State-PID Feedback for Pole Placement of LTI Systems

Abstract

Pole placement problems are especially important for disturbance rejection and stabilization of dynamical systems and regarded as algebraic inverse eigenvalue problems. In this paper, we propose gain formulae of state feedback through PID-elements to achieve desired pole placement for a delay-free LTI system with single input. Real and complex stable poles can be assigned with the proposed compact gain formulae. Numerical examples show that our proposed gain formulae can be used effectively resulting in very satisfactory responses.

1. Introduction

Pole placement has been an important design method of a linear control system [1–8]. One approach is to use state feedback in which the gain matrix is calculated via Ackermann’s formula [4]. Regarding this, the original state model can be transformed into the bidiagonal-Frobenius canonical form to achieve the desired pole placement [9]. Based on the Frobenius form, the gain matrices can be readily computed for SISO and MIMO systems [10–12]. The concept of using state-derivative feedback was introduced in 2003-2004 [13, 14]. One advantage over the conventional state feedback is that it results in smaller gains. In practical control, for example, vibration control, the derivative signal can be derived from an accelerometer output as the concept has been successfully implemented [15, 16]. A linear quadratic regulator to achieve the state-derivative feedback was also developed [17]. Recently, stabilizability and disturbance rejection issues have been investigated for an LTI system with state-derivative vector as its output [18]. The state-derivative feedback is useful for stabilization and rejection of dynamic disturbances not for set-point regulation or tracking control. The approach of state-derivative feedback leads to a possibility of using state-PID feedback.

Consider a delay-free LTI system having single input of the form 𝐱^′=𝐀𝐱+𝐁𝑢,𝐱𝑡₀=𝐱_𝟎, (1.1) where 𝐱∈𝐑^𝑛, 𝑢∈𝑅, 𝐀 and 𝐁 are (𝑛×𝑛) and (𝑛×1) real coefficient matrices, respectively.

The linear system under consideration must possess a complete controllability property. Therefore, the controllability matrix 𝐰_𝑐must have rank-𝑛 and can be formed from 𝐰_𝑐=𝐁𝐀𝐁𝐀²𝐁⋯𝐀^𝑛−1𝐁. (1.2) The Frobenius canonical form of the system is𝝃^′=010⋯0001⋯0⋮⋮⋱⋱⋮000⋯1−𝑎₀−𝑎₁⋯⋯−𝑎_𝑛−1000⋮1𝝃+𝑢, (1.3) where 𝝃=𝐓𝐱, 𝐀_𝐜=𝐓𝐀𝐓⁻¹, 𝐁_𝐜=𝐓𝐁, and 𝐓=[𝐪_𝟏𝐪_𝟏𝐀⋯𝐪_𝟏𝐀^𝑛−1]^𝑇. The vector 𝐪_𝟏=𝐞^𝑇_𝑛𝐰_𝑐⁻¹ in which 𝐞_𝑛=[00⋯1]^𝑇. For the state-PID feedback, the control 𝑢 is of the form𝑢=𝐊_𝐩𝐱+𝐊_𝐈𝐱(𝜏)𝑑𝜏+𝐊_𝐝𝐱^′, (1.4) where 𝐊_𝐩,𝐊_𝐈,𝐊_𝐝∈𝐑^𝑛 are row gain vectors for the P, I, and D feedback elements, respectively. Guo et al. (2006) [21] proposed a pole placement method consisting of 3 separated steps. The pole placement by a state-P feedback is conducted first leading to an intermediate system. Secondly, a state-I feedback is performed; another intermediate system resulted. Finally, the closed-loop system with the desired characteristic polynomial is realized by a state-D feedback incorporated as the final design stage. These steps may be utilized purposefully to design state-P, -PD, and -PI feedback elements. According to this, the control 𝑢 of the P-feedback is 𝑢=𝐊_𝐅𝝃, (1.5) where 𝐊_𝐅=[𝑘₁𝑘₂⋯𝑘_𝑛] to achieve a desired characteristic polynomialΔ_𝑑(𝑠)=𝛼₀+𝛼₁𝑠+⋯+𝛼_𝑛−1𝑠^𝑛−1+𝛼_𝑛𝑠^𝑛,𝛼_𝑛=𝑎_𝑛=1. (1.6) The closed-loop system at this interim stage has its characteristic polynomialΔ(𝑠)=𝑎₀−𝑘₁+𝑎₁−𝑘₂𝑠+⋯+𝑎_𝑛−1−𝑘_𝑛𝑠^𝑛−1+𝑠^𝑛. (1.7) Equating (1.6) and (1.7) results in the gain matrix 𝐊_𝐅. For the I-feedback, the control 𝑢 is𝑢=𝐊_𝐅^𝑡₀𝝃(𝜏)𝑑𝜏, (1.8) where 𝐊_𝐅=[𝑘₁𝑘₂⋯𝑘_𝑛] to achieve a desired characteristic polynomial Δ_𝑑(𝑠)=𝛼₀+𝛼₁𝑠+⋯+𝛼_𝑛−1𝑠^𝑛−1+𝛼_𝑛𝑠^𝑛+𝛼_𝑛+1𝑠^𝑛+1,𝛼_𝑛+1=1. (1.9) The second interim system has its closed-loop characteristic polynomial Δ(𝑠)=−𝑘₁+𝑎₀−𝑘₂𝑠+⋯+𝑎_𝑛−2−𝑘_𝑛𝑠^𝑛−1+𝑎_𝑛−1𝑠^𝑛+𝑠^𝑛+1. (1.10) Equating (1.9) and (1.10) leads to the gain matrix 𝐊_𝐅. For the D-feedback, the control 𝑢 is𝑢=𝐊_𝐅𝝃^′, (1.11) where 𝐊_𝐅=[𝑘₁𝑘₂⋯𝑘_𝑛] to achieve a desired characteristic polynomial Δ_𝑑(𝑠)=𝛼₀+𝛼₁𝑠+⋯+𝛼_𝑛−1𝑠^𝑛−1+𝛼_𝑛𝑠^𝑛. (1.12) Equating (1.12) and the closed-loop characteristic polynomial in (1.13) results in the last set of Δ(𝑠)=𝑎₀+𝑎₁−𝑘₁𝑠+⋯+𝑎_𝑛−1−𝑘_𝑛−1𝑠^𝑛−1+1−𝑘_𝑛𝑠^𝑛, (1.13) gain matrix, 𝐊_𝐅. This previous method requires three sets of poles to be assigned. Two sets are fictitious, and only the last set is the prescribed characteristic polynomial. In [21], there are no recommendations for selection of these intermediate pole sets. One may attempt arbitrarily chosen poles during the separated design phases. In due course, the calculation procedures are quite awkward. Besides, the paper [21] contains no proof of the proposed theorem.

This paper begins by presenting derivation of the gain matrices for the state-PID feedback in rigorous manner. It also presents the gain matrices for the state-PI and -PD feedback cases. Section 3 presents the analysis of disturbance rejection property of the proposed method. Such complete treatment has not appeared elsewhere before. Three numerical examples are shown in Section 4 to illustrate the effectiveness of our proposed gain formulae in comparison with the use of Ackermann’s formula [4, 19] and the methods by Guo et al. [21] and Kuo [20], respectively. Moreover, we show by simulations in the example shown in Section 4.3 that using the method [21] can result in very large controller gains although the final pole sets remain unchanged. Conclusion follows in Section 5.

2. The PID Gain Matrices

Without loss of generality, the single-input LTI system (1.1) is assumed to be completely controllable, and 𝐁 is of full column rank. The next proposition is the main result presenting the state-PID feedback gain matrices. Note that due to the integral element, one additional closed-loop pole is needed. This imposes a condition for derivation of the gain matrices and results in an increase in the order of the system by one.

Proposition 2.1. The system (1.1) with its Frobenius form of (1.3) is subject to the control input 𝑢=𝐊_𝐩𝐱+𝐊_𝐈∫𝐱(𝜏)𝑑𝜏+𝐊_𝐝𝐱^′or𝑢=𝐊_𝐅𝝃+𝐊_𝐅∫^𝑡₀𝝃(𝜏)𝑑𝜏+𝐊_𝐅𝝃^′ in which [𝐊_𝐩,𝐊_𝐈,𝐊_𝐝]=[𝐊_𝐅,𝐊_𝐅,𝐊_𝐅]𝐓. There exist the following gain matrices to achieve a desired characteristic polynomial Δ_𝑑(𝑠)=𝛼₀+𝛼₁𝑠+⋯+𝛼_𝑛−1𝑠^𝑛−1+𝛼_𝑛𝑠^𝑛+𝛼_𝑛+1𝑠^𝑛+1:
(i)for 𝑛=2, 𝐊_𝐩=𝑎₀𝑎₁𝐊𝐓,_𝐈=−𝛼₀−𝛼₁𝐊𝐓,_𝐝=−𝛼₂0𝐓, (2.1)(ii)for 𝑛≥3, 𝐊_𝐩=𝑎₀⋮𝑎₁⋮⋯⋮⋯⋮𝑎_𝑛−1𝐊𝐓,_𝐈=−𝛼₀⋮−𝛼₁⋮−2𝛼₂⋮⋯⋮−2𝛼_𝑛−1𝐊𝐓,_𝐝=𝛼₂⋮⋯⋮𝛼_𝑛−1⋮−𝛼_𝑛⋮0𝐓. (2.2)

Proof. The characteristic polynomial of the closed-loop system can be expressed as Δ_PID(𝑠)=det𝑠𝐈−𝐁_𝐜𝐊_𝐅−𝐀_𝐜−𝐁_𝐜𝐊_𝐅−𝐁_𝐜𝐊_𝐅𝑠=0, (2.3) where I is an 𝑛×𝑛 identity matrix, 𝑠𝐈−𝐁_𝐜𝐊_𝐅−𝐀_𝐜−𝐁_𝐜𝐊_𝐅=𝑎𝑠−10⋯00𝑠−1⋯000𝑠⋱⋮⋮⋮⋮⋱−1₀−𝑘₁𝑠−𝑘₁𝑎₁−𝑘₂𝑠−𝑘₂⋯⋯𝑠+𝑎_𝑛−1−𝑘_𝑛𝑠−𝑘_𝑛,𝐁_𝐜𝐊_𝐅𝑠=000⋯0000⋯0⋮⋮⋮⋱⋮000⋯0𝑘₁𝑠𝑘₂𝑠⋯⋯𝑘_𝑛𝑠, (2.4) that is, Δ_PID(𝑠)=𝑎𝑠−10⋯00𝑠−1⋯000𝑠⋱⋮⋮⋮⋮⋱−1₀−𝑘₁𝑠−𝑘₁−𝑘₁𝑠𝑎₁−𝑘₂𝑠−𝑘₂−𝑘₂𝑠⋯⋯𝑠+𝑎_𝑛−1−𝑘_𝑛𝑠−𝑘_𝑛−𝑘_𝑛𝑠 (2.5) or Δ_PID(𝑠)=−𝑘₁+𝑎₀−𝑘₁−𝑘₂𝑠+𝑎₁−𝑘₂−𝑘₁−𝑘₃𝑠²+⋯+𝑎_𝑛−2−𝑘_𝑛−1−𝑘_𝑛−2−𝑘_𝑛𝑠^𝑛−1+𝑎_𝑛−1−𝑘_𝑛−𝑘_𝑛−1𝑠^𝑛+1−𝑘_𝑛𝑠^𝑛+1. (2.6)
It can be observed that the order of the closed-loop system is increased by 1 due to the integral element. By equating (2.6) with the desired characteristic polynomial, the following relations can be obtained for an 𝑛-order system: −𝑘₁=𝛼₀,𝑎₀−𝑘₁−𝑘₂=𝛼₁,𝑎₁−𝑘₂−𝑘₁−𝑘₃=𝛼₂,⋮𝑎_𝑛−2−𝑘_𝑛−1−𝑘_𝑛−2−𝑘_𝑛=𝛼_𝑛−1,𝑎_𝑛−1−𝑘_𝑛−𝑘_𝑛−1=𝛼_𝑛,1−𝑘_𝑛=𝛼_𝑛+1. (2.7) Therefore, a desired pole placement can be achieved via the state-PID feedback using the gain matrices in (2.1) and (2.2).
This completes the proof.

The following are 2 immediate consequences of Proposition 2.1.

Corollary 2.2. The system (1.1) with its Frobenius form of (1.3) is subject to the control input 𝑢=𝐊_𝐩𝐱+𝐊_𝐈∫^𝑡₀𝐱(𝜏)𝑑𝜏or𝑢=𝐊_𝐅𝝃+𝐊_𝐅∫^𝑡₀𝝃(𝜏)𝑑𝜏 in which [𝐊_𝐩,𝐊_𝐈]=[𝐊_𝐅,𝐊_𝐅]𝐓. There exist the following gain matrices to achieve a desired characteristic polynomial Δ_𝑑(𝑠)=𝛼₀+𝛼₁𝑠+⋯+𝛼_𝑛−1𝑠^𝑛−1+𝛼_𝑛𝑠^𝑛+𝛼_𝑛+1𝑠^𝑛+1: 𝐊_𝐩=𝑎₀⋮𝑎₁⋮𝑎₂⋮⋯⋮𝑎_𝑛−1−𝛼_𝑛𝐊𝐓,_𝐈=−𝛼₀⋮−𝛼₁⋮−𝛼₂⋮⋯⋮−𝛼_𝑛−1𝐓. (2.8)

Corollary 2.3. The system (1.1) with its Frobenius form of (1.3) is subject to the control input 𝑢=𝐊_𝐩𝐱+𝐊_𝐝𝐱^′or𝑢=𝐊_𝐅𝝃+𝐊_𝐅𝝃^′ in which [𝐊_𝐩,𝐊_𝐝]=[𝐊_𝐅,𝐊_𝐅]𝐓. There exist the following gain matrices to achieve a desired characteristic polynomial Δ_𝑑(𝑠)=𝛼₀+𝛼₁𝑠+⋯+𝛼_𝑛−1𝑠^𝑛−1+𝛼_𝑛𝑠^𝑛: 𝐊_𝐩=𝑎₀−𝛼₀⋮−𝛼₁⋮−𝛼₂⋮⋯⋮−𝛼_𝑛−1𝐊𝐓,_𝐝=𝑎₁⋮𝑎₂⋮⋯⋮𝑎_𝑛−1⋮0𝐓. (2.9) Note that with the state-PD feedback, no additional pole is needed for the design. Therefore, the order of the system remains unchanged.

The design procedures are as follows:(1)calculate the transformation matrix for an 𝑛-order LTI plant using 𝐓=[𝐪_𝟏𝐪_𝟏𝐀⋯𝐪_𝟏𝐀^𝑛−1]^𝑇 where 𝐪_𝟏=𝐞^𝑇_𝑛𝐰_𝑐⁻¹, 𝐞_𝑛=[00⋯1]^𝑇, and 𝐰_𝑐=[𝐁𝐀𝐁𝐀²𝐁⋯𝐀^𝑛−1𝐁],(2)calculate the matrices 𝐀_𝐜 and 𝐁_𝐜 using 𝐀_𝐜=𝐓𝐀𝐓⁻¹ and 𝐁_𝐜=𝐓𝐁 for the Frobenius form of (1.3),(3)assign the closed-loop pole locations of an 𝑛-order:(i)for state-PID feedback, add one negative real pole having a fast time-constant (i.e., a negative real pole with a large magnitude),(ii)for state-PI feedback, add one negative real pole having a fast time constant,(iii) for state-PD feedback, no additional pole is needed,(4)determine the prescribed characteristic polynomial Δ_𝑑(𝑠) having the order of 𝑛 or 𝑛+1 corresponding to step 3,(5)calculate the gain matrices: (i)for state-PID feedback, use (2.1) or (2.2),(ii)for state-PI feedback, use (2.8),(iii)for state-PD feedback, use (2.9).

3. Disturbance Rejection

Disturbance rejection is an important property of the proposed state-PID feedback. This section provides the analysis of such property. There are three propositions, one of which has been proposed by [18] and denoted as proposition 3.1 for state-D feedback. The other propositions denoted as Propositions 3.2 and 3.3 are newly developed to confirm the disturbance rejection property accomplished by the state-P and -I feedback components, respectively.

Proposition 3.1. Consider the plant described by 𝐱^′=𝐀𝐱+𝐁(𝐮+𝜀), (3.1) where 𝜀∈𝑅, 𝜀 is an unknown but constant disturbance, and the state-D controller 𝑢=−𝐊_𝐝𝐱^′,det𝐈+𝐁𝐊_𝐝≠0. (3.2) Suppose that det(𝐀)≠0, and the equilibrium point 𝐱_𝑒=−𝐀^−𝟏𝐁𝜀 of the controller system (3.1) and (3.2) is globally asymptotically stable, then 𝐱(∞) is independent of the controller gain 𝐊_𝐝 and is given by 𝑥(∞)=lim_𝑡→∞𝐱(𝑡)=−𝐀^−𝟏𝐁𝜀. (3.3)

Proof. (see in [18], Lemma  3.2) An immediate conclusion from this proposition according to (3.3) is that the state-D feedback cannot attenuate the influence of 𝜀 in 𝐱(∞) in controlled systems because (3.3) is independent of the state-D matrix 𝐊_𝐝.

Proposition 3.2. Consider the plant, with input 𝑢 in (3.1), and the state-P controller 𝑢=−𝐊_𝐩𝐱. (3.4) Suppose that the controller system (3.1) and (3.4) is globally asymptotically stable, then 𝐱(∞) is dependent on the controller gain 𝐊_𝐩 and is given by 𝑥(∞)=lim_𝑡→∞𝐱(𝑡)=−𝐀−𝐁𝐊_𝐩^−𝟏𝐁𝜀. (3.5)

Proof. From (3.1) and (3.4), note that 𝐱^′=𝐀𝐱+𝐁−𝐊_𝐩,𝐱𝐱+𝜀_′=𝐀−𝐁𝐊_𝐩𝐱+𝐁𝜀. (3.6) Applying the Laplace transform to (3.6), observe that 𝑠𝐗(𝑠)−𝐱(0)=𝐀−𝐁𝐊_𝐩𝐗(𝑠)+𝐁𝜀𝑠⁻¹,𝑠𝐈−𝐀−𝐁𝐊_𝐩𝐗(𝑠)=𝐁𝜀𝑠⁻¹(0),𝐗(𝑠)=+𝐱𝑠𝐈−𝐀−𝐁𝐊_𝐩^−𝟏𝐁𝜀𝑠⁻¹+𝐱(0). (3.7) Thus, 𝐱(∞)=lim_𝑠→0𝑠𝐗(𝑠)=lim_𝑠→0𝑠𝑠𝐈−(𝐀−𝐁𝐊_𝐩)^−𝟏𝐁𝜀𝑠⁻¹=−(𝐀−𝐁𝐊_𝐩)⁻¹𝐁𝜀. (3.8) This completes the proof.

From Proposition 3.2, it can be concluded according to (3.8) that the state-P feedback can attenuate the influence of 𝜀 in 𝐱(∞) in controlled systems because (3.8) is dependent on the state-P matrix 𝐊_𝐩.

Proposition 3.3. Consider the plant, with input 𝑢 in (3.1), and the state-I controller 𝑢=−𝐊_𝐈𝐱(𝑡)𝑑𝑡. (3.9) Suppose that the controller system (3.1) and (3.9) is globally asymptotically stable, and the condition of nonzero 𝐊_𝐈 holds, then 𝐱(∞) is dependent on the controller gain 𝐊_𝐈 and is given by 𝑥(∞)=lim_𝑡→∞𝐱(𝑡)=0. (3.10)

Proof. From (3.1) and (3.9), note that 𝐱^′=𝐀𝐱+𝐁−𝐊_𝐈𝐱(𝑡),𝐱𝑑𝑡+𝜀_′=𝐀𝐱−𝐁𝐊_𝐈𝐱(𝑡)𝑑𝑡+𝐁𝜀. (3.11) Applying the Laplace transform to (3.11), observe that 𝑠𝐗(𝑠)−𝐱(0)=𝐀𝐗(𝑠)−𝐁𝐊_𝐈𝐗(𝑠)𝑠+𝐁𝜀𝑠⁻¹,𝑠𝐈+𝐁𝐊_𝐈𝑠−𝐀𝐗(𝑠)=𝐁𝜀𝑠⁻¹(0),𝐗(𝑠)+𝐱𝑠=𝑠²𝐈+𝐁𝐊_𝐈−𝐀𝑠⁻¹𝐁𝜀𝑠⁻¹(0),𝐗(𝑠)𝑠+𝐱=𝑠²𝐈+𝐁𝐊_𝐈−𝐀𝑠⁻¹𝐁𝜀𝑠⁻¹+𝐱(0). (3.12) Thus, 𝐱(∞)=lim_𝑠→0𝑠𝐗(𝑠)=lim_𝑠→0𝑠𝑠𝑠²𝐈+𝐁𝐊_𝐈−𝐀𝑠⁻¹𝐁𝜀𝑠⁻¹=0. (3.13) This completes the proof.

An immediate conclusion from Proposition 3.3 is that with the state-I feedback a complete rejection of disturbance in controlled systems can be achieved due to (3.13).

4. Illustrative Examples

Three illustrative examples are presented with focusing on stabilization and disturbance rejection issues. Results are compared with those designed by the previous methods including Ackermann’s formula [4, 19], Guo et al. [21], and Kuo [20], respectively.

4.1. Magnetic Ball Suspension

The magnetic ball suspension system [20] represented by the diagram in Figure 1 is adopted as the first example. This 3rd-order system is described by 𝐱^′=000109800−2.800−100𝐱+100𝑢, (4.1) where 𝑥₁=𝑦,𝑥₂=𝑦′, and 𝑥₃=𝑖. The system is originally unstable with its poles at ±31.3050 and −100. It is desirable to have the closed-loop poles at −10±𝑗10,−50, and −1000 such that the characteristic polynomial is Δ_𝑑(𝑠)=𝑠⁴+1070𝑠³+71200𝑠²+1210000𝑠+10000000. The Frobenius canonical form is𝝃^′=00101000198000980−100𝝃+𝑢. (4.2) The obtained gain matrices are 𝐊_𝐩=[03.50001], 𝐊_𝐈=10³⋅[534.11404.3214−1.4240], and 𝐊_𝐝=[−254.28573.82140].

Figure 1 

Magnetic ball suspension.

The gain matrix due to Ackermann’s formula is 𝐊=[−280−7.7857−0.3000]. Figure 2 shows the responses and the control input according to the proposed method in which the initial conditions are 𝐱(𝑡₀)=[0.00500]^𝑇, and the states are disturbed by 1 unit at the time 𝑡=1s. It can be observed that using the proposed method the states possess very good transient responses, the disturbances are completely dampened out, and the control input is reasonable. With the conventional pole placement method, some states contain a large amount of steady-state errors due to disturbance as depicted in Figure 3.

(a) Responses of system states

(b) Control signal

Figure 2 

Responses of system states with the proposed state-PID feedback.

(a) Responses of system states

(b) Control signal

Figure 3 

Responses of system states with the conventional state feedback [4, 19].

By applying the method [21] to achieve the same closed-loop poles, the design requires the following fictitious sets of poles: {−1,−2,−4} and {−5,−6,−10}. As a result, the gain matrices are 𝐊_𝑝=[24.52863.550.93], 𝐊_𝐼=[−208.66973.21560.6982], and 𝐊_𝑑=[−3.4432−0.0230.0097] with which a combined state-PID feedback controller is derived. The two intermediate systems denoted as ∑¹_Int and ∑²_Int are given in Frobenius canonical forms as follows:¹_Int=𝐳^′=001𝐊010001−8−14−7𝐳+𝑢,𝑢=_𝐹𝐳^′,_2Int=𝐳^′=001010001−299.6255−139.8240−20.9738𝐳+𝑢,𝑢=𝐊_𝐹^𝑡₀𝐳(𝜏)𝑑𝜏, (4.3) where 𝐳=𝐓𝐱. Under the same simulation situations previously described, similar state responses to those in Figure 2 are achieved due to the closed-loop poles located at the same locations −10±10𝑗,−50. Notice that some of the gains designed by the proposed method are somewhat larger but in reasonable ranges for implementation using either analog or digital technology. The proportional gains of the proposed method are smaller than those obtained using the method [21]. This means that the proposed controller draws less energy to achieve its control action.

Based on the method [20], the gains 𝐊=[−2.038−0.2278−0.68] can be obtained to place the closed-loop poles at −6±4.9𝑗, −20 with the desired characteristic polynomial of Δ_𝑑(𝑠)=𝑠³+32𝑠²+300𝑠+1200. Hence, (4.4) describes the closed-loop system 𝐱^′=0001064.40−1600−100𝐱+100𝑢. (4.4) Figure 4 shows the state responses having the initial conditions 𝐱(𝑡₀)=[100]^𝑇, and the states are disturbed by 0.05 and 1 unit at the time 𝑡=2 and 3 s. Noticeably, a large amount of steady-state errors in some states due to the disturbances still remain.

(a) Disturbance of 0.05 units at t = 2 s

(b) Disturbance of 1 unit at t = 3 s

Figure 4 

Responses of system states with the method in [20].

4.2. Inverted Pendulum

The inverted pendulum system in [19] is adopted as the second example and represented by the diagram in Figure 5. Its state model is expressed by𝐱^′=00010020.6010000001−0.4905000𝐱+−10.5𝑢, (4.5) where 𝑥₁=𝜃,𝑥₂=𝜃^′,𝑥₃=𝑥, and 𝑥₄=𝑥^′. With its poles at 0,0 and ±4.5388, the system is inherently unstable. It is desirable to place the closed-loop poles at −2±3.464𝑗,−10,−10, and −100 such that the characteristic polynomial is Δ_𝑑(𝑠)=𝑠⁵+124𝑠⁴+2.595999×10³𝑠³+2.0319915×10⁴𝑠²+7.359852×10⁴𝑠+1.5999256×10⁵. (4.6) The Frobenius canonical form is 𝝃^′=00010100001000010020.6010𝝃+𝑢. (4.7) With the proposed state-PID feedback, the gain matrices are 𝐊_𝐩=[−20.601000], 𝐊_𝐈=[7123.14901490.23861956.37811043.4560], and𝐊_𝐝=[−120.6720−24.6841−313.0368−49.3195].

Figure 5 

Inverted pendulum.

The gain matrix due to the Ackermann’s formula is𝐊=[−298.1504−60.6972−163.0989−73.3945]. Figures 6 and 7 show the responses and the control inputs in which the initial conditions are 𝐱(𝑡₀)=[0.1000]^𝑇, and the states are disturbed by 1 unit at the time 𝑡=4s. These comparative results show a similarity to those of the first example. Very good transient responses with zero steady-state errors are achieved by the proposed method. With the method [21] to achieve the same closed-loop poles, it requires two fictitious pole sets designated as {−1,−2,−4,−5} and {−5,−6,−10,−11}. The following feedback gains are obtained, respectively, as 𝐊_𝑝=[30.437715.97554.07757.9511], 𝐊_𝐼=[−1562.4506−478.4985163.0989−262.9969], and𝐊_𝑑=[−14.4536−1.1561−5.6815−0.3365] with which a combined state-PID feedback controller is derived. During the design process, the two intermediate systems ∑¹_Int and ∑²_Int are calculated and expressed in Frobenius canonical forms as follows:¹_Int=𝐳^′=0001𝐊010000100001−40−78−7.798−12𝐳+𝑢,𝑢=_𝐹𝐳^′,_2Int=𝐳^′=0001010000100001−3300−1840−371−32𝐳+𝑢,𝑢=𝐊_𝐹^𝑡₀𝐳(𝜏)𝑑𝜏. (4.8) Under the same simulation situations, similar responses to those in Figure 6 are achieved because the system possesses the same closed-loop pole locations. As a result, the proposed method gives small proportional gains meaning that the proposed controller draws less energy in comparison with that of the method [21]. The magnitude of the integral and the derivative gains are in reasonable ranges for implementation.

(a) Responses of system states

(b) Control signal

Figure 6 

Responses of system states with the proposed method.

(a) Responses of system states

(b) Control signal

Figure 7 

Responses of system states with the conventional method [4, 19].

4.3. Mechanical Vibration

The mechanical vibration system in [14] is adopted as the third example. The diagram in Figure 8 represents the system, which is described by𝐱^′=0000100001−3960360−1.20.53600−36005−5𝐱+−0.010.1𝑢, (4.9) where 𝑥₁=𝑥₁,𝑥₂=𝑥₂,𝑥₃=𝑥^′₁, and 𝑥₄=𝑥^′₂. The open-loop poles of the system are −2.1835±70.1294𝑗 and −0.9165±51.3006𝑗. It is desirable to have the closed-loop poles at −5±65𝑗, −10±55𝑗, and −1000 such that the characteristic polynomial isΔ_𝑑(𝑠)=𝑠⁵+1030𝑠⁴+37575𝑠³+7691250𝑠²[phone]𝑠+13281.2500×10⁶. The Frobenius canonical form is𝝃^′=010000100001−1.2960×10⁷000−20520−7563.5000−6.20001𝝃+𝑢. (4.10) With the proposed pole placement method, the obtained gain matrices are 𝐊_𝐩=10³⋅[−39636−0.12000.0500], 𝐊_𝐈=10⁶⋅[1166.857940−36.8923613.988644−0.352635], and𝐊_𝐝=10³⋅[317.347421.36461.00220.1002]. The gain matrix due to the Ackermann’s formula is 𝐊=[9.6338×10³892.36112774315265.7432]. Figures 9 and 10 show the responses and the control inputs in which the initial conditions are 𝐱(𝑡₀)=[0.050.050.20.2]^𝑇, and the states are disturbed by 1 unit at the time 𝑡=1.5s. Similar to the other examples, using the proposed method, the transient responses of the system states are reasonably good with moderate control input, and all the states converge to origin without steady-state errors. As shown in Figure 10, using the method [4, 19], some states cannot converge to origin properly although the control input is not high. As a result of applying the method [21], the same closed-loop pole locations can be placed through the use of two fictitious pole sets, namely, {−1,−2,−4,−5} and {−5,−6,−10,−11}. This leads to two intermediate systems, respectively, denoted as ¹_Int=𝐳^′=0001𝐊010000100001−40−60−49−12𝐳+𝑢,𝑢=_𝐹𝐳^′,_2Int=𝐳^′=0001010000100001−3300−630−371−2𝐳+𝑢,𝑢=𝐊_𝐹^𝑡₀𝐳(𝜏)𝑑𝜏. (4.11) The obtained gain matrices are 𝐊_𝑝=10³⋅[−391.102335.99991.07830.0498], 𝐊_𝐼=10³⋅[485.56441032.9861355.9761−28.0424], and𝐊_𝑑=[−1195.25610.1455−97.55200.1236]with which a combined state-PID feedback controller is derived. Again, similar simulation results to those in Figure 9 are obtained. In order to show that arbitrarily chosen intermediate poles affect the gains, more results of applying the method [21] are included. All cases aim to achieve the same closed-loop pole locations at {−5±65𝑗,−10±55𝑗}. Two fictitious pole sets being considered are {−65,−60,−55,−50} and {−45,−40,−35,−30}. This leads to two intermediate systems denoted as¹_Int=𝐳^′=010000100001−1.0725×10⁷−7.53250×10⁵000−19775−2301𝐳+𝑢,_2Int=𝐳^′=010000100001−1.89×10⁶−2.0625×10⁵000−8375−1501𝐳+𝑢. (4.12) The following three feedback gains are obtained: 𝐊_𝑝=10³⋅[12696.23−2], 𝐊_𝐼=10⁶⋅[−20.20188381−0.03689236−0.785157430.00493426], and 𝐊_𝑑=10³⋅[57.9939−1.1587−0.3011−0.0769]. Next, assume the following fictitious pole sets: {−10,−11,−12,−13} and {−10±65𝑗,−20±55𝑗} resulting in the following intermediate systems:¹_Int=𝐳^′=0001010000100001−17160−6026−791−46𝐳+𝑢,_2Int=𝐳^′=010000100001−1.4813125×10⁷−2.415×10⁵000−8550−601𝐳+𝑢. (4.13) The obtained gain matrices are𝐊_𝑝=10³⋅[−317.49377435.95233343.1270380.03327],𝐊_𝐼=10⁶⋅[−23.47560083−0.03689236−0.443681730.04083183], and 𝐊_𝑑=[−4418.315.9617−78.21392.1666]. As the final case, the fictitious pole sets are {−35±65𝑗,−25±55𝑗} and {−10±65𝑗,−20±55𝑗} resulting in two intermediate systems as follows:¹_Int=𝐳^′=010000100001−1.98925×10⁷−5.28×10⁵000−12600−1201𝐳+𝑢,_2Int=𝐳^′=010000100001−1.4813125×10⁷−2.415×10⁵000−8550−601𝐳+𝑢. (4.14) The obtained gain matrices are 𝐊_𝑝=10³⋅[301.2391−19.2569−2.6792−1.4059],𝐊_𝐼=10⁶⋅[−23.47560083−0.03689236−0.443681730.04083183], and𝐊_𝑑=[1930.8642565.8066342.029630.7740].

Figure 8 

Mechanical vibration.

(a) Responses of system states

(b) Responses of system states (continue)

(c) Control signal

Figure 9 

Responses of system states with the proposed method.

(a) Responses of system states

(b) Responses of system states (continue)

(c) Control signal

Figure 10 

Responses of system states with the method [4, 19].

The above case studies serve to show the effects of fictitious pole locations required by the method [21] on the designed gains, the magnitudes of which can be very large. Selection of fictitious poles is a critical problem of this previous method, which has been neither solved nor considered.

5. Conclusion

A new design method for pole placement via state-PID, -PI, and -PD feedback has been proposed. The method has two distinctive features: (i) compact design formulae and (ii) disturbance rejection property. The analyses of these features have been elaborated through relevant propositions. The paper also describes the design procedures and presents some illustrative examples including a magnetic ball suspension, an inverted pendulum, and mechanical vibration systems, respectively. The simulation results reflect that the proposed method is promising for a real-world application. A future work will be design optimization to achieve minimum gains subject to nonlinear restriction in control input and specified performance constraints.

Acknowledgments

The authors gratefully acknowledge the financial supports by Ratchamangkala University of Technology Isarn and Suranaree University of Technology, Thailand. The authors’ appreciations are also due to the reviewers for their constructive comments.