State Space Analysis in Control Systems

State Space Analysis

Key Concepts

State space representation
State equation solutions
Controllability & observability
Pole placement
State observers
Transfer function conversion
Canonical forms
Lyapunov stability
LQR and Kalman filtering

Advantages

Handles MIMO systems
Reveals internal states
Modern control approach
Time-varying systems
Suitable for computer implementation

State Space Model

Standard Equations

\[\dot{x}(t) = Ax(t) + Bu(t)\]

\[y(t) = Cx(t) + Du(t)\]

\(x\): state vector (\(n \times 1\))
\(u\): input vector (\(m \times 1\))
\(y\): output vector (\(p \times 1\))
\(A\): system matrix (\(n \times n\))
\(B\): input matrix (\(n \times m\))
\(C\): output matrix (\(p \times n\))
\(D\): feedforward matrix (\(p \times m\))

Example

Mass-spring-damper system:

\[A = \begin{bmatrix} 0 & 1 \\ -k/m & -b/m \end{bmatrix}, B = \begin{bmatrix} 0 \\ 1/m \end{bmatrix}\]

Transfer Function to State Space Conversion

Given Transfer Function

\[G(s) = \dfrac{Y(s)}{U(s)} = \dfrac{b_ns^n + b_{n-1}s^{n-1} + \dots + b_1s + b_0}{s^n + a_{n-1}s^{n-1} + \dots + a_1s + a_0}\]

Controller Canonical Form

\[A = \begin{bmatrix} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & 1 \\ -a_0 & -a_1 & -a_2 & \dots & -a_{n-1} \end{bmatrix}, B = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \\ 1 \end{bmatrix}\]

\[C = \begin{bmatrix} b_0 & b_1 & \dots & b_{n-1} \end{bmatrix}, D = [b_n]\]

Canonical Forms

Observer Canonical Form

\[A = \begin{bmatrix} 0 & 0 & \dots & 0 & -a_0 \\ 1 & 0 & \dots & 0 & -a_1 \\ 0 & 1 & \dots & 0 & -a_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \dots & 1 & -a_{n-1} \end{bmatrix}\]

\[B = \begin{bmatrix} b_0 \\ b_1 \\ \vdots \\ b_{n-1} \end{bmatrix}\]

\[C = \begin{bmatrix} 0 & 0 & \dots & 0 & 1 \end{bmatrix}\]

Diagonal Canonical Form

\[A = \begin{bmatrix} \lambda_1 & 0 & \dots & 0 \\ 0 & \lambda_2 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & \lambda_n \end{bmatrix}\]

Where \(\lambda_i\) are eigenvalues of \(A\)

Jordan Canonical Form

Jordan Form

For repeated eigenvalues, Jordan blocks are used:

\[A = \begin{bmatrix} \lambda & 1 & 0 & \dots & 0 \\ 0 & \lambda & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & \lambda \end{bmatrix}\]

State Space to Transfer Function

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

This is the fundamental relationship between state space and transfer function

Solving State Equations

Time Domain

\[x(t) = e^{At}x(0) + \int_0^t e^{A(t-\tau)}Bu(\tau)d\tau\]

\(e^{At}\): state transition matrix

Laplace Domain

\[X(s) = (sI - A)^{-1}x(0) + (sI - A)^{-1}BU(s)\]

\[Y(s) = CX(s) + DU(s)\]

State Transition Matrix Properties

\[\Phi(t) = e^{At} = \mathcal{L}^{-1}\{(sI - A)^{-1}\}\]

\[\Phi(0) = I, \quad \Phi(t_1 + t_2) = \Phi(t_1)\Phi(t_2)\]

\[\Phi^{-1}(t) = \Phi(-t)\]

Computing Matrix Exponential

Methods to Calculate \(e^{At}\)

Cayley-Hamilton theorem
Laplace transform method: \(e^{At} = \mathcal{L}^{-1}\{(sI - A)^{-1}\}\)
Similarity transformation
Series expansion: \(e^{At} = I + At + \dfrac{(At)^2}{2!} + \dfrac{(At)^3}{3!} + \dots\)

Example

For \(A = \begin{bmatrix} 0 & 1 \\ -2 & -3 \end{bmatrix}\):

\[e^{At} = \begin{bmatrix} 2e^{-t} - e^{-2t} & e^{-t} - e^{-2t} \\ -2e^{-t} + 2e^{-2t} & -e^{-t} + 2e^{-2t} \end{bmatrix}\]

Controllability & Observability

Controllability

Can inputs control all states?
Test: Rank \([B\ AB\ A^2B\ \dots\ A^{n-1}B] = n\)
Controllability matrix: \(\mathcal{C}_o\)
Gramian: \(W_c = \int_0^T e^{At}BB^Te^{A^Tt}dt\)

Example

\(A = \begin{bmatrix} -1 & 0 \\ 0 & -2 \end{bmatrix}, B = \begin{bmatrix} 1 \\ 1 \end{bmatrix}\): Controllable

Observability

Can states be inferred from outputs?
Test: Rank \([C^T\ A^TC^T\ \dots\ (A^T)^{n-1}C^T] = n\)
Observability matrix: \(\mathcal{O}_b\)
Gramian: \(W_o = \int_0^T e^{A^Tt}C^TCe^{At}dt\)

Example

Same \(A\), \(C = [1\ 1]\): Observable

Duality Theorem

Duality

\((A, B, C)\) is controllable if and only if \((A^T, C^T, B^T)\) is observable

PBH Test

Controllability: \((A, B)\) is controllable if and only if

\[\text{rank}[\lambda I - A\ B] = n \text{ for all } \lambda\]

Observability: \((A, C)\) is observable if and only if

\[\text{rank}\begin{bmatrix} \lambda I - A \\ C \end{bmatrix} = n \text{ for all } \lambda\]

Stabilizability & Detectability

Stabilizability

Uncontrollable modes must be stable
Condition: All uncontrollable eigenvalues in LHP
Weaker condition than controllability
Required for feedback stabilization

Detectability

Unobservable modes must be stable
Condition: All unobservable eigenvalues in LHP
Weaker condition than observability
Required for observer design

Practical Importance

Required for closed-loop stability with feedback control

Pole Placement

State Feedback

\[u = -Kx\]

\[\dot{x} = (A - BK)x\]

Poles determined by: \(\det(sI - (A - BK)) = 0\)

Ackermann’s Formula

For SISO systems:

\[K = [0\ \dots\ 0\ 1] \mathcal{C}_o^{-1} \alpha_c(A)\]

where \(\mathcal{C}_o\) is controllability matrix, \(\alpha_c(A)\) is desired characteristic polynomial

Bass-Gura Formula

Alternative method for pole placement design

State Observers

Full-Order Observer

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

Error dynamics: \(\dot{e} = (A - LC)e\), where \(e = x - \hat{x}\)

Separation Principle

Controller gain \(K\) and observer gain \(L\) designed independently
Ensures closed-loop stability
Combined system poles: \(\{A-BK\} \cup \{A-LC\}\)

Reduced-Order Observer

Estimates only unmeasurable states (order \(n-p\))
More efficient for systems with many outputs

Similarity Transformations

Transformation

Given system \((A, B, C, D)\), apply transformation \(z = Tx\):

\[\bar{A} = TAT^{-1}, \quad \bar{B} = TB, \quad \bar{C} = CT^{-1}, \quad \bar{D} = D\]

Invariant Properties

Eigenvalues (characteristic polynomial)
Transfer function
Controllability and observability
Rank of system matrices

Common Transformations

Modal transformation (diagonalization)
Balanced realization
Canonical form transformations

Lyapunov Stability Analysis

Lyapunov Equation

\[A^TP + PA + Q = 0\]

where \(Q > 0\) (positive definite)

Stability Conditions

System is asymptotically stable if \(P > 0\) exists
All eigenvalues of \(A\) have negative real parts
Lyapunov function: \(V(x) = x^TPx\)

Example

For \(A = \begin{bmatrix} -1 & 0 \\ 0 & -2 \end{bmatrix}\), choose \(Q = I\):

\[P = \begin{bmatrix} 0.5 & 0 \\ 0 & 0.25 \end{bmatrix} > 0\]

System is asymptotically stable.

Linear Quadratic Regulator (LQR)

Cost Function

\[J = \int_0^\infty (x^TQx + u^TRu) dt\]

where \(Q \geq 0\), \(R > 0\)

Optimal Control Law

\[u^* = -Kx = -R^{-1}B^TPx\]

where \(P\) solves Algebraic Riccati Equation (ARE):

\[A^TP + PA - PBR^{-1}B^TP + Q = 0\]

Properties

Guaranteed stability margins: \(\geq 60°\) phase margin
Infinite gain margin at loop breaking point
Optimal with respect to quadratic cost

Kalman Filter

Stochastic System

\[\dot{x} = Ax + Bu + w\]

\[y = Cx + v\]

where \(w \sim N(0, Q)\): process noise, \(v \sim N(0, R)\): measurement noise

Kalman Filter Equations

Prediction:

\[\hat{x}_{k+1|k} = A\hat{x}_{k|k} + Bu_k\]

Update:

\[\hat{x}_{k+1|k+1} = \hat{x}_{k+1|k} + K_{k+1}(y_{k+1} - C\hat{x}_{k+1|k})\]

Kalman gain: \(K_{k+1} = P_{k+1|k}C^T(CP_{k+1|k}C^T + R)^{-1}\)

Minimal Realization

Definitions

Minimal realization: Controllable and observable
Dimension equals McMillan degree of transfer function
Unique up to similarity transformation

Kalman Decomposition

System can be decomposed into four subsystems:

Controllable and observable
Controllable but not observable
Not controllable but observable
Neither controllable nor observable

Model Reduction

Remove uncontrollable and unobservable modes

GATE Question 1

Example

System: \(A = \begin{bmatrix} 0 & 1 \\ -2 & -3 \end{bmatrix}, B = \begin{bmatrix} 0 \\ 1 \end{bmatrix}\) is controllable because:

\(A\) is diagonal
Controllability matrix rank = 2
\(B\) has non-zero elements
System is stable

Solution

\(\mathcal{C}_o = [B\ AB] = \begin{bmatrix} 0 & 1 \\ 1 & -3 \end{bmatrix}\), rank = 2. Answer: B

GATE Question 2

Example

Transfer function \(G(s) = \dfrac{2s+1}{s^2+3s+2}\) in controller canonical form has:

\(A = \begin{bmatrix} 0 & 1 \\ -2 & -3 \end{bmatrix}\)
\(C = \begin{bmatrix} 1 & 2 \end{bmatrix}\)
\(B = \begin{bmatrix} 0 \\ 1 \end{bmatrix}\)
All of the above

Solution

From \(G(s) = \dfrac{2s+1}{s^2+3s+2}\): \(a_0 = 2, a_1 = 3, b_0 = 1, b_1 = 2\) All matrices are correct. Answer: D

GATE Question 3

Example

The state transition matrix \(\Phi(t) = e^{At}\) for \(A = \begin{bmatrix} -1 & 0 \\ 0 & -2 \end{bmatrix}\) is:

\(\begin{bmatrix} e^{-t} & 0 \\ 0 & e^{-2t} \end{bmatrix}\)
\(\begin{bmatrix} e^{t} & 0 \\ 0 & e^{2t} \end{bmatrix}\)
\(\begin{bmatrix} -e^{-t} & 0 \\ 0 & -2e^{-2t} \end{bmatrix}\)
\(\begin{bmatrix} e^{-t} & e^{-2t} \\ 0 & e^{-2t} \end{bmatrix}\)

Solution

For diagonal matrix \(A\), \(e^{At} = \text{diag}(e^{\lambda_1 t}, e^{\lambda_2 t})\). Answer: A

Key Formulas Summary

Essential Relations

Transfer function: \(G(s) = C(sI - A)^{-1}B + D\)
Controllability: rank\([B\ AB\ \dots\ A^{n-1}B] = n\)
Observability: rank\([C^T\ A^TC^T\ \dots\ (A^T)^{n-1}C^T] = n\)
Pole placement: \(\det(sI - (A - BK)) = 0\)
Observer: \(\dot{\hat{x}} = A\hat{x} + Bu + L(y - C\hat{x})\)
Lyapunov: \(A^TP + PA + Q = 0\)
LQR: \(A^TP + PA - PBR^{-1}B^TP + Q = 0\)

Summary

State space: models internal dynamics with matrices
Canonical forms: standard representations for analysis
Solutions: time/Laplace domain methods using \(e^{At}\)
Controllability/observability: fundamental system properties
Pole placement: state feedback control design
Observers: state estimation from outputs
Similarity transformations: preserve system properties
LQR: optimal quadratic control design
Kalman filter: optimal state estimation with noise
Minimal realization: controllable and observable systems

Introduction