In this part of the book we will introduce feedback control by studying the most commonly used industrial control law, the so-called proportional-integral-derivative control, or PID control. PID controllers are widely used because they are intuitive and easy to understand, they can be designed without any understanding of the underlying physics of the system, and a model of the system is not required, which is especially important in the chemical processing industry. However, the stability and performance of the closed-loop system can only be guaranteed when using PID control on second-order systems. In Part IV on observer-based control and Part V on loopshaping, we will extend the basic ideas covered in this Part to higher-order systems.

Fig. 6.1 shows a basic proportional control strategy. The output \(y\) of the physical system is subtracted from the commanded reference output \(r\) to produce the error \(e=r-y\). Proportional control determines the input to the system \(u\) to be proportional to the error, i.e., \(u=k_P e\). The constant gain \(k_P\) is called the proportional gain.

Proportional control can be augmented with integral control to produce PI control, as shown in Fig. 6.2. The basic idea is to integrate the error so that the system responds to error accrued in the past. The constant gain \(k_I\) is called the integral gain.

Alternatively, proportional control can be augmented with a derivative control as shown in Fig. 6.3. The idea is to respond to how fast the error is changing, or the derivative of the error. The constant gain \(k_D\) is called the derivative gain.

When all three elements are used, as shown in Fig. 6.4, the resulting controller is called a PID control. In Part III of this book, we will explore various aspects of PID control. Many of these concepts can be generalized to other control schemes in later chapters, and so our study of PID control will also help to motivate and understand more advanced concepts.

Intuition for PID control can be gained by considering the step response shown in Fig. 6.5. At the beginning of the response, there is a significant error between the current output \(y(t)\) and the reference command \(r(t)\). The proportional control term will tend to push the system so as to reduce the gap between \(y(t)\) and \(r(t)\). As \(y(t)\) begins to change, a fast response may lead to significant overshoot due to the momentum in the system. Therefore, if it is moving too quickly, we may want to slow down the response, or if it is moving too slowly, we may want to speed up the response. Derivative control is used to affect these changes. After the response has settled into steady state, there may be significant steady-state error, as shown in Fig. 6.5. Integral control adds up this steady error and will eventually act to reduce the steady-state error. A good example is an aircraft in a constant crosswind. An integrator can be used to correct the response of the aircraft so that it crabs into the wind to maintain its desired flight path.

Part III of the book is organized as follows. In Chapter 10 we will show how PD control can be used to specify the desired closed-loop poles of a second-order system. In Chapter 11 we introduce the concepts of rise time, natural frequency, and damping ratio for second-order systems and show how these concepts relate to the location of the closed-loop poles. We also show how these quantities can be used to design PD controllers for second-order systems and cascades of second-order systems. In Chapter 12 we use the final value theorem to show the effect of integrators on the closed-loop system. We introduce the notion of system type in the context of reference tracking and disturbance rejection. Finally, in Chapter 13 we show how to write computer code to implement PID controllers.

7 Pole Placement for Second-Order Systems

7.1 Theory

Suppose that the model for a linearized physical system is given by the second-order transfer function \[\tag{7.1} P(s) = \frac{b_0}{s^2 + a_1s + a_0}.\] The open-loop poles of the plant \(P(s)\) are defined to be the roots of the open-loop characteristic polynomial \[\Delta_{ol}(s) = s^2 + a_1 s + a_0,\] which are given by \[p_{ol} = -\frac{a_1}{2} \pm \sqrt{\left(\frac{a_1}{2}\right)^2-a_0}.\] Note that the open-loop poles are determined by the physical parameters and that the poles may be in the right half-plane. If we use PD control to regulate the output \(y\) to the reference command \(r\), then the block diagram is shown in Fig. 7.1.

The transfer function for the closed-loop system can be derived by noting from the block diagram that \begin{align} & Y(s) = \left(\frac{b_0}{s^2 + a_1s + a_0}\right)\left(k_P + k_D s\right)\left(R(s)-Y(s)\right) \notag \\ \implies & (s^2 + a_1s + a_0)Y(s) = \left(b_0k_P + b_0k_D s\right)\left(R(s)-Y(s)\right) \notag \\ \implies & \left[(s^2 + a_1s + a_0)+\left(b_0k_P + b_0k_D s\right)\right]Y(s) \\ & = \left(b_0k_P + b_0k_D s\right)R(s) \notag \\ \implies & (s^2 + (a_1 + b_0k_D)s + (a_0+b_0k_P))Y(s) \\ & = \left(b_0k_P + b_0k_D s\right)R(s) \notag \\ \implies & Y(s) = \frac{b_0k_D s + b_0k_P}{s^2 + (a_1 + b_0k_D)s + (a_0+b_0k_P)}R(s). \tag{7.4} \end{align} Equation 7.4 constitutes the closed-loop transfer function for the system under the influence of PD control. The closed-loop poles are given by the roots of the closed-loop characteristic polynomial \[\Delta_{cl}(s) = s^2 + (a_1+b_0k_D) s + (a_0+b_0k_P),\] which are \[p_{cl} = -\frac{a_1 + b_0k_D}{2} \pm \sqrt{\left(\frac{(a_1+b_0k_D)}{2}\right)^2-(a_0+b_0k_P)}.\] Note that the closed-loop poles can be specified by the designer by selecting \(k_P\) and \(k_D\). The basic idea is to select desired closed-loop poles \(-p_1^d\) and \(-p_2^d\) and then to form the desired characteristic polynomial \[\Delta_{cl}^d(s) = (s+p_1^d)(s+p_2^d) \stackrel{\triangle}{=}s^2 + \alpha_1 s + \alpha_0.\] By setting \(\Delta_{cl}(s)=\Delta_{cl}^d(s)\) we can equate the leading coefficients of each term of the polynomial and solve for the gains \(k_P\) and \(k_D\) as \[\begin{aligned} k_P &= \frac{\alpha_0-a_0}{b_0} \\ k_D &= \frac{\alpha_1-a_1}{b_0}. \end{aligned}\]

One of the disadvantages to the PD architecture shown in Fig. 7.1 is the introduction of a zero in the closed-loop transfer function. Note that while the open-loop system 7.1 does not have a zero, the closed-loop system 7.4 has a zero at \[z_{cl} = - \frac{k_P}{k_D}.\] The presence of the zero will modify the closed-loop response of the system. A simple trick that removes the zero is to use the control structure shown in Fig. 7.2. In this particular case, the differentiator acts only on the output \(y\) and not on the error \(e\). Notice the change of sign on the signal coming from the differentiator.

In this case the closed-loop transfer function is calculated as \begin{align} & Y(s) = \left(\frac{b_0}{s^2 + a_1s + a_0}\right)\left(k_P (R(s)-Y(s)) - k_D s Y(s) \right) \notag \\ \implies & (s^2 + a_1s + a_0)Y(s) = \left(b_0k_P (R(s)-Y(s)) - b_0k_D s Y(s) \right) \notag \\ \implies & \left[(s^2 + a_1s + a_0)+\left(b_0k_P + b_0k_D s\right)\right]Y(s) = b_0k_P R(s) \notag \\ \implies & (s^2 + (a_1 + b_0k_D)s + (a_0+b_0k_P))Y(s) = b_0k_P R(s) \notag \\ \implies & Y(s) = \frac{b_0k_P}{s^2 + (a_1 + b_0k_D)s + (a_0+b_0k_P)}R(s). \tag{7.5} \end{align} Note that the closed-loop transfer function 7.5 has the same poles as Equation 7.4, but that the zero has been removed. An added benefit to the structure shown in Fig. 7.2 is that when \(r(t)\) is a step input, the derivative can introduce a large signal spike in \(u(t)\). These large spikes are removed by only differentiating \(y(t)\) instead of \(e(t)=r(t)-y(t)\).

7.2 Design Study A. Single Link Robot Arm

7.2.1 Example Problem 3.1.10

(a) Given the open-loop transfer function found in problem 3.1.6, find the open-loop poles of the system, when the equilibrium angle is \(\theta_e=0\).

(b) Using the PD control architecture shown in Fig. 7.2, find the closed-loop transfer function from \(r_{\theta}\) to \(\theta\) and find the closed-loop poles as a function of \(k_P\) and \(k_D\).

(c) Select \(k_P\) and \(k_D\) to place the closed-loop poles at \(p_1=-3\) and \(p_2=-4\).

(d) Using the gains from part (c), implement the PD control for the single-link robot arm in simulation and plot the step response for an amplitude of 30 degrees.

Design Study A guided exercise

Compute the PD gains

Match the closed-loop characteristic polynomial to desired poles at \(-3\) and \(-4\).

Use this PD feedback structure to derive the closed-loop characteristic polynomial.

Step 1. Desired characteristic polynomial

\(s\) coefficient Constant term

Check Step 1

Step 2. PD gains

\(k_P\) \(k_D\)

Check Step 2

Pseudocode

The complete solution includes a PD controller class and a simulation script. Before opening it, outline each component separately.

These prompts are ungraded. They are intended to connect the gain calculation to the implementation code in the complete solution.

Show complete solution

7.3 Interactive PD Pole Placement Check

7.4 Knowledge Check

7.5 Notes and References

The concept of using feedback to place the poles of the system at desired locations is a much deeper topic than is covered in this chapter. While the method discussed in this chapter is only applicable to second-order systems, it is usually possible to arbitrarily assign the poles for an \(n^{th}\) order system. In general, to assign \(n\) poles will require \(n\) gains. We will discuss pole placement in more detail in Chapter 14 of this book.