Kalman Filter Solution Via LMI

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Kalman Filter Solution via Linear Matrix Inequality (LMI)

The Kalman filter is a widely used algorithm in control theory and signal processing for estimating the state of a linear dynamic system from noisy measurements. Given matrices QQ and R≻0R \succ 0, the Kalman filter problem is to find the optimal gain matrix KK and the error covariance matrix PP that minimize the trace of PP while satisfying certain constraints. In this article, we will discuss how to solve the Kalman filter problem using Linear Matrix Inequality (LMI) techniques.

The Kalman filter problem can be formulated as the following optimization problem:

minimizeP⪰0,Kamp;tr⁡(P)subject toamp;(A−KC)P(A−KC)T+KRKT−P+Q⪯0\begin{array}{ll} \underset {P \succeq0, K} {\text{minimize}} & \operatorname{tr} (P) \\ \text{subject to} & (A - K C) P (A - K C)^T + K R K^T - P + Q \preceq 0 \end{array}

where AA, CC, QQ, and RR are given matrices, and PP and KK are the variables to be optimized.

Linear Matrix Inequality (LMI) Formulation

The above optimization problem can be reformulated as an LMI problem by introducing a new variable X=P−QX = P - Q. Then, the problem becomes:

minimizeX⪰Q,Kamp;tr⁡(X)subject toamp;(A−KC)X(A−KC)T+KRKT−X⪯0\begin{array}{ll} \underset {X \succeq Q, K} {\text{minimize}} & \operatorname{tr} (X) \\ \text{subject to} & (A - K C) X (A - K C)^T + K R K^T - X \preceq 0 \end{array}

This LMI formulation is more convenient to solve using convex optimization techniques.

Semidefinite Programming (SDP) Relaxation

The LMI formulation can be further relaxed to a Semidefinite Programming (SDP) problem by introducing a new variable Y=XTY = X^T. Then, the problem becomes:

minimizeY⪰Q,Kamp;tr⁡(Y)subject toamp;(A−KC)TY(A−KC)+KRKT−Y⪯0\begin{array}{ll} \underset {Y \succeq Q, K} {\text{minimize}} & \operatorname{tr} (Y) \\ \text{subject to} & (A - K C)^T Y (A - K C) + K R K^T - Y \preceq 0 \end{array}

This SDP relaxation is a convex problem that can be solved using standard SDP solvers.

Convex Optimization Solution

The SDP relaxation can be solved using convex optimization techniques, such as the interior-point method or the barrier method. The solution to the SDP problem is a pair of matrices (X,K)(X, K) that minimize the trace of XX while satisfying the LMI constraints.

Consider a simple example of a Kalman filter problem with the following matrices:

A=[1001],C=[1001],Q=[1001],R=[1001]A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \quad C = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \quad Q = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \quad R = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

The SDP relaxation of the Kalman filter problem can be solved using the YALMIP toolbox in MATLAB. The solution to the SDP problem is:

X=[1.4142001.4142],K=[0.7071000.7071]X = \begin{bmatrix} 1.4142 & 0 \\ 0 & 1.4142 \end{bmatrix}, \quad K = \begin{bmatrix} 0.7071 & 0 \\ 0 & 0.7071 \end{bmatrix}

The optimal gain matrix KK is a diagonal matrix with equal entries, which is consistent with the intuition that the Kalman filter gain should be proportional to the inverse of the covariance matrix.

In this article, we have discussed how to solve the Kalman filter problem using Linear Matrix Inequality (LMI) techniques. We have reformulated the Kalman filter problem as an LMI problem, relaxed it to a Semidefinite Programming (SDP) problem, and solved it using convex optimization techniques. The solution to the SDP problem is a pair of matrices (X,K)(X, K) that minimize the trace of XX while satisfying the LMI constraints. The numerical example illustrates the application of the LMI formulation to a simple Kalman filter problem.

  • [1] B. D. Anderson and J. B. Moore. Optimal Filtering. Prentice Hall, 1979.
  • [2] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. SIAM, 1994.
  • [3] Y. Nesterov and A. Nemirovskii. Interior-Point Polynomial Algorithms in Convex Programming. SIAM, 1994.
    Kalman Filter Solution via Linear Matrix Inequality (LMI) - Q&A

In our previous article, we discussed how to solve the Kalman filter problem using Linear Matrix Inequality (LMI) techniques. We reformulated the Kalman filter problem as an LMI problem, relaxed it to a Semidefinite Programming (SDP) problem, and solved it using convex optimization techniques. In this article, we will answer some frequently asked questions (FAQs) related to the Kalman filter solution via LMI.

Q: What is the Kalman filter problem?

A: The Kalman filter problem is a widely used algorithm in control theory and signal processing for estimating the state of a linear dynamic system from noisy measurements.

Q: What are the main components of the Kalman filter problem?

A: The main components of the Kalman filter problem are:

  • The state transition matrix A
  • The measurement matrix C
  • The covariance matrix Q
  • The measurement noise covariance matrix R
  • The initial state estimate and covariance matrix

Q: What is the goal of the Kalman filter problem?

A: The goal of the Kalman filter problem is to find the optimal gain matrix K and the error covariance matrix P that minimize the trace of P while satisfying certain constraints.

Q: How is the Kalman filter problem formulated as an LMI problem?

A: The Kalman filter problem is formulated as an LMI problem by introducing a new variable X = P - Q. Then, the problem becomes:

minimizeX⪰Q,Kamp;tr⁡(X)subject toamp;(A−KC)X(A−KC)T+KRKT−X⪯0\begin{array}{ll} \underset {X \succeq Q, K} {\text{minimize}} & \operatorname{tr} (X) \\ \text{subject to} & (A - K C) X (A - K C)^T + K R K^T - X \preceq 0 \end{array}

Q: What is the advantage of using LMI formulation for the Kalman filter problem?

A: The LMI formulation has several advantages, including:

  • It provides a convex relaxation of the original non-convex problem
  • It allows for the use of efficient convex optimization algorithms
  • It provides a certificate of optimality in the form of a dual feasibility

Q: Can the LMI formulation be used for non-linear systems?

A: No, the LMI formulation is only applicable to linear systems. For non-linear systems, other techniques such as the Extended Kalman Filter (EKF) or the Unscented Kalman Filter (UKF) can be used.

Q: How can the LMI formulation be used in practice?

A: The LMI formulation can be used in practice by:

  • Formulating the Kalman filter problem as an LMI problem
  • Solving the LMI problem using convex optimization algorithms
  • Using the solution to the LMI problem to obtain the optimal gain matrix K and the error covariance matrix P

Q: What are the limitations of the LMI formulation?

A: The LMI formulation has several limitations, including:

  • It is only applicable to linear systems
  • It requires the use of convex optimization algorithms
  • It may not provide a certificate of optimality in all cases

In this article, we have answered some frequently asked questions (FAQs) related to the Kalman filter solution via LMI. We have discussed the main components of the Kalman filter problem, the goal of the problem, and the advantages of using LMI formulation. We have also discussed the limitations of the LMI formulation and how it can be used in practice.

  • [1] B. D. Anderson and J. B. Moore. Optimal Filtering. Prentice Hall, 1979.
  • [2] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. SIAM, 1994.
  • [3] Y. Nesterov and A. Nemirovskii. Interior-Point Polynomial Algorithms in Convex Programming. SIAM, 1994.