State-Space and Multivariable Control

Introduction and Context

Classical control analysis often treats one input and one output at a time. This approach is useful for simple systems, but modern robots, vehicles, manufacturing machines, and aerial systems are inherently coupled. A robot arm may have several joint angles, velocities, actuators, and sensors; changing one actuator can influence multiple outputs. Such systems are described as multivariable or multiple-input multiple-output (MIMO) systems.

State-space methods provide a systematic framework for representing, analyzing, and controlling these systems. Unlike a transfer function, which primarily describes input–output behavior, a state model represents the internal variables that determine future system behavior. State-space methods therefore support controllability analysis, observer design, state feedback, optimal control, and digital implementation.

For a continuous-time linear time-invariant (LTI) system, the standard state model is

Here, x(t) ∈ ℝⁿ is the state vector, u(t) ∈ ℝᵐ is the input vector, y(t) ∈ ℝᵖ is the output vector, A ∈ ℝⁿˣⁿ is the state matrix, B ∈ ℝⁿˣᵐ is the input matrix, C ∈ ℝᵖˣⁿ is the output matrix, and D ∈ ℝᵖˣᵐ is the direct-feedthrough matrix. The dot denotes differentiation with respect to time. The model assumes linearity and time-invariant parameters over the operating region being considered.

Learning Objectives

By the end of this chapter, the reader should be able to:

1. Formulate continuous and discrete state-space models from physical equations and transfer functions.
2. Interpret the roles of states, inputs, outputs, and system matrices.
3. Assess controllability and observability using rank tests and related interpretations.
4. Design state-feedback controllers using pole placement.
5. Formulate and solve the linear-quadratic regulator (LQR) problem.
6. Design observers and combine them with state feedback.
7. Recognize practical limitations caused by nonlinearities, saturation, uncertainty, noise, and sampling.
8. Apply state-space methods to a coupled mechanical system.

1. State Variables and State Models

A state is a minimum set of variables whose present values, together with future inputs, determine the future behavior of a system. For a mechanical system, positions and velocities commonly form the state. For an electrical system, capacitor voltages and inductor currents are typical states.

For a second-order mass–spring–damper system,

where m is mass in kilograms, c is viscous damping in N·s/m, k is stiffness in N/m, q is displacement in metres, and f is applied force in newtons. Define

Then

Thus, with u = f and y = q,

The choice of state variables is not unique. Any nonsingular transformation z = T x, where T is an invertible matrix, produces an equivalent state representation. The numerical entries of A, B, and C change, but fundamental properties such as eigenvalues, controllability, and observability are preserved.

1.1 From Transfer Functions to State Space

Consider a strictly proper transfer function

A controllable canonical realization is

This realization is one possible model, not the only one. If the transfer function is not strictly proper, polynomial division should first be used to identify the direct term D and the strictly proper remainder.

1.2 Mechanical Multivariable Models

For an n-degree-of-freedom mechanical system,

where M(q) is the inertia matrix, h represents velocity-dependent effects such as Coriolis forces, g represents gravity, F represents friction, and τ is the actuator torque vector. This equation is generally nonlinear. Around an equilibrium point (q₀, q̇₀, τ₀), define small perturbations δq, δq̇, and δτ. Linearization gives a local model of the form

where δx typically contains [δqᵀ δq̇ᵀ]ᵀ. The model is accurate only near the operating point and under the assumption that neglected higher-order terms are sufficiently small.

2. System Response and Eigenvalues

For the unforced system, u(t) = 0, the state response is

where e^{At} is the matrix exponential. For a constant input u₀, the response is

The eigenvalues of A determine the natural modes of the system. If all eigenvalues have negative real parts, the continuous-time LTI system is asymptotically stable. An eigenvalue with positive real part produces an unstable mode. Eigenvalues on the imaginary axis require further examination, particularly if repeated eigenvalues have defective eigenvectors.

For a discrete-time model,

stability requires every eigenvalue of A_d to lie strictly inside the unit circle: |λᵢ| < 1. With zero-order-hold sampling period T, the exact conversion from a continuous model is

The integral expression remains valid even when A is singular. If A is nonsingular, it may also be evaluated as B_d = A⁻¹(e^{AT} − I)B.

3. Controllability

A system is controllable if an appropriate input can transfer the state from any initial state to any desired final state in finite time. For the continuous-time pair (A, B), the controllability matrix is

The system is completely controllable if

where n is the number of states. For a single-input system, 𝒞 has n columns; for a multi-input system it has mn columns.

Controllability does not mean that every state can be reached with small or practical actuator effort. A system may be mathematically controllable but poorly controllable, meaning that reaching certain directions requires very large input. Singular values of the controllability Gramian can help reveal this issue.

For a stable continuous-time system, the controllability Gramian over an infinite horizon is

It satisfies the Lyapunov equation

If W_c is positive definite, the system is controllable. The Gramian also indicates the input energy required to excite different state directions.

Worked Example 1: Controllability

Consider

The controllability matrix is

Its determinant is −1, which is nonzero. Therefore rank(𝒞) = 2, and the system is controllable. Both modes can be influenced by the input, even though the input acts directly only on the second state.

The Popov–Belevitch–Hautus (PBH) test provides another criterion. The pair (A, B) is controllable if

for every eigenvalue λ of A. This test is especially useful for identifying whether a particular mode is inaccessible to the actuators.

4. Observability

A system is observable if its internal initial state can be uniquely reconstructed from measured inputs and outputs over a finite interval. The observability matrix is

where the semicolons indicate vertical stacking. The pair (A, C) is completely observable if

For a stable system, the observability Gramian is

and satisfies

A positive-definite W_o indicates observability. Poor observability means that some states have only a weak effect on the measured outputs and therefore require noisy or highly sensitive reconstruction.

Worked Example 2: Observability

Using the previous A matrix, suppose only the first state is measured:

Then

The rank is 2, so both position and velocity are observable. Although velocity is not measured directly, it affects the time evolution of position and can be inferred from the output history.

Controllability and observability are dual concepts. The pair (A, B) is controllable if and only if (Aᵀ, Bᵀ) is observable. Similarly, observer design is mathematically related to state-feedback design through matrix transposition.

5. State-Feedback and Pole Placement

A full-state feedback law has the form

where K is the feedback-gain matrix, r is a reference signal, and N is a reference precompensator. With N omitted for regulation, the closed-loop dynamics are

Thus, K changes the closed-loop eigenvalues. If (A, B) is controllable, the eigenvalues of A − BK can be assigned arbitrarily for an LTI system, subject to practical limitations such as actuator saturation and model uncertainty.

For a single-input system, Ackermann’s formula is

where p_d(s) is the desired characteristic polynomial. This formula is convenient for small systems but can be numerically ill-conditioned for high-order systems. Software implementations should generally use robust pole-placement algorithms based on Schur decompositions.

Worked Example 3: Pole Placement

Let

Choose K = [k₁ k₂]. Then

Its characteristic polynomial is

Suppose desired poles are −2 and −4. The desired polynomial is

Equating coefficients gives 3 + k₂ = 6 and 2 + k₁ = 8. Therefore

The closed-loop characteristic polynomial is s² + 6s + 8, confirming the selected poles. The result assumes the state vector is available and the model is sufficiently accurate.

Reference Tracking

State feedback designed for regulation does not automatically guarantee zero steady-state error to a constant reference. For a SISO system, a precompensator may be selected as

provided the inverse exists and the plant has no problematic transmission zero at the origin. In practice, integral action is often preferred when constant disturbances and modelling errors must be rejected. An augmented state may be defined by

and the integral state included in the feedback design. Integral action increases system order and may cause windup when actuators saturate, so anti-windup protection is required.

6. Linear-Quadratic Regulation

Pole placement specifies desired eigenvalues but does not directly specify how control effort should be traded against state deviations. LQR provides an optimization-based alternative. For the infinite-horizon continuous-time problem, minimize

subject to ẋ = Ax + Bu. Here Q is symmetric positive semidefinite and weights state deviations; R is symmetric positive definite and weights actuator effort. The units of Q and R must be selected consistently so that both terms in the integrand represent comparable cost dimensions.

The optimal control law is

where

and P is the symmetric positive semidefinite solution of the continuous algebraic Riccati equation (CARE):

Under standard stabilizability and detectability conditions, the stabilizing solution produces an asymptotically stable closed loop. Controllability of every state is sufficient but not necessary; unstable modes must be controllable, and unobservable modes in the state penalty must satisfy appropriate detectability conditions.

Increasing Q generally makes state regulation more important and often produces faster, more aggressive control. Increasing R penalizes actuator effort and generally produces smaller control commands. These are design tendencies, not universal monotonic guarantees for every performance measure.

For discrete-time systems, the cost is commonly written

and the gain is obtained from the discrete algebraic Riccati equation. Continuous and discrete designs should not be mixed without accounting for sampling and discretization.

Practical Weight Selection

A useful starting point is Bryson’s rule. If xᵢ,max and uⱼ,max are acceptable maximum deviations and actuator commands, choose diagonal weights approximately as

This normalizes variables relative to engineering limits. Cross weights may be included when combinations of states or inputs are important. The resulting controller must still be tested against nonlinear simulation, saturation, noise, and parameter variation.

7. State Observers

Full-state feedback is often impossible because not every state is measured. An observer estimates the state from known inputs and measured outputs. A continuous-time Luenberger observer is

where x̂ is the estimated state and L is the observer gain. Define the estimation error e = x − x̂. For the nominal model,

Therefore, observer poles are the eigenvalues of A − LC. If (A, C) is observable, L can be selected by pole placement.

The observer should not normally be made arbitrarily fast. Fast observer poles amplify measurement noise, increase sensitivity to unmodelled dynamics, and may cause excessive actuator activity because the estimated state is fed back into the controller.

A stochastic alternative is the Kalman filter. It assumes process and measurement noise models such as

where w[k] and v[k] are zero-mean random disturbances with covariance matrices W and V. The Kalman gain is computed from a Riccati equation and balances model prediction against sensor measurements. Accurate covariance modelling is important; covariance matrices are not merely arbitrary tuning knobs, although practical tuning is often needed.

8. Separation Principle and Observer-Based Control

Combining state feedback with an observer gives

For an LTI system, if (A, B) is stabilizable and (A, C) is detectable, controller and observer can be designed independently. The closed-loop eigenvalues consist of the eigenvalues of A − BK together with those of A − LC. This result is called the separation principle.

The principle does not eliminate practical interactions. Sensor noise, actuator saturation, computational delay, quantization, nonlinearities, and incorrect models can degrade the actual system. In particular, saturation invalidates the simple linear closed-loop analysis and can interact with integral or observer dynamics.

A common implementation sequence is:

1. Define the physical states, inputs, outputs, and operating point.
2. Derive and validate the nonlinear model.
3. Linearize or identify an LTI model for the intended operating region.
4. Check stabilizability and detectability.
5. Design K by pole placement or LQR.
6. Design L by pole placement or a Kalman filter.
7. Discretize consistently with the controller sampling period.
8. Simulate disturbances, noise, saturation, faults, and parameter variation.
9. Test progressively on hardware with safety limits.

9. Detailed Applied Case: Two-Link Planar Robot Arm

The following is an illustrative engineering case. Consider a two-link planar robot arm operating near a horizontal equilibrium. Each revolute joint has an encoder, and each joint is driven by a torque actuator. The small-angle linearized model is represented by

where δq = [δq₁ δq₂]ᵀ is the joint-angle perturbation vector in radians, δτ is the torque perturbation vector in N·m, M is the inertia matrix in kg·m², D is the damping matrix in N·m·s/rad, and K_s is the linearized stiffness matrix in N·m/rad. Coupling appears through the off-diagonal elements of these matrices.

Define the state

Then

with output y = [δq₁ δq₂]ᵀ. The model has four states, two inputs, and two outputs. Before controller design, engineers must verify that M is nonsingular and that the chosen equilibrium and linearization are valid.

Problem

The arm must hold both joint angles despite coupling, small external disturbances, and noisy velocity estimates. Independent single-joint PID loops may perform poorly because torque at joint 1 changes the motion of joint 2.

Analysis

The matrices are assembled from identified inertia, damping, and stiffness parameters. The eigenvalues of A reveal natural coupled modes. The controllability matrix is tested for rank four. If it has full rank but a very small singular value, the arm is theoretically controllable but may require excessive torque in one mode. The observability matrix is also tested. Position encoders can make velocities observable, but low sample rates and quantization may make velocity estimates unreliable.

Approach and Solution

An LQR controller is designed with Q weighting joint-angle and joint-velocity errors according to allowable limits. R weights the two motor torques according to their continuous ratings. The gain K is calculated from the CARE. Because only positions are measured, a discrete observer or Kalman filter estimates velocities. The implemented law is

with torque saturation and rate limits included in the software. Anti-windup is unnecessary for the basic LQR law because it has no integral state, but reference tracking may require an integral augmentation or a model-based feedforward term.

Outcome

In simulation, the coupled design should be judged using settling time, overshoot, RMS tracking error, peak torque, energy use, and robustness to parameter changes. A successful design should regulate both joints without persistent oscillation and should remain safe when one motor approaches its torque limit. If performance is poor, possible causes include poorly scaled weights, inaccurate inertia estimates, unmodelled friction, observer noise amplification, or operation too far from the linearization point.

Learning Points

The case demonstrates that state-space control handles coupling explicitly. It also shows that mathematical controllability is only a prerequisite: actuator authority, sensing quality, model validity, and safety constraints determine whether the controller is useful in practice.

10. Implementation, Limitations, and Professional Practice

Model validity and nonlinear operation

A linear model is local. Robot gravity, friction, actuator characteristics, and geometric coupling may vary substantially with configuration. Gain scheduling, feedback linearization, robust control, or nonlinear model predictive control may be required over a large workspace.

Scaling and numerical conditioning

Positions, velocities, currents, and torques may differ greatly in numerical magnitude. Poor scaling can make rank calculations and Riccati solvers unreliable. Engineers should use physically meaningful units, normalize variables when appropriate, and inspect condition numbers rather than relying only on a binary rank result.

Saturation and safety

No controller can command torque beyond actuator capability. Saturation can produce windup, slow recovery, or instability. Current limits, thermal limits, joint stops, emergency braking, watchdog timers, and fault detection must be designed independently of the nominal control law. Safety-related functions should not depend solely on an experimental software controller.

Sampling and delay

The sample period must be short enough to capture the relevant dynamics and computation must complete within the timing deadline. Sensor delay and zero-order hold alter the effective closed-loop system. Discrete-time pole placement or discrete LQR is preferable when the controller is implemented digitally.

AI and contemporary developments

Machine learning can assist system identification, friction compensation, anomaly detection, and gain adaptation. However, learned components should be constrained by physical limits and validated over representative operating conditions. For safety-critical robotics, explainability, traceability, data governance, cybersecurity, and fail-safe behavior are essential. A learned model should complement—not replace—controllability analysis, stability verification, and hardware safety mechanisms.

Best-Practice Checklist

• Define states so that they are physically interpretable and sufficient.
• Confirm units, coordinate conventions, and operating-point assumptions.
• Test stabilizability and detectability before synthesis.
• Use LQR when a transparent performance–effort trade-off is preferable to manually assigned poles.
• Design observers with explicit consideration of sensor noise and model uncertainty.
• Validate both nominal and perturbed models.
• Include saturation, delay, quantization, and faults in simulation.
• Use independent safety limits and supervisory logic.
• Revalidate the controller after mechanical, software, or sensor changes.

Chapter Summary

State-space modelling represents a dynamic system through internal states, inputs, and outputs. The matrices A, B, C, and D describe system dynamics, actuation, sensing, and direct feedthrough. Mechanical systems are often converted from second-order equations to first-order state models, while nonlinear robot dynamics are commonly linearized near an operating point.

Controllability determines whether inputs can influence all state directions, while observability determines whether internal states can be reconstructed from measured outputs. Rank tests provide fundamental criteria, and Gramians provide additional information about control and sensing strength.

State feedback changes the eigenvalues of A through the matrix A − BK. Pole placement directly assigns desired closed-loop poles when the system is controllable or stabilizable. LQR instead minimizes a quadratic cost that balances state performance against control effort through the Riccati equation. When states are not measured, an observer estimates them; its error dynamics are governed by A − LC. The separation principle permits independent nominal design of the controller and observer.

Practical robotics requires more than algebraic synthesis. Model uncertainty, nonlinear operation, saturation, sampling, delays, noise, actuator limits, and safety requirements must be included in validation. State-space control is most effective when theoretical analysis is combined with careful modelling, numerical conditioning, simulation, and staged hardware testing.

Key Terms

State: A minimum set of variables that determines future system behavior when inputs are known.

State vector: An organized column vector containing the system states.

State-space model: A first-order differential or difference-equation representation using state and output equations.

LTI system: A linear time-invariant system whose parameters do not vary with time.

MIMO system: A system with multiple inputs and multiple outputs.

Controllability: The ability to drive the state to desired values using admissible inputs.

Observability: The ability to reconstruct internal states from inputs and measured outputs.

Controllability matrix: The matrix [B AB … Aⁿ⁻¹B] used in a rank test.

Observability matrix: The stacked matrix [C; CA; …; CAⁿ⁻¹] used in a rank test.

Gramian: A matrix quantifying the strength of control or observation over a time interval.

State feedback: A control law that uses the state, commonly u = −Kx.

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

LQR: Linear-quadratic regulation using an optimization criterion for state and input effort.

Riccati equation: The matrix equation whose stabilizing solution determines the LQR gain.

Observer: A dynamic estimator that reconstructs unmeasured states.

Luenberger observer: A deterministic observer using output-estimation error feedback.

Kalman filter: A stochastic state estimator based on model and measurement-noise covariance.

Stabilizability: The property that all unstable modes can be controlled.

Detectability: The property that all unstable modes can be observed.

Separation principle: The nominal result that controller and observer poles can be designed independently.

Review Questions

Short Answer Questions

1. What is meant by a state, and why are positions and velocities common states in mechanical systems?
2. State the rank conditions for controllability and observability of an n-state LTI system.
3. What is the role of matrix D in a state-space model?
4. How does increasing R in an LQR design generally affect actuator effort?
5. Why can an observer that is excessively fast amplify measurement noise?

Long Answer Questions

1. Derive the state-space model of a mass–spring–damper system and explain the physical meaning of each matrix.
2. Explain controllability and observability using both rank tests and physical interpretations.
3. Derive the closed-loop dynamics resulting from the state-feedback law u = −Kx and discuss pole placement.
4. Explain the continuous-time LQR problem, including the cost function, CARE, and assumptions for a stabilizing solution.
5. Describe how an observer-based controller is designed and explain the separation principle and its practical limitations.

Analytical / Application Questions

1. For A = [[0,1],[-2,-3]] and B = [[0],[1]], calculate the controllability matrix and determine whether the system is controllable.
2. A two-joint robot has position measurements but no velocity sensors. Propose a state estimator and identify two practical issues that must be addressed before hardware deployment.
3. A pole-placement controller produces excellent simulation results but repeatedly saturates one motor on the real robot. Analyze at least four possible causes and propose corrective actions.

Multiple Choice Questions with Answers

Question 1

For an n-state continuous-time system, which condition establishes complete controllability?

A. rank(𝒞) = n

B. rank(𝒪) = n

C. det(A) = 1

D. All eigenvalues of B are negative

Correct Answer: A. rank(𝒞) = n

Question 2

For the system A = [[0,1],[-2,-3]] and B = [[0],[1]], what is the determinant of the controllability matrix?

A. 0

B. −1

C. 1

D. 5

Correct Answer: B. −1

Question 3

Under state feedback u = −Kx, which matrix determines the closed-loop eigenvalues?

A. A + KC

B. A − BK

C. A − LC

D. A + BD

Correct Answer: B. A − BK

Question 4

In continuous-time LQR, what does the matrix R primarily penalize?

A. Measurement noise

B. Initial state uncertainty

C. Control effort

D. System observability

Correct Answer: C. Control effort

Question 5

What condition is required for asymptotic stability of a continuous-time LTI system?

A. All eigenvalues have positive real parts

B. All eigenvalues lie on the unit circle

C. All eigenvalues have negative real parts

D. The matrix B must be diagonal

Correct Answer: C. All eigenvalues have negative real parts

Question 6

For a Luenberger observer, what matrix governs estimation-error dynamics?

A. A − BK

B. A + BK

C. A − LC

D. A + BD

Correct Answer: C. A − LC

Question 7

A discrete-time LTI system is asymptotically stable when its eigenvalues are located where?

A. Strictly outside the unit circle

B. Strictly inside the unit circle

C. On the positive real axis

D. Only at the origin

Correct Answer: B. Strictly inside the unit circle

Question 8

Which statement best describes the separation principle?

A. Input and output matrices can always be ignored

B. Controller and observer poles can be designed independently under standard conditions

C. A nonlinear plant always behaves like an LTI plant

D. Observer design removes actuator saturation

Correct Answer: B. Controller and observer poles can be designed independently under standard conditions

Question 9

If a system is controllable but has a very small controllability-Gramian singular value, what is the likely practical interpretation?

A. The system is necessarily unstable

B. Some state directions may require large control effort

C. The outputs cannot be measured

D. The model has no inputs

Correct Answer: B. Some state directions may require large control effort

Question 10

Which change generally places greater emphasis on reducing state deviations in an LQR design?

A. Increasing Q

B. Increasing R only

C. Setting B to zero

D. Removing the state vector

Correct Answer: A. Increasing Q