State-Space Analysis of Multivariable Systems

Introduction and Context

Classical control analysis commonly represents a system by a transfer function relating an input to an output. This representation is useful for single-input single-output (SISO) systems, but it becomes increasingly limited when a system has several interacting inputs and outputs, internal variables, nonzero initial conditions, or modes that are hidden by pole-zero cancellation. State-space analysis overcomes these limitations by describing the internal dynamics directly through a set of first-order differential equations.

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

x_dot(t) = A x(t) + B u(t)

y(t) = C x(t) + D u(t)

where x(t) ∈ R^n is the state vector, u(t) ∈ R^m is the input vector, y(t) ∈ R^p is the output vector, and A, B, C, and D are constant matrices of dimensions n×n, n×m, p×n, and p×m, respectively. The model is applicable when the system can be adequately approximated as linear and time invariant over the operating range under consideration.

In multivariable systems, the input-output relationships are coupled. A change in one actuator may affect several outputs, and one measured output may depend on multiple internal modes. The state-space framework provides tools for determining whether these modes can be influenced by inputs, detected through outputs, represented minimally, and transformed into useful canonical forms.

Learning Objectives

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

1. Derive and interpret the state-transition matrix for continuous-time LTI systems.

2. Analyze eigenvalues, eigenvectors, generalized eigenvectors, and modal structure.

3. Construct controllability and observability matrices for MIMO systems.

4. Distinguish controllable, observable, reachable, and reconstructible modes.

5. Apply coordinate transformations and construct useful canonical representations.

6. Determine minimal realizations and understand pole-zero cancellations.

7. Relate state-space models to transfer-function matrices.

8. Analyze a practical multivariable application and evaluate modeling limitations.

1. State-Space Representation of MIMO Systems

A state is a set of variables whose present values, together with the future input, determine the future behavior of the system. A state vector is not unique: many different coordinate choices can describe the same physical system.

The state equation

x_dot = A x + B u

specifies internal evolution. The output equation

y = C x + D u

specifies measurement or controlled variables. The matrix D represents direct feedthrough from input to output. In many physical systems D = 0, although electrical, mechanical, and digital-interface models may contain nonzero direct terms.

For a model with n states, m inputs, and p outputs:

• A determines autonomous dynamics and has units consistent with inverse time.
• B maps actuator inputs into state derivatives.
• C maps states into measured outputs.
• D maps inputs directly into outputs.

The model is called a realization of the input-output behavior. It is minimal if no realization with fewer states has the same transfer-function matrix.

Derivation from physical equations

Consider a two-degree-of-freedom mechanical system with generalized position q and velocity q_dot. Let q ∈ R^2, mass matrix M, damping matrix F, stiffness matrix K, and input force vector u. The equations are

M q_ddot + F q_dot + K q = G u.

Assume M is nonsingular. Define

x = [q; q_dot].

Then

x_dot = [0 I; -M^(-1)K -M^(-1)F]x + [0; M^(-1)G]u.

If the measured output is position, y = Hq, then

C = [H 0], D = 0.

This conversion illustrates why state-space models are especially natural for multivariable physical systems: coupled masses, dampers, and actuators appear directly in the matrices rather than being forced into separate scalar transfer functions.

2. State Transition and Complete Solution

For the homogeneous system x_dot(t) = A x(t), the state-transition matrix is

Phi(t) = e^(At).

It is defined by the convergent matrix series

e^(At) = I + At + (At)^2/2! + (At)^3/3! + ... .

The matrix exponential satisfies

Phi(0) = I,

Phi(t1 + t2) = Phi(t1)Phi(t2),

and dPhi(t)/dt = A Phi(t) = Phi(t)A.

For the forced system with initial state x(0) = x0, variation of constants gives

x(t) = e^(At)x0 + ∫ from 0 to t e^(A(t-τ))B u(τ)dτ.

Consequently,

y(t) = C e^(At)x0 + ∫ from 0 to t C e^(A(t-τ))B u(τ)dτ + D u(t).

The first term is the zero-input response, and the integral is the zero-state response. This expression is valid for piecewise continuous inputs and finite-dimensional LTI models.

Computing the matrix exponential

Several methods are available:

1. Diagonalization. If A = VΛV^(-1), where Λ is diagonal, then e^(At) = V e^(Λt)V^(-1). This is efficient conceptually but requires a complete eigenvector basis.

2. Jordan form. If A is defective, Jordan blocks produce terms such as t e^(λt), t^2 e^(λt), and so on.

3. Cayley-Hamilton reduction. Any matrix function can be reduced to a polynomial in A of degree less than n.

4. Numerical scaling and squaring. This is generally preferred in software because it is more robust than direct eigenvector calculations for ill-conditioned matrices.

Worked Example 1: State transition matrix

Let

A = [0 1; -2 -3].

The characteristic polynomial is

det(sI - A) = s^2 + 3s + 2 = (s+1)(s+2).

The eigenvalues are -1 and -2. Corresponding eigenvectors may be selected as v1 = [1; -1] and v2 = [1; -2]. Thus

V = [1 1; -1 -2], V^(-1) = [-2 -1; 1 1].

Therefore,

e^(At) = V diag(e^(-t), e^(-2t)) V^(-1)

= [2e^(-t)-e^(-2t), e^(-t)-e^(-2t);

-2e^(-t)+2e^(-2t), -e^(-t)+2e^(-2t)].

At t = 0 this matrix equals I, as required. Its eigenvalues decay to zero, confirming asymptotic stability.

3. Eigenstructure and Modal Interpretation

The eigenvalues of A are the roots of det(sI-A)=0. They determine the natural modes of the state equation. For a real matrix, complex eigenvalues occur in conjugate pairs. A mode associated with λ = σ ± jω has envelope e^(σt) and oscillation frequency |ω| rad/s.

For continuous-time LTI systems:

• Re(λ) < 0 for every eigenvalue implies asymptotic stability.
• Any eigenvalue with Re(λ) > 0 implies instability.
• Eigenvalues on the imaginary axis require additional analysis. Repeated imaginary-axis eigenvalues with nontrivial Jordan blocks produce unbounded polynomial terms and therefore instability in the Lyapunov sense.

Eigenvectors show directions in state space associated with modes. Right eigenvectors describe state patterns; left eigenvectors help quantify how inputs and outputs couple to those modes. If A is diagonalizable,

A = VΛV^(-1), z = V^(-1)x,

and the transformed state equations are

z_dot = Λz + V^(-1)Bu,

y = CVz + Du.

The input participation of mode i depends on the ith row of V^(-1)B, while its output participation depends on the ith column of CV. A mode may be dynamically present but weakly actuated or weakly measured. This distinction is important in sensor and actuator placement.

For defective matrices, generalized eigenvectors are required. Jordan form is theoretically useful, but it is often numerically ill-conditioned. In practical computation, Schur decomposition is usually preferred because it preserves orthogonality and improves numerical reliability.

4. Controllability and Reachability

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

C_c = [B AB A^2B ... A^(n-1)B].

The system is completely controllable if

rank(C_c) = n.

For finite-dimensional continuous-time LTI systems, complete controllability is equivalent to complete reachability under standard unconstrained-input assumptions. In engineering usage, the terms are often used interchangeably, although reachability emphasizes states reachable from the origin.

The controllability Gramian over [0,T] is

W_c(T) = ∫ from 0 to T e^(Aτ) B B^T e^(A^Tτ)dτ.

For a stable A, the infinite-horizon Gramian satisfies the Lyapunov equation

A W_c + W_c A^T + B B^T = 0.

If W_c is positive definite, the system is controllable. The eigenvalues of the Gramian also indicate control difficulty: very small values correspond to directions requiring large input energy.

PBH controllability test

The Popov-Belevitch-Hautus test states that (A,B) is controllable if and only if

rank[λI - A B] = n

for every eigenvalue λ of A. Equivalently, no left eigenvector w^T satisfying w^T A = λw^T may also satisfy w^T B = 0.

This test is often more informative for MIMO systems because it examines each mode directly.

5. Observability and Reconstruction

A state is observable if its initial value can be uniquely determined from the input and output histories over a finite interval. The observability matrix is

C_o = [C; CA; CA^2; ...; CA^(n-1)].

The pair (A,C) is completely observable if

rank(C_o) = n.

The observability Gramian is

W_o(T) = ∫ from 0 to T e^(A^Tτ) C^T C e^(Aτ)dτ.

For stable A, the infinite-horizon Gramian satisfies

A^T W_o + W_o A + C^T C = 0.

A positive-definite observability Gramian indicates that every state direction affects the output sufficiently to be reconstructed. Small Gramian eigenvalues indicate poorly measured modes and sensitivity to sensor noise.

The PBH observability test states that (A,C) is observable if and only if

rank[λI - A; C] = n

for every eigenvalue λ. Equivalently, no right eigenvector v satisfying Av = λv may also satisfy Cv = 0.

Controllability and observability are dual concepts: (A,B) is controllable exactly when (A^T,B^T) is observable. This duality supports parallel analysis of actuator and sensor design.

6. Coordinate Transformations and Canonical Forms

Let x = Tz, where T is nonsingular. Then

z_dot = T^(-1)AT z + T^(-1)B u,

y = CTz + Du.

Thus the transformed matrices are

A_bar = T^(-1)AT,

B_bar = T^(-1)B,

C_bar = CT,

D_bar = D.

The transfer-function matrix remains unchanged:

G(s) = C(sI-A)^(-1)B + D.

Similarity transformations preserve eigenvalues, stability, controllability, observability, and input-output behavior, but they change the numerical values and interpretation of state variables.

Important canonical representations

Modal form separates independent modes when A is diagonalizable. It is valuable for interpretation and response calculations.

Jordan form represents repeated or defective eigenvalues, but is sensitive to numerical errors.

Controllable canonical form organizes states so that controllability is explicit. For SISO systems it is directly related to the denominator polynomial. MIMO systems generally require block or staircase forms rather than a single scalar companion form.

Observable canonical form is the dual of controllable canonical form.

Kalman decomposition partitions the state into four subspaces: controllable-observable, controllable-unobservable, uncontrollable-observable, and uncontrollable-unobservable. Only the controllable-observable portion contributes to a minimal input-output realization.

Canonical forms are coordinate descriptions, not necessarily physical models. A physically meaningful position or temperature state may become a mathematically convenient but nonphysical linear combination after transformation.

7. Transfer-Function Matrices and Realizations

Taking Laplace transforms with zero initial conditions gives

Y(s) = G(s)U(s),

where

G(s) = C(sI-A)^(-1)B + D.

For p outputs and m inputs, G(s) is a p×m matrix. Its element G_ij(s) is the transfer function from input j to output i with all other inputs set to zero.

The poles of G(s) need not include every eigenvalue of A. An internal mode may cancel algebraically because it is uncontrollable, unobservable, or both. Therefore, the eigenvalues of a nonminimal realization are not always the poles of the transfer-function matrix.

A realization is minimal if and only if it is both controllable and observable. For a given proper rational transfer-function matrix, all minimal realizations have the same order, although their matrices may differ by similarity transformations.

Example of cancellation

Consider

A = diag(-1,-2), B = [1;0], C = [1 0], D=0.

The second state evolves as x2_dot = -2x2 but is neither excited by the input nor measured at the output. The transfer function is

G(s) = 1/(s+1),

although A has two eigenvalues. The realization has order two but the minimal realization has order one. Removing the second state does not change the input-output behavior, but it does remove an internal autonomous mode. This distinction matters when assessing internal stability and designing observers or controllers.

8. Minimal Realizations in MIMO Systems

Finding a minimal realization involves identifying and eliminating uncontrollable and unobservable subspaces. A practical procedure is:

1. Form the controllability and observability matrices, or use numerically stable rank-revealing decompositions.

2. Transform the model into a Kalman or staircase decomposition.

3. Retain the controllable-observable block.

4. Recompute or verify the transfer-function matrix.

Exact rank decisions are straightforward symbolically but difficult numerically. In floating-point computation, singular values are compared with a tolerance related to machine precision, matrix dimension, and scaling. Poorly scaled states can make a controllable or observable direction appear artificially singular.

Minimality should not be confused with physical completeness. A nonminimal model may be deliberately retained when an internal variable is needed for safety monitoring, fault diagnosis, energy accounting, or nonlinear extension.

9. A Detailed Applied Case: Coupled-Tank Level Control

The following is an illustrative engineering case rather than a report of a particular industrial installation.

Background

Two liquid tanks are connected so that fluid flows from tank 1 to tank 2. Two pumps provide manipulated inputs. The controlled outputs are the liquid levels. Changes in one pump influence both levels, creating a MIMO plant.

Let h1 and h2 denote deviations from an equilibrium level, and let u1 and u2 denote small deviations in pump flows. A linearized model is

x_dot = A x + B u,

y = Cx,

with x = [h1; h2], y = [h1; h2], and

A = [-a 0; a -b],

B = [k1 k2; 0 k3],

C = I,

D = 0,

where a, b, k1, k2, and k3 are positive constants determined by tank areas, outlet resistances, and pump gains.

Problem

The controller must regulate both levels despite actuator coupling. Before designing feedback, the engineer must determine whether both states can be influenced and measured, and whether the model contains redundant states.

Analysis

Because C = I, both states are directly measured, so the observability matrix has rank two. The controllability matrix is

C_c = [B AB].

Its rank depends on the actuator gains. If k1 and k3 are nonzero, the two pumps generally excite independent directions, and rank(C_c)=2 except in special parameter combinations causing exact collinearity. A numerical rank test should be performed using scaled matrices.

The eigenvalues of A are -a and -b. For positive a and b, the open-loop plant is asymptotically stable. However, stability alone does not imply good control performance. If a and b are close, the modes may be difficult to separate; if one pump has a very small gain, the corresponding controllability Gramian may be poorly conditioned.

The transfer-function matrix is

G(s) = C(sI-A)^(-1)B.

Since

sI-A = [s+a 0; -a s+b],

its inverse is

(sI-A)^(-1) = [1/(s+a) 0; a/((s+a)(s+b)) 1/(s+b)].

Thus

G(s) =

[ k1/(s+a) + 0, k2/(s+a);

a k1/((s+a)(s+b)), a k2/((s+a)(s+b)) + k3/(s+b) ].

This matrix shows direct dynamic coupling: pump 1 affects both levels, while pump 2 affects tank 1 through the coupling represented by k2 and affects tank 2 through k3.

Approach and solution

A suitable design sequence is:

1. Validate the linearization near the intended operating point.

2. Scale states and inputs so numerical rank and conditioning are meaningful.

3. Evaluate controllability and observability Gramians.

4. Use state feedback if the states are measured reliably; otherwise design an observer.

5. Consider input allocation, actuator saturation, and pump interaction.

6. Test robustness against uncertain flow coefficients and sensor bias.

For a state-feedback law u = -Kx + N r, pole placement can assign desired closed-loop eigenvalues if the pair (A,B) is controllable. For poorly conditioned plants, a quadratic optimal method such as LQR may produce a better trade-off between level error and pump effort. Integral action may be added to remove steady-state offset caused by disturbances or parameter errors, but the augmented model must itself be checked for controllability.

Outcome

The state-space analysis reveals that the main design issue is not merely the location of the plant poles. It is the ability of the available pumps to independently influence the coupled level modes. Gramian-based conditioning can expose a weak direction before controller implementation. If one sensor fails, observability must be reassessed, and a fault-tolerant estimator may be required.

Learning points

The case demonstrates that:

• MIMO transfer matrices expose cross-coupling explicitly.
• Eigenvalues describe natural dynamics but not actuator effectiveness.
• Controllability and observability are properties of system-mode pairings.
• Numerical conditioning is as important as exact rank in engineering practice.
• Physical constraints must be included after, not ignored by, mathematical design.

10. Numerical and Implementation Considerations

State-space calculations are often performed using MATLAB, Python, Julia, or specialized control software. Reliable practice includes the following:

• Use Schur-based algorithms for eigenvalue and matrix-exponential calculations when possible.
• Avoid explicitly forming matrix inverses; solve linear systems instead.
• Scale states, inputs, and outputs before rank or conditioning analysis.
• Use singular-value decompositions for numerical rank decisions.
• Verify a realization by comparing its simulated response with the original model.
• Check dimensions at every matrix multiplication.
• Distinguish continuous-time sampling models from discrete-time models.

For a sampling interval T_s, the exact zero-order-hold discretization is

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

where

A_d = e^(A T_s),

B_d = ∫ from 0 to T_s e^(Aτ)B dτ.

If A is nonsingular, B_d = A^(-1)(A_d-I)B; however, this formula should not be used when A is singular or poorly conditioned. Augmented matrix-exponential methods are safer in general.

11. Limitations, Governance, and Contemporary Developments

State-space models are approximations. Linearization is local, parameters may vary with operating conditions, and unmodeled dynamics can invalidate conclusions about stability or performance. Actuator saturation, rate limits, dead zones, quantization, delays, and sensor faults are not automatically represented in the basic LTI model.

In safety-critical applications, a controller should not be approved solely because a nominal pair (A,B) is controllable or a nominal matrix A is stable. Model uncertainty, worst-case disturbances, independent verification, logging, and fail-safe behavior must be considered. State estimators also raise governance issues when automated decisions affect physical processes: assumptions, alarms, fallback modes, and human override should be documented.

Contemporary methods extend classical state-space analysis through robust control, descriptor systems, large-scale model reduction, data-driven realization, distributed control, and nonlinear or switched models. Machine-learning methods can assist system identification and fault detection, but they do not remove the need for physical validation, stability analysis, explainability, and safe operating constraints. Balanced truncation and related reduction methods are particularly useful when high-order finite-element or network models must be converted into computationally manageable approximations while controlling input-output error.

Best Practices

1. Define states from physical energy storage or clearly justified mathematical variables.

2. State units, operating point, assumptions, and sign conventions.

3. Check dimensions and verify the matrix exponential at t=0.

4. Use both rank tests and conditioning measures.

5. Separate internal stability from input-output stability.

6. Treat canonical forms as analytical tools, not automatically as physical models.

7. Validate minimality numerically and by transfer-function comparison.

8. Include actuator, sensor, saturation, delay, and uncertainty effects in implementation studies.

9. Document model provenance and parameter uncertainty.

10. Test controllers and observers against disturbances, faults, and unmodeled dynamics.

Chapter Summary

State-space analysis represents a continuous-time MIMO system through x_dot=Ax+Bu and y=Cx+Du. The matrix exponential e^(At) is the state-transition matrix and provides the complete solution for arbitrary initial conditions and inputs. Eigenvalues determine natural modal behavior, while eigenvectors and left eigenvectors describe modal directions and input-output participation.

Controllability determines whether inputs can influence all state directions; observability determines whether states can be reconstructed from outputs. Kalman rank tests, PBH tests, and controllability and observability Gramians provide complementary exact and quantitative assessments. Similarity transformations preserve system behavior while enabling modal, canonical, and Kalman decompositions. A realization is minimal exactly when it is both controllable and observable. Transfer-function matrices describe MIMO input-output behavior, but they may conceal internal modes through cancellation.

In practice, state-space analysis must be combined with numerical conditioning, model validation, uncertainty assessment, actuator and sensor constraints, and safety governance. These principles support robust modeling, controller design, observer construction, model reduction, and fault diagnosis in modern multivariable systems.

Key Terms

State: A set of variables that, with future inputs, determines future system behavior.

State vector: Column vector containing the selected state variables.

State-space model: First-order differential and algebraic equations describing system dynamics and outputs.

MIMO system: System with multiple inputs and/or multiple outputs.

State-transition matrix: Matrix e^(At) mapping an initial state through homogeneous dynamics.

Matrix exponential: Matrix-valued exponential defined by its power series or equivalent numerical algorithms.

Eigenvalue: Scalar satisfying det(λI-A)=0 and identifying a natural system mode.

Eigenvector: Nonzero vector defining a modal direction associated with an eigenvalue.

Jordan form: Canonical representation that describes repeated or defective eigenstructure.

Controllability: Ability to drive state variables to desired values using inputs.

Observability: Ability to reconstruct state variables from measured outputs.

PBH test: Rank-based eigenvalue test for controllability or observability.

Controllability Gramian: Matrix quantifying the input energy required to excite state directions.

Observability Gramian: Matrix quantifying how strongly state directions appear in outputs.

Similarity transformation: Nonsingular coordinate change preserving input-output behavior.

Transfer-function matrix: Matrix G(s)=C(sI-A)^(-1)B+D relating multiple inputs to outputs.

Realization: A state-space model implementing a specified input-output map.

Minimal realization: Lowest-order realization having the same transfer-function matrix.

Kalman decomposition: Coordinate decomposition into controllable/observable subspaces.

Modal participation: Degree to which an input excites or an output detects a particular mode.

Review Questions

Short Answer Questions

1. What roles do the matrices A, B, C, and D play in a continuous-time state-space model?

2. Define the state-transition matrix and state its value at t=0.

3. State the Kalman rank condition for complete controllability.

4. What is the physical interpretation of a poorly conditioned controllability Gramian?

5. Why can a realization have more states than poles visible in its transfer-function matrix?

Long Answer Questions

1. Derive the complete solution x(t) for x_dot=Ax+Bu using the state-transition matrix.

2. Explain modal, Jordan, controllable canonical, observable canonical, and Kalman forms, including their uses and limitations.

3. Derive the transfer-function matrix from a state-space model and discuss the role of the direct-feedthrough matrix D.

4. Explain why controllability and observability are dual properties and compare their PBH tests.

5. Describe a systematic procedure for obtaining a minimal realization of a MIMO transfer-function matrix or nonminimal state-space model.

Analytical / Application Questions

1. For A=[0 1; -4 -5], B=[0;1], and C=[1 0], calculate the eigenvalues, controllability matrix, and observability matrix. Interpret the results.

2. A two-input plant has a controllability Gramian with singular values 12, 0.03, and 0.0002. Explain what this indicates about actuator authority and propose two engineering responses.

3. In the coupled-tank case, one level sensor fails. Formulate the new output matrix and explain how you would determine whether an observer can still reconstruct both tank levels.

Multiple Choice Questions with Answers

Question 1

For the continuous-time system x_dot=Ax+Bu, which expression gives the state-transition matrix?

A. A^t

B. e^(At)

C. e^(Bt)

D. (sI-A)^(-1)

Correct Answer: B. e^(At)

Question 2

A pair (A,B) is completely controllable when which condition holds?

A. rank(C)=n

B. rank(B)=m

C. rank([B AB ... A^(n-1)B])=n

D. det(A)=1

Correct Answer: C. rank([B AB ... A^(n-1)B])=n

Question 3

Which matrix determines the transfer-function matrix of a state-space realization?

A. A+B+C+D

B. C(sI-A)^(-1)B+D

C. B(sI-C)^(-1)A

D. (sI-D)^(-1)C

Correct Answer: B. C(sI-A)^(-1)B+D

Question 4

If every eigenvalue of A has strictly negative real part, the autonomous continuous-time system is

A. Asymptotically stable

B. Marginally unstable

C. Uncontrollable

D. Noncausal

Correct Answer: A. Asymptotically stable

Question 5

A realization is minimal if and only if it is

A. Stable and diagonalizable

B. Controllable and observable

C. Observable and strictly proper

D. Reachable and noncausal

Correct Answer: B. Controllable and observable

Question 6

Under x=Tz, with T nonsingular, the transformed A matrix is

A. TAT^(-1)

B. T^(-1)AT

C. T+A+T^(-1)

D. AT^(-1)A

Correct Answer: B. T^(-1)AT

Question 7

The observability matrix for an n-state system is

A. [B AB ... A^(n-1)B]

B. [C; CA; ...; CA^(n-1)]

C. [A C; B D]

D. [I A ... A^n]

Correct Answer: B. [C; CA; ...; CA^(n-1)]

Question 8

A mode can be absent from a transfer-function matrix even though it is an eigenvalue of A when the realization is

A. Controllable and observable

B. Uncontrollable or unobservable

C. Strictly proper

D. Time invariant

Correct Answer: B. Uncontrollable or unobservable

Question 9

For a stable A, the controllability Gramian satisfies

A. AW_c-W_cA^T=BB^T

B. AW_c+W_cA^T+BB^T=0

C. A^TW_c-W_cA=0

D. W_cA+B=0

Correct Answer: B. AW_c+W_cA^T+BB^T=0

Question 10

In a MIMO transfer-function matrix, the element G_ij(s) represents

A. The eigenvalue associated with state i

B. The response from input j to output i with other inputs set to zero

C. The response from output i to input j

D. The determinant of the plant matrix

Correct Answer: B. The response from input j to output i with other inputs set to zero