Explicit control laws for some classes of feedforward systems

Copyright © IFAC Nonlinear Control Systems, Stuttgart, Germany, 2004

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EXPLICIT CONTROL LAWS FOR SOME CLASSES OF FEEDFORWARDSYSTEMS Miroslav Krstic University o/California, San Diego, La folia, CA 92093-0411 Department 0/ Mechanical and Aerospace Engineering Abstract: In a recent paper (Krstic, "Feedforward systems linearizable by coordinate change," submitted to ACC'04) we revealed that the family of feedforward systems contains a substantial class that is linearizable by a diffeomorphic coordinate change. In this paper we present two subclasses for which explicit control formulae can be derived, which also allows us to give quantitative performance bounds. Our procedures follow the general integrator forwarding transformations of Mazenc-Praly/Sepulchre-JankovicKokotovic but avoid the requirements to solve (analytically) a series of nonlinear ODEs and to compute (analytically) a series of integrals with respect to time. Copyright © 2004 IFAC Keywords: Nonlinear control, Stabilization, Lyapunov, Forwarding, Backstepping.

Theorem 1. The diffeomorphic transformation

I. INTRODUCfION In a recent paper (Krstic, 2004) 1 we showed that the family of feedforward systems contains a substantial class that is linearizable by a diffeomorphic coordinate change. In this paper we present two subclasses for which explicit control formulae can be derived.

YI

Yi =Yi+I, Yn =u.

U

i = 2, ... ,n - I

(2)

xn =U, where 1tj{O)

(5)

i=I,2,oo.,n-1

(6) (7)

= a1 (x) = -

±(.~ )

j=1

I

I

Yi

(8)

In the next result we go even further and show that, not only does one have a closed-form formula for the control law (8) but one can even get a closed-form formula for the solutions of the system under that control law. This is not just an aesthetically pleasing result-it will allow us, in Section 6, to derive bounds on the control effort given explicitly in terms of the size of initial conditions.

j=2 Xi=Xj+l,

i=2,oo.,n

globally asymptotically stabilizes the origin of (1)(3).

n-I

(1)

(4)

1tj(s)ds

The feedback law

Consider the class of strict-feedforward systems given by

L 1tj(Xj)Xj+1 +1tn (xn )u

j=2}O

i

converts the strict-feedforward system (I )-(3) into the chain of integrators

2. LINEARlZABLE FEEDFORWARD SYSTEMS OF TYPE [

=X2 +

±r

yj=Xj,

The foundations for Lyapunov based control of feedforward systems were laid in (Mazenc and Praly, 1996). We follow a recursive integrator forwarding algorithm (Sepulchre, Jankovic, Kokotovic, 1997a) but avoid the requirements to solve (analytically) a series of nonlinear ODEs and to compute (analytically) a series of integrals with respect to time. Performance trade offs between Lyapunov-based forwarding controllers and nested saturation controllers (Teel, 1992, 1996) have been well illustrated in the literature on examples. The analytical expressions for the Lyapunov functions and the control laws derived in this paper allow us to also give quantitative performance bounds.

XI

=XI -

To prevent confusion about the notation in the theorem, before its statement we emphasize that x, which denotes the initial condition, is constant. This notation is important for a seamless use of the theorem in subsequent results.

(3)

= O.

Theorem 2. Starting from the initial condition x, the solution of the feedback system (1)-(3), (4)-(8) at time't is

I To comply with length restrictions, a literature overview is omitted. It is contained in (Krstic, 200 J).

171

[i (~:=~)

slt,x) =e- t

j=i

£

J

j X

(_'t)k

k!

k=O

Iln (Xn) =

(_I)j-i

ft· 4»n-1 (s)ds I

lli(Xn) = -

i (n-~+k)XI l=j-k 1- +k

(14)

Xn

I

Xn

J

foX. [4»i-t (0,. 00 ,O,s) 0

- j=i+1 i Ilj(S)4»i+n-j(O,.

+(-I)ii (~:=~) (:~II), j=i J J . I

for i

00

,O,S)] ds (15)

= n - l,n - 2'00 .,2, and

Xm

1tm (S)dS) ]

x Ctzfo

(9)

(16)

for i = 2, ... ,n and

(17)

~I

(_'t)k

~

. (n- j+k)

k=O

k!

I=j-k

1- J+k

r,n

Sl('t,x)=e-tu~

(n-I) j_1

X~----

~

(n _I) -~ n

j-I

(_1)1-

.

j

't -

fork=2,oo.,n-2.

I

Theorem 3. If n-l

1

VX,i = tion

(j-I)!

n

xm

1tm (S)dS)

x (];zl n

+L

1~j(t,x)

Yi =Xi -

]

L

. (20:

Yn =Xn

(10)

rrAs)ds,

converts the strict-feedforward system (12)-(13) into the chain of integrators (6)-(7). The feedback law

whereas the control signal is

u=al(x)=-i(.~ )Yi i=1 1

(21)

I

u = al ('t,x)

i (.~ ) [ij=i (~ := ~) (-I j~1 ~ (n-j+k) Xl -t

i=1

I

1

J

globally asymptotically stabilizes the origin of (12)(13). )j-i

I

Example J. To illustrate the above concepts (and notation), let us consider a fourth order example of a Type II feedforward system:

(-'t)k

X~--~

k=O

i= I,oo.,n-I (19:

lli+l+n-j(Xn)Xj,

j=i+1

j=z 0

= -e

L

'Yj-i(Xn)Xj+$;(O,oo.,O,xn) (18) j=i+1 I, ... ,n -2, then the diffeomorphic transforma-

4»i(!i+l) =

~

k!

l=j-k

.

l-J+k

+(-I)ii(~:=~)(:~-Il), j=i J J .

.

Xl

I

Xm

x Ctl

1tm (S)dS)

(11 )

] .

X

X3 4) = Xz + (XZ '2 -]'2 u

(22)

. X3 XZ=X3+'2 U

(23)

X3 =X4 +X4u

(24)

.4 =u.

(25)

The control law 3. UNEARIZABLE FEEDFORWARD SYSTEMS OF TYPE II

u = -YI-4Y2-6Y3-4Y4 = -ZI-ZZ-Z3-Z4, (26) where

Consider the class of the strict-feedforward systems given by

i=l,oo.,n-1 (12)

Xn = u,

X4Xz ~X3 x: YI =xl-T+T-24

(27)

X4X3 ~ yz=x2-T+(j

(28)

(13)

°

Y3 =X3 -

where $;(0) = and Xi denotes [XI, 00. ,x;]. In this section we construct control laws for a linearizable subclass of (12), (13).

~ '2'

Y4 =X4,

(29)

which is obtained with

To characterize the linearizable subclass, let us consider the functions 4»n-1 (xn ) and 4»i(O, ,O,xn ), i = I, 00 . ,n - 2, as given and introduce the following sequence of functions:

(30)

00.

and Zi

172

= Xi -

~i+1 with

where

(31)

j

= I, ... ,n, and the control signal is

U = Ct,(t,x)

(32)

= _e- t

.l=~ z=

o

0

~

-I 0 -I -I -1 -I -1 -I -1

X

]z

and (s + I )3YI (s) = O.

I

I

j=i

j

(-I)j-i

I

j-I(-t)* n (n-j+k) xI-, I . *=0 k. I=j-* l-j+k

(33)

achieves

±(. ~ )±(~=~)

i=1

(XI -

±

JlI+I+n-m(Xn)xm) .

(40)

m=I+'

(34)

4. TYPE 1 AND n SYSTEMS IN DIMENSIONS TWO AND THREE

o

We start by pointing out that in dimension two all strict-feedforward systems are simultaneously of Types I and n. This implies that alI second order strictfeedforward systems are linearizable.

Lemma 1. Consider the series of functions

Theorem 5. Consider the system (41)

X2 =u,

for j = n - I, ... ,2. The inverse of the diffeomorphic transformation (19) is n

Xi

= Yi +

I

Ai+l+n-j{Yn)Yj,

i

= I, ... , n -

where $1 (x) is continuous and $1 (0) law

I (37)

j=i+l Xn =Yn'

(42)

U

= -XI -

2x2

= O. The control

{'2 + lo $1 (s)ds

(43)

ensures global asymptotic stability of the origin. (38)

Example 2. Let us now consider an example with Theorem 4. Starting from the initial condition X, the solution of the feedback system (12)--(18), (21) at time t is

$, (X2) = -~. This example was worked out in (Sepulchre, Jankovlc, and Kokotovic, 1997). In this case the formula (43) gives 2 U

= -Xl

- 2x2

-1.

(44)

One should recognize that the "-XI - 2x2" portion of the control law (44) is responsible for exponential stabilization of the linearized system. To see that this linear controlIer is not sufficient for global stabilization, we plug it back into the plant and obtain a closed loop system, written in the form of a second order equation, as

X2 + (2 -~)i2 +X2 = O.

(45)

This is a Van der Pol equation with an unstable limit cycle, which exhibits a finite escape instability. Hence,

"-1,"

designed to accommodate the nonlinear term the input nonlinearity $1 (X2) = -~, is crucial for 0 global stabilization. The possibilities, as welI as the limits, of Type 1111 linearizability for strict-feedforward systems are best understood in dimension three. The folIowing class of A readec checking back the details in (Sepulchre, Jankovic. and KokOlOvic, 1997) will notice that this control law differs from (6.2.12) in (Sepulchre, Jankovic, and KokolOvic, 1997). This is due to an extra M~" term that has crept into the calculations in (Sepulchre, Jankovic, and I(okotovic, 1997), in equation (6.2.7). 2

173

is not feedback linearizable. However, the following similar (at least visually) systems, are linearizable. The system

systems, which represents a union of all Type I and Type 11 feedforward systems in dimension three, is linearized in the next theorem. Theorem 6. Consider the class of systems

Xl =X2 + 1t2(X2)X3

+ ( X32(X3) -.dJ;3 2(s)ds X2 + 1t3 ()) X3 u

(46)

(47)

X3 =u,

(48)

where 1t2 (. ), 1t3 (.) E ~ and 2 (-) E Cl are vanishing at the origin and (49) 1t2(X2)(!>2(X3) == O. Then the control law

u = -YI - 3Y2 - 3Y3

_fo

X 2

1t2(s)ds-fl3(X3)X2

_fo

X 3

(51)

x3

Y2=X2-1 2(s)ds

(52)

Y3 =X3

(53)

and

J;3 2(s)ds

fl3 (X3 ) =

,

X3 achieves global asymptotic stability of the origin.

=X2+

(1

.)

2x2+x3smx3 u

X2 =X3 +X3 U

(64)

Xl =X2 +~X3

(65)

X2 =X3

(66)

X3 =U

(67)

= X2 + xj + X2 U

(68)

X2=X3

(69)

X3 =

(70)

U

(which is temptingly close in appearance to Type I but is not in that class), is linearizable using the coordinate change

The above examples all had the last two equations actually linear. The neither-Type-I-nor-II feedforward system

0

A Type 11 example of a system from this class is

.

X3 =U

(54)

Proof On can verify that )i I + 3YI + 3YI + YI = O.

XI

(63)

XI

1t3(s)ds

I 2 1 (X3 2 +2" X3(fl3(X3)) +2"Jo (fl3(S)) ds

X2 =X3

is of Type I, and therefore linearizable. Other such systems exist, outside of Types I or Il, that are linearizable. For example,

(50)

where

Yl =Xl

(62)

is linearizable, as it is of both Type I and Type 11. The system

3

X2=X3+2(X3)U

XI =X2+xjU

XI (55)

=X2+~X3+ 1-~xju

(72)

X2=X3-xjU

(73)

X3=U,

(74)

(56)

which includes nonlinearities in both of the first two equations, is linearizable using the coordinate change

(57) which is stabilized (and feedback linearized) using

X2X3 U = -Xl - 3X2 - 3X3+ 2

3?

+ 2"X3

-~~ +X3sinX3 +COSX3 -1.

Clearly, since the systems (68)-(70) and (72)-(74) are neither of Type I nor n, the coordinate changes (71) and (75) cannot be obtained from the explicit formulae in Sections 2 and 3. However, they can be obtained following the simplified SJK procedure in (Krstic, 2004), which, we remind the reader, avoids the requirement to solve a series nonlinear ODEs.

(58)

We point out that the key restriction in this example is the boldfaced 1/2. If this value were anything else (say, I, or 0), this system would not be linearizable. The focus on third order systems is partly motivated by the fact that the celebrated "benchmark problem"

XI =X2+xj

(59)

X2 =X3

(60)

X3 =U,

(61)

5. INVERSE OPTIMALITY Definition 1. The control law n

first solved by Teel (1992) using his method of nested saturations, is of third order. The system (59)-(61)

u·

= 2uI(x) = -2 L

j=1

174

(Xj - ~j+1 (!j+I))

(76)

is said to be inverse optimal if it minimizes the cost functional

J =

f

(l(x(t)) + U(t)2) dt

large values for extended periods of time. On the other hand, there is an inherent engineering merit in having control laws that are robust to actuator saturation (by means of "caution"), which the nested saturation controllers are.

(77)

along the solutions of the plant, where

A highly desirable property arises from linearizabilitythat the performance and control effort can be quantified in terms of the original problem data (in terms of the plant vector fields). In the case of Type I and Il systems, we have closed-form solutions for the state and control which allow such quantification. In the next two theorems, which are proved using

(78)

(79)

r~ tPe-qtdt =

is a positive definite, radially unbounded function.

10 One of the most important consequences of inverse optimality is that the control law (76) remains globally asymptotically stabilizing at the origin in the presence of input unmodeled dynamics of the form

a(l + P) ,

tPe-qt

~

L

(83)

qp+t

(;r

(84)

and, respectively, Theorems 2 and 4, we calculate explicit £1 and £~ bounds on the control effort in stabilizing feedforward systems of Types I and 11.

(80)

!

where a 2: is a constant, Pu is the output of any strictly passive nonlinear system 3 with u as its input, and J denotes the identity operator (Krstic and Deng, 1998).

Theorem 9. The control law (8) applied to the plant (I)-{3), (4), (5) expends the control effort in the amount bounded by

Theorem 7. The control law (76), with


13i+I(!i+d=-

± (~=~)Xj+Oi.I±LXj1tj(S)dS

j=i+J

)

I

j=2

0

(81)

for i = I, ... , n - I, where Oi,l is the Kronecker delta.

(85)

Theorem 8. The control law (76), with
and

lIul\£- ~

±(i~ 1)

,=1

x

±.(n}.=:) J='

j-I k!' n (n-j+k) [~ekk! I=t.* l- j+k IXI(O) I (j-l)j-I

for i = 1, ... ,n -1.

+ d-I(j-I)!

In

r;r",(O)

mY::2 10

I]

1tm(s)ds, (86)

where Xi(O) are the initial conditions of the state.

6. PERFORMANCE The general performance advantages of the SJK-type integrator forwarding were thoroughly illuminated in Section 6.2.6 of (Sepulchre, Jankovic, and Kokotovic, 1997). It was shown there that "overly cautious" nested saturation designs, whose form is in many cases the same irrespective of the sign of the plant nonlinearities, don't perform as well as Lyapunovbased designs. In these, and various other simulations presented in the literature, the nested saturation (and other bounded) controllers display the trademark linear (non-exponential) decay resulting from saturating the control. Saturation itself leads to large overall control effort (at least in £].) by letting the states linger at

Theorem 10. The control law (21) applied to the plant (l2)-{ 18) expends the control effort in the amount bounded by

3 with po5sibly non-zero initial conditions

and

175

IluIIL-::;

i

1=1

t.

(i~ 1) (~=:) )-1

to

k!'

j-I

x

n

ekk! [=t-k

YI

t

2

k

i ~l+l+n-m(xn(O))xm(O)I,

I 2 ~(X3)X2 + '2X3 ~(X3))

+~ F3 ~(s))2 ds _ {X4 b(s)ds

(n-j+k) + Ix[(O) l- j

= XI -

(88)

m=[+1

lo

lo

(96)

Y2 =X2 _10'<3 a(s)ds

(97)

Y3 =X3,

(98)

Y4 =X4,

where

where Xi(O) are the initial conditions of the state.

(99)

Stabilization by bounded controls is unquestionably a major accomplishment [of (Teel, 1992) and the p~­ pers it directly inspired] especiall.y from the engIneering point of view. However, given the charact~r of open loop instability in feedforward systems, It should be less surprising that one can stabilize them with bounded controls than that one can actually indulge in controls with large nonlinear growth (in quest of performance), like those represented by the SJK Lyapunov procedure. For strict-feedback systems, due to their finite escape instabili ties, the challenge was to design bounded stabilizing controls (Freeman and Praly, 1998). By analogy, for feedforward systems, a (theoretically) worthy future challeng~ would ~e to design high-performance controllers WIth unrestricted nonlinear growth.

As a final comment, we do concede that Iinearization (by coordinate change) may be view~d .as a step backwards, if seen as a procedure that ehmmates all the nonlinearities-the 'harmful,' as well as the 'useful' ones-and applies controls with high nonlinear growth, in contrast to the nested saturation designs. To clarify what we mean by "useful" nonlinearities in the case where 'feedback linearization' amounts to just a coordinate transformation (without feedback, i.e., without direct cancellation), consider the system from Theorem 5. In Example 2 we presented a case of a harmful nonlinearity that had to be eliminated. However, if

7. DISCUSSION Type 1and 11 systems are only a su~set of linearizable feedforward systems. An example is

XI

= X2 + (~X4 -X2xj) u

(89)

X2

= X3,

(90)

= X4, X4 = U,

X3

which is linearizable via coordinate change REFERENCES

YI

= XI -

Y2 =X2,

(X2X4

-1)

Y3 =X3,

2

Y4 =X4

R. A. Freeman and L. Praly (1998), "Integrator backstepping for bounded controls and control rates," IEEE Transactions on Automatic Control, voL 43, pp. 258-262. M. Jankovic, R. Sepulchre, and P. V. Kokotovic (1996), "Constructive Lyapunov stabilization of non linear cascade systems," IEEE Transactions on Automatic Control, voL 41, pp. 1723-1735. M. Krstic (2004). "Feedforward systems linearizable by coordinate change," submitted to 2004 American Control Conference. M. Krstic and H. Deng (1998), Stabilization ofNonlinear Uncenain Systems, Springer. F. Mazenc and L. Praly (1996), "Adding integrations, saturated controls, and stabilization of feedforward systems," IEEE Transactions on Automatic Control, vo!. 41, pp. 1559-1578. R. Sepulchre, M. Jankovic, and P. KokoloVic (1997), Constructive Nonlinear Control, Springer. R. Sepulchre, M. Jankovic, and P. Kokotovic (l997a), "Integrator forwarding: A new recursive nonlinear robust design," Automatica, pp. 979-984, voL 33. A. R. Teel (1992), Feedback Stabilization: Nonlinear Solutions to Inherently Nonlinear Problems, PhD dissertation, University of California, Berkeley. A. R. Teel (1992a), "Global stabilization and restricted tracking for multiple integrators with bounded controls," Systems and Control Letters, voL 18, pp. 165-171. A. R. Teel (1996), "A Donlinear small gain theorem for the analysis of control systems with saturation," IEEE Transactions on Automatic Control, voL 41, pp. 1256-1270.

(91) (92)

but is not of Types I or II. lt would be worth exploring in the future the possibilities for combining the systems of Type I and Type 1I. Theorem 6 does this, at least notationally, for systems of order three. The condition (49) shows actually that these two classes do not mix well, i.e., that Theorem 6 is a concise statement of two results, not a statement for a mixed Type IIII class. However, while mixing is impossible in order three, it is not impossible in higher orders. For example, the fourth order system

.

XI =

X2 +

X3 a (X3) - J;3 a(s)ds ~

b() X2X4 + X4 U

(93)

X2 =X3 +a(x3)x4

(94)

X3 =X4,

(95)

X4 = u,

where a(·) and b(·) are any nonlinearities vanishing at zero (a also must be Cl), is a system that mixes the features of Types I and II and is linearizable via

176