Control Systems: Analysis, Design, and Applications

Posted on Jan 25, 2025 in Business Administration and Innovation Management

Controllability of Linear Systems

Controllability of a linear system, in both continuous and discrete time. A continuous-time system is controllable if, for any initial state and some finite time, there exists a control that transfers the state to a desired value. A discrete-time system is controllable if, for any initial state and some finite time, there exists a control sequence that transfers the state to a desired value. There are several tests for controllability.

Controllability of Time-Invariant Systems

A linear time-invariant system ẋ = Ax + Bu has a controllability matrix defined by P = [B AB A²B … A^n-1B].

Theorem: The linear time-invariant system ẋ = Ax + Bu is controllable if and only if the controllability matrix, defined by P, is positive definite for some time t.

Stability Analysis of Sampled-Data Systems

This section discusses the stability analysis of sampled-data control systems, focusing on linear time-invariant systems. It covers Bounded-Input Bounded-Output (BIBO) stability and input stability.

A linear time-invariant (LTI) system is said to be BIBO stable if every bounded input results in a bounded output. BIBO stability can be determined by the location of the closed-loop poles in the s-plane, where the roots of the characteristic equation 1 + GH(s) = 0. For a system to be stable, the poles, or roots of the characteristic equation, must lie within the unit circle. In simple terms, if a pole is outside the unit circle, the system becomes unstable. If a pair of complex conjugate poles lies on the unit circle, the system is marginally stable.

Controllers

A controller is a mechanical or electromechanical device that uses an input signal to change the condition of a process. For example, it can be used to control access to a building’s gated area or to regulate the AC voltage input from a source. Controllers analyze input conditions and adjust accordingly. Some controllers are simple manual devices that control airflow. Motion controllers are used in motor control, and drives improve motor performance.

Uses of Controllers

Control of steady-state accuracy by decreasing steady-state error.
Improve the stability of a system.
Help in reducing noise signals in a system.
Control the maximum overshoot of a system.
Make the slow response of an overdamped system faster.

Types of Controllers

Pneumatic or Hydraulic Controllers

Pneumatic controllers were used in process control before the advancement of electronics. The advantages of pneumatic controllers include their ruggedness. However, their major limitation is their slow response. These controllers are designed using mechanical components that operate according to liquid pressure. Due to the use of mechanical components, their response is slow compared to electronic components. These controllers act on the difference between air or liquid pressure, the measured signal, and the setpoint signal. A good example of a pneumatic or hydraulic controller is the speed control system of a turbine.

Electronic Hardware Controllers

Advancements in electronic components have promoted their use in control systems. The introduction of electronic control systems enhances performance, making them much faster than mechanical systems. Due to the small size of electronic components, the overall size of the controller is reduced. However, while the system response is faster, it is also very sensitive to temperature, and internal faults can easily occur. The output of an electronic controller is equal to the proportional gain multiplied by the error, plus a low-pass filter and a high-pass filter. The input computer calculates the error between the input setpoint and the actual measured value. The error is then applied to a gain amplifier, a low-pass filter, and a high-pass filter. The parallel outputs are combined at the output node and sent to an actuator to drive it.

Design of Controllers

Consider a simple schematic of a controller for liquid level control in a tank. The liquid level sensor is connected to a digital computer controller. The controller compares the liquid level value to a setpoint provided by the control panel. The error signal is processed according to control instructions stored in the controller’s memory, and an output signal is sent to a final activating device through a signal output. This signal is suitable for activating the final element. In this type of controller, the control logic can be easily modified by changing the program instructions.

Feedforward Controllers

Feedforward control is often used alongside feedback control because a feedback control system is required to track setpoint changes and suppress unmeasured disturbances that are present in real processes. A traditional block diagram of feedforward control shows the same blocks but redrawn to clearly show that the feedforward path of the system is not affected by the stability of the feedback loop. A typical application of feedforward control is a continuously stirred tank reactor under feedback control, where feedforward control is used to rapidly adjust for feed flow rate disturbances.

Design of Feedforward Control

Consider a process loop diagram where the process and disturbance transfer functions are represented by G_p and G_d, respectively.

PI Controllers

A Proportional-Integral (PI) controller combines proportional and integral control. The output is equal to the sum of the proportional and integral responses to the error signal: u(t) = K_i∫e(t)dt + K_pe(t). The advantages and disadvantages are discussed below.

Applications of Proportional-Integral Controllers

Liquid Flow Control

Processes such as liquid flow control are ideal applications for PI control. They are critical to maintaining production rates, and the extended periods of steady state associated with proportional-only (P) control can be counterproductive. These processes also have fast dynamics, which makes derivative action less beneficial. PI control effectively eliminates offset without introducing excessive oscillation in the response.

Steam Pressure Control

Generally speaking, pressure is another process that is often essential to production and exhibits highly dynamic characteristics. In steam pressure control, the primary objective is to react quickly to changes in downstream demand while limiting the negative effects on sensitive process instrumentation. PI control is often suitable, but if unable to adequately address the control needs, Proportional-Integral-Derivative (PID) control may be necessary.

Heat Exchanger Temperature Control

Heat exchanger control is among the most common process applications in the industry. The exit temperature of a heat exchanger must be controlled within relatively tight tolerances. P-only control is often insufficient, and whether plate-and-frame or shell-and-tube, heat exchanger dynamics are sufficiently fast that derivative action is not necessary. The PID form should be reserved for applications such as furnace temperature control.

PID Controllers

A Proportional-Integral-Derivative (PID) controller combines proportional, integral, and derivative control actions. It is a feedback mechanism used in control systems. PID control is considered a three-term control strategy. By adjusting the three parameters—proportional, integral, and derivative—different control actions can be achieved for specific tasks. PID control is considered one of the best control strategies in the control system family. The actuating signal consists of the proportional error signal added to the derivative and integral of the error signal. Therefore, the actuating signal for a PID controller can be expressed mathematically. There are control actions that can be achieved by using only two of the parameters while setting the third to zero. In this way, a PID controller can sometimes act as a PI, PD, P, or I controller. The derivative term (D) is responsible for noise measurement, while the integral term (I) is meant for reaching the target value. In earlier days, PID control was implemented using mechanical devices, such as pneumatic controllers. Nowadays, most complex electronic systems use PID control loops. In modern systems, it is often implemented using programmable logic controllers (PLCs). The proportional, derivative, and integral parameters can be expressed as K_p, K_d, and K_i, respectively. These three parameters affect the closed-loop system, influencing the rise time, settling time, overshoot, and steady-state error.

Characteristics of PID Controllers

Proportional Control (P): Effective in reducing rise time but never completely eliminates steady-state error.
Integral Control (I): Effective in eliminating steady-state error but may worsen the transient response.
Derivative Control (D): Increases the stability of the system, reduces overshoot, and improves the transient response.

Comparators

Comparators are used in control system design using electrical, mechanical, pneumatic, or other components. The basic idea is to determine the transfer function of a compensator by strategically placing the dominant closed-loop poles of a system. Compensated systems can be classified into six categories based on the location of the compensator:

Series or Cascade Compensation: This type of compensation involves adding a compensator in the feedforward path of the system. It adjusts the gain of the system, which reduces the response time and peak overshoot. However, it may also reduce the stability of the system.
Feedback or Parallel Compensation: In this type of system, the compensator is included in the feedback path. The addition of the compensator in the feedback path increases the response time of the system, making it more accurate and stable.
Load or Series-Parallel Compensation: This is a combination of both series and parallel compensation.
State Feedback Compensation: In this type of compensation, the control signal is fed back to the system through a constant real gain. The implementation of state feedback compensation is costly and impractical for higher-order systems.
Forward Compensation with Series Compensation: This involves placing a feedback forward controller in series with a simple closed-loop system. This results in compensation in the forward path combined with series compensation.
Feedforward Compensation: In this configuration, a feedforward controller is placed in parallel with the forward path of a simple closed-loop system. This results in a feedforward compensation system with one degree of freedom, meaning the system has a single controller. The disadvantage of having only one degree of freedom is that the performance criteria that can be achieved using this technique are limited. In simple terms, a compensator induces additional poles and zeros into the existing system to achieve the desired specifications.

Compensating Networks

A compensating network is a physical device, which may be an electrical network, a mechanical unit, or a pneumatic, hydraulic, or combination of various types of devices. Electrical networks are commonly used for series compensation.

Lead Network or Lead Compensator

If a sinusoidal input is applied to a network and the sinusoidal steady-state output has a phase lead, then the network is called a lead network. If the steady-state output leads the input, the network is called a lead network.

Lag Network or Lag Compensator

If the steady-state output lags behind the input, the network is called a lag network.

Lag-Lead Network or Lag-Lead Compensator

A lag-lead network combines both phase lag and phase lead characteristics. Phase lag occurs in the low-frequency region, while phase lead occurs in the high-frequency region.

Phase Lead Compensator

A system with one pole and one dominating zero is known as a lead compensator. If we want to add a dominating zero for compensation in a control system, we need to select a phase lead network. The basic requirement of a phase lead network is that all poles and zeros of the transfer function of the network lie on the negative real axis, with the zero closest to the origin. If we consider the frequency response of a simple proportional-derivative (PD) controller, it is evident that the magnitude of the compensator continuously grows with increasing frequency. This feature is undesirable because it amplifies the high-frequency noise typically present in real-time systems. In a lead compensator, a first-order pole is added to the denominator of the PD controller transfer function. The frequency of this pole is typically much higher than the corner frequency of the PD controller. A typical lead compensator has the following transfer function: C(s) = K(Ts + 1) / (αTs + 1), where 1/α is the ratio of the pole to zero.

Lag Compensator

A system with one zero and one dominating pole, where the pole is closer to the origin, is known as a lag compensator. If we want to add a dominating pole for compensation in a control system, we need to select a lag compensation network. The basic requirement for a phase lag network is that all poles and zeros of the transfer function of the network lie on the negative real axis, interlacing with each other. The essential feature of a lag compensator is to provide an increase in gain at low frequencies, which improves steady-state error without significantly changing the transient response. The transfer function of a lag compensator is given by C(s) = α(Ts + 1) / (αTs + 1). The typical objective of a lag compensator is to provide an additional gain, α, in the low-frequency region while maintaining a sufficient phase margin. Since the lag compensator provides maximum lag near the corner frequency, the phase margin of the system is affected. The corner frequency should be chosen such that ω = 1 / (λT).

Effects of Phase Lag Compensation

Crossover frequency decreases.
Bandwidth decreases.
Phase margin increases.
Response becomes slower due to the decrease in bandwidth.

Advantages and Disadvantages of Phase Lag Compensation

Advantages

Low-frequency gain increases.
High frequencies are attenuated.
Due to the presence of phase lag, steady-state error is reduced.

Disadvantages

Due to the presence of phase lag compensation, the speed of the system decreases.

Lag-Lead Compensator

Sometimes, a single lead or lag compensation may not satisfy the design specifications. For an unstable compensated system, lead compensation can provide a faster response but may not provide enough phase margin. Conversely, lag compensation can stabilize the system but may not provide enough bandwidth. In such cases, we may need to use multiple compensators in cascade. When a single lead or lag compensator cannot meet the specific design criteria, a lag-lead compensator is used. In a lag-lead compensator, the lag part precedes the lead part. A continuous lag-lead compensator is given by C(s) = (1 + T₁s) / (1 + αT₁s) * (1 + T₂s) / (1 + αT₂s).

Conditions for Designing a Lag-Lead Compensator

If the type of compensator is not specified, one should first check the phase margin (PM) and bandwidth (BW) of the uncompensated system. If the bandwidth is smaller than the acceptable bandwidth, one may go for a lead compensator. If the bandwidth is large, a lead compensator may not be useful since it provides high-frequency amplification. If the bandwidth is large and the phase margin is small, a lag compensator may be used. If the bandwidth is large and the phase margin is small, and adding a lag compensator results in a low bandwidth, a lag-lead compensator may be used.

Advantages of Phase Lag-Lead Compensation

Due to the presence of the phase lag-lead network, the speed of the system increases because it shifts the gain crossover frequency to a higher value.
Due to the presence of the lag-lead network, accuracy is improved.

Procedure for Designing a Phase Lead Compensator

Determine the system type and the gain needed to satisfy steady-state error specifications.
Make a Bode plot of the system with the gain determined in step 1.
Design the lead portion of the compensator. Determine the amount of phase shift needed at the gain crossover frequency and calculate the compensated phase margin.
Calculate the values of α and T that are required to raise the phase curve to the desired value.
Determine the value for the new gain crossover frequency.
Using the value of α and the new gain crossover frequency, compute the lead compensator zero and pole.
If necessary, choose appropriate resistor and capacitor values to implement the compensator design.

Procedure for Designing a Phase Lag-Lead Compensator

The procedure is almost the same as designing a lead compensator followed by designing a lag compensator. There are small differences in the steps required for the lead and lag portions. These differences take into account the interaction between the lag and lead parts of the compensator.

The following steps are followed:

Determine the system type and the gain needed to satisfy steady-state error specifications.
Make a Bode plot of the system with the gain determined in step 1.
Design the lead portion of the lag-lead compensator. Determine the desired gain crossover frequency and calculate the uncompensated phase margin.
Calculate the values of α_d and T_d that are required to raise the phase curve to the desired value.
Using the value of α_d and the specified gain crossover frequency, compute the lead compensator pole.
Design the lag portion of the lag-lead compensator. Determine the magnitude at the specified gain crossover frequency.
Determine the amount of attenuation required to drop the magnitude curve to 0 dB at the gain crossover frequency.
Using the value of α and the specified gain crossover frequency, compute the lag compensator zero and pole.
If necessary, choose appropriate resistor and capacitor values to implement the compensator design.

Nonlinear Systems

In a nonlinear system, the output signal strength does not vary in direct proportion to the input signal strength. Nonlinear systems do not follow the superposition property. They are systems that do not follow the principle of homogeneity. In practical life, most systems are not inherently linear. A well-known example of a nonlinear system is the magnetization curve or no-load curve of a DC machine. The no-load curve represents the relationship between the air gap flux and the field winding EMF. It is clear from the curve that, initially, there is a linear relationship between the winding EMF and the air gap flux. However, after saturation, the curve exhibits nonlinear behavior. This demonstrates the characteristics of a nonlinear system.

Types of Nonlinear Systems

Incidental Nonlinearities: These are inherent nonlinearities present in the system, such as saturation, dead zone, friction, etc.
Intentional Nonlinearities: These are deliberately introduced into a system to modify its characteristics, improve performance, or simplify construction.

Characteristics of Nonlinear Systems

They do not obey the rule of superposition.
Laplace transform is not applicable to nonlinear systems.
For a sinusoidal input, there is no guarantee that the output will also be sinusoidal.
To determine stability, information regarding the nature and magnitude of anticipated excitation signals and initial conditions is required.
For an unstable nonlinear system, the transient and frequency responses may exhibit peculiar features.

Saturation

Saturation occurs when a component reaches its operational limit. For example, in systems containing electromagnetic components, there is a limit to the maximum achievable magnetic field strength. In circuits with iron-core inductors, applying an AC signal can cause harmonics and intermodulation distortion due to saturation. Therefore, we need to limit the signal accordingly. Saturation is used to limit current in saturable-core transformers, arc welding transformers, and ferroresonant voltage regulators.

Friction

Friction comes into existence when mechanical surfaces come into sliding contact. Different types of friction include viscous friction, Coulomb friction, and stiction. Friction is present in voice coil motor actuators used in hard disk drives to position the read/write head assembly. The presence of friction in rotary actuators and bearings can result in large residual errors and high-frequency oscillations, which may reduce positioning accuracy and deteriorate the performance of servo systems.

Dead Zone

A dead zone is a type of nonlinearity in which the system does not respond until the input reaches a particular level. The output becomes zero when the input exceeds a limiting value. The effects of a dead zone include performance degradation, reduced positioning accuracy, and destabilization of the system.

Backlash

Backlash is the difference between the tooth space and tooth width in a mechanical system. It is essential for proper working but is often referred to as backlash. It is present in most mechanical and hydraulic systems and tends to increase with wear. When backlash is present, no torque is transmitted through the gears, which can cause impact and high-frequency noise. The main factors contributing to backlash are gear backlash and wear. Backlash can lead to sustained oscillations or chattering phenomena. A system may remain stable for small backlash but become unstable for large backlash.

Relay

A relay is an intentional nonlinearity that represents the extreme case of saturation. It can be in three states: full forward, off, or full reverse. Relays are used in temperature control systems, aircraft and missile control systems, space vehicle attitude control systems, and power systems.

Limit Cycle

A limit cycle is a distinguished geometric configuration in the phase plane portrait, representing an isolated closed path. A system may have more than one limit cycle. A limit cycle represents a steady-state oscillation to which all nearby trajectories will either converge or diverge. A nonlinear system can exhibit a limit cycle, which defines the amplitude and period of self-sustained oscillation. A limit cycle is an isolated closed curve, unlike the closed curves in conservative systems, which do not have a specific energy level. Limit cycles are periodic motions exhibited by nonlinear, non-conservative systems.

Describing Function for Nonlinear Control Systems

The describing function method provides a way to analyze nonlinear systems that is closely related to linear system techniques involving the gain of the feedback loop. It can be used to determine if limit cycles are possible in a given system. It is also possible to use the describing function to determine the response of certain nonlinear systems to sinusoidal inputs. Unfortunately, the frequency and transient responses of nonlinear systems are not directly related. However, the determination of the transient response is possible using the describing function. The describing function is an approximate technique capable of predicting the behavior of a class of nonlinear systems. It is also known as the describing function technique. In a block diagram of a nonlinear system, the nonlinear element is represented by N. If x = Xsin(ωt), then y = a₀ + a₁sin(ωt) + a₂sin(2ωt) + … + b₁cos(ωt) + b₂cos(2ωt) + ….

Describing Function for Stability Analysis of Autonomous Systems with Single Nonlinearity

Describing functions are most frequently used to determine if limit cycles are possible for a given system, to determine the amplitude of the error signal oscillations, and, in some cases, to guide the design of a compensator to eliminate oscillations. The system can often be arranged into a form similar to a unity feedback system with an added inverting block. It is required to represent the inversion conventionally by adding a negative sign to the feedback. The important feature of this topology is that the linear element appears to be in a loop with the nonlinear element. The linear element can, of course, represent the reduction of a complex interconnection of linear elements. If the original system has a single transfer function, the operation is simplified by using the describing function. In certain cases, a limiter consisting of back-to-back ideal diodes is included in the circuit. The describing function of a nonlinear element, including R₁, R₂, and the limiter, could be calculated by assuming a sinusoidal input and finding the amplitude and relative phase angle of the fundamental component. The describing function would then be a function of frequency. Once a system has been reduced to this form, it can be analyzed using the describing function. The describing function approximation states that oscillations may be possible if there are particular values of E, N, and ω such that G(jω) = -1/N(E). For example, consider the equation G(jω) = -1/N(E) = (X + j0) / (X² + 1).

Jump Resonance

When a nonlinear system is excited by a sinusoidal input, it may generate several harmonics in addition to the fundamental frequency. Usually, the fundamental is much larger than the harmonics, but in some situations, harmonics may have significant amplitude. A peculiar characteristic of nonlinear systems is the jump phenomenon. Consider the frequency response of a spring-mass-damper system with a linear spring, a softening spring, and a hardening spring. If the input frequency is gradually increased from zero, the measured response follows a certain curve. However, at a certain frequency, a discontinuous jump occurs. With further increases in frequency, the response curve follows a different path.

Phase Plane Method

The phase plane method provides insights into both the local and global behavior of nonlinear systems under various operating conditions. It is used to determine the time-domain response of a system and provides information about its stability. However, it is suitable only for second-order systems.

Basic Concepts

The phase plane method is a graphical technique for visualizing the behavior of a system. It is restricted to second-order systems with two state variables. Phase variables are a particular choice of state variables. A phase portrait is a plot of two state variables against each other. A phase trajectory describes the behavior of the system in the phase plane. The dependent variables are x₁ = x and its first derivative ẋ₁ = x₂. The time solution x(t) can be obtained by computing the time required for the system to move from point A to B, then B to C, and so on. For good accuracy, the incremental displacement Δx should be chosen as small as possible. The value of Δx can be adjusted based on the shape of the phase trajectory to obtain reasonable accuracy.

Advantages of the Phase Plane Method

It provides a graphical analysis and solution.
Phase trajectories represent the system’s qualitative behavior without the need to solve nonlinear equations.
One can study the behavior of nonlinear systems under various initial conditions.
It is not restricted to small nonlinearities and applies equally well to systems with strong nonlinearities.
Many practical systems are second-order, making phase plane analysis applicable.

Disadvantages of the Phase Plane Method

It is restricted to second-order systems.
Graphical study of higher-order systems becomes computationally and geometrically complex.

Phase Trajectory

Consider the equation m(d²x/dt²) + f(dx/dt) + kx = 0. Let x₁ = x and x₂ = dx/dt. Then, ẋ₁ = x₂ and ẋ₂ = -(k/m)x₁ – (f/m)x₂. Alternatively, we can write ẋ₁ = x₂ and ẋ₂ = (k₁/m)x₁ – (f/m)x₂ – (k₂/m)x₁.

Construction of Phase Trajectories

Given ẋ₁ = f₁(x₁, x₂) and ẋ₂ = f₂(x₁, x₂), we have dx₂/dx₁ = f₂(x₁, x₂) / f₁(x₁, x₂) = M. The location of constant slope M is given for different values. From the slope of the tangent at different points, phase trajectories can be drawn in the phase plane. For example, if ẋ₁ = x₂, then dx₂/dx₁ = (x₁ – x₂) / x₂ = M, which simplifies to x₂ = (-1/(M+1))x₁.

Procedure for Construction of a Phase Trajectory

For a given nonlinear differential equation with variables x₁ and x₂, obtain the state equations ẋ₁ = x₂ and ẋ₂ = f₂(x₁, x₂).
Draw a large number of isoclines with different slopes M.
On each isocline, draw small line segments with slope M.
Starting from the initial condition, draw a trajectory following the line segments with slope M.

Construction of Phase Trajectories for Linear Systems Using the Delta Method

The delta method can be used to construct phase trajectories for systems described by ẍ + f(x, ẋ, t) = 0. The system can be linear or nonlinear and may even be time-varying. The function f must be continuous and single-valued. With this method, the phase trajectory of a system with a step or ramp input can be conveniently drawn. The equation can be rewritten as dx₂/dx₁ = (-x₁ + Δ) / x₂. From the initial conditions, calculate the value of Δ. Draw a short arc centered at the initial point. Determine a new point on the trajectory and repeat the process.

Lyapunov Stability Method

The Lyapunov stability method is a general approach for determining the stability of nonlinear, linear, and time-invariant or time-varying control systems. Lyapunov presented two methods for studying the stability of ordinary differential equations. The first method requires knowledge of the solution to the equation and, therefore, contains much more information than just stability. The second method requires no knowledge of the solution and is a generalization of the energy concept from classical mechanics.

Concept of Lyapunov Stability

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