EE 362K - Buckman

Fall 2002

Topical Overview

This is an overview of the concepts and topics you need to know and be able to apply in order to get a good grade. For brevity, no derivations or examples and minimal explanation and background are present here.  From time to time, I will add hyperlinks to documents containing further examples and explanations.

1.  The uncontrolled system, or “plant”:  a system, described by rational polynomial transfer function, G, which is NOT subject to modification by the designer (shown in blue).  Its input and output are continuous functions of time, i(t) and o(t).

Diagram:

Flow of signals or information is left to right unless otherwise specified.

Plant Model : a (linear) differential equation.

,

where the a’s and the b’s are constants.

Transfer function:  Using LaPlace transforms, you can transform the model to a rational polynomial transfer function,

Note that whether you work with the model in the time (t) domain, or with the transfer function in the s-domain, knowledge of all of the a’s and b’s implies a complete description of the linear, time-invariant system.  Obtaining these models or transfer functions for systems of technological interest is the subject of entire other courses, such as circuit theory, electromechanics, dynamics, chemical kinetics, etc.

Text Reference: Ch. 1, Ch. 2.

Handout :  Handout 1.1.

Additional help with mechanical system modeling for ECE’s: mechsys.pdf

2.  Control system design:  adding systems to the “plant” in order to control it.  Assuming you can’t directly modify the output of the “plant”, you can modify the input, and this modification can make use of knowledge of the output.

Diagram:  A general representation of the use of additional subsystems, whose design you can modify (shown in orange) is shown below:

The subsystems, C and F have their own inputs and outputs, and hence their own transfer functions.   The control signal, r(t),  represents the output desired by the user.  By common convention, the output of F is subtracted from r(t) before being fed to C as input.

Text Reference:  Section 3.1-3.1.8.

Handout 1-2: What can you learn from the transfer function?

3. Sampled-data version of C or F.

Nearly all modern control systems use digital computers to implement the controllers.  Computers can neither accept continuous functions of time as inputs nor provide them as outputs to the external (analog) world. All they can do is operate on sequences of numbers (usually in very rapid succession). In order to interface a computer to the analog world you must implement a system like the one shown below as a replacement for subsystems C or F (or sometimes both).

Here the continuous function (solid line in the diagram) r(t), passes through a sample-and-hold subsystem, S&H.  The relationship between the input and the output of the sample-and-hold is illustrated below:

Here the white curve is r(t), a continuous time function.  The output of the S&H block is the blue line with the stair-step structure.  It is continuous everywhere except when it jumps to new values. Between the jumps, it is constant.  If you listed the vertical co-ordinates of the blue dots you would have a sequence of numbers.  The digital representations of these numbers are the output of the analog-to-digital converter, A/D, and constitute the sequence, rk, shown as a dashed line on the block diagram.

Once this sequence is formed, the computer has lost all information on the values of r(t) between sampling points.  From the picture of the input and output of the S&H block, you can see that the time between successive samples, T,  is the reciprocal of the sampling frequency, fs.

The discrete subsystem, D, transforms the input sequence, rk, into another sequence, ok, at its output.

Discrete system model: The model for the discrete system (a digital filter), is the difference equation,

.

Relationship of rk to r(t):   From the subsystem diagram and the plot, each element of the sequence is related to a sample taken at the corresponding time as

.

Relationship of ok to o(t):   From the subsystem diagram and the plot, each element of the sequence is related to the constant output of the DAC during the corresponding time interval as

,

where u(t) is the unit step function.

Discrete system transfer function:  You can use the z-transform [see Text 8.1-8.2] on the difference equation to get a discrete transfer function,

.

### It is a bit of a stretch to call this a rational polynomial, but it is indeed a rational polynomial in z-1.  You can turn in into a rational polynomial in z by multiplying numerator and denominator by zM if M>N or by zN in M<N.  The result is

4.  Loss of information in sampled-data models – modeling a system that includes sampled-data blocks:  Note that knowledge of all of the a’s and b’s provides a complete description of a linear system.  Also, solving the difference equation generates only the successive outputs, ok, without providing any information about what happens between samples. You can write a differential equation for the system that provides this information about system outputs between points in the sequence, but that equation will be non-linear.   Therefore, if you design C or F in a control system with a sampled-data replacement like the one shown above, and you want to make use of all the mathematical tools available for analyzing linear systems, you then have to analyze the entire system using a difference, rather than a differential equation.

4.1.  Effect of an upstream sample and hold on a system:  It would be convenient to find the sampled-data system that is “equivalent” to some continuous system under study: convenient, but impossible.  Sample&hold-ing will always have some effect on a system.  The effect only vanishes if the sampling frequency becomes infinite, a practical impossibility. Consider the system below, with only a sample&hold and no other digital subsystems.

A linear transfer function for this system in the s-domain cannot be written, because of the non-linear nature of the sample&hold . However, you can write its transfer function in the z-domain as

where the operation, , is a mathematical transformation from the s to the z-domain. The input to this transformation is a rational polynomial in s, while the output is a rational polynomial in z.

The key difference between H(s) and H(z) for any system is this: The a and b coefficients in H(s) depend only on the parameters of the physical system, while the a and b coefficients in H(z) depend on those parameters and also on the sampling frequency.

More details on sampling: read handout H2.

4.2.  Transfer function, H(z), for an equivalent digital filter, D(z), replacing subsystem C:  Suppose you replaced subsystem C with a digital filter having a transfer function, D(z), as shown below.

You can write the transfer function of this system as

,

which says that the digital filter transfer function you need to replicate the continuous subsystem C at a given sampling frequency is

.

More about Digital Controllers Inside Analog Feedback Loops

5. Stability from the transfer function: a stable system produces bounded (finite) outputs from any bounded input.

Continuous system:  a continuous system is stable if all the poles of H(s) lie in the left half of the complex plane.

Discrete system: a discrete system is stable of all the poles of H(z) lie within a unit circle whose center is at the origin in the complex plane.

Alternative criteria for stability exist, but are not needed as long as the poles of the transfer function can be found.  Pole locations determine stability.

6. Relative settling time from the transfer function:  In general,

Continuous systems respond faster to step-wise changes in the control signal, r(t), if all poles of H(s) lie further to the left in the complex plane.

Sampled-data systems respond faster to step-wise changes in the control signal, r(t), if all poles of H(z) lie closer to center of a unit circle at the origin in the complex plane.

This property lets you use pole locations as a rough design tool to determine stability and speed of response.

7. Frequency domain analysis and design

For systems whose transfer functions have many poles and zeros, frequency-domain analysis provides better insight into system performance than does observing pole and zero locations. You calculate the magnitude, |H|,  and phase, f,  of system transfer functions vs. angular frequency, w = 2pf,  as follows:

Continuous systems:

Sampled-data systems:

where T is the sampling period.  In the above equations, we just made the substitution, , for continuous systems and  for sampled-data systems.

The objective of frequency-domain design is to achieve a magnitude, |H(w)|, that satisfies the conditions illustrated below, and is stable.

Good designs have |H(w)| =1 within a small tolerance over a wide bandwidth.

8. State-space analysis and design:  The usefulness of state-space analysis in control system design arises primarily from three of its properties:

1) It lets you deal with systems having more than one feedback loop,

2) It lets you deal with systems having multiple inputs and outputs,

3) It lets you find a set of feedback parameters that place the system poles at any locations you want, if all the state variables are available for measurement (observable).

8.1.  Continuous systems:

Time-domain: You can model the dynamics of any linear system having:

P states (state vector x is a column vector, length P),

Q inputs(input vector i is a column vector, length Q), and

R outputs, (output vector o is a column vector, length R).

with a set of simultaneous, first-order linear differential equations in the set of state variables (termed the state vector), x, in the form,

, the state equation,

along with a linear output equation,

.

The components of these equations have the following forms,

·        A is a P x P square matrix, called the state matrix,

·        B is a P x Q rectangular matrix, called the input matrix

·        C is an R x P rectangular matrix, called the output matrix

·        D is an R x Q rectangular matrix, called the feedforward matrix.

You design the control of a state-variable system by feeding back a weighted sum of the states to each system input can control (there may be inputs you cannot control: these are disturbances).

s-Domain:  You can take the LaPlace transform of the state equation to get

,

and then solve for the output in terms of the input as

,

where I   is the square identity matrix.  This equation relates all of the inputs to all of the outputs in the s-domain, and thus plays the role of a transfer function, even though it can’t be written as a simple ratio of polynomials.  [For SISO systems (only one input and one output) you get a rational polynomial transfer function.  To see how state variable models apply to SISO systems, including some simple examples, read the handout on that subject.]

The poles of this system are the roots of the equation,

Manipulating pole locations with state feedback:    Start with the state-variable representation of a system as shown below, with blue representing the uncontrolled system or “plant” as before:

Assuming that all states are observable, you can feed back a weighted sum of these states to each input, allowing for the fact that in general, this weighting can be different for each accessible input.  Now you have the system shown below,

which can be modeled by the state equation

where K is in general a Q x P rectangular matrix.  Obviously you can rearrange this equation so it has the form of a state equation with a new state matrix, Af, as

,

whose poles are the roots of

The state-variable approach to control system design consists of adjusting the elements of the state feedback matrix, K, to produce desired pole locations consistent with other design specifications.

8.2. Discrete and sampled-data systems

Time-domain:  You could also write state equations for any system in the form of a set of difference equations, giving

for the state equation and

for the output equation.  The dimensions of the matrices and vectors are the same as for the continuous case. The vectors are now sequences, and the matrices now depend on the sampling frequency as well as on the parameters of the system.

z-Domain:  Taking the z-transform of the state equation yields

,

from which you can get the transfer function-like matrix relationship,

,

The poles of the uncontrolled system are the roots of

Manipulating pole locations with state feedback:  Similarly to state feedback for a continuous system, you can choose a feedback matrix, K, to adjust pole locations.  K has the same dimensions as for the continuous case, and the new pole locations with feedback are the roots of

8.3. Frequency-domain design for state-variable systems

There is a transfer-function relationship between each output and each input which you can evaluate in the frequency domain.  Write the matrix input-output relations for both cases as

, and substitute , OR

and substitute .

You can use transfer functions you get from these operations to design a control system with multiple inputs and outputs.