Outline Problem: identifying an ARX systems via binary sensors - PowerPoint PPT Presentation

Recursive Estimation of ARX Systems Using Binary Sensors with Adjustable Thresholds R ECURSIVE E STIMATION OF ARX S YSTEMS U SING B INARY S ENSORS WITH A DJUSTABLE T HRESHOLDS aji 1 , 2 Erik Weyer 1 Bal´ azs Csan´ ad Cs´ (1) Department of Electrical and Electronic Engineering, The University of Melbourne (2) Computer and Automation Research Institute, Hungarian Academy of Sciences 16th IFAC Symposium on System Identification, Brussels, July 11–13, 2012 Bal´ azs Csan´ ad Cs´ aji and Erik Weyer –1–

Recursive Estimation of ARX Systems Using Binary Sensors with Adjustable Thresholds Outline • Problem: identifying an ARX systems via binary sensors • Previous solutions typically assumed fully known noise characteristics • They also assumed that the input signal can be chosen by the user • We try to reduce the assumptions on the noise and the input • Full knowledge on the distribution is not needed; the input is only observed • But, the threshold of the binary sensor can be controlled ∼ dither signal • Here, two recursive identification algorithms are proposed • Algorithm I: FIR approximation; it is proved to be strongly consistent • Algorithm II: simultaneous state and parameter estimation (simulations) Bal´ azs Csan´ ad Cs´ aji and Erik Weyer –2–

Recursive Estimation of ARX Systems Using Binary Sensors with Adjustable Thresholds Structural Overview P ART I. Problem Setting (ARX System via Binary Sensors, Dithering, Assumptions) P ART II. General Form of the Algorithms (Sign-Error, Step-Sizes, Expanding Truncation Bounds) P ART III. Recursive Identification: Algorithms I and II (FIR Approximation, Strong Consistency, Simultaneous Estimation) P ART IV. Experimental Results (Simulation: Algorithms I and II on an ARX(2,2) System) P ART V. Summary and Concluding Remarks (Main Ideas, Contributions and Highlights) Bal´ azs Csan´ ad Cs´ aji and Erik Weyer –3–

Recursive Estimation of ARX Systems Using Binary Sensors with Adjustable Thresholds Problem Setting • We observe an ARX system via a binary sensor: p q ∑ ∑ � X t a ∗ i X t − i + b ∗ i U t − i + N t , i =1 i =1 � Y t I ( X t ≤ C t ) , where X t — output (hidden state), U t — input, N t — noise (at time t ) • The thresholds of the binary sensor, ( C t ) t , can be controlled at each t • Data: the inputs ( U t ) t and the binary outputs ( Y t ) t are observed ( ) • Aim: to identify (estimate) θ ∗ = ∈ R p + q a ∗ 1 , . . . , a ∗ p , b ∗ 1 , . . . , b ∗ q Bal´ azs Csan´ ad Cs´ aji and Erik Weyer –4–

Recursive Estimation of ARX Systems Using Binary Sensors with Adjustable Thresholds Adjustable Thresholds ∼ Dithering • The binary output can be rewritten as t θ ∗ + N t ≤ C t ) = I ( ϕ T t θ ∗ + N t − C t ≤ 0) , Y t = I ( ϕ T where ϕ t = ( X t − 1 , . . . , X t − p , U t − 1 , . . . , U t − q ) — random regressor • Choosing the threshold is equivalent to dithering � � � � (a) adjustable threshold (b) dither signal perspective perspective 1 1 � ∗ (�) � ∗ (�) −� � � ∗ (�) � ∗ (�) � � � � � � � � � ∗ (�) � ∗ (�) � � � � binary sensor with a binary sensor with an fixed threshold at zero adjustable threshold Bal´ azs Csan´ ad Cs´ aji and Erik Weyer –5–

Recursive Estimation of ARX Systems Using Binary Sensors with Adjustable Thresholds System Assumptions • ( N t ) t is i.i.d., continuous, zero mean, zero median, has a finite variance: σ 2 n � E [ N 2 t ] < ∞ , and has a continuous and positive density at zero • ( U t ) t is i.i.d., zero mean, ( U t ) t and ( N t ) t are independent, and 0 < σ 2 u < ∞ , where σ 2 u � E [ U 2 t ] • The system is stable, i.e., the roots of A ∗ ( z ) lie strictly inside the unit circle; additionally, the transfer function B ∗ ( z ) /A ∗ ( z ) is irreducible, 1 z − 1 − a ∗ 2 z − 2 − · · · − a ∗ p z − p , � A ∗ ( z ) 1 − a ∗ 1 z − 1 + b ∗ 2 z − 2 + · · · + b ∗ � q z − q , B ∗ ( z ) b ∗ where z − 1 is the backward shift operator, z − i x t � x t − i . • The orders p and q are known Bal´ azs Csan´ ad Cs´ aji and Erik Weyer –6–

Recursive Estimation of ARX Systems Using Binary Sensors with Adjustable Thresholds General Form of the Algorithms • The general form of both proposed algorithms is [ ( ) ] ˆ ˆ t ˆ ϕ T θ t +1 = Π M µ ( t ) θ t + α t � ϕ t 1 − 2 I ( X t ≤ � θ t , where � ϕ t is a regression vector defined differently in the two algorithms, ( α t ) t is a sequence of step-sizes and Π M µ ( t ) is a sequence of projections • Assuming that N t is continuous, we ( P -a.s.) have t ˆ t ˆ ϕ T ϕ T sign ( X t − � θ t ) = 1 − 2 I ( X t ≤ � θ t ) , • Thus, the above algorithm will behave almost surely as [ ] ˆ ˆ t ˆ ϕ T θ t +1 = Π M µ ( t ) θ t + α t � ϕ t sign ( X t − � θ t ) , which is a sign-error type algorithm with expanding truncation bounds Bal´ azs Csan´ ad Cs´ aji and Erik Weyer –7–

Recursive Estimation of ARX Systems Using Binary Sensors with Adjustable Thresholds Step-Sizes • Typical step-size assumption of stochastic approximation algorithms ∑ ∞ α t = ∞ , t =0 ∑ ∞ α 2 t < ∞ , t =0 ∀ t ≥ 0 : α t ≥ 0 . The second condition can often be weakened to lim t →∞ α t = 0 • Here, we will simply assume that α 0 = 1 ∀ t > 0 : α t = 1 /t. and Bal´ azs Csan´ ad Cs´ aji and Erik Weyer –8–

Recursive Estimation of ARX Systems Using Binary Sensors with Adjustable Thresholds Expanding Truncation Bounds • Let ( M t ) t be a sequence of (strictly) monotone increasing positive real numbers with M t → ∞ as t → ∞ , • Let I ( · ) be the indicator function and define µ ( t ) and ∆ˆ θ i as t − 1 ∑ ( ) | ˆ θ i + ∆ˆ µ ( t ) � θ i | > M µ ( i ) , I i =1 ( ) ∆ˆ i ˆ ϕ T θ i � α i � ϕ i 1 − 2 I ( X i ≤ � θ i ) . • Given a positive real M , projection Π M is   x if ∥ x ∥ ≤ M, Π M ( x ) �  0 otherwise . Bal´ azs Csan´ ad Cs´ aji and Erik Weyer –9–

Recursive Estimation of ARX Systems Using Binary Sensors with Adjustable Thresholds Algorithm I: FIR Approximation • Using impulse responses, ( c ∗ i ) ∞ i =1 and ( d ∗ i ) ∞ i =0 , we have ∑ ∑ ∞ ∞ X t = c ∗ i U t − 1 + d ∗ i N t − i , i =1 i =0 • Let’s approximate our ARX system with an FIR system of order p + q θ ∗ + W t , t ¯ ϕ T X t = ¯ θ ∗ � ( c ∗ ¯ ϕ t � ( U t − 1 , . . . , U t − p − q ) T , p + q ) T . ¯ 1 , . . . , c ∗ • W t is simply the unmodelled part of the system ∑ ∑ ∞ ∞ W t � c ∗ d ∗ i U t − i + i N t − i . i = p + q +1 i =0 Bal´ azs Csan´ ad Cs´ aji and Erik Weyer –10–

Recursive Estimation of ARX Systems Using Binary Sensors with Adjustable Thresholds Algorithm I: FIR Approximation • If we can estimate ¯ θ ∗ , we can also estimate the true parameter vector θ ∗ • There is a function f , which we use for post processing, such that θ ∗ = f (¯ θ ∗ ) , ϕ t � ¯ • Algorithm I is defined by using � ϕ t in the General Algorithm Theorem 1 (Strong Consistency of Algorithm I) . Let (ˆ θ t ) ∞ t =0 be the sequence generated by Algorithm I (i.e. � ϕ t = ¯ ϕ t ). Then, under the given assumptions, f (ˆ θ t ) converges ( P -a.s. ) to θ ∗ , as t → ∞ , for any ˆ θ 0 ∈ R p + q . √ t (ˆ θ t − ¯ • Furthermore, θ ∗ ) is approximately normal Bal´ azs Csan´ ad Cs´ aji and Erik Weyer –11–

Recursive Estimation of ARX Systems Using Binary Sensors with Adjustable Thresholds Algorithm II: Simultaneous Estimation • Main idea: to achieve a direct estimate of θ ∗ by simultaneously maintaining an estimate for the output, � X t and for the parameter, ˆ θ t , at time t . • The sequence of output estimates is defined as  ∑ p X t − 1 + ∑ q  a t,i � i =1 ˆ i =1 ˆ b t,i U t − i if t ≥ 0 � X t �  0 otherwise , i =1 and (ˆ a t,i ) p b t,i ) q where (ˆ i =1 are the estimates of the true parameters. • Algorith II: is defined by setting the General Algorithm as ( � X t − 1 , . . . , � X t − p , U t − 1 , . . . , U t − q ) T , � ϕ t � ˆ a t,p , ˆ b t, 1 , . . . , ˆ b t,q ) T . � θ t (ˆ a t, 1 , . . . , ˆ Bal´ azs Csan´ ad Cs´ aji and Erik Weyer –12–

Recursive Estimation of ARX Systems Using Binary Sensors with Adjustable Thresholds Simulation Experiment: ARX(2, 2) 0.7 0.6 0.5 parameter estimates 0.4 0.3 0.2 0.1 0 0 200 400 600 800 1000 t: time steps Figure 1: Recursive estimation with Algorithm I Bal´ azs Csan´ ad Cs´ aji and Erik Weyer –13–

Recursive Estimation of ARX Systems Using Binary Sensors with Adjustable Thresholds Simulation Experiment: ARX(2, 2) 0.7 0.6 0.5 parameter estimates 0.4 0.3 0.2 0.1 0 0 200 400 600 800 1000 t: time steps Figure 2: Recursive estimation with Algorithm II Bal´ azs Csan´ ad Cs´ aji and Erik Weyer –14–