E t = 0; E t > t = I - Semantic Scholar
2162
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 9, SEPTEMBER 2011
Control of Equilibria for Nonlinear Stochastic Discrete-Time Systems
structive method for the synthesis of stochastic attractors with required probabilistic parameters both in regular and chaotic zones.
Irina Bashkirtseva and Lev Ryashko
II. SENSITIVITY ANALYSIS OF STOCHASTIC EQUILIBRIUM Consider a deterministic system
Abstract—We consider a nonlinear discrete-time control system forced by stochastic disturbances. The problem addressed is a design of the feedback regulator that stabilizes an equilibrium of the closed-loop deterministic system and synthesizes a required dispersion of random states for corresponding stochastic system. To solve this problem we develop a method based on the stochastic sensitivity matrix technique. Main results of this technical note concern the synthesis of the stochastic sensitivity matrix, a geometric description of attainability set, and constructive design of regulator parameters. The effectiveness of the proposed approach is demonstrated on the stochastic Henon model. Using our technique we provide a low level of sensitivity for stochastically forced equilibria and suppress both regular and chaotic oscillations. Index Terms—Attainability, control, discrete-time systems, stochastic equilibria, stochastic sensitivity, suppressing chaos.
I. INTRODUCTION
x +1 = f (x ) t
x n
fx
H1
Manuscript received September 14, 2010; revised November 26, 2010 and March 26, 2011; accepted April 11, 2011. Date of publication April 19, 2011; date of current version September 08, 2011. This work was supported in part by RFBR Grants 09-01-00026, 09-08-00048, 10-01-96022ural, and FTP 02.740.11.0202. Recommended by Associate Editor Z. Wang. The authors are with the Department of Mathematics, Ural State University, Ekaterinburg 620083, Russia (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAC.2011.2142630
n
where is -vector, ( ) is a continuously differentiable -vectorfunction. It is supposed that an equilibrium t of the system (1) is exponentially stable [14]. Let "t be a solution of the system (1) with the initial condition
x
x
x
x0 = x + "v0 "
"
v n
(2)
z
"v
where is a scalar, 0 is -vector. A vector 0 = 0 defines a small initial deviation of the system (1) state "0 from the equilibrium in the direction 0 . Consider deviations t" = "t 0 of the system (1) states "t from the equilibrium and relations
v
x
z
x
x
x x
v = z" = x "0 x for t = 1; 2; . . .. For the small ", a sensitivity of the equilibrium x of " t
" t
Stochastic fluctuations and nonlinearity interaction plays an important role for understanding the corresponding dynamical phenomena in electronic generators, lasers, mechanical, chemical, and biological systems. An investigation of the various noise-induced transitions [1] from the equilibrium through periodic to more complicated chaotic regimes is a central problem in a nonlinear stochastic dynamics. The sensitivity analysis of corresponding stochastic attractors is a key for investigation of these transitions. Control theory for stochastic linear systems has received much attention [2], [3]. For instance, an optimal stochastic linear-quadratic control problem was investigated in [4]–[6]. control problem for discrete-time stochastic systems The robust was studied in [7]. Control of nonlinear systems with regular and chaotic oscillations is a challenging and fundamental problem of nonlinear engineering. After the pioneering work of Ott, Grebogi, and Yorke [8] controlling chaos attracts many researchers. Various methods are used to suppress chaos to equilibria or periodic orbits [8]–[11]. In some nonlinear dynamic systems with simple regular attractors (equilibria or limit cycles) even small random disturbances can cause unexpected transition to oscillations similar to chaotic. The underlying reason of such transformation is a high stochastic sensitivity of the initial deterministic attractor [12]. In control systems one can change a stochastic sensitivity of attractor by the corresponding choice of regulator parameters. A decrease of stochastic sensitivity implies the amplitude contraction of undesirable oscillations. This approach was used in [12] for controlling chaos in Brusselator. Results of detailed study of stochastic sensitivity and control for equilibria of nonlinear continuous-time systems are presented in [13]. In this technical note, we develop the stochastic sensitivity analysis for equilibria of nonlinear discrete-time systems and suggest a con-
(1)
t
" t
the system (1) with respect to disturbances of initial data (2) is defined by
d v = lim !0 v = d" x j =0 : t
The sequence
v
t
" t
"
" t "
satisfies the linear system
v +1 = Av ; A = @f @x (x): An inequality (A) < 1, where (A) is a spectral radius of the matrix A is a necessary and sufficient condition for the exponential stability of the equilibrium x [14]. For the exponentially stable equilibrium x for any v0 it holds that lim !1 v = 0: t
t
t
t
Along with the deterministic system (1) we consider a corresponding stochastic system in the following form
x +1 = f (x ) + "(x ) : (3) Here (x) is a n 2 m-matrix, is m-dimensional uncorrelated random t
t
t
t
t
process with parameters
E = 0; E > = I; E > = 0(t 6= k) where I is an identity m 2 m-matrix, " is a scalar parameter of the noise t
t t
t k
intensity. Let "t be a solution of the system (3) with the initial condition (2). The variable
x
x 0 x v = lim !0 " " t
t
"
x
characterizes the sensitivity of the equilibrium both to initial data disturbances (2) and random disturbances of the system (3). For the sequence t , it holds that
v
0018-9286/$26.00 © 2011 IEEE
v +1 = Av + G ; G = (x): t
t
t
(4)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 9, SEPTEMBER 2011
For small noise, random states x"t are concentrated near the stable equi and can be approximated by x"t x + "vt , where vt is a solulibrium x tion of the linear system (4). The dynamics of the moments mt = Evt , Vt = Evt vt> for the solution vt of the system (4) satisfies the equations
mt+1 = Amt ; Vt+1 = AVt A> + S;
S = GG> :
(5) (6)
Along with (6), consider an equation
W = AW A> + S:
(7)
If (A) < 1 then (7) has a unique solution W . The following Theorem explains a probabilistic sense of this solution. Theorem 1: Let (A) < 1. Then a) The solutions mt and Vt of the systems (5), (6) are stable for any m0 and V0
lim mt = 0;
lim Vt = W:
!1
!1
t
t
(8)
b) The system (4) has a stationary distributed solution vt
E vt = 0;
E vt vt> = W:
lim Ekvt 0 vt k2 = 0:
!1
(10)
Proof: For the proof of statement a) we will rewrite the (6) and (7) as
Vt+1 = A(Vt ) + S;
W = A(W ) + S
where A(V ) = AV A> . It follows from (A) = 2 (A) < 1 that the system (7) has a unique solution W and (8) is carried out. For the proof of the statement b), consider a random vector v0 with v0 = 0, Ev0 v0> = W . Then it follows from (5)–(8) that parameters E for the solution vt of the system (4) with the initial vector v0 the (9) hold. For the proof of the statement c), consider yt = vt 0 vt , where vt is an arbitrary solution of the system (4). For the sequence Yt = Eyt yt> the system Yt+1 = A(Yt ) holds. Due to the inequality (A) < 1, we have limt!1 Yt = 0. It follows from Ekvt 0 vt k2 = tr(Yt ) that (10) holds. The Theorem 1 is proved. The matrix W characterizes the response of system (3) to the small random disturbances and plays a role of the stochastic sensitivity factor . of the equilibrium x Remark: For the nonsingular noises (rank(S ) = n), the matrix W is positive definite. For the case rank(S ) < n, the matrix W can be singular. A necessary and sufficient condition of nonsingularity of the matrix W is a full controllability of the pair (A; G) [15]. A Scalar Case: In the case n = 1, the stochastic sensitivity factor is scalar and looks like
w=
s
1 0 a2
III. STOCHASTIC SENSITIVITY CONTROL Consider a controlled deterministic system
xt+1 = f (xt ; ut )
where a = f 0 ( x), s = 2 ( x). Note that for this case an inequality jf 0 (x)j < 1 is a necessary and sufficient condition of the exponential . stability for the equilibrium x x) = 0, a stochastic sensitivity of the equilibrium x is minFor f 0 ( x)j tends to unit, the value w unlimitedly inimal: w = s. As jf 0 ( creases.
(11)
where x is n-vector, f (x; u) is a continuously differentiable n-vectorfunction, u is l-vector of control parameters. It is supposed that for u = 0 the system (11) has an equilibrium xt x. A stability of x is not assumed. We shall select a stabilizing regulator from the class U of admissible feedbacks u = u(x) satisfying conditions: (a) u(x) is continuously differentiable and
u( x) = 0
(12)
(b) for the closed loop system
xt+1 = f (xt ; u(xt ))
(13)
the equilibria x is exponentially stable. For the exponential stability of the equilibria x , an exponential stability of the first approximation system
zt+1 = (A + BK )zt
(9)
c) Any solution vt of the system (4) converges in a mean-square to vt t
2163
for small deviations zt = xt 0 x is necessary and sufficient. Here
A=
@f ( x; 0); @x
B=
@f ( x; 0); @u
K=
@u ( x): @x
Consider a set of matrices = fK 2 l2n ; (A + BK ) < 1g. Here (A) is a spectral radius of the matrix A. Suppose the pair (A; B ) is stabilizable [15] that is the set is not empty. Along with the deterministic system (11) consider a stochastically forced controlled system
xt+1 = f (xt ; ut ) + "(xt ; ut )t :
(14)
Here (x; u) is a sufficiently smooth n 2 m-matrix function characterizing a dependence of disturbances on state and control, t is an uncorrelated random process with parameters
Et = 0;
Et t> = I;
Et k = 0(t 6= k)
" is a scalar parameter of the noise intensity. Consider the closed loop system
xt+1 = f (xt ; u(xt )) + " (xt ; u(xt )) t
(15)
with u 2 U . A dispersion of random states of this system depends on the choice of the control function u(x). Consider the following control problem. A Problem of Stochastic Sensitivity Synthesis: For the variable
x"t 0 x "!0 "
vt = lim the following system holds:
vt+1 = (A + BK )vt + Gt
(16)
where
A=
@f @f @u ( x; 0); B = ( x; 0); K = ( x); G = ( x; 0): @x @u @x
For any K 2 , due to Theorem 1 the system (16) has a stationary distribution with a covariance matrix W . This matrix is a unique solution of the equation
W = (A + BK )W (A + BK )> + S;
S = GG> :
(17)
2164
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 9, SEPTEMBER 2011
W
Note that the matrix is a stochastic sensitivity factor for the forced equilibrium of the system (15). As it follows from (17), this matrix =( )() of the feedback ( ) depends on local parameters only. So, capabilities of the matrix synthesis are completely determined by the linear approximation of the function ( ) and independent on higher order terms. It allows us to restrict consideration without loss of generality by more simple regulators in the following form:
x
K
@u=@x x W
ux
ux
u(x) = K (x 0 x):
(18)
Under these circumstances, we shall consider a problem of sensitivity matrix synthesis for the system (15) with a regulator (18). Consider a set = f 2 n2n j 0g
W
M
M
K
K
K
W
W W
K
WK = W:
(19)
In some cases, the Problem 1 is unsolvable. Therefore we introduce a notion of the attainability. is said to be attainable for the Definition 1: The element 2 system (14) under the feedback (18) if the equality K = is true 2 . for some Definition 2: A set of all attainable elements
W
K
W
W 2 j9K 2
W
WK = W g
=f
is called attainability set for the system (14), (18). means that Let us describe an attainability set. A symbol the matrix 0 is non-negative definite. It follows from the (17) that K for any , and . It means that
W
S
Q P
AB
V
Q P
K
V S
S
(20)
where S = f 2 j g. Let us clear up what kind of conditions guarantee a coincidence of and S . Theorem 2: Let noises in the system (14) be non-singular ( 0). If the matrix is quadratic and non-singular (rank = = ) then = S . Under these conditions, for any matrix 2 S the (17) has a solution = 01 ( 0 ) > 0 0 2 (21)
B
K B
X
K
XX > = QQ> ; (A + BK )V = X
(24) (25)
for unknown and . The set of the all solutions for the (24) can be presented [16] as
X = QU > where U is an arbitrary orthogonal n 2 n-matrix.
(26)
Now a feedback matrix representation
K = B01 (W 0 S) U >W 0 0 A
M directly follows from (23) and (25). W Since this matrix K satisfies the (17) with positive definite matrices W = W and S , a spectral radius (A + BK ) < 1. Hence, K 2 . A BK;G Thus, under theorem assumptions, an arbitrary element W 2 S is attainable. Theorem 2 is proved. Let us turn to the case rank(B ) < n. Consider the projective matrices P1 = BB + and P2 = I 0 P1 . Here a sign “+” means a pseu-
of admissible stochastic sensitivity matrices. A symbol 0 means that the matrix is symmetric and positive definite. For non-singular 2 a solution K of the (17) noises (rank = ), for any belongs to . One can weaken a requirement of non-singularity of is true [15] if the pair ( + noises. The condition K 2 ) is controllable for 2 . The aim of the proposed control design is the synthesis of the desired stochastic sensitivity matrix. Denote by K a solution of (17) with the 2 . fixed matrix Problem 1: For the assigned matrix 2 , it is necessary to find 2 guaranteeing the equality a matrix
M G n
Due to these decompositions the (22) can be rewritten as a system of two matrix equations
S B n l W
W S U W A where U is an arbitrary orthogonal n 2 n-matrix. 2 S . Rewrite the (17) Proof: Consider an arbitrary element W in the following: (A + BK )> = W 0 S: (A + BK )W (22) and non-negative definite matrix For the positive definite matrix W V = W 1=2 and W 0 S one can find [16] a positive definite1=matrix 2 non-negative definite matrix Q = (W 0 S ) , such that (23) W = V V > ; W 0 S = QQ>:
doinversion. Note some properties of these matrices
P1B = B; B> P1 = B> ; P2B = 0; B> P2 = 0; P1 = BB+ ; P1 + P2 = I; P12 = P1; P22 = P2 ; P1> = P1; P2> = P2: (27) If rank(B ) < n then P2 6= 0. Theorem 3: Let noises in the system (14) be non-singular (S 0) 2 S is attainable if and only if and rank(B ) < n. The element W is a solution of the equation the matrix W > P2 = P2 (W 0 S )P2 : P2 AWA (28) Under these conditions for matrix W 2 S , the (17) has a solution K = B+ [QU >V 01 0 A] (29) where Q and V are matrices of the decomposition (23), U is an arbitrary orthogonal n 2 n-matrix satisfying the condition P2AV = P2 QU > : (30) Proof: Necessity. Multiplying the (22) by P2 from the left and right and taking into account P2 B = 0, B > P2 = 0, we get (28). Sufficiency. It follows from (28) and the decomposition (23) that:
P2 AV V >A> P2 = P2QQ> P2:
(31)
U K BK = BB+ [QU > V 01 0 A] = P1 [QU >V 01 0 A] = (I 0 P2 )[QU > V 01 0 A] = [QU > V 01 0 A]; (32) (A + BK )V = QU > : The equality (32) implies that the matrix K satisfies the (17). As one can see from this proof, the matrix K from (29) is a solution of the (17) for any orthogonal matrix U satisfying the condition (30). Theorem 3
The equality (31) implies (30) for some orthogonal matrix . Due to (30) for the matrix from (29), the following equalities hold:
is proved.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 9, SEPTEMBER 2011
2165
For 2 [1; 1:6875) this equilibrium is exponentially stable. When the parameter passes the bifurcation value 3 = 1:6875, this equilibrium loses stability. On the interval (1.6875,2.4], Henon system demonstrates both periodic and chaotic oscillations. Under the random disturbances (" > 0), a trajectory of the system (33) leaves a deterministic attractor and forms a stochastic attractor around it. In Fig. 1, solutions of stochastically forced Henon model (33) without control (grey color) are plotted for the fixed noise intensity " = 0:005 and three values of : = 1:68, = 1:75, = 2:4. As one can see, a transition from order to chaos is accompanied by the increase of the amplitude of oscillations. The aim of the control is to provide a stability of equilibria on the whole interval 1 2:4 and synthesize the required stochastic sensitivity matrix W . We use a feedback regulator in the following form:
u = k1 (x 0 x) + k2 (y 0 y):
(34)
For attainability analysis and design of the feedback control we use Theorem 3. It follows from (28) that w22 = w11 +1. Put w12 = 0. This choice provides a non-correlatedness of random states xt , yt . Then the elements w11 , w22 of attainable matrices W are restricted by relations w11 1, w22 = w11 + 1. Using the formula (29) we get coefficients
k1 = (
9 + 16 0 3)=2;
k2 = 0:5 0
w11 0 1 w11 + 1
(35)
of the regulator (34) guaranteeing the required values of the stochastic sensitivity matrix for the system (33). Put w11 = 1. This choice gives us the minimal attainable value of the sensitivity. This regulator provides for the equilibria of the system (33) a stochastic sensitivity matrix with elements w11 = 1, w12 = 0, w22 = 2. As a result of this control, random states of the system (33) with regulator (34) are concentrated in a small neighborhood of the equilibrium. In Fig. 1, solutions of Henon model (33) without control (grey color) and with control (black color) are plotted for three values of parameter = 1:68, = 1:75, = 2:4. As we can see, the regulator (34) allows x; y) and essentially decrease a dispersion us to stabilize the equilibria ( of random states both for regular and chaotic zones.
REFERENCES
Fig. 1. Solutions of Henon model (33) without control (grey) and with control (34) (black) for " : and a) : , b) : , c) : .
= 0 005
= 1 68
= 1 75
=24
IV. STOCHASTIC HENON SYSTEM Consider the stochastically forced Henon system [17]
xt+1 = 1 0 x2t 0 0:5yt + ut + "t ; yt+1 = xt + "t ; 1 2:4
(33)
with a single control input ut . Here t ; t are sequences of the uncorrelated random disturbances with parameters Et = Et = 0, Et2 = 1 > 0, Et2 = 1 > 0, Et t = 0, and " is a scalar parameter of noise intensity. The corresponding uncontrolled deterministic model (33) (" = 0; ut = 0) has an equilibrium (x; y), where
x = y =
p9 + 16 0 3 : 4
[1] W. Horsthemke and R. Lefever, Noise-Induced Transitions. Berlin, Germany: Springer, 1984. [2] F. Kozin, “A survey of stability of stochastic systems,” Automatica J. IFAC, vol. 5, pp. 95–112, 1969. [3] H. J. Kushner, Stochastic Stability and Control. New York: Academic Press, 1967, vol. 33, Math. Sci. Eng. [4] W. M. Wonham, “Optimal stationary control of a linear system with state dependent noise,” SIAM J. Control Optim., vol. 5, pp. 486–500, 1967. [5] U. G. Hausmann, “Optimal stationary control with state and control dependent noise,” SIAM J. Control Optim., vol. 9, pp. 184–198, 1971. [6] R. F. Curtain, Stability of Stochastic Dynamical Systems. Berlin, Germany: Springer, 1972, vol. 294. [7] A. El Bouhtouri, D. Hinrichsen, and A. J. Pritchard, “ H -type control for discrete-time stochastic systems,” Int. J. Robust Nonlin. Control, vol. 9, pp. 923–948, 1999. [8] E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett., vol. 64, pp. 1196–1199, 1990. [9] A. L. Fradkov and A. Y. Pogromsky, Introduction to Control of Oscillations and Chaos, ser. Series of Nonlinear Science. Singapore: World Scientific, 1998. [10] S. Boccaletti, C. Grebogi, Y. C. Lay, H. Mancini, and D. Maza, “The control of chaos: Theory and applications,” Phys. Rep., vol. 329, pp. 103–197, 2000. [11] G. Chen and X. Yu, Chaos Control: Theory and Applications. New York: Springer-Verlag, 2003.
2166
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 9, SEPTEMBER 2011
[12] I. A. Bashkitseva and L. B. Ryashko, “Sensitivity and chaos control for the forced nonlinear oscillations,” Chaos Sol. Fractals, vol. 26, pp. 1437–1451, 2005. [13] L. B. Ryashko and I. A. Bashkirtseva, “On control of stochastic sensitivity,” Autom. Remote Control, vol. 69, pp. 1171–1180, 2008. [14] S. N. Elaydi, An Introduction to Difference Equations. New York: Springer, 1999. [15] W. M. Wonham, Linear Multivariable Control: A Geometric Approach. Berlin, Germany: Springer-Verlag, 1979. [16] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1985. [17] M. Henon, “A two-dimensional mapping with a strange attractor,” Commun. Math. Phys., vol. 50, pp. 69–77, 1976.
Multi-stage Anti-Windup Compensation for Open-loop Stable Plants Solmaz Sajjadi-Kia and Faryar Jabbari, Senior Member, IEEE
Abstract—We discuss adding a measure of scheduling to the popular Anti-Windup design. The main idea is to develop a scheme in which the Anti-Windup gains depend on how much the actuator command exceeds the saturation bound. The aim is to design and implement more aggressive Anti-windup gains in lower levels of saturation. Global stability and performance are addressed by adding an outer-loop Anti-windup compensation which become active when the system is in higher levels of saturation. We present results for both static and dynamic Anti-Windup gains, along with the convex synthesis LMIs. Benefits of the proposed design method are shown using well-known examples. Index Terms—Anti-windup (AW),
gain, saturation, scheduling.
I. INTRODUCTION Anti-windup (AW) augmentation often is used for safety and alleviation of performance degradations associated with actuator windup, when high performance linear controllers encounter actuator saturation. Generally, AW schemes are designed with two goals: 1) as long as actuators do not saturate, the closed-loop response coincides with the unconstrained response; 2) if the actuators saturate, stability is preserved and performance is recovered as much as possible ([1]). An excellent set of discussions and references on AW can be found in [2]–[5]. In most cases, the AW augmentation is a single controller (or set of gains) that is applied for all initial conditions, reference signals and disturbance levels. For such global results, typical performance guarantees are no better than those from the open-loop system. Stronger results can be obtained if reference signals or disturbances are assumed to have peak or energy bounds ([6]). Such techniques can be combined with the results here, by adding conditions that bound the reachable sets, which in turn bound the signals and states. For brevity, we focus on the global case here. The main observation, here, is that in many applications saturation can be mostly mild and the command to the actuator rarely exceeds the saturation bound by a large margin. In such cases, it seems intuitively Manuscript received November 8, 2010 revised March 25, 2011; accepted April 01, 2011. Date of publication April 25, 2011; date of current version September 08, 2011. Recommended by Associate Editor F. Wu. The authors are with the Department of Mechanical and Aerospace Engineering, University of California at Irvine (UCI), Irvine, CA 92697 USA (e-mail:[email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2011.2146930
clear that using different gains for the rare cases might allow a more aggressive and higher performance gain when the commands are only slightly larger than the actuator limitations. This leads to, in essence, a form of scheduling of the AW gains based on the level of saturation. This is the main tack in this paper. Scheduling in the traditional AW setting has been attempted before. For example, in [7] scheduling is used to improve the system performance after it re-enters the small signal domain. Another approach can be found in [8], in which a family of controllers are used to develop a scheduling approach in dealing with saturation, though the overall technique is quite different from the concept of AW used here. While the approach in [8] has the important advantage of being applicable to open-loop unstable systems, its implementation is rather involved and requires considerable on-line computation and often cannot match the high performance of nominal linear controllers in small signal regions. Other approaches to the scheduled AW design can be found, e.g., in [9] where a nonlinear state-feedback control law for the global asymptotic stabilization of non-exponentially unstable plants is proposed. Similarly, use of self (or gain)-scheduled techniques to obtain AW gains is explored in [10] and [11]. Here, we are interested in improving the performance of AW through simple and easy to implement modifications. We focus on developing an approach that provides performance guarantees for the large signal operation of open-loop stable systems, yet allows higher performance gains when the command to the actuator is only modestly above the saturation limits. This is through use of multiple sets of AW gains (as shown in Fig. 2): for moderate levels and for severe levels of saturation. By placing more weight on the gains associated with the moderate level of saturation, one can expect better performance in this level, particularly if this is the normal envelop of operation. The overall stability and some performance when the actuator command signal exceeds the saturation bound significantly are guaranteed by another set of gains. For simplicity, and to stay close to the basic concept of AW, we start by considering the case in which all gains are static. We show that the gains can be obtained from straightforward linear matrix inequalities where the complexity of the resulting search is only modestly higher than the traditional AW case. The implementation of the proposed scheme is quite straightforward and similar to the standard AW. Finally, for “all static” gains we show that the existence condition of the proposed scheduled approach is the same as the one for the traditional static AW compensation (Fig. 1). It is well known however, that in some problems the traditional static AW compensation (Fig. 1) is not feasible, while a dynamic AW augmentation only requires open-loop stability, as a sufficient condition for feasibility. In Section IV, we show that by letting one of the loops to include dynamic AW, we also can extend our approach so that open-loop stability leads to feasibility of the scheduled AW. II. PROBLEM DEFINITION AND BASIC SETUP Consider the open-loop stable plant below with state vector xp 2 Rn , control input vector u^ 2 Rn , and exogenous input vector w 2 Rn (e.g., reference signal, noise, external disturbance, etc.)
6p
x_ p = Ap xp + B1 w + B2 u^
z = C1 xp + D11 w + D12 u^
y = C2 xp + D21 w + D22 u^:
(1)
The nominal controller, designed to fulfill a specific task such as tracking or disturbance regulation, is likely to saturate. As a result, it needs to be augmented with an AW protection loop. The AW commands are introduced to the nominal controller by adding signals 1
0018-9286/$26.00 © 2011 IEEE
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 9, SEPTEMBER 2011
Control of Equilibria for Nonlinear Stochastic Discrete-Time Systems
structive method for the synthesis of stochastic attractors with required probabilistic parameters both in regular and chaotic zones.
Irina Bashkirtseva and Lev Ryashko
II. SENSITIVITY ANALYSIS OF STOCHASTIC EQUILIBRIUM Consider a deterministic system
Abstract—We consider a nonlinear discrete-time control system forced by stochastic disturbances. The problem addressed is a design of the feedback regulator that stabilizes an equilibrium of the closed-loop deterministic system and synthesizes a required dispersion of random states for corresponding stochastic system. To solve this problem we develop a method based on the stochastic sensitivity matrix technique. Main results of this technical note concern the synthesis of the stochastic sensitivity matrix, a geometric description of attainability set, and constructive design of regulator parameters. The effectiveness of the proposed approach is demonstrated on the stochastic Henon model. Using our technique we provide a low level of sensitivity for stochastically forced equilibria and suppress both regular and chaotic oscillations. Index Terms—Attainability, control, discrete-time systems, stochastic equilibria, stochastic sensitivity, suppressing chaos.
I. INTRODUCTION
x +1 = f (x ) t
x n
fx
H1
Manuscript received September 14, 2010; revised November 26, 2010 and March 26, 2011; accepted April 11, 2011. Date of publication April 19, 2011; date of current version September 08, 2011. This work was supported in part by RFBR Grants 09-01-00026, 09-08-00048, 10-01-96022ural, and FTP 02.740.11.0202. Recommended by Associate Editor Z. Wang. The authors are with the Department of Mathematics, Ural State University, Ekaterinburg 620083, Russia (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAC.2011.2142630
n
where is -vector, ( ) is a continuously differentiable -vectorfunction. It is supposed that an equilibrium t of the system (1) is exponentially stable [14]. Let "t be a solution of the system (1) with the initial condition
x
x
x
x0 = x + "v0 "
"
v n
(2)
z
"v
where is a scalar, 0 is -vector. A vector 0 = 0 defines a small initial deviation of the system (1) state "0 from the equilibrium in the direction 0 . Consider deviations t" = "t 0 of the system (1) states "t from the equilibrium and relations
v
x
z
x
x
x x
v = z" = x "0 x for t = 1; 2; . . .. For the small ", a sensitivity of the equilibrium x of " t
" t
Stochastic fluctuations and nonlinearity interaction plays an important role for understanding the corresponding dynamical phenomena in electronic generators, lasers, mechanical, chemical, and biological systems. An investigation of the various noise-induced transitions [1] from the equilibrium through periodic to more complicated chaotic regimes is a central problem in a nonlinear stochastic dynamics. The sensitivity analysis of corresponding stochastic attractors is a key for investigation of these transitions. Control theory for stochastic linear systems has received much attention [2], [3]. For instance, an optimal stochastic linear-quadratic control problem was investigated in [4]–[6]. control problem for discrete-time stochastic systems The robust was studied in [7]. Control of nonlinear systems with regular and chaotic oscillations is a challenging and fundamental problem of nonlinear engineering. After the pioneering work of Ott, Grebogi, and Yorke [8] controlling chaos attracts many researchers. Various methods are used to suppress chaos to equilibria or periodic orbits [8]–[11]. In some nonlinear dynamic systems with simple regular attractors (equilibria or limit cycles) even small random disturbances can cause unexpected transition to oscillations similar to chaotic. The underlying reason of such transformation is a high stochastic sensitivity of the initial deterministic attractor [12]. In control systems one can change a stochastic sensitivity of attractor by the corresponding choice of regulator parameters. A decrease of stochastic sensitivity implies the amplitude contraction of undesirable oscillations. This approach was used in [12] for controlling chaos in Brusselator. Results of detailed study of stochastic sensitivity and control for equilibria of nonlinear continuous-time systems are presented in [13]. In this technical note, we develop the stochastic sensitivity analysis for equilibria of nonlinear discrete-time systems and suggest a con-
(1)
t
" t
the system (1) with respect to disturbances of initial data (2) is defined by
d v = lim !0 v = d" x j =0 : t
The sequence
v
t
" t
"
" t "
satisfies the linear system
v +1 = Av ; A = @f @x (x): An inequality (A) < 1, where (A) is a spectral radius of the matrix A is a necessary and sufficient condition for the exponential stability of the equilibrium x [14]. For the exponentially stable equilibrium x for any v0 it holds that lim !1 v = 0: t
t
t
t
Along with the deterministic system (1) we consider a corresponding stochastic system in the following form
x +1 = f (x ) + "(x ) : (3) Here (x) is a n 2 m-matrix, is m-dimensional uncorrelated random t
t
t
t
t
process with parameters
E = 0; E > = I; E > = 0(t 6= k) where I is an identity m 2 m-matrix, " is a scalar parameter of the noise t
t t
t k
intensity. Let "t be a solution of the system (3) with the initial condition (2). The variable
x
x 0 x v = lim !0 " " t
t
"
x
characterizes the sensitivity of the equilibrium both to initial data disturbances (2) and random disturbances of the system (3). For the sequence t , it holds that
v
0018-9286/$26.00 © 2011 IEEE
v +1 = Av + G ; G = (x): t
t
t
(4)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 9, SEPTEMBER 2011
For small noise, random states x"t are concentrated near the stable equi and can be approximated by x"t x + "vt , where vt is a solulibrium x tion of the linear system (4). The dynamics of the moments mt = Evt , Vt = Evt vt> for the solution vt of the system (4) satisfies the equations
mt+1 = Amt ; Vt+1 = AVt A> + S;
S = GG> :
(5) (6)
Along with (6), consider an equation
W = AW A> + S:
(7)
If (A) < 1 then (7) has a unique solution W . The following Theorem explains a probabilistic sense of this solution. Theorem 1: Let (A) < 1. Then a) The solutions mt and Vt of the systems (5), (6) are stable for any m0 and V0
lim mt = 0;
lim Vt = W:
!1
!1
t
t
(8)
b) The system (4) has a stationary distributed solution vt
E vt = 0;
E vt vt> = W:
lim Ekvt 0 vt k2 = 0:
!1
(10)
Proof: For the proof of statement a) we will rewrite the (6) and (7) as
Vt+1 = A(Vt ) + S;
W = A(W ) + S
where A(V ) = AV A> . It follows from (A) = 2 (A) < 1 that the system (7) has a unique solution W and (8) is carried out. For the proof of the statement b), consider a random vector v0 with v0 = 0, Ev0 v0> = W . Then it follows from (5)–(8) that parameters E for the solution vt of the system (4) with the initial vector v0 the (9) hold. For the proof of the statement c), consider yt = vt 0 vt , where vt is an arbitrary solution of the system (4). For the sequence Yt = Eyt yt> the system Yt+1 = A(Yt ) holds. Due to the inequality (A) < 1, we have limt!1 Yt = 0. It follows from Ekvt 0 vt k2 = tr(Yt ) that (10) holds. The Theorem 1 is proved. The matrix W characterizes the response of system (3) to the small random disturbances and plays a role of the stochastic sensitivity factor . of the equilibrium x Remark: For the nonsingular noises (rank(S ) = n), the matrix W is positive definite. For the case rank(S ) < n, the matrix W can be singular. A necessary and sufficient condition of nonsingularity of the matrix W is a full controllability of the pair (A; G) [15]. A Scalar Case: In the case n = 1, the stochastic sensitivity factor is scalar and looks like
w=
s
1 0 a2
III. STOCHASTIC SENSITIVITY CONTROL Consider a controlled deterministic system
xt+1 = f (xt ; ut )
where a = f 0 ( x), s = 2 ( x). Note that for this case an inequality jf 0 (x)j < 1 is a necessary and sufficient condition of the exponential . stability for the equilibrium x x) = 0, a stochastic sensitivity of the equilibrium x is minFor f 0 ( x)j tends to unit, the value w unlimitedly inimal: w = s. As jf 0 ( creases.
(11)
where x is n-vector, f (x; u) is a continuously differentiable n-vectorfunction, u is l-vector of control parameters. It is supposed that for u = 0 the system (11) has an equilibrium xt x. A stability of x is not assumed. We shall select a stabilizing regulator from the class U of admissible feedbacks u = u(x) satisfying conditions: (a) u(x) is continuously differentiable and
u( x) = 0
(12)
(b) for the closed loop system
xt+1 = f (xt ; u(xt ))
(13)
the equilibria x is exponentially stable. For the exponential stability of the equilibria x , an exponential stability of the first approximation system
zt+1 = (A + BK )zt
(9)
c) Any solution vt of the system (4) converges in a mean-square to vt t
2163
for small deviations zt = xt 0 x is necessary and sufficient. Here
A=
@f ( x; 0); @x
B=
@f ( x; 0); @u
K=
@u ( x): @x
Consider a set of matrices = fK 2 l2n ; (A + BK ) < 1g. Here (A) is a spectral radius of the matrix A. Suppose the pair (A; B ) is stabilizable [15] that is the set is not empty. Along with the deterministic system (11) consider a stochastically forced controlled system
xt+1 = f (xt ; ut ) + "(xt ; ut )t :
(14)
Here (x; u) is a sufficiently smooth n 2 m-matrix function characterizing a dependence of disturbances on state and control, t is an uncorrelated random process with parameters
Et = 0;
Et t> = I;
Et k = 0(t 6= k)
" is a scalar parameter of the noise intensity. Consider the closed loop system
xt+1 = f (xt ; u(xt )) + " (xt ; u(xt )) t
(15)
with u 2 U . A dispersion of random states of this system depends on the choice of the control function u(x). Consider the following control problem. A Problem of Stochastic Sensitivity Synthesis: For the variable
x"t 0 x "!0 "
vt = lim the following system holds:
vt+1 = (A + BK )vt + Gt
(16)
where
A=
@f @f @u ( x; 0); B = ( x; 0); K = ( x); G = ( x; 0): @x @u @x
For any K 2 , due to Theorem 1 the system (16) has a stationary distribution with a covariance matrix W . This matrix is a unique solution of the equation
W = (A + BK )W (A + BK )> + S;
S = GG> :
(17)
2164
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 9, SEPTEMBER 2011
W
Note that the matrix is a stochastic sensitivity factor for the forced equilibrium of the system (15). As it follows from (17), this matrix =( )() of the feedback ( ) depends on local parameters only. So, capabilities of the matrix synthesis are completely determined by the linear approximation of the function ( ) and independent on higher order terms. It allows us to restrict consideration without loss of generality by more simple regulators in the following form:
x
K
@u=@x x W
ux
ux
u(x) = K (x 0 x):
(18)
Under these circumstances, we shall consider a problem of sensitivity matrix synthesis for the system (15) with a regulator (18). Consider a set = f 2 n2n j 0g
W
M
M
K
K
K
W
W W
K
WK = W:
(19)
In some cases, the Problem 1 is unsolvable. Therefore we introduce a notion of the attainability. is said to be attainable for the Definition 1: The element 2 system (14) under the feedback (18) if the equality K = is true 2 . for some Definition 2: A set of all attainable elements
W
K
W
W 2 j9K 2
W
WK = W g
=f
is called attainability set for the system (14), (18). means that Let us describe an attainability set. A symbol the matrix 0 is non-negative definite. It follows from the (17) that K for any , and . It means that
W
S
Q P
AB
V
Q P
K
V S
S
(20)
where S = f 2 j g. Let us clear up what kind of conditions guarantee a coincidence of and S . Theorem 2: Let noises in the system (14) be non-singular ( 0). If the matrix is quadratic and non-singular (rank = = ) then = S . Under these conditions, for any matrix 2 S the (17) has a solution = 01 ( 0 ) > 0 0 2 (21)
B
K B
X
K
XX > = QQ> ; (A + BK )V = X
(24) (25)
for unknown and . The set of the all solutions for the (24) can be presented [16] as
X = QU > where U is an arbitrary orthogonal n 2 n-matrix.
(26)
Now a feedback matrix representation
K = B01 (W 0 S) U >W 0 0 A
M directly follows from (23) and (25). W Since this matrix K satisfies the (17) with positive definite matrices W = W and S , a spectral radius (A + BK ) < 1. Hence, K 2 . A BK;G Thus, under theorem assumptions, an arbitrary element W 2 S is attainable. Theorem 2 is proved. Let us turn to the case rank(B ) < n. Consider the projective matrices P1 = BB + and P2 = I 0 P1 . Here a sign “+” means a pseu-
of admissible stochastic sensitivity matrices. A symbol 0 means that the matrix is symmetric and positive definite. For non-singular 2 a solution K of the (17) noises (rank = ), for any belongs to . One can weaken a requirement of non-singularity of is true [15] if the pair ( + noises. The condition K 2 ) is controllable for 2 . The aim of the proposed control design is the synthesis of the desired stochastic sensitivity matrix. Denote by K a solution of (17) with the 2 . fixed matrix Problem 1: For the assigned matrix 2 , it is necessary to find 2 guaranteeing the equality a matrix
M G n
Due to these decompositions the (22) can be rewritten as a system of two matrix equations
S B n l W
W S U W A where U is an arbitrary orthogonal n 2 n-matrix. 2 S . Rewrite the (17) Proof: Consider an arbitrary element W in the following: (A + BK )> = W 0 S: (A + BK )W (22) and non-negative definite matrix For the positive definite matrix W V = W 1=2 and W 0 S one can find [16] a positive definite1=matrix 2 non-negative definite matrix Q = (W 0 S ) , such that (23) W = V V > ; W 0 S = QQ>:
doinversion. Note some properties of these matrices
P1B = B; B> P1 = B> ; P2B = 0; B> P2 = 0; P1 = BB+ ; P1 + P2 = I; P12 = P1; P22 = P2 ; P1> = P1; P2> = P2: (27) If rank(B ) < n then P2 6= 0. Theorem 3: Let noises in the system (14) be non-singular (S 0) 2 S is attainable if and only if and rank(B ) < n. The element W is a solution of the equation the matrix W > P2 = P2 (W 0 S )P2 : P2 AWA (28) Under these conditions for matrix W 2 S , the (17) has a solution K = B+ [QU >V 01 0 A] (29) where Q and V are matrices of the decomposition (23), U is an arbitrary orthogonal n 2 n-matrix satisfying the condition P2AV = P2 QU > : (30) Proof: Necessity. Multiplying the (22) by P2 from the left and right and taking into account P2 B = 0, B > P2 = 0, we get (28). Sufficiency. It follows from (28) and the decomposition (23) that:
P2 AV V >A> P2 = P2QQ> P2:
(31)
U K BK = BB+ [QU > V 01 0 A] = P1 [QU >V 01 0 A] = (I 0 P2 )[QU > V 01 0 A] = [QU > V 01 0 A]; (32) (A + BK )V = QU > : The equality (32) implies that the matrix K satisfies the (17). As one can see from this proof, the matrix K from (29) is a solution of the (17) for any orthogonal matrix U satisfying the condition (30). Theorem 3
The equality (31) implies (30) for some orthogonal matrix . Due to (30) for the matrix from (29), the following equalities hold:
is proved.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 9, SEPTEMBER 2011
2165
For 2 [1; 1:6875) this equilibrium is exponentially stable. When the parameter passes the bifurcation value 3 = 1:6875, this equilibrium loses stability. On the interval (1.6875,2.4], Henon system demonstrates both periodic and chaotic oscillations. Under the random disturbances (" > 0), a trajectory of the system (33) leaves a deterministic attractor and forms a stochastic attractor around it. In Fig. 1, solutions of stochastically forced Henon model (33) without control (grey color) are plotted for the fixed noise intensity " = 0:005 and three values of : = 1:68, = 1:75, = 2:4. As one can see, a transition from order to chaos is accompanied by the increase of the amplitude of oscillations. The aim of the control is to provide a stability of equilibria on the whole interval 1 2:4 and synthesize the required stochastic sensitivity matrix W . We use a feedback regulator in the following form:
u = k1 (x 0 x) + k2 (y 0 y):
(34)
For attainability analysis and design of the feedback control we use Theorem 3. It follows from (28) that w22 = w11 +1. Put w12 = 0. This choice provides a non-correlatedness of random states xt , yt . Then the elements w11 , w22 of attainable matrices W are restricted by relations w11 1, w22 = w11 + 1. Using the formula (29) we get coefficients
k1 = (
9 + 16 0 3)=2;
k2 = 0:5 0
w11 0 1 w11 + 1
(35)
of the regulator (34) guaranteeing the required values of the stochastic sensitivity matrix for the system (33). Put w11 = 1. This choice gives us the minimal attainable value of the sensitivity. This regulator provides for the equilibria of the system (33) a stochastic sensitivity matrix with elements w11 = 1, w12 = 0, w22 = 2. As a result of this control, random states of the system (33) with regulator (34) are concentrated in a small neighborhood of the equilibrium. In Fig. 1, solutions of Henon model (33) without control (grey color) and with control (black color) are plotted for three values of parameter = 1:68, = 1:75, = 2:4. As we can see, the regulator (34) allows x; y) and essentially decrease a dispersion us to stabilize the equilibria ( of random states both for regular and chaotic zones.
REFERENCES
Fig. 1. Solutions of Henon model (33) without control (grey) and with control (34) (black) for " : and a) : , b) : , c) : .
= 0 005
= 1 68
= 1 75
=24
IV. STOCHASTIC HENON SYSTEM Consider the stochastically forced Henon system [17]
xt+1 = 1 0 x2t 0 0:5yt + ut + "t ; yt+1 = xt + "t ; 1 2:4
(33)
with a single control input ut . Here t ; t are sequences of the uncorrelated random disturbances with parameters Et = Et = 0, Et2 = 1 > 0, Et2 = 1 > 0, Et t = 0, and " is a scalar parameter of noise intensity. The corresponding uncontrolled deterministic model (33) (" = 0; ut = 0) has an equilibrium (x; y), where
x = y =
p9 + 16 0 3 : 4
[1] W. Horsthemke and R. Lefever, Noise-Induced Transitions. Berlin, Germany: Springer, 1984. [2] F. Kozin, “A survey of stability of stochastic systems,” Automatica J. IFAC, vol. 5, pp. 95–112, 1969. [3] H. J. Kushner, Stochastic Stability and Control. New York: Academic Press, 1967, vol. 33, Math. Sci. Eng. [4] W. M. Wonham, “Optimal stationary control of a linear system with state dependent noise,” SIAM J. Control Optim., vol. 5, pp. 486–500, 1967. [5] U. G. Hausmann, “Optimal stationary control with state and control dependent noise,” SIAM J. Control Optim., vol. 9, pp. 184–198, 1971. [6] R. F. Curtain, Stability of Stochastic Dynamical Systems. Berlin, Germany: Springer, 1972, vol. 294. [7] A. El Bouhtouri, D. Hinrichsen, and A. J. Pritchard, “ H -type control for discrete-time stochastic systems,” Int. J. Robust Nonlin. Control, vol. 9, pp. 923–948, 1999. [8] E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett., vol. 64, pp. 1196–1199, 1990. [9] A. L. Fradkov and A. Y. Pogromsky, Introduction to Control of Oscillations and Chaos, ser. Series of Nonlinear Science. Singapore: World Scientific, 1998. [10] S. Boccaletti, C. Grebogi, Y. C. Lay, H. Mancini, and D. Maza, “The control of chaos: Theory and applications,” Phys. Rep., vol. 329, pp. 103–197, 2000. [11] G. Chen and X. Yu, Chaos Control: Theory and Applications. New York: Springer-Verlag, 2003.
2166
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 9, SEPTEMBER 2011
[12] I. A. Bashkitseva and L. B. Ryashko, “Sensitivity and chaos control for the forced nonlinear oscillations,” Chaos Sol. Fractals, vol. 26, pp. 1437–1451, 2005. [13] L. B. Ryashko and I. A. Bashkirtseva, “On control of stochastic sensitivity,” Autom. Remote Control, vol. 69, pp. 1171–1180, 2008. [14] S. N. Elaydi, An Introduction to Difference Equations. New York: Springer, 1999. [15] W. M. Wonham, Linear Multivariable Control: A Geometric Approach. Berlin, Germany: Springer-Verlag, 1979. [16] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1985. [17] M. Henon, “A two-dimensional mapping with a strange attractor,” Commun. Math. Phys., vol. 50, pp. 69–77, 1976.
Multi-stage Anti-Windup Compensation for Open-loop Stable Plants Solmaz Sajjadi-Kia and Faryar Jabbari, Senior Member, IEEE
Abstract—We discuss adding a measure of scheduling to the popular Anti-Windup design. The main idea is to develop a scheme in which the Anti-Windup gains depend on how much the actuator command exceeds the saturation bound. The aim is to design and implement more aggressive Anti-windup gains in lower levels of saturation. Global stability and performance are addressed by adding an outer-loop Anti-windup compensation which become active when the system is in higher levels of saturation. We present results for both static and dynamic Anti-Windup gains, along with the convex synthesis LMIs. Benefits of the proposed design method are shown using well-known examples. Index Terms—Anti-windup (AW),
gain, saturation, scheduling.
I. INTRODUCTION Anti-windup (AW) augmentation often is used for safety and alleviation of performance degradations associated with actuator windup, when high performance linear controllers encounter actuator saturation. Generally, AW schemes are designed with two goals: 1) as long as actuators do not saturate, the closed-loop response coincides with the unconstrained response; 2) if the actuators saturate, stability is preserved and performance is recovered as much as possible ([1]). An excellent set of discussions and references on AW can be found in [2]–[5]. In most cases, the AW augmentation is a single controller (or set of gains) that is applied for all initial conditions, reference signals and disturbance levels. For such global results, typical performance guarantees are no better than those from the open-loop system. Stronger results can be obtained if reference signals or disturbances are assumed to have peak or energy bounds ([6]). Such techniques can be combined with the results here, by adding conditions that bound the reachable sets, which in turn bound the signals and states. For brevity, we focus on the global case here. The main observation, here, is that in many applications saturation can be mostly mild and the command to the actuator rarely exceeds the saturation bound by a large margin. In such cases, it seems intuitively Manuscript received November 8, 2010 revised March 25, 2011; accepted April 01, 2011. Date of publication April 25, 2011; date of current version September 08, 2011. Recommended by Associate Editor F. Wu. The authors are with the Department of Mechanical and Aerospace Engineering, University of California at Irvine (UCI), Irvine, CA 92697 USA (e-mail:[email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2011.2146930
clear that using different gains for the rare cases might allow a more aggressive and higher performance gain when the commands are only slightly larger than the actuator limitations. This leads to, in essence, a form of scheduling of the AW gains based on the level of saturation. This is the main tack in this paper. Scheduling in the traditional AW setting has been attempted before. For example, in [7] scheduling is used to improve the system performance after it re-enters the small signal domain. Another approach can be found in [8], in which a family of controllers are used to develop a scheduling approach in dealing with saturation, though the overall technique is quite different from the concept of AW used here. While the approach in [8] has the important advantage of being applicable to open-loop unstable systems, its implementation is rather involved and requires considerable on-line computation and often cannot match the high performance of nominal linear controllers in small signal regions. Other approaches to the scheduled AW design can be found, e.g., in [9] where a nonlinear state-feedback control law for the global asymptotic stabilization of non-exponentially unstable plants is proposed. Similarly, use of self (or gain)-scheduled techniques to obtain AW gains is explored in [10] and [11]. Here, we are interested in improving the performance of AW through simple and easy to implement modifications. We focus on developing an approach that provides performance guarantees for the large signal operation of open-loop stable systems, yet allows higher performance gains when the command to the actuator is only modestly above the saturation limits. This is through use of multiple sets of AW gains (as shown in Fig. 2): for moderate levels and for severe levels of saturation. By placing more weight on the gains associated with the moderate level of saturation, one can expect better performance in this level, particularly if this is the normal envelop of operation. The overall stability and some performance when the actuator command signal exceeds the saturation bound significantly are guaranteed by another set of gains. For simplicity, and to stay close to the basic concept of AW, we start by considering the case in which all gains are static. We show that the gains can be obtained from straightforward linear matrix inequalities where the complexity of the resulting search is only modestly higher than the traditional AW case. The implementation of the proposed scheme is quite straightforward and similar to the standard AW. Finally, for “all static” gains we show that the existence condition of the proposed scheduled approach is the same as the one for the traditional static AW compensation (Fig. 1). It is well known however, that in some problems the traditional static AW compensation (Fig. 1) is not feasible, while a dynamic AW augmentation only requires open-loop stability, as a sufficient condition for feasibility. In Section IV, we show that by letting one of the loops to include dynamic AW, we also can extend our approach so that open-loop stability leads to feasibility of the scheduled AW. II. PROBLEM DEFINITION AND BASIC SETUP Consider the open-loop stable plant below with state vector xp 2 Rn , control input vector u^ 2 Rn , and exogenous input vector w 2 Rn (e.g., reference signal, noise, external disturbance, etc.)
6p
x_ p = Ap xp + B1 w + B2 u^
z = C1 xp + D11 w + D12 u^
y = C2 xp + D21 w + D22 u^:
(1)
The nominal controller, designed to fulfill a specific task such as tracking or disturbance regulation, is likely to saturate. As a result, it needs to be augmented with an AW protection loop. The AW commands are introduced to the nominal controller by adding signals 1
0018-9286/$26.00 © 2011 IEEE
