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Fiftieth Annual Allerton Conference Allerton House, UIUC, Illinois, USA October 1 - 5, 2012
Constructing ρ/µ Approximations From Input/Output Snapshots for Systems Over Finite Alphabets Danielle C. Tarraf Abstract— We consider discrete-time plants that interact with their controllers by sending and receiving binary valued sensor signals and finite valued control signals, respectively. In the absence of exogenous inputs, we propose a general procedure for constructing (finite state) ρ/µ approximations, starting from finite length sequences of input and output signal pairs. We show that the proposed construction satisfies desirable properties, thus leading to a hierarchy of finite state models that can be used for certified-by-design control synthesis.
I. I NTRODUCTION High fidelity models that describe a dynamical system to a high degree of accuracy are often too complex for use in controller synthesis, particularly when the underlying dynamics of the plant are hybrid [2], [4], [5] and/or subject to quantization effects. Accordingly, the problem of approximating hybrid systems by simpler systems has been receiving much attention over the past two decades [1], [3], [29]. In particular, the problem of constructing finite state approximations of hybrid systems has been the object of intense study, due to the amenability of finite state models to analysis and control synthesis [7], [17], [27], [28]. One set of approaches make use of non-deterministic finite state automata constructed so that their input/output behavior contains that of the original model (these approximations are sometimes referred to in the literature as ‘qualitative models’) [9], [10], [16]. Controller synthesis can then be formulated as a supervisory control problem, addressed using the Ramadge-Wonham framework [18]. More recently, progress has been made in reframing these results [12], [13] in the context of Willems’ behavioral theory and lcomplete systems [30]. Another set of approaches, influenced by the theory of bisimulation in concurrent processes [11], [14], make use of bisimulation and simulation abstractions of the original plant. These approaches effectively ensure that the set of state trajectories of the original model is exactly matched by (bisimulation), contained in (simulation), matched to within some distance ϵ by (approximate bisimulation), or contained to within some distance ϵ in (approximate simulation), the set of state trajectories of the finite state abstraction [6], [15], [19], [20]. The performance objectives are typically formulated as constraints on the state trajectories of the original hybrid system, and controller synthesis is a two step procedure: A finite state supervisory D. C. Tarraf is with the Department of Electrical and Computer Engineering at the Johns Hopkins University, Baltimore, MD 21218, USA
(dtarraf)@jhu.edu
978-1-4673-4539-2/12/$31.00 ©2012 IEEE
controller is first designed, and subsequently refined to yield a certified hybrid controller for the original plant [21]. In a recent effort [22], we proposed a notion of approximation for ‘systems over finite alphabets’, namely discretetime plants that interact with their feedback controllers by sending and receiving signals taking their values in fixed, finite alphabet sets. We refer to this notion of approximation as a ‘ρ/µ approximation’, to highlight its affinity and compatibility with the analysis [26] and synthesis [27] tools previously developed for systems whose properties and performance objectives are described in terms of ρ/µ gain conditions. This new notion of approximation differs from the existing notions of qualitative approximations and simulation/bisimulation abstractions in several important respects, discussed in [22]. To summarize, one critical difference is the non-trivial state estimation problem inherent in our setup; Indeed, abstraction based methods effectively address full-state feedback problems. Another critical difference is the emphasis on quantifying the quality of approximation; Indeed, as their name indicates, qualitative models lack this quantitative characterization. It should be emphasized that the proposed notion of ρ/µ approximation explicitly identifies those properties that the approximate models need to satisfy in order to enable certified-by-design controller synthesis. However, it does not restrict us to a particular constructive algorithm, rather leaving the question of construction completely open. In particular, an intuitive state-space based construction was employed in the simple illustrative example presented in [22]. In this paper, we propose and analyze a new constructive procedure for generating ρ/µ approximations of a given plant and performance objective. In contrast to the statespace based construction, which was specifically tailored to the dynamics in question, the present construction is widely applicable to arbitrary plants over finite alphabets, provided they are not subject to exogenous inputs, and provided their outputs are a function of the state only (i.e. what would be analogous to ‘strictly proper’ transfer functions in the LTI setting). Our proposed construction effectively keeps track of finite length sequences of input-output signal pairs of the plant. The idea of using finite length sequences of inputs and outputs (innovations) is of course widely employed in system identification [8]. However, the setup of interest to us is fundamentally different: Indeed, the dynamics of the plant are assumed to be exactly known. Additionally, the data is complete and exact (i.e. uncorrupted by noise). Likewise, our construction bears some resemblance to algorithms employed
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in constructing qualitative models, surveyed earlier in our introduction. However, our notion of approximation is fundamentally different from the notion of qualitative models, as it seeks to explicitly quantify the approximation error in the spirit of robust control. The proposed construction differs significantly from our preliminary efforts reported in [23] in at least two important respects: First, it approximates the performance objectives as well as the dynamics of the plant. Second, it immediately leads to a finite nominal model with deterministic transitions, in contrast to [23] in which a nondeterministic model is contructed and the non-deterministic transitions are subsequently dropped and accounted for in quantifying the ‘worst-case’ approximation error. Finally, our proposed construction differs from our very early effort reported in [24] in several respects, the most significant of which is the structure of the corresponding approximation error. Organization of the Paper: We begin in Section II by reviewing the notion of ρ/µ approximation as well as some basic concepts that are relevant to our development. We describe the problem setup and state our objective in Section III. We revisit a previously proposed structure [25] in Section IV, and show (under mild assumptions) that it ensures desirable properties regardless of the choice of nominal model. We present our construction in Section V, and we conclude in Section VI with some directions for future work. Notation: Z+ , R+ denote the non-negative integers and non-negative reals, respectively. Given a set A, AZ+ and 2A denote the set of all infinite sequences over A (indexed by Z+ ) and the power set of A, respectively. The cardinality of a (finite) set A is denoted by |A|. Elements of A and AZ+ are denoted by a and (boldface) a, respectively. For a ∈ AZ+ , a(i) denotes its ith term. For f : A → B, C ⊂ B, f −1 (C) = {a ∈ A|f (a) ∈ C}. II. P RELIMINARIES
gain stable if there exists a finite non-negative constant γ such that inf
T ≥0
T ∑
γρ(u(t)) − µ(y(t)) > −∞.
(1)
t=0
is satisfied for all (u, y) in S. In particular, when ρ, µ are non-negative (and not identically zero), the ‘gain’ can be defined. Definition 2. Consider a system S ⊂ U Z+ × Y Z+ . Assume that S is ρ/µ gain stable for ρ : U → R+ and µ : Y → R+ , and that neither function is identically zero. The ρ/µ gain of S is the infimum of γ such that (1) is satisfied. In this paper, we are specifically interested in plants that are not subject to exogeneous inputs, and that are equipped with finite-valued actuators and sensors: Definition 3. A system over finite alphabets S is a discretetime system S ⊂ U Z+ × (Y × V)Z+ whose alphabets U and Y are finite. Here, u ∈ U Z+ represents the control input to the plant while y ∈ Y Z+ and v ∈ V Z+ represent the sensor and performance outputs of the plant, respectively. The plant dynamics may be analog, discrete or hybrid, and thus V may be finite, countable or infinite. The approximate models of the plant will be drawn from a specific class of models, namely deterministic finite state machines: Definition 4. A deterministic finite state machine (DFM) is a discrete-time system S ⊂ U Z+ × Y Z+ with finite alphabets U, Y, whose feasible input and output signals (u, y) are related by q(t + 1) = y(t) =
f (q(t), u(t)) g(q(t), u(t))
where t ∈ Z+ , q(t) ∈ Q for some finite set Q and functions f : Q × U → Q and g : Q × U → Y.
In our development, it is often convenient to view a discrete-time dynamical system as a set of feasible signals, even when a state-space description of the system is available. We thus begin this section by briefly reviewing this ‘feasible signals’ view of systems. We then present the recently proposed notion of ρ/µ approximation, slightly modified for the class of systems of interest (systems with no exogenous inputs), and state a relevant control synthesis result.
It is understood here that Q, f and g represent the set of states of the DFM, the state transition map, and the output map, respectively, in the traditional state-space sense. Given a system P ⊂ U Z+ × (Y × V)Z+ and signals uo ∈ U Z+ , yo ∈ Y Z+ , we use P |uo ,yo to denote the subset of feasible signals of P whose first component is uo and whose second component is yo . That is {( ) } P |uo ,yo = u, (y, v) ∈ P u = uo and y = yo .
A. Systems and Performance Specifications
B. ρ/µ Approximations for Control Synthesis
A discrete-time signal is an infinite sequence over some prescribed set, referred to as an alphabet. A discrete-time system S is a set of pairs of signals, S ⊂ U Z+ × Y Z+ , where U and Y are given alphabets. In this setting, system properties of interest are captured by means of ‘integral’ constraints on the feasible signals:
The following definition applies to the class of plants of interest: Definition 5. (Adapted from Definition 6 in [22]) Consider a system over finite alphabets P ⊂ U Z+ × (Y × V)Z+ and a desired closed loop performance objective
Definition 1. Consider a system S ⊂ U Z+ × Y Z+ and let ρ : U → R and µ : Y → R be given functions. S is ρ/µ
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inf
T ≥0
T ∑ t=0
−µ(v(t)) > −∞ ⇔ sup
T ∑
T ≥0 t=0
µ(v(t)) < ∞
(2)
w
z
i u
v
P y u
Fig. 1.
)
ψi P |u,y ⊆ Pˆi |u,y Wednesday, March 7, 12
for all (u, y) ∈ U Z+ × Y Z+ , where Pˆi ⊂ U Z+ × (Y × ˆ i and ∆i as Vˆi )Z+ is the feedback interconnection of M shown in Figure 1. (b) For every feasible signal (u, (y, v)) ∈ P , we have µ(v(t)) ≤ µ(ˆ vi+1 (t)) ≤ µ(ˆ vi (t)), (3) ( ) for all t ∈ Z+ , where (u, (ˆ yi , v ˆi )) = ψi (u, (y, v)) ( ) and (u, (ˆ yi+1 , v ˆi+1 )) = ψi+1 (u, (y, v)) . (c) ∆i is ρ∆ /µ∆ gain stable, and moreover, the corresponding ρ∆ /µ∆ gains satisfy γi ≥ γi+1 . We complete our review of preliminaries by reviewing a previously derived result relevant to the problem of control synthesis. This highlights the fact that a ρ/µ approximation of the plant together with an appropriately defined performance objective may be used to synthesize certifiedby-design controllers for the original plant and original performance objective: Theorem 1. (Adapted from Theorems 1 and 3 in [22]) ˆ i } as in Consider a plant P and a ρ/µ approximation {M Definition 5. If for some index i, there exists a controller K ⊂ Y Z+ × U Z+ such that the feedback interconnection of ˆ i and K, (M ˆ i , K) ⊂ W Z+ × (Vˆi × Z)Z+ , satisfies M inf
T ≥0
T ∑
yˆ
A finite state approximation of P
ˆ i }∞ of for given function µ : V → R. A sequence {M i=1 Z+ ˆ deterministic finite state machines Mi ⊂ (U × W) × (Y × Vˆi × Z)Z+ with Vˆi ⊂ V is a ρ/µ approximation of P if there exists a corresponding sequence of systems {∆i }∞ i=1 , + ∆i ⊂ Z Z ×W Z+ , and non-zero functions ρ∆ : Z Z+ → R+ , µ∆ : W Z+ → R+ , such that for every i: (a) There exists a surjective map ψi : P → Pˆi satisfying (
vˆ
ˆi M
τ µ∆ (w(t)) − µ(ˆ v (t)) − τ γi ρ∆ (z(t)) > −∞ (4)
III. P ROBLEM S ETUP Consider a discrete-time plant P described by x(t + 1)
f (x(t), u(t)) g(x(t)) h(x(t))
(5)
where t ∈ Z+ , x(t) ∈ Rn , u(t) ∈ U, y(t) ∈ Y and v(t) ∈ V. No apriori constraints are placed on alphabet set V: It may be a Euclidean space, the set of reals, or a countable or finite set. U and Y are given finite alphabets with |U| = m and |Y| = 2, respectively: They may represent quantized values of analog inputs and outputs, or may simply be symbolic inputs and outputs in general. Functions f : Rn × U → Rn , g : Rn → Y and h : Rn → V are given. Our objective in this paper is to propose a systematic procedure for constructing a ρ/µ approximation of a given plant P as in (5) and a performance objective as in (2). IV. E NSURING E XISTENCE OF ψi : A S PECIAL S TRUCTURE In [25], a special ‘observer-inspired’ structure was proposed and used in conjunction with a particular finite state machine construction (distinct from the presently proposed construction) in order to approximate and design controllers for a special class of systems, namely switched second order homogenous systems with binary outputs. In this section, we revisit this structure: We show that under mild assumptions, this structure ensures the existence of function ψi with the properties required in part (a) of Definition 5. Lemma 1. Consider a system P ⊂ U Z+ ×(Y ×Z)Z+ where U and Y are finite, and where |Y| = 2. Also consider the structure shown in Figure 2, where { 0 if y = y˜ β(y, y˜) = 1 otherwise {
and
t=0
α(y, y˜) =
for some τ > 0, then the feedback interconnection of P and K, (P, K) ⊂ V Z+ , satisfies (2).
=
y(t) = v(t) =
y˜ y˜′
if w = 0 otherwise
with y˜′ denoting the unique element of Y that is not equal to y˜. For any deterministic finite state machine Mi with fixed initial condition qo , there exists a ψi : P → Pˆi , where Pˆi is
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dnesday, March 7, 12
performance outputs of P :
v
P
y
q(t) = y(t − i) u(t − 1) .. . u(t − i)
w vˆ
Mi
vˆ(q) Y(q) y(t − 1) .. .
y˜
i
vˆ yˆ
ˆ)i and ∆i as shown, such that ψi is the interconnection ( of M surjective and ψi P |u,y ⊆ Pˆi |u,y .
Note that since the control input and sensor output alphabet sets are finite, the corresponding number of feasible inputoutput sequences of length i is also finite. The number of states of the nominal model is consequently finite by construction, and the choice of sensor output set and performance output bound follows from the choice of inputoutput sequence. As will be explained in detail, this particular interpretation of the state leads to a finite state machine with deterministic state transitions, deterministic performance output, and a sensor-like output that is set-valued in general. The relevant output function is then ‘determinised’ and the introduced error is accounted for in a worst case sense.
Proof: The proof is by construction. Consider ψ1,i : P → (U × Y) × (Y × Vˆi ) defined by
B. Details of the Construction
Mi
u
y˜
ˆi M Fig. 2.
A special structure that ensures existence of ψi
(
)
For each i ∈ Z+ , Mi is a DFM described by
ˆ )), ψ1,i (u, (y, v) = ((u, y), (˜ y, v
q(t + 1) = y˜(t) =
ˆ ) is the unique output response of Mi to input where (˜ y, v (u, y) (for fixed initial condition qo ). Note that by construction, we have ψ1,i (P ) ⊆ Mi . Also consider ψ2,i : Mi → U × (Y × Vˆi ) defined by: ( ) ˆ )) = (u, (y, v ˆ )) ψ2,i ((u, y), (˜ y, v Let ψi = ψ2,i ◦ ψ1,i . Note that ψi (P ) = Pˆi for the given choice of functions α and β, and moreover ψi (P |u,y ) ⊆ Pˆi |u,y .
vˆ(t) =
fi (q(t), u(t), y(t)) gi (q(t))
(6)
hi (q(t))
where t ∈ Z+ , q(t) ∈ Qi , u(t) ∈ U , y(t) ∈ Y, y˜(t) ∈ Y, and v(t) ∈ Vˆi . In order to specify Mi , we thus need to specify its state set Qi , its output set Vˆi , its state transition function fi : Qi × U × Y → Qi , and its two output functions gi : Qi → Y and hi : Qi → Vˆi . These are constructed as follows: The State Set Qi The state set Qi is given by
V. C ONSTRUCTION OF Mi What remains is to construct a sequence of DFM {Mi } that, when used in conjunction with the structure described in Figure 2, ensures that properties (b) and (c) of Definition 5 are satisfied. A. Overview of the Construction ˆ i at the Intuitively, the state of the nominal model M current time will keep track of the last i inputs and outputs of P , the corresponding set of all possible current sensor outputs of P , and an upper bound on the set of all possible
Qi = Qi,F easible ∪ Qi,Initial ∪ {q∅ } where Qi,F easible denotes the set of feasible states, Qi,Initial denotes the set of initial states, and {q∅ } denotes an impossible state. Introduce the notation fu (x) as shorthand for f (u, x), and fu1 ◦u2 as shorthand for fu1 ◦ fu2 . A sequence (y1 , . . . , yi , u1 , . . . , ui )′ ∈ Y i × U i is said to be a feasible input-output sequence of length i if
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there exists an x ∈ Rn satisfying: ( ) yi = g x ( ) yi−1 = g fui (xo ) ( ) yi−2 = g fui−1 ◦ui (x) .. . y1
(7)
Now consider a state q = (ˆ v (q), Y(q), y1 , . . . , yi , u1 , . . . , ui )′ ∈ Qi,F easible and an input (u, y) ∈ U × Y. We define { q∅ if y ∈ / Y(q) fi (q, u, y) = q′ otherwise where q ′ ∈ Qi,F easible is the unique state given by
. = .. ( ) = g fu2 ◦...◦ui (x)
q ′ = (ˆ v (q), Y(q), y, y1 , . . . , yi−1 , u, u1 , . . . , ui−1 ).
Notice that we can associate with every such feasible inputoutput sequence a subset of the state-space Rn , obtained by successively applying input uj and intersecting with the subset of the state-space corresponding to output yj , for j = i, i − 1, . . . , 1. To each such feasible sequence, we can then associate a unique state q in Qi,F easible given by vˆ(q) Y(q) y1 .. . q= yi u1 . .. ui Machine state q is understood to be associated with subset ˆ X(q) of the state-space Rn . We can then define
Once the transitions of the feasible states are defined, the transitions of the equivalence classes, corresponding to the initial i − 1 states, immediately follows. The Output Function hi We define { sup vˆ(q) q∈Qi,F easible hi (q) = vˆ(q)
when q ∈ Qi,Initial ∪ {q∅ } otherwise
The Output Function gi For q ∈ Qi,F easible , we arbitrarily pick gi (q) ∈ Y(q). For all other states of the machine, we simply arbitrarily pick gi (q) ∈ Y. It can be verified that for this construction, both properties (b) and (c) of Definition 5 are satisfied.
ˆ Y(q) = g(X(q)) VI. F UTURE W ORK
and vˆ(q) = sup h(x)
Future work will be in two complementary directions:
ˆ x∈X(q)
1) Understanding the advantages (or limitations) of this constructive procedure. A question of immediate interest to us is the following: Suppose that a plant admits a finite state model that can exactly match its sensor output after some finite ‘training’ period (allowing for adequate reconstruction of the state): Can the proposed constructive procedure recover it? 2) Fundamentally understanding the underlying state observation problem. Specifically, we would like to be able to identify (apriori and analytically) those classes of systems for which we can ensure that γi = 0 for some index i. Alternatively, we would like to be able to identify (apriori and analytically) those classes of systems for which γj = γi∗ for all j ≥ i∗ , for some i∗ .
We have, by construction, that Y(q) ⊆ Y and vˆ(q) ⊂ V. Note that in the first l time steps, l < i, we do not have access to i past inputs and outputs of the plant; we only have access to a sequence of length l. It is thus necessary to initialize the machine by constructing equivalence classes of feasible states sharing 0, 1, ..., i − 1 length past snapshots. Finally, since not all sequences (u, y) ∈ U Z+ × Y Z+ are feasible signals of the plant P , and since our definition of a DFM requires us to construct a state transition associated with every possible input of the machine (i.e. with every pair (u, y) ∈ U × Y, we need to include a state q∅ to transition to when a sequence that could not have been generated by P is encountered. The Output Alphabet Vˆi As noted in the construction of the state set, vˆ(q) ⊂ V. Since by construction there are only a finite number of ‘feasible states’, the set Vˆi is simply a finite subset of V. The Transition Function fi We set fi (q∅ , u, y) = q∅ for all (u, y) ∈ U × Y.
VII. ACKNOWLEDGMENTS This research was supported by NSF CAREER award ECCS 0954601 and AFOSR YIP award FA9550-11-1-0118.
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R EFERENCES [1] R. Alur, T. Henzinger, G. Lafferriere, and G. Pappas. Discrete abstractions of hybrid systems. Proceedings of the IEEE, 88(2):971– 984, 2000. [2] P. J. Antsaklis. A brief introduction to the theory and applications of hybrid systems. Proceedings of the IEEE, Special Issue on Hybrid Systems: Theory and Applications, 88(7):879–887, July 2000. [3] C. Belta, A. Bicchi, M. Egerstedt, E. Frazzoli, E. Klavins, and G. J. Pappas. Symbolic planning and control of robot motion: State of the art and grand challenges. IEEE Robotics and Automation Magazine, 14(1):61–70, March 2007. [4] M. S. Branicky. Multiple lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Transactions on Automatic Control, 43(4):475–482, April 1998. [5] M. S. Branicky, V. S. Borkar, and S. K. Mitter. A unified framework for hybrid control: Model and optimal control theory. IEEE Transactions on Automatic Control, 43(1):31–45, January 1998. [6] A. Girard and G. J. Pappas. Approximation metrics for discrete and continuous systems. IEEE Transactions on Automatic Control, 52(5):782–798, 2007. [7] M. Kloetzer and C. Belta. A fully automated framework for control of linear systems from temporal logic specifications. IEEE Transactions on Automatic Control, 53(1):287–297, February 2008. [8] L. Ljung. System Identification: Theory for the User. Information and System Sciences. Prentice Hall, second edition, 1999. [9] J. Lunze. Qualitative modeling of linear dynamical systems with quantized state measurements. Automatica, 30:417–431, 1994. [10] J. Lunze, B. Nixdorf, and J. Schr¨oder. Deterministic discrete-event representation of continuous-variable systems. Automatica, 35:395– 406, March 1999. [11] R. Milner. Communication and Concurrency. International Series in Computer Science. Prentice Hall, Upper Saddle River, NJ, 1989. [12] T. Moor and J. Raisch. Supervisory control of hybrid systems whithin a behavioral framework. Systems & Control Letters, Special Issue on Hybrid Control Systems, 38:157–166, 1999. [13] T. Moor, J. Raisch, and S. D. O’Young. Discrete supervisory control of hybrid systems by l-complete approximations. Discrete Event Dynamic Systems: Theory and Applications, 12:83–107, 2002. [14] D. Park. Concurrency and automata on infinite sequences. In Proceedings of the Fifth GI Conference on Theoretical Computer Science, number 104 in Lecture Notes in Computer Science, pages 167–183. Springer-Verlag, 1981. [15] G. Pola, A. Girard, and P. Tabuada. Approximately bisimilar symbolic models for nonlinear control systems. Automatica, 44(10):2508–2516, October 2008. [16] J. Raisch and S. D. O’Young. Discrete approximation and supervisory control of continuous systems. IEEE Transactions on Automatic Control, 43(4):569–573, April 1998. [17] P. J. Ramadge and W. M. Wonham. Supervisory control of a class of discrete event processes. SIAM Journal on Control and Optimization, 25(1):206–230, 1987. [18] P. J. Ramadge and W. M. Wonham. The control of discrete event systems. Proceedings of the IEEE, 77(1):81–98, January 1989. [19] P. Tabuada. An approximate simulation approach to symbolic control. IEEE Transactions on Automatic Control, 53(6):1406–1418, July 2008. [20] P. Tabuada, A. Ames, A. A. Julius, and G. J. Pappas. Approximate reduction of dynamical systems. Systems & Control Letters, 57(7):538– 545, 2008. [21] Paulo Tabuada. Verification and Control of Hybrid Systems: A Symbolic Approach. Springer, 2009. [22] D. C. Tarraf. A control-oriented notion of finite state approximation. IEEE Transactions on Automatic Control (in press), 57(12), December 2012. [23] D. C. Tarraf and L. A. Duffaut Espinosa. On finite memory approximations constructed from input/output snapshots. In Proceedings of the 50th IEEE Conference on Decision & Control and the European Control Conference, pages 3966–3973, Orlando, Fl, December 2011. [24] D. C. Tarraf, A. Megretski, and M. A. Dahleh. Finite automata approximations with error bounds for systems with quantized actuation and measurement: a case study. In Proceedings of the 43nd IEEE Conference on Decision and Control, pages 1436–1441, Bahamas, December 2004.
[25] D. C. Tarraf, A. Megretski, and M. A. Dahleh. Finite state controllers for stabilizing switched systems with binary sensors. In G. Buttazzo, A. Bicchi, and A. Bemporad, editors, Hybrid Systems: Computation and Control, volume 4416 of Lecture Notes in Computer Science, pages 543–557. Springer-Verlag, April 2007. [26] D. C. Tarraf, A. Megretski, and M. A. Dahleh. A framework for robust stability of systems over finite alphabets. IEEE Transactions on Automatic Control, 53(5):1133–1146, June 2008. [27] D. C. Tarraf, A. Megretski, and M. A. Dahleh. Finite approximations of switched homogeneous systems for controller synthesis. IEEE Transactions on Automatic Control, 56(5):1140–1145, May 2011. [28] D.C. Tarraf, M.A. Dahleh, and A. Megretski. Stability of deterministic finite state machines. In Proc. American Control Conference the 2005, pages 3932–3936, 8–10 June 2005. [29] A. Tiwari and G. Khanna. Series of abstractions for hybrid automata. In Hybrid Systems: Computation and Control, volume 2289 of Lecture Notes in Computer Science. Springer, 2002. [30] J. C. Willems. The behavioral approach to open and interconnected systems. IEEE Control Systems Magazine, 27:46–99, 27 2007.
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Constructing ρ/µ Approximations From Input/Output Snapshots for Systems Over Finite Alphabets Danielle C. Tarraf Abstract— We consider discrete-time plants that interact with their controllers by sending and receiving binary valued sensor signals and finite valued control signals, respectively. In the absence of exogenous inputs, we propose a general procedure for constructing (finite state) ρ/µ approximations, starting from finite length sequences of input and output signal pairs. We show that the proposed construction satisfies desirable properties, thus leading to a hierarchy of finite state models that can be used for certified-by-design control synthesis.
I. I NTRODUCTION High fidelity models that describe a dynamical system to a high degree of accuracy are often too complex for use in controller synthesis, particularly when the underlying dynamics of the plant are hybrid [2], [4], [5] and/or subject to quantization effects. Accordingly, the problem of approximating hybrid systems by simpler systems has been receiving much attention over the past two decades [1], [3], [29]. In particular, the problem of constructing finite state approximations of hybrid systems has been the object of intense study, due to the amenability of finite state models to analysis and control synthesis [7], [17], [27], [28]. One set of approaches make use of non-deterministic finite state automata constructed so that their input/output behavior contains that of the original model (these approximations are sometimes referred to in the literature as ‘qualitative models’) [9], [10], [16]. Controller synthesis can then be formulated as a supervisory control problem, addressed using the Ramadge-Wonham framework [18]. More recently, progress has been made in reframing these results [12], [13] in the context of Willems’ behavioral theory and lcomplete systems [30]. Another set of approaches, influenced by the theory of bisimulation in concurrent processes [11], [14], make use of bisimulation and simulation abstractions of the original plant. These approaches effectively ensure that the set of state trajectories of the original model is exactly matched by (bisimulation), contained in (simulation), matched to within some distance ϵ by (approximate bisimulation), or contained to within some distance ϵ in (approximate simulation), the set of state trajectories of the finite state abstraction [6], [15], [19], [20]. The performance objectives are typically formulated as constraints on the state trajectories of the original hybrid system, and controller synthesis is a two step procedure: A finite state supervisory D. C. Tarraf is with the Department of Electrical and Computer Engineering at the Johns Hopkins University, Baltimore, MD 21218, USA
(dtarraf)@jhu.edu
978-1-4673-4539-2/12/$31.00 ©2012 IEEE
controller is first designed, and subsequently refined to yield a certified hybrid controller for the original plant [21]. In a recent effort [22], we proposed a notion of approximation for ‘systems over finite alphabets’, namely discretetime plants that interact with their feedback controllers by sending and receiving signals taking their values in fixed, finite alphabet sets. We refer to this notion of approximation as a ‘ρ/µ approximation’, to highlight its affinity and compatibility with the analysis [26] and synthesis [27] tools previously developed for systems whose properties and performance objectives are described in terms of ρ/µ gain conditions. This new notion of approximation differs from the existing notions of qualitative approximations and simulation/bisimulation abstractions in several important respects, discussed in [22]. To summarize, one critical difference is the non-trivial state estimation problem inherent in our setup; Indeed, abstraction based methods effectively address full-state feedback problems. Another critical difference is the emphasis on quantifying the quality of approximation; Indeed, as their name indicates, qualitative models lack this quantitative characterization. It should be emphasized that the proposed notion of ρ/µ approximation explicitly identifies those properties that the approximate models need to satisfy in order to enable certified-by-design controller synthesis. However, it does not restrict us to a particular constructive algorithm, rather leaving the question of construction completely open. In particular, an intuitive state-space based construction was employed in the simple illustrative example presented in [22]. In this paper, we propose and analyze a new constructive procedure for generating ρ/µ approximations of a given plant and performance objective. In contrast to the statespace based construction, which was specifically tailored to the dynamics in question, the present construction is widely applicable to arbitrary plants over finite alphabets, provided they are not subject to exogenous inputs, and provided their outputs are a function of the state only (i.e. what would be analogous to ‘strictly proper’ transfer functions in the LTI setting). Our proposed construction effectively keeps track of finite length sequences of input-output signal pairs of the plant. The idea of using finite length sequences of inputs and outputs (innovations) is of course widely employed in system identification [8]. However, the setup of interest to us is fundamentally different: Indeed, the dynamics of the plant are assumed to be exactly known. Additionally, the data is complete and exact (i.e. uncorrupted by noise). Likewise, our construction bears some resemblance to algorithms employed
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in constructing qualitative models, surveyed earlier in our introduction. However, our notion of approximation is fundamentally different from the notion of qualitative models, as it seeks to explicitly quantify the approximation error in the spirit of robust control. The proposed construction differs significantly from our preliminary efforts reported in [23] in at least two important respects: First, it approximates the performance objectives as well as the dynamics of the plant. Second, it immediately leads to a finite nominal model with deterministic transitions, in contrast to [23] in which a nondeterministic model is contructed and the non-deterministic transitions are subsequently dropped and accounted for in quantifying the ‘worst-case’ approximation error. Finally, our proposed construction differs from our very early effort reported in [24] in several respects, the most significant of which is the structure of the corresponding approximation error. Organization of the Paper: We begin in Section II by reviewing the notion of ρ/µ approximation as well as some basic concepts that are relevant to our development. We describe the problem setup and state our objective in Section III. We revisit a previously proposed structure [25] in Section IV, and show (under mild assumptions) that it ensures desirable properties regardless of the choice of nominal model. We present our construction in Section V, and we conclude in Section VI with some directions for future work. Notation: Z+ , R+ denote the non-negative integers and non-negative reals, respectively. Given a set A, AZ+ and 2A denote the set of all infinite sequences over A (indexed by Z+ ) and the power set of A, respectively. The cardinality of a (finite) set A is denoted by |A|. Elements of A and AZ+ are denoted by a and (boldface) a, respectively. For a ∈ AZ+ , a(i) denotes its ith term. For f : A → B, C ⊂ B, f −1 (C) = {a ∈ A|f (a) ∈ C}. II. P RELIMINARIES
gain stable if there exists a finite non-negative constant γ such that inf
T ≥0
T ∑
γρ(u(t)) − µ(y(t)) > −∞.
(1)
t=0
is satisfied for all (u, y) in S. In particular, when ρ, µ are non-negative (and not identically zero), the ‘gain’ can be defined. Definition 2. Consider a system S ⊂ U Z+ × Y Z+ . Assume that S is ρ/µ gain stable for ρ : U → R+ and µ : Y → R+ , and that neither function is identically zero. The ρ/µ gain of S is the infimum of γ such that (1) is satisfied. In this paper, we are specifically interested in plants that are not subject to exogeneous inputs, and that are equipped with finite-valued actuators and sensors: Definition 3. A system over finite alphabets S is a discretetime system S ⊂ U Z+ × (Y × V)Z+ whose alphabets U and Y are finite. Here, u ∈ U Z+ represents the control input to the plant while y ∈ Y Z+ and v ∈ V Z+ represent the sensor and performance outputs of the plant, respectively. The plant dynamics may be analog, discrete or hybrid, and thus V may be finite, countable or infinite. The approximate models of the plant will be drawn from a specific class of models, namely deterministic finite state machines: Definition 4. A deterministic finite state machine (DFM) is a discrete-time system S ⊂ U Z+ × Y Z+ with finite alphabets U, Y, whose feasible input and output signals (u, y) are related by q(t + 1) = y(t) =
f (q(t), u(t)) g(q(t), u(t))
where t ∈ Z+ , q(t) ∈ Q for some finite set Q and functions f : Q × U → Q and g : Q × U → Y.
In our development, it is often convenient to view a discrete-time dynamical system as a set of feasible signals, even when a state-space description of the system is available. We thus begin this section by briefly reviewing this ‘feasible signals’ view of systems. We then present the recently proposed notion of ρ/µ approximation, slightly modified for the class of systems of interest (systems with no exogenous inputs), and state a relevant control synthesis result.
It is understood here that Q, f and g represent the set of states of the DFM, the state transition map, and the output map, respectively, in the traditional state-space sense. Given a system P ⊂ U Z+ × (Y × V)Z+ and signals uo ∈ U Z+ , yo ∈ Y Z+ , we use P |uo ,yo to denote the subset of feasible signals of P whose first component is uo and whose second component is yo . That is {( ) } P |uo ,yo = u, (y, v) ∈ P u = uo and y = yo .
A. Systems and Performance Specifications
B. ρ/µ Approximations for Control Synthesis
A discrete-time signal is an infinite sequence over some prescribed set, referred to as an alphabet. A discrete-time system S is a set of pairs of signals, S ⊂ U Z+ × Y Z+ , where U and Y are given alphabets. In this setting, system properties of interest are captured by means of ‘integral’ constraints on the feasible signals:
The following definition applies to the class of plants of interest: Definition 5. (Adapted from Definition 6 in [22]) Consider a system over finite alphabets P ⊂ U Z+ × (Y × V)Z+ and a desired closed loop performance objective
Definition 1. Consider a system S ⊂ U Z+ × Y Z+ and let ρ : U → R and µ : Y → R be given functions. S is ρ/µ
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inf
T ≥0
T ∑ t=0
−µ(v(t)) > −∞ ⇔ sup
T ∑
T ≥0 t=0
µ(v(t)) < ∞
(2)
w
z
i u
v
P y u
Fig. 1.
)
ψi P |u,y ⊆ Pˆi |u,y Wednesday, March 7, 12
for all (u, y) ∈ U Z+ × Y Z+ , where Pˆi ⊂ U Z+ × (Y × ˆ i and ∆i as Vˆi )Z+ is the feedback interconnection of M shown in Figure 1. (b) For every feasible signal (u, (y, v)) ∈ P , we have µ(v(t)) ≤ µ(ˆ vi+1 (t)) ≤ µ(ˆ vi (t)), (3) ( ) for all t ∈ Z+ , where (u, (ˆ yi , v ˆi )) = ψi (u, (y, v)) ( ) and (u, (ˆ yi+1 , v ˆi+1 )) = ψi+1 (u, (y, v)) . (c) ∆i is ρ∆ /µ∆ gain stable, and moreover, the corresponding ρ∆ /µ∆ gains satisfy γi ≥ γi+1 . We complete our review of preliminaries by reviewing a previously derived result relevant to the problem of control synthesis. This highlights the fact that a ρ/µ approximation of the plant together with an appropriately defined performance objective may be used to synthesize certifiedby-design controllers for the original plant and original performance objective: Theorem 1. (Adapted from Theorems 1 and 3 in [22]) ˆ i } as in Consider a plant P and a ρ/µ approximation {M Definition 5. If for some index i, there exists a controller K ⊂ Y Z+ × U Z+ such that the feedback interconnection of ˆ i and K, (M ˆ i , K) ⊂ W Z+ × (Vˆi × Z)Z+ , satisfies M inf
T ≥0
T ∑
yˆ
A finite state approximation of P
ˆ i }∞ of for given function µ : V → R. A sequence {M i=1 Z+ ˆ deterministic finite state machines Mi ⊂ (U × W) × (Y × Vˆi × Z)Z+ with Vˆi ⊂ V is a ρ/µ approximation of P if there exists a corresponding sequence of systems {∆i }∞ i=1 , + ∆i ⊂ Z Z ×W Z+ , and non-zero functions ρ∆ : Z Z+ → R+ , µ∆ : W Z+ → R+ , such that for every i: (a) There exists a surjective map ψi : P → Pˆi satisfying (
vˆ
ˆi M
τ µ∆ (w(t)) − µ(ˆ v (t)) − τ γi ρ∆ (z(t)) > −∞ (4)
III. P ROBLEM S ETUP Consider a discrete-time plant P described by x(t + 1)
f (x(t), u(t)) g(x(t)) h(x(t))
(5)
where t ∈ Z+ , x(t) ∈ Rn , u(t) ∈ U, y(t) ∈ Y and v(t) ∈ V. No apriori constraints are placed on alphabet set V: It may be a Euclidean space, the set of reals, or a countable or finite set. U and Y are given finite alphabets with |U| = m and |Y| = 2, respectively: They may represent quantized values of analog inputs and outputs, or may simply be symbolic inputs and outputs in general. Functions f : Rn × U → Rn , g : Rn → Y and h : Rn → V are given. Our objective in this paper is to propose a systematic procedure for constructing a ρ/µ approximation of a given plant P as in (5) and a performance objective as in (2). IV. E NSURING E XISTENCE OF ψi : A S PECIAL S TRUCTURE In [25], a special ‘observer-inspired’ structure was proposed and used in conjunction with a particular finite state machine construction (distinct from the presently proposed construction) in order to approximate and design controllers for a special class of systems, namely switched second order homogenous systems with binary outputs. In this section, we revisit this structure: We show that under mild assumptions, this structure ensures the existence of function ψi with the properties required in part (a) of Definition 5. Lemma 1. Consider a system P ⊂ U Z+ ×(Y ×Z)Z+ where U and Y are finite, and where |Y| = 2. Also consider the structure shown in Figure 2, where { 0 if y = y˜ β(y, y˜) = 1 otherwise {
and
t=0
α(y, y˜) =
for some τ > 0, then the feedback interconnection of P and K, (P, K) ⊂ V Z+ , satisfies (2).
=
y(t) = v(t) =
y˜ y˜′
if w = 0 otherwise
with y˜′ denoting the unique element of Y that is not equal to y˜. For any deterministic finite state machine Mi with fixed initial condition qo , there exists a ψi : P → Pˆi , where Pˆi is
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dnesday, March 7, 12
performance outputs of P :
v
P
y
q(t) = y(t − i) u(t − 1) .. . u(t − i)
w vˆ
Mi
vˆ(q) Y(q) y(t − 1) .. .
y˜
i
vˆ yˆ
ˆ)i and ∆i as shown, such that ψi is the interconnection ( of M surjective and ψi P |u,y ⊆ Pˆi |u,y .
Note that since the control input and sensor output alphabet sets are finite, the corresponding number of feasible inputoutput sequences of length i is also finite. The number of states of the nominal model is consequently finite by construction, and the choice of sensor output set and performance output bound follows from the choice of inputoutput sequence. As will be explained in detail, this particular interpretation of the state leads to a finite state machine with deterministic state transitions, deterministic performance output, and a sensor-like output that is set-valued in general. The relevant output function is then ‘determinised’ and the introduced error is accounted for in a worst case sense.
Proof: The proof is by construction. Consider ψ1,i : P → (U × Y) × (Y × Vˆi ) defined by
B. Details of the Construction
Mi
u
y˜
ˆi M Fig. 2.
A special structure that ensures existence of ψi
(
)
For each i ∈ Z+ , Mi is a DFM described by
ˆ )), ψ1,i (u, (y, v) = ((u, y), (˜ y, v
q(t + 1) = y˜(t) =
ˆ ) is the unique output response of Mi to input where (˜ y, v (u, y) (for fixed initial condition qo ). Note that by construction, we have ψ1,i (P ) ⊆ Mi . Also consider ψ2,i : Mi → U × (Y × Vˆi ) defined by: ( ) ˆ )) = (u, (y, v ˆ )) ψ2,i ((u, y), (˜ y, v Let ψi = ψ2,i ◦ ψ1,i . Note that ψi (P ) = Pˆi for the given choice of functions α and β, and moreover ψi (P |u,y ) ⊆ Pˆi |u,y .
vˆ(t) =
fi (q(t), u(t), y(t)) gi (q(t))
(6)
hi (q(t))
where t ∈ Z+ , q(t) ∈ Qi , u(t) ∈ U , y(t) ∈ Y, y˜(t) ∈ Y, and v(t) ∈ Vˆi . In order to specify Mi , we thus need to specify its state set Qi , its output set Vˆi , its state transition function fi : Qi × U × Y → Qi , and its two output functions gi : Qi → Y and hi : Qi → Vˆi . These are constructed as follows: The State Set Qi The state set Qi is given by
V. C ONSTRUCTION OF Mi What remains is to construct a sequence of DFM {Mi } that, when used in conjunction with the structure described in Figure 2, ensures that properties (b) and (c) of Definition 5 are satisfied. A. Overview of the Construction ˆ i at the Intuitively, the state of the nominal model M current time will keep track of the last i inputs and outputs of P , the corresponding set of all possible current sensor outputs of P , and an upper bound on the set of all possible
Qi = Qi,F easible ∪ Qi,Initial ∪ {q∅ } where Qi,F easible denotes the set of feasible states, Qi,Initial denotes the set of initial states, and {q∅ } denotes an impossible state. Introduce the notation fu (x) as shorthand for f (u, x), and fu1 ◦u2 as shorthand for fu1 ◦ fu2 . A sequence (y1 , . . . , yi , u1 , . . . , ui )′ ∈ Y i × U i is said to be a feasible input-output sequence of length i if
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there exists an x ∈ Rn satisfying: ( ) yi = g x ( ) yi−1 = g fui (xo ) ( ) yi−2 = g fui−1 ◦ui (x) .. . y1
(7)
Now consider a state q = (ˆ v (q), Y(q), y1 , . . . , yi , u1 , . . . , ui )′ ∈ Qi,F easible and an input (u, y) ∈ U × Y. We define { q∅ if y ∈ / Y(q) fi (q, u, y) = q′ otherwise where q ′ ∈ Qi,F easible is the unique state given by
. = .. ( ) = g fu2 ◦...◦ui (x)
q ′ = (ˆ v (q), Y(q), y, y1 , . . . , yi−1 , u, u1 , . . . , ui−1 ).
Notice that we can associate with every such feasible inputoutput sequence a subset of the state-space Rn , obtained by successively applying input uj and intersecting with the subset of the state-space corresponding to output yj , for j = i, i − 1, . . . , 1. To each such feasible sequence, we can then associate a unique state q in Qi,F easible given by vˆ(q) Y(q) y1 .. . q= yi u1 . .. ui Machine state q is understood to be associated with subset ˆ X(q) of the state-space Rn . We can then define
Once the transitions of the feasible states are defined, the transitions of the equivalence classes, corresponding to the initial i − 1 states, immediately follows. The Output Function hi We define { sup vˆ(q) q∈Qi,F easible hi (q) = vˆ(q)
when q ∈ Qi,Initial ∪ {q∅ } otherwise
The Output Function gi For q ∈ Qi,F easible , we arbitrarily pick gi (q) ∈ Y(q). For all other states of the machine, we simply arbitrarily pick gi (q) ∈ Y. It can be verified that for this construction, both properties (b) and (c) of Definition 5 are satisfied.
ˆ Y(q) = g(X(q)) VI. F UTURE W ORK
and vˆ(q) = sup h(x)
Future work will be in two complementary directions:
ˆ x∈X(q)
1) Understanding the advantages (or limitations) of this constructive procedure. A question of immediate interest to us is the following: Suppose that a plant admits a finite state model that can exactly match its sensor output after some finite ‘training’ period (allowing for adequate reconstruction of the state): Can the proposed constructive procedure recover it? 2) Fundamentally understanding the underlying state observation problem. Specifically, we would like to be able to identify (apriori and analytically) those classes of systems for which we can ensure that γi = 0 for some index i. Alternatively, we would like to be able to identify (apriori and analytically) those classes of systems for which γj = γi∗ for all j ≥ i∗ , for some i∗ .
We have, by construction, that Y(q) ⊆ Y and vˆ(q) ⊂ V. Note that in the first l time steps, l < i, we do not have access to i past inputs and outputs of the plant; we only have access to a sequence of length l. It is thus necessary to initialize the machine by constructing equivalence classes of feasible states sharing 0, 1, ..., i − 1 length past snapshots. Finally, since not all sequences (u, y) ∈ U Z+ × Y Z+ are feasible signals of the plant P , and since our definition of a DFM requires us to construct a state transition associated with every possible input of the machine (i.e. with every pair (u, y) ∈ U × Y, we need to include a state q∅ to transition to when a sequence that could not have been generated by P is encountered. The Output Alphabet Vˆi As noted in the construction of the state set, vˆ(q) ⊂ V. Since by construction there are only a finite number of ‘feasible states’, the set Vˆi is simply a finite subset of V. The Transition Function fi We set fi (q∅ , u, y) = q∅ for all (u, y) ∈ U × Y.
VII. ACKNOWLEDGMENTS This research was supported by NSF CAREER award ECCS 0954601 and AFOSR YIP award FA9550-11-1-0118.
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R EFERENCES [1] R. Alur, T. Henzinger, G. Lafferriere, and G. Pappas. Discrete abstractions of hybrid systems. Proceedings of the IEEE, 88(2):971– 984, 2000. [2] P. J. Antsaklis. A brief introduction to the theory and applications of hybrid systems. Proceedings of the IEEE, Special Issue on Hybrid Systems: Theory and Applications, 88(7):879–887, July 2000. [3] C. Belta, A. Bicchi, M. Egerstedt, E. Frazzoli, E. Klavins, and G. J. Pappas. Symbolic planning and control of robot motion: State of the art and grand challenges. IEEE Robotics and Automation Magazine, 14(1):61–70, March 2007. [4] M. S. Branicky. Multiple lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Transactions on Automatic Control, 43(4):475–482, April 1998. [5] M. S. Branicky, V. S. Borkar, and S. K. Mitter. A unified framework for hybrid control: Model and optimal control theory. IEEE Transactions on Automatic Control, 43(1):31–45, January 1998. [6] A. Girard and G. J. Pappas. Approximation metrics for discrete and continuous systems. IEEE Transactions on Automatic Control, 52(5):782–798, 2007. [7] M. Kloetzer and C. Belta. A fully automated framework for control of linear systems from temporal logic specifications. IEEE Transactions on Automatic Control, 53(1):287–297, February 2008. [8] L. Ljung. System Identification: Theory for the User. Information and System Sciences. Prentice Hall, second edition, 1999. [9] J. Lunze. Qualitative modeling of linear dynamical systems with quantized state measurements. Automatica, 30:417–431, 1994. [10] J. Lunze, B. Nixdorf, and J. Schr¨oder. Deterministic discrete-event representation of continuous-variable systems. Automatica, 35:395– 406, March 1999. [11] R. Milner. Communication and Concurrency. International Series in Computer Science. Prentice Hall, Upper Saddle River, NJ, 1989. [12] T. Moor and J. Raisch. Supervisory control of hybrid systems whithin a behavioral framework. Systems & Control Letters, Special Issue on Hybrid Control Systems, 38:157–166, 1999. [13] T. Moor, J. Raisch, and S. D. O’Young. Discrete supervisory control of hybrid systems by l-complete approximations. Discrete Event Dynamic Systems: Theory and Applications, 12:83–107, 2002. [14] D. Park. Concurrency and automata on infinite sequences. In Proceedings of the Fifth GI Conference on Theoretical Computer Science, number 104 in Lecture Notes in Computer Science, pages 167–183. Springer-Verlag, 1981. [15] G. Pola, A. Girard, and P. Tabuada. Approximately bisimilar symbolic models for nonlinear control systems. Automatica, 44(10):2508–2516, October 2008. [16] J. Raisch and S. D. O’Young. Discrete approximation and supervisory control of continuous systems. IEEE Transactions on Automatic Control, 43(4):569–573, April 1998. [17] P. J. Ramadge and W. M. Wonham. Supervisory control of a class of discrete event processes. SIAM Journal on Control and Optimization, 25(1):206–230, 1987. [18] P. J. Ramadge and W. M. Wonham. The control of discrete event systems. Proceedings of the IEEE, 77(1):81–98, January 1989. [19] P. Tabuada. An approximate simulation approach to symbolic control. IEEE Transactions on Automatic Control, 53(6):1406–1418, July 2008. [20] P. Tabuada, A. Ames, A. A. Julius, and G. J. Pappas. Approximate reduction of dynamical systems. Systems & Control Letters, 57(7):538– 545, 2008. [21] Paulo Tabuada. Verification and Control of Hybrid Systems: A Symbolic Approach. Springer, 2009. [22] D. C. Tarraf. A control-oriented notion of finite state approximation. IEEE Transactions on Automatic Control (in press), 57(12), December 2012. [23] D. C. Tarraf and L. A. Duffaut Espinosa. On finite memory approximations constructed from input/output snapshots. In Proceedings of the 50th IEEE Conference on Decision & Control and the European Control Conference, pages 3966–3973, Orlando, Fl, December 2011. [24] D. C. Tarraf, A. Megretski, and M. A. Dahleh. Finite automata approximations with error bounds for systems with quantized actuation and measurement: a case study. In Proceedings of the 43nd IEEE Conference on Decision and Control, pages 1436–1441, Bahamas, December 2004.
[25] D. C. Tarraf, A. Megretski, and M. A. Dahleh. Finite state controllers for stabilizing switched systems with binary sensors. In G. Buttazzo, A. Bicchi, and A. Bemporad, editors, Hybrid Systems: Computation and Control, volume 4416 of Lecture Notes in Computer Science, pages 543–557. Springer-Verlag, April 2007. [26] D. C. Tarraf, A. Megretski, and M. A. Dahleh. A framework for robust stability of systems over finite alphabets. IEEE Transactions on Automatic Control, 53(5):1133–1146, June 2008. [27] D. C. Tarraf, A. Megretski, and M. A. Dahleh. Finite approximations of switched homogeneous systems for controller synthesis. IEEE Transactions on Automatic Control, 56(5):1140–1145, May 2011. [28] D.C. Tarraf, M.A. Dahleh, and A. Megretski. Stability of deterministic finite state machines. In Proc. American Control Conference the 2005, pages 3932–3936, 8–10 June 2005. [29] A. Tiwari and G. Khanna. Series of abstractions for hybrid automata. In Hybrid Systems: Computation and Control, volume 2289 of Lecture Notes in Computer Science. Springer, 2002. [30] J. C. Willems. The behavioral approach to open and interconnected systems. IEEE Control Systems Magazine, 27:46–99, 27 2007.
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