3
Model-Based PID and Resonant Controller Design

3.1 Introduction

The models used in the design of PID controllers are limited to two particular types. One is a first order model while the other is a second order model. If the plant dynamics yield a higher order model, an approximation is often involved to obtain a first order or a second order model so that a PID controller can be designed using a model based approach.

When using model based design methods, a desired closed-loop performance specification is required before commencing. The desired performance is chosen in terms of the locations of the desired closed-loop poles, which reflect the closed- loop response time to reference change and disturbance rejection or in the frequency domain the bandwidth of the desired closed- loop control system. The desired closed-loop performance is often adjusted several times using closed-loop simulation and experimental validation before the designer finds the suitable closed-loop performance.

3.2 PI Controller Design

To design a PI controller, a first order model is used. Although the first order dynamics is the basic unit to form a system, it can also be used to describe a number of commonly encountered physical systems, such as the dynamic relationship between motor torque and angular velocity in the motor control problem, and fluid in-flow rate and fluid level in a fluid vessel control problem.

3.2.1 Desired Closed-loop Performance Specification

In the model-based design, a desired closed-loop performance is required to be specified. In the PI controller case, a second order transfer function is used in the specification,

(3.1)

where images and images are the natural frequency and damping coefficient for the second order transfer function. These are the free parameters to be selected by the designer as the desired performance specification.

The parameter images is often chosen as 1 or 0.707. When images , the poles of the desired closed-loop transfer function (3.1) are the solutions of the polynomial equation,

(3.2)

which are images . Namely, we have two identical poles when images . With the second choice of images , the poles are a pair of complex conjugate numbers determined by

(3.3)

With the parameter images chosen (either 1 or 0.707), the natural frequency images becomes a closed-loop performance parameter that the user specifies according to the desired closed-loop response requirement. In general, when images is larger, the desired closed-loop response is faster. The parameter images is directly related to the closed-loop response time and the band limit of the closed-loop system, which provide us with the guidelines on its selection. These two aspects are examined.

From the simulation of a step response (3.1) (see Figure 3.1), the response time is inversely proportional to the parameter images . Figure 3.1(a) shows that with the damping coefficient images , the total step response time is about images and with images , as shown in Figure 3.1(b), the total step response time is about images . There is another estimate of images that can be used as a guideline for the designer. Here the parameter images is related to the bandwidth of the desired closed-loop control system. For the desired closed-loop transfer function given by (3.1), when images , it can be verified that images at the frequency images . Hence, with the special choice of the damping coefficient images , the natural frequency images is the bandwidth of the closed-loop system, which we can directly use for the closed-loop performance specification. With the choice of images , the bandwidth of the closed-loop system is slightly smaller.

3.2.2 Model and Controller Structures

For the first order model, we assume that a first order time constant images and a steady state gain images are known to form the Laplace transfer function,

(3.4)

Image described by caption and surrounding text. — **Figure 3.1** Step response of the desired closed-loop transfer function. (a) . (b) . Key: line (1) ; line (2) .

images — **Figure 3.1** Step response of the desired closed-loop transfer function. (a) . (b) . Key: line (1) ; line (2) .

which can also be expressed in the pole–zero form,

(3.5)

where images and images

For a PI controller, its transfer function is given by

(3.6)

which can be written in the transfer function form,

(3.7)

where images and images . We will first find the coefficients images and images based on the model (3.5), then convert these coefficients into the standard PI controller parameters images and images .

The key to the solution of the PI controller parameters is to equate the desired closed-loop poles to the actual closed-loop poles. The locations of the closed-loop poles determine whether the closed-loop system is stable, the closed-loop response time, and the band limit of the closed-loop system.

To this end, we calculate the actual closed-loop system using the design model (3.5) and the controller model (3.7) via the closed-loop transfer function

(3.8)

The closed-loop poles of the actual system are the solutions of the polynomial equation with respect to images :

(3.9)

Equation 3.9 is called the closed-loop characteristic equation. Since the model parameters images and images are given, the free parameters in (3.9) are the controller parameters images and images . To find the controller parameters images and images , the following polynomial equation is set:

(3.10)

where the left-hand side of the equation 3.10 is the polynomial that determines the actual closed-loop poles and the right-hand side is the polynomial that determines the desired closed-loop poles. By equating these two polynomials, the actual closed-loop poles are assigned to the desired closed-loop poles. This controller design technique is called pole assignment controller design.

Now, we compare the coefficients of the polynomial equation 3.10 on both sides:

(3.11)

(3.12)

(3.13)

Solving (3.12) gives

(3.14)

and solving (3.13) gives

(3.15)

With the relationships between images , images and images , images (see Equation 3.7), we find the PI controller parameters as

(3.16)

(3.17)

Example 3.1

A first order system is used to describe the dynamic relationship between voltage change and velocity of a DC motor. Assume that a particular motor has the Laplace transfer function model,

(3.18)

Find the PI controller parameters for velocity control, where the desired closed-loop performance is specified by two performance levels: one fast response and one slow response ; in both cases. Simulate the closed-loop step responses with proportional control on the output only and compare the results.

Solution. The model parameters needed for the PI controller design are and . With these parameters and the closed-loop performance specification, based on equations (3.16 and 3.17) we find the controller parameters as, for and ,

for and

As we can see, when is larger, the proportional gain is larger whilst the integral time constant is smaller.

We simulate the closed-loop unit step responses with the results shown in Figure 3.2. The closed-loop response speed is much faster when in comparison with the case of , and the control signal has a much larger amplitude.

3.2.3 Closed-loop Transfer Functions for Different Configurations

With the PI controller designed, now we study the closed-loop transfer functions for using the traditional PI controller configuration and the IP controller configuration.

For the traditional PI controller configuration (see Figure 1.5), the Laplace transform of the control signal, images , is expressed as the function of feedback error signal images using the relation

(3.19)

where images and images . The closed- loop transfer function between the reference signal images and the output signal images is then

(3.20)

where images is the first order transfer function images . By substituting the transfer functions of controller ( images ) and the plant transfer function images into (3.20), we obtain the closed-loop transfer function for the PI control system using the traditional implementation:

(3.21)

Note that the denominator of (3.21) is used in the design of the PI controller (see (3.10)), which is made equal to the desired closed-loop characteristic polynomial images . By substituting the controller parameters (see (3.14) and (3.15) and the desired closed-loop characteristic polynomial into (3.21), the closed-loop transfer function becomes:

(3.22)

Using the traditional PI controller configuration, the actual closed-loop transfer function between the reference signal and the output is not equal to the desired closed-loop transfer function specified in the design (see images in (3.8)). Instead, there is a zero in the closed-loop transfer function at the location determined by the polynomial equation:

(3.23)

which is at images . The existence of this zero could cause some overshoot in the step response. One can verify that the closed-loop transfer function (3.22) has a bandwidth larger than images for the choice of images .

For the IP controller configuration (see Figure 1.7), the control signal is expressed as

(3.24)

The output signal images is expressed as

(3.25)

By substituting (3.24) into (3.25), we find the closed-loop transfer function for the alternative configuration as

(3.26)

By the design procedure, the denominator of this transfer function is images and the numerator images , therefore the closed-loop transfer function is

(3.27)

which is equal to the desired closed-loop transfer function we specified in the performance specification (see (3.1)).

As for disturbance rejection and measurement noise attenuation, both PI control system configurations have the identical closed- loop transfer function because the structural change introduced in the alternative configuration is only related to how the reference signal is introduced in the feedback loop. We illustrate this point by calculating the closed-loop transfer function between input disturbance and the plant output.

Suppose that an input disturbance has a Laplace transform images and it enters the system at the position of plant input. In this case, the plant output is expressed as

(3.28)

To derive the closed-loop transfer function between the input disturbance and the plant output, we assume that the reference signal images so to concentrate on the disturbance rejection. When the reference signal images , the control signals from both configurations (see (3.19) and (3.24)) become identical to the transform:

(3.29)

Substituting (3.29) into (3.28), we obtain the closed-loop transfer function between the input disturbance images and output images as

(3.30)

where we have used the design equation in (3.10).

Note that there is a factor images in the numerator of the closed-loop transfer function. This factor ensures that the closed-loop control system will reject a step input disturbance without steady- state error. This point will be made clear through the examples in the following section.

Example 3.2

A continuous-time system is described by the following differential equation:

(3.31)

where and are the control and output signal, is the unknown disturbance, and , , are the physical parameters. The control objective is to maintain a desired output of the system while rejecting the disturbance. The desired closed-loop band limit is specified as () with the damping coefficient . Design a PI controller to achieve the control objective and simulate the closed-loop response using both PI control system configurations.

Solution. In order to obtain the transfer function model between the input and the output , we take the Laplace transform of the dynamic model (3.31):

(3.32)

where , and are the Laplace transforms of the continuous time variables.

Equation 3.32 can also be expressed as

(3.33)

The PI controller parameters are calculated as

where and , and .

The closed-loop response is simulated using Simulink. In the closed-loop simulation, a unit step input signal is used as the reference change at time and a step input signal with magnitude 1.5 is used as the input disturbance, which occurred at time . Figures 3.3(a) and (b) compare the closed-loop output and control signals from the original PI controller configuration with those from the IP controller configuration. It is seen that the IP controller configuration avoids the overshoot that exhibits in the original configuration; however, both controller configurations provide the same responses in disturbance rejection.

3.2.4 Food for Thought

Would you use a second order model to design a PI controller?
If you found that the closed-loop response speed was too slow, would you decrease the parameter ?
Is it correct to say that the closed-loop bandwidth when the closed-loop system transfer function is characterized by if the damping coefficient ?

Figure 3.3 Closed-loop response of PI control system (Example 3.2). (a) Control signal. (b) Output response. Key: line (1) IP control system; line (2) PI control system.
Would you be able to list three physical systems that can be described by a first order model?
Has the IP controller configuration changed the characteristics of disturbance rejection when comparing with the original PI controller?

3.3 Model Based Design for PID Controllers

Second order models are used directly for the design of a PD or a PID controller. In addition, a first order plus delay model is approximated using a second order transfer function model, where the irrational transfer function images is approximated using a rational transfer function images that is called first order Padé approximation. If the mathematical model is of higher order, then approximation is made to obtain a second order model in order for the PID controller design to be carried out.

3.3.1 PD Controller Design

The combination of proportional and derivative (PD) control could be useful in the situation where stabilization of an unstable system is of main concern or in the situation where the system is severely oscillatory.

Because the derivative action will amplify measurement noise (see Chapter 2), a derivative filter is required for the implementation of a PD controller. Thus, the general form of a PD controller is given as

(3.34)

where images , images and images are the proportional gain, the derivative control gain and the filter time constant, respectively.

For this type of controller design, we assume that the continuous time system is a second order with the transfer function:

(3.35)

It is not a straightforward task to choose the parameters images , images and images based on the second order model (3.35). However, the PD controller can be converted into the classical lead-lag compensator that has the following form:

(3.36)

where the parameters images , images are images are related to images , images and images via the relations below:

(3.37)

(3.38)

(3.39)

The lead-lag compensator in (3.36) can be designed by positioning the desired closed-loop poles on the left half of the complex plane.

With the lead-lag compensator, the actual closed-loop characteristic polynomial is a third order, which is

By choosing a third order desired closed-loop characteristic polynomial having the following form,

and letting images , we obtain the following linear equations:

The parameters are found via the solution of the linear equations as

(3.40)

The polynomial equation images is called Diophantine equation, which is the essential step for finding the controller parameters in the pole assignment controller design. The matrix with the dimensions images in (3.40) is called the Sylvester matrix, which is required to be invertible in the pole assignment controller design.

The following tutorial summarizes the computational procedure for finding the parameters for the PD controller with filter. We will use this program later on for applications.

Tutorial 3.1

In this tutorial, we will write a simple MATLAB program to calculate the PD controller parameters. The desired closed-loop characteristic polynomial is specified as

and the model is a second order system with the transfer function

Step by Step

Create a new file called PDplace.m for the MATLAB function.
Define the input and output variables for the MATLAB function, where , , , and are the model parameters and is the desired closed-loop characteristic polynomial. Enter the following program into the file:
```
function [Kc,Kd,tauf]=PDplace(a1,a0,b1,b0,Acl); 
```
Find the closed-loop performance parameters. Continue entering the following program into the file:
```
ac_2=Acl(2);
ac_1=Acl(3);
ac_0=Acl(4); 
```
Form the following matrix and vector for the solution of the PD controller parameters, and solve the linear equation. Continue entering the following program into the file:
```
S_matrix=[1 b1 0; a1 b0 b1; a0 0 b0];
Vec=[ac_2-a1;ac_1-a0;ac_0];
contr_p=inv(S_matrix)*Vec; 
```
Convert the parameters into a PD controller with derivative filter. Continue entering the following program into the file:
```
L0=contr_p(1);
p1=contr_p(2);
p0=contr_p(3);
tauf=1/L0;
Kc=p0/L0;
Kd=p1/L0-p0/L0^2; 
```

Test this program using the double integrator system presented in Example 3.3, where we use the following codes:

a1=0;
a0=0;
b1=0;
b0=0.1;
Ac=conv([1 1],[1 1]);
Acl=conv(Ac,[1 1]);
[Kc,Kd,tauf]=PDplace(a1,a0,b1,b0,Acl)

For many applications, the second order model is simplified as

(3.41)

and the PD controller parameters have the following solutions:

(3.42)

(3.43)

(3.44)

3.3.2 Analytical Examples for Ideal PID with Pole-zero Cancellation

When designing a PID controller, the pole–zero cancellation technique is widely used in the application field. The main reason behind this practice is that with the pole–zero cancellation technique, the controller parameter calculations become very simple and can be performed using pencil on the back of an envelope. However, there are two important rules regarding the pole–zero cancellation technique. Firstly, an unstable pole or zero in the system should not be canceled because the cancellation will lead to an internally unstable system, for the reason that the canceled pole or zero is still part of desired closed-loop poles (see Section 2.5). Secondly, a stable pole close to the imaginary axis, which is a pole corresponding to a large time constant, in the system should not be canceled because the slow pole will re-appear to become the dominant pole in the closed-loop response to input disturbance, and as a result, slow disturbance rejection will occur (see the sensitivity analysis in Chapter 2). As a general rule, a stable pole or zero could be canceled if its position is on the left-hand side of the desired closed-loop poles on the complex plane. Additionally, it is the faster stable pole in the plant model that gets canceled.

We assume that there are two poles in a second order model and both are real, stable poles. With these assumptions, the transfer function is expressed as

(3.46)

where images is positive and images .

The PID controller is assumed to have the ideal structure,

(3.47)

and an implementation filter will be added at the implementation stage with a small images parameter chosen by the designer (see Section 1.2). This means that the parameter images will not be considered in the design stage. We will deploy a technique called pole–zero cancellation in the PID controller design, and with this technique the controller parameter solutions become very simple.

We re-write the PID controller given in (3.47) into the transfer function form,

(3.48)

By comparing (3.48) with (3.47), we have the relationships,

(3.49)

Thus, we find the parameters in (3.48) first, then convert them into the PID controller parameters required in the implementation stage.

When using the pole–zero cancellation technique, we assume that the numerator of the controller images is factored and the controller now has the form,

(3.50)

By choosing the zero of the controller images equal to the pole of the model images (i.e. images ), we canceled the pole in the model with the zero in the controller. The net effect of this is that the relationship images is simplified into

(3.51)

the closed-loop transfer function between the reference signal images and the output images becomes

(3.52)

Note that the free parameters in (3.52) are images and images , respectively, and its denominator is a second order polynomial. The design becomes identical to the case when we designed the PI controller using pole assignment technique. Thus, by choosing the desired closed-loop characteristic polynomial as

(3.53)

with images and images or images , and equating the desired closed-loop characteristic polynomial with the denominator of (3.52), we obtain the polynomial equation,

(3.54)

By comparing both sides of (3.54), the free parameters are determined as

(3.55)

With the parameters, the PID controller is re-constructed with the information images as

(3.56)

Its actual parameters are expressed using (3.49) as

(3.57)

(3.58)

(3.59)

Example 3.4

Assume that a dynamic system has the second order transfer function

(3.60)

Design a PID controller using the pole assignment controller design technique with pole–zero cancellation. The closed-loop performance is specified by with . Simulate the closed-loop response and observe the characteristics of input disturbance rejection and the effect of measurement noise.

Solution. First, we write the transfer function model (3.60) in the form used for the derivation of PID controller parameters:

(3.61)

From this equation, we find , , and . Therefore, in the pole–zero cancellation, we will cancel the pole at . Using (3.57)–(3.59), we calculate the PID controller parameters as

With the implementation parameter , in the closed-loop simulation a unit step signal as the reference signal is introduced at time and an input step disturbance with amplitude of 1.5 is introduced at Figure 3.4 (a) shows the closed-loop response in a noise-free environment. In order to observe the characteristics of the PID controller in a noisy environment, a band limit white noise with variance 0.01 is added to the closed-loop simulation and the closed-loop simulation results are shown in Figure 3.4 (b). In comparison with the noise free case, it is seen that the measurement noise has caused significant fluctuations in the control signal. This observation leads to some caution when we use a PID controller in a noisy environment.

Four graphs depicting Time on the horizontal axes, Control and Output on the vertical axes, and curves plotted for PID controller in a noise free environment and noisy environment. — **Figure 3.4** Closed-loop response of PID control system (Example 3.4). (a) PID controller in a noise free environment. (b) PID controller in noisy environment.

3.3.3 Analytical Examples for PID Controllers with Filters

As the system becomes more complicated, a derivative filter needs to be used as part of the PID controller design to give an extra degree of freedom in the solution of the controller parameters. This is important in eliminating the approximation introduced by choosing a filter time constant in the implementation stage, hence enhancing the robustness of the PID control system.

The general form of PID controller is defined by the transfer function

(3.62)

where images is a parameter that needs to be utilized as part of the derivative filter. However, the industrial control systems are often defined in terms of the PID controller parameters, images , images , images , and a filter time constant images . 1

Thus, we need to find PID controller parameters that lead to a completely identical configuration between the controller images defined by (3.62) and an industrial PID controller. With this in mind, we will choose the PID controller parameters such that

(3.63)

is identical to the PID controller in (3.62). The problem is solved using reverse engineering by expressing (3.63) as

(3.64)

which should be exactly equal to (3.62). By comparing these two expressions, we obtain,

(3.65)

(3.66)

(3.67)

(3.68)

We solve the PID controller parameters using these four linear equations and their values are,

(3.69)

(3.70)

(3.71)

(3.72)

Example 3.5

We consider the Padé approximation of irrational transfer function for a first order plus delay model:

(3.73)

Find the analytical solution of the PID controller parameters with filter using pole-zero cancelation.

Solution. The first order plus delay model is stable with . The transfer function model (3.73) can also be written in a second order model pole–zero form,

(3.74)

where , .

To find the analytical solution of the PID controller with filter, we will use the pole–zero cancellation technique. For this purpose, if , then we choose and . In the case of a plant with a dominant time delay when , we let and .

Because the zero is unstable located at , this zero should not be canceled in the controller design. However, we will cancel the pole , which has a faster dynamics response. The PID controller structure has the transfer function form,

(3.75)

which is an ideal PID controller with a filter. The filter pole will be used in the design. We also assume that the PID with filter has the zeros located at and , where corresponds to one of the poles in the model. The open- loop transfer function has the quantity,

(3.76)

The closed-loop transfer function is

(3.77)

Note that the denominator of (3.77) is a third order polynomial and it has three unknown controller parameters, , and . Thus, the desired closed-loop characteristic polynomial must be a third order polynomial with its order to match the denominator of (3.77). To this end, we select

(3.78)

where is a positive parameter. As before, we select the damping coefficient and the natural frequency to reflect the design requirements, such as closed-loop response time and bandwidth. The extra pole located at is often chosen to be far away to the left of the dominant poles (). For example, we could select .

The closed-loop characteristic polynomial equation is expressed as

(3.79)

where the left-hand side of this equation is the closed-loop characteristic polynomial and the right-hand side is the desired one. For an exercise, we work out the exact quantities for both sides as

(3.80)

By comparing the coefficients of the both sides of the polynomials, three linear equations are obtained,

(3.81)

(3.82)

(3.83)

We solve for based on (3.83):

(3.84)

Then, the value of is substituted into (3.82), which becomes,

(3.85)

Note that both (3.81) and (3.85) contain the same pair of unknown variables (, ), so we will solve these two together using these two equations. From (3.81), we find the value of ,

(3.86)

Substituting this into (3.85) and collecting the terms, we find as

(3.87)

where we assume that . The value of is found using (3.86) with given by (3.87). From , we also find as

(3.88)

With the parameters , , and calculated, the PID controller with filter is re-constructed as,

(3.89)

where corresponds to the location of the pole in the model that we chose to cancel.

Equivalently, (3.89) is expressed in the more general form,

(3.90)

where is calculated using (3.87), and .

Example 3.6

Given a first order plus delay system with the transfer function

(3.91)

find the PID controller parameters using the pole assignment design technique. The desired closed-loop performance is specified by , . To understand that the approximation of time delay using the transfer function model causes error between the actual plant and the model used for the design, find the PID controller parameters for and then reduce it to , and simulate the closed-loop performance with a unit step reference signal and disturbance rejection of step signal with amplitude 0.2.

Solution. The first order plus delay model is approximated using Padé approximation, leading to

(3.92)

In the design, we cancel the pole from the time delay, and assign the values of and as and Also from (3.92), we find and

In the calculation, we first use . We calculate the value of as

(3.93)

Four graphs depicting Time on the horizontal axes, Control and Output on the vertical axes, and curves plotted for PID controller in 0.707, 1 and wn=0.4 and 0.707, 1 and wn=0.2. — **Figure 3.5** Closed-loop response of PID control system (Example 3.6). (a) PID control with , and . (b) PID control with , and .

and as

(3.94)

, and . From these parameters, we calculate the PID controller parameters using (3.69)–(3.72):

Figure 3.5(a) shows the closed-loop responses. It is seen that both control signal and the plant output signal are oscillatory, which is due to the modeling error introduced by the approximation of the time delay. To demonstrate that reducing the desired bandwidth of the closed-loop system will reduce the oscillation, we reduce the performance parameter from 0.4 to 0.2. Following the same procedure, we obtain the PID controller parameters for , as

Figure 3.5(b) shows the closed-loop responses with this reduced . It is seen indeed that the closed-loop oscillation is eliminated.

The following example will examine the effect of pole–zero cancellation on reference following and disturbance rejection.

Example 3.7

AC motor angular position control is required in the applications of high precision machining. Assume that an AC motor has the Laplace transfer function model,

(3.95)

where the input to the model is the torque current and the output is the angular position of the motor. Design a PID controller with filter using the pole assignment design technique with pole–zero cancellation, where all the desired closed-loop poles are placed at . Verify that the output of the closed-loop control system will track a constant reference input without steady- state error. Also investigate the dynamic response of the closed- loop system to an input disturbance with unknown amplitude of .

Solution. Write the model in a pole and zero form,

(3.96)

and the controller in the form,

(3.97)

Since we are going to cancel the pole in the AC motor model, the controller is parameterized as

(3.98)

With this choice, we have the open-loop transfer function,

(3.99)

The closed-loop transfer function from the reference signal to the output is

(3.100)

The denominator of (3.100) is a third order polynomial, which is the actual closed-loop characteristic polynomial determining the closed- loop poles. Thus the number of closed-loop poles is 3. With all closed-loop poles placed at , we select the desired closed- loop polynomial as

(3.101)

The actual closed-loop characteristic polynomial is made equal to the desired one:

(3.102)

This equation is expressed as

(3.103)

Comparing both sides of the equation, we have

(3.104)

(3.105)

(3.106)

These equations give the solution of the PID controller parameters, , , and . The controller is constructed with these parameters and the canceled pole:

(3.107)

The PID controller parameters are calculated based on as

The transfer function between the reference signal and the output signal is

(3.108)

With the pole–zero cancellation, we obtain the closed-loop transfer function as

(3.109)

where we have used the Diophantine equation 3.102.

Assume that the reference is a step signal with the Laplace transform , then the output is

(3.110)

Applying final value theorem, we calculate the final value of the output as

(3.111)

So, the output of the closed-loop control system will follow the step reference signal with zero steady-state error, and all closed- loop poles for this relationship are all positioned at as the design required.

Figure 3.6 shows the reference responses for the three different controller configurations. Because the controller has a filter, the order of its denominator equals the order of its numerator and it can be directly implemented on the feedforward path, namely . The output is shown by line (1) in Figure 3.6. It is seen that there is a large overshoot occurring in the output response. The output response illustrated by line (2) is the closed-loop response by putting both the proportional and derivative terms on the output only. In this implementation approach, there is no overshoot, but the response is slower than the first approach. When only the derivative term is implemented on the output, the closed-loop response is shown by line (3), which has a smaller amount of overshoot and the closed- loop response speed is faster than the case indicated by line (2).

To investigate the dynamic characteristics of the output response with respect to the input disturbance, we write the closed-loop transfer function between the input disturbance and the output as

(3.112)

which is, by substituting the values of , , and , and the Diophantine equation into (3.112),

(3.113)

The denominator of (3.113) indicates that there are four poles for this closed-loop relationship: three of them are at the desired location and one of them is at , which is the plant pole we canceled in the design. This is not good because the plant pole has re-appeared in the closed-loop system for input disturbance response and the magnitude of the plant pole was 10 times smaller than the desired closed-loop poles we selected in the design. This means that the actual closed-loop response speed for input disturbance rejection is much slower than the closed-loop performance we designed for, despite that the dynamic response to reference following has achieved the closed-loop performance we have designed for.

At the steady-state, using the final value theorem, we can verify that

(3.114)

Therefore, the PID controller will completely reject the disturbance with zero steady-state error. Figure 3.7 shows the dynamic response to an input disturbance with unknown amplitude. The disturbance occurred at time 50 s, and it is seen that the closed-loop response to the disturbance is very slow and it takes about 50 s to bring the output response back to 0, which is the steady-state in this case.

Graph depicting Time (sec) on the horizontal axis, Output on the vertical axis, and a curve plotted for Input disturbance rejection. — **Figure 3.7** Input disturbance rejection (Example 3.7). All controller structures produce identical disturbance responses.

When the plant is underdamped, the pole–zero cancellation technique we used before will be avoided for the reason that the plant pole that was canceled in the design will re-appear in the closed-loop system for input disturbance rejection.

3.3.4 PID Controller Design without Pole–Zero Cancellation

We assume that the PID controller has the form,

(3.115)

and the second order model has the form,

(3.116)

Without pole–zero cancellation, the open-loop transfer function is

(3.117)

The closed-loop transfer function is

(3.118)

Note that the denominator of the closed-loop transfer function is a fourth order polynomial and there are four unknown controller parameters to be determined in the design. Thus, the desired closed-loop characteristic polynomial images must be a fourth order polynomial with all zeros on the left half of the complex plane. For instance, we can assume that images has the form,

(3.119)

where the dominant poles are images ( images or 1), and images . For simplicity, the desired closed-loop characteristic polynomial images is denoted as images . To assign the closed-loop poles to the desired locations, we solve the Diophantine equation,

(3.120)

By multiplication and collecting terms, we find the exact quantity on the left-hand side of (3.120), which is equal to its right-hand side:

(3.121)

By comparing both sides of (3.121), a set of linear equations is formed,

(3.122)

(3.123)

(3.124)

(3.125)

This set of linear equations is expressed in matrix and vector form for convenience of solution,

(3.126)

where we assume that the square matrix images (called the Sylvester matrix) is invertible.

Example 3.8

An inverted pendulum on a cart has external force as input and its output is the angular position . The Laplace transfer function to describe the dynamics of a laboratory pendulum test bed has the form,

(3.127)

Design a PID controller for this test bed, where the closed-loop performance is specified by , and . Simulate a closed-loop step response using a PID controller with the configuration of the derivative term on output while the proportional and integral terms are on the error signal, and with the configuration of both derivative term and proportional term on the output while only the integral term is on the error signal.

Solution. For this example, , , , and . From (3.126), the linear equation formulated based on the pole assignment controller design technique is

(3.128)

Solution of this set of linear equations gives the controller designed using the pole assignment technique: , , , . We convert these parameters into PID controller parameters as

(3.129)

(3.130)

(3.131)

(3.132)

Note that the proportional gain is negative, not positive, as one might think. This is because the system is unstable with a pole on the right-hand side of the complex plane and the proportional controller gain does not follow the rules established for stable systems.

The closed-loop step response is simulated for both structures. Figure 3.8 shows both control signal and output signal for a unit step change at time . In comparison, the IPD controller structure by implementing the proportional term on the output only significantly reduced the overshoot exhibited in the original structure.

3.3.5 MATLAB Tutorial on Solution of a PID Controller with Filter

Tutorial 3.2

In this tutorial, we will write a simple MATLAB program to calculate the PID controller parameters. The desired closed-loop characteristic polynomial is specified as

and the model is a second order system with the transfer function

Step by Step

Create a new file called PIDplace.m for the MATLAB function.
Define the input and output variables for the MATLAB function, where , , , and are the model parameters and is the desired closed-loop characteristic polynomial. Enter the following program into the file:
```
function [Kc,tauI,tauD,tauf]=PIDplace(a1,a0,b1,b0,Acl); 
```
Find the closed-loop performance parameters. Continue entering the following program into the file:
```
ac_3=Acl(2);
ac_2=Acl(3);
ac_1=Acl(4);
ac_0=Acl(5); 
```
Form the following matrix and vector for the solution of the PID controller parameters, and solve the linear equation. Continue entering the following program into the file:
```
S_matrix=[1 b1 0 0; a1 b0 b1 0; a0 0 b0 b1; 0 0 0 b0];
Vec=[ac_3-a1;ac_2-a0;ac_1;ac_0];
contr_p=inv(S_matrix)*Vec; 
```

Convert the parameters into a PID controller with derivative filter. Continue entering the following program into the file:

L0=contr_p(1);
c2=contr_p(2);
c1=contr_p(3);
c0=contr_p(4);
tauf=1/L0;
tauI=c1/c0-tauf;
Kc=tauI*tauf*c0;
tauD=(c2*tauI*tauf-Kc*tauI*tauf)/(Kc*tauI);

Test this program using the inverted pendulum system presented in Example 3.8, where we use the following codes:

b1=0;
b0=-0.1;
a1=0;
a0=-1;
Ac1=[1 0.707*20 100];
Ac2=[1 20 100];
Acl=conv(Ac1,Ac2);
[Kc,tauI,tauD,tauf]=PIDplace(a1,a0,b1,b0,Acl)

3.3.6 Food for Thought

Would you design a PD or a PID controller for a first order system?
If a system has a pole located on the right half of the complex plane (unstable pole), can you choose a controller zero to cancel this unstable pole?
Is it correct to say that the open-loop pole that was cancelled in the design will re-appear at the output response to input disturbance?
Is it correct to say that the open-loop pole that was cancelled will re-appear at the output response to output disturbance?
Is it correct to say that the open-loop pole that was cancelled will re-appear at the output response to reference signal?
How would you deal with time delay when using pole-assignment controller design technique?
In the general pole-assignment controller design (without pole-zero cancellation), what is the condition required to obtain unique solutions of controller parameters?

3.4 Resonant Controller Design

The resonant controller, different from the PI or PID controller, incorporates the factor images in the denominator of the controller. With the embedded mode, the closed-loop feedback control system is designed to be stable, and at the steady-state the output of the control system will completely track the sinusoidal signal and/or reject a sinusoidal disturbance signal that contains the frequency images without any steady-state errors (see Section 2.5.3).

3.4.1 Resonant Controller Design

Consider a first order transfer function that is used to describe the dynamics of an AC motor with the form,

(3.133)

where the input is the torque current and the output is velocity. The task is to design a controller images to reject a sinusoidal disturbance with frequency images ( images ). Pole-assignment controller design will be used here.

For the first order system, the controller structure is chosen as

(3.134)

Here, the denominator of the controller is second order ( images ), the numerator is also chosen to be second order to allow proportional control action. This choice leads to three unknown coefficients images , images , and images to be determined. The actual closed-loop characteristic polynomial is

(3.135)

This is a third order polynomial. Therefore, the desired closed- loop characteristic polynomial should be third order and the number of desired closed-loop poles should be 3 accordingly. For instance, we can assume that the desired closed-loop characteristic polynomial images has the form ( images )

(3.136)

where the dominant poles are images ( images or 0.707), and images . For simplicity, the desired closed-loop characteristic polynomial images is denoted as images .

With the pole assignment controller design technique, we let the actual closed-loop characteristic polynomial equal the desired closed-loop characteristic polynomial, which leads to,

(3.137)

This equation 3.137 is called the Diophantine equation. By substituting the expressions of images , images , images , images , and images into (3.137), the Diophantine equation is written as

(3.138)

In order for the left-hand side of the equation to be equal to the right-hand side of the equation, we have the following linear equations:

(3.139)

(3.140)

(3.141)

Solving these linear equations gives the coefficients of the controller as

(3.142)

(3.143)

(3.144)

3.4.2 Steady-state Error Analysis

To show that the output of the closed-loop control system will follow a sinusoidal signal with frequency images , we calculate the closed-loop feedback error signal images in relation to the sinusoidal reference signal images . Here, the control signal is

(3.145)

and the output signal is

(3.146)

Noting that images , by substituting this into (3.146), we obtain

(3.147)

The relationship between the reference signal images and the output signal images is

(3.148)

where we have used the Diophantine equation 3.137. When the reference signal images , its Laplace transform is images . From (3.148), we have the Laplace transform of the feedback error signal,

(3.149)

where we have canceled the factor images . Since the denominator of (3.149) contains all zeros on the left half of the complex plane, by applying final value theorem, we obtain

(3.150)

Because images and images , we conclude that the output images will converge to the reference signal images .

To show that the closed-loop control system will completely reject a sinusoidal disturbance, we will find the relationship between the input disturbance and the output. Here, the transfer function between the input disturbance images and the output images is

(3.151)

Assume that the disturbance signal is a sinusoidal signal images with unknown amplitude images , and it has the Laplace transform images . Thus, the output in response to the input disturbance images is

(3.152)

The zeros of the denominator are all on the left half of the complex plane as the parameters images , images , or 1 and images . Thus, by applying final value theorem, we obtain

(3.153)

Therefore, the output images in response to the input disturbance images is zero at the steady-state. This means that the input disturbance images will be completely rejected by the closed- loop feedback control system.

Example 3.9

For an AC motor with parameters and , if the disturbance is at frequency , and all three desired closed-loop poles are selected as , design the feedback controller for this AC motor so that the closed-loop control signal will reject the sinusoidal disturbance signal.

Solution. The desired closed-loop characteristic polynomial is

(3.154)

Four graphs depicting Time (sec) on the horizontal axis, y and r, disturbance, and Error on the vertical axis, and curves plotted for Disturbance rejection and Reference following. — **Figure 3.9** Closed-loop response of resonant control (Example 3.9). (a) Disturbance rejection. (b) Reference following.

Here, , , and . From the equations given in (3.138), we find the controller parameters as

(3.155)

(3.156)

(3.157)

First, we simulate the closed-loop disturbance rejection by operating the system at the steady-state value 0.3 and at time , inject a disturbance signal . Figure 3.9(a) shows the output response to the disturbance signal. It is seen that the disturbance is completely rejected and the output returns to the reference signal. The same control system will also follow a sinusoidal reference signal, as shown in Figure 3.9(b), which shows that the output tracks a sinusoidal signal . Thus, the control system we designed for sinusoidal disturbance rejection will automatically follow a sinusoidal signal with the same frequency.

3.4.3 Pole–Zero Cancellation in the Design of a Resonant Controller

Example 3.10

An electric circuit has a second order transfer function:

(3.158)

This system has a sinusoidal disturbance with frequency . Design a resonant controller to completely reject this disturbance at the steady-state. All desired closed-loop poles are positioned at .

Solution. Since the positions of the closed-loop poles are on the right-hand side of the fast pole of the system , we will choose to use the pole–zero cancellation technique in the design. Cancellation of this pole will not significantly alter the response speed of the input disturbance. However, because we do not know the exact order of the controller, we will start the design using the lowest order possible and see if we can find the unique controller parameter solutions.

The simplest controller structure for a resonant controller is

(3.159)

This structure is chosen because we know that in order to reject the sinusoidal disturbance the denominator of the controller will contain the factor , and the order of the numerator is to match the order of the denominator.

To implement the pole–zero cancellation, the controller is re- written as

(3.160)

The open-loop transfer with the plant and the controller is

(3.161)

The closed-loop transfer function between the reference signal and the output is

(3.162)

Here, from (3.162), we can see that the denominator of the closed-loop transfer function is a third order polynomial and therefore there are three closed-loop poles in the system. When we look closely at (3.162), there are only two unknown controller coefficients, and . If we were to examine the Diophantine equation,

(3.163)

where the right-hand side of the equation is the desired closed- loop polynomial, there are three linear equations in order to match the actual closed-loop poles to the desired closed-loop poles. Thus, there is no unique solution to determine the two unknown controller coefficients and when there are three linear equations. This means that the controller structure is too simple for the system.

We increase the controller complexity by choosing

(3.164)

where both denominator and numerator have the same order of 3 for maintaining proportional control and practical implementation. With the pole–zero cancellation, we re-write the controller structure as

(3.165)

The open-loop transfer function is

(3.166)

With the cancellation of the factor , the closed-loop transfer function is

(3.167)

From (3.167), it is seen that the closed-loop characteristic polynomial is fourth order and there are four closed-loop poles, and there are four independent unknown controller parameters: , , , and . As per the design specification, all closed-loop poles are positioned at , hence the desired closed-loop characteristic polynomial is . This leads to the Diophantine equation:

(3.168)

which is

(3.169)

Comparing both sides of the equation 3.169, we obtain the linear equations:

(3.170)

(3.171)

(3.172)

(3.173)

Solving these linear equations gives the values of the controller parameters: , , , and . With these parameters, we re-construct the resonant controller as

(3.174)

The closed-loop output response to an input disturbance is simulated with the system operating at the steady state of 1, the disturbance entering the system at . Figure 3.10 shows the output response to the disturbance input. It is seen that it takes about three seconds for the closed-loop system to completely reject the sinusoidal disturbance.

With the resonant controller, as shown in Section 2.5.3, the complementary sensitivity function is required to satisfy images at the frequency images . Thus, this indicates that the bandwidth of the closed-loop resonant control system is to be greater than images , at least. If images is large, from the analysis given in Chapter 2, this implies in general that a resonant control system requires a better mathematical model for robustness and better sensors for noise attenuation.

3.4.4 Food for Thought

A periodic reference signal has a known frequency 50 Hz. What is the frequency parameter you would use in the design of resonant controller so that the reference signal could be tracked without steady-state error?

Figure 3.10 Sinusoidal input disturbance rejection (Example 3.10).
If a resonant controller is designed to track a sinusoidal signal with frequency without steady-state error, will the same resonant controller reject a periodic input disturbance signal with the same frequency?
If the sinusoidal reference signal is known to have the frequency and the disturbance is known to have a frequency , is it correct to choose the denominator of the resonant controller to contain the factor ?
If the reference signal is a sinusoidal signal with frequency plus a constant, what would be the key factors in the denominator of the resonant controller you would choose so that the closed-loop control system will track this reference signal?

3.5 Feedforward Control

Feed forward control is widely used in combination with either PID or resonant controllers. The starting point for feedforward compensation is that the feedforward variables used are either directly measured or estimated. Their effect is captured at the control signal from which it can be subtracted. It has various forms; however, the basic idea remains the same.

3.5.1 Basic Ideas about Feedforward Control

Assume that the output of a dynamic system is described by the Laplace transform:

(3.175)

where images is the Laplace transform of the control signal and images is the disturbance that is measured and used in the feedforward compensation. To introduce feedforward compensation, (3.175) is re-written as

(3.176)

where an intermediate control signal images is defined as

Based on (3.176), a PID controller images is designed using the transfer function images to generate the intermediate control signal images from which the effect of the disturbance images is subtracted. This explicitly leads to the actual control signal images to be expressed as

(3.177)

Clearly, the assumption is that the transfer function images is stable and realizable in addition to images being measured. Figure 3.11 illustrates the feedback and feedforward control system. As an example, if we assume that images and images are represented by the following first order plus delay transfer functions:

then the control signal images is expressed as

Because the time delay of the disturbance model is larger than the time delay of the plant model, the transfer function images is realizable, hence the feedforward compensation can be implemented for the application.

3.5.2 Three Springs and Double Mass System

To illustrate how to use a feedforward controller in conjunction with a PID controller, we will examine the PID control system design for a three springs and double mass system.

A three springs and double mass system Tongue (2002) is illustrated in Figure 3.12. In this figure, the two blocks are with mass images and images , and there are two spring constants images for the right and left springs, images for the middle spring. The manipulated variables are the two applied forces images and images and the output variables are the distances of the mass block movement images corresponding to block one and images corresponding to block two.

We consider the simplified spring force images where images and images is the distance of the mass movement, and apply Newton's law 2 to the first block to obtain the following equation:

(3.178)

where the first term represents the first external force, the second and third terms represent the forces from spring 1 and spring 2, respectively. The right-hand side of (3.178) is the multiplication of the mass and acceleration of block 1. Similarly, by applying Newton's law to the second block, we obtain

(3.179)

where the first term represents the second external force, and the second and the third terms are the forces generated from spring 3 and spring 2. The right-hand side of (3.179) is the multiplication of mass and acceleration of block 2.

To have a clear view of the dynamics, we re-write (3.178) and (3.179) as

(3.180)

(3.181)

We have two second order systems for the three springs and double mass system. Additionally, there are interactions between the two systems.

As in Tongue (2002), the physical parameters are images kg, images kg, images N images and images N images . As an exercise, the parameter is varied to images N images .

Example 3.11

In this example, we assume that the second block is not controlled. The force from the second block acts as a disturbance on the closed-loop control of the first block. Find the natural frequency of the uncontrolled system and choose all closed-loop poles in relation to this natural frequency when designing the PID controller.

Solution. The Laplace transforms of the differential equations 3.180 and (3.181) are obtained as

(3.182)

(3.183)

where we have assumed zero initial conditions of the variables and . With these two transfer function models, we can build the Simulink simulator for the three springs double mass system using transfer function blocks.

To find the natural frequency for the first mass, (3.182) is considered, where the Laplace transform is the external disturbance. Thus, the open-loop system has a pair of poles on the imaginary axis, determined by the solution of the polynomial equation:

which is

The natural frequency is 8.3667 N .

3.6.2.1 PID Controller with Filter

The PID controller with filter is designed based on the transfer function model (3.182) using the MATLAB program PIDplace.m (see Tutorial 3.2). The term images is considered as disturbance. Two pairs of complex poles are used in the pole assignment PID controller design. With damping coefficient images , the parameter images is used as a performance tuning parameter. Because of the interactions between the two outputs, the closed-loop dynamics from the first mass block are more complex. As a result, there is a minimum images required for the stabilization of the closed-loop system. Through simulation studies, this minimum images is approximately five times the natural frequency of the original system. This behavior is similar to that of controlling unstable systems where a minimum controller gain is required for robustness of the feedback system.

By selecting images and images for the two pairs of complex poles, the program PIDplace.m calculates the PID controller parameters as

Closed-loop Simulation Studies

In the first simulation, we assume images for images . However, because the second mass is connected to the first mass, the output images is not zero. With sampling interval images (s), a step reference signal with amplitude 0.1 is added to the closed-loop control. Figure 3.13(a) shows the control signal used to move the first mass from the origin to 0.1 m and Figure 3.13(b) shows the output signal. It is clearly seen from Figure 3.13(a) that the second mass generates a sinusoidal disturbance and the feedback control signal tries to compensate it. The effect of the sinusoidal disturbance is hardly noticeable. Another simulation scenario is presented. Assume that there is a constant negative force acting on the second mass block, which is images . Additionally, the control signal images is constrained to positive values ( images ). Figure 3.14 shows the closed-loop responses for the three springs and double mass system with images . The control signal (see Figure 3.14(a)) still tries to reduce the effect of the disturbance, however because of its larger amplitude, the effect of the disturbance can be seen from the output response (see Figure 3.14(b)).

Two graphs depicting Time (sec) on the horizontal axis, Output and Control on the vertical axis, and two curves plotted for output and control signals with u2 equals 0. — **Figure 3.13** Closed-loop responses for three springs and double mass system with (Example 3.11). (a) Control signal. (b) Output signal.

Example 3.12

To overcome the sinusoidal disturbance generated by the second mass, we will design a disturbance feedforward control together with the PID feedback control and simulate the closed-loop control performance. We assume that the output is measured.

Solution. To design the feedforward together with the PID controller, we consider the physical equation 3.180 that describes the dynamics of the first mass block:

(3.184)

where we have defined an intermediate control variable

Now, the PID controller presented in Example (3.11) is unchanged and is used here to calculate the intermediate control signal . The feedforward control is then realized as

(3.185)

The PID controller with disturbance feedforward compensation is evaluated with the case of . Figure 3.15(a) shows the closed-loop control signal, from which we can see that the control signal captures the oscillations caused by the second mass block. More importantly, because of the characteristics of the control signal, the output no longer has the periodic oscillations when compared with that shown in Figure 3.14(b). This means that the periodic disturbance is completely eliminated.

One needs to be cautious when using feedforward compensation. The effectiveness of feedforward compensation is dependent on the accuracy of the model used in the compensation. It is reasonable to say that this dependence is more severe than the cases when feedback control strategies are used. Here, the parameter plays an important role. To illustrate, we consider the following three cases:

(3.186)

Figure 3.16 compares the output responses for the three feedforward compensations. It is clearly seen that the periodic disturbance is no longer completely compensated and the responses are different from each other in the steady-state behavior. Also, the worse case is when the sign of the compensation was wrong.

3.5.3 Food for Thought

In the feedforward control, if the feedforward variable is bounded, as shown in the three springs and double mass system, would you believe that the feedforward control will not cause closed-loop instability?
In most of the applications when using feedforward compensation, the feedforward variable is captured as part of the control signal as shown in the three springs and double mass system so it is conveniently subtracted. Can we extend this idea of feedforward compensation to the reference signal?
What are the main advantages of using feedforward control?
What are the drawbacks of using feedforward control?

3.6 Summary

This chapter has discussed one of the most widely used control system design methods for PID and resonant controllers. The central ideas behind the pole-assignment controller design are to specify the desired closed-loop performance via the locations of the closed-loop poles and to match the desired closed-loop poles with the actual closed-loop poles by finding the solutions of a polynomial equation. These methods are conceptually and computationally simple, and the design methodology can be extended to various controller structures and models. The other important aspects are summarized as follows.

For a first order model, we can design a P controller or a PI controller or a resonant controller.
For a second order model, we can design a PD controller or a PID controller or a resonant controller with a first order filter.
For a higher order model, the structure of the controller needs to be selected so that the controller parameters can be determined uniquely.
We can use the pole-zero cancellation technique for the purpose of getting simple analytical solutions. We can only cancel the poles corresponding to fast dynamic response. The canceled poles will re-appear in the closed-loop system in response to input disturbance. We can not cancel unstable poles or zeros in the design.
For a higher order system, model order reduction is used to obtain first order or second order model for the PID controller design.
The locations of the desired closed-loop poles are chosen, and they need to be adjusted to suit the requirement of closed-loop response speed for reference following, disturbance rejection, noise attenuation and robustness of the closed-loop system. It will be desirable to adjust them according to the Nyquist diagram and the sensitivity analysis presented in Chapter 2.

3.7 Further Reading

Internal model control is introduced in the book Morari and Zafiriou (1989).
Pole-assignment PID controller design together with several other design methods are introduced in Goodwin et al. (2000),Cominos and Munro (2002).
Robust pole-assignment controller design is proposed in Soh et al. (1987), Nurges (2006), Wang et al. (2009).
Pole-assignment in combination with sensitivity shaping is proposed in Langer and Landau (1999).
Self-tuning control using pole-assignment design is one of the original self-tuning controllers Wellstead et al. (1979).
Pole-assignment PID controller design with dominant pole consideration and analysis is introduced in Zítek et al. (2013).
With approximation of time-delay and specification of a desired closed-loop time constant, PI and PID controllers are found analytically using direct synthesis method for disturbance rejection (Chen and Seborg (2002)).
A simple relay nonlinear PD controller is proposed for controlling high precision motion system with frictions (Zheng et al. (2018)).
Current control in electrical AC drives and power converters typically uses PI control structures in the synchronous reference frame ( reference frame) because their reference signals are constant (Wang et al. (2015)). If the stationary reference frame is used, then current control for the AC drives and power converters utilizes resonant controllers because their reference signals are sinusoidal (Wang et al. (2015)).

Problems

3.1 Design PI controllers for the systems with the transfer function , where the parameters and are given as follows:
1. and ;
2. and ;
3. and .
In the design, the desired closed-loop characteristic polynomial is selected as with and choosing the natural frequency parameter as , and . Compare the proportional controller gain and the integral time constant for the three choices of the bandwidth. What are your observations?
3.2 PD controllers are designed based on a second order model, which has the following form:
1. Find the PD controllers with a derivative filter for the following systems:
  1. , , , .
  2. , , , .
  3. , , , .
  where all the desired closed-loop desired closed-loop polynomial is chosen to be , where and .
2. Write a Simulink simulation program to evaluate the performance of the three closed-loop PD control systems with a unit step reference signal using sampling interval (sec), where the derivative control is implemented on the output only. Will the closed-loop output follow the reference signal without steady-state error?
3.3 In the design of a PI controller, it is often to approximate a higher order system with a first order transfer function where the dominant time constant is identified (largest time constant) and kept, while the other smaller time constants are neglected. This follows the procedure of writing the transfer function using the form

where we assume that .
1. Design PI controllers for the following systems with appropriate approximation of the complex dynamics. The desired closed-loop performance is specified using the polynomial , where and is .
  1. ;
  2. ;
  3. . (Hint: use Pade approximation to the time-delay in order to find the dominant dynamics, )
2. Evaluate the closed-loop stability by using Nyquist diagram. If the closed-loop system is unstable, reduce the parameter to achieve stable closed-loop system.
3. Simulate the closed-loop step response for the PI controlled systems with sampling interval .
4. If the closed-loop response were oscillatory, reduce the parameter until satisfactory performance is achieved.
3.4 Find the PID controller parameters using pole-assignment controller design technique and apply pole-zero cancellation to simplify the parameter solutions. Here, the PID controller structure is assumed to be . Take approximation of the complex dynamics if necessary. The desired closed-loop characteristic polynomial is where and is adjusted for each case. The transfer functions with performance specification are given as below:
1. , and
2. , and .
3. , and .
4. , and .
3.5 Find the parameters for PID controller with a filter using pole-assignment controller design technique. Use pole-zero cancellation technique to simplify the parameter solution and take approximation of the complex dynamics if necessary. With a pole-zero cancellation, the desired closed-loop polynomial is specified as . The transfer functions with performance specification are given as below:
1. , and .
2. , and .
3.6 For a system with the transfer function

design a controller with the structure

where all three desired closed-loop poles are chosen to be . Convert this controller into an ideal PID controller that has the structure so to find the proportional gain , integral time constant and the derivative time . Use final value theorem to show that for a step reference signal with amplitude , the closed-loop output response is as . The following three systems are used in this exercise.
1. , , .
2. , , .
3. , , .
3.7 For a system with the transfer function

design a resonant controller with the structure

where all three desired closed-loop poles are chosen to be . Supposing that the reference signal is a sinusoidal signal , show that as , the feedback error . The following three systems are used in this exercise.
1. , , and .
2. , , and .
3. , , and .
3.8 In order for the closed-loop system to track a reference signal , the controller needs to contain the factors and in its denominator. For a system with the transfer function

design a resonant controller plus integral action with the structure

where all four desired closed-loop poles are chosen to be . Supposing that the reference signal is a sinusoidal signal , show that as , the feedback error . The following three systems are used in this exercise.
1. , , and .
2. , , and .
3. , , and
3.9 A robot arm is described by the second order transfer function:
(3.187)

where the input is the voltage and the output is the position of the arm on x-axis.
1. Design a resonant control system such that the output of the robot arm will follow a sinusoidal reference signal without steady-state error. All desired closed-loop poles are positioned at (hint: use pole-zero cancellation technique to simplify the computation).
2. Verify your design by showing that the error signal () converges to zero as time . That is
  
  Write a Simulink simulation program to evaluate the closed-loop system performance for reference following of the sinusoidal signal where the sampling interval .
3. Now, suppose that the reference signal contains a constant 10, leading to the new reference signal . Show that with this resonant controller, the error signal () also converges to zero as time . Modify the Simulink simulation program to include the new reference signal and confirm that the closed-loop output tracks the new reference signal without steady-state error.
4. Assume that the input disturbance enters the system at half of the simulation time, where and . Modify the Simulink simulation program to include the disturbance and confirm that the resonant controller can not completely reject this input disturbance. What would you propose for the resonant controller structure so that it will reject this disturbance without steady-state error? Verify your answer using final value theorem.
3.10 An unstable system is described by the transfer function:
(3.188)

where and .
1. Design a PI controller to control this system, where all desired closed-loop poles are positioned at .
2. In order to guarantee the closed-loop stability of the PI control system, the magnitude of the stable pole has a relationship with the unstable pole . Use Routh-Hurwitz stability criterion to find this relationship.