Nonlinear Stabilization under Sampled and Delayed ... - arXiv

Nonlinear Stabilization under Sampled and Delayed Measurements, and with Inputs Subject to Delay and Zero-Order Hold Iasson Karafyllis1 and Miroslav Krstic2 Abstract Sampling arises simultaneously with input and output delays in networked control systems. When the delay is left uncompensated, the sampling period is generally required to be sufficiently small, the delay sufficiently short, and, for nonlinear systems, only semiglobal practical stability is generally achieved. For example, global stabilization of strict-feedforward systems under sampled measurements, sampled-data stabilization of the nonholonomic unicycle with arbitrarily sparse sampling, and sampled-data stabilization of LTI systems over networks with long delays, are open problems. In this paper we present two general results that address these example problems as special cases. First, we present global asymptotic stabilizers for forward complete systems under arbitrarily long input and output delays, with arbitrarily long sampling periods, and with continuous application of the control input. Second, we consider systems with sampled measurements and with control applied through a zero-order hold, under the assumption that the system is stabilizable under sampled-data feedback for some sampling period, and then construct sampled-data feedback laws that achieve global asymptotic stabilization under arbitrarily long input and measurement delays. All the results employ “nominal” feedback laws designed for the continuous-time systems in the absence of delays, combined with “predictorbased” compensation of delays and the effect of sampling.

Keywords: feedback stabilization, time-delay systems, sampled-data systems, nonlinear control.

1. Introduction Motivation. Sampling arises simultaneously with input and output delays in many control problems, most notably in control over networks. In the absence of delays, in sampled-data control of nonlinear systems semiglobal practical stability is generally guaranteed [5,27,28,29], with the desired region of attraction achieved by sufficiently fast sampling. Alternatively, global results are achieved under restrictive conditions on the structure of the system [4,7,11,12,14,31].

1

Dept. of Environmental Eng., Technical University of Crete, 73100, Chania, Greece, email: [email protected].

2

Dept. of Mechanical and Aerospace Eng., University of California, San Diego, La Jolla, CA 92093-0411, U.S.A.,

email: [email protected].

1

On the other hand, in purely continuous-time nonlinear control, input delays of arbitrary length can be compensated [15,19,20] but no sampled-data extensions of such results are available. Simultaneous consideration to sampling and delays (either physical or sampling-induced) is given in

the

literature

on

control

of

linear

and

nonlinear

systems

over

networks

[2,3,6,26,30,31,34,35,37], but all available results rely on delay-dependent conditions for the existence of stabilizing feedback. Despite the remarkable accomplishments in the fields of sampled-data, networked, and nonlinear delay systems, the following example problems remain open: global stabilization of strict-feedforward systems under sampled measurements and continuous control, sampled-data stabilization of the nonholonomic unicycle with inputs applied via zero-order hold and under arbitrarily sparse sampling, and sampled-data stabilization of LTI systems over networks with long delays. In this paper we introduce two frameworks for solving such problems: 1. We present global asymptotic stabilizers for forward complete systems under arbitrarily long input and output delays, with arbitrarily long sampling periods, and with continuous application of the control input. 2. We consider systems with sampled measurements and with control applied through a zeroorder hold, under the assumption that the system is stabilizable under sampled-data feedback for some sampling period, and then construct sampled-data feedback laws that achieve global asymptotic stabilization under arbitrarily long input and measurement delays. In both frameworks we employ “nominal” feedback laws designed in the absence of delays, combined with “predictor-based” compensation of delays. Problem Statement. As in [15,19,20,21,22,23,24,36,38], we consider systems with input delay, x& (t ) = f ( x(t ), u (t − τ ))

(1.1)

where x(t ) = ( x1 (t ),...., x n (t )) ′ ∈ ℜ n , u (t ) ∈ ℜ m , f : ℜ n × ℜ m → ℜ n is a locally Lipschitz mapping with f (0,0) = 0

and τ ≥ 0 is a constant. In [15,19,20,21,38], the feedback design problem for system

(1.1) is addressed by assuming a feedback stabilizer u = k (x) for system (1.1) with no delay, i.e. (1.1) with τ = 0 , or x& (t ) = f ( x(t ), u (t ))

2

(1.2)

and applying a delay compensator (predictor) methodology based on the knowledge of the delay. In this paper, we incorporate also a consideration of measurement delay, namely, we address the problem of stabilization of (1.1) with output y (t ) = x(t − r ) ∈ ℜ n

(1.3)

where r ≥ 0 is a constant, i.e., we consider delayed measurements. The motivation for a simultaneous consideration of input and measurement delays is that in many chemical process control problems the measurement delay of concentrations of chemical species can be large. We also assume that the output is available at discrete time instants τ i (the sampling times) with τ i +1 − τ i = T > 0 , where T > 0 is the sampling period. Very few papers have studied this problem (an exception is [8] where input and measurement delays are considered for linear systems but the measurement is not sampled). The problem of stabilization of (1.1) with output given by (1.3) is intimately related to the stabilization of system (1.1) alone. To see this, notice that the output y (t ) of (1.1), (1.3) satisfies the following system of differential equations for all t ≥ r : y& (t ) = f ( y (t ), u (t − r − τ ))

Consider the comparison between two problems described by the same differential equations: the problem of stabilization of (1.1) with input delay r > 0 and no measurement delay (i.e., x& (t ) = f ( x (t ), u (t − r ))

for all t ≥ 0 ) and the problem of stabilization of (1.1), (1.3) with no input delay

and measurement delay r > 0 (i.e., y& (t ) = f ( y (t ), u (t − r )) for all t ≥ r ). The two problems are not identical: in the first stabilization problem the applied input values for t ∈ [0, r ] are given (as initial conditions), while in the second stabilization problem the applied input values for t ∈ [0, r ] must be computed based on an arbitrary initial condition x(θ ) = x 0 (θ ) , θ ∈ [−r ,0] (irrespective of the current value of the state). Therefore, serious technical issues concerning the existence of the solution for t ∈ [0, r ]

arise for the second stabilization problem (see Remark 2.2(b) below).

Results of the paper. We establish two general results: 1. A solution for the stabilization of (1.1) with output given by (1.3) under the assumption that system (1.2) is globally stabilizable and forward complete and the input can be continuously adjusted (Theorem 2.1). The proposed dynamic sampled-data controller uses values of the output (1.3) at the discrete time instants τ i = t0 + iT , i ∈ Z + , where T > 0 is the sampling period 3

and t0 ≥ 0 is the initial time. This justifies the term “sampled-data”. No restrictions for the values of the delays r , τ ≥ 0 or the sampling period T > 0 are imposed. In general, we show that there is no need for continuous measurements for global asymptotic stabilization of any stabilizable forward complete system with arbitrary input and output delays. 2. A solution for the stabilization of (1.1) with output given by (1.3) under the assumption that system (1.2) is globally stabilizable and forward complete and the control action is implemented with zero order hold (Theorem 3.2). Again, the proposed sampled-data controller uses values of the output (1.3) at the discrete time instants τ i = t 0 + iT , i ∈ Z + , where T > 0 is the sampling period and t0 ≥ 0 is the initial time. In this case, we can solve the stabilization problem for systems with both delayed inputs and measurements provided that the user chooses the sampling period as the ratio of the input delay and any integer. Our delay compensation methodology guarantees that any controller (continuous or sampleddata) designed for the delay-free case can be used for the regulation of the delayed system with input/measurement delays and sampled measurements. For example, all sampled-data feedback designs proposed in [4,5,11,14,27,28,29,31] which guarantee global stabilization can be exploited for the stabilization of a delayed system with input/measurement delays, sampled measurements and input applied with zero order hold. The results are applied to •

the Linear Time Invariant (LTI) case, where f ( x, u ) = Ax + Bu , A ∈ ℜ n×n , B ∈ ℜ n×m . This case has been recently studied extensively in the context of linear Networked Control Systems, where various delays arise. Delay-dependent and/or sampling period-dependent sufficient conditions for the stabilization of Networked Control Systems have been proposed in the literature [2,3,6,26,30,31,34,35,37]. Here, we propose a linear delay compensator that guarantees exponential stability of the closed-loop system under the mild restriction that the user chooses the sampling period as the ratio of the input delay and any integer, with no additional restrictions for the delays (Corollary 3.4). The compensator is designed based on the knowledge of linear feedback stabilizer for the delay-free case.

•

strict-feedforward systems [18,20,33], which are studied in Examples in 2.4 and 3.8.

•

the stabilization of the nonholonomic integrator x&1 = u1

,

x& 2 = x1u 2

,

x& 3 = u 2

(1.4)

with both delayed inputs and measurements. The problem was recently studied in [17] in the presence of delays and in [4,29] in the presence of sampling. Here, our proposed 4

dynamic sampled-data controller is applied with no restrictions for the value of the delays or the size of the sampling period. The stabilization problem is solved for the case where the inputs can be continuously adjusted (Corollary 4.1), as well as for the case where the inputs are applied with zero order hold (Proposition 4.2). Organization of the paper. In Section 2 the main results concerning the case of the continuously adjusted input are stated and many comments and explanations are provided. In Section 3 the main results concerning the case of input applied with zero order hold are provided. Special results are provided for the case of linear autonomous systems and for the case of nonlinear systems which are diffeomorphically equivalent to a chain of integrators. Section 4 is devoted to the application of the obtained results to the stabilization of a three-wheeled vehicle with two independent rear motorized wheels (the nonholonomic integrator). Finally, in Section 5 we present the concluding remarks of the present work. The Appendix contains the proofs of certain results.

Notation. Throughout the paper we adopt the following notation: ∗ For a vector x ∈ ℜ n we denote by x its usual Euclidean norm, by x ′ its transpose. For a real

matrix A ∈ ℜ n×m , A′ ∈ ℜ m×n denotes its transpose and A := sup{ Ax ; x ∈ ℜ n , x = 1 } is its induced norm. I ∈ ℜ n×n denotes the identity matrix. ∗ ℜ + denotes the set of non-negative real numbers. Z + denotes the set of non-negative integers.

For every t ≥ 0 , [t ] denotes the integer part of t ≥ 0 , i.e., the largest integer being less or equal to t ≥0. ∗ For the definition of the class of functions KL , see [16]. ∗ By C j ( A) ( C j ( A ; Ω) ), where j ≥ 0 is a non-negative integer, we denote the class of functions

(taking values in Ω ) that have continuous derivatives of order j on A . ∗

Let x : [a − r , b) → ℜ n with b > a ≥ 0 and r ≥ 0 . By Tr (t ) x we denote the “history” of x from t − r (

to t , i.e., (Tr (t ) x )(θ ) := x(t + θ ) ; θ ∈ [−r ,0] , for t ∈ [a, b) . By Tr (t ) x we denote the “open history” of x from t − r to t , i.e., (Tr (t ) x )(θ ) := x(t + θ ) ; θ ∈ [−r ,0) , for t ∈ [a, b) . (

∗

∞ Let I ⊆ ℜ + := [0,+∞) be an interval. By L ∞ ( I ; U ) ( Lloc ( I ; U ) ) we denote the space of measurable

and (locally) bounded functions u ( ⋅ ) defined on I and taking values in U ⊆ ℜ m . Notice that we do not identify functions in L ∞ ( I ; U ) which differ on a measure zero set. For x ∈L ∞ ([−r ,0]; ℜ n ) or x ∈L ∞ ([− r ,0); ℜ n )

we define x r := sup x(θ ) or x r := sup x(θ ) . Notice that sup x(θ ) is not θ ∈[ − r , 0 ]

θ ∈[ − r , 0 )

5

θ ∈[ − r , 0 ]

the essential supremum but the actual supremum and that is why the quantities sup x(θ ) and θ ∈[ − r , 0 ]

sup x(θ ) do not coincide in general. We will also use the notation M U for the space of

θ ∈[ − r , 0 )

measurable and locally bounded functions u : ℜ + → U . ∗

We say that a system of the form (1.2) is forward complete if for every x 0 ∈ ℜ n , u ∈ M U the solution x(t ) of (1.2) with initial condition x(0) = x 0 ∈ ℜ n corresponding to input u ∈ M U exists for all t ≥ 0 .

Throughout the paper we adopt the convention L∞ ([−r ,0]; ℜ n ) = ℜ n and C 0 ([−r ,0]; ℜ n ) = ℜ n for r = 0 . Finally, for reader’s convenience, we mention the following fact, which is a direct consequence of Lemma 2.2 in [1] and Lemma 3.2 in [10]. The fact is used extensively throughout the paper. FACT: Suppose that system (1.2) is forward complete. Then for every ∞ u ∈Lloc ([−τ ,+∞); ℜ m )

x0 ∈ ℜ n ,

the solution x(t ) of (1.1) with initial condition x(0) = x 0 ∈ ℜ n corresponding to

∞ input u ∈Lloc ([−τ ,+∞); ℜ m ) exists for all t ≥ 0 . Moreover, for every T > 0 there exists a function

a ∈ K∞

∞ such that for every x 0 ∈ ℜ n , u ∈Lloc ([−τ ,+∞); ℜ m ) the solution x(t ) of (1.1) with initial

condition

x ( 0) = x 0 ∈ ℜ n

corresponding

to

input

∞ u ∈Lloc ([−τ ,+∞); ℜ m )

satisfies

⎞ ⎛ x(t ) ≤ a⎜ x 0 + sup u ( s) ⎟ , for all t ∈ (0, T ] . −τ ≤ s 0 in the following way: 6

“for every x 0 ∈ ℜ n , u ∈L ∞ ([−r − τ ,0); ℜ m ) the solution x(t ) of (1.1) with initial condition x(−r ) = x 0

corresponding to input u ∈L ∞ ([−r − τ ,0); ℜ m ) satisfies x(τ ) = Φ ( x 0 , u ) ”

By virtue of the Fact, we can guarantee the existence of a ∈ K ∞ such that

(

Φ ( x, u ) ≤ a x + u

r +τ

) , for all ( x, u) ∈ ℜ

n

(

×L ∞ [− r − τ ,0); ℜ m

)

(2.1)

Using (2.1) and the fact that f : ℜ n × ℜ m → ℜ n is a locally Lipschitz mapping, we can guarantee the existence of a non-decreasing function L : ℜ + → ℜ + such that

( + v τ )( x − y + u − v τ ) , τ ([−r − τ ,0); ℜ ), ( y, v) ∈ ℜ ×L ([−r − τ ,0); ℜ )

Φ ( x, u ) − Φ ( y , v ) ≤ L x + y + u

for all ( x, u ) ∈ ℜ n ×L∞

r+

r+

m

r+

n

∞

m

(2.2)

We assume next that (1.2) is globally stabilizable. Hypothesis (H2) (continuously adjusted input): There exists k ∈ C 1 (ℜ + × ℜ n ; ℜ m ) , g ∈ K ∞ with k (t , x) ≤ g ( x ) ,

for all (t , x) ∈ ℜ + × ℜ n

(2.3)

such that 0 ∈ ℜ n is Uniformly Globally Asymptotically Stable for system (1.2) with u = k (t , x) , i.e., there exists a function σ ∈ KL such that for every (t 0 , x 0 ) ∈ ℜ + × ℜ n the solution x(t ) of (1.2) with u = k (t , x)

and initial condition x(t 0 ) = x0 ∈ ℜ n satisfies the following inequality: x(t ) ≤ σ ( x0 , t − t 0 ) , ∀t ≥ t 0

(2.4)

Consider system (1.1) under hypotheses (H1), (H2) for system (1.2). Our proposed dynamic sampled-data feedback has states ( z (t ), Tr +τ (t )u ) ∈ ℜ n ×L∞ ([−r − τ ,0]; ℜ m ) and inputs y (t ) ∈ ℜ n and for each t 0 ≥ 0 , ( z 0 , u 0 ) ∈ ℜ n ×L∞ ([−r − τ ,0]; ℜ m ) the states are computed by the interconnection of two subsystems: 1) A sampled-data subsystem (see [10]) with inputs ( y (t ), Tr +τ (t )u ) ∈ ℜ n ×L∞ ([−r − τ ,0]; ℜ m ): z& (t ) = f ( z (t ), u (t )) , t ∈ [τ i , τ i +1 ) , i ∈ Z + ( z (τ i +1 ) = Φ y (τ i +1 ), Tr +τ (τ i +1 )u

(

z (t 0 ) = z 0 ∈ ℜ

)

(2.5)

n

where τ i = t 0 + iT , i ∈ Z +

are the sampling times and T > 0 is the sampling period. We stress that the proposed sampleddata dynamic controller uses only values of the output y (t ) = x(t − r ) ∈ ℜ n at the discrete time instants τ i = t0 + iT , where i ∈ Z + .

7

2) A subsystem described by Functional Difference Equations (see [13]) with inputs z (t ) ∈ ℜ n : u (t ) = k (t + τ , z (t )) , t > t 0

(

Tr +τ (t 0 )u = u 0 ∈L ∞ [− r − τ ,0]; ℜ m

(2.6)

)

Our first main result is now stated. Theorem 2.1: Let T > 0 , r , τ ≥ 0 with r + τ > 0 and suppose that hypotheses (H1), (H2) hold for system (1.2). Then the closed-loop system (1.1), (1.3) (2.5), (2.6) is Uniformly Globally Asymptotically Stable, in the sense that there exists a function σ~ ∈ KL such that for every t 0 ≥ 0 ,

(

( x 0 , z 0 , u 0 ) ∈ C 0 ([− r ,0]; ℜ n ) × ℜ n ×L ∞ [− r − τ ,0]; ℜ m

closed-loop

system

(2.5),

(

(2.6),

),

the solution ( x(t ), z (t ), u (t )) ∈ ℜ n × ℜ n × ℜ m of the

(1.3),

)

(1.1)

with

(

initial

condition

z (t 0 ) = z 0 ∈ ℜ n ,

)

Tr +τ (t 0 )u = u 0 ∈L ∞ [− r − τ ,0]; ℜ m , Tr (t 0 ) x = x 0 ∈ C 0 [−r ,0]; ℜ n satisfies the following inequality for all t ≥ t0 : z (t ) + Tr (t ) x r + Tr +τ (t )u

r +τ

(

≤ σ~ z 0 + x0

r

+ u0

r +τ

, t − t0

)

(2.7)

Some remarks for the dynamic sampled-data feedback given by (2.5), (2.6) are in order before we proceed to the proof of Theorem 2.1. Remark 2.2: (a) The dynamic sampled-data controller (2.5), (2.6) is time-varying if k is time-varying. If k is T − periodic then the dynamic sampled-data controller (2.5), (2.6) is T − periodic too.

(b) Since the output y (t ) given by (1.3) of system (1.1) satisfies the system of differential equations y& (t ) = f ( y (t ), u (t − r − τ )) , for all t ≥ t 0 + r , where t 0 ≥ 0 is the initial time, we can in principle apply the predictor-based delay compensation approach described in [20] (extended for (

time-varying feedback laws), which gives the static feedback law u (t ) = k (t + τ , Φ ( y (t ), Tr +τ (t )u )) . This is the inspiration for the construction of the sampled-data dynamic feedback (2.5), (2.6): for all t = t 0 + iT , where i ∈ Z + , the value of u (t )

computed by (2.5), (2.6) is exactly

( u (t ) = k (t + τ , Φ( y (t ), Tr +τ (t )u )) . However, there are technical problems with the application of the

static

feedback

(

law

)

( u (t ) = k (t + τ , Φ ( y (t ), Tr +τ (t )u )) :

(

given

initial

conditions

)

Tr (t 0 ) x = x 0 ∈ C 0 [−r ,0]; ℜ n , Tr +τ (t 0 )u = u 0 ∈L ∞ [− r − τ ,0]; ℜ m , we cannot guarantee existence of a

measurable and essentially bounded u : [t 0 − r − τ , t 0 + r ] → ℜ m satisfying the integral equation ( u (t ) = k (t + τ , Φ ( x(t − r ), Tr +τ (t )u ))

for all t ∈ (t 0 , t 0 + r ] . For the case τ = 0 , one sufficient condition for 8

the existence of a solution of the integral equation is that the initial condition

(

)

(

Tr (t 0 )u = u 0 ∈L∞ [−r ,0]; ℜ m

Tr (t 0 ) x = x 0 ∈ C 0 [−r ,0]; ℜ n ,

)

satisfies

the

equation

x& (t − r ) = f ( x(t − r ), u (t − r )) for all t ∈ [t 0 , t 0 + r ] : in this case the solution is u (t ) = k (t , z (t )) for t ∈ (t 0 , t 0 + r ] , where z (t ) is the solution of the initial value problem z& (t ) = f ( z (t ), k (t , z (t ))) with z (t 0 ) = x(t 0 ) . Other restrictive sufficient conditions for the existence of a solution of the integral

equation can be obtained by using fixed point theory. The proof of Theorem 2.1 shows that this issue can be completely avoided for the dynamic sampled-data feedback (2.5), (2.6). (c) For every initial condition the value of u (t ) computed by (2.5), (2.6) is exactly ( u (t ) = k (t + τ , Φ( y (t ), Tr +τ (t )u )) for all t ≥ t 0 + iT with i ∈ Z + satisfying iT ≥ r , so our dynamic sampled-

data feedback is based on the predictor principle. (d) For the implementation of the controller (2.5), (2.6), we must know the “predictor” mapping

(

)

Φ : ℜ n ×L ∞ [− r − τ ,0); ℜ m → ℜ n .

(i)

This mapping can be explicitly computed for

Linear systems x& = Ax + Bu , with x ∈ ℜ n , u ∈ ℜ m . In this case (Corollary 3.4 below) the predictor mapping

(

)

Φ : ℜ n ×L ∞ [−r − τ ,0); ℜ m → ℜ n

is given by the explicit equation

0

Φ ( x, u ) := exp( A(τ + r ) )x +

∫ exp(− Aw)Bu(w)dw .

− r −τ

(ii)

Bilinear systems x& = Ax + Bu + uCx , with x ∈ ℜ n , u ∈ ℜ and AC = CA . In this case the predictor

(

)

Φ : ℜ n ×L ∞ [− r − τ ,0); ℜ m → ℜ n

mapping

is

given

by

the

explicit

equation

0 ⎛ 0 ⎞ ⎛ 0 ⎞ Φ ( x, u ) := exp( A(τ + r ) ) exp⎜ C u ( s )ds ⎟ x + exp(− Aw) exp⎜ C u ( s )ds ⎟ Bu ( w)dw . ⎜ ⎟ ⎜ ⎟ ⎝ − r −τ ⎠ − r −τ ⎝ w ⎠

∫

∫

∫

(iii) Nonlinear systems of the following form: x&1 = a1 (u ) x1 + f 1 (u ) x& 2 = a 2 (u , x1 ) x 2 + f 2 (u, x1 ) M x& n = a n (u , x1 ,..., x n −1 ) x n + f n (u , x1 ,..., x n −1 ) x = ( x1 ,..., x n ) ′ ∈ ℜ n , u ∈ ℜ m

where all mappings ai , f i ( i = 1,..., n ) are locally Lipschitz. In this case the predictor mapping

(

)

Φ : ℜ n ×L ∞ [− r − τ ,0); ℜ m → ℜ n can be constructed inductively. For example, for n = 1 the 0 ⎛ 0 ⎞ ⎛0 ⎞ ⎜ ⎟ predictor mapping is given by Φ( x, u ) = exp⎜ a1 (u ( s))ds ⎟ x + exp⎜ a1 (u ( s ))ds ⎟ f 1 (u ( w))dw . ⎜ ⎟ ⎝ − r −τ ⎠ − r −τ ⎝w ⎠

∫

∫

∫

Example 2.4 below applies Theorem 2.1 to a three-dimensional nonlinear system of the above class. Moreover, the nonholonomic integrator (1.4) belongs to the above class and Theorem 2.1 can be applied (see Corollary 4.1). 9

(iv) Nonlinear systems x& = f ( x, u ) , for which there exists a global diffeomorphism Θ : ℜ n → ℜ n such that the change of coordinates z = Θ(x) transforms the system to one of the above cases (Corollary 3.7 below). For globally Lipschitz systems, one can utilize approximate “predictor” mappings

(

)

Φ : ℜ n ×L ∞ [− r − τ ,0); ℜ m → ℜ n as shown in [15] under additional and more restrictive hypotheses.

Proof of Theorem 2.1: We start with the following claim, which we prove in the Appendix. Claim

1:

There

exists

a

function

(

( x 0 , z 0 , u 0 ) ∈ C 0 ([− r ,0]; ℜ n ) × ℜ n ×L ∞ [− r − τ ,0]; ℜ m

closed-loop

system

(2.5),

(

(2.6),

such

G ∈ K∞

)

for

every

t0 ≥ 0

and

the solution ( x(t ), z (t ), u (t )) ∈ ℜ n × ℜ n × ℜ m of the

(1.3),

(1.1)

)

that

with

(

initial

condition

) ), for all t ∈ [t , t τ

z (t 0 ) = z 0 ∈ ℜ n ,

Tr +τ (t 0 )u = u 0 ∈L ∞ [− r − τ ,0]; ℜ m , Tr (t 0 ) x = x 0 ∈ C 0 [−r ,0]; ℜ n exists for all t ∈ [t 0 , t 0 + T ] and satisfies Tr +τ (t )u

r +τ

+ z (t ) + Tr (t ) x

r

(

≤ G z 0 + x0

r

+ u0

0

r+

0

(2.8)

+T]

By virtue of induction and Claim 1, the following claim holds. Claim 2: There exists a function G ∈ K ∞ such that for every t0 ≥ 0 , p ∈ Z + , p ≥ 1 and

(

( x 0 , z 0 , u 0 ) ∈ C 0 ([−r ,0]; ℜ n ) × ℜ n ×L ∞ [−r − τ ,0]; ℜ m

closed-loop

system

(2.5),

(

(2.6),

)

the solution ( x(t ), z (t ), u (t )) ∈ ℜ n × ℜ n × ℜ m of the

(1.3),

)

(1.1)

with

(

initial

condition

z (t 0 ) = z 0 ∈ ℜ n ,

)

Tr +τ (t 0 )u = u 0 ∈L ∞ [− r − τ ,0]; ℜ m , Tr (t 0 ) x = x 0 ∈ C 0 [−r ,0]; ℜ n exists for all t ∈ [t 0 , t 0 + pT ] and satisfies

Tr +τ (t )u

r +τ

(

r

≤ G ( p ) z 0 + x0

all

t ≥ t 0 + iT

+ z (t ) + Tr (t ) x

r

+ u0

r +τ

) , for all t ∈ [t , t 0

0

(2.9)

+ pT ]

oK o3 where G ( p ) ( s ) := G G. 14 24 p times

We

notice

that

for

with

i∈Z+

satisfying

iT ≥ r

the

solution

( x(t ), z (t ), u (t )) ∈ ℜ n × ℜ n × ℜ m of the closed-loop system (2.5), (2.6), (1.3), (1.1) satisfies:

(2.10)

z (t ) = x(t + τ )

Consequently, for all t ≥ t 0 + iT + τ with i ∈ Z + satisfying iT ≥ r it holds that: (2.11)

u (t − τ ) = k (t , x(t ))

Hypothesis (H2) in conjunction with inequality (2.4) and equation (2.11) implies that the following inequality holds: x(t ) ≤ σ ( x(t 0 + iT + τ ) , t − t 0 − iT − τ ) , ∀t ≥ t 0 + iT + τ

10

(2.12)

τ

Define p = ⎡⎢ ⎤⎥ + ⎡⎢ ⎤⎥ + 2 . Using (2.9), (2.10) and (2.12), it follows that the following inequality ⎣T ⎦ ⎣T ⎦ r

holds:

(

(

x(t ) + z (t ) ≤ σ G ( p ) Tr +τ (t 0 )u

Define

(

r +τ

) ((

σ~ ( s, t ) := σ G ( p ) ( s), t − pT − r + g σ G ( p )

(

σ~ ( s, t ) := G ( p ) ( s ) + g G ( p ) ( s )

)

), t − t ( s ), t − pT − r ))

+ z (t 0 ) + Tr (t 0 ) x

r

0

)

− pT , ∀t ≥ t 0 + pT

for

all

(2.13) and

t ≥ pT + r

for all t ∈ [0, pT + r ) . Using (2.3), (2.10), (2.11) and (2.13) we can

conclude that (2.7) holds. The proof is complete.


0 , where T > 0 is the sampling period. Hypothesis (H1) holds for system (2.15) and the predictor mapping can be explicitly expressed by the equations: 0 0 s 0 ⎡ ⎤ (1 + u ( s)) u (q)dq ds , x3 + u (s)ds ⎥ Φ ( x, u ) := ⎢φ1 ( x, u ) , x 2 + (τ + r )x 3 + x 3 u ( s )ds + ⎢ ⎥ − r −τ − r −τ − r −τ − r −τ ⎣ ⎦

∫

∫

where 11

∫

∫

′

(2.16)

φ1 ( x, u ) = x1 + (τ + r )x 2 + (τ + r )x 32 + 0

+

0

s

∫ ∫ u(q)dq ds

− r −τ − r −τ 2

(2.17)

⎛ s ⎞ (1 + u ( w) ) u (q)dq dw ds + ⎜⎜ u (q)dq ⎟⎟ ds − r −τ − r −τ − r −τ ⎝ − r −τ ⎠ s

∫ ∫

− r −τ

1 (τ + r )2 x3 + 3x 3 2 0

w

∫

∫ ∫

Moreover, hypothesis (H2) holds as well with the smooth, time-independent feedback law: k ( x) := − x1 − 3 x 2 −

2 3 2 3 ⎛⎜ 1 1 5 1 3⎛ 1 ⎞ ⎞ x 2 + x 3 − 4 − x1 − 2 x 2 + x 3 + x 2 x 3 + x 32 − x 33 − ⎜ x 2 − x 32 ⎟ ⎟ 8 4 ⎜⎝ 2 2 8 4 8⎝ 2 ⎠ ⎟⎠

(2.18)

It follows from Theorem 2.1 that the dynamic sampled-data controller u (t ) = k ( z (t )) with z&1 (t ) = z 2 (t ) + z 32 (t ),

z& 2 (t ) = z 3 (t ) + z 3 (t )u (t ),

z (t ) = ( z1 (t ), z 2 (t ), z 3 (t )) ′ ∈ ℜ

z& 3 (t ) = u (t )

3

, for t ∈ [τ i , τ i +1 )

(2.19)

and ( z (τ i +1 ) = Φ ( y (τ i +1 ), Tr +τ (τ i +1 )u ) , i ∈ Z +

(2.20)

where Φ : ℜ 3 ×L∞ ([−r − τ ,0); ℜ m ) → ℜ 3 is defined by (2.16), (2.17) and k : ℜ 3 → ℜ is defined by (2.18), guarantees global asymptotic stability for system (2.15). The reader should notice that the dynamic sampled-data controller (2.19), (2.20), (2.21) can still be used even if no delays are present but the state is available only at the discrete time instants τ i (the sampling times) with τ i +1 − τ i = T > 0 , where T > 0 is the sampling period. Hence, in this section we have provided, as a

special case, the first solution to the problem of global asymptotic stabilization of strict feedforward systems with arbitrarily sparse in time sampling of the state and with continuous control.


0 such that k ( x) ≤ g ( x ) ,

for all x ∈ ℜ n 12

(3.1)

and such that 0 ∈ ℜ n is Uniformly Globally Asymptotically Stable for the sampled-data system x& (t ) = f ( x(t ), k ( x(τ i ))) , t ∈ [τ i , τ i +1 ) x(τ i +1 ) = lim− x(t ) t →τ i +1

(3.2)

τ i +1 = τ i + T τ 0 = 0 ≥ 0 , x ( 0) = x 0 ∈ ℜ n

in the sense that there exists a function σ ∈ KL such that for every x 0 ∈ ℜ n the solution x(t ) of (3.2) with initial condition x(0) = x 0 ∈ ℜ n satisfies inequality (2.4) with t 0 = 0 for all t ≥ 0 . Remark 3.1: Hypothesis (H3) seems like a restrictive hypothesis, because it demands global stabilizability by means of sampled-data feedback with positive sampling rate. However, hypothesis (H3) can be satisfied for: (i) Linear stabilizable systems, where f ( x, u ) = Ax + Bu , A ∈ ℜ n×n , B ∈ ℜ n×m (see Corollary 3.4 and Remark 3.5 below), (ii) Nonlinear systems of the form x& = f ( x) + g ( x)u , x ∈ ℜ n , u ∈ ℜ , where the vector field f : ℜn → ℜn

is globally Lipschitz and the vector field g : ℜ n → ℜ n is locally Lipschitz and

bounded, which can be stabilized by a globally Lipschitz feedback law u = k (x) (see [7]). (iii) Nonlinear

systems

x& n = f n ( x, u ) + g n ( x, u )u ,

of

the

form

x& i = f i ( x, u ) + g i ( x, u ) x i +1

for

i = 1,..., n − 1

and

where the drift terms f i ( x, u ) ( i = 1,..., n ) satisfy the linear growth

conditions f i ( x) ≤ L x1 + ... + L x i ( i = 1,..., n ) for certain constant L ≥ 0 and there exist constants b ≥ a > 0 such that a ≤ g i ( x, u ) ≤ b for all i = 1,..., n , x ∈ ℜ n , u ∈ ℜ (see [12]).

(iv) Asymptotically controllable homogeneous systems with positive minimal power and zero degree (see [4]). (v) Systems satisfying the reachability hypotheses of Theorem 3.1 in [14], or hypotheses (A1), (A2), (A3) in Section 4 of [11], (vi) Nonlinear systems x& = f ( x, u ) , for which there exists a global diffeomorphism Θ : ℜ n → ℜ n such that the change of coordinates z = Θ(x) transforms the system to one of the above cases.

Consider system (1.1) under hypotheses (H1), (H3) for system (1.2). In this case we propose a feedback law that is simply a composition of the feedback stabilizer and the delay compensator:

( (

))

( u (t ) = k Φ y (τ i ), Tr +τ (τ i )u , t ∈ [τ i , τ i +1 )

(3.3)

where τ i = iT , i ∈ Z + are the sampling times and Φ : ℜ n ×L∞ ([−r − τ ,0); ℜ m ) → ℜ n is the predictor mapping involved in (2.1), (2.2). The control action is applied with zero order hold, i.e., it is

13

constant on [τ i ,τ i +1 ) ; however the control action affecting system (1.1) remains constant on the interval [τ i + τ ,τ i +1 + τ ) . Our main result is stated next. Theorem 3.2: Let T > 0 , r , τ ≥ 0 with r + τ > 0 and suppose that there exists l ∈ Z + such that τ = l T . Moreover, suppose that hypotheses (H1), (H2) hold for system (1.2). Then the closed-loop system (1.1) with (3.3), i.e., the following sampled-data system x& (t ) = f ( x(t ), u (t − τ )) ( u (t ) = k Φ ( x(τ i − r ), Tr +τ (τ i )u ) , t ∈ [τ i , τ i +1 ) , i ∈ Z +

(

)

(3.4)

τ i +1 = τ i + T , τ 0 = 0

is Uniformly Globally Asymptotically Stable, in the sense that there exists a function σ~ ∈ KL such that for every ( x0 , u0 ) ∈ C 0 ([−r ,0]; ℜ n ) ×L∞ ([−r − τ ,0); ℜ m ) , the solution ( x(t ), u (t )) ∈ ℜ n × ℜ m of system (3.4) with initial condition Tr +τ (0)u = u 0 ∈L∞ ([−r − τ ,0); ℜ m ) , Tr (0) x = x0 ∈ C 0 ([−r ,0]; ℜ n ) satisfies the (

following inequality for all t ≥ 0 : ( Tr (t ) x r + Tr +τ (t )u

r +τ

(

≤ σ~ x0

r

+ u0

r +τ

,t

)

(3.5)

Finally, if system (3.2) satisfies the dead-beat property of order jT , where j ∈ Z + is positive, i.e., for all x 0 ∈ ℜ n the solution x(t ) of (3.2) with initial condition x(0) = x 0 ∈ ℜ n satisfies x(t ) = 0 for all t ≥ jT

⎛

⎞

then system (3.4) satisfies the dead-beat property of order ⎜⎜ j + l + ⎡⎢ ⎤⎥ + 1⎟⎟T , where ⎡⎢ ⎤⎥ is ⎣T ⎦ ⎣T ⎦ ⎠ ⎝

(

(

r

)

r , i.e., for every ( x0 , u 0 ) ∈ C 0 ([−r ,0]; ℜ n ) ×L∞ [−r − τ ,0); ℜ m , the solution T ( of system (3.4) with initial condition Tr +τ (0)u = u0 ∈L∞ [−r − τ ,0); ℜ m ,

the integer part of ( x(t ), u (t )) ∈ ℜ n × ℜ m

r

(

)

)

⎛ ⎡r⎤ ⎞ Tr (0) x = x0 ∈ C 0 [− r ,0]; ℜ n satisfies x(t ) = 0 for all t ≥ ⎜⎜ j + l + ⎢ ⎥ + 1⎟⎟T . ⎣T ⎦ ⎠ ⎝

Remark 3.3: (a) If we denote T1 ≥ 0 to be the delay in receiving the measured data, T2 ≥ 0 the computation time for the quantity v = k (Φ( x, u ) ) , where ( x, u ) ∈ ℜ n ×L∞ ([−r − τ ,0); ℜ m ), and T3 ≥ 0 the time for the data to reach the actuator, then one should notice that r = T1 + T2 and τ = T3 . (b) In practice, when r ,τ ≥ 0 are given, the control practitioner should look for sampled-data stabilizers satisfying hypothesis (H3) for certain sampling period T > 0 with τ = l T for some l ∈ Z + . Therefore, the value of the sampling period is determined after the estimation of the input

14

delay. If τ = 0 then one can choose any sampling period T > 0 (the condition τ = l T holds with l = 0 ). However, if τ > 0 then the sampling period is constrained to be less or equal to τ > 0 . This

may seem to be impractical for the cases where τ > 0 is small. However, in practice, a delay that is smaller than a reasonable sampling period would be typically ignored, or the control engineer can induce an input delay of magnitude equal to the sampling period (for example, by delaying the transmission of the control action). Proof of Theorem 3.2: Using the Fact, we can guarantee the existence of b ∈ K ∞ such that for every ( x0 , u0 ) ∈ C 0 ([−r ,0]; ℜ n ) ×L∞ ([−r − τ ,0); ℜ m ) the solution x(t ) ∈ ℜ n of (1.1) with initial condition

(

( Tr +τ (0)u = u 0 ∈L∞ [− r − τ ,0); ℜ m

) , T ( 0) x = x r

0

(

) exists for all t ∈ [0,τ ] and satisfies ) , for all t ∈ [0,τ ] τ

∈ C 0 [− r ,0]; ℜ n

(

x(t ) ≤ b x(0) + u 0

r+

(3.6)

If τ > 0 then the input u (t ) takes exactly l values on the interval t ∈ [0, τ ) with input values u i = u (t ) for t ∈ [(i − 1)T , iT ) , i = 1,..., l . We also set u l +1 = u (lT ) = u (τ ) . Using (3.1) and (2.1), we obtain

the following estimates:

((

u1 ≤ g a x(−r ) + u 0 ⎛ ⎛ u i ≤ g ⎜⎜ a⎜ x((i − 1)T − r ) + u 0 ⎝ ⎝

r +τ

r +τ

⎞⎞ + max u q ⎟ ⎟⎟ , for i = 2,..., l + 1 1≤ q ≤i −1 ⎠⎠

Using the above inequalities, the trivial inequality x((i − 1)T − r ) ≤ x 0

r

(

+ b x0

r

+ u0

r +τ

) for

))

x(−r ) ≤ x 0

r

and the inequalities

i = 2,..., l + 1 (which are direct consequences of (3.6)), we

can construct a function h ∈ K ∞ such that

(

u i ≤ h x0

r

+ u0

r +τ

) , for i = 1,..., l + 1

(3.7)

Thus we may conclude that there exists H ∈ K ∞ such that Tr (t ) x

r

( + Tr +τ (t )u

r +τ

(

≤ H x0

r

+ u0

r +τ

), for all t ∈ [0,τ ]

(3.8)

Inequality (3.8) holds trivially for the case τ = 0 . We next continue with the following claim. Its proof is provided in the Appendix. Claim 3: There exists a function G ∈ K ∞ such that the solution ( x(t ), u (t )) ∈ ℜ n × ℜ m of system (3.4) exists for all t ∈ [τ , T + τ ] and satisfies ( Tr +τ (t )u

r +τ

+ Tr (t ) x

r

(

( ≤ G Tr +τ (τ )u

r +τ

+ Tr (τ ) x

r

), for all t ∈ [τ , T + τ ]

By virtue of induction and Claim 3, the following claim holds.

15

(3.9)

Claim 4: There exists a function G ∈ K ∞ such that for every p ∈ Z + , p ≥ 1 the solution ( x(t ), u (t )) ∈ ℜ n × ℜ m of system (3.4) exists for all t ∈ [τ , τ + pT ] and satisfies

( Tr +τ (t )u

r +τ

+ Tr (t ) x

r

(

( ≤ G ( p ) Tr +τ (τ )u

r +τ

+ Tr (τ ) x

r

), for all t ∈ [τ , pT + τ ]

(3.10)

where G ( p ) ( s) := G G. o2 K4 o3 14 p times

We notice that for all i ∈ Z + with iT ≥ r the solution ( x(t ), u (t )) ∈ ℜ n × ℜ m of system (3.4) satisfies: u (t − τ ) = k ( x(iT + τ )) , ∀t ∈ [iT + τ , (i + 1)T + τ )

(3.11)

Hypothesis (H3) in conjunction with inequality (2.4) with t 0 = 0 and equation (3.11) implies that the following inequality holds for all i ∈ Z + with iT ≥ r : x(t ) ≤ σ ( x(iT + τ ) , t − iT − τ ) , ∀t ≥ iT + τ

(3.12)

Define p = [r / T ] + 1 . Using (3.10) and (3.12), the following inequality holds:

( (

x(t ) ≤ σ G ( p ) Tr +τ (τ )u

Define t ≥ ( p + 1)T + r + τ

(

r +τ

( + Tr (τ ) x

) (( ( H ( s )) + g (G ( H ( s )) )

r

), t − pT − τ ), ∀t ≥ pT + τ

σ~ ( s, t ) := σ G ( p ) ( H ( s )), t − ( p + 1)T − r + g σ G ( p ) ( H ( s )), t − ( p + 1)T − r − τ

and σ~( s, t ) := G ( p )

( p)

))

(3.13) for

all

for all t ∈ [0, ( p + 1)T + r + τ ) . Using (3.1),

(3.8), (3.10), (3.11) and (3.13) we can conclude that (3.5) holds. Notice that if σ ( s, t ) = 0 for all s ≥ 0 and t ≥ jT (i.e., the dead-beat property of order jT ) then (3.13) implies that x(t ) = 0 for all t ≥ ( j + p + l )T (i.e., the dead-beat property of order ( j + p + l )T ). The proof is complete.


0 such that all the eigenvalues of the matrix T ⎛ ⎞ ⎜ exp( AT ) I + exp(− Aw)dw BK ⎟ are strictly inside the unit circle on the complex plane. ⎜ ⎟ 0 ⎝ ⎠

∫

The use of the sampled-data feedback law ui = Kyi , where ui = u (iT ) and yi = y (iT ) , does not in general guarantee global asymptotic stability for the delayed case, where r + τ > 0 . However, Theorem 3.2 can be used for the design of a delay compensator, which guarantees global asymptotic stability for the corresponding closed-loop system. The following corollary is a direct consequence of Theorem 3.2 and its proof is omitted. Corollary 3.4 (Stabilization of Linear Networked Control Systems with Delays): Let T > 0 , ⎡r⎤ r , τ ≥ 0 with r + τ > 0 and suppose that there exists l ∈ Z + such that τ = l T . Define q := ⎢ ⎥ and ⎣T ⎦ T ⎛ ⎞ ~ ⎜ r = r − qT . Moreover, suppose that all the eigenvalues of the matrix exp( AT ) I + exp(− Aw)dw BK ⎟ ⎜ ⎟ 0 ⎝ ⎠

∫

are strictly inside the unit circle on the complex plane. Then the closed-loop LTI system x& (t ) = Ax(t ) + Bu (t − τ )

(3.14)

u (t ) = ui , t ∈ [iT , (i + 1)T ) , i ∈ Z +

with input applied with zero order hold given by ui = K exp( A(r + τ ) ) yi + K

l + q +1

∑Q

(3.15)

p B ui − p

p =1

where yi = y(iT ) = x(iT − r ) and T

Q p = exp( ApT ) exp(− As )ds , p = 1,..., l + q

∫ 0

(3.16)

~

Ql + q +1

⎛r ⎞ = exp( A(l + q)T )⎜ exp( As )ds ⎟ ⎜ ⎟ ⎝0 ⎠

∫

is Globally Exponentially Stable, in the sense that there exist constants M , σ > 0 such that for every ( x0 , u0 ) ∈ C 0 ([−r ,0]; ℜ n ) ×L∞ ([−r − τ ,0); ℜ m ) , the solution ( x(t ), u (t )) ∈ ℜ n × ℜ m of system (3.14), (3.15) with initial condition Tr +τ (0)u = u0 ∈L∞ ([−r − τ ,0); ℜ m ) , Tr (0) x = x0 ∈ C 0 ([−r ,0]; ℜ n ) satisfies the (

following inequality for all t ≥ 0 : ( Tr (t ) x r + Tr +τ (t )u

r +τ

(

≤ M x0

17

r

+ u0

r +τ

)exp(− σ t )

(3.17)

Remark 3.5: If the pair of matrices ( A, B) is stabilizable then there exists K ∈ ℜ m×n such that the matrix ( A + BK ) is Hurwitz, a symmetric positive definite matrix P ∈ ℜ n×n and a constant μ > 0 such that P( A + BK ) + ( A + BK )′P + μ I < 0 . Using Corollary 4.3 in [12] with V ( x) = x′Px , a( s ) := λ s , ⎧

arbitrary λ ∈ (0,1) , and the fact that A(T , x) ⊆ ⎪⎨ x0 ∈ ℜ n : x − x0 ≤ T BK x0 + T A ⎪⎩

k 2 := max x′Px , x =1

k1 := min x′Px , x =1

T ⎛ ⎞ exp( AT )⎜ I + exp(− Aw)dw BK ⎟ ⎜ ⎟ 0 ⎝ ⎠

∫

⎫⎪ k2 x⎬, λ k1 ⎭⎪

where

one can show that all the eigenvalues of the matrix are strictly inside the unit circle on the complex plane for all T > 0

satisfying

T BK < 1

and

⎛ k T ⎜ A 2 + BK ⎜ k1 2 ⎝ 1 − T BK

⎞ ⎟ ⎟ ⎠ PBK < μ

Of course, the estimate for the maximum allowable sampling period provided by the above inequalities is conservative in most cases. Other estimates for the maximum allowable sampling period can be found in [31]. Example 3.6: We consider the scalar control system x& (t ) = x(t ) + u (t )

(3.18)

where x(t ) ∈ ℜ , u (t ) ∈ ℜ . The system can be exponentially stabilized by the linear feedback u = −kx with k > 1 applied with zero order hold, i.e., u (t ) = u i , t ∈ [iT , (i + 1)T ) u i = −kx(iT ) , i ∈ Z +

(3.19)

where the sampling period T > 0 must satisfy 2 ⎞ ⎛ T < ln⎜1 + ⎟ ⎝ k −1 ⎠

(3.20)

The use of the same feedback law for the case where measurement delays are present is described by the equations: x& (t ) = x(t ) + u (t ) u (t ) = u i , t ∈ [iT , (i + 1)T )

(3.21)

u i = −kx(iT − r ) , i ∈ Z +

where r ≥ 0 is the measurement delay. Numerical experiments for the closed-loop system (3.21) show that for each pair of k > 1 and T > 0 satisfying (3.20), there exists rc > 0 such that •

if r < rc then system (3.21) is globally exponentially stable, 18

•

if r > rc then system (3.21) admits exponentially growing solutions.

For the case k = 2 , T = 1 the value of the critical measurement delay satisfies rc ∈ (0.20,0.21) . Figure 1 shows the evolution of the state for system (3.21) with k = 2 , T = 1 , r = 0.1 and initial condition x(θ ) = 1

for θ ∈ [−1,0] . The state converges exponentially to zero. Figure 2 shows the evolution of

the state for system (3.21) with k = 2 , T = 1 , r = 0.3 and same initial condition x(θ ) = 1 for θ ∈ [−1,0] . In this case, the state grows exponentially, indicating instability.

x(t)

1 0,8 0,6 0,4 0,2 0 -0,2 0

5

10

15

t

20

-0,4 -0,6 -0,8 -1

Figure 1: The evolution of the state for system (3.21) with k = 2 , T = 1 , r = 0.1 and initial condition x(θ ) = 1 for θ ∈ [−1,0]

x(t)

40 30 20 10 0 -10

0

5

10

15

t

20

-20 -30

Figure 2: The evolution of the state for system (3.21) with k = 2 , T = 1 , r = 0.3 and initial condition x(θ ) = 1 for θ ∈ [−1,0] 19

It is clear that for the case r > rc one needs a delay compensator. Notice that for the case k = 2 , T =1

the critical measurement delay rc ∈ (0.20,0.21) is only a small fraction of the sampling period.

The usual practice would be to ignore the delay and this would give rise to completely unacceptable results. Corollary 3.4 shows that the feedback law: u (t ) = ui , t ∈ [iT , (i + 1)T )

(3.22)

ui = − k exp(r ) x(iT − r ) − k (exp(r ) − 1)ui −1

will guarantee global exponential stability for the closed-loop system (3.18) with (3.22) when r 1 and T > 0 satisfying (3.20).


0 there exists K ∈ ℜ n , such that all the eigenvalues of the matrix T ⎛ ⎞ exp( A0T )⎜ I + exp(− A0 w)dw B0 K ′ ⎟ ⎜ ⎟ 0 ⎝ ⎠

∫

are zero. For example, for n = 2 the vector K ∈ ℜ 2 is defined by

T ⎛ ⎞ ⎛ 1 3 ⎞ ⎜ ( ) + . If all eigenvalues of the matrix exp A T I exp(− A0 w)dw B0 K ′ ⎟ are zero then the K ′ = −⎜ 2 , ⎟ 0 ⎜ ⎟ ⎝ T 2T ⎠ 0 ⎝ ⎠

∫

sampled-data controller with zero order hold u (t ) = K ′z (iT ), t ∈ [iT , (i + 1)T ) , i ∈ Z +

applied to the linear system z& = A0 z + B0u will guarantee the dead-beat property of order nT for the resulting closed-loop system, i.e., z (t ) = 0 ,

for all t ≥ nT and for all initial conditions z (0) ∈ ℜ n

Thus, we can conclude that the sampled-data controller with zero order hold u (t ) = K ′ Θ( x(iT )), t ∈ [iT , (i + 1)T ) , i ∈ Z +

applied to the nonlinear system x& = f ( x, u ) will guarantee the dead-beat property of order nT for the resulting closed-loop system. Therefore, Theorem 3.2 and Corollary 3.4 lead us to the following corollary. Corollary 3.7 (Predictor for Linearizable Controllable Systems): Let T > 0 , r , τ ≥ 0 with r +τ > 0

r and suppose that there exists l ∈ Z + such that τ = l T . Define q := ⎡⎢ ⎤⎥ and ~r = r − qT . ⎣T ⎦

Consider system (1.1) with m = 1 and suppose that there exists a global diffeomorphism Θ : ℜn → ℜn

such that DΘ( x) f ( x, u ) = A0 Θ( x) + B0u ,

for all x ∈ ℜ n , u ∈ ℜ

(3.23)

where DΘ( x) is the Jacobian of Θ , B0′ = (0,...,0,1) , A0 = {ai , j , i, j = 1,..., n } with a i,i +1 = 1 for i = 1,..., n − 1 and

ai, j = 0

if

j ≠ i +1 .

Let

T ⎛ ⎞ exp( A0T )⎜ I + exp(− A0 w)dw B0 K ′ ⎟ ⎜ ⎟ 0 ⎝ ⎠

∫

K ∈ℜn

be such that all eigenevalues of the matrix

are strictly inside the unit circle on the complex plane. Then the

closed-loop system (1.2) with input applied with zero order hold given by u (t ) = ui , t ∈ [iT , (i + 1)T ) , i ∈ Z +

21

(3.24)

l + q +1

ui = K ′ exp( A0 (r + τ ) )Θ( yi ) + K ′

∑Q

p B0

(3.25)

ui − p

p =1

where yi = y (iT ) = x(iT − r ) and the matrices Q p ( p = 1,..., l + q + 1 ) are defined by (3.16) with A0 in place of A , is Globally Asymptotically Stable. Moreover, if all eigenevalues of the matrix T ⎛ ⎞ exp( A0T )⎜ I + exp(− A0 w)dw B0 K ′ ⎟ ⎜ ⎟ 0 ⎝ ⎠

∫

solution

( x(t ), u (t )) ∈ ℜ n × ℜ m

(

( Tr +τ (0)u = u 0 ∈L∞ [− r − τ ,0); ℜ m

are zero then for every ( x0 , u0 ) ∈ C 0 ([−r ,0]; ℜ n ) ×L∞ ([−r − τ ,0); ℜ m ) , the of

system

) , T ( 0) x = x

(1.1),

(

(3.24),

∈ C 0 [− r ,0]; ℜ n

(3.25)

with

initial

condition

) satisfies:

r

0

x(t ) = 0 ,

for all t ≥ (l + q + 1 + n)T

(3.26)

Example 3.8: Dead-beat control with a predictor can be applied to any delayed 2-dimensional strict feedforward system, i.e., any system of the form: x&1 (t ) = x 2 (t ) + p( x 2 (t ))u (t − τ ),

(3.27)

x& 2 (t ) = u (t − τ )

where p : ℜ → ℜ is a smooth function and the measurements are sampled and given by (1.3). The diffeomorphism given by (see [18]) x2 ⎡ ⎤ Θ( x) = ⎢ x1 − p( w)dw , x 2 ⎥ ⎢ ⎥ 0 ⎣ ⎦

′

∫

(3.28)

transforms system (3.27) with τ = 0 to a chain of two integrators. Therefore, the feedback law u=−

1 T2

x1 +

1 T2

x2

3

∫ p(w)dw − 2T x

(3.29)

2

0

applied with zero order hold and sampling period T > 0 achieves global stabilization of system (3.27) with τ = 0 when no measurement delays are present. Moreover, the dead-beat property of order 2T is guaranteed for the corresponding closed-loop system. Interestingly, the feedback law (3.29) is also globally asymptotically stabilizing as a continuous-time controller, placing the −3±i 7 4T

closed-loop poles in the Θ -coordinates at

for any T > 0 .

We next consider the case where we have measurement delay r > 0 satisfying r < T . In this case ( l = q = 0 , ~r = r ) we apply Corollary 3.7 and we can conclude that the feedback law (3.24) with ui = −

1 T

2

x1 (τ i − r ) +

1 T

2

x2 (τ i − r )

∫ p(w)dw − 0

3T + 2r 2T

2

x2 (τ i − r ) −

r (r + 3T ) 2T 2

ui −1

(3.30)

guarantees the dead-beat property of order 3T for the corresponding closed-loop system. Similar formulas to (3.30) are obtained for other cases, where τ > 0 or r ≥ T . 22


0 ( i = 1,2,3 ), are available at discrete time instants which differ by a

constant T > 0 . Moreover, we assume that there is a time delay τ ≥ 0 between the computed control action and the applied input (communication delay). In this case the equations of the vehicle are: x& (t ) = v(t − τ ) cos(θ (t )),

y& (t ) = v(t − τ ) sin(θ (t )), θ&(t ) = ω (t − τ )

(4.4)

with measurements x(t − r ) , y (t − r ) and θ (t − r ) .

The reader should notice that hypotheses (H1), (H2) hold for system (4.1). Particularly, there exist smooth time-periodic feedback stabilizers for system (1.4) (see [25,32]) and consequently we can guarantee that hypothesis (H2) holds. The predictor mapping for system (4.4) is given by the following equation: 0 s ⎡ ⎛ ⎞ Φ ( x, y, θ , v, ω ) := ⎢ x + v( s ) cos⎜θ + ω ( p)dp ⎟ds , ⎜ ⎟ ⎢ − r −τ ⎝ ⎠ ⎣ − r −τ

∫

∫

s 0 ⎤ ⎛ ⎞ y+ v ( s ) sin ⎜θ + ω ( p )dp ⎟ds , θ + ω ( s )ds ⎥ ⎜ ⎟ ⎥ − r −τ − r −τ − r −τ ⎝ ⎠ ⎦ 0

∫

∫

′

∫

(4.5)

23

Using any stabilizing feedback from [25,32] for the nonholonomic integrator (1.4), the coordinate transformation (4.2) and the input transformation (4.3) and Theorem 2.1, we arrive at the following corollary. Corollary 4.1: Assume that k ∈ C 1 (ℜ + × ℜ 3 ; ℜ 2 ) with k (t ,0) = 0 for all t ≥ 0 , is a time periodic uniform

stabilizer

for

u 2 (t ) = k 2 (t , x1 (t ), x2 (t ), x3 (t )) r ,τ ≥ 0 , T > 0

(1.4),

i.e.,

the

feedback

law

u1 (t ) = k1 (t , x1 (t ), x2 (t ), x3 (t )) ,

uniformly, globally stabilizes 0 ∈ ℜ 3 for system (1.4). Then for every

the sampled-data dynamic feedback z& x (t ) = v(t ) cos(zθ (t ) ) z& y (t ) = v(t ) sin (zθ (t ) ) , t ∈ [τ i ,τ i +1 )

(4.6)

z&θ (t ) = ω (t ) s ⎛ ⎞ ⎜ ⎟ v( s ) cos⎜θ (τ i +1 − r ) + ω ( p )dp ⎟ds ⎜ ⎟ τ i +1 − r −τ τ i +1 − r −τ ⎝ ⎠

τ i +1

z x (τ i +1 ) = x (τ i +1 − r ) +

∫

∫

s ⎛ ⎞ ⎜ ⎟ z y (τ i +1 ) = y (τ i +1 − r ) + v( s ) sin ⎜θ (τ i +1 − r ) + ω ( p)dp ⎟ds ⎜ ⎟ τ i +1 − r −τ τ i +1 − r −τ ⎝ ⎠

τ i +1

∫

∫

(4.7)

τ i +1

zθ (τ i +1 ) = θ (τ i +1 − r ) +

∫ ω (s)ds

τ i +1 − r −τ

where τ i = t0 + iT , i ∈ Z + and v(t ) = k1 (t + τ , z x (t ) cos( zθ (t )) + z y (t ) sin( zθ (t )), z x (t ) sin( zθ (t )) − z y (t ) cos( zθ (t )), zθ (t )) +

(

)

+ z x (t ) sin( zθ (t )) − z y (t ) cos( zθ (t )) ω (t )

(4.8)

ω (t ) = k 2 (t + τ , z x (t ) cos( zθ (t )) + z y (t ) sin( zθ (t )), z x (t ) sin( zθ (t )) − z y (t ) cos( zθ (t )), zθ (t ))

achieves uniform global stabilization of 0 ∈ ℜ 3 for system (4.4). At this point it should be emphasized that if the smooth, time-varying feedback proposed in [9] for the stabilization of the nonholonomic integrator were used in (4.8) then the closed-loop system (4.4) with (4.6), (4.7), (4.8) would be non-uniformly in time globally asymptotically stable (see Remark 2.3 above). Moreover, the reader should compare the result of Corollary 4.1 with the results in [17]: no restrictions for the magnitudes of the delays are imposed in the present work. If the control action is implemented with zero order hold, then a different procedure has to be applied. The reader should notice that for every sampling period T > 0 the discontinuous feedback stabilizer

24

2 2 1/ 2 sgn( x2 ) x2 − x1 T T , 1 1/ 2 k 2 ( x) = x2 T k1 ( x) = −

k1 ( x) = − k 2 ( x) =

(

x1 x32

T x12 + x32

T

x3 x12 x12 + x32

(

for x ∈ C1 := {ξ ∈ ℜ3 :ξ 2 (2ξ 2 − ξ1ξ 3 ) ≠ 0}

) , for x ∈ C := {ξ ∈ ℜ 2

3

}

(4.9b)

:ξ 2 = 0 , ξ1ξ 3 ≠ 0

)

x1 T , for x ∈ C3 := ξ ∈ ℜ 3 : 2ξ 2 = ξ1ξ 3 x3 k 2 ( x) = − T k1 ( x) = −

(4.9a)

{

}

(4.9c)

satisfies hypothesis (H3) for system (1.4). To see this notice that inequality (3.1) holds with g ( s ) :=

•

2 3 s+ s . Furthermore, by explicit computation of the solution one can show that: T T

if x0 ∈ C1 then the solution x(t ) of (1.4) with u (t ) = k ( x0 ) satisfies x2 (T ) = 0 , i.e., x(T ) ∈ C 2 ∪ C3 ,

•

if x0 ∈ C2 then the solution x(t ) of (1.4) with u (t ) = k ( x0 ) satisfies 2 x2 (T ) = x1 (T ) x3 (T ) , i.e., x(T ) ∈ C3 ,

•

if x0 ∈ C3 then the solution x(t ) of (1.4) with u (t ) = k ( x0 ) satisfies x(T ) = 0 .

It follows that the sampled-data implementation of the feedback (4.9) with sampling period T > 0 guarantees the dead-beat property of order 3T for the corresponding closed-loop system. The inequality sup x(t ) ≤ 3 x(0) + T (2 + x(0) ) sup u (t ) +

0≤t 0 and assume that τ = lT for some l ∈ Z + . Then 0 ∈ ℜ 3 is uniformly globally asymptotically stable for the closed-loop system (4.4) with ω (t ) = k 2 ( X cos(Θ) + Y sin(Θ), X sin(Θ) − Y cos(Θ), Θ) , for t ∈ [τ i ,τ i +1 ) v(t ) = k1 ( X cos(Θ) + Y sin(Θ), X sin(Θ) − Y cos(Θ), Θ) + ( X sin(Θ) − Y cos(Θ) )ω (t ) ,

for t ∈ [τ i ,τ i +1 )

(4.10) (4.11)

where τ i = iT , i ∈ Z + , k : ℜ3 → ℜ 2 is defined by (4.9) and s ⎞ ⎛ ⎟ ⎜ v( s ) cos⎜θ (τ i − r ) + ω ( p)dp ⎟ds ⎟ ⎜ τ i − r −τ τ i − r −τ ⎠ ⎝

τi

X = x(τ i − r ) +

∫

∫

s ⎞ ⎛ ⎟ ⎜ Y = y (τ i − r ) + v( s ) sin ⎜θ (τ i − r ) + ω ( p)dp ⎟ds ⎟ ⎜ τ i − r −τ τ i − r −τ ⎠ ⎝

τi

∫

∫

(4.12)

τi

Θ = θ (τ i − r ) +

∫ ω (s)ds

τ i − r −τ

Moreover, for every initial condition the solution of the closed-loop system (4.4) with (4.10), (4.11), (4.12), where τ i = iT , i ∈ Z + , k : ℜ3 → ℜ 2 is defined by (4.9) satisfies x(t ) = y (t ) = θ (t ) = 0 for ⎛ r ⎞ r r all t ≥ ⎜⎜ 4 + l + ⎡⎢ ⎤⎥ ⎟⎟T , where ⎡⎢ ⎤⎥ is the integer part of . T ⎣T ⎦ ⎣T ⎦ ⎝

⎠

It should be noticed that the control action computed by (4.10), (4.11), (4.12) is applied with zero order hold, i.e., v(t ) and ω (t ) are constant on each interval [τ i ,τ i +1 ) , i ∈ Z + , and hence they are piecewise constant over the integration intervals [τ i − r − τ , τ i ] in (4.12).

5. Concluding Remarks Stabilization is studied for nonlinear systems with input and measurement delays, and with measurements available only at discrete time instants (sampling times). Two different cases are considered: the case where the input can be continuously adjusted and the case where the input is applied with zero order hold. Under the assumption of forward completeness and certain 26

additional stabilizability assumptions, it is shown that sampled-data feedback laws with a predictor-based delay compensation can guarantee global asymptotic stability for the closed-loop system with no restrictions for the magnitude of the delays. Additionally, when the control is applied continuously and only the measurements are sampled, the sampling time can be arbitrarily long. Applications to the stabilization of linear networked control systems, strict feedforward systems and a nonholonomic mobile robot over a long-distance communication network are presented. Future work will address the issue of robustness of the proposed feedback laws with respect to actuator and measurement errors, as well as the extension of the obtained results to the case where the delayed and sampled measured output does not necessarily coincide with the state vector.

Acknowledgments: The authors would like to thank Professor Costas Kravaris for bringing to their attention the fact that the predictor mapping can be explicitly computed for bilinear systems x& = Ax + Bu + uCx ,

with x ∈ ℜ n , u ∈ ℜ and AC = CA (see Remark 2.2(d)).

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[7] Herrmann, G., S.K. Spurgeon and C. Edwards, “Discretization of Sliding Mode Based Control Schemes”, Proceedings of the 38th Conference on Decision and Control, Phoenix, Arizona, U.S.A., 1999, 4257-4262. [8] Jankovic, M., “Recursive Predictor Design for State and Output Feedback Controllers for Linear Time Delay Systems”, Automatica, 46, 2010, 510-517. [9] Karafyllis, I. and J. Tsinias, “Global Stabilization and Asymptotic Tracking for a Class of Nonlinear Systems by Means of Time-Varying Feedback”, International Journal of Robust and Nonlinear Control, 13(6), 2003, 559-588. [10] Karafyllis, I., “A System-Theoretic Framework for a Wide Class of Systems I: Applications to Numerical Analysis”, Journal of Mathematical Analysis and Applications, 328(2), 2007, 876-899. [11] Karafyllis, I. and Z.-P. Jiang, “A Small-Gain Theorem for a Wide Class of Feedback Systems with Control Applications”, SIAM Journal Control and Optimization, 46(4), 2007, 1483-1517. [12] Karafyllis, I., and C. Kravaris, “Global Stability Results for Systems under Sampled-Data Control”, International Journal of Robust and Nonlinear Control, 19(10), 2009, 1105-1128. [13] Karafyllis, I., P. Pepe and Z.-P. Jiang, “Stability Results for Systems Described by Coupled Retarded Functional Differential Equations and Functional Difference Equations”, Nonlinear Analysis, Theory, Methods and Applications, 71(7-8), 2009, 3339-3362. [14] Karafyllis, I. and C. Kravaris, “Robust Global Stabilizability by Means of Sampled-Data Control with Positive Sampling Rate”, International Journal of Control, 82(4), 2009, 755-772. [15] Karafyllis, I., “Stabilization By Means of Approximate Predictors for Systems with Delayed Input”, submitted to SIAM Journal on Control and Optimization. [16] Khalil, H. K., Nonlinear Systems, 2nd Edition, Prentice-Hall, 1996. [17] Kojima, K., T. Oguchi, A. Alvarez-Aguirre and H. Nijmeijer, “Predictor-Based Tracking Control of a Mobile Robot With Time-Delays”, Proceedings of NOLCOS 2010, Bologna, Italy, 2010, 167-172. [18] Krstic, M., “Feedback Linearizability and Explicit Integrator Forwarding Controllers for Classes of Feedforward Systems”, IEEE Transactions on Automatic Control, 49, 2004, 16681682. [19] Krstic, M., Delay compensation for nonlinear, adaptive, and PDE systems. Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA, 2009. [20] Krstic, M., “Input Delay Compensation for Forward Complete and Feedforward Nonlinear Systems”, IEEE Transactions on Automatic Control, 55(2), 2010, 287-303. [21] Krstic, M. “Lyapunov stability of linear predictor feedback for time-varying input delay”, IEEE Transactions on Automatic Control, 55(2), 2010, 554–559. 28

[22] Mazenc, F., S. Mondie and R. Francisco, “Global Asymptotic Stabilization of Feedforward Systems with Delay at the Input”, IEEE Transactions on Automatic Control, 49, 2004, 844850. [23] Mazenc, F. and P.-A. Bliman, “Backstepping Design for Time-Delay Nonlinear Systems”, IEEE Transactions on Automatic Control, 51, 2006, 149-154. [24] Mazenc, F., M. Malisoff and Z. Lin, “Further Results on Input-to-State Stability for Nonlinear Systems with Delayed Feedbacks”, Automatica, 44, 2008, 2415-2421. [25] Morin, P., J.-B. Pomet and C. Samson, “Design of Homogeneous Time-Varying Stabilizing Control Laws for Driftless Controllable Systems Via Oscillatory Approximation of Lie Brackets in Closed Loop”, SIAM Journal on Control and Optimization, 38(1), 1999, 22-49. [26] Naghshtabrizi, P., J. Hespanha and A. R. Teel, “Stability of Delay Impulsive Systems With Application to Networked Control Systems”, Proceedings of the 26th American Control Conference, New York, U.S.A., 2007. [27] Nešić, D., A.R. Teel and P.V. Kokotovic, “Sufficient Conditions for Stabilization of Sampled-Data Nonlinear Systems via Discrete-Time Approximations”, Systems and Control Letters, 38(4-5), 1999, 259-270. [28] Nešić, D. and A.R. Teel, “Sampled-Data Control of Nonlinear Systems: An Overview of Recent Results”, Perspectives on Robust Control, R.S.O. Moheimani (Ed.), Springer-Verlag: New York, 2001, 221-239. [29] Nešić, D. and A. Teel, “A Framework for Stabilization of Nonlinear Sampled-Data Systems Based on their Approximate Discrete-Time Models”, IEEE Transactions on Automatic Control, 49(7), 2004, 1103-1122. [30] Nešić, D. and D. Liberzon, “A unified framework for design and analysis of networked and quantized control systems”, IEEE Transactions on Automatic Control, 54(4), 2009, 732-747. [31] Nešić, D., A. R. Teel and D. Carnevale, “Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems”, IEEE Transactions on Automatic Control, 54(3), 2009, 619-624. [32] Pomet, J.-P., “Explicit Design of Time-Varying Stabilizing Control Laws for a Class of Controllable Systems Without Drift", Systems and Control Letters, 18(2), 1992, 147-158. [33] Sepulchre, R., M. Jankovic, and P. Kokotovic, Constructive Nonlinear Control. New York: Springer, 1997. [34] Tabbara, M., D. Nešić and A. R. Teel, “Networked control systems: emulation based design”, in Networked Control Systems (Eds. D. Liu and F.-Y. Wang) Series in Intelligent Control and Intelligent Automation, World Scientific, 2007.

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Appendix Proof

of

Claim

1

in

the

(

( x 0 , z 0 , u 0 ) ∈ C 0 ([−r ,0]; ℜ n ) × ℜ n ×L ∞ [−r − τ ,0]; ℜ m ( x(t ), z (t ), u (t )) ∈ ℜ n × ℜ n × ℜ m

proof

)

be

of

arbitrary

Theorem and

2.1:

consider

the

Let solution

of the closed-loop system (2.5), (2.6), (1.3), (1.1) with initial

condition z (t 0 ) = z 0 ∈ ℜ n , Tr +τ (t 0 )u = u 0 ∈L∞ ([− r − τ ,0]; ℜ m ) , Tr (t 0 ) x = x 0 ∈ C 0 ([−r ,0]; ℜ n ) . It is crucial to notice that the solution of z&(t ) = f ( z (t ), k (t + τ , z (t ))) with z (t 0 ) = z 0 ∈ ℜ n satisfies z (t ) = ξ (t + τ ) , where ξ (s) is the solution of ξ&( s ) = f (ξ ( s ), k ( s, ξ ( s ))) with ξ (t 0 + τ ) = z 0 ∈ ℜ n . Inequality (2.4) implies that

the solution z (t ) ∈ ℜ n exists for all t ∈ [t 0 , t 0 + T ) and that the following inequality holds z (t ) ≤ σ ( z 0 , t − t 0 ) , ∀t ∈ [t 0 , t 0 + T )

(A1)

It follows that the solution of (2.6) exists for all t ∈ [t 0 , t 0 + T ) . Continuity of k ∈ C 1 (ℜ + × ℜ n ; ℜ m ) and inequalities (2.3), (A1) imply that the mapping t → u (t ) is continuous on (t 0 , t 0 + T ) and bounded with lim+ u (t ) = k (t 0 + τ , z 0 ) and t →t 0

that the limit z ∗ = [t 0 , t 0 + T ) .

lim

t → ( t 0 +T ) −

z (t )

lim

t → ( t 0 +T ) −

u (t ) = k (t 0 + T + τ , z ∗ ) ,

where z ∗ =

lim

t → ( t 0 +T ) −

z (t ) .

Notice

exists by virtue of uniform continuity of the mapping t → z (t ) on

By virtue of inequalities (2.3) and (A1) we obtain the inequality sup

t 0 −τ − r ≤ s