Recent Developments on the Stability and Control of Stochastic SystemsView this Special Issue
Study on Index of Stochastic Linear Continuous-Time Systems
This paper studies the index problem. We obtain a necessary and sufficient condition of index larger than . A generalized differential equation is introduced and it is proved that its solvability and the feasibility of the index are equivalent. We extend the deterministic cases to the stochastic models. Our results can be used to fault detection filter analysis. Finally, the effectiveness of the proposed results is illustrated by an example.
It is well known that many control and filtering problems have been discussed based on a certain performance index of a system, such as norm, norm, and index; see [1–9]. norm is the measure of the worst-case disturbance inputs on the controlled outputs [1–4]. The index is a measure of the minimum sensitivity of system outputs to system inputs. norm and index with specific application to fault detection filter have been carried out in [10–17]. To ensure robustness, index should be maximized and norm should be minimized. Using performance can make certain that the residual signal is maximally sensitive to faults and highly robust to disturbance inputs; see [16, 17].
In , index was defined as the minimum nonzero singular value in zero frequency. In , the authors extended the results of  to all frequency range. By means of LMIs, a necessary and sufficient condition was given for the infinite frequency range. The case for finite frequency range was concluded through frequency weighting. In recent decades, a great deal of attention has been attracted to index in time domain. A fault residual generator was designed to maximize the fault sensitivity in the finite time domain [16–20]. Based on index, results on optimal fault detection can be found in [17, 18] and the references. The lower bound of index for linear time-varying systems was proposed in [19, 20]. A sliding mode observer was designed for sensor fault diagnosis of nonlinear time-delay systems; see . In , a fault-tolerant controller was projected to compensate nonlinear faults by using a fuzzy adaptive fault observer.
Although there is much work on the index problem, to the best of our knowledge, very little work was concerned with the index in stochastic systems. In this paper, the index for stochastic linear continuous-time systems is discussed. The definition of the index is extended to the stochastic case. We present a necessary and sufficient condition of the index. A generalized differential equation is introduced and it is proved that its solvability and the feasibility of the index are equivalent. Comparing our results with the bounded real lemma [2, 9], it shows that the index is not completely dual to norm. The index discussed in this paper is only for tall or square systems. The reason for this is that index is zero for wide systems. But bounded real lemma for is applicable to any systems. Finally, the effectiveness of the given methods is illustrated by numerical example.
The outline of the paper is organized as follows. In Section 2, some efficient criteria are given for the index of stochastic linear systems in finite horizon. Section 3 contains an example provided to show the efficiency of the proposed results. Finally, we conclude this paper in Section 4.
Notations. is the field of real numbers. is the vector space of all matrices with entries in . is the set of all real symmetric matrices . is the transpose of matrix . is the inverse of . Given positive semidefinite (positive definite) matrix , we denote it by (). is the mathematical expectation. is identity matrix. is zero matrix. is the space of nonanticipative stochastic process with respect to increasing -algebras () satisfying , where . A square (wide or tall) system denotes a system when the number of inputs equals (is more than or less than) the outputs number.
2. Finite Horizon Stochastic Index
In this section, we will discuss the index problem of stochastic linear continuous-time systems. We give a necessary and sufficient condition of the index larger than in finite horizon.
Consider the following stochastic linear time-varying system :In the above, is the one-dimensional standard Wiener process defined on the complete probability space , with the natural filter generated by up to time . Consider , , and are the system state, control input, and regulated output, respectively. , , , , , and are coefficients with appropriate dimensions. For any and , there exists unique solution of (1).
The finite horizon stochastic index of system (1) can be stated as follows.
Definition 1. For stochastic system (1), given , its index in is defined aswhere .
Remark 2. If is fault signal and is the residual, then the index describes the smallest fault sensitivity of system (1). In this paper, we suppose that system (1) is tall or square because the index is zero for wide system.
Given and , letWe will study the following optimal problem:
Remark 3. It can be shown that is equivalent to the following inequality, .
Remark 4. When , (2) corresponds to the infinite horizon case.
Lemma 5. Suppose is continuously differentiable, . Then, for every , ,where ,
Proof. Let , , and denote the corresponding solution of (1). Applying Ito’s formula to and taking expectations, we have that, for any ,whereSowhich ends the proof.
Below, we prove the following theorem which is necessary in this paper.
Theorem 6. For (1) and some given , if the following differential Riccati equationadmits solution on , then .
Proof. By Lemma 5, for every , , , we conclude thatBy using completion of squares argument and the first equality in (11), we havewhere .
From , , to show , we define the operator : with its realization:Then exists, which is determined bywhere .
We assume that , , so there exists constant , such thatThat is, .
Now, we consider the following equation:where and this equation has unique solution , .
It is easy to see that (17) satisfies the following equation:
Lemma 7. Suppose and is the solution of (18). Then if , one obtainswhere is the solution ofwith andAs , then
Proof. Let us first prove that . Suppose this is false; then there exists , , such that for some . Then, for sufficiently small ,DefineUsing Lemma 7 with this and , we can derive that for andSince is continuous and , (27) is negative. Moreover, the condition implies . As a result, this is a contradiction. If , we can replace by and use a similar proof.
Next, let for any and . Replacing with in (18), we obtain the corresponding solution . Applying the previous step, we can deduce that . For any , set , . Let be the solution of (18) with replaced by and replaced by on . Then, , . By (22), for any , ,and so . By continuity, for all . As this holds for arbitrary , it follows that . This completes the proof.
Now, we present the following theorem which is important in this paper.
Theorem 11. Suppose system (1) is time-invariant and square and satisfies for given . Then (11) has a unique solution on for every . Moreover, is minimized by the feedback control:where satisfies and the optimal cost is
Proof. We prove that implies the existence of solution of (11) on . Using a contradiction argument, we suppose that (11) does not admit a solution. By the standard theory of differential equations, there exists unique solution backward in time on maximal interval (), and as , becomes unbounded.
Let , , , , by completing the squares; thenObviously,where .
Furthermore, we can see thatFrom Remark 10,where . Considering (35), we getwhich implies thatBy linearity, the solution of (1) with initial state satisfiesSuppose is the solution ofand we haveTake ,and it is easy to show thatIt follows thatIt is obvious that there exists constant such thatSo, there is constant such thatIn addition,From (45), we haveIn view of (35) and (39), it yieldsSo, can not become unbounded as , which means that (11) has unique solution on .
Setting , , in (17), from (31), we obtainHence satisfies (17), or equivalently (18). So By (31), and, in terms of Lemma 7, But by Lemma 8, Hence, minimizes and .
Theorem 12. If system (1) is time-invariant and square, for given , the following are equivalent:(i)Consider .(ii)The following equationhas unique solution on . Moreover, .
Remark 13. For given , if we replace , , , and with , , , and , respectively, and with in (1), we deduce the corresponding index andWhen , then . Using Theorem 12 to the modified data, it is easy to see that the following equationhas unique solution on . Moreover, .
Now, we are to show what happens as increases.
Proof. Suppose , , and . Let be optimal for on , and setBy the time invariance of , . Then,This means that decreases as increases for every .
3. A Numerical Example
Example 1. In system (1), we consider a two-dimensional linear stochastic system with the following parameters:Set , ; by solving (56), we can obtain the solutions of for which their trajectories are shown in Figure 1. If we set , then it yieldsIt is easy to see that , which verifies the rightness of Theorem 14.
In this paper, we have solved the index problem where both stochastic and deterministic perturbations are present. Necessary and sufficient condition for the lower bound of index is given by means of the solvability of a generalized differential equation. The proposed results are not completely dual to norm, and the effectiveness of the given methods is illustrated by numerical example.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by National Natural Science Foundation of China (Grants nos. 61174078, 61170054, and 61402265) and the Research Fund for the Taishan Scholar Project of Shandong Province of China.
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