An Iterative Algorithmic Implementation of Input-Output Finite State ...

51st IEEE Conference on Decision and Control December 10-13, 2012. Maui, Hawaii, USA

An Iterative Algorithmic Implementation of Input-Output Finite State Approximations Fereshteh Aalamifar and Danielle C. Tarraf Abstract— The problem of finding finite state models of systems with quantized inputs and outputs has received much deserved attention. In particular, a notion of ρ/µ approximation was proposed and shown to be relevant to the problem of control synthesis. In this paper, we revisit a recently developed constructive algorithm for generating ρ/µ approximations for a class of systems, and we propose and analyze several algorithms for improving the computational efficiency and memory requirements of the construction. We demonstrate the use of this approach for synthesizing certified-by-design controllers for a simple illustrative example with reachability type specifications.

I. I NTRODUCTION 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 control synthesis. One set of results makes use of non-deterministic finite state automata constructed so that their input/output behavior contains that of the original model [2], [7], [3], [4], [5]. Controller synthesis can then be formulated as a supervisory control problem, addressed using the RamadgeWonham framework [8], [9]. Another set of results makes use of bisimulation and simulation abstractions (either exact or approximate) of the original plant [1], [10], [12], [6]. Controller synthesis is then a two step procedure in which a finite state supervisory controller is designed and subsequently refined into a certified hybrid controller [11]. In recent efforts, we proposed a notion of ρ/µ approximation for ‘systems over finite alphabets’ [13], basically plants that interact with their feedback controllers by sending and receiving signals taking values in fixed, finite (alphabet) sets. We showed that if sequences of finite state machines and corresponding approximation errors are constructed to satisfy three critical properties, controllers that are designed for the finite state machines and subsequently implemented in the original plant offer performance guarantees by design. The construction of these approximations, on the other hand, was left open in [13]. In [16], we considered plants with no exogenous inputs, and proposed a procedure for constructing finite state machines and corresponding approximation errors that satisfy two of the three properties of a ρ/µ approximation: The proposed procedure focused on the question of approximating the dynamics, while completely disregarding the question of approximating the performance objectives. In this paper, we focus once again on plants over finite alphabets in the absence of exogeneous input. Building on our The authors are with the Department of Electrical and Computer Engineering at the Johns Hopkins University, Baltimore, MD 21218, USA

{fereshteh,dtarraf}@jhu.edu 978-1-4673-2064-1/12/$31.00 ©2012 IEEE

previous work, we revisit a recently proposed constructive procedure for generating ρ/µ approximations [15], [14]: We propose a set of algorithms allowing us to implement this constructive procedure in an iterative manner, thus leading to a significant reduction in computational burden. Organization: We review the preliminaries and relevant results in Section II. The main contribution of the paper is presented in Section III, where we propose and analyze a set of algorithms leading to a more computationally efficient iterative implementation of this construction. We present a simple illustrative example in Section IV, and we conclude with directions for future work in Section V. 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. 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 We review in this Section the relevant background. Interested readers are referred to [18] for an in-depth treatment of the material in Section II-A, to [13], [19] for an in-depth treatement of the material in Section II-B, and to [15], [14] for an in-depth treatment of the material in Section II-C. A. Systems and Performance Specifications A (discrete-time) signal is understood to be an infinite sequence over some prescribed alphabet set, while a (discretetime) system S is simply a set of pairs of feasible signals, S ⊂ U Z+ × Y Z+ , where U and Y are given input and output alphabets. System properties of interest are captured by means of ‘integral’ constraints on the feasible signals. Definition 1: Consider a system S ⊂ U Z+ × Y Z+ and given function ρ : U → R, µ : Y → R. S is ρ/µ gain stable if there exists a finite non-negative constant γ such that the following inequality is satisfied for all (u, y) in S: inf

T ≥0

T X t=0

γρ(u(t)) − µ(y(t)) > −∞.

(1)

In particular, when ρ and µ are non-negative (and not identically zero), a notion of ‘gain’ can be defined as the infimum of γ such that (1) is satisfied. We are interested in plants with finite-valued actuators and sensors, which we refer to as ‘systems over finite alphabets’. More precisely, a system over finite alphabets S is a discretetime system S ⊂ U Z+ × (Y × V)Z+ whose alphabets U and

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Y are finite. u ∈ U Z+ , v ∈ V Z+ , and y ∈ Y Z+ represent the control input, the performance output and the sensor output of the plant, respectively. No restrictions are placed a-priori on the internal dynamics of the plant or on alphabet V. We will use the notation S|uo ,yo to denote the subset of feasible signals of S defined by n  o S|uo ,yo = u, (y, v) ∈ S u = uo and y = yo .

The nominal models of interest to us are deterministic finite state machine (DFM) models, which are simply discretetime systems with finite input and output alphabets and whose feasible signals are related by q(t + 1) = y(t) =

f (q(t), u(t)) g(q(t), u(t))

for some f and g, and some finite state set Q, q(t) ∈ Q. B. Notion of ρ/µ Approximation The underlying assumption generally is that the purpose of deriving a DFM approximation of a system over finite alphabets P is to simplify the process of synthesizing a controller K such that the closed loop system (P, K) is ρ/µ gain stable for some given functions ρ and µ. The notion of approximation of interest to us is given by Definition 2. w

z

∆i v u

P

≈ y u

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)) . 3) ∆i is ρ∆ /µ∆ gain stable, and moreover, the corresponding ρ∆ /µ∆ gains satisfy γi ≥ γi+1 . We continue our review of preliminaries by stating two results relevant to the problem of control synthesis. The first highlights the fact that a ρ/µ approximation of the plant together with an appropriately defined performance objective may be used to synthesize certified-by-design controllers for the original plant and original performance objective: Theorem 1: (Adapted from Theorems 1 and 3 in [13]) ˆ i } as in Consider a plant P and a ρ/µ approximation {M Definition 2. 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 X

τ µ∆ (w(t)) − µ(ˆ v (t)) − τ γi ρ∆ (z(t)) > −∞ (4)

t=0

for some τ > 0, then the feedback interconnection of P and K, (P, K) ⊂ V Z+ , satisfies (2). The second provides an algorithmic approach for synthesizing a full state feedback controller for a given DFM in order to satisfy given performance objectives: Theorem 2: (Adapted from Theorem 4 in [19]) Consider a DFM M with control input u(t) ∈ U, disturbance input w(t) ∈ W, and state transition equation q(t + 1) = f (q(t), u(t), w(t)).

vˆ

ˆi M

Let σ : Q×U ×W → R be given. There exists a ϕ : Q → U such that the closed loop system (M, ϕ) satisfies

yˆ Fig. 1.

2) For every feasible signal (u, (y, v)) ∈ P , we have

A finite state approximation of P

inf

Definition 2: (Adapted from Definition 6 in [13]) Consider a system over finite alphabets P ⊂ U Z+ × (Y × V)Z+ and a desired closed loop performance objective sup

T X

µ(v(t)) < ∞

T ≥0

J0 Jk+1

T ≥0 t=0

for all (u, y) ∈ U Z+ × Y Z+ , where Pˆi ⊂ U Z+ × (Y × ˆ i and ∆i Vˆi )Z+ is the feedback interconnection of M as shown in Figure 1.

σ(q(t), ϕ(q(t)), w(t)) > −∞.

(5)

t=0

iff the sequence of functions Jk : Q → R, k ∈ Z+ , defined recursively by

(2)

ˆ i }∞ of for given function µ : V → R. A sequence {M i=1 ˆ i ⊂ (U × W)Z+ × (Y × deterministic finite state machines M Vˆi × Z)Z+ with Vˆi ⊂ V is a ρ/µ approximation of P if there exists a corresponding sequence of systems {∆i }∞ i=1 , Z+ Z+ Z+ ∆i ⊂ Z ×W , and non-zero functions ρ∆ : Z → R+ , µ∆ : W Z+ → R+ , such that for every i: 1) There exists a surjective map ψi : P → Pˆi satisfying   ψi P |u,y ⊆ Pˆi |u,y

T X

= 0 = max{0, T(Jk )}

where T(J(q)) = min max {−σ(q, u, w) + J(f (q, u, w))}, u∈U w∈W

converges. Note that in particular, a gain condition such as (4), can ˆ i , vˆ and be written in the form (5) as the two outputs of M ˆ z, are each a function of the state of Mi and its inputs. C. Construction of the Approximate Models ˆ i and ∆i shown in Figure 2, Consider the structure for M first proposed in [17]. β(y, y˜) is 0 if y = y˜ and 1 otherwise, while α(˜ y , w) = y˜ iff w = 0. We will make use of this structure in the construction that follows. As such, we will

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2) Letting X o (q) = {x ∈ Rn |x satisfies (6)}, Y(q) is the smallest subset of Y such that   g fu1 ◦...◦ui (X o (q)) ∈ Y(q).

v

P

y

β

w

vˆ

Mi

y˜

∆i

y˜

Mi

u

vˆ yˆ

α ˆi M

ˆ i and ∆i Structures of M

Fig. 2.

focus on simply constructing Mi in an appropriate manner: ˆ i and ∆i immediately follow. M Given a discrete-time plant P described by x(t + 1)

=

q ′ = (Y(q ′ ), y, y1 , . . . , yi−1 , u, u1 , . . . , ui−1 ).

g(x(t)) h(x(t))

where t ∈ Z+ , x(t) ∈ Rn , u(t) ∈ U , y(t) ∈ Y and v(t) ∈ V. U and Y are given finite alphabets with |U| = m and |Y| = p, respectively. Functions f : Rn × U → Rn , g : Rn → Y and h : Rn → V are given, as is a performance objective (2). For each i ∈ Z+ , Mi is a DFM described by

ˆ X(q) = fu1 ◦...◦ui (X o (q)).   We can then define hi (q) = h arg max µ(h(x)) . ˆ x∈X(q)

hi (q(t))

where t ∈ Z+ , q(t) ∈ Qi , u(t) ∈ U , y(t) ∈ Y, y˜(t) ∈ Y, and v(t) ∈ Vˆi . The construction of Qi , Vˆi , fi : Qi ×U ×Y → Qi , gi : Qi → Y and hi : Qi → Vˆi is detailed below. 1) Construction of State Set Qi : The state set Qi is given by 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 . An element q = (Y(q), y1 , . . . , yi , u1 , . . . , ui )′ ∈ 2Y × Y i × U i is a feasible state (q ∈ Qi,F easible ) iff: 1) There exists an x ∈ Rn satisfying:   yi = g x   yi−1 = g fui (x)   yi−2 = g fui−1 ◦ui (x) y1

Once the transitions of the feasible states are defined, the transitions of the equivalence classes, corresponding to the initial i − 1 states, immediately follows. 3) Construction of Output Alphabet Vˆi and Output Function hi : We can associate with every feasible state q ∈ ˆ Qi,F easible a subset X(q) of Rn defined by

fi (q(t), u(t), y(t)) gi (q(t))

vˆ(t) =

.. .

where q ′ ∈ Qi,F easible is the unique state given by

f (x(t), u(t))

y(t) = v(t) =

q(t + 1) = y˜(t) =

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 for P , and since a DFM should have a state transition associated with every possible input of the machine, we include a state q∅ to transition to when a sequence that could not have been generated by P is encountered. 2) Construction of Transition Function fi : We set fi (q∅ , u, y) = q∅ for all (u, y) ∈ U × Y. Now consider a state 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

= =

.. .

(6) 

g fu2 ◦...◦ui (x)



Finally, we define hi (q) = vo for all q ∈ Qi,Initial ∪ {q∅ }, where   vo = hi arg max µ(hi (q)) . q∈Qi,F easible

4) Constuction of Ouput 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. III. A LGORITHMIC I MPLEMENTATION A. The Need for Lowering the Computational Burden An important first step in the process of constructing the ˆ i , is the process ith component of the ρ/µ approximation, M of constructing the set of feasible states of Mi , Qi,F easible . The brute force way consists of listing all possible sequences of i + 1 outputs and i inputs: (˜ y , y 1 , . . . , yi , u 1 , . . . , u i ) T ; where y˜, y1 , . . . , yi ∈ Y and u1 , . . . , ui ∈ U (call the set of all such sequences Si,P otential ), explicitly checking their feasibility (the feasible subset is denoted by Si,F easible ), and then merging feasible sequences with identical inputs and identical last i outputs into a single feasible state. Since the computational burden of this procedure grows exponentially with i, it is advantageous to find efficient algorithms for reducing the set of possible sequences whose feasibility needs

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to be checked. In Section III-B, we propose several such algorithms that make use of the computed data of component i − 1 of the sequence, thus leading to a streamlined iterative construction. In a similar vein, the computation of the gain bound γi of ∆i as well as the construction of the feedback controller grow with the number of states of Mi . It is therefore imperative to ensure that the realization of Mi is minimal, in order to alleviate the computational burden. In our initial implementations, we noticed significant redundancy in the states of Mi . In a bid to alleviate this redundancy, we propose to carry out state aggregation in two stages: The first ensures that the smallest realization possible is used for computing the gain bound, while the second ensures the most compact representation for the synthesis procedure. The details of these algorithms are given in Section III-C. B. Construction of Potential Feasible States In this section, we introduce a transition table, T T i , for step i, which contains feasible transitions among elements of the set, Si,F easible . Each row of the T T i corresponds to one element in the set, Si,F easible and each column of that row stores the destination state given a specified input of the form (u, y). The number of columns of the T T i is equal to |U| × |Y|. When a feasible sequence could not be found as a destination state, given an input, that cell of table is left empty to represent q∅ . As explained in Section III-A, the brute force way of constructing the set of potential feasible sequences is to list all possible permutations of i input and i+1 output alphabets, do feasibility check on each permutation, and eliminate unfeasible ones. However, the expense of computation required for truncating Si,P otentail to Si,F easible depends upon the size of Si,P otentail , which means a smaller Si,P otentail is desired. In fact, there are more efficient ways of constructing Si,P otentail . For instance, one way of constructing the set of potential feasible sequences of length i is using the information of feasible sequences of length i − 1. When a specific permutation of i − 1 input and i output alphabets are not feasible in step i − 1, it cannot be feasible in the next step as well. Therefore, a smaller Si,P otentail can be constructed immediately by injecting a pair of inputoutput to each feasible sequence of length i − 1. However, the set Si,P otential could be constructed even in a more computationally efficient way: Proposition 1. Let Si+1,F easible be the set of feasible sequences of step i + 1, T T i be the transition table of step i, 3 and Si+1,P otential be the set of potential feasible sequences of step i + 1 derived from T T i as follow: 3 s ∈ Si+1,P otential iff s = sa ⊎ sb for some (sa → sb ) ∈ i T T (meaning that sa has a transition to sb ) where sa ⊎ sb is defined as below: sa ⊎ sb := (˜ yb , y1b , . . . , yib , yia , u1b , . . . , uib , uia )T if sb = (˜ yb , y1b , . . . , yib , u1b , . . . , uib )T , sa = (˜ ya , y1a , . . . , yia , u1a , . . . , uia )T then: 3 Si+1,F easible ⊆ Si+1,P otential

Proof: Pick a s ∈ Si+1,F easible , s = (˜ y , y1 , . . . , yi , yi+1 , u1 , . . . , ui , ui+1 )T . Since s is a feasible sequence, sa := (y1 , y2 , . . . , yi+1 , u2 , . . . , ui+1 )T and sb := (˜ y , y1 , . . . , yi , u1 , . . . , ui )T are feasible and belong to Si,F easible . In addition, (sa → sb ) is in T T i because (y1b , . . . , yib , u2b , . . . , uib )T = (˜ ya , y1a . . . , yi−1a , u1a , . . . , ui−1a )T . (See Section IIC) Moreover: s = sa ⊎ sb 3 Therefore, s ∈ Si+1,P otentail . Since we picked s arbitrarily 3 from Si+1,F easible and showed that s ∈ Si+1,P otential : 3 Si+1,F easible ⊆ Si+1,P otential . Proposition 1 provides an alternative way of constructing the set of potential feasible sequences. Proposition 2 clarifies the advantage of the proposed method. 1 Proposition 2. Let Si+1,P otential be the potential set of feasible sequences of step i + 1 constructed with the brute 2 force method, Si+1,P otential be the one constructed using 3 the set of feasible sequences of step i, and Si+1,P otential be the one constructed using proposition 1 ( using the T T i ). Then: 1 i+1 i+2 |Si+1,P p otential | = m

(7)

2 |Si+1,P otential | = mp|Si,F easible |

(8)

3 m|Si,F easible | ≤ |Si+1,P otential | ≤ mp|Si,F easible |

(9)

where m = |U| and p = |Y|. Proof: The statement can be proved directly from the definition of the sets: T 1 Si+1,P otential = {s = (y0 , y1 , . . . , yi+1 , u1 , . . . , ui+1 ) |

uj ∈ U , yj ∈ Y, 0 ≤ j ≤ i + 1}

2 T Si+1,P otential = {s = (y0 , y1 , . . . , yi+1 , u1 , . . . , ui+1 ) |

u1 ∈ U , y0 ∈ Y, (y1 , y2 , . . . , yi+1 , u2 , . . . , ui+1 )T ∈ Si,F easible } 3 T Si+1,F easible = {s = (y0 , y1 , . . . , yi+1 , u1 , . . . , ui+1 ) |

(y0 , y1 , . . . , yi , u1 , . . . , ui )T , (y1 , y2 , . . . , yi+1 , u2 , . . . , ui+1 )T ∈ Si,F easible } 2 3 ≤ ≤ |Si+1,P Proposition 3. |Si+1,P otential | otential | 1 |Si+1,P |. otential Proof: The left inequality is trivial. It remains to show the right inequality. From Equation 7, it can be concluded that |Si,F easible | ≤ mi pi+1 (due to the fact that the potential set is a superset of the feasible set). Applying Equation 8: 2 1 |Si+1,P otential | ≤ |Si+1,P otential |. To illustrate the significance of this improvement, the size of the three sets of potential feasible sequences are compared in Table I for the system presented in Section IV.

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TABLE I P OTENTIAL FEASIBLE SEQUENCES FOR TANK EXAMPLE , T=60 i 1 2 3 4 5 6

1 Si,P otential 12 72 432 592 15552 93312

2 Si,P otential 12 48 156 480 1452 4368

3 Si,P otential 12 32 86 248 734 2192

i=i+1

Si,F easible 8 26 80 242 728 2186

Construct Si,F easible

Construct T T i

1 Si,P otential : Constructed by brute force 2 Si,P otential : Constructed using Si−1,F easible 3 i−1 Si,P otential : Constructed using T T

Merge to get Qi,F easible C. Multi-Stage Aggregation

Add Qi,initial and qφ

Aggregation stage 1

Calculate γ

Compute vˆ

Aggregation stage 2

D. A Flowchart of the Algorithmic Implementation The flowchart of the algorithm is shown in Figure 3. IV. I LLUSTRATIVE E XAMPLE

Control Synthesis

Consider a tank in which the level of the water is controlled by a bimodal pump. The tank has a cross-sectional area A=100 cm2 and height h=30 cm. The pump controls the water level with three operating modes: mode ’1’ in which the pump feeds water at a rate of 1 liter/minute, mode ’-1’ in which the pump removes water at a rate of 1 liter/minute, and ’0’ in which the pump does not add or remove water. A sensor is attached to the tank that can only determine whether the level of the water is below 15 cm or above. The dynamics of the system with sampling rate T are described by:

Achieved performance satisfactory?

Refine performance objective

The constructed DFM can be truncated to an equivalent DFM (with a smaller number of states) without affecting the approximation error. To construct the equivalent DFM, the states that have the same transitions for every input, and produce the same set of outputs are aggregated into one state. Our simulations show that the aggregated DFM has a noticeably smaller number of states and as expected, produces the same results. Since the performance output, vˆ(q), is calculated using ˆ X(q), and also depends on the range of performance objective, we need to repeat aggregation for each desired range. In order to reduce such repeating computation, we divided the aggregation algorithm into two parts: agg1 and agg2. ˆ Agg1 aggregates all states having the same X(q) and the same transitions while agg2 aggregates those having the same performance output and the same transitions. The comparison of the number of states of the DFMs and its aggregated versions are shown in Table II for the example explained in Section IV.

No

Yes   min(x(t) + T6 , 30), u(t) = 1 x(t), u(t) = 0 x(t + 1) =  max(x(t) − T6 , 0), u(t) = −1

and the output is given by:  F, y(t) = E,

Finish

Fig. 3.

x(t) ∈ (15, 30] x(t) ∈ [0, 15] 6739

Flowchart of the algorithmic implementation

where F stands for full and E stands for empty. The ultimate objective is to design a controller which can impose input sequences on the pump such that the water level converges to a desired range. For a given i, the approximated DFM is constructed, the approximation error, γ, is calculated, and then a value iteration approach is used to synthesize the controller. Table II shows the simulation results for different values of T and i in the construction of the deterministic finite state machine. Note that, the algorithm is sensitive to boundaries of sensor output and may produce numerical errors which may slightly change the numbers shown in the table. The labels used in the table can be described as below: • Nf is the number of feasible sequences • N0 corresponds to the number of initial states • N shows the total number of states after merging and before aggregation • Nagg1 shows the number of states after aggregation 1 • Nagg2 shows the number of states after aggregation 2 It is worth noting that the computed γ for all the deterministic finite state machines shown in the table is 1 which is acceptable in terms of controller design. In other words, a finite γ shows that we can attempt to design a controller for the real plant but using the approximated machine. Table III,

higher amount of memory (i). The reason is that when T is small, the water level changes with a smaller step. Thus, more accurate ranges of water level are achievable. However, in comparison with a larger T , each state of the deterministic finite state machine provides less information about the real plant’s next output. During simulations, we noticed that, having exactly the same constructed feasible states but possibly different output functions, and the same simulation parameters, the control synthesis algorithm may converge in one run and not in another one. We conjecture that, this is due to the arbitrary choice of output function and needs further investigation. Figure 4, shows the system’s state trajectories for T=42 and TABLE III S IMULATION

RESULTS FOR DIFFERENT EXPERIMENTAL SETUPS

T

midpoint 10

15 14

10 30

TABLE II

14

S IMULATION RESULTS OF CONSTRUCTING DETERMINISTIC FINITE STATE MACHINE

T 15

30

60

states Nf N0 N 0 + Nf N Nagg1 Nagg2 Nf N0 N 0 + Nf N Nagg1 Nagg2 Nf N0 N 0 + Nf N Nagg1 Nagg2

i=2 28 7 35 31 27 17 28 7 35 31 27 16 26 7 33 31 25 12

i=3 94 31 125 115 71 38 92 31 123 115 65 39 80 31 111 109 55 23

i=4 308 115 423 397 177 86 292 115 407 391 137 89 242 109 351 349 110 39

i=5 994 397 1391 1321 408 199 908 391 1299 1267 254 168 728 349 1077 1075 201 59

i 2 3 4 2 3 4 5 2 3 4 2 3 4 5

ε Not Feasible Not Feasible 10 14 11 6 4 10 10 11 9 6 8 8

different lengths of memory, i. Figure 5 shows the sequences of control inputs for Figure 4. Figure 6 shows the state trajectory for some other values of T and v. T=42, midpoint=20 30

25

20

15

10

5

0

shows the simulation results for different experiment setups. The minimum achievable range is shown by ε, i.e. achieved range is (midpoint − ε, midpoint + ε). The value iteration is assumed to converge, yielding a feasible controller, when the difference between the two iterations becomes less than a predefined tolerance (θ). The values of θ are between 0 and 1 in the presented simulations. ’Not Feasible’ indicates that for this specific setup, a controller, that can drive the approximate model to the desired range for all initial water levels, could not be synthesized. As i increases, ε decreases until it levels off. In addition, when T decreases, the minimum achievable range decreases but becomes achievable with a

i=3 i=4 i=5

0

5

Fig. 4.

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30

35

40

45

50

Comparing different selections of i

V. FUTURE WORK Future work will focus on developing more interesting illustrative examples, as well as developing better constructive algorithms that would allow us to avoid the expensive state explosion followed by aggregation procedure that is observed at every iteration ’i’ in the current algorithm.

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VI. ACKNOWLEDGMENTS This research was supported by NSF CAREER award ECCS 0954601 and AFOSR YIP award FA9550-11-1-0118.

2

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Fig. 5.

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Comparing the state trajectory for different initial values

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