DIGITAL CONTROL (0301-741)Dr. AbdullaIsmailProfessor of Electrical Engineeringaxicad@rit.edu

Introduction to Digital ControlDiscrete Time SystemsThe Z-transformDifference EquationsImpulse Response and Step response of Discrete-Time SystemsFrequency Response of Discrete-Time SystemsModeling Digital Control SystemsStability of Digital Control SystemsDigital Control System DesignState-Space Analysis of Discrete-Time SystemsCourse Outline (Topics):Digital Control

Digital Control

Digital Control*Stability of Digital Control SystemsStability is a basic requirement for digital and analog control systems. Digital control is based on samples and is updated every sampling period, and there is a possibility that the system will become unstable between updates. This obviously makes stability analysis different in the digital case. We examine different definitions and tests of the stability of linear time-invariant (LTI) digital systems based on transfer function models. In particular, we consider input-output stability and internal stability. We provide two tests for stability: the Routh-Hurwitz criterion and the Jury criterion.

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Digital Control*Stability of Digital Control SystemsSystem Response and pole location for CTS

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Digital Control*Stability of Digital Control SystemsMapping regions of the s-plane onto the z-plane

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Digital Control*Stability of Digital Control SystemsSystem Response and pole location for DTS

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Digital Control*Real PolesComplex PolesPoles and Time Response

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Digital Control*Transient responses corresponding to various pole locations of Y(s) in the s-plane (complex-conjugate poles only). Transient-response sequence corresponding to various pole locations of Y(z) in the z-plane.Transient responses and Pole Loci

Digital Control

Digital Control*Stability of Digital Control SystemsDefinitions of StabilityThe most commonly used definitions of stability are based on the. magnitude of the system response in the steady stateIf the steady-state response is unbounded, the system is said to be unstable. Here, we discuss two stability definitions that concern the boundedness or exponential decay of the system output.The first stability definition considers the system output due to its initial conditions. To apply it to transfer function models, we need the assumption that no pole-zero cancellation occurs in the transfer function.

Digital Control

Digital Control*Stability of Digital Control SystemsDefinitions of StabilityDefinition 4.1: Asymptotic Stability.A system is said to be asymptotically stable if its response y(k) to any initial conditions decays to zero asymptotically in the steady statethat is, the response due to the initial conditions satisfiesIf the response due to the initial conditions remains bounded but does not decay to zero, the system is said to be marginally stable.

Digital Control

Digital Control*The second definition of stability concerns the forced response of the system for a bounded input. A bounded input satisfies the conditionStability of Digital Control SystemsDefinitions of StabilityFor example, a bounded sequence satisfying the constraint |u(k)| < 3 is shown in Figure here

Digital Control

Digital Control*Stability of Digital Control SystemsDefinitions of StabilityA system is said to be bounded-inputbounded-output (BIBO) stable if its response to any bounded input remains boundedthat is, for any input satisfying , the output satisfiesDefinition 4.2: Bounded-InputBounded-Output Stability.

Digital Control

Digital Control*Stability of Digital Control SystemsStable z-Domain Pole LocationsThe locations of the poles of linear discrete time systems z-transfer function determine its time response. The implications of this fact for system stability are now examined more closely. Consider the sampled exponential and its z-transformwhere p is real or complex.Then the time sequence for large k is given by

Digital Control

Digital Control*Stability of Digital Control SystemsStable z-Domain Pole LocationsAny time sequence can be described byHence, we conclude that the sequence is bounded if its poles lie in the closed unit disc (i.e., on or inside the unit circle) and decays exponentially if its poles lie in the open unit disc (i.e., inside the unit circle). This conclusion allows us to derive stability conditions based on the locations of the system poles. Note that the case of repeated poles on the unit circle corresponds to an unbounded time sequence (see, for example, the transform of the sampled ramp).where Ai are partial fraction coefficients and pi are z-domain poles.

Digital Control

Digital Control*Stability of Digital Control SystemsStability ConditionsAsymptotic StabilityThe following theorem gives conditions for asymptotic stability.Theorem 4.1: Asymptotic Stability. In the absence of pole-zero cancellation, an LTI digital control system is:Asymptotically stable if all system poles are inside the unit disc and Marginally stable if at least one of the simple(not repeated) poles is in on the unit disc or circle and no poles outside the unit disc.

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Digital Control*Stability of Digital Control SystemsStability ConditionsAsymptotic StabilityExample 4.1Determine the asymptotic stability of the following systems:

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Digital Control*Stability of Digital Control SystemsStability ConditionsAsymptotic StabilityExample 4.1 SolutionTheorem 4.1 can only be used for transfer functions (a) and (b) if their poles and zeros are not canceled. Ignoring the zeros, which do not affect the response to the initial conditions, (a) has a pole outside the unit circle and the poles of (b) are inside the unit circle. Hence, (a) is unstable, whereas (b) is asymptotically stable.Theorem 4.1 can be applied to the transfer functions (c) and (d). The poles of (c) are all inside the unit circle, and the system is therefore asymptotically stable. However, (d) has one pole on the unit circle and is only marginally stable.

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Digital Control*Stability of Digital Control SystemsStability ConditionsBIBO StabilityBIBO stability concerns the response of a system to a bounded input. The response of the system to any input is given by the convolution summationwhere h(k) is the impulse response sequence.It may seem that a system should be BIBO stable if its impulse response is bounded. To show that this is generally false, let the impulse response of a linear system be bounded with lower bound bh1 and upper bound bh2that is,

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Digital Control*Stability of Digital Control SystemsStability ConditionsBIBO StabilityThen using this bound in gives the inequalityThe following theorem establishes necessary and sufficient conditions for BIBO stability of a discrete-time linear system.

Digital Control

Digital Control*Stability of Digital Control SystemsStability ConditionsBIBO StabilityTheorem 4.2: A discrete-time linear system is BIBO stable if and only if its impulse response sequence is absolutely summablethat is,Because the z-transform of the impulse response is the transfer function, BIBO stability can be related to pole locations as follows.Theorem 4.3: A discrete-time linear system is BIBO stable if and only if the poles of its transfer function lie inside the unit circle.

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Digital Control*Stability of Digital Control SystemsStability ConditionsBIBO StabilityExample 4.2Investigate the BIBO stability of the class of systems with the impulse responsewhere K is a finite constant.SolutionThe impulse response satisfiesUsing condition , the systems are all BIBO stable. This is the class of finite impulse response (FIR) systems (i.e., systems whose impulse response is nonzero over a finite interval). Thus, we conclude that all FIR systems are BIBO stable.

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Digital Control*Stability of Digital Control SystemsStability ConditionsBIBO StabilityExample 4.3Investigate the BIBO stability of the systems discussed in Example 4.1:

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Digital Control*Stability of Digital Control SystemsStability ConditionsBIBO StabilityExample 4.3

SolutionAfter pole-zero cancellation, the transfer functions (a) and (b) have all poles inside the unitcircle and are therefore BIBO stable. The transfer function (c) has all poles inside the unit circle and is stable; (d) has a pole on the unit circle and is not BIBO stable.

Digital Control

Digital Control*Stability of Digital Control SystemsStability ConditionsBIBO StabilityThe preceding analysis and examples show that for LTI systems, with no pole/zero cancellation, BIBO and asymptotic stability are equivalent and can be investigated using the same tests.

Hence, the term stability is used in the sequel to denote either BIBO or asymptotic stability with the assumption of no unstable pole-zero cancellation.

Pole locations for a stable system (inside the unit circle) are shown in Figure here.

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Digital Control*Stability of Digital Control SystemsStability ConditionsInternal StabilitySo far, we have only considered stability as applied to an open-loop system. For closed-loop systems, these results are applicable to the closed-loop transfer function.However, the stability of the closed-loop transfer function is not always sufficient for proper system operation because some of the internal variables may be unbounded. In a feedback control system, it is essential that all the signals in the loop be bounded when bounded external inputs are applied to the system.

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Digital Control*Stability of Digital Control SystemsStability ConditionsInternal StabilityConsider the unity feedback digital control scheme of Figure shown below where, for simplicity, a disturbances input is added to the controller output before the ADC.We consider that system as having two outputs, Y and U, and two inputs, R and D. Thus, the transfer functions associated with the system are given by

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Digital Control*Stability of Digital Control SystemsStability ConditionsInternal StabilityClearly, it is not sufficient to prove that the output of the controlled system Y is bounded for bounded reference input R because the controller output U can be unbounded. This suggests the following definition of stability.

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Digital Control*Stability of Digital Control SystemsStability ConditionsInternal StabilityDefinition 4.3: Internal Stability. If all the transfer functions that relate system inputs (R and D) to the possible system outputs (Y and U) are BIBO stable, then the system is said to be internally stable.Because internal stability guarantees the stability of the transfer function from R to Y, among others, it is obvious that an internally stable system is also externally stable (i.e., the system output Y is bounded when the reference input R is bounded).However, external stability does not, in general, imply internal stability.We now provide some results that allow us to test internal stability.

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Digital Control*Stability of Digital Control SystemsStability ConditionsInternal StabilityTheorem 4.4: The system shown in Figure below is internally stable if and only if all its closed-loop poles are in the open unit disc.Theorem 4.5: The system of Figure below is internally stable if and only if the following two conditions hold:1. The characteristic polynomial 1 + C(z)GZAS(z) has no zeros on or outside the unit circle.2. The loop gain C(z)GZAS(z) has no pole-zero cancellation on or outside the unit circle.

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Digital Control*Stability of Digital Control SystemsStability ConditionsInternal StabilityExample 4.4An isothermal chemical reactor where the product concentration is controlled by manipulating the feed flow rate is modeled by the following transfer function:Determine GZAS(Z) with a sampling rate T = 0.1, and then verify that the closed-loop system with the feedback controlleris not internally stable.

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Digital Control*Stability of Digital Control SystemsStability ConditionsInternal StabilityThe discretized process transfer function isThe transfer function from the reference input to the system output is given byThe system appears to be asymptotically stable with all its poles inside the unit circle.However, the system is not internally stable as seen by examining the transfer functionExample 4.4 .. Solution

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Digital Control*Stability of Digital Control SystemsStability ConditionsInternal StabilityExample 4.4 .. Solution which has a pole at 1.334 outside the unit circle. The control variable is unbounded even when the reference input is bounded. In fact, the system violates condition 2 of Theorem 4.5 because the pole at 1.334 cancels in the loop gain

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Digital Control*Stability of Digital Control SystemsStability DeterminationRouth-Hurwitz CriterionThe Routh-Hurwitz stability criterion determines conditions for left half plane (LHP) polynomial roots of continuous-time systems.But, it cannot be directly used to investigate the stability of discrete-time systems. However, the given bilinear transformation transforms the inside of the unit circle to the LHP.

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Digital Control*Stability of Digital Control SystemsFor the general z-polynomial,This allows the use of the Routh-Hurwitz criterion for the investigation of discrete-time system stability. Routh-Hurwitz Criterion

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Digital Control*The Routh-Hurwitz approach becomes progressively more difficult as the order of the z-polynomial increases. But for low-order polynomials, it easily gives stability conditions. The Routh-Hurwitz approach is demonstrated in the following example.

Stability of Digital Control SystemsRouth-Hurwitz Criterion

Digital Control

Digital Control*Stability of Digital Control SystemsStability DeterminationExample 4.5Find stability conditions forSolution1. The stability of the first-order polynomial can be easily determined by solving for its root. Hence, the stability condition is2. The roots of the second-order polynomial are in general given by

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Digital Control*Stability of Digital Control SystemsStability DeterminationExample 4.5 .. SolutionThus, it is not easy to determine the stability of the second-order polynomial by solving for its roots. Hence, for pole magnitudes less than unity, we obtain the necessary stability conditionor equivalently

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Digital Control*Stability of Digital Control SystemsStability DeterminationExample 4.5 .. SolutionBy the Routh-Hurwitz criterion, it can be shown that the poles of the second-order w-polynomial remain in the LHP if and only if its coefficients are all positive. Hence, the stability conditions are given byAdding the first and third conditions gives

Digital Control

Digital Control*Stability of Digital Control SystemsStability DeterminationExample 4.5 .. SolutionThis condition, obtained earlier, is therefore satisfied if the three conditions of are satisfied.

The reader can verify through numerical examples that if real roots satisfying the above three condition are substituted in , we obtain roots between 1 and +1.

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Digital Control*Stability of Digital Control SystemsStability DeterminationWithout loss of generality, the coefficient a2 can be assumed to be unity, and the stable parameter range can be depicted in the a0 versus a1 parameter plane as shown in Figure here.Example 4.5 .. Solution

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Digital Control*Stability of Digital Control SystemsExample 4.6Solution

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Digital Control*Routh-Hurwitz array can now be developed from the transformed characteristic equation.

Since there are no sign changes in the first column of the Routh array therefore the system is stable. Stability of Digital Control SystemsExample 4.6 .. Solution

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Digital Control*SolutionStability of Digital Control SystemsExample 4.7

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Digital Control*Stability of Digital Control SystemsRouth array can now be developed from the transformed characteristic equation.

From the table above, since there is one sign change in the first column above equation has one root in the right-half of the w-plane.

This, in turn, implies that there will be one root of the characteristic equation outside of the unit circle in the z-plane.Example 4.7 .. Solution

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Digital Control*Jurys Stability TestStability test method presented by Eliahu Ibraham Jury.

It is possible to investigate the stability of z-domain polynomials directly using the Jury test.

These tests involve determinant evaluations as in the Routh-Hurwitz test for s-domain polynomials but are more time consuming.

Digital Control

Digital Control*Stability of Digital Control SystemsStability DeterminationJury TestTheorem 4.6: the roots of the polynomial are inside the unit circle if and only if the following conditions are met.For the polynomialwhere the terms in the n + 1 conditions are calculated from Table 4.1. given next.(C)

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Digital Control*Stability of Digital Control SystemsStability DeterminationJury TestTheorem 4.6:

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Digital Control*Stability of Digital Control SystemsStability DeterminationJury TestTheorem 4.6: Based on the Jury table and the Jury stability conditions, we make the following observations:1. The first row of the Jury table is a listing of the coefficients of the polynomial F(z) in order of increasing power of z.2. The number of rows of the table (2n 3) is always odd, and the coefficients of each even row are the same as the odd row directly above it with the order of the coefficients reversed.

Digital Control

Digital Control*Stability of Digital Control SystemsStability DeterminationJury Test3. There are n + 1 conditions in (C) that correspond to the n + 1 coefficients of F(z).4. Conditions 3 through 2n 3 of (C) are calculated using the coefficient of the first column of the Jury table together with the last coefficient of the preceding row. The middle coefficient of the last row is never used and need not be calculated.5. Conditions 1 and 2 of (C) are calculated from F(z) directly. If one of the first two conditions is violated, we conclude that F(z) has roots on or outside the unit circle without the need to construct the Jury table or test the remaining conditions.

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Digital Control*Stability of Digital Control SystemsStability Determination6. Condition 3 of (C), with an = 1, requires the constant term of the polynomial to be less than unity in magnitude. The constant term is simply the product of the roots and must be smaller than unity for all the roots to be inside the unit circle.

7. Conditions (C) reduce to conditions and

for first and second order systems respectively where the Jury table is simply one row.8. For higher-order systems, applying the Jury test by hand is laborious, and it is preferable to test the stability of a polynomial F(z) using a computer-aided design (CAD) package.Jury Test

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Test the stability of the polynomial.

Develop Jurys Table [(2n-3) rows].*SolutionStability of Digital Control SystemsJury TestExample 4.8

*Stability of Digital Control SystemsJury TestExample 4.8

*4rth row is same as 3rd row in reverse orderStability of Digital Control SystemsJury TestExample 4.8

*Stability of Digital Control SystemsJury TestExample 4.8

*6th row is same as 5th row in reverse orderStability of Digital Control SystemsJury TestExample 4.8

*Stability of Digital Control SystemsJury TestExample 4.8

nth order system5th order System*Now we need to evaluate following conditionsStability of Digital Control SystemsJury TestExample 4.8

*The first two conditions require the evaluation of F(z) at z = 1.SatisfiedNot SatisfiedStability of Digital Control SystemsJury TestExample 4.8

*Next four conditions require Jurys tableSatisfiedSatisfiedNot SatisfiedSatisfiedStability of Digital Control SystemsJury TestExample 4.8

Test the stability of the polynomial.

Develop Jurys Table [(2n-3) rows].*SolutionSatisfiedSatisfiedStability of Digital Control SystemsJury TestExample 4.9

Next four conditions require Jurys table

Since all the conditions are satisfied, the system is stable.SatisfiedStability of Digital Control SystemsJury TestExample 4.9

ExerciseDetermine the stability of a discrete data system described by the following CE by using Jurys Stability criterion.

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Digital Control*Stability of Digital Control SystemsStability DeterminationMATLABThe roots of a polynomial are obtained using one of the MATLAB commands,>> roots(den)>> zpk(g)where den is a vector of denominator polynomial coefficients.The command zpk factorizes the numerator and denominator of the transfer function g and displays it.

The poles of the transfer function can be obtained with the command pole and then sorted with the command dsort in order of decreasing magnitude.

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Digital Control*Stability of Digital Control SystemsStability DeterminationMATLABAlternatively, one may use the command ddamp, which yields the pole locations (eigenvalues), the damping ratio, and the undamped natural frequency.

For example, given a sampling period of 0.1 s and the denominator polynomial with coefficients>> den = [1.0, 0.2, 0.0, 0.4]the command is >> ddamp(den, 0.1)The command yields the output.

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Digital Control*Stability of Digital Control SystemsStability DeterminationMATLABThe MATLAB command>> T = feedback(g, gf, 1)calculates the closed-loop transfer function T using the forward transfer function g and the feedback transfer function gf. For negative feedback, the third argument is -1 or is omitted. For unity feedback, we replace the argument gf with 1. We can solve for the poles of the closed-loop transfer function as before using zpk or ddamp.

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Digital Control*Modeling Digital Control SystemsBACKUP SLIDES

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*Zero-Pole MatchingExact map between poles (z = esT) exists.Basic idea, simply use same map for zeros.Rules1. All poles are mapped by z = esT.For example, s = -a maps to z = e-aT2. All finite zeros are mapped by z = esT.For example, s = -b maps to z = e-bT3. Basically, zeros at infinity maps to z = ejp = -1 (representing the highest frequency)4. Identical gain at some critical freq. (typically, s=0)H(s) at s=0 = H(z) at z=1

*Zero-Pole MatchingExample (1)Compute the discrete equivalent by zero-pole matching

*Zero-Pole Matchings-planez-plane

Digital Control*Stability of Digital Control SystemsJury TestExample 4.6We compute the entries of the Jury table using the coefficients of the polynomial

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Digital Control*Stability of Digital Control SystemsStability DeterminationJury TestExample 4.6

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Digital Control*Stability of Digital Control SystemsStability DeterminationJury TestExample 4.7Find the stable range of the gain K for the unity feedback digital cruise control system of last lecture set with the analog plant transfer functionand with digital-to-analog converter (DAC) and analog-to-digital converter (ADC) if the sampling period is 0.02 s.

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Digital Control*Stability of Digital Control SystemsStability DeterminationJury TestExample 4.7 .. Solution

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Digital Control*Stability of Digital Control SystemsStability DeterminationJury TestExample 4.7 .. Solution

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Digital Control*Stability of Digital Control SystemsStability DeterminationJury TestExample 4.8Find the stable range of the gain K for the vehicle position control system, given in the lecture set, with the analog plant transfer functionand with DAC and ADC if the sampling period is 0.05 s.

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Digital Control*Stability of Digital Control SystemsStability DeterminationJury TestExample 4.8 .. Solution

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Digital Control*Stability of Digital Control SystemsStability DeterminationJury TestExample 4.8 .. Solution

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