Abstract
This article is concerned with the stability analysis of a conservation law with PI and PDP boundary feedback control. In the PI controller case, the authors establish a sufficient and necessary stability condition for feedback parameters by Walton–Marshall criterion. Moreover, in PDP controller case, the characteristic equation is an exponential polynomial which includes two exponential arguments. Based on Ruan’s results on stability for quasi-polynomials and Schur–Cohn criterion on the roots inside the unit circle for a real coefficient polynomial, a complete characterization for system parameters and the feedback delay is presented. Also, the delay-independent and delay-dependent stability intervals are obtained in this article.
1. Introduction and Problem Formulation
For the feedback control problem of hyperbolic systems, the PI (proportional-integral) controller may be one of the most widely used controller (see [1–3]). With PI controller, the system is directed to a designated equilibrium based on the present state and the accumulated past error. Moreover, the main approach to stability analysis is the Lyapunov function technique (see [4, 5]). In [6], the authors propose a strict Lyapunov functional for the hyperbolic systems with conservation laws. And then, in [7], the quadratic Lyapunov function is used to demonstrate the local exponential stability in norm. Moreover, the boundary dissipative conditions are presented explicitly. By using the Lyapunov direct method and spectral analysis, the local exponential stability was proved in [8]. Necessary and sufficient stability conditions are obtained in [9], which improves the result in [8].
However, there is little reference applying the characteristic method to investigate the stability of a hyperbolic system. The exponential stability of the quasilinear wave equation is established in [10] by studying the solution along the characteristic curves. The feedback stabilization problem of a scalar conservation law with PI controlleris discussed in [11], where is a positive constant, and are feedback parameters, and the characteristic equation is a first-order exponential polynomial:And then, the sufficient and necessary stability conditions for the parameters are deduced by the Walton–Marshall stability criterion. It is noted that it poses an open problem in dealing with hyperbolic systems of conservation or balance laws in [11].
In this article, we are mainly concerned with the stability analysis of a hyperbolic system with the form:with the following PI controller and initial conditions:where are the system states over the space interval and time domain . Here, and denote the velocity of the system state, and we assume that and are two positive constants. The parameters , and denote the proportional and integral control gain, respectively. Furthermore, we ignore the memory term and substitute it by the delayed position ; i.e., the PDP (position and delayed position) controller is as follows:where denotes the feedback time delay. Similarly, the PDP controller means that the system is designated to the equilibrium based upon the information of the present state and some past states. Our question is as follows: can system (3) be stabilized by PDP feedback (6)? In [12, 13], the PDP feedback control method was used to stabilize an inverted pendulum system, and it shows that the inverted equilibrium position is asymptotically stable for . In this paper, the delay-independent stability and instability for systems (3) and (6) can also be analyzed.
The contribution of this research is as follows: (i) we construct the characteristic equation for systems (3) and (4) and (3) and (6): a kind of second-order exponential polynomial with single delay and multiple delays, respectively. (ii) We study the root distribution of the characteristic equation and give the sufficient and necessary conditions about the feedback parameters such that both the closed-loop systems (3) and (4) and (3) and (6) are asymptotically stable. Additionally, the delay-independent and delay-dependent stability intervals are obtained in this article.
The structure of the article is as follows: we deduce the characteristic equation of (3) (4), and then, by Walton–Marshall stability criterion, we obtain the stability conditions for the system parameters in Section 2. In Section 3, the sufficient and necessary stability conditions for (3) and (6) are given by Schur–Cohn criterion. Next, we present some numerical simulations in Section 4. Finally, several concluding remarks are presented in Section 5.
2. Stability of the Closed-Loop System (3) and (4)
2.1. The Characteristic Equation
Given the formal solution of (4) can be calculated as follows:
Substitute (7) into the PI boundary (4), and assume is a solution. By a series of calculation, we can get the characteristic equation of the closed-loop system (3) with PI control (4):where
Remark 1. We can also obtain the characteristic equation (8) as follows. Consider the characteristic problem for system (3)whose fundamental solution matrix under PI boundary (4) isHence, let the determinant of (11) equal to 0, we can have (8).
2.2. Walton–Marshall Stability Criterion
In this section, firstly, we state the Walton–Marshall stability criterion. It gives the sufficient and necessary conditions under which all the roots of the characteristic equation with an exponential argumenthave negative real parts, where denotes time delay, , is a order polynomial with real coefficients, is a order polynomial with real coefficients, and .
The Walton–Marshall stability criterion ([14, 15] (subsection 5.6)) can be summarized in three steps as follows: Step 1: We show the stability if . Step 2: We increase from zero to and analyze how the eigenvalues move with the increasing . The process can be divided into two parts:(i)If , for an infinitely small , the new roots must appear at infinity; otherwise, would be near 1 and there cannot be any new roots. Hence, if , (12) can be satisfied for large if and only if is large, i.e., . So, in this situation, we can omit this step.(ii)If , we study the potential crossings of the imaginary axis, i.e., we analyze the purely imaginary roots . Given to conjugate symmetry of (12), we get that if is a solution, then is also a solution. Substitute into (12), we get a real polynomial in , and for the sake of simplicity, we denote it by The location of new roots is determined by the sign of for large . A necessary condition for exponential stability is which can be found in [15] (subsection 5.6). Step 3 We study the positive real roots of . Moreover, according to , the corresponding critical values for time-delay can be calculated: When we got a value of such that a pure imaginary root appears, we study the roots’ behavior as changes. To be specific, we study whether, as is increasing, this root crosses the imaginary axis from the right half plane into the left plane (a stabilizing root), or vice versa, if it goes from the left half plane into the right half plane (a destabilizing root). And then, the stability switching would be analyzed. A nonnegative root of is stabilizing if the first derivative and destabilizing if . If , one should consider the higher-order derivatives.
2.3. Stability Conditions for System Parameters
For characteristic equation (8), we assumewhere , and . Now, we analyze according to the three steps of Walton–Marshall stability criterion.
Step 1. If , (8) is equivalent towhich is a quadratic equation in . The discriminant of (17) isHence, the system without delay is asymptotically stable if and only if one of the following cases holds. Case 1 , , and Case 2 , , and
Remark 2. Since in step 2, we have holds. Here, the case can be omitted.
Step 2. Let is a pure imaginary root of (8), by (13), we have thatIt is obvious that , for large if and only if
Step 3. Assume , we can haveand the discriminant isunder the condition (20). Thus, we can have one positive root and one negative root :Since , so we only need to consider . Noticed thatandthe root is destabilizing. By (15), time delay satisfiesand then,Assume is the smallest , so we havewhereIn conclusion, we can have the following results.
Theorem 1. Suppose . The closed-loop system (3) and (4) are asymptotically stable for the-norm if and only if the feedback parameters satisfy and one of the following conditions:(i), , and (ii), and Furthermore,satisfies, whereis given by (28).
Remark 3. If , from the characteristic equation (8), we havewhere . Hence, . By Walton–Marshall stability criterion, we can have results as follows.
Theorem 2. Let us assume , and
The closed-loop system (3) and (4) is asymptotically stable for if and only if the feedback parameters satisfy one of the following conditions:(i), and (ii), and
Remark 4. If , the characteristic equation (8) changes towhich meansmust be an eigenvalue, and then, systems (3) and (4) cannot be stable.
3. Stability of System (3) with PDP Control (6)
In this part, we will analyze the stability for the hyperbolic system (3) under PDP (position and delayed position) feedback instead of PI controller:where denotes the feedback delay. By a simple calculation as in Section 2, we can get the characteristic equation of the closed-loop system (33) aswhere .
Remark 5. Here, the Walton–Marshall stability criterion in Section 2.2 cannot be applied, since there are two delays: and .
3.1. Delay-Independent Stability and Instability Results
According to Corollary 2.4 in [16], we can get the following delay-independent stability and instability results.
Letand
Theorem 3. Assume the feedback parameters satisfy the following conditions:
Then,(i)If , system (33) is always asymptotically stable for (ii)If , and , system (33) is always asymptotically stable for (iii)If , and (or ), system (33) is always unstable for
Proof. Firstly, we consider the limit case , and then, the characteristic equation (34) turns toHence, all roots of (38) have negative real parts if and only if the condition (37) holds.
As the delay increases , let is a pure imaginary root of (34). Hence, we have thatSeparating the real part and imaginary part, we getBy squaring both sides of the (40), we haveand then,which is a quadratic equation of .
Let , and denotes the two roots of (42). If (40) has no real root (i.e., ), or the absolute values of both roots are bigger than 1 (i.e. and ), it means that the characteristic equation has no pure imaginary roots. Hence, system (33) is always stable for any positive value of the delay , if and only if (37) holds. On the other hand, if , and (or ), no matter what the time delay is, the characteristic equation (34) must have pure imaginary root. Hence, system (33) is always unstable for any positive value of the delay .
3.2. Schur–Cohn Criterion and Stability Analysis
In this section, we discuss a special case: is an integral multiple of . Letwhere is an integral number. Then, the characteristic (34) can be written as follows:
Remark 6. It should be noted that the Walton–Marshall stability criterion for systems with multiple commensurate delayscannot be applied here, where denotes the delay, and are polynomials in with real coefficients. In fact, in (41), are constant, and then, the recursive procedure for finding the pure imaginary roots in Walton–Marshall stability criterion cannot be used.
On the other hand, denoteand then, the characteristic equation (34) can be written as follows:which is a real order polynomial in . Hence, in (34) if and only if in (47).
Next, we shall apply the Schur–Cohn criterion (see, e.g., [17] on pp. 34–36 or Proposition 5.3 of [18] on p. 27) to check that the roots of the polynomial are inside the unit circle.
Theorem 4. Suppose that all the roots of in (43) lie inside the unit circle. Then,and
More accurately, there are two cases: Case 1: if is an even number, then Case 2: if is an odd number, then
Proof. By Schur–Cohn Criterion, we haveSubstitute this into (43), we can get (44) and (45), respectively.
Furthermore, when is an even number, (45) becomesCombining (44) with (48), we have (46) directly. Similarly, if is an odd number, we get (47).
Next, for the given , we show the necessary and sufficient conditions of the stability of the feedback parameters .
Theorem 5. (i)For , all the roots of lie inside the unit circle if and only if(ii)For , all the roots of lie inside the unit circle if and only if
Proof. If ,By Theorem 4, we have the first inequality of (54): . Noting thatall the determinants of all inners areSo yields the last two equalities of (54).
If ,Since , and , we have the first inequality of (55) by Theorem 4. Noting thatSo the determinants of all inners and yield the last three inequalities in (55).
4. Simulation Results
In this part, we will give some simulation results by Matlab software.
Example 1. Let the feedback parameters in (4). Then, we haveandwhich satisfy the conditions of Theorem 1 (ii). Furthermore, by a series of calculations, we can get the critical value. Therefore, systems (3) and (4) is asymptotically stable if. Hence, we can choose, and then. Figure 1shows the stability of (3) and (4) with the initial conditions.
Example 2. Let the feedback parameters in (33). Then, we haveandin (35). Hence, by Theorem 3 (i), system (33) is always stable for. Figure 2shows the stability of (33) with the parametersand the initial conditions.
5. Concluding Remarks
In this paper, the PI and PDP boundary feedback controllers are applied to a hyperbolic system with conservation law. By using Walton–Marshall stability criterion and Schur–Cohn criterion, the sufficient and necessary stability and instability conditions for feedback parameters are established individually. And especially, the delay-dependent and delay-independent stabilities are analyzed.
It is worth noting that a hyperbolic system with balance lawswhich can depict the phenomenon of biological population affected by age structure in population ecology and describe the traffic dynamics in a freeway segment, which can also be discussed by the methods of the Walton–Marshall stability criterion and Schur–Cohn criterion, where the right-hand side of (62) can be regarded as a nonhomogeneous disturbance term.
In the future, our objective is to study the stability of a coupled hyperbolic systemwhich can describe the dynamics of a single open-channel system with nonzero bottom slope and bottom friction. The main difficulty is to find the characteristic equation under PI and PDP boundary feedback controller and choose a suitable method to analyze the stability and instability conditions for system parameters and the time delay.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Fundamental Research Program of Shanxi Province (no. 20210302123046).