D. F. Thompson Ford Motor Company,'

Transmission and Chassis Operations, Livonia, Ml 48150

0. D. I. Nwokah School of Mechanical Engineering,

Purdue University, West Lafayette, IN 47907-1288

Analytic Loop Shaping Methods in Quantitative Feedback Theory Quantitative Feedback Theory (QFT), a robust control design method introduced by Horowitz, has been shown to be useful in many cases of multi-input, multi-output (MIMO) parametrically uncertain systems. Prominent is the capability for direct design to closed-loop frequency response specifications. In this paper, the theory and development of optimization-based algorithms for design of minimum-gain controllers is presented, including an illustrative example. Since MIMO QFT design is reduced to a series of equivalent single-input, single-output (SISO) designs, the emphasis is on the SISO case.

1 Introduction Quantitative Feedback Theory (QFT), introduced by Ho­

rowitz (1963, 1979), Horowitz and Sidi (1972, 1978), is a feed­back design method allowing direct design to closed-loop robust performance and stability specifications. In this paper, new techniques for the systematic design of fixed-order, minimum-gain controllers within the traditional QFT paradigm will be presented.

The historical practice for QFT controller synthesis has been that of systematic cut-and-try using finite-order rational con­trollers, typically multiple lead and lag filters (Houpis, 1987; D'Azzo and Houpis, 1988). In this case, QFT controller syn­thesis is approached as a constraint satisfaction problem, the goal of which is to obtain a nominal open-loop transfer func­tion L0(ju) which satisfies open-loop frequency response con­straints (based upon closed-loop specifications and plant uncertainty), ideally on w € [0, oo), but practically at most at a discrete set of frequencies « 6 {«i, . . . , « „ } . For existence of solutions, the plant set must meet conditions for internal stabilizability by a single, fixed controller, as well as certain compatibility conditions involving the uncertain plant set and the closed loop performance bounds (e.g., in terms of allowable sensitivity reduction); see Nordgren et al. (1993b).

Given a collection of discrete, open loop frequency response constraints and a choice of finite-order, rational controllers, the use of cut-and-try methods by an experienced designer most often leads to a solution to the constraint satisfaction problem. However, a result of this type is not unique. Within the con­straints, the designer may adjust controller parameters and/ or controller order depending upon gain, bandwidth, and sta­bility considerations. Within this framework a more systematic optimal design methodology is attractive, one which reduces or eliminates trial-and-error steps in the synthesis stage.

The original notion of optimality for QFT controllers, sug­gested by Horowitz (1963), Horowitz and Sidi (1972, 1978), was that of minimum high frequency controller gain. Since all

Contributed by the Dynamics Systems and Control Division for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL, Manuscript

received by the DSCD August 16, 1990; revised manuscript received April 23, 1993. Associate Technical Editor: S. Jayasuriya.

performance and stability objectives are specified (subject to existence conditions) as constraints in the QFT problem, this notion of optimality was based upon controller bandwidth considerations (Horowitz and Sidi, 1972). A significant con­clusion from this work was that a controller satisfying the open loop frequency response constraints with equality on the con­tinuum oi € [0, oo) is unique, and furthermore possesses the minimum high frequency gain property; i.e., the constraint satisfaction problem and the optimization problem are one and the same. However, the authors observed that this property can only be realized by controllers of infinite order. Upon discretization of the constraint set and choice of finite-order controllers, this uniqueness property is lost.

As a result, the historical QFT loop shaping techniques of systematic cut-and-try could only address the constraint sat­isfaction problem; questions of optimality with respect to high frequency controller gain (or any other measure, for that mat­ter) could not be directly answered. An optimal design meth­odology to this end is developed in this paper. The philosophy of constrained, parametric optimization has not previously been developed for QFT controllers, although related fre­quency domain optimization methods have been shown by Polak et al. (1984), Boyd and Barratt (1991).

This paper consists of five sections, of which this is the first. In Section 2, we develop a complete description of the rep­resentation of plant uncertainty. In Section 3, we state the QFT optimal design problem and provide the nonlinear optimization framework for the solution of the problem. In Section 4, we consider an illustrative design example, followed by some con­cluding remarks in Section 5.

2 Representation of Plant Uncertainty Let the plant family under consideration be described in a

combined parametric and non-parametric, unstructured form. The overall plant set {P{s) j is given by

{P(s)} = {P(a,s)[\+An(s)], a€fi, A„€A) (2.1) where

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ln|L(jci))|

Fig. 1 Illustration of the mapping from parameter space SJ, to plant template P(u) in the gain-phase plane, to closed-loop uncertainty

A=(A„eRH° lA„(yco)l<m(co)) (2.2)

and m(oi) is a given function in V°. Note that A„(^) is the unstructured uncertainty associated with each member of the parametric family P (a, s) rather than with one nominal model. Also observe the strict inequality in (2.2) used to define the radius of admissible uncertainty.

The parametric plant subset is defined as [P{a, s): a £ fl C IRm) where the compact parameter space fl may be given as a collection of closed intervals fl = (a,-: a,€ [a,-, aj], /= 1, . . . , m]. More generally, it is assumed that fl is a dif­ferentiate manifold with the numerator and denominator coefficients of P(a, s), and by implication the poles and zeros of P(ot, s), bounded continuous functions of a € 0 (Nordgren et al., 1993b). The parametric model is represented as:

P(a,s) =

i * , CO < CO/,

— , co>co/,

(2.3)

where e is the relative degree of P(a, s). To characterize the parametric uncertainty, the mapping P (., y'co): fi—P generates, at each frequency, a compact set P(co) in the gain-phase plane at each frequency, commonly called the plant template (see Fig. 1). Here, «A is Horowitz's universal high frequency; i.e., the frequency above which the phase variation of the para­metric plant set is essentially zero (Horowitz and Sidi, 1972). Such an assumption is analogous to the "split-frequency" model suggested by Astrom et al. (1990).

In addition, the following assumptions are made: (1) The parametric modelP(a, s) may be unstable for some

or all a € fl, but must be robustly internally stabilizable by a single, fixed controller G(s). A necessary condition for robust stabilizability is that the plant set {P(s)} is topological^ path connected (Nwokah, 1988; Foo and Postlethwaite, 1988). In the SISO case, an equivalent condition for path connectedness is that the gain of the high frequency plant k„ has the same sign for all a € 0. A sufficient condition for robust internal stabilizability is given in terms of the zero exclusion principle (Nordgren et al., 1993b).

(2) In traditional QFT practice (Horowitz and Sidi, 1972), the functions A„(/'co) and m(co) describing the unstructured plant set are not expressed explicitly; the effect on system stability is recognized implicitly by imposition of one of various relative stability exclusion regions (see Section 3.2). Conse­quently, traditional QFT explicitly considers only parametric models (i.e., An{s)=0), as will be the case in the following developments; however the effect of unstructured uncertainty

- Q -

Fig. 2 The two degree-of-freedom feedback structure assumed in SISO QFT

is recognized, as described above; see also Jayasuriya and Zhao (1993).

(3) In addition to stability issues, realizability and existence questions arise, given the assumption of arbitrary performance specifications. Subsequent discussion will focus upon design of QFT controllers to meet "traditional" (i.e., pointwise) bounds on closed loop frequency response for minimum phase plants. Under appropriate assumptions on the closed loop spec­ifications (i.e., sensitivity reduction cannot be demanded as co— 00), QFT controllers must always exist for parametrically uncertain minimum phase plants (Horowitz and Sidi, 1972). For plant models explicitly including parametric and nonpar-ametric uncertainty (2.1), additional'existence conditions are required; see Bailey and Hui (1991), Bailey and Cockburn (1991), Jayasuriya and Zhao (1993).

(4) For nonminimum phase plants, arbitrary performance specifications are not achievable, and thus QFT controllers may not exist (Horowitz and Sidi, 1978). In such a case, the specifications can normally be relaxed such that a feasible QFT design is obtained; however, necessary and sufficient condi­tions for this are not known. General necessary and sufficient conditions for the existence of QFT controllers based upon L2-equivalent time domain bounds (Krishnan and Cruick-shanks, 1977), for combined parametric and non-parametric, non-minimum phase plants, are given by Nordgren et al. (1993b) in terms of frequency domain sensitivity inequalities.

3 Design and Optimal Loop Shaping The QFT problem can now be stated as follows: There is

given an uncertain family of linear time-invariant, finite-di­mensional plants (P(s)) as described in Section 2, but now considering parametric uncertainty exclusively. We are re­quired to find (if possible) an admissible pair of strictly proper, rational, stable functions {G(s'), F(s)} in the two degree-of-freedom feedback arrangement shown in Fig. 2, such that the following conditions are satisfied.

(1 Robust Sability): Define the loop transmission (open-loop transfer function) asL (a, s) =P(a, s)G(s). It is required that the closed-loop system,

L(a,s) T(a, s) (3.1)

l+L(a, s)

is stable for all a e fl. (2 Robust Performance): The specification for robust closed

loop performance is given as

a(u)<\F{j^)T(a,ju)\<b(u>), cog [0, 00), (3.2)

where the frequency response bounds a (co) and b(o>) are spec­ified a priori by the designer.

For design purposes, the closed-loop constraints (3.2) are mapped into bounds on the open-loop L(s) which are to be met by choice of G(s). Let the set of all G(y'co) which yield L(y'co) satisfying the open-loop constraint for some co be de­noted as G(co). The optimal G(s) will be defined to be that G, G(y'co) € (G(co), co€ [0, 00) J having the minimum high fre­quency gain.

3.1 MIMO and SISO Design: Introduction. Since MIMO design consists of a sequence of equivalent SISO designs (Ho­rowitz, 1979), it is sufficient to consider the SISO case. For fixed co, let the image of a € fl under the mapping P{., y'co):

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ln|L(j»)|

iL(jco)

(M*.9°)

Fig. 3 Illustration of the plant template P and its rectangular approx­imation P

A—P be denoted as the ordered pair (M, 0), where P denotes the image space, or plant template. Let the image of the nom­inal a0 be denoted as (A/0, 0°) (see e.g., Fig. 1 and Fig. 3). For the purpose of analysis (Fig. 3), assume that the plant template P is approximated by the smallest possible super­scribed rectangle in the gain-phase plane (Polak et al., 1984) denoted as P.

For fixed co, denote the open-loop gain phase (Nichols Chart) ordinate and abscissa as M(co)=ln \L(ju)\, and 0(co) = zZ,(y'co). Now define the template height and width,

AM(co) = max In iP(y'w) I - m i n l n \P(joi) I

A0(co) = maxxP(,/'co)-o ( 8

min z P ( / u ) .

(3.3)

(3.4)

The argument co will be suppressed when fixed. For illustration, choose a0 such that its image lies in the lower left corner of P at some frequency (the "minimum plant")- For a given (A/0, 0°), the symbol M C i denotes closed-loop magnitude in loga­rithmic units:

MCL(M, 0 ) i l n 1+L

(3.5)

(M+jO) where L = e alternative form

The relationship (3.5) may be put in the

MCL(M, 0) = M - - l n [ l + 2 e M c o s 0 + e2M], (3.6)

Define for a nominal template position (M°, 0°) the closed-loop extrema

MCL(M°, 0°) = min {MCL(M, 8): M e [M°, M° + AM],

0e[0°, 0° + A0]) (3.7)

MCL(M°, 0°) = max [MCL(M, 8): M e [M°, M° + AM],

0€[0°, 0° + A0]) (3.8)

The closed-loop magnitude uncertainty, denoted as 5(M°, 0°), is then given as 5(M°, 0°)=Mcz.(M°, e0)-MCL(M°, 8°).

3.2 Formulation of Open-Loop Constraints. Define the closed-loop tolerance for the bounds of (3.2), denoted as S^co), as67<w) = ln IZ>(«) I - I n la(u) I. For a given template P at the frequency co, the position of the nominal point (M°, 8°) is said to be a feasible template location if and only if <5 (M°, 0°) < 5-jioi). We may now define the open-loop constraints.

Definition: At a given frequency co, the contour Mr (u ) is defined as follows:

MAco) = I (M°, 0°): <5(M°, 0°) = 57(co)). (3.9)

The Mr(to) is the locus, in the gain-phase plane, of all template nominal point locations (M°, 0°) for which the closed-loop magnitude uncertainty tolerance is met with equality. In general, there exists some frequency oiH below which the bounds are single-valued functions of phase. In this case, the bound may be cast as a real inequality: \L0(ja>) I >MT( z.L0(Joo), to), co e [0, a;/,). This frequency uH is distinct from the Horowitz universal high frequency to/, of (2.3). In many cases, however, these characteristic frequencies would lie close together. For the purposes of this paper, consider an effective high frequency coeff = minjcj//, ooh] as the effective upper limit on both the parametric plant model and single-valued open-loop robust performance bounds.

In addition to the performance constraints (3.2), the closed-loop stability condition of (3.1) dictates a constraint of the form

<MS for all (M°, 0°), a e Q , co e [0, oo) (3.10) \+L(jw)

It is often more convenient to modify (3.10) and instead employ the alternative condition

L(j<*) l+L(jo>)

<M„ for all (M°, 0°), a€f i , coe [0, oo).

(3.11)

This condition has the interpretation of the Nichols chart that (3.11) is satisfied if and only if no point of the template P penetrates the Nichols chart M-contour Mp for all (M°, 0°), toe [0, oo). An exclusion region based upon (3.11), consisting of the locus (M°, 0°) for which this constraint is active as co— oo (considering only the high frequency template of the parametric model) is referred to in traditional QFT literature as the ul­timate high frequency boundary or U-contour. It is through this type of high frequency stability bound that the presence of unstructured plant uncertainty A„(,/co) is recognized, as dis­cussed in Section 2. In the subsequent design example, this type of exclusion region is used. A similar stability bound for QFT, of the form

sup IIT(a, 5)11 a fS l

J<sup6(co)=M / ! (3.12)

which generates a forbidden disk around the critical point, was proposed by Jayasuriya and Zhao (1993). A rigorous treat­ment of robust stability exclusion regions of QFT explicitly including non-parametric uncertainty is given in Bailey and Hui (1991).

3.3 Solution to Finite-Order Problem. Semi-infinite op­timization algorithms have been developed by Polak et al. (1984) for infinitely constrained control synthesis problems. However, these algorithms require optimization over compact time or frequency intervals. Fortunately, simplification of the QFT synthesis problem is achieved through discretization of the constraint set. This permits solution as a finite-dimensional nonlinear program (Luenberger, 1973):

min f(x)

h(x)=0

g(x)<0

f: TR"G~ml, g: 1R"G_IR»«> n: JR"G-*JR"» (3.13)

Let the design vector x e IR"G be taken as the vector of all free controller parameters (poles, zeros, and gain constant) for G(s). The cost function f is then taken as

f(x)=k2G (3.14)

where kG is the high frequency gain of the controller G(y'co), equivalent to the gain constant of the transfer function in Evans root locus form. Constraints due to the single-valued contours MT(8, co), co<coeff, would be cast in the form

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gi(x) = MT(zL(x, jwi), co,)-In \L(x, y'co,) I < 0 (3.15) where dependence of the loop transmission L upon the para­meter vector x has now been explicitly indicated. Multiply-valued constraints could be handled through an analogous procedure in which the locus is divided into two single-valued parts.

The formulation of (3.15) is especially advantageous since the gradient "7x\L(x, joi)\ can be computed analytically, a substantial numerical benefit. The element of the gradient of constraint g, with respect to design variable xk is given as follows:

dgj dMT d z. L (x, yco,)

dXi In \L(x,M)\. (3.16)

The components of this expression follow straightforwardly from (3.6). It may be shown that for the rectangular template approximation, the constraint function MT( zL, ju), and its derivative dMT/d z. L, each take on one of 18 possible analytical expressions. This result will be outlined here as follows. To simplify notation, consider a compensator G(x,joi) comprised of real poles and zeros only. Then,

L(x, y'co) = P(yco)G(x, yco) = / > ( »

n Uu+Zi)

"p

H U^+Pk)

(3.17)

Thus (3.15) is reduced to

1 "z

g(x)=MT{zL(x,jw), «)- ln l P f » l - r X > (co2 + z,2) ,= i

+ -J]ln(o>2+p2k). (3.18)

Derivatives of g(x) follow as in (3.16): dg dMT dzL 1 d , , , a ~ a r a i a l n ( " +Z OZm OA.L dZm 2 dzm

dMT

~dzL -co

2 2 Z,„ + W

Zm

Z2m + o>2

and similarly,

dg dMT

dpm dzL CO

pi+w 2 , Pm ' 2 2

Pm + <»

(3.19)

(3.20)

(3.21)

3.4 Analytic Expressions for Open-Loop Constraints. From (3.15), (3.16), and subsequently (3.20), (3.21), it remains to find expressions for MT(6, co), and dMT/38, in terms of 8, co, and the template dimensions AM and Ad. By standard meth­ods of nonlinear programming, it is possible to show that extrema of MCL{M, 8) over a rectangular region in the (M, 6) plane must occur at a vertex of this region or at a possible tangency point with a closed ' 'M' contour in the Nichols chart plane. These candidate extrema are numbered (for identifi­cation purposes) 1 through 8, respectively (see Fig. 4). Recall that robust stability requirements (3.1) dictate that parameter variations must not induce a change in the number of encir­clements of the critical (0 db, - 180 deg) point by the Nyquist locus; thus a nominal point (M°, 8°) for which the template contains the critical point as an interior point would be in­admissible. Elimination of prohibited template locations and other trivial combinations leads to a total of 18 feasible max-min pairings of template vertices, and thus 18 possible analytic expressions for (3.9); for additional details, see Thompson (1990).

Direct computation of the closed-loop robustness bounds may now proceed straightforwardly. Algebraic expressions for MCL(M, 6)-MCL(M, d) = 8T(p>) may be computed by direct

Fig. 4 Illustration of the Nichols Chart of rotational convention for candidate closed-loop magnitude extrema, assuming rectangular tem­plate

substitution of the closed-loop magnitude expressions. Collec­tion of terms in powers of eM indicates that the expressions are quadratic, a substantial benefit. These equations are listed in Table 1. Furthermore, the quadratic expressions define relations F(M, 8) = 0 from which the required derivatives 3M/d8\M=MT

can be computed more easily by implicit differentiation. The notation of Table 1 is explained as follows: the label in

Column 1 (e.g., "2-1") designates that the expression yields the solution MT=Min terms of 8, AM, A8, and co, when Vertex 2 of the template yields the closed loop maximum, MCu and Vertex 1 of the template yields the closed-loop minimum, M_CL. Sorting and exhaustive evaluation of all candidate extrema at arbitrary points is not necessary, as simplified algebraic expres­sions (three, in fact) involving M0, 80, AM, and A8, may be obtained which, in the form of a truth table, determine the applicable equation (Thompson, 1990).

Note that the quadratic equations of Table 1 are applicable to both cases of single-valued, as well as multiply valued bounds. In the case of a single-valued bound, the applicable equation yields one positive real root (note that the solution is a log-magnitude M which must be positive). In the case of a multiply-valued bound, the applicable equation yields either no positive real roots, or two positive real roots, depending upon the phase angle 8. From this point, the constraint set can be considered within the standard nonlinear programming framework. Strictly speaking, the constraint set may possess a finite number of isolated, nonsmooth points (i.e., at tran­sition points for quadratic equations), however this does not appear to be a limitation for practical design.

Although discussion here has focused upon gradient-based minimization using piecewise quadratic constraint equations, the primary intent is to demonstrate the general feasibility of finite-dimensional, constrained parameter optimization for the loop shaping problem of QFT; it is not intended that these assumptions be so limiting. The most general form for QFT would utilize the open loop robustness bounds MT(8, co) in numerical form, without template approximation. Such a for­mulation, which must employ non-differential optimization methods, is the subject of current research. Various nondif-ferential optimization methods have been proposed by Polak et al. (1984), Boyd and Barratt (1991).

3.5 Stability Bounds and Well-Posed Problems. The nonlinear programming formulation, as given in (3.13)-(3.16), does not incorporate an explicit means of enforcement of a relative stability constraint such as (3.11) and, with no addi­tional restrictions, convergence to an unstable optimum could

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Table 1 Quadratic expressions F(M, 0) = O defining MT(6, co)

Condition Equation F(M, 0) = 0

2-1

1-2

4-1

1-4

2-3

3-2

4-3

3-4

5-2

5-4

6-1

6-3

7-1

7-2

7-3

7-4

8-1

8-2

[1 - e26r]e2M + 2[cos 0 - e2 i rcos (0 + A0)]eM + [1 - e M r ] = 0 2i7-cos(0 + A 0 ) ] e M + [ l - e - 2 6 r ] [1 -e"2 5 r]e2 M+2[cos6

[ 1 -[ 1 - e

n e 2 M + 2[cos0-e 2 6 nr A M cos(0 + A 0 ) ] e M + [ l - e

> ^ + 2[cos0-e_2 ' i 7e~AA 'cos(0 + A 0 ) ] e M + [ l - e

2(f j -4M)

- 2{ST- AM)

e2M,ll-e2ST]e^ + 2[e'1Mcose-e2i^+AM)cos(e + Ae)]eM+ll-e2i>T*AM)]

[ 1 - ?2M+2[eAMc r,2(-5T+AM)

„ 2 ! T

cos(0 + A0) ]e M +[ l -e 2(-6r+AA/)-i

e2AM[l-e2l'T]e2M + 2eAM[cos6-e2''Tcos(e + A8)]e

T

• [ ! • „2Sx

e 2 ^ u _ g - 2 8 . e 2 M + 2 e A M [ c Q s 6 | _ e - 2 « r c o s ( 6 , + A 9 ) ] e M + [ 1 _ e - 2 « r ]

[ l -e 2 6 r s in 2 0]e 2 M + 2cos(0 + A0)eA 1 -eM7"sin20]e2M+2eAMcos( -A0)eM+l

[l-e2 S 7 's in2(0 + A0)]e2M+2[cos0]eM+l

- eM rsin2 (0 + A0)]e2M + le^ [cos 0]eM + 1

[1 -e2ST]e2M- 2[cos0 + e 2 i r ] e M + [ l - e 25 r j

[1 - e2iT)e2M + 2[cos (0 + AS) + e ^ e + [1 - e24r]

e ™ , [ 1 _ e 2 J T ] e 2 M + 2 [ e A M c o s e + e 2 ( A M + 6 7 - ) ] e M + [ 1 _ e 2 ( A W + 5 r ) ]

< ' [ l - eM 7 - ] e 2 M + 2 [ e A M c o s ( f l + A 0 ) + e 2 ( 4 M + S r ) ] e M + [ 1 _ e 2 ( 4 A / + « ;

[! ^ e 2 S 7 . ] e 2 M + 2 [ c o s 0 + e 2 5 r e - A M ] e A / + [ j ^ e 2 , S 7 - A W ]

[ l - e 2 S r ] e 2 M + 2[cos(0 + A0) + eM ' "e- ' i M ]e 'w+[l-e 2 ( 5 ' " - ' u / ) ]

g 2 A M ,

„2AMi

easily occur. Furthermore, the constraint set described by (3.15) does not , in itself, generate a compact domain (Luenberger, 1973). Because of this, as well as the non-convex nature of the constraint set, the cost function (3.13) may possess local ex-trema a n d / o r an infimum which is approached only as some controller parameters are driven to infinity (see also Thomp­son, 1990).

Within this framework, the most expedient solution is to choose an initial design which meets all stability criteria, and then to solve a sequence of restricted optimization problems in which the design variables (controller parameters) are limited to a neighborhood of the initial point x°:

Xj£\Xj, xi\, / = 1 , . . . , n (3.22)

where the interval \xh xj\ might be given as, say Xj=(l/a,)xf, Xj = cijX?. As an example, the choice a, = 2, / = 1, . . . , n cor­responds to a limitation of ± one octave on the poles and zeros from their initial values. A n algorithm for sequential restricted minimization may then be given as follows: For some x°,

x = min / ( * ) (3.23)

where, in general, xk would serve as the initial point at the kth stage. The optimization sequence can be continued until rel­ative stability requirements (3.11) are violated. Various opti­mization strategies in which algorithms are reset after a number of iterations have also been suggested by Polak et al. (1984).

Moreover, given a procedure in which optimization is per­formed in successive stages in an interactive C A D environment, the " t r anspa rency" of Q F T is retained, inasmuch as the de­signer can study tradeoffs in increasing or decreasing controller order, as well as the effects of additional frequency constraints. Viewed in this larger scope, the principal role of the parametric optimization for Q F T controller synthesis is that at each stage, the designer can be assured that he /she has obtained the " b e s t " controller (in terms of high frequency gain) of a given order.

As an additional condition to create a well-posed optimi­zation problem, it is important to place designable controller parameters (e.g., lead a n d / o r lag filters) at suitable frequency intervals such that an even covering of the system bandwidth is obtained. In addit ion, the frequency grid [w\ oi„] should be chosen such that a reasonably uniform covering of

the gain-phase (M, 8) plane is generated by the system of constraints MT(6, co). These observations are important in the context of initialization, which we shall discuss subsequently.

3.6 Traditional QFT Controller Design Initialization. A problem which is somewhat beyond the scope of the design formulation itself is that of initialization. The historical meth­ods of systematic cut-and-try for Q F T controller synthesis can be viewed as a legitimate means of initialization, subject to local optimization. In many such cases, the high frequency controller gain of otherwise acceptable Q F T designs can be reduced, often substantially, by parameter optimization over a relatively modest neighborhood (e.g., ± 2 0 percent) of the initial point.

It is relevant to note that most cut-and-try methods actually incorporate a formal procedure in which, for many problems, open loop bounds (particularly those near zero slope) are trans­ferred to a Bode magnitude plot, creating an approximate envelope for open loop magnitude (D'Azzo and Houpis , 1988). From this, straight-line Bode synthesis techniques employing lead and lag controllers can be used to generate an acceptable design.

A more rigorous extension of these ideas is demonstrated in formal system identification approaches to Q F T controller synthesis. As in the straight-line Bode synthesis technique, it remains for the designer to evaluate the tradeoff between con­troller order and "goodness of f i t " in determination of an acceptable design. Such methods have been developed by Bai­ley and Cockburn (1991), and Sobhani and Jayasuriya (1992), the latter of which makes contact with the related Hilbert boundary-value solution developed by Gera and Horowitz (1980). The use of combined identification and optimization approaches for Q F T design is to be the subject of future work.

3.7 H°°-Based QFT Design Initialization. As an alter­native to identification-based methods for generating initial QFT controllers, we explore the use of exact sensitivity-based solutions of the Z2-equivalent Q F T problem due to Nordgren et al. (1993b), based upon the formulation of Krishnan and Cruickshanks (1977).

For the closed-loop system of Fig. 2 , consider the t ime do­main inequality

a(t)<y(a, t)<b(t), a6 0 , t£[0, <x) (3.24)

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or equivalently,

(y(a,t)-b(t)+a(t) bV)-a(t)

aefl, /e[0, oo).

(3.25)

An exact frequency domain equivalent to the above time do­main inequality is unknown. However, a relaxed condition analogous to (3.25) is the following:

b(t)+a(t)^ C(-y(a,t)- dt

n-(t)-a(t) dt ae f i . (3.26)

Let y0(t)±[b(t)+a(t)]/2 and v(t)±[b(t)-a(t)]/2. Then, by Parseval's theorem a sufficient condition for satisfaction of (3.26) is the following (Nordgren et al., 1993b):

\y(jo>)-y0(ju)\<\v(jo))\, a€f i , co€[0, oo) (3.27)

where it is assumed that a(t), b(t), and y(t) possess corre­sponding Laplace transforms. It is then shown that (3.27) is equivalent to the following sensitivity inequality:

IS(a,yco)l<- - = MT(oi), a €Q, co€[0, oo) ^o(y'w)l6p0(co)

(3.28)

where the parametrically uncertain sensitivity function is de­fined as S{a, y'co) = [1 +L(a, y'wjp1 and the maximum nor­malized plant uncertainty, denoted as 5Po, is given as

P(a,ja)-P0Uw) <W<°) = max

U alii

(3.29) Po(jo>)

In addition to the tracking specification (3.27), or equivalently (3.28), a closed-loop disturbance response bound MD(CO) may be specified:

lS(a,yco)l<M f l(co), a€fi , a>€[0, oo), (3.30)

and define the composite boundM{co)= mm{MT(u), MD(w)) 6L00. Furthermore, the robust stability condition \T(a, jio) m(o))\ <1, a€Q is added, analogous to the traditional con­straint (3.11). The sensitivity-based robust performance spec­ification (3.28) is then shown to be equivalent to the following (Nordgren et al., 1993b):

lM_1(co)S(a,yco)l + \m(u)T(u, yco) I < 1,

a€f i , co€[0, oo). (3.31)

This is in fact an extension of the exact H°° robust performance problem (Nordgren et al., 1993a), parameterized by the un­certain sensitivity and complementary sensitivity functions S and T, respectively. Choice of a particular a0 in (3.31) yields a nominal H°" robust performance problem which is not an­alytic, but which admits a loop shaping solution (Nordgren et al., 1993a). One can, however, obtain an analytic solution to the nominal FT mixed sensitivity problem

I ^(yo))S(a 0 , yco) l2+ I ^2(y'co)T(a0, yco) l2< 1,

where W\ (s) and W2(s) are suitable RH'

co€[0, oo). (3.32)

' weighting functions whose magnitudes upper bound M~ (co) and m(u), respec­tively. The solution of (3.32) may be used to initialize a tra­ditional QFT design, which is the methodology employed in the following example.

4 Illustrative Design Example The following illustrative example for optimization-based

QFT design is developed from a flight control problem due to Blakelock (1965). The problem is to design outer loop corn-

Table 2 Parameter uncertainty, longitudinal autopilot (Blakelock, 196S)

Parameter Nominal Value Range k z P

2.0 0.5

10.0 6.0 0.8

£€[0.2, 2.0] Z6[0.5, 0.75] pd[l.0, 10.0]

oo„€[5.0, 6.0] f€ [0.8, 0.9]

phase (deq )

Fig. 5 Gain-phase plot of QFT tracking boundaries MT(u) = B(u), nom­inal plant P0(ju)

pensation for a basic longitudinal autopilot. The model given is for a four-engine jet transport of unspecified type in the landing configuration.

4.1 Plant Model and Performance Specifications. The longitudinal motion of the aircraft in open-loop has the basic structure

P(s)-

k\\+-

1+P iA+

(4.1)

From aerodynamic data supplied by Blakelock, appropriate ranges for the uncertain parameters are identified. This in­formation is summarized in Table 2.

Closed-loop response I Y(a, yco) I will be specified as a(co) < I Y(a, y'co) I <&(co) for the following bounding func­tions tf(co), b(o):

1 a(co) =

(1 +yco)(l +yco) 1 + , yco (4.2)

Z>(to) =

1 + 0.35

1 + yw 0.5

1 + J" (4.3)

Design frequencies are chosen as co€ {0.01, 0.05, 0.1, 0.2, 1.0, 5.0, 10.0); the effective high frequency is limited to coeff = 10.0 rad/s for this problem. In Fig. 5 are shown the resulting single-valued open loop bounds MT( z. L, co), as well as the traditional high frequency (U-contour) stability boundary, based upon (3.11), for Mp = 6 db. Also shown in Fig. 5 is the nominal plant transfer function Po{ju).

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phase fdeg )

Fig. 6 Gain-phase plot of Ufa), initial Wbased design (4.4)

4.2 Initial Controller Design. As discussed in Section 3.6, we shall attempt to use the solution to the induced nominal H°° mixed sensitivity problem in order to obtain an initial controller. Based upon the performance specifications (4.2) and (4.3), as well as the plant model and choice of nominal described in Table 2, the following H°° controller is obtained:

phase (deq )

Fig. 7 Gain-phase plot of i 0 U» with modified H°°-based controller, in­tegral factor (4.5) removed.

0.8654

G,(s)= ('•£) 1+:

2.22 1+:

2.22 ••£ K o s\ 1+-

0.51 1 +

0.615 1 +

2.60 1 +

1000

The resulting nominal loop transmission La(jw) = Pa(jui) GiO'u) is given in Fig. 6.

As can be seen in Fig. 6, the controller given in (4.4) is not adequate to meet the constraints of the traditional QFT prob­lem; both tracking bounds MT(u), as well as the [/-contour constraint, are violated. However, it is possible to modify the controller of (4.4) in order to obtain an acceptable QFT design.

In order to increase stability margin and avoid the [/-con­tour, the factor

1 + 0.55

(4.5)

is removed from Gi(s). The resulting nominal loop transmis­sion function is shown in Fig. 7. Adequate stability margin has now been achieved; however, the tracking constraints Mr(<j)) are not met. A straight-line Bode estimate indicates that an acceptable design can be obtained by addition of a single lead factor of approximately 70 degrees at co = 5.0 rad/ s, in conjunction with 19 db of additional DC gain. Such a lead factor is given as

8.9125 . • 0.882

1+;

(4.6)

28.36,)

The controller Gi(s) is then given as

7.709 1 +

G2(s)--0.882 ' • £ " ia '•£ '•£

(4.4)

phase (deq >

Fig. 8 Gain-phase plot of Z.0(M initial design (4.6)

and the resulting loop transmission function LQ(jio) =P0(jo>) G2(ju>) is shown in Fig. 8. Note that the QFT tracking con­straints MT(oi) are met with near-equality at u=1 .0 rad/s, with generally less than 5.0 db overdesign at most other fre­quencies. This design is judged acceptable and will be used as the initial controller for optimization.

,1+oii 1+oii5 1 + 2 . 6 0 / l 1 + 28.36 1+-

1000

(4.6)

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-270 00 -225 00 135 00 -90 000 -45 000 00 000

phase (deg )

Fig. 9 Gain-phase plot of L*0(jw), optimized controller (4.7)

log frequency ( r a d ' s )

Fig. 11 Closed-loop frequency responses, prefilter (4.8), extreme plant parameters

phase (deg >

Fig. 10 Gain-phase plot comparison of L0(]u), initial (4.6), optimized (4.7) controllers

4.3 Optimization of Initial Design. The controller ob­tained in (4.6) is now subjected to constrained minimization of high frequency gain, according to the methodology de­scribed in Section 3. The sequential quadratic programming routine DNCONG from the International Mathematical and Statistical Libraries (IMSL), Version 10.0, is used in conjunc­tion with specialized software; for additional details, see Thompson (1990).

In order to maintain stability margins, each controller pa­rameter is limited to a ±40 percent neighborhood of its initial value in (4.6) by use of simple bounds, as discussed in (3.28). After 17 iterations of DNCONG the following controller, de­noted as G* (s), is obtained:

3.355 1 +

G*(s)--0.630

1+ : 2.994

1 + 2.994

1 + 7.035

1 + 0.714

1 + 0.822

1 + 3.640

1 + 20.254

Fig. 12 Closed-loop step responses, prefilter (4.8), extreme plant parameters

L%O'w) =P0(jw)G* (ju) given in Fig. 9. In Fig. 10, a direct comparison between the initial and optimal designs is shown.

In order to meet the closed loop tracking requirement (3.2) in the two degree-of-freedom structure, a suitable prefilter must be obtained. This is designed according to the method­ology described in D'Azzo and Houpis (1988). This prefilter is given as

F(s)

1+ : 0.5

'•£ K (4.8)

1 + 7.035

1+ : 714.29

(4.7)

As a result, the high frequency gain of LQ(s) was reduced to *" ' 5.344(105) over its initial value of 1.205(106), a reduction over A collection of closed-loop Bode plots for the extreme plant 55 percent and, accordingly, overdesign in the loop was es- parameter conditions, as described in Table 2, are given in Fig. sentially eliminated. This is shown by the plot of the optimized 11. Although there is no direct implication for time domain

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performance under these assumptions, it is seen that the cor­responding time responses are favorable, as shown in Fig. 12.

5 Summary In this paper, we have introduced topics specific to imple­

mentation of the nonlinear programming problem for QFT design. The plant models and significant features of the QFT problem were reviewed, and the objectives of a controller syn­thesis methodology for QFT were outlined. We arrived at a specialized nonlinear programming formulation in which the design space is restricted at each major step to a suitable neigh­borhood of the initial point in order to guarantee relative stability. The advantages of this method, in light of the tra­ditional advantages of QFT design, such as the ability to ana­lyze controller order tradeoffs, were brought forward. In addition to this discussion, the major points of this develop­ment were illustrated in the controller synthesis example.

Acknowledgments This research was sponsored in part by a grant from the Air

Force Office of Scientific Research/Universal Energy Systems under contract No. F49620-88-C-0053, and by the School of Mechanical Engineering, Purdue University.

The authors wish to thank Mr. Richard Nordgren of the School of Mechanical Engineering, Purdue University, for his valuable assistance in developing initial controllers for use in the design example.

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