Introduction
State Space Analysis
Key Concepts
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State space representation
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State equation solutions
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Controllability & observability
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Pole placement
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State observers
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Transfer function conversion
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Canonical forms
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Lyapunov stability
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LQR and Kalman filtering
Advantages
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Handles MIMO systems
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Reveals internal states
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Modern control approach
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Time-varying systems
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Suitable for computer implementation
State Space Representation
State Space Model
Standard Equations
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\(x\): state vector (\(n \times 1\))
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\(u\): input vector (\(m \times 1\))
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\(y\): output vector (\(p \times 1\))
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\(A\): system matrix (\(n \times n\))
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\(B\): input matrix (\(n \times m\))
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\(C\): output matrix (\(p \times n\))
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\(D\): feedforward matrix (\(p \times m\))
Example
Mass-spring-damper system:
Transfer Function to State Space
Transfer Function to State Space Conversion
Given Transfer Function
Controller Canonical Form
Canonical Forms
Canonical Forms
Observer Canonical Form
Diagonal Canonical Form
Jordan Canonical Form
Jordan Form
For repeated eigenvalues, Jordan blocks are used:
State Space to Transfer Function
Solution of State Equations
Solving State Equations
Time Domain
Laplace Domain
State Transition Matrix Properties
Matrix Exponential
Computing Matrix Exponential
Methods to Calculate \(e^{At}\)
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Cayley-Hamilton theorem
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Laplace transform method: \(e^{At} = \mathcal{L}^{-1}\{(sI - A)^{-1}\}\)
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Similarity transformation
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Series expansion: \(e^{At} = I + At + \dfrac{(At)^2}{2!} + \dfrac{(At)^3}{3!} + \dots\)
Example
For \(A = \begin{bmatrix} 0 & 1 \\ -2 & -3 \end{bmatrix}\):
Controllability & Observability
Controllability & Observability
Controllability
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Can inputs control all states?
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Test: Rank \([B\ AB\ A^2B\ \dots\ A^{n-1}B] = n\)
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Controllability matrix: \(\mathcal{C}_o\)
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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
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Can states be inferred from outputs?
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Test: Rank \([C^T\ A^TC^T\ \dots\ (A^T)^{n-1}C^T] = n\)
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Observability matrix: \(\mathcal{O}_b\)
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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
Stabilizability & Detectability
Stabilizability & Detectability
Stabilizability
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Uncontrollable modes must be stable
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Condition: All uncontrollable eigenvalues in LHP
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Weaker condition than controllability
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Required for feedback stabilization
Detectability
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Unobservable modes must be stable
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Condition: All unobservable eigenvalues in LHP
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Weaker condition than observability
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Required for observer design
Practical Importance
Required for closed-loop stability with feedback control
Pole Placement
Pole Placement
State Feedback
Ackermann’s Formula
For SISO systems:
Bass-Gura Formula
Alternative method for pole placement design
State Observers
State Observers
Full-Order Observer
Separation Principle
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Controller gain \(K\) and observer gain \(L\) designed independently
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Ensures closed-loop stability
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Combined system poles: \(\{A-BK\} \cup \{A-LC\}\)
Reduced-Order Observer
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Estimates only unmeasurable states (order \(n-p\))
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More efficient for systems with many outputs
Similarity Transformations
Similarity Transformations
Transformation
Given system \((A, B, C, D)\), apply transformation \(z = Tx\):
Invariant Properties
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Eigenvalues (characteristic polynomial)
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Transfer function
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Controllability and observability
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Rank of system matrices
Common Transformations
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Modal transformation (diagonalization)
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Balanced realization
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Canonical form transformations
Lyapunov Stability
Lyapunov Stability Analysis
Lyapunov Equation
Stability Conditions
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System is asymptotically stable if \(P > 0\) exists
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All eigenvalues of \(A\) have negative real parts
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Lyapunov function: \(V(x) = x^TPx\)
Example
For \(A = \begin{bmatrix} -1 & 0 \\ 0 & -2 \end{bmatrix}\), choose \(Q = I\):
Linear Quadratic Regulator
Linear Quadratic Regulator (LQR)
Cost Function
Optimal Control Law
Properties
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Guaranteed stability margins: \(\geq 60°\) phase margin
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Infinite gain margin at loop breaking point
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Optimal with respect to quadratic cost
Kalman Filter
Kalman Filter
Stochastic System
Kalman Filter Equations
Prediction:
Minimal Realization
Minimal Realization
Definitions
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Minimal realization: Controllable and observable
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Dimension equals McMillan degree of transfer function
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Unique up to similarity transformation
Kalman Decomposition
System can be decomposed into four subsystems:
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Controllable and observable
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Controllable but not observable
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Not controllable but observable
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Neither controllable nor observable
Model Reduction
Remove uncontrollable and unobservable modes
GATE Questions
GATE Question 1
Example
System: \(A = \begin{bmatrix} 0 & 1 \\ -2 & -3 \end{bmatrix}, B = \begin{bmatrix} 0 \\ 1 \end{bmatrix}\) is controllable because:
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\(A\) is diagonal
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Controllability matrix rank = 2
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\(B\) has non-zero elements
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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:
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\(A = \begin{bmatrix} 0 & 1 \\ -2 & -3 \end{bmatrix}\)
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\(C = \begin{bmatrix} 1 & 2 \end{bmatrix}\)
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\(B = \begin{bmatrix} 0 \\ 1 \end{bmatrix}\)
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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:
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\(\begin{bmatrix} e^{-t} & 0 \\ 0 & e^{-2t} \end{bmatrix}\)
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\(\begin{bmatrix} e^{t} & 0 \\ 0 & e^{2t} \end{bmatrix}\)
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\(\begin{bmatrix} -e^{-t} & 0 \\ 0 & -2e^{-2t} \end{bmatrix}\)
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\(\begin{bmatrix} e^{-t} & e^{-2t} \\ 0 & e^{-2t} \end{bmatrix}\)
Solution
For diagonal matrix \(A\), \(e^{At} = \text{diag}(e^{\lambda_1 t}, e^{\lambda_2 t})\). Answer: A
Important Formulas
Key Formulas Summary
Essential Relations
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Transfer function: \(G(s) = C(sI - A)^{-1}B + D\)
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Controllability: rank\([B\ AB\ \dots\ A^{n-1}B] = n\)
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Observability: rank\([C^T\ A^TC^T\ \dots\ (A^T)^{n-1}C^T] = n\)
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Pole placement: \(\det(sI - (A - BK)) = 0\)
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Observer: \(\dot{\hat{x}} = A\hat{x} + Bu + L(y - C\hat{x})\)
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Lyapunov: \(A^TP + PA + Q = 0\)
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LQR: \(A^TP + PA - PBR^{-1}B^TP + Q = 0\)
Summary
Summary
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State space: models internal dynamics with matrices
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Canonical forms: standard representations for analysis
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Solutions: time/Laplace domain methods using \(e^{At}\)
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Controllability/observability: fundamental system properties
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Pole placement: state feedback control design
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Observers: state estimation from outputs
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Similarity transformations: preserve system properties
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LQR: optimal quadratic control design
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Kalman filter: optimal state estimation with noise
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Minimal realization: controllable and observable systems