Flight Control Using Physical Dynamic Inversion Fubiao Zhang1 and Florian Holzapfel.2 Institute of Flight System Dynamics (FSD), Technische Universität München (TUM),Garching, Germany,D-85748
This paper presents a new architecture of dynamic inversion based flight control law, which decouples desired dynamics specification and stabilizing feedback controller design. The decoupling is achieved by using a physically integrated reference model, which mimics desired aircraft response, and an incrementally cascaded error controller, which shapes the extra dynamics beyond the desired dynamics. This approach makes best use of flight dynamics both in feed forward and feedback design, in an attempt to provide design transparency and to ease the feedback control efforts, and thus named Physical Dynamic Inversion here. Pilots commands are filtered out by the explicit reference model when getting close to the envelope boundaries, together with reducing feedback commands at the same time, flight envelope protection are completed. The novel features are demonstrated by comparison with the existing architectures in the literature in the aspects of desired dynamic specification, loop closure and ideal closed loop dynamics. Rules of gain tuning are also introduced, the benefits of auto gain scheduling of dynamic inversion based control law preserves, while providing satisfactory flying qualities.
Nomenclature A = B = O = E = K = ̅ 𝐾 = 𝛾𝐾 = 𝜇𝐾 = 𝜃 = 𝜂 = 𝜁 = 𝑉𝐴 = 𝑉𝐾 = 𝜌̅̇𝐾 = 𝜈 = Subscripts abs = act = c = des = ff = n = r = sp = extra = alw =
1 2
Aerodynamic frame Body-fixed frame NED frame Earth centered earth fixed frame Kinematic frame Rotated kinematic frame Flight path angle Flight path bank angle Pitch angle Elevator deflection Damping ratio Aerodynamic speed of the Center of gravity Kinematic speed of the Center of gravity Modified Flight path angle rate, 𝜌̅̇𝐾 = 𝛾̇𝐾 /cos(𝜇𝐾 ) Pseudo Control Absolute Actuator Command signal Desired Feed forward signal Nominal Reference signal Short period Extra command compared to signal limit, or extra dynamics Allowed magnitude of the signal
Ph.D Candidate, Institute of Flight System Dynamics(FSD), Email:
[email protected], Student Member AIAA. Professor, Institute of Flight System Dynamics(FSD), Email:
[email protected], Senior Member AIAA. 1 American Institute of Aeronautics and Astronautics
I. Introduction
D
YNAMIC inversion is used as a systematic approach in designing controllers. In aerospace applications, the general procedure directly starts from the rigid body equations of motion, using measurement information, onboard aerodynamic, mass and propulsion models to cancel the known nonlinear dynamics, then the desired dynamics can be imposed to follow, combining with control allocation, a mapping from the required response and control effector deflections is achieved. This concept may date back to 1970s by the work of Singh1,2, where decoupling was the main problem and had already been applied to control rapidly maneuvering aircraft3, theoretical foundation of dynamic inversion in differential geometry can be found, for example, in the work by Isidori 4 or Slotine5. As maneuverability continued to be pursued in the 1980s, high performance fighter aircrafts were expected to work well at high angle of attack or rapid roll maneuvers, this posed difficulties for traditional design based on trim and linearization: (1) excess gain scheduling might be employed even with angle of attack, resulting in high workload both on controller synthesis and analysis. (2) Synthesis models based on Jacobian linearization is not sufficient to represent the nonlinear dynamics, because this classical methods inevitably neglect couplings (inertial, aerodynamic or kinematic) between axes, which become more significant at rapid maneuvers. The advantages of nonlinear dynamic inversion (NDI) based control law dealing inherent nonlinearities attracted researchers’ attention, Honeywell implemented an NDI controller to the stage of full hardware in the loop simulation for the F-18 High Angle of attack Research Vehicle (HARV), to execute highly nonlinear and uncertain post stall maneuver. Another program, the development of X-38 Crew Rescue Vehicle in the 1990s, involved using the flexibility of NDI controller for a series of test vehicles by only updating onboard aerodynamic models. The progresses are summarized in Ref 10 and 11. All the efforts contributed to the maturation of dynamic inversion concept, finally lead to its application even in the production F-35, predictable response of transition phase from conventional aircraft flight mode to hover was reported 6,7. Nowadays dynamic inversion based controller seems to be one of the standard design approaches for vehicles from standard fixed wings to quad copters, ducted fan UAVs, helicopters and tilt-roters12-18. The power of NDI in various applications roots in its ability to cancel the known nonlinear dynamics, providing the designer a highly simple and structured new “plant” The drawback of this approach comes also from the cancellation, which usually requires high fidelity models. This assumption can not be always fulfilled because either it is too difficult to model the physical process like unsteady aerodynamics, or it is too time consuming and expensive to obtain an exact model within project schedule or budget. Generally speaking, NDI can not be used alone, controller must be used to compensate the model error. Three approaches might be used to reduce the model error sensitivity associated with NDI: 1) Approximate the model error online and cancel it further by adaptive augmentation, Neural Networks (NN) can be used as a universal approximater and trained in real time with tracking error excitation 11-19, .input dynamics like the common actuator saturation, which may cause misadaptation, can be considered and hidden by Pseudo Control Hedging (PCH), the NDI based adaptive NN controller plus PCH architecture has been successfully flight tested in a number of applications17,18, new development under this framework includes efficient online model substitution20 and nonparametric adaptive nonlinearity representation19 2) Robust mu controller augmentation. Uncertainties in model, sensor and actuator dynamics and considered in the mu framework by weighting functions, a robust controller is designed such that the closed loop stability retains with bounded perturbation21,22. 3) Latest development also suggest sensor based approach. Instead of using online aerodynamic tables, this approach completely gets rid of aerodynamic dependencies except in input effectiveness. Measurements or estimations (like pitch acceleration) are used to replace uncertain model information in the moment equation23,24. Another equally important but not so widely covered point in NDI design is the desired dynamics specification. With dynamic inversion, the original system can be reduced to as simple as an integrator or a series of integrators depending on the input output selection. This topic has already been noticed by authors in Ref 25 and 26, the authors considered desired dynamic selection from the aspects of error compensation structure, flying qualities and ride qualities. However, take the pitch rate loop for example, although second order dynamics are desired dynamics in their architecture for flying qualities consideration, the closed loop is dynamics in third order, not exactly desired as specified in the military standard39, the order itself is not a problem, further insights must be gained by nested loop analysis. Especially for multiple loops design, it is even less straightforward to specify considering requirements like flying qualities, although bandwidth separation is always available as a general rule. In this paper, we present a reference model based design philosophy. A physically integrated reference model33 is used to provide consistent tracking signals to reach desired dynamics, with envelope protection intuitively embedded also in the reference model, excess command can be filtered out before forming the closed loop system. The noncascaded reference model structure, as opposed to many designs, enjoys the benefits like requirements validation34 and straight forward desired dynamics specification. A cascaded feedback controller with command inversion based 2 American Institute of Aeronautics and Astronautics
on equations of motion is then designed for the purpose of stabilization. The closed loop contains always the desired dynamics in series connection with the same extra dynamics for different loops. One criteria or motivation that justifying doing so is that the extra dynamics must be within the MUAD envelope for keeping the flying qualities specified in the reference model. The organization of the paper is as follows. Section II provides the principles of dynamic inversion process. Section III reviews the candidate architecture commonly used in the literature. Then in section IV new architecture named Physical Dynamic Inversion is introduced, including controller structure design, gain tuning and assessment. Finally section VI summarizes the conclusions.
II. Dynamic Inversion Preliminaries Dynamic inversion, also named feedback linearization, is a method of state transformation. As the name indicates, the method uses feedback signals from the sensors and actuation power to cancel known nonlinearities, such that linearity is obtained between the new input called pseudo control and the controlled output in a new state representation. In fact, the resulting linear system turns out to be an integrator or a series of integrators, which can be seen as a “new transformed plant” from the controller. With this dynamic inversion as a mask, widely available and sophiscated linear techniques can be applied. For illustration purpose, a single input single output system is shown here. Consider an affine in input nonlinear system in the form 𝒙̇ = 𝑓(𝒙) + 𝑔(𝒙) ∙ 𝑢 (1) 𝑦 = ℎ(𝒙) (2) where 𝒙 ∈ 𝑹𝑛×1 is a state vector, 𝑢 ∈ 𝑹1×1 is single control input, 𝑦 ∈ 𝑹1×1 is single control output, The control objective is to let output 𝑦 track the command 𝑦𝑐 . The design procedure is as follows. Differentiate the selected output variable 𝑦 𝜕ℎ(𝒙) 𝑦̇ = (𝑓(𝒙) + 𝑔(𝒙) ∙ 𝑢) (3) 𝜕𝒙 Using the notation of Lie derivative 𝑦̇ = 𝐿𝑓 ℎ(𝒙) + 𝐿𝑔 ℎ(𝒙) ∙ 𝑢 (4) Where 𝐿𝑓 ℎ(𝒙) defines the first order Lie derivative of ℎ(𝒙) along the direction of 𝑓(𝒙). If 𝐿𝑔 ℎ(𝒙) = 0, continue the differentiation 𝑦̈ = 𝐿2𝑓 ℎ(𝒙) + 𝐿𝑔 𝐿𝑓 ℎ(𝒙) ∙ 𝑢 (5) … 𝑦 (𝑟) = 𝐿𝑟𝑓 ℎ(𝒙) + 𝐿𝑔 𝐿𝑟−1 (6) 𝑓 ℎ(𝒙) ∙ 𝑢 𝑟−1 Until 𝐿𝑔 𝐿𝑓 ℎ(𝒙) ≠ 0. The number of times one need to differentiate the output in order to have the control input 𝑢 explicitly appears in the equation, is defined as relative degree 𝑟. Define new states 𝑧𝑖 = 𝜙𝑖 (𝒙) = 𝐿𝑖−1 (7) 𝑓 ℎ(𝒙) with 1 ≤ 𝑖 ≤ 𝑟 Remember the original system dimension is 𝑛, the 𝑛 − 𝑟 can not be seen from the differenciation of output 𝑦, One can always find 𝑛 − 𝑟 linear independent functions such that 𝑧𝑖 = 𝜙𝑖 (𝒙) for 𝑟 < 𝑖 ≤ 𝑛. Then with the nonlinear transformation 𝒛 = Φ(𝒙) = [𝜙1 (𝒙) 𝜙2 (𝒙) ⋯ 𝜙𝑛 (𝒙)]′ (8) 𝑟 𝑟 𝑟−1 −𝟏 −𝟏 Denote also 𝐿𝑓 ℎ(𝒙) = 𝐿𝑓 ℎ(Φ (𝒛)) = 𝑎(𝒛), 𝐿𝑔 𝐿𝑟−1 𝑓 ℎ(𝒙) = 𝐿𝑔 𝐿𝑓 ℎ(Φ (𝒛)) = 𝑏(𝒛) The original system described by equation (1) and (2) is transformed in to new linearizing state representation in 𝒛 as follows 𝑧̇1 = 𝑧2 (9) 𝑧̇2 = 𝑧3 (10) … 𝑧̇𝑟 = 𝑎(𝒛) + 𝑏(𝒛) ∙ 𝑢 (11) 𝑧̇𝑟+1 = 𝐿𝑓 𝜙𝑖 (Φ−𝟏 (𝒛)) + 𝐿𝑔 𝜙𝑖 (Φ−𝟏 (𝒛)) ∙ 𝑢 (12) … 𝑧̇𝑛 = 𝐿𝑓 𝜙𝑛 (Φ−𝟏 (𝒛)) + 𝐿𝑔 𝜙𝑢 (Φ−𝟏 (𝒛)) ∙ 𝑢 (13) They dynamics describe by Eq. (12) to Eq. (13) is not observable from the selected output, it is called internal dynamics. The internal dynamics depends on inherent dynamics of the original system as well as the selected output variable, local zero dynamics analysis can be found in Ref 10 and 38 Now define the new control input 𝜐, called pseudo control, select control law in the form of 3 American Institute of Aeronautics and Astronautics
𝑢 = 𝑏 −1 (𝒛)(−𝑎(𝒛) + 𝜐)
(14)
Substitute in Eq. (11) 𝑦 (𝑟) = 𝜐 (15) The result in perfect inversion, if = is provided as pseudo control input, we get output tracks the input ideally, that is 𝑦 = 𝑦𝑐 . The dynamic inversion process is illustrated in Fig. 1. (𝑟) 𝑦𝑑𝑒𝑠
(𝑟) 𝑦𝑐
Original System
System in New States
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h
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Internal Dynamics
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f DI Process
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Linear Design
integrators
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Figure 1. Dynamic Inversion Concept.
III. Candidate NDI Architectures in Flight Control The following section summarizes the architecture commonly used in the literature with NDI involved as control strategy. Depending on the selection of output variable to be controller, feedback loop closure. Motivation of desired dynamics selection, linear error controller structure and gain design are included. With all these in mind, the Physical Dynamic Inversion architecture can be better understood. A. Design for Good Flying Qualities Having level 1 flying qualities is a subset of the requirements to flight control system design, flying qualities of manned aircraft are well defined in standard38-41, several criteria must be checked and fulfilled in order to fly with acceptable pilot workload and safety. Table 1 summarizes the desired dynamics structure from the pilot input to angle of attack and pitch rate respectively, basically second order form. They are obtained from the matching structure of Low Order Equivalent System (LOES) approach, which was developed for assessing flying qualities of high order control system. The time delay is intentionally omitted, because it is used originally to approximate the high order effect, as a byproduct of complex flight control system like fly by wire. The damping and natural frequency, or CAP to be achieved can de designed according to the specification, donated as 𝜁𝑑𝑒𝑠 and 𝜔𝑑𝑒𝑠 here. However, selecting dynamics alone according Table 1 can not guarantee but necessary for good flying qualities Table 1. Desired form of flying qualities specifications
Angle of Attack Dynamics 𝐾𝑙𝑜𝑛,𝛼
Pitch Rate Dynamics 𝐾𝑙𝑜𝑛,𝑞 (𝑠 − 𝑍𝛼 )
2 ∙ 𝜔𝑠𝑝,𝑑𝑒𝑠
𝑠2 + 2𝜁𝑠𝑝,𝑑𝑒𝑠 ∙ 𝜔𝑠𝑝,𝑑𝑒𝑠 ∙ 𝑠 + 𝜔2𝑠𝑝,𝑑𝑒𝑠
𝑠2 + 2𝜁𝑠𝑝,𝑑𝑒𝑠 ∙ 𝜔𝑠𝑝,𝑑𝑒𝑠 ∙ 𝑠 + 𝜔2𝑠𝑝,𝑑𝑒𝑠
B. Implicit Reference Model and Explicit Reference Model Generally speaking, candidate architecture can be split into two main categories: 1) control with implicit reference model and 2) control with explicit reference model. A pitch rate control task is assumed to illustrate in Fig. 1, LC stands for linear controller, denoted as 𝐺𝐿𝐶 (𝑠) and RM stands for reference model, it is already clear with the 4 American Institute of Aeronautics and Astronautics
background of Section II, after dynamic inversion process, the aircraft dynamic behavior is reduced approximately to an integrator. In both cases linear controller is used to stabilize the tracking error dynamics. 1. Implicit Reference Model As show in Fig.1(a), the structure was used in many design from Singh’s work in 1970s and also in Ref 10 and 11, it seems to be extensively used in Honeywell applications12. In case 1, the linear controller together with the integrator formed the desired closed loop dynamics, in case of perfect inversion 𝐺𝐿𝐶 (𝑠) (16) 𝐺𝐷𝑒𝑠 (𝑠) = 𝑠 + 𝐺𝐿𝐶 (𝑠) For example selecting 𝐺𝐿𝐶 (𝑠) as a proportional controller with gain 𝐾, first order closed loop dynamics can be expected, ideally for roll axis. If steady state accuracy is required, by specially selecting the gains, PI structure can also be used to get first order closed loop dynamics12,24 .For pitch rate control, selecting 𝐺𝐿𝐶 (𝑠) first order transfer function will result in second order dynamics, matching the second column of Table 1. More variant of 𝐺𝐿𝐶 (𝑠) section can be found in Ref.12 and 24. Implicit model architecture in case 2 fits 𝐺𝐿𝐶 (𝑠) design into 𝜇 synthesis, aerodynamic uncertainties can be included and captured by analyzing the the transfer function from 𝑞̇ 𝑑𝑒𝑠 to 𝑞, sensor and actuator model fidelity can be considered by the corresponding weighting filters. An implicit reference model following is used to produce a robust controller under the aforementioned uncertainties. Desired dynamics can certainly be specified in this implicit reference model. 2. Explicit Reference Model. In this approach, a reference model specifying the desired dynamics is carried onboard to provide the reference command and its derivatives. These reference command derivatives can be fed forward as part of the pseudo control, or be used to form feedback loops. The baseline controller will normally work depending on the complexity of the problem, although the tracking error dynamics will be excited by the inversion error, leading to time response different from the reference signal, they will be damped out by the linear controller. In general, the smaller the norm of inversion error is, the closer the closed loop system is closer to the reference model. One of the benefits of explicit reference model approach is that the reference command tracking error can be directly build up to drive the adaptation, inversion error can be approximated online and reduced further, this architecture is the case 2 shown in Fig. 1 (b), it is also adapted in this paper.
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1. Closed Loop Forming Desired Dynamics DI
1. RM Following by Baseline Controller
LC
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RM
RM
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LC
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2. Implicit Model Following by
Adaptation
Synthesis
2. Adaptive Augmentation
(b) Explicit Reference Model Architecture
(a) Implicit Reference Model Architecture
Figure 2. Candidate Architecture Involving NDI Based Control Design C. Using Explicit Reference Model and Relative Degree Consideration 1. Pitch rate control If pitch rate is chosen to be the control variable, only one differentiation is needed before the input appears, so it is a relative degree 1 design. An often used architecture is shown in Fig. 3, the linear controller appears in the PI form. Reference model is selected as in Table 1 second column, written in time domain is 2 𝑞̈ 𝑟 = −2𝜁𝑠𝑝,𝑑𝑒𝑠 𝜔𝑠𝑝,𝑑𝑒𝑠 ∙ 𝑞̇ 𝑟 − 𝜔𝑠𝑝,𝑑𝑒𝑠 ∙ 𝑞𝑟 + 𝐾𝑙𝑜𝑛,𝑞 (𝑞̇ 𝑐 − 𝑍𝛼 𝑞𝑐 ) (17) The system response after dynamic inversion is (18) 𝑞̇ = 𝐾𝑃 (𝑞𝑟 − 𝑞) + 𝐾𝐼 ∫((𝑞𝑟 − 𝑞)) + 𝑞̇ 𝑟 + ∆ 5 American Institute of Aeronautics and Astronautics
Where ∆ denotes the dynamic inversion error Then the equation governing the tracking error 𝑒𝑞 = 𝑞𝑟 − 𝑞 is (19)
𝑒̇𝑞 = −𝐾𝑃 𝑒𝑞 − 𝐾𝐼 ∫ 𝑒𝑞 − ∆ Select
2 𝐾𝑃 = 2𝜁𝑠𝑝,𝑑𝑒𝑠 𝜔𝑠𝑝,𝑑𝑒𝑠 , 𝐾𝐼 = 𝜔𝑠𝑝,𝑑𝑒𝑠 (20) the transfer function from ∆ to tracking error 𝑒𝑞 is 𝑒𝑞 (𝑠) 1 1 (21) =− 2 = − 𝐺𝑞,𝑑𝑒𝑠 (𝑠) ∆(𝑠) 𝑠 + 𝐾𝑃 𝑠 + 𝐾𝐼 𝐾𝐼 This means, 1 (22) 𝑞(𝑠) = 𝑞𝑟 (𝑠) − 𝑒𝑞 (𝑠) = 𝐺𝑞,𝑑𝑒𝑠 (𝑠)(𝑞𝑟 (𝑠) + ∆(𝑠)) 𝐾𝐼 Indicating the two parts contributes to the response of pitch rate, one part from pilots command, the other part from inversion error. By selecting as in Eq. (21), the tracking error dynamics is influencing the response the same way as the pilot command, and might not be noticeable by the pilot. The same conclusion can be made by seeing in time domain, Integrate both sides of Eq.(17), and insert to Eq. (18) will give (23) 2 𝑞̇ = (𝐾𝑃 − 2𝜁𝑠𝑝,𝑑𝑒𝑠 𝜔𝑠𝑝,𝑑𝑒𝑠 )𝑞𝑟 + (𝐾𝐼 − 𝜔𝑠𝑝,𝑑𝑒𝑠 ) ∫ 𝑞𝑟 − 𝐾𝑃 𝑞 − 𝐾𝐼 ∫ 𝑞 + 𝐾𝑙𝑜𝑛,𝑞 (𝑞̇ 𝑐 − 𝑍𝛼 𝑞𝑐 ) + ∆
Forcing the coupling term with reference command zero gives the same result.
RM
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DI RM
+ - + -
DI
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Figure 4. Angle of Attack Control with Pitch Rate Control with Explicit RM Explicit RM, Relative 2 Design.
Figure 3. Pitch Rate Control with Explicit RM
2. Angle of attack control-relative degree 2 design A relative 2 design for angle of attack control is shown in Fig.4, similar idea as pitch rate control applies, with a PD controller, and by selecting, 2 𝐾𝑃 = 𝜔𝑠𝑝,𝑑𝑒𝑠 , 𝐾𝐷 = 2𝜁𝑠𝑝,𝑑𝑒𝑠 𝜔𝑠𝑝,𝑑𝑒𝑠 (24) The error dynamics hides in the desired dynamics 3. Angle of attack control - two relative degree 1 loops design One flaw of the architecture design in Fig.4 is requiring 𝛼̇ feedback, which needs to be estimated. Another approach to realize the goal is to design two relative degree 1 loops based on time scale separation concept. That is pitch rate is seen as control input for translational equation, dynamic inversion is applied only to this force equation, desired pitch rate to reach angle of attach tracking is denoted 𝑞𝑐 , which serves as inner loop command. However selecting reference model for angle of attack and pitch rate loop respectively as second order will not lead to closed loop second order system, insights for its equivalence to a second order and the vision for reference model gains and controller gains are lost due to the nested structure. To explain the difficulty, two simple first order reference model is used with simple P controller for each dynamic inversion loop as shown in Fig.5. for the pitch rate loop, the best can be achieved is the same as the reference model, which means the required pith rate command described below (25) 𝑞𝑐 = −𝑍𝛼 𝛼 + 𝛼̇ 𝑑𝑒𝑠 can be only achieved exactly to the bandwidth of the reference model. Assume perfect inversion in both loops, the closed loop transfer function is describe as (26) 𝛼(𝑠) 𝑐1 (𝑠 + 𝐾1 )𝐺𝑞𝑐,𝑞 (𝑠) = 𝛼𝑐 (𝑠) 𝑠 + 𝑐1 1 + 𝐾1 𝐺𝑞𝑐,𝑞 (𝑠) 6 American Institute of Aeronautics and Astronautics
Where 𝐺𝑞𝑐,𝑞 is the closed loop transfer function for pitch rate loop, in perfect inversion, it is the same as the reference model. It can be seen, the transfer function appears both in the numerator and denominator, suppose there is another loop nested or choosing a more complex structure for angle of attack, the intuitiveness is lost immediately. But still, after a few trial and error, by selecting 𝑐2 𝐺𝑞,𝑟𝑚 (𝑠) = 𝑠 + 𝑐2 Then 𝛼(𝑠) 𝑐1 (𝑠 + 𝐾1 )𝑐2 = ∙ 2 (𝑐 𝛼𝑐 (𝑠) 𝑠 + 𝑐1 𝑠 + 2 − 𝑍𝛼 )𝑠 + 𝐾1 𝑐2 By further choosing, 𝐾1 = 𝑐1 The closed loop is 𝛼(𝑠) 𝑐1 𝑐2 = 2 𝛼𝑐 (𝑠) 𝑠 + (𝑐2 − 𝑍𝛼 )𝑠 + 𝑐1 𝑐2 Now by matching with Table 1 reference model gains can be determined.
(27)
(28)
(29) (30)
DI
+ -
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Figure 5. Two Relative Degree 1 Loops Architecture for Angle of Attack Control D. Conclusion of Section III Dynamic inversion architecture based on implicit reference model following and explicit modeling are summarized in this section, the latter is explained in detail with implementation in different control variables and relative degrees, guidelines and motivations of gain selection for reaching good flying qualities are considered. For the multiple loop design, the nested loops architecture reduces the intuitiveness of structure and gain selection, despite the tedious analysis process, it is still possible to gain insight.
IV. Physical Dynamic Inversion The physical dynamic inversion architecture is shown in Fig.6. The reference model takes a non-cascaded structure, although the feedback loops are nested to take a physical conversion of commands, it is not necessary depending on design considerations, non-nested feedback loops are possible if the outer loop are predicted accurately enough in the PIRM43. As far as this paper is concerned, the most inner pitch rate loop can be implemented with nonlinear dynamic inversion, but the other two outer loops build only incremental type command to the relative inner loop. The reason is that, since the commands from the integrated reference model are consistent, the required pitch rate acceleration for the commanded load factor (and also change of flight path angle) is already fedfoward, only an extra command based on this is required for trim command compensation, uncertainty toleration and disturbance rejection. Another way to see it is linearization based inversion around the reference trajectory. For illustration and analysis purpose, linearized longitudinal dynamics is considered here in Eq. 31 0 ∆𝛾̇𝐴 [∆𝑛̇ 𝑧,𝐵 ] = 0 ∆𝑞̇ 𝐾 0 [
𝑔
0
𝐺 𝑉𝐾,0
𝑍𝛼 −
𝑔 𝐺 𝑉𝐾,0 ∙𝑍𝛼
− ∙ 𝑀𝛼
7 American Institute of Aeronautics and Astronautics
∆𝛾𝐴 0 𝑍𝛼 [∆𝑛𝑧,𝐵 ] + [ 0 ] ∆𝜂 𝑔 𝑀𝜂 ∆𝑞𝐾 𝑀𝑞 ]
𝐺 𝑉𝐾,0
(31)
Figure 6. Overall Architecture of Physical Dynamic Inversion A. Physically Integrated Nonlinear Reference Model The proposed integrated reference model is nonlinear one, in a way that reference commands are formulated by the summation of two parts: trim term and the maneuver part. The maneuver part is the output of the desired linear dynamics. The trim term remains when pilot’s stick centered to realize a certain function like turn compensation. Another three common features could be seen later in the reference model: 1) all the output signals are limited by its own upper and lower bound based on physical knowledge. 2) all the inputs of the integrators are limited for the sake of anti-windup. 3) all the reference signals are forward propagated based on the flight dynamics, whereas the saturation (when a certain signal hits the envelope boundary) and boundary overshoot information are backward propagated to the previous integrators to avoid overloading, which is also based on flight dynamics. The physically integrated model can be either totally nonlinear to account for significant nonlinearity in the plant43,44, and fully exporing its capability, or nonlinear feedback inside the reference model can be designed to reshape the desired dynamics, as the application in this paper is to general aviation aircraft, linear maneuver dynamics are sufficient. Saturation consideration in the reference model can also be found in Ref 35 with strong stability proof. Here only the longitudinal reference model is introduced, more detailed design including the lateral axis could be found in Ref. 33
Longitudinal Envelope Limitation
Lon Trim Part
Longitudinal Axis : Commanded Maneuver Part
Figure 7. Architecture of the Longitudinal Reference Model with Limitation 𝐾𝑠,𝑙𝑜𝑛 Stick scaling gain 𝐾𝑐,𝑙𝑜𝑛 Dynamic Gain 𝑀𝑎𝑐𝑡𝑢𝑎𝑡𝑜𝑟 Actuator dynamic model with rate and position limits 𝑀𝑙𝑜𝑛,𝑠𝑝 Desired aircraft short period dynamics 𝑀𝑙𝑜𝑎𝑑_𝑓𝑎𝑐𝑡𝑜𝑟 Algebraic Model of load factor output 𝑀𝑝𝑎𝑡ℎ Dynamic model to generate flight path reference command 4. Preparations before Short Period Dynamics Stick scaling gain is designed to reach the goal that at speed below corner speed, full stick deflection is commanding maximum angle of attack, whereas at speed higher than corner speed, full stick deflection is commanding 8 American Institute of Aeronautics and Astronautics
maximum load factor. The dynamic gain converts the flight path angle rate command to the load factor command, and then to the AoA command, with the DC gain of short period dynamics divided by the 𝑀𝜂 , the AoA command is converted to elevator command, thus actuator limitations can be integrated here. The dynamic gain is expressed in (32) 𝑉𝐾𝐺 𝑔 𝑍𝛼 𝑀𝑞,𝑑𝑒𝑠 − 𝑀𝛼,𝑑𝑒𝑠 𝐾𝑐,𝑙𝑜𝑛 = − ∙ 𝐺∙ 𝑔 𝑉𝐴 𝑀𝜂 5. Desired Short Period Dynamics The linearized short period dynamics is used here, the subscript “des” indicates they can be specified to reach the desired dynamics. 𝑀𝛼,𝑑𝑒𝑠 and 𝑀𝑞,𝑑𝑒𝑠 can be designed according to flying qualities requirements (mainly from military standard40). The core of the reference model resembles the plant in structure, leaving out actuator dynamics for the moment. 𝑔 𝐺 𝑉𝐾,0
0 ∆𝛾̇𝐴,𝑟 [∆𝑛̇ 𝑧,𝐵,𝑟 ] = 0 ∆𝑞̇ 𝐾,𝑟 0 [
0
𝑍𝛼,𝑛 −
−
𝑔 ∙ 𝑀𝛼,𝑑𝑒𝑠 𝐺 𝑉𝐾,0 ∙ 𝑍𝛼,𝑛
∆𝛾𝐴,𝑟 0 𝐺 𝑉𝐾,0 0 ] 𝛿𝑛,𝑐,𝑝𝑖𝑙𝑜𝑡 𝑍𝛼,𝑛 [∆𝑛𝑧,𝐵,𝑟 ] + [ 𝑔 𝑍𝛼,𝑛 𝑀𝑞,𝑑𝑒𝑠 − 𝑀𝛼,𝑑𝑒𝑠 ∆𝑞𝐾,𝑟 𝑀𝑞,𝑑𝑒𝑠 ]
(33)
This linear dynamics seems to make sense because when the pilot commands a maneuver from a certain trimmed flight, the response of the aircraft is quite predictable. In Fig. 8, the overshoot information, that excessive command
Figure 8. Desired Short Period Dynamics of pitch rate Δ𝑝𝐾,𝑟,𝑒𝑥𝑡𝑟𝑎 is propagated back with a gain 𝑀𝛼,𝑑𝑒𝑠 ⁄𝑀𝜂 , which would convert Δ𝑝𝐾,𝑟,𝑒𝑥𝑡𝑟𝑎 to Δ𝜂Δ𝑞,𝑒𝑥𝑡𝑟𝑎 , and subtracted from the actuator command shown in Fig. 8. In this way, output signals are not hard limited. The trim part in the reference model provide turn compensation and turn coordination33. 6. Actuator Dynamics in the Reference Model A second order actuator model with limitations in the position, rate and acceleration can be included, similarly to the one shown in Fig. 8. , the limiting information combines that from modelling and also that backpropagated from the pitch rate limitation. A comment is that, the bandwidth of this actuator does not need to be the same as the real actuator, a slower “model” can act as a pilot command filter such that abrupt command will not go into the system 7. Envelope Protection in the Reference Model Envelope protection can be done on the level of the reference model based on phase plane protection concept, details can be found in Ref. Here an example of load factor limitation combining limitation and back propagation using flight dynamics described by the equation of motion Input to the load factor limitation is the conservative one from absolute structural load limit and the limitation back propagated from flight path angle. (34) 𝑛𝑧,𝑚𝑎𝑥,𝑎𝑙𝑤 = min(𝑛𝑧,𝑚𝑎𝑥,𝑎𝑏𝑠 , 𝑛𝑧,𝑚𝑎𝑥,𝛾,𝑎𝑙𝑤 ) Using phase plane protection concept 𝑔 (35) ∆𝛼̇ = 𝐾 (𝑛 −𝑛 ) 𝑟,𝑚𝑎𝑥,𝑛𝑧 ,𝑎𝑙𝑤
𝑉𝐺𝐾,0 (−𝑍𝛼,𝑛 )
𝑛𝑧 ,𝑚𝑎𝑥,𝑝𝑟𝑜𝑡
𝑧,𝑚𝑎𝑥,𝑎𝑙𝑤
Limitation is also active when the angle of attach is close to the absolute limit ∆𝛼̇ 𝑟,𝑚𝑎𝑥,𝛼,𝑎𝑙𝑤 = 𝐾𝛼,𝑚𝑎𝑥,𝑝𝑟𝑜𝑡 (𝛼𝑚𝑎𝑥,𝑎𝑏𝑠 − 𝛼𝐴,𝑓𝑓 ) 9 American Institute of Aeronautics and Astronautics
𝑧,𝐵,𝑓𝑓
(36)
Again, taking conservative limit of the two ∆𝛼̇ 𝑟,𝑚𝑎𝑥,𝑎𝑙𝑤 = min(∆𝛼̇ 𝑟,𝑚𝑎𝑥,𝑛𝑧,𝑎𝑙𝑤 , ∆𝛼̇ 𝑟,𝑚𝑎𝑥,𝛼,𝑎𝑙𝑤 ) Further propagation to the pitch rate limitation 𝑞𝐾𝑚𝑎𝑥,𝛼,𝑎𝑙𝑤 = ∆𝛼̇ 𝑟,𝑚𝑎𝑥,𝑎𝑙𝑤 − 𝑍𝛼 ∙ ∆𝛼𝑟 − 𝑍𝜂 ∙ Δ𝜂𝑐 This serves as one source of limitation on the actuator position in the reference model
(37) (38)
B. Cascaded Feedback Structure 1. Error Controller Structure Evolution and Command Conversion The general error controller takes the form of PI controller with feed forward gain, as 𝛾 error controller shown in Fig.9, it is widely employed in the industry due to its simplicity and effectiveness 37. Usually, if steady state accuracy is required, an integrator can not be avoided in the loop. Then the proportional gain and integral gain are used to determine the bandwidth and place the pole of the integrator, while the feed forward gain generates a zero to cancel of slow pole introduced by the integrator. The output of the error controller is seen as the desired rate change related to the feedback variable. Command inversion basically inverted the equations of motions from the reference state trajectory, then the incremental command for the next inner loop is obtained. EC
Com. Conv. from
reset
+ + -
+ -
Figure 9. 𝜸 Error Controller with Incremental Command Conversion from 𝜸 to 𝒏𝒛 It is desirable to maintain the kinematic flight path angle unchanged if the pilot does not actively command, so the general error controller. Required incremental command is obtained by inverting the first row of Eq.(31) Suppose the same structure is applied for designing the error controller of the load factor, recall the dynamics describe by the second row of Eq. (31), we ask for the required Δ𝑞𝑐 given ∆𝑛̇ 𝑧,𝐵,𝑑𝑒𝑠 , in frequency domain, 𝑔 𝑔 1 (39) Δ𝑞𝑐 (𝑠) = ( 𝐺 + 𝐺 ∙ )∆𝑛̇ 𝑧,𝐵,𝑑𝑒𝑠 𝑉𝐾,0 (−𝑍𝛼,𝑛 ) 𝑉𝐾,0 𝑠 As shown in Fig. 10, a natural integral action is created from command inversion and totally based on inverting the equations of motion. This means the integrator is in the error controller is not necessary anymore. Then the feed forward path must be avoided to build the error latterly feeding to the integrator. Therefore only a P controller is used for the load factor error controller. As for the pitch rate error controller, the integrator is not required and also deleted because two much integration will slow down the control loop, and may suffer from resetting and ant wind up problem in practice. But the feed forward gain is kept for design freedom, by can simply made equal to the proportional gain if undesirable. The structure is shown in Fig. 11. EC
+
-
EC
Com. Conv. From
+ +
Figure 10.𝒏𝒛 Error Controller with Incremental Command Converion from 𝒏𝒛 to 𝒒
Figure 11.𝒒 Error Controller
2. Envelope Protection in the Feedback Path Similar phase plane based concept in the reference model can be applies to the feedback path to complete the flight envelope protection, The difference is that envelope protection in the reference model is predicted by modeling, 10 American Institute of Aeronautics and Astronautics
whereas envelope protection in the feedback using measurement of real aircraft response, on the basis that dangerous and aggressive command has already been filtered out by the reference model. For fast changing variable like load factor and AoA, direct feeding these two to the envelope protections in PIRM might need to take the nonlinear PIRM dynamics in to account, although instability was not observed in simulation. A easy way to decouple is to implement a separate feedback protection. When the protection signal is inserted after the integrator as for the flight path angle part, correct resetting must be done to unload the integrator during protection. 3. PCH for actuator saturation with feedback signal Although actuator dynamics is considered in the feedforward part in PIRM, actuator saturation due to excess controller efforts is still possible, a remedy is to build the actuator model, and feed the achiveble to the PIRM, then substract the unachievable difference in the actuator model in PIRM. C. Gain Tuning Controller tuning are done in two steps. Firstly a load factor controller is designed focusing on tracking performance, controller gains for flight path control are designed with fixed load factor control. In case of flight path angle is not available for feedback, which is likely when Satellite Navigation System is temporally not working, the control system degrade to load factor control system . Then the requirement for flight path control is ensure steady state accuracy and wind disturbance rejection, in addition the degradation should not cause noticeable change in the closed loop dynamics. In this paper, only load factor controller is considered. 1. Ideal Closed Loop Dynamics Using the control law generated by the control structure, the ideal closed loop dynamics can be first written transfer function form in Eq.40-42, Note that they are highly structured in a homogeneous form ∆𝛾𝐾 (40) = 𝐺𝑟𝑚,𝛾 (𝑠) ∙ 𝐺𝑒𝑥𝑡𝑟𝑎 (𝑠) 𝛿𝑛,𝑐,𝑝𝑖𝑙𝑜𝑡 ∆𝑛𝑧,𝐵
𝛿𝑛,𝑐,𝑝𝑖𝑙𝑜𝑡 ∆𝑞 𝛿𝑛,𝑐,𝑝𝑖𝑙𝑜𝑡
= 𝐺𝑟𝑚,𝑛𝑧,𝐵 (𝑠) ∙ 𝐺𝑒𝑥𝑡𝑟𝑎 (𝑠)
(41)
= 𝐺𝑟𝑚,𝑞 (𝑠) ∙ 𝐺𝑒𝑥𝑡𝑟𝑎 (𝑠)
(42)
Where 𝐺𝑟𝑚,𝛾 (𝑠), 𝐺𝑟𝑚,𝑛𝑧,𝐵 (𝑠) and 𝐺𝑟𝑚,𝑞 (𝑠) are exactly the desired dynamics specified by the reference model, and all the extra dynamics 𝐺𝑒𝑥𝑡𝑟𝑎 (𝑠) in the three loops are exactly the same, with the detailed structure dependant on the error controller structure. Table 2. Extra Dynamics with Difference Error Controller
NO: 1
Controller Structure P in 𝑛𝑧 , P in 𝑞 with feedforward
2
P in 𝑛𝑧 , PI in 𝑞 with feedforward
Extra Dynamics 𝐺𝑒𝑥𝑡𝑟𝑎 (𝑠) 𝑠 2 + 𝐾ℎ,𝑞 𝑠 + 𝐾𝑃,𝑛 𝐾ℎ,𝑞 𝑠 2 + 𝐾𝑃,𝑞 𝑠 + 𝐾𝑃,𝑛 𝐾ℎ,𝑞 3 2 𝑠 + 𝐾ℎ,𝑞 𝑠 + (𝐾𝑃,𝑛 𝐾ℎ,𝑞 + 𝐾𝐼,𝑞 )𝑠 + 𝐾𝑃,𝑛 𝐾𝐼,𝑞 𝑠 3 + 𝐾𝑃,𝑞 𝑠 2 + (𝐾𝑃,𝑛 𝐾ℎ,𝑞 + 𝐾𝐼,𝑞 )𝑠 + 𝐾𝑃,𝑛 𝐾𝐼,𝑞 4 𝑠 + 𝐾ℎ,𝑞 𝑠 3 + 𝐾𝑃,𝑛 𝐾ℎ,𝑞 𝑠 2 + 𝐾ℎ,𝛾 𝐾𝑃,𝑛 𝐾ℎ,𝑞 𝑠 + 𝐾𝐼,𝛾 𝐾𝑃,𝑛 𝐾ℎ,𝑞 𝑠 4 + 𝐾𝑃,𝑞 𝑠 3 + 𝐾𝑃,𝑛 𝐾ℎ,𝑞 𝑠 2 + 𝐾𝑃,𝛾 𝐾𝑃,𝑛 𝐾ℎ,𝑞 𝑠 + 𝐾𝐼,𝛾 𝐾𝑃,𝑛 𝐾ℎ,𝑞
PI in 𝛾 with feedforward P in 𝑛𝑧 , P in 𝑞 with feedforward 4 P in 𝛾 𝑠 3 + 𝐾ℎ,𝑞 𝑠 2 + 𝐾𝑃,𝑛 𝐾ℎ,𝑞 𝑠 + 𝐾𝑃,𝛾 𝐾𝑃,𝑛 𝐾ℎ,𝑞 P in 𝑛𝑧 , 𝑠 3 + 𝐾𝑃,𝑞 𝑠 2 + 𝐾𝑃,𝑛 𝐾ℎ,𝑞 𝑠 + 𝐾𝑃,𝛾 𝐾𝑃,𝑛 𝐾ℎ,𝑞 P in 𝑞 with feedforward By observing the analytical structure of 𝐺𝑒𝑥𝑡𝑟𝑎 (𝑠), it can be concluded by correctly selecting the error controller structure, a DC gain of 1 can be achieved. The role of difference gains in their structure can be identified. In the following design, controller form NO. 1 is selected in design and analysis 3
2. Close Loop in Case of Parametric Uncertainty With controller No. 1, consider uncertainty ONLY in 𝑀𝛼,𝑛 , 𝑀𝑞,𝑛 , with ∆𝑀𝛼,𝑛 = 𝑀𝛼,𝑛 − 𝑀𝛼 and ∆𝑀𝑞,𝑛 = 𝑀𝑞,𝑛 − 𝑀𝑞 𝐺𝑒𝑥𝑡𝑟𝑎 (𝑠) =
(𝑠−𝑍𝛼,𝑛 )(𝑠 2 +𝐾ℎ,𝑞 𝑠+𝐾𝑃,𝑛 𝐾ℎ,𝑞 ) 2 𝑠 3 +[(𝐾𝑃,𝑞 +∆𝑀 ⏟ 𝛼,𝑛 )−(𝐾𝑃,𝑞 +∆𝑀 ⏟ 𝑞,𝑛 )−𝑍𝛼,𝑛 ]𝑠 +[(𝐾𝑃,𝑛𝐾ℎ,𝑞 +∆𝑀 ⏟ 𝑞,𝑛 )𝑍𝛼 ]𝑠−𝐾𝑃,𝑛 𝐾ℎ,𝑞 𝑍𝛼,𝑛
Uncertainty ONLY in 𝑀𝛼,𝑛 , 𝑀𝑞,𝑛 , 𝑍𝛼,𝑛 , with ∆𝑀𝛼,𝑛 = 𝑀𝛼,𝑛 − 𝑀𝛼 , ∆𝑀𝑞,𝑛 = 𝑀𝑞,𝑛 − 𝑀𝑞 and ∆𝑍𝛼,𝑛 = 𝑍𝛼,𝑛 − 𝑍𝛼 11 American Institute of Aeronautics and Astronautics
(43)
𝐺𝑒𝑥𝑡𝑟𝑎 (𝑠) =
∆𝑍𝛼,𝑛
(1− )(𝑠−𝑍𝛼,𝑛)(𝑠 2 +𝐾ℎ,𝑞 𝑠+𝐾𝑃,𝑛 𝐾ℎ,𝑞 ) ⏟ 𝑍𝛼,𝑛 ∆𝑍𝛼,𝑛
2 𝑠 3 +[(𝐾𝑃,𝑞 +∆𝑀 )𝐾 𝐾 +𝑍 ∆𝑀 −𝑀𝛼,𝑛 ∆𝑍𝛼,𝑛)−(𝐾𝑃,𝑞 +∆𝑀 ⏟ 𝛼,𝑛)]𝑠 +[((1− ⏟ 𝛼,𝑛)]𝑠−𝐾𝑃,𝑛 𝐾ℎ,𝑞 (𝑍𝛼,𝑛 −∆𝑍 ⏟ 𝛼,𝑛 ) ⏟ 𝑞,𝑛)−(𝑍𝛼,𝑛−∆𝑍 ⏟ 𝑞,𝑛 )(𝑍𝛼,𝑛−∆𝑍 ⏟ 𝑍𝛼,𝑛 𝑃,𝑛 ℎ,𝑞 ⏟𝛼,𝑛 𝛼,𝑛
(44) It can be observed how the uncertainty enter the extra dynamics and the physical meaning of controller gains is to suppress these perturbations, stability properties can also be concluded by Routh Hurwiz Criteria when observing the denominator. The numerator zeroes are independent of parameter uncertainties. Note also that the hidden zero becomes visible when inversion error is nonzero (compare Eq. 43 or Eq.44 with extra dynamics in No.1 Controller Structure in Table 2), as long as the aircraft is away from stall, the pole and zero cancellation will not lead to stability problem. When the aircraft does stall, the command inversion should not schedule with angle of attach to the post stall region. In terms of state space form, choosing controller structure No.1, (45) 𝑥̇ 𝑐𝑙 = 𝐴𝑐𝑙 𝑥𝑐𝑙 + 𝐵𝑐𝑙 𝛿𝑛,𝑐,𝑝𝑖𝑙𝑜𝑡
𝑍𝛼,𝑛 Δ𝑛𝑧,𝐵,𝑟 Δ𝑞𝐾,𝑟 𝑥𝑐𝑙 = Δ𝑛𝑧,𝐵 , 𝐴𝑐𝑙 = Δ𝑞𝐾 [ 𝑒𝐼,𝑛 ]
−
𝑔 𝑉𝐺𝐾,0 ∙ 𝑍𝛼,𝑛
− 𝑀𝛼,𝑑𝑒𝑠
0 −
[
𝑔(𝑀𝛼,𝑑𝑒𝑠 + 𝐾𝑃,𝑛 𝐾ℎ,𝑞 ) 𝑉𝐺𝐾,0 ∙ 𝑍𝛼,𝑛
𝐾𝑃,𝑛
𝑉𝐺𝐾,0
0
0
0
𝑀𝑞,𝑑𝑒𝑠
0
0
0
0
𝑍𝛼
𝑔
𝑍𝛼,𝑛
𝐾ℎ,𝑞 + 𝑀𝑞,𝑑𝑒𝑠
𝑔(𝐾𝑃,𝑛 𝐾ℎ,𝑞 ) 𝑉𝐺𝐾,0 ∙ 𝑍𝛼,𝑛
0
−𝐾𝑃,𝑛
−
𝑉𝐺𝐾,0 𝑔
𝑍𝛼
−𝐾𝑃,𝑞 0
0 𝑔(𝐾ℎ,𝑞 ) 𝑉𝐺𝐾,0
0
]
0 −𝑀𝛼,𝑑𝑒𝑠 + 𝑀𝑞,𝑑𝑒𝑠 𝑍𝛼,𝑛 0 𝐵𝑐𝑙 = −𝑀𝛼,𝑑𝑒𝑠 + 𝑀𝑞,𝑑𝑒𝑠 𝑍𝛼,𝑛 [ ] 0
(46)
(47)
3. Rule of Thumb Tuning From the structure of No.1 error controller, 𝐾𝑃,𝑛 𝐾ℎ,𝑞 maily determine the bandwidth of the extra dynamics, and 𝐾𝑃,𝑞 determines the damping of the etra dynamics, temporarily let, (48) 𝐾𝑃,𝑞 = 𝐾ℎ,𝑞 it can be concluded 2 𝐾𝑃,𝑞 (2𝜁𝜔)2 𝐾𝑃,𝑞 = = = 4𝜁 2 𝐾𝑃,𝑛 𝐾ℎ,𝑞 𝜔2 𝐾𝑃,𝑛
(49)
Choose 𝜁 = √2⁄2, then (50) 𝐾𝑃,𝑞 = 2𝐾𝑃,𝑛 Since the actuator bandwidth is about 30 𝑟𝑎𝑑 ⁄𝑠, and pilot is mainly working in frequency range from 1~10 𝑟𝑎𝑑 ⁄𝑠, select the bandwidth of 10 𝑟𝑎𝑑 ⁄𝑠 appears be a good estimation. Translating into equations is (51) 𝐾𝑃,𝑛 𝐾ℎ,𝑞 = 100 Solving Eq.50 to Eq.51, 𝐾𝑃,𝑛 = 7.07, 𝐾𝑃,𝑞 = 14.14 Choose 𝐾𝑃,𝑞 = 14.14 would lead to the damping of √2⁄2, but a higher damping can be obtained, which might help to reduce the tracking error. 12 American Institute of Aeronautics and Astronautics
(52)
Table 3. Rule of Thumb Tuning According to Extra Dynamics Bandwidth
[𝐾𝑃,𝑛
Bandwidth (rad/s)
5 6 7 8 9 10 11 12
√2 𝐾𝑃,𝑞 𝜁= , =2 2 𝐾𝑃,𝑛 [3.5355 7.0711] [4.2426 8.4853] [4.9497 9.8995] [5.6569 11.3137] [6.3640 12.7279] [7.0711 14.1421] [7.7782 15.5563] [8.4853 16.9706]
𝐾ℎ,𝑞 = 𝐾𝑃,𝑞 ]
√3 𝐾𝑃,𝑞 , =3 2 𝐾𝑃,𝑛 [2.8868 8.6603] [3.4641 10.3923] [4.0415 12.1244] [4.6188 13.8564] [5.1962 15.5885] [5.7735 17.3205] [6.3509 19.0526] [6.9282 20.7846] 𝜁=
√3 𝐾𝑃,𝑞 , =4 2 𝐾𝑃,𝑛 [2.5000 10.0000] [3.0000 12.0000] [3.5000 14.0000] [4.0000 16.0000] [4.5000 18.0000] [5.0000 20.0000] [5.5000 22.0000] [6.0000 24.0000] 𝜁=
4. Tuning for Hiding Error Dynamics Let the coupling term from reference command to the aircraft response to be zero in the 𝐴𝑐𝑙 matrix in Eq. 46 gives (53) 𝐾𝑃,𝑛 𝐾ℎ,𝑞 = −𝑀𝛼,𝑑𝑒𝑠 , 𝐾ℎ,𝑞 = −𝑀𝑞,𝑑𝑒𝑠 This will lead to low gain design. If there is no augmentation such as adaptive controller to reduce the inversion error, less uncertainty can be tolerated, as can be concluded also from the uncertainty analysis. 5. Time Domain Gain Optimization The optimization was set up by minimizing the load factor tracking error using differential evolution genetic algorithm, the stairs command magnitude was distributed from 0.1 to 1.2, command to activated load factor protection is also included. Here the protection limit was set to from -1 to 1.5, only optimizing the gain for small amplitude command. Time history of the closed loop response was recorded as input for the optimizer The advantage of using time domain optimization is nonlinearities such as actuator situation can be included Table 4. Properties Incorporated by Time Domain Optimization
Actuator Torque Limit 900 [𝑑𝑒𝑔⁄𝑠 2 ]
Actuator Rate Limit 60 [𝑑𝑒𝑔⁄𝑠 ]
Actuator Position Limit 14 [𝑑𝑒𝑔]
The cost function to be minimized is of the form of min 𝑓 = 𝑒 𝑇 𝑄𝑒 Where 𝑒 = [𝑒𝑛 𝑒𝑞 𝜂 𝜂̇ ]′ , 𝑄 = diag(
1
,
1
(54) 1
,
Sensor Time Delay 10 [ms]
,
1
0.0012 0.0052 (𝑘𝑟𝑎𝑡𝑒 60∙𝜋/180)2 (𝑘𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 14∙𝜋/180)2
) , hence the error is
normalized with their maximum allowed error. In configuration 1, actuation power in the plant actuator is full, meaning 𝑘𝑟𝑎𝑡𝑒 = 𝑘𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 = 1, the actuator power considered in the reference model is also full, meaning 𝑘𝑟𝑎𝑡𝑒,𝑟𝑚 = 𝑘𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛,𝑟𝑚 = 1 In configuration 2, actuation power in the plant actuator is full, meaning 𝑘𝑟𝑎𝑡𝑒 = 𝑘𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 = 0.8, the actuator power considered in the reference model is also full, meaning 𝑘𝑟𝑎𝑡𝑒,𝑟𝑚 = 𝑘𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛,𝑟𝑚 = 0.6. Flight conditions at the four corners of the flight envelope are used for optimization separately, inversion model as assumed to be known exactly, actuator dynamics and limitation, and sensor delay is the main reason for being not “optimal” , as can be seen from Table 3, the optimal gains with respect to cost function in Eq. 54 are mostly aligned with the rule of thumb tuning in trend. Config.2 tuning has high gain than Config. 1 due to more strictly limiting on actuator dynamics and thus on the reference command from the reference model. The greedy nature of the optimization setup will generate higher gains for better tracking if the reference command is easier to track. The case of setting 𝐾ℎ,𝑞 = 𝐾𝑃,𝑞 is also investigated, a freedom of tuning damping of extra dynamics is lost, but the two-gain structure is simple. Since the Physical Dynamic Inversion structure is supposed to be auto gain scheduled. A fixed controller gains of FC1 in Config. 2 , 13 American Institute of Aeronautics and Astronautics
[𝐾𝑃,𝑛 𝐾ℎ,𝑞 𝐾𝑃,𝑞 ] = [7.5700 14.8121 17.1583] (55) is used for all four flight conditions, the simulation is shown in Fig 12. , basically well tracking performance in load factor is achieved, extra dynamics can be seen from the pitch rate tracking, due to the excitation from imperfect sensor and actuator dynamics. Although pilot is commanding load factor that goes beyond the limit, the reference model filtered out the excess command, tracking without overshooting load factor boundary can be seen from 9 to 13 seconds in Fig. 12. Fig. 13 shows the actuation power consumed, they are well below the limit. Table 5. Optimization Results by Genetic Algorithm
[𝐾𝑃,𝑛 𝐾ℎ,𝑞 𝐾𝑃,𝑞 ] Config. 2
[𝐾𝑃,𝑛 𝐾𝑃,𝑞 ] Config. 2
[12.1794 7.9858 10.5084]
[7.5700 14.8121 17.1583]
[8.1149 11.8847]
4000
[9.2761 13.2834 16.3419]
[𝟖. 𝟐𝟔𝟒𝟔 𝟏𝟓. 𝟒𝟖𝟕𝟑]
150
4000
[8.0662 15.8434 19.6513]
[9.6526 15.4133 18.6827] [8.2244 18.0022 21.9079]
150
200
[5.4023 23.7755 28.2159]
[6.1039
[7.4097 27.2738]
Flight Cond.
Speed [knots]
Altitude [meter]
1
70
200
2 3 4
110
[𝐾𝑃,𝑛
𝐾ℎ,𝑞 𝐾𝑃,𝑞 ] Config. 1
22.4946 26.9874]
[𝟕. 𝟗𝟕𝟔𝟏 𝟏𝟖. 𝟕𝟔𝟒𝟎]
Figure 12.Linear Simulation with Fixed Controller Gain for Different Flight Condition
Table 6. Uncertainty Level in Monte Carlo Simulation
Para. 𝑀𝛼 𝑀𝑞 𝑀𝜂 𝑍𝛼
Figure 13.Actuator Response
14 American Institute of Aeronautics and Astronautics
Uncer. Level (%) 20% 20% 20% 20%
Para.
Uncer. Level (%) 𝜏𝑠𝑒𝑛𝑠𝑜𝑟 𝑑𝑒𝑙𝑎𝑦 20% 20% 𝜂̇ 𝑚𝑎𝑥 20% 𝜂̈ 𝑚𝑎𝑥 -
6. LQR Output Feedback Tuning Controller design based on LQR is usually considered to have good stability margins, when the controller structure is fixed, output feedback is required to design the controller gains. In this application, numerical algorithm from Ref 42 is used for gain calculation. The first step for LQR gain tuning is to have a design model. As far as this paper is concerned, actuator dynamics in both the plant and reference model are included in the design model, with the original states as 𝑥 = [𝑛𝑧,𝑟 𝑞𝑟 𝜂𝑟 𝜂̇ 𝑟 𝑛𝑧 𝑞 𝜂 𝜂̇ 𝜖] Where 𝜖 = ∫(𝑛𝑧,𝑟 − 𝑛𝑧 ) The original plant plus the dynamic inversion is now seen as the “plant” for LQR An transformation has been done such that 𝑔 𝑔 (𝑛𝑧,𝑟 − 𝑛𝑧 ) 𝑞𝑟 − 𝑞 𝜂 𝜂̇ 𝜖 ] 𝑥̅ = 𝑇𝑥 = [𝑛𝑧,𝑟 𝑞𝑟 𝜂𝑟 𝜂̇ 𝑟 𝑉𝐺 𝜖 + 𝐺 𝑉𝐾,0 (−𝑍𝛼,𝑛 ) 𝐾,0 𝑥̅̇ = 𝐴̅𝑥̅ + 𝐵̅ 𝑢 Control law is in the form of 𝑢 = −𝐾𝐶𝑥̅ Here we design controller gains for with 𝐾ℎ,𝑞 = 𝐾𝑃,𝑞 , then 0 0 0 0 1 0 0 0 0 𝐶=[ ] 0 0 0 0 0 1 0 0 0 Only the fifth and sixth states in 𝑥̅ are used for feedback, the gains for the rest are made smaller enough by penalizing in the cost function with 𝑔𝑖 being large 𝐽 = ∫ 𝑥 𝑇 𝑄𝑥 + 𝑢𝑇 𝑅𝑢 + ∑ 𝑔𝑖 ∙ 𝑘𝑖2 Where 𝑄 = 𝐻𝑇 𝐻 and 𝐻 = [0
0
0
0
𝑔 𝑉𝐺𝐾,0 (−𝑍𝛼,𝑛 )
𝑖
1
1
0.01
10 ( 𝑝𝑖) 180
0
0
0]
𝑔1 = 𝑔2 = 𝑔3 = 𝑔4 = 𝑔7 = 𝑔8 = 𝑔9 = 5000, 𝑔5 = 𝑔6 = 0 1 7 ( 𝑝𝑖)2 180 𝑄 and 𝑅 are chosen by the principle weight more if the acceptable error is small Then the final controller gain for this structure is given by 𝐾(1) 𝐾𝑃,𝑛 = 𝑎𝑏𝑠 ( ) , 𝐾𝑃,𝑞 = 𝐾ℎ,𝑞 = 𝐾(2) 𝐾(2) The results are shown in Table 7 𝑅=
Table 7. Gain Design by LQR Output Feedback
Altitude [meter]
𝐾𝑃,𝑞 ] 𝐾𝑃,𝑛
𝐾𝑃,𝑞 ] 𝐾𝑃,𝑛
Flight Cond.
Speed [knots]
𝐻1 = 𝐻
𝐻1 = 3𝐻
1
70
200
[3.2936 11.2198]; [3.4066]
[4.0413 12.2672];[3.0354]
2 3 4
110
4000
[3.8345 9.6673];[2.5211]
150
4000
150
200
[4.3089 13.2533]; [3.0758] [4.6192 15.9555];[3.4542]
[5.2458 12.7040];[2.4217] [6.4549 17.2420];[2.6711]
[𝐾𝑃,𝑛
𝐾ℎ,𝑞 = 𝐾𝑃,𝑞 ]; [
[𝐾𝑃,𝑛
𝐾ℎ,𝑞 = 𝐾𝑃,𝑞 ]; [
[7.1630 21.7926];[3.0424]
7. Robustness Analysis by Monte Carlo Simulation To analyze the performance robustness, uncertainties are introduced in model accuracy of dynamic derivatives, actuator and sensor properties, as shown in Table 6.
15 American Institute of Aeronautics and Astronautics
Figure 15.Monte Carlo Sim. at FC2 Response
Figure 14.Monte Carlo Sim. at FC1 Response
Figure 17.Monte Carlo Sim. at FC2 Response
Figure 16.Monte Carlo Sim. at FC3 Response
It can be observed from Fig.14 to Fig.16 that acceptatble performance can be achived under 20% perturbation in the main model parameters using fixed gain error controller optimized for FC1. However as it is shown in Fig.18, Gains obtained by GA optimization has quite small stability margins. Therefore in the following assessment is donw by using LQR gains fixed at designed for FC1.
Figure 19.Stability Margin by LQR Response
Figure 18.Stability Margin by Optimization Response 16 American Institute of Aeronautics and Astronautics
Figure 20.Linear Simulation with Fixed Controller Gain for Different Flight Condition--LQR
Figure 21.Actuator Response--LQR
Fig.20 and Fig.21 show performance and actuation power usage with fixed error controller gain for the norminal case, relative smaller acutation is required for LQR design due to the relatively low gains
8. Robustness Analysis by Monte Carlo Simulation for LQR design Monte Carlo simulation show that the performace degrades due to uncertainty, and at large uncertainties, overshoot can be seen.
17 American Institute of Aeronautics and Astronautics
Figure 22.Monte Carlo Sim. at FC1 Response
Figure 24.Monte Carlo Sim. at FC3 Response
Figure 23.Monte Carlo Sim. at FC2 Response
Figure 25.Monte Carlo Sim. at FC2 Response
D. Flying Qualities Assessment Low Order Equivalent System method is used in analyzing both the reference model and the ideal closed loop dynamics. Fig 26,27 show the matching result of closed loop dynamics for LQR design, the mismatch is well within the Maximum Unnoticeable Added Dynamics (MUAD) envelope, indicating the matching is well done, then the result can be reliably used for modal criteria like CAP and damping ratio shown in Fig. 28 it can be observed both reference model with actuator dynamics and ideal closed dynamics are in the Level 1 flying qualities region Bandwidth Criteria is also used to illustrate Flying Qualities are preserved.
V. Conclusion In the physical dynamic inversion approach, what we want as desired dynamics specified by the reference model is separated from how we achieve by feedback tracking controller. The decoupling encourages flexibility of validating reference dynamics in the first step without propagating unachievable or unwanted dynamics to the next stage. Analytical form of error controller and ideal closed loop dynamics under physical dynamic inversion architecture are derived, insights of gain tuning can be gained from the controller structure. Due to the special form, error controller dynamics can be separated and seen as added dynamics. Pilot are expected to be tolerable to this extra or added dynamics once they are in the MUAD envelope. The design procedure is then quite systematic using this architecture by specifying desired in the reference model, and shaping structured extra dynamics with error controller. The intuitiveness also comes from that all the signals in both the physically integrated reference model and the feedback are physically interpretable, giving more design transparency. Little or no scheduling is needed because of the extensive use of equations of motion to physically provide tracking command, and physically generate incremental error controller command. Fixed error controller gains work well with difference flight conditions, and provide satisfactory flying qualities. LQR output feedback controller has descent performance with good stability margins compared to direct gain design GA optimized.. 18 American Institute of Aeronautics and Astronautics
Figure 26.Extra Dynamics in MUAD Envelope--LQR
Figure 27.LOES CAP and Damping-- LQR Flight
Figure 28.Bandwidth Criteria--LQR
Figure 29.Bandwidth Criteria--LQR Flight
Figure 30.PIO Bandwidth Criteria--LQR 19 American Institute of Aeronautics and Astronautics
Acknowledgments The author appreciates the insights from Jian Wang, Miguel Leitao, TUM, and Dr. Girish Chowdhary, Oklahoma State University in the discussions.
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