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Slide 1
Good morning, everyone. My name is hongru Liu.Thank you all for being here today. I'm here to present my topic named "A Unified Bayesian Framework for Jointly Inferring System Order and Parameters of LTI State-Space Models", supervised by associate Professor Pengfei Wei.
Slide 2
My presentation is structured as follows: First, the Background and Methodology, then the core Results and their Discussion, and finally, our Conclusion.
Slide 3
Let's start with some background.
Slide 4
Since I'm a doctoral student in control theory, I will begin by the usage of sytem identification in control theory. Control theory has envolved from classical control like pole placemet to modern control like optimal control and robust control and then to model predictive control, which in my view doesn't belong to modern control anymore. In all these control methods, a precise mathematical model of the system is required in the process of controller design. However, in practical engineering applications, obtaining an accurate mathematical model can be challenging due to the complexity of real-world systems. Such as the control of spacecrafts, airplanes and aeroengines, the dynamics of these systems are often complex and not fully understood, making it difficult to derive accurate models from first principles.
Therefore, to apply a model-based control method, it is nessary to identify the black-box system from the input and output data, which is known as system identifycation.
Slide 5
In a Bayesian perspective, system identification can be viewed as an inference problem, where parameters of a model are treated as random variables with prior distributions. We infer the posterior distribution of these parameters given the observed data using Bayes's theorem. This approach allows us to quantify uncertainty in the parameter estimates and incorporate prior knowledge into the identification process.
From a state-space perspective, we often represent dynamic systems using state-space models, which consist of a set of first-order differential or difference equations with unknown system order and matrices, as shown in the slide A B C D are the system matrices, x is the state vector, u is the input vector, and y is the output vector. The system order n, which is the dimension of the state vector x, is also unknown and needs to be inferred from the data.
By combining the Bayesian approach with state-space models, we can formulate the system identification problem as jointly inferring the system order and parameters of LTI state-space models from observed input-output data. As shown in the slide, the estimated system state-space model with the estimated matrices Ahat, Bhat, Chat, Dhat can be used for various control applications.
Slide 6
In previous works, various methods have been proposed for system identification, trying to infer the A B C D matrices in LTI state-space models. Bryutkin demonstrate that by leveraging canonical forms, we can represent linear systems with a minimal set of parameters, thereby eliminating redundancies and resolving the identifiability issue. However, in these methods, the system order n is typically assumed to be known, and the HMC algorithm used in Bruytkin's work is not suitable for trans-dimensional sampling required for jointly inferring system order and parameters.
In this work, we used an RJ-MCMC framework to infer model order and model parameters at the same time. By performing a random walk directly on the number and ratio of real and complex eigenvalues, we can identify the system order and parameters simultaneously. This approach allows us to explore models of varying complexity and select the most appropriate one based on the observed data.
Slide 7
Now, let's move on to the methodology.
Slide 8
To better illustrate RJ-MCMC, let's first gain an intuitive grasp of probability transition in MCMC. The detailed balance condition for a standard MCMC algorithm is defined on a fixed-measure space. It compares the transition densities between two points without involving a change of coordinates or a transformation of the underlying measure.
As shown in the slide, x and x prime are two points both in the state space, meaning that they have the same probability measure. The transition density from x to x prime is equal to the transition density from x prime to x, weighted by their respective target densities pi of x and x prime. Here, the differential elements dx and dx prime represent infinitesimal volumes around the points x and x prime, respectively.
Thus, the detailed balance condition implies that the probability flux from state x to state x' equals the probability flux from x' to x.
Slide 9
However, RJ-MCMC uses auxiliary variables to achieve "dimension matching" for moves between spaces. Since this mapping distorts the underlying measure the detailed balance must be corrected, ultimately introducing a Jacobian determinant into the acceptance rate.
As shown in the slide, x is a point in the state space with probability measure dx, while mu omiga is point in another space with probability measure dmu omiga. Omiga represents the auxiliary variables used to match the dimensions between the two spaces. To ensure that the detailed balance condition holds when transitioning between these two spaces, the Jacobian determinant is introduced in the acceptance ratio.
In simple terms, the Jacobian determinant accounts for the change in volume when mapping between spaces of different dimensions, ensuring that the probability flux remains balanced.
Slide 10
In this slide, two types of random moves are illustrated: birth-death moves of complex conjugate eigenvalue pairs and real eigenvalue. In the birth move, a new complex conjugate eigenvalue pair or a real eigenvalue is added to the model, increasing the system order by one ortwo. Conversely, in the death move, an existing complex conjugate eigenvalue pair or a real eigenvalue is removed from the model, decreasing the system order by one or two. These moves allow the RJ-MCMC algorithm to explore models of varying complexity by adding or removing eigenvalues as needed.
Here in the slide, the auxiliary variables in the birth and death moves of complex conjugate eigenvalue pairs making the detail balance condition unhold, therefore, the Jacobian determinant is required in the acceptance ratio to correct for this distortion in the measure.
Slide 11
In this slide, two types of random moves are illustrated: Split move of complex conjugate eigenvalue pair and Merge move of two real eigenvalues. In the Split move, an existing complex conjugate eigenvalue pair is divided into two separate real eigenvalues, effectively changing the nature of the eigenvalues while keeping the system order the same. Conversely, in the Merge move, two existing real eigenvalues are combined to form a single complex conjugate eigenvalue pair, again maintaining the system order. These moves allow the RJ-MCMC algorithm to explore different configurations of eigenvalues without altering the overall system order.
Here in the slide, although the order of the system remains unchanged during Split and Merge moves, the auxiliary variables involved in these moves still distort the underlying measure. Therefore, the Jacobian determinant is required in the acceptance ratio to correct for this distortion and ensure that the detailed balance condition holds.
Slide 12
In this slide, we summerize the previous four types of random moves and additionally include the random walk move, which updates the parameters of existing eigenvalues without changing the system order or the number of real and complex eigenvalues. The random walk move allows for fine-tuning of the eigenvalue parameters, enabling the RJ-MCMC algorithm to explore the parameter space more thoroughly while maintaining the current model structure.
So in total, we have five types of random moves in our RJ-MCMC framework: in the early stages of training, mainly focus on birth-death processes to infer the model order; in the later stages, we emphasize split-merge processes to refine the configuration of eigenvalues; throughout the entire process, random walk moves are employed to optimize the parameters of existing eigenvalues.
Slide 13
To apply the RJ-MCMC framework, the posterior distribution or unnormalized posterior distribution of model parameters given by the observed data is required in the acceptance ratio of each random move. Here in this slide, we present the likelihood function of the observed input-output data given the model parameters using kalman filter to compute the one-step-ahead prediction errors. Here the notation of errors are mu t, which is the difference between the real output y real and the filtered output y filtered at time t. Thus the likelihood function can be computed as shown in the slide. Next, the total log-likelihood function over the entire data set is given by summing the log-likelihoods at each time step.
Slide 14
Originally, next we will cover the prior distributions and the proposal distributions of model parameters used in our RJ-MCMC framework. To make the presentation more intuitive, I have moved the example setup to the Results section, so we can directly see how these prior and proposal distributions are applied in practice.
Now, let's move on to the Results section.
Slide 15
In this slide, we will present the proposal distributions of birth-death moves of real eigenvalues and complex conjugate eigenvalue pairs.
For the birth move of a real eigenvalue, a new auxiliary variable u is sampled from a uniform distribution between -1 and 1, representing a stable real eigenvalues lambda k plus one. As shown in the slide, the original eigenvalue sets x are augmented with the new eigenvalue lambda k plus one to form the new eigenvalue sets x prime.
For the birth move of a complex conjugate eigenvalue pair, two new auxiliary variables \rho and \theta are sampled from uniform distributions. To ensure stability, rho is sampled between 0 and 1, while \theta is sampled between 0 and pi. These variables represent the magnitude and angle of the complex conjugate eigenvalue pair lumbda k plus one prime and lambda k plus two prime. Here in the slide, the original eigenvalue sets x are augmented with the new auxiliary variables \rho and \theta which have different probability measures, to form the new eigenvalue sets x prime. Therefore, the Jacobian determinant is required in the acceptance ratio to correct for this distortion in the measure.
Similarly, for the death moves of real eigenvalues and complex conjugate eigenvalue pairs, the corresponding auxiliary variables are removed from the eigenvalue sets to form the new eigenvalue sets x prime. Again, the Jacobian determinant is included in the acceptance ratio to account for the change in measure.
Slide 16
In this slide, we will present the proposal distributions of split-merge moves of real eigenvalues, complex conjugate eigenvalue pairs and the random walk move. For the merge move of two real eigenvalues, the two ramdom selected real eigenvalues lambda i and lambda p are merged into a new complex conjugate eigenvalue pair lambda k plus one prime and lambda k plus two prime. Here in the slide, the original eigenvalue set x is updated by removing the two selected real eigenvalues and adding the new auxiliary varables rho and theta, which have different probability measures, to form the new eigenvalue sets x prime. Therefore, the Jacobian determinant is required in the acceptance ratio to correct for this distortion in the measure.
For the split move of a complex conjugate eigenvalue pair, the selected complex conjugate eigenvalue pair lambda i and lambda i plus one is split into two new real eigenvalues lambda k plus one prime and lambda k plus two prime. Here in the slide, the original eigenvalue set x is updated by removing the selected complex conjugate eigenvalue pair and adding the new auxiliary variables u and v to form the new eigenvalue sets x prime.
Slide 17
In this slide, we present the prior distributions of model parameters and the order selection domain used in our RJ-MCMC framework. To ensure system stability, we assume that all the eigenvalues lie within the unit circle in the complex plane. And other parameters such as the output matrix C are assumed to follow Gaussian prior distributions with specified means and variances.
The two graph in the slide illustrate the posterior distribution of eigenvalues and the matrices C obtained from our purposed algorithm. The sheet shows that the coefficients of variation CV of the inferred eigenvalues and matrices C are relatively small. Moreover, the system order will be shown in the next section.
Slide 18
Now, let's move on to the discussion of results. In this part, we will focus on the convergence analysis of the system order.
Slide 19
Here in the slide, we present the convergence process of the system order n during our sampling procedure. From the praph, we can observe that after the burn-in process, the model order remains at 3, and its eigenvalues consist of one complex conjugate pair and one real value. This indicates that our RJ-MCMC framework successfully converges to the true system order and accurately identifies the eigenvalue structure of the underlying system. But the system order converges seems to be due to the convergence of other parameters, which is not as expected.
Slide 20
What's more, to prevent the system order from getting to high, or to say, to avoid overfitting, a Bayesian information criterion BIC penalty term is added to the log-likelihood function in the acceptance ratio of each random move. Here in the slide, we present the estimated likelihood of the observed data given the model order M. After adding the BIC penalty term, the estimated overall likelihood is shown in the slide.
Slide 21
Next, the conclusion section.