Lecture 5 Root-Locus PDF

MOD007954 Automation & Control S-Plane: Poles and Zeros A linear system can be represented by the following transfer function: Zeros: zi (the roots of the numerator) Poles: pi (the roots of the denominator or of the system or characterestic equation) These can only be real or occur in complex con...

MOD007954 Automation & Control S-Plane: Poles and Zeros A linear system can be represented by the following transfer function: Zeros: zi (the roots of the numerator) Poles: pi (the roots of the denominator or of the system or characterestic equation) These can only be real or occur in complex conjugate pairs, as all parameters of a real system must be real. Locating poles and zeros with MATLAB The roots of a polynomial (numerator or denominator of a transfer function) can be found using the MATLAB command roots (P) where P is a vector containing the coefficients of the polynomial. The roots of the polynomial then represent the poles and zeros of the transfer function. Alternatively, a map of the poles and zeros of the LTI system sys=tf(num,den) in the s-plane can be obtained with MATLAB using the command: pzmap(sys) or directly: pzmap(num,den) For example, to identify the locations of the poles and zeros of the LTI system given by: on the s-plane, we type on the MATLAB command line: sys = tf([0 2 1 3],[1 4 2 -1]) pzmap(sys) The following output will then be produced. Note that poles are identified with crosses, while zeros are identified with open circles. Lecture 5_ Root- Locus Page | 1 MOD007954 Automation & Control The Stability of Linear Feedback Systems The concept of stability Fundamentally important. An unstable closed-loop system is of no use in practice. Many physical systems are inherently open-loop unstable (e.g. aircraft, inverse pendulum, etc.). In designing a control system, we must be able to predict the dynamic behaviour of the system from knowledge of the components. A linear time-invariant control system is stable if the output eventually comes back to its equilibrium state when the system is subjected to an initial condition. It is marginally stable if oscillations of the output continue forever. It is unstable if the output diverges without bound from its equilibrium state when the system is subjected to an initial condition. Mathematical description Consider the first-order differential equation: where 'a' is a constant and the initial condition is y(0) = 0. Laplace transform with impulse input: Time response: y(t) = e-at. There are three cases: a > 0, y(∞) = 0 stable a = 0, y(∞) = 1 neutral (marginally stable) a < 0, y(∞) = ∞ unstable Lecture 5_ Root- Locus Page | 2 MOD007954 Automation & Control Graphical representation Location of the pole Time response FACT: A system is stable if ALL the closed-loop poles are in the LEFT HALF of the s-plane and have negative real parts, i.e. all the roots of the characteristic equation are in the left-hand splane. Lecture 5_ Root- Locus Page | 3 MOD007954 Automation & Control A general closed-loop system transfer function: To use the above fact, we need to solve the characteristic equation: A necessary condition for stability of the system is that all the coefficients of the closed-loop characteristic equation ai are positive. This is to say that if at least one of the coefficients of the characteristic equation q(s) is negative, we can conclude that the system is unstable. However, it does not mean that if all the coefficients of q(s) are positive, the system is stable. For example: q(s) = s3 + 3s2 + 2s + 3 ® stable All poles in left half of s-plane (s = -2.67, s = -0.16 ± j1.05) q(s) = s3 + 3s2 + 2s + 7 ® unstable Two poles in right half of s-plane (s = -3.09, s = 0.04 ± j1.50) Lecture 5_ Root- Locus Page | 4 MOD007954 Automation & Control 2. Root Locus Closed Loop Feedback Systems + k G - H G = Plant H = Feedback K = Gain Compensator The closed loop transfer function of the above system = kG 1 + kGH What is a compensator (controller)? To achieve a stable feedback control system compensators may need to be included in the forward path. The compensators job is to modify the performance characteristic of a system so that the required characteristics are obtained. For example a pre-amplifier in an electronic measurement system or a damper in a suspension system. The term compensator or controller receive an error signal at its input and output signal to modify the output. Before we look at the design process a detailed look at closed loop systems is required. Lecture 5_ Root- Locus Page | 5 MOD007954 Automation & Control What are the effects on System Characteristics? Consider an open loop system with a transfer function and a series compensator below. 1 k ( s + 4)( s + 20) We can see that the open loop poles are at s = - 4 and s = - 20, the gain k is simply a multiplier. If we now close the loop with unity feedback (H = 1) and apply kG 1 + kGH k ×1 ( s + 4)( s + 20) = k ×1 1+ ×1 ( s + 4)( s + 20) Simplify by multiplying top and bottom by (s+4)(s+20) ( s + 4)( s + 20)k ( s + 4)( s + 20) = ( s + 4)( s + 20)k ( s + 4)( s + 20) + ( s + 4)( s + 20) After cancelling above common terms we get k k = = 2 ( s + 4)( s + 20) + k ( s + 24 s + 80) + k Lecture 5_ Root- Locus Page | 6 MOD007954 Automation & Control We will now see the effect on the system response (pole locations) when we vary the gain between the settings 20, 40 and 80. Each time we re-calculate the quadratic in the denominator and then re-calculate the roots and hence plot the new pole locations. The plot that we create is called a Root Locus. Class Exercise: Plotting the root locus Calculate the closed loop pole locations for k = 20, k = 40 and k = 80 and plot the locations on the s-plane (use graph paper) Analysis of Root Locus Plot At a certain value of gain the poles break away from the real axis and become complex. This point is known as the break point. As gain is increased the closed loop poles change the transient response of the system by making it underdamped, however the real part of the complex pole remains unchanged meaning that the overall settling time remains unchanged. As gain increases %overshoot increases and the damped frequency of oscillations which reduces peak time. The properties of the root locus can be derived from the closed loop transfer function kG T ( s) = 1 + kGH A pole exists when the characteristic polynomial in the denominator becomes zero 1 + kGH = 0 ∴ kGH = −1 Lecture 5_ Root- Locus Page | 7 MOD007954 Automation & Control For a unity feedback system (ie H = 1), therefore:- kG = −1 It is important to note that the locations of closed loop poles can lie anywhere on the complex plane and therefore will have a real and imaginary coordinates. Gain Margin from the Root-Locus The gain margin is the factor by which the design value of the gain factor K can be multiplied before the closed loop system becomes unstable. It can be determined from the root-locus using the following formula: 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝐾 𝑎𝑡 𝑗 𝑎𝑥𝑖𝑠 𝑐𝑟𝑜𝑠𝑠𝑜𝑣𝑒𝑟 𝑔𝑎𝑖𝑛 𝑚𝑎𝑟𝑔𝑖𝑛 = 𝑑𝑒𝑠𝑖𝑔𝑛 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝐾 If the root locus does not cross the jω axis the gain margin is infinite. Exercise: Consider the unity negative feedback control system whose open-loop transfer function is: K ( s + 2) G= ( s + 1)( s + 4) (i) What is the closed-loop transfer function of this system? (ii) What is the characteristic equation of the system? (iii) What is the gain margin of this system? (iv) Plot the root-locus of this system. (v) For a gain factor K=2, calculate the closed-loop poles and plot them on the graph from question (i). Lecture 5_ Root- Locus Page | 8