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# First-Order Open-Loop and Closed-Loop Systems

An open loop control system for a first order system allows us to increase or decrease the static gain k of the system, but it does not allow us to change its time constant T, which represents a great limitation for the design of a system that fulfill specific tasks where, perhaps, a faster response is necessary (for a review of the k and T parameters see Sistema de primer orden). In contrast, with a closed-loop control system for a first-order system, we can vary both parameters. Let’s see it by simulating the response of the system to the unit step input.

Let us assume both cases, represented by the following Block Diagrams for a control system consisting of a controller and a plant. The transfer function (FT) of the first order plant is Gp(s), while the FT of the adjustable proportional controller isGc(s):

(See: Diagrama de Bloques)

Let’s see what happens considering the following values:

`First-Order Open-Loop System`

For the open-loop system it is satisfied that:

Atención: No confundir la K del controlador con la k (ganancia estática) del sistema (ver Sistema de primer orden).

The following Matlab script shows how the response (output) of the open-loop system to the unit step input varies as the gain of the K controller acquires the following values:

G=tf([2.9276],[1 0.2336]);
K=[1 2 3 4];
G1=K(1)*G; G2=K(2)*G;
G3=K(3)*G; G4=K(4)*G;
step(G1,G2,G3,G4)

legend(‘K=1′,’K=2′,’K=3′,’K=4’) Gráfica 1. Respuesta en el tiempo del sistema de primer orden a lazo abierto, a la entrada escalón unitario para diferentes valores de K del controlador proporcional.

In graph 1 we can see how the output of the system varies as the gain K of the controller changes. We can see that the static gain k of the first order system increases as the K of the controller increases. However, in each case, the time constant T remains constant. According to the First-Order System, the value of the time constant T is equal to;

In graph 2 we can see that the constant T, the time each system reaches 63.2% of its final value, remains constant for the 4 values of K considered: Gráfica 2. El valor de la constante de tiempo se mantiene constante para un sistema de primer orden a lazo abierto a medida que se varía la ganancia K del controlador proporcional.

Graph 3 allows us to see that the static gain k of each system as the gain K of the proportional controller increases is: Gráfica 3. La ganancia estática para un sistema de primer orden a lazo abierto a medida que se varía la ganancia K del controlador proporcional.

Most simple systems are zero, first, or second order. But then these simple systems interact with each other, generating higher order systems (third order onwards). An example is a solenoid, considered as a hybrid (electromechanical) system, represented by the following block diagram, where the series connection of three systems of first degree (electrical part), zero degree (transducer) and second degree ( mechanical part), respectively. It is also a good example of where in practice, we can find a first order open loop system: Definición de Sistema Electromecánico

`First-Order Closed-Loop System`

For the closed loop system it is satisfied that:

The following Matlab script shows how the response (output) of the closed-loop system to the unit step input varies as the gain of the K controller acquires the values indicated above:

G=tf([2.9276],[1 0.2336]); K=[1 2 3 4]; G1=K(1)*G; G2=K(2)*G; G3=K(3)*G; G4=K(4)*G;

sys1=feedback(G1,1);
sys2=feedback(G2,1);
sys3=feedback(G3,1);
sys4=feedback(G4,1);
step(sys1,sys2,sys3,sys4)
legend(‘K1=1′,’K2=2′,’K3=3′,’K4=4’)

Graph 4 shows how the system is faster as the gain K of the controller increases. That is, the time constant T of the closed-loop first-order system decreases as the gain K of the controller increases.:

The above results show that for the closed-loop system, we can use the gain K of the proportional controller to adjust the system in such a way that it responds at a given speed. Observe that the pole of the system moves to the left of the real axis as K increases.

sys=feedback(G1,1);
rlocus(sys) Gráfica 5. El Lugar de las raíces para un sistema de primer orden a lazo cerrado. El polo del sistema se desplaza hacia la izquierda del eje real a medida que aumenta la ganancia K del controlador proporcional.

Graph 6 shows the constant T of each system, the time each system reaches 63.2% of its final value: Gráfica 6. Para un sistema de primer orden a lazo cerrado el valor de la constante de tiempo disminuye a medida que aumenta la ganancia K del controlador proporcional.

Sources:

• Introducción a los sistemas de control con Matlab – Ricardo Gaviño
• Control Systems Engineering, Nise
• Sistemas de Control Automatico Benjamin C Kuo
• Modern_Control_Engineering, Ogata 4t

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