4), cannot track the references. At 100 s when the decrease of lift occurs, the baseline controller cannot reject the external disturbances effectively, which is clearly seen from Figures 1 and 3. Control inputs of elevator deflection and throttle setting are given in Figures 5 and 6. It is shown that the control inputs can implement quickly in the presence of external disturbance, so as to ensure the good performance. System states of angle of attack and pitch rate are given in Figures 7 and 8, respectively, from which we can see these flight states are within the admissible ranges.

Figure 1.
Velocity tracking.

Figure 2.
Velocity tracking error.

Figure 3.
Altitude tracking.

Figure 4.
Altitude tracking error.

Figure 5.
Elevator defection.

Figure 6.
Command of throttle setting.

Figure 7.
Angle of attack.

Figure 8.
Pitch rate.

Conclusion

A tracking controller based on DOBC is proposed for a class of MIMO systems with mismatched disturbances. The nonlinear disturbance observer can estimate the mismatched disturbances effectively without adding extra complexity to the controller. The proposed controller inherits the merit of DOBC, that is, the ability to maintain the nominal performance and strong robustness. As one highlight of our work, the proposed controller has a simpler structure compared with the resulting papers. The compensation gain matrix reduces to be constant by the proper observer design. An application to the HFV shows the ability of this method.

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) received no financial support for the research, authorship, and/or publication of this article.

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Disturbance observer-based control for nonlinear systems subject to mismatched disturbances with application to hypersonic flight vehicles

Abstract

Keywords

Introduction

Problem formulation and preliminaries

Problem formulation

Assumption 1

Assumption 2

Assumption 3

Assumption 4

The nonlinear disturbance observer

Lemma 1 (Chen and Guo 36 )

The tracking control design

Theorem 1

Proof

Remark 1

Remark 2

Application to the control of HFVs

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References

Lemma 1 (Chen and Guo³⁶)