Controllability & Observability - Control Systems - Electrical Engineering

Controllability

Controllability of a dynamical system is the property that its state can be transferred from any initial value to any desired final value in finite time by means of a suitable control input. For linear time-invariant (LTI) state-space models the equations are written as

x˙(t) = A x(t) + B u(t)

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

Here A is the system (state) matrix, B the input matrix, C the output matrix and D the feed-through matrix. The vector x(t) contains the state variables, u(t) the inputs and y(t) the outputs.

Kalman's test for controllability

For an n×n matrix A and an input matrix B (for single-input systems B is n×1), form the controllability matrix Qc as

Qc = [ B AB A^2B ... A^{n-1}B ]

The matrix Qc has n rows. For a single-input system it has n columns; for a multi-input system it has n·m columns where m is the number of inputs.

The Kalman rank condition is:

• If rank(Qc) = n the system is controllable (every state is reachable).
• If rank(Qc) < n the system is uncontrollable; only rank(Qc) independent combinations of the states are reachable.

Equivalently, for a single-input system if det(Qc) ≠ 0 then Qc is full rank and the system is controllable. When rank(Qc) < n one can determine how many independent state directions are controllable by examining non-zero determinants of square submatrices of Qc:

• If any (n-1)×(n-1) submatrix has non-zero determinant then rank(Qc) = n-1 and there are n-1 controllable state directions.
• If all (n-1)×(n-1) determinants are zero, examine (n-2)×(n-2) submatrices, and so on, until a non-zero determinant is found.

Note: for multi-input systems use the rank condition (not a single determinant) because Qc will usually be rectangular with more than n columns.

Gilbert's test for controllability

Gilbert's test is a simple rule applicable when A is in Jordan canonical form or diagonal form. It inspects the structure of B relative to Jordan blocks of A.

It states that the system is controllable if the row(s) of B corresponding to the last row of each Jordan block are non-zero. For diagonal A (each Jordan block is 1×1) this reduces to the requirement that all rows of B are non-zero (for a single-input system, each row must have some non-zero entry in the column of B).

The method is illustrated below using a Jordan canonical form example.

Here the matrix A is in Jordan canonical form composed of Jordan blocks JB₁ and JB₂.

To check controllability we inspect the last row of each Jordan block. The last row of JB₁ corresponds to the second row of A and the last row of JB₂ corresponds to the third row of A. According to Gilbert's test we check the corresponding rows of B (the second and third rows). If these rows are non-zero then the system is controllable.

A diagonal canonical form may be treated as a Jordan form with 1×1 blocks. For example:

Each Jordan block JB₁, JB₂, JB₃ has a single row; Gilbert's rule requires that the corresponding rows of B are non-zero for controllability.

Remarks and consequences

• Controllability is necessary for state-feedback design that places closed-loop poles arbitrarily. If a system is not fully controllable, only the controllable portion of the system can be moved by state feedback.
• For systems that are not fully controllable, the concept of stabilisability is useful: an unstable mode that is uncontrollable makes the system not stabilisable, while uncontrollable stable modes can be acceptable in some designs.
• Controllability is a structural property of (A,B); it does not depend on C or D.

Worked example (single-input 3×3)

Check controllability for the system with

A = [ [0,1,0], [0,0,1], [-2,-3,-4] ], B = [0,0,1]^T

Form the controllability matrix Qc.

B = [0,0,1]^T

AB = A·B = [1,1,-4]^T

A^2B = A·(AB) = [1,-3,10]^T

Qc = [ B AB A^2B ] = [ [0,1,1]; [0,1,-3]; [1,-4,10] ] (columns shown)

Compute rank(Qc). Evaluate whether the three columns are linearly independent. If they are, rank(Qc)=3 and the system is controllable. (Omitted routine arithmetic steps; perform determinant or row reduction to confirm rank.)

Observability

Observability of a system is the property that the initial state (or current state) of the system can be uniquely determined from measurements of the outputs over a finite time interval and knowledge of the inputs. For the LTI state model given earlier the question is whether x(0) can be inferred from y(t) and u(t) for 0 ≤ t ≤ T for some finite T.

Kalman's test for observability

For an n×n matrix A and an output matrix C (for single-output systems C is 1×n) form the observability matrix Q_o as

In algebraic form

Qo = [ C^T (CA)^T (CA^2)^T ... (CA^{n-1})^T ]^T

Equivalently Qo is the vertical stacking

Qo = [ C ; CA ; CA^2 ; ... ; CA^{n-1} ]

The Kalman observability condition is:

• If rank(Qo) = n the system is observable (all state components can be determined from the outputs).
• If rank(Qo) < n the system is unobservable; only rank(Qo) independent state combinations can be reconstructed.

If rank(Qo) < n one may inspect submatrices of decreasing order to find how many independent states are observable (same procedure as for controllability, using determinants of square submatrices where applicable).

Gilbert's test for observability

Gilbert's test for observability is the dual of the controllability version and applies when A is in Jordan or diagonal canonical form. It inspects columns of C corresponding to the first row of each Jordan block.

It states that the system is observable if the column(s) of C corresponding to the first row of each Jordan block are non-zero. For diagonal A this reduces to requiring that all columns of C are non-zero (for single-output systems, each column must have a non-zero entry in the corresponding position).

Illustration for a Jordan form:

Suppose A is in Jordan canonical form with Jordan blocks JB₁ and JB₂.

To check observability we inspect the first row of each Jordan block. The first row of JB₁ corresponds to the first row of A and the first row of JB₂ corresponds to the third row of A. According to Gilbert's test we check the corresponding columns of C (the first and third columns). If these columns are non-zero then the system is observable.

For diagonal canonical form (1×1 blocks):

Each Jordan block JB₁, JB₂, JB₃ yields a first row corresponding to the first, second and third rows of A respectively, and Gilbert's rule requires that the corresponding columns of C be non-zero for full observability.

Remarks and consequences

• Observability is necessary to design state observers (estimators) such as the Luenberger observer or Kalman filter. If the system is not observable, a full state observer cannot reconstruct all states from outputs.
• Observability is a property of the pair (A,C); it does not depend on B or D.
• The concepts of detectability and stabilisability are weaker forms useful in design: a system is detectable if unobservable modes are stable, and stabilisable if uncontrollable modes are stable.

Duality between controllability and observability

Controllability and observability are dual properties. The pair (A,B) is controllable if and only if the pair (A^T,C^T) is observable. This duality can be used to translate tests and results from one property to the other.

Practical applications

• State-feedback design and pole placement require controllability to place closed-loop eigenvalues arbitrarily.
• Design of observers and state estimators requires observability to reconstruct internal states from measured outputs.
• Model reduction and minimal realisation: to obtain a minimal state-space model, remove unobservable and uncontrollable modes; a minimal realisation must be both controllable and observable.
• Fault detection and system monitoring often rely on observability to detect changes in state from measurements.

Summary

Controllability asks whether inputs can move the system state anywhere in finite time; observability asks whether outputs reveal the full state. Kalman's rank tests give algebraic conditions using the controllability matrix Qc and the observability matrix Qo. Gilbert's tests give quick structural checks when A is in Jordan or diagonal form. These concepts are fundamental for state-space control design, observer design, and model reduction.