0018-9286 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAC.2016.2558156, IEEE Transactions on Automatic Control JOURNAL OF L A T E X CLASS FILES, VOL. 11, NO. 4, DECEMBER 2012 1 A Study on Stability of the Interacting Multiple Model Algorithm Inseok Hwang, Member, IEEE, Chze Eng Seah, and Sangjin Lee Abstract—The Interacting Multiple Model (IMM) algorithm has been used in various applications with good performance. However, to the best of our knowledge, stability conditions of the IMM algorithm have not been shown in the control literature. This paper presents a sufficient condition for the exponential stability of the IMM algorithm for a class of Markov jump linear systems. We derive a lower bound and an upper bound for the error covariances of the IMM algorithm, and then derive the stability condition by the Lyapunov approach. Index Terms—Interacting Multiple Model algorithm, Multiple Model Kalman Filter, Hybrid Systems, Stability Analysis I. I NTRODUCTION T HE Interacting Multiple Model (IMM) algorithm [1] is a hybrid state estimation algorithm for Markov jump linear systems. It has been shown to yield good performance with low computational cost in various applications, such as target tracking [2], fault diagnosis [3], and classifications [4]. In spite of its popularity, the stability of the IMM algorithm has been barely analyzed. To the best of our knowledge, no conditions which ensure the stability of the IMM algorithm have been given in the control literature. Only few studies have been done for evaluating the performance of the IMM algorithm [5], [6]. Finding stability conditions is crucial, as they can be used for designing appropriate filter parameters and guaranteeing the estimation performance. In this paper, we present a sufficient condition for the stability of the IMM algorithm for a class of Markov jump linear systems, motivated by the Lyapunov approach in [7]. The IMM algorithm uses a set of Kalman filters, each of which uses a mixed initial condition, that is each filter uses a different combination of the Kalman filter estimates computed at the previous time-step. The resulting coupling among the means and covariance update equations of the Kalman filters makes the stability analysis of the IMM algorithm extremely challenging. We develop a sequence of steps to overcome this complexity. Comparing to our previous work [6], [11], this paper presents the main theoretical results more rigorously with formal theorems and the corresponding mathematical proofs. II. BACKGROUND AND MOTIVATION A. Review of the IMM Algorithm The Interactive Multiple Model (IMM) algorithm uses a bank of Kalman filters, each matched to a mode of the I. Hwang is an Associate Professor in the School of Aeronautics and Astronautics, Purdue University, Email: ihwang@purdue.edu C. E. Seah is a Principal Member of Technical Staff in DSO National Laboratories, Singapore, Email: schzeeng@dso.org.sg S. Lee is a PhD Student in the School of Aeronautics and Astronautics, Purdue University, Email: lee997@purdue.edu following Markov jump linear system: x(k)= A m(k) x(k − 1) + B m(k) w m(k) (k) (1) z(k)= C m(k) x(k)+ v m(k) (k) (2) where x ∈ R n is the state of the system and z ∈ R p is the measurement vector; A m(k) , B m(k) , C m(k) are the system matrices correspond to a mode m(k) ∈{1, 2,...,r} at time k; w m(k) (k) and v m(k) (k) are uncorrelated white zero-mean Gaussian noise vectors with covariance matrices Q m(k) and R m(k) respectively. The evolution of the mode m(k) is given by p[m(k)= j |m(k − 1) = i]= π ij for i, j =1,...,r, where π ij is a constant; p[·|·] denotes a conditional probability. We assume that, for all i, j =1,...,r, A j is non-singular and 0 <ξ 1 I ≤ Q j ≤ ξ 2 I, 0 <ξ 3 I ≤ R j ≤ ξ 4 I (3) where I is the identity matrix. The matrix inequality A>B (A ≥ B) means that A − B is positive definite (semi-definite). Let Z k := {z(1),z(2),...,z(k)} denote the set of mea- surements up to time k. The IMM algorithm computes the approximate posterior mean ˆ x j (k) and covariance P j (k) for each Kalman filter j , and the mode probability α j (k) := p[m(k)= j |Z k ] recursively as follows [1]. Let us assume that, α j (k − 1), ˆ x j (k − 1), and P j (k − 1) for j =1, 2,...,r are computed from the last iteration at time k − 1. Then, at time k, the IMM algorithm computes the followings: 1. Mixing: Compute the mixing probability γ ji (k − 1) := p[m(k − 1) = i|m(k)= j, Z k-1 ]= 1 ∑ r l=1 π lj α l (k-1) π ij α i (k − 1). The initial conditions to Kalman filter j are then given by ˆ x j0 (k − 1) = r  i=1 γ ji (k − 1)ˆ x i (k − 1) (4) P j0 (k − 1) = r  i=1  P i (k − 1) + [ˆ x i (k − 1)− ˆ x j0 (k − 1)][ˆ x i (k − 1) − ˆ x j0 (k − 1)] T  γ ji (k − 1) (5) 2. Filtering: Each Kalman filter j computes ˆ x j (k)= A j ˆ x j0 (k − 1) + K j (k)ν j (k) (6) K j (k)= P j (k|k − 1)C T j S -1 j (k) (7) P j (k|k − 1) = A j P j0 (k − 1)A T j + B j Q j B T j (8) P j (k)=[P -1 j (k|k − 1) + C T j R -1 j C j ] -1 (9) where ν j (k)= z(k) −C j A j ˆ x j0 (k −1) is the residual, K j (k) is the Kalman filter gain, S j (k)= C j P j (k|k − 1)C T j + R j is the residual covariance, and P j (k) (P j (k|k − 1)) is the posterior (prior) state covariance.