vision electricity: 10/11/09

Sunday, October 11, 2009

PID controller

A proportional–integral–derivative controller (PID controller) is a generic control loop feedback mechanism (controller) widely used in industrial control systems. A PID controller attempts to correct the error between a measured process variable and a desired setpoint by calculating and then outputting a corrective action that can adjust the process accordingly and rapidly, to keep the error minimal.

General

A block diagram of a PID controller

The PID controller calculation (algorithm) involves three separate parameters; the proportional, the integral and derivative values. The proportional value determines the reaction to the current error, the integral value determines the reaction based on the sum of recent errors, and the derivative value determines the reaction based on the rate at which the error has been changing. The weighted sum of these three actions is used to adjust the process via a control element such as the position of a control valve or the power supply of a heating element.
By tuning the three constants in the PID controller algorithm, the controller can provide control action designed for specific process requirements. The response of the controller can be described in terms of the responsiveness of the controller to an error, the degree to which the controller overshoots the setpoint and the degree of system oscillation. Note that the use of the PID algorithm for control does not guarantee optimal control of the system or system stability.
Some applications may require using only one or two modes to provide the appropriate system control. This is achieved by setting the gain of undesired control outputs to zero. A PID controller will be called a PI, PD, P or I controller in the absence of the respective control actions. PI controllers are particularly common, since derivative action is very sensitive to measurement noise, and the absence of an integral value may prevent the system from reaching its target value due to the control action.
Note: Due to the diversity of the field of control theory and application, many naming conventions for the relevant variables are in common use.

Control loop basics

A familiar example of a control loop is the action taken to keep one's shower water at the ideal temperature, which typically involves the mixing of two process streams, cold and hot water. The person feels the water to estimate its temperature. Based on this measurement they perform a control action: use the cold water tap to adjust the process. The person would repeat this input-output control loop, adjusting the hot water flow until the process temperature stabilized at the desired value.
Feeling the water temperature is taking a measurement of the process value or process variable (PV). The desired temperature is called the setpoint (SP). The output from the controller and input to the process (the tap position) is called the manipulated variable (MV). The difference between the measurement and the setpoint is the error (e), too hot or too cold and by how much.
As a controller, one decides roughly how much to change the tap position (MV) after one determines the temperature (PV), and therefore the error. This first estimate is the equivalent of the proportional action of a PID controller. The integral action of a PID controller can be thought of as gradually adjusting the temperature when it is almost right. Derivative action can be thought of as noticing the water temperature is getting hotter or colder, and how fast, anticipating further change and tempering adjustments for a soft landing at the desired temperature (SP).
Making a change that is too large when the error is small is equivalent to a high gain controller and will lead to overshoot. If the controller were to repeatedly make changes that were too large and repeatedly overshoot the target, the output would oscillate around the setpoint in either a constant, growing, or decaying sinusoid. If the oscillations increase with time then the system is unstable, whereas if they decay the system is stable. If the oscillations remain at a constant magnitude the system is marginally stable. A human would not do this because we are adaptive controllers, learning from the process history, but PID controllers do not have the ability to learn and must be set up correctly. Selecting the correct gains for effective control is known as tuning the controller.
If a controller starts from a stable state at zero error (PV = SP), then further changes by the controller will be in response to changes in other measured or unmeasured inputs to the process that impact on the process, and hence on the PV. Variables that impact on the process other than the MV are known as disturbances. Generally controllers are used to reject disturbances and/or implement setpoint changes. Changes in feed water temperature constitute a disturbance to the shower process.
In theory, a controller can be used to control any process which has a measurable output (PV), a known ideal value for that output (SP) and an input to the process (MV) that will affect the relevant PV. Controllers are used in industry to regulate temperature, pressure, flow rate, chemical composition, speed and practically every other variable for which a measurement exists. Automobile cruise control is an example of a process which utilizes automated control.
Due to their long history, simplicity, well grounded theory and simple setup and maintenance requirements, PID controllers are the controllers of choice for many of these applications.

PID controller theory

This section describes the parallel or non-interacting form of the PID controller. For other forms please see the Section "Alternative notation and PID forms".
The PID control scheme is named after its three correcting terms, whose sum constitutes the manipulated variable (MV). Hence:

$\mathrm{MV(t)}=\,P_{\mathrm{out}} + I_{\mathrm{out}} + D_{\mathrm{out}}$

where

P out

I out

, and

D out

are the contributions to the output from the PID controller from each of the three terms, as defined below.

Proportional term

Plot of PV vs time, for three values of K_p (K_i and K_d held constant)

The proportional term (sometimes called gain) makes a change to the output that is proportional to the current error value. The proportional response can be adjusted by multiplying the error by a constant K_p, called the proportional gain.
The proportional term is given by:

$P_{\mathrm{out}}=K_p\,{e(t)}$

where

P out

: Proportional term of output

K p

: Proportional gain, a tuning parameter

e

: Error

= S P - P V

t

: Time or instantaneous time (the present)

A high proportional gain results in a large change in the output for a given change in the error. If the proportional gain is too high, the system can become unstable (See the section on loop tuning). In contrast, a small gain results in a small output response to a large input error, and a less responsive (or sensitive) controller. If the proportional gain is too low, the control action may be too small when responding to system disturbances.
In the absence of disturbances, pure proportional control will not settle at its target value, but will retain a steady state error that is a function of the proportional gain and the process gain. Despite the steady-state offset, both tuning theory and industrial practice indicate that it is the proportional term that should contribute the bulk of the output change.

Integral term

Plot of PV vs time, for three values of K_i (K_p and K_d held constant)

The contribution from the integral term (sometimes called reset) is proportional to both the magnitude of the error and the duration of the error. Summing the instantaneous error over time (integrating the error) gives the accumulated offset that should have been corrected previously. The accumulated error is then multiplied by the integral gain and added to the controller output. The magnitude of the contribution of the integral term to the overall control action is determined by the integral gain,

K i

.
The integral term is given by:
$I_{\mathrm{out}}=K_{i}\int_{0}^{t}{e(\tau)}\,{d\tau}$
where

I out

: Integral term of output

K i

: Integral gain, a tuning parameter

e

: Error

= S P - P V

t

: Time or instantaneous time (the present)

τ

: a dummy integration variable

The integral term (when added to the proportional term) accelerates the movement of the process towards setpoint and eliminates the residual steady-state error that occurs with a proportional only controller. However, since the integral term is responding to accumulated errors from the past, it can cause the present value to overshoot the setpoint value (cross over the setpoint and then create a deviation in the other direction). For further notes regarding integral gain tuning and controller stability, see the section on loop tuning.

Derivative term

Plot of PV vs time, for three values of K_d (K_p and K_i held constant)

The rate of change of the process error is calculated by determining the slope of the error over time (i.e., its first derivative with respect to time) and multiplying this rate of change by the derivative gain

K d

. The magnitude of the contribution of the derivative term (sometimes called rate) to the overall control action is termed the derivative gain,

K d

.
The derivative term is given by:

$D_{\mathrm{out}}=K_d\frac{d}{dt}e(t)$

where

D out

: Derivative term of output

K d

: Derivative gain, a tuning parameter

e

: Error

= S P - P V

t

: Time or instantaneous time (the present)

The derivative term slows the rate of change of the controller output and this effect is most noticeable close to the controller setpoint. Hence, derivative control is used to reduce the magnitude of the overshoot produced by the integral component and improve the combined controller-process stability. However, differentiation of a signal amplifies noise and thus this term in the controller is highly sensitive to noise in the error term, and can cause a process to become unstable if the noise and the derivative gain are sufficiently large.

Summary

The proportional, integral, and derivative terms are summed to calculate the output of the PID controller. Defining

u (t)

as the controller output, the final form of the PID algorithm is:

$\mathrm{u(t)}=\mathrm{MV(t)}=K_p{e(t)} + K_{i}\int_{0}^{t}{e(\tau)}\,{d\tau} + K_{d}\frac{d}{dt}e(t)$

where the tuning parameters are:

Proportional gain, $K p$: larger values typically mean faster response since the larger the error, the larger the Proportional term compensation. An excessively large proportional gain will lead to process instability and oscillation.
Integral gain, $K i$: larger values imply steady state errors are eliminated more quickly. The trade-off is larger overshoot: any negative error integrated during transient response must be integrated away by positive error before we reach steady state.
Derivative gain, $K d$: larger values decrease overshoot, but slows down transient response and may lead to instability due to signal noise amplification in the differentiation of the error.

Loop tuning

If the PID controller parameters (the gains of the proportional, integral and derivative terms) are chosen incorrectly, the controlled process input can be unstable, i.e. its output diverges, with or without oscillation, and is limited only by saturation or mechanical breakage. Tuning a control loop is the adjustment of its control parameters (gain/proportional band, integral gain/reset, derivative gain/rate) to the optimum values for the desired control response.
The optimum behavior on a process change or setpoint change varies depending on the application. Some processes must not allow an overshoot of the process variable beyond the setpoint if, for example, this would be unsafe. Other processes must minimize the energy expended in reaching a new setpoint. Generally, stability of response (the reverse of instability) is required and the process must not oscillate for any combination of process conditions and setpoints. Some processes have a degree of non-linearity and so parameters that work well at full-load conditions don't work when the process is starting up from no-load. This section describes some traditional manual methods for loop tuning.
There are several methods for tuning a PID loop. The most effective methods generally involve the development of some form of process model, then choosing P, I, and D based on the dynamic model parameters. Manual tuning methods can be relatively inefficient.
The choice of method will depend largely on whether or not the loop can be taken "offline" for tuning, and the response time of the system. If the system can be taken offline, the best tuning method often involves subjecting the system to a step change in input, measuring the output as a function of time, and using this response to determine the control parameters.

Choosing a Tuning Method
Method	Advantages	Disadvantages
Manual Tuning	No math required. Online method.	Requires experienced personnel.
Ziegler–Nichols	Proven Method. Online method.	Process upset, some trial-and-error, very aggressive tuning.
Software Tools	Consistent tuning. Online or offline method. May include valve and sensor analysis. Allow simulation before downloading.	Some cost and training involved.
Cohen-Coon	Good process models.	Some math. Offline method. Only good for first-order processes.

Manual tuning

If the system must remain online, one tuning method is to first set

K i

and

K d

values to zero. Increase the

K p

until the output of the loop oscillates, then the

K p

should be left set to be approximately half of that value for a "quarter amplitude decay" type response. Then increase

K i

until any offset is correct in sufficient time for the process. However, too much

K i

will cause instability. Finally, increase

K d

, if required, until the loop is acceptably quick to reach its reference after a load disturbance. However, too much

K d

will cause excessive response and overshoot. A fast PID loop tuning usually overshoots slightly to reach the setpoint more quickly; however, some systems cannot accept overshoot, in which case an "over-damped" closed-loop system is required, which will require a

K p

setting significantly less than half that of the

K p

setting causing oscillation.

Effects of increasing parameters
Parameter	Rise time	Overshoot	Settling time	Error at equilibrium
$K p$	Decrease	Increase	Small change	Decrease
$K i$	Decrease	Increase	Increase	Eliminate
$K d$	Indefinite (small decrease or increase)^[1]	Decrease	Decrease	None

Ziegler–Nichols method

Another tuning method is formally known as the Ziegler–Nichols method, introduced by John G. Ziegler and Nathaniel B. Nichols. As in the method above, the

K i

and

K d

gains are first set to zero. The P gain is increased until it reaches the critical gain,

K c

, at which the output of the loop starts to oscillate.

K c

and the oscillation period

P c

are used to set the gains as shown:

Ziegler–Nichols method
Control Type	$K p$	$K i$	$K d$
P	$0.50 K c$	-	-
PI	$0.45 K c$	$1.2 K p / P c$	-
PID	$0.60 K c$	$2 K p / P c$	$K p P c / 8$

PID tuning software

Most modern industrial facilities no longer tune loops using the manual calculation methods shown above. Instead, PID tuning and loop optimization software are used to ensure consistent results. These software packages will gather the data, develop process models, and suggest optimal tuning. Some software packages can even develop tuning by gathering data from reference changes.
Mathematical PID loop tuning induces an impulse in the system, and then uses the controlled system's frequency response to design the PID loop values. In loops with response times of several minutes, mathematical loop tuning is recommended, because trial and error can literally take days just to find a stable set of loop values. Optimal values are harder to find. Some digital loop controllers offer a self-tuning feature in which very small setpoint changes are sent to the process, allowing the controller itself to calculate optimal tuning values.
Other formulas are available to tune the loop according to different performance criteria.

Modifications to the PID algorithm

The basic PID algorithm presents some challenges in control applications that have been addressed by minor modifications to the PID form.
One common problem resulting from the ideal PID implementations is integral windup. This problem can be addressed by:

Initializing the controller integral to a desired value
Increasing the setpoint in a suitable ramp
Disabling the integral function until the PV has entered the controllable region
Limiting the time period over which the integral error is calculated
Preventing the integral term from accumulating above or below pre-determined bounds

Freezing the integral function in case of disturbances: If a PID loop is used to control the temperature of an electric resistance furnace, the system has stabilized and then the door is opened and something cold is put into the furnace the temperature drops below the setpoint. The integral function of the controller tends to compensate this error by introducing another error in the positive direction. This can be avoided by freezing of the integral function after the opening of the door for the time the control loop typically needs to reheat the furnace.

Replacing the integral function by a model based part: Often the time-response of the system is approximately known. Then it is an advantage to simulate this time-response with a model and to calculate some unknown parameter from the actual response of the system. If for instance the system is an electrical furnace the response of the difference between furnace temperature and ambient temperature to changes of the electrical power will be similar to that of a simple RC low-pass filter multiplied by an unknown proportional coefficient. The actual electrical power supplied to the furnace is delayed by a low-pass filter to simulate the response of the temperature of the furnace and then the actual temperature minus the ambient temperature is divided by this low-pass filtered electrical power. Then, the result is stabilized by another low-pass filter leading to an estimation of the proportional coefficient. With this estimation it is possible to calculate the required electrical power by dividing the set-point of the temperature minus the ambient temperature by this coefficient. The result can then be used instead of the integral function. This also achieves a control error of zero in the steady-state but avoids integral windup and can give a significantly improved control action compared to an optimized PID controller. This type of controller does work properly in an open loop situation which causes integral windup with an integral function. This is an advantage if for example the heating of a furnace has to be reduced for some time because of the failure of a heating element or if the controller is used as an advisory system to a human operator who may or may not switch it to closed-loop operation or if the controller is used inside of a branch of a complex control system where this branch may be temporarily inactive.

Many PID loops control a mechanical device (for example, a valve). Mechanical maintenance can be a major cost and wear leads to control degradation in the form of either stiction or a deadband in the mechanical response to an input signal. The rate of mechanical wear is mainly a function of how often a device is activated to make a change. Where wear is a significant concern, the PID loop may have an output deadband to reduce the frequency of activation of the output (valve). This is accomplished by modifying the controller to hold its output steady if the change would be small (within the defined deadband range). The calculated output must leave the deadband before the actual output will change.
The proportional and derivative terms can produce excessive movement in the output when a system is subjected to an instantaneous step increase in the error, such as a large setpoint change. In the case of the derivative term, this is due to taking the derivative of the error, which is very large in the case of an instantaneous step change. As a result, some PID algorithms incorporate the following modifications:

Derivative of output: In this case the PID controller measures the derivative of the output quantity, rather than the derivative of the error. The output is always continuous (i.e., never has a step change). For this to be effective, the derivative of the output must have the same sign as the derivative of the error.
Setpoint ramping: In this modification, the setpoint is gradually moved from its old value to a newly specified value using a linear or first order differential ramp function. This avoids the discontinuity present in a simple step change.
Setpoint weighting: Setpoint weighting uses different multipliers for the error depending on which element of the controller it is used in. The error in the integral term must be the true control error to avoid steady-state control errors. This affects the controller's setpoint response. These parameters do not affect the response to load disturbances and measurement noise.

Limitations of PID control

While PID controllers are applicable to many control problems, they can perform poorly in some applications.
PID controllers, when used alone, can give poor performance when the PID loop gains must be reduced so that the control system does not overshoot, oscillate or hunt about the control setpoint value. The control system performance can be improved by combining the feedback (or closed-loop) control of a PID controller with feed-forward (or open-loop) control. Knowledge about the system (such as the desired acceleration and inertia) can be fed forward and combined with the PID output to improve the overall system performance. The feed-forward value alone can often provide the major portion of the controller output. The PID controller can then be used primarily to respond to whatever difference or error remains between the setpoint (SP) and the actual value of the process variable (PV). Since the feed-forward output is not affected by the process feedback, it can never cause the control system to oscillate, thus improving the system response and stability.
For example, in most motion control systems, in order to accelerate a mechanical load under control, more force or torque is required from the prime mover, motor, or actuator. If a velocity loop PID controller is being used to control the speed of the load and command the force or torque being applied by the prime mover, then it is beneficial to take the instantaneous acceleration desired for the load, scale that value appropriately and add it to the output of the PID velocity loop controller. This means that whenever the load is being accelerated or decelerated, a proportional amount of force is commanded from the prime mover regardless of the feedback value. The PID loop in this situation uses the feedback information to effect any increase or decrease of the combined output in order to reduce the remaining difference between the process setpoint and the feedback value. Working together, the combined open-loop feed-forward controller and closed-loop PID controller can provide a more responsive, stable and reliable control system.
Another problem faced with PID controllers is that they are linear. Thus, performance of PID controllers in non-linear systems (such as HVAC systems) is variable. Often PID controllers are enhanced through methods such as PID gain scheduling or fuzzy logic. Further practical application issues can arise from instrumentation connected to the controller. A high enough sampling rate, measurement precision, and measurement accuracy are required to achieve adequate control performance.
A problem with the Derivative term is that small amounts of measurement or process noise can cause large amounts of change in the output. It is often helpful to filter the measurements with a low-pass filter in order to remove higher-frequency noise components. However, low-pass filtering and derivative control can cancel each other out, so reducing noise by instrumentation means is a much better choice. Alternatively, the differential band can be turned off in many systems with little loss of control. This is equivalent to using the PID controller as a PI controller.

Cascade control

One distinctive advantage of PID controllers is that two PID controllers can be used together to yield better dynamic performance. This is called cascaded PID control. In cascade control there are two PIDs arranged with one PID controlling the set point of another. A PID controller acts as outer loop controller, which controls the primary physical parameter, such as fluid level or velocity. The other controller acts as inner loop controller, which reads the output of outer loop controller as set point, usually controlling a more rapid changing parameter, flowrate or acceleration. It can be mathematically proven^{[citation needed]} that the working frequency of the controller is increased and the time constant of the object is reduced by using cascaded PID controller.^[vague].

Physical implementation of PID control

In the early history of automatic process control the PID controller was implemented as a mechanical device. These mechanical controllers used a lever, spring and a mass and were often energized by compressed air. These pneumatic controllers were once the industry standard.
Electronic analog controllers can be made from a solid-state or tube amplifier, a capacitor and a resistance. Electronic analog PID control loops were often found within more complex electronic systems, for example, the head positioning of a disk drive, the power conditioning of a power supply, or even the movement-detection circuit of a modern seismometer. Nowadays, electronic controllers have largely been replaced by digital controllers implemented with microcontrollers or FPGAs.
Most modern PID controllers in industry are implemented in programmable logic controllers (PLCs) or as a panel-mounted digital controller. Software implementations have the advantages that they are relatively cheap and are flexible with respect to the implementation of the PID algorithm.

Alternative nomenclature and PID forms

Ideal versus standard PID form

The form of the PID controller most often encountered in industry, and the one most relevant to tuning algorithms is the standard form. In this form the

K p

gain is applied to the

I out

, and

D out

terms, yielding:

$\mathrm{MV(t)}=K_p\left(\,{e(t)} + \frac{1}{T_i}\int_{0}^{t}{e(\tau)}\,{d\tau} + T_d\frac{d}{dt}e(t)\right)$

where

T i

is the integral time

T d

is the derivative time

In the ideal parallel form, shown in the controller theory section

$\mathrm{MV(t)}=K_p{e(t)} + K_i\int_{0}^{t}{e(\tau)}\,{d\tau} + K_d\frac{d}{dt}e(t)$

the gain parameters are related to the parameters of the standard form through $K_i = \frac{K_p}{T_i}$ and $K_d = K_p T_d \,$ . This parallel form, where the parameters are treated as simple gains, is the most general and flexible form. However, it is also the form where the parameters have the least physical interpretation and is generally reserved for theoretical treatment of the PID controller. The standard form, despite being slightly more complex mathematically, is more common in industry.

Laplace form of the PID controller

Sometimes it is useful to write the PID regulator in Laplace transform form:

$G(s)=K_p + \frac{K_i}{s} + K_d{s}=\frac{K_d{s^2} + K_p{s} + K_i}{s}$

Having the PID controller written in Laplace form and having the transfer function of the controlled system, makes it easy to determine the closed-loop transfer function of the system.

Series/interacting form

Another representation of the PID controller is the series, or interacting form

$G(s) = K_c \frac{(\tau_i{s}+1)}{\tau_i{s}} (\tau_d{s}+1)$

where the parameters are related to the parameters of the standard form through

$K_p = K_c \cdot \alpha$ , $T_i = \tau_i \cdot \alpha$ , and

$T_d = \frac{\tau_d}{\alpha}$

with

$\alpha = 1 + \frac{\tau_d}{\tau_i}$ .

This form essentially consists of a PD and PI controller in series, and it made early (analog) controllers easier to build. When the controllers later became digital, many kept using the interacting form.

Discrete implementation

The analysis for designing a digital implementation of a PID controller in a Microcontroller (MCU) or FPGA device requires the standard form of the PID controller to be discretised ^[2]. Approximations for first-order derivatives are made by backward finite differences. The integral term is discretised, with a sampling time

Δ t

,as follows,

$\int_{0}^{t_k}{e(\tau)}\,{d\tau} = \sum_{i=1}^k e(t_i)\Delta t$

The derivative term is approximated as,

$\dfrac{de(t_k)}{dt}=\dfrac{e(t_k)-e(t_{k-1})}{\Delta t}$

Thus, a velocity algorithm for implementation of the discretised PID controller in a MCU is obtained,

$u(t_k)=u(t_{k-1})+K_p\left[\left(1+\dfrac{\Delta t}{T_i}+\dfrac{T_d}{\Delta t}\right)e(t_k)+\left(-1-\dfrac{2T_d}{\Delta t}\right)e(t_{k-1})+\dfrac{T_d}{\Delta t}e(t_{k-2})\right]$

state observer

In control theory, a state observer is a system that models a real system in order to provide an estimate of its internal state, given measurements of the input and output of the real system. It is typically a computer-implemented mathematical model.
Knowing the system state is necessary to solve many control theory problems; for example, stabilizing a system using state feedback. In most practical cases, the physical state of the system cannot be determined by direct observation. Instead, indirect effects of the internal state are observed by way of the system outputs. A simple example is that of vehicles in a tunnel: the rates and velocities at which vehicles enter and leave the tunnel can be observed directly, but the exact state inside the tunnel can only be estimated. If a system is observable, it is possible to fully reconstruct the system state from its output measurements using the state observer.

Typical observer model

The state of a physical discrete-time system is assumed to satisfy

$\mathbf{x}(k+1) = A \mathbf{x}(k) + B \mathbf{u}(k)$ $\mathbf{y}(k) = C \mathbf{x}(k) + D \mathbf{u}(k)$

where, at time

k

, $\mathbf{x}(k)$ is the plant's state; $\mathbf{u}(k)$ is its inputs; and $\mathbf{y}(k)$ is its outputs. These equations simply say that the plant's current outputs and its future state are both determined solely by its current state and the current inputs. (Although these equations are expressed in terms of discrete time steps, very similar equations hold for continuous systems). If this system is observable then the output of the plant, $\mathbf{y}(k)$ , can be used to steer the state of the state observer.
The observer model of the physical system is then typically derived from the above equations. Additional terms may be included in order to ensure that, on receiving successive measured values of the plant's inputs and outputs, the model's state converges to that of the plant. In particular, the output of the observer may be subtracted from the output of the plant and then multiplied by a matrix

L

; this is then added to the equations for the state of the observer to produce a so-called Luenberger observer, defined by the equations below. Note that the variables of a state observer are commonly denoted by a "hat": $\mathbf{\hat{x}}(k)$ and $\mathbf{\hat{y}}(k)$ to distinguish them from the variables of the equations satisfied by the physical system.

$\mathbf{\hat{x}}(k+1) = A \mathbf{\hat{x}}(k) + L \left[\mathbf{y}(k) - \mathbf{\hat{y}}(k)\right] + B \mathbf{u}(k)$ $\mathbf{\hat{y}}(k) = C \mathbf{\hat{x}}(k) + D \mathbf{u}(k)$

The observer is called asymptotically stable if the observer error $\mathbf{e}(k) = \mathbf{\hat{x}}(k) - \mathbf{x}(k)$ converges to zero when $k \rightarrow \infty$ . For a Luenberger observer, the observer error satisfies $\mathbf{e}(k+1) = (A - LC) \mathbf{e}(k)$ . The Luenberger observer for this discrete-time system is therefore asymptotically stable when the matrix

A - L C

has all the eigenvalues inside the unit circle.
For control purposes the output of the observer system is fed back to the input of both the observer and the plant through the gains matrix

K

$\mathbf{u(k)}= -K \mathbf{\hat{x}}(k)$

The observer equations then become:

$\mathbf{\hat{x}}(k+1) = A \mathbf{\hat{x}}(k) + L \left(\mathbf{y}(k) - \mathbf{\hat{y}}(k)\right) - B K \mathbf{\hat{x}}(k)$ $\mathbf{\hat{y}}(k) = C \mathbf{\hat{x}}(k) - D K \mathbf{\hat{x}}(k)$

or, more simply,

$\mathbf{\hat{x}}(k+1) = \left(A - B K \right) \mathbf{\hat{x}}(k) + L \left(\mathbf{y}(k) - \mathbf{\hat{y}}(k)\right)$ $\mathbf{\hat{y}}(k) = \left(C - D K\right) \mathbf{\hat{x}}(k)$

Due to the separation principle we know that we can choose

K

and

L

independently without harm to the overall stability of the systems. As a rule of thumb, the poles of the observer

A - L C

are usually chosen to converge 10 times faster than the poles of the system

A - B K

Continuous-time case

The previous example was for an observer implemented in a discrete-time LTI system. However, the process is similar for the continuous-time case; the observer gains

L

are chosen to make the continuous-time error dynamics converge to zero asymptotically (i.e., when

A - L C

is a Hurwitz matrix).
For a continuous-time linear system

$\mathbf{\dot{x}} = A \mathbf{x}+ B \mathbf{u},$

$\mathbf{y} = C \mathbf{x},$ ,

where $\mathbf{x} \in \mathbb{R}^n, \mathbf{u} \in \mathbb{R}^m ,\mathbf{y} \in \mathbb{R}^r$ , the observer looks similar to discrete-time case described above:

$\mathbf{\dot{\hat{x}}} = A \mathbf{\hat{x}}+ B \mathbf{u} + L \left(\mathbf{y} - C \mathbf{\hat{x}}\right)$ .

The observer error $\mathbf{e}=\mathbf{\hat{x}}-\mathbf{x}$ satisfies the equation

$\mathbf{\dot{e}} = (A - LC) \mathbf{e}$ .

The eigenvalues of the matix

A - L C

can be made arbitrary by appropriate choice of the observer gain

L

when the pair

[A, C]

is observable, i.e. observability condition holds. In particular, it can be made Hurwitz, so the observer error $e(t) \rightarrow 0$ when $t \rightarrow \infty$ .

Peaking and other observer methods

When the observer gain

L

is high, the linear Luenberger observer converges to the system states very quickly. However, high observer gain leads to a peaking phenomenon in which initial estimator error can be prohibitively large (i.e., impractical or unsafe to use).^[1] As a consequence, nonlinear high gain observer methods are available that converge quickly without the peaking phenomenon. For example, sliding mode control can be used to design an observer that brings one estimated state's error to zero in finite time even in the presence of measurement error; the other states have error that behaves similarly to the error in a Luenberger observer after peaking has subsided. Sliding mode observers also have attractive noise resilience properties that are similar to a Kalman filter.^[2]^[3]

State observers for nonlinear systems

Sliding mode observers can be designed for the non-linear systems as well. For simplicity, first consider the no-input non-linear system:

$\dot{\mathbf{x}} = f(\mathbf{x})$

where $\mathbf{x} \in \mathbb{R}^n$ . Also assume that there is a measurable output $\mathbf{y} \in \mathbb{R}$ given by

$\mathbf{y} = h(\mathbf{x}).$

There are several non-approximate approaches for designing an observer. The two observers given below also apply to the case when the system has an input. That is,

$\dot{\mathbf{x}} = f(\mathbf{x}) + B(\mathbf{x}) \mathbf{u},$

$\mathbf{y} = h(\mathbf{x}),$ .

Linearizable error dynamics

One suggested by Kerner and Isidori^[4] and Krener and Respondek^[5] can be applied in a situation when there exists a linearizing transformation (i.e., a diffeomorphism, like the one used in feedback linearization) $\mathbf{z}=\Phi(\mathbf{x})$ such that in new variables the system equations read

$\dot{\mathbf{z}} = A \mathbf{z}+ \phi(\mathbf{y}),$

$\mathbf{y} = C\mathbf{z}.$

The Luenberger observer is then designed as

$\dot{\hat{\mathbf{z}}} = A \hat{\mathbf{z}}+ \phi(\mathbf{y}) - L \left(C \hat{\mathbf{z}}-\mathbf{y} \right)$ .

The observer error for the transformed variable $\mathbf{e}=\mathbf{\hat{z}}-\mathbf{z}$ satisfies the same equation as in classical linear case.

$\mathbf{\dot{e}} = (A - LC) \mathbf{e}$ .

As shown by Gauthier, Hammouri, and Othman^[6] and Hammouri and Kinnaert,^[7] if there exists transformation $\mathbf{z}=\Phi(\mathbf{x})$ such that the system can be transformed into the form

$\dot{\mathbf{z}} = A(u(t)) \mathbf{z}+ \phi(\mathbf{y},u(t) ),$

$\mathbf{y} = C\mathbf{z},$

then the observer is designed as

$\mathbf{\dot{\hat{z}}} = A(u(t)) \mathbf{\hat{z}}+ \phi(\mathbf{y},u(t) ) - L(t) \left(C \mathbf{\hat{z}}-\mathbf{y} \right)$ ,

where

L (t)

is a time-varying observer gain.

Sliding mode observer

As discussed for the linear case above, the peaking phenomenon present in Luenberger observers justifies the use of a sliding mode observer. The sliding mode observer uses non-linear high-gain feedback to drive estimated states to a hypersurface where there is no difference between the estimated output and the measured output. The non-linear gain used in the observer is typically implemented with a scaled switching function, like the signum (i.e., sign) of the estimated–measured output error. Hence, due to this high-gain feedback, the vector field of the observer has a crease in it so that observer trajectories slide along a curve where the estimated output matches the measured output exactly. So, if the system is observable from its output, the observer states will all be driven to the actual system states. Additionally, by using the sign of the error to drive the sliding mode observer, the observer trajectories become insensitive to many forms of noise. Hence, some sliding mode observers have attractive properties similar to the Kalman filter but with simpler implementation.^[2]^[3]
As suggested by Drakunov,^[8] a sliding mode observer can also be designed for a class of non-linear systems. Such an observer can be written in terms of original variable estimate $\mathbf{\hat{x}}$ and has the form

$\mathbf{\dot{\hat{x}}} = \left [ \frac{\partial H(\mathbf{\hat{x}})}{\partial \mathbf{x}} \right]^{-1} M(\mathbf{\hat{x}}) \, \operatorname{sign}( V(t) - H(\mathbf{\hat{x}}) )$

where:

The $\operatorname{sign}(\mathord{\cdot})$ vector extends the scalar signum function to $n$ dimensions. That is,

$\operatorname{sign}(\mathbf{z}) = \begin{bmatrix} \operatorname{sign}(z_1)\\ \operatorname{sign}(z_2)\\ \vdots\\ \operatorname{sign}(z_i)\\ \vdots\\ \operatorname{sign}(z_n) \end{bmatrix}$

for the vector $\mathbf{z} \in \mathbb{R}^n$ .

The vector $H(\mathbf{x})$ has components that are the output function $h(\mathbf{x})$ and its repeated Lie derivatives. In particular,

$H(\mathbf{x}) \triangleq \begin{bmatrix} h_1(\mathbf{x})\\ h_2(\mathbf{x})\\ h_3(\mathbf{x})\\ \vdots\\ h_n(\mathbf{x}) \end{bmatrix} \triangleq \begin{bmatrix} h(\mathbf{x})\\ L_{f}h(\mathbf{x})\\ L_{f}^2 h(\mathbf{x})\\ \vdots\\ L_{f}^{n-1}h(\mathbf{x}) \end{bmatrix}$

where $L^i_f h$ is the i^th Lie derivative of output function

h

along the vector field

f

(i.e., along $\mathbf{x}$ trajectories of the non-linear system). In the special case where the system has no input or has a relative degree of n, $H(\mathbf{x}(t))$ is a collection of the output $\mathbf{y}(t)=h(\mathbf{x}(t))$ and its

n - 1

derivatives. Because the inverse of the Jacobian linearization of $H(\mathbf{x})$ must exist for this observer to be well defined, the transformation $H(\mathbf{x})$ is guaranteed to be a local diffeomorphism.

The diagonal matrix $M(\hat{\mathbf{x}})$ of gains is such that

$M(\hat{\mathbf{x}}) \triangleq \operatorname{diag}( m_1(\hat{\mathbf{x}}), m_2(\hat{\mathbf{x}}), \ldots, m_n(\hat{\mathbf{x}}) ) = \begin{bmatrix} m_1(\hat{\mathbf{x}}) & & & & & \\ & m_2(\hat{\mathbf{x}}) & & & & \\ & & \ddots & & & \\ & & & m_i(\hat{\mathbf{x}}) & &\\ & & & & \ddots &\\ & & & & & m_n(\hat{\mathbf{x}}) \end{bmatrix}$

where, for each $i \in \{1,2,\dots,n\}$ , element $m_i(\hat{\mathbf{x}}) > 0$ and suitably large to ensure reachability of the sliding mode.

The observer vector $V (t)$ is such that

$V(t) \triangleq \begin{bmatrix}v_{1}(t)\\ v_2(t)\\ v_3(t)\\ \vdots\\ v_i(t)\\ \vdots\\ v_{n}(t) \end{bmatrix} \triangleq \begin{bmatrix} \mathbf{y}(t)\\ \{ m_1(\hat{\mathbf{x}}) \operatorname{sign}( v_1(t) - h_1(\hat{\mathbf{x}}(t)) ) \}_{\text{eq}}\\ \{ m_2(\hat{\mathbf{x}}) \operatorname{sign}( v_2(t) - h_2(\hat{\mathbf{x}}(t)) ) \}_{\text{eq}}\\ \vdots\\ \{ m_{i-1}(\hat{\mathbf{x}}) \operatorname{sign}( v_{i-1}(t) - h_{i-1}(\hat{\mathbf{x}}(t)) ) \}_{\text{eq}}\\ \vdots\\ \{ m_{n-1}(\hat{\mathbf{x}}) \operatorname{sign}( v_{n-1}(t) - h_{n-1}(\hat{\mathbf{x}}(t)) ) \}_{\text{eq}} \end{bmatrix}$

where $\operatorname{sign}(\mathord{\cdot})$ here is the normal signum function defined for scalars, and $\{ \ldots \}_{\text{eq}}$ denotes an "equivalent value operator" of a discontinuous function in sliding mode.

The idea can be briefly explained as follows. According to the theory of sliding modes, in order to describe the system behavior, once sliding mode starts, the function $\operatorname{sign}( v_{i}(t)\!-\! h_{i}(\hat{\mathbf{x}}(t)) )$ should be replaced by equivalent values (see equivalent control in the theory of sliding modes). In practice, it switches (chatters) with high frequency with slow component being equal to the equivalent value. Applying appropriate lowpass filter to get rid of the high frequency component on can obtain the value of the equivalent control, which contains more information about the state of the estimated system. The observer described above uses this method several times to obtain the state of the nonlinear system ideally in finite time.
The modified observation error can be written in the transformed states $\mathbf{e}=H(\mathbf{x})-H(\mathbf{\hat{x}})$ . In particular,

$\begin{align} \dot{\mathbf{e}} &= \frac{\operatorname{d}}{\operatorname{d}t} H(\mathbf{x}) - \frac{\operatorname{d}}{\operatorname{d}t} H(\hat{\mathbf{x}})\\ &= \frac{\operatorname{d}}{\operatorname{d}t} H(\mathbf{x}) - M(\hat{\mathbf{x}}) \, \operatorname{sign}( V(t) - H(\hat{\mathbf{x}}(t)) ), \end{align}$

and so

$\begin{align} \begin{bmatrix} \dot{\mathbf{e}}_1\\ \dot{\mathbf{e}}_2\\ \vdots\\ \dot{\mathbf{e}}_i\\ \vdots\\ \dot{\mathbf{e}}_{n-1}\\ \dot{\mathbf{e}}_n \end{bmatrix} &= \mathord{\overbrace{ \begin{bmatrix} \dot{h}_1(\mathbf{x})\\ \dot{h}_2(\mathbf{x})\\ \vdots\\ \dot{h}_i(\mathbf{x})\\ \vdots\\ \dot{h}_{n-1}(\mathbf{x})\\ \dot{h}_n(\mathbf{x}) \end{bmatrix} }^{\tfrac{\operatorname{d}}{\operatorname{d}t} H(\mathbf{x})}} - \mathord{\overbrace{ M(\hat{\mathbf{x}}) \, \operatorname{sign}( V(t) - H(\hat{\mathbf{x}}(t)) ) }^{\tfrac{\operatorname{d}}{\operatorname{d}t} H(\mathbf{\hat{x}})}} = \begin{bmatrix} h_2(\mathbf{x})\\ h_3(\mathbf{x})\\ \vdots\\ h_{i+1}(\mathbf{x})\\ \vdots\\ h_n(\mathbf{x})\\ L_f^n h(\mathbf{x}) \end{bmatrix} - \begin{bmatrix} m_1 \operatorname{sign}( v_1(t) - h_1(\hat{\mathbf{x}}(t)) )\\ m_2 \operatorname{sign}( v_2(t) - h_2(\hat{\mathbf{x}}(t)) )\\ \vdots\\ m_i \operatorname{sign}( v_i(t) - h_i(\hat{\mathbf{x}}(t)) )\\ \vdots\\ m_{n-1} \operatorname{sign}( v_{n-1}(t) - h_{n-1}(\hat{\mathbf{x}}(t)) )\\ m_n \operatorname{sign}( v_n(t) - h_n(\hat{\mathbf{x}}(t)) ) \end{bmatrix}\\ &= \begin{bmatrix} h_2(\mathbf{x}) - m_1(\hat{\mathbf{x}}) \operatorname{sign}( \mathord{\overbrace{ \mathord{\overbrace{v_1(t)}^{v_1(t) = y(t) = h_1(\mathbf{x})}} - h_1(\hat{\mathbf{x}}(t)) }^{\mathbf{e}_1}} )\\ h_3(\mathbf{x}) - m_2(\hat{\mathbf{x}}) \operatorname{sign}( v_2(t) - h_2(\hat{\mathbf{x}}(t)) )\\ \vdots\\ h_{i+1}(\mathbf{x}) - m_i(\hat{\mathbf{x}}) \operatorname{sign}( v_i(t) - h_i(\hat{\mathbf{x}}(t)) )\\ \vdots\\ h_n(\mathbf{x}) - m_{n-1}(\hat{\mathbf{x}}) \operatorname{sign}( v_{n-1}(t) - h_{n-1}(\hat{\mathbf{x}}(t)) )\\ L_f^n h(\mathbf{x}) - m_n(\hat{\mathbf{x}}) \operatorname{sign}( v_n(t) - h_n(\hat{\mathbf{x}}(t)) ) \end{bmatrix}. \end{align}$

So:

As long as $m_1(\mathbf{\hat{x}}) \geq |h_2(\mathbf{x}(t))|$ , the first row of the error dynamics, $\dot{\mathbf{e}}_1 = h_2(\hat{\mathbf{x}}) - m_1(\hat{\mathbf{x}}) \operatorname{sign}( \mathbf{e}_1 )$ , will meet sufficient conditions to enter the $e 1 = 0$ sliding mode in finite time.
Along the $e 1 = 0$ surface, the corresponding $v_2(t) = \{m_1(\hat{\mathbf{x}}) \operatorname{sign}( \mathbf{e}_1 )\}_{\text{eq}}$ equivalent control will be equal to $h_2(\mathbf{x})$ , and so $v_2(t) - h_2(\hat{\mathbf{x}}) = h_2(\mathbf{x}) - h_2(\hat{\mathbf{x}}) = \mathbf{e}_2$ . Hence, so long as $m_2(\mathbf{\hat{x}}) \geq |h_3(\mathbf{x}(t))|$ , the second row of the error dynamics, $\dot{\mathbf{e}}_2 = h_3(\hat{\mathbf{x}}) - m_2(\hat{\mathbf{x}}) \operatorname{sign}( \mathbf{e}_2 )$ , will enter the $e 2 = 0$ sliding mode in finite time.
Along the $e i = 0$ surface, the corresponding $v_{i+1}(t) = \{\ldots\}_{\text{eq}}$ equivalent control will be equal to $h_{i+1}(\mathbf{x})$ . Hence, so long as $m_{i+1}(\mathbf{\hat{x}}) \geq |h_{i+2}(\mathbf{x}(t))|$ , the $(i + 1)$ ^th row of the error dynamics, $\dot{\mathbf{e}}_{i+1} = h_{i+2}(\hat{\mathbf{x}}) - m_{i+1}(\hat{\mathbf{x}}) \operatorname{sign}( \mathbf{e}_{i+1} )$ , will enter the $e i + 1 = 0$ sliding mode in finite time.

So, for sufficiently large

m i

gains, all observer estimated states reach the actual states in finite time. In fact, increasing

m i

allows for convergence in any desired finite time so long as each $|h_i(\mathbf{x}(0))|$ function can be bounded with certainty. Hence, the requirement that the map $H:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is a diffeomorphism (i.e., that its Jacobian linearization is invertible) asserts that convergence of the estimated output implies convergence of the estimated state. That is, the requirement is an observability condition.
In the case of the sliding mode observer for the system with the input, additional conditions are needed for the observation error to be independent of the input. For example, that

$\frac{\partial H(\mathbf{x})}{\partial \mathbf{x}} B(\mathbf{x})$

does not depend on time. The observer is then

$\dot{\mathbf{\hat{x}}} = \left[ \frac{\partial H(\mathbf{\hat{x}})}{\partial \mathbf{x}} \right]^{-1} M(\mathbf{\hat{x}}) \operatorname{sign}(V(t) - H(\mathbf{\hat{x}}))+B(\mathbf{\hat{x}})u.$

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Sunday, October 11, 2009

PID controller

General

Control loop basics

PID controller theory

Proportional term

Integral term

Derivative term

Summary

Loop tuning

Manual tuning

Ziegler–Nichols method

PID tuning software

Modifications to the PID algorithm

Limitations of PID control

Cascade control

Physical implementation of PID control

Alternative nomenclature and PID forms

Ideal versus standard PID form

Laplace form of the PID controller

Series/interacting form

Discrete implementation

state observer

Typical observer model

Continuous-time case

Peaking and other observer methods

State observers for nonlinear systems

Linearizable error dynamics

Sliding mode observer

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