I am using the PID block in Simulink and limiting the output to (+/-)5. The plant becomes unstable and untunable after putting these limits. The plant is stable and tuneable otherwise if no limits are applied.
The output from the PID tuner goes beyond 10 when no limits are applied and I want to limit the output to 5. 
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2$\begingroup$ What exactly is the question? If you limit the controller output to $\pm 5$ you don't have enough control input - but so what? This is a bit like saying "if I only press the gas pedal half way down my car won't climb a hill because the engine slows down too much, and if I only press the brake pedal half way I can't stop when going down the hill." $\endgroup$– alephzeroCommented Jan 2, 2020 at 14:42
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2$\begingroup$ Please edit your question with your actuator transfer function (preferably broken down into single-order blocks), and let us know what integrator anti-windup method you used (see "Anti-windup method" in the PID block documentation). If the default behavior of the PID block is to limit the output without limiting the integrator, that would actually encourage problems with integrator windup. $\endgroup$– TimWescottCommented Jan 2, 2020 at 18:16
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Not all systems are tunable in all ranges. You can find via the Nyquist plot the stability ranges for the ranges of a gain the system can be stable. This can be plotted and analyzed in matlab.
H = tf([2 5 1],[1 2 3]);
nyquist(H)
