Proceedings of the 2007 American Control Conference Marriott Marquis Hotel at Times Square New York City, USA, July 11-13, 2007
FrC09.5
On external semi-global stochastic stabilization of linear systems with input saturation Anton A. Stoorvogel1
Ali Saberi2
Abstract— This paper introduces the notion of external semiglobal stochastic stabilization for linear plants with saturating actuators, driven by a stochastic external disturbance, and having random Gaussian-distributed initial conditions. The aim of this stabilization is to control such plants by a possibly nonlinear static state feedback law that achieves global asymptotic stability in the absence of disturbances, while guaranteeing a bounded variance of the state vector for all time in the presence of disturbances and Gaussian distributed initial conditions. We report complete results for both continuous- and discrete-time open-loop critically stable plants.
I. I NTRODUCTION Internal and external stabilization of linear plants with actuators subject to saturation has been the subject of intense renewed interest among the control research community for the past two decades. The number of recent books and special issues of control journals devoted to this subject matter evidence this intense research focus, see for instance [1], [4], [11], [15] and the references therein. It is now considered a classical fact that both continuous- and discrete-time linear plants with saturating actuators can be globally internally stabilized if and only if all of the open-loop poles are located in the closed left half plane (in the continuous-time case), or in the closed unit disc (in the discrete-time case). These conditions on open-loop plants can be equivalently stated as a requirement that the plant be asymptotically null controllable with bounded control (ANCBC). It is also a classical fact that, in general, global internal stabilization of ANCBC plants requires nonlinear feedback laws. See for instance. [2], [10], [16]. By weakening the notion of global internal stabilization to a semiglobal framework, Saberi et al. showed that ANCBC plants with saturating actuators can be internally stabilized in the semiglobal sense using only linear feedback laws. Such a relaxation from a global to semiglobal framework is, from an engineering standpoint, both sensible and attractive. A body of work exists proposing design methodologies (lowgain, low-high gain, and variations) for both continuous- and 1 Department of Mathematics, and Computing Science, Eindhoven Univ. of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. Department of Electrical Engineering, Mathematics and Computer Science, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands. E-mail:
[email protected]. 2 School of Electrical Engineering and Computer Science, Washington State University, Pullman, WA 99164-2752, U.S.A. E-mail:
[email protected]. 3 Department of Electrical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands, E-mail:
[email protected].
1-4244-0989-6/07/$25.00 ©2007 IEEE.
Siep Weiland3
discrete-time ANCBC plants with saturating actuators. See for instance [3], [5], [6], [17] With regard to external stabilization, the picture is complicated. Unlike linear plants, internal stability does not necessarily guarantee external stability when saturation is present. Hence, stabilization must be done simultaneously in both the internal and external sense. There is a body of work that examines the standard notion of L p stability for ANCBC plants with saturating actuators when the external input (disturbance) is input-additive and at the same time requiring either global or semiglobal internal stability in the absence of disturbance, see [7]. For the more general case in which the disturbance is not necessarily input-additive, there are surprising results in the literature which point to the complexity of the notion of external stability for ANCBC plants with saturating actuators. Notable among these is the result from [13] that for a double integrator with saturating actuators, “all linear internally stabilizing control laws achieve L p stability without finite gain for p ∈ [1, 2], but no linear internally stabilizing control law can achieve L p stability for any p ∈ (2, ∞]”. Moreover, considering the recently developed notion of input-to-state stability (ISS) as a framework for simultaneous external and internal stability, such a double-integrator system cannot achieve stability with a linear feedback law, see [8]. All these results point toward the delicacy of the external stability concept for linear plants with saturating actuators. All these considerations lead us to believe that a suitable notion of external stability for linear plants with saturating actuators, and indeed for general nonlinear systems in the presence of disturbances and nonzero initial conditions, is yet to be developed. In this paper we look at the simultaneous external and internal stabilization of ANCBC plants with saturating actuators when the external input is a stochastic disturbance. Specifically, we consider a linear time-invariant system subject to input saturation, stochastic external disturbances and random Gaussian distributed initial conditions, independent of the external disturbances. The aim will be to control this system by a possibly nonlinear static state feedback law that achieves global asymptotic stability in the absence of disturbances, while guaranteeing a bounded variance of the state vector for all time.This problem seems a natural extension of the results in [9], [12]. In [14], a result was claimed. In this paper we give a complete proof for the case that the system is neutrally stable, i.e., the eigenvalues of the system matrix of the continuous-time (discrete-time) system are in the closed left half plane (open unit disc) and the
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FrC09.5 eigenvalues on the imaginary axis (unit circle) have equal algebraic and geometric multiplicity. The paper is organized as follows. For both discrete- and continuous-time systems, a formal problem formulation and the main results are stated in Section II. A formal proof of the discrete-time case for neutrally stable systems is given in Section III while an outline of the proof for the continuoustime case is given in Section IV.
Theorem II.3 Consider the system (1) and suppose that (A, B) is stabilizable while A is neutrally stable, i.e. the eigenvalues of A are in the closed unit disc and the eigenvalues on the unit circle have equal geometric and algebraic multliplicity. In the above case, there exists a linear feedback which solves the semiglobal external stochastic stabilization problem as defined in Problem II.1.
II. P ROBLEM FORMULATION AND MAIN RESULTS
We claim that the condition above is only sufficient and that we can weaken the neutral stability condition of A to the requirement that all eigenvalues of A are in the closed unit disc. However, this does, in general, require a nonlinear feedback.
A. The discrete-time case In discrete time, we consider systems of the form x(k + 1) = Ax(k) + Bσ (u(k)) + Ew(k)
(1)
where the state x, the control u and the disturbance w are vector valued signals of dimension n, m and , respectively. Here, k ∈ Z+ , w is a white noise stochastic process with variance Q, the initial condition x 0 of (1) is a Gaussian random vector independent of w(k) for all k ≥ 0. Moreover σ is the standard saturation function given by: ⎧ ⎪ ⎨−1 if u < −1 σ (u) = u if − 1 ≤ u ≤ 1 ⎪ ⎩ 1 if u > 1 An admissible feedback is a nonlinear feedback of the form u(k) = f (x(k))
(2)
where f : Rn → Rm is a continuous map with f (0) = 0. We therefore consider nonlinear static state feedbacks. We will be interested in the following problem. Problem II.1 Given the system (1), the semiglobal external stochastic stabilization problem is to find an admissible feedback (2) such that the following properties hold: (i) in the absence of the external input w, the equilibrium point x = 0 of the controlled system (1)-(2) is globally asymptotically stable. (ii) the variance Var(x(k)) of the state of the controlled system (1)-(2) is bounded over k ≥ 0. Remark II.2 We call this semiglobal external stabilization because the controller explicitly depends on the variance Q of the disturbance w. We can find a controller which achieves a bounded variance for the state for any given Q but we have not been able to find a controller which is independent of Q and achieves a bounded variance for the state irrespective of the variance of the disturbance. The latter problem we would call global external stabilization. The fact that controllers exist that achieve global asymptotic stability in the absence of disturbances as described in condition (i) is well-known. The main objective of this paper is to look at the additional requirement on the variance. The following is the main result of this paper for discrete time systems:
Conjecture II.1 Consider the system (1). There exists a feedback (2) which solves the semiglobal external stochastic stabilization problem as defined in Problem II.1 if and only if (A, B) is stabilizable while the eigenvalues of A are in the closed unit disc. B. The continuous time case In continuous time we consider the differential equation dx(t) = Ax(t)dt + Bσ (u(t))dt + Edw(t)
(3)
where the state x, the control u and the disturbance w are vector valued signals of dimension n, m and , respectively. Here w is a Wiener process (a process of independent Brownian motions) with rate Q, that is, Var[W (t)] = Qt and the initial condition x 0 of (3) is a Gaussian random vector, independent of w. Its solution x is rigorously defined through Wiener integrals and is a Gauss-Markov process. Like in the discrete time case, σ denotes the standard saturation function and admissible feedbacks are possibly nonlinear static state feedbacks of the form (2) where f is a Lipschitz-continuous mapping with f (0) = 0. Problem II.4 Given the system (3), the semiglobal external stochastic stabilization problem is to find an admissible feedback (2) such that the following properties hold: (i) in the absence of the external input w, the equilibrium point x = 0 of the controlled system (2)-(3) is globally asymptotically stable. (ii) the variance Var(x(t)) of the state of the controlled system (2)-(3) is bounded over t ≥ 0. Like in the discrete time, the fact that controllers exist that achieve global asymptotic stability in the absence of disturbances as described in condition (i) is well-known. The main objective of this paper is to look at the additional requirement on the variance. The following is the main result of this paper for continuous time systems:
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FrC09.5 Theorem II.5 Consider the system (3) and suppose that (A, B) is stabilizable while A is neutrally stable, i.e. the eigenvalues of A are in the closed left half plane and the eigenvalues on the imaginary axis have equal geometric and algebraic multliplicity. In the above case, there exists a linear feedback which solves the semiglobal external stochastic stabilization problem as defined in Problem II.4. Finally, like in discrete time, we conjecture that the above conditions can be weakened: Conjecture II.2 Consider the system (3). There exists a feedback (2) which solves the semiglobal external stochastic stabilization problem as defined in Problem II.4 if and only if (A, B) is stabilizable while the eigenvalues of A are in the closed left half plane. III. P ROOFS FOR THE DISCRETE - TIME CASE We first present a little lemma that we need later:
The general case of the theorem can then easily be established using the reduction to the system (4) as described earlier. If the system is not affected by noise, it is well-known that the feedback (6) achieves asymptotic stability which is also easily verified by noting that V (x) = x T x is a suitable Lyapunov function. Next, we note that due to the Markov property of the state, we have:
E x(k + r )T x(k + r ) | x(k + s) =
E E x(k + r )T x(k + r ) | x(k + t) | x(k + s) for r ≥ t ≥ s. Next we consider:
E x(k + 1)T x(k + 1) | x(k) Using the feedback (6), the dynamics (1) and the fact that AT A = I we obtain:
E x(k + 1)T x(k + 1) | x(k) = x(k)T x(k) − 2x(k)T AT Bσ (ρ B T Ax(k)) + σ (ρ B T Ax(k))T B T Bσ (ρ B T Ax(k)) + trace E Q E T
Lemma III.1 For any a ∈ Rm and b ∈ Rm and n > 1 we have: 1 σ (a)2 − 8nb2 σ (a + b)2 ≥ 1 − 2n
where σ is the standard saturation function.
+ σ (ρ B T Ax(k))T B T Bσ (ρ B T Ax(k)) + trace E Q E T 2 ≤ x(k)T x(k) − 1 σ (ρ B T Ax(k)) + trace E Q E T (7) ρ
Using a simple basis transformation we can assume without loss of generality that: 0 A11 B1 E1 A= , B= , E= 0 A22 B2 E2 with A11 asymptotically stable while A22 A22 = I . This is guaranteed by the fact that A is neutrally stable. Next we consider the subsystem: T
x 2 (t + 1) = A22 x 2 (t) + B2 u(t) + E 2 w(t)
≤ x(k)T x(k) − ρ2 σ (ρ B T Ax(k))T σ (ρ B T Ax(k))+
provided ρ is small enough such that ρ B T B < I . Next, we consider:
2 E σ (ρ B T At x(k + 1)) | x(k) for some integer t ≥ 1. We get:
2 E σ (ρ B T At x(k + 1)) | x(k)
= E σ (ρ B T At +1 x(k)
2 −ρ B T At Bσ (ρ B T Ax(k)) + ρ B T At Ew(k)) | x(k)
(4)
≥ (1 −
If we find an admissible feedback of the form u(t) = f (x 2 (t))
and we will establish that for any given variance Q of the white noise there exists ρ small enough such that the resulting system has a bounded variance for the state. The choice of ρ is independent of the variance of the initial condition x 0 . Proof of Theorem II.3 : To keep notation simple, we assume the original system satisfies AT A = I and we consider a feedback of the form: u(k) = −ρ B T Ax(k)
− 8n E − ρ B A Bσ (ρ B Ax(k))
(5)
which solves the semiglobal stochastic stabilization problem for the system (4) then it is easily verified that this controller also solves the semiglobal stochastic stabilization problem for the original system (1). In other words we can restrict attention to the system (4). We will use a controller of the form: u(k) = −ρ B2T A22 x 2 (k)
T t +1 1 x(k)2 2n )σ (ρ B A T t T
+ρ B T At Ew(k))2 | x(k) ≥ (1 −
T t +1 1 x(k))2 2n )σ (ρ B A 2 T t 2 T
− 16nρ B A B σ (ρ B Ax(k))2 − 16nρ 2 B T At 2 trace E Q E T where the first inequality follows from Lemma III.1. If we choose ρ ∗ such that ρ ∗ B B T < I [as used in deriving (7)] and 16n ρ ∗ B T At 2 ≤ 1, 16n ρ ∗ B T At B2 ≤ 1 for all t, then it is easy to see that for all ρ < ρ ∗ and any positive integer t we have:
2 E σ (ρ B T At x(k + 1)) | x(k)
(6)
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T t +1 1 x(k))2 2n )σ (ρ B A 3/2 T 2 3/2
≥ (1 − −ρ
σ (ρ B Ax(k)) − ρ
trace E Q E T
FrC09.5 The following lemma presents a crucial inequality: Lemma III.2 We have for ρ small enough that for all s ≤ n :
E x(k + s)T x(k + s) | x(k) s t 1 σ (ρ B T At x(k)) 2 1− ≤ x(k)T x(k) − ρ 2n t =1 s trace E Q E T (8) +s 1+ n Proof: We will establish this inequality recursively. Note that we already established this inequality for s = 1. Next, consider any s > 1 and assume the above inequality is true for s (and any k). Then we obtain:
E x(k + s + 1)T x(k + s + 1) | x(k)
= E E x(k + s + 1)T x(k + s + 1) | x(k + 1) | x(k)
≤ E x(k + 1)T x(k + 1) s 1 t σ (ρ B T At x(k + 1)) 2 − 1− ρ 2n t =1 s +s 1 + trace E Q E T | x(k) n Using the previously obtained inequality we have: s 2 t T t σ (ρ B A x(k + 1)) | x(k) 1− E 2n t =1 s 2 t 1 1− ≥ 1− σ (ρ B T At +1 x(k)) 2n 2n
Next we note that since (A, B) is controllable while A is invertible, the matrix ⎛ T ⎞ B A ⎜ B T A2 ⎟ ⎜ ⎟ ⎜ .. ⎟ ⎝ . ⎠ B T An is injective which implies that there exists α (the smallest singular value of this matrix divided by n) such that for any x there exists a positive integer t ≤ n such that B T At x ≥ αx and hence 1 σ (ρ B T At x)2 ≥ min ρα 2 x2 , ρ1 ρ But then (8) yields:
E x(k + n)T x(k + n) | x(k) ≤ x(k)T x(k) − 12 min ρα 2 x(k)2 , ρ1 + 2n trace E Q E T We define M= and
y(k) = max x(k)T x(k), M
t =1
3/2
2
3/2
− sρ σ (ρ B Ax(k)) − sρ trace E Q E s 2 t +1 ≥ 1− σ (ρ B T At +1 x(k)) 2n T
T
t =1
− sρ 3/2 σ (ρ B T Ax(k))2 − sρ 3/2 trace E Q E T We then obtain:
E x(k + s + 1)T x(k + s + 1) | x(k)
= E E x(k + s + 1)T x(k + s + 1) | x(k + 1) | x(k) 2 ≤ x(k)T x(k) − ρ1 σ (ρ B T Ax(k)) + trace E Q E T s 2 t +1 1 1− − σ (ρ B T At +1 x(k)) ρ 2n t =1 √ + s ρσ (ρ B T Ax(k))2 s √ trace E Q E T + s ρ trace E Q E T + s 1 + n s+1 1 t T σ (ρ B T At x(k)) 2 ≤ x(k) x(k) − 1− ρ 2n t =1 s+1 + (s + 1) 1 + trace E Q E T n
1 ρ 2 α2
The following lemma presents another crucial result: Lemma III.3 For all k ≥ 1 there holds:
E y(k + n) | y(k) ≤ y(k)
(9)
provided ρ is small enough such that the inequality of Lemma III.2 is valid for s = 1, . . . , n and: 1 > 2n trace E Q E T , 2ρ
and ρα 2 < 1
Moreover, E[y(k + pn)] ≤ E[y(k)],
p = 1, 2, . . .
(10)
Proof: This follows from the fact that if y(k) = x(k)T x(k) and hence x(k)T x(k) ≥ M we have:
E x(k + n)T x(k + n) | x(k) ≤ y(k) − 12 min ρα 2 x(k)2 , ρ1
provided ρ < ρ ∗ (recall that s ≤ n) satisfies: √ 1 ρ ≤ n1 , nρ 3/2 ≤ 2n .
+ 2n trace E Q E T ≤ y(k) − ≤ y(k)
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1 2ρ
+ 2n trace E Q E T
FrC09.5 while for the case y(k) = M and hence x(k)T x(k) ≤ M we have:
E x(k + n)T x(k + n) | x(k) ≤ x(k)T x(k) − 12 min ρα 2 x(k)2 , ρ1 + 2n trace E Q E T ≤ (1 − 12 ρα 2 )x(k)T x(k) + 2n trace E Q E T ≤ (1 − 12 ρα 2 )y(k) + 2n trace E Q E T ≤ y(k) − 12 ρα 2 M + 2n trace E Q E T ≤ y(k)
If we find an admissible feedback of the form u(t) = f (x 2 (t))
which solves the semiglobal stochastic stabilization problem for the system (11) then it is easily verified that this controller also solves the semiglobal stochastic stabilization problem for the original system (3). In other words we can restrict attention to the system (11). We will use a controller of the form: u(t) = −ρ B2T x 2 (t) and we will establish that for any given rate Q of the Wiener process there exists ρ small enough such that the resulting system has a bounded variance for the state. The choice of ρ is independent of the variance of the initial condition x 0 . Proof of Theorem II.5 : To keep notation simple, we assume the original system satisfies A + AT = 0 and we consider a feedback of the form:
but then (using that y(k) ≥ M):
E y(k + n) | x(k) ≤ y(k) and finally,
E y(k + n) | y(k) = E E y(k + n) | x(k) | y(k)
≤ E y(k) | y(k) = y(k)
u(t) = −ρ B T x(t)
(13)
The general case of the theorem can then easily be established using the reduction to the system (11) as described earlier. We first note that: τ e A(τ −s) Bσ (u(s))ds x(τ ) = e A(τ −t ) x(t) + t τ A(τ −s) + e Edw(s)
Similarly,
E y(k + 2n) | y(k)
= E E y(k + 2n) | x(k + n), x(k) | y(k)
= E E E y(k + 2n) | x(k + n), x(k) | x(k) | y(k)
= E E E y(k + 2n) | x(k + n) | x(k) | y(k)
≤ E E y(k + n) | x(k) | y(k)
≤ E y(k) | y(k) = y(k)
t
= e A(τ −t ) x(t) + vt (τ ) We note that σ (u) is bounded by 1 and the second integral has bounded variance for all τ such that τ ∈ [t, t + 1]. This implies that there exists a constant N such that:
Using a simple recursion, we obtain (10). We have, using (7): E[x(k + 1)T x(k + 1)|x(k)] ≤ x(k)T x(k) + trace E Q E T
Var[vt (τ )] < N
which yields:
(14)
for all τ such that τ ∈ [t, t + 1]. Next, we note that σ (ρ B T x(τ ))2 = σ ρ B T e A(τ −t ) x(t) + ρ B T vt (τ ) 2
E[x(k)T x(k)] ≤ Var[x(0)] + k trace E Q E T for k = 1, . . . , n − 1 and hence: E[y(k)] ≤ max M, Var[x(0)] + k trace E Q E T
≥ 12 σ (ρ B T e A(τ −t ) x(t))2 − 8ρ 2 B2 vt (τ )2 (15)
for k = 1, . . . , n − 1. This implies that the expectation of y(0), . . . , y(n − 1) is bounded. Using (10) we find that the expectation of y(k) is bounded in k. Using that y(k) ≥ x(k)T x(k) this yields that the variance of x(k) is bounded in k. IV. P ROOFS FOR THE CONTINUOUS - TIME CASE
where we applied Lemma III.1 with n = 1. Next we note that: d x T x = (2x T Bσ (u) + trace E T E)dt + 2x T Edw using that A + AT = 0 and Itô’s lemma. We can then show that: E[x(t + 1)T x(t + 1) | x(t)] t +1 T T = x(t) x(t) + E 2x(τ ) Bσ (u(τ ))dτ | x(t)
Using a simple basis transformation we can assume without loss of generality that: A11 0 B1 E1 A= , B= , E= 0 A22 B2 E2 with A11 asymptotically stable while A22 + AT22 = 0. This is guaranteed by the fact that A is neutrally stable. Next we consider the subsystem: dx 2 (t) = A22 x 2 (t)dt + B2 σ (u(t))dt + E 2 dw(t)
(12)
t
+ trace E T E On the other hand, we have:
(11)
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2x(τ )T Bσ (u(τ )) = 2x(τ )T Bσ (−ρ B T x(τ )) ≤ − ρ2 σ (ρ B T x(τ ))2
FrC09.5 Combining the last two equations we obtain:
V. D ISCUSSION AND CONCLUSIONS
E[x(t + 1)T x(t + 1) | x(t)] ≤ x(t)T x(t) + trace E T E t +1 T 2 2 −ρE σ (ρ B x(τ )) dτ | x(t) t
Using (14), (15) and the fact that x(t) and vt (τ ) are independent for τ ∈ [t, t + 1] we obtain a crucial inequality: E[x(t + 1)T x(t + 1) | x(t)] ≤ x(t)T x(t) + V t +1 − ρ1 σ (ρ B T e A(τ −t ) x(t))2 dτ
where V = + trace E E. It can be shown that controllability of (A, B) implies that t +1 2 T A(τ −t ) 1 (ρ B e x(t)) dτ ≥ min{ραx(t)2 , ρR } σ ρ T
t
for suitably defined constants α and R (independent of ρ). The latter yields with the help of (16): E[x(t + 1)T x(t + 1) | x(t)] ≤ x(t)T x(t) − min{ραx(t)2 , ρR } + V We define: M= and
R ρ2α
y(t) = max x(t)T x(t), M
The following lemma presents another crucial result: Lemma IV.1 We have:
E y(t + 1) | y(t) ≤ y(t)
(17)
provided ρ is small enough such that: ρ
