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V. ROOTS OF SECOND-ORDER AND THIRD-ORDER POLYNOMIALS
We consider conditions that second- and third-order polynomials have roots in the left half of the complex s-plane.
1/ Second-order polynomials
Second-order polynomials with real coefficients have either real or complex roots of the form
$(s+a)(s+b)=\text{0}\begin{array}{cc}& \end{array}\text{or}\begin{array}{cc}& \end{array}(s+a+\text{jc})(s+\text{a - jc})\text{= 0}$
where a>0 and b>0. The polynomials can be expressed as
${s}^{2}+(a+b)s+\text{ab}\begin{array}{cc}& \end{array}\text{or}\begin{array}{cc}& \end{array}{s}^{2}+2\text{as}+{a}^{2}+{c}^{2}=0$
Thus, both polynomials have the form
${s}^{2}+\mathrm{\alpha s}+\beta =0$
where α>0 and β>0. These conditions are both necessary and sufficient.
2/ Third-order polynomials
Third-order polynomials must have one real root and either a pair of real or complex roots of the form
$(s+a)(s+b)(s+c)=\text{0}\begin{array}{cc}& \end{array}\text{or}\begin{array}{cc}& \end{array}(s+b)(s+a+\text{jd})(s+\text{a - jd})\text{= 0}$
where a>0, b>0, and c>0. The polynomials can be expressed as
$\begin{array}{}{s}^{3}+(a+b+c){s}^{2}+(\text{ab}+\text{ac}+\text{bc})s+\text{abc}=0\begin{array}{cc}& \end{array}\\ \text{or}\begin{array}{cc}& \end{array}{s}^{3}+(\mathrm{2a}+b){s}^{2}+({a}^{2}+{d}^{2}+2\text{ab})s+({a}^{2}+{d}^{2})b=0\end{array}$
Thus, both can be put in the form
${s}^{3}+{\mathrm{\alpha s}}^{2}+\mathrm{\beta s}+\gamma =0$
Note that α>0, β>0, and γ>0. But in addition, β>γ/α. These conditions are both necessary and sufficient.
VI. ROOT LOCUS PLOTS FOR POSITION CONTROL SYSTEMS
1/ Proportional controller
$H(s)=\frac{K}{{s}^{2}+\text{101}s+\text{100}+K}$
Recall the step response of the position control system with proportional controller.
The poles of the closed-loop system function are at
${s}_{\mathrm{1,2}}=-\frac{\text{101}}{2}\pm {\left[{\left[\frac{\text{101}}{2}\right]}^{2}-\text{100}-K\right]}^{1/2}$
The root locus plot is
2/ Integral controller
$H(s)=\frac{K}{{s}^{3}+\text{101}{s}^{2}+\text{100}s+K}$
Recall the step response of the position control system with integral controller.
The poles are the roots of the polynomial
${s}^{3}+\text{101}{s}^{2}+\text{100}s+K=0$
The root locus plot is
Note that all the poles are in the lhp for K>0 and 100>K/101 or 0<K<10100. For K>10100 two poles move into the rhp and the system is unstable.
VII. STABILIZATION OF UNSTABLE SYSTEMS
1/ Many common systems are unstable
Some common systems are annoyingly unstable (adapted from Figure 11.7 in Oppenheim&Willsky, 1983).
We can model the audio feedback system with SIMULINK as follows.
Another example of an unstable system is an inverted pendulum. For example, balancing a broomstick in your hand is an example of an inherently unstable system that is stabilized by your motor control system.
Figure adapted from Figure 11.2 in Oppenheim&Willsky, 1983.
2/ Inverted pendulum
We will analyze an inverted pendulum attached to a cart. A schematic diagram is shown on the left and a free-body diagram showing the forces on the pendulum is shown on the right.
The forces of attachment of the cart and pendulum on the pendulum are obtained from the equations of rectilinear motion.
The equation of rotational motion about the center of mass is
$J\frac{{d}^{2}\theta (t)}{{\text{dt}}^{2}}=\text{mgl}\text{sin}\theta (t)-\text{ml}\text{cos}\theta (t)\frac{{d}^{2}x(t)}{{\text{dt}}^{2}}$
where J is the moment of inertia of the mass about the mass, m is the mass, and g is acceleration of gravity. For small θ, this differential equation is linearized by noting that sinθ≈θ and cos θ ≈ 1. Therefore,
$J\frac{{d}^{2}\theta (t)}{{\text{dt}}^{2}}-\text{mgl}\text{sin}\theta (t)=-\text{ml}\frac{{d}^{2}x(t)}{{\text{dt}}^{2}}$
The system function is
$H(s)=\frac{\theta (s)}{X(s)}=\frac{-(\text{ml}/J){s}^{2}}{{s}^{2}-(\text{mgl}/J)}$
Therefore, the poles occur at
${s}_{\mathrm{1,2}}=\pm {\left[\frac{\text{mgl}}{J}\right]}^{1/2}$
Hence, the system is inherently unstable since one of its poles (natural frequencies) is in the rhp.
3/ Stabilization of the inverted pendulum
To stabilize the inverted pendulum, a rotary potentiometer is used to measure θ(t). A current proportional to θ(t) − θo, where θo is the desired angle, drives a motor so as to increase x(t) (adapted from Figure 6.4-1, Siebert, 1986).
This inverted pendulum is connected in a feedback configuration as follows.
M(s) represents the system function of the motor dynamics, 1 + (a/s) is the system function of the proportional plus integral controller for θ(t), and c + bs is the system function of the proportional plus derivative controller for x(t).
Demo of stabilization of an inverted pendulum.
4/ Stabilize by pole cancellation?
Why not cascade an unstable system with another system that cancels the unstable pole?
VIII. CONCLUSIONS
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