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Die Arbeit widmet sich der Konstruktion gefährlicher Störungen linearer Konfliktkontrollprobleme. Mit Hilfe der Technik der sequentiellen Linearisierung können gefährliche Störungen auch für nichtlineare Systeme wie Flugzeugdynamikgleichungen konstruiert werden, einschließlich Filter, Servomechanismen usw. Das vorgeschlagene Verfahren basiert auf einer dynamischen Programmiermethode und besteht in der Rückwärtsintegration gewöhnlicher Matrix-Differentialgleichungen, die Zentren, Größen und Orientierungen zeitabhängiger Parallelotope definieren, die eine Abstoßungsröhre im Raum-Zeit-Bereich bilden. Eine Rückkopplungsstörungsstrategie kann den Zustandsvektor des Konfliktkontrollsystems außerhalb der Abstoßungsröhre für alle zulässigen Eingänge der Steuerung halten.
Mit KI übersetzt
Abstract
The paper is devoted to the construction of dangerous disturbances in linear conflict control problems. Using the technique of sequential linearization, dangerous disturbances can also be constructed for nonlinear systems such as aircraft dynamics equations, including filters, servomechanisms, etc. The procedure proposed is based on a dynamic programming method and consists in the backward integration of ordinary matrix differential equations defining centers, sizes, and orientations of time-dependent parallelotopes forming a repulsive tube in the time-space domain. A feedback disturbance strategy can keep the state vector of the conflict control system outside the repulsive tube for all admissible inputs of the control.
The original version of this chapter was revised: This chapter has been changed to open access under a CC BY 4.0 license. The correction to this chapter is available at https://doi.org/10.1007/978-3-030-56534-3_14
1 Introduction
One of the important problems in control engineering is generation of extremal disturbances for various types of dynamical systems. This is of interest in many application areas because such disturbances can be used to evaluate the robustness of models and quality of controllers.
This paper concerns with generation of feedback disturbances for linear conflict control systems where the aim of the disturbance is to deflect the state vector from a target set at a fixed termination time for all admissible controls. It is assumed that the target set and the constraints imposed on the control and disturbance variables are represented by parallelotopes. Starting with the parallelotope representing the target set and integrating backward in time a system of ordinary vector-matrix differential equations yield parallelotopes forming a repulsive tube in the time-space domain. It is proven that a certain feedback disturbance can keep all trajectories outside the repulsive tube, and therefore outside the target set at the termination time.
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It should be noted that the minimal repulsive tube can be computed using general grid methods for solving differential games [3, 4, 8]. Nevertheless, such methods require large computation resources on multiprocessor computer platforms. More appropriate for linear conflict control problems are methods proposed in [5, 12] where repulsive tubes are approximated by polyhedrons, which however involves solving a lot of linear programming problems. Therefore, such methods also require significant computer resources. In contrast, the scheme suggested in the current paper is computationally cheap so that it can run in real time on a common computer. Moreover, high-dimensional models can be effectively treated with this method. Finally, disturbances for nonlinear models can be constructed by applying techniques of sequential linearization. Thus, the approach presented in this paper is rather general and can be used in various areas. As a demonstration of the method, generation of dangerous disturbances for aircraft control problems is considered.
The paper is structured as follows: In Sect. 2, a formal statement of the problem and some definitions are given. Section 3 contains a detailed description of the method for constructing repulsive feedback disturbances and provides a proof of their correctness. In Sect. 4, some numerical aspects of the method are addressed. It is shown that the method can be implemented in the discrete-time scheme. In Sect. 5, the method is applied to a three-dimensional linear differential game. This simple example allows us to visualize and clearly demonstrate in which extent the constructed repulsive tube is minimal. Section 6 considers the problem of aircraft take-off under windshear conditions. This example demonstrates a technique of generating dangerous disturbances for nonlinear models. Section 7 describes the construction of disturbances for a linearized aircraft closed-loop system for the lateral dynamics.
2 Problem Formulation
First, introduce the following notation. For a set \(\mathcal {V} \subset [0,\theta ] \times \mathbb {R}^d\) and \(t\in [0,\theta ]\), the set \(\mathcal {V}(t):=\{x\in \mathbb {R}^d: (t,x)\in \mathcal {V}\}\) is called cross section of \(\mathcal {V}\) at t. For a vector \(x\in \mathbb {R}^d\), the norm \(\Vert x \Vert _\infty \) is defined as \(\max \{|x_i|,i=1,...,d\}\). Let the superscript T denotes the transposition operation.
Here, u and v, respectively, denote the control and disturbance variables constrained as follows: \( u(t) \in \mathcal {R} \subset \mathbb {R}^d,\ v(t) \in \mathcal {Q} \subset \mathbb {R}^d\). The problem is considered on a time interval \([0,\theta ] \). The aim of the control is to meet the target set \(\mathcal {M}\) at the termination time \(\theta \), whereas the aim of the disturbance is opposite. The objective of this paper is to propose a method of constructing a feedback disturbance v(t, x) which deflects all trajectories from the target set at the termination time. More precisely, the problem is formulated as follows:
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Problem 1
Find a tube \( \mathcal {V} \subset [0,\theta ] \times \mathbb {R}^d,\, \mathcal {V}(\theta ) = \mathcal {M}\) such that there exists a feedback disturbance v(t, x) fulfilling the following condition: If \((t_0,x(t_0)) \not \in int(\mathcal {V})\), then \((t,x(t)) \not \in int(\mathcal {V}),\ t \in [t_0,\theta ]\), for all possible controls.
Remark 1
In what follows, \(\mathcal {V}\) and v(t, x) from the formulation of Problem 1 are called repulsive tube and repulsive disturbance, respectively. It will be shown below that the knowledge of a repulsive tube allows us to find explicitly a repulsive disturbance appearing in the formulation of Problem 1.
The main property of repulsive tubes is illustrated in Fig. 1.
Fig. 1
Repulsive tube \(\mathcal {V}\) with a sample trajectory x(t)
This section describes the computation of time-dependent parallelotopes that form a repulsive tube in \([0,\theta ]\times \mathbb {R}^d\) and define a repulsive feedback disturbance. This approach raises from the idea by E. K. Kostousova to use parallelotopes for constructing feedback controls, see a detailed description in [7].
A parallelotope is defined as
$$\begin{aligned} \mathcal {V_P} [p,P] := \{ x \in \mathbb {R}^d | x = p + P\,\varepsilon ,\ \Vert \varepsilon \Vert _{\infty } \le 1 \}, \end{aligned}$$
(2)
where \(p \in \mathbb {R}^d \) and \(P \in \mathbb {R}^{d \times \hat{d}},\ \hat{d} \le d \), are its center and shape matrix, respectively. Note that \(\hat{d}=d\) in our consideration. The columns of the matrix P are called axes of the parallelotope \(\mathcal {V_P}\) and denoted as \(p^1,...,p^{\hat{d}}\in \mathbb {R}^d\). Furthermore, let \(h_i(\mathcal {V_P})\) be the euclidean distance between two opposite faces of \(\mathcal {V_P}\) along the axis \(p^{i}\), and \(h_{min}(\mathcal {V_P}) = \text {min} \{ h_i (\mathcal {V_P})\ |\, 1 \le i \le \hat{d}\}\). Figure 2 shows \(p^i\) and \(h_i\) for a two-dimensional parallelotope.
Fig. 2
Two-dimensional parallelotope \(\mathcal {V_P}\) with the axes \(p^1,\ p^2\) and the corresponding distance \(h_1,\ h_2\)
The system matrix A as well as the constraints on the control and disturbance inputs may depend on time. Thus, in general, \(A=A(t), \mathcal {R} = \mathcal {V_P} [r(t),R(t)]\), and \(\mathcal {Q} = \mathcal {V_P} [q(t),Q(t)]\). In the following, this time-dependence is not shown explicitly in order to simplify the notation.
Remark 3
Parallelotope-shaped representation of the control and disturbance constraints is fairly generic and allows to capture different common types of constraints. For example, consider a control \(u \in \mathbb {R}^2\) subject to
With the assumptions introduced in (3), the following system of ODEs defines a repulsive tube \(\mathcal {V_P}(t) = \mathcal {V_P}[p(t),P(t)]\), \(t \in [0,\theta ]\) :
In (6) and (7), the matrices diag\(\, \beta \) and \(\varGamma \), respectively, represent the influence of the control and disturbance capacities on the repulsive tube. Note that the time evolution of the matrix \(\varGamma \), satisfying the condition (7), should be chosen in such a way that the repulsive tube maximally decreases backward in time. Below, this principle will be discussed more exactly.
A repulsive feedback disturbance appearing in the statement of Problem 1 may be defined as follows:
Let Eqs. (4)–(5), with relations (6)–(7), be solvable on \([0,\theta ]\), and \(\det \big (P(t)\big ) \ne 0,~ t \in [0,\theta ]\), then the tube \(\mathcal {V_P}(\cdot )\) and the disturbance strategy (8) provide a solution to Problem 1.
Proof
Observe that the condition \(\det \big (P(t)\big ) \ne 0,~ t \in [0,\theta ]\), define the vector function
and note that the vector \(\xi (t,x) \in \mathbb {R}^d\) defines relative coordinates of any point x in the parallelotope \(\mathcal {V_P}(t)\). It is easily seen that a point x lies outside the interior of the parallelotope \(\mathcal {V_P}(t)\) whenever \(\Vert \xi (t,x) \Vert _\infty \ge 1\).
Let \(x(\cdot )\) be a trajectory of (1) corresponding to the disturbance (8) and starting from a position \((t_0,x_0)\) such that \(\Vert \xi (t_0,x_0) \Vert _\infty \ge 1\). Denote \(K(t):=\text{ cl } \Big (\mathbb {R}^d\setminus \mathcal {V_P}(t)\Big )\) and prove that \(x(t) \in K(t),~ t \in [t_0,\theta ].\) Bearing in mind that \(\Vert \xi (t,x) \Vert _\infty = \max \limits _{j\in \overline{1,d}} |\xi _j(t,x)|\) introduce the functions
According to [1, Theorem 11.1.3], the condition \((1,\dot{x}(t)) \in T_K\big (t,x(t)\big )\), \(t \in [t_0, \theta )\), guarantees the inclusion \(x(t) \in K(t),\, t \in [t_0,\theta ]\). Let us prove the validity of that condition.
If \(\Vert \xi (t,x(t)) \Vert _\infty > 1\), one of the first two relations of (11) holds for some index \(j\in J\big (t,x(t)\big )\), which provides the desired result due to (10).
The “boundary” case, \(\Vert \xi (t,x(t)) \Vert _\infty = 1\), is being treated as follows: Obviously, there exists an index \(j_0\in J(t,x(t))\) such that the third relation of (11) holds. Assume that \(j_0\in \overline{1,d}\) (the case \(j_0\in \overline{d+1,2d}\) is considered analogously). The full time derivative of the vector function \(\xi \big (t,x(t)\big )\) reads
if formulas (1), (4), (5), and the definition of \(\xi \) are used. Note that every admissible control u satisfies the relation \(u - r = R \alpha \) at time t, where \(\alpha \) is a vector such that \(\Vert \alpha \Vert _{\infty } \le 1\). Additionally, using (8) yields
which implies that \((1,\dot{x}(t)) \in T_K(t,x(t))\) according to (10) and (11). Thus, in all cases, \((1,\dot{x}(t)) \in T_K(t,x(t)),\, t\in [t_0,\theta )\), and therefore, \(x(t) \in K(t),\, t\in [t_0,\theta ]\), because of the continuity of x(t) and K(t). Finally, since \(K(t) \cap int\big (\mathcal {V_P}(t)\big ) = \emptyset \), the condition \(x(t) \notin int\big (\mathcal {V_P}(t)\big ), t \in [t_0, \theta ]\), holds.
Remark 4
Note that the repulsive tube \(\mathcal {V_P}\) can degenerate so that \(\det \big (P(\hat{t})\big ) = 0\) for some \(\hat{t} \in [0,\theta )\), and \(P(\hat{t})\) is no longer invertible. In this case, the tube \(\mathcal {V_P}\) is constructed only on \([\hat{t},\theta ]\), and the disturbance may be set as \(v(t) \equiv q,\ t \le \hat{t}\). Obviously, \(x(\hat{t})\not \in \mathcal {V_P}(\hat{t})\), and the rule (8) can be used for \(t > \hat{t}\).
As it was mentioned after formula (7), the choice of \(\varGamma \) is crucial for obtaining a possibly smaller repulsive tube, which allows for the application of (8) to a possibly larger set of initial conditions. The following choice is used in the numerical simulations in Sects. 5–7: The whole time interval \([0,\theta ]\) is divided into subintervals \((\tau _i,\tau _{i+1}],\, i = 0,...,N\), with \(\tau _0 = 0\) and \(\tau _N=\theta \). The system (4)–(5) is then integrated backward in time from \(\theta \) to 0, and a constant matrix \(\varGamma _k\) satisfying (7) is chosen for each subinterval \((\tau _{k-1},\tau _k]\) to minimize the minimum distance between the opposite faces of \(\mathcal {V_P}(\tau _{k-1})\). Intuitively, such a choice of \(\varGamma \) yields the strongest contraction of the parallelotope tube along the direction of its shortest axis.
Note that the resulting \(\varGamma \) may be discontinuous at time instants \(\tau _i\). However, the number of discontinuities is finite, and solutions of (4)–(5) remain continuous and unique.
4 Numerical Implementation of Repulsive Feedback Disturbances
The proof of appropriateness of the repulsive disturbance (8) is done in Sect. 3 under the assumption of continuous-time scheme. In a discrete-time scheme, the feedback repulsive disturbance (8) may not properly work because the condition (12) holds only on the boundary of \(\mathcal {V_P}\). In this section, an extended discrete-time control scheme is presented, and a bound on the time step length of this procedure is evaluated.
Assume for simplicity that the discrete-time scheme involves equidistant time instants \(t_i\) corresponding to the step length \(\varDelta t\). As it was declared in the introduction, the disturbance is basically associated with wind, and the maximum expected wind speed can hardly be exactly predicted. Therefore, the extension of disturbance bounds along all parallelotope axes by the factor \(1+\delta \), where \(\delta >0\) is a small parameter, is not prohibited. Thus, it is now assumed that \(v \in \mathcal {V_P}[q,(1+\delta )\,Q],\) and the repulsive disturbance v(t, x) is computed by the formula
if the matrix \(\varGamma \) is constant on each interval \([t_i,t_{i+1})\). Let \(x(\cdot )\) be a trajectory started from a position \((t_0,x_0)\) such that \(\Vert \xi (t_0,x_0) \Vert _\infty \ge 1+\delta \) and computed in the continuous-time scheme using the disturbance (14). The same argumentation as in the proof of Theorem 1 implies that \(\Vert \xi (t,x(t)) \Vert _\infty \ge 1+\delta ,~t\in [t_0,\theta ]\).
Let \(x_\varDelta (\cdot )\) be the corresponding trajectory (the same control \(u(\cdot )\) and the same initial position \((t_0,x_0)\)) computed in the discrete-time scheme using the disturbance (14). In virtue of condition (15), it is possible to prove that
and therefore, \(\Vert \xi (t,x_\varDelta (t)) \Vert _\infty \ge 1+\delta - M G \varDelta t,~t\in [t_0,\theta ]\), where M is the Lipschitz constant of the function \(\Vert \xi (t,x) \Vert _\infty \) in x. It remains to set \(\varDelta t \le \delta /(M G)\).
Remark 5
The theoretical bound on the step size \(\varDelta t\) may be too small. However, for simulations presented in the following sections, it is possible to maintain the property \(\Vert \xi (t,x_\varDelta (t)) \Vert _\infty \ge 1, ~t\in [t_0,\theta ],\) for much larger time steps.
Finally, note that for any given problem dimension d (i.e., the state \(x \in \mathbb {R}^d\)), the computational complexity of the proposed scheme is \(O(d^3)\) per time step \(\varDelta t\) as it involves matrix equations of dimension d, which can be solved with, e.g., LU-decomposition. Even for fairly low-dimensional problems, this dependency is far superior to complexity of other common methods for construction of disturbances, such as
grid methods, e.g., [3], that scale as \(O(N^d)\) per time step, where N is the grid resolution per dimension,
methods that represent repulsive tubes with arbitrary convex polygons, e.g., [5], that scale as O(d!m) per time step, where m is the number of inequalities describing the polygon.
Clearly, the difference in complexity between these methods and the presented approach quickly grows with the increasing problem dimension. Thus, the presented method allows us to consider problems that would not be accessible with many other techniques.
5 Application: Simple Example
In this section, the techniques developed in Sects. 3 and 4 are applied to compute a repulsive disturbance in a linear three-dimensional differential game. This example is appropriate to visualize repulsive tubes and demonstrate the proper work of repulsive disturbances.
The repulsive sets \(\mathcal {V_P}[p(t_i),P(t_i)]\) are constructed on the uniform time grid \(\{t_i=i \varDelta t\}\) with \(\varDelta t = 10^{-3}\). The same time sampling is used in the forward integration of the system including the repulsive disturbance (14).
It follows from the general theory of differential games (see [8]) that, in particular, for linear problems there exists a minimal repulsive set \(\mathcal {V}_0 \subset [0,\theta ] \times \mathbb {R}^d\). This set is also the maximal solvability set and, therefore, it has the following property. If \((t_0,x(t_0)) \notin \mathcal {V}_0\) then there exists a feedback disturbance v(t, x) that prevents any trajectory \(x(\cdot )\) from the penetration into \(\mathcal {V}_0\). In the opposite case, there exists a feedback control u(t, x) ensuring the condition \((t,x(t)) \in \mathcal {V}_0,\ t \in [t_0,\theta ],\) for all trajectories. This alternative is sketched in Fig. 3.
For low-dimensional problems, \(\mathcal {V}_0\) can be approximated using grid methods (see, for example, [3] and [4]). In the following simulation, such a grid scheme is used to approximate the cross sections \(\mathcal {V}_0(t_i)\) for all time instants \(t_i=i\varDelta t\). For each current state \(x(t_i) \in \mathcal {V}_0(t_i)\) it is possible to compute a control \(u(t_i,x(t_i))\) which pushes the state vector into the next cross section \(\mathcal {V}_0(t_{i+1})\) so that the feedback control \(u(t_i,x(t_i))\) can approximately keep (in the discrete-time scheme) all trajectory inside \(\mathcal {V}_0\) if the initial state lies there. This control is used to implement the strategy of the first player in the simulation.
To test the constructed repulsive disturbance, twenty-five initial conditions were generated in the proximity of origin but outside of \(\mathcal {V_P}[p(0),P(0)]\). Resulting trajectories as well as cross sections of the repulsive tubes \(\mathcal {V_P}\) and \(\mathcal {V}_0\) are shown in Figs. 4–6. The results are consistent with the theory: \(\mathcal {V}_0(t_i) \subset \mathcal {V_P}(t_i)\) for all \(t_i\), and none of the trajectories penetrates into the tube \(\mathcal {V_P}\). Furthermore, one can see that the parallelotope tube \(\mathcal {V_P}\) provides a rather good upper estimate of the minimal repulsive tube \(\mathcal {V}_0\) along the shortest axis of the parallelotope. This is in agreement with the previously discussed choice of the matrix \(\varGamma \) involved in the construction of \(\mathcal {V_P}\).
Remark 6
Note that the view direction in Figs. 4, 5, and 6 is always chosen orthogonal to the minimum width face of \(\mathcal {V_P}(t_i)\). Therefore, the view direction is rotating together with the tube \(\mathcal {V_P}\). In this way, it is possible to visually demonstrate that all trajectories remain outside of \(\mathcal {V_P}\) throughout the whole simulation.
Fig. 4
The sets \(\mathcal {V_P}(t)\) (green) and \(\mathcal {V}_0(t)\) (red) as well as the current state vectors (for various initial conditions) at \(t = 0.0\) (left) and \(t=0.2\) (right)
The sets \(\mathcal {V_P}(t)\) (green) and \(\mathcal {V}_0(t)\) (red) as well as the current state vectors (for various initial conditions) at \(t = 0.4\) (left) and \(t=0.6\) (right)
The sets \(\mathcal {V_P}(t)\) (green) and \(\mathcal {V}_0(t)\) (red) as well as the current state vectors (for various initial conditions) at \(t = 0.8\) (left) and \(t=1.0\) (right)
In the following sections, the construction of a repulsive disturbance in a nonlinear model of aircraft take-off is presented. The model has already been considered in several papers devoted to aircraft control (cf. [10, 11]). In contrast to the mentioned works, the problem of finding a dangerous wind disturbance is now considered. More precisely, it is necessary to find a wind disturbance that maximizes the deviation of aerodynamic velocity and kinematic path inclination angle from their reference values.
6.1 Model Equations
A simplified aircraft model is under consideration.
First, the motion in a vertical plane is assumed. Second, the rigid body rotations are neglected to obtain a point-mass model. Third, the thrust force of the engine is kept constant.
The following notation is used:
V\({\mathop {=}\limits ^{\rm {\tiny def}}}\) aerodynamic velocity of the aircraft, [m/s];
Here, the angle of attack, \(\alpha \), is the single control input governed by the pilot. The coefficients \(A_i,~ i=0,1,2,\) depend on the altitude and air temperature, whereas \(B_i\) and \(C_i,~i=0,1,2,\) are influenced by the position of flaps and chassis. Finally, \(m,\ S,\ \rho ,\ \delta ,\ \alpha _{**},\ A_i,\ B_i\), and \(C_i\) are constant parameters corresponding to Boeing-727 on take-off. The exact values of them can be found in [10].
The dynamics (16)–(19) is considered on the time interval \([0,\theta ]\) with \(\theta = 14\) s, and appropriate initial conditions are chosen.
The target set \(\mathcal {M}\) is defined by maximum permissible deviation of V and \(\gamma \) from their reference values \(V_0\) and \(\gamma _0\) at \(t=\theta \). That is,
The reference values \(V_0\) and \(\gamma _0\) will be discussed below in more detail.
6.2 Relaxed Nonlinear Model
It can be observed that the right-hand sides of Eqs. (16), (17) do not depend on x and h. Therefore, these state variables and the corresponding Eqs. (18), (19) will be excluded from the consideration, keeping in mind that x(t) and h(t) can be reconstructed from V(t) and \(\gamma (t)\).
Moreover, jumps in the wind velocity components will be smoothed using first-order filters defined by PT1 transfer functions, which assumes the introduction of artificial disturbances \(v_1\) and \(v_2\), the inputs of these filters.
Thus, similar to [11], we arrive at the following nonlinear model:
$$\begin{aligned}&m\dot{V} = P cos(\alpha + \sigma ) - D - mg\, \sin \gamma - m \dot{W_x} \cos \gamma - m \dot{W_h} \sin \gamma , \end{aligned}$$
(22)
$$\begin{aligned}&mV\dot{\gamma } = P \sin (\alpha + \sigma ) + L - mg\, \cos \gamma + m \dot{W_x} \sin \gamma - m \dot{W_h} \cos \gamma , \end{aligned}$$
Here, the coefficient \(k = 0.5\,\text {s}^{-1}\) defines the smoothing rate of the wind velocity components. The time derivatives \(\dot{W}_x\) and \(\dot{W}_h\) in (22), (23) are assumed to be replaced by the right-hand sides of (24), (25). The constraints on the artificial disturbances, \(v_1\) and \(v_2\), are chosen as follows:
Note that any wind disturbance designed for the relaxed system (22)–(25) produces, using (24) and (25), the same performance of V and \(\gamma \) in the original system (16)–(19). Therefore, repulsive disturbances will be designed for the relaxed system.
6.3 Linearization of the Relaxed Model
The relaxed system (22)–(25) is linearized around the reference values (cf. [11]) \(V=V_0= 84.1\) m/s, \(\gamma =\gamma _0=6.989^{\circ }\), \(\alpha =\alpha _0= 10.367^{\circ }\), \(W_x=W_{x0} = 0\), \(W_h=W_{h0} = 0\), \(v_1=0\), and \(v_2=0\). Here, the values of \(V_0, \gamma _0\), and \(\alpha _0\) are chosen such that the right-hand sides of (22) and (23) are equal to zero. Note that the above reference values define a straight ascending trajectory. Such a line would be a perfect take-off path in the absence of wind disturbances. Denote \(x_{ref} := (V_0,\ \gamma _0,\ W_{x0},\ W_{h0})^T\) and \(u_{ref}:=\alpha _0\).
6.4 Linear Conflict Control Problem
Having chosen the reference values, the linearization of the relaxed model yields the following linear conflict control problem (cf. [11]):
$$\begin{aligned} \dot{x} =A (x-x_{ref})&+ B (u-u_{ref}) + C v,\ \text { for } t \in [0,\theta ], \end{aligned}$$
All partial derivatives are computed at \(x_{ref}\), \(u_{ref}\), and \(v=(0,0)^T.\) Note that the state vector, control parameter, and disturbance inputs are the same as in the nonlinear relaxed model (22)–(25). Therefore, the target set \(\mathcal {M}\) and the constraints on the control and disturbance inputs remain the same as in the nonlinear relaxed model.
Remark 8
The system (28)–(30) can be reduced to the form (1) by setting \(\overline{x} := x-x_{ref},\ \overline{u} := B (u-u_{ref})\) and \(\overline{v} := C v\). Obviously, the new target set \(\overline{\mathcal {M}}\) and the constraints on the new control \(\overline{u}\) and disturbance vector \(\overline{v}\) are of the parallelotope type so that the new system satisfies the requirements of Sect. 3.
6.5 Generation of Disturbances
To construct a repulsive disturbance for the relaxed nonlinear model (22)–(25), a parallelotope tube \(\mathcal {V_P}\) is constructed for the linearized problem (28)–(30). More precisely, the cross sections \(\mathcal {V_P}(t_i)=\mathcal {V_P}[p(t_i),P(t_i)]\) are computed for a time sampling. The disturbance in the relaxed nonlinear model at each time instant \(t_i\) is being chosen according to (8) based on the cross section \(\mathcal {V_P}(t_i)\).
It should be noted that the condition \(x(0) \notin \mathcal {V_P}[p(0),P(0)]\) is required for the application of the feedback rule (8). To satisfy this condition, a scheme with multiple target sets \(\overline{\mathcal {M}}^\mu \) can be used. Here, \(\mu \in \mathbb {R}^+\) is a scaling factor applied to the original target set \(\overline{\mathcal {M}} = \mathcal {V_P}[p_f,P_f]\). Therefore,
Further, a set of scaling factors \(\mu _1<\mu _2< ... <\mu _M\) is chosen, and multiple target sets \(\overline{\mathcal {M}}^{\mu _1} , ... , \overline{\mathcal {M}}^{\mu _M}\) are defined according to formula (31). For each \(\overline{\mathcal {M}}^{\mu _s}, s\in \overline{1,M}\), the corresponding parallelotope repulsive tube \(\mathcal {V_P}^{\mu _s}\) is constructed. At the current position \((t_i,x(t_i))\) an index \(s\in \overline{1,M}\) is chosen in such a way that \(x(t_i) \notin \mathcal {V_P}^{\mu _s}(t_i)\) and \(x(t_i) \in \mathcal {V_P}^{\mu _{s+1}}(t_i)\). The repulsive disturbance is computed according to formula (8), based on \(\mathcal {V_P}^{\mu _s}(t_i) = [p(t_i),\ P^{\mu _s}(t_i)]\) .
Remark 9
It is clear that \(\mathcal {V_P}^{\mu _k}(t_i) \subset \mathcal {V_P}^{\mu _s}(t_i)\) whenever \(\mu _k < \mu _s\). Therefore, for the linearized system (28)–(30), the repulsive property guarantees that the trajectory does not penetrate into the sets \(\mathcal {V_P}^{\mu _k}\) with \(\mu _k \le \mu _s\) in future time steps. On the other hand, if the control (pilot) plays nonoptimally, the disturbance can achieve that \(x(t_r) \notin \mathcal {V_P}^{\mu _j}(t_r)\) with \(\mu _j > \mu _s\) at some \(t_r > t_i\). In such a case, the repulsive cross section \(\mathcal {V_P}^{\mu _j}(t_r)\) should be used at \(t_r\) to increase the deviation of the trajectory from the reference path.
The simulation results for the nonlinear relaxed model (22)–(25) with constraints on the disturbance and control given by (26) and (27) are shown in Figs. 7, 8, 9. Multiple target sets \(\overline{\mathcal {M}}^{\mu _s}, s\in \overline{1,25}\), with \(\mu _s\) uniformly distributed in the interval [0.04; 1], are used. The right-hand sides of inequalities (20) and (21) are chosen as \(\varDelta V = 15.2\, {\text {m/s}}\) and \(\varDelta \gamma =5^{\circ }\), respectively. The repulsive tubes are constructed with the uniform time sampling \(t_{i+1}-t_i = 10^{-3}{\text {s}}\). To play against the repulsive disturbance, a quasi-optimal feedback control strategy u(t, x) based on parallelotope approximations of solvability tubes (see [7]) is used. Such a strategy has already been successfully applied to problems of aircraft control (see [9]).
Fig. 7
Left: Aerodynamic velocity V of the aircraft and the reference value \(V_0\) (thin horizontal line). Right: Kinematic path inclination angle \(\gamma \) and the reference value \(\gamma _0\) (thin horizontal line). The vertical lines at \(t = 14\) s show the corresponding projections of the target set
Left: Horizontal wind velocity \(W_x\) along the trajectory (yielded by the disturbance command \(v_1\)). Right: Vertical wind velocity \(W_h\) along the trajectory (yielded by the disturbance command \(v_2\))
Simulation results show that the repulsive disturbance provides evasion from the target set, whereas constant disturbances whose values coincide with the vertices of the rectangle given by (26) cannot solve this problem. Figure 10 shows the comparison between the repulsive disturbance and the strongest constant disturbance, \(v_1 \equiv -13.7\) m/s and \(v_2 \equiv 5.5\) m/s, providing the largest deviation among all constant disturbances.
Fig. 10
Left: Aerodynamic velocity V for the repulsive (solid) and optimal constant (dashed) disturbances. Right: Kinematic path inclination angle \(\gamma \) for the repulsive (solid) and optimal constant (dashed) disturbances. The vertical lines at \(t = 14\) s show the corresponding projections of the target set. The thin horizontal lines depict the reference values \(V_0\) and \(\gamma _0\)
7 Application: Linear Model of Aircraft Lateral Dynamics
In this section, a repulsive disturbance for a linearized aircraft closed-loop dynamics of lateral motion (see [6]) is constructed. Such a model is derived under the assumption of horizontal balanced flight, which results in decoupling the longitudinal and lateral motions after the linearization.
7.1 Model Equations
The rigid body states for the linearized model of lateral motion are the yaw rate r, roll rate p, side-slip angle \(\beta \), and roll angle \(\varPhi \). Furthermore, second-order transfer functions of the form
with natural frequency \(\omega _0\) and damping constant d are employed to model the actuator dynamics of the primary control surfaces in the lateral plane. This results in additional states for the aileron position \(\xi _{pos}\) and angular rate \(\xi _{vel}\), as well as the rudder position \(\zeta _{pos}\) and angular rate \(\zeta _{vel}\). Moreover, a wind disturbance \(V_{W,cmd}\) is introduced by using the following first-order lag filter
with \(\tau _W=2\), which produces smooth wind profiles for the wind state \(V_W\). Besides this wind disturbance, we additionally consider worst case pilot commands as disturbance inputs, which are the side load factor command \(\delta _{n_y}\) and the roll angle command \(\delta _\varPhi \). As the control structure under investigation features a proportional and integral part for both the roll angle command and the side load force command, we also include the corresponding states of the integral parts denoted by \(e_\varPhi \) and \(e_{n_y}\) as states. Summarizing, the state vector for the linear system
$$\begin{aligned} \dot{x} = A x + C v, \quad \text {with } x(0) = 0 \end{aligned}$$
(34)
comprises nine states, \(x = \left[ e_\varPhi ,e_{n_y},r,\beta ,p,\varPhi ,\xi _{pos},\xi _{vel},\zeta _{pos},\zeta _{vel}\right] ^T\), and the disturbance vector includes three components, \(v=\left[ \delta _{n_y},\delta _\varPhi ,V_{W,cmd}\right] ^T\), for the pilot and wind disturbance commands. These components are constrained as follows:
In (34), the first two components of the state vector x stands for the integrated errors. Therefore, the aim of the disturbance is to maximize the functional \(|x_1(\theta )| + |x_2(\theta )|\). This objective is associated with two-dimensional parallelotope target sets
defined for different positive values of the parameter c.
Note that the approach of Sect. 3 requires the full dimensionality of the target set, i.e., it should involve all components of the state vector of system (34). In order to remain in two dimensions, equations (34) will be transformed using the following substitution:
The properties of X imply that \(y(\theta ) = x(\theta )\), and therefore, only the two first equations of (40) and the two-dimensional target sets \(\mathcal {M}^c\) defined by (36) should be used. Similar to Sect. 6.5, a repulsive disturbance will be constructed using the technique of multiple target sets obtained by varying the parameter c in (36).
7.3 Validation Using Optimal Control Theory
It is interesting to compare the result obtained using the repulsive disturbance with that gained from solving an appropriate optimal control problem. In this comparison, the criterion to be maximized is the Mayer cost function \(J_M = x_1(\theta ) + x_2(\theta )\) which is evaluated at the fixed time instant \(\theta = 4\) s. In order to solve this optimal control problem numerically, the following trapezoidal collocation scheme, which assumes the uniformly spaced time grid with the discretization step length \(t_{i+1}-t_i = \varDelta t=0.004\) s, is used:
$$\begin{aligned} x_{i+1} = x_i + \varDelta t \cdot \frac{f(x_i,v_i)+f(x_{i+1},v_{i+1})}{2}. \end{aligned}$$
(41)
Here \(f(x,v)=A x +Cv\) according to the notation (34), and the low indices correspond to the time sampling instants, e.g., \(x_i = x(t_i)\) and \(v_i=v(t_i)\). The initial state \(x(t_0)=0\) is enforced as equality constraint at the beginning of the time interval and the final state is free. The parameter optimization problem resulting from the discretization of the continuous-time optimal control problem is solved using an interior point solver with a feasibility and optimality tolerance of \(10^{-7}\). See [6] for more details.
7.4 Simulation Results
Simulation results for the time interval \([0,\theta ],\, \theta = 4\,{\text {s}},\) are shown in Figs. 11 and 12. As discussed in Sect. 6.5, the repulsive disturbance can be compared with extreme constant disturbances. In virtue of (35), there are eight extreme points of the disturbance constraint. However, only four of them should be considered due to the symmetry of the system equations. Figure 13 presents the comparison of the extreme and repulsive disturbances. Note that the extreme disturbances perform well, but the repulsive disturbance yields a better result.
Finally, the parallelotope-based repulsive disturbance is compared with that obtained from optimal control theory (see Sect. 7.3). Theoretically, the parallelotope-based repulsive disturbance cannot outperform the optimal one. Nevertheless, the results produced by the both disturbances are very close to each other as it is shown in Fig. 14. Furthermore, Figs. 14 and 15 demonstrate that the parallelotope-based repulsive disturbance and the optimal one produce very similar input signals.
Fig. 11
Left: The absolute values sum of the error components \(e_\varPhi \) and \(e_{n_y}\) obtained with the repulsive disturbance. Right: Disturbance \(\delta _{n_y}\)
The absolute values sum of the error components \(e_\varPhi \) and \(e_{n_y}\) obtained with the repulsive disturbance (solid line) and all possible constant extreme disturbances (dashed lines)
Left: The absolute values sum of the error components \(e_\varPhi \) and \(e_{n_y}\) obtained with the repulsive disturbance (solid line) and the optimal control-based one (dashed line). Right: Disturbance \(\delta _{n_y}\), comparison of the repulsive disturbance (solid line) and the optimal control-based one (dashed line)
Left: Disturbance \(\delta _\varPhi \), comparison of the repulsive disturbance (solid line) and the optimal control-based one (dashed line). The lines coincide. Right: Disturbance \(V_{W,cmd}\), comparison of the repulsive disturbance (solid line) and the optimal control-based one (dashed line)
The results of Sects. 5–7 demonstrate that the method presented can be successfully applied to various types of control systems. In particular, promising results are obtained for a nonlinear model considered in Sect. 6 and a complex linear system treated in Sect. 7. As it is shown, the parallelotope-based repulsive disturbance is expected to provide a near-optimal result. In any case, it significantly outperforms constant extreme disturbances.
The main advantage of the method proposed is its applicability to high-dimensional conflict control problems. The computational efforts are relatively low so that the method may run in real time. Therefore, advanced aircraft models comprising numerous state variables, controllers, filters, etc. can be tested with this approach. One of the main future objectives is the implementation of the method on a real flight simulator.
Acknowledgments
This work is supported by the DFG grant TU427/2-1 and HO4190/8-1 as well as TU427/2-2 and HO4190/8-2. Computer resources for this project have been provided by the Gauss Centre for Supercomputing/Leibniz Supercomputing Centre under grant: pr74lu.
Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.
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