## 841 Controllability and observability

The concepts of controllability and observability were introduced by Kalman (1960) and play an important role in the control of multivariable systems. A system is said to be controllable if a control vector u(i) exists that will transfer the system from any initial state x(z0) to some final state x(i) in a finite time interval. A system is said to be observable if at time z0, the system state x(z0) can be exactly determined from observation of the output y(i) over a finite time interval. If a...

## 243 Mass in mechanical systems

The force to accelerate a body is the product of its mass and acceleration (Newton's second law). T(t) a(t) , < (2.17) In equation (2.17) is the moment of inertia about the rotational axis. When analysing mechanical systems, it is usual to identify all external forces by the use of a 'Free-body diagram', and then apply Newton's second law of motion in the form y M a for rotational systems (2.18) Find the differential equation relating the displacements xi(t) and xo(t) for the...

## Lead compensator design one

Place wm at the modulus crossover frequency of 2 rad s and position the compensator corner frequencies an octave below, and an octave above this frequency. Set the compensator gain to unity. Hence wm 2rad s 1 T1 1 rad s 1 T2 4rad s K 1.0 < m 36.9 Fig. 6.34 Nichols chart for lead compensator, design one. Fig. 6.34 Nichols chart for lead compensator, design one. The Nichols chart for the uncompensated and compensated system (curve (a)) is shown in Figure 6.34 (see also Appendix 1,From Figure...

## 442 Linear hydraulic actuators

Hydraulic actuators are employed in such areas as the aerospace industry because they possess a good power to weight ratio and have a fast response. Figure 4.19 shows a spool-valve controlled linear actuator. When the spool-valve is moved to the right, pressurized hydraulic oil flows into chamber (1) causing the piston to move to the left, and in so doing forces oil in chamber (2) to be expelled to the exhaust port. The following analysis will be linearized for small perturbations of the...

## The defuzzification process

Defuzzification is the procedure for mapping from a set of inferred fuzzy control signals contained within a fuzzy output window to a non-fuzzy (crisp) control signal. The centre of area method is the most well known defuzzification technique, which in linguistic terms can be expressed as . , . Sum of first moments of area For a continuous system, equation (10.30) becomes or alternatively, for a discrete system, equation (10.30) can be expressed as For the case when e(t) 2.5 and ce -0.2, as a...

## A17 Tutorial 6 Digital control system design

This tutorial looks at the application of MATLAB to digital control system design, using the problems in Chapter 7 as design examples. To obtain the z-transform of a first-order sampled data system in cascade with a zero-order hold (zoh), as shown in Figure 7.10. Example 7.3 Transfer Funetion to z-Transform Continuous and Diserete Step Response num 1 den 1 1 Ts 0.5 numd,dend e2dm(num,den,Ts,'zoh') printsys(num,den,'s') printsys(numd,dend,'z') subplot(121), step(num,den) subplot(122),...

## 42 Block diagram reduction 421 Control systems with multiple loops

A control system may have several feedback control loops. For example, with a ship autopilot, the rudder-angle control loop is termed the minor loop, whereas the heading control loop is referred to as the major loop. When analysing multiple loop systems, the minor loops are considered first, until the system is reduced to a single overall closed-loop transfer function. To reduce complexity, in the following examples the function of s notation (V) used for transfer functions is only included in...

## 453 Proportional plus Integral PI control

Including a term that is a function of the integral of the error can, with the type of plant shown in Figure 4.23, eliminate steady-state errors. Consider a control law of the form u(t) K1e(t) + K2 j edt (4.67) In equation (4.68), Ti is called the integral action time, and is formally defined as 'The time interval in which the part of the control signal due to integral action increases by an amount equal to the part of the control signal due to proportional action when the error is unchanging'....

## 455 The Ziegler Nichols methods for tuning PID controllers

The selection of the PID controller parameters K, Ti and Td can be obtained using the classical control system design techniques described in Chapters 5 and 6. In the 1940s, when such tools were just being developed, Ziegler and Nichols (1942) devised two empirical methods for obtaining the controller parameters. These methods are still in use. (a) The Process Reaction Method This is based on the assumption that the open-loop step response of most process control systems has an S-shape, called...

## 83 Discretetime solution of the state vector differential equation

The discrete-time solution of the state equation may be considered to be the vector equivalent of the scalar difference equation method developed from a z-transform approach in Chapter 7. The continuous-time solution of the state equation is given in equation (8.47). If the time interval (t t0) in this equation is T, the sampling time of a discrete-time system, then the discrete-time solution of the state equation can be written as x (k + 1)T eATx(kT) + jji eArBdr Ju(kT) (8.75) Equation (8.75)...

## The Nichols chart

The Nichols chart shown in Figure 6.26 is a rectangular plot of open-loop phase on the x-axis against open-loop modulus (dB) on the j-axis. M and N contours are superimposed so that open-loop and closed-loop frequency response characteristics can be evaluated simultaneously. Like the Bode diagram, the effect of increasing the open-loop gain constant K is to move the open-loop frequency response locus in the j-direction. The Nichols chart is one of the most useful tools in frequency domain...

## A15 Tutorial 4 Classical design in the splane

The tutorial demonstrates how MATLAB is used to generate root locus diagrams, and hence how to design control systems in the s-plane. Examples given in Chapter 5 are used to illustrate the MATLAB commands. The roots of the characteristic equation (or any polynomial) can be found using the roots command. Check the stability of the system that has the characteristic equation ce 12 14 2 roots(ce) ans -2.1877 0.3516+1.2843i 0.3516-1.2843i -0.5156 Two roots have positive real parts, hence the system...

## A18 Tutorial 7 Statespace methods for control system design

This tutorial looks at how MATLAB commands are used to convert transfer functions into state-space vector matrix representation, and back again. The discrete-time response of a multivariable system is undertaken. Also the controllability and observability of multivariable systems is considered, together with pole placement design techniques for both controllers and observers. The problems in Chapter 8 are used as design examples. see equation (7.115) see equation (7.128) This converts a...

## 28 Further problems

A solenoid valve is shown in Figure 2.18. The coil has an electrical resistance of 4 fi, an inductance of 0.6 H and produces an electromagnetic force Fc(t) of Kc times the current i(t). The valve has a mass of 0.125 kg and the linear bearings produce a resistive force of C times the velocity u(t). The values of Kc and C are 0.4 N A and 0.25 Ns m respectively. Develop the differential equations relating the voltage V(t) and current i(t) for the electrical circuit, and also for the current i(t)...

## 47 Further problems

For the block diagrams shown in Figure 4.39 find an expression for the complete output when all inputs act simultaneously. (a) CM _ (Gi (s)G2(s)G3(s))Ri (s) + G3(s)(1 + G2(s)H3(s))R2(s) (a) C(s) i + G3(s)H2(s) + G2(s)H3(s) + Gi(s)G2(s)Gs(s)Hi(s) (b) C(s) (Gi (s)G2(s)Gs(s)G4(s))Ri (s) - (Gi (s)G2(s)G3(s)G4(s)Hi (s))R2(s) - (G3(s)G4(s))R3 (s) (b) C(S) i + G3(s)H2(s) + Gi(s)G2(s)G3(s)G4(s)Hi(s) The speed control system shown in Figure 4.40 consists of an amplifier, a field-controlled DC servomotor...

## References and further reading

Ackermann, J. 1972 Der Entwurf Linearer Regelungssysteme im Zustandsraum, Regelungstechnik und Prozessdatenverarbeitung, 7, pp. 297-300. Anderson, J.A. 1972 A Simple Neural Network Generating an Interactive Memory, Mathematical Biosciences, 14, pp. 197-220. Anderson, B.D.O. and Moore, J.B. 1979 Optimal Filtering, Prentice-Hall, Englewood Cliffs, NJ. Anderson, B.D.O. and Moore, J.B. 1990 Optimal Control, Prentice-Hall, Englewood Cliffs, NJ. Astrom, K.J. and Wittenmark, B. 1984 Computer...

## 994 The weighted mixedsensitivity approach

Multivariate loop shaping in robust control system design may be achieved using a weighted mixed sensitivity approach. As with the SISO systems described in section 9.8.2, the sensitivity function S s given in equation 9.166 and the complementary sensitivity function T s given in equation 9.167 may be combined with weights Ws s and WT s to give where the infinity norm of Ty1u1 is lt 1 as given in equation 9.173 . Equation 9.174 defines a mixed-sensitivity cost function since both S s and T s...

## 761 Mapping from the splane into the zplane

Just as transient analysis of continuous systems may be undertaken in the s-plane, stability and transient analysis on discrete systems may be conducted in the z-plane. It is possible to map from the s to the z-plane using the relationship z e a Jw r earejwr using the positive jw value 7.62 Fig. 7.16 Mapping from the s to the z-plane. If eaT z and T equation 7.62 can be written Equation 7.63 results in a polar diagram in the z-plane as shown in Figure 7.16. Figure 7.17 shows mapping of lines of...

## 372 Step response performance specification

The three parameters shown in Figure 3.21 are used to specify performance in the a Rise time tr The shortest time to achieve the final or steady-state value, for the first time. This can be 100 rise time as shown, or the time taken for example from 10 to 90 of the final value, thus allowing for non-overshoot response. b Overshoot The relationship between the percentage overshoot and damping ratio is given in equation 3.68 . For a control system an overshoot of between 0 and 10 1 lt C gt 0.6 is...

## 46 Case study examples

Example 4.6.1 CNC Machine-Tool Positional Control See also Appendix 1, examp461.m The physical configuration and block diagram representation of a CNC machinetool is shown in Figures 1.10 and 1.11. The fundamental control problem here is that, by design, the lead-screw by the use of re-circulating ball-bearings is friction-free. This means that the positional control system will have no damping, and will oscillate continuously at the undamped natural frequency of the closed-loop system. Damping...

## 911 Types of optimal control problems

a The terminal control problem This is used to bring the system as close as possible to a given terminal state within a given period of time. An example is an automatic aircraft landing system, whereby the optimum control policy will focus on minimizing errors in the state vector at the point of landing. b The minimum-time control problem This is used to reach the terminal state in the shortest possible time period. This usually results in a 'bang-bang' control policy whereby the control is...

## 354 Experimental determination of system time constant using step response

Method one The system time constant is the time the system takes to reach 63.2 of its final value see Table 3.2 . Method two The system time constant is the intersection of the slope at t 0 with the final value line see Figure 3.13 since This also applies to any other tangent, see Figure 3.13. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Number of Time Constants Fig. 3.13 Unit step response of a first-order system. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Number of Time Constants Fig. 3.13 Unit step response of a...

## 355 Ramp response of firstorder systems

Find an expression for the response of a first-order system to a ramp function of slope Q. Zo s _ s2 1 Ts s2 s 1 T _7 2 i T T 3 2 See partial fraction expansion equation 3.13 . Multiplying both sides by s2 s 1 T , we get i.e. Q- As2 As Bs B Cs2 3.33 Equating coefficients on both sides of equation 3.33 Substituting into 3.35 Hence from 3.34 Fig. 3.14 Ramp response of a first-order system see also Figure A1.1 . Fig. 3.14 Ramp response of a first-order system see also Figure A1.1 . Fig. 3.15 Unit...

## 422 Block diagram manipulation

There are occasions when there is interaction between the control loops and, for the purpose of analysis, it becomes necessary to re-arrange the block diagram configuration. This can be undertaken using Block Diagram Transformation Theorems. Table 4.1 Block Diagram Transformation Theorems Table 4.1 Block Diagram Transformation Theorems Moving a summing point ahead of a block. Z GX Y Z X G Y G 4.9 A complete set of Block Diagram Transformation Theorems is given in Table 4.1. Example 4.3 Find the...

## 441 DC servomotors

One of the most common devices for actuating a control system is the DC servomotor shown in Figure 4.13, and can operate under either armature or field control. a Armature control This arrangement is shown in schematic form in Figure 4.14. Now air gap flux is proportional to z'f, or where Kfd is the field coil constant. Also, torque developed Tm is proportional to the product of the air gap flux and the armature current Fig. 4.14 DC servo-motor under armature control. ea t Armature excitation...

## Predictive Self Organizing Fuzzy Logic Control PSOFLC

This is an extension of the SOFLC strategy discussed in section 10.2.5 and illustrated in Figure 10.17. Predictive Self-Organizing Fuzzy Logic Control is particularly useful when the plant dynamics are time-varying, and the general architecture is shown in Figure 10.32. In Figure 10.30 the predictive neural network model tracks the changing dynamics of the plant. Following a suitable time delay, em kT is passed to the performance index table. If this indicates poor performance as a result of...

## Case study

The laser guided missile shown in Figure 5.26 has an open-loop transfer function combining the fin dynamics and missile dynamics of Fig. 6.33 Nichols chart for uncompensated laser guided missile. Fig. 6.33 Nichols chart for uncompensated laser guided missile. Design a cascade lead compensator that will ensure stability and provide a phase margin of at least 30 , a bandwidth greater than 5rad s and a peak closed-loop modulus Mp of less than 6dB. The open-loop transfer function is third-order...

## 431 Principle of superposition

A dynamic system is linear if the Principle of Superposition can be applied. This states that 'The response y t of a linear system due to several inputs x t , x2 t , , xn t , acting simultaneously is equal to the sum of the responses of each input acting alone'. Find the complete output for the system shown in Figure 4.9 when both inputs act simultaneously. The block diagram shown in Figure 4.9 can be reduced and simplified to the form given in Figure 4.10. Putting R2 s 0 and replacing the...

## The Adaptive Network based Fuzzy Inference System ANFIS

The ANFIS neurofuzzy controller was implemented by Jang 1993 and employs a Takagi-Sugeno-Kang TSK fuzzy inference system. The basic ANFIS architecture is shown in Figure 10.31. Square nodes in the ANFIS structure denote parameter sets of the membership functions of the TSK fuzzy system. Circular nodes are static non-modifiable and perform operations such as product or max min calculations. A hybrid learning rule is used to accelerate parameter adaption. This uses sequential least squares in the...

## The magnitude criterion

If a point 1 lies on a locus, then the value of the open-loop gain constant K at that point may be evaluated by using the magnitude criterion. Equation 5.56 can be expressed as Product of pole vector magnitudes Product of zero vector magnitudes Fig. 5.13 Application of the magnitude criterion. Fig. 5.13 Application of the magnitude criterion. For Example 5.7, if s1 lies on a locus, then the pole and zero magnitudes are shown in Figure 5.13. From Figure 5.13 and equation 5.61 , the value of the...

## 134 Ship autopilot control system

A ship autopilot is designed to maintain a vessel on a set heading while being subjected to a series of disturbances such as wind, waves and current as shown in Figure 1.3. This method of control is referred to as course-keeping. The autopilot can also be used to change course to a new heading, called course-changing. The main elements of the autopilot system are shown in Figure 1.12. The actual heading is measured by a gyro-compass or magnetic compass in a smaller vessel , and compared with...