The General Formulation of the Problem

7 .1 Introduction

7.2 The General Formulation of the Problem

7.2.1 The General Interconnection Structure

The general problem is based on the idealized linear design given in terms of the stan- dard feedback problem shown in Figure 2 [32,41,12]. The interconnection structure, P( s ), is fixed and linear time invariant and describes the interconnection between ex- ogenous system inputs, outputs, and the controller. It includes a model of the plant, G( s ), and performance and noise weights. The individual blocks of P( s ), denoted PiJ ( .s ), a.re obtained by pa.rtitiouiug P( .s) to wrresµoml. Lo the <limeusium of w, z, u, and Ym• K(s) is the LTI controller produced in the linear controller synthesis step of the overall design.

The exogenous input, w, includes all signals which enter the system from its environment including commands, disturbances, and sensor noises. The other inter- connection input, u, represents the control effort applied to the plant by the controller K(s). The interconnection outputs, z and Ym, represent the controlled output, con- sisting of signals which the controller is designed to keep small ( typically tracking errors and weighted control efforts), and all measurements available to the controller (including commands, measured disturbances, measured plant outputs) respectively.

Any feedforward/feedback interconnection of linear system elements can be brought

into this general interconnection form. Examples include, but are not limited to, cascade, feedforward, and multiple degree of freedom structures in addition to the traditional error feedback configuration.

As an example of the rearrangement of a particular feedback arrangement into the standard framework of Figure 2, we consider the error feedback system of Figure la.

The exogenous inputs are the command, r, and output disturbance, d. Thus we define w

= [d].

The controlled output is the tracking error, e, so we define z

=

^{e. The}

information made available to the controller, J<(s), is the tracking error, so Ym

=

^e.

The output of K(s) is the plant input, u. The interconnection corresponding to these definitions is given by

[

J -I -G(s)

l

P(s)

=

I - I -G(.s)

(7.1) The interested reader is encouraged to verify that with these definitions the input- output behavior, from exogenous input to controlled output, of the system in Figure 2 is equivalent to that in Figure la.

The distinction between the blocks P( s) and K ( s) is that the components in P( s) are assumed Lo be fixed a priori, i.e., they are realized in hardware which we are not free to modify. On the other hand, K(s) is the controller design we wish to implement and its physical realization is unspecified.

It is assumed that both P(s) and K(s) are finite dimensional and that state space realizations for them are available. We will use the notation

[ ; I;]

^ii, C(sl - At' B

+

D

to represent the transfer function arising from the state space realization

x

=

^{Ax+ Bu.}

y

=

Cx

+

Du.

where x is the state, u the input, and y the output of the system of interest.

(7 .2)

(7.3) ( 7.4)

The closed loop transfer function from w( s) to z( s) in Figure 2 is denoted T zw(s) an<l is given by the linear fractional transformation

(7.5) We assume that performance specifications are provided for the linear design and that the controller design, K ( s ), meets these specifications in the absence of lim- itations and substitutions. For the purposes of this paper we assume that these specifications are of the form

(7.6)

where the norm,

II • II,

is either the H⁰⁰ norm,

IIZ(s)lloo

~ supa-[Z(jw)]

wen ^{(7. 7)}

where a-(Z) represents the largest singular value of Z, or the H² norm,

I

JJZ(-,)Jl2 - Ll11" 1.:

trace[Z*(jw)Z(jw)]Jw]

2

(7.8)

These frequency domain performance specifications are standard in fl= and H ² op- timal control theory. By including suitable weights in the interconnection structure P( s), the performance requirement ( 6) allows very general specification of the fre- quency domain characteristics of the closed loop transfer function. In the remainder of the paper we will use the notation

II • ll

2

oroo

in situations where either the H² or H⁰⁰ norm may be used.

The general AWBT problem is based on Figure 3. The interconnection .P(s) is obtained from P( s) by adding an additional output Um. Thus

Pu P12

F(s)

=

^{P21 P22 (7 .9)}

w "' ^z

'

P(s)

u

"' y=[:~

K(s)

Figure 3: The standard AWBT design problem.

and

(7.10) The new signal, um, is the measured or estimated value of the actual plant input u'.

We allow the general relation (10) so that measurement noises, entering through w (i.e., P₃₁

¥

⁰⁾ and non-trivial measurement dynamics ( P₃₂

¥

I) may be considered.

The situation where a perfect estimate of u' is available corresponds to P₃₁ = 0, P₃₂ = I. As in the error feedback example (Figure lb), the plant input estimate is made available to the AWBT compensated controller

k (

s), in this case as a component of the measurement vector y. Note that ^Um need not represent a raw measurement signal but may include appropriate pre-compensation and filtering. We do not address the design of this pre-compensation but will generally assume that it is such that P_{32 (} s) ~ / over the closed loop bandwidth of the idealized linear design. As we will see, if this is not the case achievable AWBT performance will be limited.

Also included. in Figure 3 is the input limitation/substitution mechanism, repre- sented by the nonlinear block N. The nonlinear limitation/substitution map, N, is assumed to be cone bounded and of fixed structure. We will discuss the implica- tions of these assumptions, and the type of "real world" limitation and substitution mechanisms which admit such a description, in Section 7.6.

Given this framework the general A\VRT problem amounts to the synthesis of

w z

A