To stabilize the inherently unstable system in the given block diagram, we can analyze and design a feedforward compensator and/or a phase compensator. Below is the step-by-step process for achieving stability:

1. Analyze the Open-Loop Transfer Function

The open-loop transfer function is the product of the components in the forward path: $G(s) = k_c \cdot \frac{1}{s(s-1)}$

• The pole at $s=1$ is in the right-half plane (RHP), making the system unstable.

2. Introduce Feedforward Compensation

We need to modify the open-loop transfer function using a feedforward compensator $C(s)$. A suitable approach is to use pole-zero cancellation or adjust the dynamics of the system.

Feedforward compensator design:

Let $C(s) = \frac{s+a}{s+b}$, where $a$ and $b$ are constants.

To counteract the instability introduced by the pole at $s=1$, set $b > 1$. For simplicity: $C(s) = \frac{s+a}{s+b}, \quad b > 1$

This introduces a new stable pole at $s = -b$. The zero at $s = -a$ can be adjusted to improve transient response.

3. Add Phase Compensation

To ensure stability and improve the phase margin, you can implement a phase lead compensator of the form: $C_{\text{lead}}(s) = k_{\text{lead}} \cdot \frac{s+z}{s+p}, \quad |z| > |p|$ This compensator increases the phase of the system at critical frequencies and helps stabilize the feedback system.

Typical tuning steps:

1. Choose zero $z$: Place it near the unstable pole (e.g., $z = 1$).
2. Choose pole $p$: Place it farther left in the complex plane (e.g., $p = 5$).

Example: $C_{\text{lead}}(s) = k_{\text{lead}} \cdot \frac{s+1}{s+5}$

The compensator stabilizes the phase response of the system.

4. Closed-Loop Stability Verification

After adding the feedforward and phase compensator, verify the closed-loop transfer function stability. The new open-loop transfer function becomes: $G'(s) = C(s) \cdot G(s) = \left(\frac{s+a}{s+b}\right) \cdot k_c \cdot \frac{1}{s(s-1)}$ Analyze the poles of the closed-loop transfer function to ensure all poles are in the left-half plane (LHP).

5. Practical Guidelines

1. Use software tools like MATLAB to simulate and verify the step response and stability.
2. Adjust $k_c$, $a$, $b$, $z$, and $p$ to fine-tune the response.
3. Ensure sufficient phase margin (typically $45^\circ$ or higher) for robustness.

Would you like a MATLAB simulation or further details on compensator design?

Related Questions to Explore:

1. What is the root-locus plot of the original system?
2. How does the feedforward compensator affect the system’s stability?
3. What phase margin is required to stabilize this system?
4. How can the lead compensator improve the transient response?
5. What is the effect of varying the controller gain $k_c$ on stability?

Tip:

Always verify the closed-loop poles using the characteristic equation $1 + G'(s)H(s) = 0$.