1. Automatic Control Theory School of Automation NWPU Teaching Group of Automatic Control Theory

2. Automatic Control Theory Excecies (11) 3 — 15, 16, 17,18

3. Review （ 1 ） （ 1 ） The methods to improve the second order system’s performances （ 2 ） Addition of a zero to the open-loop transfer function Increasing damping （ 3 ） Addition of a zero to the closed-loop transfer function With tachometer feedback Change ： Coefficients of the characteristic equation →Characteristic roots →Modes→Step response→Performance Change ： Coefficients of partial fraction →Coefficient of modes→ Step response→ Performance With proportional-differential compensation Advance control

4. Review （ 2 ） §3.4.1 The Step Response of Higher Order Systems §3.4.2 Dominant Poles of Closed-Loop Transfer-Functions §3.4.3 Performance Estimation by Locations of Higher- Order System Poles and Zeros

5. Automatic Control Theory （ Lecture 11 ） §3 Time-Domain Analysis and Adjustments of Linear Systems §3.1 Introduction §3.2 First-order System §3.3 Second-order System §3.4 Higher-order System §3.5 Stability Analysis of Linear Systems §3.6 The Steady-State Error of Linear Systems §3.7 Time-Domain Compensation

6. Automatic Control Theory （ Lecture 11 ） §3.5 Stability Analysis of Linear Systems

7. §3.5 Stability Analysis of Linear Systems (1) §3.5.1 The Concept of Stability Stability is the most important requirement for a control system. Determining whether a system is stable or unstable and obtaining the conditions for stability are the basic tasks of the automatic control theory. Definition ： When a system is offset from its equilibrium by a disturbance, the system is stable if it can return to the original equilibrium with sufficient accuracy. Otherwise, it is unstable.

8. §3.5 Stability Analysis of Linear Systems (2) §3.5.2 The Necessary and Sufficient Condition for System Stability Necessary and sufficient condition for the system stability ： All the characteristic roots of the closed-loop transfer function have negative real parts, or all the poles of the closed-loop transfer function lie in the left half s-plane.. By the definition of the stability, if the impulse response of the system decays, that is, the system is stable. : Necessity : Sufficiency:

9. §3.5 Stability Analysis of Linear Systems (3) §3.5.3 Stability Criterion (1) A necessary condition Note ： Example1 Unstable May be stable

10. §3.5 Stability Analysis of Linear Systems (4) (2) Routh Criterion Routh’s array The system is stable if and only if all elements of the first column of the Routh’s array are positive and the number of changes of signs in the elements of the first column equals the number of roots with positive real parts or in the right-half s-plane.

11. §3.5 Stability Analysis of Linear Systems (5) s4s3s2s1s0s4s3s2s1s0 Solution. Routh’s array is carried as follows 1 7 10 5 2 Since there are two sign changes in the first column of Routh’s array. The system is unstable. 10 Example2 ： D(s)=s 4 +5s 3 +7s 2 +2s+10=0

12. §3.5 Stability Analysis of Linear Systems (6) s3s2s1s0s3s2s1s0 1 -3  2 Since there are two sign changes in the first column of Routh’s array. The system is unstable. 0 Example3 ： D(s)=s 3 -3s+2=0 determine the number of roots in the right half s-plane. (3) Special cases of Routh’s array The first element in any row of Routh’s array is zero, but the others are not. To remedy the situation, we replace the zero element in the first column by an arbitrary small positive number of , and then proceed with Routh’s array. Solution. Routh’s array is carried as follows

13. §3.5 Stability Analysis of Linear Systems (7) Solution. Routh’s array is carried as follows 1 12 35 3 20 25 s5s4s3s2s1s0s5s4s3s2s1s0 5 25 00 10 25 0 Auxiliary equation ： Example4 D(s)=s 5 +3s 4 +12s 3 +20s 2 +35s+25=0 The elements in one row of Routh’s array are all zero. The situation can be remedied by using the auxiliary equation, which is formed from the coefficients of the row just above the row of zeros in Routh’s array. Replace the row of zeros with the coefficients of the auxiliary equation.

14. Special Cases Routh Criterion In the second case, it indicates that one or more of the following conditions may exist: 1. 1.The equation has at least one pair of real roots with equal magnitude but opposite signs. 2. 2.The equation has one or more pairs of imaginary roots. 3. 3.The Equation has pairs of complex-conjugate roots forming symmetry about the origin of the s-plane, for example 出现全零行时，系统可能出现一对纯虚根；或一对符号 相反的实根；或两对实部符号相异、虚部相同的复根。

15. §3.5 Stability Analysis of Linear Systems (8) Solution. Routh’s array is carried as follows 1 0 -1 2 0 -2 s5s4s3s2s1s0s5s4s3s2s1s0 0 -2 16 /  0 8 -2 0 Auxiliary equation ： Example5 D(s)=s 5 + 2s 4 -s-2=0  Since there are one sign change in the first column of Routh’s array. The system is unstable. =(s+2)(s+1)(s-1)(s+j5)(s-j5)

16. §3.5 Stability Analysis of Linear Systems (9) (4) The application of the Routh’s criterion Example6 The location of the open-loop zero and poles of a unity feedback system is shown in Figure. Determine if the system could be stable. If it could be stable, obtain the range of the K. Solution: This example indicates that the closed-loop stability does not depend on the open-loop stability.

17. §3.5 Stability Analysis of Linear Systems (10) Example7 The block diagram of a control system is shown in the figure. (1) Determine the ranges of (K, ξ) for stability (2) When ξ=2 determine the range of K for all closed-loop poles to be located in the left hand side of the line s=-1 。 Solution (1)

18. §3.5 Stability Analysis of Linear Systems (11) When   ， carrying a coordinate transformation ：

19. §3.5 Stability Analysis of Linear Systems (12) Remarks:. (1) Stability is the system’s own attributes. It does not depend on the inputs, but depends on the structure and parameters of a control system. (2) Only the closed-loop poles (not the closed-loop zeros) determines the system stability. Closed-loop zeros have an effect on the coefficients C i to change the performance but not the stability. Closed-loop poles determines mode shapes, thus to determine the stability and the performance. (3) There are no direct relation between the stabilities of closed-loop systems and open-loop systems.

20. Summary §3.5.1 The Concept of Stability §3.5.2 The Necessary and Sufficient Condition for System Stability §3.5.3 Stability Criterion （ 1 ） A necessary condition （ 2 ） Routh Criterion （ 3 ） Special cases of Routh’s array （ 4 ） The application of the Routh’s criterion Characteristic roots of the closed-loop transfer function have negative real parts or all the poles of the closed-loop transfer function lie in the left half s-plane

21. Automatic Control Theory Excecies(7) 3 — 15, 16, 17, 18