Dynamic modes of aircraft demonstrate transient behavior of various flight states, at certain initial conditions, under the influence of disturbances, or under control surface inputs. Compared with static stability and control, a topic covered in the last chapter, dynamic modes and responses to input provide further insight of the flight characteristics. Aircraft dynamic modes and dynamic responses are core to understand dynamic flight behavior, they are associated with the stability and control concepts we have learned in the last chapter, but provide a detailed insight how the flight states converge to their equilibrium operating point. Therefore, we have the full and complete definition of stability by the location of roots of flight dynamics characteristic equation in LHP. In addition, the representative longitudinal modes and lateral modes reveal dynamic flight performance and corresponding flight states. The impact of control surface deflection is also addressed. Flight modes and dynamic responses are based on the foundation of linear systems and feedback control theory, the focus is placed on its special features representing aircraft dynamics.

Aircraft performance addresses quantitative measurement of the flying vehicle’s capabilities, seeks its operation optimization as well as sets its boundary. In the first chapter of the performance part, a steady level flight is sustained by the power plant to generate propulsion to balance applied aerodynamic forces. We focus on the force-related performance measurements when aircraft is engaged in steady flight operation, no acceleration is involved, therefore the statics of flight serves as the governing principle for technical analysis. Aerodynamic forces of lift and drag are first introduced, followed by propulsion thrust and power required to sustain the steady level flight, depending on the engine type and property. Thrust and power available indicate the propulsion capacity. Of course, the performance focuses on the optimal flying speed to achieve best-possible performance from thrust/power perspective.

Kinematics is the “branch of classical mechanics which describes the motion of particles, bodies, and systems of bodies without consideration of the masses of those objects nor the forces that may have caused the motion,” according to the popular Wikipedia. As such, in this chapter, we will address the geometric movement with respect to the subject of aircraft. We adopt the mathematical tools of vectors and matrices to provide the systematic analysis of flight motion, coining the name of vectorial flight kinematics. Three main reference frames are built, one as the inertial reference frame, one in the body-attached format for rigid-body motion analysis, and one to follow flight path (trajectory) for performance analysis. The velocity vector covers the 3DOF translational motion, while the angular rate vector covers the other 3DOF movement showing orientation change. It provides the foundation for the full 6DOF aircraft motion. In addition, through rotation matrix defined by aircraft Euler angles, other states associated with the motion analysis, such as position displacement, orientation can be derived accordingly.

Aircraft dynamics develops the equations of motion, treating a fixed-wing flying vehicle as a rigid body. The special attention is placed on the aerodynamic impact that makes solving the equations challenging due to the strong interactions with aircraft motion. In flight mechanics, the treatment is suggested through some engineering approximations. Aircraft equations of motion are governed by rigid-body dynamics, where the expression in body-fixed frame provides an opportunity of solving 6DOF motion variables through six sets of differential equations. Further, we shall recognize that the applying inputs of forces and moments are interacting with aircraft motion due to aerodynamic effect. Therefore, we use Taylor series expansion to approximate perturbed aerodynamics through the aerodynamic and control derivatives. By connecting these derivatives to the nondimensional parameters that are assumed to be available, we are ready to conduct dynamic analysis.

Static flight stability and control addresses the stability concepts in flight, and impact of flight control effectors on attributions of flight characteristics. Based on static force and moment equations in steady flight conditions, the stability and control trends are revealed without going through dynamic modes or solving dynamic equations. The representative longitudinal stability is the pitch stiffness and the longitudinal control is through the control surface of elevator. The representative lateral stability are addressed by the directional yaw stiffness and the roll stiffness, correspondingly, the yaw control is through the rudder, while the roll control is through the aileron. It is noteworthy that aircraft is treated as a system, that we need to integrate component contributions. In that sense, the wing-horizontal tail configuration is considered for longitudinal analysis, while the body-wing-vertical tail configuration is taken into account for lateral analysis. Stability and control derivatives are estimated and identified as key parameters for the static stability analysis.

This is the first chapter of a new part, “state-space based aircraft dynamics and control,” where a so-called state-space description based modern control is introduced and applied to solve flight dynamics and control problems. We will first officially introduce the concept of state-space model, followed by a model-based design method to systematically calculate feedback control gains to place representative characters to their desired positions, in order to achieve the desired dynamic performance. The placement in flight control introduces two design approaches. In terms of Learning Objectives, the pole placement calculates state (or output) feedback control gain K to place the closed-loop poles to desired positions. For a scalar input, there are various formulas to calculate the control gain vector. For an MIMO system, the placement leads to algebraic matrix manipulation, illustrated by a two-dimensional flight control example. On the other hand, the eigenstructure assignment enables closed-loop desired eigen values and eigenvectors to be placed simultaneously, where eigenvalues are the same as the closed-loop poles, and eigenvectors represent desired modes.

State space based modern flight control has the distinctive feature of systematic design depending on the linearized aircraft flight dynamics model and measurement of feedback state or output signals. In this chapter, we present basic concepts addressing the model uncertainty or disturbance challenges by introducing state estimation (observer) as well as sensitivity in flight control. In the presence of external disturbance (for example, the gust), measurement or process noises, or uncertainty in modelling (linearization approximation, variations of models, or un- modelled modes), the follow-up discussions associated with state-based design address the estimation and robustness in flight control. The linear observer design becomes a companion tool similar to the linear quadratic control design that guarantees the convergence of estimation to the ground truth. Further, the linear quadratic Gaussian design (LQG), based on stochastic process concepts, shows that control and observer design can be decoupled according to the separation principle, each will deal with control performance and estimation performance, respectively.