## PI controller design method with desired phase margin and settling time In this paper controller design method for SISO systems with performance specification in term of phase margin and desired settling time is presented. Mentioned method is based on modification of Neimark D-partition method which ensure desired phase margin of open loop and desired settling time by setpoint change. The developed frequency domain design technique is graphical, interactive and insightful. Theoretical results are demonstrated on examples.

1. Introduction

Frequency domain techniques for analysis and controller design dominate SISO control system theory. Bode, Nyquist, Nichols, and root locus are the usual tools for SISO system analysis.
Frequency methods are often used for controller design because it is easy to ensure performance of closed loop system through phase margin in open loop (Nagurka and Yaniv, 2003). To achieve the desired phase margin, controllers are usually designed using Bode characteristics (Fung, et al 1998),( Ho, et al 1995).

In this paper Neimarks method of D-partition is used (Neimark, 1992). This method is usually used for controller design which ensures closed loop stability with desired stability degree. In this paper is presented modification of this method aimed on phase margin instead of stability degree. Settling time is ensured using crossover frequency which has significant influence on this second performance specification.

2. Preliminaries and problem formulation

Consider classical feedback system depicted in Fig. 1. Figure 1: Classical feedback system

The aim is to design PI controller for plant using Neimark D-partition method so, that not only stability will be ensured but performance in term of phase margin and desired settling time too.

3. Theoretical results

Consider closed loop feedback system with PI controller and SISO plant. $R(s) = k + \frac{k_i}{s} \hspace{30pt} G(s) = \frac{B(s)}{A(s)}$ (1)

Substituting (1) into characteristic function (2) $1 + R(s)G(s) = 0$ (2)

and after small manipulations is possible to obtain: $k + \frac{k_i}{s} = -1 \frac{A(s)}{B(s)}$ (3)

After other substitution $s=j \omega$ and division into real and imaginary part: $k = real \left \{ (-1+0j)\frac{A(j \omega)}{B(j \omega)} \right \}$ $k_i = imag \left \{ (-1+0j)\frac{A(j \omega)}{B(j \omega)}(- \omega) \right \}$ (4)

If $\omega$ is changing step by step in interval $\omega \in (0, \infty)$ from real part of (4) is possible to calculate frequency dependent vector of complex numbers which plotted in complex plane create D-curve for parameter k. Similar is it with imaginary part of (4) from which is possible obtain ki. D-curve plot for parameters k and ki (PI controller) can be done in one step. This way is possible to calculate and paint marginal D-curves. If PI controller parameters will be chosen from this D-curves system with controller will have Nyquist plot intersecting point -1+0j in complex plane (Figure. 2). Figure. 2 Nyquist plot intersecting point -1+0j

To obtain any phase margin it is necessary to move intersection point with unit circle form point -1+0j to point situated in lower left quadrant on the same circle. To obtain desired phase margin $\varphi$, (2) have to be modified as follows: $(x+yj)+R(s)G(s)=0$ (5)

where values of x and y depends on $\varphi$ (Figure 3) Figure. 3 Nyquist plot with intersection ensuring phase margin $\varphi$.

According to situation on Figure 3 and known radius of unit circle, values of x and y can be calculate as follows: $y=sin \varphi$ $x=\sqrt{1-y^2}$ (6)

For calculating D-curves of PI controller parameters k and ki, ensuring desired phase margin $\varphi$ (4) is modified to (7). $k = real \left \{ (-x+yj) \frac{A(j \omega)}{B(j \omega)} \right \}$ $k_i = imag \left \{ (-x+yj)\frac{A(j \omega)}{B(j \omega)}(- \omega) \right \}$ (7)

Settling time of controlled system depends on plant gain and values of controller parameters. Higher values of ki controller parameter provide shorter settling time for the price of higher controller output. D-curves of plants with possitive gain, starts $( \omega=0)$ in negative values of both PI controller parameters not far from origin. If plant is possible to stabilize with chosen phase margin using PI control structure, with increasing frequency D-curves continues to area where both PI controller parameters has possitive values (Fig. 4). Fig. 4 PI controller D-curves for different phase margin

Chosing controller parameters from D-cruve also crossover frequency $\omega_c$ is chosen (frequency when open-loop system reaches unity circle). Controller parameters from D-curves maximum point M, yields shortest settling time. Consider parameters with higher frequency as maximum point frequency $\omega_M$ has no practical purpose. By increasing of crossover freqeuncy $\omega_c$ from $\omega_{min}$ (lowest frequency which yields possitive controller parameters) to $\omega_c$, both controller parameters will increase and settling time Ts will decrease. $\omega_{c2} > \omega_{c1} \Rightarrow T_{s2} < T_{s1} \hspace{20pt} \omega_{c1} , \omega_{c2} \in < \omega_{min} , \omega_M>$ (8)

According to (8) it is possible to define controller design algorithm, ensuring desired phase margin and settling time consisting of following steps:

1. Chose desired phase margin $\varphi$ and settling time Ts
2. Calculate x and y according to (6)
3. Calculate $k(\omega )$ and $k_i(\omega )$ frequency vectors from (7) and find $\omega_M$ as frequency where $k_i(\omega )$ has maximum value $max = k_i(\omega_M )$
4. Plot D-curve with k parameter on x-axis and ki on y-axis and design temporary PI controller for any crossover frequency $\omega_{c_tmp} \in (\omega_{min} , \omega_M)$
5. Measure settling time Ts_tmp of system with designed temporary controller
6. Calculate $\omega_c$ frequency ensuring desired settling time Ts as $\omega_c = \frac{T_{s_tmp}}{T_s} \omega_{c_tmp}$

7. If $\omega_c > \omega_M$ back to a) and chose higher Ts or lower $\varphi$, else design final PI controller for crossover frequency $\omega_c$

4. Case study

Consider stable 3-th order plant $G(s)=\frac{2s+1}{s^3 + 3s^2 + 3s+1}$

a) For this plant PI controller ensuring phase margin $\varphi=50^o$ and settling time Ts=6s will be designed.

b) According to (6) and desired phase margin: $y=sin \varphi = 0.766$ $x=\sqrt{1-y^2}=0.6428$

c) D-curves for parameters k and ki can be calculated in one step and plotted in one figure, because characteristic function is divided into real and imaginary part and each parameter is calculated from another part. $k = real \left \{ (-0.6428-0.766j) \frac{A(j \omega)}{B(j \omega)} \right \}$ $k_i = imag \left \{ (-0.6428-0.766j) \frac{A(j \omega)}{B(j \omega)}(-\omega) \right \}$ (8)

d) In fig. 5 is plotted D-curve for parameters k and ki calculated for $\varphi$. Parameter vector $k_i(\omega )$ reaches the maximum value 1.5569 by frequency $\omega_M = 1.73$ rad/s. Fig. 5 D-curve for phase margin $\varphi=50^o$

Temporary controller has following parameters: $R_{tmp}(s) = \frac{0.532s+0.9876}{s}$

and from bode plot Fig. 6 it is possible to see that phase margin is really $\varphi=50^o$ and crossover frequency $\omega_c = 0.92$ rad/s. Fig. 6 Bode plot for open-loop system with temporary PI controller

e) Settling time is the time required for the output to reach and remain within a 2% error band following input stimulus. This way for system with temporary controller was measured settling time Ts_tmp=7.3s (Fig. 7). Fig. 7 Step response for closed loop system with temporary PI controller

f) Final crossover frequency $\omega_c$ is now calculated according to (9) $\omega_c = \frac{7.3}{6}0.92 = 1.193$ rad/s

g) 1.193 < 1.557 \Rightarrow \omega_c < \omega_M[/latex] and final PI controller has following parameters
 R(s) = \frac{0.864s+1.205}{s}

From Fig. 8 is possible to see that phase margin is equal to desired phase margin $\varphi$ and also crossover frequency is very close to calculated frequency $\omega_c$. Fig. 8 Bode plot for open-loop system with final PI controller

Settling time was measured from unity step response (Fig. 9) and the value is Ts_fin=6.15s what is very close to desired settling time Ts=6s. Fig. 9 Step response for closed loop system with final PI controller

5. Conclusions

In this paper modification of Neimark D-partition method was presented. This controller design approach ensures not only stability but performance in term of phase margin and desired settling time too. The developed frequency domain design technique is graphical, interactive and insightful and it is useful for stable systems. Theoretical results have been verified on case study.

Acknowledgments

This work is supported by the Slovak Research and Development Agency (APVV) using financial support no. APVV-0772-12.

References

1. Fung, H.W., Wang, Q.G., Lee T.H.: PI Tuning in terms of gain and phase margins, Automatica, 34, pp. 1145-1149, 1998.
2. Ho, W.K., Hang, C.C.,Cao, L.: Tuning of PID controllers based on gain and phase margin specifications, Automatica, 31, pp. 497-502, 1995.
3. Hudzovič, P.: Teória riadenia I., Alfa, 1980, Bratislava
4. Nagurka M., Yaniv, O.: Robust PI Controller Design Satisfying Gain and Phase Margin Constraints. Proceedings of the American Control Conference, Denver, Colorado June 4-6, 2003
5. Neimark, Y.I.: Robust stability and D-partition, Automation and Remote Control 53 (1992) (7), pp. 957–965

Coauthor of this paper is Stanislav Števo, Institute of Control and Industrial Informatics, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology, Ilkovičova 3, 812 19 Bratislava, Slovak Republic