Why Do We Need Another Control Method?

What Influences Closed Loop Pole Position?

A More General Approach

A Model-Reference Approach

Problems

        There are many different ways you can control systems, and you should have learned about most of them by now.  There are PID controllers of all sorts, and you can use lead or lag compensation.  Why bother with another method?

        The best answer to the question is that even with all of the methods we have considered there are still things that we want to do that we don't have good insight for.  For example, pilots flying aircraft often want the aircraft to respond as though it has a certain natural frequency and a certain damping ratio.  Flying an aircraft with those parameters seems somehow easier to them.  The "raw" aircraft - without any control system - may have parameters that are violently different from what a pilot wants.  The control system designer has the problem of making the aircraft feel right, and for the aircraft to feel right it has to have closed loop poles at particular points in the s plane.

        As you proceed through this lesson, keep this goal in mind.  Your goal is the following:

• Given a system in which some given closed loop pole locations are desired,
• Know the structure of a controller which will produce the desired pole locations, and be able to implement that controller for low order systems.

        Let us consider a simple system for which you need to design a controller.  We've looked at this same system in the lesson on lead compensation, so you can review the system there.

        Next we will look at the root locus that is obtained when the controller is a proportional controller.  Before we do that, let us assume that we want to have closed loop poles at these locations:

 Here is the root locus.

• Can we get the closed loop poles to the desired location?
• Although the closed loop poles wander away from the open loop poles, they never pass through the point where we want them to be - not for any gain value!

• Shown here is the formula for the closed loop transfer function for the situation where the controller is proportional controller with a gain, K_p .

We can view the problem to be one in which the denominator never has the correct factors - not for any value of the proportional controller gain.

• Here is a block diagram model of the new system.

• We can compute the closed loop transfer function for this system.  Here is the closed loop transfer function.

        The question here is "Can we put the poles where we want with this system?".  To understand the answer to that question, consider the following.

• If we want poles at -4 + j4 and -4 - j4, we are really saying that the denominator should be this polynomial, which has the roots we want.
• s² + 8s + 32
• The closed loop denominator is:
• s² + 5s + 4 + 10(K_p + sK_D)
• And, we can collect terms to get this form:
• s² + s(5 + 10K_D) + (4 + 10K_p)

• Both constant terms must be equal:
• 10K_p + 4 = 32
• Both "s" terms must be equal:
• 10*K_D  + 5 = 8

• 10K_p + 4 = 32 gives
• Kp  = 2.8
• 10*K_D  + 5  gives
• Kd   = 0.3

        Let us think about what we have done.

• We have taken a second order system with two real poles, and forced the closed loop poles to an arbitray location using P-D control.
• Looking back, we could have forced the poles to any position.
• We should wonder what happens if the open loop poles are complex?.
• We should wonder what happens if the desired closed loop poles are complex?
• We should wonder what happens in higher order systems?

• If the open loop poles are complex, the coefficients of the open loop denominator will still be real, so we can set up the equations for the proportional and derivative gain terms and still get real number for the gains when we solve.

• If the desired clopsed loop poles are complex, the coefficients of the closed loop denominator will still be real, so we can set up the equations for the proportional and derivative gain terms and still get real number for the gains when we solve.

• We have a third order system shown below.

• The open loop denominator, (s + 1)(s + 4)(s + 5), expands to a cubic polynomial shown at the right.
• s³ + 10s² + 29s + 20 + 10(K_p + sK_D)
• The complete closed loop denominator is:
• s³ + 10s² + s(29 + K_D) + (20 + 10K_p)

• We can adjust the gains to get whatever constant and "s" terms we want.
• We can't do anything about the 10 in front of the quadratic term.
• If we don't want 10 there, we're stuck.
• We can't put all three poles where we want them, although we still haven't ruled out putting two of them where we want.

        Let us look at what we have to do to get all three poles where we want them.  We'll continue to work with the last system we used, shown below.  We will modify our approach.  The controller will now include a second derivative term.

• Here's the new denominator of the closed loop transfer function:
• s³ + 10s² + 29s + 20 + 10(K_p + sK_D + s²K_DD)
• Here is the same denoninator expanded:
• s³ + s²(10 + K_DD) + s(29 + K_D) + (20 + 10K_p)
• It is clear that this controller allows us to set all three coefficients in the closed loop denominator.
• Since we can set all three terms in the denominator, we can put the closed loop poles wherever we want them to be.

• Can we actually build any of these controllers?

        There are some systems, like motors, where it is possible to use something like a tachometer to measure rotational velocity.  So, at least for those systems we can get a voltage signal proportional to a time derivative.  Are there any others.

• In aircraft control systems, it is possible to use rate gyros to measure rotational rates of aircraft around the various aircraft axes (yaw, pitch and roll, or is it shake, rattle and roll?).
• In other systems it might be harder.  No sensor comes to mind when we try to measure the time derivative of temperature, for example.
• We may have to differentiate a signal numerically, using either an analog differentiator or a numerica/digital differentiator.
• Both of the proccesses in the last bullet can get ugly.  You are now usually encouraged to do analog or digital differentiation, because both proccesses are usually prone to having excessive noise.

        Actually, there has been a lot of thought put into devising answers to that question.  Some answers are the following:

• Build a model of the system, and measure the derivatives internally.
• Build a system the uses the system output(s) and input(s) to compute those derivatives.

 In another lesson you should have learned about state representations for systems where you had a given transfer function.  (Click here to review that section.)  The block diagram below has a transfer function of:

This is the block diagram.

We'll discuss a strategy that uses this block diagram.

        Here is an approach that might be used for controlling a third order system.

• Determine the transfer function of the system being controlled.
• Build a model of the system.  With the block diagram above, it's a natural idea to use analog integrators, but you can do the integration digitally as well.
• For the input, use the same input as enters the system being controlled.

• The states are really derivatives of the outputs.  Follow that through on the block diagram.  Going backward through an integrator gets you the derivative.
• You can use the signals from the model instead of the system!

• Build the model.
• Use the signals from the model instead of the system.
• Otherwise, everything is just like it would be using the pole placement method, and you shouldn't be able to tell the difference.
• What could possibly go wrong?

• You don't implement the model exactly.
• You don't even know what the model is all that well.  After all, systems change in time.  Many systems - like aircraft - have different transfer function models for different operating conditions.
• Other than that, everything will be fine.

        Clearly, this approach requires that you really have a very good handle on your system.  If you don't you will need something that adapts to varying conditions, and that's taking us beyond the scope of this lesson.

        What if the model approach doesn't work.  There are other approaches, and in future versions of these lessons they will be covered in more detail.  For now, you will have to read up on things like observers - which are systems designed to estimate states within a system using observed/measured inputs and outputs.  Observers are not exactly model-based systems, and can be used when your system description is not exact.

• Problem Comp4A01 - Designing a PID to produce specific pole locations.
• Problem Comp4A02 - Designing a PID to produce specific pole locations.
• Problem Comp4A03 - Designing a Model Reference Controller to produce specific pole locations.

• PID Controllers
• General PID Controllers
• Compensators
• Lag Compensators
• Lead Compensators
• General Compensation Techniques