9. State-Space and Digital Control

Introduction and Context

Classical control methods based on transfer functions are powerful for single-input, single-output systems, but many modern mechanical systems are multivariable, dynamically coupled, constrained, and implemented through computers. Examples include robotic manipulators, active suspension systems, machine-tool drives, unmanned vehicles, turbomachinery, and precision positioning stages. For such systems, it is useful to represent the internal dynamic condition of the plant explicitly rather than describing only the input–output relationship.

The state-space approach provides this representation. It expresses a system through first-order differential or difference equations involving a vector of state variables. This formulation supports systematic treatment of multiple inputs and outputs, time-varying models, initial conditions, state feedback, observers, controllability, observability, and optimal control. When the controller is executed digitally, the continuous model must additionally be sampled and converted into a discrete-time representation. Sampling, quantization, computation time, delays, and numerical limitations then become part of the control problem.

This chapter develops the connection between physical modeling, state-space analysis, controller synthesis, and real-time implementation. The emphasis is not only on formulae but also on design judgment: selecting sampling rates, interpreting poles, recognizing model limitations, and ensuring that a mathematically satisfactory controller remains safe and effective on actual hardware.

Learning Objectives

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

1. formulate continuous-time and discrete-time state-space models of mechanical systems;

2. distinguish states, inputs, outputs, controllability, and observability;

3. design state-feedback controllers using pole placement;

4. explain the purpose and operation of full-order and reduced-order observers;

5. design reference tracking systems with integral action or precompensation;

6. discretize continuous models and assess sampling, aliasing, quantization, and delay effects;

7. use the Z-transform and discrete pole locations to analyze digital control systems;

8. identify implementation risks and select suitable engineering practices for real-time control.

From Mechanical Equations to State Variables

Consider a linear time-invariant mechanical system described by

M q̈ + C q̇ + Kq = Fu,

y = Hq,

where q is the vector of generalized coordinates, u is the actuator input, and y is the measured output. If M is nonsingular, define the state vector

x = [qᵀ q̇ᵀ]ᵀ.

The model can then be written as

ẋ = Ax + Bu,

y = Cx + Du,

where A is the system matrix, B the input matrix, C the output matrix, and D the direct-transmission matrix. In general, the state need not consist only of positions and velocities. Electrical currents, actuator states, flexible-mode coordinates, thermal variables, and integrator states may also be included.

A state is a minimum set of variables that, together with the future input, determines the future behavior of the system. The state is therefore a dynamic memory of the system. Different state choices can describe the same physical plant. For example, modal coordinates and physical coordinates may generate different matrices but equivalent input–output behavior.

For a single-degree-of-freedom mass–spring–damper system,

m q̈ + c q̇ + kq = bu,

choose x₁ = q and x₂ = q̇. Then

ẋ = [ [0, 1],[-k/m, -c/m] ]x + [0, b/m]u,

y = [1, 0]x.

The eigenvalues of A are the natural poles of the unforced system. They reveal stability, oscillation frequency, and damping. State-space analysis is therefore closely connected to the physical modes of a mechanical structure.

State-Space Concepts and Structural Properties

The solution of the continuous model is

x(t) = e^{At}x(0) + ∫₀ᵗ e^{A(t-τ)}Bu(τ)dτ.

The matrix exponential e^{At} describes the free response, while the integral describes the response generated by the input. This expression shows why the state formulation naturally incorporates nonzero initial conditions.

A system is controllable if an admissible input can move the state from any initial condition to any desired final state within finite time. For an n-state, single-input or multi-input linear system, the controllability matrix is

𝒞 = [B AB A²B … Aⁿ⁻¹B].

The system is controllable when rank(𝒞) = n. Physically, every dynamic mode must be reachable through the actuators. An uncontrollable unstable mode cannot be stabilized by state feedback, and an uncontrollable lightly damped mode may limit achievable performance.

Observability concerns the reverse question: can the internal state be reconstructed from measured inputs and outputs? The observability matrix is

𝒪 = [Cᵀ AᵀCᵀ … (Aⁿ⁻¹)ᵀCᵀ]ᵀ.

The system is observable when rank(𝒪) = n. An unobservable unstable mode cannot be estimated reliably from the available sensors. In mechanical systems, poor sensor placement can make a flexible mode nearly unobservable even when the ideal mathematical model is technically observable.

Controllability and observability are coordinate-independent properties, although their numerical conditioning depends strongly on the chosen coordinates and scaling. Poorly scaled models can cause computational errors in rank tests, pole placement, and observer design. Singular-value analysis and balanced realizations are often more reliable than direct rank calculations for high-order systems.

State Feedback and Pole Placement

If all states are available, a linear state-feedback law may be written as

u = -Kx + Nr,

where K is the feedback-gain matrix, r is a reference signal, and N is a reference precompensator. The closed-loop dynamics become

ẋ = (A - BK)x + BNr.

Thus, state feedback changes the system matrix from A to A − BK. The eigenvalues of this matrix are the closed-loop poles. Pole placement selects K so that these eigenvalues have desired locations. For a controllable system, arbitrary closed-loop poles can theoretically be assigned, subject to practical limitations.

Pole locations should not be selected solely by demanding the fastest possible response. Very fast poles require large actuator effort, amplify measurement noise, increase sensitivity to unmodeled high-frequency dynamics, and may expose sampling and delay limitations. A useful design begins with requirements on settling time, overshoot, bandwidth, actuator limits, and robustness. For a dominant second-order pair,

s₁,₂ = −ζωₙ ± jωₙ√(1−ζ²),

where ζ is damping ratio and ωₙ is natural frequency. Additional poles are often placed several times farther into the left half-plane, but this rule must be checked against noise and neglected dynamics.

For single-input systems, Ackermann’s formula provides a direct computational method:

K = [0 … 0 1]𝒞⁻¹ϕ(A),

where ϕ(A) is the desired characteristic polynomial evaluated at A. In practical numerical work, dedicated Schur-based algorithms are preferable because Ackermann’s formula can be ill-conditioned for higher-order systems.

State feedback alone does not guarantee zero steady-state error to a constant reference. A common remedy is integral augmentation. Introduce an integrator state

x_İ = r − y.

The augmented state is xₐ = [xᵀ x_I]ᵀ, and the control law may be

u = −Kx − K_Ix_I.

The integrator removes constant tracking error when the augmented system is controllable and actuator saturation is managed. During saturation, the integral state may continue to grow, producing windup. Conditional integration, back-calculation, or integrator clamping should therefore be included.

Observers: Estimating What Cannot Be Measured

Mechanical systems frequently have fewer sensors than state variables. A position sensor may be available while velocity is obtained by estimation rather than numerical differentiation, because differentiation magnifies noise. A Luenberger observer uses the model and output error to estimate the state:

x̂̇ = Ax̂ + Bu + L(y − Cx̂).

The estimation error e = x − x̂ obeys

ė = (A − LC)e.

Consequently, observer poles are the eigenvalues of A − LC. If the pair (A,C) is observable, L can be selected by pole placement. The controller then uses u = −Kx̂ rather than the unavailable x.

The separation principle states that, for a linear time-invariant system with appropriate controllability and observability, controller and observer poles can be designed independently. This principle simplifies design but does not eliminate practical interaction. An observer chosen excessively fast may amplify sensor noise and generate high-frequency control action. A common engineering choice is to make observer dynamics faster than the dominant controller dynamics, but not so fast that noise, sampling, and neglected actuator dynamics dominate.

A Kalman filter provides a statistically based observer when process and measurement noise are modeled. It balances confidence in the model against confidence in the measurements through covariance matrices. In mechanical applications, Kalman filtering is useful for sensor fusion, inertial navigation, flexible-body estimation, and disturbance rejection. Its performance depends on credible noise models; arbitrary covariance tuning can hide modeling errors rather than solve them.

Reference Tracking and Disturbance Rejection

Reference tracking has two distinct requirements: transient shaping and elimination of steady-state error. A precompensator can provide exact tracking for a specified model, but its accuracy deteriorates with parameter uncertainty. Integral action is more robust for constant disturbances and plant-gain errors, although it introduces an additional state and possible windup.

Disturbance rejection can also be improved by augmenting the state with a disturbance model. For example, a constant unknown disturbance d may be represented by ḋ = 0 and estimated as an additional state. Sinusoidal disturbances can be represented by a pair of imaginary-axis states. Such augmentation is effective only when the disturbance model is plausible and the resulting system remains observable and controllable.

In industrial motion systems, feedforward and feedback are often combined. Feedforward uses a plant model to anticipate the force required for acceleration, friction, or gravity, while feedback corrects model error. This arrangement can reduce tracking error without requiring extremely high feedback gains.

Discrete-Time Models for Digital Controllers

A digital controller samples signals at times t = kT, computes a control action, and holds that action until the next update. With a zero-order hold, the exact discrete model corresponding to

ẋ = Ax + Bu

is

x[k+1] = A_d x[k] + B_d u[k],

y[k] = C_d x[k] + D_d u[k],

where

A_d = e^{AT},

B_d = ∫₀ᵀ e^{Aτ}B dτ.

If A is nonsingular, B_d = A⁻¹(A_d − I)B, although the integral or an augmented matrix exponential is numerically safer in singular cases. C_d and D_d depend on the measurement definition and hold arrangement.

The continuous and discrete stability regions differ. A continuous pole s maps to a discrete pole z through

z = e^{sT}.

The left half of the s-plane maps inside the unit circle. Therefore, a discrete-time linear system is asymptotically stable when all poles satisfy |z_i| < 1. Poles near z = 1 produce slow responses; complex poles near the unit circle produce lightly damped oscillations.

The Z-transform is the discrete counterpart of the Laplace transform. For a sequence x[k],

X(z) = Σₖ₌₀^∞ x[k]z⁻ᵏ.

Time delay by one sample is represented by multiplication by z⁻¹. Difference equations can therefore be transformed into algebraic expressions. For example,

y[k] − ay[k−1] = bu[k−1]

has transfer function

Y(z)/U(z) = bz⁻¹/(1 − az⁻¹).

The region of convergence and pole locations are important when interpreting Z-transform results. The discrete final-value theorem, under appropriate stability conditions, is

limₖ→∞ x[k] = lim_{z→1}(1 − z⁻¹)X(z).

Sampling, Aliasing, and Reconstruction

Sampling converts a continuous signal into a sequence. If the highest significant frequency in a signal is f_max, a theoretical requirement is f_s > 2f_max. This is the Nyquist condition, but it is not a sufficient design rule by itself. Frequencies above half the sampling frequency fold into lower frequencies and appear as false components, a phenomenon called aliasing. Once aliasing has occurred, digital processing cannot uniquely recover the original signal.

Anti-aliasing low-pass filters are therefore placed before analog-to-digital conversion. Their cutoff must be below the Nyquist frequency, with enough transition band for attenuation. Mechanical systems with resonances, switching ripple, and sensor noise require particular care. Sampling frequency should also be selected in relation to closed-loop bandwidth, computational delay, actuator dynamics, and available processing resources. A sampling rate several times the desired closed-loop bandwidth is a starting point, not a universal law.

The zero-order hold introduces a phase lag and modifies the effective plant. Although its magnitude effect may be modest at low frequency, the phase delay can reduce stability margins near the control bandwidth. Computational delay and communication delay add further phase loss. In networked or distributed machines, deterministic timing is often more important than merely increasing nominal processor speed.

Quantization, Numerical Effects, and Delay

An analog-to-digital converter maps a continuous amplitude to one of finite levels. Quantization error is commonly modeled as bounded noise of approximately one-half the least significant bit under suitable assumptions. This model becomes inaccurate for low-level periodic signals, limit cycles, or strongly correlated errors. Quantization can cause oscillations in digital controllers, especially when high gain is combined with small position increments.

Finite word length affects stored states, coefficients, sensor values, and intermediate calculations. Scaling, saturation arithmetic, coefficient quantization, and implementation structure must be examined. Direct-form realizations can be sensitive to round-off in high-order systems; cascaded second-order sections or state-space realizations are often better conditioned.

Delays may be represented by additional states or factors of z⁻ᵈ. A one-sample delay can substantially alter phase margin in a fast loop. Controller design should include the expected worst-case delay rather than assuming a negligible computation time. Timestamping, synchronized clocks, priority scheduling, and execution-time monitoring are important in safety-critical systems.

Digital Controller Design Workflow

A robust workflow begins with a physically justified continuous model and a clear definition of measured and controlled variables. The designer then evaluates operating-point validity, actuator limits, sensor noise, neglected modes, and uncertainty. Controllability and observability are assessed before synthesis.

The next step is selection of a sampling period based on bandwidth, resonances, delay, and computation. The model is discretized using the actual hold and sampling arrangement. A discrete controller may then be designed directly by placing poles in the z-plane or by designing in continuous time and mapping the desired poles using z = e^{sT}. Direct discrete design is preferable when sampling is not fast relative to the plant dynamics.

Simulation should include nonlinearities and nonidealities: saturation, friction, backlash, sensor noise, quantization, delays, parameter variation, and disturbances. Hardware-in-the-loop testing can reveal timing and interface problems before full deployment. Finally, commissioning should proceed with conservative gains, fault detection, output limiting, and a safe fallback mode.

Illustrative Design Example

Consider a rotary inertia with angle θ and angular velocity ω:

Jω̇ = −bω + u,

θ̇ = ω,

where J = 0.02 kg·m² and b = 0.1 N·m·s/rad. Let the state be x = [θ ω]ᵀ and suppose angle is measured. The continuous matrices are

A = [ [0,1],[0,−5] ], B = [0,50]ᵀ, C = [1,0].

The controllability matrix [B AB] has full rank, and the observability matrix [Cᵀ AᵀCᵀ]ᵀ also has full rank. Thus, the state can be controlled and reconstructed.

Suppose the desired controller poles are −10 and −12. The state feedback u = −Kx can be obtained by pole placement. Since velocity is not measured, an observer is introduced with poles, for example, at −30 and −36. The observer estimate is used in the feedback law. If exact constant angle tracking is required despite a constant load torque, an integral state based on θ_ref − θ is added.

For digital implementation with sampling period T, the matrices are obtained from the matrix exponential and hold integral. The designer then checks the mapped poles, computation time, encoder resolution, and motor torque limit. If the calculated control action frequently saturates, the poles are too aggressive or the actuator is undersized; merely increasing observer speed will not correct the problem.

Applied Case: Digital Active Suspension Controller

Background

The following is an illustrative engineering case involving a quarter-car active suspension. The system contains sprung mass displacement, unsprung mass displacement, and their velocities. A hydraulic actuator applies force between the masses, while an accelerometer and displacement sensor provide measurements.

Situation and Problem

A passive suspension meets basic ride-comfort requirements but permits excessive body acceleration near a resonance. The objective is to reduce acceleration while maintaining tire contact and respecting actuator force limits. The plant is multivariable in practice because road displacement acts as a disturbance and actuator force affects both body and wheel dynamics.

Analysis

A state model is formed from the two displacements and velocities. Controllability analysis shows that the actuator strongly influences the body mode but has weaker authority over the high-frequency wheel mode. Observability analysis reveals that body acceleration alone is insufficient for accurate reconstruction of all states; a wheel or suspension-deflection measurement is added. The expected control bandwidth is compared with the wheel-hop resonance and sensor noise spectrum.

A continuous state-feedback controller with integral action is initially designed. The model is then discretized using a sampling period selected to provide adequate resolution of the dominant dynamics. The design includes a zero-order hold, measured computation delay, actuator saturation, and an observer. Observer poles are faster than the principal closed-loop poles but are limited to avoid excessive acceleration-noise amplification.

Approach and Solution

The controller uses estimated states, feedforward compensation for measured road disturbance where available, and anti-windup for the integral channel. An anti-aliasing filter is installed before the accelerometer input. Simulations are performed over varying vehicle mass, damping, tire stiffness, road profiles, and sensor noise. Hardware-in-the-loop testing then verifies timing, saturation behavior, and fault responses.

Outcome

The design process demonstrates that a controller cannot be judged by nominal pole locations alone. A mathematically stable design may produce unacceptable actuator demand, noise amplification, or loss of tire contact when nonidealities are included. The final implementation is selected by balancing comfort improvement, robustness, force demand, and fail-safe behavior.

Learning Points

The case emphasizes four principles: actuator authority limits achievable pole placement; sensor selection determines observer quality; sampling and delay are part of the plant; and validation must include physical constraints rather than only linear nominal simulations.

Limitations, Risks, and Professional Considerations

State-space models are abstractions. Linearization is valid only near the operating condition for which it was performed. Friction, backlash, saturation, hysteresis, structural flexibility, and actuator dynamics may invalidate a low-order model. Pole placement can produce fragile gains when the model is poorly conditioned or uncertain. Robust or optimal methods may be preferable when uncertainty is significant.

A controller should not be deployed solely because a simulation is stable. Sensor failure, communication loss, arithmetic overflow, watchdog timeout, and actuator faults must be considered. Safety-related machinery may require documented verification, independent limits, emergency shutdown paths, and compliance with applicable functional-safety standards. The control algorithm should never be allowed to override mechanical guards or established safety interlocks.

Ethically and professionally, designers are responsible for communicating assumptions, uncertainty, operating limits, and failure modes. Data-driven observers and adaptive controllers require additional governance: training data, validation ranges, explainability of abnormal behavior, cybersecurity, and protection against unauthorized parameter changes. In connected industrial systems, authentication, secure updates, network segmentation, and logging are control-system concerns, not merely information-technology concerns.

Emerging Directions

Current practice increasingly combines state-space control with model predictive control, disturbance observers, digital twins, and data-based fault diagnosis. Model predictive control handles constraints explicitly by solving an optimization problem at each sample, although computational burden and model quality remain important. Digital twins support virtual commissioning and condition monitoring, but their predictions must be validated against measured plant behavior.

Machine-learning methods are being investigated for residual dynamics, parameter estimation, and anomaly detection. In safety-critical motion control, these methods are more commonly used alongside a verified model-based controller than as an unrestricted replacement. Event-triggered and networked control seek to reduce communication while maintaining performance, but introduce irregular sampling and timing analysis challenges.

Best Practices for Engineering Implementation

A reliable implementation generally follows these practices:

• scale states, inputs, and outputs so numerical calculations are well conditioned;
• test controllability and observability using numerically robust methods;
• include actuator dynamics, delays, saturation, and sensor filtering in the design model;
• select sampling frequency from bandwidth, resonances, delay, and computation—not from a rule of thumb alone;
• use anti-aliasing filters and verify their phase effect;
• include anti-windup and explicit output limits;
• validate nominal and uncertain models with nonlinear, hardware-in-the-loop, and fault tests;
• monitor timing, sensor plausibility, estimator residuals, and actuator health;
• document assumptions, pole choices, units, scaling, safety limits, and revision history.

Chapter Summary

State-space modeling represents the internal dynamics of mechanical systems through first-order equations and supports multivariable analysis, initial-condition response, state feedback, and estimation. Controllability determines whether actuators can influence all relevant modes, while observability determines whether sensors provide enough information to reconstruct them. State feedback changes closed-loop poles, but reference tracking commonly requires precompensation or integral action. Observers estimate unavailable states, and the separation principle permits independent nominal placement of controller and observer poles.

Digital implementation requires a discrete model that includes the sampling interval and zero-order hold. The Z-transform describes discrete sequences and delays, while stability requires all discrete poles to lie inside the unit circle. Sampling must address aliasing, filtering, quantization, finite word length, and computation delay. Practical design therefore combines mathematical synthesis with actuator limits, uncertainty analysis, timing verification, safety engineering, and hardware testing.

Key Terms

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

State-space model: A first-order vector description of system dynamics and outputs.

System matrix A: Matrix describing autonomous state evolution.

Controllability: Ability to drive the complete state using the available inputs.

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

State feedback: Control law using measured or estimated states to modify closed-loop dynamics.

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

Observer: Dynamic estimator that reconstructs unmeasured states.

Luenberger observer: Model-based observer corrected by output estimation error.

Kalman filter: Statistical state estimator based on process and measurement-noise covariances.

Integral action: Control action that integrates tracking error to remove constant steady-state error.

Discretization: Conversion of a continuous-time model or controller into discrete form.

Zero-order hold: Device that maintains a computed input constant between samples.

Sampling: Measurement of a continuous signal at discrete time instants.

Aliasing: Frequency-folding distortion caused by inadequate sampling.

Quantization: Representation of a continuous amplitude using finite digital levels.

Z-transform: Transform-domain representation of discrete-time sequences.

Discrete stability: Condition in which all system poles lie strictly inside the unit circle.

Anti-windup: Method preventing excessive integrator accumulation during saturation.

Separation principle: Linear result allowing nominal controller and observer pole design to be treated independently.

Review Questions

Short Answer Questions

1. What is meant by a state in a mechanical system?

2. State the rank conditions for controllability and observability.

3. Why may state feedback fail to provide zero steady-state tracking error?

4. What is the purpose of an anti-aliasing filter?

5. State the discrete-time stability condition in terms of pole locations.

Long Answer Questions

1. Derive a state-space model for a two-degree-of-freedom mass–spring–damper system and explain the physical meaning of each state.

2. Explain pole-placement design, including the role of controllability and the practical limitations of very fast poles.

3. Develop the equations of a full-order Luenberger observer and explain how observer-pole selection affects noise sensitivity.

4. Derive the exact zero-order-hold discretization of a continuous state-space model and discuss its engineering significance.

5. Explain how sampling, quantization, computation delay, and actuator saturation influence a digital control loop.

Analytical / Application Questions

1. A two-state plant has a controllable pair (A,B), but its actuator saturates for the desired pole locations. What design changes would you investigate, and why?

2. A measured position signal is clean, but the estimated velocity is extremely noisy. Analyze likely causes and propose an observer-based remedy.

3. A digital suspension controller is stable in simulation but oscillates on hardware. Formulate a diagnostic sequence involving sampling, delay, quantization, model uncertainty, and timing verification.

22. Multiple Choice Questions with Answers

Question 1

For the continuous-time model ẋ = Ax + Bu, which condition indicates complete controllability of an n-state system?

A. rank([C; CA; …; CAⁿ⁻¹]) = n

B. rank([B AB … Aⁿ⁻¹B]) = n

C. All eigenvalues of C are positive

D. The matrix D is nonsingular

Correct Answer: B. rank([B AB … Aⁿ⁻¹B]) = n

Question 2

Under state feedback u = −Kx, the closed-loop system matrix is:

A. A + KC

B. A − BK

C. A − CK

D. B − AK

Correct Answer: B. A − BK

Question 3

Why is an observer often used in a mechanical control system?

A. To increase actuator power

B. To eliminate the sampling process

C. To estimate states that are not directly measured

D. To guarantee zero quantization error

Correct Answer: C. To estimate states that are not directly measured

Question 4

For exact zero-order-hold discretization, the discrete state matrix is:

A. A_d = I + AT only

B. A_d = A/T

C. A_d = e^{AT}

D. A_d = T A⁻¹

Correct Answer: C. A_d = e^{AT}

Question 5

A stable continuous pole s maps into the discrete-time plane according to:

A. z = s + T

B. z = s/T

C. z = e^{sT}

D. z = 1 − sT exactly for all T

Correct Answer: C. z = e^{sT}

Question 6

What is the primary consequence of aliasing?

A. High-frequency components appear as false lower-frequency components

B. All sensor noise is removed

C. The actuator becomes uncontrollable

D. The system automatically becomes stable

Correct Answer: A. High-frequency components appear as false lower-frequency components

Question 7

A discrete-time linear system is asymptotically stable when:

A. All poles are outside the unit circle

B. All poles lie on the unit circle

C. At least one pole is at z = 1

D. All poles lie strictly inside the unit circle

Correct Answer: D. All poles lie strictly inside the unit circle

Question 8

Why is integral action commonly added to a state-feedback controller?

A. To remove constant steady-state tracking error

B. To eliminate the need for sensors

C. To make all poles equal to zero

D. To prevent aliasing before conversion

Correct Answer: A. To remove constant steady-state tracking error

Question 9

If an observer is designed with excessively fast poles, a likely practical consequence is:

A. Reduced sensitivity to measurement noise in every case

B. Amplification of sensor noise and high-frequency control activity

C. Loss of the plant's physical inertia

D. Automatic removal of actuator saturation

Correct Answer: B. Amplification of sensor noise and high-frequency control activity

Question 10

Which measure directly reduces the risk of aliasing before analog-to-digital conversion?

A. Increasing integral gain

B. Using an anti-aliasing low-pass filter

C. Moving controller poles closer to z = 1

D. Removing the zero-order hold

Correct Answer: B. Using an anti-aliasing low-pass filter