· There are a few terms to be defined,
identification - the process of determining a state space (or other) model of a process.
parameter identification - used to estimate model parameter values.
estimatior - this tries to determine state variable values given available data.
observer - a special type of estimator that uses available input and output data to estimate a system state given unknown initial conditions. This is simalar to running a system simulation in parallel with the real system.
compensator - a feedback control system designed to drive a system to an arbitrary input.
regulator - a feedback control system designed to drive a system to a given operating point. A special case of a compensator with a zero input.
· The basic definition of a feedback controller is below,
· An example of a controller design follows,
Figure 21.1 Block diagram of a state based controller
· The value of K in the previous example could also be picked using Ackermann's formula.
· Consider the previous example,
· Observers are used to estimate the next system state. They can also be used to predict future state values.
· As an example, an observer might be used for a targeting system that is tracking a moving object. The estimator can be used to direct a missle to an estimated location so that the missle and target arrive at the same time.
· An observer is shown in block diagram form
Figure 21.2 Block diagram of an observer
· A closed loop estimatior is shown below. It uses a learning matrix L to adjust the convergence rate. A larger L value will result in faster convergence, but the exponent must now be checked to ensure observability.
· Estimator gain selection
· Consider the example below, for a second order system. The learning rate values are selected to determine how quickly the model converges.
· This can also be done with Ackermann's formula,
· The open loop estimator will normally ignore the response to initial conditions and will have long term errors resulting from modeling errors and random disturbances in the system.
XXXXXXXXXX Example of learning parameter estimator to obtain desired responses.
· In many cases some state variables can be measured directly, but others cannot. In these cases it is more efficient to only estimate the unmeasurable variable values [Eronini].
· This can be
· For a regulated system the control output 'u' can be estimated using the desired system state.
· The system state is often difficult to measure directly, so an estimator can be used. .
· A regulator can use an estimate of the system state to
· The estimators and regulators can be designed separately and then combined. This is the separation principle.
· The simplified block diagram for the system is,
· The compensator,
· This can be converted to a transfer function for the regulator.
· The system can also be set up for a compensator by assuming the setpoint input is not zero,
· Consider an example of a regulator/estimator design,
· This system can be implemented with a C program as shown below,
- Linear Quadratic Regulator (LQR) allows the design of a regulating system that tends to zero, but also considers the control effort.
- The basic function for evaluating the controller is defined below. The integral is evaluated to reduce the value. Ideally the state value will go to zero (it is a regulator), and the plant input will be minimized. It is assumed that D=0. Here a new state 'z' is defined. It should reflect the system state values that should be regulated to zero.
- the design process involves adjusting the r value to obtain the desired result.
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· Uses an optimal regulator and estimator.
· Stability is guaranteed if the model is accurate. Verification is needed
· The design focuses on,
A cost function for the system state, the design tries to minimize this
The relative weighting for the process and sensor noise
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· Introduce random errors to the system model and test for stability
· Use the Nyquist Stability Theorem,
· The stability function is defined using the Nyquist plot.
· In practical terms system gain is limited by real components. As a result it is important to ensure that the design will not exceed the maximum values available.
· There are two basic types of adaptive control,
Self Tuining Regulators
Model Reference Adaptive Control (MRAC)
· A self tuning regulator
· Parameter Adaptive MRAC
· Signal Synthesis MRAC
· Parallel MRAC
· Series MRAC
· The basic MRAC system is defined below,
H-infinity and mu methods are designed to shape the sensitivity plot
· Kalman Filter helps settle estimators using estimates of snesor and process noise.
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