GATE Electrical Engineering Control Systems Syllabus

State the approximate settling time $t_s$ (2% criterion) for a second-order system.

$$t_s \approx \frac{4}{\zeta\omega_n}$$ For the 5% criterion, $t_s \approx \dfrac{3}{\zeta\omega_n}$.

Give the peak time $t_p$ expression for an underdamped second-order system.

$$t_p = \frac{\pi}{\omega_d} = \frac{\pi}{\omega_n\sqrt{1-\zeta^{2}}}$$

What is the rule for blocks connected in cascade (series) in a block diagram?

The equivalent transfer function is the product of the individual transfer functions: $G_{eq}(s) = G_1(s)\,G_2(s)\,\cdots\,G_n(s)$ (assuming no loading effects).

What is the rule for blocks connected in parallel in a block diagram?

The equivalent transfer function is the algebraic sum of the individual transfer functions: $G_{eq}(s) = G_1(s) \pm G_2(s) \pm \cdots$ depending on the summing-junction signs.

State the block-diagram reduction rule for a basic negative feedback loop.

A forward block $G$ with feedback block $H$ reduces to $$\frac{G}{1+GH}$$ (use $1-GH$ for positive feedback).

When moving a summing junction ahead of (before) a block $G$, how must the moved signal be modified?

The signal entering the relocated summing point must be passed through $\frac{1}{G}$ to keep the diagram equivalent.

When moving a takeoff (pickoff) point from after a block $G$ to before it, what modification is needed?

The branch taken off must be multiplied by $G$ (insert a block $G$ in that branch) to preserve the original signal value.

Define a signal flow graph (SFG).

A graphical representation of a set of linear algebraic equations using nodes (variables) connected by directed branches; each branch has a gain (transmittance), and signals flow only in the arrow direction.

In a signal flow graph, what is a 'forward path' and a 'forward path gain'?

A forward path is a path from the input node to the output node along which no node is encountered more than once. The forward path gain is the product of the branch gains along that path.

Define a 'loop' and 'loop gain' in a signal flow graph.

A loop is a closed path that starts and ends at the same node, encountering no node more than once. The loop gain is the product of all branch gains around that loop.

What are two loops called when they share no common node?

Non-touching loops. Two paths or loops are non-touching if they have no node in common; this concept is central to Mason's gain formula.

State Mason's gain formula for the overall transfer function of a signal flow graph.

$$T = \frac{\sum_{k} P_k \Delta_k}{\Delta}$$ where $P_k$ = gain of the $k$-th forward path, $\Delta$ = graph determinant, and $\Delta_k$ = value of $\Delta$ with all loops touching path $k$ removed.

Write the expression for the graph determinant $\Delta$ in Mason's formula.

$$\Delta = 1 - \sum L_i + \sum L_i L_j - \sum L_i L_j L_k + \cdots$$ i.e. 1 minus the sum of all individual loop gains, plus the sum of products of gains of all pairs of non-touching loops, minus triples, and so on.

In Mason's gain formula, how is the path cofactor $\Delta_k$ obtained?

$\Delta_k$ is $\Delta$ evaluated by eliminating all loops that touch the $k$-th forward path (i.e. set the gains of loops touching path $k$ to zero in the $\Delta$ expression).

Compare block diagrams and signal flow graphs as system representations.

Both represent the same equations. Block diagrams use blocks and summing junctions and require step-by-step reduction; SFGs use nodes and branches and allow a one-shot solution via Mason's gain formula without successive reduction.

What is the general state-space representation of a continuous LTI system?

$$\dot{\vec{x}}(t) = A\vec{x}(t) + B\vec{u}(t)$$ $$\vec{y}(t) = C\vec{x}(t) + D\vec{u}(t)$$ where $A$ is the system matrix, $B$ input matrix, $C$ output matrix, $D$ feedforward matrix.

Define the state transition matrix $\phi(t)$ of an LTI system.

$\phi(t) = e^{At}$, the matrix exponential. It maps the initial state to the state at time $t$ in the zero-input case: $\vec{x}(t)=\phi(t)\vec{x}(0)$.

Give the complete solution of the state equation $\dot{\vec{x}}=A\vec{x}+B\vec{u}$ with initial state $\vec{x}(0)$.

$$\vec{x}(t) = e^{At}\vec{x}(0) + \int_{0}^{t} e^{A(t-\tau)} B\,\vec{u}(\tau)\,d\tau$$ The first term is the zero-input response; the integral (convolution) is the zero-state response.

How is the state transition matrix obtained using the Laplace transform?

$$\phi(t) = e^{At} = \mathcal{L}^{-1}\big\{(sI - A)^{-1}\big\}$$ where $I$ is the identity matrix.

State the Laplace-domain solution of the state equation including initial conditions.

$$\vec{X}(s) = (sI-A)^{-1}\vec{x}(0) + (sI-A)^{-1} B\,\vec{U}(s)$$

List three key properties of the state transition matrix $\phi(t)=e^{At}$.

$\phi(0)=I$; $\phi(t_1+t_2)=\phi(t_1)\phi(t_2)$; $\phi^{-1}(t)=\phi(-t)$. Also $\dot{\phi}(t)=A\phi(t)$.

How do you obtain the transfer function from a state-space model $(A,B,C,D)$?

$$G(s) = C(sI-A)^{-1}B + D$$ The poles of $G(s)$ are the eigenvalues of $A$.

What is the relationship between the eigenvalues of matrix $A$ and the system's stability?

The eigenvalues of $A$ equal the system poles. The LTI system is asymptotically stable if and only if all eigenvalues of $A$ have strictly negative real parts (lie in the left half of the s-plane).

What is the characteristic equation of a state-space system?

$$\det(sI - A) = |sI - A| = 0$$ Its roots are the eigenvalues of $A$ and equal the system poles.

For the homogeneous (zero-input) system $\dot{\vec{x}}=A\vec{x}$, what is the solution?

$$\vec{x}(t) = e^{At}\vec{x}(0) = \phi(t)\vec{x}(0)$$ It is determined entirely by the initial conditions and the state transition matrix.

State the Cayley-Hamilton theorem and one of its uses in state-space analysis.

Every square matrix satisfies its own characteristic equation: if $|sI-A|=0$ gives the characteristic polynomial, then substituting $A$ for $s$ yields the zero matrix. It is used to compute $e^{At}$ and matrix powers as finite polynomials in $A$.

Give the series-expansion definition of the matrix exponential $e^{At}$.

$$e^{At} = I + At + \frac{A^{2}t^{2}}{2!} + \frac{A^{3}t^{3}}{3!} + \cdots = \sum_{k=0}^{\infty}\frac{(At)^{k}}{k!}$$

Why is the state-space representation of a system non-unique?

Any similarity (linear) transformation $\vec{z}=P^{-1}\vec{x}$ gives a new model $(\hat A=P^{-1}AP,\ \hat B=P^{-1}B,\ \hat C=CP,\ \hat D=D)$ with the same input-output transfer function. The choice of state variables is arbitrary.

What two system properties are preserved under a similarity transformation of the state-space model?

The eigenvalues (poles) and the transfer function $G(s)$ are invariant. Controllability and observability are also preserved.

Compare the transfer-function approach with the state-space approach to system modeling.

Transfer functions describe only the input-output (external) behavior of SISO LTI systems with zero initial conditions and hide internal states. State-space captures internal (state) variables, handles MIMO systems, nonzero initial conditions, and time-varying/nonlinear extensions.