DIGITAL CONTROL SYSTEM – EEE011
B.Tech (SEM VI) | Section-wise Solved Answers
SECTION – A
(10 × 2 = 20 marks)
(a) State-space representation of digital control system
A digital control system can be represented in state-space form as
x(k+1)=Ax(k)+Bu(k),y(k)=Cx(k)+Du(k)x(k+1)=Ax(k)+Bu(k), \quad y(k)=Cx(k)+Du(k)x(k+1)=Ax(k)+Bu(k),y(k)=Cx(k)+Du(k)
where x(k)x(k)x(k) is the state vector, u(k)u(k)u(k) the input, and y(k)y(k)y(k) the output at sampling instant kkk.
(b) Controller design from continuous to digital system
A continuous controller is converted to digital form using methods such as ZOH equivalent, bilinear (Tustin) transformation, or matched pole–zero method, ensuring similar dynamic behavior.
(c) Acquisition time for sample and hold
Acquisition time is the time required by the sample-and-hold circuit to acquire and hold the input signal accurately within a specified error band.
(d) Shifting property of Z-transform
If Z{x(k)}=X(z)Z\{x(k)\}=X(z)Z{x(k)}=X(z), then
Z{x(k−n)}=z−nX(z)Z\{x(k-n)\}=z^{-n}X(z)Z{x(k−n)}=z−nX(z)
(e) Initial and final values
Given
X(z)=2+3z−1+4z−2X(z)=2+3z^{-1}+4z^{-2}X(z)=2+3z−1+4z−2
Initial value: x(0)=limz→∞X(z)=2x(0)=\lim_{z\to\infty}X(z)=2x(0)=limz→∞X(z)=2
Final value: x(∞)=limz→1(1−z−1)X(z)=0x(\infty)=\lim_{z\to1}(1-z^{-1})X(z)=0x(∞)=limz→1(1−z−1)X(z)=0
(f) Cayley–Hamilton theorem
Every square matrix satisfies its own characteristic equation. It is widely used to compute matrix powers and state transition matrices.
(g) Pulse transfer function of ZOH
The pulse transfer function of a zero-order hold is
G(z)=Z{1−e−sTsG(s)}G(z)=Z\left\{\frac{1-e^{-sT}}{s}G(s)\right\}G(z)=Z{s1−e−sTG(s)}
(h) Equilibrium point of nonlinear system
An equilibrium point is obtained by setting state derivatives (or differences) equal to zero and solving the resulting equations.
(i) Euler–Lagrange equation
It provides the necessary condition for optimal control problems and is derived from the calculus of variations.
(j) Asymptotic stability
A system is asymptotically stable if its state variables not only remain bounded but also approach zero as time goes to infinity.
SECTION – B
(Attempt any five – 10 marks each)
(a) Digital control system block diagram
A basic digital control system consists of sampler, A/D converter, digital controller, D/A converter, hold device, plant, and feedback path.
Laplace vs Z-transform: Laplace is used for continuous-time systems, while Z-transform is used for discrete-time systems.
(b) Controllability and observability
A system is controllable if the state can be driven to any desired value using input.
It is observable if the initial state can be determined from output measurements.
Sample and hold circuits discretize continuous signals for digital processing.
(c) Jury stability test
Jury’s test is applied to discrete systems to determine the range of gain KKK for which all roots lie inside the unit circle, ensuring stability.
(d) Pulse transfer function conditions
Controllability and observability are determined using rank conditions of controllability and observability matrices.
The bilinear transformation maps the z-plane to the w-plane for stability analysis.
(e) Jury stability criteria (numerical)
For
F(z)=z3−1.25z2−1.375z−0.25F(z)=z^3-1.25z^2-1.375z-0.25F(z)=z3−1.25z2−1.375z−0.25
Jury’s table is constructed to check stability. The system is found unstable due to violation of necessary conditions.
(f) Relation between G(s)G(s)G(s) and G(z)G(z)G(z)
The pulse transfer function is obtained using ZOH and relates discrete response to continuous dynamics.
(g) Principle of optimality and dynamic programming
The principle states that an optimal policy has the property that remaining decisions must be optimal regardless of previous decisions.
(h) Design in W-plane
W-plane design simplifies discrete stability analysis by mapping the z-plane into an equivalent continuous domain.
SECTION – C
(Attempt any two – 15 marks each)
(3) Optimal control problem
The optimal control u0(k)u_0(k)u0(k) is obtained by minimizing the performance index subject to system constraints using dynamic programming or Euler–Lagrange equations.
Solutions differ based on whether the final state is fixed or free.
(4)
(i) Z-transform of f(k)=(1/2)kf(k)=(1/2)^kf(k)=(1/2)k is
Z{(1/2)k}=zz−12Z\{(1/2)^k\}=\frac{z}{z-\frac{1}{2}}Z{(1/2)k}=z−21z
(ii) State transition matrix is computed using Cayley–Hamilton theorem.
(5) Liapunov stability
A discrete system is stable if a suitable Liapunov function exists.
For
X1(k+1)=−0.5X1(k),X2(k+1)=−0.5X2(k)X_1(k+1)=-0.5X_1(k),\quad X_2(k+1)=-0.5X_2(k)X1(k+1)=−0.5X1(k),X2(k+1)=−0.5X2(k)
Both eigenvalues lie inside the unit circle, hence the system is asymptotically stable.
