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CONTROL SYSTEMS OF MOVING OBJECTS

Methods for Suppressing Nonlinear Oscillations in Astatic Auto-Piloted Aircraft Control Systems B. R. Andrievskya, b*, N. V. Kuznetsova, c**, and G. A. Leonov c*** a

b

St. Petersburg State University, St. Petersburg, Russia Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, Russia c University of Jyväskylä, Jyväskylä, Finland *e-mail: [email protected] ** e-mail: [email protected] *** e-mail: [email protected] Received September 1, 2015; in final form, December 6, 2016

Abstract—This review is devoted to the control problem with constraints on the magnitude and rate of change of the control action in aircraft control systems. In engineering practice, it is accepted to use actuators with a sufficiently large level of magnitude and energy supply. Theoretical investigations in the field of systems with saturation in the control loop should provide engineers with design methods for the minimization (as much as possible) of the weight–size and energy characteristics of actuators. Therefore, such investigations are highly relevant for practice. The review is focused on a specific effect that can occur when there is an integral component in the control law, the so-called integrator windup, that leads to the deterioration of the quality of system processes, a significant increase in control errors, sometimes to the loss of function associated with the occurrence of large amplitude limit cycles, and even to a loss of stability. The occurrence of this effect in aircraft flights can have dramatic consequences. Thus, the suppression of this effect is an urgent task for designing aircraft control systems. The main part of this review is devoted to the corresponding methods. DOI: 10.1134/S1064230717030042

INTRODUCTION The urgent nature of the control problem with constraints on the magnitude, rate, and energy of the control has long attracted the attention of scientists and developers of automatic control systems. Thus, investigations in this area have been carried out intensively for decades, and the bibliography on the subject is quite extensive. First of all, the fundamental works of L.S. Pontryagin et al. on the maximum principle in optimal control problems with constraints carried out in the 1960s should be noted [1–3]. The analysis of the controllability and stability of nonlinear systems with limited control resources is also considered in [4], in which the effect of constraints on the level, energy, and pulse of the control action is investigated. Various aspects of the nonlinear plant control with constraints are also discussed in [5, 6]. Methods for optimizing and reducing the effect of constraints on the magnitude and rate of change of the control action are considered in a recently published work [7]. The combination of control and state constraints in aircraft control problems is analyzed in [8] in adaptive control systems with a predictive model. In the case of certain combinations of aircraft characteristics and external actions in the system with constraints on the rate and magnitude of the displacement of steering controls, self-oscillations of significant amplitude can occur, the so-called loss of stability “in the large.” The analysis of this phenomenon for aircraft control systems (both manned and unmanned) based on the harmonic linearization method can be found in [9, 10]. In particular, in [9] it is shown that for an aerodynamically stable aircraft in the angular longitudinal motion there can be a stable limit cycle with a small amplitude and an unstable limit cycle with a large amplitude. If the aircraft is aerodynamically unstable, one of two stable limit cycles with a small amplitude can be realized. In addition, there is also an unstable limit cycle. Its presence makes it necessary to investigate the stability of the aircraft with the longitudinal controller in the large, i.e., when the aircraft is affected by large disturbances (including pilot commands) that put it outside the amplitude boundary of the unstable limit cycle. One way to prevent such motion is to install a prefilter that limits the 455

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control signal’s rise rate. Similar results are obtained in [10], where the main focus is on the saturation nonlinearity that is characteristic of drives of aircraft control systems. The effect of saturation nonlinearities can cause so-called pilot-induced oscillations (PIOs) that impede piloting an aircraft [9, 11–19]. This phenomenon is characterized by diverging oscillations with an increasing amplitude of the angular velocities, accelerations, and angular motions of the manned aircraft. As noted in [11, 20], PIOs typically occur in cases when the pilot attempts to accurately maneuver the aircraft. There a number of well-known plane crashes caused by PIOs. During these events, the flutter pitch effect was observed in the airplane landing mode (i.e., there were pitch angular oscillations with increasing amplitude). Cases of a spacecraft’s uncontrolled rotation are also known [21, 22]. An investigation of the transient modes of such motion makes it necessary to develop a mathematical theory of the global analysis of orientation systems. The need to develop such a theory was pointed out by Academician B.V. Rauschenbach, who noted the complexity of spacecraft control during fast turns. The possibility of the occurrence of hidden periodic oscillations and hidden chaotic attractors in nonlinear systems that do not have a stable equilibrium state or in systems with a single equilibrium state is presented in [23–25], where the classification of different types of attractors and examples of the emergence of hidden oscillations, including aircraft control systems, are given (see also [26, 27]). Since automation devices have a significant influence on the dynamic characteristics of modern aircraft, it is necessary to take into account the interaction with the pilot in their development. The criteria for assessing the performance of highly-automated aircraft, the occurrence of PIOs, and techniques for modeling pilot behavior in the aircraft control loop are presented in [28–31] and the references contained there. Further in this review, the buckling phenomenon (oscillations of unacceptably large amplitude), which is particular but important from the practical point of view, in automatic systems with an integral component in the control law and saturation in the magnitude or speed of the control action is considered of the variety of nonlinear oscillations in control loops of manned and unmanned aircraft. In international publications, this phenomenon is called windup. Accordingly, measures to counter this phenomenon by introducing additional feedback and/or compensators are called anti-windup (AW). For brevity, hereinafter, this term is used in this paper. 1. LOSS OF STABILITY IN THE CASE OF THE CONTROL CONSTRAINT IN SYSTEMS WITH AN INTEGRATED CONTROLLER As noted in [7], in engineering practice, it is accepted to use actuators with significant reserves in terms of the level and energy of control in an attempt to avoid the negative effect of control constraints during the system operation. Theoretical investigations in the field of systems with saturation in the control loop should provide engineers with design methods for the minimization (as much as possible) of the weight– size and power characteristics of the actuators. Therefore, such investigations are highly relevant in practice. Around the middle of the last century, it was noted that when using integral control laws (integral (I), proportional integral (PI), and proportional-integral-derivative (PID)) in constrained control systems there can be a specific effect that leads to a significant increase in control errors and sometimes to the loss of system function. The occurrence of undesirable situations can be represented especially clearly if the system describes a plant in the form of a pure integrator with constraints on the input saturation covered by the negative feedback with the PI- or PID-controllers. In this case, the control error is integrated by the controller, but in the case of large error terms, it cannot be countered because of the saturation which leads to oscillations in the system that corresponds to the maximum possible amplitudes of the input action for the plant (integrator), deterioration of system performance, occurrence of limit cycles, and even to the loss of stability [17, 18, 32–36]. In practice, the saturation of controllers breaks the feedback loop of the control system (CS) with subsequent (primarily in the control of unstable plants) negative consequences. The windup phenomenon poses a serious threat for the automatic control of an aircraft. In the case of external disturbances in the aircraft steering system, along with the desired stable solution corresponding to the desired flight there can be other stable and unstable solutions that correspond to undesirable and dangerous aircraft behavior. In addition, for large input signal amplitudes, the desired solution can lose its stability, which can lead to catastrophic consequences [32]. There are many publications on the windup problem. The Scopus citation index returns about 1700 publications on the key words “windup & control,” including 190 to 630 citations in [24, 37–42]. Several papers, such as [7, 41, 43–47], are devoted to the synthesis of controllers that prevent windup (antiwindup). This review aims to give an idea of the available results, focusing mainly on the description of the windup phenomenon and approaches to suppressing it in relation to aircraft. Given the extensive bibliography and the limited scope of this article, many notable works are outside the scope of this review. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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2. EARLY WORKS ON PREVENTING INTEGRATOR EXCITATION Early methods for preventing integrator excitation were primarily heuristic. They lacked mathematical rigor. Reviews of these methods are given in [40, 43, 48]. One of the first attempts at theoretical justification of the then known methods of suppressing integrator excitation in the controller in the case of the saturation of the control action is made in [49] (in this paper, we use the abbreviation ARW (antireset windup), because the integral component in PI- and PIDcontrollers is sometimes called reset in the technical literature). In that paper, it is pointed out that at that time the main idea of suppressing integrator excitation consisted in limiting the output signal of the linear controller using additional feedbacks, so that the controlling variable (for example, the command to move the actuator rod) did not exceed the specified limits. The authors of [49] pose the following problem: to show the robustness of the system with ARW proposed in [50] compared with the nominal linear CS for scalar control using methods of the theory of nonlinear systems. Systems with ARW are considered in which the control error signal arrives at the integrator input via a nonlinear link that comprises operations of selecting the minimum and maximum values and output feedback of the PI-controller in the following manner. The control signal u(t ) supplied to the actuator is generated by the PI law with an ARW circuit: t

u(t ) = k P e(t ) + k I uI (t ),

∫

uI (t ) = σ I (τ)d τ,

(2.1)

0

where e(t ) = r (t ) − y(t ) is the control error, r (t ) is the setting (control) action, y(t ) is the plant output, k P and k I are proportional and integral controller transmission coefficients, and σ I (t ) is the signal at the integrator input calculated by the following formula

σ I = min ( k(u − u), max(e, k(u − u))) ,

(2.2)

where u and u are the upper and lower threshold values selected so that in the steady-state mode the control input of the plant is within the specified limits, k is a high transfer coefficient (in order to avoid a closed algebraic loop, an aperiodic link with a small time constant is additionally introduced in the implementation). Thus, in this circuit, for u ∈ [u , u], the correction signal is not supplied, and the nominal (usual in linear PI-controllers) error signal is fed to the integrator input, i.e., σ I = e . When the output control signal u overshoots the specified limits, the controller is covered by a deep negative feedback acting on the integrator input. The limits of the parameter values in (2.2) are obtained based on the circular criterion [51, 52]. These limits provide absolute stability depending on the type of Nyquist plot (amplitudephase-frequency characteristic) of the CS’s reduced linear part.1 Following [53], it is also recommended to estimate the CS performance stability η (own motion in the closed CS decay not slower than the function exp(−η t )) based on the circular stability criterion using the substitution in the transfer function of the linear part of the CS variable s = −η + i ω (i 2 = − 1) instead of s = i ω in the construction of the Nyquist plot. The authors of [54] investigate the antiwindup correction in the CS with a cascade (dual-loop) discrete controller, in which the control signal quantity generated by the PI-controller of the outer loop is limited (e.g., the constraint on the intensity of the current flowing through the engine winding in the speed control system). It is noted that because of discretization a high correction feedback coefficient leads to loss of stability. The investigation of stability in [54] is based on the Popov criterion for discrete systems [53, 55, 56]. A numerical example of the CS with the inertial plant of the first order is considered. In [57], an ARW method is proposed for a CS with several actuators with saturation and vector control u(t ) ∈ R p , in the formation of which the control law with an integral component is used. It is assumed that nonlinear blocks in the actuators are described by the static dependences with the upper and lower boundary values U iH and U iL , respectively (i = 1, 2, … , p ), which set the saturation levels. Within these limits, the outputs of the nonlinear blocks coincide with the input signals (as in (2.2)). In the proposed ARW circuit, the integrators in the controller are covered by simple nonlinear feedbacks with insensitivity: if the i th integrator output signal is in the range [U iL ,U iH ], the correction feedback signal is zero, and outside this range a negative feedback signal proportional to the integrator output with some coefficient α i occurs. The parameter α i is selected in the case of the CS synthesis. It is easy to note that in the saturation case the aperiodic link of the first order W I (s) = 1/(s + α) occurs in the control law instead of the integrator 1Note that in the nominal mode the nonlinear correction is inactive; therefore, it is natural to assume that the reduced linear

system designed for the unsaturated case is stable. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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with the transfer function W I (s) = 1/ s (here, s ∈ C is the Laplace transform argument). Thus, by each i th component of the control action the CS takes the Lurie form with two nonlinear blocks with a common input (the system’s linear output is the controller integrating link output). The CS’s linear part has two inputs: the control one (output of the link with saturation) and another input, the correction feedback nonlinear link output. In addition, the system is affected by the setting (control) action r (t ) generated outside the feedback. Assuming that the control signals (to the nonlinear block with saturation) generated by the controller are limited, it is easy to show that the nonlinearities considered in this problem belong to the class of nonlinear dependences f ( x) with one input and output that belong to the conical sector with 2 the radius ρ and center c : for all x ≠ 0 and for some ε > 0 the inequality ( f ( x) − cx)/ x < ρ 2 − ε holds. Sufficient CS stability conditions are derived from the multiloop (vector) circular criterion [58]. For illustration, the GE F404 jet engine model operating at an altitude of 10.7 km is considered in [57]. The model is of the third order. It has two control signals (fuel flow rate and nozzle area) and two output variables (flow rate under low pressure and temperature sensor readings). The linear control law is synthesized based on the LQR/LTR-method [59]. The simulation results given in the paper show a slight increase in the quality of control processes because of the introduction of the antiwindup correction. A systematic methodology for the synthesis of multivariable (with vector inputs and outputs) CSs for stable plants if there are several saturation blocks is presented in [60]. In that paper, it is noted that in the case of the vector control the saturation can lead not only to a windup of the integrator but also to a change in the direction of the vector control signal, which also leads to the CS malfunctioning. The method proposed in [60] is based on the introduction of a supervisory feedback so that when the setting and disturbing actions are sufficiently small, the CS operates as a nominal linear CS (synthesized without saturation). When relatively large actions leading to saturation occur, the control law is modified in order to ensure stability and to maintain, if possible, the nominal linear CS characteristics. In order to solve this problem, in [60] it is proposed to introduce a so-called error governor (EG) in the error signal circuit. It prevents the control signal from reaching saturation under any setting or disturbing actions (despite the fact that it is possible in principle). The following system is considered

x(t ) = Ax(t ) + B λ(t )u(t ),

y(t ) = Cx(t ),

(2.3)

where x(t ) ∈ R n , y(t ) ∈ R m , u(t ) ∈ R p , and λ(t ) is the scalar factor to be determined in the synthesis. The matrix A is considered neutrally stable (its eigenvalues have negative real parts, probably except for simple eigenvalues with a zero real part). For system (2.3), the problem of finding the coefficient λ(t ) that provides the finiteness of the output y(t ) (understood elementwise: yi (t ) ≤ 1 for all i = 1, … , m, t > t 0 ). The procedure for determining λ(t ) ∈ [0, 1] as a function of the current state x(t ) is obtained. This result is applied to the synthesis of the controller (dynamic compensator) of the following form

x c(t ) = Ac x c (t ) + Bcλ(t )e(t ),

u(t ) = C c x c (t ),

e(t ) = r (t ) − y(t ),

(2.4)

where x c (t ) is the vector of the controller’s state, r (t ) is the setting action, y(t ) is the plant output, and e(t ) is the control error. In order to generate the factor λ(t ), the logical feedback by the state x c (t ) of controller (2.4) is used. The F8 fighter’s longitudinal motion control problem is considered in [60] as one of the examples. From the graphs given in the paper, it can be seen that the parameter λ(t ) reaches the unit value for about half the duration of the transients. When the control errors become small relative to the saturation levels, the CS quite quickly reaches the nominal (linear) mode. 3. ACCOUNTING FOR THE LIMITED LEVEL AND RATE OF CHANGE OF THE CONTROL SIGNAL Apparently, [61] is the first paper in which the effect of saturation by the level and rate of change of the control signal are considered. As in [60], in [61] a systematic method for the synthesis of controllers, which ensures the stability and acceptable behavior of a closed system, is proposed for neutrally stable multivariable open linear CSs. The following designation is used to describe the saturation by level and rate of change

u s (t ) = rsat ( sat(u)) , JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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where sat(u) is the standard saturation function with one level. Equation (3.1) denotes the componentwise operation for the vector variable u(t ) ∈ R p (control action by the controller) as a result of which the control action u s (t ) ∈ R p entering a plant, limited by both the level and rate of change is calculated. The operator rsat(⋅) expands as follows. Components u si (t ) , i = 1, … , p of the vector u s (t ) ∈ R p are calculated by integrating the first order differential equations with saturation on the right side:

⎧k −1, ε i (t ) ≥ k −1, ⎪ −1 −1 u si(t ) = ⎨ε i (t ), − k < ε i (t ) < k , ⎪−k −1, ε (t ) ≤ −k −1, i ⎩

(3.2)

where ε i (t ) = sat(ui (t )) − u si (t ) is the error between the control signal (taking into account saturation) and the output of the tracking actuator of the first order with the rate limit by the ith control channel (i = 1, … , p ), k is a parameter that can be selected sufficiently large so that, when using saturation in the linear region, u(t ) would be approximately equal to u s (t ). The authors assume that there exists a linear system with the desired properties. As in [60], in the paper, an EG block is introduced in order to replace the error signal e(t ) = r (t ) − y(t ) with eλ (t ) by multiplying e(t ) by the variable parameter λ(t ). The signal e(t ) is modified only when the setting action is sufficiently large to cause the control saturation u(t ) in the magnitude or rate of change. In [62], an indirect reconfigurable CS of an aircraft is synthesized, in which they are recurrently identified in real time based on the least squares method with constraints that take into account the a priori information about the parameters (coefficients of static stability and efficiency of rudders). The identification results are used to redesign control laws using the Hopfield networks [63], in which the system dynamics take into account the introduction of the penalty function. The neural network generates the optimal program control law with the reference model. A variable that reduces the transfer factor in the closed loop when there is a risk of saturation is introduced in the feedback in order to prevent saturating the controllers. A similar approach is used in [64], where a combination of methods of the linear-quadratic optimal control and linear programming are used to construct an aircraft’s adaptive and reconfigurable CS taking into account the constraints in the rudder. In contrast to [62], in [64] a more rigorously substantiated method is considered for optimizing the tracking algorithm in which the saturation of the rudders is taken into account in the optimization to avoid exceeding the limits. Since the future behavior should be known for the optimization (which is unknown), in the paper it is extrapolated into some subsequent receding horizon [65], and therefore the obtained solution is suboptimal, not optimal. In [37, 42, 66], anti-windup bumpless transfer (AWBT) methods are presented in which taking into account the nonlinearities of the input signal is implemented by the following two-step synthesis procedure. An initial synthesis of the linear controller without taking into account the nonlinearities at the plant output and then the introduction of the AWBT correction in order to minimize the harmful effects of the input nonlinearities on the behavior of the closed CS. In this way, the standard controllers designed for standard CSs without constraints are adjusted to the occurrence of constraints. The controller’s synthesis is based on the passivity concept [67] and the multiplier theory [68, 69]. Sufficient stability conditions are reduced to equivalent linear matrix inequalities (LMIs) for the correct selection of multipliers [70, 71]. In the publications, sector nonlinearities, primarily, saturation nonlinearities, are considered. An attempt to translate the stability investigation methods based on solving the LMI in the controller synthesis methods was made earlier in [72], where a modified control task is used to synthesize the static and dynamic antiwindup corrections, and the connections between the synthesis and the solution of nonlinear matrix inequalities is observed. 4. DYNAMIC ANTI-WINDUP CORRECTION The authors of [73] considered the quadratic stability of closed CSs and the quality of linear CSs by the criterion + 2 in the case of input signal saturation. In greater detail, the system characteristics are investigated under a widely used linear antiwindup correction method, in which the linear filter is added to the linear CS synthesized based on the specifications to “the behavior of a closed system without constraints on the control.” It is noted that this approach is justified in practice, since usually systems operate mostly in the nominal mode, in which the control signal does not reach the constraints, and saturation manifests JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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itself on relatively short time intervals. Therefore, first of all, it is necessary to achieve the desired operating quality of the nominal system, while ensuring an acceptable behavior if saturation manifests itself. The plants described by the following linear equations of state are considered:

P

⎧ x p = A p x p + B p,uu + B p,w w, ⎪ ⎨ y = C p, y x p + D p, yuu + D p, yw w, ⎪z = C p, z x p + D p, zuu + D p, zw w, ⎩

(4.1)

where x p ∈ R p is the plant state, u ∈ R nu is the control, w ∈ R nw is the vector of setting actions (including n

disturbances, measurement errors, setting action, etc.), y ∈ R y is the vector of measured variables, n z ∈ R z is the vector of variables relative to which the control objective is formulated (it can also include control errors), and A p , B p,u , B p,w , C p, y , C p, z , D p, yu , D p, yw , D p, zu , and D p, zw are constant matrices of appropriate dimensions. In [73], it is assumed that a controller without constraints is constructed in the form of a dynamic output feedback n

C

⎧ x c = Ac x c + Bc, y y + Bc,w w + v 1, ⎨ ⎩ y c = C c x c + Dc, y y + D c, w w + v 2 ,

(4.2)

where x c ∈ R nc is the state of the controller, yc ∈ R nu is the controller output, v 1 and v 2 are additional inputs for introducing the antiwindup correction, and the constant matrices Ac , Bc, y , Bc,w , C c , Dc, y , and Dc,w have the appropriate dimensions. It is assumed that controller (4.2) is selected so that the closure of the linear system by the following constraints

u = yc ,

v 1 = 0,

v2 = 0

(4.3)

can be uniquely solved and ensure the stability of the closed linear CS (4.1)–(4.3). In [73], it is assumed that the control input signal of the plant depends linearly on the controller output as follows:

u = ϕ( yc ),

(4.4)

where the vector-function ϕ(⋅) is locally Lipschitzian and belongs to a certain incremental sector of [74] and ϕ(0) = 0 . In particular, the following saturation function understood componentwise can act as ϕ(⋅) :

ϕ( yc ) ≡ sat( yc ) = [sat1( yc1),...,sat nu ( ycnu )] , yci sat i ( yci ) = , max(1, yci / M i ) T

where M i > 0 are the given saturation levels (i = 1, … , nu ). The dynamic antiwindup compensator [73] is set by the following linear equations:

antiwind up

⎧ x aw = Λ1 x aw + Λ 2( yc − u), ⎪ ⎡v 1 ⎤ ⎨ ⎪v = ⎢v 2 ⎥ = Λ 3 x aw + Λ 4 ( yc − u), ⎩ ⎣ ⎦

(4.5)

where x aw ∈ R naw is the compensator state vector, v ∈ R nv is the compensator output, and Λ j , j = 1, … , 4 , are matrices of the appropriate dimensions. The desired properties of the stability and quality of the system with an antiwindup correction are presented in terms of Lyapunov functions. It is necessary to construct an antiwindup correction so that system (4.1), (4.2), (4.4), and (4.5) for any ϕ(⋅) of the given class has a positive-definite quadratic Lyapunov function V ( x) = x T Px (where x = col{ x p, x c , x aw }, P = P T > 0 ) such that for some ε > 0 and the given quadratic performance level γ its time derivative V by virtue of (4.1), (4.2), (4.4), and (4.5) satisfies the inequality T T T V < ε x x − 1 z z + γ w w γ

∀( x, w) ≠ 0 .

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qd uc

Nominal controller

yc u + +

Rudder gear

δ

Aircraft

461

[α, q]T

− v1 v2

+ δaw AW

[z+, u]T

+ + Fig. 1. Block diagram of an aircraft’s control system with antiwindup correction according to [76]. 2 Thus, the resulting system should be quadratically stable (for w ≡ 0 from (11) it follows that V ≤ −ε x ) and have an L2 -transfer coefficient of γ from w to z for any nonlinearity ϕ(⋅) of a given class (which can be easily seen by the time integration of Eq. (4.6)). In order to investigate the properties of system (4.1), (4.2), (4.4), and (4.5) and the antiwindup correction synthesis, the LMI method is used in the paper. It is shown that the linear antiwindup correction of the order coinciding with the plant order is always feasible for sufficiently large values of the L2 -norm and that there is a static antiwindup correction, which provides a common Lyapunov function for the open system and closed CS without constraints, if and only if the plant is asymptotically stable. The F8 fighter’s longitudinal angular motion control is considered as an example [60, 75]. The results are compared with those from [60] and with the controller without an antiwindup correction. The simulation results show that all of the above circuits are characterized by a significant transient overshoot. In order to improve the control quality, it is proposed to use an objective function that includes the pitch error and angular acceleration, which makes it possible to significantly improve the control quality in comparison with that obtained in [60]. The authors of [76] used the structure of the controller with an antiwindup correction proposed in [77], in which the connection between the nominal controller and the plant is supplemented with two antiwindup compensator outputs, to the input of which signals from the outputs of the nominal controller and saturation unit (aircraft rudder deflection δ ), as well as the values of the aircraft’s state variables are fed. The constraints on both the magnitude M of the rudder deflection, δ ≤ M , and on its rate R , δ(t ) ≤ R , are taken into account. The dynamics of the rudder gear are described by the following equation:

δ(t ) = R sgn ( M sat(u/ M − δ(t ))) ,

(4.7)

where sgn is the sign function and sat is the saturation function. The block diagram of the aircraft control system with an antiwindup correction by [76] is given in Fig. 1. In this figure, the angle of attack and the pitch rate of the aircraft are designated by α and q . The nominal controller is a dynamic system of the same order as the aircraft. The controller is described by the following equations

x c = g( x c , uc , q d ), yc = κ( x c , uc , q d ),

(4.8)

where yc is the controller output, and g(⋅) and κ(⋅) are functions to be selected during the controller synthesis. The nominal controller is synthesized without taking into account the limitations in the rudder gear based on the predefined nominal mode specifications. In [78], the controller’s synthesis procedure for a linear asymptotically stable plant is obtained based on the LMI solution and on the structure of the system with the antiwindup correction proposed in [77]. In addition, a nonlinear algorithm with the hysteretic switching of the controller coefficients from a set in order to improve the control quality is proposed. The authors of [79, 80] considered the antiwindup correction mechanism for linear stationary CSs with nonlinear constraints on the magnitude and deflection rate of the rudders. The procedure for the synthesis of a convex antiwindup control is developed based on the extended version of the circular criterion for the so-called LFT systems, which are a class of linear time-varying systems with a linear fractional dependence of the parameters. The compensator equations are obtained in an explicit form, which facilJOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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− e

+ K umrs

s

+

. δ

+η + v2

1 s

δ

ymrs

Fig. 2. Block diagram of an actuator (4.9).

itates their implementation. The effectiveness of the proposed control method is demonstrated for the F8’s linearized model [60]. The paper [81] is devoted to an antiwindup correction for a linear system with a strictly proper controller in the case of signal saturation by the level and rate of change. The selection procedure of the linear antiwindup correction coefficient is obtained based on the LMI solution and on the generalized sector condition. The coefficient affects the controller’s behavior to provide the local stability of a closed system and optimize the corresponding quality indicators. In [81], a new model, which is different from the previously used model in [80, 82–85], of the actuator of the CS with constraints on the magnitude and rate of change of the output, which has the following form

⎧δ = sat r (u mrs + K (umrs − δ) + v 2 ) , ⎨ ⎩ ymrs = sat m(δ),

(4.9)

where umrs , δ , and ymrs are vectors of inputs, states, and outputs of the actuator, respectively, m ∈ R p and r ∈ R p are vectors with strictly positive components setting constraints on the magnitude and rate of change of outputs ymrs and K = diag k i , k i > 0, i = 1, … , p . The signal umrs is considered to be differentiable. The saturation function at the level of a is denoted by sat a(v ) :

⎧a sgn(v ), sa t a(v ) = ⎨ ⎩a,

if v > a, ot h erw ise,

(4.10)

where sgn(v ) is the sign function. The additional input v 2 is not designed for the actuator simulation and is used in [81] to organize antiwindup correction. The structural diagram of the actuator model (4.9) by [81] is shown in Fig. 2. The basic idea used in the synthesis is to transform a closed system in which the modified controller is connected to the extended plant through the derivative of the source controller output that makes it possible to use the approach proposed in [39, 86] to take into account the magnitude and output rate, in which it is not necessary to take into account the rate at which the signal changes. For this purpose, the model of the dynamics of the CS is reduced to the following equivalent form:

⎧ x p = A p x p + B p,uu p + B p,w w, ⎪ 3 ⎨ y p = C p, y x p + D p, yuu p w + D p, yw w, ⎪z = C x + D u + D w, p, z p p, zu p p, zw ⎩ p

(4.11)

⎧ x c = Ac x p + Bc, y y p + Bc,w w + v 1, #⎨ ⎩ yc = C c, y x c + Dc, yu p + Dc,w w + Dc,v 2v 2,

(4.12)

v = L( yc − sat b ( yc )),

where

v = [v 1T,v 2T ]T,

(4.13)

u p = sat b ( yc ),

(4.14)

in which y p = col{y p, δ} , x p = col{x p, δ}. The vector variable w(t ) ∈ R nw refers to external actions, i.e., disturbances and measurement noise. The parameter L is the static antiwindup correction coefficient selected during the synthesis. The block diagram of system (4.11)–(4.14) is given in Fig. 3. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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− #

w

−y c

−xc

−u p +

v L

− 3

−xp

463

−y p

− q

Fig. 3. Block diagram of a transformed control system [81].

This paper discusses the following problems. Problem 1. For the given plant, nominal control law, and constraints m and r , to synthesize a modified CS (with an antiwindup correction) so that the following hold conditions hold: (a) For the given initial conditions on the plant and controller, as well as for the predetermined external action w , to ensure the coincidence of processes in the nominal and modified systems for all t , if under these conditions, saturation does not occur in the nominal system (i.e., the constraints on yc and y c are inactive). (b) For the given level s of the input signal,2 to find the smallest γ > 0 such that for the zero initial conditions and any input signal w(t ) such that w 2 ≤ s , z p ≤ γ w 2 hold. Problem 2. For the given plant, nominal control law, and constraints m and r , to synthesize the modified CS so that the following conditions hold: (a) Item (a) of Problem 1 holds. (b) The smallest parameter λ < 0 is found for w(t ) ≡ 0 . For this parameter, there is a region Ω of initial conditions in which the equilibrium state of a modified closed CS is locally exponentially stable with the damping coefficient λ . In order to solve the problems set in the paper, the LMI method is used. The theoretical results from [81] are illustrated by the problem of controlling the F8 fighter’s longitudinal angular motion considered in [60, 75]. The authors of [87] proposed the synthesis method of a static or dynamic antiwindup controller based on the LMI method. The synthesis method extends the existing LMI applications for developing robust filters or regulators in the forward loop controllers. The problem of controlling the angle of attack of a highly maneuverable missile, the model of the dynamics of which is shown in [88], is used as an example. The problem of a static antiwindup correction for an unstable aircraft in the case of saturation in the control channel is investigated in [89]. The standard approaches based on quadratic Lyapunov functions, the S-procedure, and nonlinearities with sector constraints are used (see [52]). The proposed approach is analyzed in terms of the expansion of the class of acceptable setting actions and increase of the region of the safe initial conditions for which the stability of a closed CS can be ensured. In [90], a robust antiwindup correction for improving the control quality in the aircraft’s lateral control channel is proposed, and its effectiveness is demonstrated. The antiwindup correction problem, in which a compromise is found between the control quality in the presence and absence of saturation in the actuator, is considered in [91]. The results are applicable to the F8 fighter’s lateral control. The case of large parametric uncertainties in the aircraft model with saturation in the actuators is given in [92, 93]. The authors proposed a robust adaptive linear-quadratic synthesis of the control law with an adaptive antiwindup correction for countering the temporal changes of the aircraft’s parameters. In [94], the problem of an antiwindup correction in discrete time is discussed. In [95], the results of [94] are used for the model example of controlling an advanced fighter. The authors of [96] proposed an antiwindup correction procedure based on a nonlinear CS reaction with nonlinearity of the saturation type in the step input action. This procedure is applied to the control problem of the longitudinal motion of a M2000 fighter. This approach is further developed in [97, 98] for the fastest tracking of the desired angle of attack without the associated steady-state error and with a high 2The inequality

w

2

≤ s , where w

2 2

t 0

= lim ∫ | w(τ)| 2d τ , is assumed to hold. t →∞

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level of control quality. In [99, 100], the LMI-based solution of the antiwindup correction problem is proposed for the construction of dynamic compensators of the complete and reduced orders using the modified sector condition for nonlinearities with a dead band. 5. SPACECRAFT CONTROL AND ANTIWINDUP CORRECTION The features of spacecraft control (SC) under constraints are investigated in [21, 22, 101–112]. The authors of [103, 104] presented a solution for the optimal control problem of the SC’s orbital orientation using a propulsion system limited by thrust. Rendezvous problems of SCs in circular orbits are considered in [105, 106]. In [113, 114], the unloading of the angular momentum of the SC’s inertial actuators for circular orbits is considered. In [114], the results obtained in [113] for solving the pitch channel problem are extended to the synthesis of the three-channel unloading system. The authors of [114] noted that because of the independence of the pitch channel from the other two channels the synthesis of the control for this channel is much easier than the synthesis of the control of the gyroscopically connected roll-yaw channels. A similar problem is also investigated in [107] devoted to the stabilization of an orbital SC’s angular position by three channels using inertia flywheels and the unloading of their angular momentum by jet engines or a magnetic system (current coils). Antiwindup correction methods are used in [107] for the synthesis of a nonlinear controller of the unloading system. This controller is introduced only when the flywheel’s angular momentum reaches a specified level which ensures the preservation of the stabilization quality at low angular momentum values, as well as a limited quality decrease at a high level of angular momentum. A static nonlinear controller with a dead band is synthesized by the example of the DEMETER satellite [115] using the LMI method. The objectives and methods of construction an antiwindup correction when controlling an SC are considered in [108–112]. In [108], an antiwindup problem when controlling satellites using jet microengines is considered. It is noted that a high degree of accuracy is required in advanced systems, which leads to the need for microengines that can generate an almost continuous thrust between zero and the maximum value [116, 117]. Although such engines meet the high accuracy requirements, their maximum power can be critically low and lead to the saturation of the control action. An antiwindup correction method for systems with actuators of this type is proposed in [108]. LMI methods, Lyapunov quadratic functions, and modified sector conditions are used for the synthesis. The application of the proposed method to control the SC along a single axis is illustrated. The investigation is continued in [110], where it is noted that since the microengine creates force along one direction, the following distribution problem arises: how can the required generalized force (or torque) be obtained given a redundant set of microengines? The distribution function is usually quite nonlinear, which greatly complicates finding the mathematical conditions that ensure the required quality and stability of the system. The authors of [111] also considered the saturation effect of a microengine on SC control processes. The basic idea for the antiwindup correction construction is to ensure the closeness of the system processes to the processes inherent in the linear models. Two methods are usually used for solving the problem: the first one is based on the calculation of the saturation feedback signal [46], and the second is associated with the selection of a dynamic antiwindup correction that includes the plant model [77, 118]. A numerical example of controlling the satellite by the pitch is given as an illustration. For this example, a detailed description of the procedure of the synthesis of a antiwindup correction and the results of a comparative analysis with other methods are given in [111]. In [112], the controlling the motion of satellites in formation is investigated. The control of the relative position of two satellites using a centralized controller that receives data on their coordinates and generates the vector yc that contains control signals for each satellite is considered. The approach from [110, 111] is used for the synthesis of the control system. 6. CONVERGENCE-BASED ANTIWINDUP CORRECTION In most of the papers mentioned above, the antiwindup correction problem is understood as the provision of the global asymptotic stability of the equilibrium state of an autonomous system with saturation and the constrained paths of a nonautonomous system with a small transfer coefficient + 2 between the input and the output of the system (see, e.g., [7, 73, 119]). In some papers a different approach is used based on the convergence property introduced by Demidovich [120]. The use of this approach in investigating the windup effect and its suppression are suggested in [121–123]. In [121, 122], an example is given showing the insufficiency of the global asymptotic stability of an unforced response of a nonlinear system with saturation for the stability of its behavior in the presJOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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ence of external actions (harmonic, in the example). In addition, the possibility is shown of the existence of different stable solutions depending on the initial conditions for the same input action. Similar results for the problem of controlling an aircraft’s yawing motion are presented in [35, 36, 124, 125]. It should be noted that for all the considered systems, the lack of convergence in the solution is caused by the integrator windup when controlling a neutral-resistant plant by the PI- and PID-controllers in the case of the saturation of the control action. Hence the approach according to which the windup is suppressed by converging the nonlinear system, which in turn can be checked by testing the frequency condition, is presented in [126]. The use of this method for the analysis of the robustness of an aircraft’s control system in relation to the antiwindup correction delay is shown in [127]. A similar problem is considered in [98], where the Padé approximant of the transfer function of the delay element and the LMI method are used, while the frequency approach used in [127] makes it possible to save the infinite-dimensional delay model with a transcendental transfer function. CONCLUSIONS A review of papers on controlling aircraft with constraints on the magnitude and deflection rate of the aircraft controllers indicates a long-term and growing interest of both design engineers and scientists in this subject. This interest is caused by the pressing and highly urgent problems inextricably linked with aircraft safety, as well as with the development of new, advanced aeronautical samples with a high level of energy efficiency and reliability, and the required handling characteristics. The scientific investigations carried out in the world should provide a compromise between these often conflicting characteristics at a higher quality level. ACKNOWLEDGMENTS This work was supported by the Russian Science Foundation (project no. 14-21-00041) and St. Petersburg State University. REFERENCES 1. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, Mathematical Theory of Optimal Processes, 4th ed. (Nauka, Moscow, 1983) [in Russian]. 2. V. G. Boltyanskii, Mathematical Method of Optimal Control (Nauka, Moscow, 1969) [in Russian]. 3. V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Optimal Control (Nauka, Moscow, 1979) [in Russian]. 4. A. M. Formal’skii, Controlability and Stability of Systems with Limited Resources (Nauka, Moscow, 1974) [in Russian]. 5. F. L. Chernous’ko, I. M. Anan’evskii, and S. A. Reshmin, Control Methods of Nonlinear Mechanical Systems (Fizmatlit, Moscow, 2006) [in Russian]. 6. A. M. Formalskii, Stabilization and Motion Control of Unstable Objects (Moscow, Fizmatlit, 2012; Walter de Grüter, Berlin, Boston, 2015). 7. S. Tarbouriech, G. Garcia, J. Gomes da Silva, Jr., and I. Queinnec, Stability and Stabilization of Linear Systems with Saturating Actuators (Springer, London, 2011). 8. V. N. Bukov, Adaptive Predictive Flight Control Systems (Nauka, Moscow, 1987) [in Russian]. 9. G. S. Byushgens and R. V. Studnev, Aerodynamics of an Aircraft: Dynamics of Longitudinal and Lateral Motion (Moscow, Mashinostroenie, 1979), p. 352 [in Russian]. 10. V. S. Berko, Yu. G. Zhivov, and A. M. Poedinok, “Approximate criteria of forced vibration stability of regulated objects with nonlinear actuator,” Uch. Zap. TsAGI 25 (4), 72–80 (1984). 11. Aviation Safety and Pilot Control: Understanding and Preventing Unfavorable Pilot-Vehicle Interactions, Ed. by D. T. McRuer and J. D. Warner (Committee on the Effects of Aircraft-Pilot Coupling on Flight Safety Aeronautics and Space Engineering Board Commission on Engineering and Technical Systems National Research Council National Academy Press, Washington, DC, 1997). 12. O. Brieger, M. Kerr, D. Leiβling, et al., “Anti-windup compensation of rate saturation in an experimental aircraft,” in Proceedings of the American Control Conference ACC 2007 (IEEE, New York, 2007), pp. 924–929. 13. O. Brieger, M. Kerr, J. Postlethwaite, et al., “Flight testing of low-order anti-windup compensators for improved handling and PIO suppression,” in Proceedings of the American Control Conference ACC 2008 (AACC, Seattle, USA, 2008), pp. 1776–1781. 14. O. Brieger, M. Kerr, D. Leiβling, et al., “Flight testing of a rate saturation compensation scheme on the ATTAS aircraft,” Aerospace Sci. Technol. 13, 92–104 (2009). JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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Translated by O. Pismenov

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