In this paper we develop an optimality-based framework for designing controllers for discrete-time nonlinear cascade systems. Specifically, using a nonlinear-nonquadratic optimal control framework we develop a family of globally stabilizing backstepping-type controllers parameterized by the cost functional that is minimized. Furthermore, it is shown that the control Lyapunov function guaranteeing closed-loop stability is a solution to the steady-state Bellman equation for the controlled system and thus guarantees both optimality and stability

Abdallah, Chaouki T.

Chellaboina, V.

Fausz, J. L.

Haddad, W. M.

English

University of New Mexico

University of New Mexico
UNM Digital Repository
Electrical & Computer Engineering Faculty
Publications Engineering Publications
7-31-1998
Optimal discrete-time control for non-linear
cascade systems
Chaouki T. Abdallah
W. M. Haddad
J. L. Fausz
V. Chellaboina
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Recommended Citation
Abdallah, Chaouki T.; W. M. Haddad; J. L. Fausz; and V. Chellaboina. "Optimal discrete-time control for non-linear cascade systems."
Journal of the Franklin Institute 335, 5 (1998): 827-839. http://digitalrepository.unm.edu/ece_fsp/67
Proceedings of the American Control Conference 
Albuquerque, New Mexico June 1997 
0-7803-3832-4/97/$10.00 0 1997 AACC 
Optimal Discrete-Time Control for Nonlinear Cascade Systems 
W. M. Haddad, J. L. Fausz, V. Chellaboina C. T .  Abdallah 
School of Aerospace Engineering 
Georgia Institute of Technology 
Atlanta, GA 30332-0150 
Abstract 
In this paper we develop an optimality-based framework 
for designing controllers for discrete-time nonlinear cas- 
cade systems. Specifically, using a nonlinear-nonquadratic 
optimal control framework we develop a family of globally 
stabilizing backstepping-type controllers parameterized by 
the cost functional that is minimized. Furthermore, it is 
shown that the control Lyapunov function guaranteeing 
closed-loop stability is a solution to the steady-state Bell- 
man equation for the controlled system and thus guaran- 
tees both optimality and stability. 
1. In t roduct ion  
Since most physical processes evolve naturally in con- 
tinuous time, it is not surprising that the bulk of non- 
linear control theory has been developed for continuous- 
time systems. Nevertheless, it is the overwhelming trend 
to implement controllers digitally. Despite this fact the 
development of nonlinear control theory for discrete-time 
systems has lagged its continuous-time counterpart. This 
is in part due to  the fact that concepts such as zero dy- 
namics, normal forms, and minimum phase are much more 
intricate for discrete-time systems. For example, in con- 
trast to the continuous-time case, technicalities involving 
passivity analysis tools needed to  prove global stability via 
smooth feedback controllers [l] as well as system relative 
degree requirements [2] are more involved in the discrete- 
time case. 
Recent work involving differential geometric methods [3] 
employing concepts of zero dynamics and feedback lin- 
earization have been applied to discrete-time systems. In 
particular, these results parallel continuous-time results 
on linearization of nonlinear systems via state and out- 
put feedback. However, as in the continuous-time case, 
these techniques cancel out system nonlinearities and may 
therefore lead to  inefficient designs since the resulting feed- 
back linearizing controller may generate large control ef- 
fort to cancel beneficial nonlinearities. 
Backstepping control for continuous-time systems has 
recently received a great deal of attention in the non- 
linear control literature [4]. The popularity of this con- 
trol methodology can be explained in a large part due to 
the fact that it provides a framework for designing sta- 
bilizing nonlinear controllers for a large class of nonlin- 
ear cascade systems. Even though discrete-time recur- 
sive backstepping techniques have not been developed, 
the closest discrete-time analog to backstepping is given 
in [2,5 . Specifically, in [2,5] discrete-time passivity analy- 
guaranteeing global asymptotic stability for block cascade 
discrete-time systems. 
In this paper we develop an optimality-based control 
design theory for nonlinear discrete-time cascade systems. 
The key motivation for developing an optimal nonlinear 
control theory framework for discrete-time cascade sys- 
sis too 1 s are used to  construct control Lyapunov functions 
~~~ ~ 
This research was supported in part by NSF under Grant 
ECS-9496249 and AFOSR under Grant F49620-96-1-0125. 
Electrical Engineering Department 
University of New Mexico 
Albuquerque, NM 87131 
tems is that it provides a family of candidate controllers 
parameterized by the cost functional that is minimized. 
In order to address the optimality-based nonlinear con- 
trol problem we use the nonlinear-nonquadratic optimal 
control framework developed in [6]. The basic underly- 
ing ideas of the results in [6 rely on the fact that the 
steady-state solution of the d iscrete-time Bellman equa- 
tion is a control Lyapunov function for the nonlinear con- 
trolled system thus guaranteeing both optimality and sta- 
bility. Finally, we use the following standard notation. 
Let N n x n  (resp., Pnx") denote the set of n x n nonneg- 
ative (resp., positive) definite matrices and let N denote 
the set of nonnegative integers. 
2. Opt imal  Control for Nonlinear Systems 
In this section we consider the nonlinear system 
z ( k +  1) = f (z(k))  + g ( z ( k ) ) ~ ( k ) ,  4 0 )  = 50, k E NI (1) 
where z E R", U E R", f : R" -+ R" such that f(0) = 0, 
g : R" 4 Rnx", with performance criterion 
J(zo,.(.)) = cr=lJ [Ll(Z(k)) + L2(4k))U(k) 
+uT ( I C )  R s ~ ( k ) ] .  (2) 
where L1 : Rn -+ R,L2 : Rn -+ RIx", and R2 E IF'"'".
Furthermore, define the set of asym totically stabilizing 
controllers for the nonlinear system &) by 
S(Q) f {U : R" x N -+ R" : x(.) given by (1) 
satisfies z ( k )  -+ 0 as IC -+ CO}. 
Theorem 2.1. Consider the controlled system (1) with 
performance functional (2). Assume there exist functions 
V : Rn -+ EX, L2 : R" -+ RWlxm, P12 : Rn -+ Rlxm,  and a 
nonnegative-definite function P2 : R" 4 N m x m  such that 
Lz(0) = 0,  P12(0) = 0,  V(0) = 0, (3) 
V(z) > 0, 5 E R", z # 0, (4) 
V[f(.) - &7(z)(R2 + P z ( 4 - l  
(5) 
V(f(.) + d.) = V(f(.)) + k ! ( Z ) U  + .TP2(4U, (6) 
.LT(Z)] - V ( 5 )  < 0, 5 E R", z # 0, 
where V ( x )  a CO as [lxv 4 00. Then the solution z ( k )  = 
0, k E NI of the closed- oop system 
z ( k  + 1) = f ( z ( k ) )  + g(z(k))4(z(k)),  z(0) = 2 0 1  k E N ,  
(7) 
is globally asymptotically stable with the feedback control 
law 4(z) = - f ( &  + Pz(z))-'[L2(5) + P12(z)lT, and the 
performance functional (2), with 
Ll(5) = dTb)(R2 + P2(5))4(2) - V ( f ( z ) )  + V ( z ) ,  (8) 
21 75 
is minimized in the sense that 
Finally, J ( z0 ,  $!)(z(.))) = V(zo), for all 20 E R". 
3. Optimal Block Backstepping Controllers 
with nonlinear input subsystems of the form 
In this section we consider nonlinear cascade systems 
z(k  + 1) = f (z(k))  + g(z(k))y(k), 4 0 )  = 20, (10) 
2 ( k  + 1) = f ( i ( k ) )  + g(? (k ) )U(k ) ,  2(0)  = PO, (11) 
(12) y(k) = h(?(k))  + J ( 2 ( k ) ) U ( k ) ,  
where k E N, 2 E RQ, u ,y  E R", f : Rq + Rq, g : 
Rq -+ RQxm,  h : Rq -+ R", and J : RQ + EXWmxm such 
that f ( 0 )  = 0 and h(0) = 0. Here, we consider the case in 
which the nonlinear input subsystem (11), (12 is feedback 
Rq 4 R such that V , ( O )  = 0 and 
strictly passive with positive definite storage i unction V ,  : 
K ( f ( 2 )  + P ( ~ ) u )  = V,(f(i)) + P 1 2 ( 2 ) u + ~ ~ P 2 ( 2 ) ~ ,  (13) 
where $12 : Rq -+ Rlxm and P2 : RQ -+ Pmxm. Specif- 
ically, we assume there exist functions k : RQ -+ R", 
I : Rq -+ RP, and W : Rq -+ RPx" such that l (0)  = 0, 
k ( 0 )  = 0, 
0 > K ( f ( 2 )  + 3(2)k(2) )  - K(2) + lT(2)1(2), 2 # 0, (14) 
0 = p;(2) + WT(2)Z(2) - (h(2)  + J ( i ) k ( 2 ) )  
0 = 9 ( 2 )  + WT(2)W(2) - ( J ( 2 )  + JT(2)). 
(15) 
(16) 
+ P 2  ( 2 )  I C ( ? ) ,  
Theorem 3.1. Consider the cascade system (10)-(12) 
with performance functional 
M 
J(z0, 2 0 ,  U ( - ) )  [ L l ( z ( k ) , w )  
k=O 
+ Z 2 ( z ( k ) ,  Z ( k ) ) u ( k )  + UT(k)R2.(k)] , (17) 
where ( z ( k ) , ! t ( k ) ) ,  k E N, solves (lo),  (11). Assume that 
the input subsystem (ll), (12) is feedback strictly passive 
and the subsystem (10) has a globally stable equilibrium 
at z ( k )  = 0, k E N, and Lyapunov function K u b ( 2 )  so 
that KUb( f (z ) )  < Kub(z), for all z E R" such that z # 0. 
Furthermore, assume there exist functions : R" x Rq 4 
1Wlxm, P12 : R" + Rlxm,  and P2 : Rn -+ Nmxm such that 
Z 2 ( 0 , 0 )  = 0,  &(0) = 0, 
+YTP2(4Y, 
(18) 
x u b ( f ( 2 )  + g ( z ) Y )  = &b(f(z)) f P 1 2 ( z ) y  (I9) 
p{ PZ(z) + Pz(z)h(f) - 2/42) - (I" + p2(z)J(2)) 
+JT(qr ' ,T , ( z )]}  5 0, (z,?) E R" x R4, (20) 
.&((z ,2) [P;(2)  + 2JT($)F2(2)h(2) + e:(,, 2 )  
where Raa(z,2) e R2+P2(2)+JT(2)P2(z)J(x) and k ( 2 )  
satisfies (14)-(16). Then the solution ( z ( k ) , 2 ( k ) )  = (0,0), 
k E N, of the cascade system (lo),  (11) is globally asymp- 
totically stable with the feedback control law U = &z, ?), 
where 
$(.,a) = -;R;:(z,2)[F;(2) + 2JT(?)P2(z)h(2) 
+LT(z, 2 )  + JT(2)P3.)]. (21) 
Furthermore, for (zo,?~) E R" x Rm 
&o, 20, d(z(*), 2 ( . ) ) )  = V(z0, ZO), (22) 
where V(x,2)  = Vsub(x) + vs(?), and the performance 
functional (17), with 
L 1 ( Z ,  2 )  = JT(z, 2)R2,(z, ?)$(z, 2 )  f K u b ( z )  
- K u b ( f ( z )  f g(z)h(?)) + - &(f (? ) ) ,  (23) 
is minimized in the sense that 
(24) 
Remark 3.1. Assuming det(Im + $P2(z)J(2))  # 0 for 
all (z,2) E R" x RQ, a particular choice of z 2 ( z , 2 )  satis- 
fying conditions (18) and (20) is given by 
L, ( z ,  2 )  = [PZ(z) + P2(z)h(2) - 2k(2)IT 
.(Im + +P2(z)J(2))-TR2,(2,  )- P12(z)J(2) 
- P 1 2  ( 2 )  - 2hT ( 2 )  P2 (z) J ( 2 ) .  (25) 
In this case, 
$(z,2) = k ( 2 )  - +(IWm + $ & ( z ) 4 2 ) ) - l  [P&) 
+ 9 ( z )  (h(2)  + J ( P ) k ( f ) ) ] .  (26) 
References 
[l] W. Lin and C. I. Byrnes, "Passivity and absolute stabi- 
lization of a class of discrete-time nonlinear systems," 
Automatica, vol. 31, pp. 263-268, 1995. 
[2]  C. I. Byrnes and W. Lin, "Losslessness, feedback 
equivalence, and the' global stabilization of discrete- 
time nonlinear systems," IEEE Trans. Autom. Contr., 
[3] J. W. Grizzle, Feedback Linearizatzon of Discrete- Time 
Systems. Berlin: Springer-Verlag, 1986. 
[4] M. Krstib, I. Kanellakopoulos, and P. V. Kokotovib, 
Nonlinear and Adaptive Control Design. New York, 
NY: John Wiley and Sons, 1995. 
[5] C. I. Byrnes and W. Lin, "Stabilization of discrete- 
time nonlinear systems by smooth state feedback," 
Sys. Contr. Lett., vol. 21, pp. 255-263, 1993. 
[6] W. M. Haddad and V. S. Chellaboina, "Discrete- 
time nonlinear analysis and feedback control with non- 
quadratic performance criteria," in Proc. IFAC World 
Congress, vol. F, San Francisco, CA, pp. 31-36, 1996. 
vol. 39, pp. 833-898, 1994. 
21 76 


Optimal discrete-time control for non-linear cascade systems

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Optimal discrete-time control for non-linear cascade systems

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