The selection criteria of the closed-loop control and estimator poles (or roots, have been encountered throughout the examples in Chapter 8tothis point. Also see the review of state-space design for continuous systems in Section 2.6 as well
8.3.2
Figure 8.13
Sketch of root locus for Example 8.10 Figure 8.12
Time histories of system with reduced-order estimator, Example 8.10
308 Chapter 8 Deslgn Lsing State-Space Methods
8.4 Introductionof the Reference Input 311
I that a unique and although other situations tudied.
See Trankle and (1978) for a more discussion and of the (0inputs other than often "model ing.-·
(8.69) u =N r.
than one input and output (MIMO), and. therefore. we will allow for that in the development here and in the following subsection. We will, however, require that the number of inputs in u and desired outputs in be the same.'2 The basic idea in detennining is that it should transform the reference input. r, to a reference state that an equilibrium one for thatr. For example, for a step command to Example8.1. = [l 0):that we wish to command the position state element. but the velocity state element will be zero in steady state. For the double system of Example 8.3. if we desire that d =r-that is.
that = [1 0 0 OJ-then we should set = [I 0 1 0]because that provides a state reference. that, if matched by the actual state. is at equilibrium for the desired output.
More specifically. we have definedN, so that
N,r
=
and u=
- (8.68)Ifthe system is Type I or higher andr is a step. there will be no steady-state error. and the final state
= =
For Type0systems, there will be an error because some control is required to maintain the system at the desiredx .
Often the designer has knowledge of the plant to know what the equilibrium state is for the desired output, in which case the determination ofN is complete. For complex plants, however. this can be difficult. In these cases,
to solve for the equilibrium condition that = r."
In order for the solution to be valid for all system types. they require a steady-state control input or not, we will include the possibility of a steady-state control term that is proportional to the reference input step. that is.
as shown in Fig. 8.15(a). The proportionality constant. will be solved for in the formulation.
Ifthe resulting is aClllally computed and implemented in the reference input structure, we refer to it as "feedforward." but the feedforward component of the input is often not used. Instead, the preferred method of providing for zero steady-state error through integral control or bias estimation. which essentially replaces the in Fig. 8.15(a) with an integral that is an estimate of the steady-state control, a topic that is the subject of Section 8.5. In some cases. it is difficult to achieve a high enough bandwidth when replacing feedforward with integral control: therefore, feed forward sometimes used to reduce the demands on the integral so that it need only provide the error in the feedforward control. thus
8.4 Introduction of the Reference Input
estimation by using a large estimator gain (fast response) in order to use sensed information to correct the model errors as quickly as possible.
The compensation obtained by combining the control law studied in Section 8.1 with any of the estimators of Section 8.2 is a regulator design in that goal was to drive all states tozero. We designed the characteristic
of the control and the estimator to give satisfactory natural mode transients to initial conditions disturbances. but no mention was made of howtostructure a reference input or of the considerations necessary to obtain good transielll response to reference inputs. To study these matters we will consider first one introduces a reference input to the full-state feedback case, and then we will proceed to the case with estimators. We then tum to a discussion of output elTur command. a structure that occurs when the sensor is capable of providing an error signal. for example. an attitude error from a gyro or the pointing error from a radar signal. The output error structure is also the one that results if one is designing the compensation using transfer-function methods and the reference input is structured according to Fig. 7,5. the typical case. It is. therefore. of interest to study this structure in order to understand the impact of its use on the dynamic response of the In conclusion. we will discuss the implication, of this section's results and compare the relative advantages of the structure made possible by the state-space control/estimation approach with the approach.
8.4.1 Reference Inputs for Full-State Feedback
Let us first consider a reference input for a full-state feedback system as in Eq. (8.5). The structure is shown in Fig. and consists of a state command matrixN that defines the desired value of the state.x .We wish to find so that system output.Y,.= at a desired refe;ence value. This desired output. might not be the same quantity that sense and feed to an estimator that has been called and determined byH in the previous two sections.
Although so far in this book we have only considered systems with a single control input and single output (SISO). Chapter9considers the case of
310 Chapler Design Using \Ielhods
Figure 8.14 Block diagram for full-state feedback with reference Input
(8.73)
(8.74)
= +KN,.
Introductionllfthe Reference lnpul 313
Collecting Eqs. (8.71) and into one matrix equation
[ I [ = [
yields the desired result"
also possible to enter the reference input after the gain multiplication according to Fig. 8.15(b) by combining and according to
Compute the referenl'e for the of Example where it desiredto commandd a new Compare rhe two (a)and(hiinFig.8.15.
Calculation ofN. and N can be carried out the MATLAB function refi.m contained in the Digital Toolbox. •
Solution. The state defined to be x=
rd
,i Therefore. to command a desired ofd. = [I 0 0 0]. and the evaluation ofEq. (8.731 leads to = [I 0 I 0]and = O. thc phy,ical Fig. A.8. The fact that
= 0 makes sense because this system is in equilibrium without the of any control as long d - which ensured by the \'alue of fact. be zero for
of Type I or higher. as already' discussed. The elemeuts do not depend on
values of plant are therefore not to madding of the
plant.
llse of Eq. to =0.005 the K from and =0.011
the K from Eq. 18.191. note that this input structure can be sensitive to errors in K [n thi, panic"lar example. is the result of a difference of two numbers (K
jand that
are to each other in absolute salue. for the first I also exhibited poor response I. thus producing extreme sensit;\ Specifically. if one of the elements of K in Eg. 18.18) were in error I the resulting error in would To avoid the high for cases like this. it is advisable to structure the reference as 8.15(al.
ol\'edif planlha a =I
This example that there are some systems where it is better to use the structure of Fig. 8.l5(a). most do not exhibit this sensitivity and Fig. preferred due to its simplicity.
• Example8.11
. _
-- -- -- ' -- -- -- ' --- - - - - - - - - . - -
-.
(8.701
(8.71 )
(8.72) H N =
= rand (bl
(al
speeding up the system response. On the other hand. if the system Type Ior higher. the steady-state value uf cuntrol for a step will be zero. and the solution will simply give us N" = 0 and which defines the desired value of the state,
Continuing on then. the steady state requirements for the system are that Nrr= xr =
H,x,.,
=
Y,=
r.which reduce to
x(k
+
I)= +
ru(k)= +
- I)N,
+
rN" = O.Furthermore. we are assuming the system is at steady therefore.
or
which reduces to
+
ru" = and, from Eqs. (8.69) and (8.70)I)N,r
+
rN"r= O.312 Chapter8 Design State-Space
Figure 8.15 (a)Blockdiagram and (b) modif'ed block diagram for full-state feedback with reference input and feedforward
(8.75)
(876)
(878)
(8.80)
_OK] [
]1 -I x(k)
x(k+ 1)=
+
fu(k)+y(k) = Hx(k)
+
lurk).K, ] [ [) ] - x ]r
+
r= - [K/'
[ u(k) ]
u(k)
= +
N"r=
+Nr.u(k)=
+
= -Kx(k)+
u(k) = -[K"
or. with the current estimator. Eq. (8.33) is used with
or, for the reduced-order estimator. Eq. (8,45) should be used
K,J [ ] +
(8.77) Under ideal conditions where the model in the estimator perfect and the inputu applied to plant and estimator is identical. no estimator error will be excited. We use the feedback to the estimator through y only to correct for imperfections in the estimator model. input scale factor errors. and unknown plant disturbances.
To analyze the response of a system. we must combine the estimator equations with the model of the system to be controlled. It is often lIseful to analyze the effect of disturbances. so the system equations (8.3) are augmented to include the disturbance.II',as
thus minimizing estimation errors. Therefore. the form of the estimator given by Eq. should be used withu(k) shown by Fig. 8.16. that is
Introduction or the Input 315
For the predictor estimator, (8.62)is augmented with the input command and disturbance as follows:
[ x] -fK ] [ x ] [ ] ,. [ 1'1 ]
= L"H -
rK -
LI'H+ +
0 w(k).(8.79) Note that the term on the right with r introduces the command input in an identical way to both the plant and estimator equations. The term on the right with II'introduces the disturbance into the plant only: the estimator is unaware of it.
ltmay be useful to inspect the performance of the system in terms of the de-sired output the controlu. and the estimator error This can be accomplished with the output equation
I I
(a)
If the system in the example had been Type O. we would have found that was nonzero and that its value was inversely proportional to the plant gain However. the plant gain can vary considerably in practice: therefore. designer, usually choose not to use any feedforward for Type
°
systems. relying instead the feedback to keep errors acceptably small or by implementing integral contrll]as discussed in Section 8.5. If this is the desired action for a control design state space. the designer can simply ignore the computed value of and rely on for guidance on how to command the state vector to the desired
Ib)