- Email: [email protected]

A method is presented for the design of optimal feedback controllers for large multivariable systems with subsystem sensitivity constraints. The weighted sum of subsystem and / or operational mode relative performance index s e nsitiviti e s is defined as the overall performance ind e x. The method is developed for lin e ar s y ste ms with quadratic p e rformance criteria and either full or partial state feedback. An e xa mple concerning the design of a stability augmentation system for a VTOL aircraft in the transition mode demonstrates the effectiveness of the design method.

INTRODUCTIO N In recent years much attention has b e en focused upon the application of modern control theory to the design of large multivariable systems and processes. Investigations of this type require an adequate definition of the problem in physical terms and a translation of this physical description into mathematical terms. Similarly, the objective or goal of the physical process must be mathematically tractable. The subsequent search for the control mechanism to attain the desired objective constitutes the fundamental problem of optimal control th e ory. In many design problems a single unifying objective function cannot be easily defined. Situations of this type arise most frequently in the analysis of large systems consisting of a complex interconnection of many subsystems, each possessing a separate design objective. In other situations, a suitable control mechanism is to be designed for a specific system for operation over a wide range of environmental conditions. Thus, the primary concern is not necessarily the optimal design at any given point, but rather a blending of the control mechanisms at each point into a common fixed structure providi~g acceptable overall system performance. The objective of this paper is to present a unifying approach to the evaluation and subsequent implementation of control * Dept. of Electrical Engineering, University of Toledo, Toledo, Ohio U.S.A. **National Aeronautic and Space Administration, Lewis Research Center, Cleveland, Ohio U.S.A.

339

mechanisms for large systems sensitive to the design limitations of major subsystems. Problems of this type occur in such diverse areas as power generation, transportation, economics, man-machine interaction, process control and aircraft flight dynamics. MULTIVARIABLE SENSITIVITY ALGORITHM Consider a system composed of N subsystems each characterized by a set of n first order linear time invariant differential equations X("t) == R X(t) + B Add Jet) ~ C X(iJ (1) where x(t) is the n dimensional state vector, u(t) the m dimensional input vector and y(t) the p vector of outputs. Formulating each subsystem optimization problem as an output regulator (1), a feedback control law is obtained, u(t)=-Ly(t). The gain matrix is determined by minimizing ~ ( 2) J( l) .=; t TR, [p] =x'Qx + .A-(TR, A..]J-c

±5 ! o

r [A-BLC] P

subject to

-t P[fl-BL...C.]

+G( +

r C LTRI-.c.. -=0

(3 )

where Q and R are positive semidefinite and definite symmetric matrices respectively. The optimum subsystem gain matrix and performance are denoted by L* and J* respectively. For any gain matrix, L1L*, a measure of the deviation of the performance index from the optimal point is indicated using relative sensitivity (2)

(J"(L.)-r(L)1/J"(L*j

SR(.L.)::;

(4)

which is always positive, hence system performance is always compared with an attainable value. Since each subsystem may be of different dimension the subscript i(i=1,2,···,N) will be used for identification purposes. To optimize the composite system, a performance index reflecting the interests and concerns of each individual subsystem is chosen to be of the form N K ~ ~ J(L, Ll. .. . L,...\ = S. (I... .) ?-rl.· == I (5) J...." ... .:al )

.)

J

)

LA' •

where N is the number of ;ubsystems, S~ is the relative sensitivity of the ith system and Ai is a weighting factor (or probability factor) associated with the ith system. Clearly, small relative sensitivity assures a design close to the optimum and, hence, a smaller influence in the final optimization procedure. Three specific cases concerning the gain optimization of equation (5) are apparent: (I) No Gains in Common - this represents a degenerate system whereby the subsystems are totally uncoupled. The solution of the optimization problem under these circumstances is equivalent to the optimization of each subsystem independently. The total N

num b er

0 f

"-z::.. un k nown ga1ns 1S n I - i=lmiPi.

(11) All gains in Common - This is representative of a multioperating point system where each subsystem is essentially the original system under different operating conditions. The total

340

number of unknown gains

1S

nII=mp.

(Ill) Complex gain structure - let Li and Lj represent the gain matrices for subsystem i and j respectively. If some of the gain elements of L . and L . are common then

L· = L·. J.J J

1

~r.. J

J

LJ · .=-L·· -rL.J IJ

where Lij represents the gains in common and Li and Lj are gains unique to subsystems i and j respectively. This concept can be extended to all subsystem groupings. The total number of unknown gain elements is IV '" N '. m rrr == z::::..Mt.. D. - ..L L: L.. "...,:" J (.J' t: J) ........... IJ:z.... IlL: where

ni~I

represent;-='the numbe; =~i~ains in common between sub-

systems i and j. The minimization of equation (5) with respect to the unknown gain matrices is accomplished using any of the well known algorithms (Powell's method, Conjugate gradients, etc.). APPLICATION The problem of designing the stability augmentation system (SAS) for a turbojet/lift fan powered VTOL aircraft in the transition mode is considered. The set of subsystems consists of five operating points along the transition flight path: hover, 30 knots, 60 knots, 90 knots and 120 knots (cruise). For simplicity, only the longitudinal axis is considered. The VTOL aircraft equations for small perturbations about the operating points were developed and linearized. The state variables for the most general case are forward velocity, vertical velocity, pitch angle, pitch rate, engine moment and total thrust. The control inputs are the inputs to the engine, the elevator, thrust magnitude and thrust vector angle. The first step in the analysis was to assume total state feedback for each subsystem. The five subsystems were then independently optimized to the twenty-four gain elements to establish (i=1,2,···,5). Then, several of the gains were constrained to be constant over the flight path while the others were scheduled. Finally, the full state feedback assumption was removed and only those states which are most easily measured were considered. The subsequent analysis clearly demonstrated the effectiveness of the sensitivity method in providing an alternative to the total gain scheduling problem. The algorithm was then modified slightly to incorporate random input disturbances.

Ji

The numerical solution to the VTOL minimization problem was performed on an IBM 360-75 using the Powell-Zanquill Method. REFERENCES 1.

Athans, M. and Falb, P., 1966, "Optimal Control", McGraw Hill, New York.

2.

Kreindler, E., 1969, IEEE Auto Control 14, 206.

341