GATE EE

State Space Analysis in Control Systems – Quick GATE EE Notes

Lecture Notes

SEC 01

Introduction

1State Space Analysis
1Key Concepts

  • State space representation

  • State equation solutions

  • Controllability & observability

  • Pole placement

  • State observers

  • Transfer function conversion

  • Canonical forms

  • Lyapunov stability

  • LQR and Kalman filtering

1Advantages

  • Handles MIMO systems

  • Reveals internal states

  • Modern control approach

  • Time-varying systems

  • Suitable for computer implementation

SEC 02

State Space Representation

1State Space Model
1Standard Equations
\[\dot{x}(t) = Ax(t) + Bu(t)\]
\[y(t) = Cx(t) + Du(t)\]
1Example

Mass-spring-damper system:

\[A = \begin{bmatrix} 0 & 1 \\ -k/m & -b/m \end{bmatrix}, B = \begin{bmatrix} 0 \\ 1/m \end{bmatrix}\]
SEC 03

Transfer Function to State Space

1Transfer Function to State Space Conversion
1Given Transfer Function
\[G(s) = \dfrac{Y(s)}{U(s)} = \dfrac{b_ns^n + b_{n-1}s^{n-1} + \dots + b_1s + b_0}{s^n + a_{n-1}s^{n-1} + \dots + a_1s + a_0}\]
1Controller Canonical Form
\[A = \begin{bmatrix} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & 1 \\ -a_0 & -a_1 & -a_2 & \dots & -a_{n-1} \end{bmatrix}, B = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \\ 1 \end{bmatrix}\]
\[C = \begin{bmatrix} b_0 & b_1 & \dots & b_{n-1} \end{bmatrix}, D = [b_n]\]
SEC 04

Canonical Forms

1Canonical Forms
1Observer Canonical Form
\[A = \begin{bmatrix} 0 & 0 & \dots & 0 & -a_0 \\ 1 & 0 & \dots & 0 & -a_1 \\ 0 & 1 & \dots & 0 & -a_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \dots & 1 & -a_{n-1} \end{bmatrix}\]
\[B = \begin{bmatrix} b_0 \\ b_1 \\ \vdots \\ b_{n-1} \end{bmatrix}\]
\[C = \begin{bmatrix} 0 & 0 & \dots & 0 & 1 \end{bmatrix}\]
1Diagonal 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 \(A\)
1Jordan Canonical Form
1Jordan Form

For repeated eigenvalues, Jordan blocks are used:

\[A = \begin{bmatrix} \lambda & 1 & 0 & \dots & 0 \\ 0 & \lambda & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & \lambda \end{bmatrix}\]
1State Space to Transfer Function
\[G(s) = C(sI - A)^{-1}B + D\]
This is the fundamental relationship between state space and transfer function
SEC 05

Solution of State Equations

1Solving State Equations
1Time Domain
\[x(t) = e^{At}x(0) + \int_0^t e^{A(t-\tau)}Bu(\tau)d\tau\]
\(e^{At}\): state transition matrix
1Laplace Domain
\[X(s) = (sI - A)^{-1}x(0) + (sI - A)^{-1}BU(s)\]
\[Y(s) = CX(s) + DU(s)\]
1State Transition Matrix Properties
\[\Phi(t) = e^{At} = \mathcal{L}^{-1}\{(sI - A)^{-1}\}\]
\[\Phi(0) = I, \quad \Phi(t_1 + t_2) = \Phi(t_1)\Phi(t_2)\]
\[\Phi^{-1}(t) = \Phi(-t)\]
SEC 06

Matrix Exponential

1Computing Matrix Exponential
1Methods to Calculate \(e^{At}\)

  1. Cayley-Hamilton theorem

  2. Laplace transform method: \(e^{At} = \mathcal{L}^{-1}\{(sI - A)^{-1}\}\)

  3. Similarity transformation

  4. Series expansion: \(e^{At} = I + At + \dfrac{(At)^2}{2!} + \dfrac{(At)^3}{3!} + \dots\)

1Example

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

\[e^{At} = \begin{bmatrix} 2e^{-t} - e^{-2t} & e^{-t} - e^{-2t} \\ -2e^{-t} + 2e^{-2t} & -e^{-t} + 2e^{-2t} \end{bmatrix}\]
SEC 07

Controllability & Observability

1Controllability & Observability
1Controllability

  • Can inputs control all states?

  • Test: Rank \([B\ AB\ A^2B\ \dots\ A^{n-1}B] = n\)

  • Controllability matrix: \(\mathcal{C}_o\)

  • Gramian: \(W_c = \int_0^T e^{At}BB^Te^{A^Tt}dt\)

1Example

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

1Observability
1Example

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

1Duality Theorem
1Duality

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

1PBH 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
\[\text{rank}\begin{bmatrix} \lambda I - A \\ C \end{bmatrix} = n \text{ for all } \lambda\]
SEC 08

Stabilizability & Detectability

1Stabilizability & Detectability
1Stabilizability

  • Uncontrollable modes must be stable

  • Condition: All uncontrollable eigenvalues in LHP

  • Weaker condition than controllability

  • Required for feedback stabilization

1Detectability
1Practical Importance

Required for closed-loop stability with feedback control

SEC 09

Pole Placement

1Pole Placement
1State Feedback
\[u = -Kx\]
\[\dot{x} = (A - BK)x\]
Poles determined by: \(\det(sI - (A - BK)) = 0\)
1Ackermann’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
1Bass-Gura Formula

Alternative method for pole placement design

SEC 10

State Observers

1State Observers
1Full-Order Observer
\[\dot{\hat{x}} = A\hat{x} + Bu + L(y - C\hat{x})\]
Error dynamics: \(\dot{e} = (A - LC)e\), where \(e = x - \hat{x}\)
1Separation Principle
1Reduced-Order Observer
SEC 11

Similarity Transformations

1Similarity Transformations
1Transformation

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

\[\bar{A} = TAT^{-1}, \quad \bar{B} = TB, \quad \bar{C} = CT^{-1}, \quad \bar{D} = D\]
1Invariant Properties
1Common Transformations
SEC 12

Lyapunov Stability

1Lyapunov Stability Analysis
1Lyapunov Equation
\[A^TP + PA + Q = 0\]
where \(Q > 0\) (positive definite)
1Stability Conditions
1Example

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.
SEC 13

Linear Quadratic Regulator

1Linear Quadratic Regulator (LQR)
1Cost Function
\[J = \int_0^\infty (x^TQx + u^TRu) dt\]
where \(Q \geq 0\), \(R > 0\)
1Optimal Control Law
\[u^* = -Kx = -R^{-1}B^TPx\]
where \(P\) solves Algebraic Riccati Equation (ARE):
\[A^TP + PA - PBR^{-1}B^TP + Q = 0\]
1Properties
SEC 14

Kalman Filter

1Kalman Filter
1Stochastic System
\[\dot{x} = Ax + Bu + w\]
\[y = Cx + v\]
where \(w \sim N(0, Q)\): process noise, \(v \sim N(0, R)\): measurement noise
1Kalman 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: \(K_{k+1} = P_{k+1|k}C^T(CP_{k+1|k}C^T + R)^{-1}\)
SEC 15

Minimal Realization

1Minimal Realization
1Definitions

  • Minimal realization: Controllable and observable

  • Dimension equals McMillan degree of transfer function

  • Unique up to similarity transformation

1Kalman Decomposition

System can be decomposed into four subsystems:

1Model Reduction

Remove uncontrollable and unobservable modes

SEC 16

GATE Questions

1GATE Question 1
1Example

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

  1. \(A\) is diagonal

  2. Controllability matrix rank = 2

  3. \(B\) has non-zero elements

  4. System is stable

1Solution

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

1GATE Question 2
1Example

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

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

  2. \(C = \begin{bmatrix} 1 & 2 \end{bmatrix}\)

  3. \(B = \begin{bmatrix} 0 \\ 1 \end{bmatrix}\)

  4. All of the above

1Solution

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

1GATE Question 3
1Example

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

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

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

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

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

1Solution

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

SEC 17

Important Formulas

1Key Formulas Summary
1Essential 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: \(A^TP + PA - PBR^{-1}B^TP + Q = 0\)

SEC 18

Summary

1Summary