National achievements in control theory: The aerospace perspective

It is well known that among the first motivations for modern control theory were dynamic optimization problems in rocket launching and navigation in aerospace. These problems had become especially important in the forties and fifties due to requirement to minimize various costly resources and design parameters, such as flight time, amount (mass) of fuel, weight of the spacecraft, the drag forces and other items. This had to be done under various restrictions on control capacities and other complicating factors , such for example, as incomplete information on the system. In the precious funds of applied mathematical techniques there had long been stored an adequate tool for such problems: it is the Calculus of Variations. Problems in flight dynamics had become the earliest serious technical object for its application. A large number of new basic ideas for adapting Calculus of Variations to modern control problems and synthesizing them into modern control theory were elaborated in the course of investigations in flight dynamics. This presentation traces some seminal investigations, which were crucial for related theoretical developments in former Soviet Union and present Russia and had also influenced related research beyond national borders. Such investigations had good historical precursors in the earlier mathematical works of P.L.Chebyshev, A.M.Lyapunov, A.A.Markov, the works in mechanics by N.E.Zhukovski and S.A.Chaplygin and the activities in dynamic systems theory of the thirties (A.A.Andronov, L.S.Pontryagin et al.). The present paper is confined only to deterministic problems in trajectory analysis, control and optimization within the framework ofmathematical theory of controlled processes. The national community of researchers involved in these topics was enormous, including those in the Academy of Sciences, the Universities and the numerous institutions and plants supervised by related industrial ministries. While giving tribute to all those involved, this paper does not claim to give a full review of available publications, concentrating on what the authors believe to be the seminal issues in the field and their role in future directions of research. This publication will therefore inevitably have a subjective flavour. We sincerely apologize to all those whose contributions may have been missed. 1. CONTROL IN FLIGHT DYNAMICS As mentioned in the above, the early motivations for modern control theory were problems of dynamic optimization in rocket launching and navigation in aerospace. The first mathematical techniques to be used for these problems were sought for within the Calculus of Variations. Indeed, flight dynamics problems seem to be the earliest serious technical object for its application. And indeed, a large number of new basic ideas for adapting Calculus of Variations to problems in control and synthesizing these within a framework for control theory were elaborated precisely in the course of investigations in flight dynamics. However the new applied problems of optimal control for dynamic processes essentially differed from canonical propositions in Calculus of Variations. The equations of motion for aircraft’s mass center in an inertial frame of reference have the form mV̇ = R +G+ P , ṙ = V, ṁ = −βf , (1) where r and V, are, respectively, the radius vector and the velocity vector of the aircraft’s mass center; m is the mass; βf is the mass (propellant) consumption per second; R is the resultant vector of the aerodynamic forces, P is the engine thrust vector; G = mg is the aircraft weight; and g is the gravitational acceleration. As a rule, the motion of aircraft is considered in the wind system of coordinates: V̇ = (1/m)[P cos(u− θ) cosβ− −X cos(−β) + Z sin(β)−G sin θ], θ̇ = (1/mV ){[sin(α− θ) cos γ+ + cos(α− θ) sinβ sin γ]−X sinβ sin γ+ + Y cos γ − Z cosβ sin γ −G cos θ}, ψ̇ = (1/mV cos θ){P [sin(α− θ) sin γ− − cos(α− θ) sinβ cos γ] +X sinβ cos γ+ + Y sin γ + Z cosβ cos γ}, ḣ = V sin θ, ṁ = −βf , ẋ = V cos θ (2) where V is the velocity, h is the altitude, x — the range on the Earth’s surface; θ — the angle of inclination of the trajectory to the local horizon; α — the angle of attack; ψ — the yaw angle; β — the angle of slide slip; γ is the angle of bank; θ — the angle between the thrust vector and the velocity vector; P — the engine thrust, which is a deterministic function of h, V and βf , X, Y , Z are the aerodynamic drag, the lift, and the side force respectively: 1 Supported by RFFI (No. 03-01-00663), the program “Universities of Russia” (project No. 03.03.007) and the program of the President of Russian Federation for the support of the scientific research of the leading scientific schools (No. NSh-1889.2003.1) X = cx(M,α)(ρ(h)V /2)S; Y = cy(M,α)(ρ(h)V /2)S; Z = cz(M,α)(ρ(h)V /2)S, (3) where ρ(h) is the atmospheric density; S is the effective wing area; M the Mach number, equal to the ratio of the flight velocity to the velocity of sound a(h) at given altitude; the cx, cy, cz are the aerodynamic coefficients. Equations (1) and (2) relate two essentially different groups of variables. The variables r, V, m with components h, V , θ, ψ, x, z and m in the wind system of coordinates, enter the equations together with their first derivatives and thus characterize the state of the process; the number of these variables is equal to the order of the system. The variables α, β, γ, φ and βf enter the equations without their derivatives and thus are the controls. Classification of the variables into phase coordinates and control elements is closely linked with the particular choice of mathematical model for the system being controlled. In some problems the mathematical model (system (1)) does not provide a sufficiently accurate description of the actual behaviour of the aircraft, and it can be improved by supplementing it with an equation of the angular motion of the aircraft about its mass center. The variables α, β, and γ become the phase coordinates, and the rudder deflection angles assume the role of control elements. On the other hand, in some problems, certain phase coordinates may be upgraded to the status of control elements without detrimental effects; this would involve dropping the corresponding differential equations from the mathematical model. This approach was actually applied by some authors in solving the problems of powered ascent of aircraft when the trajectory inclination angle θ was used as a control element. In addition to differential equations, we have to consider a variety of constraints on the variables, which stem from the particular properties of the system being controlled. The following typical constraints are imposed on aircraft flying in the denser atmospheric layers: the altitude h ≥ 0, the angle of attack αmin ≤ α ≤ αmax; the dynamic head q = 1/2ρV 2 ≤ qmax, the total overload N = [X2(h, V, a) + Y2(h, V, a)]1/2G ≤ Nmax, and the surface temperature Tw(h, , V, a) ≤ Tw,max. These constraints define conditions for actual (current) time t < T . Certain additional conditions are also imposed on the initial and final states of the system. For example, a vehicle may be designed to transport a payload from starting point h0, x on the ground, where it was at rest (V0 = 0) with starting mass m0, to a circular orbit at assigned altitude h1 (θ1 = 0, V1 = Vcir). The latter conditions define the terminal state set M in (??). Examples of optimality criteria for aircraft are that the flight range ∫ θ τ V cos θdt should achieve maximum, the flight duration T = θ − τ or fuel consumption m(τ) − m(θ) should achieve minimum, the final altitude h(θ) or the final velocity V (θ) should achieve maximum. The early work in this field was due to D. E. Okhotsimsky, T.M.Eneyev, V.A.Egorov and their colleagues (see Okhotsimsky, 1946; Okhotsimsky and Eneyev, 1957; Egorov, 1958), as well as to I. V. Ostoslavskii and A. A. Lebedev (1946). We shall return to their contributions after a tour to basic theory. 2. THEORETICAL ACHIEVEMENTS. CONTROL THEORY. PONTRYAGIN’S MAXIMUM PRINCIPLE The process of formalizing and analyzing applied problems of control generated an array of new mathematical ideas. In the nineteen-fifties and sixties this let to the initiation of a new branch of applied mathematics, namely, the “mathematical theory of optimally controlled processes” or, a broader engineering version known simply as “the theory of control”. Loosely speaking, control theory provides us with two basic methods to investigate optimal processes: the Maximum Principle of L.S.Pontryagin — a generalization for nonsmooth functions and constraints of the Euler–Lagrange variation method and the method of Dynamic Programming due to R. Bellman — a generalization of the classical Hamilton–Jacobi method which has been recently propagated to nonsmooth functions as well in the form of generalized “viscosity solutions” and their equivalents. The first method is adequate for the problem of open-loop programmed control while the second method is for the problem of feedback “closed-loop” control synthesis. We now present a description of Pontryagin’s Maximum Principle which was published in a series of pioneering publications (Pontryagin, 1958; Pontryagin et al., 1962), and at the ICM (International Congress of Mathematicians) in 1958 (Edinburgh) and at the First IFAC Congress in Moscow in 1960. Pontryagin’s Maximum Principle is a proposition which gives relations for solving the variational problem of optimal open-loop control. In general this is a nonclassical variational problem which allows to treat functions and constraints that are beyond those considered in classical theory, but are very natural for applied problems. The Maximum Principle was formulated in 1956 by L. S. Pontryagin, and further developed by himself and his associates V. G. Boltyanski, R. V. Gamkrelidze, E. F. Mischenko followed by many other investigations. It was motivated by new problems in automation and aerospace engineering, initiating the “mathematical theory of controlled processes”. The maximum principle was and is broadly used for solving applied problems of control and other problems of dynamic optimization. It has triggered numerous generalizations and applications. The basic necessary conditions from classical Calculus of Variations follow from the Maximum Principle. In many Western publications the Maximum Principle of Pontryagin is also referred to as the “the Minimum Principle” (By changing signs in some of the upcoming relations the “maximum condition” of the sequel may be rewritten in the form of a minimum condition). We now proceed with a more detailed formulation. The Typical Problem of Open-Loop Control. One of the typical problems of optimal open-loop control is as follows. Given are the vector-valued equations of system model dx/dt = f(x, u), (4) where x ∈ IR is the n-dimensional state of the system and u ∈ IR — the m-dimensional control. Also given are the starting point x and the terminal point x: x(0) = x, x(τ) = x. (5) Relations (??) are the boundary conditions. The range of the control is the constraint set P. Problem OOLC of Optimal Open-Loop Control is to find such a function u(t) which would steer the system from starting point x to terminal point x(τ) = x under constraint