- - - - . - - - - - - - - - - - . - - - - - - - - - - . -
-GIS)= (lOs+1((60s+I)' where the duetothe sensor.
(a) Write state cquations for this
(b) Compute the zero-order-hold model with a sample period of 5 (e) Design the compensation including the input with the control
0.8 jO.25and the estimator at ± O.
(dl Compute the step responsetoa reference input andtoa disturbance inpul the control. Vcrify that is no delay for the command mpul.
Franklin. andEmamiI Example
(gl Plot root of the system (plant controller) and comment on the of this to changes in the ol'erallioop
8.25 A heat has the transfer function
358 Chapter 8 Space
(9.1 )
(92)
(9.3) x(k+I)= +ru(k).
u
=[ . x=
and p=
A comral of the standard form
Kll
] [ ]
Solution. The aircraft lateral equations are multivariable and of the fonn
control surfaces affect longitudinal motion: the aileron and rudder primarily af-fect lateral motion. Although there is a small amount of coupling of lateral motion into longitudinal motion. this is ignored with no serious consequences when the control. or "stability-augmentation:' systems are designed independently for the two founh-order systems.
Further decoupling of the equations is also possible.
Decouple the lateral aircraft founh-order equations two second-order and
todesign the control.
9.1 Decouplmg 361
that there are eight elements in the gain matrix to be selected. and the specification of four closed-loop roots clearly causes the problem to be underdetennined and man) possible ofKthat will meet the specifications. This shows that the pole-placement approach to multi\'ariable design poses difficulties and additional criteria need to be introduced.
Adecoupling that removes the ambiguity is to restrict contrallaw10
good physical the rudder primarily the aircraft about a axis (r-motion). thus directly causing l·).and the ailerons primarily roll the aircraft about an axis through the nose. thus causing changes in the roll angle. and the roll rate. p.
GII'en an set of desired pole there are unique values of the four nonzero components ofK: the governing equations cannot be cast in the same form as in Eq. (8.5) and therefore can be difficult to
• Example 9.1
Velocity components:
rate
= rate
Dateral ,peed)
= altitude rate Angularvelocity
p = roll rate q pitch rate r ::; yaw rate
The first in any multivariable be an attempt eithertofind of two or more single input-output models or todecouple thc control gain matrix K the estimator gain matrix L. This will give physical in,ight iIllo the important variables and
toa plant description that substantially simpler for design purposes et yields no degradation an analysis on the full
- For example. the linearized equations of motion of (Fig.9.1) of eighth order are almost into two fourth-order
longitudinal motion and lateral motion r. The elevator
always it to the control in
that optimization of the system will better Section
the optimal control solution that directly the optimal control problem Section shows howtofind the
lalue of the optimal gain that toimplement
and the one in control implementations. Section
companion optimal problem. for the control the time-I ing
gain solution found firs\. the gain that
implemented. The final 9.5. how to these in
design of
9.1 Decoupling
Ele,,'ator Figure9.1
Schematic of an aircraft shOWing
19.71
(k+ = x,lk)+
=[1
0][
J-DecllUpling 363
(k+I) = + + +L I \'Ik)
=
I , " ,
and the measurements
which can be done the methods in Chapter 8. There is one-way coupling into the equation. but this just acts like an additional control input and can be ignored in the calculation of the stick estimator gain matrix. using the pole-placement methods of Chapter 8. However. the coupling should not be ignored in the estimator equations. and there is no reason10ignore the weak coupling. The final (predictOr) estimator would be of the form
The stick pictured in Fig. 9.2 is substantially lighter than the cart. This means that motion has a small dynamic effect on cart motion. which in turn that O.This does not imply that = in fact. motion the mechanism for stick motion and stabili7ing it.
An estimator for the system de.scribed by Eq.(9.6)and Eq.19.7)requires the determination of eight elements of an estimator gain matrix, L Hence. specifying the four estimator roots and using the methods of Chapter would not determine thisLuniquely-another example of an underdetermined system caused by the multi\'ariable nature of the problem.
But because we can assume that =O.the cart equation in Eq. uncouple, from
stick equation, and simply an for
where and are both 2 x I matrices.
without the \'ery weak one-way coupling in that obvious for this example.
one could assume this to be the case. then check the resulting full-system characteristic roots using a method similartothe previous airplane example. Note that ignoring the coupling only causes approximations in the gain-matrix calculation and thus the root locations. There is no Figure 9.2
Hingedstick and cart
(9.6!
I
x, [ ] cart position and velocity.
x., [ ] stick angle and angular rate.
The equations of motion can be written as
=
[ ].
+ [ ]u(k).[
r ]
= 0 0
p 0 0
P
where Solution.
Decouple the estimator for the inverted pendulum on a motorized cart IFig. 9.2).
The same ideas apply equally well to the decoupling of the estimator into SISO parts.
=
_ f K ]
r K - . , . -- - f K
- K - [11K "
-. ' \ " . - -- K [ K
- f K"
-which resulis from combining Eq. (9.31 and is important the rOOI' If the plant coupling that was ignored in the gam computa Ion . . I obtained from Eq.(9.5)will differ from those used to com:utedthe In
(9.4), In many cases. the method be until the correct
other cases. one could the deSIred h thods of optimal control to be describedIn fromEq. (9.5) satisfactory or else tum to t e me
the following sections.
and that the control law is given Eq. (9.3).. between the two This makes some physical sense but ignores Important coupI f 'h' the
d 1 the into second-order or Ie .
modes. It however. ecoup e J . h i ' I oed loop
. . I bt' the T e resu tmg c os
-of Chapter 8 can be applied to0 k the eigenvalues of the roots of the full lateral equatIOns can ec e e
loop matrix: (seeelg.minMATLAB)
A further decoupling that would pennit an easy gain calculation is to assume that Eq.
is of the
• Example 9.2
362 Chapter 9 and Optimal Control
Chapter9 Multivariable and Optimal Control
subject to the constraint that
, [I
TI
= (klQlx(k)
+
+
(k+
1)( -x(k+
I)+ +
ru(k))] . (9.12) find the minimum of with respect to x(kj.u(k).and Note that for an opltmal that obeys Eq. (9, 10). the two cost functions •.1' and.1. are identical magnttude. on is arbitrary conceptually. but we let it bek+
I because thIS chOIce WIll yIeld a particularly easy form f th . IP . , . 0 e equatIOns ater on
roceedIng wtth the minimization leads to .
9.2 Time-Varying Optimal Control 365
easily accomplished by picking the Q's to be diagonal with all diagonal elements . Another of stating the problem given by Eqs. (9.10) and (9.11) '. th
we WIsh to IS at
-x(k
+
I)+ +
ruCk)= O. k= 0.1. . ". N. [9.101 This a standard constrained-minima problem which can be solved usin the meth,od of multipliers. There will be one Lagrange vector. we call+
I).for each value ofk, The proc d .rewnte Eqs. (9.10) and (9.11) as e ureISto
+
= -x(k+
I)+ +
rUCk)= O. state equations. and [9.IOJax(k) = xT
(k)QI - (k)
+
(k+
= O. adjoint equations. (9.14) The set of the equations. the adjoint equations, can be written as the back-ward difference equation=
+
I)+
Qlx(k).Restating the results in more convenient fonns, we have from Eq. (9.10) x(k
+
IJ=+
ru(k).I If the T, long. a control that moves the state along rapidly as mightbefeasible. Such comrols are called because they the state to a dead stop in most They correspondtoplacement of all poles o.See Problem 8.23.
2 equivalenr ofa number; it ensures thatxTQ1Xand are nonnegative for all possible x and u.
u=-Kx
that was used in Chapter8and illustrated by Eq. (9.2) in Example 9.t.
Given a discrete plant
x(k
+
1)=+
ruCk).•
we wish to picku(k)so that a cost function
= -I (k)Q1X(k)
+
uT 2is minimized. Q, and Q, are symmetric weighting matrices to be selected by the designer. who bases the choice on the relative importance of the various states and controls. Some weight will almost always be selected for the control 0):
otherwise the solution will include large components in the control gains. and the states would be driven to zero at a very fast rate. which could saturate the actuator device.I The must also be nonnegative definite,' which most approximation in the system model used in the estimator: therefore. the estimation errors will still approach zero for stable estimator roots.
In short. it often useful to apply your knowledge of the physical aspects of the system at hand to break the design into simpler and more tractable subsets.
With luck. the whole job can be finished this way. At worst. insight will be gained that will aid in the design procedures to follow and in the implementation and checkout of the control system.
Optimal control methods are attractive because they handle MIMO systems easily and aid in the selection of the desired pole locations for5150systems. They also allow the designer to determine many good candidate values of the feedback gain,K.using very efficient computation tools. We will develop the time-varying optimal control solution first and then reduce it to a steady-state solution in the following section. The result amounts to another method of computingK in the centroIlaw Eq.(8.5)