Control System Design Methodology

Our approach to control system design is summarized below. To address each of the points within [4, page 39, HYP.2.04.200], our control system design approach combines worst-case analysis with classical control to establish a starting point for the application of more powerful modern multivariable control methods.

1. Classical Approach. We propose to begin our design phase using a classical gain scheduled hierarchical control structure similar to that used on the X-43A [6]-[7] and found on many functioning aerospace systems [223]. While root locus ideas were used on the X-43A, classical frequency domain loop shaping ideas are likely to be much more insightful [211], [220]-[222] - particularly since such ideas extend to the multivariable setting [213]-[217], [228]. It is reasonable to question why we should try a classical approach first.
   
   It is understood that the classic sequential loop closing root locus approach used for X-43A is limited in that it does not (directly) take into account critical multivariable cross-coupling effects as do truly multivariable techniques [213]-[217] such as the ones we intend to use (see below). Nevertheless, such an approach can provide considerable insight regarding fundamental performance limitations associated with a fixed decoupled SISO controller structure (order). This typically yields valuable insight (qualitative and quantitative) about required control coupling that has been omitted. Such classically derived information, in addition to crucial bandwidth information, can serve as a starting point for the selection of design parameters associated with the advanced design methods to be exploited in our work (e.g. H∞ [158]-[160], [213]-[217], quasi-LPV [69]-[78], [207]-[209], GPC [166]-[170], [171]-[186], see below). Likewise, the more advanced methods (described below) can also provide insight into improving the classical design (i.e. introducing specific cross-coupling terms within the classical decoupled controller that can be optimized).

2. Modern Gain Scheduling: It is well known that scheduling on key aerodynamic variables (e.g. Mach number, AOA, SSA) can yield very good performance across a vehicle’s flight regime. This scheduling approach - a form of adaptive control [61]-[66], [80], [101], [135]-[156], [205] - has proven itself across a wide range of dynamical systems (e.g. aircraft, missiles). This approach, roughly speaking, relies on the gluing of individual linear designs based upon linear time invariant (LTI) models obtained by linearizing the nonlinear system being controlled (plant) at specific operating points; i.e. equilibria. Several issues associated with the approach (i.e. unresolved open questions) are as follows [61]:
   
   

   How do we select the operating points? How many operating (linearization) points should be selected? How do we design the individual controllers? How do we glue the resulting control designs to one another?

   
   Because these are formidable theoretical questions (which have contributed to increased design cycle time), researchers have proposed a new framework which directly uses the nonlinear model.
   
   The quasi-linear parameter varying (LPV) framework has received considerable attention over the last 15 years [43], [69]-[78], [207]-[209]. In short, it can be viewed as a form of modern gain scheduling which directly exploits the non- linear system dynamics and convex optimization [218] concepts which exploit linear matrix inequalities (LMIs) [217]. The key to this framework is to write the nonlinear model in state space form where the matrices are functions of key scheduling parameters (e.g. Mach, AOA, SSA) as follows:
   
   
   where θ represents a vector of signals that are measured in real-time (e.g. Mach, AOA, SSA). Under certain conditions, stability can be proven as well as performance guarantees [43], [69]-[78], [207]-[209]. Given this, it can be shown that in many cases efficient LMI-based methods can be used to design model-based multivariable controllers that satisfy a myriad of time and frequency domain specifications [217]. Moreover, the resulting model-mimicking ( i.e. model-based) controller is automatically scheduled - taking the form:
   
   

   A particularly attractive feature of the LPV approach is that it can be used to ensure that closed loop modes (rigid, flexible body, servoelastic, etc.) are in pre-specified regions of the left half plane (e.g. sector/wedge to control damping and settling time) while maintaining desirable closed loop sensitivity properties over a range of flight conditions;
   
   When combined with
   
   (1) performance objectives that inherently yield stabilizing controllers with nominal robustness properties; e.g. H∞,
   
   (2) frequency-dependent weighting functions that can be used to achieve meaningful design specifications [158]-160], [213]-[217]; e.g. bandwidth,
   
   (3) parameter-dependent weighting functions that can be used to alter the nonlinear characteristics of the closed loop system and accommodate fast or slow parameter changes (e.g. uncertain aero-thermo-elastic dynamics),
   
   (4) worst-case analysis/robustness concepts [213]-[217],
   
   the quasi-LPV design framework can be quite powerful for generating high performance robust control systems [43], [69]-[78], [207]-[209].

3. Constraint Enforcement Subsystems. Methods for smoothly altering the nominal control design to ensure that variable limiting, constraint enforcement, envelope protection, and regime protection have also received considerable attention over the past 15 years [187]-[198], [199], [213]. These include model-based look-ahead (simulation) methods that use approximate future predictions to appropriately scale-back control effort when it is necessary (e.g. governor methods [187]-[198], [213]). These methods also include controller augmentation methods (e.g. observer based [199], [213]) that alter control effort on the basis of constraint conditions.
   
   Figure 6 shows how governor-based constraint enforcement systems can work.
   
   

   Figure 6: Visualization of Constraint Enforcement: Quasi-LPV Missile (top), X-29 (bottom)

   • Missile Normal Acceleration Limiting. The subplot on the left within Figure 6 corresponds to an unstable (nonlinear) quasi-LPV missile [189] operating with an LPV controller designed to follow normal acceleration commands. An error governor system is used to scale back the error signal (input to controller) so as to ensure that control limits are not exceeded. Since the missile is unstable over the flight regime, a limit exists on the maximum achievable acceleration. The governor ensures that the control limit is enforced, even for large commands, without an appreciable sacrifice in performance.
   
   
   
   • X-29 AOA Limiting. The subplot on the right within Figure 6 corresponds to an unstable forward-swept wing X-29 aircraft with a canard, symmetric flaps, and strake flap [157], [213] in which pitch is commanded. A reference governor is used to scale back the pilot pitch command so that angle-of-attack (AOA) is suitably limited based on the flight condition. Here, we note that the reference command is scaled back only when necessary - maintaining the high bandwidth properties of the original multivariable controller.
   
   
   

   
   When used together, the above constraint enforcement methods can ensure that the controlled system (i.e. uncertain plant) remains within a flight regime over which the model is known to be valid. As such, meaningful performance guarantees become possible. Such methods are particularly helpful in applications such as ours where key aerodynamic variables must be tightly controlled in order to ensure proper performance (e.g. AOA deviations can significantly impact air-breathing propulsion, control surface rate limits can significantly impact effective bandwidth and overall system performance [199]). Finally, the ideas associated with constraint enforcement can be used to address mode switching transients (e.g. improve X-43A results by 20% as directed within [4]). Although mode switching is not a major focus of this work, collaboration with NASA ARC can alter priorities .
   


4. Generalized Predictive Control (GPC). Generalized predictive control (GPC) - an extension of model predictive control (MPC) - can be thought of a method that attempts to do the former (2)-(3) via constrained real-time moving-horizon (non-convex) optimization that typically penalizes error signals as well as control effort. This method has showed great promise in many aerospace applications [166]-[186]. In short, the method can be thought of as a “generalized” moving horizon real-time “linear quadratic regulator (LQR) type” optimization:
   where f and g are typically selected to be quadratic functions in x and u and T is a finite (look-ahead) planning horizon, respectively. The optimization is computed every Tu time units. Here, (f, g, T, Tu) are ``design knobs'' used to achieve desirable closed loop properties. Also, specific variable constraints can be readily added.
   
   To date, results indicate that GPC shows great promise. When used in conjunction with well established LQR [212]-[215], [228] and moving horizon LQR, significant insight can be obtained and GPC can be powerful and systematic. It should be noted that like our quasi-LPV paradigm, GPC also comes with nominal stability (and robustness) guarantees under certain conditions [166]-[186]. This is not surprising, since it is LQR-like.
   
   
Other Methods and Concepts. While the above methods represent the core methods to be used and comprehensively compared in terms of performance, robustness, and design cycle time, we also plan borrow relevant ideas from Dynamic Inversion [81]-[106] and Adaptive Control [135]-[156]. Roughly speaking, dynamic inversion is expected to provide insight into the inversion of (invertible) nonlinearities and dynamics that are not too uncertain, while adaptive control will provide insight into adaptation (scheduling) mechanisms that have shown promise and methods that have exhibited sensitivity to unmodeled (typically high frequency) dynamics, and disturbances (e.g. atmospheric turbulence within the troposphere and stratosphere), and measurement noise.

Fundamental Goals. The following represent fundamental goals to be accomplished over the course of the proposed research.

• Design Cycle-Time Reduction. Overall, the goal is to reduce design cycle time by 50% by reducing the number of required analysis design cycles [4]. This is fundamentally tied to how easy it is to synthesize a suitable control system. Some methods for certain problems (and operating regimes) are “less transparent” then others and require far more “creativity.” In this sense, one fundamental goal, here, is to reveal the limitations of each approach in handling aero-thermo-elastic-propulsion coupling and uncertainty issues [22]-[27], [28]-[29], [30]-[32], [43], [53].
  


• Integration of Methodologies. More importantly, we intend to show how the above design methods can be used in unison to obtain insight and significantly reduce design cycle time by reducing the number of analysis/optimization cycles. Given the system-specific class of vehicles we are considering (i.e. waverider/gliderlike vehicles such as X-43A, X-51A, SOAREX) [5]-[27], we strongly feel that a systematic method which bridges the aero-thermo (fluid-kinetic) and control worlds is indeed possible.
  
  

We intend to show how the proposed methods can be used to systematically develop a robust Hi-C system which improves reliability in accordance with the 20% NASA directive (see earlier discussion). This can be addressed indirectly by providing enhanced multivariable robustness margins, but will require hi-fidelity simulation models from other research efforts (see [4, page 39, HYP.2.04.300], beyond the scope of the proposed work). For this reason, the PI plans to spend 1.5 months each summer (over the proposed 3 year grant period) to work closely with members of the Advanced Control Methods team at the NASA ARC (Dr. Marcus Murbach, lead). It should also be emphasized that the developed methods will be compared with ongoing parallel efforts at NASA ARC (e.g. their own GPC controller).