Disturbance attenuation via state feedback for systems with a ...
Ammorica. Vol. 32. No. 6. pp. 929-931. 1996 Copyright 0 19% Ekvier Science Ltd Printed in Great Britain. All rights reserved ms-lG98/% $15.00 + 0.00
PII: SOOO5-1098(%)00019-2
Brief Paper
Disturbance
Attenuation
via State Feedback for Systems
with a Saturation Nonlinearity WE1 ZHANt Key Words-H”
in the Control Channel*
and LE YI WANG?
control; state feedback; nonlinear systems: stability; saturation.
global results. In Helton and Zhan (1995) the H^ control problem was considered in which the nonlinearities take some simple forms such as piecewise-linear functions and saturation in the control channel. In these special cases, the general conditions could be somewhat simplified. In fact, some sufficient conditions were given for solvability, which only involves algebraic inequalities, in contrast to the partial differential inequalities in the general case. This shows that the design is quite different from the linear design, and much more difficult to solve numerically. In Zhan et al. (1995), the problem of H” control for systems with sector-bound nonlinearities was considered. The energy function (Ball et al., 1993) was assumed to have a special form in order to obtain practical sufficient conditions. Inspired by these developments in nonlinear H” control, in this article, we study the H’ control problem for systems with saturation in the control channel. Our goal is to obtain sufficient conditions that are numerically easy to check (at least easier than those in Helton and Zhan, 1995). The result should be valid globally or in a large region in state space. Because of the limitation in the magnitude of the control signal, it would be almost impossible to guarantee the system performance if the disturbance were not bounded, unless the open-loop system itself (i.e. the system with zero input) satisfied the performance criteria or somehow one could guarantee that the trajectories of the closed loop system stayed inside a relatively small neighborhood of the origin. To guarantee that the trajectories stay inside a given set for arbitrary disturbance is certainly not easy. Here we propose to consider a more practical case where the magnitude of the disturbance is bounded from above by a known positive number over the time period of interest, i.e. Iw(t)l~ w(, Vt E [0, T]. We shall see that this assumption amounts to adding a saturation nonlinearity in the disturbance channel. This key assumption allows us to get a firm result which is valid globally. It is worth noting that the main contribution of this article is on H” performance. The stability result comes as a by-product, and could be somewhat more conservative than those obtained in Lin er al. (1994) and Liu et al. (1993). The article is organized as follows. In Section 2, we introduce the notation used, and present some well-known results on H” control for nonlinear systems, which will be the starting point for our main result. The latter is presented in Section 3. Conclusions are drawn in Section 4.
Abstract-We
study the problem of disturbance attenuation via state feedback for asymptotically stable systems with a saturation nonlinearity in the control channel. The disturbance is assumed to be bounded, with the bound known. This amounts to adding a saturation nonlinearity in the disturbance channel. A sufficient condition is derived in terms of algebraic matrix inequalities, which are much simpler to solve than the general Hamilton-Jacobi-Isaacs partial differential inequality. A suboptimal control law is given based on the solution to the Riccati inequalities. The computational complexity of solving the optimal controller is similar to that of liner H’ controller design. Copyright 0 19% Elsevier Science Ltd. 1. Inrroduction This article is concerned with the problem of disturbance attenuation via state feedback for systems with a saturation nonlinearity in the control channel. The control of such systems has been studied by many researchers using different approaches. A great deal of work has been done on stability analysis and stabilization of such systems (Boyd et al., 1994). While stability is one of the most important issues in control, often it is also required that the closed-loop systems satisfy certain performance criterion. During the past few years, there have been some major developments in H” control of nonlinear systems (see e.g. Bapr and Bernhard, 1991; Van der Schaft, 1992; Isidori and Astolfi, 1992; Ball er al., 1993; Isidori and Kang, 1993; Pan and Bapr, 1993) following the breakthrough in the linear H’ theory, where the time-domain analysis resulted in elegant state-space formulas for the solution of the standard linear H” control problem in terms of the solutions of the two Riccati equations (Doyle et al., 1989). The results in Willems (1972) and Hill and Moylan (1980) were instrumental to the development of the time-domain approach of nonlinear H” control. In the H” control design, in addition to stability, the closed-loop system is required to satisfy the H” performance criterion. As pointed out in Ball et al. (1993), H” design has its unique features and is advantageous in many situations. While some neat theoretical results have appeared, unlike its linear counterpart, most of the results on H” control for nonlinear systems involve solving partial differential inequalities or equations. For this reason, it is usually extremely difficult to actually design a controller using the existing results on H” control for nonlinear systems. Also, most results on nonlinear H” control are local. Recently there has been increasing interest in reducing the complexity of the conditions involved in nonlinear H” design and obtaining
2. Preliminaries In this article, we shall more or less follow the notation used in Ball ef al. (1993). 08” denotes the n-dimensional Euclidean space of real numbers. For a vector u E R”, we denote its transpose by uT. For a differentiable function e(v) of u, Ve(u) denotes its gradient, namely
*Received 7 February 1995; revised 28 August 1995; received in final form 8 January 1996. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Matthew James under the direction of Editor Tamer author Professor Wei Zhan. BaFr. Corresponding Tel. +I 313 577 0403; Fax +1313 577 1101; E-mail wzhan@ ece.eng.wayne.edu. t Department of Electrical and Computer Engineering, Wayne State University, Detroit, MI 48202, U.S.A.
$n1. T
Ve(u)=
[
$
,
The following function will be used: Sat [u] =
929
1 u -1
if u>l, if -1sull. if u 0 are weighting functions.
wT(s)wjw(s) ds 2 jr(r)w, J
zl'(y ‘.(s)W,.v(s) +
holds for all
2.1. Suppose there exist a positive-definite
Prooj
By Theorem 2.1 in Ball er al. (1993), (2) follows from follows from the fact that e(x) is a Lyapunov function of the closed-loop 0 system with w = 0.
(6)
Choose e(x) =x’-Ex. Since H,(w. u) is a second-order polynomial in w and Sat [u/us], by completion of squares, it can be written as 1
+x’C”W,Cx
+x’EAx
+xTATEx
- (1 - W;‘)x’EBBTEx. The
H(w, u) < 0 VW. The stability conclusion
- W,w’
+ Ve.r(x){Ax + Bu,, Sat [a/~~,] + Bw}.
performance
requirement
H2(w, u) < 0 V /WI5 w,, is
apparently equivalent to HZ(~,,,,,t, u) < 0, where w,,,,, = w,)Sat [BTEx/(w0W3)]. And the optimal control law is given by u = u,,r,. where u,,~, = - BTEx. We discuss two cases: (i) u. 2 woW7; (ii) ug < w,,W,.
3. Main result Consider the system
Cuse (i)
When
@I’Ex/W31 5 w,,, we have IB ‘-Ex/u(,l 5 IBTEx/(woW,)l I 1. Thus H~(w, wo, we have Hz(w,,,,~. u,d 5 x-~R,x + (u,) - IBTExl)’ - W’,(w,, - IBTExl/W,)? If p = ug - W&s0 then Hz(w,.,,,,t, u _,) 5 xTR,x < 0. If p > 0 and lB*Exl~ ug then H2(wworst, uopt) 5 x’R,x + p( 1 -~B7E~I/~0)2 0) is the saturation nonlinearity in the control channel. &I u,) Sat [u/u”] =
u -u
if u 1 u,,, if -u,) 5 u 5 u,), if u < -u,,.
Cuse (ii). Note that $ > ul,/4wo W?; thus h > 0
Without loss of generality, we assume (Y= 1, since it can be easily incorporated into the disturbance signal w. Clearly. the system (4) is a special case of (1). Therefore Theorem 2.1 holds for the system (4), with a corresponding modification in the Hamiltonian. Constructive solutions to the following disturbance attenuation problem are sought.
l
l
When IBTEx/uOl 5 1, we have 1 2 IB’rEx/u,ji 1 IBTEx/w0LV31,1.Thus H*(w,,,,,, u& = xTR,x i 0.
When /B1‘Ex/uO/ > 1 and IB’Ex/w0W31< Hz(wworrtrn ,,J = xT$x + (u,) - lB“Exl)‘;
1. we that
Sincew H2(w
+
lBTzt;;u~ orst, u
control
attenuation
law u = k(x)
problem
such that,
Find a feedback under the zero initial
(DAP).
is.
IBTEx121. (I, we get ~2(ww,,,t, uopt)<x*&x + R2x
( -N,, IB’Exl + IBTEx12). Disturbance
have
+
bo(uo
-
2 IB*W
We also have jBTExl < w,,W,, Therefore uo ’ MY,%) IP*W. + (1 - uo/wOW,) IB=Ex12 u0, we get that is. we) lBTErl = (&I + W,,)Uo’ u:,>p; w,)) IB’Exl/p 2 1, which implies
1
(uo+ (u. +
2
[
1 - ; (u,, + WC,)IBTExl
PII: SOOO5-1098(%)00019-2
Brief Paper
Disturbance
Attenuation
via State Feedback for Systems
with a Saturation Nonlinearity WE1 ZHANt Key Words-H”
in the Control Channel*
and LE YI WANG?
control; state feedback; nonlinear systems: stability; saturation.
global results. In Helton and Zhan (1995) the H^ control problem was considered in which the nonlinearities take some simple forms such as piecewise-linear functions and saturation in the control channel. In these special cases, the general conditions could be somewhat simplified. In fact, some sufficient conditions were given for solvability, which only involves algebraic inequalities, in contrast to the partial differential inequalities in the general case. This shows that the design is quite different from the linear design, and much more difficult to solve numerically. In Zhan et al. (1995), the problem of H” control for systems with sector-bound nonlinearities was considered. The energy function (Ball et al., 1993) was assumed to have a special form in order to obtain practical sufficient conditions. Inspired by these developments in nonlinear H” control, in this article, we study the H’ control problem for systems with saturation in the control channel. Our goal is to obtain sufficient conditions that are numerically easy to check (at least easier than those in Helton and Zhan, 1995). The result should be valid globally or in a large region in state space. Because of the limitation in the magnitude of the control signal, it would be almost impossible to guarantee the system performance if the disturbance were not bounded, unless the open-loop system itself (i.e. the system with zero input) satisfied the performance criteria or somehow one could guarantee that the trajectories of the closed loop system stayed inside a relatively small neighborhood of the origin. To guarantee that the trajectories stay inside a given set for arbitrary disturbance is certainly not easy. Here we propose to consider a more practical case where the magnitude of the disturbance is bounded from above by a known positive number over the time period of interest, i.e. Iw(t)l~ w(, Vt E [0, T]. We shall see that this assumption amounts to adding a saturation nonlinearity in the disturbance channel. This key assumption allows us to get a firm result which is valid globally. It is worth noting that the main contribution of this article is on H” performance. The stability result comes as a by-product, and could be somewhat more conservative than those obtained in Lin er al. (1994) and Liu et al. (1993). The article is organized as follows. In Section 2, we introduce the notation used, and present some well-known results on H” control for nonlinear systems, which will be the starting point for our main result. The latter is presented in Section 3. Conclusions are drawn in Section 4.
Abstract-We
study the problem of disturbance attenuation via state feedback for asymptotically stable systems with a saturation nonlinearity in the control channel. The disturbance is assumed to be bounded, with the bound known. This amounts to adding a saturation nonlinearity in the disturbance channel. A sufficient condition is derived in terms of algebraic matrix inequalities, which are much simpler to solve than the general Hamilton-Jacobi-Isaacs partial differential inequality. A suboptimal control law is given based on the solution to the Riccati inequalities. The computational complexity of solving the optimal controller is similar to that of liner H’ controller design. Copyright 0 19% Elsevier Science Ltd. 1. Inrroduction This article is concerned with the problem of disturbance attenuation via state feedback for systems with a saturation nonlinearity in the control channel. The control of such systems has been studied by many researchers using different approaches. A great deal of work has been done on stability analysis and stabilization of such systems (Boyd et al., 1994). While stability is one of the most important issues in control, often it is also required that the closed-loop systems satisfy certain performance criterion. During the past few years, there have been some major developments in H” control of nonlinear systems (see e.g. Bapr and Bernhard, 1991; Van der Schaft, 1992; Isidori and Astolfi, 1992; Ball er al., 1993; Isidori and Kang, 1993; Pan and Bapr, 1993) following the breakthrough in the linear H’ theory, where the time-domain analysis resulted in elegant state-space formulas for the solution of the standard linear H” control problem in terms of the solutions of the two Riccati equations (Doyle et al., 1989). The results in Willems (1972) and Hill and Moylan (1980) were instrumental to the development of the time-domain approach of nonlinear H” control. In the H” control design, in addition to stability, the closed-loop system is required to satisfy the H” performance criterion. As pointed out in Ball et al. (1993), H” design has its unique features and is advantageous in many situations. While some neat theoretical results have appeared, unlike its linear counterpart, most of the results on H” control for nonlinear systems involve solving partial differential inequalities or equations. For this reason, it is usually extremely difficult to actually design a controller using the existing results on H” control for nonlinear systems. Also, most results on nonlinear H” control are local. Recently there has been increasing interest in reducing the complexity of the conditions involved in nonlinear H” design and obtaining
2. Preliminaries In this article, we shall more or less follow the notation used in Ball ef al. (1993). 08” denotes the n-dimensional Euclidean space of real numbers. For a vector u E R”, we denote its transpose by uT. For a differentiable function e(v) of u, Ve(u) denotes its gradient, namely
*Received 7 February 1995; revised 28 August 1995; received in final form 8 January 1996. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Matthew James under the direction of Editor Tamer author Professor Wei Zhan. BaFr. Corresponding Tel. +I 313 577 0403; Fax +1313 577 1101; E-mail wzhan@ ece.eng.wayne.edu. t Department of Electrical and Computer Engineering, Wayne State University, Detroit, MI 48202, U.S.A.
$n1. T
Ve(u)=
[
$
,
The following function will be used: Sat [u] =
929
1 u -1
if u>l, if -1sull. if u 0 are weighting functions.
wT(s)wjw(s) ds 2 jr(r)w, J
zl'(y ‘.(s)W,.v(s) +
holds for all
2.1. Suppose there exist a positive-definite
Prooj
By Theorem 2.1 in Ball er al. (1993), (2) follows from follows from the fact that e(x) is a Lyapunov function of the closed-loop 0 system with w = 0.
(6)
Choose e(x) =x’-Ex. Since H,(w. u) is a second-order polynomial in w and Sat [u/us], by completion of squares, it can be written as 1
+x’C”W,Cx
+x’EAx
+xTATEx
- (1 - W;‘)x’EBBTEx. The
H(w, u) < 0 VW. The stability conclusion
- W,w’
+ Ve.r(x){Ax + Bu,, Sat [a/~~,] + Bw}.
performance
requirement
H2(w, u) < 0 V /WI5 w,, is
apparently equivalent to HZ(~,,,,,t, u) < 0, where w,,,,, = w,)Sat [BTEx/(w0W3)]. And the optimal control law is given by u = u,,r,. where u,,~, = - BTEx. We discuss two cases: (i) u. 2 woW7; (ii) ug < w,,W,.
3. Main result Consider the system
Cuse (i)
When
@I’Ex/W31 5 w,,, we have IB ‘-Ex/u(,l 5 IBTEx/(woW,)l I 1. Thus H~(w, wo, we have Hz(w,,,,~. u,d 5 x-~R,x + (u,) - IBTExl)’ - W’,(w,, - IBTExl/W,)? If p = ug - W&s0 then Hz(w,.,,,,t, u _,) 5 xTR,x < 0. If p > 0 and lB*Exl~ ug then H2(wworst, uopt) 5 x’R,x + p( 1 -~B7E~I/~0)2 0) is the saturation nonlinearity in the control channel. &I u,) Sat [u/u”] =
u -u
if u 1 u,,, if -u,) 5 u 5 u,), if u < -u,,.
Cuse (ii). Note that $ > ul,/4wo W?; thus h > 0
Without loss of generality, we assume (Y= 1, since it can be easily incorporated into the disturbance signal w. Clearly. the system (4) is a special case of (1). Therefore Theorem 2.1 holds for the system (4), with a corresponding modification in the Hamiltonian. Constructive solutions to the following disturbance attenuation problem are sought.
l
l
When IBTEx/uOl 5 1, we have 1 2 IB’rEx/u,ji 1 IBTEx/w0LV31,1.Thus H*(w,,,,,, u& = xTR,x i 0.
When /B1‘Ex/uO/ > 1 and IB’Ex/w0W31< Hz(wworrtrn ,,J = xT$x + (u,) - lB“Exl)‘;
1. we that
Sincew H2(w
+
lBTzt;;u~ orst, u
control
attenuation
law u = k(x)
problem
such that,
Find a feedback under the zero initial
(DAP).
is.
IBTEx121. (I, we get ~2(ww,,,t, uopt)<x*&x + R2x
( -N,, IB’Exl + IBTEx12). Disturbance
have
+
bo(uo
-
2 IB*W
We also have jBTExl < w,,W,, Therefore uo ’ MY,%) IP*W. + (1 - uo/wOW,) IB=Ex12 u0, we get that is. we) lBTErl = (&I + W,,)Uo’ u:,>p; w,)) IB’Exl/p 2 1, which implies
1
(uo+ (u. +
2
[
1 - ; (u,, + WC,)IBTExl
