Robust quantized feedback stabilization of linear systems

Automatica 44 (2008) 2458–2462

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Robust quantized feedback stabilization of linear systemsI M.L. Corradini a,∗ , G. Orlando b a Dipartimento di Matematica e Informatica, Università di Camerino, Italy b DIIGA, Università Politecnica delle Marche, Italy

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Article history: Received 16 January 2007 Received in revised form 25 September 2007 Accepted 28 January 2008 Available online 7 May 2008

a b s t r a c t This paper investigates the feedback stabilization problem for SISO linear uncertain control systems with saturating quantized measurements. In the fixed quantization sensitivity framework, we propose a time varying control law able to effectively account for the presence of saturation, which is often the main source of instability, designed using sliding mode techniques. Such controller is proved able to stabilize the plant both in the presence and in the absence of quantization. © 2008 Elsevier Ltd. All rights reserved.

Keywords: Quantized systems Saturating actuators Sliding mode control Robust control

1. Introduction The growing interest in networked control systems is currently bringing, with increasing persistence, the problem of stabilizing plants with saturating quantized measurements to the attention of the research community. In fact, when measurements to be used for feedback are transmitted by a digital communication channel, data are quantized before transmission. In other words, real valued signals are mapped into piecewise constant signals taking values in a finite set. The hybrid system resulting from quantization cannot be stabilized, in general, by a feedback controller (if any) able to stabilize the plant without measurement quantization. As discussed in Liberzon (2003), this is mainly due to the presence of saturation and to the performance degradation near equilibrium, which makes asymptotic convergence impossible. A number of results have been given in the past considering the quantizer parameters fixed in advance (i.e. memoryless and with fixed quantization levels) (Delchamps, 1990; Feng & Loparo, 1997), as is usually the case in practice, mainly addressing discrete-time linear plants. The seminal paper by Elia and Mitter (2001) studied the problem of quadratic stabilization of discrete time SISO linear systems using quantized feedback. Further results along the same

I This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Zongli Lin under the direction of Editor André Tits. ∗ Corresponding author. E-mail addresses: [email protected] (M.L. Corradini), [email protected] (G. Orlando).

0005-1098/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2008.01.027

research direction have been given by Fu (2003) and Fu and Xie (2003), all addressing discrete-time state feedback stabilizable plants, while other works in the discrete-time framework have been provided by Sznaier and Damborg (1989) and Turner and Postlethwaite (2004). Scalar linear discrete-time systems, again controlled by a static feedback, have been studied by Fagnani and Zampieri (2003). In the continuous-time framework, it is worth mentioning the paper by Lee and Haddad (1999), which addressed plants actuated by a quantized static state feedback control in the presence of additive Gaussian noise. The remarkable work by Brockett and Liberzon (2000) showed that a linear system stabilizable by a linear feedback law can also be globally asymptotically stabilized by a hybrid quantized feedback control policy, under the assumption that quantizer sensitivity can be changed dynamically. These results have also been extended to the case when quantization parameters are updated at discrete instants of time (Liberzon, 2003). Note that, in the case of vanishing quantizer sensitivity, the property that the equilibrium point is stable in the sense of Lyapunov, and that the state tends to zero as time t → ∞ is loosely referred to as “asymptotic stability” (Brockett & Liberzon, 2000). Turning back to a fixed quantization sensitivity framework, which is indeed a more feasible approach from the practical viewpoint, this paper investigates the use of time-varying controllers for the stabilization of linear plants with saturating quantized measurements. In fact, it is likely to expect that the time invariance requirement is one of the main hindrance to cope with saturation, which is in turn the main source of instability. The recent paper by Corradini and Orlando (2007) showed that a timevarying controller can be easily designed, under mild conditions,

M.L. Corradini, G. Orlando / Automatica 44 (2008) 2458–2462

to stabilize uncertain plants with saturating actuators. The present work extends that approach in the direction of considering state and output quantized feedback controllers. Note that, differently from most of the available literature on the subject, the plant is supposed affected by matched disturbances. A few comments are in order. The definition of ”asymptotic stability” in Brockett and Liberzon (2000) will be borrowed here, but with the difference that it is required that the state tends to zero as time t → ∞, in the case the quantizer sensitivity could be considered vanishing (sensitivity is fixed here). Secondly, by means of a constructive proof, it will be proved that a same time-varying controller is able to asymptotically stabilize the plant both in the presence and in the absence of quantization. Results herewith reported deal with SISO plants. Indeed, current investigations seem to show an easy extension to MIMO systems, along with a possible relevant role assumed by the sliding surface decay rate. 2. Problem statement Consider the following continuous-time, time invariant, uncer-

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hybrid system, the control problem addressed in this paper can be stated as follows: the problem here considered consists in finding stabilizing quantized state and output feedback controller, i.e. control laws guaranteeing the robust stabilization of the system (1) using saturating quantized measurements.

3. A time varying sliding surface def

Define D =[d1 . . . dn ], with di

∈

R, i

=

1, . . . , n to be def

chosen according to the discussion below, and define C¯ (t) = (C + D exp(−λ¯ t)). Consider the following time-varying sliding surface (Corradini & Orlando, 2007) h i ¯ (t) x(t) − x(0) exp(−λ¯ t) s(x(t), x(0), t) = C h i = (c1 + d1 exp(−λ¯ t)) . . . (cn + dn exp(−λ¯ t)) h i × x(t) − x(0) exp(−λ¯ t) = 0, λ¯ > 0; (3)

def

tain, controllable SISO plant S ={A, B, 0} described by:  x˙ = Ax + B(u + d(x)) y = 0x     0 In−1 0 A= ; B= , a1 . . . an 1

(1)

where: x ∈ Rn is the state vector, u, y ∈ R are the control input and the plant output respectively, 0 is the output matrix, In−1 is the identity matrix of size n − 1 and 0 denotes a null matrix of appropriate dimension. The uncertain term d(x) is an additive perturbation term affecting the system. Assumption 1. The nominal plant is observable. Moreover, the uncertain term d(x) is bounded (in absolute value) by a known ¯ kxk), i.e. |d(x)| ≤ ρ( ¯ kxk) ≤ ρ( ¯ M) for kxk ≤ M. function ρ( As well known (Emelyanov, Korovin, & Mamedov, 1995), a vector C ∈ R1×n can be chosen such that CB 6= 0 and that, when a sliding motion is achieved on the following sliding surface: sˆ(x) = Cx = [c1 c2 . . . cn ]x = 0;

ci

> 0, i = 1, . . . , n

(2)

the reduced order system has assigned stable eigenvalues, and system (1) is stable, too. As a consequence, the straightforward state¯ kxk)sign(sˆ(x))] feedback controller uc (x) = −(CB)−1 [CAx + |CB|ρ( ensures the achievement of such a motion, hence plant stabilization. It is considered here that the given system evolves in continuous time, but state variables are quantized, i.e. they are filtered by a device that converts a real-valued signal into a piecewise constant one taking values on a finite set. According to Liberzon (2003), a quantizer can be defined as a piecewise constant function q(ξ) : Rr → Q, where ξ ∈ Rr is the variable being quantized and Q is a finite subset of Rr . When ξ does not belong to the union of the quantization regions of finite size, the quantizer saturates. Define M as the range of q(·) and ∆ as the quantization error, both quantities being fixed a priori. The quantizer can be described by the following analytical expression:  kq(ξ) − ξk ≤ ∆ if kξk ≤ M q(ξ) : kq(ξ)k > M − ∆ if kξk > M, where the operator k · k denotes the Euclidean norm. It is also assumed that q(ξ) = 0 for ξ in some neighborhood of the origin, i.e. the origin lies in the interior of the set {ξ : q(ξ) = 0}. As discussed, in networked control systems measurements to be used for feedback are transmitted by a digital communication channel, in other words data are quantized before transmission. Since the presence of the quantizer makes the plant become a

where C ∈ Rn is chosen such that C¯ (t)B 6= 01 and that, when a sliding motion is achieved on (3), the reduced order system has assigned stable eigenvalues. The decay rate λ¯ > 0 can be chosen arbitrarily. It is straightforward that, for any values of di Q 0, i = 1, . . . , n, constraining the system to the surface s(x(t), x(0), t) = 0 implies plant asymptotical stabilization. Remark 1. As proved in Corradini and Orlando (2007), the adoption of the time-varying sliding surface (3) allows one to ”shape” the control activity and, with a proper choice of the coefficients di , makes it able to comply with the amplitude restrictions imposed by saturating actuators. Indeed, the surface contains a vanishing term depending on some design parameters. Suitably choosing these parameters, it is possible to modulate the control input during the initial transient, avoiding the violation of the saturation bounds. As discussed before, we are interested in the case when only quantized measurements q(x) of the state x are available. Since quantization implies saturation, it is useful, for the sake of readability, to consider first quantization without saturation. The following Lemma (given without proof) provides a stabilizing controller in such a case. Lemma 2. The following control input u = ueq (x(0), t) + un (q(x), t), built using quantized measures q(x) of the state vector, with quantization error ∆ and without saturation (M = ∞):

¯ (t)B)−1 ϕ(x(0), t) ueq (x(0), t) = −(C h ¯ (t)B)−1 |β(t)|(∆ + |q(x)|) un (q(x), t) = −(C i ¯ kxk) sign(s(q(x), x(0), t)) + |C¯ (t)B|ρ(

(4)

(5) def

¯ C¯ (t) + 2D exp(−λ¯ t))x(0) exp(−λ¯ t), β(t) = where ϕ(x(0), t) = λ( (λ¯ D exp(−λ¯ t)−CA−DA exp(−λ¯ t)) ensures the achievement of stable sliding motion on the surface (3), i.e. the Ultimate Boundedness of the trajectories of the plant (1).

1 This is always possible since the presence of saturation induces constraints on the absolute value of the coefficients di , as shown in Corradini and Orlando (2007), hence the terms CB and DB can be always chosen with the same sign.

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4. Quantized state feedback stabilization Due to the Ultimate Boundedness property guaranteed by Lemma 2, for initial states belonging to a given bounded set D , bounding functions exist for system trajectories, the reaching phase being absent due to the presence of the term (i ) C¯ (t)x(0) exp(−λ¯ t) in (3). In other words, functions ∆F (t), i = (max) 1, . . . , n and ∆F can be found such that: (i )

|xi (t)| ≤ ∆F (t);

i = 1, . . . , n;

(max) def

∆F

(i )

= sup ∆F (t).

(6)

i

The following lemma shows that the imposition of the constraint kxk ≤ M can be ensured imposing an analogous requirement on the control input. Lemma 3. Consider the plant (1) in the presence of the state quantizer. If the controller

¯ (t)B)−1 ϕ(x(0), t) u¯ eq (x(0), t) = −(C h ¯ (t)B)−1 |β(t)|(∆ + |q(x)|) u¯ n ((x), t) = −(C i ¯ M) sign(s(q(x), x(0), t)) + |C¯ (t)B|ρ(

(7)

(8)

def

satisfies |¯ueq (x(0), t)| + |¯un (x(0), t)| ≤ N(t) = |(C¯ (t)B)−1 ||β(t)|M + ¯ M) + |(C¯ (t)B)−1 ||ϕ(x(0))|, then it both ensures the Ultimate ρ( Boundedness of system trajectories, and guarantees that kxk ≤ M. Proof. According to Lemma 2, it is immediate to verify that the controller (7) and (8) ensures the achievement of stable sliding ¯ kxk) ≤ ρ( ¯ M). Moreover, if motion on the surface (3), as ρ( |¯ueq (x(0), t)| + |¯un (x(0), t)| ≤ N(t), it follows

|¯ueq (x(0), t)| + |¯un (x(0), t)| ¯ M) + |(C¯ (t)B)−1 ||ϕ(x(0))| = |(C¯ (t)B)−1 ||β(t)|(∆ + |q(x)|) + ρ( ¯ M) + |(C¯ (t)B)−1 ||ϕ(x(0))|. < |(C¯ (t)B)−1 ||β(t)|M + ρ( It follows immediately that ∆ + kq(x)k < M, hence kxk < M, and the assertion is proved.  Exploiting our previous results (Corradini & Orlando, 2007), it can be now easily proved that the coefficients of the sliding surface (3) can always be chosen to satisfy the constraint |u| ≤ N(t) for the control variable, which in turn implies the fulfillment of the quantization constraint kxk < M. This proves that the state quantized feedback controller (7) and (8) actually guarantees the robust practical stabilization of the hybrid system (1) using saturating quantized measurements. Theorem 1. It is given the uncertain system (1) in the presence of the state quantizer under Assumption 1. There exists a suitable region D of the state space such that, for any kx(0)k ∈ D proper coefficients di , i = 1, . . . , n can always be chosen such that the quantized state feedback controller (7)- (8) guarantees the robust Ultimate Boundedness of system trajectories of the plant. Proof. The proof is fully analogous to that reported in Corradini and Orlando (2007). Following the constructive procedure reported there, it can be shown that coefficients di can always be found. Recalling Lemma 3, the use of (7) and (8) guarantees plant trajectories Ultimate Boundedness instead of asymptotical stability as in Corradini and Orlando (2007).  Remark 4. It is well known that a feedback law which asymptotically stabilizes a given system in the absence of quantization will in general fail to provide asymptotical stability of the closed loop system arising in the presence of quantization (Bullo & Liberzon, 2006). Indeed, a controller built using the time invariant surface (2) would not be able to fulfill the saturation requirement, which is one of the two phenomena inducing changes as a consequence of quantization (see the discussion in Bullo and Liberzon (2006)).

Remark 5. It may be worthwhile noting that, according to Theorem 1, a same time-varying controller (7) and (8) is able to stabilize the plant (1) both in the presence (∆ 6= 0, M < ∞) and in the absence (∆ = 0, M = ∞) of state quantization. In the former case Ultimate Boundedness of plant trajectories is ensured, while asymptotic stability is guaranteed in the latter case. 5. Quantized output feedback stabilization We now turn to the problem of stabilizing the SISO plant (1) with quantized measurements of the output. The following assumptions are introduced Edwards and Spurgeon (1995): Assumption 2. It is assumed that 0 B 6= 0, (1) is stabilizable by linear output static feedback K , and that the invariant zeros of (A, B, 0 ) have negative real parts. Remark 6. The above assumptions, indeed restrictive, are due to the well known limitations associated to the use of sliding mode techniques in an output feedback framework. Note however that the output static linearizability hypothesis has been introduced also in Brockett and Liberzon (2000), albeit then relaxed by the use of state estimators. Under the above assumption, define the following sliding surface

¯ t))y(0) exp(−λ¯ t) − (F + G exp(−λ¯ t))y = 0, (9) s˜(y, t) = (F + G exp(−λ where the coefficient F is fixed (typically F = 1) and the scalar G has to be determined. Note that F and G can be related in terms in sign like C and D. Such surface ensures output vanishing and plant asymptotical stabilization, due to Assumption 1, when a sliding motion is achieved on it. Moreover, state trajectories are bounded, and bounds (6) still hold. Note, however, that only quantized measurements of the output are available, hence (9) cannot be measured. We have to consider instead

¯ t))y(0) exp(−λ¯ t) s¯(q(y), t) = (F + G exp(−λ − (F + G exp(−λ¯ t))q(y) = 0,

(10)

where the output initial condition has been assumed known. As before, it is useful, for the sake of readability, to consider first quantization without saturation in the following Lemma (given without proof). Lemma 7. The control input u = Kq(y) + ν, built using quantized measures q(y) of the output variable, with quantization error ∆ and without saturation (M = ∞), with ν defined as:  −1 ν = − (F + G exp(−λ¯ t))0 B n ¯ 2G exp(−λ¯ t) + F )y(0) exp(−λ¯ t) − λ¯ Gq(y) × λ( h   + |F + G exp(−λ¯ t)| k0 A0 k∆(Fmax) + |0 B|ρ(x) i o + Ψ∆ sign(s¯(q(y), t)) (11) def

def

with A0 = A + BK 0 , Ψ = |F 0 BK + λ¯ G| + |G0 BK |, ensures the achievement of stable sliding motion on the surface (10), i.e. the Ultimate Boundedness of trajectories of the plant (1), kxk ≤ η for some η > 0. We need now to prove that the coefficients of the sliding surface (10) can always be chosen as to fulfill the constraint |y| ≤ M, due to the saturation threshold, for the output variable. Recalling that A0 has stable eigenvalues, denote by λm the slowest one. Define def

T = k/λm = kτm , with k a suitable positive integer (usually chosen

as 4 − 5, see the constructive proof below).

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Assumption 3. Define ∆1 = (K ∆ + ρ(x)). Fixing a region D of the state space, the threshold M is such that inf kx(0)k∈D (M − k0k · def

ˆ (t) > 0.. kx(0)k − T Ψ∆1 ) = M Theorem 2. It is given the uncertain system (1) under Assumptions 1–3. There exists a suitable region D of the state space such that, for any kx(0)k ∈ D the coefficient G of (10) can always be chosen such that the quantized output feedback controller of Theorem 2 guarantees the robust η-stabilization of the plant, for some η > 0. Proof. The proof is constructive. The plant output, when driven by the input u = Kq(y) = Ky + , || ≤ ∆, has to fulfill |y| ≤ MR. Taking the worst case one has: |y| ≤ k0k · kx(0)k + k0kkBk 0t k exp A0 (t − τ)k(|ν|+ K ∆ + d(x))dτ ≤ M. Recalling that A0 has stable eigenvalues, denote by λm the smallest (and slowest) one and by τm the relative time constant. Suitably choosing the positive constant k to guarantee the vanishing of the slowest natural mode, Rt one can write: 0 k exp A0 (t − τ)k(|ν| + ∆)dτ ≤ (νM + ∆1 )kτm = (νM + ∆1 )T ; where

Fig. 1. Quantized state feedback controller: State variable x1 .

νM = sup ν = [(kF | − |Gk) · |0 B|]−1 t

h

¯ 2|G| + |F |)|y(0)| + λ¯ |G|q(y) + (|F | + |G|) × λ( × k0 A0 k∆(Fmax) + k0k|B|ρ(x) + Ψ∆1 . 



i

Coupling the previous relations, one gets h ¯ 2|G| + |F |)|y(0)| + λ¯ |G|q(y) + (|F | + |G|) T λ(   i × k0 · A0 k ∆(Fmax) + |0 B|ρ(x) k0k kBk ∆1 kF | − |Gk

ˆ (t). ≤ kF | − |Gk M Set F = Gα and assume |F | > |G|, i.e. |α| > 1 (This assumption will be verified later). One has:   ¯ 2 + |α|)|y(0)| + λ¯ q(y) + (1 + |α|) |G| λ(    × k0 A0 k∆(Fmax) + k0kkBkρ(x)  ˆ (t)  |F |M ˆ (t) M + |0 B|∆1 (|α| − 1) + ≤ T  T

Fig. 2. Quantized state feedback controller: Control input u.

n o ˆ (t) ≤ |F |M ˆ (t) with Γ1 = or equivalently |G| Γ1 T |α| + Γ2 T + M   λ¯ |y(0)| + k0 A0 k∆(Fmax) + k0k|B|ρ(x) + |0 B|∆1 , Γ2 = 2λ¯ |y(0)| +   λ¯ q(y) + k0 A0 k∆(Fmax) + k0kkBkρ(x) − |0 B|∆1 . Note that Γ2 is

always positive, using eventually the arbitrary parameter λ¯ > 0. Sincen we need to impose o |G| < |F |, it is enough to choose |G| < ˆ (t ) F |M min |F |, Γ T |α||+ for an arbitrary |α| > 1.  ˆ Γ T +M(t) 1

2

6. Simulation results In order to validate previous theoretical results, the proposed control approach has been applied by simulation on a simple unstable third-order system described by:  x˙ 1 = x2   ˙ x =x 2

3

 x˙ = −x1 + 2x2 + 3x3 + u + d(x)   3 y = x1 + x2 + x3

with a disturbance term d(x) = α sin(ωt), |α| ≤ 1, ω = 1. In a first simulation study, the state variables have been assumed quantized with range M = 1.5 and error ∆ = 0.2. The following vectors

Fig. 3. Quantized output feedback controller: State variable x1 .

have been used C = [1 3 2], D = [1 4 0] (for details about the determination of D see Corradini and Orlando (2007)), with η = 0.5, λ¯ = 3 and a boundary layer of width  = 0.02. Results are reported in Figs. 1 and 2, showing the first state variable and the control input u respectively. In a further simulation study, the output variable has been assumed quantized, with range M = 0.52 and error ∆ = 0.01. The sliding surface (10) with F = 1, λ = 0.1G = 0.035 has been used to stabilize the plant, which started from y(0) = 0.5. A boundary layer of width  = 0.05 has been included in simulations. Results are reported in Figs. 3 and 4 showing the first state variable and the output variable y respectively.

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Fig. 4. Quantized output feedback controller: Output variable y.

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