Generation of Dangerous Disturbances for Flight Systems

Abstract

This paper addresses the generation of dangerous disturbances in problems of aircraft control. The method is based on the construction of repulsive disturbances in linear differential games using a dynamic programming approach. The computation involves solving a large number of linear programs. For this purpose, a fast algorithm efficiently working in low dimensions is proposed. Several examples of constructing dangerous disturbances are given. First, a simple linear differential game is considered to demonstrate the main features of the method. Second, a linearized model of the longitudinal motion of an aircraft is considered. The model includes flight and servomechanism dynamics, pilot and disturbance inputs, and a controller supporting the piloting process. This model is investigated in two setups: as a control law clearance problem and as a conflict control task. Third, a common nonlinear model of aircraft dynamics with reference to flight in a vertical plane is considered, and the problem of takeoff in wind shear conditions is addressed. For all above-listed problems, simulations showing the efficiency of the constructed disturbances are presented.

Appendix: Feedback Controls for Linear Problems

This section briefly outlines a method of constructing feedback controls for linear problems of the following form:

$$\begin{aligned} \dot{x} = A\,x + u + v,\ t \in [0,\theta ],\ x(\theta ) \in \mathcal {M} \subset \mathbb {R}^d, \end{aligned}$$

(42)

Here, $\theta$ is a terminal time instant, $\mathcal {M}$ a target set, and u and v are control and disturbance variables, respectively, constrained as follows: $u \in \mathcal {R} \subset \mathbb {R}^d,\ v \in \mathcal {Q} \subset \mathbb {R}^d$. The problem is considered for $t \in [0,\theta ]$.

Problem 1

Find a solvability tube $\mathcal {U} \subset [0;\theta ] \times \mathbb {R}^d,~ \mathcal {U}(\theta ) = \mathcal {M},$ with the following property: there exists a feedback control u(t, x) that provides the condition: If $(t_0,x(t_0)) \in \mathcal {U}$, then $(t,x(t)) \in \mathcal {U},\ t \in [t_0,\theta ]$, for all possible disturbances.

Remark 12

Below, it is shown that the computation of solvability tubes is closely related to finding feedback controls mentioned in the formulation of Problem 1.

The technique discussed here is based on a parallelotope approximation of solvability tubes. This method is developed by Kostousova, see a detailed description in [11]. A parallelotope is defined as: $\mathcal {P} [p,P] := \{ x \in \mathbb {R}^d | x = p + P\,\varepsilon ,\ ||\varepsilon ||_{\infty } \le 1 \},$ where $p \in \mathbb {R}^d$ and $P \in \mathbb {R}^{d \times \hat{d}},\ \hat{d} \le d$. To apply the method, assume that the following problem data are of the parallelotope type:

$$\begin{aligned} \mathcal {M}= & {} \mathcal {P} [p_f,P_f],\ P_f \in \mathcal {R}^{n \times n},\ {\text {det}}\,P_f \ne 0, \\ \mathcal {R}= & {} \mathcal {P} [r,R],\ R \in \mathbb {R}^{d \times d_1 },\quad \mathcal {Q} = \mathcal {P} [q,Q],\ Q \in \mathbb {R}^{d \times d_2 }. \end{aligned}$$

It is proven in [11] that the following ODE system defines a solvability tube $\mathcal {P}(t) = \mathcal {P}[p(t),P(t)]$ :

$$\begin{aligned}&\frac{{\text {d}}p}{{\text {d}}t} = A\,p + r + q,\quad p(\theta ) = p_f, \end{aligned}$$

(43)

$$\begin{aligned}&\frac{{\text {d}}P}{{\text {d}}t} = A\,P + R\,\varGamma (t) + P\, \text {diag}\,\gamma (t,P),\quad P(\theta ) = P_f, \end{aligned}$$

(44)

$$\begin{aligned}&\gamma = \text {Abs}\big (P^{-1}\, Q\big )\, e,\ \text {where } (\text {Abs}(G))_{ij} = |G_{ij}|,\ e = (1,1,\ldots ,1)^{\prime } \in \mathbb {R}^{d}, \end{aligned}$$

(45)

$$\begin{aligned}&\qquad \quad \varGamma (t) \in \mathbb {R}^{d_1 \times d},\ \text {max}_{1 \le i \le d_1} \sum \limits _{j=1}^d |\varGamma _{ij}| \le 1. \end{aligned}$$

(46)

In (45) and (46), the matrices diag$\,\gamma$ and $\varGamma$, respectively, define the influence of the disturbance and control resources on the size of solvability tube. Note that the solvability tube depends on the concrete choices of the matrix $\varGamma$ satisfying constraints (46).

The feedback control appearing in the statement of Problem 1 is defined as follows:

$$\begin{aligned} u(t,x) = r(t) + R(t)\, \varGamma (t) \, P(t)^{-1} \, (x-p(t)) := p_u(t) + P_u(t) \, x. \end{aligned}$$

(47)

Theorem 1

(see [11] ) Let equations (43)–(44), with relations (45)–(46), be solvable on the time interval $[0,\theta ]$, and $det\,P(t) \ne 0,~ t \in [0,\theta ]$, then the tube $\mathcal {P}(\cdot )$ and the control strategy (47) provide a solution to Problem 1.

Remark 13

For the numerical integration of matrix Eq. (44), a time sampling $\tau _k = k\, \varDelta \tau$ is used. Then, the matrices $\varGamma (\tau _k)$ can be chosen as minimizers of the following linear programs:

$$\begin{aligned} \text {min}\ tr(P(\tau _k)^{-1} \, R(\tau _k) \, \varGamma ),\ \text {where } \sum \limits _{j=1}^n|\varGamma _{ij}| \le 1. \end{aligned}$$

(48)

As discussed in [11], such a heuristic choice locally maximizes the volume growth rate of the solvability tube in backward time.