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**Design of Discrete-Time Control Systems by Conventional Methods**

Chapter 4 Design of Discrete-Time Control Systems by Conventional Methods

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**Introduction Mapping from the s plane to the z plane**

Stability of closed-loop control systems in z plane The design methods Root-locus based technique Frequency-response method in the plane Analytical method The chapter covers: Stability criterion for closed-loop control systems in the z plane Transient and steady-state response characteristics of discrete-time control systems Design technique based on the root-locus method Frequency-response techniques Analytical design method Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Mapping between the s plane and the z plane**

The absolute and the relative stability Determined by the locations of the closed-loop poles in the s plane. Relation between z and s The pole and zero locations in the z plane are related to the pole and zero locations in the s plane. The dynamic behavior of the discrete-time control system depends on the sampling period T. Locations of poles and zeros in the z plane depend on T Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Mapping between the s plane and the z plane**

Mapping of the LH of the s plane into the z plane The location of the poles and zeros in the z plane are important in predicting the dynamic behavior of the system Poles and zeros in the s plane, where frequencies differ in integral multiples of the sampling frequency 2/T, are mapped into the same locations in the z plane. The left half of the s plane The j axis in the s plane corresponds to (interior of the unit circle) (unit circle) Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Mapping between the s plane and the z plane**

Primary strip and Complementary strips As the point in the s plane moves from – to on the j axis, we trace the unit circle in the z plane an infinite number of times. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Mapping between the s plane and the z plane**

Primary strip and Complementary strips (cont.) The area enclosed by any of the complementary strips is mapped into the same unit circle in the z plane : the correspondence between the z plane and the s plane is not unique. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Mapping between the s plane and the z plane**

Primary strip and Complementary strips (cont.) Mapping regions of the s-plane onto the z-plane Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Mapping between the s plane and the z plane**

Constant-Attenuation Loci A constant-attenuation line (=constant) in the s plane maps into a circle of radius z=eT centered at the origin in the z plane. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**The Laplace Transform (2nd order system)**

The general Second-order System Natural frequency(n): the frequency of oscillation of the system without damping. Damping ratio(): General second-order transfer function The poles of the transfer function: Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Mapping between the s plane and the z plane**

Settling Time ts The settling time is determined by the value of attenuation of the dominant closed-loop poles. The region to the left of the line = -1 in the s plane corresponds to the inside of a circle with radius e -1T in the z plane. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Mapping between the s plane and the z plane**

Constant-Frequency Loci A constant-frequency locus = 1 in the s plane is mapped into a radial line of constant angle T1 in the z plane. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Mapping between the s plane and the z plane**

Constant-Frequency Loci (cont.) The region bounded by constant-frequency lines = 1 and constant-attenuation lines = -1 and = -2 is mapped into a region bounded by two radial lines and to circular arcs. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Mapping between the s plane and the z plane**

Constant-Damping-Ratio Loci A constant-damping-ratio line in the s plane is mapped into a spiral in the z plane. A constant-damping-ratio line in the s plane: Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Mapping between the s plane and the z plane**

Constant-Damping-Ratio Loci (cont.) In the z plane this line becomes Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Mapping between the s plane and the z plane**

Constant-Damping-Ratio Loci (cont.) Constant-damping-ratio loci in the z plane Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Mapping between the s plane and the z plane**

Constant-Damping-Ratio Loci (cont.) Constant loci and constant n loci Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Mapping between the s plane and the z plane**

s Plane and z Plane Regions for > 1 If all the poles in the s plane are specified as having a damping ratio not less than a specified value 1 , then the poles must lie to the left of the constant-damping ratio line in the s plane. In the z plane, the poles must lie in the region bounded by logarithmic spirals. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Mapping between the s plane and the z plane**

Example 4-1 Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Mapping between the s plane and the z plane**

Comments It is necessary to pay particular attention to the sampling period T. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Stability analysis of closed-loop systems in the z plane**

Stability Analysis of a closed-loop system Closed-loop pulse-transfer function system: The stability of the system defined by the equation may be determined from the locations of the closed-loop poles in the z plane, or the roots of the characteristic equation: For the system to be stable, the closed-loop poles or the roots of the characteristic equation must lie within the unit circle in the z plane. If a simple pole lies at z=1, then the system becomes critically stable. Also, the system becomes critically stable if a single pair of conjugate complex poles lies on the unit circle in the z plane. Any multiple pole on the unit circle makes the system unstable. Closed-loop zeros do not affect the absolute stability. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Stability analysis of closed-loop systems in the z plane**

Methods for Testing Absolute Stability Three stability tests can be applied directly to the characteristic equation P(z)=0 without solving for the roots. Schur-Cohn stability test Jury stability test Bilinear transformation coupled with the Routh stability criterion Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Stability analysis of closed-loop systems in the z plane**

The Jury Stability Test To construct a table whose elements are based on the coefficients of P(z). Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Stability analysis of closed-loop systems in the z plane**

The Jury Stability Test (cont.) The first row: the coefficients in P(z) in ascending order The second row: the coefficients in P(z) in descending order Rows 3 through 2n – 3 are given by the following determinants: The last row in the table consists of three elements. The elements in any even-numbered row are simply the reverse of the immediately preceding odd-numbered row. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Stability analysis of closed-loop systems in the z plane**

Stability Criterion by the Jury Test A system with the characteristic equation is stable if the following conditions are all satisfied: 1. 2. 3. 4. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Stability analysis of closed-loop systems in the z plane**

Example 4-4 First condition is satisfied: Second condition is satisfied: Third condition is satisfied: The Jury stability table: (next page) Fourth condition is satisfied: Therefore, the given characteristic equation is stable, or all roots lie inside the unit circle in the z plane. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Stability analysis of closed-loop systems in the z plane**

Example 4-4 (cont.) Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Stability analysis of closed-loop systems in the z plane**

Stability Analysis by use of the Bilinear Transformation and Routh Stability Criterion The method requires transformation from the z plane to another complex plane, the w plane. The bilinear transformation maps the inside of the unit circle in the z plane into the left half of the w plane. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Stability analysis of closed-loop systems in the z plane**

Stability Analysis by use of the Bilinear Transformation and Routh Stability Criterion (cont.) We first substitute (w+1)/(w-1) for z in the characteristic equation : Then, it is possible to apply the Routh stability criterion in the same manner as in continuous-time systems. The method will indicate exactly how many roots of the characteristic equation lie in the right half of the w plane ad how many lie on the imaginary axis. The amount of computation required is much more than that required in the Jury stability test. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Transient and Steady-state Response Analysis**

The transient response portion of the response due to the closed-loop poles of the system The steady-state response portion of the response due to the poles of the input or forcing function Standard inputs will be used to analyze discrete-time control systems Step inputs, ramp inputs, or sinusoidal inputs Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Transient and Steady-state Response Analysis**

Transient Response Specifications Transient response to a unit-step input : easy to generate and sufficiently drastic to provide useful information depending on the initial conditions Unit-step response Discrete-time output in the unit-step response Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Transient and Steady-state Response Analysis**

Transient Response Specifications (cont.) Delay time td the time required for the response to reach half the final value the very first time Rise time tr the time required for the response to rise from 10% to 90%, or 5% to 95%, or 0% to 100% of its final value. Peak time tp the time required for the response to reach the first peak of the overshoot Maximum overshoot Mp the maximum peak value of the response curve measured from unity. It is common to use the maximum percent overshoot. Settling time ts the time required for the response curve to reach and stay within a range about the final value of a size specified as an absolute percentage of the final value, usually 2%. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Transient and Steady-state Response Analysis**

Transient Response Specifications (cont.) Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Transient and Steady-state Response Analysis**

Transient Response Specifications (cont.) Assume that the sampling theorem is satisfied and no frequency folding occurs. The nature of the transient response of a discrete-time control system to a given input depends on the actual locations of the closed-loop poles and zeros in the z plane. Example 4-8 Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Transient and Steady-state Response Analysis**

Steady-State Error Analysis The steady-state performance of a stable control system is generally judged by the steady-state error due to step, ramp, and acceleration inputs. Any physical control system inherently suffers steady-state error in response to certain types of inputs. Open-loop transfer function G(s)H(s) of a continuous-time control system It is customary to classify the system according to the number of integrators in the open-loop transfer function. A system is said to be of type 0, type 1, type 2,…, if N=0, N =1, N =2, …, respectively. A compromise between steady-state accuracy and relative stability is always necessary increasing type number: improved accuracy, but aggravation of the stability problem) Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Transient and Steady-state Response Analysis**

The concept of static error constants Discrete-time control systems can be classified according to the number of open-loop poles at z=1 (corresponds to an integrator in the loop). The system can be classified as a type 0, a type 1, or a type 2 according to whether N=0, N =1, or N =2, …, respectively. The system type specifies the steady-state characteristics or steady-state accuracy. The physical meaning of the static error constants for discrete-time control system is the same as that for continuous-time control systems. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Transient and Steady-state Response Analysis**

The concept of static error constants (cont.) Final value theorem Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Transient and Steady-state Response Analysis**

Static Position Error Constant For a unit-step input The steady-state actuating error in response to a unit-step input becomes zero if Kp=,which requires that GH(z) have at least one pole at z=1. We define the static position error constant Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Transient and Steady-state Response Analysis**

Static Velocity Error Constant For a unit-ramp input If Kv=, the the steady-state actuating error in response to a unit-ramp input Is zero. This requires GH(z) to posses a double pole at z=1. We define the static velocity error constant Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Transient and Steady-state Response Analysis**

Static Acceleration Error Constant For a unit-acceleration input The steady-state actuating error in response to a unit-acceleration input becomes zero if Ka=. This requires GH(z) to posses a triple pole at z=1. We define the static acceleration error constant Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Transient and Steady-state Response Analysis**

System types and the corresponding steady-state errors Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Transient and Steady-state Response Analysis**

Static error constants for typical closed-loop configurations of discrete-time control systems Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Transient and Steady-state Response Analysis**

Response to Disturbances The effects of disturbances are explored. The larger the gain of GD(z) is, the smaller the error E(z). Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Transient and Steady-state Response Analysis**

Response to Disturbances (cont.) If a linear system is subject to both the reference input and a disturbance input, then the resulting error is the sum of the errors due to the reference input and the disturbance input. For a constant disturbance of magnitude N if GD(z) involves a pole at z=1, Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Transient and Steady-state Response Analysis**

Response to Disturbances (cont.) Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Design based on the Root-Locus Method**

The root-locus method developed for continuous-time systems can be extended to discrete-time systems without modifications, except that the stability boundary is changed from the jw axis in the s plane to the unit circle in the z plane. Characteristic equation is 1+G(z)H(z)=0 which is of exactly the same form as the equation for root-locus analysis in the s plane. However, the pole locations for closed-loop systems in the z plane must be interpreted differently from those in the s plane Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Design based on the Root-Locus Method**

Angle and Magnitude Conditions The values of z that fulfill both the angle and the magnitude conditions are the roots of the characteristic equation, or the closed-loop poles. Angle condition: Magnitude condition: Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Design based on the Root-Locus Method**

General Procedure for Constructing Root Loci Obtain the characteristic equation and rearrange this equation so that the parameter of interest, K, appears as the multiplying factor in the form. Find the starting points and terminating points of the root loci. Find also the number of separate branches of the root loci. The point on the root loci corresponding to K =0 are open-loop poles and those corresponding to K= are open-loop zeros. A root-locus plot will have as many branches as there are roots of the characteristic equation. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Design based on the Root-Locus Method**

General Procedure for Constructing Root Loci. (cont.) Determine the root loci on the real axis. Root loci on the real axis are determined by open-loop poles and zeros lying on it. In constructing the root loci on the real axis, choose a test point on it. If the total number of real poles and zeros to the right of this test point is odd, the this point lies on a root locus. The root locus and its complement form alternate segments along the real axis. Determine the asymptotes of the root loci. The abscissa of the intersection of the asymptotes and the real axis. n=number of finite poles of F(z) m=number of finite zeros of F(z) Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Design based on the Root-Locus Method**

General Procedure for Constructing Root Loci. (cont.) Find the breakaway and break-in points. Because of the conjugate symmetry of the root loci, the breakaway points and break-in points either lie on the real axis or occur in conjugate complex pairs. If the value of K corresponding to a root z=z0 of dK/dz=0 is positive, point z=z0 is an actual breakaway or break-in point. Since K is assumed to be non negative, if the value of K is negative, then point z=z0 is neither a breakaway nor a break-in point. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Design based on the Root-Locus Method**

General Procedure for Constructing Root Loci. (cont.) Determine the angle of departure (or angle of arrival) of the root loci from the complex poles (or at the complex zeros). The angle of departure (or angle of arrival) of the root locus from a complex pole (or at a complex zero) can be found by subtracting from 180 the sum of all the angles of lines from all other poles and zeros to the complex pole in question. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Design based on the Root-Locus Method**

General Procedure for Constructing Root Loci. (cont.) Find the points where the root loci cross the imaginary axis. The points where the root loci intersect the imaginary axis can be found by setting z=jv in the characteristic equation, equating both the real part and the imaginary part to zero, and solving for v and K. The values of v and K found give the location at which the root loci cross the imaginary axis and the value of the corresponding gain K, respectively. Any point on the root loci is a possible closed-loop pole. A particular point will be a closed-loop pole when the value of gain K satisfies the magnitude condition. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Design based on the Root-Locus Method**

Cancellation of Poles of G(z) with Zeros of H(z) If F(z)=G(z)H(z) and the denominator of G(z) and the numerator of H(z) involve common factors then the corresponding open-loop poles and zeros will cancel each other, reducing the degree of the characteristic equation by one or more. Example: The closed-loop pulse transfer function z=-a, the canceled pole of G(z)H(z),is a closed-loop pole Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Design based on the Root-Locus Method**

Cancellation of Poles of G(z) with Zeros of H(z) (cont.) Note that if pole-zero cancellation occurs in the feed-forward pulse transfer function, then the same pole-zero cancellation occurs in the closed-loop pulse transfer function. In the previous example, assume The effect of pole-zero cancellation in G(z) and H(z) is different from that of pole-zero cancellation in the feed-forward pulse transfer function. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Design based on the Root-Locus Method**

Root-Locus Diagrams of Digital Control Systems The effects of gain K and sampling period T on the relative stability of the closed-loop control system Char. Eq. : Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Design based on the Root-Locus Method**

Root-Locus Diagrams of Digital Control Systems Sampling period T =0.5 sec: The closed-loop poles corresponding to K=2: The critical gain KC: (z=-1) KC =8.165 Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Design based on the Root-Locus Method**

Root-Locus Diagrams of Digital Control Systems (cont.) 2. Sampling period T =1.0 sec: The closed-loop poles corresponding to K=2: The breakaway and break-in points The critical gain KC: KC =4.328 Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Design based on the Root-Locus Method**

Root-Locus Diagrams of Digital Control Systems (cont.) 3. Sampling period T =2.0 sec: The closed-loop poles corresponding to K=2: The breakaway and break-in points The critical gain KC: KC =2.626 Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Design based on the Root-Locus Method**

Effects of Sampling Period T on Transient Response Characteristics For a given value of gain K, increasing the sampling period T will make the discrete-time control system less stable and eventually will make it unstable. Making T shorter and shorter tends to make the system behave more like the continuous-time system. The damping ratios for the closed-loop poles corresponding to T=0.5, 1, and 2 are determined approximately as 0.24, 0.32, and 0.37 respectively. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Design based on the Root-Locus Method**

Effects of Sampling Period T on Transient Response Characteristics (cont.) Analytical determination of the damping ratio from the location of the closed-loop pole in the z plane. In the second-order system the damping ratio is indicative of the relative stability only if the sampling frequency is sufficiently high. If the sampling frequency is not high enough the maximum overshoot in the unit-step response will be much higher than would be predicted by the damping ratio. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Design based on the Root-Locus Method**

Effects of Sampling Period T on Transient Response Characteristics (cont.) Comparison of the unit-step response sequences for the three values of T considered in the preceding analysis. For T=0.5 sec and K=2 Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Design based on the Root-Locus Method**

Effects of Sampling Period T on Transient Response Characteristics (cont.) For T=1.0 sec and K=2 For T=2.0 sec and K=2 Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Design based on the Root-Locus Method**

Effects of Sampling Period T on Transient Response Characteristics (cont.) The effect of T on the steady-state accuracy. (the unit-ramp response for each of the three cases) T=0.5 sec and K=2 T=1.0 sec and K=2 T=2.0 sec and K=2 Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Design based on the Root-Locus Method**

Effects of Sampling Period T on Transient Response Characteristics (cont.) Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Design based on the Root-Locus Method**

Example 4-9 (page 221) Design a digital controller such that the dominant closed-loop poles have a damping ratio() of 0.5 and settling time(ts) of 2 sec.We assume T=0.2 sec. Obtain the unit-step response of the designed digital system. Obtain Kv of the system. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Design based on the Root-Locus Method**

Example 4-9 (page 221) (cont.) For the standard second-order system Location of the desired dominant closed-loop poles the T=0.2 is satisfactory Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Design based on the Root-Locus Method**

Example 4-9 (page 221) (cont.) The pulse transfer function of ZOH and the plant: G(z) poles : z=1, z=0.6703 zero : z= The sum of the angle contribution at P The angle deficiency is – = and needs to be provided by the controller. 17.10 109.84 135.52 Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Design based on the Root-Locus Method**

Example 4-9 (page 221) (cont.) The controller may be assumed to be If we decide to cancel the pole at z= by the zero of GD(z) at z=-, then the pole of the controller can be determined as z= The open-loop PTF The gain K can be determined 17.10 109.84 135.52 Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Design based on the Root-Locus Method**

Example 4-9 (page 221) (cont.) The designed digital controller The closed-loop pulse transfer function The response to the unit-step input Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Design based on the Root-Locus Method**

Example 4-9 (page 221) (cont.) The static velocity error constant KV If it is required to have a large value of KV, then we may include a lag compensator. The pole and zero fo the compensator lie on a finite number of allocable discrete points. – the pole and the zero are very closed to each other and do not significantly change the root locus near the dominant closed-loop poles. Even the designed system is of the third order, it acts as a second-order system, since one pole of the plant has been canceled by the zero of the controller. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Frequency Response Methods Spring 2001**

Chapter 8 Frequency Response Methods Spring 2001

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Preview The steady-state response of a system to a sinusoidal input test signal. The steady-state response of the system to a sinusoidal input as the frequency varies. Methods for graphical displaying the complex number G(j) as varies: The Bode Plot: one of the most powerful graphical tools for analyzing and designing control systems. Polar plots and log magnitude and phase diagrams. Several time-domain performance measures in terms of the frequency response of the system as well as the concept of system bandwidth will be introduced. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Introduction Consider the system Y(s)=T(s)R(s) with r(t)=A sin t.**

The frequency response of a system is defined as the steady-state response of the system to a sinusoidal input signal. The sinusoid is a unique input signal, and the resulting output signal for a linear system, as well as signals throughout the system, is sinusoidal in the steady state; it differs from the input waveform only in amplitude and phase angle. Consider the system Y(s)=T(s)R(s) with r(t)=A sin t. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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Introduction The steady-state output signal depends only on the magnitude and phase of T(j) at a specific frequency . Advantages of the frequency response method Availability of sinusoid test signals for various ranges of frequencies and amplitudes. The transfer function describing the sinusoidal steady-state behavior of a system can be obtained by replacing s with j in the system transfer function T(s). Disadvantage of the method The indirect link between the frequency and the time domain: In practice, the frequency response characteristic is adjusted by using various design criteria that will normally result in a satisfactory transient response. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Frequency Response Plots**

Transfer function of a system G(s) where where Polar plot representation of the frequency response Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Frequency Response Plots**

Example 8.1 Frequency response of an RC filter where Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Frequency Response Plots**

Bode plots The logarithmic plots simplifies the determination of the graphical portrayal of the frequency response. The logarithmic gain in dB and the angle () can be plotted versus the frequency by utilizing several different arrangements. The transfer function in the frequency domain Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Frequency Response Plots**

Example 8.3 Bode diagram of an RC filter where Corner frequency Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Frequency Response Plots**

A logarithmic scale of frequency An interval of two frequencies with a ratio equal to 10 is called a decade. The difference between the logarithmic gains, for >>1/, over a decade of frequency is (2=101) Asymptotic curve for (j+1)-1 Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Frequency Response Plots**

A logarithmic scale of frequency The frequency interval 2=21 is often used and is called an octave of frequencies. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Frequency Response Plots**

The primary advantage of the logarithmic plot is the conversion of multiplicative factors into additive factors. 1 3 4 2 3 Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Frequency Response Plots**

Constant Gain Kb If the gain is a negative value,the logarithmic gain remains 20logKb. The negative sign is accounted for by the phase angle, -180. Poles (or Zeros) at the Origin (j) A pole at the origin Multiple pole at the origin A zero at the origin Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Frequency Response Plots**

Poles (or Zeros) at the Origin (j) (cont.) Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Frequency Response Plots**

Poles or Zeros on the Real Axis The asymptotic curve for <<1/ is 20log1=0dB, and for >>1/ is –20log, which has a slope of –20dB/decade. The intersection of the two asymptotes occurs when The break frequency: = 1/ The actual logarithmic gain when = 1/ is –3dB. The phase angle is ()=-tan-1. The Bode diagram of a zero factor is obtained in the same manner as that of the pole. The slope is +20dB/decade, and the phase angle is ()=+tan-1. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Frequency Response Plots**

Poles or Zeros on the Real Axis (cont.) Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Frequency Response Plots**

Complex Conjugate Poles or Zeros When u << 1, the magnitude is and the phase angle approaches 0. When u >> 1, the magnitude approaches which results in a curve with a slope 0f –40dB/decade. The phase angle approaches –180. normalized form u= /n Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Frequency Response Plots**

Complex Conjugate Poles or Zeros (cont.) Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Frequency Response Plots**

Asymptotic Curves for Basic Terms of a Transfer Function Gain, Zero, Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Frequency Response Plots**

Asymptotic Curves for Basic Terms of a Transfer Function Pole, Pole at the origin, Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**Frequency Response Plots**

Asymptotic Curves for Basic Terms of a Transfer Function Two complex poles, Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**An Example of Drawing The Bode Diagram**

The factors, in order of their occurrence as frequency increases: A constant gain K = 5 A pole at the origin A pole at =2 A zero at =10 A pair of complex poles at =n=50 Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**An Example of Drawing The Bode Diagram**

Magnitude characteristic Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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**An Example of Drawing The Bode Diagram**

Phase characteristic Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

93
**Frequency Response Measurements**

A sine wave can be used to measure the open-loop frequency response of a control system. From an amplitude and phase plot, the open-loop transfer function, GH(j) , can be deduced. Similarly the closed-loop frequency response of a control system, T(j), may be obtained and actual transfer function deduced. A wave analyzer: to measure the amplitude and phase variations as the frequency of the input sine wave is altered. A transfer function analyzer: to measure the open-loop and closed-loop transfer functions. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

94
**Performance Specifications in The Frequency Domain**

How does the frequency response of a system relate to the expected transient response of the system? Given a set of time-domain specifications, how do we specify the frequency response? Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

95
**Performance Specifications in The Frequency Domain**

At the resonant frequency, r, a maximum value of the frequency response, Mp, is attained. The bandwidth is the frequency, B, at which the frequency response has declined 3 dB from its low-frequency value. The r and the –3dB bandwidth can be related to the speed of the transient response. As the B increases, the rise time of the step response of the system will decrease. The overshoot to a step input can be related to Mp through the damping ratio. As Mp increases in magnitude, the overshoot to a step input increases. In general, Mp indicates the relative stability of a system. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

96
**Performance Specifications in The Frequency Domain**

The bandwidth B can be approximately related to the natural frequency of the system. The greater the magnitude of n when is constant, the more rapidly the response approaches the desired steady-state value. Desirable frequency-domain specifications: Relatively small resonant magnitude. Relatively large bandwidths so that the system time constant is sufficiently small. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

97
**Performance Specifications in The Frequency Domain**

The steady-state error specification can be related to the frequency response of a closed-loop system. If the open-loop transfer function of a feedback system is written as Then the system is type N and the gain K is the gain constant for the steady-state error. Thus for a type-zero system, we have K=Kp that appears as the low-frequency gain on the Bode diagram. The gain K=Kv for the type-one system appears as the gain of the low-frequency section of the magnitude characteristic. The frequency response characteristics represent the performance of a system, and they are useful for analysis and design of feedback systems. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

98
**Relative Stability and The Nyquist Criterion**

The gain and phase margins from the Bode diagram. The critical point, u=-1, v=0, in the GH(j) –plane is equivalent to a logarithmic magnitude of 0 dB and a phase angle of 180 on the Bode diagram. Phase margin = 180 -137 =43 Gain margin = 15 dB Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

99
**Design based on the Frequency-Response Method**

Response of a linear time-invariant discrete system to a sinusoidal input Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

100
**Design based on the Frequency-Response Method**

Response of a linear time-invariant discrete system to a sinusoidal input (cont.) Let’s define Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

101
**Design based on the Frequency-Response Method**

Bilinear Transformation and the w Plane Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

102
**Design based on the Frequency-Response Method**

Bilinear Transformation and the w Plane (cont.) Once the pulse transfer function G(z) is transformed into G(w), it may be treated as a conventional transfer function in w. Conventional frequency-response techniques can be used in the w plane. By replacing w by jv, conventional frequency-response techniques may be used to draw the Bode diagram for the transfer function in w. If wT is samll, v= w. See Example 4-11. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

103
**Design based on the Frequency-Response Method**

Bilinear Transformation and the w Plane (cont.) To summarize, The w transformation, a bilinear transformation, maps the inside of the unit circle of the z plane into the left half of the w plane. The overall result due to the transformations from the s plane into the z plane and from the z plane into the w plane is that the w plane and the s plane are similar over the region of interest in the s plane. In Bode diagram, the conventional straight-line approximation to the magnitude curve as well as the concept of the phase margin and gain margin apply to G(jv). Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

104
**Design based on the Frequency-Response Method**

Bode Diagrams The conventional frequency-response methods apply to the transfer functions in the w plane. There may be a difference in the high-frequency magnitudes for G(j) and G(jv). The high-frequency asymptote of the logarithmic magnitude curve for G(jv) may be a constant-decibel line. If limsG(s)=0, the magnitude of G(j) always approaches zero (-dB) as approaches infinity. Note that we are interested only in the frequency range 0 1/2s which corresponds to 0v . When |G(jv)| is a nonzero constant at v= it is implied that G(w) contains the same number of poles and zeros. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

105
**Design based on the Frequency-Response Method**

Advantages of the Bode diagram approach to the design The low-frequency asymptote of the magnitude curve is indicative of one of the static error constants Kp, Kv, or Ka. Specifications of the transient response can be translated into those of the frequency response in terms of the phase margin, gain margin, bandwidth, and so forth. The design of a digital compensator to satisfy the given specifications can be carried out in the Bode diagram in a simple and straightforward manner. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

106
**Design based on the Frequency-Response Method**

Phase lead, Phase lag, and Phase Lag-Lead Compensation Phase lead compensation Used for improving stability margins. Increasing system bandwidth – faster speed to respond. May be subject to high-frequency noise problems due to its increased high-frequency gains. Phase lag compensation Reducing the system gain at higher frequencies without reducing the system gain at lower frequencies. Reducing the system bandwidth – slower speed of response Steady-state accuracy can be improved. Any high-frequency noises involved in the system can be attenuated. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

107
**Design based on the Frequency-Response Method**

Phase lead, Phase lag, and Phase Lag-Lead Compensation (cont.) Phase lead-lag compensation A phase lag compensator is cascaded with a phase lead compensator Increasing low-freq. gain, while at the same time increasing the system bandwidth and stability margins PID controller is a special case of a phase lag-lead controller. PD controller behaves in much the same way as a phase lead compensator. PI controller acts as a phase lag compensator. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

108
Compensation The design of a control system is concerned with the arrangement of the system structure and the selection of suitable components and parameters. It is often not sufficient to adjust a system parameter and thus obtain the desired performance. Compensation adjustment of a system in order to make up for deficiencies or inadequacies. Compensator: A compensating device A compensator is an additional component or circuit that is inserted into a control system to compensate for a deficient performance. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

109
**Compensation Types of compensation: cascade feedback output input**

Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

110
**Cascade Compensation Networks (see Dorf. P.557)**

Consider the first-order compensator When |z| <|p|, the network is called a phase-lead network Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

111
**Cascade Compensation Networks (see Dorf. P.557)**

Consider the first-order compensator When |z| >|p|, the network is called a phase-lag network Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

112
**Relative Stability and The Nyquist Criterion**

The gain and phase margins from the Bode diagram. The critical point, u=-1, v=0, in the GH(j) –plane is equivalent to a logarithmic magnitude of 0 dB and a phase angle of 180 on the Bode diagram. Phase margin = 180 -137 =43 Gain margin = 15 dB Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

113
**Design based on the Frequency-Response Method**

Some Remarks on the coefficient quantization problem From the viewpoint of microprocessor implementation of phase lag and phase lag-lead compensators, the phase lag network presents a coefficient quantization problem because the locations of poles and zeros are close to each other. Because of the limited numbers of bits, if the number of bits employed is insufficient, the pole and zero locations of the filter may not be realized exactly as desired. To minimize the problem, it is necessary to structure the filter so that it is least subject to coefficient inaccuracies. Because the sensitivity of the roots of polynomials to the parameter variations becomes severe as the order of the polynomial increases, direct realization of a higher-order filter is not desirable. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

114
**Design based on the Frequency-Response Method**

Design Procedure in the w plane Obtain G(z). Then transform G(z) into a transfer function G(w) through the bilinear transformation: It is important that the sampling period T be chosen properly. A rule of thumb: 10 times that of the bandwidth of the closed-loop system. Substitute w=jv into G(w) and plot the Bode diagram for G(jv). Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

115
**Design based on the Frequency-Response Method**

Design Procedure in the w plane (cont.) Read from the Bode diagram the static error constants, the phase margin, and the gain margin. By assuming that the low-freq. gain of the discrete-time controller transfer function GD(w) is unity, determine the system gain by satisfying the requirement for a given static error constant. Then, using conventional design techniques, determine the pole(s) and zero(s) – GD(w).The the open-loop transfer function is given by GD(w)G(w). Transform the controller transfer function GD(w) into GD(z). Realize the pulse transfer function GD(z) by a computational algorithm. Design Example 4-12 (page236) Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

116
**Design based on the Frequency-Response Method**

Example 4-12 Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

117
**Analytical Design Method**

Design of digital controllers for minimum settling time with zero steady-state error It is desired to design a digital controller GD(z) such that the closed-loop control system will exhibit the minimum possible settling time with zero steady-state error in response to test inputs. It is required that the output not exhibit intersampling ripples after the steady state is reached. ….. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

118
**Analytical Design Method**

Design of digital controllers for minimum settling time with zero steady-state error (cont.) To exhibit a finite settling time with zero steady-state error, the system must exhibit a finite impulse response Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

119
**Analytical Design Method**

Design of digital controllers for minimum settling time with zero steady-state error (cont.) The conditions for physical realizability: The order of the numerator of GD(z) must be equal to or lower than the order of the denominator. Otherwise, the controller requires future input data to produce the current input. If the plant Gp(s) involves a transportation lag e-Ls, the the designed closed-loop system must involve at least the same magnitude of the transportation lag. Otherwise, the closed-loop system would have to respond before an input was given, which is impossible for a physically realizable system. If G(z) is expanded into a series in z-1, the lowest-power term of the series expansion of F(z) in z-1 must be at least as large as that of G(z). The response comes at a delay of at least one sampling period if the series expansion of G(z) begins with a term in z-1. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

120
**Analytical Design Method**

Design of digital controllers for minimum settling time with zero steady-state error (cont.) The stability aspects of the system must be considered: If G(z) involves an unstable pole, a pole z= outside the unit circle, what will happen? 1- F(z) must have z= as a zero. If zeros of G(z) do not cancel poles of GD(z), the zeros of G(z) become zeros of F(z). Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

121
**Analytical Design Method**

Design of digital controllers for minimum settling time with zero steady-state error (cont.) Summary related to the stability issues: Since the digital controller GD(z) should not cancel unstable poles of G(z), all unstable poles of G(z) must be included in 1- F(z) as zeros. Zeros of G(z) that lie inside the unit circle may be canceled with poles of GD(z). However, zeros of G(z) that lie on or outside the unit circle must not be canceled with poles of GD(z). Hence, all zeros of G(z) that lie on or outside the unit circle must be included in F(z) as zeros. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

122
**Analytical Design Method**

Design of digital controllers for minimum settling time with zero steady-state error (cont.) Issues related to the error in general unit-step input unit-ramp input unit-acceleration input Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

123
**Analytical Design Method**

Design of digital controllers for minimum settling time with zero steady-state error (cont.) The pulse transfer function is determined. By letting F(z) satisfy the physical realizability and stability conditions and substituting previous equations, we obtain The equation gives the pulse transfer function of the digital controller that will produce zero steady-state error after a finite number of sampling periods. Chap. 4 Design of Discrete-Time Control Systems by Conventional Methods 41-142

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1 Chapter 9 Mapping Contours in the s-plane The Nyquist Criterion Relative Stability Gain Margin and Phase Margin PID Controllers in the Frequency Domain.

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