Kiểm soát và ổn định thích ứng dự toán cho các hệ thống phi tuyến P12 pdf

Thus, the system is not uniformly completely observable and the tools presented that the dynamic projection used in Chapter 10 to eliminate the peaking phenomenon can be used here to bou

Chapter 12

Applications

of nonlinear systems both via non-adaptive and adaptive methods In this chapter we apply these techniques to three illustrative examples

In the first example we consider the problem of controlling stall and surge in a jet engine compressor and we seek to find a controller which only employs the measurement of the differential pressure across the compressor

to reject stall and surge while regulating the pressure at a desired value This design presents an interesting challenge in that, when there is no mass flow through the compressor, the system looses observability Thus, the system is not uniformly completely observable and the tools presented

that the dynamic projection used in Chapter 10 to eliminate the peaking phenomenon can be used here to bound the observer states away from the unobservable region of the state space

The second example considers the problem of controlling the horizontal position of a beam by using two electromagnets at its sides in the presence

of an unknown force representing, for example, friction acting on the beam

We first solve the problem assuming that the unknown force is zero and then develop an adaptive controller to recover the closed-loop performance

we show that a simple linear high-gain observer and control saturation allow

us to define an adaptive output feedback controller

The last example of the chapter focuses on the tracking problem for

cha’llenge in solving the tracking problem is to find a practical internal model, as defined in Definition 10.2, and a full information controller (i.e.,

a controller which exploits the knowledge of the state of the plant and its stable inverse) Once this is done, one can use the tools of Chapter 10

to define estimators for the state of the plant and the stable inverse, and

Stable Adaptive Control and Estimation for Nonlinear Systems:

Neural and Fuzzy Approximator Techniques.

Jeffrey T Spooner, Manfredi Maggiore, Ra´ul Ord´o˜nez, Kevin M Passino

Copyright  2002 John Wiley & Sons, Inc ISBNs: 0-471-41546-4 (Hardback); 0-471-22113-9 (Electronic)

invoke a separation principle to find a partial information controller Some design details in this example are skipped and are left to the reader as an exercise

12.2 Nonadaptive Stabilization: Jet Engine

[ 1151 for an analogous exposition)

li = gR(l - a2 - m R(O) > 0, - where ip represents the mass flow, Q is the plenum pressure rise, R > 0 is -

the throttle, 0 = 7, and ,0 = I/& The functions !I!&%) and @&I!) are

our control input and use

to define y Our control objective is to stabilize system (12.1) around the

the change of variables 4 = (a - l,$ = KU - X!!C,, - 2 System (12.1) then

becomes

ri = -oR2 - aR(2$ + 4”)

(P = -$-3/2c,b" - l/2@ -3R+3R (12.2)

ti - = -&(a* - l- 4) The pressure rise (and hence Y/I) is the only measurable state variable

traction D = {x E R3/R > O} -

Proof: For the sake of simplicity, redefine the control input to be u’ =

- -& (u- c$), so that the last equation in (12.2) becomes Y+!I = u’ Next, notice that system (12.2) can be viewed as the interconnection of two subsystems:

(12.8)

where we are ignoring the interconnection term for now A Lyapunov func-

of S1 is an asymptotically stable equilibrium point of S1, and its domain

control law for S2 is found to be U’ = -cl+ + C.LT,LJ, where cr and c2 are two

In the following we will show that, in order to stabilize the interconnec-

consider the following Lyapunov function candidate for system (12.2)

u’), letting xi = kr - 9/8, and using the definition of 4, we get

since it is negative definite, and that the term !$+3$ cancels out After

By using Young’s inequality five times we have

-(2Co + 3)&b 5 fR2 + (2c02+ 3)2&2

we obtain Ca - $ > O! (Ca - 2) $klkl - i (Ca + 3 - j$k1)” > 0, while by imposing the positivity of the coefficients of the remaining two terms we get j& > (2C~+3)~ + 1, k2 > kl + ikf + 2 + (kfi1)2 By using the definition of

&I, inequilities (12.4), (12.5), (12.6), and (12.7) follow In conclusion, if kl ,

kz, and C are chosen so that (12.4)-(12.7) hold, we have that v is negative definite on D which contains the origin This leads to the conclusion that {R = 0, 4 = O,$ = O> is an asymptotically stable equilibrium point, which

in turn implies that {R = 0, + = 0, $ = 0} is an asymptotically stable equilibrium point

Our next objective is to show that D is a region of attraction for the origin This, however, is not immediately evident from our result, since the

the term CR in V, it is not completely contained in D In other words, it

negative, and thus the state trajectory exits the set D, where V is guaran- teed to be negative definite Therefore, in order to complete our analysis,

we need to show that D is invariant, which, together with V < 0, implies that the set {Z E R3 : V _< I(, K > 0} R D is a region of attraction of the origin for any K > 0 This is readily seen by noticing that, on the boundary of D, R = 0 From (12.2), R = 0 implies k = 0, thus proving that no trajectory of the system can cross the boundary of D, and there-

there exists a constant K > 0 such that the initial condition is contained

system (12.2) is an asymptotically stable equilibrium point with domain of

We conclude this section by remarking that inequalities (12.4)-( 12.7) repre- sent conservative bounds on kr and JQ In practical implementation, these parameters may be chosen significantly smaller after some tuning

Recalling that the plant (12.2) is SISO with input u = @T - 1 and output $, the observability mapping is given by

$ I

-+u - 4)

3 -ti-?i,-242 - ;gb3 -3Rqb-3R >

where ye = [y, G, yl’ and X, = [xT,u, iLfT In what follows, we will im- plement the output feedback control scheme illustrated in Section 10.4.4 Specifically, since 3c depends on u and ti, we augment the system with two integrators at the input side

a,nd, using backstepping, we redesign the control input for the augmented

v = cl - 51 - Ii& e 8(x,), (12.16) where

a=- k3iG1 - g&l~x~ + grfw + 9(x> 4 (12.17)

and ks, /Q are arbitrary positive constants The Lyapunov function of the closed-loop augmented system is

V(xJ = V(x) + 5”: + p

Notice that, following the same reasoning as in the proof of Lemma 12.1

by using the design outlined in Section 10.6, one could avoid the dynamic

the reader

We now seek to design a nonlinear observer to estimate the state of the

isfied for system (12.2) To this end, we are interested in checking whether

and for all [u, iLIT E R2 Noting that when 4 = -1 7-f does not depend on

R, we have that R cannot be calculated as a function of ye In other words,

formly completely observable Note however that ?f is a diffeomorphism on

observability mapping is singular corresponds to the situation when @ = 0, i.e., there is no mass flow through the compressor, which is a condition one would like to avoid during normal engine operation Note also that the singularity in the observability mapping poses a theoretical problem in that the separation principle illustrated in Theorem 10.2 requires the plant to be uniformly completely observable When the UC0 assumption is violated, one may have that during the transient the observer-generated estimate crosses the region where the observability mapping has a singularity, i.e.,

the observer (10.39) becomes undefined because [ZY(?, s)/EG!]-’ does not exist whenever x E 8X

In conclusion, in order to achieve separation between the controller and observer designs when the plant in not UCO, one must make sure that the observer estimate does not cross the unobservable boundary dX during transient or, in other words, one must constrain the observer estimate to lie inside the observable set X Keeping this in mind, recall that the dy- namic projection (10.63)) when applied to the nonlinear observer (10.39), constrains its state in within the compact set 7 /-l (C ), where C is a convex

0,, are defined by

i-I,, = {xa E R5 : V(xa) 5 c2, R > O}

contained in within the observable region X In conclusion, we seek to find

Given a pair of scalars ai, bi, i = 1, ,5, it is easy to see that the set

[y,‘) sTIT E R5 : ye,1 E [al,h],ye,2 E

P -s2 + a3), -y(-52 P + b3) 1 , Sl E [a4, bill, s2 6 [a5, b51

is contained in F(X x R2) for all u2 < 1 and it is compact Furthermore, the set C is convex In order to see that, note that C can be written as the cross product of three sets C 1, C2, and Cs, i.e., C = Cl x C2 x C3, where

C 2 = [Ye,3, S2] E R2 : Ye,3 E -“>T u3 7

-s2 + b3 p2 1 ,s2 E [a:,$51 ,

>

CS = {ye,1 E R : Ye,1 E [al 7 h]}

The sets C 1, C 2, C 3 are immediately seen to be convex and, hence, the resulting product set C is convex as well (it is easy to show that the product set of any finite number of convex sets is a convex set) Hence, in order to satisfy the condition in (12.18)) it remains to use the Lyapunov function v

to find the largest value of c2 such that a,, c X x R” and subsequently pick values for the scalars ui, bi, i = 1, ,5 such that u2 < 1 and ?(S&,) C C

A more practical way to address the design of C entails running a number of simulations for the closed-loop system under state feedback corresponding

to several initial conditions x, (0) and calculating upper and lower bounds

al = -1.15, bl = 0.5, a2 = 0.3, b2 = -0.1, a3 = -0.75, b3 = 0.4, a4 = -2,

have a smooth boundary Its boundary is continuous but not differentiable

at some corners This, in general, may generate some numerical problems in the projection, since at the points when the boundary is not differentiable the normal vector to C is not uniquely defined Should numerical problems a,rise, one may slightly modify the definition of C by smoothing out the corners

Having chosen the set C , we define the nonlinear observer

is chosen to be [3,3, I], which is Hurwitz Next, we calculate the solution

(A,, C,) is a canonical observable pair The dynamic projection confining

ip = [g]-l {P (1/,,i,,s,i) - y$}

B, - r Nl?le(Ge7s) (N$/e(ije~s)Tlie +Ns(!i,;s)TS)

~(8e&,s,S) = N(Ce7 S)TrN(iie7S)

if N(fie, s)TSe + N,(Qe, s)TS > 0 and $e E dC

I & otherwise,

& = ‘J-@&s), & = 1 a-f- XX + XS dW

> and Ny,($e, s), N&j,, s) denote the ye

[L 0, OIT if $e,l = bl [-I, O,OIT if jje,l = al [o, 1: qT

[0, -l,OIT

if ce,z = -&(sI - b2) [o, 0, 1lT

and

I ;2 10 if iL,2 = -&+l - a2)

if &,3 = &4 -s2 + as)

if s1 = bd

L-1, OIT

Lo 11 T

P’ 11 T 7-

where the choice of 3cr = 25 and k2 = 1.1 lo5 fulfills inequalities (12.4)- (12.7) in Lemma 12.1

as predicted by the theory in Chapter 10 In Figure 12.3 the trajectories

of the system under state feedback are compared to the ones under out- put feedback for decreasing values of v The smaller q is, the closer the output-feedback trajectories are to the state feedback ones

12.3 Adaptive Stabilization: Electromagnet Control

In this section we consider the problem of using a pair of electromagnets

electroma’gnet is independently driven by a linear power amplifier providing

a voltage ua and ub across its windings to electromagnet a and b, respec- tively In order to model the dynamics of the plant, we start by defining

x as the beam position with x = 0 when the beam is centered between the electromagnets Let G denote the nominal gaps of the electromagnets ( i.e., G is the distance between each electromagnet and the beam when

proportional to the square of the corresponding phase current and inversely proportional to the squared distance to the beam

Figure 12.4 Electromagnet control of a beam

With this framework we obtain the system model defined by

b, respectively Here we also have the gaps gn(x) = G - x, gb(x) = G + x,

beam a,t some desired location r (where 1~1 < G) while keeping the state of the system bounded Notice that if either x = G or x = -G, that is, when there is physical contact between the beam and one of the electromagnets,

and invokes the separation principle developed in Chapter 11 to recover the

12.3.1 Ideal Controller Design

Figure 12.5 Block diagram of the magnetic levitation system Rewrite (12.19) as the interconnection of two subsystems with the topology depicted in Figure 12.5, where Sr , S2 are given by

(12.20)

Notice that the pair (Sr , S,) is cascade-connected and hence has a trian- gular structure: we will utilize this structure to simplify the control design procedure Specifically, referring to Figure 12.5, we will first design a con- troller for Sr assuming that i, and ib are control inputs Then, applying the

ba.ckstepping methodology to Ss, we will derive stabilizing voltage inputs

where rln(z) and r&z) are chosen such that Q(Z) - v&z) = x and such that

%dzhb(x) 2 0 f or all x where x is defined by (12.25) Then the closed-loop system has an equilibrium at (x, v) = (T, 0) which is exponentially stable Proof: Define the first error term as er = x - r so that

Ideally we could choose v = -Krer with ~1 > 0 so that tii = -Krei has

er = 0 as an exponentially stable equilibrium point But v is not an input (it is the velocity), so instead we choose e2 = v + Krer (so if e2 -+ 0, then the er dynamics have the desired form) Taking the derivative of e2, we find

kai”,

where & = -krer + e2

Notice that this is achieved by letting

(12.26)

Sec 12.3 Adaptive Stabilization: Electromagnet Control

where qn (x) and Q(Z) are chosen such that qn (2) Q,(Z) = x for all Z Keep-

positive as long as m, kn, kb > 0, which must be the case for any physically rea,liza,ble system

All that is left to do for the Sr controller is to find some r), a.nd qb

to (12.27) Notice that even though this is an approximation to our orig- ina, choice of v~, qb, the smooth version does satisfy both requirements of Lemma 12.2 (i.e., qa(z) - q)(z) = +Z and qa, qb > 0 for all Z) so that we still

Having found a controller for subsystem Sr , we proceed to the derivation

of a controller for the cascade (Sr , S,) when n = 0 (see Figure 12.5) Lemma 12.3: Consider the cascade connection of system Sz and Sl and let es = i, - Y,, e4 = ib - vb with u, and ub defined by (12.26) The controller (12.33) guarantees that e = 0 is an exponentially stable equilibrium point and the set

D = {e E R4 : )e12 5 je(O)l") ,

is positively invariant when A = 0

Notice that Lemma 12.3 implies that all trajectories of the cascade con- nection (Sr , S2) originating in D at t = 0 remain in D for all future time Furthermore, noting that e E D implies that

I 2 - ?-I2 = ef 5 le(O)l",

we conclude that the initial conditions ma’y be chosen such that the closed-

Proof: Consider the cascade interconnection of Sr and Sz and the fol- lowing Lyapunov function candidate

(12.29)

Taking the derivative of this Lyapunov candidate we find

p = & + (i, - v,)(& - tin) + (ib - @) (;;b - fib) (12.30) Using the cascaded system definition we find

k&ii

- + Klifl

+ (2, - u,)& - fin) + (ib - u&b - i/b)

Adding and subtracting the ideal control laws va and vb for the first sub- system

so that

r;; = -245 + e2 it&~ - u,“> kb(iE - u,“)

+ (ia - ua)(ia - tin) + (ib - ub)(& - i/b)

Using the electrical dynamics for each phase, we then choose the controllers

12.3.2 Adaptive Controller Design

that ]LQ$ + F-(x$)] < lit/ for all x E [-G, G] -

Lemma 12.4: Consider the cascade connection of system Sz and S1 and let e3 = i, - un(i), e4 = ib - L&??) with V, and Yb defined by (12.26) with 2 = -~1k.l - el - (~1 + q)e2 + F(x,@/m

Proof: Assume that for a

8 such that In(x) + F(x,8)]

to define a stabilizing signal

given approximator F(x, 8) there exists some

5 VV for all x E [-G, G] Then one may want for the Si subsystem using