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A pseudo-sliding mode control synthesis procedure discussed previously in the literature is applied to the design of a control system for a nonlinear model of the NASA Langley Generic Transport Model. The complete vehicle model is included as an appendix. The goal of the design effort is the synthesis of a robust control system to minimize aircraft loss-of-control by preserving fundamental pilot input—system response characteristics across the flight envelope, here including the possibility of actuator damage. The design is carried out completely in the frequency domain and is described by a ten-step synthesis procedure, also previously introduced it the literature. Five different flight tasks are considered in computer simulations of the completed design demonstrating the stability and performance robustness of the control system.

Transport aircraft loss-of-control (LOC) incidents are a continuing problem in aviation safety [

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The NASA generic transport model.

The paper is organized as follows:

The appendix provides a nonlinear model of the vehicle in question, the NASA Langley GTM. These dynamics are taken directly from [

The design procedure for the frequency-domain sliding mode control approach has been thoroughly described in the literature, e.g., [

A vehicle model is obtained. If the vehicle in question is nonlinear, a linearized version is obtained for initiating the SMC design. Actuator dynamics are ignored at this stage. The linearization here is only evoked to allow classical frequency-domain design techniques to be employed in the synthesis procedure to be described.

A square control structure is identified. The design requires that no transmission zeros lie in the right-half plane and any uncontrollable states must be asymptotically stable. These requirements are similar to those employed in feedback linearization designs. Observability of the linear model must be in evidence to accommodate the observer design to be discussed in Step (7). The control loop structure in the square system is based upon a simple premise. Consider a square structure containing “m” control loops eventually driving a single control variable u_{c}. Define the inner-most loop as loop “1”, with the next being “m − 1”,

Rules (3)–(10) refer to the inner-most control loops in the square structure.

Sliding mode control is limited to the inner-most control loops in the structure. The control law for each of these loops is given by

σ for any channel is derived from a tracking error expression such as
^{ξ-i} refers the (ξ-i)^{th} derivative of e(t). An integral term also appears in Equation (2) to counter the steady-state bias often created with the inclusion of a boundary layer.

Recognizing that a boundary layer is to be implemented, the control law is expressed as a linear transfer function,
_{i} are chosen to provide “desirable” properties in the frequency domain. This means creating a loop transmission with broad K/s-like characteristics around crossover. The minimum K_{ρ} is that which permits pure gain or PI compensation in the frequency range at and beyond which ξ = 1, while exhibiting adequate gain and phase margins.

With the K_{( - )} values just determined, the existence of sliding behavior is verified through computer simulation. A convenient and practical choice for ρ is the amplitude limit of the actuator for the loop in question. Thus u_{c}(t) is the output of a relay element with limits of ±ρ.

A boundary layer is included in the controller to eliminate the high-frequency switching in the control variable(s) u_{c}(t).

Actuator dynamics are now included in the computer simulation. Instability will typically result. This condition highlights the effect of so-called “parasitic dynamics” in the SMC design procedure.

Asymptotic observers are created for each channel with actuator dynamics still eliminated in the observer design. The eigenvalues of the observers are selected as real and (approximately) equal, and are determined by either maximizing the stability margins evident in “effective” unity feedback loop transmissions L_{eq}, e.g., [

To increase robustness, “hedge” dynamics can be created in which an additional loop is closed in parallel with the observer. Briefly, the hedge dynamics further decrease the destabilizing effects of the actuator in the loop in question. The method for choosing hedge dynamics can be found in [_{hedge} as follows: The magnitude plot of the hedge dynamics should exhibit (a) a +20 dB/dec slope at low frequencies, (b) a −20ξ dB/dec slope at frequencies near the actuator bandwidth, where ξ is the relative order of the vehicle dynamics in the loop in question (excluding actuators), and (c) a (−20ξ – 20) dB/dec slope at high frequencies. The gain value is chosen so that the magnitude of the Bode diagram of the hedge dynamics equals that of the transfer function of the appropriate observer output to SMC output near the natural frequency of the appropriate actuator [

In some applications, the addition of a rate-feedback loop in the inner-most control loops to which FDSMC is being applied can be considered. This was done, for example in [

If desired, the observer(s) of Step (7) can be scheduled with flight condition.

The vehicle model in the Appendix was mechanized in Simulink^{®} and linearized about a flight condition of steady-wings-level flight at a trim velocity of 110 ft/s. This step required an estimation of trim values of pitch attitude, θ_{0}, angle of attack, α_{0}, thrust level, T_{0}, and elevator angle, δe_{0}. These were estimated as: θ_{0} = 5/57.3 rad, α_{0} = 5/57.3 rad, T_{0} = 5 lbf, δe_{0} = 0 rad. The results of [

The reader will recognize typical modal characteristics in Equations (4) and (5),

Bode Diagrams for the Transfer Functions of Equations (4) and (5).

The control structure to be used is shown in simplified form in

The control system architecture.

The sliding manifolds for the q and p loops, as interpreted in the frequency domain can be given by

By introducing initial conditions of 5/57.3 rad/s in the q-loop and −5/57.3 rad/s in the p-loop sliding behavior was verified. That is, “reaching” and “sliding” behavior was noted along with “infinite frequency” switching in the outputs of the signum elements.

Reaching and sliding behavior in the p loop.

A boundary layer was included in the q and p loops to eliminate the high-frequency switching characteristic of SMC systems. The signum elements in the q and p loops were removed. As in other applications of the methodology, e.g., [

Actuator dynamics were next included. Due to the relative high-frequency bandwidth of the actuators, instability was avoided in this case. Nonetheless, asymptotic observers were created for the q and p loops to ensure robustness in the presence of the actuator dynamics. A single observer was utilized with inputs consisting of φ, θ, ψ, p, q, r and the outputs of all the inner-loop compensators. The eigenvalues were selected as λ = −75, −75.1, −75.2,…,−75.8, just beyond the actuator bandwidths of 62.8 rad/s.

Hedge dynamics were created in the q and p loops as follows:
_{cq} and u_{cp} are the output of the elements in Equations (6) and (7).

Rate feedback loops were included in the q and p loops as follows:

Hedge and rate dynamics placement.

Effect of rate-loop closure in loop transmission of p loop.

For the purposes of this study, two flight conditions were considered, both at sea level, one at 110 ft/s and one at 220 ft/s. The asymptotic observers were scheduled with respect to these flight conditions.

The four non-SMC compensators in

The simplicity of these elements is worthy of note as each is either a pure gain or PI controller.

Additive sensor noise was included in each of the response variables used in the observers. The noise was created by passing white noise second-order filters with break frequencies of 10 rad/s. The root-mean square noise levels were 0.1 deg, 0.1 deg/s on vehicle attitudes and attitude rates, respectively. In addition, unmodeled time delays of 0.2 s were introduced at the input of each actuator.

A computer simulation of the nonlinear vehicle and control system was undertaken with the following task:

Command flight path angle γ = 0 deg.

Command velocity V_{c} = 110 ft/s

Command velocity perturbations around trim ΔV_{c} = 15 sin(0.1 t) ft/s

Command roll attitude φ_{c} = 10 sin(t) deg and simultaneous rudder input δ_{r} = 5 sin(t) deg. The roll and rudder commands were deliberately designed to produce positive (negative) sideslip β, with positive (negative) roll attitude φ. These inputs were included to excite the nonlinearities in the vehicle model. While decidedly atypical pilot inputs, they have been seen in piloted simulations involving crosswind landings [

To emulate damage to the elevator and aileron actuators, at 30 s into a 180 s run the effectiveness of the actuators was reduced by 25% and a 0.5 deg backlash element was introduced in the outer loop of the actuator mechanization. Simultaneously a −10 deg elevator bias (trailing edge up) was introduced.

Vehicle responses for Task 1.

Elevator and thrust inputs for Task 1.

Vehicle responses for Task 1 with no actuator damage.

Elevator and thrust inputs for Task 1 with no actuator damage.

The scenario of Task 1 was repeated, with the command velocity V_{c} = 220 ft/s. _{e} – 10 (deg) has been plotted to prevent overlap with the thrust time history. The very active elevator inputs are attributable to the effect of the actuator backlash element at the higher airspeed.

Vehicle responses for Task 2.

Elevator and thrust inputs for Task 2.

Task 3 consisted of the following

Command flight path angle γ = 2.5 deg.

Command velocity V_{c} = 110 ft/s

Command velocity perturbation around trim ΔV_{c} = −22.5 ft/s

Command roll attitude φ_{c} = 10 sin(t) deg and simultaneous rudder input δ_{r} = 5 sin(t) deg.

The actuator damage described in

Vehicle responses for Task 3.

Elevator and thrust inputs for Task 3.

Task 4 was identical to Task 3 with the following two exceptions:

Command velocity perturbation around trim ΔV_{c} = −7.5 ft/s

Air density in the vehicle model was set to a value appropriate for an altitude of 10,000 ft (ρ = 0.0017556 slugs/ft^{3}). The author is aware that the vehicle in question was not intended to operate at this altitude. The inclusion of this flight condition, however, provides an opportunity to evaluate the robustness of the SMC design.

The necessity of reducing the magnitude ΔV_{c} from that of Task 3 was necessary to prevent stall at the higher simulated altitude.

Vehicle responses for Task 4.

Elevator and thrust inputs for Task 4.

Task 5 was identical to Task 4 save that the trim airspeed was 220 ft/s.

Vehicle responses for Task 5.

Elevator and thrust inputs for Task 5.

Quoting from the Introduction: The philosophy behind the approach espoused herein is that the preservation of fundamental pilot input—system response characteristics throughout the flight envelope is important, indeed vital, in preventing LOC events. The computer simulations of

With two exceptions, the compensation elements that have been employed in the design were, at most, PI controllers or first-order bandwidth limited rate controllers. The two exceptions were the “hedge dynamics” for the pitch-rate and roll-rate loops. The form of these latter elements, however, is explicitly called out in the design Step (8) of

Based upon the study summarized herein, the following conclusions can be drawn:

The frequency-domain based pseudo-sliding mode design procedure introduced previously in the literature can be successfully applied to the design of a highly nonlinear model of the GTM aircraft.

The sliding-mode design can be limited to the inner-most loops of a square control architecture in which nested control loops are closed in sequential fashion.

Simple loop-shaping principles are used in each of the loop closures.

The majority of the compensators obtained in the design are simple in form,

Computer simulations of the vehicle and control system exhibited stability and performance robustness in different flight conditions and including simulated actuator damage.

The overall goal of the study, to demonstrate that fundamental pilot input—system response characteristics could be preserved, with significant variations in flight condition and actuator characteristics, was met.

Elevator, aileron and rudder:

Unity dynamics

Maximum thrust = 40 lbf

GTM wing span, 6.85 ft

GTM mean aerodynamic chord 0.92 ft

_{L}

Rolling moment coefficient

_{M}

Pitching moment coefficient

_{N}

Yawing moment coefficient

_{X}

X-force coefficient

_{Z}

Z-force coefficient

Acceleration due to gravity, 32.17 ft/s^{2}

_{p}

Compensation in p loop

_{q}

Compensation in q loop

_{V}

Compensation in V loop

_{γ}

Compensation in γ loop

_{θ}

Compensation in θ loop

_{φ}

Compensation in φ loop

GTM moment of inertia, 0.12 slugs/ft^{3}

GTM moment of inertia, 4.254 slugs/ft^{3}

GTM moment of inertia, 5.454 slugs/ft^{3}

GTM product of inertia, 0.12 slugs/ft^{3}

GTM mass, 1.54 slugs

Roll rate (rad/s)

Pitch rate (rad/s)

Yaw rate (rad/s)

GTM wing area, 5.9 ft^{2}

_{cp}

Output of Gc_{p} (rad)

_{cq}

Output of Gc_{q} (rad)

Velocity (ft/s)

Change in velocity from trim value, ft/s

_{c}

Change in velocity command from trim value, ft/s

_{cg}

GTM center of gravity position, 0.15 ft

GTM reference center of gravity position, 0.25 ft

Angle of attack (rad)

Angle of sideslip (rad)

Angle of flight path (rad)

_{a}

Aileron deflection (rad)

_{c}

Actuator command (rad)

_{e}

Elevator deflection (rad)

_{r}

Rudder deflection (rad)

Pitch attitude (rad)

Air density (slugs/ft^{3})

Roll attitude (rad)

Heading (rad)

The author declares no conflict of interest.

_{1}

_{1}Adaptive Control Law: Offset Landings and Large Flight Envelope Modeling Work