Subspace State-Space Identification and Model Predictive Control of Nonlinear Dynamical Systems Using Deep Neural Network with Bottleneck

1 INTRODUCTION . Model predictive control ( MPC ) has been widely used as a practical method to control nonlinear dynamical systems ( Qin & Badgwell , 2000 ; Vazquez et al. , 2014 ; Karamanakos et al. , 2020 ) . As the name implies , MPC requires a prediction model , and in most cases , state-space models are used . When we want to apply MPC to a complex system whose structure and state are unknown , model construction from input-output data , i.e. , system identification , is required . For linear state-space system identification , the subspace methods ( Favoreel et al. , 2000 ; Katayama , 2005 ; van der Veen et al. , 2013 ) are widely used . Its fundamental idea is to reduce the dimensionality of the measured past and future input and output data , and it strongly depends on the linearity of the system . Although there have been continuous attempts to extend the subspace methods to nonlinear systems with specific structures such as Wiener and Hammerstein structures ( e.g. , Verhaegen & Westwick , 1996 ; Goethals et al. , 2005 ; Katayama & Ase , 2016 ) and related research has also been conducted in the field of machine learning as spectral learning of Hidden Markov Models ( Kawahara et al. , 2007 ; Hsu et al. , 2012 ; Kandasamy et al. , 2016 ) , there is still a lot of room for research in extending the subspace method to general nonlinear dynamical systems . On the other hand , recent advances in machine learning techniques have had an impact on the research field of system identification ( Ljung et al. , 2020a ; b ) . For example , deep state space models ( Rangapuram et al. , 2018 ) , and Kalman variational auto-encoder ( Fraccaro et al. , 2017 ) apply deep learning to state space models . Among those technologies , autoencoder is a notable dimensional reduction or feature extraction technique with deep neural networks , whose traits derives from topological constraints of own bottleneck structure . Although several pieces of research apply it to state-space system identification ( Masti & Bemporad , 2018 ; Beintema et al. , 2021 ) , these works do not provide a theoretical basis for obtaining the state sequence . This paper aims to propose a theoretically supported and interpretable nonlinear dynamical system identification method by simply extending the subspace method with neural networks . For the proposed method , it can be shown based on the observability of the target system that when a neural network with an appropriate bottleneck is trained , the state estimator and predictor are formed across the bottleneck layer . The obtained model can be interpreted based on the knowledge from linear system identification methods , and this aspect is explained through an illustrative example . In this paper , the set of integers and the natural numbers are denoted by Z and N , respectively . The set { sa , sa+1 , . . . , sb−1 , sb } is denoted as { sk } bk=a for conciseness . The n ×m zero matrix is denoted as 0n×m , and 0n : = 0n×1 . For a matrix M , M † is the Moore–Penrose inverse of M . 2 PROBLEM FORMULATION & NOTATION . We consider the following discrete-time nonlinear dynamical system : xt+1 = f ( xt , ut ) , yt = h ( xt ) ( 1 ) with t ∈ Z the time index , xt ∈ Rnx the state , yt ∈ Rny the output , ut ∈ Rnu the input , f : Rnx × Rnu → Rnx the state-transition map and h : Rnx → Rny the output map . As in the usual setting for system identification , xt is a latent variable and ut and yt are measurable with noise . In the following discussion , the fundamental properties of the proposed method for nonlinear target systems are presented in the absence of measurement noise and disturbances . The effects of noise and disturbance are indirectly provided for linear target systems in section 4 by showing that the proposed method is equivalent to the subspace identification method . For convenience , We define the sequence of input and output signals in the past and future from a certain time t as follows : upt : = ut−hp ut−hp+1 ... ut−1 , ypt : = yt−hp yt−hp+1 ... yt−1 , uft : = ut ut+1 ... ut+hf−1 , yft : = yt yt+1 ... yt+hf−1 ( 2 ) where the superscripts p and f indicate past and future , and hp ∈ N and hf ∈ N are horizons for corresponding directions , respectively . Using these notations , we formulate the problem as follows . Problem 1 . Given the measured input-output data { ( ut , yt ) } T+hf−1t=−hp+1 , and the design parameters hp , hf , nx̂ ∈ N , construct a model that consists of a state estimator E , which maps ( upt , y p t ) 7→ x̂t , and a predictor P , which maps ( x̂t , uft ) 7→ yft . Here , x̂t ∈ Rnx̂ is a state equivalent to xt but in an arbitrary coordinate system and T ∈ N indicates the size of the dataset . Remark 1 . In the usual state-space system identification , the goal is to construct a model of f and h , i.e. , a state-space model . However , in many applications , including model predictive control , the state estimator and multi-step ahead predictor are often reconstructed from the resulting state-space model . In particular , for nonlinear systems that are difficult to analyze , the merits of going through state-space models are questionable . Therefore , we focus here on the construction of the state estimator E and predictor P and only briefly discuss the identification of f and h in Appendix A . To guarantee Problem 1 to be solvable , we make some assumptions . With the vectors defined above ( 2 ) , we can write the system ( 1 ) as xt = f hp ( xt−hp , u p t ) , ypt = h hp ( xt−hp , u p t ) ( 3 ) where fk ( xt , [ u > t , . . . , u > t+k ] > ) : = f ( · · · f ( f ( xt , ut ) , ut+1 ) · · · , ut+k ) ( 4a ) hk ( xt , [ u > t , . . . , u > t+k ] > ) : = h ( xt ) h ◦ f ( xt , ut ) ... h ◦ fk−1 ( xt , [ u > t , . . . , u > t+k−1 ] > ) . ( 4b ) And define the uniform k-observability as follows . Definition 1 ( uniform k-observability ( Moraal & Grizzle , 1995 ) ) . If the mapping Rnx × ( Rnu ) k → ( Rny ) k × ( Rnu ) k by ( x , u ) 7→ ( hk ( x , u ) , u ) ( 5 ) is injective , the system ( 1 ) is said to be uniformly k-observable ( Moraal & Grizzle , 1995 ) . Then , we make the following assumptions and theorem . Assumption 1 . The system ( 1 ) is uniformly k-observable , where k = min ( hp , hf ) . Remark 2 . According to Stark et al . ( 1997 ) , typical dynamical systems are k-observable for k ≥ 2nx + 1 . Assumption 2 . The dimension of the state estimate is large enough , that is , nx̂ ≥ nx . Theorem 1 . Under Assumptions 1 and 2 , Problem 1 has a solution . Proof . From Assumption 1 , there exists the map Φhp : { ( up , hhp ( xp , up ) ) ∣∣ xp ∈ Rnx , up ∈ ( Rnu ) hp } → Rnx ( 6 ) that maps ( upt , y p t ) 7→ xt−hp , and the map Ψ : { ( up , hhp ( xp , up ) ) ∣∣ xp ∈ Rnx , up ∈ ( Rnu ) hp } → Rnx ( 7 ) that maps ( upt , y p t ) 7→ fhp ( Φhp ( upt , y p t ) , u p t ) also exists . Since the pair of maps Ψ and hhf is a pair of state estimator and predictor that is a solution to Problem 1 , it is guaranteed that there is a solution when nx̂ ≥ nx . 3 PROPOSED METHOD . This section presents the proposed nonlinear system identification approach . Also , nonlinear MPC based on the obtained model is described . 3.1 TRAINING STATE ESTIMATOR & PREDICTOR . In the proposed method , we train the neural network which has the structure of figure 1 , where the state estimator E : ( Rnu ) hp × ( Rny ) hp → Rnx̂ maps the past input and output to the current state and the predictor P : Rnx̂ × Rhfnu → Rhf×ny maps the current state and the future input to the future output . We set nx̂ < hpny for the bottleneck structure of the network to take a lower-dimensional state , and the training problem is formulated as the following minimization problem min E , P 1 T T∑ t=1 ∥∥∥yft − P ( E ( upt , ypt ) ︸ ︷︷ ︸ estimate of xt , uft ) ︸ ︷︷ ︸ prediction of yft ∥∥∥2 . ( 8 ) Since the state xt can not be measured , the state estimator E and predictor P are coupled in series and trained in the same way as the encoder and decoder in autoencoder . To solve ( 8 ) , we can utilize modern batch optimization algorithms for deep learning ( e.g . Adam by Kingma & Ba , 2014 ) . Here , we show that x̂t : = E ( u p t , y p t ) is also a state of the system in the ideal situation . Theorem 2 . Assume the training ( 8 ) is perfectly done , that is , for all xp ∈ Rnx , up ∈ ( Rnu ) hp , uf ∈ ( Rnu ) hf , the equation P ( E ( up , yp ) , uf ) = yf , ( 9 ) where yf ( xp , up , uf ) : = hhf ( x , uf ) , x ( xp , up ) : = fhp ( xp , up ) , yp ( xp , up ) : = hhp ( xp , up ) and arguments are omitted for conciseness , holds . Then , for all xp ∈ Rnx and up ∈ ( Rnu ) hp , E ( up , hhp ( xp , up ) ) is a state equivalent to fhp ( xp , up ) . Proof . From hf -observability of the system , the map Φhf : { ( uf , hhf ( x , uf ) ) ∣∣ x ∈ Rnx , uf ∈ ( Rnu ) hf } → Rnx that maps ( uf , yf ) 7→ x ( 10 ) exists , and the map Γ : Rnx̂ × ( Rnu ) hf → Rnx that maps ( x̂ , uf ) 7→ Φhf ( uf , P ( x̂ , uf ) ) ( 11 ) also exists . Then , a map obtained by fixing the second argument of Γ , e.g. , Γ0 ( x̂ ) : = Γ ( x̂,0hfnu ) ( 12 ) maps x̂ to x , and x̂ is proved to be a state . Remark 3 . The existence of the map from x̂ to x is ensured by the predictor P trained with a common input 0hfnu for all states as in ( 12 ) . Although this is just one way of constructing a map that is possible under the strong assumption of Theorem 2 , this implies that to obtain an appropriate state estimator , the training data must be rich enough to include situations where common inputs are applied in diverse states . Corollary 1 . When the assumption of Theorem 2 holds , P is the predictor that predicts the sequence of future output from a state and a sequence of future input .

Summary: This paper proposes a new system identification and model predictive control method based on neural networks for non-linear dynamical systems. The basic idea behind the proposed method comes from subspace identification for linear systems. In this sense, the proposed method can be regarded as a non-linear extension of subspace identification by deep neural networks.