Stability test for real-time control systems using

Stability test for real-time control systems using interval algebra Manel Velasco, Pau Mart´ı, Ricard Vill`a and Josep M. Fuertes Automatic Control Department, Technical University of Catalonia Pau Gargallo, 5, 08028 Barcelona, Spain {manel.velasco,pau.marti,ricard.villa,josep.m.fuertes}@upc.edu

Abstract— In a real-time system, executing successive jobs of a controller task, subject to sampling and latency jitter, may bring the controlled plant to instability. Hence, stability analysis techniques are required to prevent run-time system failures. First, in this paper, we will represent the effects of jitters as non-static parametric uncertainties in the discretetime closed-loop model. Then we will apply robust control techniques to study system stability. We will present a sufficient stability test based on interval algebra methods, that allows assessing the stability of a closed-loop system, whose controller jobs suffer jitters, by simply evaluating a time invariant matrix.

We will show that just checking the stability of one timeinvariant matrix is enough to guarantee stability of the original system. The rest of the paper is organized as follows. Section II motivates the stability test we present. In III the problem to be solved is analyzed. The model that wraps the timevarying system is described in IV. Section V presents the stability test and illustrates its applicability with an academic example. Finally, conclusions are given in VI. II. M OTIVATION

I. I NTRODUCTION In a real-time system, executing successive jobs of a controller task, subject to sampling and latency jitter, may bring the controlled plant to instability [1]. This situation appears because, at the controller design stage, the specific timing of the computer implementation is not usually considered. For example, controllers are designed assuming a constant sampling period h and time delay τ [2]. But at run-time, sampling instants may not be equidistant and time delays (i.e., time elapsed from sampling to actuation) may vary. In a real-time multitasking system, this is mainly caused by the scheduling policy (i.e., a set of rules that determine the serialization of task-job executions [3]). This can not ensure constant inter-sampling and execution times (i.e., delays) for all control tasks in the system. Such effects are known as sampling and latency jitter, respectively [4]. However, for a given control task, the sampling period and time delay are bounded by a maximum and minimum values (as we will explain in greater detail in Section III). As a consequence, h and τ can be considered bounded uncertain time-varying parameters. Then, the discrete-time closed-loop model used in the controller design stage is no longer time-invariant, and can be treated as an interval model [5] that wraps the original. By applying robust control techniques to the wrapping model we present a sufficient stability test that allows us to assess the stability of a closed-loop system whose controller jobs suffer jitters. From the wrapping system, we construct a looser interval model that simplifies the stability test. This work is partially supported by Spanish Ministerio de Ciencia y Tecnologia Project ref. DPI2002-01621.

The problem of studying the stability of a plant whose controller is subject to jitters can be treated using different techniques. For example, in [6] stability conditions (based on linear algebra [7]) are established in terms of the spectral radius of the set of matrices that can appear in the system dynamics according to its time-varying model (when jitters are considered). Similarly, [8] presents stability conditions based on switching control theory that are also applicable. Both methods, which require to know offline (at the controller design stage) the exact values for the sampling period and time delay that will apply at run-time, are useful when the number of matrices to account for is relatively small. If the number of matrices increases (which depend on the set of values for h and τ ), the computation complexity of those stability tests may prevent their application. An alternative approach is to use probabilistic techniques. For example, in [9], irregular sampling and varying time delays are considered as random variables with known expectation. With this model, the system evolution can be seen as a sequence of random state vectors generated by the system closed-loop matrix. Accordingly, stability conditions are presented in terms of convergence of a sequence of random variables. This kind of approach eliminates the need for knowing the exact values for h and τ , but still requires knowing their expectations, which may not be possible in the controller design stage. In this paper, we have approached the stability problem using interval algebra methods. Interval arithmetic extends computation on real numbers to intervals in a natural and intuitive way and is the natural tool to use when dealing with interval models. The basis for this applied mathematical tool can be found in [10] or [11].

Since the theorem of Kharitonov on robust stability of interval polynomials appeared ([12] and [13]), a number of papers on robustness analysis of uncertain systems have been published in the past few years [14], [15], [16], [17], [18], [19], and [20]. All these methods to study stability are for systems with static parametric uncertainties. Therefore, they are not suitable for our case, because our system model presents time-varying uncertainties, that is, the specific value of each parameter changes at each controller job execution (see [21] for an explanation of static versus time-varying uncertainties for dynamics systems described by intervals models, in the context of simulation).

the closed-loop system requirements, the system specified by (8) can be controlled using state feedback (9), where L(h0 , τ0 ) is the gain matrix.   x(kh) (9) u(kh) = −L(h0 , τ0 ) z(kh)

The discrete-time closed-loop model to be considered will include the effects of sampling and latency jitter.

At the end, the closed-loop time invariant system (given by equation (8) with the control signal given by (9)) is characterized by a closed-loop matrix (10) that depends on Φ(h), Γ0 (h, τ ) and Γ1 (h, τ ) (constant matrices in terms of the period h and time delay τ ), and on the constant matrix gain L(h0 , τ0 ) expressed in terms of fixed h 0 and τ0 .   Φ(h) Γ1 (h, τ ) Φcl (h, τ ) = − 0 0   Γ0 (h, τ ) (10) L(h0 , τ0 ) I

A. System representation in the absence of jitters

B. Jitters in job executions

Let (1) and (2) be the system equations that describe the linear time invariant dynamics of the controlled plant.

(5)

In a real-time system, each job of a feasible periodic task executes within its period (h 0 ). If the sampling is performed at the beginning of each job execution, the sampling given by a feasible control task differs from the timing expected by discrete-time control theory: given the nominal period h0 , the k th job of a control task should sample at kh 0 whereas with the scheduling practice, it can start sampling at any time t ∈ [kh0 , kh0 + h0 ). Therefore, the actual (real) sampling (hr ) of a controller vary (i.e., sampling jitter) and does not comply with the assumptions taken in the controller design stage (h 0 ). Similarly, if the actuation is performed at the end of each job execution, variation in job execution times c (for example, due to preemptions of other higher priority tasks) causes variation in the sample-to-actuation latency (i.e., latency jitter). The actual (real) latency (τ r ) will differ from the delay assumed in the design of the controller (τ 0 ). For an extended description of jitters in control job executions due to real-time scheduling methods, see [22].

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C. Modeling jitters as discrete sets

III. P ROBLEM FORMULATION

x˙ = Ax(t) + Bu(t − τ )

t ∈ R+

y(t) = Cx(t) + Du(t)

(1) (2)

Equation (1) includes a time delay, τ , that will correspond to the execution time of each controller job. To meet realtime scheduling feasibility constraints, τ is assumed to be less than or equal to the sampling period h that will be used in the design of the discrete-time controller. In the sampling instants, systems represented by (1) and (2) can be described by their discrete-time model (equations (3) and (4)) using transformations (5), (6) and (7) [2]. x(kh+h) = Φ(h)x(kh)+Γ0 (h, τ )u(kh)+Γ1 (h, τ )u(kh−h) (3) y(kh) = Cx(kh) + Du(kh) (4) Φ(h) = eAh
h−τ Γ0 = eAs ds B 0

Γ1 = e

A(h−τ




τ

e

As

ds B

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0

A state space model of (3) and (4) is given by (8), where an extra state variable, z(kh) = u(kh−h), which represents the past values of the control signal, is introduced [2].      x(kh) x(kh + h) Φ(h) Γ1 (h, τ ) + = 0 0 z(kh) z(kh + h)   Γ0 (h, τ ) u(kh) (8) I For each control task in a real-time system, a nominal (fixed) sampling period, h 0 , is specified, which corresponds to the task period, and is equal to the period used in the controller design. Let us call τ 0 the nominal (fixed) time delay also used when designing the controller. To meet

Extreme situations for sampling and latency jitter are when: • the separation between two successive jobs is the minimum when a job finishes at the end of its period, kh0 + h− 0 , while the following job starts executing at the beginning of its period, (k + 1)h 0 . The real intersampling time is given by (11). hr = c •

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the separation between two successive jobs is the maximum when a job starts executing at the beginning of its period, kh 0 , while the following job finishes at the end of its period, (k + 1)h 0 + h− 0 . The real intersampling time is given by (12). hr = 2h0 − c

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•

the execution time of a job is the maximum, that is, it lasts all the period. The real latency is given by (13). τr = h0

•

(13)

the execution time of a job is the minimum, that is, the one assumed in the controller design stage. The real latency is thus given by (14). τr = τ0

(14)

Therefore, jitters are bounded and can be specified using the (discrete) sets given by (15) and (16). Note that the number of possible values for h r and τr depends on the processor clock granularity g. hr ∈ {hmin , . . . , hmax }

hmin = c, hmax = 2h0 − c hr = k · g, k ∈ N

τr ∈ {τmin , . . . , τmax }

(15)

τmin = τ0 , τmax = h0 τr = k · g, k ∈ N (16)

F. Problem statement In the controller design stage, the exact sequence of parameters hi and τi that will apply at run-time in (18) is not known. Therefore, applying stability results from switching control theory (e.g., by assessing the stability of the set of matrices [8]) may not be desirable because two requirements must be fulfilled: 1) To know the processor speed in order to determine the overall possible values for h r and τr 2) To compute all possible closed-loop matrices in order to be able to study system stability. And this information may not be available at the controller design stage or may require intensive computations (with a polynomial cost). Since the closed-loop dynamics are expressed in terms of (17) and sets (15) and (16), we can transform these sets into intervals and apply interval algebra methods. This becomes then a more natural approach to tackle the stability analysis problem. IV. I NTERVAL MODEL

D. System representation including jitter effects By including sets (15) and (16) in the closed-loop matrix specified in (10), the dynamic system will be given by the closed-loop matrix specified in (17).   Φ(hr ) Γ1 (hr , τr ) Φcl (hr , τr ) = − 0 0   Γ0 (hr , τr ) (17) · L(h0 , τ0 ) I Remark 1: In (17), the controller gain L, is the one obtained in the controller design stage supposing h 0 constant, and will not be changed at runtime regardless of jitters on h0 . Model (17) must be considered as a system that depends on uncertain parameters that are time-varying. Hence, it is expressed in terms of h r and τr . E. Closed-loop system dynamics The dynamics of the closed-loop system depend on the multiplication of a sequence of closed-loop matrices (17) taken from the set of matrices that can be obtained according to the sets of values for the sampling period (15) and for the time delay (16). Considering that the matrix gain is time-invariant, the system evolution is given by (18) xk+n = Φcl (hk+n , τk+n ) . . . Φcl (hk+1 , τk+1 )Φcl (hk , τk )xk (18) where hi ∈ {hmin , . . . , hmax }, i ∈ {k . . . k + n};

(19)

τj ∈ {τmin , . . . , τmax }, j ∈ {k . . . k + n};

(20)

To transform sets (15) and (16) into intervals, we use transformations (21) and (22). That is, we choose the worst case scenario where h r and τr can take any value within the intervals given by the extreme values of those sets. g→0

hr ∈ {hmin , . . . , hmax } −→ hr ∈ [hmin , hmax ] g→0

τr ∈ {τmin , . . . , τmax } −→ τr ∈ [τmin , hmax ]

(21)

(22)

Afterwards, we construct a new closed-loop system model where we substitute the functional relation of each closed-loop matrix coefficient in (17) by the maximum and minimum values of each of these functions (that depend on hr and τr ) evaluated using intervals (21) and (22). The new closed-loop matrix model will be denoted by [Φ cl ], and each of its elements is a new closed interval φ ij = [φij , φij ], as specified in (23). φij = min f (hr , τr ) and φij = max f (hr , τr ) (23) with hr ∈ [hmin , hmax ], τr ∈ [τmin , hmax ] The new system dynamics will be given by (24). [xk+1 , xk+1 ] = [Φcl ]xk

(24)

Remark 2: To study the stability of system (24) is not n def n enough to evaluate [Φ cl ]n = [[Φcl ] , [Φcl ] ]. This could be done if the uncertain parameters h r and τr of our system were static. However in our model, they are time-varying. Therefore, due to the multi-incidence property of intervals

with varying parameters 1 , a different approach to assess system stability is required. Remark 3: The interval system specified in (24) is not equivalent to the original system whose dynamics were given by (17), (15) and (16). However, one of the possible dynamics of the interval system will coincide with the dynamics of the original system. Therefore, the interval system is more generic, and wraps the original. Although this may introduce some pessimism in the stability analysis, it highly simplifies the analysis. Remark 4: Since the interval system wraps the original system, instability of the interval system does not imply instability of the original system. Therefore, the test that we present next is sufficient though not necessary. V. S TABILITY TEST The stability test is based on studying the stability of a time invariant matrix Θ that is the upper-bound of an interval obtained when looking at the convergence of a looser interval system [Θ]. The later is constructed from the interval system [Φcl ] (given by (23)), which wraps the original system Φcl (hr , τr ) (equations (17), (15) and (16)). Lema 1: Stability of a wrapping system implies the stability of any of the systems that it includes. If we denote the set of dynamics of a system [A] (original system) by (25) k=∞ 

χ = {[

Ak ]x0 | Ak ∈ [A]}

(25)

k=1

and the set of dynamics of a system [B] (wrapping system) by (26) k=∞ 

Ξ = {[

Bk ]x0 | Bk ∈ [B]},

(26)

k=1

we can state that if [A] ∈ [B], then χ ∈ Ξ. And according to Lema 1, if [B] is stable, then [A] is stable. Definition 1: An interval system [A] is included in [B] if ∀i, j: aij ≥ bij and aij ≤ bij (27) where aij and aij are, respectively, the maximum and minimum values that can take the elements of the matrix under analysis and bij and bij are their respective equivalents in the wrapping system. Lema 2: Given an interval [a, b], we can construct another interval including the former and symmetric respect 1 It

is well known in interval algebra that for an interval def [A] = [a, a] with static uncertainties, [A]k = [ak , ak ]. = However, if [A] has time-varying uncertainties, [A]k def [a, a][a, a] = [min(a a, a a, a a, a a), max(a a, a a, a a, a a)] which may differ from [ak , ak ]. To illustrate the difference let us simply compute [−2, 3]2 and [−2, 3][−2, 3]: [−2, 3]2 = [4, 9], whereas [−2, 3][−2, 3] = [−6, 9]. Therefore, with time varying parameters, the computation of the power of an interval matrix becomes much more complex.

to 0 as follows: we choose the biggest absolute value of the original interval extremes and construct the newer interval: if abs(a) > abs(b) → [a, b] ∈ [−abs(a), abs(a)] if abs(a) < abs(b) → [a, b] ∈ [−abs(b), abs(b)] Theorem 1: The system defined by (17), (15) and (16) is stable if matrix Θ, whose elements are θ ij = max(abs(φij ), abs(φij )), is stable. Proof: To prove the theorem we will start by constructing an interval matrix called [Θ] whose upper bound is Θ. We will show that [Φcl ] ∈ [Θ]. We construct [Θ] by applying Lema 2 to each element of the original system matrix [Φ cl ]. The elements of the looser wrapping system [Θ] are specified by (28). θij = [θij , θij ] =

(28)

= [−max(abs(φij ), abs(φij )), max(abs(φij ), abs(φij ))] As explained in Definition 1 and according to Lema 2, [Φcl ] ∈ [Θ]. Now we will show that the stability of [Θ] is equivalent to the stability of Θ, and according to Lema 1, this will imply that [Φcl ] is stable. Therefore, since Φ cl (hr , τr ) ∈ [Φcl ], the stability of Φcl (hr , τr ) will have been proved. Each element of [Θ] is a symmetric interval respect to 0. To simplify notation in equation (28), we will use θ ij = [−aij , aij ], where aij = max(abs(φij ), abs(φij )). To evaluate the stability of the system dynamics given by [Θ], we check whether the infinite product of the interval matrices tends toward 0. The first product [Θ][Θ], that we denote by [Θ] 2 (as specified in (29)), has the structure specified in (30). Remember that by construction, all bounds in each interval in matrix [Θ] are equal (due to its symmetry) in absolute value but with different sign. Hence, when performing the product of two intervals, the upper bound of the resulting interval will be the product of each of the former upper bounds. And the lower bound will be the product of the lower bound of the first interval by the upper bound of the second interval (or equivalent, the upper bound of the first interval and the lower bound of the second interval). Therefore, the equation (31) holds. We express (31), which is a matrix of intervals, as an interval of matrices, as specified in (32). Note that any element of the upper bound matrix of (32) is an upper bound of any of the intervals of matrix (31). Similarly for lower bounds. Interval (32) can be reordered as in (33), where each bound matrix in (32) can be expressed as a product of identical matrices. Interval (33) can be rewritten as in (34) It can be seen in (34) that the product of the interval matrices gives an interval with squared power matrices in its bounds. 2 In fact, [Θ]2 = [−(Θ2 ), Θ ], where the symmetry respect to 0 has been kept. The same transformation that goes from a matrix of intervals to an interval with matrices in its bounds (equations ((29),(30),(31),(32),(33) and (34))) can be extended

⎤⎡ [−a1n , a1n ] [−a11 , a11 ] · · · ⎥⎢ .. .. .. ⎦⎣ . . . [−ann , ann ] [−an1 , an1 ] · · ·

⎡

[−a11 , a11 ] · · · ⎢ .. .. [Θ] [Θ] = [Θ]2 = ⎣ . . [−an1 , an1 ] · · · ⎡

n 

⎢ i=1 ⎢ ⎢ [Θ]2 = ⎢ ⎢ n ⎣  i=1

[−a1i , a1i ] [−ai1 , ai1 ]

n 

⎢ i=1 ⎢ ⎢ [Θ]2 = ⎢ ⎢  n ⎣  i=1

⎢⎢ ⎢⎢ ⎢⎢ [Θ]2 = ⎢⎢ ⎢⎢ ⎣⎣

⎤ [−a1i , ain ] [−ai1 , ain ] ⎥ i=1 ⎥ ⎥ .. ⎥ . ⎥ n ⎦  [−ani , ani ] [−ain , ain ]

.

[−ani , ani ] [−ai1 , ai1 ] · · ·

⎡ 

⎡⎡

..

−a1i ai1 , .. . −ani ai1 ,

n  i=1 n  i=1

 ···

 ani ai1

..

.

···

n  i=1



n  i=1

−a1i ain , .. . −ani ain ,

n  i=1 n  i=1

 ⎤ a1i ain

ani ain

⎥ ⎥ ⎥ ⎥  ⎥ ⎦

⎤ ⎡ n n   −a1i ain ⎥ ⎢ a1i ai1 · · · a1i ain i=1 i=1 i=1 ⎥ ⎢ i=1 ⎥ ⎢ .. . . .. .. . . .. ⎥,⎢ . .. . .. ⎢ ⎥ n n n n    ⎦ ⎣  −ani ai1 · · · −ani ain ani ai1 · · · ani ain n 

i=1

−a1i ai1 · · ·

n 

i=1

to the product of k matrices [Θ], [Θ] k = [Θ][Θ] . k. . [Θ], as specified in (35). As it can be seen in equation (35), k k [Θ]k = [−(Θk ), Θ ]. Therefore, if Θ tends toward 0 , k that is, if Θ is stable, the interval [−(Θk ), Θ ] will also k tend toward 0 (due to its symmetry, Θk = Θ ), implying stability of [Θ]. Corollary 1: If Θ is stable, then the system given by equations (17), (15) and (16) will be stable. Remark 5: The application of the stability test can be summarized as follows: 1) we start by the original system whose discrete-time closed-loop matrix is given by Φ cl (hr , τr ) (17) with hr and τr in sets (15) and (16) 2) we construct an interval system [Φ cl ] (24) using (23) with bounded time-varying uncertainties given by (21) and (22), that wraps the original system. 3) we construct a looser interval system [Θ] as specified in (28). Theorem 1 states that to check the stability of [Θ] is equivalent to check the stability of matrix Θ (whose elements are defined in the theorem) because k Θ is the upper bound of the interval obtained at computing [Θ] k A. Example To illustrate the proposed method, let us take the academic example: a system whose closed-loop matrix Φ cl is given by (36).

(30)

i=1

 a1i ai1

(29)

n 

···

.. .

⎤ [−a1n , a1n ] ⎥ .. ⎦ . [−ann , ann ]

i=1

(31)

⎤⎤ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎦⎦

(32)

i=1

In (37) (which corresponds to eq. (17)) the nominal sampling period and time delay used in the controller design stage are h0 = 0.01s and τ0 = 0.0003s respectively. According to (15) and (16), h r ∈ {0.0003, . . . , 0.0197} and τr ∈ {0.0003, . . . , 0.01}, which transform to intervals hr ∈ [0.0003, 0.0197] and τr ∈ [0.0003, 0.01] using (21) and (22). From these intervals, and applying (23) we obtain the interval matrix [Φ cl ] specified in (37). ⎤ ⎡ [−0.8, −0.1]

[0.01, 0.2]

[0.05, 0.1]

⎥

⎢ [Φcl ] = ⎣ [−0.1, −0.09]

[−0.8, −0.5]

[−0.02, 0] ⎦ (37)

[0.002, 0.01]

[0.001, 0.09]

[0.5, 0.9]

Next, by using (28), we construct the wrapping matrix [Θ], given in (38), whose elements are symmetric intervals respect to 0. ⎤ ⎡ [−0.8, 0.8]

[−0.2, 0.2]

⎢ [Θ] = ⎣ [−0.1, 0.1]

[−0.8, 0.8]

[−0.01, 0.01]

[−0.09, 0.09]

[−0.1, 0.1]

⎥

[−0.02, 0.02] ⎦ [−0.9, 0.9]

(38)

And from (38) we obtain matrix Θ (39). ⎤ ⎡ 0.8

⎢ Θ = ⎣ 0.1 0.01

0.2

0.1

0.8

0.02 ⎦

0.09

0.9

⎥

(39)

Finally, applying Theorem 1, we can affirm that the system, whose closed-loop matrix is (36) and whose parameter variability is hr ∈ [0.0003, 0.0197] and τr ∈ [0.0003, 0.01],

⎤⎡ ⎤ ⎡ ⎤⎡ ⎤⎤ a11 · · · a1n a11 · · · a1n a11 · · · a1n a11 · · · a1n ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎥ ⎢ ⎢ .. . . .. .. . . .. .. . . .. .. . . .. [Θ]2 = ⎣− ⎣ ⎦⎣ ⎦, ⎣ ⎦⎣ ⎦⎦ . .. . .. . .. . .. an1 · · · ann an1 · · · ann an1 · · · ann an1 · · · ann ⎡ ⎡

⎡ ⎡ a11 ⎢ ⎢ ⎢ ⎢ a21 ⎢ [Θ]2 = ⎢− ⎢ . ⎢ ⎢ ⎣ ⎣ .. an1 ⎡ ⎡ a11 ⎢ ⎢ ⎢ ⎢ a21 ⎢ [Θ]k = ⎢− ⎢ . ⎢ ⎢ ⎣ ⎣ .. an1 ⎡

a12

···

a22 ..

.

···

a12

···

a22 .. ···

51.54 τr − 0.12 + 20.61 hr

⎢ Φcl (hr , τr ) = ⎣ 3.09 τr − 0.11 − 5314 hr 2 + 54.7 hr 341.4 τr 2 − 3.51 τr + 0.011

.

⎤2 ⎡ a1n a11 .. ⎥ ⎢ ⎢ . ⎥ ⎥ , ⎢ a21 ⎥ ⎢ . ⎦ ⎣ .. ann an1 ⎤k ⎡ a11 a1n .. ⎥ ⎢ ⎢ . ⎥ ⎥ , ⎢ a21 ⎥ ⎢ .. ⎦ ⎣ . ann an1

a12

···

a22 ..

.

···

a12

···

a22 .. ···

.

⎤2 ⎤ a1n .. ⎥ ⎥ ⎥ . ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎥ ⎦ ann ⎤k ⎤ a1n .. ⎥ ⎥ ⎥ . ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎥ ⎦ ann

(33)

(34)

(35)

9.79 hr + 0.00706

0.103 τr + 0.049 + 579.2 hr 2 + 1.35 hr

8.24 τr + 0.49 + 11.3 hr

10.6 hr 2 + 0.81 hr − 0.0202

7.21 τr + 0.0076 + 0.51 hr

4251 hr 2 − 85 hr + 0.92

is stable because the absolute values of the eigenvalues of Θ, 0.6675, 0.9905, and 0.8420, are less than 1. VI. C ONCLUSIONS We have presented a stability test for real-time control systems that permits to asses the stability of a closed-loop system whose controller job executions are subject to jitters. The proposed test is based on interval algebra methods, and reduce the stability analysis problem to study the stability of a single time invariant matrix although the original closedloop system model is time-varying. R EFERENCES [1] P. Marti, R. Vill`a, J. M. Fuertes, and G. Fohler, “On real-time control tasks schedulability,” in Proceedings of the European Control Conference, Porto, Portugal, Sept. 2001. ˚ om and B. Wittenmark, Computer-Controlled Systems. [2] K. J. Astr¨ Third Edition. Prentice–Hall, 1997. [3] G. Buttazzo, Hard Real-Time Computer Systems. Predictable Scheduling Algorithms and Applications. Kluwer Academic Publishers, 1997. ˚ en, A. Cervin, J. Eker, and L. Sha, “An introduction to [4] K.-E. Arz´ control and scheduling co-design,” in Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia, Dec. 2000. [5] R. E. Moore, “Methods and applications of interval analysis,” in SIAM studies in applied mathematics, 1979. [6] P. Marti, R. Vill`a, J. M. Fuertes, and G. Fohler, “Stability of on-line compensated real-time scheduled control tasks,” in IFAC Conference on New Technologies for Computer Control, Hong Kong, P.R. China, Nov. 2001. [7] G. Strang, Linear Algebra and its Applications. 2 Ed. MIT. Academic Press, Inc, USA, 1980. ¨ uner, “Stability of a set of matrices: A control [8] M. Dogruel and U. Ozg¨ theoretic approach,” in Proceedings of the 34th IEEE Conference on Decision and Control, Sept. 1995.

⎤ ⎥ ⎦

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