State Feedback, Observers and Servo Design

Introduction and Context

Classical control design commonly uses transfer functions and output feedback. These methods are powerful for single-input single-output systems, but they become less convenient when a system has several interacting variables, multiple actuators, or internal states that are not directly measured. State-space methods address these limitations by representing the plant through its internal state variables and by designing feedback directly in the state space.

State feedback permits the designer to assign the closed-loop poles of a linear time-invariant system, subject to controllability constraints. Because not every state is normally measured, an observer can estimate unavailable states from known inputs and measured outputs. Combining an observer with state feedback produces an observer-based compensator. Integral action extends this framework to achieve accurate tracking in the presence of constant disturbances and model uncertainty. The separation principle allows controller and observer dynamics to be designed independently under appropriate assumptions.

These ideas are central to modern control applications, including aircraft flight control, robotics, process control, automotive systems, power converters, and mechatronics.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Formulate state-space models suitable for state-feedback design.

2. Test controllability and use pole placement for state-feedback synthesis.

3. Design reference precompensators and integral controllers for servo operation.

4. Test observability and construct full- and reduced-order observers.

5. Combine observers and controllers while explaining the separation principle.

6. Evaluate practical limitations such as noise, saturation, uncertainty, and sampling.

7. Select an appropriate architecture for an engineering application.

1. State-Space Foundation

Consider the continuous-time linear time-invariant model

\[

\dot{x}(t)=Ax(t)+Bu(t), \qquad y(t)=Cx(t)+Du(t),

\]

where \(x\in\mathbb{R}^{n}\) is the state vector, \(u\in\mathbb{R}^{m}\) is the input, and \(y\in\mathbb{R}^{p}\) is the measured output. The matrices have dimensions \(A\in\mathbb{R}^{n\times n}\), \(B\in\mathbb{R}^{n\times m}\), \(C\in\mathbb{R}^{p\times n}\), and \(D\in\mathbb{R}^{p\times m}\). The model assumes linearity, time invariance, and a sufficiently accurate representation around an operating point.

A state is a minimum set of variables that, together with future inputs, determines future system behavior. A state need not be a directly physical quantity; it may be a mathematical coordinate. Different state transformations can describe the same input-output behavior.

For zero initial conditions, the transfer matrix is

\[

G(s)=C(sI-A)^{-1}B+D,

\]

provided \(sI-A\) is invertible. State-feedback design uses the internal model rather than only this input-output expression.

2. State Feedback and Pole Placement

2.1 Basic feedback law

For a single-input system, the state-feedback law is commonly written as

\[

u(t)=-Kx(t)+Nr(t),

\]

where \(K\in\mathbb{R}^{1\times n}\) is the feedback-gain vector, \(r(t)\) is the reference, and \(N\) is a reference precompensator. With \(D=0\), the closed-loop state matrix is

\[

A_{cl}=A-BK.

\]

The closed-loop poles are the eigenvalues of \(A-BK\). For a continuous-time design, desired poles normally lie in the open left half-plane. Faster poles generally produce faster response, but excessive speed can increase actuator demand, sensitivity to unmodelled dynamics, and measurement-noise amplification.

For a multi-input system, \(K\in\mathbb{R}^{m\times n}\). Pole placement may not determine a unique gain when there are multiple inputs; different gains can produce the same eigenvalues, and additional criteria such as control effort or robustness can be used to select among them.

2.2 Controllability

The pair \((A,B)\) is controllable if any initial state can be transferred to any final state in finite time using an admissible input. The Kalman controllability matrix is

\[

\mathcal{C}=\begin{bmatrix}B&AB&A^2B&\cdots&A^{n-1}B\end{bmatrix}.

\]

The system is completely controllable if

\[

\operatorname{rank}(\mathcal{C})=n.

\]

For a controllable single-input system, arbitrary closed-loop pole locations can be assigned through suitable state feedback. If the system is not completely controllable, only the controllable modes can be moved. An equivalent eigenvalue-based test is the Popov–Belevitch–Hautus (PBH) test:

\[

\operatorname{rank}\begin{bmatrix}\lambda I-A & B\end{bmatrix}=n

\]

for every eigenvalue \(\lambda\) of \(A\). Stabilizability is weaker than controllability: every uncontrollable mode must already be stable for continuous-time stabilization to be possible.

2.3 Ackermann’s formula

For a controllable single-input system, let the desired monic characteristic polynomial be

\[

p_d(s)=s^n+a_{n-1}s^{n-1}+\cdots+a_1s+a_0.

\]

Define the matrix polynomial

\[

p_d(A)=A^n+a_{n-1}A^{n-1}+\cdots+a_1A+a_0I.

\]

Ackermann’s formula gives

\[

K=\begin{bmatrix}0&0&\cdots&1\end{bmatrix}\mathcal{C}^{-1}p_d(A).

\]

This formula is theoretically useful and convenient for small systems. In numerical software, robust pole-placement algorithms are generally preferred because direct matrix inversion can be ill-conditioned.

2.4 Worked example: state-feedback pole placement

Consider

\[

A=\begin{bmatrix}0&1\\-2&-3\end{bmatrix},\qquad

B=\begin{bmatrix}0\\1\end{bmatrix}.

\]

The controllability matrix is

\[

\mathcal C=\begin{bmatrix}0&1\\1&-3\end{bmatrix},

\]

whose determinant is \(-1\); hence the system is controllable. Let \(K=[k_1\ k_2]\). Then

\[

A-BK=

\begin{bmatrix}0&1\\-2-k_1&-3-k_2\end{bmatrix}.

\]

Its characteristic polynomial is

\[

s^2+(3+k_2)s+(2+k_1).

\]

Suppose the desired poles are \(-4\) and \(-5\). The desired polynomial is

\[

(s+4)(s+5)=s^2+9s+20.

\]

Equating coefficients gives

\[

3+k_2=9,\qquad 2+k_1=20,

\]

so

\[

K=[18\quad 6].

\]

The feedback law is therefore \(u=-18x_1-6x_2\), apart from any reference or integral term.

3. Reference Tracking and Servo Design

Regulation means driving a system toward an equilibrium, often the origin. Servo design means tracking a commanded reference. A feedback law designed only to stabilize the origin does not necessarily provide zero steady-state tracking error for a nonzero reference.

3.1 Reference precompensation

For the law

\[

u=-Kx+Nr,

\]

assume the closed loop is stable, \(r\) is constant, and \(D=0\). At steady state,

\[

0=(A-BK)x_{ss}+BNr,

\qquad y_{ss}=Cx_{ss}.

\]

If \(A-BK\) is nonsingular, then

\[

x_{ss}=-(A-BK)^{-1}BNr,

\]

and exact unit steady-state tracking requires

\[

N=-\left[C(A-BK)^{-1}B\right]^{-1},

\]

provided the scalar inverse exists. For a square multi-input/multi-output system, the inverse is replaced by an appropriate nonsingular matrix inverse. If \(D\neq0\), the steady-state gain is

\[

C[-(A-BK)^{-1}B]N+D N,

\]

and \(N\) must be selected accordingly. Precompensation depends on the model and does not automatically reject constant disturbances or eliminate errors caused by parameter uncertainty.

3.2 Integral control

A more robust servo approach introduces the integral of tracking error. For reference \(r\) and output \(y\), define

\[

\dot{x}_I=r-y.

\]

For state feedback with integral action,

\[

u=-Kx-K_Ix_I.

\]

For a constant reference and a stable augmented closed loop, \(\dot{x}_I\to0\), which implies \(r-y\to0\). Thus integral action provides zero steady-state error for constant references and constant matched disturbances, subject to actuator feasibility and model assumptions.

For regulation about the origin, \(r=0\), the augmented dynamics are

\[

\begin{bmatrix}\dot{x}\\\dot{x}_I\end{bmatrix}

=\begin{bmatrix}A&0\\-C&0\end{bmatrix}

\begin{bmatrix}x\\x_I\end{bmatrix}

+\begin{bmatrix}B\\0\end{bmatrix}u.

\]

The augmented pair must be controllable, or at least stabilizable, for pole placement. Integral augmentation can fail if the plant has invariant zeros at the origin or if the actuator cannot provide the required steady input.

3.3 Integral windup and anti-windup

When an actuator saturates, the commanded input cannot equal the calculated input. The integrator may continue accumulating error, producing overshoot and long recovery times after the actuator leaves saturation. This is integral windup. Common remedies include conditional integration, integrator clamping, and back-calculation. Anti-windup parameters should be selected with regard to actuator dynamics rather than treated as arbitrary tuning constants.

4. Observability and State Observers

4.1 Observability

The pair \((A,C)\) is observable if the initial state can be uniquely reconstructed from the input and output over a finite observation interval. The observability matrix is

\[

\mathcal O=\begin{bmatrix}C\\CA\\CA^2\\\vdots\\CA^{n-1}\end{bmatrix}.

\]

Complete observability requires

\[

\operatorname{rank}(\mathcal O)=n.

\]

The PBH observability test is

\[

\operatorname{rank}\begin{bmatrix}\lambda I-A\\C\end{bmatrix}=n

\]

for every eigenvalue \(\lambda\) of \(A\). Detectability is the weaker condition that every unobservable mode is stable. Detectability is sufficient for designing an observer whose estimation error converges to zero.

4.2 Full-order Luenberger observer

A full-order observer estimates all \(n\) state variables:

\[

\dot{\hat{x}}=A\hat{x}+Bu+L(y-C\hat{x}),

\]

where \(L\in\mathbb{R}^{n\times p}\) is the observer gain. Define the estimation error \(e=x-\hat{x}\). For the model with the same input matrix and no direct-feedthrough complication,

\[

\dot e=(A-LC)e.

\]

The observer poles are the eigenvalues of \(A-LC\). If \((A,C)\) is observable, these poles may be assigned arbitrarily in theory. In practice, observer poles are often selected faster than controller poles, but making them excessively fast amplifies sensor noise and increases sensitivity to sampling, quantization, and model uncertainty.

Observer design is dual to state-feedback design. Specifically, observer gain placement for \((A,C)\) can be computed as state-feedback placement for the transposed pair \((A^T,C^T)\): if \(K_o\) is designed so that \(A^T-C^TK_o\) has the desired poles, then

\[

L=K_o^T.

\]

4.3 Worked observer example

Use the plant from the earlier example and suppose only the first state is measured:

\[

C=\begin{bmatrix}1&0\end{bmatrix}.

\]

The observability matrix is

\[

\mathcal O=\begin{bmatrix}1&0\\0&1\end{bmatrix},

\]

so the system is observable. Let \(L=[l_1\ l_2]^T\). Then

\[

A-LC=\begin{bmatrix}-l_1&1\\-2-l_2&-3\end{bmatrix}.

\]

Its characteristic polynomial is

\[

s^2+(l_1+3)s+(3l_1+2+l_2).

\]

Choose observer poles at \(-8\) and \(-9\), giving

\[

(s+8)(s+9)=s^2+17s+72.

\]

Equating coefficients gives

\[

l_1+3=17 \Rightarrow l_1=14,

\]

and

\[

3(14)+2+l_2=72 \Rightarrow l_2=28.

\]

Thus

\[

L=\begin{bmatrix}14\\28\end{bmatrix}.

\]

The observer error decays according to poles at \(-8\) and \(-9\), assuming the plant model and measurement are sufficiently accurate.

5. Reduced-Order Observers

A full-order observer estimates both measured and unmeasured states. If some state components are measured directly, only the unmeasured components need to be estimated. A reduced-order observer can therefore have order \(n-p\) in the ideal case of \(p\) independent measured state combinations.

After a suitable coordinate transformation, write

\[

z=T x=\begin{bmatrix}z_m\\z_u\end{bmatrix},

\qquad y=z_m,

\]

where \(z_m\) contains measured coordinates and \(z_u\) contains unmeasured coordinates. The transformed model can be partitioned as

\[

\begin{bmatrix}\dot z_m\\\dot z_u\end{bmatrix}

=\begin{bmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{bmatrix}

\begin{bmatrix}z_m\\z_u\end{bmatrix}

+\begin{bmatrix}B_1\\B_2\end{bmatrix}u.

\]

Because \(z_m=y\) is known, an observer for \(z_u\) can be constructed, for example,

\[

\dot{\hat z}_u=A_{22}\hat z_u+A_{21}y+B_2u+L_r( y-z_m),

\]

but this expression is not useful when \(z_m=y\) exactly, since the innovation term shown is zero. In practice, the reduced-order observer is derived using a coordinate transformation and an auxiliary variable so that the measured-state derivative is not required directly. One standard form is obtained by defining

\[

\eta=z_u-L_r z_m,

\]

and selecting \(L_r\) so that the error dynamics have matrix \(A_{22}-L_rA_{12}\). The resulting implementation uses \(y\), \(u\), and \(\hat\eta\), and reconstructs \(\hat z_u=\hat\eta+L_r y\). The exact equations depend on the chosen partition and must be derived consistently.

Reduced-order observers lower computational burden and may reduce noise injection, but they require reliable measurement of the selected state combinations and careful coordinate transformations. They are especially useful in embedded systems with constrained processing resources.

6. Observer-Based State Feedback and the Separation Principle

When the complete state is unavailable, replace \(x\) in the control law with its estimate:

\[

u=-K\hat{x}+Nr.

\]

For the plant and observer,

\[

\dot{x}=Ax-BK\hat{x}+BNr,

\]

\[

\dot{\hat{x}}=A\hat{x}+Bu+L(y-C\hat{x}).

\]

Using the estimation error \(e=x-\hat{x}\), the combined dynamics in coordinates \([x^T\ e^T]^T\) have a block-triangular system matrix:

\[

\begin{bmatrix}

A-BK & BK\\

0 & A-LC

\end{bmatrix}.

\]

Consequently, the closed-loop eigenvalues are the union of the eigenvalues of \(A-BK\) and \(A-LC\). This is the separation principle. It states that, for a linear time-invariant plant with appropriate controllability and observability properties, controller and observer gains can be designed independently, and the combined system is stable if both component matrices are stable.

The principle does not mean that controller and observer can always be tuned independently in practice. Noise, nonlinearities, saturation, uncertain parameters, finite sampling, delays, and unmodelled dynamics couple their practical performance. It is a structural result, not a guarantee of robustness.

7. Digital Implementation and Practical Design

Most modern controllers are implemented digitally. A continuous-time design must be sampled and discretized, or designed directly using

\[

x_{k+1}=A_dx_k+B_du_k,

\qquad y_k=C_dx_k+D_du_k.

\]

For zero-order-hold sampling with period \(T_s\),

\[

A_d=e^{AT_s},

\qquad B_d=\int_0^{T_s}e^{A\tau}B\,d\tau.

\]

Discrete-time state feedback is \(u_k=-K_dx_k+N_dr_k\), with closed-loop matrix \(A_d-B_dK_d\). Stability requires all discrete-time poles to lie strictly inside the unit circle. Continuous-time pole locations do not transfer directly to discrete time; a continuous pole \(s_i\) maps under exact sampling to \(z_i=e^{s_iT_s}\).

Important implementation practices include:

• scale states and inputs so numerical conditioning is acceptable;
• verify controllability and observability numerically, not only symbolically;
• account for sensor bias and actuator limits;
• filter noisy measurements without introducing unacceptable delay;
• use anti-windup with integral control;
• validate the design under parameter variation and disturbances;
• test initialization and estimator transients;
• include computation delay in the design model when it is significant.

Quadratic optimal control, such as linear-quadratic regulation, is often used instead of arbitrary pole placement when a systematic trade-off between state deviation and control effort is desired. Kalman filtering provides a statistically motivated observer when process and measurement noise can be characterized. A linear-quadratic-Gaussian controller combines these ideas, while retaining a separation result under standard assumptions.

8. Applied Case: Servo Control of a Two-Axis Positioning Stage

Background

Consider an illustrative two-axis positioning stage used to place electronic components. Each axis has motor position and velocity as dominant states. Encoders measure position, while velocity measurement is either unavailable or too noisy for direct use.

Problem

A conventional proportional controller produces acceptable motion near the nominal operating point but exhibits coupling-related overshoot and a persistent position error under a constant load torque. The design must achieve accurate reference tracking without requiring a velocity sensor.

Analysis

A linearized model is assembled in the form

\[

\dot{x}=Ax+Bu,\qquad y=Cx,

\]

where \(x\) includes positions and velocities for both axes. Controllability analysis confirms that the motor voltages can influence all relevant modes. Observability analysis confirms that encoder positions make the velocity states detectable. Because a constant load produces a nonzero steady-state error, a reference prefilter alone is not sufficient.

Approach and solution

First, state-feedback gains are selected to place the dominant closed-loop poles with adequate damping. Integral states are then added for the two position errors, and the augmented system is checked for controllability. The integral gains are selected together with the state gains rather than appended without analysis.

A full-order observer estimates velocities and compensates for the fact that position measurements are noisy. Observer poles are placed faster than the dominant servo poles but not so fast that encoder quantization is amplified. An anti-windup mechanism limits integral accumulation when the motor voltage saturates during large moves.

The implementation is discretized using the actual sampling period. Performance is evaluated using settling time, peak overshoot, steady-state error, RMS control voltage, and sensitivity to changes in payload.

Outcome

The resulting architecture provides zero steady-state position error for constant commands and constant load disturbances within actuator capability. The observer removes the need for velocity sensors, while the pole locations provide a predictable transient response. Under large commands, anti-windup significantly reduces recovery time after voltage saturation.

Learning points

1. State feedback is valuable when dynamic interactions are important.

2. Integral action addresses constant tracking errors but introduces windup risk.

3. An observer can reduce sensor count, but its bandwidth must respect noise and sampling limitations.

4. The separation principle supports modular design, whereas practical validation must still consider nonlinear and hardware effects.

9. Challenges, Limitations and Design Governance

State-space designs are only as reliable as the models and measurements supporting them. Poorly scaled states can cause numerical problems. Nearly uncontrollable or nearly unobservable modes may technically satisfy rank tests but produce very large gains and fragile implementations. Pole placement can hide poor robustness because eigenvalue assignment alone does not optimize sensitivity.

Model uncertainty is particularly important when observer poles are made very fast. High-gain observers can respond strongly to sensor noise and unmodelled high-frequency dynamics. Integral action can destabilize systems with delays or nonminimum-phase behavior if introduced without checking the augmented dynamics.

Engineering governance requires traceable assumptions, documented operating ranges, independent simulation checks, hardware-in-the-loop testing, and protection against unsafe commands. In safety-critical systems, a state estimate should not be treated as an unquestionable measurement; fault detection, sensor plausibility checks, redundancy, and safe fallback modes may be required. AI-based estimators can complement model-based observers in highly nonlinear systems, but they introduce validation, explainability, cybersecurity, and distribution-shift concerns. They should not replace stability analysis and safety mechanisms without compelling evidence.

Best-Practice Design Workflow

1. Define the operating point, performance requirements, constraints, and measured variables.

2. Derive and validate the state-space model, including units and sign conventions.

3. Test controllability, stabilizability, observability, and detectability.

4. Select controller poles or an optimization criterion using settling-time, damping, and actuator requirements.

5. Add a precompensator or integral states according to the tracking and disturbance-rejection objective.

6. Design the observer using realistic noise and uncertainty assumptions.

7. Form the observer-based compensator and verify separation-theorem conditions.

8. Discretize using the actual sampling and computation architecture.

9. Simulate reference changes, disturbances, saturation, sensor noise, initialization errors, and parameter variation.

10. Validate experimentally with safety limits, logging, and a staged commissioning plan.

Chapter Summary

State feedback uses the internal state vector to shape closed-loop dynamics. For a controllable pair \((A,B)\), the gain \(K\) in \(u=-Kx\) can assign the closed-loop poles of \(A-BK\). Controllability and stabilizability determine whether such design objectives are feasible. Reference precompensation can provide nominal steady-state tracking, whereas integral augmentation provides stronger rejection of constant tracking errors and disturbances, subject to stability and actuator constraints.

Observability and detectability characterize whether internal states can be reconstructed from measured outputs. A full-order Luenberger observer estimates every state, while a reduced-order observer estimates only unmeasured states. Replacing the true state by its estimate produces an observer-based compensator. The separation principle shows that, under linear and appropriate controllability-observability conditions, controller and observer poles appear independently in the combined dynamics. Practical design must nevertheless account for noise, uncertainty, saturation, sampling, delays, scaling, and safety governance.

Key Terms

State: A minimum set of variables that determines future system behavior with the input.

State-space model: A first-order matrix description of system dynamics and output generation.

State feedback: Control based directly on measured or estimated state variables.

Pole placement: Selection of feedback gains to assign specified closed-loop eigenvalues.

Controllability: Ability to move the state through the available inputs.

Stabilizability: Ability to stabilize all unstable modes through control.

Observability: Ability to reconstruct the state from inputs and measured outputs.

Detectability: Condition in which every unobservable mode is stable.

Luenberger observer: A model-based state estimator corrected by output error.

Full-order observer: An observer estimating all plant states.

Reduced-order observer: An observer estimating only unmeasured state components.

Servo system: A control system designed to track a reference command.

Reference precompensator: A gain used to achieve nominal steady-state reference tracking.

Integral action: Integration of tracking error to remove constant steady-state error.

Integral windup: Excessive integrator accumulation during actuator saturation.

Separation principle: Result allowing controller and observer poles to be designed independently under stated conditions.

Observer-based compensator: A controller that uses estimated states in place of unavailable states.

PBH test: Rank-based controllability or observability test evaluated at system eigenvalues.

Pole sensitivity: Dependence of closed-loop poles and performance on parameter or modelling errors.

Anti-windup: A mechanism that limits or corrects integrator accumulation during saturation.

Review Questions

Short Answer Questions

1. What is the difference between controllability and stabilizability?

2. Why does state feedback require an estimate when all states are not measured?

3. State the rank condition for complete observability.

4. What is the main purpose of integral action in a servo system?

5. Why can excessively fast observer poles be undesirable?

Long Answer Questions

1. Derive the closed-loop state matrix for the feedback law \(u=-Kx+Nr\), and explain how pole placement is performed.

2. Explain the construction and error dynamics of a full-order Luenberger observer.

3. Compare reference precompensation with integral control for reference tracking and disturbance rejection.

4. Explain the separation principle and identify the assumptions under which it applies.

5. Describe a complete engineering workflow for designing a digital observer-based servo controller.

Analytical / Application Questions

1. For \(A=\begin{bmatrix}0&1\\-2&-3\end{bmatrix}\), \(B=\begin{bmatrix}0\\1\end{bmatrix}\), and desired poles at \(-4,-5\), calculate the state-feedback gain \(K\).

2. A plant has accurate position measurements but no velocity sensor. Explain how you would determine whether a reduced-order or full-order observer is appropriate, and identify the tests required before implementation.

3. An integral servo controller tracks small commands accurately but recovers slowly after large commands. Diagnose the likely mechanism and propose two design or implementation remedies.

Multiple Choice Questions with Answers

Question 1

For the continuous-time pair (A,B), complete controllability requires which condition?

A. rank(O)=n

B. rank(C)=n

C. rank(𝒞)=n

D. All eigenvalues of A equal zero

Correct Answer: C. rank(𝒞)=n

Question 2

For u=-Kx in a plant with state matrix A and input matrix B, the closed-loop state matrix is:

A. A+BK

B. A-BK

C. B-AK

D. A-KB

Correct Answer: B. A-BK

Question 3

The desired polynomial for closed-loop poles at -4 and -5 is:

A. s²+ s+20

B. s²+9s+20

C. s²-9s+20

D. s²+20s+9

Correct Answer: B. s²+9s+20

Question 4

For the system in the chapter example, A=[[0,1],[-2,-3]] and B=[[0],[1]], what gain places the poles at -4 and -5?

A. [4, 5]

B. [18, 6]

C. [6, 18]

D. [-18, -6]

Correct Answer: B. [18, 6]

Question 5

The estimation-error dynamics of a Luenberger observer are:

A. ė=(A+LC)e

B. ė=(A-BK)e

C. ė=(A-LC)e

D. ė=(C-A L)e

Correct Answer: C. ė=(A-LC)e

Question 6

Complete observability of (A,C) requires:

A. rank(𝒪)=n

B. rank(𝒞)=m

C. det(A)=1

D. C=0

Correct Answer: A. rank(𝒪)=n

Question 7

Integral action is primarily introduced to:

A. Guarantee zero control effort

B. Remove constant steady-state tracking error

C. Eliminate measurement noise

D. Make every plant controllable

Correct Answer: B. Remove constant steady-state tracking error

Question 8

Under the separation principle, the observer-based compensator’s eigenvalues are:

A. Only the observer eigenvalues

B. Only the controller eigenvalues

C. The union of controller and observer eigenvalues

D. The products of controller and observer eigenvalues

Correct Answer: C. The union of controller and observer eigenvalues

Question 9

In continuous-time observer design, placing observer poles excessively far left can:

A. Eliminate all actuator saturation

B. Amplify measurement noise and model uncertainty

C. Make an unobservable system observable

D. Guarantee robust stability

Correct Answer: B. Amplify measurement noise and model uncertainty

Question 10

For a discrete-time closed loop, asymptotic stability requires all poles to be:

A. Outside the unit circle

B. On the imaginary axis

C. Inside the unit circle

D. Equal to one

Correct Answer: C. Inside the unit circle