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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 296701, 13 pages
D-Optimal Design for Parameter Estimation in Discrete-Time Nonlinear Dynamic Systems
1The Control Science and Engineering Department, Harbin Institute of Technology, Harbin 150001, China
2The Department of Engineering, Faculty of Engineering and Science, University of Agder, 4898 Grimstad, Norway
Received 21 June 2012; Accepted 7 August 2012
Academic Editor: Bo Shen
Copyright © 2012 Yu Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
An optimal input design method for parameter estimation in a discrete-time nonlinear system is presented in the paper to improve the observability and identification precision of model parameters. Determinant of the information matrix is used as the criterion function which is generally a nonconvex function about the input signals to be designed. To avoid the locally optimizing problem, a randomized design method is proposed by which a globally optimizing test plan other than input signals may be obtained. Then the randomized design can be approximated by a nonrandomized design about optimal inputs. An iterative algorithm integrated with dynamic programming is given and verified by a numerical example on experimental design for self-calibration tests of ISP system.
Model parameters applied to computation or compensation in science and engineering, such as error model coefficients in INS (inertial navigation system), generally require much higher identification precision than in other applications. However, haphazard experiments not only lead to poor accuracy in parameter estimating, but also would make some parameters unobservable. A good experimental design can increase both the precision and the efficiency of a test  and then improve the precision of system identification or state estimation [2–5].
The field of system identification and filtering are relatively mature [6–10]; relevant experimental design methods have not made substantial advance yet. D-optimal design which allocates the experimental input variables by maximizing the determinant of information matrix of the system is recognized as the most effective method for an experimental design [11, 12]. For model parameter identification of a dynamic system, the D-optimal design problem has a similar mathematical expression as the optimal control problem, but cannot be solved by the Pontryagin’s maximum principle and dynamic programming method, due to particularity in the form of performance index .
On the other hand, even for a dynamic system with linear or low-order nonlinear model, the D-optimal design problems may involve global optimizing of nonconvex function and cannot achieve analytical solutions by traditional nonlinear programming methods. Although many numerical searching algorithms have been proposed to solve the nonconvex problem in global optimizing, such as genetic algorithm, simulated annealing algorithm, and so forth, most of them are either time-consuming or no guarantee of global optimization of searching results [14–16].
D-optimal design for randomized inputs is a convex optimization technique, in which the experimental variables are transformed to test plans. A test plan specifies different probability measures to each input variable in admissible set and one selects inputs for a particular trial of the experiment via randomization. The randomized design method is mainly used in regression design problems. Mehra introduces the method to optimal input design for parameter identification in a discrete-time MIMO linear system with process noise .
Morelli and Klein consider input design problem for LTI systems in aircraft flight tests, and the specific goals with test time optimization are achieved using principles of dynamic programming . Neto et al. generalize the results to nonlinear dynamic systems and consider additive colored noises in measurement . The cost function selected by Neto et al. is the trace of a dispersion matrix in which the autocorrelation matrix of the colored measurement noises is introduced. They solve the optimization problem by genetic algorithm.
Lintereur studies optimal trajectories for a 2-axis gimbaled test table by which errors in inertial systems caused by angular motion are calibrated . The trace of covariance matrix computed by Kalman filtering is minimized using a conjugate gradient algorithm, but local minimum may be obtained.
In this paper, we propose a randomized design method for parameter estimation in discrete-time nonlinear dynamic systems with constraints on inputs. By this design method, the original nonconvex optimization problem can be solved by the convex optimization technique, and the global optimal maximum is guaranteed. An iterative algorithm is given and verified by a numerical example on experimental design for continuous tumbling self-calibration tests of ISP system.
2. Problem Statement
In this section we give a mathematical formulation of the D-optimal design problem, in which a time-varying MIMO nonlinear system with unknown model parameters is considered. For simplification in notation and deduction, the process noise is assumed to be zero here.
Consider the following nonlinear dynamic system: where is a state vector, is a constant vector, is a input vector, is a sampling output vector at moment k, and is a measurement noise vector. is the Gaussian white noise sequences with and , is the number of output samples observed and is fixed. Since denotes the vector of constant identifiable parameters, we estimate from the knowledge of and give an unbiased efficient estimator with covariance , where is the Fisher information matrix. Therefore, the design problem is to select a series of inputs such that a suitable criterion function corresponding to the objectives of the identification experiment is optimized.
The Fisher information matrix is defined as follows: where denotes the set of observations and the expectation in (2.2) is taken over by the sample space of observations and the parameter space of .
Using conditional expectations, may be evaluated in two steps, first by computing and then . The second step is generally more tedious and an a priori distribution should be known exactly. Here a Taylor-series approximation is used to simplify the computation: where is the a priori mean of .
Retaining terms up to second order, where is the a priori covariance of .
The second term is typically small compared to the first term either because is small or is insensitive to .
The conditional likelihood function for system (2.1) is given as follows:
The matrix has elements
The sensitivity function is where is the partial derivative of about at moment k, that is, which meets the following equation:
Since and all depend on the elements of , where is the -dimensional vector to be designed, we denote as .
From (2.6), it is easy to get
Also from (2.4), the Fisher information matrix is generally
There are many formulations of criterion function that measures the degree of observability about parameter , such as or . A design which minimizes the scalar measure or maximizes is called D-optimal, and it is equivalent to minimizing the volume of the uncertainty ellipsoid about parameter estimators. An important advantage of D-optimality is that it is invariant under scale changes in the parameters and linear transformations of the output.
Now, we choose as the criterion function and formulate the D-optimal design problem as follows:
It should be pointed out that the problem in (2.11) cannot be solved by typical methods such as the Pontryagin’s maximum principle and dynamic programming method since or cannot be transformed to the index form in multistage decision process. In fact, there also exists great difficulty in getting a numerical solution with global optimization since is not a convex function of .
In the next section, we will present a randomized design method based on test plan theories.
3. D-Optimal Design Method
For a randomized input with probability measure defined for all Borel sets and points of , the definition of the information matrix is where .
If the probability measure is purely discrete, the information matrix is defined as follows: where is the number of spectrums, , .
is linear in , so the criteria or are convex functions of , and optimization with respect to gives globally optimizing design. However, we cannot find directly the optimal design which minimizes . Here, an iterative algorithm is proposed for searching based on the following theorem .
Theorem 3.1. Let be the optimal design then the following are equivalent:(i) maximizes ,(ii) minimizes ,(iii).
The D-optimal design may be computed with the following algorithm.
Algorithm 3.2. Step 1. Start with any design such that is nonsingular and let k=0.
Step 2. Compute and using (2.1), (2.6)–(2.10).
Step 3. Maximize over and get .
Step 4. If , stop. Otherwise, let , where is the design at the single point .
Choose such that , , .
Step 5. Set and go to step 2.
The convergence of the above algorithm to the global maximum is proved in the appendix.
Remarks. The optimal design can be depicted by the following set:
This can be used in a manner of randomized strategies when the experiment can be repeated. If the experiment is to be conducted only once, a nonrandomized design involving only one input should be preferable, that is, it assigns probability one to a particular . Since the randomized design (3.3) has been derived, we can seek nonrandomized design so that approximates .
Step 3 is most time-consuming computationally, and the criterion function is generally not a convex function of . Only if model (2.1) can be reduced to a linear discrete-time system, it would be a quadratic functional of . Using (2.9) and (2.10), we get
Unlike the computing of determinant, the operations with trace and sum of matrix can exchange order. By (3.4), the optimization problem can be solved by using maximum principle or dynamic programming methods since the above equation possesses the form of criterion function in multistage decision process.
Therefore, by using randomization and Theorem 3.1 the solution to a highly nonlinear and nonconvex optimization problem, that is, minimization of is reduced to solving a relatively simpler optimization problem. This is mainly due to the fact that randomization produces convexity.
In this section, we present a numerical example to verify the effectiveness of the proposed design method. The experiment to be designed is the continuous tumbling self-calibration test of an ISP system. Choose the determinant of information matrix as observability index and select the currents of gyro torquers or the command angular speed to the ISP as the experimental input variables, then the idea of D-optimal design can be applied to program the rotational trajectories of platform which represent the attitude and angular speed of platform at each moment.
First, we give the model equations of accelerometers and gyroscopes in the ISP system. The output equations of accelerometers that are also the observation equations are: where represent the misalignment angles of accelerometers’ input axes with respect to platform frame.
and represent the bias and scale factors of accelerometers.
represents outputs of accelerometers; denotes the observation noises in outputs with zero mean and covariance matrix at each moment.
, , are projections of gravitational acceleration on the ideal platform frame which is defined based on accelerometer’s input axis and initially aligned to north, west, vertical direction, unit , where is the local gravitational acceleration value: where represent three ideal angular positions of platform gimbals from outer to inner, respectively, and meet the following differential equations: where represent north and vertical components of rotational speed of the earth , respectively.
in (4.1) represent the attitude errors between the practical platform frame and ideal one: where represent the misalignment angles of gyroscopes’ input axes with respect to platform frame.
and represent the fixed drifts and scale factor errors of gyro torquers.
represent equivalent command angular speed to the ISP in an ideal platform frame and are the input variables to be designed.
Choose and as the state vector ; note that only the state of (4.4) is related to partial elements of unknown parameter vector , where
Therefore, the equations for can be derived by making partial derivative of (4.4) about .
Then the equations for are as follows: where denotes the th column of matrix , and is the dimension of . Where .
Thus, in this problem for any and state equation for can be deleted from the constraint conditions in (2.11).
Now we reformulate the design problem for self-calibration tests of ISP system as follows: where is the amplitude constraint on the precise command angular speed.
Since three state equations should be added to the constraint equations in (4.10) if one parameter in is to be estimated, the dynamic programming algorithm in Step 3 will be very time-consuming. Therefore, only 9 parameters in are considered in this example.
The initial design is chosen from a single-point design which maximizes via a rough search by confining each element of the input signals to a four-segment square wave varying between −10 , 0, and 10. Figure 1 shows the convergence of after eleven times of iterations, and tends to one which means converges to a local maximum . Since is a concave function about , is global optimizing as well.
The values of at each supporting point and measure are listed in Table 1 with . In addition, and . It shows that the randomized design has better performance than any single-point design , whereas the linear combination , although it is also an admissible control input in , shows poor performance and cannot be used as a nonrandomized approximation to the optimal test plan here.
In Table 1 the 4th supporting point has the maximum which approximates mostly to . So single-point design is chosen as the nonrandomized approximation to . Comparison of estimation errors between tests with and (the initial rough design) is in Table 2, which shows some improvement in budgets of estimation precision. The control input curves of are plotted in Figure 2 with unit . The whole test time is three hours, and the sampling time is 22.5 minutes. None of , , and in is the bang-bang type mainly because a constraint on angle () is also assumed. The projection of gravitational acceleration in the ideal platform frame is plotted in Figure 3.
The profiles of , , and show that input axis of each accelerometer completes nearly a whole tumble in gravitational field which can provide sufficient stimulations to the error terms of accelerometers in practical testing.
A D-optimal design method for parameter estimation in nonlinear dynamic systems is presented based on test plan design theories. The corresponding iterative algorithm is proposed with a dynamic programming algorithm imbedded. The proof for the convergence of the algorithm is given as well. Simulation results on an optimal trajectory design problem in self-calibration test of ISP system demonstrate the effectiveness of the proposed algorithm.
Proof of the Convergence of Algorithm 3.2
Proof. In the algorithm, is chosen such that
Since any bounded monotone nondecreasing sequence converges, the sequence converges to some limit .
To prove , we assume the contrary . Then, by Theorem 3.1, for any there is a constant such that
It follows that
For the smoothness of the function about and by the assumption , there is a positive integer such that for any , .
Now integrating both sides of the above inequality over from 0 to , one obtains
On the other hand, in view of the convergence of the sequence , for any small positive number , there is a positive integer such that for any integer , the following inequality holds:
Let , then for any integer ,
This means which contradicts the assumption .
Therefore, , the global maximum is obtained by the algorithm.
This work was supported by the Fundamental Research Funds for the Central Universities (Grant no. HIT. NSRIF. 2013037).
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