Unit3 - Subjective Questions

Common test signals are standard mathematical functions used to analyze the transient and steady-state behavior of control systems. They provide a predictable input against which system performance can be evaluated.

• Unit Step Function ():

  • Definition: Represents an instantaneous change in input from zero to a constant value (usually 1) at .
  • Application: Used to evaluate the system's ability to reach and maintain a desired constant output. It tests the system's transient response (e.g., rise time, overshoot) and steady-state error.

• Unit Ramp Function ():

  • Definition: Represents an input that increases linearly with time starting from zero at .
  • Application: Used to evaluate the system's ability to follow a constantly changing input (e.g., a tracking radar antenna following a moving target). It primarily tests the system's steady-state error for velocity inputs.

• Unit Parabolic Function ():

  • Definition: Represents an input that increases quadratically with time starting from zero at .
  • Application: Used to evaluate the system's ability to follow an input with constant acceleration. It tests the system's steady-state error for acceleration inputs, often used for higher-order tracking systems.

• Unit Impulse Function ():

  • Definition: Represents a very short duration, high-amplitude input, whose area is unity. It is an idealization of a shock or sudden disturbance.
  • has unit area, defined such that .
  • Application: Used to determine the system's impulse response, which is crucial for characterizing its inherent dynamics. The Laplace transform of an impulse response gives the system's transfer function. It reveals how the system reacts to abrupt, short-duration disturbances.

Standard test signals are crucial for control system analysis due to several key reasons:

• Predictable and Reproducible Behavior: These signals (step, ramp, parabolic, impulse) have well-defined mathematical forms and known Laplace transforms. This predictability allows for consistent analysis and comparison of different system designs.
• Characterization of System Dynamics: The response of a system to these standard inputs reveals fundamental characteristics about its performance:
  • Transient Response: How quickly and smoothly the system reacts to a change (e.g., rise time, peak overshoot, settling time from a step input).
  • Steady-State Response: The accuracy of the system in reaching and maintaining a desired output over time (e.g., steady-state error for step, ramp, or parabolic inputs).
  • Stability: The system's inherent ability to return to equilibrium after a disturbance, which can be inferred from the transient behavior.
• Benchmarking and Comparison: Using standard inputs provides a common benchmark. Engineers can easily compare the performance of different control systems or different controllers for the same system by evaluating their responses to the same standard inputs.
• Design and Tuning: The specifications derived from standard test responses (like peak overshoot, settling time, steady-state error) are directly used in design objectives and controller tuning. For example, a controller might be designed to meet specific rise time and overshoot requirements for a step input.
• Mathematical Tractability: The simple mathematical forms of these signals make analytical solutions (e.g., using Laplace transforms) feasible, allowing for theoretical understanding and derivation of performance equations.
• Approximation of Real-World Inputs: While real-world inputs are complex, they can often be approximated or decomposed into combinations of these standard signals. For instance, an abrupt change can be modeled as a step, and a continuously varying input as a ramp or parabolic function.

Arbitrary inputs, while more realistic, are difficult to analyze mathematically, make system comparison challenging, and do not easily yield quantifiable performance metrics. Therefore, standard test signals serve as essential tools for systematic analysis, design, and performance evaluation in control systems.

A standard first-order system is characterized by a single energy storage element and has a transfer function of the form:

Where:

• is the Laplace transform of the output.
• is the Laplace transform of the input.
• is the DC gain of the system.
• is the time constant.

For a unit step input, the Laplace transform is .

The output is then:

To find the time response , we use partial fraction expansion:

Solving for :

Solving for :

Substituting and back into the partial fraction expansion:

Taking the inverse Laplace transform, we get the time response :

Definition of Time Constant ():

The time constant () of a first-order system is a measure of the speed of its response. It is defined as the time required for the system's output to reach approximately of its final steady-state value when subjected to a unit step input.

From the derived response :

• The final steady-state value is .
• At , the response is:
  

Significance of Time Constant ():

• Speed of Response: The time constant directly dictates how quickly a first-order system responds to an input change. A smaller means the system reaches its steady state faster, indicating a faster response.
• System Dynamics: It is an inherent property of the first-order system, determined by its physical parameters (e.g., resistance and capacitance in an RC circuit, mass and damping coefficient in a mechanical system).
• Settling Time: For practical purposes, the system is considered to have settled to its final value after approximately (within of final value) or (within of final value). For example, at , .
• Pole Location: For , the system has a single pole at . The time constant is the reciprocal of the absolute value of the pole's location. A pole further to the left in the s-plane implies a smaller time constant and a faster response.

A standard first-order system has a transfer function . The pole of this system is found by setting the denominator to zero: . This means a first-order system has a single real pole.

The location of this pole in the s-plane significantly affects the system's time response characteristics:

1. Pole on the Negative Real Axis (Stable System):

   • If , the pole is located at on the negative real axis. This indicates a stable system. The time response to a step input is an exponential rise to a final steady-state value, given by .
   • Effect of Distance from Origin:
     • Pole Far from the Origin (Large negative value, e.g., , so ): A pole located further to the left on the negative real axis corresponds to a smaller time constant (). A smaller time constant means the exponential term decays more rapidly, leading to a faster system response. The system reaches its steady state quickly.
     • Pole Close to the Origin (Small negative value, e.g., , so ): A pole located closer to the origin corresponds to a larger time constant (). A larger time constant means the exponential term decays slowly, leading to a slower system response. The system takes a longer time to reach its steady state.

2. Pole at the Origin (Marginally Stable System):

   • If , the pole is at . The system's transfer function becomes (an integrator). For a step input, the output would be a ramp, increasing indefinitely. This indicates a marginally stable or unstable system depending on the input; it cannot reach a steady-state for a step input.

3. Pole on the Positive Real Axis (Unstable System):

   • If , the pole is located at on the positive real axis. This indicates an unstable system. The exponential term would grow unboundedly with time, causing the system's output to diverge to infinity, even for a bounded input.

In summary, for a stable first-order system, the closer the pole is to the imaginary axis (origin), the slower the system's response. Conversely, the further the pole is into the left-half plane, the faster the system response. The pole's position directly defines the time constant, which governs the rate of exponential decay or growth.

The characteristic equation of a standard second-order system is , where is the natural frequency and is the damping ratio. The nature of the poles of this equation, and consequently the system's time response to a unit step input, depends critically on the value of . The poles are given by .

1. Undamped ():

   • Pole Locations: The poles are purely imaginary, . They lie on the imaginary axis of the s-plane.
   • Time Response: The system oscillates indefinitely at its natural frequency () without any decay. The oscillations do not diminish over time, and the system never settles to a steady-state value. There is no damping present.
   • Sketch (Description): The response would be a continuous sine wave oscillating around the final steady-state value, with constant amplitude.

2. Underdamped ():

   • Pole Locations: The poles are complex conjugate pairs with negative real parts, . They lie in the left-half of the s-plane.
   • Time Response: The system exhibits oscillatory behavior, but these oscillations gradually decay to zero. The response reaches its steady-state value after a transient period, typically with some overshoot. This is the most common and often desired response for practical control systems, offering a good balance between speed and stability.
   • Sketch (Description): The response would show oscillations that decrease in amplitude over time, eventually settling smoothly to the final value. It would typically have an initial overshoot.

3. Critically Damped ():

   • Pole Locations: The poles are two identical real poles, . They lie on the negative real axis of the s-plane.
   • Time Response: The system returns to the steady-state as quickly as possible without any oscillation or overshoot. It represents the fastest possible non-oscillatory response. This condition is often sought when oscillations are strictly undesirable.
   • Sketch (Description): The response would be a smooth, non-oscillatory curve rising to the final value without exceeding it. It reaches the steady-state faster than an overdamped system.

4. Overdamped ():

   • Pole Locations: The poles are two distinct real poles, . Both poles are distinct and lie on the negative real axis of the s-plane.
   • Time Response: The system returns to the steady-state without any oscillation, similar to critically damped, but it does so more slowly (sluggishly). The response is slower because the damping forces are excessively high, hindering the system's ability to respond quickly.
   • Sketch (Description): The response would be a slow, smooth, non-oscillatory curve rising to the final value without exceeding it. It is slower than a critically damped response.

The transfer function of a standard second-order system is:

For a unit step input, . So the output in the Laplace domain is:

For an underdamped system (), the quadratic term in the denominator can be written by completing the square:

Let be the damped natural frequency. Then:

Now, we perform partial fraction expansion:

To find , we use the cover-up method:

So, .
Multiply by and equate coefficients:

Comparing coefficients:

• For :
• For :

Substitute and back:

To facilitate inverse Laplace transform, we split the second term:

We know that and .

For the last term, we need in the numerator, so we multiply and divide by :

Now, recall that .

Taking the inverse Laplace transform term by term:

Factor out :

This expression can be further simplified using trigonometric identities. Let and (this defines an angle such that ). Then the bracketed term can be written as .

Therefore, the unit step response for an underdamped second-order system is:

where .

Time-domain specifications are quantitative measures used to characterize the transient response of a control system, particularly for underdamped second-order systems. They provide insight into how well a system responds to a sudden input.

a) Rise Time ():

• Definition: The time required for the response to rise from a small percentage (e.g., or ) to a large percentage (e.g., or ) of its final value. For underdamped systems, it is often defined as the time taken to rise from to of the final value for the first time.
• Significance: It indicates the speed of the system's initial response. A smaller rise time generally implies a faster system. However, a very small rise time might come at the cost of higher overshoot or oscillations.

b) Peak Time ():

• Definition: The time required for the response to reach the first peak of the overshoot after the step input is applied. This is the first time the derivative of the response is zero after .
• Expression (for unit step response):
• Significance: It indicates how quickly the system reaches its maximum deviation from the steady-state value. A shorter peak time means a quicker initial response, but it is often associated with a higher overshoot if other parameters are not tuned.

c) Maximum Overshoot ():

• Definition: The maximum peak value of the response curve measured from the unity (steady-state) value, expressed as a percentage of the steady-state value. It represents how much the response exceeds the final desired value.
• Expression (for unit step response):
• Significance: It represents the relative stability of the system. A smaller overshoot indicates better relative stability (less oscillation and less severe transients). Large overshoot can be undesirable as it might saturate actuators, damage components, or cause instability in certain applications. It is often a critical design constraint.

d) Settling Time ():

• Definition: The time required for the response curve to reach and stay within a specified small percentage (typically or ) of the final value. It indicates when the transient oscillations have essentially died out.
• Expressions (approximations):
  • For criterion:
  • For criterion:
• Significance: It indicates how long it takes for the transient oscillations to die out and for the system to settle to its steady-state within an acceptable error band. A smaller settling time implies a faster and more damped response, making the system reach its final desired state more quickly and with less lingering oscillation. It reflects the overall duration of the transient period.

The unit step response for an underdamped second-order system is given by:

where is the damped natural frequency, and (angle in the second quadrant for ). Equivalently, we often use and .

1. Peak Time ()

The peak time occurs at the first instant when the derivative of with respect to time is zero. The derivative is:

It's easier to differentiate the form directly.

Setting and factoring out (which is never zero for finite ):

Recall . Substitute this into the equation:

The terms cancel out:

Since and the term in parentheses is non-zero for , we must have .

This occurs when , where is an integer. The first peak occurs when (since for is the start of the step, not a peak).

Therefore, for peak time :

2. Maximum Overshoot ()

The maximum overshoot occurs at . We substitute into the step response equation:

Substitute :

We know that . Also, from the definition of , .

The maximum overshoot is defined as the maximum value of the response minus the final value (which is 1 for a unit step input), expressed as a percentage:

These derivations provide the fundamental relationships for peak time and maximum overshoot in terms of the system's natural frequency and damping ratio.

Definition of Steady-State Error ():

The steady-state error () is defined as the difference between the desired output (reference input) and the actual output of a control system as time approaches infinity (). In mathematical terms:

where is the input and is the output. It represents the accuracy of a control system in tracking or maintaining a desired setpoint once all transient effects have died out and the system has reached a stable equilibrium.

Importance in Control System Design:

The steady-state error is a crucial performance metric in control system design for several reasons:

1. Accuracy and Precision: A small steady-state error indicates high accuracy and precision in the system's ability to follow the input or maintain a setpoint. In many applications (e.g., robotics, aerospace, process control), high precision is paramount.
2. Performance Evaluation: Along with transient response specifications (like rise time, overshoot, settling time), steady-state error provides a comprehensive view of overall system performance. A system might have a good transient response but an unacceptable steady-state error, or vice-versa.
3. System Type and Input Matching: The value of the steady-state error depends heavily on the 'type' of the control system (number of integrators in the open-loop transfer function) and the nature of the input signal (step, ramp, parabolic). Designers must select a system type appropriate for the expected input to achieve desired steady-state accuracy.
4. Controller Design: Minimizing or eliminating steady-state error is a primary objective in controller design. Techniques like adding integral control (I-action) are specifically employed to reduce or eliminate steady-state errors for certain input types.
5. Economic and Safety Implications: In industrial processes, even small steady-state errors can lead to product quality issues, material waste, or safety hazards. Therefore, accurate steady-state performance is vital.

Derivation of General Expression for Steady-State Error (Unity Feedback System):

Consider a unity feedback control system, where the feedback signal is directly the output . The block diagram consists of an input , an error detector, an open-loop transfer function , and an output .

The error signal in the Laplace domain is given by:

The output is related to the error signal by the open-loop transfer function :

Substitute into the error equation:

Rearrange to solve for :

According to the Final Value Theorem, the steady-state value of a time function can be found from its Laplace transform by:

Substitute the expression for into the final value theorem:

This is the general expression for the steady-state error for a unity feedback system. It allows us to calculate given the input and the open-loop transfer function . This formula is fundamental for analyzing the steady-state accuracy of control systems.

The steady-state error () of a unity feedback control system is profoundly affected by both the nature of the input signal and the 'type' of the system. The system type is defined by the number of pure integrators (poles at the origin, ) in its open-loop transfer function . If , then is the system type.

The general formula for steady-state error is .

Let's analyze the effect for different input signals and system types:

Impact of Input Signal

1. Unit Step Input ():

   • .
   • This error constant is related to the Static Position Error Constant, . So, .
   • A system's ability to track a constant input without error depends on its type.

2. Unit Ramp Input ():

   • .
   • This error constant is related to the Static Velocity Error Constant, . So, .
   • A system's ability to track a linearly increasing input without error depends on its type.

3. Unit Parabolic Input ():

   • .
   • This error constant is related to the Static Acceleration Error Constant, . So, .
   • A system's ability to track a quadratically increasing input without error depends on its type.

Categorization of Steady-State Error based on System Type

Let the open-loop transfer function be written as , where is the system type.

In essence, for a given input, to achieve zero steady-state error, the system type must be equal to or greater than the order of the input. For example, to track a ramp input with zero steady-state error, the system must be Type 2 or higher.

Static error constants (, , ) are specific values derived from the open-loop transfer function of a unity feedback control system. They quantify the system's ability to minimize steady-state error for standard step, ramp, and parabolic inputs, respectively. They are essentially measures of the gain of the system at steady-state for different types of error.

1. Static Position Error Constant ():

• Definition: is defined for a unit step input ().
  
• Relationship with Steady-State Error:
  For a unit step input, the steady-state error is given by:
  
• Physical Significance: indicates the system's ability to maintain a constant position (output) when subjected to a constant input (step). A higher means a smaller steady-state error for a step input. For Type 0 systems, is a finite constant, resulting in a finite . For Type 1 or higher systems, , leading to for a step input. It measures the gain of the system at DC (zero frequency) for proportional errors.

2. Static Velocity Error Constant ():

• Definition: is defined for a unit ramp input ().
  
• Relationship with Steady-State Error:
  For a unit ramp input, the steady-state error is given by:
  
• Physical Significance: indicates the system's ability to follow a linearly increasing input (constant velocity) with minimal error. A higher means a smaller steady-state error for a ramp input. For Type 0 systems, , leading to . For Type 1 systems, is a finite constant, resulting in a finite . For Type 2 or higher systems, , leading to . It measures the gain of the system when tracking an input that is changing at a constant rate.

3. Static Acceleration Error Constant ():

• Definition: is defined for a unit parabolic input ().
  
• Relationship with Steady-State Error:
  For a unit parabolic input, the steady-state error is given by:
  
• Physical Significance: indicates the system's ability to follow an input that is accelerating at a constant rate. A higher means a smaller steady-state error for a parabolic input. For Type 0 or Type 1 systems, , leading to . For Type 2 systems, is a finite constant, resulting in a finite . For Type 3 or higher systems, , leading to . It measures the gain of the system when tracking an input that is accelerating.

In summary, these static error constants characterize the open-loop transfer function's behavior at low frequencies (or near ) and are crucial indicators of a system's steady-state tracking performance against common reference inputs. A designer often strives to achieve sufficiently large values of these constants to meet steady-state accuracy requirements for specific applications.

Given the open-loop transfer function of a unity feedback system:

where and .

1. Determine the Type of the System:

The type of a unity feedback system is determined by the number of poles at the origin () in its open-loop transfer function .

In the given , there is a term in the denominator, which means there is one pole at .

Therefore, the system is a Type 1 system.

2. Determine the Steady-State Error for a Unit Ramp Input:

For a unity feedback system, the steady-state error () for a unit ramp input (, so ) is given by:

where is the static velocity error constant.

is defined as:

Substitute the given :

Now, substitute :

Finally, calculate the steady-state error:

Thus, for the given system, the steady-state error for a unit ramp input is . This error will be a finite, non-zero value, which is characteristic of a Type 1 system tracking a ramp input. As increases, the steady-state error decreases, improving tracking accuracy.

Concept of Stability in Control Systems:

Stability is a fundamental concept in control systems, referring to the system's ability to maintain equilibrium or return to a desired state after being disturbed. A stable system will produce a bounded output for a bounded input, and if left undisturbed, its output will eventually return to zero (for zero input) or settle to a constant value. Conversely, an unstable system will produce an unbounded output, or its output will oscillate with increasing amplitude, even for a bounded or zero input.

Stability ensures that the system operates predictably and within acceptable limits, preventing damage, runaway behavior, or unsafe conditions. It is the most crucial requirement for any practical control system.

BIBO (Bounded-Input Bounded-Output) Stability:

BIBO stability is a specific and widely used definition of stability for linear time-invariant (LTI) systems. A system is said to be Bounded-Input Bounded-Output (BIBO) stable if and only if every bounded input produces a bounded output.

• Bounded Input: An input signal is bounded if there exists a finite positive number such that for all .
• Bounded Output: An output signal is bounded if there exists a finite positive number such that for all .

If even one bounded input can produce an unbounded output, the system is considered BIBO unstable.

Necessary and Sufficient Conditions for BIBO Stability:

For a linear time-invariant (LTI) system, the necessary and sufficient conditions for BIBO stability are related to the system's impulse response or its poles:

1. Condition in terms of Impulse Response ():
   A continuous-time LTI system is BIBO stable if and only if its impulse response is absolutely integrable.
   
   This means that the area under the magnitude of the impulse response must be finite.

2. Condition in terms of Poles of the Transfer Function ():
   For an LTI system represented by a rational transfer function (where and are polynomials in ) with no pole-zero cancellations in the Right-Half Plane (RHP) or on the imaginary axis, the system is BIBO stable if and only if all the poles of its closed-loop transfer function have negative real parts.

   • In the s-plane, this means all poles must lie strictly in the left-half plane (LHP).

   • No poles are allowed on the imaginary axis (-axis), and no poles are allowed in the right-half plane (RHP).

   • Implications:

     • If there are any poles in the RHP, the system is unstable.
     • If there are any simple (non-repeated) poles on the imaginary axis (e.g., ), the system is marginally stable (not BIBO stable in the strict sense, as a sinusoidal input at that frequency would produce an unbounded output due to resonance).
     • If there are any repeated poles on the imaginary axis (e.g., in the denominator), the system is unstable.

These conditions ensure that transient responses decay to zero and the system maintains predictable behavior under bounded inputs.

The stability of a control system is determined by the location of its closed-loop poles in the complex s-plane. Based on pole locations, systems can be categorized as absolutely stable, conditionally stable, or marginally stable:

1. Absolutely Stable System:

   • Definition: An absolutely stable system is one that remains stable for all valid ranges of its system parameters (e.g., gain ). Its stability does not depend on specific values of these parameters, as long as they are within their physically meaningful bounds.
   • Pole Location: All closed-loop poles lie strictly in the left-half of the s-plane (LHP). This means the real part of every pole is negative ().
   • Response: For any bounded input, the output will be bounded, and the transient response will decay to zero as . For zero input, the output returns to zero.
   • Example: A simple first-order system with transfer function is absolutely stable, as its pole is at .

2. Conditionally Stable System:

   • Definition: A conditionally stable system is one that is stable only for a certain restricted range of system parameters (e.g., gain ), but becomes unstable if the parameters move outside this specific range. The system might be stable for high gain, unstable for intermediate gain, and then stable again for low gain (or vice versa).
   • Pole Location: The closed-loop poles lie in the LHP only for specific, non-contiguous intervals of the system parameter. For other parameter values, poles may move into the right-half plane (RHP) or onto the imaginary axis.
   • Response: Its transient response will decay to zero only when the parameters are within the stable range. Outside this range, the response will be unbounded or oscillatory.
   • Example: A high-order system that might become unstable if the gain is too low or too high, indicating that its stability is 'conditional' on the specific operating point or parameter settings.

3. Marginally Stable System:

   • Definition: A marginally stable system is one that is on the boundary between stability and instability. It exhibits undamped oscillations for a bounded input. While it does not produce an unbounded output in response to all bounded inputs, it also does not return to a specific equilibrium point for specific inputs (e.g., an oscillatory input at the natural frequency).
   • Pole Location: The system has at least one pair of non-repeated (simple) poles on the imaginary axis () and no poles in the right-half plane. All other poles (if any) must be in the LHP.
   • Response: For a bounded input, the output might oscillate with constant amplitude (sustained oscillation) rather than decaying to zero. For a zero input, the output might continue to oscillate at a constant amplitude.
   • Example: An ideal LC circuit (undamped second-order system) is marginally stable. A control system that is at the edge of its stable gain range (as determined by Routh-Hurwitz) might exhibit marginal stability.

In summary, absolute stability is the most robust, conditional stability requires careful parameter tuning, and marginal stability represents an undesirable boundary condition that often precedes instability in practical systems.

Both absolute and relative stability are crucial aspects of analyzing control systems, but they address different facets of system behavior.

Absolute Stability

• Definition: Absolute stability refers to whether a system is stable or unstable in a binary (yes/no) sense. It determines if all transient responses eventually die out or if they grow unbounded. It's the most fundamental requirement for any practical control system.
• Assessment in s-plane:
  • A system is absolutely stable if and only if all its closed-loop poles lie strictly in the left-half of the s-plane (LHP). This means all poles must have negative real parts.
  • If any closed-loop pole lies in the right-half of the s-plane (RHP) or if there are repeated poles on the imaginary axis, the system is unstable.
  • If there are simple (non-repeated) poles on the imaginary axis and no poles in the RHP, the system is marginally stable, which is typically considered not absolutely stable in a strict sense for robust design.
  • Tools like the Routh-Hurwitz criterion and the Nyquist stability criterion are used to determine absolute stability without explicitly finding the pole locations.
• Concern: Simply whether the system will operate without oscillations growing to infinity.

Relative Stability

• Definition: Relative stability provides a quantitative measure of how stable a system is. It indicates how close the system is to becoming unstable and how quickly the transient response decays. It addresses the degree of stability and the robustness of the system to parameter variations.
• Assessment in s-plane:
  • Relative stability is assessed by considering the distance of the closed-loop poles from the imaginary axis in the s-plane.
  • Damping Ratio () and Damped Natural Frequency (): For second-order and dominant pole systems, the damping ratio () and natural frequency () are direct measures of relative stability. A higher damping ratio (closer to 1) means less oscillatory and more stable response. Poles farther to the left in the LHP imply faster decay of transients and thus better relative stability.
  • Distance from Imaginary Axis: Poles closer to the imaginary axis (smaller negative real part) indicate a less stable system with slower decaying transients or more pronounced oscillations. Poles farther to the left (larger negative real part) indicate better relative stability (faster decay).
  • Angle with Negative Real Axis: For complex conjugate poles (), the damping ratio , where is the angle between the pole vector from the origin and the negative real axis. Smaller (larger ) means better relative stability.
  • Gain Margin (GM) and Phase Margin (PM): These frequency-domain metrics (often derived from Bode or Nyquist plots) are direct measures of relative stability, indicating how much gain or phase can be added before instability occurs.
• Concern: How well the system performs, specifically how well it damps oscillations and its resilience to changes or disturbances.

In summary, absolute stability is a prerequisite for a usable control system, ensuring basic functionality. Relative stability then refines this, providing insights into the quality and robustness of that stable performance. A system can be absolutely stable but have poor relative stability if it is highly oscillatory or very close to the instability boundary.

The Routh-Hurwitz stability criterion is a powerful mathematical test used to determine the absolute stability of a linear time-invariant (LTI) system. It allows us to ascertain whether any roots (poles) of the system's characteristic equation lie in the right-half of the s-plane (RHP) or on the imaginary axis, without explicitly calculating the roots. This criterion applies to polynomials with real coefficients.

Statement of the Routh-Hurwitz Criterion:

A linear time-invariant system is absolutely stable if and only if two conditions are met concerning the coefficients of its characteristic polynomial:

1. All coefficients of the characteristic polynomial must be positive and non-zero. If any coefficient is zero or negative, the system is either unstable or marginally stable, and further analysis with the Routh array is not strictly necessary to conclude instability, though constructing the array can confirm pole locations.

2. All the elements in the first column of the Routh array must be positive and non-zero. If all coefficients are positive, but any element in the first column of the Routh array is zero or negative, then the system is unstable or marginally stable. The number of sign changes in the first column indicates the number of roots in the RHP.

Conditions for System Stability (Summary):

Given the characteristic equation of a system as , the system is stable if and only if the following conditions are satisfied:

• Condition 1 (Necessary Condition):

  • All coefficients must be present (non-zero).
  • All coefficients must have the same sign (typically, all positive).
  • If this condition is violated, the system is unstable (or marginally stable if some coefficients are zero). No need to proceed to Routh array if this fails.

• Condition 2 (Necessary and Sufficient Condition):

  • All elements in the first column of the Routh array, constructed from the coefficients of the characteristic equation, must be positive (i.e., greater than zero).
  • If any element in the first column is zero, or if there is a sign change, the system is not stable. The number of sign changes in the first column corresponds to the number of roots located in the right-half of the s-plane, indicating instability.

The Routh-Hurwitz criterion is not only useful for determining the absolute stability of a system for fixed parameters but also for finding the range of an adjustable system parameter (commonly a gain ) for which the system remains stable. This is critical in control system design, as it allows engineers to select an appropriate gain that ensures stable operation.

Steps to Determine the Range of a Parameter for Stability using Routh-Hurwitz:

1. Formulate the Closed-Loop Characteristic Equation:

   • Start with the closed-loop transfer function. For a unity feedback system, the characteristic equation is . If contains the parameter , the characteristic equation will be a polynomial in whose coefficients are functions of .
   • Let the characteristic equation be , where are expressions involving .

2. Apply the Necessary Condition for Stability:

   • All coefficients () of the characteristic polynomial must be positive. Set each coefficient that depends on to be greater than zero ().
   • Solve these inequalities to obtain an initial range for . This step often provides a lower bound for (e.g., ).

3. Construct the Routh Array:

   • Construct the Routh array using the coefficients of the characteristic polynomial. The elements in the array will also be expressions involving . The first two rows are directly from the polynomial coefficients.
   • Subsequent elements are calculated recursively based on the elements in the two preceding rows.
     
     Where , etc.

4. Apply the Sufficient Condition for Stability:

   • All elements in the first column of the Routh array must be positive. Set each expression in the first column (that depends on ) to be greater than zero.
   • Solve these inequalities to obtain additional constraints or ranges for . This step typically provides an upper bound for (e.g., ) and may refine the lower bound.

5. Determine the Overall Stable Range for K:

   • The stable range for is the intersection of all inequalities obtained from Steps 2 and 4. This will yield a specific interval, such as , for which the system is absolutely stable.

Example: If Step 2 yields and Step 4 yields , then the overall stable range for is .

By following these steps, engineers can systematically determine the allowable parameter variations that maintain system stability, which is essential for robust and reliable control system operation.

The Routh-Hurwitz criterion relies on all elements in the first column of the Routh array being non-zero. A special case arises when the first element of a row (say, the row) is zero, but at least one other element in that same row is non-zero. This situation typically indicates the presence of poles on the imaginary axis or in the right-half plane, but the standard calculation formula for the next row would involve division by zero.

Procedure to Handle the Special Case (Zero First Element):

When the first element of a row in the Routh array is zero (), but not all elements in that row are zero, the following steps are taken:

1. Replace the Zero Element with a Small Positive Number (): Substitute a very small positive number, denoted by (where ), for the zero first element in that row.

2. Continue Routh Array Construction: Proceed with the calculation of the subsequent rows of the Routh array as usual, treating as a small positive number. The elements in the following rows will then be functions of .

3. Analyze the Signs as : After completing the array, examine the signs of the elements in the first column by taking the limit as . If there are any sign changes in the first column, the system is unstable. The number of sign changes indicates the number of roots in the RHP.

   • If, upon taking the limit, any term becomes negative, it implies a sign change, and thus instability.
   • If all terms remain positive, the system might be stable, but this case usually points to instability due to poles on the imaginary axis or RHP.

Illustrative Example:

Consider a characteristic equation:

Construct the Routh array:

Calculate :

Here, the first element of the row is zero (), but the rest of the row is not entirely zero (since , but this calculation would use the term after '2' which is 0, so is 0).
Let's apply the procedure by replacing with :

Calculate :

Now, examine the first column as :

• (positive)
• (positive)
• (positive)
• (positive)

In this specific example, all elements in the first column are positive. This indicates that there are no roots in the RHP. When this special case (zero in the first column, but not a full row of zeros) leads to all positive first column elements, it implies that there is a pair of complex conjugate roots on the imaginary axis, making the system marginally stable. In fact, for , the roots are and . The presence of confirms marginal stability.

If, for example, had turned out to be and was very small, would approach , which is positive. But if it was , it would approach , which is negative, indicating instability due to a sign change.

The Routh-Hurwitz criterion encounters another special case when an entire row of zeros appears in the Routh array. This scenario has significant implications for system stability and requires a modified procedure to continue the analysis.

What an Entire Row of Zeros Indicates:

When an entire row of zeros appears in the Routh array, it signifies that there are pairs of roots that are symmetrically located with respect to the origin of the s-plane. These pairs can be:

• Purely Imaginary Conjugate Pairs: Roots like .
• Real and Equal Magnitude, Opposite Sign Pairs: Roots like .
• Complex Conjugate Pairs with Real and Imaginary Symmetry: Roots like and .

The presence of an entire row of zeros immediately implies that the system is either marginally stable or unstable, because it means there are roots on the imaginary axis or in the right-half plane, or both. Specifically, it means the characteristic equation has factors that are symmetric with respect to the origin.

Procedure to Handle the Special Case (Entire Row of Zeros):

When an entire row of zeros is encountered (let's say row is all zeros), the following steps are performed:

1. Form an Auxiliary Polynomial (): Construct an auxiliary polynomial using the elements of the row immediately preceding the row of zeros. The highest power of for this polynomial corresponds to the power of of the row preceding the row of zeros. The coefficients of the auxiliary polynomial are the elements of that preceding row, taken alternately.

   • For example, if the row is all zeros, the auxiliary polynomial is formed from the row: .

2. Differentiate the Auxiliary Polynomial: Differentiate the auxiliary polynomial with respect to .
   

3. Replace the Zero Row: Replace the row of zeros with the coefficients of the differentiated auxiliary polynomial. These new coefficients become the elements of the first column for the row that was all zeros.

4. Continue Routh Array Construction: Continue building the rest of the Routh array using these new coefficients.

5. Analyze Stability:

   • Roots of the Auxiliary Polynomial: The roots of the auxiliary polynomial are also roots of the original characteristic equation. These roots represent the symmetrically located poles that caused the row of zeros.
   • Routh Array Analysis: Continue the Routh-Hurwitz criterion with the modified array. The number of sign changes in the first column of the modified array (starting from the original top of the array) indicates the number of roots in the Right-Half Plane (RHP). If there are no sign changes after modifying the array, all the remaining roots are in the LHP, and the roots of the auxiliary polynomial are on the imaginary axis (marginally stable).

Example:

Consider a characteristic equation:

Initial Routh Array:

An entire row of zeros (the row) is encountered. This means there are roots symmetric about the origin.

Step 1: Form Auxiliary Polynomial from the row ():

Step 2: Differentiate Auxiliary Polynomial:

Step 3: Replace Zero Row: The coefficients of are $8, 8, 0$. Replace the row with these coefficients.

Modified Routh Array:

- another zero in the first column here! This means we have another zero in the first element of row. Since it's only a single zero, we would replace with and continue, but usually, this pattern indicates repeated roots on imaginary axis. In this specific case, the auxiliary polynomial , implies roots twice. So the system is unstable due to repeated poles on the imaginary axis.

By following this procedure, the Routh-Hurwitz criterion can still be used to determine the exact number of RHP roots and characterize the stability even in the presence of symmetrically located poles.

Given a unity feedback system with an open-loop transfer function .

a) Determine the range of K for which the system is stable using the Routh-Hurwitz criterion.

Step 1: Formulate the Closed-Loop Characteristic Equation.
For a unity feedback system, the characteristic equation is .

Multiply by the denominator:

Expand the denominator:

Step 2: Apply the Necessary Condition for Stability.
All coefficients of the characteristic equation must be positive.

• Coefficient of : (Positive)
• Coefficient of : (Positive)
• Coefficient of : (Positive)
• Coefficient of : (Must be Positive)

From this, we establish the first condition: .

Step 3: Construct the Routh Array.

Using the coefficients :

Calculate :

Calculate :

Step 4: Apply the Sufficient Condition for Stability.
All elements in the first column of the Routh array must be positive.

1. (Satisfied)
2. (Satisfied)
4. (This matches the necessary condition from Step 2)

Step 5: Determine the Overall Stable Range for K.
Combining all conditions ( and ), the range of for which the system is stable is .

b) For the stable system, find the steady-state error for a unit ramp input in terms of K.

For a unity feedback system, the steady-state error () for a unit ramp input (, so ) is given by:

where is the static velocity error constant.

is defined as:

Given :

Substitute :

Therefore, the steady-state error for a unit ramp input is:

For the system to be stable, must be in the range . Within this stable range, the steady-state error for a unit ramp input is . As increases within the stable range, the steady-state error decreases, indicating improved tracking accuracy.