PDF_Chương 3_Linear Control Theory (I).pdf
- 1. Ho Chi Minh City University of Technology Linear Control Theory Dynamic Systems and Control Phuong-Tung Pham, Ph.D. Department of Mechatronics Faculty of Mechanical Engineering 1
- 2. Ho Chi Minh City University of Technology CONTENTS Poles and Zeros Stability of LTI Systems Routh’s Stability Criterion Root Locus Method 2
- 3. Ho Chi Minh City University of Technology Poles and Zeros 3
- 4. Ho Chi Minh City University of Technology Poles and zeros Transfer Function Poles of a Transfer Function The poles of a transfer function are the values of the Laplace transform variables, s, that cause the transfer function to become infinite Zeros of a Transfer Function The zeros of a transfer function are the values of the Laplace transform variable, s, that cause the transfer function to become a zero, 4
- 5. Ho Chi Minh City University of Technology Poles and zeros Example A linear system is described by the differential equation Find the system poles and zeros? From the differential equation the transfer function is The system therefore has a single real zero at s = -1/2, and a pair of real poles at s = -3 and s = -2 5
- 6. Ho Chi Minh City University of Technology Poles and zeros Remark: The poles and zeros are properties of the transfer function, and therefore of the differential equation describing the input-output system dynamics. A system is characterized by its poles and zeros in the sense that they allow reconstruction of the input/output differential equation. 6 Example: A system has a pair of complex conjugate poles 𝑝𝑝1, 𝑝𝑝2 = -1 ± 2𝑗𝑗, a single real zero 𝑧𝑧1 = -4, and a gain factor 𝐾𝐾 = 3. Find the differential equation representing the system. The transfer function is: The differential equation is
- 7. Ho Chi Minh City University of Technology Pole-Zero Plot • The poles and zeros of a transfer function may be complex • The system dynamics may be represented graphically by plotting their locations on the complex s-plane, whose axes represent the real and imaginary parts of the complex variable s. • Such plots are known as pole-zero plots. It is usual to mark a zero location by a circle (◦) and a pole location a cross (×). The location of the poles and zeros provide qualitative insights into the response characteristics of a system. 7
- 8. Ho Chi Minh City University of Technology Pole-zero plot 8 The system poles directly define the components in the homogeneous response. The unforced response of a linear SISO system to a set of initial conditions is Ci are determined from the given set of initial conditions λi are the roots of the characteristic equation. The characteristic equation is
- 9. Ho Chi Minh City University of Technology Pole-zero plot 9
- 10. Ho Chi Minh City University of Technology Effect of Zeros 10 Zeros affect the contribution of each pole, that is, the coefficients of the partial fraction expansion A zero near a pole reduces the amount of that term in the total response.
- 11. Ho Chi Minh City University of Technology Effect of Zeros 11 Zeros tend to have the effect of increasing overshoot and decreasing rise/peak time The derivative has a large hump in the early part of the curve, and adding this to the H0(s) response lifts up the total response of H(s) to produce the overshoot. Zero: z = -αζ
- 12. Ho Chi Minh City University of Technology Effect of Zeros 12 If zero has positive real part, then the response will initially move in the reverse direction, called a nonminimum phase zero
- 13. Ho Chi Minh City University of Technology Stability of LTI Systems 13
- 14. Ho Chi Minh City University of Technology Concept of Stability 14 The stability of a linear system may be determined directly from its transfer function. An nth order linear system is asymptotically stable only if all of the components in the homogeneous response from a finite set of initial conditions decay to zero as time increases, or where the pi are the system poles. In a stable system all components of the homogeneous response must decay to zero as time increases. If any pole has a positive real part there is a component in the output that increases without bound, causing the system to be unstable.
- 15. Ho Chi Minh City University of Technology Concept of Stability 15 • In order for a linear system to be stable, all of its poles must have negative real parts, that is they must all lie within the left-half of the s- plane. • An “unstable” pole, lying in the right half of the s-plane, generates a component in the system homogeneous response that increases without bound from any finite initial conditions. • A system having one or more poles lying on the imaginary axis of the s- plane has nondecaying oscillatory components in its homogeneous response, and is defined to be marginally stable
- 16. Ho Chi Minh City University of Technology Stability 16
- 17. Ho Chi Minh City University of Technology 17 A system is said to have bounded input–bounded output (BIBO) stability if every bounded input results in a bounded output Bounded Input–Bounded Output Stability It turns out that a continuous time LTI system with impulse response h(t) is BIBO stable if and only if Continuous-Time Condition for BIBO Stability This is to say that the impulse response is absolutely integrable. ( ) h t dt ∞ −∞ < ∞ ∫
- 18. Ho Chi Minh City University of Technology Routh-Hurwitz Criterion 18
- 19. Ho Chi Minh City University of Technology Routh-Hurwitz Criterion 19 Routh-Hurwitz Criterion is a technique that one can use to check the stability of the system from the characteristic equation without solving it. Determine the closed-loop stability of this system Find k such that the closed-loop system is stable
- 20. Ho Chi Minh City University of Technology Routh’s Stability Criterion 20 Routh-Hurwitz Criterion A necessary (but not sufficient) condition for stability is that all the coefficients of the characteristic polynomial be positive. A system is stable if and only if all the elements in the first column of the Routh array are positive.
- 21. Ho Chi Minh City University of Technology Example 21 The Routh table can be formulated as follows: s3 + s2 + 2s +24 = 0 s3 s2 s1 s0 1 𝑎𝑎1 = b1 = c1 = 𝑎𝑎2 = a3 = b2 =
- 22. Ho Chi Minh City University of Technology Assignment 22 Consider the system Determine the stability of the system as a function of the parameter K.
- 23. Ho Chi Minh City University of Technology Assignment 23 Consider the system Determine the stability of the system as a function of parameters K and KI. -
- 24. Ho Chi Minh City University of Technology Special case 24 Case 1: There is a zero in the first column, but some other elements of the row containing the zero in the first column are nonzero Replace zero with a small positive number
- 25. Ho Chi Minh City University of Technology Special case 25 Case 2: There is a zero in the first column, some other elements of the row containing the zero in the first column are also zero
- 26. Ho Chi Minh City University of Technology Root-Locus Method 26
- 27. Ho Chi Minh City University of Technology Root locus and transient response 27 The characteristic equation of the closed-loop system is The root locus (quỹ đao nghiệm số) is essentially the trajectories of roots of the characteristic equation as the parameter K is varied from 0 to infinity.
- 28. Ho Chi Minh City University of Technology Example 28 A camera control system: How the dynamics of the camera changes as K is varied ?
- 29. Ho Chi Minh City University of Technology Example 29 Pole location:
- 30. Ho Chi Minh City University of Technology Example 30
- 31. Ho Chi Minh City University of Technology Guidelines for Determining a Root Locus 31 Consider that for the system with transfer function given below we have to sketch the root locus and predict its stability. n is the number of the poles m is the number of the zeros 0 2 ( ) ( 2 2) K G s s s s = + +
- 32. Ho Chi Minh City University of Technology Guidelines for Determining a Root Locus 32 Rules for sketching the root locus Rule 1: The number of branches of loci is equal to the order of the system, i.e. the number of open-loop poles Rule 2: Branches start at the open-loop poles and end at the open-loop zeros or at infinity Example: How many branches? Start? End? • 3 root locus branches • Start at: 0,-2, -5 • One branch ends at -1 (zero), two branches go to infinity • Two zeros at infinity
- 33. Ho Chi Minh City University of Technology Guidelines for Determining a Root Locus 33 Rules for sketching the root locus Rule 3: Root locus is always symmetric with respect to real axis Rule 4: The section of the real axis is a part of locus if and only if the sum of the number of poles and zeros to its right is odd
- 34. Ho Chi Minh City University of Technology Guidelines for Determining a Root Locus 34 Rules for sketching the root locus Rule 5: The branches going to infinity asymptotically approach the straight lines defined by the angle: Rule 6: The intersection of the asymptotes with the real axis can be determined by: Example: Find the asymptotes for ( ) 2 1 , 0, 1, 2,... k k n m π θ + = = ± ± − 1 1 n m i i i i p z n m α = = − = − ∑ ∑ ( ) 2 1 , 0, 1 2 3 k k π θ α = ± + = ± = −
- 35. Ho Chi Minh City University of Technology Guidelines for Determining a Root Locus 35 Rules for sketching the root locus Rule 7: (Điểm tách nhập) Breakaway points (points of departure from the real axis) correspond to maxima of K, while breaking points (points of arrival at the real axis) correspond to minima of K. Solving the equation dK/ds = 0 Rule 8: Routh-Hurwitz criteria can be used to determine the gain value at which the locus goes across the imaginary axis
- 36. Ho Chi Minh City University of Technology Guidelines for Determining a Root Locus 36 Rules for sketching the root locus Rule 9: The angle of departure from a complex pole pj is 1 1 180 arg( ) arg( ) m n o j j i j i i i p z p p θ = = = + − − − ∑ ∑